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Properties

Label 77.8.a
Level $77$
Weight $8$
Character orbit 77.a
Rep. character $\chi_{77}(1,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $4$
Sturm bound $64$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 77.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(64\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(77))\).

Total New Old
Modular forms 58 36 22
Cusp forms 54 36 18
Eisenstein series 4 0 4

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
\(+\)\(+\)$+$\(9\)
\(+\)\(-\)$-$\(9\)
\(-\)\(+\)$-$\(7\)
\(-\)\(-\)$+$\(11\)
Plus space\(+\)\(20\)
Minus space\(-\)\(16\)

Trace form

\( 36 q + 16 q^{2} - 26 q^{3} + 2328 q^{4} - 634 q^{5} + 132 q^{6} - 2124 q^{8} + 31318 q^{9} + O(q^{10}) \) \( 36 q + 16 q^{2} - 26 q^{3} + 2328 q^{4} - 634 q^{5} + 132 q^{6} - 2124 q^{8} + 31318 q^{9} + 1448 q^{10} + 5324 q^{11} + 17380 q^{12} + 4832 q^{13} + 10976 q^{14} - 5778 q^{15} + 184824 q^{16} - 37220 q^{17} + 227256 q^{18} - 143844 q^{19} - 251468 q^{20} + 21296 q^{22} + 102278 q^{23} - 434596 q^{24} + 860454 q^{25} - 88564 q^{26} - 288446 q^{27} - 491016 q^{29} + 841456 q^{30} + 243150 q^{31} - 412276 q^{32} - 220946 q^{33} + 470600 q^{34} - 58996 q^{35} + 1199552 q^{36} + 378298 q^{37} + 2162060 q^{38} + 1863888 q^{39} - 2685984 q^{40} + 1341980 q^{41} - 862988 q^{42} + 533712 q^{43} + 851840 q^{44} - 2140384 q^{45} - 6609016 q^{46} + 1519880 q^{47} + 9726692 q^{48} + 4235364 q^{49} + 2082200 q^{50} - 1704980 q^{51} + 4575980 q^{52} - 5042512 q^{53} - 4261288 q^{54} + 1349634 q^{55} + 2107392 q^{56} - 3892012 q^{57} - 4302608 q^{58} + 8264122 q^{59} - 7723640 q^{60} + 9349676 q^{61} - 8474572 q^{62} - 993328 q^{63} + 10291128 q^{64} - 4397144 q^{65} + 4947714 q^{67} - 22271324 q^{68} - 13294822 q^{69} - 3126788 q^{70} + 9649234 q^{71} + 39244204 q^{72} - 8394268 q^{73} + 17633200 q^{74} - 7649376 q^{75} - 12624744 q^{76} + 1826132 q^{77} - 29522040 q^{78} - 5886012 q^{79} - 29281780 q^{80} + 35605628 q^{81} - 5935504 q^{82} - 924728 q^{83} - 16547692 q^{84} - 9965588 q^{85} + 9845640 q^{86} - 7958820 q^{87} + 846516 q^{88} - 12584966 q^{89} + 37844016 q^{90} + 13747440 q^{91} - 40848424 q^{92} + 17026602 q^{93} - 12534708 q^{94} - 7719044 q^{95} - 78044636 q^{96} - 32342722 q^{97} + 1882384 q^{98} + 4948658 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(77))\) 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
77.8.a.a 77.a 1.a $7$ $24.054$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(-46\) \(-580\) \(2401\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-7+\beta _{1}-\beta _{2})q^{3}+(33+\cdots)q^{4}+\cdots\)
77.8.a.b 77.a 1.a $9$ $24.054$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-8\) \(-129\) \(13\) \(-3087\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(-14-\beta _{5})q^{3}+(61+\cdots)q^{4}+\cdots\)
77.8.a.c 77.a 1.a $9$ $24.054$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(116\) \(-244\) \(-3087\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(13+\beta _{2})q^{3}+(68+\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
77.8.a.d 77.a 1.a $11$ $24.054$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(24\) \(33\) \(177\) \(3773\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(2+\beta _{1})q^{2}+(3-\beta _{3})q^{3}+(85+\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(77))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(77)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)