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

• Label: 400.2.l
• Level: $400$
• Weight: $2$
• Character orbit: 400.l
• Rep. character: $\chi_{400}(101,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $70$
• Newform subspaces: $9$
• Sturm bound: $120$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.l (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 16 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(120\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

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

	Total	New	Old
Modular forms	132	82	50
Cusp forms	108	70	38
Eisenstein series	24	12	12

Trace form

70 * q + 2 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^8 - 2 * q^11 + 16 * q^12 + 2 * q^13 + 24 * q^14 - 20 * q^16 + 4 * q^17 - 2 * q^18 - 6 * q^19 - 12 * q^21 + 20 * q^22 - 20 * q^24 - 12 * q^26 - 16 * q^27 - 4 * q^28 + 10 * q^29 - 8 * q^32 + 4 * q^33 - 40 * q^34 + 16 * q^36 + 10 * q^37 - 8 * q^38 - 60 * q^42 - 18 * q^43 - 44 * q^44 - 32 * q^46 + 24 * q^47 + 32 * q^48 - 30 * q^49 + 28 * q^51 - 52 * q^52 - 6 * q^53 - 44 * q^54 - 52 * q^56 + 24 * q^58 + 30 * q^59 + 10 * q^61 - 8 * q^62 - 44 * q^63 + 76 * q^64 + 28 * q^66 - 30 * q^67 + 48 * q^68 + 36 * q^69 + 36 * q^72 + 36 * q^74 + 12 * q^76 - 20 * q^77 + 20 * q^78 - 34 * q^81 + 76 * q^82 - 38 * q^83 + 156 * q^84 - 28 * q^86 - 8 * q^88 - 60 * q^91 + 76 * q^92 + 32 * q^93 - 56 * q^94 - 20 * q^96 + 4 * q^97 - 54 * q^98 + 18 * 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.l.a	$400$	$2$	400.l	16.e	$4$	$2$	$1$	$3.194$	\(\Q(\sqrt{-1}) \)	None					80.2.q.a	$2$	$1$	\(-2\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i-1)q^{2}+(i-1)q^{3}-2 i q^{4}+\cdots\)
400.2.l.b	$400$	$2$	400.l	16.e	$4$	$2$	$1$	$3.194$	\(\Q(\sqrt{-1}) \)	None					80.2.q.a	$2$	$0$	\(2\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i+1)q^{2}+(-i+1)q^{3}-2 i q^{4}+\cdots\)
400.2.l.c	$400$	$2$	400.l	16.e	$4$	$2$	$1$	$3.194$	\(\Q(\sqrt{-1}) \)	None					16.2.e.a	$2$	$0$	\(2\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i+1)q^{2}+(-i+1)q^{3}+2 i q^{4}+\cdots\)
400.2.l.d	$400$	$2$	400.l	16.e	$4$	$4$	$2$	$3.194$	\(\Q(i, \sqrt{11})\)	None		✓			400.2.l.d	$2$	$0$	\(-4\)	\(-2\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\beta _{1})q^{2}+(-\beta _{1}+\beta _{2})q^{3}+2\beta _{1}q^{4}+\cdots\)
400.2.l.e	$400$	$2$	400.l	16.e	$4$	$4$	$2$	$3.194$	\(\Q(i, \sqrt{11})\)	None		✓			400.2.l.d	$2$	$0$	\(4\)	\(2\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1+\beta _{1})q^{2}+(\beta _{1}-\beta _{2})q^{3}+2\beta _{1}q^{4}+\cdots\)
400.2.l.f	$400$	$2$	400.l	16.e	$4$	$12$	$6$	$3.194$	12.0.\(\cdots\).1	None		✓			400.2.l.f	$2$	$0$	\(-4\)	\(-2\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{1}q^{2}-\beta _{6}q^{3}+\beta _{2}q^{4}+(\beta _{1}-\beta _{2}+\cdots)q^{6}+\cdots\)
400.2.l.g	$400$	$2$	400.l	16.e	$4$	$12$	$6$	$3.194$	12.0.\(\cdots\).1	None		✓			400.2.l.f	$2$	$0$	\(4\)	\(2\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{2}+\beta _{6}q^{3}+\beta _{2}q^{4}+(\beta _{1}-\beta _{2}+\cdots)q^{6}+\cdots\)
400.2.l.h	$400$	$2$	400.l	16.e	$4$	$16$	$8$	$3.194$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None					80.2.l.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{8}q^{2}+(\beta _{4}+\beta _{11})q^{3}+(\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
400.2.l.i	$400$	$2$	400.l	16.e	$4$	$16$	$8$	$3.194$	16.0.\(\cdots\).1	None					80.2.q.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{10}q^{2}+(-\beta _{2}+\beta _{9}+\beta _{12}-\beta _{14}+\cdots)q^{3}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(400, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)