Space of modular forms of level 740, weight 4, and trivial character

• Label: 740.4.a
• Level: $740$
• Weight: $4$
• Character orbit: 740.a
• Rep. character: $\chi_{740}(1,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $5$
• Sturm bound: $456$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	740.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(456\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(740))\).

	Total	New	Old
Modular forms	348	36	312
Cusp forms	336	36	300
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.

\(2\)	\(5\)	\(37\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(-\)	\(8\)
\(-\)	\(+\)	\(-\)	\(+\)	\(11\)
\(-\)	\(-\)	\(+\)	\(+\)	\(11\)
\(-\)	\(-\)	\(-\)	\(-\)	\(6\)
Plus space			\(+\)	\(22\)
Minus space			\(-\)	\(14\)

Trace form

\( 36 q + 4 q^{3} - 10 q^{5} + 364 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(740))\) 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
740.4.a.a	$740$	$4$	740.a	1.a	$1$	$1$	$1$	$43.661$	\(\Q\)	None	✓	✓			740.4.a.a	$1$	$1$	\(0\)	\(-8\)	\(5\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}+5q^{5}+4q^{7}+37q^{9}-20q^{11}+\cdots\)
740.4.a.b	$740$	$4$	740.a	1.a	$1$	$5$	$5$	$43.661$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		740.4.a.b	$1$	$1$	\(0\)	\(4\)	\(25\)	\(-11\)		$-$	$-$	$-$	$-$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+5q^{5}+(-3+2\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
740.4.a.c	$740$	$4$	740.a	1.a	$1$	$8$	$8$	$43.661$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	740.4.a.c	$1$	$1$	\(0\)	\(0\)	\(-40\)	\(7\)		$-$	$+$	$+$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-5q^{5}+(1+\beta _{6})q^{7}+(7+\beta _{2}+\cdots)q^{9}+\cdots\)
740.4.a.d	$740$	$4$	740.a	1.a	$1$	$11$	$11$	$43.661$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	✓	✓	✓	740.4.a.d	$1$	$0$	\(0\)	\(2\)	\(55\)	\(-7\)		$-$	$-$	$+$	$+$	$2^{9}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+5q^{5}+(-1-\beta _{6})q^{7}+(15+\cdots)q^{9}+\cdots\)
740.4.a.e	$740$	$4$	740.a	1.a	$1$	$11$	$11$	$43.661$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	✓	✓	✓	740.4.a.e	$1$	$0$	\(0\)	\(6\)	\(-55\)	\(7\)		$-$	$+$	$-$	$+$	$2^{9}\cdot 3$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}-5q^{5}+(1+\beta _{5})q^{7}+(12+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(740))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(740)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(148))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(370))\)\(^{\oplus 2}\)