Properties

Label 315.2.d
Level $315$
Weight $2$
Character orbit 315.d
Rep. character $\chi_{315}(64,\cdot)$
Character field $\Q$
Dimension $14$
Newform subspaces $5$
Sturm bound $96$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(96\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(11\), \(29\)

Dimensions

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

Total New Old
Modular forms 56 14 42
Cusp forms 40 14 26
Eisenstein series 16 0 16

Trace form

\( 14 q - 8 q^{4} + O(q^{10}) \) \( 14 q - 8 q^{4} - 8 q^{10} + 6 q^{11} - 4 q^{14} + 12 q^{16} + 20 q^{20} + 10 q^{25} - 28 q^{26} - 10 q^{29} - 28 q^{31} - 4 q^{34} - 2 q^{35} + 4 q^{40} - 12 q^{41} + 20 q^{44} + 8 q^{46} - 14 q^{49} + 24 q^{50} + 4 q^{55} + 24 q^{56} + 36 q^{59} + 8 q^{61} - 52 q^{64} - 22 q^{65} + 8 q^{70} - 16 q^{71} - 72 q^{74} + 16 q^{76} + 10 q^{79} - 60 q^{80} + 22 q^{85} + 16 q^{86} + 40 q^{89} - 2 q^{91} + 92 q^{94} - 16 q^{95} + O(q^{100}) \)

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.d.a 315.d 5.b $2$ $2.515$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-2q^{4}+(2-i)q^{5}+iq^{7}+\cdots\)
315.2.d.b 315.d 5.b $2$ $2.515$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+q^{4}+(-2+i)q^{5}+iq^{7}+\cdots\)
315.2.d.c 315.d 5.b $2$ $2.515$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+q^{4}+(-1-2i)q^{5}-iq^{7}+\cdots\)
315.2.d.d 315.d 5.b $2$ $2.515$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+q^{4}+(2+i)q^{5}-iq^{7}+3iq^{8}+\cdots\)
315.2.d.e 315.d 5.b $6$ $2.515$ 6.0.350464.1 None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+(-1+\beta _{3}+\beta _{5})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(315, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(315, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)