Space of modular forms of level 400, weight 10, and trivial character

• Label: 400.10.a
• Level: $400$
• Weight: $10$
• Character orbit: 400.a
• Rep. character: $\chi_{400}(1,\cdot)$
• Character field: $\Q$
• Dimension: $84$
• Newform subspaces: $33$
• Sturm bound: $600$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 33 \)
Sturm bound:			\(600\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(400))\).

	Total	New	Old
Modular forms	558	87	471
Cusp forms	522	84	438
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	Dim
\(+\)	\(+\)	$+$	\(20\)
\(+\)	\(-\)	$-$	\(23\)
\(-\)	\(+\)	$-$	\(20\)
\(-\)	\(-\)	$+$	\(21\)
Plus space		\(+\)	\(41\)
Minus space		\(-\)	\(43\)

Trace form

\( 84 q - 82 q^{3} + 1374 q^{7} + 517572 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(400))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
400.10.a.a	$400$	$10$	400.a	1.a	$1$	$1$	$1$	$206.014$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-204\)	\(0\)	\(5432\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-204q^{3}+5432q^{7}+21933q^{9}+\cdots\)
400.10.a.b	$400$	$10$	400.a	1.a	$1$	$1$	$1$	$206.014$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-156\)	\(0\)	\(-952\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-156q^{3}-952q^{7}+4653q^{9}+56148q^{11}+\cdots\)
400.10.a.c	$400$	$10$	400.a	1.a	$1$	$1$	$1$	$206.014$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-114\)	\(0\)	\(4242\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-114q^{3}+4242q^{7}-6687q^{9}+46208q^{11}+\cdots\)
400.10.a.d	$400$	$10$	400.a	1.a	$1$	$1$	$1$	$206.014$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-60\)	\(0\)	\(-4344\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-60q^{3}-4344q^{7}-16083q^{9}-93644q^{11}+\cdots\)
400.10.a.e	$400$	$10$	400.a	1.a	$1$	$1$	$1$	$206.014$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-48\)	\(0\)	\(-532\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-48q^{3}-532q^{7}-17379q^{9}+33180q^{11}+\cdots\)
400.10.a.f	$400$	$10$	400.a	1.a	$1$	$1$	$1$	$206.014$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-21\)	\(0\)	\(-1882\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-21q^{3}-1882q^{7}-19242q^{9}-8007q^{11}+\cdots\)
400.10.a.g	$400$	$10$	400.a	1.a	$1$	$1$	$1$	$206.014$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(21\)	\(0\)	\(1882\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+21q^{3}+1882q^{7}-19242q^{9}-8007q^{11}+\cdots\)
400.10.a.h	$400$	$10$	400.a	1.a	$1$	$1$	$1$	$206.014$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(46\)	\(0\)	\(-10318\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+46q^{3}-10318q^{7}-17567q^{9}+\cdots\)
400.10.a.i	$400$	$10$	400.a	1.a	$1$	$1$	$1$	$206.014$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(68\)	\(0\)	\(10248\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+68q^{3}+10248q^{7}-15059q^{9}+\cdots\)
400.10.a.j	$400$	$10$	400.a	1.a	$1$	$1$	$1$	$206.014$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(174\)	\(0\)	\(4658\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+174q^{3}+4658q^{7}+10593q^{9}+\cdots\)
400.10.a.k	$400$	$10$	400.a	1.a	$1$	$1$	$1$	$206.014$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(228\)	\(0\)	\(-6328\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+228q^{3}-6328q^{7}+32301q^{9}+\cdots\)
400.10.a.l	$400$	$10$	400.a	1.a	$1$	$2$	$2$	$206.014$	\(\Q(\sqrt{79}) \)	None	✓	$1$	$0$	\(0\)	\(-260\)	\(0\)	\(-380\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-130+\beta )q^{3}+(-190+69\beta )q^{7}+\cdots\)
400.10.a.m	$400$	$10$	400.a	1.a	$1$	$2$	$2$	$206.014$	\(\Q(\sqrt{319}) \)	None	✓	$1$	$1$	\(0\)	\(-152\)	\(0\)	\(-5784\)		$-$	$-$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+(-76+\beta )q^{3}+(-2892-43\beta )q^{7}+\cdots\)
400.10.a.n	$400$	$10$	400.a	1.a	$1$	$2$	$2$	$206.014$	\(\Q(\sqrt{22}) \)	None	✓	$1$	$1$	\(0\)	\(-116\)	\(0\)	\(11284\)		$+$	$+$	$+$	$2^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-58+\beta )q^{3}+(5642-35\beta )q^{7}+\cdots\)
400.10.a.o	$400$	$10$	400.a	1.a	$1$	$2$	$2$	$206.014$	\(\Q(\sqrt{6049}) \)	None	✓	$1$	$1$	\(0\)	\(-92\)	\(0\)	\(-6908\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-46-\beta )q^{3}+(-3454+37\beta )q^{7}+\cdots\)
400.10.a.p	$400$	$10$	400.a	1.a	$1$	$2$	$2$	$206.014$	\(\Q(\sqrt{1009}) \)	None	✓	$1$	$0$	\(0\)	\(-68\)	\(0\)	\(-56\)		$-$	$+$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+(-34-\beta )q^{3}+(-28-42\beta )q^{7}+\cdots\)
