Space of modular forms of level 75, weight 8, and Character 75.b

• Label: 75.8.b
• Level: $75$
• Weight: $8$
• Character orbit: 75.b
• Rep. character: $\chi_{75}(49,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $6$
• Sturm bound: $80$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(80\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(75, [\chi])\).

	Total	New	Old
Modular forms	76	20	56
Cusp forms	64	20	44
Eisenstein series	12	0	12

Trace form

20 * q - 1100 * q^4 - 540 * q^6 - 14580 * q^9 - 13780 * q^11 + 9420 * q^14 - 460 * q^16 + 68090 * q^19 - 23490 * q^21 + 137700 * q^24 + 945980 * q^26 - 784480 * q^29 - 114830 * q^31 - 1181640 * q^34 + 801900 * q^36 + 1115910 * q^39 - 348140 * q^41 - 373720 * q^44 - 1421400 * q^46 + 2153250 * q^49 + 532440 * q^51 + 393660 * q^54 + 7474560 * q^56 - 1475060 * q^59 - 2275550 * q^61 - 12142580 * q^64 + 4844880 * q^66 + 9349020 * q^69 - 2175200 * q^71 - 11841160 * q^74 - 26034080 * q^76 + 16046240 * q^79 + 10628820 * q^81 + 33362280 * q^84 - 16626580 * q^86 - 43242180 * q^89 - 4353070 * q^91 + 102377400 * q^94 - 53248860 * q^96 + 10045620 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(75, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
75.8.b.a	$75$	$8$	75.b	5.b	$2$	$2$	$2$	$23.429$	\(\Q(\sqrt{-1}) \)	None					15.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+22 i q^{2}+27 i q^{3}-356 q^{4}-594 q^{6}+\cdots\)
75.8.b.b	$75$	$8$	75.b	5.b	$2$	$2$	$2$	$23.429$	\(\Q(\sqrt{-1}) \)	None					15.8.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+13 i q^{2}-27 i q^{3}-41 q^{4}+351 q^{6}+\cdots\)
75.8.b.c	$75$	$8$	75.b	5.b	$2$	$2$	$2$	$23.429$	\(\Q(\sqrt{-1}) \)	None					3.8.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+6 i q^{2}+27 i q^{3}+92 q^{4}-162 q^{6}+\cdots\)
75.8.b.d	$75$	$8$	75.b	5.b	$2$	$4$	$4$	$23.429$	\(\Q(i, \sqrt{601})\)	None					15.8.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-3\beta _{2})q^{2}+3^{3}\beta _{2}q^{3}+(-38+\cdots)q^{4}+\cdots\)
75.8.b.e	$75$	$8$	75.b	5.b	$2$	$4$	$4$	$23.429$	\(\Q(i, \sqrt{31})\)	None		✓			75.8.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-4\beta _{1}-\beta _{3})q^{2}-3^{3}\beta _{1}q^{3}+(-12+\cdots)q^{4}+\cdots\)
75.8.b.f	$75$	$8$	75.b	5.b	$2$	$6$	$6$	$23.429$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None		✓	✓		75.8.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{2}+2\beta _{4})q^{2}-3^{3}\beta _{4}q^{3}+(-51+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(75, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)