$G_2$ Forms

We refer to [7] and [3] for the definition of a Siegel modular cusp form of degree $3$ of type $G_2$. We list the cases where the dimension of $S_{j,k,l}$ with $4\leq l \leq (86-j-2k)/3$ is $1$ and where the normalized spinor $L$-factor at $2$ satisfies the condition for being of type $G_2$. Assuming that this condition (see below) holds also for the other primes, we are able to predict the characteristic polynomial at $3$ and $5$.

The following is a list of all weights $(j,k,l)$ with $j+2k+3l-6 \leq 80$ and $l \geq 4$, such that the space of non-lifted forms in $S_{j,k,l}$ is one-dimensional:

[6, 3, 6], [3, 3, 7], [7, 5, 5], [10, 2, 6], [6, 4, 6], [2, 6, 6], [9, 1, 7], [5, 3, 7], [4, 2, 8],
[13, 3, 5], [11, 4, 5], [9, 5, 5], [7, 6, 5], [5, 7, 5], [1, 9, 5], [12, 2, 6], [10, 3, 6], 
[8, 4, 6], [6, 5, 6], [4, 6, 6], [11, 1, 7], [9, 2, 7], [7, 3, 7], [5, 4, 7], [3, 5, 7], [8, 1, 8],
[6, 2, 8], [4, 3, 8], [2, 4, 8], [12, 6, 4], [8, 8,4], [0, 12, 4], [15, 3, 5], [13, 4, 5], 
[9, 6, 5], [5, 8, 5], [3, 9, 5], [18, 0, 6], [16, 1, 6], [4, 7, 6], [13, 1, 7], [11, 2, 7], 
[7, 4, 7], [3, 6, 7], [12, 0, 8], [10, 1, 8], [6, 3, 8], [0, 6, 8], [7, 1, 9], [3, 3, 9], 
[6, 0, 10], [2, 2, 10], [16, 5, 4], [14, 6, 4], [12, 7, 4], [10, 8, 4], [8, 9, 4], [6, 10, 4], 
[4, 11, 4], [19, 2, 5], [3,10, 5], [18, 1, 6], [17, 0, 7], [1, 8, 7], [14, 0, 8], [7, 2, 9], 
[3, 4, 9], [1, 5, 9], [8, 0, 10], [6, 1, 10], [4, 2, 10], [0, 4, 10], [20, 4, 4], [18, 5, 4], 
[10, 9, 4], [0, 14, 4], [23, 1, 5], [21, 2, 5], [22, 0, 6], [20, 1, 6], [5, 4, 9], [1, 6, 9], 
[4, 3, 10], [5, 1, 11], [1, 3, 11], [4, 0, 12], [0, 2, 12], [22, 4, 4], [2, 14, 4], [0, 15, 4], 
[25, 1, 5], [24, 0, 6], [5, 2, 11], [1, 4, 11], [4, 1, 12], [26, 3, 4], [27, 1, 5], [17, 0, 9], 
[2, 3, 12], [28, 3, 4], [31, 0, 5], [0, 11, 8], [2, 1, 14], [32, 2, 4], [0, 15, 6], [0, 9, 10], 
[1, 1, 15], [0, 0, 16], [34, 2, 4], [35, 0, 5], [0, 7, 12], [1, 2, 15], [7, 0, 15], [2, 1, 16], 
[1, 1, 17], [42, 1, 4], [44, 1, 4], [46, 1, 4], [0, 3, 18], [0, 0, 20], [56, 0, 4], [1, 0, 23], 
[60, 0, 4], [62, 0, 4], [64, 0, 4], [66, 0, 4]

In all these cases we can compute the (normalized) spinor $L$-factor for $p=2$, say, $x^8-Ax^7+Bx^6-Cx^5+Dx^4-Cx^3+Bx^2-Ax+1$ and they all fulfill the "Spin7"-condition $A^2(D+2B+1)=C^2+2AC+A^4$.

For the following weights $(j,k,l)$ the (normalized) spinor $L$-factor for $p=2$, also fulfills the "$G_2$"-condition $2A-2B+2C-D-2=0$. And assuming that this condition holds also for other primes, we compute the spinor $L$-factor for $p=3$ and $p=5$. For details see [7] and [3].

