Properties

Label 77.2.a
Level $77$
Weight $2$
Character orbit 77.a
Rep. character $\chi_{77}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $4$
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.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(16\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(77))\).

Total New Old
Modular forms 10 5 5
Cusp forms 7 5 2
Eisenstein series 3 0 3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(7\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(3\)
Plus space\(+\)\(1\)
Minus space\(-\)\(4\)

Trace form

\( 5 q + q^{2} + 2 q^{3} + q^{4} - 4 q^{5} - 8 q^{6} + q^{7} - 3 q^{8} + 11 q^{9} + O(q^{10}) \) \( 5 q + q^{2} + 2 q^{3} + q^{4} - 4 q^{5} - 8 q^{6} + q^{7} - 3 q^{8} + 11 q^{9} - 2 q^{10} - 3 q^{11} + 8 q^{12} - 2 q^{13} - q^{14} - 2 q^{15} + 5 q^{16} - 2 q^{17} - 19 q^{18} - 14 q^{20} + 4 q^{21} + q^{22} - 10 q^{23} - 16 q^{24} - 3 q^{25} + 14 q^{26} + 2 q^{27} + 7 q^{28} + 6 q^{29} + 16 q^{30} + 6 q^{31} + 5 q^{32} + 2 q^{33} - 6 q^{34} + 2 q^{35} + 9 q^{36} - 8 q^{37} + 20 q^{38} + 8 q^{39} + 6 q^{40} - 10 q^{41} - 12 q^{42} + 28 q^{43} - 3 q^{44} - 26 q^{45} + 16 q^{46} + 8 q^{47} - 12 q^{48} + 5 q^{49} - q^{50} + 4 q^{51} + 18 q^{52} - 10 q^{53} - 24 q^{54} + 3 q^{56} + 4 q^{57} + 14 q^{58} - 2 q^{59} - 20 q^{60} - 22 q^{61} + 20 q^{62} - 3 q^{63} - 35 q^{64} - 20 q^{65} + 12 q^{66} + 30 q^{67} - 2 q^{68} - 14 q^{69} + 2 q^{70} - 14 q^{71} - 23 q^{72} - 2 q^{73} - 26 q^{74} + 12 q^{75} + 20 q^{76} - 3 q^{77} + 8 q^{78} + 4 q^{79} + 14 q^{80} + 21 q^{81} - 6 q^{82} + 28 q^{83} - 24 q^{85} + 12 q^{86} - 60 q^{87} - 3 q^{88} - 20 q^{89} + 38 q^{90} - 2 q^{91} - 4 q^{92} + 2 q^{93} - 20 q^{94} + 4 q^{95} + 40 q^{96} - 8 q^{97} + q^{98} - 9 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(77))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 7 11
77.2.a.a 77.a 1.a $1$ $0.615$ \(\Q\) None \(0\) \(-3\) \(-1\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-2q^{4}-q^{5}-q^{7}+6q^{9}-q^{11}+\cdots\)
77.2.a.b 77.a 1.a $1$ $0.615$ \(\Q\) None \(0\) \(1\) \(3\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}+3q^{5}+q^{7}-2q^{9}-q^{11}+\cdots\)
77.2.a.c 77.a 1.a $1$ $0.615$ \(\Q\) None \(1\) \(2\) \(-2\) \(-1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}-q^{4}-2q^{5}+2q^{6}-q^{7}+\cdots\)
77.2.a.d 77.a 1.a $2$ $0.615$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(-4\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(1+\beta )q^{3}+3q^{4}-2q^{5}+(-5+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(77))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(77)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)