Space of modular forms of level 400, weight 6, and trivial character

• Label: 400.6.a
• Level: $400$
• Weight: $6$
• Character orbit: 400.a
• Rep. character: $\chi_{400}(1,\cdot)$
• Character field: $\Q$
• Dimension: $46$
• Newform subspaces: $27$
• Sturm bound: $360$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 27 \)
Sturm bound:			\(360\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(400))\).

	Total	New	Old
Modular forms	318	49	269
Cusp forms	282	46	236
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
\(+\)	\(+\)	$+$	\(11\)
\(+\)	\(-\)	$-$	\(13\)
\(-\)	\(+\)	$-$	\(11\)
\(-\)	\(-\)	$+$	\(11\)
Plus space		\(+\)	\(22\)
Minus space		\(-\)	\(24\)

Trace form

\( 46 q - 10 q^{3} + 110 q^{7} + 3462 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\) 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
400.6.a.a	$400$	$6$	400.a	1.a	$1$	$1$	$1$	$64.154$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-26\)	\(0\)	\(-22\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-26q^{3}-22q^{7}+433q^{9}+768q^{11}+\cdots\)
400.6.a.b	$400$	$6$	400.a	1.a	$1$	$1$	$1$	$64.154$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-18\)	\(0\)	\(242\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-18q^{3}+242q^{7}+3^{4}q^{9}-656q^{11}+\cdots\)
400.6.a.c	$400$	$6$	400.a	1.a	$1$	$1$	$1$	$64.154$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-14\)	\(0\)	\(-158\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-14q^{3}-158q^{7}-47q^{9}+148q^{11}+\cdots\)
400.6.a.d	$400$	$6$	400.a	1.a	$1$	$1$	$1$	$64.154$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-12\)	\(0\)	\(-88\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-12q^{3}-88q^{7}-99q^{9}-540q^{11}+\cdots\)
400.6.a.e	$400$	$6$	400.a	1.a	$1$	$1$	$1$	$64.154$	\(\Q\)	None	✓	$1$	$2$	\(0\)	\(-11\)	\(0\)	\(-142\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-11q^{3}-142q^{7}-122q^{9}-777q^{11}+\cdots\)
400.6.a.f	$400$	$6$	400.a	1.a	$1$	$1$	$1$	$64.154$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-8\)	\(0\)	\(-108\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}-108q^{7}-179q^{9}+604q^{11}+\cdots\)
400.6.a.g	$400$	$6$	400.a	1.a	$1$	$1$	$1$	$64.154$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(192\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+192q^{7}-227q^{9}+148q^{11}+\cdots\)
400.6.a.h	$400$	$6$	400.a	1.a	$1$	$1$	$1$	$64.154$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-62\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-62q^{7}-239q^{9}+12^{2}q^{11}+\cdots\)
400.6.a.i	$400$	$6$	400.a	1.a	$1$	$1$	$1$	$64.154$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(6\)	\(0\)	\(-118\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}-118q^{7}-207q^{9}-192q^{11}+\cdots\)
400.6.a.j	$400$	$6$	400.a	1.a	$1$	$1$	$1$	$64.154$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(11\)	\(0\)	\(142\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+11q^{3}+142q^{7}-122q^{9}-777q^{11}+\cdots\)
400.6.a.k	$400$	$6$	400.a	1.a	$1$	$1$	$1$	$64.154$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(14\)	\(0\)	\(158\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+14q^{3}+158q^{7}-47q^{9}+148q^{11}+\cdots\)
400.6.a.l	$400$	$6$	400.a	1.a	$1$	$1$	$1$	$64.154$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(20\)	\(0\)	\(-24\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+20q^{3}-24q^{7}+157q^{9}-124q^{11}+\cdots\)