[3, 3, 7]
2, (x^6+7112*x^5+34431488*x^4+176085008384*x^3+577664511377408*x^2+2001850034366185472*x+4722366482869645213696)*(x-4096)^2
3, (x^6+1244322*x^5+814612073823*x^4+445663920046456764*x^3+230070510421656043636863*x^2+99255139982300155218197086242*x+22528399544939174411840147874772641)*(x-531441)^2
5, (x^6+856793450*x^5+403899977523359375*x^4+91028277903038940429687500*x^3+24074314685068093240261077880859375*x^2+3043941809721673052990809082984924316406250*x+
211758236813575084767080625169910490512847900390625)*(x-244140625)^2

[2, 6, 6]
2, (x^6+11200*x^5+98631680*x^4+992103890944*x^3+6619059999211520*x^2+50440315826549555200*x+302231454903657293676544)*(x-8192)^2
3, (x^6+3061638*x^5+6657012250263*x^4+13741701692529607956*x^3+16921231957711060743100527*x^2+19781493833168173585596308470758*x+16423203268260658146231467800709255289)*(x-\
1594323)^2
5, (x^6+4568059954*x^5+12054956471908304375*x^4+13934325053441049663085937500*x^3+17963284957272267900407314300537109375*x^2+10143130677597866906580748036503791809082031250
*x+3308722450212110699485634768279851414263248443603515625)*(x-1220703125)^2

[4, 2, 8]
2, (x^6+6880*x^5+45916160*x^4+188433301504*x^3+3081381336842240*x^2+30984765436309012480*x+302231454903657293676544)*(x-8192)^2
3, (x^6+4158918*x^5+8311870710423*x^4+11964960584285600916*x^3+21127660128312912588973167*x^2+26871109768578817663048677875238*x+16423203268260658146231467800709255289)*(x-\
1594323)^2
5, (x^6+2335354354*x^5+3806045130352784375*x^4+5956453587658194194335937500*x^3+5671449199844575487077236175537109375*x^2+5185528348938817089219810441136360168457031250*x+
3308722450212110699485634768279851414263248443603515625)*(x-1220703125)^2

[1, 9, 5]
2, (x^6+29528*x^5+527886848*x^4+10069172289536*x^3+141703546759282688*x^2+2127716636751936094208*x+19342813113834066795298816)*(x-16384)^2
3, (x^6+11665458*x^5+71132197283343*x^4+344464392236631357276*x^3+1627276514116378483883014623*x^2+6105089832480087197721655680741618*x+
11972515182562019788602740026717047105681)*(x-4782969)^2
5, (x^6+12808780250*x^5+108208768264499609375*x^4+819116576251853561401367187500*x^3+4031090746242538443766534328460693359375*x^2+
17775753438642460935170674929395318031311035156250*x+51698788284564229679463043254372678347863256931304931640625)*(-6103515625+x)^2

[3, 5, 7]
2, (x^6+26072*x^5+550650368*x^4+12107906023424*x^3+147814082630647808*x^2+1878685591756857147392*x+19342813113834066795298816)*(x-16384)^2
3, (x^6+19980594*x^5+217743990034959*x^4+868794665038388751708*x^3+4981284068344853221822981599*x^2+10456796576380681785685065015423474*x+
11972515182562019788602740026717047105681)*(x-4782969)^2
5, (x^6+14552816090*x^5+80213085121331609375*x^4+316055840516326061401367187500*x^3+2988170278121963492594659328460693359375*x^2+
20196089370316805400307202944532036781311035156250*x+51698788284564229679463043254372678347863256931304931640625)*(-6103515625+x)^2

[4, 3, 8]
2, (x^6-5248*x^5+277741568*x^4-2963795869696*x^3+74555684456235008*x^2-378158253511045808128*x+19342813113834066795298816)*(x-16384)^2
3, (x^6+8199954*x^5+35061784014159*x^4+170134938083958778908*x^3+802101155992584814793792799*x^2+4291426516833237146566082650670034*x+
11972515182562019788602740026717047105681)*(x-4782969)^2
5, (x^6+21151784570*x^5+202866608643419609375*x^4+866881811896781403198242187500*x^3+7557370090612009516917169094085693359375*x^2+
29353997801906395181958941975608468055725097656250*x+51698788284564229679463043254372678347863256931304931640625)*(-6103515625+x)^2

