Properties

Label 78.3.f
Level $78$
Weight $3$
Character orbit 78.f
Rep. character $\chi_{78}(31,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $12$
Newform subspaces $2$
Sturm bound $42$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 78.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(42\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 64 12 52
Cusp forms 48 12 36
Eisenstein series 16 0 16

Trace form

\( 12 q - 4 q^{2} + 12 q^{5} + 8 q^{8} + 36 q^{9} + 24 q^{11} - 24 q^{13} - 16 q^{14} - 24 q^{15} - 48 q^{16} - 12 q^{18} - 24 q^{20} + 24 q^{21} - 48 q^{22} + 52 q^{26} - 96 q^{29} - 48 q^{31} + 16 q^{32}+ \cdots + 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{3}^{\mathrm{new}}(78, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
78.3.f.a 78.f 13.d $4$ $2.125$ \(\Q(\zeta_{12})\) None 78.3.f.a \(4\) \(0\) \(12\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\beta_1+1)q^{2}+\beta_{3} q^{3}+2\beta_1 q^{4}+\cdots\)
78.3.f.b 78.f 13.d $8$ $2.125$ 8.0.\(\cdots\).1 None 78.3.f.b \(-8\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\beta _{6})q^{2}+\beta _{1}q^{3}+2\beta _{6}q^{4}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(78, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)