Space of modular forms of level 700, weight 5, and Character 700.s

• Label: 700.5.s
• Level: $700$
• Weight: $5$
• Character orbit: 700.s
• Rep. character: $\chi_{700}(101,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $102$
• Newform subspaces: $5$
• Sturm bound: $600$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	700.s (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(600\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(700, [\chi])\).

	Total	New	Old
Modular forms	996	102	894
Cusp forms	924	102	822
Eisenstein series	72	0	72

Trace form

102 * q + 9 * q^3 - 44 * q^7 + 1422 * q^9 - 135 * q^11 + 243 * q^17 - 825 * q^19 + 145 * q^21 + 423 * q^23 + 924 * q^29 + 1113 * q^31 + 2247 * q^33 + 545 * q^37 - 1644 * q^39 - 7808 * q^43 - 3897 * q^47 - 2646 * q^49 - 1831 * q^51 + 2121 * q^53 + 6606 * q^57 - 6867 * q^59 - 25599 * q^61 - 6322 * q^63 - 5501 * q^67 + 28560 * q^71 + 8007 * q^73 + 2625 * q^77 + 10755 * q^79 - 44015 * q^81 - 9756 * q^87 + 8307 * q^89 - 19046 * q^91 + 5299 * q^93 - 66200 * q^99

Decomposition of \(S_{5}^{\mathrm{new}}(700, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
700.5.s.a	$700$	$5$	700.s	7.d	$6$	$6$	$3$	$72.359$	6.0.11337408.1	None					28.5.h.a	$2$	$0$	\(0\)	\(-9\)	\(0\)	\(-66\)	$2^{2}\cdot 3^{3}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\beta _{1}-\beta _{3})q^{3}+(-9-4\beta _{1}+\cdots)q^{7}+\cdots\)
700.5.s.b	$700$	$5$	700.s	7.d	$6$	$20$	$10$	$72.359$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None					140.5.r.a	$2$	$0$	\(0\)	\(18\)	\(0\)	\(22\)	$2^{4}\cdot 3^{2}\cdot 5^{4}\cdot 7^{5}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\beta _{3}-\beta _{4})q^{3}+(2-\beta _{1}+2\beta _{3}+\cdots)q^{7}+\cdots\)
700.5.s.c	$700$	$5$	700.s	7.d	$6$	$22$	$11$	$72.359$		None		✓			700.5.s.c	$2$	$0$	\(0\)	\(-9\)	\(0\)	\(-66\)		$\mathrm{SU}(2)[C_{6}]$
700.5.s.d	$700$	$5$	700.s	7.d	$6$	$22$	$11$	$72.359$		None		✓			700.5.s.c	$2$	$0$	\(0\)	\(9\)	\(0\)	\(66\)		$\mathrm{SU}(2)[C_{6}]$
700.5.s.e	$700$	$5$	700.s	7.d	$6$	$32$	$16$	$72.359$		None			✓		140.5.n.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{5}^{\mathrm{old}}(700, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(700, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(350, [\chi])\)\(^{\oplus 2}\)