Space of modular forms of level 75, weight 7, and Character 75.c

• Label: 75.7.c
• Level: $75$
• Weight: $7$
• Character orbit: 75.c
• Rep. character: $\chi_{75}(26,\cdot)$
• Character field: $\Q$
• Dimension: $35$
• Newform subspaces: $6$
• Sturm bound: $70$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	75.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 3 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(70\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(75, [\chi])\).

	Total	New	Old
Modular forms	66	41	25
Cusp forms	54	35	19
Eisenstein series	12	6	6

Trace form

35 * q + 7 * q^3 - 1088 * q^4 - 354 * q^6 + 126 * q^7 + 651 * q^9 - 4492 * q^12 + 894 * q^13 + 31752 * q^16 + 14920 * q^18 - 16338 * q^19 - 22686 * q^21 + 40620 * q^22 + 60702 * q^24 - 51497 * q^27 - 77196 * q^28 + 6550 * q^31 + 55040 * q^33 - 3868 * q^34 - 168054 * q^36 + 61566 * q^37 + 7026 * q^39 - 1140 * q^42 - 197346 * q^43 + 440152 * q^46 + 566612 * q^48 + 425897 * q^49 - 137544 * q^51 - 133104 * q^52 - 539724 * q^54 + 9046 * q^57 - 1227780 * q^58 + 712150 * q^61 + 1300134 * q^63 - 1081896 * q^64 - 1999290 * q^66 + 978606 * q^67 + 499956 * q^69 - 1935720 * q^72 - 2397906 * q^73 + 4345052 * q^76 + 4499000 * q^78 - 333858 * q^79 - 4279785 * q^81 + 2395080 * q^82 + 2621376 * q^84 - 2525000 * q^87 - 6660780 * q^88 + 1849108 * q^91 + 4592454 * q^93 - 2256328 * q^94 - 8047482 * q^96 + 3215406 * q^97 - 885420 * q^99

Decomposition of \(S_{7}^{\mathrm{new}}(75, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
75.7.c.a	$75$	$7$	75.c	3.b	$2$	$1$	$1$	$17.254$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				3.7.b.a	$2$	$0$	\(0\)	\(27\)	\(0\)	\(286\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+3^{3}q^{3}+2^{6}q^{4}+286q^{7}+3^{6}q^{9}+\cdots\)
75.7.c.b	$75$	$7$	75.c	3.b	$2$	$2$	$2$	$17.254$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-15}) \)					15.7.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+11 i q^{2}-27 i q^{3}-57 q^{4}+297 q^{6}+\cdots\)
75.7.c.c	$75$	$7$	75.c	3.b	$2$	$8$	$8$	$17.254$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					15.7.c.a	$2$	$0$	\(0\)	\(-20\)	\(0\)	\(-160\)	$2^{2}\cdot 3^{5}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+(-3+\beta _{3})q^{3}+(-40-\beta _{3}+\cdots)q^{4}+\cdots\)
75.7.c.d	$75$	$7$	75.c	3.b	$2$	$8$	$8$	$17.254$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			75.7.c.d	$2$	$0$	\(0\)	\(-10\)	\(0\)	\(280\)	$2^{3}\cdot 3^{5}\cdot 5^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-1-\beta _{3})q^{3}+(-40+\beta _{2}+\cdots)q^{4}+\cdots\)
75.7.c.e	$75$	$7$	75.c	3.b	$2$	$8$	$8$	$17.254$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					15.7.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{6}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(-9-\beta _{3}+\cdots)q^{4}+\cdots\)
75.7.c.f	$75$	$7$	75.c	3.b	$2$	$8$	$8$	$17.254$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			75.7.c.d	$2$	$0$	\(0\)	\(10\)	\(0\)	\(-280\)	$2^{3}\cdot 3^{5}\cdot 5^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(-40+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(75, [\chi]) \simeq \) \(S_{7}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)