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Properties

Label 78.2.e
Level $78$
Weight $2$
Character orbit 78.e
Rep. character $\chi_{78}(55,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $4$
Newform subspaces $2$
Sturm bound $28$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 78.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(28\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(78, [\chi])\).

Total New Old
Modular forms 36 4 32
Cusp forms 20 4 16
Eisenstein series 16 0 16

Trace form

\( 4 q + 2 q^{2} - 2 q^{4} + 4 q^{5} - 4 q^{8} - 2 q^{9} + O(q^{10}) \) \( 4 q + 2 q^{2} - 2 q^{4} + 4 q^{5} - 4 q^{8} - 2 q^{9} + 2 q^{10} - 8 q^{11} - 2 q^{13} - 4 q^{15} - 2 q^{16} - 2 q^{17} - 4 q^{18} - 2 q^{20} + 8 q^{21} + 8 q^{22} + 2 q^{26} + 6 q^{29} + 4 q^{30} - 16 q^{31} + 2 q^{32} - 4 q^{33} - 4 q^{34} - 8 q^{35} - 2 q^{36} + 18 q^{37} + 12 q^{39} - 4 q^{40} - 2 q^{41} + 4 q^{42} + 16 q^{44} - 2 q^{45} + 16 q^{47} + 6 q^{49} - 16 q^{51} + 4 q^{52} + 4 q^{53} - 16 q^{55} + 8 q^{57} - 6 q^{58} + 8 q^{59} + 8 q^{60} + 18 q^{61} - 8 q^{62} + 4 q^{64} - 26 q^{65} - 8 q^{66} + 8 q^{67} - 2 q^{68} + 12 q^{69} - 16 q^{70} + 8 q^{71} + 2 q^{72} - 52 q^{73} - 18 q^{74} - 8 q^{75} + 16 q^{77} - 16 q^{79} - 2 q^{80} - 2 q^{81} + 2 q^{82} - 4 q^{84} + 14 q^{85} - 12 q^{87} + 8 q^{88} - 20 q^{89} - 4 q^{90} + 8 q^{94} - 8 q^{95} - 12 q^{97} - 6 q^{98} + 16 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(78, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
78.2.e.a 78.e 13.c $2$ $0.623$ \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(6\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
78.2.e.b 78.e 13.c $2$ $0.623$ \(\Q(\sqrt{-3}) \) None \(1\) \(1\) \(-2\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(78, [\chi])\) into lower level spaces

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