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Properties

Label 77.4.f
Level $77$
Weight $4$
Character orbit 77.f
Rep. character $\chi_{77}(15,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $72$
Newform subspaces $2$
Sturm bound $32$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 77.f (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 2 \)
Sturm bound: \(32\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(77, [\chi])\).

Total New Old
Modular forms 104 72 32
Cusp forms 88 72 16
Eisenstein series 16 0 16

Trace form

\( 72 q + 6 q^{2} - 82 q^{4} - 8 q^{5} + 60 q^{6} + 14 q^{7} - 10 q^{8} - 182 q^{9} + O(q^{10}) \) \( 72 q + 6 q^{2} - 82 q^{4} - 8 q^{5} + 60 q^{6} + 14 q^{7} - 10 q^{8} - 182 q^{9} + 144 q^{10} - 136 q^{11} - 256 q^{12} + 72 q^{13} + 105 q^{14} + 32 q^{15} + 198 q^{16} - 260 q^{17} + 249 q^{18} - 318 q^{19} - 698 q^{20} - 448 q^{21} - 52 q^{22} - 108 q^{23} + 1456 q^{24} - 126 q^{25} + 554 q^{26} - 6 q^{27} + 427 q^{28} - 20 q^{29} + 640 q^{30} - 112 q^{31} - 944 q^{32} + 2170 q^{33} + 428 q^{34} + 280 q^{35} - 1314 q^{36} - 444 q^{37} + 46 q^{38} - 28 q^{39} - 2266 q^{40} - 180 q^{41} + 848 q^{43} + 999 q^{44} - 1880 q^{45} + 281 q^{46} - 1452 q^{47} + 864 q^{48} - 882 q^{49} - 1228 q^{50} + 2602 q^{51} + 2494 q^{52} - 496 q^{53} + 7456 q^{54} + 164 q^{55} - 462 q^{56} - 862 q^{57} - 481 q^{58} - 1990 q^{59} + 1192 q^{60} - 984 q^{61} - 8042 q^{62} + 630 q^{63} + 2966 q^{64} - 5992 q^{65} - 5314 q^{66} + 2136 q^{67} - 8726 q^{68} - 1812 q^{69} - 756 q^{70} - 954 q^{71} + 2477 q^{72} + 1404 q^{73} + 4458 q^{74} + 5730 q^{75} + 7172 q^{76} - 364 q^{77} + 1692 q^{78} + 3834 q^{79} + 7748 q^{80} - 4476 q^{81} + 4860 q^{82} + 582 q^{83} + 3696 q^{84} - 5140 q^{85} + 3531 q^{86} - 1208 q^{87} - 13332 q^{88} - 4452 q^{89} - 4752 q^{90} - 756 q^{91} + 11699 q^{92} + 68 q^{93} + 8254 q^{94} + 444 q^{95} - 16292 q^{96} - 4290 q^{97} - 686 q^{98} + 1198 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(77, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
77.4.f.a 77.f 11.c $32$ $4.543$ None \(-2\) \(18\) \(16\) \(-56\) $\mathrm{SU}(2)[C_{5}]$
77.4.f.b 77.f 11.c $40$ $4.543$ None \(8\) \(-18\) \(-24\) \(70\) $\mathrm{SU}(2)[C_{5}]$

Decomposition of \(S_{4}^{\mathrm{old}}(77, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(77, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(11, [\chi])\)\(^{\oplus 2}\)