Properties

Label 77.2.e
Level $77$
Weight $2$
Character orbit 77.e
Rep. character $\chi_{77}(23,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $12$
Newform subspaces $2$
Sturm bound $16$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 77.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(16\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(77, [\chi])\).

Total New Old
Modular forms 20 12 8
Cusp forms 12 12 0
Eisenstein series 8 0 8

Trace form

\( 12 q - 2 q^{3} - 4 q^{4} - 4 q^{5} - 8 q^{6} + 2 q^{7} - 12 q^{8} + O(q^{10}) \) \( 12 q - 2 q^{3} - 4 q^{4} - 4 q^{5} - 8 q^{6} + 2 q^{7} - 12 q^{8} + 6 q^{10} + 6 q^{12} - 16 q^{13} + 4 q^{15} + 4 q^{16} + 6 q^{17} + 2 q^{18} + 2 q^{19} + 40 q^{20} - 2 q^{21} - 12 q^{23} - 8 q^{24} - 6 q^{25} - 10 q^{26} + 16 q^{27} + 16 q^{28} - 24 q^{29} + 4 q^{30} - 6 q^{31} + 12 q^{32} - 4 q^{33} - 32 q^{34} - 6 q^{35} - 12 q^{36} + 4 q^{37} - 8 q^{38} + 2 q^{39} + 8 q^{41} - 20 q^{42} + 4 q^{43} + 4 q^{44} + 6 q^{45} + 34 q^{46} + 6 q^{47} - 16 q^{48} - 24 q^{49} + 24 q^{50} + 16 q^{51} - 2 q^{52} - 26 q^{53} + 26 q^{54} + 16 q^{55} + 6 q^{56} + 40 q^{57} - 2 q^{58} - 8 q^{59} - 12 q^{60} + 12 q^{61} + 20 q^{62} + 18 q^{63} - 20 q^{64} - 30 q^{65} + 2 q^{66} + 16 q^{67} + 16 q^{68} + 36 q^{69} + 18 q^{70} - 4 q^{71} - 22 q^{72} + 14 q^{73} - 40 q^{74} - 34 q^{75} - 96 q^{76} - 2 q^{77} + 18 q^{80} + 2 q^{81} - 38 q^{82} - 52 q^{83} + 12 q^{84} - 32 q^{85} + 30 q^{86} - 6 q^{87} - 12 q^{88} + 14 q^{89} - 52 q^{90} - 6 q^{91} + 44 q^{92} + 20 q^{93} + 10 q^{94} + 8 q^{95} - 30 q^{96} + 108 q^{97} - 30 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(77, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
77.2.e.a 77.e 7.c $6$ $0.615$ \(\Q(\zeta_{18})\) None \(0\) \(-3\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\zeta_{18}+\zeta_{18}^{2}+\zeta_{18}^{4}-\zeta_{18}^{5})q^{2}+\cdots\)
77.2.e.b 77.e 7.c $6$ $0.615$ 6.0.1783323.2 None \(0\) \(1\) \(2\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{3}+\beta _{5})q^{2}+\beta _{1}q^{3}+(-1-\beta _{1}+\cdots)q^{4}+\cdots\)