Space of modular forms of level 800, weight 4, and trivial character

• Label: 800.4.a
• Level: $800$
• Weight: $4$
• Character orbit: 800.a
• Rep. character: $\chi_{800}(1,\cdot)$
• Character field: $\Q$
• Dimension: $57$
• Newform subspaces: $28$
• Sturm bound: $480$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(480\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(800))\).

	Total	New	Old
Modular forms	384	57	327
Cusp forms	336	57	279
Eisenstein series	48	0	48

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
\(+\)	\(+\)	\(+\)		\(99\)	\(15\)	\(84\)		\(87\)	\(15\)	\(72\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)		\(93\)	\(14\)	\(79\)		\(81\)	\(14\)	\(67\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(93\)	\(12\)	\(81\)		\(81\)	\(12\)	\(69\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)		\(99\)	\(16\)	\(83\)		\(87\)	\(16\)	\(71\)		\(12\)	\(0\)	\(12\)
Plus space		\(+\)		\(198\)	\(31\)	\(167\)		\(174\)	\(31\)	\(143\)		\(24\)	\(0\)	\(24\)
Minus space		\(-\)		\(186\)	\(26\)	\(160\)		\(162\)	\(26\)	\(136\)		\(24\)	\(0\)	\(24\)

Trace form

57 * q + 533 * q^9 - 26 * q^13 + 2 * q^17 - 16 * q^21 + 198 * q^29 - 592 * q^33 + 174 * q^37 - 710 * q^41 + 2737 * q^49 + 1086 * q^53 - 1056 * q^57 + 1670 * q^61 + 1312 * q^69 + 2250 * q^73 + 2224 * q^77 + 8689 * q^81 + 1802 * q^89 - 1648 * q^93 + 2114 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(800))\) 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
800.4.a.a	$800$	$4$	800.a	1.a	$1$	$1$	$1$	$47.202$	\(\Q\)	None	✓				32.4.a.a	$1$	$1$	\(0\)	\(-8\)	\(0\)	\(-16\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}-2^{4}q^{7}+37q^{9}-40q^{11}+\cdots\)
800.4.a.b	$800$	$4$	800.a	1.a	$1$	$1$	$1$	$47.202$	\(\Q\)	None	✓	✓			800.4.a.b	$1$	$1$	\(0\)	\(-5\)	\(0\)	\(10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}+10q^{7}-2q^{9}-15q^{11}+8q^{13}+\cdots\)
800.4.a.c	$800$	$4$	800.a	1.a	$1$	$1$	$1$	$47.202$	\(\Q\)	None	✓	✓			800.4.a.b	$1$	$1$	\(0\)	\(-5\)	\(0\)	\(10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}+10q^{7}-2q^{9}+15q^{11}-8q^{13}+\cdots\)
800.4.a.d	$800$	$4$	800.a	1.a	$1$	$1$	$1$	$47.202$	\(\Q\)	None	✓				160.4.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+6q^{7}-23q^{9}-60q^{11}-50q^{13}+\cdots\)
800.4.a.e	$800$	$4$	800.a	1.a	$1$	$1$	$1$	$47.202$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				160.4.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-3^{3}q^{9}-92q^{13}+104q^{17}+130q^{29}+\cdots\)
800.4.a.f	$800$	$4$	800.a	1.a	$1$	$1$	$1$	$47.202$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.4.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-3^{3}q^{9}+18q^{13}+94q^{17}-130q^{29}+\cdots\)
800.4.a.g	$800$	$4$	800.a	1.a	$1$	$1$	$1$	$47.202$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				160.4.c.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-3^{3}q^{9}+92q^{13}-104q^{17}+130q^{29}+\cdots\)
800.4.a.h	$800$	$4$	800.a	1.a	$1$	$1$	$1$	$47.202$	\(\Q\)	None	✓				160.4.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-6q^{7}-23q^{9}+60q^{11}-50q^{13}+\cdots\)
800.4.a.i	$800$	$4$	800.a	1.a	$1$	$1$	$1$	$47.202$	\(\Q\)	None	✓	✓			800.4.a.b	$1$	$1$	\(0\)	\(5\)	\(0\)	\(-10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-10q^{7}-2q^{9}-15q^{11}-8q^{13}+\cdots\)
800.4.a.j	$800$	$4$	800.a	1.a	$1$	$1$	$1$	$47.202$	\(\Q\)	None	✓	✓			800.4.a.b	$1$	$1$	\(0\)	\(5\)	\(0\)	\(-10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-10q^{7}-2q^{9}+15q^{11}+8q^{13}+\cdots\)
800.4.a.k	$800$	$4$	800.a	1.a	$1$	$1$	$1$	$47.202$	\(\Q\)	None	✓				32.4.a.a	$1$	$0$	\(0\)	\(8\)	\(0\)	\(16\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}+2^{4}q^{7}+37q^{9}+40q^{11}+\cdots\)
