## Congruences

Here we give congruences that we found between degree 3 Siegel modular forms and elliptic modular or degree 2 eigenforms. Many of these congruences can be found in the paper [3], but some others (numbers 6 and 7 below) were found on instigation by Neil Dummigan whom we thank for his suggestions.

We refer to [3], especially equations (10.3)-(10.6).

Say that we have elliptic eigenforms of weight $m_1,m_2,m_3$, with trace of $T(q)$ equal to $\lambda_1(q),\lambda_2(q),\lambda_3(q)$.

Say also that we have genus 2 eigenforms of weight $j_1,k_1$ (for a weight such that the space of forms is one-dimensional) with trace of $T(q)$ equal to $\mu(q)$.

The following lists will contain all congruences we have found between a genus 3 eigenform of weight $j=a-b, k=b-c,l=c+4$ for which the space is one-dimensional and with trace of $T(q)$ equal to $\Lambda(q)$ such that (the norm of, if the space of elliptic eigenforms is more than one-dimensional) $\Lambda(q)-\operatorname{tr}(q)$ is divisible by $\ell$ for all $q \leq 25$, where the demand on $\ell$ is that it should contain a prime factor larger than 13.

### Eisenstein congruence 1

Equation (10.3) in [3]

Here $m_1:=a+4$, $m_2:=b+c+4$, $r_1:=b+2$, $r_2:=c+1$ and $\operatorname{tr}:=\lambda_1(q)(\lambda_2(q)+q^{r_1}+q^{r_2})$.

$[j,k,l]$ $[m_1,m_2,r_1,r_2]$ $\ell$
[6, 4, 6][16, 12, 8, 3]$2^6\cdot 3^2\cdot 101$
[7, 6, 5][18, 12, 9, 2]$2^5\cdot 3^3\cdot 17^2$
[8, 8, 4][20, 12, 10, 1]$2^4\cdot 3^3\cdot 5^2\cdot 43$
[0, 12, 4][16, 16, 14, 1]$2^4\cdot 3^5\cdot 5^2\cdot 37$
[9, 6, 5][20, 12, 9, 2]$2^5\cdot 3^3\cdot 263$
[11, 2, 7][20, 12, 7, 4]$2^4\cdot 3^2\cdot 5\cdot 127$
[3, 6, 7][16, 16, 11, 4]$2^4\cdot 3^3\cdot 7\cdot 137$
[12, 0, 8][20, 12, 6, 5]$2^{10}\cdot 3^4\cdot 5^2\cdot 29$
[6, 0, 10][16, 16, 8, 7]$2^6\cdot 3^2\cdot 229$
[3, 10, 5][18, 16, 13, 2]$2^9\cdot 3^2\cdot 37$
[7, 2, 9][18, 16, 9, 6]$2^6\cdot 3^2\cdot 71$
[4, 2, 10][16, 18, 10, 7]$2^5\cdot 3^2\cdot 7\cdot 73$
[0, 14, 4][18, 18, 16, 1]$2^5\cdot 3^3\cdot 5^2\cdot 59$
[4, 0, 12][16, 20, 10, 9]$2^7\cdot 3^2\cdot 5\cdot 61$
[0, 0, 16][16, 28, 14, 13]$2^{15}\cdot 3^4\cdot 5\cdot 107$
[1, 2, 15][18, 28, 15, 12]$2^{21}\cdot 3^4\cdot 41$
[0, 0, 20][20, 36, 18, 17]$2^{24}\cdot 3^6\cdot 5\cdot 157$

### Eisenstein congruence 2

Equation (10.4) in [3]

Here $m_1:=c+2$, $m_2:=a+b+6$, $r_1:=a+3$, $r_2:=b+2$ and $\operatorname{tr}:=\lambda_1(q)(\lambda_2(q)+q^{r_1}+q^{r_2})$.

$[j,k,l]$$[m_1,m_2,r_1,r_2]$$\ell$
[2, 1, 14][12, 30, 16, 13]$2^{18}\cdot 3^{4}\cdot 5\cdot 11\cdot 199$

### Eisenstein congruence 3

Equation (10.5) in [3]

Here $m_1:=a+b+6$, $m_2:=a-b+2$, $r_1:=b+2$, $r_2:=c+1$ and $\operatorname{tr}:=(\lambda_1(q)+q^{r_1}\lambda_2(q))(1+q^{r_2})$.

