Space of modular forms of level 800, weight 6, and trivial character

• Label: 800.6.a
• Level: $800$
• Weight: $6$
• Character orbit: 800.a
• Rep. character: $\chi_{800}(1,\cdot)$
• Character field: $\Q$
• Dimension: $95$
• Newform subspaces: $28$
• Sturm bound: $720$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(720\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(800))\).

	Total	New	Old
Modular forms	624	95	529
Cusp forms	576	95	481
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
\(+\)	\(+\)	\(+\)		\(153\)	\(21\)	\(132\)		\(141\)	\(21\)	\(120\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)		\(159\)	\(26\)	\(133\)		\(147\)	\(26\)	\(121\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(159\)	\(24\)	\(135\)		\(147\)	\(24\)	\(123\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)		\(153\)	\(24\)	\(129\)		\(141\)	\(24\)	\(117\)		\(12\)	\(0\)	\(12\)
Plus space		\(+\)		\(306\)	\(45\)	\(261\)		\(282\)	\(45\)	\(237\)		\(24\)	\(0\)	\(24\)
Minus space		\(-\)		\(318\)	\(50\)	\(268\)		\(294\)	\(50\)	\(244\)		\(24\)	\(0\)	\(24\)

Trace form

95 * q + 7739 * q^9 + 354 * q^13 + 398 * q^17 - 2608 * q^21 - 3678 * q^29 - 5680 * q^33 + 11562 * q^37 - 4058 * q^41 + 211575 * q^49 + 41466 * q^53 - 50400 * q^57 + 16930 * q^61 + 100192 * q^69 - 92074 * q^73 - 45680 * q^77 + 343831 * q^81 - 146314 * q^89 + 515120 * q^93 - 278098 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(800))\) 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
800.6.a.a	$800$	$6$	800.a	1.a	$1$	$1$	$1$	$128.307$	\(\Q\)	None	✓				32.6.a.a	$1$	$1$	\(0\)	\(-8\)	\(0\)	\(-208\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}-208q^{7}-179q^{9}+536q^{11}+\cdots\)
800.6.a.b	$800$	$6$	800.a	1.a	$1$	$1$	$1$	$128.307$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				160.6.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-3^{5}q^{9}-244q^{13}-808q^{17}-2950q^{29}+\cdots\)
800.6.a.c	$800$	$6$	800.a	1.a	$1$	$1$	$1$	$128.307$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				160.6.c.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-3^{5}q^{9}+244q^{13}+808q^{17}-2950q^{29}+\cdots\)
800.6.a.d	$800$	$6$	800.a	1.a	$1$	$1$	$1$	$128.307$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.6.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-3^{5}q^{9}+1194q^{13}-2242q^{17}+\cdots\)
800.6.a.e	$800$	$6$	800.a	1.a	$1$	$1$	$1$	$128.307$	\(\Q\)	None	✓				32.6.a.a	$1$	$0$	\(0\)	\(8\)	\(0\)	\(208\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}+208q^{7}-179q^{9}-536q^{11}+\cdots\)
800.6.a.f	$800$	$6$	800.a	1.a	$1$	$2$	$2$	$128.307$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-5}) \)	✓				160.6.c.b	$2$	$1$	\(0\)	\(-38\)	\(0\)	\(366\)		$-$	$-$	$+$	$2\cdot 5$	$N(\mathrm{U}(1))$	\(q+(-19-\beta )q^{3}+(183+\beta )q^{7}+(3^{5}+\cdots)q^{9}+\cdots\)
800.6.a.g	$800$	$6$	800.a	1.a	$1$	$2$	$2$	$128.307$	\(\Q(\sqrt{70}) \)	None	✓				160.6.a.a	$1$	$0$	\(0\)	\(-8\)	\(0\)	\(-104\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{3}+(-52-\beta )q^{7}+(53+\cdots)q^{9}+\cdots\)
800.6.a.h	$800$	$6$	800.a	1.a	$1$	$2$	$2$	$128.307$	\(\Q(\sqrt{10}) \)	None	✓				160.6.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+7\beta q^{7}-203q^{9}+114\beta q^{11}+\cdots\)
800.6.a.i	$800$	$6$	800.a	1.a	$1$	$2$	$2$	$128.307$	\(\Q(\sqrt{5}) \)	None	✓				160.6.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3\beta q^{3}-31\beta q^{7}-63q^{9}+58\beta q^{11}+\cdots\)
800.6.a.j	$800$	$6$	800.a	1.a	$1$	$2$	$2$	$128.307$	\(\Q(\sqrt{85}) \)	None	✓				160.6.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+3\beta q^{7}+97q^{9}-26\beta q^{11}+\cdots\)
800.6.a.k	$800$	$6$	800.a	1.a	$1$	$2$	$2$	$128.307$	\(\Q(\sqrt{3}) \)	None	✓				32.6.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-6\beta q^{7}+525q^{9}-3\beta q^{11}+\cdots\)
800.6.a.l	$800$	$6$	800.a	1.a	$1$	$2$	$2$	$128.307$	\(\Q(\sqrt{70}) \)	None	✓				160.6.a.a	$1$	$1$	\(0\)	\(8\)	\(0\)	\(104\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{3}+(52-\beta )q^{7}+(53+8\beta )q^{9}+\cdots\)
800.6.a.m	$800$	$6$	800.a	1.a	$1$	$2$	$2$	$128.307$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-5}) \)	✓				160.6.c.b	$2$	$0$	\(0\)	\(38\)	\(0\)	\(-366\)		$+$	$-$	$-$	$2\cdot 5$	$N(\mathrm{U}(1))$	\(q+(19+\beta )q^{3}+(-183-\beta )q^{7}+(3^{5}+\cdots)q^{9}+\cdots\)
