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Properties

Label 315.3.h
Level $315$
Weight $3$
Character orbit 315.h
Rep. character $\chi_{315}(181,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $4$
Sturm bound $144$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 315.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(144\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(315, [\chi])\).

Total New Old
Modular forms 104 28 76
Cusp forms 88 28 60
Eisenstein series 16 0 16

Trace form

\( 28 q + 2 q^{2} + 58 q^{4} - 6 q^{7} - 2 q^{8} + O(q^{10}) \) \( 28 q + 2 q^{2} + 58 q^{4} - 6 q^{7} - 2 q^{8} + 14 q^{11} + 62 q^{14} + 162 q^{16} + 80 q^{22} - 4 q^{23} - 140 q^{25} + 14 q^{28} - 142 q^{29} + 294 q^{32} - 30 q^{35} - 156 q^{37} - 120 q^{43} - 180 q^{44} - 172 q^{46} - 176 q^{49} - 10 q^{50} - 256 q^{53} + 238 q^{56} - 136 q^{58} + 186 q^{64} + 90 q^{65} + 80 q^{67} + 60 q^{70} - 184 q^{71} - 44 q^{74} - 40 q^{77} + 326 q^{79} - 90 q^{85} - 140 q^{86} + 52 q^{88} + 54 q^{91} - 36 q^{92} - 622 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(315, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
315.3.h.a 315.h 7.b $2$ $8.583$ \(\Q(\sqrt{-5}) \) None \(-4\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q-2q^{2}+\beta q^{5}+(-2-3\beta )q^{7}+8q^{8}+\cdots\)
315.3.h.b 315.h 7.b $2$ $8.583$ \(\Q(\sqrt{-5}) \) None \(2\) \(0\) \(0\) \(14\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{2}-3q^{4}+\beta q^{5}+7q^{7}-7q^{8}+\cdots\)
315.3.h.c 315.h 7.b $12$ $8.583$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(2-\beta _{2})q^{4}+\beta _{7}q^{5}+(-1+\cdots)q^{7}+\cdots\)
315.3.h.d 315.h 7.b $12$ $8.583$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(4\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+(4-\beta _{1})q^{4}-\beta _{8}q^{5}+(-1+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(315, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(315, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)