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Properties

Label 325.2.x
Level $325$
Weight $2$
Character orbit 325.x
Rep. character $\chi_{325}(7,\cdot)$
Character field $\Q(\zeta_{12})$
Dimension $76$
Newform subspaces $3$
Sturm bound $70$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.x (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 65 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 3 \)
Sturm bound: \(70\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(325, [\chi])\).

Total New Old
Modular forms 164 92 72
Cusp forms 116 76 40
Eisenstein series 48 16 32

Trace form

\( 76 q + 6 q^{2} + 2 q^{3} + 34 q^{4} - 8 q^{6} + 2 q^{7} - 24 q^{9} + O(q^{10}) \) \( 76 q + 6 q^{2} + 2 q^{3} + 34 q^{4} - 8 q^{6} + 2 q^{7} - 24 q^{9} + 8 q^{11} + 24 q^{12} + 4 q^{13} - 30 q^{16} - 4 q^{17} - 12 q^{21} - 16 q^{22} + 10 q^{23} - 16 q^{24} + 24 q^{26} - 4 q^{27} - 18 q^{28} + 16 q^{31} - 48 q^{32} - 18 q^{33} + 2 q^{34} - 108 q^{36} + 4 q^{37} + 8 q^{38} - 8 q^{39} - 74 q^{41} - 40 q^{42} - 10 q^{43} + 48 q^{44} - 32 q^{46} + 40 q^{47} + 56 q^{48} - 22 q^{49} + 30 q^{52} + 10 q^{53} + 108 q^{56} + 32 q^{59} + 20 q^{61} + 44 q^{62} + 36 q^{63} - 36 q^{64} + 16 q^{66} - 18 q^{67} - 22 q^{68} + 32 q^{69} + 8 q^{71} - 4 q^{72} - 126 q^{74} + 72 q^{76} + 28 q^{77} - 68 q^{78} - 22 q^{81} - 56 q^{82} - 48 q^{83} + 96 q^{84} - 144 q^{86} + 34 q^{87} - 82 q^{88} - 18 q^{89} - 60 q^{91} + 8 q^{92} - 32 q^{93} + 24 q^{94} - 16 q^{96} - 66 q^{97} + 30 q^{98} - 156 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(325, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
325.2.x.a 325.x 65.t $16$ $2.595$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+\beta _{8}q^{2}+(-\beta _{1}+\beta _{4}-\beta _{6}-\beta _{8}-\beta _{11}+\cdots)q^{3}+\cdots\)
325.2.x.b 325.x 65.t $20$ $2.595$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(6\) \(2\) \(0\) \(2\) $\mathrm{SU}(2)[C_{12}]$ \(q+\beta _{2}q^{2}+(\beta _{2}+\beta _{4}+\beta _{8}-\beta _{12}+\beta _{15}+\cdots)q^{3}+\cdots\)
325.2.x.c 325.x 65.t $40$ $2.595$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$

Decomposition of \(S_{2}^{\mathrm{old}}(325, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(325, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 2}\)