Space of modular forms of level 400, weight 8, and trivial character

• Label: 400.8.a
• Level: $400$
• Weight: $8$
• Character orbit: 400.a
• Rep. character: $\chi_{400}(1,\cdot)$
• Character field: $\Q$
• Dimension: $65$
• Newform subspaces: $38$
• Sturm bound: $480$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 38 \)
Sturm bound:			\(480\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(400))\).

	Total	New	Old
Modular forms	438	68	370
Cusp forms	402	65	337
Eisenstein series	36	3	33

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(111\)	\(16\)	\(95\)		\(102\)	\(16\)	\(86\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(-\)		\(109\)	\(17\)	\(92\)		\(100\)	\(17\)	\(83\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(-\)		\(108\)	\(16\)	\(92\)		\(99\)	\(15\)	\(84\)		\(9\)	\(1\)	\(8\)
\(-\)	\(-\)	\(+\)		\(110\)	\(19\)	\(91\)		\(101\)	\(17\)	\(84\)		\(9\)	\(2\)	\(7\)
Plus space		\(+\)		\(221\)	\(35\)	\(186\)		\(203\)	\(33\)	\(170\)		\(18\)	\(2\)	\(16\)
Minus space		\(-\)		\(217\)	\(33\)	\(184\)		\(199\)	\(32\)	\(167\)		\(18\)	\(1\)	\(17\)

