Properties

Label 78.3.d
Level $78$
Weight $3$
Character orbit 78.d
Rep. character $\chi_{78}(77,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $2$
Sturm bound $42$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 78.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 39 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(42\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 32 8 24
Cusp forms 24 8 16
Eisenstein series 8 0 8

Trace form

\( 8 q + 16 q^{4} - 8 q^{9} + O(q^{10}) \) \( 8 q + 16 q^{4} - 8 q^{9} + 16 q^{10} - 40 q^{13} + 32 q^{16} - 120 q^{25} - 64 q^{30} - 16 q^{36} + 128 q^{39} + 32 q^{40} - 80 q^{42} - 8 q^{49} + 320 q^{51} - 80 q^{52} + 192 q^{55} + 64 q^{64} - 192 q^{66} + 80 q^{69} - 128 q^{75} + 160 q^{78} + 160 q^{79} - 632 q^{81} + 80 q^{82} + 160 q^{87} - 16 q^{90} - 160 q^{91} + 192 q^{94} + O(q^{100}) \)

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.d.a 78.d 39.d $4$ $2.125$ \(\Q(\sqrt{2}, \sqrt{-5})\) None \(0\) \(-8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-2-\beta _{3})q^{3}+2q^{4}+3\beta _{1}q^{5}+\cdots\)
78.3.d.b 78.d 39.d $4$ $2.125$ \(\Q(\sqrt{2}, \sqrt{-5})\) None \(0\) \(8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(2+\beta _{3})q^{3}+2q^{4}-\beta _{1}q^{5}+\cdots\)

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

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