800.4.a.l	$800$	$4$	800.a	1.a	$1$	$2$	$2$	$47.202$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-5}) \)	✓				160.4.c.c	$2$	$1$	\(0\)	\(-14\)	\(0\)	\(-18\)		$+$	$-$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+(-7-\beta )q^{3}+(-9-11\beta )q^{7}+(3^{3}+\cdots)q^{9}+\cdots\)
800.4.a.m	$800$	$4$	800.a	1.a	$1$	$2$	$2$	$47.202$	\(\Q(\sqrt{6}) \)	None	✓				160.4.a.c	$1$	$0$	\(0\)	\(-8\)	\(0\)	\(-8\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{3}+(-4-5\beta )q^{7}+(13+\cdots)q^{9}+\cdots\)
800.4.a.n	$800$	$4$	800.a	1.a	$1$	$2$	$2$	$47.202$	\(\Q(\sqrt{29}) \)	None	✓				160.4.c.b	$2$	$0$	\(0\)	\(-8\)	\(0\)	\(8\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-4q^{3}+4q^{7}-11q^{9}+2\beta q^{11}+\beta q^{13}+\cdots\)
800.4.a.o	$800$	$4$	800.a	1.a	$1$	$2$	$2$	$47.202$	\(\Q(\sqrt{5}) \)	None	✓				160.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+7\beta q^{7}-7q^{9}-2\beta q^{11}+62q^{13}+\cdots\)
800.4.a.p	$800$	$4$	800.a	1.a	$1$	$2$	$2$	$47.202$	\(\Q(\sqrt{10}) \)	None	✓				160.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+3\beta q^{7}+13q^{9}+2\beta q^{11}+\cdots\)
800.4.a.q	$800$	$4$	800.a	1.a	$1$	$2$	$2$	$47.202$	\(\Q(\sqrt{13}) \)	None	✓				160.4.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-\beta q^{7}+5^{2}q^{9}+6\beta q^{11}-34q^{13}+\cdots\)
800.4.a.r	$800$	$4$	800.a	1.a	$1$	$2$	$2$	$47.202$	\(\Q(\sqrt{29}) \)	None	✓				160.4.c.b	$2$	$1$	\(0\)	\(8\)	\(0\)	\(-8\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+4q^{3}-4q^{7}-11q^{9}-2\beta q^{11}+\beta q^{13}+\cdots\)
800.4.a.s	$800$	$4$	800.a	1.a	$1$	$2$	$2$	$47.202$	\(\Q(\sqrt{6}) \)	None	✓				160.4.a.c	$1$	$1$	\(0\)	\(8\)	\(0\)	\(8\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{3}+(4-5\beta )q^{7}+(13+8\beta )q^{9}+\cdots\)
800.4.a.t	$800$	$4$	800.a	1.a	$1$	$2$	$2$	$47.202$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-5}) \)	✓				160.4.c.c	$2$	$0$	\(0\)	\(14\)	\(0\)	\(18\)		$-$	$-$	$+$	$2$	$N(\mathrm{U}(1))$	\(q+(7-\beta )q^{3}+(9-11\beta )q^{7}+(3^{3}-14\beta )q^{9}+\cdots\)
800.4.a.u	$800$	$4$	800.a	1.a	$1$	$3$	$3$	$47.202$	3.3.5685.1	None	✓	✓	✓		800.4.a.u	$1$	$1$	\(0\)	\(-5\)	\(0\)	\(2\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{3}+(1+\beta _{1}+\beta _{2})q^{7}+\cdots\)
800.4.a.v	$800$	$4$	800.a	1.a	$1$	$3$	$3$	$47.202$	3.3.5685.1	None	✓	✓			800.4.a.u	$1$	$0$	\(0\)	\(-5\)	\(0\)	\(2\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{3}+(1+\beta _{1}+\beta _{2})q^{7}+\cdots\)
800.4.a.w	$800$	$4$	800.a	1.a	$1$	$3$	$3$	$47.202$	3.3.5685.1	None	✓	✓			800.4.a.u	$1$	$1$	\(0\)	\(5\)	\(0\)	\(-2\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{3}+(-1-\beta _{1}-\beta _{2})q^{7}+\cdots\)
800.4.a.x	$800$	$4$	800.a	1.a	$1$	$3$	$3$	$47.202$	3.3.5685.1	None	✓	✓			800.4.a.u	$1$	$0$	\(0\)	\(5\)	\(0\)	\(-2\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{3}+(-1-\beta _{1}-\beta _{2})q^{7}+\cdots\)
800.4.a.y	$800$	$4$	800.a	1.a	$1$	$4$	$4$	$47.202$	4.4.37485.2	None	✓		✓		160.4.c.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{7}+(19-3\beta _{2}+\cdots)q^{9}+\cdots\)
800.4.a.z	$800$	$4$	800.a	1.a	$1$	$4$	$4$	$47.202$	4.4.37485.2	None	✓				160.4.c.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{7}+(19-3\beta _{2}+\cdots)q^{9}+\cdots\)
800.4.a.ba	$800$	$4$	800.a	1.a	$1$	$4$	$4$	$47.202$	4.4.2106005.1	None	✓	✓			800.4.a.ba	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(\beta _{1}-\beta _{3})q^{7}+(24+\beta _{2})q^{9}+\cdots\)
800.4.a.bb	$800$	$4$	800.a	1.a	$1$	$4$	$4$	$47.202$	4.4.2106005.1	None	✓	✓	✓		800.4.a.ba	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(\beta _{1}-\beta _{3})q^{7}+(24+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(800))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(800)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(200))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(400))\)\(^{\oplus 2}\)