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Properties

Label 78.3.l
Level $78$
Weight $3$
Character orbit 78.l
Rep. character $\chi_{78}(7,\cdot)$
Character field $\Q(\zeta_{12})$
Dimension $16$
Newform subspaces $3$
Sturm bound $42$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 78.l (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 3 \)
Sturm bound: \(42\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(78, [\chi])\).

Total New Old
Modular forms 128 16 112
Cusp forms 96 16 80
Eisenstein series 32 0 32

Trace form

\( 16 q + 4 q^{2} - 12 q^{5} + 20 q^{7} + 16 q^{8} - 24 q^{9} + O(q^{10}) \) \( 16 q + 4 q^{2} - 12 q^{5} + 20 q^{7} + 16 q^{8} - 24 q^{9} + 12 q^{10} - 32 q^{14} - 48 q^{15} + 32 q^{16} - 72 q^{17} - 24 q^{18} + 116 q^{19} - 24 q^{20} - 60 q^{21} + 48 q^{22} + 48 q^{23} - 52 q^{26} + 40 q^{28} - 24 q^{29} + 100 q^{31} - 16 q^{32} - 72 q^{33} + 24 q^{34} - 192 q^{35} + 40 q^{37} + 144 q^{39} + 48 q^{40} - 72 q^{41} - 108 q^{43} - 48 q^{44} - 192 q^{46} + 312 q^{47} - 300 q^{49} - 80 q^{50} - 8 q^{52} + 216 q^{53} + 264 q^{55} + 192 q^{56} + 48 q^{57} + 84 q^{58} + 240 q^{59} - 48 q^{60} - 36 q^{61} + 144 q^{62} - 48 q^{63} - 264 q^{65} + 192 q^{66} + 32 q^{67} - 72 q^{68} - 384 q^{70} + 72 q^{71} - 24 q^{72} - 352 q^{73} - 52 q^{74} + 420 q^{75} - 232 q^{76} + 192 q^{78} - 192 q^{79} - 48 q^{80} - 72 q^{81} + 276 q^{82} + 432 q^{83} + 120 q^{84} - 492 q^{85} + 240 q^{86} + 120 q^{87} + 588 q^{89} - 244 q^{91} - 288 q^{92} + 12 q^{93} + 48 q^{94} - 480 q^{95} - 344 q^{97} + 124 q^{98} - 72 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(78, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
78.3.l.a 78.l 13.f $4$ $2.125$ \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(-12\) \(-10\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(\zeta_{12}+\cdots)q^{3}+\cdots\)
78.3.l.b 78.l 13.f $4$ $2.125$ \(\Q(\zeta_{12})\) None \(2\) \(0\) \(6\) \(20\) $\mathrm{SU}(2)[C_{12}]$ \(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-\zeta_{12}+\cdots)q^{3}+\cdots\)
78.3.l.c 78.l 13.f $8$ $2.125$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(4\) \(0\) \(-6\) \(10\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-\beta _{4}-\beta _{5})q^{2}+(2\beta _{2}-\beta _{5})q^{3}+(2\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(78, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(78, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)