Space of modular forms of level 351, weight 2, and trivial character

• Label: 351.2.a
• Level: $351$
• Weight: $2$
• Character orbit: 351.a
• Rep. character: $\chi_{351}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $6$
• Sturm bound: $84$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 351 = 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	351.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(84\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(351))\).

	Total	New	Old
Modular forms	48	16	32
Cusp forms	37	16	21
Eisenstein series	11	0	11

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

\(3\)	\(13\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(8\)	\(2\)	\(6\)		\(6\)	\(2\)	\(4\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(14\)	\(6\)	\(8\)		\(11\)	\(6\)	\(5\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(16\)	\(6\)	\(10\)		\(13\)	\(6\)	\(7\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(10\)	\(2\)	\(8\)		\(7\)	\(2\)	\(5\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(18\)	\(4\)	\(14\)		\(13\)	\(4\)	\(9\)		\(5\)	\(0\)	\(5\)
Minus space		\(-\)		\(30\)	\(12\)	\(18\)		\(24\)	\(12\)	\(12\)		\(6\)	\(0\)	\(6\)

Trace form

16 * q + 20 * q^4 + 4 * q^7 + 12 * q^10 + 8 * q^19 - 16 * q^22 + 36 * q^25 - 4 * q^28 + 16 * q^31 - 4 * q^34 + 20 * q^37 + 16 * q^40 - 20 * q^43 - 52 * q^46 + 8 * q^49 + 4 * q^52 - 40 * q^55 - 32 * q^58 + 40 * q^61 - 24 * q^64 - 52 * q^67 - 76 * q^70 + 4 * q^73 - 84 * q^76 - 68 * q^79 - 48 * q^82 + 32 * q^85 - 60 * q^88 + 4 * q^91 + 56 * q^94 + 56 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(351))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	13
351.2.a.a	$351$	$2$	351.a	1.a	$1$	$2$	$2$	$2.803$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	351.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(-3\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+(-1-\beta )q^{5}+\cdots\)
351.2.a.b	$351$	$2$	351.a	1.a	$1$	$2$	$2$	$2.803$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓	✓	351.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(-5\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1+\beta )q^{4}+(-3+\beta )q^{5}-q^{7}+\cdots\)
351.2.a.c	$351$	$2$	351.a	1.a	$1$	$2$	$2$	$2.803$	\(\Q(\sqrt{5}) \)	None	✓	✓			351.2.a.a	$1$	$0$	\(1\)	\(0\)	\(3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(1+\beta )q^{5}+(-1+\cdots)q^{7}+\cdots\)
351.2.a.d	$351$	$2$	351.a	1.a	$1$	$2$	$2$	$2.803$	\(\Q(\sqrt{13}) \)	None	✓	✓			351.2.a.b	$1$	$0$	\(1\)	\(0\)	\(5\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1+\beta )q^{4}+(3-\beta )q^{5}-q^{7}+\cdots\)
351.2.a.e	$351$	$2$	351.a	1.a	$1$	$4$	$4$	$2.803$	4.4.65712.1	None	✓	✓	✓		351.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}-\beta _{3}q^{5}+2q^{7}+\cdots\)
351.2.a.f	$351$	$2$	351.a	1.a	$1$	$4$	$4$	$2.803$	4.4.7600.1	None	✓	✓	✓		351.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+(\beta _{1}+\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(351))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(351)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 2}\)