Space of modular forms of level 350, weight 3, and Character 350.p

• Label: 350.3.p
• Level: $350$
• Weight: $3$
• Character orbit: 350.p
• Rep. character: $\chi_{350}(93,\cdot)$
• Character field: $\Q(\zeta_{12})$
• Dimension: $96$
• Newform subspaces: $7$
• Sturm bound: $180$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	350.p (of order \(12\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 35 \)
Character field:			\(\Q(\zeta_{12})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(180\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(350, [\chi])\).

	Total	New	Old
Modular forms	528	96	432
Cusp forms	432	96	336
Eisenstein series	96	0	96

Trace form

\( 96 q + 32 q^{6} + 4 q^{7} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(350, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
350.3.p.a	$350$	$3$	350.p	35.l	$12$	$8$	$2$	$9.537$	8.0.303595776.1	None		$4$	$0$	\(-4\)	\(-2\)	\(0\)	\(-18\)	$2^{2}$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{2}+\beta _{4}-\beta _{5})q^{2}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
350.3.p.b	$350$	$3$	350.p	35.l	$12$	$8$	$2$	$9.537$	8.0.303595776.1	None		$4$	$0$	\(-4\)	\(-2\)	\(0\)	\(12\)	$2^{2}$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{4}-\beta _{5})q^{2}+(-1+\beta _{1}+\beta _{2}-\beta _{4}+\cdots)q^{3}+\cdots\)
350.3.p.c	$350$	$3$	350.p	35.l	$12$	$8$	$2$	$9.537$	8.0.3317760000.2	None		$4$	$0$	\(-4\)	\(4\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-1-\beta _{2}+\beta _{4})q^{2}+(\beta _{2}+2\beta _{3}+\beta _{4}+\cdots)q^{3}+\cdots\)
350.3.p.d	$350$	$3$	350.p	35.l	$12$	$8$	$2$	$9.537$	8.0.303595776.1	None		$4$	$0$	\(4\)	\(2\)	\(0\)	\(18\)	$2^{2}$	$\mathrm{SU}(2)[C_{12}]$	\(q+(\beta _{2}-\beta _{4}-\beta _{5})q^{2}+(1+\beta _{2}+\beta _{3}+\cdots)q^{3}+\cdots\)
350.3.p.e	$350$	$3$	350.p	35.l	$12$	$16$	$4$	$9.537$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(8\)	\(-2\)	\(0\)	\(-12\)	$2^{2}$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1+\beta _{4}+\beta _{6}-\beta _{8})q^{2}+(\beta _{1}-\beta _{7}+\cdots)q^{3}+\cdots\)
350.3.p.f	$350$	$3$	350.p	35.l	$12$	$24$	$6$	$9.537$		None		$4$	$0$	\(-12\)	\(-4\)	\(0\)	\(-24\)		$\mathrm{SU}(2)[C_{12}]$
350.3.p.g	$350$	$3$	350.p	35.l	$12$	$24$	$6$	$9.537$		None		$4$	$0$	\(12\)	\(4\)	\(0\)	\(24\)		$\mathrm{SU}(2)[C_{12}]$

Decomposition of \(S_{3}^{\mathrm{old}}(350, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(350, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)