Properties

Label 315.2.r
Level $315$
Weight $2$
Character orbit 315.r
Rep. character $\chi_{315}(184,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $88$
Newform subspaces $2$
Sturm bound $96$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.r (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 315 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(96\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(315, [\chi])\).

Total New Old
Modular forms 104 104 0
Cusp forms 88 88 0
Eisenstein series 16 16 0

Trace form

\( 88 q - 84 q^{4} + q^{5} - 8 q^{9} + O(q^{10}) \) \( 88 q - 84 q^{4} + q^{5} - 8 q^{9} + 2 q^{10} + 16 q^{14} + 4 q^{15} + 68 q^{16} - 4 q^{19} - 12 q^{20} - 8 q^{21} + q^{25} - 32 q^{26} - 14 q^{29} + 5 q^{30} - 4 q^{31} - 8 q^{34} + 24 q^{35} - 12 q^{36} - 32 q^{39} - 8 q^{40} - 34 q^{41} - 16 q^{44} + 27 q^{45} + 10 q^{46} + 4 q^{49} + 34 q^{50} - 52 q^{51} + 24 q^{54} - 30 q^{55} + 6 q^{56} - 100 q^{59} - 21 q^{60} - 16 q^{61} - 56 q^{64} + 16 q^{65} + 40 q^{66} - 32 q^{69} - 9 q^{70} + 12 q^{71} + 38 q^{74} + 42 q^{75} + 12 q^{76} + 8 q^{79} - 7 q^{80} + 20 q^{81} + 74 q^{84} - 7 q^{85} + 44 q^{86} + 40 q^{89} + 5 q^{90} - 36 q^{91} + 4 q^{94} + 74 q^{95} + 48 q^{96} - 30 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(315, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
315.2.r.a 315.r 315.r $4$ $2.515$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}^{3}q^{2}+(\zeta_{12}+\zeta_{12}^{3})q^{3}+q^{4}+\cdots\)
315.2.r.b 315.r 315.r $84$ $2.515$ None \(0\) \(0\) \(3\) \(0\) $\mathrm{SU}(2)[C_{6}]$