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Properties

Label 78.4.g
Level $78$
Weight $4$
Character orbit 78.g
Rep. character $\chi_{78}(5,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $28$
Newform subspaces $1$
Sturm bound $56$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 78.g (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 39 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(56\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(78, [\chi])\).

Total New Old
Modular forms 92 28 64
Cusp forms 76 28 48
Eisenstein series 16 0 16

Trace form

\( 28 q + 20 q^{7} + O(q^{10}) \) \( 28 q + 20 q^{7} + 120 q^{13} + 168 q^{15} - 448 q^{16} - 96 q^{18} - 124 q^{19} + 264 q^{21} + 900 q^{27} - 80 q^{28} + 364 q^{31} - 1188 q^{33} - 672 q^{34} + 796 q^{37} - 912 q^{39} - 360 q^{42} - 252 q^{45} + 384 q^{46} + 592 q^{52} - 72 q^{54} + 144 q^{55} - 540 q^{57} + 1056 q^{58} + 672 q^{60} - 816 q^{61} + 252 q^{63} + 3072 q^{66} - 124 q^{67} + 192 q^{70} - 384 q^{72} + 1028 q^{73} - 496 q^{76} + 984 q^{78} - 10632 q^{79} - 3888 q^{81} - 1056 q^{84} + 9288 q^{85} + 1632 q^{87} - 844 q^{91} + 1788 q^{93} + 5232 q^{94} + 6628 q^{97} - 2052 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(78, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
78.4.g.a 78.g 39.f $28$ $4.602$ None 78.4.g.a \(0\) \(0\) \(0\) \(20\) $\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{4}^{\mathrm{old}}(78, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(78, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)