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Properties

Label 78.2.a
Level $78$
Weight $2$
Character orbit 78.a
Rep. character $\chi_{78}(1,\cdot)$
Character field $\Q$
Dimension $1$
Newform subspaces $1$
Sturm bound $28$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 78.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(28\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(78))\).

Total New Old
Modular forms 18 1 17
Cusp forms 11 1 10
Eisenstein series 7 0 7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(13\)FrickeDim
\(+\)\(+\)\(-\)$-$\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(1\)

Trace form

\( q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} + 4 q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} + q^{4} + 2 q^{5} + q^{6} + 4 q^{7} - q^{8} + q^{9} - 2 q^{10} - 4 q^{11} - q^{12} + q^{13} - 4 q^{14} - 2 q^{15} + q^{16} + 2 q^{17} - q^{18} - 8 q^{19} + 2 q^{20} - 4 q^{21} + 4 q^{22} + q^{24} - q^{25} - q^{26} - q^{27} + 4 q^{28} + 6 q^{29} + 2 q^{30} - 4 q^{31} - q^{32} + 4 q^{33} - 2 q^{34} + 8 q^{35} + q^{36} - 2 q^{37} + 8 q^{38} - q^{39} - 2 q^{40} - 10 q^{41} + 4 q^{42} + 4 q^{43} - 4 q^{44} + 2 q^{45} + 8 q^{47} - q^{48} + 9 q^{49} + q^{50} - 2 q^{51} + q^{52} - 10 q^{53} + q^{54} - 8 q^{55} - 4 q^{56} + 8 q^{57} - 6 q^{58} + 4 q^{59} - 2 q^{60} - 2 q^{61} + 4 q^{62} + 4 q^{63} + q^{64} + 2 q^{65} - 4 q^{66} - 16 q^{67} + 2 q^{68} - 8 q^{70} - 8 q^{71} - q^{72} + 2 q^{73} + 2 q^{74} + q^{75} - 8 q^{76} - 16 q^{77} + q^{78} + 8 q^{79} + 2 q^{80} + q^{81} + 10 q^{82} + 12 q^{83} - 4 q^{84} + 4 q^{85} - 4 q^{86} - 6 q^{87} + 4 q^{88} + 14 q^{89} - 2 q^{90} + 4 q^{91} + 4 q^{93} - 8 q^{94} - 16 q^{95} + q^{96} + 10 q^{97} - 9 q^{98} - 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(78))\) into newform subspaces

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

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(78))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(78)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 2}\)