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

• Label: 400.6.n
• Level: $400$
• Weight: $6$
• Character orbit: 400.n
• Rep. character: $\chi_{400}(143,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $90$
• Newform subspaces: $8$
• Sturm bound: $360$
• Trace bound: $21$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.n (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(360\)
Trace bound:			\(21\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	636	90	546
Cusp forms	564	90	474
Eisenstein series	72	0	72

Trace form

\( 90 q + O(q^{10}) \)

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	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
400.6.n.a	$400$	$6$	400.n	20.e	$4$	$2$	$1$	$64.154$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q-3^{5}iq^{9}+(-475+475i)q^{13}+(1525+\cdots)q^{17}+\cdots\)
400.6.n.b	$400$	$6$	400.n	20.e	$4$	$4$	$2$	$64.154$	\(\Q(i, \sqrt{155})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{3}-11\beta _{3}q^{7}+67\beta _{1}q^{9}+(-5\beta _{2}+\cdots)q^{11}+\cdots\)
400.6.n.c	$400$	$6$	400.n	20.e	$4$	$4$	$2$	$64.154$	\(\Q(i, \sqrt{5})\)	\(\Q(\sqrt{-5}) \)		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 5^{2}$	$\mathrm{U}(1)[D_{4}]$	\(q-\beta _{3}q^{3}+\beta _{2}q^{7}-7\beta _{1}q^{9}+250q^{21}+\cdots\)
400.6.n.d	$400$	$6$	400.n	20.e	$4$	$4$	$2$	$64.154$	\(\Q(i, \sqrt{195})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{3}+9\beta _{3}q^{7}+147\beta _{1}q^{9}+(15\beta _{2}+\cdots)q^{11}+\cdots\)
400.6.n.e	$400$	$6$	400.n	20.e	$4$	$16$	$8$	$64.154$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{68}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{4}q^{3}+(-\beta _{3}+\beta _{8})q^{7}+(83\beta _{2}+\beta _{12}+\cdots)q^{9}+\cdots\)
400.6.n.f	$400$	$6$	400.n	20.e	$4$	$16$	$8$	$64.154$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{44}\cdot 5^{16}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{4}q^{3}+\beta _{7}q^{7}+(-182\beta _{1}+\beta _{14}+\cdots)q^{9}+\cdots\)
400.6.n.g	$400$	$6$	400.n	20.e	$4$	$20$	$10$	$64.154$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{67}\cdot 3^{4}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{6}q^{3}+(-\beta _{5}+\beta _{15})q^{7}+(103\beta _{4}+\cdots)q^{9}+\cdots\)
400.6.n.h	$400$	$6$	400.n	20.e	$4$	$24$	$12$	$64.154$		None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

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

\( S_{6}^{\mathrm{old}}(400, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)