Properties

Label 675.4.b
Level $675$
Weight $4$
Character orbit 675.b
Rep. character $\chi_{675}(649,\cdot)$
Character field $\Q$
Dimension $72$
Newform subspaces $17$
Sturm bound $360$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 17 \)
Sturm bound: \(360\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(2\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(675, [\chi])\).

Total New Old
Modular forms 288 72 216
Cusp forms 252 72 180
Eisenstein series 36 0 36

Trace form

\( 72 q - 276 q^{4} + O(q^{10}) \) \( 72 q - 276 q^{4} + 1116 q^{16} + 12 q^{19} - 264 q^{31} - 2460 q^{34} + 528 q^{46} - 1620 q^{49} + 936 q^{61} - 1920 q^{64} - 7668 q^{76} - 4296 q^{79} + 6660 q^{91} - 4980 q^{94} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(675, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
675.4.b.a 675.b 5.b $2$ $39.826$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3iq^{2}-q^{4}-5^{2}iq^{7}+21iq^{8}+\cdots\)
675.4.b.b 675.b 5.b $2$ $39.826$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3iq^{2}-q^{4}+5^{2}iq^{7}+21iq^{8}+\cdots\)
675.4.b.c 675.b 5.b $2$ $39.826$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}+4q^{4}+24iq^{8}-10q^{11}+\cdots\)
675.4.b.d 675.b 5.b $2$ $39.826$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}+4q^{4}+24iq^{8}+10q^{11}+\cdots\)
675.4.b.e 675.b 5.b $2$ $39.826$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+7q^{4}-6iq^{7}+15iq^{8}-47q^{11}+\cdots\)
675.4.b.f 675.b 5.b $2$ $39.826$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+7q^{4}+6iq^{7}+15iq^{8}+47q^{11}+\cdots\)
675.4.b.g 675.b 5.b $2$ $39.826$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+8q^{4}-17iq^{7}-70iq^{13}+2^{6}q^{16}+\cdots\)
675.4.b.h 675.b 5.b $2$ $39.826$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+8q^{4}-37iq^{7}+70iq^{13}+2^{6}q^{16}+\cdots\)
675.4.b.i 675.b 5.b $4$ $39.826$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}^{2}q^{2}-10q^{4}-11\zeta_{8}q^{7}-2\zeta_{8}^{2}q^{8}+\cdots\)
675.4.b.j 675.b 5.b $4$ $39.826$ \(\Q(i, \sqrt{13})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}-5q^{4}+9\beta _{1}q^{7}+3\beta _{2}q^{8}+\cdots\)
675.4.b.k 675.b 5.b $6$ $39.826$ 6.0.2033649216.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-7-\beta _{2}+\beta _{4})q^{4}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
675.4.b.l 675.b 5.b $6$ $39.826$ 6.0.2033649216.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-7-\beta _{2}+\beta _{4})q^{4}+(-2\beta _{1}+\cdots)q^{7}+\cdots\)
675.4.b.m 675.b 5.b $6$ $39.826$ 6.0.12559936.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(2\beta _{2}-\beta _{4})q^{2}+(-7+3\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)
675.4.b.n 675.b 5.b $6$ $39.826$ 6.0.12559936.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(2\beta _{2}-\beta _{4})q^{2}+(-7+3\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)
675.4.b.o 675.b 5.b $8$ $39.826$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+(-7+\beta _{6})q^{4}+(-7\beta _{1}-\beta _{4}+\cdots)q^{7}+\cdots\)
675.4.b.p 675.b 5.b $8$ $39.826$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+(-5+\beta _{5})q^{4}+(\beta _{1}-3\beta _{2}+\cdots)q^{7}+\cdots\)
675.4.b.q 675.b 5.b $8$ $39.826$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+(-5+\beta _{5})q^{4}+(-\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(675, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(675, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)