Space of modular forms of level 7400, weight 2, and trivial character

• Label: 7400.2.a
• Level: $7400$
• Weight: $2$
• Character orbit: 7400.a
• Rep. character: $\chi_{7400}(1,\cdot)$
• Character field: $\Q$
• Dimension: $171$
• Newform subspaces: $31$
• Sturm bound: $2280$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 7400 = 2^{3} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7400.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 31 \)
Sturm bound:			\(2280\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(7400))\).

	Total	New	Old
Modular forms	1164	171	993
Cusp forms	1117	171	946
Eisenstein series	47	0	47

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

\(2\)	\(5\)	\(37\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(19\)
\(+\)	\(+\)	\(-\)	\(-\)	\(23\)
\(+\)	\(-\)	\(+\)	\(-\)	\(25\)
\(+\)	\(-\)	\(-\)	\(+\)	\(19\)
\(-\)	\(+\)	\(+\)	\(-\)	\(20\)
\(-\)	\(+\)	\(-\)	\(+\)	\(19\)
\(-\)	\(-\)	\(+\)	\(+\)	\(21\)
\(-\)	\(-\)	\(-\)	\(-\)	\(25\)
Plus space			\(+\)	\(78\)
Minus space			\(-\)	\(93\)

Trace form

171 * q + 2 * q^3 - 4 * q^7 + 161 * q^9 + 2 * q^11 + 2 * q^13 + 2 * q^17 + 12 * q^19 + 16 * q^21 + 8 * q^23 + 8 * q^27 + 10 * q^29 - 4 * q^33 + q^37 + 16 * q^39 - 8 * q^41 + 8 * q^43 - 8 * q^47 + 175 * q^49 + 72 * q^51 - 2 * q^53 + 16 * q^57 + 32 * q^59 - 22 * q^61 + 12 * q^63 + 22 * q^67 - 24 * q^69 + 28 * q^71 - 12 * q^73 - 32 * q^77 - 28 * q^79 + 179 * q^81 + 12 * q^83 + 20 * q^87 - 18 * q^89 + 12 * q^91 - 48 * q^93 - 18 * q^97 + 64 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(7400))\) 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	5	37
7400.2.a.a	$7400$	$2$	7400.a	1.a	$1$	$1$	$1$	$59.089$	\(\Q\)	None	✓				1480.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-3q^{7}+q^{9}+3q^{11}-5q^{17}+\cdots\)
7400.2.a.b	$7400$	$2$	7400.a	1.a	$1$	$1$	$1$	$59.089$	\(\Q\)	None	✓				1480.2.d.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+4q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
7400.2.a.c	$7400$	$2$	7400.a	1.a	$1$	$1$	$1$	$59.089$	\(\Q\)	None	✓				1480.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}-2q^{9}-3q^{11}+8q^{17}+\cdots\)
7400.2.a.d	$7400$	$2$	7400.a	1.a	$1$	$1$	$1$	$59.089$	\(\Q\)	None	✓	✓			7400.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}-2q^{9}-3q^{11}+7q^{17}+\cdots\)
7400.2.a.e	$7400$	$2$	7400.a	1.a	$1$	$1$	$1$	$59.089$	\(\Q\)	None	✓	✓			7400.2.a.d	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}-2q^{9}-3q^{11}-7q^{17}+\cdots\)
7400.2.a.f	$7400$	$2$	7400.a	1.a	$1$	$1$	$1$	$59.089$	\(\Q\)	None	✓				296.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}+q^{11}+6q^{13}+\cdots\)
7400.2.a.g	$7400$	$2$	7400.a	1.a	$1$	$1$	$1$	$59.089$	\(\Q\)	None	✓				296.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{7}-2q^{9}-3q^{11}-2q^{17}+\cdots\)
7400.2.a.h	$7400$	$2$	7400.a	1.a	$1$	$1$	$1$	$59.089$	\(\Q\)	None	✓				1480.2.d.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-4q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
7400.2.a.i	$7400$	$2$	7400.a	1.a	$1$	$1$	$1$	$59.089$	\(\Q\)	None	✓				1480.2.a.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{7}+q^{9}+6q^{13}+2q^{17}+\cdots\)
7400.2.a.j	$7400$	$2$	7400.a	1.a	$1$	$1$	$1$	$59.089$	\(\Q\)	None	✓				1480.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{7}+6q^{9}-3q^{11}+4q^{13}+\cdots\)
7400.2.a.k	$7400$	$2$	7400.a	1.a	$1$	$3$	$3$	$59.089$	3.3.229.1	None	✓				296.2.a.c	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-7\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(-2-\beta _{1}+\beta _{2})q^{7}+\cdots\)
7400.2.a.l	$7400$	$2$	7400.a	1.a	$1$	$3$	$3$	$59.089$	3.3.316.1	None	✓				1480.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{1}q^{7}+\beta _{2}q^{9}+(-1-\beta _{2})q^{11}+\cdots\)
7400.2.a.m	$7400$	$2$	7400.a	1.a	$1$	$3$	$3$	$59.089$	3.3.568.1	None	✓				1480.2.a.f	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{1}q^{7}+(1+2\beta _{1}+\beta _{2})q^{9}+\cdots\)
7400.2.a.n	$7400$	$2$	7400.a	1.a	$1$	$4$	$4$	$59.089$	4.4.48389.1	None	✓				296.2.a.d	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(\beta _{2}-\beta _{3})q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)
7400.2.a.o	$7400$	$2$	7400.a	1.a	$1$	$5$	$5$	$59.089$	5.5.998068.1	None	✓				1480.2.a.j	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{3}+\beta _{2}q^{7}+(2-\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{9}+\cdots\)
