Space of modular forms of level 310, weight 2, and trivial character

• Label: 310.2.a
• Level: $310$
• Weight: $2$
• Character orbit: 310.a
• Rep. character: $\chi_{310}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $5$
• Sturm bound: $96$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 310 = 2 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	310.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(96\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(310))\).

	Total	New	Old
Modular forms	52	9	43
Cusp forms	45	9	36
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.

\(2\)	\(5\)	\(31\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(3\)	\(2\)	\(1\)		\(3\)	\(2\)	\(1\)		\(0\)	\(0\)	\(0\)
\(+\)	\(+\)	\(-\)	\(-\)		\(10\)	\(0\)	\(10\)		\(9\)	\(0\)	\(9\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(7\)	\(2\)	\(5\)		\(6\)	\(2\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(6\)	\(0\)	\(6\)		\(5\)	\(0\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(9\)	\(1\)	\(8\)		\(8\)	\(1\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(6\)	\(0\)	\(6\)		\(5\)	\(0\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(7\)	\(3\)	\(4\)		\(6\)	\(3\)	\(3\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(24\)	\(3\)	\(21\)		\(21\)	\(3\)	\(18\)		\(3\)	\(0\)	\(3\)
Minus space			\(-\)		\(28\)	\(6\)	\(22\)		\(24\)	\(6\)	\(18\)		\(4\)	\(0\)	\(4\)

Trace form

9 * q + q^2 + 9 * q^4 + q^5 + 4 * q^6 - 8 * q^7 + q^8 + 13 * q^9 + q^10 + 4 * q^11 - 14 * q^13 + 4 * q^15 + 9 * q^16 - 6 * q^17 - 3 * q^18 + 4 * q^19 + q^20 - 16 * q^21 - 16 * q^23 + 4 * q^24 + 9 * q^25 - 10 * q^26 - 8 * q^28 + 14 * q^29 - q^31 + q^32 - 8 * q^33 + 10 * q^34 + 13 * q^36 - 14 * q^37 - 4 * q^38 + 16 * q^39 + q^40 - 14 * q^41 + 8 * q^42 - 24 * q^43 + 4 * q^44 + 5 * q^45 - 8 * q^46 + 32 * q^47 + 17 * q^49 + q^50 - 24 * q^51 - 14 * q^52 - 30 * q^53 + 16 * q^54 + 4 * q^55 - 8 * q^57 - 14 * q^58 + 28 * q^59 + 4 * q^60 - 18 * q^61 + 7 * q^62 - 24 * q^63 + 9 * q^64 + 6 * q^65 - 24 * q^66 - 20 * q^67 - 6 * q^68 + 32 * q^69 + 8 * q^70 + 8 * q^71 - 3 * q^72 - 14 * q^73 - 2 * q^74 + 4 * q^76 + 8 * q^77 - 32 * q^78 + 32 * q^79 + q^80 - 39 * q^81 + 10 * q^82 + 24 * q^83 - 16 * q^84 + 6 * q^85 - 12 * q^86 - 8 * q^87 - 6 * q^89 - 3 * q^90 + 8 * q^91 - 16 * q^92 + 8 * q^94 + 12 * q^95 + 4 * q^96 - 6 * q^97 + 9 * q^98 + 20 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(310))\) 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	31
310.2.a.a	$310$	$2$	310.a	1.a	$1$	$1$	$1$	$2.475$	\(\Q\)	None	✓	✓	✓	✓	310.2.a.a	$1$	$1$	\(1\)	\(-2\)	\(-1\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}+q^{4}-q^{5}-2q^{6}-4q^{7}+\cdots\)
310.2.a.b	$310$	$2$	310.a	1.a	$1$	$1$	$1$	$2.475$	\(\Q\)	None	✓	✓	✓	✓	310.2.a.b	$1$	$0$	\(1\)	\(2\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}-q^{5}+2q^{6}+q^{8}+\cdots\)
310.2.a.c	$310$	$2$	310.a	1.a	$1$	$2$	$2$	$2.475$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	310.2.a.c	$1$	$1$	\(-2\)	\(-2\)	\(-2\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta )q^{3}+q^{4}-q^{5}+(1+\cdots)q^{6}+\cdots\)
310.2.a.d	$310$	$2$	310.a	1.a	$1$	$2$	$2$	$2.475$	\(\Q(\sqrt{6}) \)	None	✓	✓	✓	✓	310.2.a.d	$1$	$0$	\(-2\)	\(0\)	\(2\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}+q^{5}-\beta q^{6}-2q^{7}+\cdots\)
310.2.a.e	$310$	$2$	310.a	1.a	$1$	$3$	$3$	$2.475$	3.3.148.1	None	✓	✓	✓	✓	310.2.a.e	$1$	$0$	\(3\)	\(2\)	\(3\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+q^{5}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(310))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(310)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 2}\)