Properties

Label 312.2.bf
Level $312$
Weight $2$
Character orbit 312.bf
Rep. character $\chi_{312}(49,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $12$
Newform subspaces $2$
Sturm bound $112$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 312.bf (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(112\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(312, [\chi])\).

Total New Old
Modular forms 128 12 116
Cusp forms 96 12 84
Eisenstein series 32 0 32

Trace form

\( 12 q - 2 q^{3} + 6 q^{7} - 6 q^{9} - 12 q^{11} - 6 q^{13} + 8 q^{17} + 12 q^{19} + 4 q^{23} + 4 q^{25} + 4 q^{27} + 4 q^{29} + 4 q^{35} + 12 q^{37} - 24 q^{41} + 6 q^{43} - 12 q^{45} + 12 q^{49} - 32 q^{51}+ \cdots - 66 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{2}^{\mathrm{new}}(312, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
312.2.bf.a 312.bf 13.e $4$ $2.491$ \(\Q(\zeta_{12})\) None 312.2.bf.a \(0\) \(2\) \(0\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-\zeta_{12}^{2})q^{3}+(1-2\zeta_{12}^{2})q^{5}+(2+\cdots)q^{7}+\cdots\)
312.2.bf.b 312.bf 13.e $8$ $2.491$ 8.0.649638144.4 None 312.2.bf.b \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{2}q^{3}+(-\beta _{2}-\beta _{6})q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(312, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(312, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(156, [\chi])\)\(^{\oplus 2}\)