[0, 12, 4]
2, (x^6+75904*x^5+2690220032*x^4+80149995323392*x^3+2888601764121018368*x^2+87511353885678112866304*x+1237940039285380274899124224)*(x-32768)^2
3, (x^6+23303862*x^5+348340967920071*x^4+5891655255005195839476*x^3+71720316240009227983038800079*x^2+987877702465796426861032498684322262*x+
8727963568087712425891397479476727340041449)*(x-14348907)^2
5, (x^6+99749025250*x^5+5472501330219842234375*x^4+159322717736893774032592773437500*x^3+5096664028446974451071582734584808349609375*x^2+
86518487903489149637437094497727230191230773925781250*x+807793566946316088741610050849573099185363389551639556884765625)*(-30517578125+x)^2

[3, 6, 7]
2, (x^6-12656*x^5+1446889472*x^4-7972541431808*x^3+1553585740791676928*x^2-14591374562304255328256*x+1237940039285380274899124224)*(x-32768)^2
3, (x^6+33179382*x^5+572318870066631*x^4+8382054920458868704116*x^3+117835380077148980749228557519*x^2+1406512433835859548990517459649461782*x+
8727963568087712425891397479476727340041449)*(x-14348907)^2
5, (x^6+61671292450*x^5+2370146147469290234375*x^4+86621576052227531356811523437500*x^3+2207370612276056999689899384975433349609375*x^2+
53491319403423109868356277729617431759834289550781250*x+807793566946316088741610050849573099185363389551639556884765625)*(-30517578125+x)^2

[6, 0, 10]
2, (x^6+67840*x^5+1424752640*x^4+17882096336896*x^3+1529816498422415360*x^2+78214194872528498851840*x+1237940039285380274899124224)*(x-32768)^2
3, (x^6+35944182*x^5+596039267182023*x^4+7459013848363689657204*x^3+122719199492971686119547294927*x^2+1523715508235177308466838708563586582*x+
8727963568087712425891397479476727340041449)*(x-14348907)^2
5, (x^6+131652729250*x^5+9584458802703290234375*x^4+244423713309158727645874023437500*x^3+8926222848429614896303974092006683349609375*x^2+
114190540053196731840046140860067680478096008300781250*x+807793566946316088741610050849573099185363389551639556884765625)*(-30517578125+x)^2

[0, 14, 4]
2, (x^6+178624*x^5+10062774272*x^4+89717034975232*x^3+172877145621080834048*x^2+52720499414756719165702144*x+5070602400912917605986812821504)*(x-131072)^2
3, (x^6+214073766*x^5+32795401912621047*x^4+5385132555518325855508308*x^3+546934876610373720408112751677743*x^2+59539991759726114904610561056166366081926*x+
4638397686588101979328150167890591454318967698009)*(x-129140163)^2
5, (x^6+1230158736850*x^5+1049492952510280358084375*x^4+990764951172931361410140991210937500*x^3+610885299107921518201692379079759120941162109375*x^2+
416793992185873112335525725313800649018958210945129394531250*x+197215226305252951352932141320696557418301608777255751192569732666015625)*(-762939453125+x)^2

[1, 6, 9]
2, (x^6+135424*x^5+14401667072*x^4+1521028308140032*x^3+247418756328480309248*x^2+39970109911008677088722944*x+5070602400912917605986812821504)*(x-131072)^2
3, (x^6+175323366*x^5+18076986111085047*x^4+2239794367903911924023508*x^3+301473181956914285416034701693743*x^2+48762405417427213865986234312875134627526*x+
4638397686588101979328150167890591454318967698009)*(x-129140163)^2
5, (x^6+217492892050*x^5+562476483414271396484375*x^4+187198186268100750387191772460937500*x^3+327404404183774835246367729268968105316162109375*x^2+
73689458143989155017306025285961368354037404060363769531250*x+197215226305252951352932141320696557418301608777255751192569732666015625)*(-762939453125+x)^2

[0, 15, 4]
2, (x^6+455168*x^5+70181715968*x^4+9014002482937856*x^3+4822850797755535720448*x^2+2149470107274810672627580928*x+324518553658426726783156020576256)*(x-262144)^2
3, (x^6+1049813298*x^5+529350757861699503*x^4+182026668058245613515998556*x^3+79452708945441877168074996957136863*x^2+23650613424934293898691115889222757248380018*x+
3381391913522726342930221472392241170198527451848561)*(x-387420489)^2
5, (x^6+6539036937050*x^5+3907782464714041888109375*x^4-54009921814875965087709426879882812500*x^3+56865719157417215874147586873732507228851318359375*x^2+
1384694932248548574155525003881450629705796018242835998535156250*x+3081487911019577364889564708135883709660962637144621112383902072906494140625)*(-3814697265625+x)^2