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Properties

Label 312.4.bf
Level $312$
Weight $4$
Character orbit 312.bf
Rep. character $\chi_{312}(49,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $44$
Newform subspaces $2$
Sturm bound $224$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(224\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(312, [\chi])\).

Total New Old
Modular forms 352 44 308
Cusp forms 320 44 276
Eisenstein series 32 0 32

Trace form

\( 44 q + 6 q^{3} + 54 q^{7} - 198 q^{9} + O(q^{10}) \) \( 44 q + 6 q^{3} + 54 q^{7} - 198 q^{9} + 84 q^{11} - 54 q^{13} + 4 q^{17} + 204 q^{19} - 76 q^{23} - 1156 q^{25} - 108 q^{27} + 8 q^{29} + 260 q^{35} + 24 q^{37} - 432 q^{39} - 876 q^{41} - 402 q^{43} - 216 q^{45} + 1784 q^{49} + 1344 q^{51} + 1368 q^{53} + 288 q^{55} - 180 q^{59} - 1294 q^{61} - 486 q^{63} + 232 q^{65} - 846 q^{67} + 552 q^{69} - 1500 q^{71} - 738 q^{75} - 1192 q^{77} + 1788 q^{79} - 1782 q^{81} - 2448 q^{85} + 888 q^{87} - 6912 q^{89} - 2310 q^{91} + 1386 q^{93} + 3352 q^{95} + 1098 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(312, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
312.4.bf.a 312.bf 13.e $20$ $18.409$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(0\) \(-30\) \(0\) \(12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-3-3\beta _{1})q^{3}+(-\beta _{2}-\beta _{5})q^{5}+\cdots\)
312.4.bf.b 312.bf 13.e $24$ $18.409$ None \(0\) \(36\) \(0\) \(42\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{4}^{\mathrm{old}}(312, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(312, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(156, [\chi])\)\(^{\oplus 2}\)