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Properties

Label 325.2.o
Level $325$
Weight $2$
Character orbit 325.o
Rep. character $\chi_{325}(74,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $36$
Newform subspaces $3$
Sturm bound $70$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.o (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 65 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(70\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 84 44 40
Cusp forms 60 36 24
Eisenstein series 24 8 16

Trace form

\( 36 q + 16 q^{4} + 6 q^{6} + 8 q^{9} + O(q^{10}) \) \( 36 q + 16 q^{4} + 6 q^{6} + 8 q^{9} - 6 q^{11} + 40 q^{14} + 4 q^{16} - 8 q^{19} - 20 q^{24} - 30 q^{26} - 10 q^{29} - 32 q^{31} - 84 q^{34} + 42 q^{36} + 20 q^{39} - 36 q^{41} - 8 q^{44} - 14 q^{46} - 2 q^{49} - 48 q^{51} + 58 q^{54} + 2 q^{56} - 10 q^{61} + 20 q^{64} + 68 q^{66} + 8 q^{69} + 2 q^{71} - 16 q^{74} + 4 q^{76} + 72 q^{79} - 18 q^{81} - 104 q^{84} - 48 q^{86} - 52 q^{89} - 6 q^{91} - 70 q^{94} + 76 q^{96} - 176 q^{99} + O(q^{100}) \)

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.o.a 325.o 65.n $8$ $2.595$ 8.0.12960000.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{6}q^{2}+(\beta _{3}+2\beta _{6}+\beta _{7})q^{3}+(-\beta _{2}+\cdots)q^{4}+\cdots\)
325.2.o.b 325.o 65.n $8$ $2.595$ 8.0.592240896.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(1-2\beta _{3}+\beta _{4}+\beta _{6}+\cdots)q^{4}+\cdots\)
325.2.o.c 325.o 65.n $20$ $2.595$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}-\beta _{10}q^{3}+(\beta _{7}+\beta _{11})q^{4}+\cdots\)

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

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