400.10.a.q	$400$	$10$	400.a	1.a	$1$	$2$	$2$	$206.014$	\(\Q(\sqrt{1009}) \)	None	✓	$1$	$1$	\(0\)	\(68\)	\(0\)	\(56\)		$-$	$-$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+(34-\beta )q^{3}+(28-42\beta )q^{7}+(6698+\cdots)q^{9}+\cdots\)
400.10.a.r	$400$	$10$	400.a	1.a	$1$	$2$	$2$	$206.014$	\(\Q(\sqrt{46}) \)	None	✓	$1$	$1$	\(0\)	\(108\)	\(0\)	\(-908\)		$+$	$+$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(54+\beta )q^{3}+(-454+13\beta )q^{7}+(9729+\cdots)q^{9}+\cdots\)
400.10.a.s	$400$	$10$	400.a	1.a	$1$	$2$	$2$	$206.014$	\(\Q(\sqrt{319}) \)	None	✓	$1$	$1$	\(0\)	\(152\)	\(0\)	\(5784\)		$-$	$-$	$+$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+(76+\beta )q^{3}+(2892-43\beta )q^{7}+(17993+\cdots)q^{9}+\cdots\)
400.10.a.t	$400$	$10$	400.a	1.a	$1$	$2$	$2$	$206.014$	\(\Q(\sqrt{1009}) \)	None	✓	$1$	$0$	\(0\)	\(260\)	\(0\)	\(1700\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(130-\beta )q^{3}+(850-107\beta )q^{7}+(1253+\cdots)q^{9}+\cdots\)
400.10.a.u	$400$	$10$	400.a	1.a	$1$	$3$	$3$	$206.014$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-89\)	\(0\)	\(-5258\)		$-$	$-$	$+$	$2^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-30+\beta _{1})q^{3}+(-1744-26\beta _{1}+\cdots)q^{7}+\cdots\)
400.10.a.v	$400$	$10$	400.a	1.a	$1$	$3$	$3$	$206.014$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-73\)	\(0\)	\(-1482\)		$-$	$-$	$+$	$2^{4}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-24+\beta _{1})q^{3}+(-495-4\beta _{1}+\beta _{2})q^{7}+\cdots\)
400.10.a.w	$400$	$10$	400.a	1.a	$1$	$3$	$3$	$206.014$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(73\)	\(0\)	\(1482\)		$-$	$+$	$-$	$2^{4}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(24-\beta _{1})q^{3}+(495+4\beta _{1}-\beta _{2})q^{7}+\cdots\)
400.10.a.x	$400$	$10$	400.a	1.a	$1$	$3$	$3$	$206.014$	3.3.7117.1	None	✓	$1$	$1$	\(0\)	\(84\)	\(0\)	\(-5520\)		$+$	$+$	$+$	$2^{9}\cdot 3^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+(28+\beta _{1})q^{3}+(-1840+17\beta _{1}-\beta _{2})q^{7}+\cdots\)
400.10.a.y	$400$	$10$	400.a	1.a	$1$	$3$	$3$	$206.014$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(89\)	\(0\)	\(5258\)		$-$	$+$	$-$	$2^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q+(30-\beta _{1})q^{3}+(1744+26\beta _{1}-23\beta _{2})q^{7}+\cdots\)
400.10.a.z	$400$	$10$	400.a	1.a	$1$	$4$	$4$	$206.014$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-236\)	\(0\)	\(-592\)		$+$	$+$	$+$	$2^{10}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-59-\beta _{1})q^{3}+(-148+6\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
400.10.a.ba	$400$	$10$	400.a	1.a	$1$	$4$	$4$	$206.014$	4.4.49740556.1	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{8}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(7^{2}\beta _{1}+21\beta _{2})q^{7}+(-2907+\cdots)q^{9}+\cdots\)
400.10.a.bb	$400$	$10$	400.a	1.a	$1$	$4$	$4$	$206.014$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{8}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(\beta _{1}-3\beta _{2})q^{7}+(2261+7\beta _{3})q^{9}+\cdots\)
400.10.a.bc	$400$	$10$	400.a	1.a	$1$	$4$	$4$	$206.014$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(236\)	\(0\)	\(592\)		$+$	$-$	$-$	$2^{10}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(59+\beta _{1})q^{3}+(148-6\beta _{1}-\beta _{2})q^{7}+\cdots\)
400.10.a.bd	$400$	$10$	400.a	1.a	$1$	$5$	$5$	$206.014$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-1618\)		$+$	$-$	$-$	$2^{17}\cdot 3^{3}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-322-10\beta _{1}+\beta _{2})q^{7}+\cdots\)
400.10.a.be	$400$	$10$	400.a	1.a	$1$	$5$	$5$	$206.014$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(1618\)		$+$	$+$	$+$	$2^{17}\cdot 3^{3}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(322+10\beta _{1}-\beta _{2})q^{7}+(5861+\cdots)q^{9}+\cdots\)
400.10.a.bf	$400$	$10$	400.a	1.a	$1$	$7$	$7$	$206.014$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-10\)	\(0\)	\(-1802\)		$+$	$-$	$-$	$2^{25}\cdot 3^{2}\cdot 5^{7}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-2^{8}+3\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
400.10.a.bg	$400$	$10$	400.a	1.a	$1$	$7$	$7$	$206.014$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(10\)	\(0\)	\(1802\)		$+$	$-$	$-$	$2^{25}\cdot 3^{2}\cdot 5^{7}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(2^{8}-3\beta _{1}-\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(400))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(400)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 9}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 5}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 2}\)