800.6.a.n	$800$	$6$	800.a	1.a	$1$	$3$	$3$	$128.307$	3.3.39180.1	None	✓				160.6.a.f	$1$	$1$	\(0\)	\(-10\)	\(0\)	\(6\)		$+$	$+$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{3}+(2+\beta _{1}-\beta _{2})q^{7}+\cdots\)
800.6.a.o	$800$	$6$	800.a	1.a	$1$	$3$	$3$	$128.307$	3.3.39180.1	None	✓				160.6.a.f	$1$	$0$	\(0\)	\(10\)	\(0\)	\(-6\)		$-$	$+$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{3}+(-2-\beta _{1}+\beta _{2})q^{7}+\cdots\)
800.6.a.p	$800$	$6$	800.a	1.a	$1$	$4$	$4$	$128.307$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			800.6.a.p	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{6}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-2\beta _{1}-\beta _{2})q^{7}+(56-\beta _{3})q^{9}+\cdots\)
800.6.a.q	$800$	$6$	800.a	1.a	$1$	$4$	$4$	$128.307$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			800.6.a.p	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{6}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-2\beta _{1}-\beta _{2})q^{7}+(56-\beta _{3})q^{9}+\cdots\)
800.6.a.r	$800$	$6$	800.a	1.a	$1$	$4$	$4$	$128.307$	4.4.81998080.1	None	✓				160.6.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{11}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-6\beta _{1}+\beta _{2})q^{7}+(181+\beta _{3})q^{9}+\cdots\)
800.6.a.s	$800$	$6$	800.a	1.a	$1$	$5$	$5$	$128.307$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			800.6.a.s	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-110\)		$-$	$-$	$+$	$2^{10}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-22+\beta _{2})q^{7}+(11^{2}-3\beta _{1}+\cdots)q^{9}+\cdots\)
800.6.a.t	$800$	$6$	800.a	1.a	$1$	$5$	$5$	$128.307$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			800.6.a.s	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-110\)		$-$	$+$	$-$	$2^{10}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-22+\beta _{2})q^{7}+(11^{2}-3\beta _{1}+\cdots)q^{9}+\cdots\)
800.6.a.u	$800$	$6$	800.a	1.a	$1$	$5$	$5$	$128.307$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		800.6.a.s	$1$	$1$	\(0\)	\(1\)	\(0\)	\(110\)		$+$	$+$	$+$	$2^{10}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(22-\beta _{2})q^{7}+(11^{2}-3\beta _{1}+\cdots)q^{9}+\cdots\)
800.6.a.v	$800$	$6$	800.a	1.a	$1$	$5$	$5$	$128.307$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			800.6.a.s	$1$	$0$	\(0\)	\(1\)	\(0\)	\(110\)		$+$	$-$	$-$	$2^{10}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(22-\beta _{2})q^{7}+(11^{2}-3\beta _{1}+\cdots)q^{9}+\cdots\)
800.6.a.w	$800$	$6$	800.a	1.a	$1$	$6$	$6$	$128.307$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				160.6.c.d	$2$	$0$	\(0\)	\(-20\)	\(0\)	\(452\)		$+$	$-$	$-$	$2^{15}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{3}+(77-5\beta _{1}+\beta _{4})q^{7}+\cdots\)
800.6.a.x	$800$	$6$	800.a	1.a	$1$	$6$	$6$	$128.307$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		800.6.a.x	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{15}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{7}+(38+\beta _{4})q^{9}+\cdots\)
800.6.a.y	$800$	$6$	800.a	1.a	$1$	$6$	$6$	$128.307$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			800.6.a.x	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{15}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{7}+(38+\beta _{4})q^{9}+\cdots\)
800.6.a.z	$800$	$6$	800.a	1.a	$1$	$6$	$6$	$128.307$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				160.6.c.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{13}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{3}q^{7}+(12^{2}-9\beta _{2}+4\beta _{4}+\cdots)q^{9}+\cdots\)
800.6.a.ba	$800$	$6$	800.a	1.a	$1$	$6$	$6$	$128.307$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				160.6.c.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{13}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{3}q^{7}+(12^{2}-9\beta _{2}+4\beta _{4}+\cdots)q^{9}+\cdots\)
800.6.a.bb	$800$	$6$	800.a	1.a	$1$	$6$	$6$	$128.307$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				160.6.c.d	$2$	$1$	\(0\)	\(20\)	\(0\)	\(-452\)		$-$	$-$	$+$	$2^{15}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{3}+(-77+5\beta _{1}-\beta _{4})q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(800))\) into lower level spaces

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