Defining parameters
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.a (trivial) |
Character field: | \(\Q\) | ||
Newform subspaces: | \( 6 \) | ||
Sturm bound: | \(96\) | ||
Trace bound: | \(2\) | ||
Distinguishing \(T_p\): | \(2\) |
Dimensions
The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(315))\).
Total | New | Old | |
---|---|---|---|
Modular forms | 56 | 10 | 46 |
Cusp forms | 41 | 10 | 31 |
Eisenstein series | 15 | 0 | 15 |
The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.
\(3\) | \(5\) | \(7\) | Fricke | Dim |
---|---|---|---|---|
\(+\) | \(+\) | \(+\) | \(+\) | \(2\) |
\(+\) | \(-\) | \(+\) | \(-\) | \(2\) |
\(-\) | \(+\) | \(+\) | \(-\) | \(2\) |
\(-\) | \(+\) | \(-\) | \(+\) | \(1\) |
\(-\) | \(-\) | \(-\) | \(-\) | \(3\) |
Plus space | \(+\) | \(3\) | ||
Minus space | \(-\) | \(7\) |
Trace form
Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(315))\) into newform subspaces
Label | Dim | $A$ | Field | CM | Traces | A-L signs | $q$-expansion | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
$a_{2}$ | $a_{3}$ | $a_{5}$ | $a_{7}$ | 3 | 5 | 7 | |||||||
315.2.a.a | $1$ | $2.515$ | \(\Q\) | None | \(-1\) | \(0\) | \(-1\) | \(1\) | $-$ | $+$ | $-$ | \(q-q^{2}-q^{4}-q^{5}+q^{7}+3q^{8}+q^{10}+\cdots\) | |
315.2.a.b | $1$ | $2.515$ | \(\Q\) | None | \(0\) | \(0\) | \(1\) | \(1\) | $-$ | $-$ | $-$ | \(q-2q^{4}+q^{5}+q^{7}+3q^{11}+5q^{13}+\cdots\) | |
315.2.a.c | $2$ | $2.515$ | \(\Q(\sqrt{2}) \) | None | \(-2\) | \(0\) | \(-2\) | \(-2\) | $+$ | $+$ | $+$ | \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}-q^{5}-q^{7}+\cdots\) | |
315.2.a.d | $2$ | $2.515$ | \(\Q(\sqrt{5}) \) | None | \(0\) | \(0\) | \(2\) | \(2\) | $-$ | $-$ | $-$ | \(q-\beta q^{2}+3q^{4}+q^{5}+q^{7}-\beta q^{8}-\beta q^{10}+\cdots\) | |
315.2.a.e | $2$ | $2.515$ | \(\Q(\sqrt{17}) \) | None | \(1\) | \(0\) | \(-2\) | \(-2\) | $-$ | $+$ | $+$ | \(q+\beta q^{2}+(2+\beta )q^{4}-q^{5}-q^{7}+(4+\beta )q^{8}+\cdots\) | |
315.2.a.f | $2$ | $2.515$ | \(\Q(\sqrt{2}) \) | None | \(2\) | \(0\) | \(2\) | \(-2\) | $+$ | $-$ | $+$ | \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+q^{5}-q^{7}+\cdots\) |
Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(315))\) into lower level spaces
\( S_{2}^{\mathrm{old}}(\Gamma_0(315)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)