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Properties

Label 325.2.f
Level $325$
Weight $2$
Character orbit 325.f
Rep. character $\chi_{325}(18,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $38$
Newform subspaces $4$
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.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 65 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 4 \)
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 82 46 36
Cusp forms 58 38 20
Eisenstein series 24 8 16

Trace form

\( 38 q + 4 q^{3} - 34 q^{4} - 4 q^{6} + 4 q^{7} + O(q^{10}) \) \( 38 q + 4 q^{3} - 34 q^{4} - 4 q^{6} + 4 q^{7} - 20 q^{11} - 10 q^{13} + 18 q^{16} - 14 q^{17} - 18 q^{18} - 12 q^{19} - 8 q^{22} + 20 q^{23} + 52 q^{24} - 36 q^{26} - 20 q^{27} + 12 q^{28} - 16 q^{31} - 2 q^{34} + 44 q^{37} - 8 q^{38} + 8 q^{39} + 14 q^{41} - 20 q^{42} + 4 q^{43} + 48 q^{44} - 28 q^{46} - 28 q^{47} + 16 q^{48} - 50 q^{49} + 42 q^{52} + 14 q^{53} - 36 q^{54} - 24 q^{58} + 16 q^{59} - 8 q^{61} + 40 q^{62} + 6 q^{64} + 56 q^{66} - 2 q^{68} + 16 q^{69} + 4 q^{71} + 22 q^{72} - 12 q^{76} + 20 q^{77} - 4 q^{78} + 34 q^{81} - 34 q^{82} - 60 q^{83} + 24 q^{84} + 84 q^{86} - 16 q^{87} + 16 q^{88} - 6 q^{89} - 48 q^{91} - 44 q^{92} + 20 q^{93} - 212 q^{96} + 36 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
325.2.f.a 325.f 65.f $2$ $2.595$ \(\Q(\sqrt{-1}) \) None 65.2.f.a \(0\) \(-2\) \(0\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q-iq^{2}+(-1+i)q^{3}+q^{4}+(1+i)q^{6}+\cdots\)
325.2.f.b 325.f 65.f $8$ $2.595$ 8.0.619810816.2 None 65.2.f.b \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{6}q^{2}+(1+\beta _{2}-\beta _{3})q^{3}+(-1-\beta _{4}+\cdots)q^{4}+\cdots\)
325.2.f.c 325.f 65.f $12$ $2.595$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 325.2.f.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{7}q^{2}-\beta _{4}q^{3}+(-2+\beta _{5})q^{4}+(-1+\cdots)q^{6}+\cdots\)
325.2.f.d 325.f 65.f $16$ $2.595$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 325.2.f.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{13}q^{2}+\beta _{3}q^{3}+\beta _{1}q^{4}+(1+\beta _{4}+\cdots)q^{6}+\cdots\)

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

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