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

• Label: 400.2.q
• Level: $400$
• Weight: $2$
• Character orbit: 400.q
• Rep. character: $\chi_{400}(149,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $68$
• Newform subspaces: $8$
• Sturm bound: $120$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.q (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 80 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(120\)
Trace bound:			\(2\)
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 + 4 * q^4 - 4 * q^6 - 4 * q^11 - 20 * q^14 - 12 * q^16 + 20 * q^19 + 8 * q^21 + 28 * q^24 - 56 * q^26 + 4 * q^29 - 32 * q^31 - 12 * q^34 - 60 * q^36 - 8 * q^44 - 28 * q^46 + 52 * q^49 + 4 * q^54 + 12 * q^56 - 28 * q^59 + 28 * q^61 - 44 * q^64 + 128 * q^66 - 40 * q^69 + 60 * q^74 + 120 * q^76 - 64 * q^79 - 44 * q^81 - 4 * q^84 + 52 * q^86 - 8 * q^91 + 104 * q^94 - 84 * q^96 - 84 * 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.q.a	$400$	$2$	400.q	80.q	$4$	$2$	$1$	$3.194$	\(\Q(\sqrt{-1}) \)	None					16.2.e.a	$2$	$1$	\(-2\)	\(-2\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i-1)q^{2}+(-i-1)q^{3}-2 i q^{4}+\cdots\)
400.2.q.b	$400$	$2$	400.q	80.q	$4$	$2$	$1$	$3.194$	\(\Q(\sqrt{-1}) \)	None					16.2.e.a	$2$	$0$	\(2\)	\(2\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i+1)q^{2}+(i+1)q^{3}-2 i q^{4}+\cdots\)
400.2.q.c	$400$	$2$	400.q	80.q	$4$	$4$	$2$	$3.194$	\(\Q(i, \sqrt{11})\)	None		✓			400.2.l.d	$2$	$0$	\(-4\)	\(2\)	\(0\)	\(-4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\beta _{1})q^{2}+(\beta _{1}-\beta _{2})q^{3}+2\beta _{1}q^{4}+\cdots\)
400.2.q.d	$400$	$2$	400.q	80.q	$4$	$4$	$2$	$3.194$	\(\Q(i, \sqrt{11})\)	None		✓			400.2.l.d	$2$	$0$	\(4\)	\(-2\)	\(0\)	\(4\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{1})q^{2}+(-1-\beta _{2})q^{3}+2\beta _{1}q^{4}+\cdots\)
400.2.q.e	$400$	$2$	400.q	80.q	$4$	$12$	$6$	$3.194$	12.0.\(\cdots\).1	None		✓			400.2.l.f	$2$	$0$	\(-2\)	\(2\)	\(0\)	\(-12\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{6}q^{2}-\beta _{7}q^{3}+\beta _{10}q^{4}+(-\beta _{3}+\cdots)q^{6}+\cdots\)
400.2.q.f	$400$	$2$	400.q	80.q	$4$	$12$	$6$	$3.194$	12.0.\(\cdots\).1	None		✓			400.2.l.f	$2$	$0$	\(2\)	\(-2\)	\(0\)	\(12\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{6}q^{2}+\beta _{7}q^{3}+\beta _{10}q^{4}+(-\beta _{3}+\cdots)q^{6}+\cdots\)
400.2.q.g	$400$	$2$	400.q	80.q	$4$	$16$	$8$	$3.194$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					80.2.l.a	$2$	$0$	\(-4\)	\(0\)	\(0\)	\(-8\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{2}+(-\beta _{6}-\beta _{11})q^{3}+(-\beta _{2}+\cdots)q^{4}+\cdots\)
400.2.q.h	$400$	$2$	400.q	80.q	$4$	$16$	$8$	$3.194$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					80.2.l.a	$2$	$0$	\(4\)	\(0\)	\(0\)	\(8\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{2}q^{2}+(\beta _{6}+\beta _{11})q^{3}+(-\beta _{2}-\beta _{12}+\cdots)q^{4}+\cdots\)

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

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