Properties

Label 78.3.j
Level $78$
Weight $3$
Character orbit 78.j
Rep. character $\chi_{78}(17,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $20$
Newform subspaces $1$
Sturm bound $42$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 78.j (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 39 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(42\)
Trace bound: \(0\)

Dimensions

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

Total New Old
Modular forms 64 20 44
Cusp forms 48 20 28
Eisenstein series 16 0 16

Trace form

\( 20 q - 20 q^{4} + 18 q^{7} - 4 q^{9} + O(q^{10}) \) \( 20 q - 20 q^{4} + 18 q^{7} - 4 q^{9} + 8 q^{10} - 42 q^{13} + 60 q^{15} - 40 q^{16} - 84 q^{19} + 260 q^{25} - 36 q^{27} - 36 q^{28} + 4 q^{30} - 258 q^{33} - 8 q^{36} - 192 q^{37} + 46 q^{39} - 32 q^{40} + 32 q^{42} + 26 q^{43} + 180 q^{45} + 144 q^{46} + 264 q^{49} - 188 q^{51} + 12 q^{52} + 324 q^{54} - 120 q^{55} - 168 q^{58} - 98 q^{61} + 252 q^{63} + 160 q^{64} + 144 q^{66} - 498 q^{67} - 146 q^{69} - 144 q^{72} - 556 q^{75} + 168 q^{76} - 220 q^{78} + 492 q^{79} + 212 q^{81} + 16 q^{82} + 168 q^{84} + 540 q^{85} + 302 q^{87} - 512 q^{90} + 10 q^{91} + 750 q^{93} + 48 q^{94} - 498 q^{97} + O(q^{100}) \)

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.j.a 78.j 39.h $20$ $2.125$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(0\) \(0\) \(18\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{5}q^{2}-\beta _{1}q^{3}+(-2-2\beta _{6})q^{4}+\cdots\)

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

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