Space of modular forms of level 400, weight 6, and Character 400.c

• Label: 400.6.c
• Level: $400$
• Weight: $6$
• Character orbit: 400.c
• Rep. character: $\chi_{400}(49,\cdot)$
• Character field: $\Q$
• Dimension: $44$
• Newform subspaces: $16$
• Sturm bound: $360$
• Trace bound: $9$

Defining parameters

Level:	$N$	$=$	$400 = 2^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	400.c (of order $2$ and degree $1$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$5$
Character field:			$\Q$
Newform subspaces:			$16$
Sturm bound:			$360$
Trace bound:			$9$
Distinguishing $T_p$:			$3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{6}(400, [\chi])$.

	Total	New	Old
Modular forms	318	46	272
Cusp forms	282	44	238
Eisenstein series	36	2	34

Trace form

44 * q - 3400 * q^9 - 724 * q^11 - 196 * q^19 - 2128 * q^21 - 12240 * q^29 + 12928 * q^31 - 10376 * q^39 - 8404 * q^41 - 85604 * q^49 - 115444 * q^51 + 77136 * q^59 + 2288 * q^61 + 43648 * q^69 + 25016 * q^71 + 58888 * q^79 + 156724 * q^81 - 132588 * q^89 + 160104 * q^91 + 244272 * q^99