$[j,k,l]$$[m_1,m_2,r_1,r_2]$$\ell$
[10, 2, 6][24, 12, 6, 3]$2^{9}\cdot 3^{4}\cdot 5\cdot 7\cdot 103$
[16, 1, 6][28, 18, 5, 3]$2^{9}\cdot 3^{8}\cdot 5\cdot 7\cdot 17$
[10, 1, 8][26, 12, 7, 5]$2^{4}\cdot 3^{2}\cdot 691$
[14, 0, 8][28, 16, 6, 5]$2^{12}\cdot 3^{6}\cdot 5^{2}\cdot 17$
[22, 0, 6][32, 24, 4, 3]$2^{18}\cdot 3^{8}\cdot 5^{2}\cdot 7^{2}\cdot 13\cdot 17\cdot 31$
[20, 1, 6][32, 22, 5, 3]$2^{9}\cdot 3^{4}\cdot 5\cdot 7\cdot 13\cdot 19$
[24, 0, 6][34, 26, 4, 3]$2^{13}\cdot 3^{6}\cdot 5\cdot 7\cdot 17$
[34, 2, 4][44, 36, 4, 1]$2^{45}\cdot 3^{23}\cdot 5^{5}\cdot 7\cdot 31$
[44, 1, 4][52, 46, 3, 1]$2^{63}\cdot 3^{36}\cdot 5^{6}\cdot 11\cdot 19$
[56, 0, 4][62, 58, 2, 1]$2^{98}\cdot 3^{32}\cdot 5^{5}\cdot 7^{4}\cdot 11\cdot 13\cdot 17\cdot 19$
[60, 0, 4][66, 62, 2, 1]$2^{112}\cdot 3^{46}\cdot 5^{7}\cdot 11\cdot 19$
[64, 0, 4][70, 66, 2, 1]$2^{162}\cdot 3^{50}\cdot 5^{9}\cdot 7^{7}\cdot 19\cdot 23$

### Eisenstein congruence 4

Here $m_1:=a+c+5$, $m_2:=a-c+3$, $r_1:=c+1$, $r_2:=b+2$ and $\operatorname{tr}:=(\lambda_1(q)+q^{r_1}\lambda_2(q))(1+q^{r_2})$.

$[j,k,l]$$[m_1,m_2,r_1,r_2]$$\ell$
[44, 1, 4][50, 48, 1, 3]$2^{61}\cdot 3^{39}\cdot 5^{6}\cdot 11^{2}\cdot 19$

### Eisenstein congruence 5

Equation (10.6) in [3]

Here $j_1:=a-b$, $k_2:=b+4$, $r_1:=c+1$ and $\operatorname{tr}:=\mu(q)(1+q^{r_1})$.

$[j,k,l]$$[j_1,k_1,r_1]$$\ell$
[6, 4, 6][6, 10, 3]$2^{4}\cdot 3^{2}\cdot 149$
[4, 2, 8][4, 10, 5]$2^{4}\cdot 3^{2}\cdot 11\cdot 41$
[4, 6, 6][4, 12, 3]$2^{5}\cdot 3^{2}\cdot 59$
[8, 1, 8][8, 9, 5]$2^{4}\cdot 3^{2}\cdot 601$
[6, 2, 8][6, 10, 5]$2^{4}\cdot 3^{2}\cdot 379$
[18, 0, 6][18, 6, 3]$2^{4}\cdot 3^{3}\cdot 5\cdot 37$
[10, 1, 8][10, 9, 5]$2^{8}\cdot 3^{2}\cdot 29$
[6, 3, 8][6, 11, 5]$2^{5}\cdot 3^{2}\cdot 5\cdot 23$
[4, 11, 4][4, 15, 1]$2^{6}\cdot 3^{2}\cdot 5\cdot 7\cdot 17$
[6, 1, 10][6, 11, 7]$2^{5}\cdot 3^{3}\cdot 1621$
[4, 2, 10][4, 12, 7]$2^{9}\cdot 3^{3}\cdot 7\cdot 13\cdot 53$
[4, 0, 12][4, 12, 9]$2^{5}\cdot 3^{2}\cdot 5\cdot 89$
[0, 0, 20][0, 20, 17]$2^{8}\cdot 3^{2}\cdot 5\cdot 691$

### Eisenstein congruence 6

Here $m_1:=a+b+6$, $r_1:=a+3$, $r_2:=b+2$, $r_3:=c+1$ and $\operatorname{tr}:=(\lambda_1(q)+q^{r_1}+q^{r_2})(1+q^{r_3})$.

