Space of modular forms of level 750, weight 2, and trivial character

• Label: 750.2.a
• Level: $750$
• Weight: $2$
• Character orbit: 750.a
• Rep. character: $\chi_{750}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $8$
• Sturm bound: $300$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	750.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(300\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(750))\).

	Total	New	Old
Modular forms	170	16	154
Cusp forms	131	16	115
Eisenstein series	39	0	39

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

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(16\)	\(2\)	\(14\)		\(12\)	\(2\)	\(10\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)		\(26\)	\(2\)	\(24\)		\(21\)	\(2\)	\(19\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)		\(24\)	\(4\)	\(20\)		\(19\)	\(4\)	\(15\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)	\(+\)		\(19\)	\(0\)	\(19\)		\(14\)	\(0\)	\(14\)		\(5\)	\(0\)	\(5\)
\(-\)	\(+\)	\(+\)	\(-\)		\(19\)	\(2\)	\(17\)		\(14\)	\(2\)	\(12\)		\(5\)	\(0\)	\(5\)
\(-\)	\(+\)	\(-\)	\(+\)		\(24\)	\(2\)	\(22\)		\(19\)	\(2\)	\(17\)		\(5\)	\(0\)	\(5\)
\(-\)	\(-\)	\(+\)	\(+\)		\(21\)	\(0\)	\(21\)		\(16\)	\(0\)	\(16\)		\(5\)	\(0\)	\(5\)
\(-\)	\(-\)	\(-\)	\(-\)		\(21\)	\(4\)	\(17\)		\(16\)	\(4\)	\(12\)		\(5\)	\(0\)	\(5\)
Plus space			\(+\)		\(80\)	\(4\)	\(76\)		\(61\)	\(4\)	\(57\)		\(19\)	\(0\)	\(19\)
Minus space			\(-\)		\(90\)	\(12\)	\(78\)		\(70\)	\(12\)	\(58\)		\(20\)	\(0\)	\(20\)

Trace form

16 * q + 16 * q^4 + 16 * q^9 - 4 * q^11 - 4 * q^14 + 16 * q^16 + 28 * q^19 + 32 * q^21 - 4 * q^26 - 8 * q^29 + 20 * q^31 - 4 * q^34 + 16 * q^36 + 28 * q^39 - 12 * q^41 - 4 * q^44 - 8 * q^46 + 28 * q^49 - 8 * q^51 - 4 * q^56 - 20 * q^59 + 16 * q^61 + 16 * q^64 - 4 * q^66 - 8 * q^69 - 24 * q^71 - 12 * q^74 + 28 * q^76 + 4 * q^79 + 16 * q^81 + 32 * q^84 - 16 * q^86 - 36 * q^89 + 32 * q^91 - 16 * q^94 - 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(750))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	5
750.2.a.a	$750$	$2$	750.a	1.a	$1$	$2$	$2$	$5.989$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	750.2.a.a	$1$	$0$	\(-2\)	\(-2\)	\(0\)	\(-6\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}+(-2-2\beta )q^{7}+\cdots\)
750.2.a.b	$750$	$2$	750.a	1.a	$1$	$2$	$2$	$5.989$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	750.2.a.b	$1$	$1$	\(-2\)	\(-2\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}+(-1+\beta )q^{7}+\cdots\)
750.2.a.c	$750$	$2$	750.a	1.a	$1$	$2$	$2$	$5.989$	\(\Q(\sqrt{5}) \)	None	✓	✓			750.2.a.c	$1$	$0$	\(-2\)	\(2\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}+2\beta q^{7}-q^{8}+\cdots\)
750.2.a.d	$750$	$2$	750.a	1.a	$1$	$2$	$2$	$5.989$	\(\Q(\sqrt{5}) \)	None	✓	✓			750.2.a.d	$1$	$0$	\(-2\)	\(2\)	\(0\)	\(7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}+(4-\beta )q^{7}+\cdots\)
750.2.a.e	$750$	$2$	750.a	1.a	$1$	$2$	$2$	$5.989$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	750.2.a.d	$1$	$1$	\(2\)	\(-2\)	\(0\)	\(-7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+(-3-\beta )q^{7}+\cdots\)
750.2.a.f	$750$	$2$	750.a	1.a	$1$	$2$	$2$	$5.989$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	750.2.a.c	$1$	$0$	\(2\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}-2\beta q^{7}+q^{8}+\cdots\)
750.2.a.g	$750$	$2$	750.a	1.a	$1$	$2$	$2$	$5.989$	\(\Q(\sqrt{5}) \)	None	✓	✓			750.2.a.b	$1$	$0$	\(2\)	\(2\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}+(1-\beta )q^{7}+\cdots\)
750.2.a.h	$750$	$2$	750.a	1.a	$1$	$2$	$2$	$5.989$	\(\Q(\sqrt{5}) \)	None	✓	✓			750.2.a.a	$1$	$0$	\(2\)	\(2\)	\(0\)	\(6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}+(4-2\beta )q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(750))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(750)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(250))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(375))\)\(^{\oplus 2}\)