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Properties

Label 312.2.bt
Level $312$
Weight $2$
Character orbit 312.bt
Rep. character $\chi_{312}(19,\cdot)$
Character field $\Q(\zeta_{12})$
Dimension $112$
Newform subspaces $4$
Sturm bound $112$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 312.bt (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 104 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 4 \)
Sturm bound: \(112\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 240 112 128
Cusp forms 208 112 96
Eisenstein series 32 0 32

Trace form

\( 112 q - 56 q^{9} + O(q^{10}) \) \( 112 q - 56 q^{9} - 32 q^{14} + 8 q^{16} - 12 q^{20} + 28 q^{22} - 12 q^{24} + 40 q^{26} - 28 q^{28} - 40 q^{32} + 64 q^{34} + 24 q^{40} - 16 q^{41} - 12 q^{42} - 24 q^{44} - 44 q^{46} - 16 q^{48} + 48 q^{49} - 92 q^{50} + 56 q^{52} - 108 q^{56} - 32 q^{57} - 56 q^{58} + 64 q^{59} - 64 q^{60} - 60 q^{62} + 16 q^{65} - 16 q^{66} - 24 q^{68} + 24 q^{70} + 24 q^{73} - 24 q^{74} + 108 q^{76} + 44 q^{78} + 4 q^{80} - 56 q^{81} - 60 q^{82} + 80 q^{83} + 132 q^{88} - 40 q^{89} - 176 q^{91} - 24 q^{92} + 12 q^{94} - 40 q^{96} - 8 q^{97} - 4 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
312.2.bt.a 312.bt 104.u $4$ $2.491$ \(\Q(\zeta_{12})\) None \(-4\) \(-2\) \(-2\) \(-6\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-1+\zeta_{12}^{3})q^{2}-\zeta_{12}^{2}q^{3}-2\zeta_{12}^{3}q^{4}+\cdots\)
312.2.bt.b 312.bt 104.u $4$ $2.491$ \(\Q(\zeta_{12})\) None \(2\) \(-2\) \(2\) \(6\) $\mathrm{SU}(2)[C_{12}]$ \(q+(1-\zeta_{12}-\zeta_{12}^{2})q^{2}-\zeta_{12}^{2}q^{3}+\cdots\)
312.2.bt.c 312.bt 104.u $48$ $2.491$ None \(2\) \(-24\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$
312.2.bt.d 312.bt 104.u $56$ $2.491$ None \(0\) \(28\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$

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

\( S_{2}^{\mathrm{old}}(312, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 2}\)