Space of modular forms of level 75, weight 8, and trivial character

• Label: 75.8.a
• Level: $75$
• Weight: $8$
• Character orbit: 75.a
• Rep. character: $\chi_{75}(1,\cdot)$
• Character field: $\Q$
• Dimension: $23$
• Newform subspaces: $10$
• Sturm bound: $80$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(80\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(75))\).

	Total	New	Old
Modular forms	76	23	53
Cusp forms	64	23	41
Eisenstein series	12	0	12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(6\)
\(+\)	\(-\)	\(-\)	\(6\)
\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(+\)	\(7\)
Plus space		\(+\)	\(13\)
Minus space		\(-\)	\(10\)

Trace form

23 * q + 22 * q^2 - 27 * q^3 + 1392 * q^4 + 162 * q^6 - 2200 * q^7 + 3756 * q^8 + 16767 * q^9 + 720 * q^11 - 12852 * q^12 + 16586 * q^13 + 25452 * q^14 + 61876 * q^16 + 3646 * q^17 + 16038 * q^18 + 4706 * q^19 + 52326 * q^21 + 25332 * q^22 - 154800 * q^23 + 63828 * q^24 - 164664 * q^26 - 19683 * q^27 + 159860 * q^28 + 30282 * q^29 + 348874 * q^31 + 871820 * q^32 - 128412 * q^33 + 772564 * q^34 + 1014768 * q^36 - 114910 * q^37 + 821152 * q^38 - 29160 * q^39 + 413634 * q^41 - 413100 * q^42 - 264772 * q^43 - 251472 * q^44 - 1589536 * q^46 - 1793816 * q^47 - 1522368 * q^48 + 1354153 * q^49 + 1888434 * q^51 + 1677176 * q^52 - 1147430 * q^53 + 118098 * q^54 + 2322360 * q^56 + 2904228 * q^57 + 1967544 * q^58 + 1365864 * q^59 - 4464428 * q^61 - 3913080 * q^62 - 1603800 * q^63 + 1759084 * q^64 + 5158728 * q^66 + 5543876 * q^67 + 12046016 * q^68 - 3870828 * q^69 - 6492264 * q^71 + 2738124 * q^72 - 3986290 * q^73 - 10689708 * q^74 - 337104 * q^76 - 6793920 * q^77 - 16185204 * q^78 + 1742360 * q^79 + 12223143 * q^81 - 9224820 * q^82 + 21698004 * q^83 + 10750968 * q^84 - 54203580 * q^86 - 4738554 * q^87 - 24400524 * q^88 - 6440334 * q^89 + 22761954 * q^91 - 16885920 * q^92 - 11751480 * q^93 - 27352400 * q^94 + 20437596 * q^96 - 2696074 * q^97 + 26165174 * q^98 + 524880 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(75))\) 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					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5
75.8.a.a	$75$	$8$	75.a	1.a	$1$	$1$	$1$	$23.429$	\(\Q\)	None	✓				3.8.a.a	$1$	$1$	\(-6\)	\(27\)	\(0\)	\(64\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{2}+3^{3}q^{3}-92q^{4}-162q^{6}+\cdots\)
75.8.a.b	$75$	$8$	75.a	1.a	$1$	$1$	$1$	$23.429$	\(\Q\)	None	✓				15.8.a.b	$1$	$1$	\(13\)	\(27\)	\(0\)	\(-1380\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+13q^{2}+3^{3}q^{3}+41q^{4}+351q^{6}+\cdots\)
75.8.a.c	$75$	$8$	75.a	1.a	$1$	$1$	$1$	$23.429$	\(\Q\)	None	✓				15.8.a.a	$1$	$0$	\(22\)	\(-27\)	\(0\)	\(420\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+22q^{2}-3^{3}q^{3}+356q^{4}-594q^{6}+\cdots\)
75.8.a.d	$75$	$8$	75.a	1.a	$1$	$2$	$2$	$23.429$	\(\Q(\sqrt{31}) \)	None	✓	✓	✓		75.8.a.d	$1$	$1$	\(-8\)	\(54\)	\(0\)	\(-86\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{2}+3^{3}q^{3}+(12-8\beta )q^{4}+\cdots\)
75.8.a.e	$75$	$8$	75.a	1.a	$1$	$2$	$2$	$23.429$	\(\Q(\sqrt{601}) \)	None	✓				15.8.a.c	$1$	$0$	\(-7\)	\(-54\)	\(0\)	\(-1304\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{2}-3^{3}q^{3}+(31+7\beta )q^{4}+\cdots\)
75.8.a.f	$75$	$8$	75.a	1.a	$1$	$2$	$2$	$23.429$	\(\Q(\sqrt{31}) \)	None	✓	✓			75.8.a.d	$1$	$1$	\(8\)	\(-54\)	\(0\)	\(86\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{2}-3^{3}q^{3}+(12+8\beta )q^{4}+\cdots\)
75.8.a.g	$75$	$8$	75.a	1.a	$1$	$3$	$3$	$23.429$	3.3.717484.1	None	✓	✓	✓		75.8.a.g	$1$	$0$	\(-6\)	\(-81\)	\(0\)	\(-83\)		$+$	$+$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}-3^{3}q^{3}+(51+4\beta _{1}+\cdots)q^{4}+\cdots\)
75.8.a.h	$75$	$8$	75.a	1.a	$1$	$3$	$3$	$23.429$	3.3.717484.1	None	✓	✓			75.8.a.g	$1$	$0$	\(6\)	\(81\)	\(0\)	\(83\)		$-$	$-$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}+3^{3}q^{3}+(51+4\beta _{1}-\beta _{2})q^{4}+\cdots\)
75.8.a.i	$75$	$8$	75.a	1.a	$1$	$4$	$4$	$23.429$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		15.8.b.a	$1$	$1$	\(-9\)	\(-108\)	\(0\)	\(-1188\)		$+$	$-$	$-$	$2^{2}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}-3^{3}q^{3}+(83+3\beta _{1}+\cdots)q^{4}+\cdots\)
75.8.a.j	$75$	$8$	75.a	1.a	$1$	$4$	$4$	$23.429$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		15.8.b.a	$1$	$0$	\(9\)	\(108\)	\(0\)	\(1188\)		$-$	$-$	$+$	$2^{2}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}+3^{3}q^{3}+(83+3\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(75))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(75)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)