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.
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] | $2^{24}\cdot 3^6\cdot 5\cdot 157$ |
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] | $2^{18}\cdot 3^{4}\cdot 5\cdot 11\cdot 199$ |
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] | $2^{162}\cdot 3^{50}\cdot 5^{9}\cdot 7^{7}\cdot 19\cdot 23$ |
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] | $2^{61}\cdot 3^{39}\cdot 5^{6}\cdot 11^{2}\cdot 19$ |
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] | $2^{8}\cdot 3^{2}\cdot 5\cdot 691$ |
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] | $2^{13}\cdot 3^{5}\cdot 5^{2}\cdot 17\cdot 37$ |
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] | $2^{3}\cdot 3^{3}\cdot 5^{2}\cdot 19$ |
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] | $2^{8}\cdot 3^{2}\cdot 5\cdot 17$ |
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] | $2^{6}\cdot 3^{4}\cdot 71$ |