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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(6\)	\(1\)	\(5\)		\(4\)	\(1\)	\(3\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(12\)	\(2\)	\(10\)		\(9\)	\(2\)	\(7\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(11\)	\(2\)	\(9\)		\(8\)	\(2\)	\(6\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(7\)	\(0\)	\(7\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(9\)	\(1\)	\(8\)		\(6\)	\(1\)	\(5\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(9\)	\(0\)	\(9\)		\(6\)	\(0\)	\(6\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(10\)	\(1\)	\(9\)		\(7\)	\(1\)	\(6\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(8\)	\(3\)	\(5\)		\(5\)	\(3\)	\(2\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(32\)	\(2\)	\(30\)		\(21\)	\(2\)	\(19\)		\(11\)	\(0\)	\(11\)
Minus space			\(-\)		\(40\)	\(8\)	\(32\)		\(28\)	\(8\)	\(20\)		\(12\)	\(0\)	\(12\)

Trace form

10 * q + 2 * q^3 + 10 * q^4 - 2 * q^6 + 18 * q^9 + 2 * q^12 + 10 * q^13 + 2 * q^14 + 10 * q^16 - 8 * q^17 + 4 * q^18 - 2 * q^19 + 6 * q^21 - 4 * q^22 - 2 * q^24 - 18 * q^26 - 4 * q^27 - 16 * q^29 + 4 * q^31 - 8 * q^34 + 18 * q^36 + 8 * q^37 + 2 * q^38 + 24 * q^39 - 12 * q^41 - 2 * q^42 - 12 * q^43 + 8 * q^46 + 4 * q^47 + 2 * q^48 + 10 * q^49 + 10 * q^52 - 4 * q^53 - 32 * q^54 + 2 * q^56 + 4 * q^57 - 12 * q^58 + 2 * q^59 - 6 * q^61 - 12 * q^62 - 4 * q^63 + 10 * q^64 - 12 * q^66 + 16 * q^67 - 8 * q^68 - 48 * q^69 - 48 * q^71 + 4 * q^72 - 4 * q^73 + 12 * q^74 - 2 * q^76 + 4 * q^77 + 8 * q^78 + 32 * q^79 - 18 * q^81 + 4 * q^82 - 2 * q^83 + 6 * q^84 - 20 * q^86 - 12 * q^87 - 4 * q^88 - 88 * q^89 + 10 * q^91 - 8 * q^93 - 4 * q^94 - 2 * q^96 + 8 * q^97 - 84 * q^99

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	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	7
350.2.a.a	$350$	$2$	350.a	1.a	$1$	$1$	$1$	$2.795$	\(\Q\)	None	✓	✓	✓	✓	350.2.a.a	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{7}-q^{8}+\cdots\)
350.2.a.b	$350$	$2$	350.a	1.a	$1$	$1$	$1$	$2.795$	\(\Q\)	None	✓				70.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}-3q^{9}+4q^{11}+\cdots\)
350.2.a.c	$350$	$2$	350.a	1.a	$1$	$1$	$1$	$2.795$	\(\Q\)	None	✓	✓			350.2.a.c	$1$	$0$	\(-1\)	\(3\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+3q^{3}+q^{4}-3q^{6}+q^{7}-q^{8}+\cdots\)
350.2.a.d	$350$	$2$	350.a	1.a	$1$	$1$	$1$	$2.795$	\(\Q\)	None	✓	✓	✓	✓	350.2.a.c	$1$	$1$	\(1\)	\(-3\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}+q^{4}-3q^{6}-q^{7}+q^{8}+\cdots\)
350.2.a.e	$350$	$2$	350.a	1.a	$1$	$1$	$1$	$2.795$	\(\Q\)	None	✓	✓			350.2.a.a	$1$	$0$	\(1\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{7}+q^{8}+\cdots\)
350.2.a.f	$350$	$2$	350.a	1.a	$1$	$1$	$1$	$2.795$	\(\Q\)	None	✓		✓	✓	14.2.a.a	$1$	$0$	\(1\)	\(2\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}+2q^{6}-q^{7}+q^{8}+\cdots\)
350.2.a.g	$350$	$2$	350.a	1.a	$1$	$2$	$2$	$2.795$	\(\Q(\sqrt{6}) \)	None	✓		✓	✓	70.2.c.a	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{7}-q^{8}+\cdots\)
350.2.a.h	$350$	$2$	350.a	1.a	$1$	$2$	$2$	$2.795$	\(\Q(\sqrt{6}) \)	None	✓		✓		70.2.c.a	$1$	$0$	\(2\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+q^{7}+q^{8}+\cdots\)

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

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