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Properties

Label 325.2.z
Level $325$
Weight $2$
Character orbit 325.z
Rep. character $\chi_{325}(8,\cdot)$
Character field $\Q(\zeta_{20})$
Dimension $264$
Newform subspaces $2$
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.z (of order \(20\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 325 \)
Character field: \(\Q(\zeta_{20})\)
Newform subspaces: \( 2 \)
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 296 296 0
Cusp forms 264 264 0
Eisenstein series 32 32 0

Trace form

\( 264 q - 8 q^{2} - 16 q^{3} - 62 q^{4} - 10 q^{5} - 6 q^{6} - 4 q^{8} - 20 q^{9} + O(q^{10}) \) \( 264 q - 8 q^{2} - 16 q^{3} - 62 q^{4} - 10 q^{5} - 6 q^{6} - 4 q^{8} - 20 q^{9} + 8 q^{10} - 6 q^{11} - 14 q^{13} - 20 q^{14} - 50 q^{15} - 66 q^{16} + 14 q^{17} + 10 q^{19} + 30 q^{20} + 24 q^{21} - 28 q^{22} - 20 q^{23} + 20 q^{24} - 6 q^{25} - 16 q^{26} - 40 q^{27} + 30 q^{28} - 40 q^{29} - 16 q^{30} - 6 q^{31} + 44 q^{32} + 72 q^{33} - 12 q^{34} - 4 q^{35} + 40 q^{36} + 70 q^{37} + 8 q^{38} - 50 q^{39} - 48 q^{40} - 46 q^{41} - 40 q^{42} - 4 q^{43} - 2 q^{44} - 72 q^{45} - 6 q^{46} - 60 q^{47} - 4 q^{48} - 168 q^{49} + 106 q^{50} - 172 q^{52} - 6 q^{53} + 146 q^{54} - 36 q^{55} + 140 q^{56} + 40 q^{57} - 130 q^{58} - 10 q^{59} - 54 q^{60} - 12 q^{61} - 40 q^{62} + 130 q^{63} - 62 q^{64} - 26 q^{65} - 12 q^{66} + 10 q^{67} - 22 q^{68} - 48 q^{70} - 6 q^{71} + 60 q^{72} + 94 q^{73} - 96 q^{75} + 120 q^{77} + 150 q^{78} - 20 q^{79} + 176 q^{80} + 22 q^{81} + 74 q^{82} - 80 q^{83} + 28 q^{84} + 38 q^{85} + 14 q^{86} - 36 q^{87} - 16 q^{88} - 50 q^{89} + 90 q^{90} + 124 q^{91} - 64 q^{92} + 140 q^{94} - 48 q^{95} - 136 q^{96} - 26 q^{97} + 2 q^{98} + 12 q^{99} + O(q^{100}) \)

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.z.a 325.z 325.z $8$ $2.595$ \(\Q(\zeta_{20})\) None \(-6\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{20}]$ \(q+(-\zeta_{20}^{2}+\zeta_{20}^{4}-\zeta_{20}^{6})q^{2}+(1+\cdots)q^{3}+\cdots\)
325.2.z.b 325.z 325.z $256$ $2.595$ None \(-2\) \(-16\) \(-10\) \(0\) $\mathrm{SU}(2)[C_{20}]$