Space of modular forms of level 370, weight 2, and trivial character

• Label: 370.2.a
• Level: $370$
• Weight: $2$
• Character orbit: 370.a
• Rep. character: $\chi_{370}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $7$
• Sturm bound: $114$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	370.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(114\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(370))\).

	Total	New	Old
Modular forms	60	11	49
Cusp forms	53	11	42
Eisenstein series	7	0	7

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
\(+\)	\(+\)	\(+\)	\(+\)		\(4\)	\(1\)	\(3\)		\(4\)	\(1\)	\(3\)		\(0\)	\(0\)	\(0\)
\(+\)	\(+\)	\(-\)	\(-\)		\(10\)	\(1\)	\(9\)		\(9\)	\(1\)	\(8\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(8\)	\(1\)	\(7\)		\(7\)	\(1\)	\(6\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(7\)	\(2\)	\(5\)		\(6\)	\(2\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(9\)	\(3\)	\(6\)		\(8\)	\(3\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(7\)	\(0\)	\(7\)		\(6\)	\(0\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(9\)	\(0\)	\(9\)		\(8\)	\(0\)	\(8\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(6\)	\(3\)	\(3\)		\(5\)	\(3\)	\(2\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(27\)	\(3\)	\(24\)		\(24\)	\(3\)	\(21\)		\(3\)	\(0\)	\(3\)
Minus space			\(-\)		\(33\)	\(8\)	\(25\)		\(29\)	\(8\)	\(21\)		\(4\)	\(0\)	\(4\)

Trace form

11 * q + q^2 + 11 * q^4 + q^5 + 4 * q^6 - 8 * q^7 + q^8 + 15 * q^9 - q^10 + 8 * q^11 - 2 * q^13 + 4 * q^14 + 4 * q^15 + 11 * q^16 + 10 * q^17 + 13 * q^18 - 8 * q^19 + q^20 + 12 * q^22 - 8 * q^23 + 4 * q^24 + 11 * q^25 + 10 * q^26 - 24 * q^27 - 8 * q^28 - 10 * q^29 - 16 * q^31 + q^32 - 24 * q^33 - 6 * q^34 - 4 * q^35 + 15 * q^36 + q^37 + 4 * q^38 - 40 * q^39 - q^40 - 14 * q^41 - 8 * q^42 - 4 * q^43 + 8 * q^44 - 3 * q^45 + 8 * q^46 - 8 * q^47 - 9 * q^49 + q^50 - 2 * q^52 + 38 * q^53 - 8 * q^54 - 12 * q^55 + 4 * q^56 + 8 * q^57 + 10 * q^58 + 16 * q^59 + 4 * q^60 - 10 * q^61 - 20 * q^62 - 32 * q^63 + 11 * q^64 + 2 * q^65 - 8 * q^66 + 10 * q^68 - 16 * q^69 + 4 * q^70 + 16 * q^71 + 13 * q^72 - 2 * q^73 - q^74 - 8 * q^76 + 16 * q^77 - 40 * q^78 - 24 * q^79 + q^80 + 3 * q^81 - 6 * q^82 - 24 * q^83 + 6 * q^85 - 8 * q^86 - 24 * q^87 + 12 * q^88 + 6 * q^89 - 13 * q^90 - 16 * q^91 - 8 * q^92 - 16 * q^93 - 12 * q^94 - 4 * q^95 + 4 * q^96 + 10 * q^97 + 9 * q^98 + 56 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(370))\) 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
370.2.a.a	$370$	$2$	370.a	1.a	$1$	$1$	$1$	$2.954$	\(\Q\)	None	✓	✓	✓	✓	370.2.a.a	$1$	$0$	\(-1\)	\(-2\)	\(-1\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}-q^{7}+\cdots\)
370.2.a.b	$370$	$2$	370.a	1.a	$1$	$1$	$1$	$2.954$	\(\Q\)	None	✓	✓	✓	✓	370.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}-q^{8}-3q^{9}+q^{10}+\cdots\)
370.2.a.c	$370$	$2$	370.a	1.a	$1$	$1$	$1$	$2.954$	\(\Q\)	None	✓	✓	✓	✓	370.2.a.c	$1$	$0$	\(-1\)	\(2\)	\(1\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2q^{3}+q^{4}+q^{5}-2q^{6}+q^{7}+\cdots\)
370.2.a.d	$370$	$2$	370.a	1.a	$1$	$1$	$1$	$2.954$	\(\Q\)	None	✓	✓			370.2.a.d	$1$	$0$	\(1\)	\(-2\)	\(1\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}+q^{4}+q^{5}-2q^{6}+2q^{7}+\cdots\)
370.2.a.e	$370$	$2$	370.a	1.a	$1$	$2$	$2$	$2.954$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	370.2.a.e	$1$	$1$	\(-2\)	\(-2\)	\(2\)	\(-6\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta )q^{3}+q^{4}+q^{5}+(1+\cdots)q^{6}+\cdots\)
370.2.a.f	$370$	$2$	370.a	1.a	$1$	$2$	$2$	$2.954$	\(\Q(\sqrt{33}) \)	None	✓	✓	✓		370.2.a.f	$1$	$0$	\(2\)	\(4\)	\(2\)	\(-3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}+q^{5}+2q^{6}+(-1+\cdots)q^{7}+\cdots\)
370.2.a.g	$370$	$2$	370.a	1.a	$1$	$3$	$3$	$2.954$	3.3.892.1	None	✓	✓	✓	✓	370.2.a.g	$1$	$0$	\(3\)	\(0\)	\(-3\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta _{2}q^{3}+q^{4}-q^{5}-\beta _{2}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(370))\) into lower level spaces

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