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

• Label: 400.2.s
• Level: $400$
• Weight: $2$
• Character orbit: 400.s
• Rep. character: $\chi_{400}(107,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $68$
• Newform subspaces: $5$
• Sturm bound: $120$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.s (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 80 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(120\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	132	76	56
Cusp forms	108	68	40
Eisenstein series	24	8	16

Trace form

68 * q + 2 * q^2 + 4 * q^3 - 8 * q^4 - 4 * q^6 + 4 * q^7 + 8 * q^8 + 60 * q^9 - 4 * q^11 + 4 * q^16 + 4 * q^17 + 26 * q^18 + 16 * q^19 - 4 * q^21 - 12 * q^22 + 4 * q^23 - 12 * q^24 + 28 * q^26 + 16 * q^27 - 28 * q^28 - 28 * q^32 + 4 * q^33 - 48 * q^34 - 52 * q^36 - 24 * q^38 + 16 * q^42 + 20 * q^44 - 36 * q^46 - 24 * q^47 - 20 * q^48 - 20 * q^51 - 16 * q^52 + 4 * q^53 - 52 * q^54 + 8 * q^56 + 12 * q^57 - 48 * q^58 - 32 * q^59 - 36 * q^61 + 36 * q^62 + 12 * q^63 - 32 * q^64 - 20 * q^66 + 56 * q^68 + 20 * q^69 - 72 * q^71 + 64 * q^72 + 8 * q^73 + 96 * q^74 - 4 * q^76 + 32 * q^77 + 28 * q^78 + 28 * q^81 - 40 * q^82 - 36 * q^83 - 132 * q^84 + 44 * q^86 - 52 * q^87 + 16 * q^88 - 36 * q^91 + 4 * q^92 + 56 * q^94 + 100 * q^96 + 4 * q^97 + 78 * q^98 - 76 * q^99

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.s.a	$400$	$2$	400.s	80.s	$4$	$2$	$1$	$3.194$	\(\Q(\sqrt{-1}) \)	None					80.2.j.a	$2$	$0$	\(2\)	\(4\)	\(0\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i+1)q^{2}+2 q^{3}-2 i q^{4}+(-2 i+2)q^{6}+\cdots\)
400.2.s.b	$400$	$2$	400.s	80.s	$4$	$8$	$4$	$3.194$	\(\Q(\zeta_{24})\)	None		✓			400.2.j.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta_{5}+\beta_1)q^{2}+\beta_{7} q^{3}-2 q^{4}+\cdots\)
400.2.s.c	$400$	$2$	400.s	80.s	$4$	$16$	$8$	$3.194$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			400.2.j.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{6}q^{2}+\beta _{5}q^{3}-\beta _{15}q^{4}+(-1-\beta _{11}+\cdots)q^{6}+\cdots\)
400.2.s.d	$400$	$2$	400.s	80.s	$4$	$18$	$9$	$3.194$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None					80.2.j.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)	$2^{7}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{2}+\beta _{4}q^{3}-\beta _{14}q^{4}+(-1-\beta _{3}+\cdots)q^{6}+\cdots\)
400.2.s.e	$400$	$2$	400.s	80.s	$4$	$24$	$12$	$3.194$		None		✓	✓		400.2.j.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

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

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