Trace form

65 * q + 26 * q^3 + 662 * q^7 + 46317 * q^9 - 3388 * q^11 + 3278 * q^13 - 11258 * q^17 - 60448 * q^19 + 3356 * q^21 + 7282 * q^23 + 12188 * q^27 - 68378 * q^29 + 8340 * q^31 - 166544 * q^33 - 104322 * q^37 + 268748 * q^39 - 133454 * q^41 - 142462 * q^43 - 1589618 * q^47 + 6831653 * q^49 + 1514416 * q^51 - 392442 * q^53 + 725144 * q^57 - 5391876 * q^59 - 5518 * q^61 + 4296854 * q^63 + 4606486 * q^67 + 2445284 * q^69 - 3556636 * q^71 + 1901070 * q^73 + 7780240 * q^77 + 9605360 * q^79 + 27940077 * q^81 - 3397918 * q^83 - 19863444 * q^87 + 10608630 * q^89 - 18655020 * q^91 + 9069672 * q^93 + 8946158 * q^97 - 46333544 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(400))\) 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					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5
400.8.a.a	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				50.8.a.d	$1$	$1$	\(0\)	\(-87\)	\(0\)	\(-1366\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-87q^{3}-1366q^{7}+5382q^{9}+1083q^{11}+\cdots\)
400.8.a.b	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				8.8.a.a	$1$	$0$	\(0\)	\(-84\)	\(0\)	\(-456\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-84q^{3}-456q^{7}+4869q^{9}+2524q^{11}+\cdots\)
400.8.a.c	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				200.8.a.a	$1$	$0$	\(0\)	\(-69\)	\(0\)	\(174\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-69q^{3}+174q^{7}+2574q^{9}-7111q^{11}+\cdots\)
400.8.a.d	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				50.8.a.c	$1$	$0$	\(0\)	\(-57\)	\(0\)	\(1174\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-57q^{3}+1174q^{7}+1062q^{9}+7563q^{11}+\cdots\)
400.8.a.e	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				5.8.a.a	$1$	$1$	\(0\)	\(-48\)	\(0\)	\(-1644\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-48q^{3}-1644q^{7}+117q^{9}-172q^{11}+\cdots\)
400.8.a.f	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				50.8.a.a	$1$	$0$	\(0\)	\(-43\)	\(0\)	\(-974\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-43q^{3}-974q^{7}-338q^{9}-87q^{11}+\cdots\)
400.8.a.g	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				40.8.a.a	$1$	$0$	\(0\)	\(-36\)	\(0\)	\(776\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-6^{2}q^{3}+776q^{7}-891q^{9}+124q^{11}+\cdots\)
400.8.a.h	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				40.8.c.a	$1$	$1$	\(0\)	\(-34\)	\(0\)	\(-106\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-34q^{3}-106q^{7}-1031q^{9}+1324q^{11}+\cdots\)
400.8.a.i	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				200.8.a.d	$1$	$1$	\(0\)	\(-9\)	\(0\)	\(694\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+694q^{7}-2106q^{9}-4901q^{11}+\cdots\)
400.8.a.j	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				20.8.a.a	$1$	$1$	\(0\)	\(-6\)	\(0\)	\(-706\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{3}-706q^{7}-2151q^{9}+3840q^{11}+\cdots\)
400.8.a.k	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				200.8.a.d	$1$	$0$	\(0\)	\(9\)	\(0\)	\(-694\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}-694q^{7}-2106q^{9}-4901q^{11}+\cdots\)
400.8.a.l	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				2.8.a.a	$1$	$1$	\(0\)	\(12\)	\(0\)	\(1016\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+12q^{3}+1016q^{7}-2043q^{9}-1092q^{11}+\cdots\)
400.8.a.m	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				10.8.a.a	$1$	$1$	\(0\)	\(28\)	\(0\)	\(104\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+28q^{3}+104q^{7}-1403q^{9}+5148q^{11}+\cdots\)
400.8.a.n	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				40.8.c.a	$1$	$1$	\(0\)	\(34\)	\(0\)	\(106\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+34q^{3}+106q^{7}-1031q^{9}+1324q^{11}+\cdots\)
400.8.a.o	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				50.8.a.a	$1$	$1$	\(0\)	\(43\)	\(0\)	\(974\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+43q^{3}+974q^{7}-338q^{9}-87q^{11}+\cdots\)
400.8.a.p	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				8.8.a.b	$1$	$0$	\(0\)	\(44\)	\(0\)	\(-1224\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+44q^{3}-1224q^{7}-251q^{9}+3164q^{11}+\cdots\)
400.8.a.q	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				50.8.a.c	$1$	$1$	\(0\)	\(57\)	\(0\)	\(-1174\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+57q^{3}-1174q^{7}+1062q^{9}+7563q^{11}+\cdots\)
400.8.a.r	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				200.8.a.a	$1$	$1$	\(0\)	\(69\)	\(0\)	\(-174\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+69q^{3}-174q^{7}+2574q^{9}-7111q^{11}+\cdots\)
400.8.a.s	$400$	$8$	400.a	1.a	$1$	$1$	$1$	$124.954$	\(\Q\)	None	✓				50.8.a.d	$1$	$0$	\(0\)	\(87\)	\(0\)	\(1366\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+87q^{3}+1366q^{7}+5382q^{9}+1083q^{11}+\cdots\)
