Space of modular forms of level 400, weight 4, and trivial character

• Label: 400.4.a
• Level: $400$
• Weight: $4$
• Character orbit: 400.a
• Rep. character: $\chi_{400}(1,\cdot)$
• Character field: $\Q$
• Dimension: $27$
• Newform subspaces: $24$
• Sturm bound: $240$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(240\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(400))\).

	Total	New	Old
Modular forms	198	30	168
Cusp forms	162	27	135
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
\(+\)	\(+\)	\(+\)		\(51\)	\(7\)	\(44\)		\(42\)	\(7\)	\(35\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(-\)		\(49\)	\(7\)	\(42\)		\(40\)	\(7\)	\(33\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(-\)		\(48\)	\(7\)	\(41\)		\(39\)	\(6\)	\(33\)		\(9\)	\(1\)	\(8\)
\(-\)	\(-\)	\(+\)		\(50\)	\(9\)	\(41\)		\(41\)	\(7\)	\(34\)		\(9\)	\(2\)	\(7\)
Plus space		\(+\)		\(101\)	\(16\)	\(85\)		\(83\)	\(14\)	\(69\)		\(18\)	\(2\)	\(16\)
Minus space		\(-\)		\(97\)	\(14\)	\(83\)		\(79\)	\(13\)	\(66\)		\(18\)	\(1\)	\(17\)

Trace form

27 * q + 2 * q^3 - 26 * q^7 + 207 * q^9 - 20 * q^11 - 22 * q^13 + 26 * q^17 - 136 * q^19 + 92 * q^21 - 30 * q^23 + 236 * q^27 + 2 * q^29 + 84 * q^31 + 208 * q^33 + 170 * q^37 - 100 * q^39 + 214 * q^41 + 378 * q^43 + 430 * q^47 + 911 * q^49 - 896 * q^51 - 526 * q^53 + 728 * q^57 + 292 * q^59 - 754 * q^61 - 2170 * q^63 - 1186 * q^67 - 1372 * q^69 + 2132 * q^71 - 1198 * q^73 - 368 * q^77 + 784 * q^79 + 279 * q^81 + 3178 * q^83 + 2748 * q^87 - 454 * q^89 - 2828 * q^91 + 1608 * q^93 - 766 * q^97 + 2752 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(400))\) 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
400.4.a.a	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				200.4.a.b	$1$	$1$	\(0\)	\(-9\)	\(0\)	\(-26\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}-26q^{7}+54q^{9}+59q^{11}+\cdots\)
400.4.a.b	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				10.4.a.a	$1$	$1$	\(0\)	\(-8\)	\(0\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}-4q^{7}+37q^{9}-12q^{11}+58q^{13}+\cdots\)
400.4.a.c	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				25.4.a.a	$1$	$1$	\(0\)	\(-7\)	\(0\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}-6q^{7}+22q^{9}+43q^{11}-28q^{13}+\cdots\)
400.4.a.d	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				50.4.a.a	$1$	$0$	\(0\)	\(-7\)	\(0\)	\(34\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}+34q^{7}+22q^{9}-3^{3}q^{11}+\cdots\)
400.4.a.e	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				40.4.a.a	$1$	$0$	\(0\)	\(-6\)	\(0\)	\(-34\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-6q^{3}-34q^{7}+9q^{9}-2^{4}q^{11}-58q^{13}+\cdots\)
400.4.a.f	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				200.4.a.c	$1$	$0$	\(0\)	\(-5\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}-2q^{7}-2q^{9}-39q^{11}+84q^{13}+\cdots\)
400.4.a.g	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				8.4.a.a	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(24\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+24q^{7}-11q^{9}+44q^{11}+\cdots\)
400.4.a.h	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				10.4.b.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-26q^{7}-23q^{9}+28q^{11}+\cdots\)
400.4.a.i	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				100.4.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-26q^{7}-26q^{9}-45q^{11}+44q^{13}+\cdots\)
400.4.a.j	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				200.4.a.e	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+6q^{7}-26q^{9}+19q^{11}+12q^{13}+\cdots\)
400.4.a.k	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				200.4.a.e	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-6q^{7}-26q^{9}+19q^{11}-12q^{13}+\cdots\)
400.4.a.l	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				100.4.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(26\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+26q^{7}-26q^{9}-45q^{11}-44q^{13}+\cdots\)
400.4.a.m	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				5.4.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+6q^{7}-23q^{9}-2^{5}q^{11}+38q^{13}+\cdots\)
400.4.a.n	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				10.4.b.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+26q^{7}-23q^{9}+28q^{11}+\cdots\)
400.4.a.o	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				20.4.a.a	$1$	$1$	\(0\)	\(4\)	\(0\)	\(-16\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}-2^{4}q^{7}-11q^{9}+60q^{11}+\cdots\)
400.4.a.p	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				40.4.a.b	$1$	$0$	\(0\)	\(4\)	\(0\)	\(16\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}+2^{4}q^{7}-11q^{9}-6^{2}q^{11}+\cdots\)
400.4.a.q	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				200.4.a.c	$1$	$1$	\(0\)	\(5\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}+2q^{7}-2q^{9}-39q^{11}-84q^{13}+\cdots\)
400.4.a.r	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				50.4.a.a	$1$	$1$	\(0\)	\(7\)	\(0\)	\(-34\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}-34q^{7}+22q^{9}-3^{3}q^{11}+\cdots\)
400.4.a.s	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				25.4.a.a	$1$	$0$	\(0\)	\(7\)	\(0\)	\(6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+7q^{3}+6q^{7}+22q^{9}+43q^{11}+28q^{13}+\cdots\)
400.4.a.t	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				200.4.a.b	$1$	$0$	\(0\)	\(9\)	\(0\)	\(26\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}+26q^{7}+54q^{9}+59q^{11}+\cdots\)
400.4.a.u	$400$	$4$	400.a	1.a	$1$	$1$	$1$	$23.601$	\(\Q\)	None	✓				40.4.a.c	$1$	$0$	\(0\)	\(10\)	\(0\)	\(-18\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+10q^{3}-18q^{7}+73q^{9}+2^{4}q^{11}+\cdots\)
400.4.a.v	$400$	$4$	400.a	1.a	$1$	$2$	$2$	$23.601$	\(\Q(\sqrt{6}) \)	None	✓				40.4.c.a	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(-4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{3}+(-2-3\beta )q^{7}+(1-4\beta )q^{9}+\cdots\)
400.4.a.w	$400$	$4$	400.a	1.a	$1$	$2$	$2$	$23.601$	\(\Q(\sqrt{19}) \)	None	✓		✓		20.4.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{7}+7^{2}q^{9}-20q^{11}+6\beta q^{13}+\cdots\)
400.4.a.x	$400$	$4$	400.a	1.a	$1$	$2$	$2$	$23.601$	\(\Q(\sqrt{6}) \)	None	✓				40.4.c.a	$1$	$1$	\(0\)	\(4\)	\(0\)	\(4\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{3}+(2-3\beta )q^{7}+(1+4\beta )q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(400))\) into lower level spaces

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