Space of modular forms of level 675, weight 5, and Character 675.d

• Label: 675.5.d
• Level: $675$
• Weight: $5$
• Character orbit: 675.d
• Rep. character: $\chi_{675}(674,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $14$
• Sturm bound: $450$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	675.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(450\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(675, [\chi])\).

	Total	New	Old
Modular forms	378	96	282
Cusp forms	342	96	246
Eisenstein series	36	0	36

Trace form

\( 96 q + 740 q^{4} + 5500 q^{16} - 890 q^{19} + 538 q^{31} - 172 q^{34} - 8088 q^{46} - 21758 q^{49} + 15974 q^{61} + 20976 q^{64} - 44484 q^{76} - 7792 q^{79} + 24350 q^{91} + 152804 q^{94}+O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(675, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
675.5.d.a	$675$	$5$	675.d	15.d	$2$	$2$	$2$	$69.775$	\(\Q(\sqrt{-1}) \)	None					27.5.b.c	$2$	$0$	\(-6\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-3 q^{2}-7 q^{4}-19 i q^{7}+69 q^{8}+\cdots\)
675.5.d.b	$675$	$5$	675.d	15.d	$2$	$2$	$2$	$69.775$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)		✓			675.5.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-16 q^{4}-94 i q^{7}+337 i q^{13}+\cdots\)
675.5.d.c	$675$	$5$	675.d	15.d	$2$	$2$	$2$	$69.775$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)					27.5.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-16 q^{4}+71 i q^{7}+337 i q^{13}+\cdots\)
675.5.d.d	$675$	$5$	675.d	15.d	$2$	$2$	$2$	$69.775$	\(\Q(\sqrt{-1}) \)	None					27.5.b.c	$2$	$0$	\(6\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 q^{2}-7 q^{4}-19 i q^{7}-69 q^{8}+\cdots\)
675.5.d.e	$675$	$5$	675.d	15.d	$2$	$4$	$4$	$69.775$	\(\Q(i, \sqrt{5})\)	None					135.5.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}-11q^{4}-24\beta _{1}q^{7}-3^{3}\beta _{3}q^{8}+\cdots\)
675.5.d.f	$675$	$5$	675.d	15.d	$2$	$4$	$4$	$69.775$	\(\Q(i, \sqrt{5})\)	None					135.5.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{2}+4q^{4}+51\beta _{1}q^{7}+12\beta _{3}q^{8}+\cdots\)
675.5.d.g	$675$	$5$	675.d	15.d	$2$	$4$	$4$	$69.775$	\(\Q(i, \sqrt{6})\)	None					27.5.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+38q^{4}+17\beta _{1}q^{7}+22\beta _{3}q^{8}+\cdots\)
675.5.d.h	$675$	$5$	675.d	15.d	$2$	$8$	$8$	$69.775$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			675.5.c.j	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(9-\beta _{2})q^{4}+(-11\beta _{4}-2\beta _{7})q^{7}+\cdots\)
675.5.d.i	$675$	$5$	675.d	15.d	$2$	$10$	$10$	$69.775$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		✓			675.5.c.r	$2$	$0$	\(-6\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{4}\cdot 5^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{1})q^{2}+(9+2\beta _{1}-\beta _{3})q^{4}+\cdots\)
675.5.d.j	$675$	$5$	675.d	15.d	$2$	$10$	$10$	$69.775$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		✓			675.5.c.r	$2$	$0$	\(6\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{4}\cdot 5^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{1})q^{2}+(9+2\beta _{1}-\beta _{3})q^{4}+(2\beta _{2}+\cdots)q^{7}+\cdots\)
675.5.d.k	$675$	$5$	675.d	15.d	$2$	$12$	$12$	$69.775$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					135.5.c.d	$2$	$0$	\(-6\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3^{6}\cdot 5^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(12+\beta _{2}-\beta _{4})q^{4}-\beta _{7}q^{7}+\cdots\)
675.5.d.l	$675$	$5$	675.d	15.d	$2$	$12$	$12$	$69.775$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					135.5.c.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3^{6}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}+(8-\beta _{1})q^{4}+(-\beta _{5}+3\beta _{9}+\cdots)q^{7}+\cdots\)
675.5.d.m	$675$	$5$	675.d	15.d	$2$	$12$	$12$	$69.775$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓			675.5.c.o	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(8-\beta _{2})q^{4}+(2\beta _{6}+\beta _{9}+\beta _{10}+\cdots)q^{7}+\cdots\)
675.5.d.n	$675$	$5$	675.d	15.d	$2$	$12$	$12$	$69.775$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					135.5.c.d	$2$	$0$	\(6\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3^{6}\cdot 5^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+(12+\beta _{2}-\beta _{4})q^{4}+\beta _{7}q^{7}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(675, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(675, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)