Space of modular forms of level 675, weight 4, and Character 675.b

• 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: $19$

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

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

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

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
675.4.b.a	$675$	$4$	675.b	5.b	$2$	$2$	$2$	$39.826$	\(\Q(\sqrt{-1}) \)	None					27.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{2}-q^{4}-25 i q^{7}+21 i q^{8}+\cdots\)
675.4.b.b	$675$	$4$	675.b	5.b	$2$	$2$	$2$	$39.826$	\(\Q(\sqrt{-1}) \)	None					27.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{2}-q^{4}+25 i q^{7}+21 i q^{8}+\cdots\)
675.4.b.c	$675$	$4$	675.b	5.b	$2$	$2$	$2$	$39.826$	\(\Q(\sqrt{-1}) \)	None					135.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}+4 q^{4}+24 i q^{8}-10 q^{11}+\cdots\)
675.4.b.d	$675$	$4$	675.b	5.b	$2$	$2$	$2$	$39.826$	\(\Q(\sqrt{-1}) \)	None					135.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}+4 q^{4}+24 i q^{8}+10 q^{11}+\cdots\)
675.4.b.e	$675$	$4$	675.b	5.b	$2$	$2$	$2$	$39.826$	\(\Q(\sqrt{-1}) \)	None					135.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+7 q^{4}-6 i q^{7}+15 i q^{8}+\cdots\)
675.4.b.f	$675$	$4$	675.b	5.b	$2$	$2$	$2$	$39.826$	\(\Q(\sqrt{-1}) \)	None					135.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+7 q^{4}+6 i q^{7}+15 i q^{8}+\cdots\)
675.4.b.g	$675$	$4$	675.b	5.b	$2$	$2$	$2$	$39.826$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		✓			675.4.a.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+8 q^{4}-17 i q^{7}-70 i q^{13}+64 q^{16}+\cdots\)
675.4.b.h	$675$	$4$	675.b	5.b	$2$	$2$	$2$	$39.826$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		✓			675.4.a.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+8 q^{4}-37 i q^{7}+70 i q^{13}+64 q^{16}+\cdots\)
675.4.b.i	$675$	$4$	675.b	5.b	$2$	$4$	$4$	$39.826$	\(\Q(\zeta_{8})\)	None					27.4.a.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{2} q^{2}-10 q^{4}-11\beta_1 q^{7}-2\beta_{2} q^{8}+\cdots\)
675.4.b.j	$675$	$4$	675.b	5.b	$2$	$4$	$4$	$39.826$	\(\Q(i, \sqrt{13})\)	None		✓			675.4.a.l	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}-5q^{4}+9\beta _{1}q^{7}+3\beta _{2}q^{8}+\cdots\)
675.4.b.k	$675$	$4$	675.b	5.b	$2$	$6$	$6$	$39.826$	6.0.2033649216.1	None					135.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-7-\beta _{2}+\beta _{4})q^{4}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
675.4.b.l	$675$	$4$	675.b	5.b	$2$	$6$	$6$	$39.826$	6.0.2033649216.1	None					135.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-7-\beta _{2}+\beta _{4})q^{4}+(-2\beta _{1}+\cdots)q^{7}+\cdots\)
675.4.b.m	$675$	$4$	675.b	5.b	$2$	$6$	$6$	$39.826$	6.0.12559936.1	None					135.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2\beta _{2}-\beta _{4})q^{2}+(-7+3\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)
675.4.b.n	$675$	$4$	675.b	5.b	$2$	$6$	$6$	$39.826$	6.0.12559936.1	None					135.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2\beta _{2}-\beta _{4})q^{2}+(-7+3\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)
675.4.b.o	$675$	$4$	675.b	5.b	$2$	$8$	$8$	$39.826$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			675.4.a.w	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(-7+\beta _{6})q^{4}+(-7\beta _{1}-\beta _{4}+\cdots)q^{7}+\cdots\)
675.4.b.p	$675$	$4$	675.b	5.b	$2$	$8$	$8$	$39.826$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			675.4.a.u	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(-5+\beta _{5})q^{4}+(\beta _{1}-3\beta _{2}+\cdots)q^{7}+\cdots\)
675.4.b.q	$675$	$4$	675.b	5.b	$2$	$8$	$8$	$39.826$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			675.4.a.u	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(-5+\beta _{5})q^{4}+(-\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots\)

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

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