Decomposition of $S_{6}^{\mathrm{new}}(400, [\chi])$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
400.6.c.a	$400$	$6$	400.c	5.b	$2$	$2$	$2$	$64.154$	$\Q(\sqrt{-1})$	None					10.6.a.a	$2$	$0$	$0$	$0$	$0$	$0$	$2$	$\mathrm{SU}(2)[C_{2}]$	$q+13\beta q^{3}-11\beta q^{7}-433 q^{9}+768 q^{11}+\cdots$
400.6.c.b	$400$	$6$	400.c	5.b	$2$	$2$	$2$	$64.154$	$\Q(\sqrt{-1})$	None					10.6.a.b	$2$	$0$	$0$	$0$	$0$	$0$	$2$	$\mathrm{SU}(2)[C_{2}]$	$q+12\beta q^{3}+86\beta q^{7}-333 q^{9}-132 q^{11}+\cdots$
400.6.c.c	$400$	$6$	400.c	5.b	$2$	$2$	$2$	$64.154$	$\Q(\sqrt{-1})$	None					20.6.a.a	$2$	$0$	$0$	$0$	$0$	$0$	$2$	$\mathrm{SU}(2)[C_{2}]$	$q+11\beta q^{3}-109\beta q^{7}-241 q^{9}+\cdots$
400.6.c.d	$400$	$6$	400.c	5.b	$2$	$2$	$2$	$64.154$	$\Q(\sqrt{-1})$	None					8.6.a.a	$2$	$0$	$0$	$0$	$0$	$0$	$2$	$\mathrm{SU}(2)[C_{2}]$	$q+10\beta q^{3}+12\beta q^{7}-157 q^{9}-124 q^{11}+\cdots$
400.6.c.e	$400$	$6$	400.c	5.b	$2$	$2$	$2$	$64.154$	$\Q(\sqrt{-1})$	None					40.6.a.a	$2$	$0$	$0$	$0$	$0$	$0$	$2$	$\mathrm{SU}(2)[C_{2}]$	$q+9\beta q^{3}+121\beta q^{7}-81 q^{9}-656 q^{11}+\cdots$
400.6.c.f	$400$	$6$	400.c	5.b	$2$	$2$	$2$	$64.154$	$\Q(\sqrt{-1})$	None					4.6.a.a	$2$	$0$	$0$	$0$	$0$	$0$	$2$	$\mathrm{SU}(2)[C_{2}]$	$q+6\beta q^{3}-44\beta q^{7}+99 q^{9}-540 q^{11}+\cdots$
400.6.c.g	$400$	$6$	400.c	5.b	$2$	$2$	$2$	$64.154$	$\Q(\sqrt{-1})$	None					50.6.a.a	$2$	$0$	$0$	$0$	$0$	$0$	$1$	$\mathrm{SU}(2)[C_{2}]$	$q+11 i q^{3}-142 i q^{7}+122 q^{9}+\cdots$
400.6.c.h	$400$	$6$	400.c	5.b	$2$	$2$	$2$	$64.154$	$\Q(\sqrt{-1})$	None					40.6.a.b	$2$	$0$	$0$	$0$	$0$	$0$	$2$	$\mathrm{SU}(2)[C_{2}]$	$q+4\beta q^{3}-54\beta q^{7}+179 q^{9}+604 q^{11}+\cdots$
400.6.c.i	$400$	$6$	400.c	5.b	$2$	$2$	$2$	$64.154$	$\Q(\sqrt{-1})$	None					10.6.a.c	$2$	$0$	$0$	$0$	$0$	$0$	$2$	$\mathrm{SU}(2)[C_{2}]$	$q+3\beta q^{3}+59\beta q^{7}+207 q^{9}-192 q^{11}+\cdots$
400.6.c.j	$400$	$6$	400.c	5.b	$2$	$2$	$2$	$64.154$	$\Q(\sqrt{-1})$	None					5.6.a.a	$2$	$0$	$0$	$0$	$0$	$0$	$2$	$\mathrm{SU}(2)[C_{2}]$	$q+2\beta q^{3}+96\beta q^{7}+227 q^{9}+148 q^{11}+\cdots$
400.6.c.k	$400$	$6$	400.c	5.b	$2$	$2$	$2$	$64.154$	$\Q(\sqrt{-1})$	None					40.6.a.c	$2$	$0$	$0$	$0$	$0$	$0$	$2$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta q^{3}-31\beta q^{7}+239 q^{9}+144 q^{11}+\cdots$
400.6.c.l	$400$	$6$	400.c	5.b	$2$	$4$	$4$	$64.154$	$\Q(i, \sqrt{129})$	None					40.6.a.d	$2$	$0$	$0$	$0$	$0$	$0$	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	$q+(-3\beta _{1}+\beta _{2})q^{3}+(-13\beta _{1}-3\beta _{2}+\cdots)q^{7}+\cdots$
400.6.c.m	$400$	$6$	400.c	5.b	$2$	$4$	$4$	$64.154$	$\Q(i, \sqrt{409})$	None					100.6.a.c	$2$	$0$	$0$	$0$	$0$	$0$	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+(-\beta _{1}+2\beta _{2})q^{3}+(-6\beta _{1}+4\beta _{2}+\cdots)q^{7}+\cdots$
400.6.c.n	$400$	$6$	400.c	5.b	$2$	$4$	$4$	$64.154$	$\Q(i, \sqrt{241})$	None					25.6.a.b	$2$	$0$	$0$	$0$	$0$	$0$	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+(\beta _{1}-2\beta _{2})q^{3}+(-2\beta _{1}+20\beta _{2})q^{7}+\cdots$
400.6.c.o	$400$	$6$	400.c	5.b	$2$	$4$	$4$	$64.154$	$\Q(i, \sqrt{241})$	None					200.6.a.e	$2$	$0$	$0$	$0$	$0$	$0$	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+(-4\beta _{1}-\beta _{2})q^{3}+(-4\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots$
400.6.c.p	$400$	$6$	400.c	5.b	$2$	$6$	$6$	$64.154$	$\mathbb{Q}[x]/(x^{6} + \cdots)$	None			✓		200.6.a.h	$2$	$0$	$0$	$0$	$0$	$0$	$2^{8}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{3}q^{3}+(-24\beta _{1}-\beta _{3}-\beta _{4})q^{7}+\cdots$

Decomposition of $S_{6}^{\mathrm{old}}(400, [\chi])$ into lower level spaces

$S_{6}^{\mathrm{old}}(400, [\chi]) \simeq$ $S_{6}^{\mathrm{new}}(5, [\chi])$$^{\oplus 10}$$\oplus$$S_{6}^{\mathrm{new}}(10, [\chi])$$^{\oplus 8}$$\oplus$$S_{6}^{\mathrm{new}}(20, [\chi])$$^{\oplus 6}$$\oplus$$S_{6}^{\mathrm{new}}(25, [\chi])$$^{\oplus 5}$$\oplus$$S_{6}^{\mathrm{new}}(40, [\chi])$$^{\oplus 4}$$\oplus$$S_{6}^{\mathrm{new}}(50, [\chi])$$^{\oplus 4}$$\oplus$$S_{6}^{\mathrm{new}}(80, [\chi])$$^{\oplus 2}$$\oplus$$S_{6}^{\mathrm{new}}(100, [\chi])$$^{\oplus 3}$$\oplus$$S_{6}^{\mathrm{new}}(200, [\chi])$$^{\oplus 2}$