Space of modular forms of level 400, weight 3, and Character 400.bg

• Label: 400.3.bg
• Level: $400$
• Weight: $3$
• Character orbit: 400.bg
• Rep. character: $\chi_{400}(17,\cdot)$
• Character field: $\Q(\zeta_{20})$
• Dimension: $232$
• Newform subspaces: $6$
• Sturm bound: $180$
• Trace bound: $1$

Defining parameters

Level:	$N$	$=$	$400 = 2^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	400.bg (of order $20$ and degree $8$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$25$
Character field:			$\Q(\zeta_{20})$
Newform subspaces:			$6$
Sturm bound:			$180$
Trace bound:			$1$
Distinguishing $T_p$:			$3$

Dimensions

The following table gives the dimensions of various subspaces of $M_{3}(400, [\chi])$.

	Total	New	Old
Modular forms	1008	248	760
Cusp forms	912	232	680
Eisenstein series	96	16	80

Trace form

232 * q + 8 * q^3 - 8 * q^5 + 8 * q^7 - 10 * q^9 + 6 * q^11 + 4 * q^13 - 40 * q^15 - 12 * q^17 + 10 * q^19 - 6 * q^21 + 56 * q^23 + 4 * q^25 + 122 * q^27 - 10 * q^29 + 6 * q^31 + 58 * q^33 + 56 * q^35 + 12 * q^37 + 10 * q^39 - 6 * q^41 - 56 * q^43 - 118 * q^45 - 232 * q^47 + 16 * q^51 - 36 * q^53 - 138 * q^55 + 6 * q^57 + 10 * q^59 - 6 * q^61 + 232 * q^63 - 132 * q^65 + 344 * q^67 - 10 * q^69 + 6 * q^71 - 180 * q^73 + 312 * q^75 + 90 * q^77 + 10 * q^79 + 372 * q^81 + 696 * q^83 - 568 * q^85 + 954 * q^87 + 490 * q^89 + 6 * q^91 - 374 * q^93 + 250 * q^95 + 92 * q^97

Decomposition of $S_{3}^{\mathrm{new}}(400, [\chi])$ 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
400.3.bg.a	$400$	$3$	400.bg	25.f	$20$	$16$	$2$	$10.899$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	None					50.3.f.a	$2$	$0$	$0$	$-2$	$0$	$2$	$5^{2}$	$\mathrm{SU}(2)[C_{20}]$	$q+(-1-\beta _{1}+\beta _{2}-\beta _{3}-\beta _{8}-\beta _{9}+\cdots)q^{3}+\cdots$
400.3.bg.b	$400$	$3$	400.bg	25.f	$20$	$24$	$3$	$10.899$		None					50.3.f.b	$2$	$0$	$0$	$-2$	$0$	$2$		$\mathrm{SU}(2)[C_{20}]$
400.3.bg.c	$400$	$3$	400.bg	25.f	$20$	$32$	$4$	$10.899$		None					25.3.f.a	$2$	$0$	$0$	$10$	$-10$	$10$		$\mathrm{SU}(2)[C_{20}]$
400.3.bg.d	$400$	$3$	400.bg	25.f	$20$	$40$	$5$	$10.899$		None					100.3.k.a	$2$	$0$	$0$	$2$	$6$	$-14$		$\mathrm{SU}(2)[C_{20}]$
400.3.bg.e	$400$	$3$	400.bg	25.f	$20$	$56$	$7$	$10.899$		None					200.3.u.a	$2$	$0$	$0$	$0$	$-10$	$4$		$\mathrm{SU}(2)[C_{20}]$
400.3.bg.f	$400$	$3$	400.bg	25.f	$20$	$64$	$8$	$10.899$		None			✓		200.3.u.b	$2$	$0$	$0$	$0$	$6$	$4$		$\mathrm{SU}(2)[C_{20}]$

Decomposition of $S_{3}^{\mathrm{old}}(400, [\chi])$ into lower level spaces

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