$[j,k,l]$$[m_1,r_1,r_2,r_3]$$\ell$
[6, 4, 6][24, 15, 8, 3]$2^{9}\cdot 3^{4}\cdot 7\cdot 73$
[2, 6, 6][24, 13, 10, 3]$2^{9}\cdot 3^{6}\cdot 7\cdot 11\cdot 179$
[4, 2, 8][22, 13, 8, 5]$2^{6}\cdot 3^{2}\cdot 41$
[12, 2, 6][26, 19, 6, 3]$2^{4}\cdot 3^{2}\cdot 7\cdot 43$
[10, 3, 6][26, 18, 7, 3]$2^{4}\cdot 3^{3}\cdot 17$
[8, 4, 6][26, 17, 8, 3]$2^{4}\cdot 3^{2}\cdot 5\cdot 97$
[4, 6, 6][26, 15, 10, 3]$2^{4}\cdot 3^{2}\cdot 7\cdot 11\cdot 29$
[4, 3, 8][24, 14, 9, 5]$2^{13}\cdot 3^{5}\cdot 5\cdot 11\cdot 13\cdot 17$
[2, 4, 8][24, 13, 10, 5]$2^{11}\cdot 3^{4}\cdot 11\cdot 73$
[4, 7, 6][28, 16, 11, 3]$2^{7}\cdot 3^{8}\cdot 5^{2}\cdot 157$
[10, 1, 8][26, 18, 7, 5]$2^{4}\cdot 3^{3}\cdot 29\cdot 691$
[6, 3, 8][26, 16, 9, 5]$2^{4}\cdot 3^{2}\cdot 19$
[0, 6, 8][26, 13, 12, 5]$2^{4}\cdot 3^{3}\cdot 11\cdot 29$
[18, 1, 6][30, 24, 5, 3]$2^{10}\cdot 3^{6}\cdot 5\cdot 7\cdot 97$
[4, 2, 10][26, 15, 10, 7]$2^{4}\cdot 3^{3}\cdot 43$
[0, 4, 10][26, 13, 12, 7]$2^{4}\cdot 3^{3}\cdot 5\cdot 97$
[0, 2, 12][26, 13, 12, 9]$2^{4}\cdot 3^{2}\cdot 5\cdot 43$
[2, 3, 12][30, 16, 13, 9]$2^{12}\cdot 3^{4}\cdot 5\cdot 19\cdot 593$
[2, 1, 14][30, 16, 13, 11]$2^{12}\cdot 3^{4}\cdot 5\cdot 7\cdot 97$
[0, 0, 16][30, 15, 14, 13]$2^{12}\cdot 3^{4}\cdot 5^{3}\cdot 13\cdot 17$
[2, 1, 16][34, 18, 15, 13]$2^{12}\cdot 3^{6}\cdot 7^{2}\cdot 103$
[0, 0, 20][38, 19, 18, 17]$2^{13}\cdot 3^{5}\cdot 5^{2}\cdot 17\cdot 37$

### Eisenstein congruence 7

Here $m_1:=a+c+5$, $r_1:=a+3$, $r_2:=c+1$, $r_3:=b+2$ and $\operatorname{tr}:=(\lambda_1(q)+q^{r_1}+q^{r_2})(1+q^{r_3})$.

$[j,k,l]$$[m_1,r_1,r_2,r_3]$$\ell$
[26, 3, 4][34, 32, 1, 5]$2^{2}\cdot 3^{2}\cdot 5\cdot 79$
[44, 1, 4][50, 48, 1, 3]$2^{3}\cdot 3^{3}\cdot 5^{2}\cdot 19$

### Endoscopic congruence 1

Equation (10.7) in [3]

Here $m_1:=b+3$, $m_2:=a+c+5$, $m_3:=a-c+3$, $r_1:=c+1$ and $\operatorname{tr}:=\lambda_1(q)(\lambda_2(q)+q^{r_1}\lambda_3(q))$.

$[j,k,l]$$[m_1,m_2,m_3,r_1]$$\ell$
[3, 6, 7][12, 20, 4, 12]$2^{4}\cdot 3^{3}\cdot 5\cdot 37$
[0, 15, 4][18, 20, 1, 18]$2^{7}\cdot 3^{2}\cdot 5\cdot 43$
[0, 15, 6][20, 24, 3, 18]$2^{14}\cdot 3^{5}\cdot 5\cdot 317$
[0, 9, 10][18, 26, 7, 12]$2^{8}\cdot 3^{2}\cdot 5\cdot 17$

### Endoscopic congruence 2

Equation (10.8) in [3]

Here $m_1:=a+4$, $m_2:=b+c+4$, $m_3:=b-c+2$ and $r_1:=c+1$.

$[j,k,l]$$[m_1,m_2,m_3,r_1]$$\ell$
[3, 10, 5][18, 16, 2, 12]$2^{6}\cdot 3^{2}\cdot 5\cdot 19$
[0, 14, 4][18, 18, 1, 16]$2^{6}\cdot 3^{3}\cdot 19$
[2, 14, 4][20, 18, 1, 16]$2^{6}\cdot 3^{4}\cdot 71$