Space of modular forms of level 370, weight 4, and trivial character

• Label: 370.4.a
• Level: $370$
• Weight: $4$
• Character orbit: 370.a
• Rep. character: $\chi_{370}(1,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $10$
• Sturm bound: $228$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	370.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(228\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(370))\).

	Total	New	Old
Modular forms	176	36	140
Cusp forms	168	36	132
Eisenstein series	8	0	8

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
\(+\)	\(+\)	\(+\)	$+$	\(5\)
\(+\)	\(+\)	\(-\)	$-$	\(5\)
\(+\)	\(-\)	\(+\)	$-$	\(5\)
\(+\)	\(-\)	\(-\)	$+$	\(4\)
\(-\)	\(+\)	\(+\)	$-$	\(3\)
\(-\)	\(+\)	\(-\)	$+$	\(6\)
\(-\)	\(-\)	\(+\)	$+$	\(6\)
\(-\)	\(-\)	\(-\)	$-$	\(2\)
Plus space			\(+\)	\(21\)
Minus space			\(-\)	\(15\)

Trace form

\( 36 q - 4 q^{2} + 4 q^{3} + 144 q^{4} - 10 q^{5} + 32 q^{6} - 16 q^{8} + 268 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(370))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5	37
370.4.a.a	$370$	$4$	370.a	1.a	$1$	$1$	$1$	$21.831$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(-2\)	\(5\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-2q^{3}+4q^{4}+5q^{5}-4q^{6}+\cdots\)
370.4.a.b	$370$	$4$	370.a	1.a	$1$	$1$	$1$	$21.831$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(1\)	\(5\)	\(-25\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+4q^{4}+5q^{5}+2q^{6}+\cdots\)
370.4.a.c	$370$	$4$	370.a	1.a	$1$	$1$	$1$	$21.831$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(6\)	\(5\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+6q^{3}+4q^{4}+5q^{5}+12q^{6}+\cdots\)
370.4.a.d	$370$	$4$	370.a	1.a	$1$	$3$	$3$	$21.831$	3.3.4860.1	None	✓	$1$	$1$	\(6\)	\(-3\)	\(-15\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-1+\beta _{2})q^{3}+4q^{4}-5q^{5}+\cdots\)
370.4.a.e	$370$	$4$	370.a	1.a	$1$	$4$	$4$	$21.831$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-8\)	\(3\)	\(20\)	\(16\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+5q^{5}+\cdots\)
370.4.a.f	$370$	$4$	370.a	1.a	$1$	$5$	$5$	$21.831$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-10\)	\(-9\)	\(25\)	\(-33\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-2+\beta _{1})q^{3}+4q^{4}+5q^{5}+\cdots\)
370.4.a.g	$370$	$4$	370.a	1.a	$1$	$5$	$5$	$21.831$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(-10\)	\(-3\)	\(-25\)	\(4\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-\beta _{1}+\beta _{2})q^{3}+4q^{4}-5q^{5}+\cdots\)
370.4.a.h	$370$	$4$	370.a	1.a	$1$	$5$	$5$	$21.831$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-10\)	\(3\)	\(-25\)	\(11\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(1-\beta _{1})q^{3}+4q^{4}-5q^{5}+\cdots\)
370.4.a.i	$370$	$4$	370.a	1.a	$1$	$5$	$5$	$21.831$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(10\)	\(11\)	\(25\)	\(23\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(2+\beta _{1})q^{3}+4q^{4}+5q^{5}+\cdots\)
370.4.a.j	$370$	$4$	370.a	1.a	$1$	$6$	$6$	$21.831$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(12\)	\(-3\)	\(-30\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-1+\beta _{1})q^{3}+4q^{4}-5q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(370))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(370)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 2}\)