Space of modular forms of level 400, weight 2, and Character 400.bi

• Label: 400.2.bi
• Level: $400$
• Weight: $2$
• Character orbit: 400.bi
• Rep. character: $\chi_{400}(47,\cdot)$
• Character field: $\Q(\zeta_{20})$
• Dimension: $120$
• Newform subspaces: $4$
• Sturm bound: $120$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.bi (of order \(20\) and degree \(8\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 100 \)
Character field:			\(\Q(\zeta_{20})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(120\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(400, [\chi])\).

	Total	New	Old
Modular forms	528	120	408
Cusp forms	432	120	312
Eisenstein series	96	0	96

Trace form

\( 120 q + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(400, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
400.2.bi.a	$400$	$2$	400.bi	100.l	$20$	$8$	$1$	$3.194$	\(\Q(\zeta_{20})\)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{U}(1)[D_{20}]$	\(q+(-\zeta_{20}+\zeta_{20}^{3}+2\zeta_{20}^{4}-\zeta_{20}^{5}+\cdots)q^{5}+\cdots\)
400.2.bi.b	$400$	$2$	400.bi	100.l	$20$	$16$	$2$	$3.194$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{20}]$	\(q+(\beta _{11}+\beta _{15})q^{3}+(1-\beta _{1}+\beta _{2}-2\beta _{3}+\cdots)q^{5}+\cdots\)
400.2.bi.c	$400$	$2$	400.bi	100.l	$20$	$16$	$2$	$3.194$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{20}]$	\(q+(\beta _{13}-\beta _{15})q^{3}+(-1+2\beta _{3}-2\beta _{5}+\cdots)q^{5}+\cdots\)
400.2.bi.d	$400$	$2$	400.bi	100.l	$20$	$80$	$10$	$3.194$		None		$4$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$\mathrm{SU}(2)[C_{20}]$

Decomposition of \(S_{2}^{\mathrm{old}}(400, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(400, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)