Defining parameters
Level: | \( N \) | \(=\) | \( 56 = 2^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 56.a (trivial) |
Character field: | \(\Q\) | ||
Newform subspaces: | \( 4 \) | ||
Sturm bound: | \(80\) | ||
Trace bound: | \(3\) | ||
Distinguishing \(T_p\): | \(3\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(56))\).
Total | New | Old | |
---|---|---|---|
Modular forms | 76 | 14 | 62 |
Cusp forms | 68 | 14 | 54 |
Eisenstein series | 8 | 0 | 8 |
The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.
\(2\) | \(7\) | Fricke | Dim |
---|---|---|---|
\(+\) | \(+\) | $+$ | \(3\) |
\(+\) | \(-\) | $-$ | \(4\) |
\(-\) | \(+\) | $-$ | \(4\) |
\(-\) | \(-\) | $+$ | \(3\) |
Plus space | \(+\) | \(6\) | |
Minus space | \(-\) | \(8\) |
Trace form
Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(56))\) into newform subspaces
Label | Dim | $A$ | Field | CM | Traces | A-L signs | $q$-expansion | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
$a_{2}$ | $a_{3}$ | $a_{5}$ | $a_{7}$ | 2 | 7 | |||||||
56.10.a.a | $3$ | $28.842$ | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) | None | \(0\) | \(-92\) | \(274\) | \(-7203\) | $+$ | $+$ | \(q+(-31-\beta _{1})q^{3}+(92+\beta _{1}-\beta _{2})q^{5}+\cdots\) | |
56.10.a.b | $3$ | $28.842$ | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) | None | \(0\) | \(84\) | \(-2958\) | \(7203\) | $-$ | $-$ | \(q+(28-\beta _{1})q^{3}+(-986+5\beta _{1}-\beta _{2})q^{5}+\cdots\) | |
56.10.a.c | $4$ | $28.842$ | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) | None | \(0\) | \(70\) | \(1022\) | \(9604\) | $+$ | $-$ | \(q+(17-\beta _{1})q^{3}+(255-\beta _{1}-\beta _{2})q^{5}+\cdots\) | |
56.10.a.d | $4$ | $28.842$ | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) | None | \(0\) | \(84\) | \(1540\) | \(-9604\) | $-$ | $+$ | \(q+(21-\beta _{1})q^{3}+(385-2\beta _{1}+\beta _{2})q^{5}+\cdots\) |
Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(56))\) into lower level spaces
\( S_{10}^{\mathrm{old}}(\Gamma_0(56)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 2}\)