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

• Label: 400.2.n
• Level: $400$
• Weight: $2$
• Character orbit: 400.n
• Rep. character: $\chi_{400}(143,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $18$
• Newform subspaces: $4$
• Sturm bound: $120$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.n (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(120\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	156	18	138
Cusp forms	84	18	66
Eisenstein series	72	0	72

Trace form

\( 18 q + 6 q^{13} + 6 q^{17} + 48 q^{21} - 24 q^{33} - 30 q^{37} - 48 q^{41} + 18 q^{53} + 48 q^{57} + 18 q^{73} - 24 q^{77} - 66 q^{81} + 24 q^{93} + 18 q^{97}+O(q^{100}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
400.2.n.a	$400$	$2$	400.n	20.e	$4$	$2$	$1$	$3.194$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)					80.2.n.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q-3 i q^{9}+(-5 i+5)q^{13}+(5 i+5)q^{17}+\cdots\)
400.2.n.b	$400$	$2$	400.n	20.e	$4$	$4$	$2$	$3.194$	\(\Q(\zeta_{12})\)	None					80.2.n.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta_{2} q^{3}+\beta_{3} q^{7}+3\beta_1 q^{9}+(\beta_{3}-\beta_{2})q^{11}+\cdots\)
400.2.n.c	$400$	$2$	400.n	20.e	$4$	$4$	$2$	$3.194$	\(\Q(i, \sqrt{5})\)	\(\Q(\sqrt{-5}) \)		✓			400.2.n.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{4}]$	\(q-\beta _{3}q^{3}+\beta _{2}q^{7}-7\beta _{1}q^{9}+10q^{21}+\cdots\)
400.2.n.d	$400$	$2$	400.n	20.e	$4$	$8$	$4$	$3.194$	\(\Q(\zeta_{24})\)	None		✓	✓		400.2.n.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta_1 q^{3}+4\beta_{5} q^{7}-2\beta_{3} q^{9}+3\beta_{4} q^{11}+\cdots\)

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

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