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	Dim
\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(-\)	$-$	\(6\)
\(-\)	\(+\)	$-$	\(6\)
\(-\)	\(-\)	$+$	\(2\)
Plus space		\(+\)	\(4\)
Minus space		\(-\)	\(12\)

Trace form

\( 16 q + 20 q^{4} + 4 q^{7} + O(q^{10}) \)

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

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

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