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

• Label: 400.2.u
• Level: $400$
• Weight: $2$
• Character orbit: 400.u
• Rep. character: $\chi_{400}(81,\cdot)$
• Character field: $\Q(\zeta_{5})$
• Dimension: $56$
• Newform subspaces: $7$
• Sturm bound: $120$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.u (of order \(5\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 25 \)
Character field:			\(\Q(\zeta_{5})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(120\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	264	64	200
Cusp forms	216	56	160
Eisenstein series	48	8	40

Trace form

\( 56 q + 3 q^{3} - 5 q^{5} + 8 q^{7} - 15 q^{9} + O(q^{10}) \)

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.u.a	$400$	$2$	400.u	25.d	$5$	$4$	$1$	$3.194$	\(\Q(\zeta_{10})\)	None		$2$	$1$	\(0\)	\(-5\)	\(-5\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-2+2\zeta_{10}+\zeta_{10}^{3})q^{3}+(-2+\zeta_{10}+\cdots)q^{5}+\cdots\)
400.2.u.b	$400$	$2$	400.u	25.d	$5$	$4$	$1$	$3.194$	\(\Q(\zeta_{10})\)	None		$2$	$0$	\(0\)	\(1\)	\(-5\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+\zeta_{10}^{3}q^{3}+(-2+\zeta_{10}-2\zeta_{10}^{2}+\cdots)q^{5}+\cdots\)
400.2.u.c	$400$	$2$	400.u	25.d	$5$	$4$	$1$	$3.194$	\(\Q(\zeta_{10})\)	None		$2$	$0$	\(0\)	\(1\)	\(5\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(1-\zeta_{10}-2\zeta_{10}^{3})q^{3}+(2-2\zeta_{10}+\cdots)q^{5}+\cdots\)
400.2.u.d	$400$	$2$	400.u	25.d	$5$	$8$	$2$	$3.194$	8.0.58140625.2	None		$2$	$0$	\(0\)	\(3\)	\(0\)	\(-4\)	$5$	$\mathrm{SU}(2)[C_{5}]$	\(q+(\beta _{2}+\beta _{3}+\beta _{5}+\beta _{6})q^{3}+(\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
400.2.u.e	$400$	$2$	400.u	25.d	$5$	$8$	$2$	$3.194$	8.0.58140625.2	None		$2$	$0$	\(0\)	\(6\)	\(5\)	\(-4\)	$5$	$\mathrm{SU}(2)[C_{5}]$	\(q+(1-\beta _{3})q^{3}+(-\beta _{4}+\beta _{6})q^{5}+(-1+\cdots)q^{7}+\cdots\)
400.2.u.f	$400$	$2$	400.u	25.d	$5$	$12$	$3$	$3.194$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(-2\)	\(-4\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q-\beta _{3}q^{3}+(-\beta _{1}+\beta _{2}+\beta _{3}-2\beta _{5}+\cdots)q^{5}+\cdots\)
400.2.u.g	$400$	$2$	400.u	25.d	$5$	$16$	$4$	$3.194$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(0\)	\(-1\)	\(-1\)	\(6\)	$5$	$\mathrm{SU}(2)[C_{5}]$	\(q-\beta _{1}q^{3}+(-1+\beta _{2}-\beta _{4}+\beta _{5}+\beta _{14}+\cdots)q^{5}+\cdots\)

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

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