Space of modular forms of level 740, weight 2, and trivial character

• Label: 740.2.a
• Level: $740$
• Weight: $2$
• Character orbit: 740.a
• Rep. character: $\chi_{740}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $6$
• Sturm bound: $228$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	740.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(228\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(740))\).

	Total	New	Old
Modular forms	120	12	108
Cusp forms	109	12	97
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.

\(2\)	\(5\)	\(37\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(13\)	\(0\)	\(13\)		\(12\)	\(0\)	\(12\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(17\)	\(0\)	\(17\)		\(15\)	\(0\)	\(15\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(17\)	\(0\)	\(17\)		\(15\)	\(0\)	\(15\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(13\)	\(0\)	\(13\)		\(11\)	\(0\)	\(11\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(17\)	\(4\)	\(13\)		\(16\)	\(4\)	\(12\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(13\)	\(1\)	\(12\)		\(12\)	\(1\)	\(11\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(13\)	\(1\)	\(12\)		\(12\)	\(1\)	\(11\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(17\)	\(6\)	\(11\)		\(16\)	\(6\)	\(10\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(52\)	\(2\)	\(50\)		\(47\)	\(2\)	\(45\)		\(5\)	\(0\)	\(5\)
Minus space			\(-\)		\(68\)	\(10\)	\(58\)		\(62\)	\(10\)	\(52\)		\(6\)	\(0\)	\(6\)

Trace form

12 * q + 4 * q^3 + 2 * q^5 + 16 * q^9 - 4 * q^11 - 4 * q^15 + 16 * q^17 + 12 * q^19 + 20 * q^21 + 4 * q^23 + 12 * q^25 + 16 * q^27 + 8 * q^29 + 4 * q^31 + 24 * q^33 + 4 * q^35 + 2 * q^37 + 4 * q^39 + 4 * q^41 + 28 * q^43 + 10 * q^45 + 4 * q^47 + 24 * q^49 + 12 * q^51 - 4 * q^53 - 20 * q^57 - 20 * q^59 + 28 * q^61 - 36 * q^63 - 8 * q^65 + 8 * q^67 - 4 * q^69 - 20 * q^71 + 12 * q^73 + 4 * q^75 - 8 * q^77 - 16 * q^79 + 28 * q^81 - 12 * q^85 - 48 * q^87 - 24 * q^89 + 4 * q^93 - 8 * q^95 + 12 * q^97 - 16 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(740))\) 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
740.2.a.a	$740$	$2$	740.a	1.a	$1$	$1$	$1$	$5.909$	\(\Q\)	None	✓	✓	✓	✓	740.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{7}-2q^{9}-3q^{11}-6q^{13}+\cdots\)
740.2.a.b	$740$	$2$	740.a	1.a	$1$	$1$	$1$	$5.909$	\(\Q\)	None	✓	✓	✓	✓	740.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-q^{7}-2q^{9}-3q^{11}-4q^{13}+\cdots\)
740.2.a.c	$740$	$2$	740.a	1.a	$1$	$1$	$1$	$5.909$	\(\Q\)	None	✓	✓			740.2.a.c	$1$	$0$	\(0\)	\(3\)	\(-1\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-q^{5}-3q^{7}+6q^{9}+5q^{11}+\cdots\)
740.2.a.d	$740$	$2$	740.a	1.a	$1$	$2$	$2$	$5.909$	\(\Q(\sqrt{3}) \)	None	✓	✓			740.2.a.d	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+q^{5}+(3+\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
740.2.a.e	$740$	$2$	740.a	1.a	$1$	$3$	$3$	$5.909$	3.3.148.1	None	✓	✓	✓		740.2.a.e	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}-q^{5}+(2\beta _{1}-\beta _{2})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
740.2.a.f	$740$	$2$	740.a	1.a	$1$	$4$	$4$	$5.909$	4.4.286164.1	None	✓	✓	✓		740.2.a.f	$1$	$0$	\(0\)	\(3\)	\(4\)	\(-5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+q^{5}+(-1-\beta _{1})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(740))\) into lower level spaces

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