Space of modular forms of level 675, weight 3, and Character 675.d

• Label: 675.3.d
• Level: $675$
• Weight: $3$
• Character orbit: 675.d
• Rep. character: $\chi_{675}(674,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $12$
• Sturm bound: $270$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	675.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(270\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(675, [\chi])\).

	Total	New	Old
Modular forms	198	48	150
Cusp forms	162	48	114
Eisenstein series	36	0	36

Trace form

\( 48 q + 100 q^{4} + 140 q^{16} + 50 q^{19} + 134 q^{31} + 196 q^{34} + 216 q^{46} - 586 q^{49} - 518 q^{61} + 672 q^{64} + 1068 q^{76} + 256 q^{79} - 230 q^{91} - 812 q^{94}+O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(675, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
675.3.d.a	$675$	$3$	675.d	15.d	$2$	$2$	$2$	$18.392$	\(\Q(\sqrt{-1}) \)	None					27.3.b.b	$2$	$0$	\(-6\)	\(0\)	\(0\)	\(0\)	$5$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 q^{2}+5 q^{4}+\beta q^{7}-3 q^{8}-3\beta q^{11}+\cdots\)
675.3.d.b	$675$	$3$	675.d	15.d	$2$	$2$	$2$	$18.392$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		✓			675.3.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-4 q^{4}+2 i q^{7}+i q^{13}+16 q^{16}+\cdots\)
675.3.d.c	$675$	$3$	675.d	15.d	$2$	$2$	$2$	$18.392$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)					27.3.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-4 q^{4}-13 i q^{7}+i q^{13}+16 q^{16}+\cdots\)
675.3.d.d	$675$	$3$	675.d	15.d	$2$	$2$	$2$	$18.392$	\(\Q(\sqrt{-1}) \)	None					27.3.b.b	$2$	$0$	\(6\)	\(0\)	\(0\)	\(0\)	$5$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 q^{2}+5 q^{4}+\beta q^{7}+3 q^{8}+3\beta q^{11}+\cdots\)
675.3.d.e	$675$	$3$	675.d	15.d	$2$	$4$	$4$	$18.392$	\(\Q(i, \sqrt{5})\)	None					135.3.c.c	$2$	$0$	\(-6\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{2})q^{2}+(-2-3\beta _{2})q^{4}+3\beta _{1}q^{7}+\cdots\)
675.3.d.f	$675$	$3$	675.d	15.d	$2$	$4$	$4$	$18.392$	\(\Q(i, \sqrt{5})\)	None					135.3.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+q^{4}+12\beta _{1}q^{7}-3\beta _{3}q^{8}+\cdots\)
675.3.d.g	$675$	$3$	675.d	15.d	$2$	$4$	$4$	$18.392$	\(\Q(i, \sqrt{5})\)	None		✓			675.3.c.j	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+q^{4}+3\beta _{1}q^{7}-3\beta _{3}q^{8}+\cdots\)
675.3.d.h	$675$	$3$	675.d	15.d	$2$	$4$	$4$	$18.392$	\(\Q(i, \sqrt{11})\)	None		✓			675.3.c.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+7q^{4}+9\beta _{1}q^{7}+3\beta _{2}q^{8}+\cdots\)
675.3.d.i	$675$	$3$	675.d	15.d	$2$	$4$	$4$	$18.392$	\(\Q(i, \sqrt{5})\)	None					135.3.c.c	$2$	$0$	\(6\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{2})q^{2}+(-2-3\beta _{2})q^{4}+3\beta _{1}q^{7}+\cdots\)
675.3.d.j	$675$	$3$	675.d	15.d	$2$	$6$	$6$	$18.392$	6.0.60217600.1	None		✓			675.3.c.r	$2$	$0$	\(-6\)	\(0\)	\(0\)	\(0\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{2})q^{2}+(2-2\beta _{2}-\beta _{4})q^{4}+\cdots\)
675.3.d.k	$675$	$3$	675.d	15.d	$2$	$6$	$6$	$18.392$	6.0.60217600.1	None		✓			675.3.c.r	$2$	$0$	\(6\)	\(0\)	\(0\)	\(0\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{2})q^{2}+(2-2\beta _{2}-\beta _{4})q^{4}+(-\beta _{3}+\cdots)q^{7}+\cdots\)
675.3.d.l	$675$	$3$	675.d	15.d	$2$	$8$	$8$	$18.392$	8.0.\(\cdots\).1	None			✓		135.3.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(4+\beta _{6})q^{4}+(-\beta _{2}+4\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(675, [\chi])\) into lower level spaces

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