↑

Properties

Label 315.3.o
Level $315$
Weight $3$
Character orbit 315.o
Rep. character $\chi_{315}(127,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $60$
Newform subspaces $3$
Sturm bound $144$
Trace bound $2$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 315.o (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 3 \)
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 208 60 148
Cusp forms 176 60 116
Eisenstein series 32 0 32

Trace form

\( 60 q - 4 q^{2} - 8 q^{5} + 24 q^{8} + O(q^{10}) \) \( 60 q - 4 q^{2} - 8 q^{5} + 24 q^{8} - 4 q^{10} + 12 q^{11} - 20 q^{13} - 328 q^{16} - 12 q^{17} - 132 q^{20} + 188 q^{22} + 8 q^{23} + 128 q^{25} + 136 q^{26} + 96 q^{31} - 88 q^{32} - 184 q^{37} - 136 q^{38} - 380 q^{40} - 112 q^{41} - 4 q^{43} + 272 q^{46} + 100 q^{47} + 308 q^{50} - 264 q^{52} + 276 q^{53} + 340 q^{55} + 168 q^{56} + 292 q^{58} + 152 q^{61} + 208 q^{62} - 120 q^{65} + 36 q^{67} - 752 q^{68} + 112 q^{70} - 88 q^{73} - 184 q^{76} - 56 q^{77} + 468 q^{80} + 184 q^{82} + 256 q^{83} - 564 q^{85} - 696 q^{86} - 552 q^{88} + 84 q^{91} - 880 q^{92} - 428 q^{95} + 84 q^{97} + 28 q^{98} + O(q^{100}) \)

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.o.a 315.o 5.c $12$ $8.583$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(4\) \(0\) \(8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{3}+\beta _{4}+\beta _{10})q^{4}+\cdots\)
315.3.o.b 315.o 5.c $24$ $8.583$ None \(-8\) \(0\) \(-16\) \(0\) $\mathrm{SU}(2)[C_{4}]$
315.3.o.c 315.o 5.c $24$ $8.583$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$

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

\( S_{3}^{\mathrm{old}}(315, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 2}\)