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Properties

Label 675.3.j
Level $675$
Weight $3$
Character orbit 675.j
Rep. character $\chi_{675}(251,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $70$
Newform subspaces $5$
Sturm bound $270$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 675.j (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 5 \)
Sturm bound: \(270\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(675, [\chi])\).

Total New Old
Modular forms 396 82 314
Cusp forms 324 70 254
Eisenstein series 72 12 60

Trace form

\( 70 q - 3 q^{2} + 65 q^{4} + O(q^{10}) \) \( 70 q - 3 q^{2} + 65 q^{4} - 27 q^{11} + 6 q^{13} - 84 q^{14} - 111 q^{16} + 30 q^{19} + 21 q^{22} - 102 q^{23} - 36 q^{28} + 132 q^{29} - 4 q^{31} + 243 q^{32} - 5 q^{34} + 24 q^{37} + 219 q^{38} + 243 q^{41} + 63 q^{43} - 112 q^{46} - 300 q^{47} - 155 q^{49} - 66 q^{52} - 378 q^{56} - 12 q^{58} - 75 q^{59} + 38 q^{61} - 270 q^{64} - 45 q^{67} - 315 q^{68} + 138 q^{73} + 834 q^{74} + 127 q^{76} + 708 q^{77} + 72 q^{79} + 162 q^{82} + 438 q^{83} - 603 q^{86} - 159 q^{88} - 52 q^{91} - 1284 q^{92} + 118 q^{94} + 27 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(675, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
675.3.j.a 675.j 9.d $2$ $18.392$ \(\Q(\sqrt{-3}) \) None \(-3\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+(2-2\zeta_{6})q^{7}+\cdots\)
675.3.j.b 675.j 9.d $16$ $18.392$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{1}+\beta _{3})q^{2}+(2-\beta _{1}-2\beta _{5}-\beta _{14}+\cdots)q^{4}+\cdots\)
675.3.j.c 675.j 9.d $16$ $18.392$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{1}+\beta _{3})q^{2}+(2-\beta _{2}+2\beta _{4}+\beta _{10}+\cdots)q^{4}+\cdots\)
675.3.j.d 675.j 9.d $16$ $18.392$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(1\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}-\beta _{3})q^{2}+(2-\beta _{2}+2\beta _{4}+\beta _{10}+\cdots)q^{4}+\cdots\)
675.3.j.e 675.j 9.d $20$ $18.392$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{12}q^{2}+(2-\beta _{3}+2\beta _{6}-\beta _{9})q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(675, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(675, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)