400.6.a.m	$400$	$6$	400.a	1.a	$1$	$1$	$1$	$64.154$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(22\)	\(0\)	\(218\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+22q^{3}+218q^{7}+241q^{9}+480q^{11}+\cdots\)
400.6.a.n	$400$	$6$	400.a	1.a	$1$	$1$	$1$	$64.154$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(24\)	\(0\)	\(-172\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+24q^{3}-172q^{7}+333q^{9}-132q^{11}+\cdots\)
400.6.a.o	$400$	$6$	400.a	1.a	$1$	$2$	$2$	$64.154$	\(\Q(\sqrt{241}) \)	None	✓	$1$	$1$	\(0\)	\(-20\)	\(0\)	\(-200\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-10-\beta )q^{3}+(-10^{2}-2\beta )q^{7}+\cdots\)
400.6.a.p	$400$	$6$	400.a	1.a	$1$	$2$	$2$	$64.154$	\(\Q(\sqrt{409}) \)	None	✓	$1$	$0$	\(0\)	\(-20\)	\(0\)	\(40\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-10-\beta )q^{3}+(20+6\beta )q^{7}+(266+\cdots)q^{9}+\cdots\)
400.6.a.q	$400$	$6$	400.a	1.a	$1$	$2$	$2$	$64.154$	\(\Q(\sqrt{129}) \)	None	✓	$1$	$1$	\(0\)	\(-12\)	\(0\)	\(52\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-6-\beta )q^{3}+(26-3\beta )q^{7}+(309+\cdots)q^{9}+\cdots\)
400.6.a.r	$400$	$6$	400.a	1.a	$1$	$2$	$2$	$64.154$	\(\Q(\sqrt{241}) \)	None	✓	$1$	$1$	\(0\)	\(-8\)	\(0\)	\(8\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{3}+(4+2\beta )q^{7}+(14+8\beta )q^{9}+\cdots\)
400.6.a.s	$400$	$6$	400.a	1.a	$1$	$2$	$2$	$64.154$	\(\Q(\sqrt{31}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-11\beta q^{7}-119q^{9}+10^{2}q^{11}+\cdots\)
400.6.a.t	$400$	$6$	400.a	1.a	$1$	$2$	$2$	$64.154$	\(\Q(\sqrt{11}) \)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+3\beta q^{3}-9\beta q^{7}+153q^{9}-252q^{11}+\cdots\)
400.6.a.u	$400$	$6$	400.a	1.a	$1$	$2$	$2$	$64.154$	\(\Q(\sqrt{241}) \)	None	✓	$1$	$0$	\(0\)	\(8\)	\(0\)	\(-8\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{3}+(-4-2\beta )q^{7}+(14+8\beta )q^{9}+\cdots\)
400.6.a.v	$400$	$6$	400.a	1.a	$1$	$2$	$2$	$64.154$	\(\Q(\sqrt{409}) \)	None	✓	$1$	$1$	\(0\)	\(20\)	\(0\)	\(-40\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(10-\beta )q^{3}+(-20+6\beta )q^{7}+(266+\cdots)q^{9}+\cdots\)
400.6.a.w	$400$	$6$	400.a	1.a	$1$	$2$	$2$	$64.154$	\(\Q(\sqrt{241}) \)	None	✓	$1$	$0$	\(0\)	\(20\)	\(0\)	\(200\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(10-\beta )q^{3}+(10^{2}-2\beta )q^{7}+(98-20\beta )q^{9}+\cdots\)
400.6.a.x	$400$	$6$	400.a	1.a	$1$	$3$	$3$	$64.154$	3.3.47217.1	None	✓	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(70\)		$+$	$-$	$-$	$2^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(24-\beta _{1}-\beta _{2})q^{7}+(50+5\beta _{1}+\cdots)q^{9}+\cdots\)
400.6.a.y	$400$	$6$	400.a	1.a	$1$	$3$	$3$	$64.154$	3.3.47217.1	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-70\)		$+$	$+$	$+$	$2^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-24+\beta _{1}+\beta _{2})q^{7}+(50+\cdots)q^{9}+\cdots\)
400.6.a.z	$400$	$6$	400.a	1.a	$1$	$4$	$4$	$64.154$	4.4.1595208.1	None	✓	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(-148\)		$+$	$-$	$-$	$2^{8}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(-37-\beta _{1}+\beta _{3})q^{7}+\cdots\)
400.6.a.ba	$400$	$6$	400.a	1.a	$1$	$4$	$4$	$64.154$	4.4.1595208.1	None	✓	$1$	$0$	\(0\)	\(4\)	\(0\)	\(148\)		$+$	$-$	$-$	$2^{8}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(37+\beta _{1}-\beta _{3})q^{7}+(5^{3}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(400))\) into lower level spaces

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