Space of modular forms of level 675, weight 2, and Character 675.b

• 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: $19$

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

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

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

Label	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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
675.2.b.a	$675$	$2$	675.b	5.b	$2$	$2$	$2$	$5.390$	\(\Q(\sqrt{-1}) \)	None					135.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-2 q^{4}+3 i q^{7}-2 q^{11}+\cdots\)
675.2.b.b	$675$	$2$	675.b	5.b	$2$	$2$	$2$	$5.390$	\(\Q(\sqrt{-1}) \)	None					135.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-2 q^{4}-3 i q^{7}+2 q^{11}+\cdots\)
675.2.b.c	$675$	$2$	675.b	5.b	$2$	$2$	$2$	$5.390$	\(\Q(\sqrt{-1}) \)	None		✓			675.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+q^{4}+3 i q^{8}-5 q^{11}+5 i q^{13}+\cdots\)
675.2.b.d	$675$	$2$	675.b	5.b	$2$	$2$	$2$	$5.390$	\(\Q(\sqrt{-1}) \)	None		✓			675.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+q^{4}+3 i q^{8}+5 q^{11}-5 i q^{13}+\cdots\)
675.2.b.e	$675$	$2$	675.b	5.b	$2$	$2$	$2$	$5.390$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		✓			675.2.a.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+2 q^{4}+4 i q^{7}+5 i q^{13}+4 q^{16}+\cdots\)
675.2.b.f	$675$	$2$	675.b	5.b	$2$	$2$	$2$	$5.390$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)					27.2.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+2 q^{4}+i q^{7}+5 i q^{13}+4 q^{16}+\cdots\)
675.2.b.g	$675$	$2$	675.b	5.b	$2$	$4$	$4$	$5.390$	\(\Q(i, \sqrt{7})\)	None		✓			675.2.a.n	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}-5q^{4}+3\beta _{1}q^{7}-3\beta _{3}q^{8}+\cdots\)
675.2.b.h	$675$	$2$	675.b	5.b	$2$	$4$	$4$	$5.390$	\(\Q(i, \sqrt{13})\)	None					135.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-2+\beta _{3})q^{4}+(-2\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
675.2.b.i	$675$	$2$	675.b	5.b	$2$	$4$	$4$	$5.390$	\(\Q(i, \sqrt{13})\)	None					135.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-2+\beta _{3})q^{4}+(2\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)

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

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