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Properties

Label 65.2.e
Level $65$
Weight $2$
Character orbit 65.e
Rep. character $\chi_{65}(16,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $8$
Newform subspaces $2$
Sturm bound $14$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 65 = 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 65.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(14\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(65, [\chi])\).

Total New Old
Modular forms 16 8 8
Cusp forms 8 8 0
Eisenstein series 8 0 8

Trace form

\( 8 q - 2 q^{3} - 2 q^{4} + 6 q^{6} - 2 q^{7} - 12 q^{8} + O(q^{10}) \) \( 8 q - 2 q^{3} - 2 q^{4} + 6 q^{6} - 2 q^{7} - 12 q^{8} - 2 q^{10} - 4 q^{12} - 4 q^{13} - 4 q^{14} + 2 q^{15} + 6 q^{16} - 10 q^{17} + 8 q^{18} + 4 q^{20} + 16 q^{21} + 4 q^{22} - 6 q^{23} - 4 q^{24} + 8 q^{25} - 6 q^{26} - 20 q^{27} + 10 q^{28} + 4 q^{29} + 4 q^{30} - 16 q^{31} + 2 q^{32} + 14 q^{33} - 12 q^{34} - 6 q^{35} + 8 q^{36} - 6 q^{37} + 24 q^{38} + 12 q^{40} - 12 q^{41} + 4 q^{42} + 14 q^{43} + 20 q^{44} - 8 q^{45} - 14 q^{46} + 24 q^{47} - 12 q^{48} + 8 q^{49} - 24 q^{51} + 18 q^{52} + 40 q^{53} - 10 q^{54} - 8 q^{55} - 16 q^{56} + 12 q^{57} - 32 q^{58} - 12 q^{59} - 16 q^{60} - 4 q^{62} - 12 q^{63} - 4 q^{65} - 52 q^{66} + 6 q^{67} - 34 q^{68} + 16 q^{69} - 8 q^{70} + 20 q^{71} - 12 q^{72} - 56 q^{73} + 10 q^{74} - 2 q^{75} + 26 q^{76} + 4 q^{77} + 12 q^{78} - 8 q^{79} + 20 q^{81} + 20 q^{82} - 8 q^{83} + 18 q^{84} + 6 q^{85} + 60 q^{86} + 18 q^{87} - 2 q^{88} + 16 q^{89} - 20 q^{91} - 20 q^{92} + 8 q^{93} + 4 q^{94} - 8 q^{95} + 4 q^{96} + 22 q^{97} + 12 q^{98} + 32 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(65, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
65.2.e.a 65.e 13.c $4$ $0.519$ \(\Q(\sqrt{-3}, \sqrt{5})\) None \(-1\) \(0\) \(4\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(-1+2\beta _{1}-\beta _{3})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
65.2.e.b 65.e 13.c $4$ $0.519$ \(\Q(\sqrt{-3}, \sqrt{13})\) None \(1\) \(-2\) \(-4\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+(-1+\beta _{1}+\cdots)q^{4}+\cdots\)