Space of modular forms of level 808, weight 2, and trivial character

• Label: 808.2.a
• Level: $808$
• Weight: $2$
• Character orbit: 808.a
• Rep. character: $\chi_{808}(1,\cdot)$
• Character field: $\Q$
• Dimension: $25$
• Newform subspaces: $6$
• Sturm bound: $204$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 808 = 2^{3} \cdot 101 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	808.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(204\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(808))\).

	Total	New	Old
Modular forms	106	25	81
Cusp forms	99	25	74
Eisenstein series	7	0	7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(101\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(25\)	\(4\)	\(21\)		\(24\)	\(4\)	\(20\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(28\)	\(9\)	\(19\)		\(26\)	\(9\)	\(17\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(28\)	\(5\)	\(23\)		\(26\)	\(5\)	\(21\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(25\)	\(7\)	\(18\)		\(23\)	\(7\)	\(16\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(50\)	\(11\)	\(39\)		\(47\)	\(11\)	\(36\)		\(3\)	\(0\)	\(3\)
Minus space		\(-\)		\(56\)	\(14\)	\(42\)		\(52\)	\(14\)	\(38\)		\(4\)	\(0\)	\(4\)

Trace form

25 * q - 4 * q^7 + 19 * q^9 - 4 * q^13 - 12 * q^15 - 6 * q^17 + 6 * q^19 - 4 * q^21 + 25 * q^25 + 6 * q^29 - 16 * q^31 + 12 * q^35 - 4 * q^37 - 16 * q^39 - 10 * q^41 + 6 * q^43 + 16 * q^45 + 8 * q^47 + 9 * q^49 - 6 * q^53 - 16 * q^55 + 16 * q^57 + 8 * q^59 + 10 * q^61 - 24 * q^63 + 12 * q^65 - 4 * q^67 + 8 * q^69 - 12 * q^71 + 6 * q^73 + 36 * q^77 - 28 * q^79 + q^81 + 16 * q^85 - 12 * q^87 + 14 * q^89 - 8 * q^91 + 32 * q^93 - 16 * q^95 - 2 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(808))\) 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	101
808.2.a.a	$808$	$2$	808.a	1.a	$1$	$1$	$1$	$6.452$	\(\Q\)	None	✓	✓			808.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-q^{7}-3q^{9}+4q^{13}+5q^{17}+\cdots\)
808.2.a.b	$808$	$2$	808.a	1.a	$1$	$1$	$1$	$6.452$	\(\Q\)	None	✓	✓			808.2.a.b	$1$	$0$	\(0\)	\(2\)	\(3\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+3q^{5}+2q^{7}+q^{9}-2q^{11}+\cdots\)
808.2.a.c	$808$	$2$	808.a	1.a	$1$	$3$	$3$	$6.452$	3.3.361.1	None	✓	✓	✓		808.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{1}-\beta _{2})q^{5}+(2-\beta _{2})q^{7}+\cdots\)
808.2.a.d	$808$	$2$	808.a	1.a	$1$	$4$	$4$	$6.452$	4.4.2777.1	None	✓	✓	✓	✓	808.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(-4\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{2})q^{5}+(\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
808.2.a.e	$808$	$2$	808.a	1.a	$1$	$7$	$7$	$6.452$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	808.2.a.e	$1$	$1$	\(0\)	\(-2\)	\(-3\)	\(-10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{1}-\beta _{4})q^{5}+(-1+\cdots)q^{7}+\cdots\)
808.2.a.f	$808$	$2$	808.a	1.a	$1$	$9$	$9$	$6.452$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓	✓	808.2.a.f	$1$	$0$	\(0\)	\(2\)	\(5\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{4})q^{5}+\beta _{8}q^{7}+(1+\beta _{5}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(808))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(808)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(101))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(202))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(404))\)\(^{\oplus 2}\)