Properties

Label 675.2.b
Level $675$
Weight $2$
Character orbit 675.b
Rep. character $\chi_{675}(649,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $9$
Sturm bound $180$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(180\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(2\), \(7\), \(11\)

Dimensions

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

Total New Old
Modular forms 108 24 84
Cusp forms 72 24 48
Eisenstein series 36 0 36

Trace form

\( 24 q - 28 q^{4} + O(q^{10}) \) \( 24 q - 28 q^{4} + 28 q^{16} - 2 q^{19} + 10 q^{31} + 20 q^{34} - 48 q^{46} - 50 q^{49} + 62 q^{61} + 84 q^{76} - 40 q^{79} - 94 q^{91} - 4 q^{94} + O(q^{100}) \)

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.b.a 675.b 5.b $2$ $5.390$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-2q^{4}+3iq^{7}-2q^{11}-5iq^{13}+\cdots\)
675.2.b.b 675.b 5.b $2$ $5.390$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-2q^{4}-3iq^{7}+2q^{11}+5iq^{13}+\cdots\)
675.2.b.c 675.b 5.b $2$ $5.390$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+q^{4}+3iq^{8}-5q^{11}+5iq^{13}+\cdots\)
675.2.b.d 675.b 5.b $2$ $5.390$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+q^{4}+3iq^{8}+5q^{11}-5iq^{13}+\cdots\)
675.2.b.e 675.b 5.b $2$ $5.390$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{4}+4iq^{7}+5iq^{13}+4q^{16}+\cdots\)
675.2.b.f 675.b 5.b $2$ $5.390$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{4}+iq^{7}+5iq^{13}+4q^{16}+7q^{19}+\cdots\)
675.2.b.g 675.b 5.b $4$ $5.390$ \(\Q(i, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{2}-5q^{4}+3\beta _{1}q^{7}-3\beta _{3}q^{8}+\cdots\)
675.2.b.h 675.b 5.b $4$ $5.390$ \(\Q(i, \sqrt{13})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-2+\beta _{3})q^{4}+(-2\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
675.2.b.i 675.b 5.b $4$ $5.390$ \(\Q(i, \sqrt{13})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-2+\beta _{3})q^{4}+(2\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(675, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)