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Properties

Label 715.2.z
Level $715$
Weight $2$
Character orbit 715.z
Rep. character $\chi_{715}(56,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $96$
Newform subspaces $3$
Sturm bound $168$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 715 = 5 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 715.z (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(168\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(715, [\chi])\).

Total New Old
Modular forms 176 96 80
Cusp forms 160 96 64
Eisenstein series 16 0 16

Trace form

\( 96 q - 4 q^{3} + 52 q^{4} + 36 q^{6} + 12 q^{7} - 52 q^{9} + O(q^{10}) \) \( 96 q - 4 q^{3} + 52 q^{4} + 36 q^{6} + 12 q^{7} - 52 q^{9} - 40 q^{12} - 4 q^{13} + 8 q^{14} - 56 q^{16} - 12 q^{19} - 12 q^{20} - 4 q^{22} + 20 q^{23} + 72 q^{24} - 96 q^{25} - 28 q^{26} - 16 q^{27} - 36 q^{28} - 72 q^{32} + 36 q^{36} - 12 q^{37} + 24 q^{38} + 8 q^{39} - 24 q^{41} - 20 q^{42} - 12 q^{43} + 108 q^{46} - 44 q^{48} + 36 q^{49} + 48 q^{51} - 32 q^{52} + 48 q^{53} - 36 q^{54} - 8 q^{55} + 16 q^{56} + 12 q^{58} + 48 q^{59} + 24 q^{61} - 12 q^{62} - 48 q^{63} - 16 q^{64} + 8 q^{65} - 40 q^{66} - 12 q^{67} - 16 q^{68} - 16 q^{69} + 108 q^{72} + 8 q^{74} + 4 q^{75} - 48 q^{76} - 32 q^{77} + 100 q^{78} - 24 q^{79} + 24 q^{80} - 96 q^{81} + 32 q^{82} + 12 q^{84} + 16 q^{87} + 12 q^{88} - 48 q^{89} - 72 q^{90} - 44 q^{91} - 56 q^{92} + 84 q^{93} - 36 q^{94} + 36 q^{97} - 96 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(715, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
715.2.z.a 715.z 13.e $8$ $5.709$ 8.0.22581504.2 None 715.2.z.a \(0\) \(2\) \(0\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{3}+\beta _{5}+\beta _{7})q^{2}+(2-\beta _{1}-\beta _{3}+\cdots)q^{3}+\cdots\)
715.2.z.b 715.z 13.e $32$ $5.709$ None 715.2.z.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
715.2.z.c 715.z 13.e $56$ $5.709$ None 715.2.z.c \(0\) \(-6\) \(0\) \(6\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{2}^{\mathrm{old}}(715, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(715, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(143, [\chi])\)\(^{\oplus 2}\)