Space of modular forms of level 400, weight 2, and trivial character

• Label: 400.2.a
• Level: $400$
• Weight: $2$
• Character orbit: 400.a
• Rep. character: $\chi_{400}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $8$
• Sturm bound: $120$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(120\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(400))\).

	Total	New	Old
Modular forms	78	11	67
Cusp forms	43	8	35
Eisenstein series	35	3	32

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
\(+\)	\(+\)	\(+\)		\(18\)	\(2\)	\(16\)		\(10\)	\(2\)	\(8\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)		\(20\)	\(3\)	\(17\)		\(11\)	\(3\)	\(8\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(-\)		\(21\)	\(3\)	\(18\)		\(12\)	\(2\)	\(10\)		\(9\)	\(1\)	\(8\)
\(-\)	\(-\)	\(+\)		\(19\)	\(3\)	\(16\)		\(10\)	\(1\)	\(9\)		\(9\)	\(2\)	\(7\)
Plus space		\(+\)		\(37\)	\(5\)	\(32\)		\(20\)	\(3\)	\(17\)		\(17\)	\(2\)	\(15\)
Minus space		\(-\)		\(41\)	\(6\)	\(35\)		\(23\)	\(5\)	\(18\)		\(18\)	\(1\)	\(17\)

Trace form

8 * q - 2 * q^3 - 2 * q^7 + 8 * q^9 + 8 * q^11 + 4 * q^17 - 4 * q^19 - 4 * q^21 + 10 * q^23 + 4 * q^27 - 8 * q^29 - 12 * q^31 - 8 * q^37 + 20 * q^39 - 8 * q^41 - 18 * q^43 - 2 * q^47 - 12 * q^49 + 24 * q^51 - 8 * q^57 + 4 * q^61 + 14 * q^63 + 10 * q^67 + 4 * q^69 - 4 * q^71 + 4 * q^73 + 16 * q^77 - 32 * q^79 - 4 * q^81 - 10 * q^83 - 12 * q^87 + 12 * q^89 - 28 * q^91 - 8 * q^93 + 12 * q^97 - 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(400))\) 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
400.2.a.a	$400$	$2$	400.a	1.a	$1$	$1$	$1$	$3.194$	\(\Q\)	None	✓				200.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+2q^{7}+6q^{9}-q^{11}-4q^{13}+\cdots\)
400.2.a.b	$400$	$2$	400.a	1.a	$1$	$1$	$1$	$3.194$	\(\Q\)	None	✓				40.2.c.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
400.2.a.c	$400$	$2$	400.a	1.a	$1$	$1$	$1$	$3.194$	\(\Q\)	None	✓				20.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{7}+q^{9}-2q^{13}+6q^{17}+\cdots\)
400.2.a.d	$400$	$2$	400.a	1.a	$1$	$1$	$1$	$3.194$	\(\Q\)	None	✓		✓	✓	50.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}-2q^{9}+3q^{11}-4q^{13}+\cdots\)
400.2.a.e	$400$	$2$	400.a	1.a	$1$	$1$	$1$	$3.194$	\(\Q\)	None	✓				40.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}-3q^{9}-4q^{11}+2q^{13}-2q^{17}+\cdots\)
400.2.a.f	$400$	$2$	400.a	1.a	$1$	$1$	$1$	$3.194$	\(\Q\)	None	✓				50.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}-2q^{9}+3q^{11}+4q^{13}+\cdots\)
400.2.a.g	$400$	$2$	400.a	1.a	$1$	$1$	$1$	$3.194$	\(\Q\)	None	✓				40.2.c.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{7}+q^{9}+4q^{11}-4q^{13}+\cdots\)
400.2.a.h	$400$	$2$	400.a	1.a	$1$	$1$	$1$	$3.194$	\(\Q\)	None	✓				200.2.a.a	$1$	$0$	\(0\)	\(3\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-2q^{7}+6q^{9}-q^{11}+4q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(400))\) into lower level spaces

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