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	Dim.
\(+\)	\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)	\(3\)
Plus space			\(+\)	\(3\)
Minus space			\(-\)	\(6\)

Trace form

\( 9 q + q^{2} + 9 q^{4} + q^{5} + 4 q^{6} - 8 q^{7} + q^{8} + 13 q^{9} + O(q^{10}) \)

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

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

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