Space of modular forms of level 700, weight 3, and Character 700.d

• Label: 700.3.d
• Level: $700$
• Weight: $3$
• Character orbit: 700.d
• Rep. character: $\chi_{700}(601,\cdot)$
• Character field: $\Q$
• Dimension: $26$
• Newform subspaces: $6$
• Sturm bound: $360$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	700.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(360\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(700, [\chi])\).

	Total	New	Old
Modular forms	258	26	232
Cusp forms	222	26	196
Eisenstein series	36	0	36

Trace form

\( 26 q - 4 q^{7} - 66 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(700, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
700.3.d.a	$700$	$3$	700.d	7.b	$2$	$2$	$2$	$19.074$	\(\Q(\sqrt{-6}) \)	None					28.3.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-10\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+(-5-\beta )q^{7}-15q^{9}-6q^{11}+\cdots\)
700.3.d.b	$700$	$3$	700.d	7.b	$2$	$4$	$4$	$19.074$	\(\Q(i, \sqrt{105})\)	\(\Q(\sqrt{-35}) \)					140.3.h.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{3}-7\beta _{2}q^{7}+(-18+\beta _{3})q^{9}+\cdots\)
700.3.d.c	$700$	$3$	700.d	7.b	$2$	$4$	$4$	$19.074$	\(\Q(\sqrt{-2}, \sqrt{41})\)	None					140.3.h.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(2\beta _{1}+\beta _{2})q^{7}+7q^{9}-2q^{11}+\cdots\)
700.3.d.d	$700$	$3$	700.d	7.b	$2$	$4$	$4$	$19.074$	\(\Q(\sqrt{-5}, \sqrt{-13})\)	None					140.3.d.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(6\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(2+2\beta _{1}-\beta _{2}+\beta _{3})q^{7}+\cdots\)
700.3.d.e	$700$	$3$	700.d	7.b	$2$	$6$	$6$	$19.074$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		✓			700.3.d.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-\beta _{3}q^{7}+(-1+\beta _{2})q^{9}+(1+\cdots)q^{11}+\cdots\)
700.3.d.f	$700$	$3$	700.d	7.b	$2$	$6$	$6$	$19.074$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None		✓			700.3.d.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{4}q^{7}+(-1+\beta _{2})q^{9}+(1+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(700, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(700, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(350, [\chi])\)\(^{\oplus 2}\)