Space of modular forms of level 372, weight 2, and trivial character

• Label: 372.2.a
• Level: $372$
• Weight: $2$
• Character orbit: 372.a
• Rep. character: $\chi_{372}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $5$
• Sturm bound: $128$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 372 = 2^{2} \cdot 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	372.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(128\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(372))\).

	Total	New	Old
Modular forms	70	6	64
Cusp forms	59	6	53
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.

\(2\)	\(3\)	\(31\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(8\)	\(0\)	\(8\)		\(7\)	\(0\)	\(7\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(9\)	\(0\)	\(9\)		\(7\)	\(0\)	\(7\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(10\)	\(0\)	\(10\)		\(8\)	\(0\)	\(8\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(9\)	\(0\)	\(9\)		\(7\)	\(0\)	\(7\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(9\)	\(2\)	\(7\)		\(8\)	\(2\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(8\)	\(1\)	\(7\)		\(7\)	\(1\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(8\)	\(1\)	\(7\)		\(7\)	\(1\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(9\)	\(2\)	\(7\)		\(8\)	\(2\)	\(6\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(33\)	\(2\)	\(31\)		\(28\)	\(2\)	\(26\)		\(5\)	\(0\)	\(5\)
Minus space			\(-\)		\(37\)	\(4\)	\(33\)		\(31\)	\(4\)	\(27\)		\(6\)	\(0\)	\(6\)

Trace form

6 * q - 4 * q^7 + 6 * q^9 + 4 * q^11 - 4 * q^15 - 4 * q^17 + 4 * q^19 + 4 * q^23 + 6 * q^25 + 8 * q^29 + 12 * q^35 + 8 * q^37 + 16 * q^41 - 12 * q^43 - 20 * q^47 + 10 * q^49 - 4 * q^51 - 4 * q^53 - 20 * q^55 - 8 * q^57 - 24 * q^59 - 20 * q^61 - 4 * q^63 + 12 * q^65 + 8 * q^67 - 16 * q^73 + 8 * q^75 - 28 * q^77 - 8 * q^79 + 6 * q^81 - 8 * q^83 + 32 * q^85 + 12 * q^87 - 28 * q^89 + 12 * q^91 + 2 * q^93 + 4 * q^95 - 16 * q^97 + 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(372))\) 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	31
372.2.a.a	$372$	$2$	372.a	1.a	$1$	$1$	$1$	$2.970$	\(\Q\)	None	✓	✓	✓	✓	372.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-q^{7}+q^{9}-6q^{13}+q^{15}+\cdots\)
372.2.a.b	$372$	$2$	372.a	1.a	$1$	$1$	$1$	$2.970$	\(\Q\)	None	✓	✓	✓	✓	372.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(-5\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}-5q^{7}+q^{9}+2q^{11}+\cdots\)
372.2.a.c	$372$	$2$	372.a	1.a	$1$	$1$	$1$	$2.970$	\(\Q\)	None	✓	✓			372.2.a.c	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+4q^{7}+q^{9}+2q^{13}+\cdots\)
372.2.a.d	$372$	$2$	372.a	1.a	$1$	$1$	$1$	$2.970$	\(\Q\)	None	✓	✓			372.2.a.d	$1$	$0$	\(0\)	\(1\)	\(3\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{5}-q^{7}+q^{9}+2q^{13}+3q^{15}+\cdots\)
372.2.a.e	$372$	$2$	372.a	1.a	$1$	$2$	$2$	$2.970$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	372.2.a.e	$1$	$0$	\(0\)	\(-2\)	\(3\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1+\beta )q^{5}+(-1+\beta )q^{7}+q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(372))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(372)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(93))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(124))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(186))\)\(^{\oplus 2}\)