Properties

Label 78.2.i
Level $78$
Weight $2$
Character orbit 78.i
Rep. character $\chi_{78}(43,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $8$
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.i (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
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 8 28
Cusp forms 20 8 12
Eisenstein series 16 0 16

Trace form

\( 8q + 4q^{4} - 4q^{9} + O(q^{10}) \) \( 8q + 4q^{4} - 4q^{9} + 4q^{10} - 4q^{13} - 16q^{14} - 12q^{15} - 4q^{16} - 8q^{17} - 8q^{23} + 8q^{26} + 8q^{29} + 4q^{30} + 12q^{33} + 16q^{35} + 4q^{36} - 12q^{37} + 4q^{39} + 8q^{40} + 48q^{41} - 4q^{42} + 4q^{49} + 24q^{50} + 16q^{51} - 8q^{52} - 8q^{56} - 12q^{58} - 48q^{59} - 12q^{61} - 16q^{62} - 8q^{64} - 8q^{65} + 24q^{66} - 48q^{67} + 8q^{68} + 4q^{69} + 24q^{71} - 8q^{74} + 8q^{75} - 48q^{77} - 16q^{78} - 32q^{79} - 4q^{81} - 4q^{82} - 12q^{84} + 36q^{85} - 4q^{87} - 24q^{89} - 8q^{90} + 32q^{91} - 16q^{92} - 24q^{93} + 24q^{95} + 24q^{98} + O(q^{100}) \)

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

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
78.2.i.a \(4\) \(0.623\) \(\Q(\zeta_{12})\) None \(0\) \(-2\) \(0\) \(6\) \(q+\zeta_{12}q^{2}+(-1+\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
78.2.i.b \(4\) \(0.623\) \(\Q(\zeta_{12})\) None \(0\) \(2\) \(0\) \(-6\) \(q+\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)

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

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