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Properties

Label 312.2.t
Level $312$
Weight $2$
Character orbit 312.t
Rep. character $\chi_{312}(187,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $56$
Newform subspaces $6$
Sturm bound $112$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 312.t (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 104 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 6 \)
Sturm bound: \(112\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

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

Total New Old
Modular forms 120 56 64
Cusp forms 104 56 48
Eisenstein series 16 0 16

Trace form

\( 56 q + 56 q^{9} + O(q^{10}) \) \( 56 q + 56 q^{9} + 32 q^{14} - 8 q^{16} - 36 q^{20} + 32 q^{22} + 12 q^{24} + 20 q^{26} - 32 q^{28} - 20 q^{32} - 16 q^{34} - 48 q^{40} - 8 q^{41} - 24 q^{42} - 36 q^{44} - 16 q^{46} - 32 q^{48} - 16 q^{50} - 56 q^{52} - 16 q^{57} + 8 q^{58} - 64 q^{59} + 4 q^{60} + 8 q^{65} - 8 q^{66} + 24 q^{68} - 96 q^{70} - 24 q^{73} + 24 q^{74} - 48 q^{76} - 44 q^{78} - 4 q^{80} + 56 q^{81} - 80 q^{83} + 48 q^{86} + 40 q^{89} - 16 q^{91} + 96 q^{92} + 48 q^{94} + 40 q^{96} + 8 q^{97} + 112 q^{98} + O(q^{100}) \)

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.t.a 312.t 104.m $2$ $2.491$ \(\Q(\sqrt{-1}) \) None \(-2\) \(-2\) \(-4\) \(2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+i)q^{2}-q^{3}-2iq^{4}+(-2+\cdots)q^{5}+\cdots\)
312.2.t.b 312.t 104.m $2$ $2.491$ \(\Q(\sqrt{-1}) \) None \(-2\) \(2\) \(-4\) \(-6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{2}+q^{3}+2iq^{4}+(-2+\cdots)q^{5}+\cdots\)
312.2.t.c 312.t 104.m $2$ $2.491$ \(\Q(\sqrt{-1}) \) None \(-2\) \(2\) \(4\) \(6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{2}+q^{3}+2iq^{4}+(2-2i)q^{5}+\cdots\)
312.2.t.d 312.t 104.m $2$ $2.491$ \(\Q(\sqrt{-1}) \) None \(2\) \(-2\) \(4\) \(-2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-i)q^{2}-q^{3}-2iq^{4}+(2-2i)q^{5}+\cdots\)
312.2.t.e 312.t 104.m $24$ $2.491$ None \(0\) \(-24\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
312.2.t.f 312.t 104.m $24$ $2.491$ None \(4\) \(24\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$

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

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