Space of modular forms of level 185, weight 2, and trivial character

• Label: 185.2.a
• Level: $185$
• Weight: $2$
• Character orbit: 185.a
• Rep. character: $\chi_{185}(1,\cdot)$
• Character field: $\Q$
• Dimension: $13$
• Newform subspaces: $5$
• Sturm bound: $38$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 185 = 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	185.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(38\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(185))\).

	Total	New	Old
Modular forms	20	13	7
Cusp forms	17	13	4
Eisenstein series	3	0	3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(5\)	\(37\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(3\)	\(2\)	\(1\)		\(3\)	\(2\)	\(1\)		\(0\)	\(0\)	\(0\)
\(+\)	\(-\)	\(-\)		\(7\)	\(5\)	\(2\)		\(6\)	\(5\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(7\)	\(5\)	\(2\)		\(6\)	\(5\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(6\)	\(3\)	\(3\)		\(5\)	\(3\)	\(2\)		\(1\)	\(0\)	\(1\)
Minus space		\(-\)		\(14\)	\(10\)	\(4\)		\(12\)	\(10\)	\(2\)		\(2\)	\(0\)	\(2\)

Trace form

13 * q + q^2 + 15 * q^4 - q^5 - 12 * q^6 + 8 * q^7 - 3 * q^8 + 5 * q^9 - q^10 + 4 * q^12 + 6 * q^13 + 4 * q^14 - 4 * q^15 + 23 * q^16 - 14 * q^17 + q^18 - 7 * q^20 - 4 * q^21 - 12 * q^22 - 24 * q^24 + 13 * q^25 - 22 * q^26 + 12 * q^28 + 6 * q^29 + 8 * q^30 + 12 * q^31 + 5 * q^32 - 8 * q^33 - 10 * q^34 - 21 * q^36 - q^37 - 24 * q^38 + 12 * q^39 - 9 * q^40 - 2 * q^41 - 36 * q^42 + 4 * q^43 - 12 * q^44 - 5 * q^45 - 32 * q^46 + 12 * q^47 - 56 * q^48 + 41 * q^49 + q^50 + 4 * q^51 - 22 * q^52 - 22 * q^53 - 8 * q^54 + 4 * q^55 - 4 * q^57 - 18 * q^58 - 32 * q^59 - 14 * q^61 + 28 * q^62 + 68 * q^63 + 23 * q^64 + 6 * q^65 + 32 * q^66 + 20 * q^67 + 14 * q^68 - 12 * q^69 + 4 * q^70 - 4 * q^71 + 13 * q^72 - 2 * q^73 + 3 * q^74 + 36 * q^76 - 24 * q^77 + 76 * q^78 + 32 * q^79 + q^80 - 19 * q^81 - 14 * q^82 + 44 * q^83 - 8 * q^84 - 10 * q^85 + 52 * q^86 + 24 * q^87 - 20 * q^88 - 14 * q^89 - 13 * q^90 - 16 * q^91 + 64 * q^92 - 28 * q^93 - 8 * q^94 + 12 * q^95 - 44 * q^96 - 18 * q^97 - 27 * q^98 + 12 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(185))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	37
185.2.a.a	$185$	$2$	185.a	1.a	$1$	$1$	$1$	$1.477$	\(\Q\)	None	✓	✓			185.2.a.a	$1$	$1$	\(-2\)	\(1\)	\(-1\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}-q^{5}-2q^{6}-5q^{7}+\cdots\)
185.2.a.b	$185$	$2$	185.a	1.a	$1$	$1$	$1$	$1.477$	\(\Q\)	None	✓	✓	✓	✓	185.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}+q^{5}-3q^{7}-2q^{9}-5q^{11}+\cdots\)
185.2.a.c	$185$	$2$	185.a	1.a	$1$	$1$	$1$	$1.477$	\(\Q\)	None	✓	✓			185.2.a.c	$1$	$1$	\(1\)	\(-2\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}-q^{4}-q^{5}-2q^{6}-2q^{7}+\cdots\)
185.2.a.d	$185$	$2$	185.a	1.a	$1$	$5$	$5$	$1.477$	5.5.368464.1	None	✓	✓	✓	✓	185.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(5\)	\(7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}-\beta _{3}q^{3}+(2-\beta _{1}-\beta _{4})q^{4}+\cdots\)
185.2.a.e	$185$	$2$	185.a	1.a	$1$	$5$	$5$	$1.477$	5.5.973904.1	None	✓	✓	✓	✓	185.2.a.e	$1$	$0$	\(2\)	\(3\)	\(-5\)	\(11\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{2}+(1-\beta _{1})q^{3}+(2+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(185))\) into lower level spaces

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