Properties

Label 315.2.a
Level $315$
Weight $2$
Character orbit 315.a
Rep. character $\chi_{315}(1,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $6$
Sturm bound $96$
Trace bound $2$

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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.

\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(3\)
Plus space\(+\)\(3\)
Minus space\(-\)\(7\)

Trace form

\( 10q + 12q^{4} - 2q^{7} + 12q^{8} + O(q^{10}) \) \( 10q + 12q^{4} - 2q^{7} + 12q^{8} + 4q^{10} - 2q^{11} - 4q^{13} - 2q^{14} + 16q^{16} + 4q^{17} - 8q^{19} - 20q^{22} - 8q^{23} + 10q^{25} + 20q^{26} - 6q^{28} + 2q^{29} + 12q^{31} + 4q^{32} - 28q^{34} + 4q^{35} - 8q^{37} - 32q^{38} + 20q^{41} - 12q^{43} - 12q^{44} + 32q^{46} - 20q^{47} + 10q^{49} - 40q^{52} - 4q^{53} + 8q^{55} - 6q^{56} + 4q^{58} + 4q^{59} + 32q^{61} - 24q^{62} - 48q^{64} + 6q^{65} - 20q^{67} + 24q^{68} - 2q^{70} - 24q^{71} - 16q^{73} - 32q^{74} + 16q^{76} + 6q^{79} + 6q^{85} + 24q^{86} - 8q^{88} + 16q^{89} + 2q^{91} - 40q^{92} - 20q^{94} + 20q^{95} - 44q^{97} + O(q^{100}) \)

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

\( S_{2}^{\mathrm{old}}(\Gamma_0(315)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)