400.8.a.t	$400$	$8$	400.a	1.a	$1$	$2$	$2$	$124.954$	\(\Q(\sqrt{31}) \)	None	✓				10.8.b.a	$1$	$0$	\(0\)	\(-56\)	\(0\)	\(-408\)		$-$	$-$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+(-28+\beta )q^{3}+(-204-7\beta )q^{7}+\cdots\)
400.8.a.u	$400$	$8$	400.a	1.a	$1$	$2$	$2$	$124.954$	\(\Q(\sqrt{2521}) \)	None	✓				100.8.a.b	$1$	$0$	\(0\)	\(-40\)	\(0\)	\(-280\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-20-\beta )q^{3}+(-140-6\beta )q^{7}+\cdots\)
400.8.a.v	$400$	$8$	400.a	1.a	$1$	$2$	$2$	$124.954$	\(\Q(\sqrt{649}) \)	None	✓				25.8.a.c	$1$	$1$	\(0\)	\(-40\)	\(0\)	\(600\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-20-\beta )q^{3}+(300+42\beta )q^{7}+(-1138+\cdots)q^{9}+\cdots\)
400.8.a.w	$400$	$8$	400.a	1.a	$1$	$2$	$2$	$124.954$	\(\Q(\sqrt{1129}) \)	None	✓				20.8.a.b	$1$	$1$	\(0\)	\(-20\)	\(0\)	\(1660\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-10-\beta )q^{3}+(830+9\beta )q^{7}+(2429+\cdots)q^{9}+\cdots\)
400.8.a.x	$400$	$8$	400.a	1.a	$1$	$2$	$2$	$124.954$	\(\Q(\sqrt{3889}) \)	None	✓				200.8.a.l	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-1144\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{3}+(-572-2\beta )q^{7}+(1706+\cdots)q^{9}+\cdots\)
400.8.a.y	$400$	$8$	400.a	1.a	$1$	$2$	$2$	$124.954$	\(\Q(\sqrt{29}) \)	None	✓				5.8.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3\beta q^{3}-39\beta q^{7}-1143q^{9}+6828q^{11}+\cdots\)
400.8.a.z	$400$	$8$	400.a	1.a	$1$	$2$	$2$	$124.954$	\(\Q(\sqrt{46}) \)	None	✓				40.8.a.b	$1$	$0$	\(0\)	\(4\)	\(0\)	\(844\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{3}+(422+17\beta )q^{7}+(761+\cdots)q^{9}+\cdots\)
400.8.a.ba	$400$	$8$	400.a	1.a	$1$	$2$	$2$	$124.954$	\(\Q(\sqrt{3889}) \)	None	✓				200.8.a.l	$1$	$0$	\(0\)	\(4\)	\(0\)	\(1144\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{3}+(572+2\beta )q^{7}+(1706+\cdots)q^{9}+\cdots\)
400.8.a.bb	$400$	$8$	400.a	1.a	$1$	$2$	$2$	$124.954$	\(\Q(\sqrt{19}) \)	None	✓				5.8.a.b	$1$	$1$	\(0\)	\(20\)	\(0\)	\(-100\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(10+\beta )q^{3}+(-50-7\beta )q^{7}+(2777+\cdots)q^{9}+\cdots\)
400.8.a.bc	$400$	$8$	400.a	1.a	$1$	$2$	$2$	$124.954$	\(\Q(\sqrt{6}) \)	None	✓				40.8.a.c	$1$	$0$	\(0\)	\(36\)	\(0\)	\(-404\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(18+\beta )q^{3}+(-202-63\beta )q^{7}+(-1479+\cdots)q^{9}+\cdots\)
400.8.a.bd	$400$	$8$	400.a	1.a	$1$	$2$	$2$	$124.954$	\(\Q(\sqrt{649}) \)	None	✓				25.8.a.c	$1$	$0$	\(0\)	\(40\)	\(0\)	\(-600\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(20-\beta )q^{3}+(-300+42\beta )q^{7}+(-1138+\cdots)q^{9}+\cdots\)
400.8.a.be	$400$	$8$	400.a	1.a	$1$	$2$	$2$	$124.954$	\(\Q(\sqrt{2521}) \)	None	✓				100.8.a.b	$1$	$1$	\(0\)	\(40\)	\(0\)	\(280\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(20-\beta )q^{3}+(140-6\beta )q^{7}+(734+\cdots)q^{9}+\cdots\)
400.8.a.bf	$400$	$8$	400.a	1.a	$1$	$2$	$2$	$124.954$	\(\Q(\sqrt{31}) \)	None	✓				10.8.b.a	$1$	$0$	\(0\)	\(56\)	\(0\)	\(408\)		$-$	$-$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+(28+\beta )q^{3}+(204-7\beta )q^{7}+(1697+\cdots)q^{9}+\cdots\)
400.8.a.bg	$400$	$8$	400.a	1.a	$1$	$2$	$2$	$124.954$	\(\Q(\sqrt{601}) \)	None	✓				40.8.a.d	$1$	$0$	\(0\)	\(76\)	\(0\)	\(796\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(38-\beta )q^{3}+(398-7\beta )q^{7}+(1661+\cdots)q^{9}+\cdots\)
400.8.a.bh	$400$	$8$	400.a	1.a	$1$	$3$	$3$	$124.954$	3.3.40101.1	None	✓				200.8.a.o	$1$	$1$	\(0\)	\(-97\)	\(0\)	\(1630\)		$+$	$-$	$-$	$2^{8}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-2^{5}-\beta _{1})q^{3}+(543+2\beta _{1}+\beta _{2})q^{7}+\cdots\)
400.8.a.bi	$400$	$8$	400.a	1.a	$1$	$3$	$3$	$124.954$	3.3.40101.1	None	✓		✓		200.8.a.o	$1$	$0$	\(0\)	\(97\)	\(0\)	\(-1630\)		$+$	$+$	$+$	$2^{8}\cdot 5$	$\mathrm{SU}(2)$	\(q+(2^{5}+\beta _{1})q^{3}+(-543-2\beta _{1}-\beta _{2})q^{7}+\cdots\)
400.8.a.bj	$400$	$8$	400.a	1.a	$1$	$4$	$4$	$124.954$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				40.8.c.b	$1$	$1$	\(0\)	\(-32\)	\(0\)	\(-1632\)		$+$	$-$	$-$	$2^{8}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-8-\beta _{1})q^{3}+(-408+3\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
400.8.a.bk	$400$	$8$	400.a	1.a	$1$	$4$	$4$	$124.954$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		20.8.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{8}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(6\beta _{1}+\beta _{2})q^{7}+(509+\beta _{3})q^{9}+\cdots\)
400.8.a.bl	$400$	$8$	400.a	1.a	$1$	$4$	$4$	$124.954$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				40.8.c.b	$1$	$1$	\(0\)	\(32\)	\(0\)	\(1632\)		$+$	$-$	$-$	$2^{8}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(8+\beta _{1})q^{3}+(408-3\beta _{1}+2\beta _{2}-\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(400))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(400)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)