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

• Label: 850.2.c
• Level: $850$
• Weight: $2$
• Character orbit: 850.c
• Rep. character: $\chi_{850}(749,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $10$
• Sturm bound: $270$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	850.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(270\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\), \(13\)

Dimensions

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

	Total	New	Old
Modular forms	148	24	124
Cusp forms	124	24	100
Eisenstein series	24	0	24

Trace form

24 * q - 24 * q^4 + 4 * q^6 - 24 * q^9 + 4 * q^11 - 8 * q^14 + 24 * q^16 + 8 * q^19 + 8 * q^21 - 4 * q^24 + 8 * q^26 + 20 * q^29 - 40 * q^31 + 4 * q^34 + 24 * q^36 + 64 * q^39 - 24 * q^41 - 4 * q^44 + 24 * q^46 - 40 * q^49 - 4 * q^51 + 8 * q^54 + 8 * q^56 + 16 * q^59 + 20 * q^61 - 24 * q^64 + 8 * q^69 - 40 * q^71 - 20 * q^74 - 8 * q^76 + 40 * q^79 - 56 * q^81 - 8 * q^84 - 8 * q^86 + 40 * q^89 - 72 * q^94 + 4 * q^96 + 116 * q^99

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.c.a	$850$	$2$	850.c	5.b	$2$	$2$	$2$	$6.787$	\(\Q(\sqrt{-1}) \)	None					170.2.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+3 i q^{3}-q^{4}-3 q^{6}-2 i q^{7}+\cdots\)
850.2.c.b	$850$	$2$	850.c	5.b	$2$	$2$	$2$	$6.787$	\(\Q(\sqrt{-1}) \)	None					34.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+2 i q^{3}-q^{4}-2 q^{6}-4 i q^{7}+\cdots\)
850.2.c.c	$850$	$2$	850.c	5.b	$2$	$2$	$2$	$6.787$	\(\Q(\sqrt{-1}) \)	None					170.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+i q^{3}-q^{4}-q^{6}-2 i q^{7}+\cdots\)
850.2.c.d	$850$	$2$	850.c	5.b	$2$	$2$	$2$	$6.787$	\(\Q(\sqrt{-1}) \)	None		✓			850.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+i q^{3}-q^{4}-q^{6}+5 i q^{7}+\cdots\)
850.2.c.e	$850$	$2$	850.c	5.b	$2$	$2$	$2$	$6.787$	\(\Q(\sqrt{-1}) \)	None					170.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+i q^{3}-q^{4}+q^{6}-2 i q^{7}+\cdots\)
850.2.c.f	$850$	$2$	850.c	5.b	$2$	$2$	$2$	$6.787$	\(\Q(\sqrt{-1}) \)	None					170.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+2 i q^{3}-q^{4}+2 q^{6}-2 i q^{7}+\cdots\)
850.2.c.g	$850$	$2$	850.c	5.b	$2$	$2$	$2$	$6.787$	\(\Q(\sqrt{-1}) \)	None					170.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+2 i q^{3}-q^{4}+2 q^{6}+2 i q^{7}+\cdots\)
850.2.c.h	$850$	$2$	850.c	5.b	$2$	$2$	$2$	$6.787$	\(\Q(\sqrt{-1}) \)	None		✓			850.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+3 i q^{3}-q^{4}+3 q^{6}-i q^{7}+\cdots\)
850.2.c.i	$850$	$2$	850.c	5.b	$2$	$4$	$4$	$6.787$	\(\Q(i, \sqrt{17})\)	None					170.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+\beta _{1}q^{3}-q^{4}+(-1+\beta _{3})q^{6}+\cdots\)
850.2.c.j	$850$	$2$	850.c	5.b	$2$	$4$	$4$	$6.787$	\(\Q(\zeta_{8})\)	None		✓			850.2.a.m	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta_1 q^{2}+(\beta_{2}+\beta_1)q^{3}-q^{4}+(\beta_{3}+1)q^{6}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(850, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(170, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(425, [\chi])\)\(^{\oplus 2}\)