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

• Label: 351.2.b
• Level: $351$
• Weight: $2$
• Character orbit: 351.b
• Rep. character: $\chi_{351}(298,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $5$
• Sturm bound: $84$
• Trace bound: $14$

Defining parameters

Level:	\( N \)	\(=\)	\( 351 = 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	351.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(84\)
Trace bound:			\(14\)
Distinguishing \(T_p\):			\(2\), \(5\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	48	18	30
Cusp forms	36	18	18
Eisenstein series	12	0	12

Trace form

18 * q - 16 * q^4 + 20 * q^10 + 7 * q^13 + 8 * q^16 + 8 * q^22 - 34 * q^25 - 80 * q^40 + 36 * q^43 - 24 * q^49 + 10 * q^52 - 16 * q^55 + 50 * q^61 + 16 * q^64 + 2 * q^79 - 64 * q^82 - 68 * q^88 - 39 * q^91 + 104 * q^94

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
351.2.b.a	$351$	$2$	351.b	13.b	$2$	$2$	$2$	$2.803$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			351.2.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3$	$\mathrm{U}(1)[D_{2}]$	\(q+2 q^{4}+(2\beta-1)q^{7}+(\beta-3)q^{13}+\cdots\)
351.2.b.b	$351$	$2$	351.b	13.b	$2$	$4$	$4$	$2.803$	4.0.8112.1	\(\Q(\sqrt{-39}) \)		✓			351.2.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{2}+(-4+\beta _{3})q^{4}+\beta _{2}q^{5}+(-2\beta _{1}+\cdots)q^{8}+\cdots\)
351.2.b.c	$351$	$2$	351.b	13.b	$2$	$4$	$4$	$2.803$	4.0.8112.1	\(\Q(\sqrt{-39}) \)		✓			351.2.b.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}-\beta _{3}q^{5}+\beta _{3}q^{8}+\cdots\)
351.2.b.d	$351$	$2$	351.b	13.b	$2$	$4$	$4$	$2.803$	4.0.8112.1	None		✓			351.2.b.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}+(-\beta _{1}+\beta _{3})q^{5}+\cdots\)
351.2.b.e	$351$	$2$	351.b	13.b	$2$	$4$	$4$	$2.803$	4.0.8112.1	None		✓			351.2.b.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}+(-\beta _{1}+\beta _{3})q^{5}+\cdots\)

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

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