Properties

Label 675.2.e
Level $675$
Weight $2$
Character orbit 675.e
Rep. character $\chi_{675}(226,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $32$
Newform subspaces $5$
Sturm bound $180$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 5 \)
Sturm bound: \(180\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(675, [\chi])\).

Total New Old
Modular forms 216 44 172
Cusp forms 144 32 112
Eisenstein series 72 12 60

Trace form

\( 32 q - 2 q^{2} - 12 q^{4} + 2 q^{7} + O(q^{10}) \) \( 32 q - 2 q^{2} - 12 q^{4} + 2 q^{7} + 6 q^{11} + 2 q^{13} - 6 q^{14} - 8 q^{16} + 4 q^{17} + 8 q^{19} - 6 q^{22} - 6 q^{23} + 24 q^{26} - 16 q^{28} + 6 q^{29} + 4 q^{31} - 22 q^{32} - 14 q^{34} - 4 q^{37} + 10 q^{38} + 6 q^{41} + 2 q^{43} - 72 q^{44} - 16 q^{46} - 20 q^{47} + 10 q^{49} - 10 q^{52} - 8 q^{53} - 48 q^{56} - 18 q^{58} + 42 q^{59} + 10 q^{61} + 84 q^{62} + 28 q^{64} + 8 q^{67} + 26 q^{68} - 24 q^{71} + 8 q^{73} + 48 q^{74} - 30 q^{76} - 6 q^{77} + 2 q^{79} + 48 q^{82} + 6 q^{83} - 12 q^{86} - 18 q^{88} + 60 q^{89} - 60 q^{91} - 36 q^{92} - 2 q^{94} - 16 q^{97} - 76 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(675, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
675.2.e.a 675.e 9.c $2$ $5.390$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+\zeta_{6}q^{4}+(-3+3\zeta_{6})q^{7}+\cdots\)
675.2.e.b 675.e 9.c $6$ $5.390$ 6.0.954288.1 None \(-1\) \(0\) \(0\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{4}-\beta _{5})q^{2}+(-2-\beta _{1}-\beta _{2}+2\beta _{3}+\cdots)q^{4}+\cdots\)
675.2.e.c 675.e 9.c $8$ $5.390$ 8.0.1223810289.2 None \(-2\) \(0\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(\beta _{1}-\beta _{2}-\beta _{3}+\beta _{6})q^{4}+\cdots\)
675.2.e.d 675.e 9.c $8$ $5.390$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{24}^{7}q^{2}-\zeta_{24}^{2}q^{4}+(-2\zeta_{24}^{3}+\cdots)q^{7}+\cdots\)
675.2.e.e 675.e 9.c $8$ $5.390$ 8.0.1223810289.2 None \(2\) \(0\) \(0\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(\beta _{1}-\beta _{2}-\beta _{3}+\beta _{6})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(675, [\chi])\) into lower level spaces

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