Space of modular forms of level 800, weight 2, and trivial character

• Label: 800.2.a
• Level: $800$
• Weight: $2$
• Character orbit: 800.a
• Rep. character: $\chi_{800}(1,\cdot)$
• Character field: $\Q$
• Dimension: $19$
• Newform subspaces: $14$
• Sturm bound: $240$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(240\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(800))\).

	Total	New	Old
Modular forms	144	19	125
Cusp forms	97	19	78
Eisenstein series	47	0	47

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
\(+\)	\(+\)	\(+\)		\(33\)	\(3\)	\(30\)		\(22\)	\(3\)	\(19\)		\(11\)	\(0\)	\(11\)
\(+\)	\(-\)	\(-\)		\(39\)	\(6\)	\(33\)		\(27\)	\(6\)	\(21\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(39\)	\(6\)	\(33\)		\(27\)	\(6\)	\(21\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)		\(33\)	\(4\)	\(29\)		\(21\)	\(4\)	\(17\)		\(12\)	\(0\)	\(12\)
Plus space		\(+\)		\(66\)	\(7\)	\(59\)		\(43\)	\(7\)	\(36\)		\(23\)	\(0\)	\(23\)
Minus space		\(-\)		\(78\)	\(12\)	\(66\)		\(54\)	\(12\)	\(42\)		\(24\)	\(0\)	\(24\)

Trace form

19 * q + 15 * q^9 + 10 * q^13 - 10 * q^17 + 16 * q^21 + 10 * q^29 + 16 * q^33 + 18 * q^37 + 14 * q^41 + 43 * q^49 - 30 * q^53 + 32 * q^57 + 10 * q^61 - 32 * q^69 - 2 * q^73 - 48 * q^77 + 11 * q^81 - 34 * q^89 - 16 * q^93 - 42 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(800))\) 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
800.2.a.a	$800$	$2$	800.a	1.a	$1$	$1$	$1$	$6.388$	\(\Q\)	None	✓				160.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots\)
800.2.a.b	$800$	$2$	800.a	1.a	$1$	$1$	$1$	$6.388$	\(\Q\)	None	✓	✓			800.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}-2q^{9}-5q^{11}+5q^{17}+\cdots\)
800.2.a.c	$800$	$2$	800.a	1.a	$1$	$1$	$1$	$6.388$	\(\Q\)	None	✓	✓			800.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{7}-2q^{9}+5q^{11}-5q^{17}+\cdots\)
800.2.a.d	$800$	$2$	800.a	1.a	$1$	$1$	$1$	$6.388$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				32.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}-6q^{13}-2q^{17}-10q^{29}+\cdots\)
800.2.a.e	$800$	$2$	800.a	1.a	$1$	$1$	$1$	$6.388$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				160.2.c.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}-4q^{13}-8q^{17}+10q^{29}+\cdots\)
800.2.a.f	$800$	$2$	800.a	1.a	$1$	$1$	$1$	$6.388$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				160.2.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}+4q^{13}+8q^{17}+10q^{29}+\cdots\)
800.2.a.g	$800$	$2$	800.a	1.a	$1$	$1$	$1$	$6.388$	\(\Q\)	None	✓	✓			800.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}-2q^{9}-5q^{11}-5q^{17}+\cdots\)
800.2.a.h	$800$	$2$	800.a	1.a	$1$	$1$	$1$	$6.388$	\(\Q\)	None	✓	✓			800.2.a.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{7}-2q^{9}+5q^{11}+5q^{17}+\cdots\)
800.2.a.i	$800$	$2$	800.a	1.a	$1$	$1$	$1$	$6.388$	\(\Q\)	None	✓				160.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
800.2.a.j	$800$	$2$	800.a	1.a	$1$	$2$	$2$	$6.388$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-5}) \)	✓		✓		160.2.c.b	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(-6\)		$-$	$-$	$+$	$2$	$N(\mathrm{U}(1))$	\(q+(-1-\beta )q^{3}+(-3+\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
800.2.a.k	$800$	$2$	800.a	1.a	$1$	$2$	$2$	$6.388$	\(\Q(\sqrt{5}) \)	None	✓	✓			800.2.a.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-2\beta q^{7}+2q^{9}+\beta q^{11}-4q^{13}+\cdots\)
800.2.a.l	$800$	$2$	800.a	1.a	$1$	$2$	$2$	$6.388$	\(\Q(\sqrt{5}) \)	None	✓	✓			800.2.a.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-2\beta q^{7}+2q^{9}-\beta q^{11}+4q^{13}+\cdots\)
800.2.a.m	$800$	$2$	800.a	1.a	$1$	$2$	$2$	$6.388$	\(\Q(\sqrt{2}) \)	None	✓				160.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{7}+5q^{9}+2\beta q^{11}+2q^{13}+\cdots\)
800.2.a.n	$800$	$2$	800.a	1.a	$1$	$2$	$2$	$6.388$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-5}) \)	✓				160.2.c.b	$2$	$0$	\(0\)	\(2\)	\(0\)	\(6\)		$+$	$-$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+(1+\beta )q^{3}+(3-\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(800))\) into lower level spaces

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