Space of modular forms of level 75, weight 22, and trivial character

• Label: 75.22.a
• Level: $75$
• Weight: $22$
• Character orbit: 75.a
• Rep. character: $\chi_{75}(1,\cdot)$
• Character field: $\Q$
• Dimension: $66$
• Newform subspaces: $14$
• Sturm bound: $220$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(220\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(75))\).

	Total	New	Old
Modular forms	216	66	150
Cusp forms	204	66	138
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.
\(+\)	\(+\)	\(+\)	\(15\)
\(+\)	\(-\)	\(-\)	\(18\)
\(-\)	\(+\)	\(-\)	\(17\)
\(-\)	\(-\)	\(+\)	\(16\)
Plus space		\(+\)	\(31\)
Minus space		\(-\)	\(35\)

Trace form

\( 66 q + 1598 q^{2} + 68549728 q^{4} - 105225318 q^{6} + 559598932 q^{7} + 6582404412 q^{8} + 230127770466 q^{9} + O(q^{10}) \)

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
75.22.a.a	$75$	$22$	75.a	1.a	$1$	$1$	$1$	$209.608$	\(\Q\)	None	✓	$1$	$0$	\(-1728\)	\(59049\)	\(0\)	\(-538429808\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-12^{3}q^{2}+3^{10}q^{3}+888832q^{4}+\cdots\)
75.22.a.b	$75$	$22$	75.a	1.a	$1$	$1$	$1$	$209.608$	\(\Q\)	None	✓	$1$	$1$	\(-544\)	\(-59049\)	\(0\)	\(-1277698380\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-544q^{2}-3^{10}q^{3}-1801216q^{4}+\cdots\)
75.22.a.c	$75$	$22$	75.a	1.a	$1$	$1$	$1$	$209.608$	\(\Q\)	None	✓	$1$	$0$	\(2844\)	\(59049\)	\(0\)	\(-363303920\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2844q^{2}+3^{10}q^{3}+5991184q^{4}+\cdots\)
75.22.a.d	$75$	$22$	75.a	1.a	$1$	$2$	$2$	$209.608$	\(\Q(\sqrt{649}) \)	None	✓	$1$	$1$	\(-666\)	\(-118098\)	\(0\)	\(-679896112\)		$+$	$+$	$+$	$2\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-333-\beta )q^{2}-3^{10}q^{3}+(589618+\cdots)q^{4}+\cdots\)
75.22.a.e	$75$	$22$	75.a	1.a	$1$	$3$	$3$	$209.608$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(-803\)	\(-177147\)	\(0\)	\(1577598316\)		$+$	$+$	$+$	$2^{2}\cdot 3\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(-268+\beta _{1})q^{2}-3^{10}q^{3}+(2238318+\cdots)q^{4}+\cdots\)
75.22.a.f	$75$	$22$	75.a	1.a	$1$	$3$	$3$	$209.608$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(-702\)	\(177147\)	\(0\)	\(2072418204\)		$-$	$+$	$-$	$2^{5}\cdot 3^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-234-\beta _{1})q^{2}+3^{10}q^{3}+(1748076+\cdots)q^{4}+\cdots\)
75.22.a.g	$75$	$22$	75.a	1.a	$1$	$3$	$3$	$209.608$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(2300\)	\(-177147\)	\(0\)	\(-465666872\)		$+$	$+$	$+$	$2^{5}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(767+\beta _{1})q^{2}-3^{10}q^{3}+(421908+\cdots)q^{4}+\cdots\)
75.22.a.h	$75$	$22$	75.a	1.a	$1$	$4$	$4$	$209.608$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(897\)	\(236196\)	\(0\)	\(234577504\)		$-$	$+$	$-$	$2^{5}\cdot 3^{4}\cdot 5^{2}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(224+\beta _{1})q^{2}+3^{10}q^{3}+(-290854+\cdots)q^{4}+\cdots\)
75.22.a.i	$75$	$22$	75.a	1.a	$1$	$6$	$6$	$209.608$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(-92\)	\(354294\)	\(0\)	\(-1327143454\)		$-$	$-$	$+$	$2^{18}\cdot 3^{6}\cdot 5^{7}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(-15+\beta _{1})q^{2}+3^{10}q^{3}+(731822+\cdots)q^{4}+\cdots\)
75.22.a.j	$75$	$22$	75.a	1.a	$1$	$6$	$6$	$209.608$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(92\)	\(-354294\)	\(0\)	\(1327143454\)		$+$	$+$	$+$	$2^{18}\cdot 3^{6}\cdot 5^{7}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(15-\beta _{1})q^{2}-3^{10}q^{3}+(731822+\cdots)q^{4}+\cdots\)
75.22.a.k	$75$	$22$	75.a	1.a	$1$	$8$	$8$	$209.608$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(-666\)	\(472392\)	\(0\)	\(-134034472\)		$-$	$+$	$-$	$2^{21}\cdot 3^{10}\cdot 5^{14}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(-83-\beta _{1})q^{2}+3^{10}q^{3}+(1335201+\cdots)q^{4}+\cdots\)
75.22.a.l	$75$	$22$	75.a	1.a	$1$	$8$	$8$	$209.608$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(666\)	\(-472392\)	\(0\)	\(134034472\)		$+$	$-$	$-$	$2^{21}\cdot 3^{10}\cdot 5^{14}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(83+\beta _{1})q^{2}-3^{10}q^{3}+(1335201+\cdots)q^{4}+\cdots\)
75.22.a.m	$75$	$22$	75.a	1.a	$1$	$10$	$10$	$209.608$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(-645\)	\(590490\)	\(0\)	\(-115357290\)		$-$	$-$	$+$	$2^{24}\cdot 3^{14}\cdot 5^{24}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(-65+\beta _{1})q^{2}+3^{10}q^{3}+(1004337+\cdots)q^{4}+\cdots\)
75.22.a.n	$75$	$22$	75.a	1.a	$1$	$10$	$10$	$209.608$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(645\)	\(-590490\)	\(0\)	\(115357290\)		$+$	$-$	$-$	$2^{24}\cdot 3^{14}\cdot 5^{24}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(65-\beta _{1})q^{2}-3^{10}q^{3}+(1004337+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(75))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(75)) \cong \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)