Space of modular forms of level 351, weight 2, and Character 351.bd

• Label: 351.2.bd
• Level: $351$
• Weight: $2$
• Character orbit: 351.bd
• Rep. character: $\chi_{351}(80,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $76$
• Newform subspaces: $5$
• Sturm bound: $84$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 351 = 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	351.bd (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 39 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(84\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(351, [\chi])\).

	Total	New	Old
Modular forms	192	76	116
Cusp forms	144	76	68
Eisenstein series	48	0	48

Trace form

\( 76 q + 4 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(351, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
351.2.bd.a	$351$	$2$	351.bd	39.k	$12$	$4$	$1$	$2.803$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{U}(1)[D_{12}]$	\(q+2\zeta_{12}q^{4}+(1+\zeta_{12}+2\zeta_{12}^{2}-3\zeta_{12}^{3})q^{7}+\cdots\)
351.2.bd.b	$351$	$2$	351.bd	39.k	$12$	$16$	$4$	$2.803$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{1}-\beta _{14})q^{2}+(-2\beta _{7}-\beta _{9}+\beta _{13}+\cdots)q^{4}+\cdots\)
351.2.bd.c	$351$	$2$	351.bd	39.k	$12$	$16$	$4$	$2.803$	16.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$2^{4}$	$\mathrm{SU}(2)[C_{12}]$	\(q-\beta _{15}q^{2}+(1-\beta _{5}-\beta _{7})q^{4}+(\beta _{1}+\beta _{13}+\cdots)q^{5}+\cdots\)
351.2.bd.d	$351$	$2$	351.bd	39.k	$12$	$20$	$5$	$2.803$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q-\beta _{4}q^{2}+(-\beta _{3}-\beta _{6}-\beta _{13}+\beta _{15}+\cdots)q^{4}+\cdots\)
351.2.bd.e	$351$	$2$	351.bd	39.k	$12$	$20$	$5$	$2.803$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+\beta _{4}q^{2}+(-\beta _{3}-\beta _{6}-\beta _{13}+\beta _{15}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(351, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(351, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(117, [\chi])\)\(^{\oplus 2}\)