Space of modular forms of level 75, weight 2, and trivial character

• Label: 75.2.a
• Level: $75$
• Weight: $2$
• Character orbit: 75.a
• Rep. character: $\chi_{75}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $3$
• Sturm bound: $20$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(20\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(75))\).

	Total	New	Old
Modular forms	16	3	13
Cusp forms	5	3	2
Eisenstein series	11	0	11

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
\(+\)	\(-\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(2\)
Plus space		\(+\)	\(0\)
Minus space		\(-\)	\(3\)

Trace form

3 * q + q^2 + q^3 + 3 * q^4 - 3 * q^6 - 3 * q^8 + 3 * q^9 - q^12 + 2 * q^13 - 12 * q^14 - 9 * q^16 - 2 * q^17 + q^18 - 6 * q^19 + 6 * q^21 - 4 * q^22 - 3 * q^24 + 6 * q^26 + q^27 + 18 * q^29 - 6 * q^31 + 5 * q^32 - 4 * q^33 + 6 * q^34 + 3 * q^36 + 10 * q^37 + 4 * q^38 - 6 * q^41 - 4 * q^43 + 12 * q^44 + 24 * q^46 - 8 * q^47 - q^48 - 3 * q^49 - 6 * q^51 - 2 * q^52 + 10 * q^53 - 3 * q^54 + 4 * q^57 - 2 * q^58 - 24 * q^59 + 12 * q^61 - 9 * q^64 - 12 * q^66 - 12 * q^67 + 2 * q^68 - 12 * q^69 - 24 * q^71 - 3 * q^72 - 10 * q^73 + 18 * q^74 - 24 * q^76 + 2 * q^78 + 3 * q^81 + 10 * q^82 - 12 * q^83 + 12 * q^84 - 2 * q^87 + 12 * q^88 - 6 * q^89 - 6 * q^91 + 21 * q^96 - 2 * q^97 - 7 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(75))\) 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
75.2.a.a	$75$	$2$	75.a	1.a	$1$	$1$	$1$	$0.599$	\(\Q\)	None	✓	✓			75.2.a.a	$1$	$0$	\(-2\)	\(1\)	\(0\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}-2q^{6}+3q^{7}+\cdots\)
75.2.a.b	$75$	$2$	75.a	1.a	$1$	$1$	$1$	$0.599$	\(\Q\)	None	✓				15.2.a.a	$1$	$0$	\(1\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}+q^{9}+\cdots\)
75.2.a.c	$75$	$2$	75.a	1.a	$1$	$1$	$1$	$0.599$	\(\Q\)	None	✓	✓	✓	✓	75.2.a.a	$1$	$0$	\(2\)	\(-1\)	\(0\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}-2q^{6}-3q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(75))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(75)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)