Space of modular forms of level 350, weight 2, and trivial character

• Label: 350.2.a
• Level: $350$
• Weight: $2$
• Character orbit: 350.a
• Rep. character: $\chi_{350}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $8$
• Sturm bound: $120$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(120\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(350))\).

	Total	New	Old
Modular forms	72	10	62
Cusp forms	49	10	39
Eisenstein series	23	0	23

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

\(2\)	\(5\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(-\)	\(+\)	$-$	\(2\)
\(-\)	\(+\)	\(+\)	$-$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(3\)
Plus space			\(+\)	\(2\)
Minus space			\(-\)	\(8\)

Trace form

\( 10 q + 2 q^{3} + 10 q^{4} - 2 q^{6} + 18 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(350))\) 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	7
350.2.a.a	$350$	$2$	350.a	1.a	$1$	$1$	$1$	$2.795$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{7}-q^{8}+\cdots\)
350.2.a.b	$350$	$2$	350.a	1.a	$1$	$1$	$1$	$2.795$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}-3q^{9}+4q^{11}+\cdots\)
350.2.a.c	$350$	$2$	350.a	1.a	$1$	$1$	$1$	$2.795$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(3\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+3q^{3}+q^{4}-3q^{6}+q^{7}-q^{8}+\cdots\)
350.2.a.d	$350$	$2$	350.a	1.a	$1$	$1$	$1$	$2.795$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-3\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}+q^{4}-3q^{6}-q^{7}+q^{8}+\cdots\)
350.2.a.e	$350$	$2$	350.a	1.a	$1$	$1$	$1$	$2.795$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{7}+q^{8}+\cdots\)
350.2.a.f	$350$	$2$	350.a	1.a	$1$	$1$	$1$	$2.795$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(2\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}+2q^{6}-q^{7}+q^{8}+\cdots\)
350.2.a.g	$350$	$2$	350.a	1.a	$1$	$2$	$2$	$2.795$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{7}-q^{8}+\cdots\)
350.2.a.h	$350$	$2$	350.a	1.a	$1$	$2$	$2$	$2.795$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(350))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(350)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)