7400.2.a.p	$7400$	$2$	7400.a	1.a	$1$	$5$	$5$	$59.089$	5.5.6397264.1	None	✓				1480.2.a.i	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{1}+\beta _{3})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
7400.2.a.q	$7400$	$2$	7400.a	1.a	$1$	$5$	$5$	$59.089$	5.5.935504.1	None	✓				1480.2.a.h	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{3}q^{7}+(\beta _{1}+\beta _{2}-\beta _{3})q^{9}+\cdots\)
7400.2.a.r	$7400$	$2$	7400.a	1.a	$1$	$5$	$5$	$59.089$	5.5.583504.1	None	✓				1480.2.a.g	$1$	$0$	\(0\)	\(5\)	\(0\)	\(5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1+\beta _{2}-\beta _{4})q^{7}+(1+\cdots)q^{9}+\cdots\)
7400.2.a.s	$7400$	$2$	7400.a	1.a	$1$	$6$	$6$	$59.089$	6.6.693982032.1	None	✓				1480.2.a.k	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-8\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{2})q^{7}+(2+\beta _{3}-\beta _{4}+\cdots)q^{9}+\cdots\)
7400.2.a.t	$7400$	$2$	7400.a	1.a	$1$	$8$	$8$	$59.089$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		7400.2.a.t	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-6\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{7})q^{7}+(1+\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
7400.2.a.u	$7400$	$2$	7400.a	1.a	$1$	$8$	$8$	$59.089$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			7400.2.a.u	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{7}q^{7}+(-1+\beta _{3}-\beta _{4}+\cdots)q^{9}+\cdots\)
7400.2.a.v	$7400$	$2$	7400.a	1.a	$1$	$8$	$8$	$59.089$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		7400.2.a.u	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{7}q^{7}+(-1+\beta _{3}-\beta _{4}+\cdots)q^{9}+\cdots\)
7400.2.a.w	$7400$	$2$	7400.a	1.a	$1$	$8$	$8$	$59.089$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			7400.2.a.t	$1$	$0$	\(0\)	\(4\)	\(0\)	\(6\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{7})q^{7}+(1+\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
7400.2.a.x	$7400$	$2$	7400.a	1.a	$1$	$9$	$9$	$59.089$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		7400.2.a.x	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{7}q^{7}+(1+\beta _{1}+\beta _{3}+\beta _{4}+\cdots)q^{9}+\cdots\)
7400.2.a.y	$7400$	$2$	7400.a	1.a	$1$	$9$	$9$	$59.089$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓			7400.2.a.x	$1$	$0$	\(0\)	\(2\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{7}q^{7}+(1+\beta _{1}+\beta _{3}+\beta _{4}+\cdots)q^{9}+\cdots\)
7400.2.a.z	$7400$	$2$	7400.a	1.a	$1$	$10$	$10$	$59.089$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓			7400.2.a.z	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{5}q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
7400.2.a.ba	$7400$	$2$	7400.a	1.a	$1$	$10$	$10$	$59.089$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓		✓		1480.2.d.b	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(7\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{4})q^{7}+(\beta _{2}+\beta _{3})q^{9}+\cdots\)
7400.2.a.bb	$7400$	$2$	7400.a	1.a	$1$	$10$	$10$	$59.089$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				1480.2.d.b	$1$	$1$	\(0\)	\(4\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{4})q^{7}+(\beta _{2}+\beta _{3}+\cdots)q^{9}+\cdots\)
7400.2.a.bc	$7400$	$2$	7400.a	1.a	$1$	$10$	$10$	$59.089$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		7400.2.a.z	$1$	$0$	\(0\)	\(4\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{5}q^{7}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
7400.2.a.bd	$7400$	$2$	7400.a	1.a	$1$	$16$	$16$	$59.089$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓		✓		1480.2.d.c	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-9\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{8})q^{7}+(1+\beta _{2})q^{9}+\cdots\)
7400.2.a.be	$7400$	$2$	7400.a	1.a	$1$	$16$	$16$	$59.089$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓		✓		1480.2.d.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(9\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{8})q^{7}+(1+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(7400))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(7400)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(148))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(296))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(370))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(740))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(925))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1480))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1850))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3700))\)\(^{\oplus 2}\)