Space of modular forms of level 700, weight 2, and Character 700.c

• Label: 700.2.c
• Level: $700$
• Weight: $2$
• Character orbit: 700.c
• Rep. character: $\chi_{700}(699,\cdot)$
• Character field: $\Q$
• Dimension: $68$
• Newform subspaces: $11$
• Sturm bound: $240$
• Trace bound: $12$

Defining parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	700.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 140 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(240\)
Trace bound:			\(12\)
Distinguishing \(T_p\):			\(3\), \(11\), \(13\), \(19\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(700, [\chi])\).

	Total	New	Old
Modular forms	132	76	56
Cusp forms	108	68	40
Eisenstein series	24	8	16

Trace form

\( 68 q + 4 q^{4} - 52 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(700, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
700.2.c.a	$700$	$2$	700.c	140.c	$2$	$4$	$4$	$5.590$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(-4\)	\(0\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{1})q^{2}+\beta _{2}q^{3}-2\beta _{1}q^{4}+\cdots\)
700.2.c.b	$700$	$2$	700.c	140.c	$2$	$4$	$4$	$5.590$	\(\Q(\zeta_{12})\)	None		$2$	$1$	\(-2\)	\(0\)	\(0\)	\(-8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\zeta_{12}^{3})q^{2}+(\zeta_{12}^{2}-\zeta_{12}^{3})q^{3}+\cdots\)
700.2.c.c	$700$	$2$	700.c	140.c	$2$	$4$	$4$	$5.590$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(-2\)	\(0\)	\(0\)	\(-8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\zeta_{12}^{3})q^{2}+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots\)
700.2.c.d	$700$	$2$	700.c	140.c	$2$	$4$	$4$	$5.590$	\(\Q(i, \sqrt{7})\)	\(\Q(\sqrt{-7}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-2\beta _{1}+\beta _{3})q^{7}+\cdots\)
700.2.c.e	$700$	$2$	700.c	140.c	$2$	$4$	$4$	$5.590$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(2\)	\(0\)	\(0\)	\(8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\zeta_{12}^{3})q^{2}+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots\)
700.2.c.f	$700$	$2$	700.c	140.c	$2$	$4$	$4$	$5.590$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(2\)	\(0\)	\(0\)	\(8\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\zeta_{12}^{3})q^{2}+(\zeta_{12}^{2}-\zeta_{12}^{3})q^{3}+\cdots\)
700.2.c.g	$700$	$2$	700.c	140.c	$2$	$4$	$4$	$5.590$	\(\Q(i, \sqrt{6})\)	None		$4$	$0$	\(4\)	\(0\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1-\beta _{1})q^{2}+\beta _{2}q^{3}-2\beta _{1}q^{4}+(\beta _{2}+\cdots)q^{6}+\cdots\)
700.2.c.h	$700$	$2$	700.c	140.c	$2$	$8$	$8$	$5.590$	8.0.49787136.1	\(\Q(\sqrt{-7}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 5^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}+(\beta _{1}-\beta _{7})q^{7}+\cdots\)
700.2.c.i	$700$	$2$	700.c	140.c	$2$	$8$	$8$	$5.590$	8.0.342102016.5	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{2}+(\beta _{3}+\beta _{4})q^{3}+(\beta _{3}+\beta _{4}+\beta _{7})q^{4}+\cdots\)
700.2.c.j	$700$	$2$	700.c	140.c	$2$	$8$	$8$	$5.590$	8.0.342102016.5	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{2}+(-\beta _{3}-\beta _{4})q^{3}+(\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
700.2.c.k	$700$	$2$	700.c	140.c	$2$	$16$	$16$	$5.590$	16.0.\(\cdots\).7	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{7}q^{2}+(-\beta _{5}+\beta _{6}-\beta _{7}+\beta _{8}-\beta _{9}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(700, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(700, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 2}\)