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Properties

Label 69.2.e
Level $69$
Weight $2$
Character orbit 69.e
Rep. character $\chi_{69}(4,\cdot)$
Character field $\Q(\zeta_{11})$
Dimension $40$
Newform subspaces $3$
Sturm bound $16$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 69.e (of order \(11\) and degree \(10\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 23 \)
Character field: \(\Q(\zeta_{11})\)
Newform subspaces: \( 3 \)
Sturm bound: \(16\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(69, [\chi])\).

Total New Old
Modular forms 100 40 60
Cusp forms 60 40 20
Eisenstein series 40 0 40

Trace form

\( 40 q - 4 q^{2} - 12 q^{4} - 4 q^{5} - 4 q^{6} - 8 q^{7} - 12 q^{8} - 4 q^{9} + O(q^{10}) \) \( 40 q - 4 q^{2} - 12 q^{4} - 4 q^{5} - 4 q^{6} - 8 q^{7} - 12 q^{8} - 4 q^{9} - 8 q^{10} - 24 q^{11} - 8 q^{13} - 12 q^{14} - 8 q^{15} + 16 q^{16} + 10 q^{17} - 4 q^{18} - 10 q^{19} + 52 q^{20} - 4 q^{21} + 4 q^{22} + 20 q^{23} - 12 q^{24} + 16 q^{25} - 12 q^{26} + 24 q^{28} - 10 q^{29} - 28 q^{30} - 10 q^{31} - 24 q^{32} - 8 q^{33} - 26 q^{34} - 12 q^{35} + 10 q^{36} + 4 q^{37} + 58 q^{38} + 36 q^{39} - 58 q^{40} + 4 q^{41} + 78 q^{42} - 12 q^{43} + 50 q^{44} - 4 q^{45} + 20 q^{46} + 48 q^{47} + 56 q^{48} - 52 q^{49} + 46 q^{50} + 12 q^{51} + 6 q^{52} + 8 q^{53} + 18 q^{54} + 2 q^{55} - 6 q^{56} + 32 q^{57} - 6 q^{58} - 12 q^{59} - 26 q^{60} - 12 q^{61} - 120 q^{62} - 8 q^{63} - 20 q^{64} - 64 q^{65} - 40 q^{66} - 42 q^{67} - 92 q^{68} - 20 q^{69} - 56 q^{70} - 64 q^{71} - 12 q^{72} - 34 q^{73} - 74 q^{74} - 24 q^{75} + 30 q^{76} - 8 q^{77} - 16 q^{78} + 36 q^{79} + 10 q^{80} - 4 q^{81} + 8 q^{82} - 16 q^{83} - 52 q^{84} + 114 q^{85} + 48 q^{86} - 32 q^{87} + 8 q^{88} + 56 q^{89} - 8 q^{90} + 72 q^{91} + 30 q^{92} - 40 q^{93} + 54 q^{94} + 78 q^{95} - 24 q^{96} + 36 q^{97} + 116 q^{98} - 2 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(69, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
69.2.e.a 69.e 23.c $10$ $0.551$ \(\Q(\zeta_{22})\) None \(-4\) \(1\) \(5\) \(-8\) $\mathrm{SU}(2)[C_{11}]$ \(q+(-1+\zeta_{22}-\zeta_{22}^{2}-\zeta_{22}^{4}-\zeta_{22}^{6}+\cdots)q^{2}+\cdots\)
69.2.e.b 69.e 23.c $10$ $0.551$ \(\Q(\zeta_{22})\) None \(4\) \(1\) \(-3\) \(6\) $\mathrm{SU}(2)[C_{11}]$ \(q+(-\zeta_{22}^{4}+\zeta_{22}^{5}-\zeta_{22}^{8}+\zeta_{22}^{9})q^{2}+\cdots\)
69.2.e.c 69.e 23.c $20$ $0.551$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-4\) \(-2\) \(-6\) \(-6\) $\mathrm{SU}(2)[C_{11}]$ \(q+(\beta _{1}+\beta _{3}-\beta _{4}+\beta _{5}+2\beta _{6}+2\beta _{7}+\cdots)q^{2}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(69, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(69, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(23, [\chi])\)\(^{\oplus 2}\)