Space of modular forms of level 850, weight 2, and Character 850.l

• Label: 850.2.l
• Level: $850$
• Weight: $2$
• Character orbit: 850.l
• Rep. character: $\chi_{850}(151,\cdot)$
• Character field: $\Q(\zeta_{8})$
• Dimension: $116$
• Newform subspaces: $9$
• Sturm bound: $270$
• Trace bound: $18$

Defining parameters

Level:	\( N \)	\(=\)	\( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	850.l (of order \(8\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 17 \)
Character field:			\(\Q(\zeta_{8})\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(270\)
Trace bound:			\(18\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(850, [\chi])\).

	Total	New	Old
Modular forms	592	116	476
Cusp forms	496	116	380
Eisenstein series	96	0	96

Trace form

\( 116 q - 4 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(850, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
850.2.l.a	$850$	$2$	850.l	17.d	$8$	$4$	$1$	$6.787$	\(\Q(\zeta_{8})\)	None					34.2.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{8}]$	\(q-\zeta_{8}^{3}q^{2}+(-\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}-\zeta_{8}^{2}q^{4}+\cdots\)
850.2.l.b	$850$	$2$	850.l	17.d	$8$	$8$	$2$	$6.787$	8.0.18939904.2	None		✓			850.2.l.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{8}]$	\(q-\beta _{5}q^{2}+\beta _{4}q^{3}-\beta _{7}q^{4}-\beta _{3}q^{6}+\cdots\)
850.2.l.c	$850$	$2$	850.l	17.d	$8$	$8$	$2$	$6.787$	8.0.18939904.2	None		✓			850.2.l.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{8}]$	\(q+\beta _{5}q^{2}+(\beta _{4}-\beta _{5}+\beta _{7})q^{3}-\beta _{7}q^{4}+\cdots\)
850.2.l.d	$850$	$2$	850.l	17.d	$8$	$8$	$2$	$6.787$	\(\Q(\zeta_{16})\)	None					170.2.k.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{8}]$	\(q-\zeta_{16}^{6}q^{2}+(\zeta_{16}+\zeta_{16}^{3}+\zeta_{16}^{4}+\cdots)q^{3}+\cdots\)
850.2.l.e	$850$	$2$	850.l	17.d	$8$	$16$	$4$	$6.787$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					170.2.k.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{8}]$	\(q-\beta _{5}q^{2}-\beta _{13}q^{3}+\beta _{1}q^{4}+\beta _{12}q^{6}+\cdots\)
850.2.l.f	$850$	$2$	850.l	17.d	$8$	$16$	$4$	$6.787$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			850.2.l.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{8}]$	\(q+\beta _{5}q^{2}-\beta _{4}q^{3}-\beta _{7}q^{4}+\beta _{1}q^{6}+\cdots\)
850.2.l.g	$850$	$2$	850.l	17.d	$8$	$16$	$4$	$6.787$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			850.2.l.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{8}]$	\(q-\beta _{5}q^{2}+\beta _{4}q^{3}-\beta _{7}q^{4}+\beta _{1}q^{6}+\cdots\)
850.2.l.h	$850$	$2$	850.l	17.d	$8$	$20$	$5$	$6.787$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None					170.2.n.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{8}]$	\(q+\beta _{8}q^{2}-\beta _{1}q^{3}-\beta _{14}q^{4}+\beta _{2}q^{6}+\cdots\)
850.2.l.i	$850$	$2$	850.l	17.d	$8$	$20$	$5$	$6.787$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None					170.2.n.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{8}]$	\(q+\beta _{11}q^{2}+\beta _{2}q^{3}+\beta _{14}q^{4}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(850, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(850, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(17, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(34, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(170, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(425, [\chi])\)\(^{\oplus 2}\)