Space of modular forms of level 400, weight 5, and Character 400.p

• Label: 400.5.p
• Level: $400$
• Weight: $5$
• Character orbit: 400.p
• Rep. character: $\chi_{400}(193,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $70$
• Newform subspaces: $17$
• Sturm bound: $300$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	400.p (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(300\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(400, [\chi])\).

	Total	New	Old
Modular forms	516	74	442
Cusp forms	444	70	374
Eisenstein series	72	4	68

Trace form

70 * q - 2 * q^3 - 2 * q^7 + 4 * q^11 + 122 * q^13 + 122 * q^17 + 1212 * q^21 - 1442 * q^23 + 1120 * q^27 + 260 * q^31 - 1276 * q^33 - 1398 * q^37 + 2492 * q^41 + 4478 * q^43 - 4322 * q^47 + 6532 * q^51 - 2518 * q^53 - 3680 * q^57 - 7556 * q^61 - 11202 * q^63 - 12002 * q^67 - 18812 * q^71 + 1002 * q^73 + 4804 * q^77 - 44426 * q^81 - 5282 * q^83 + 33440 * q^87 - 8060 * q^91 - 14236 * q^93 + 7722 * q^97

Decomposition of \(S_{5}^{\mathrm{new}}(400, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
400.5.p.a	$400$	$5$	400.p	5.c	$4$	$2$	$1$	$41.348$	\(\Q(\sqrt{-1}) \)	None					5.5.c.a	$2$	$0$	\(0\)	\(-12\)	\(0\)	\(-52\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(6 i-6)q^{3}+(-26 i-26)q^{7}+\cdots\)
400.5.p.b	$400$	$5$	400.p	5.c	$4$	$2$	$1$	$41.348$	\(\Q(\sqrt{-1}) \)	None					10.5.c.b	$2$	$0$	\(0\)	\(2\)	\(0\)	\(-38\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i+1)q^{3}+(-19 i-19)q^{7}+\cdots\)
400.5.p.c	$400$	$5$	400.p	5.c	$4$	$2$	$1$	$41.348$	\(\Q(\sqrt{-1}) \)	None					10.5.c.a	$2$	$0$	\(0\)	\(18\)	\(0\)	\(58\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-9 i+9)q^{3}+(29 i+29)q^{7}+\cdots\)
400.5.p.d	$400$	$5$	400.p	5.c	$4$	$2$	$1$	$41.348$	\(\Q(\sqrt{-1}) \)	None					40.5.l.a	$2$	$0$	\(0\)	\(20\)	\(0\)	\(-84\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-10 i+10)q^{3}+(-42 i-42)q^{7}+\cdots\)
400.5.p.e	$400$	$5$	400.p	5.c	$4$	$4$	$2$	$41.348$	\(\Q(i, \sqrt{6})\)	None					50.5.c.c	$2$	$0$	\(0\)	\(-24\)	\(0\)	\(-144\)	$5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-6+\beta _{1}-6\beta _{2})q^{3}+(-6^{2}+6^{2}\beta _{2}+\cdots)q^{7}+\cdots\)
400.5.p.f	$400$	$5$	400.p	5.c	$4$	$4$	$2$	$41.348$	\(\Q(i, \sqrt{61})\)	None					200.5.l.b	$2$	$0$	\(0\)	\(-16\)	\(0\)	\(48\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-4+4\beta _{1}-\beta _{3})q^{3}+(12+12\beta _{1}+\cdots)q^{7}+\cdots\)
400.5.p.g	$400$	$5$	400.p	5.c	$4$	$4$	$2$	$41.348$	\(\Q(i, \sqrt{29})\)	None					40.5.l.b	$2$	$0$	\(0\)	\(-12\)	\(0\)	\(44\)	$2^{5}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-3+3\beta _{1})q^{3}+(11+11\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
400.5.p.h	$400$	$5$	400.p	5.c	$4$	$4$	$2$	$41.348$	\(\Q(i, \sqrt{241})\)	None					20.5.f.a	$2$	$0$	\(0\)	\(-10\)	\(0\)	\(110\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-2+2\beta _{1}+\beta _{3})q^{3}+(26+29\beta _{1}+\cdots)q^{7}+\cdots\)
400.5.p.i	$400$	$5$	400.p	5.c	$4$	$4$	$2$	$41.348$	\(\Q(i, \sqrt{69})\)	None					100.5.f.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{3}+7\beta _{2}q^{7}-57\beta _{1}q^{9}-180q^{11}+\cdots\)
400.5.p.j	$400$	$5$	400.p	5.c	$4$	$4$	$2$	$41.348$	\(\Q(i, \sqrt{6})\)	None					25.5.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+7\beta _{1}q^{3}+28\beta _{3}q^{7}+66\beta _{2}q^{9}-117q^{11}+\cdots\)
400.5.p.k	$400$	$5$	400.p	5.c	$4$	$4$	$2$	$41.348$	\(\Q(i, \sqrt{6})\)	None					100.5.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{1}q^{3}-12\beta _{3}q^{7}-78\beta _{2}q^{9}-45q^{11}+\cdots\)
400.5.p.l	$400$	$5$	400.p	5.c	$4$	$4$	$2$	$41.348$	\(\Q(i, \sqrt{21})\)	None					25.5.c.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{3}-9\beta _{2}q^{7}+39\beta _{1}q^{9}+108q^{11}+\cdots\)
400.5.p.m	$400$	$5$	400.p	5.c	$4$	$4$	$2$	$41.348$	\(\Q(i, \sqrt{61})\)	None					200.5.l.b	$2$	$0$	\(0\)	\(16\)	\(0\)	\(-48\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4+4\beta _{1}-\beta _{2})q^{3}+(-12+12\beta _{1}+\cdots)q^{7}+\cdots\)
400.5.p.n	$400$	$5$	400.p	5.c	$4$	$4$	$2$	$41.348$	\(\Q(i, \sqrt{6})\)	None					50.5.c.c	$2$	$0$	\(0\)	\(24\)	\(0\)	\(144\)	$5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(6+\beta _{1}+6\beta _{2})q^{3}+(6^{2}-6^{2}\beta _{2}+4\beta _{3})q^{7}+\cdots\)
400.5.p.o	$400$	$5$	400.p	5.c	$4$	$6$	$3$	$41.348$	6.0.313431616.3	None					40.5.l.c	$2$	$0$	\(0\)	\(-8\)	\(0\)	\(-40\)	$2^{6}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\beta _{1}-\beta _{3})q^{3}+(-6+6\beta _{1}+\cdots)q^{7}+\cdots\)
400.5.p.p	$400$	$5$	400.p	5.c	$4$	$8$	$4$	$41.348$	8.0.\(\cdots\).18	None					200.5.l.f	$2$	$0$	\(0\)	\(-8\)	\(0\)	\(-48\)	$2^{6}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-\beta _{1}+\beta _{3})q^{3}+(-6+6\beta _{1}+\cdots)q^{7}+\cdots\)
400.5.p.q	$400$	$5$	400.p	5.c	$4$	$8$	$4$	$41.348$	8.0.\(\cdots\).18	None					200.5.l.f	$2$	$0$	\(0\)	\(8\)	\(0\)	\(48\)	$2^{6}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(1-\beta _{1}-\beta _{2})q^{3}+(6+6\beta _{1}-2\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(400, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(400, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)