Space of modular forms of level 400, weight 9, and Character 400.h

• Label: 400.9.h
• Level: $400$
• Weight: $9$
• Character orbit: 400.h
• Rep. character: $\chi_{400}(399,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $6$
• Sturm bound: $540$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	400.h (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 20 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(540\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(400, [\chi])\).

	Total	New	Old
Modular forms	498	72	426
Cusp forms	462	72	390
Eisenstein series	36	0	36

Trace form

\( 72 q + 157464 q^{9} - 1025088 q^{21} + 4001904 q^{29} + 6910128 q^{41} + 53045112 q^{49} - 45922800 q^{61} - 46335744 q^{69} + 710296776 q^{81} + 136561680 q^{89}+O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(400, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
400.9.h.a	$400$	$9$	400.h	20.d	$2$	$4$	$4$	$162.951$	\(\Q(\zeta_{12})\)	None					16.9.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{2} q^{3}-238\beta_{2} q^{7}-6369 q^{9}+\cdots\)
400.9.h.b	$400$	$9$	400.h	20.d	$2$	$4$	$4$	$162.951$	\(\Q(i, \sqrt{35})\)	None					16.9.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{9}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}-18\beta _{2}q^{7}+13599q^{9}-3^{3}\beta _{3}q^{11}+\cdots\)
400.9.h.c	$400$	$9$	400.h	20.d	$2$	$8$	$8$	$162.951$	8.0.12960000.1	None					80.9.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{17}\cdot 3^{2}\cdot 5^{16}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(28\beta _{1}-7\beta _{4})q^{7}+(-3111+\cdots)q^{9}+\cdots\)
400.9.h.d	$400$	$9$	400.h	20.d	$2$	$12$	$12$	$162.951$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓			400.9.b.h	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{30}\cdot 3^{4}\cdot 5^{22}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{6}q^{3}+(7\beta _{6}-\beta _{7})q^{7}+(1794+\beta _{1}+\cdots)q^{9}+\cdots\)
400.9.h.e	$400$	$9$	400.h	20.d	$2$	$20$	$20$	$162.951$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None		✓			400.9.b.j	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{90}\cdot 3^{12}\cdot 5^{24}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-\beta _{4}q^{7}+(2994+\beta _{5})q^{9}+\cdots\)
400.9.h.f	$400$	$9$	400.h	20.d	$2$	$24$	$24$	$162.951$		None			✓		80.9.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{9}^{\mathrm{old}}(400, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(400, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)