Space of modular forms of level 350, weight 7, and Character 350.b

• Label: 350.7.b
• Level: $350$
• Weight: $7$
• Character orbit: 350.b
• Rep. character: $\chi_{350}(251,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $5$
• Sturm bound: $420$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	350.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(420\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(350, [\chi])\).

	Total	New	Old
Modular forms	372	76	296
Cusp forms	348	76	272
Eisenstein series	24	0	24

Trace form

\( 76 q + 2432 q^{4} - 388 q^{7} - 20532 q^{9} + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(350, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
350.7.b.a	$350$	$7$	350.b	7.b	$2$	$4$	$4$	$80.519$	4.0.211968.1	None					14.7.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-308\)	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{2}-\beta _{1}q^{3}+2^{5}q^{4}+(5\beta _{1}+\beta _{2}+\cdots)q^{6}+\cdots\)
350.7.b.b	$350$	$7$	350.b	7.b	$2$	$16$	$16$	$80.519$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓			350.7.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-680\)	$2^{25}\cdot 5^{4}\cdot 7^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+\beta _{1}q^{3}+2^{5}q^{4}+(-\beta _{1}-\beta _{5}+\cdots)q^{6}+\cdots\)
350.7.b.c	$350$	$7$	350.b	7.b	$2$	$16$	$16$	$80.519$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					70.7.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-80\)	$2^{18}\cdot 3^{2}\cdot 5^{8}\cdot 7^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}-\beta _{2}q^{3}+2^{5}q^{4}+(-\beta _{2}+\beta _{3}+\cdots)q^{6}+\cdots\)
350.7.b.d	$350$	$7$	350.b	7.b	$2$	$16$	$16$	$80.519$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓			350.7.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(680\)	$2^{25}\cdot 5^{4}\cdot 7^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{2}+\beta _{1}q^{3}+2^{5}q^{4}+(\beta _{1}+\beta _{5}+\cdots)q^{6}+\cdots\)
350.7.b.e	$350$	$7$	350.b	7.b	$2$	$24$	$24$	$80.519$		None			✓		70.7.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{7}^{\mathrm{old}}(350, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(350, [\chi]) \simeq \) \(S_{7}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)