Space of modular forms of level 3808, weight 2, and trivial character

• Label: 3808.2.a
• Level: $3808$
• Weight: $2$
• Character orbit: 3808.a
• Rep. character: $\chi_{3808}(1,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $18$
• Sturm bound: $1152$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3808.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(1152\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3808))\).

	Total	New	Old
Modular forms	592	96	496
Cusp forms	561	96	465
Eisenstein series	31	0	31

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

\(2\)	\(7\)	\(17\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(65\)	\(12\)	\(53\)		\(62\)	\(12\)	\(50\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(83\)	\(14\)	\(69\)		\(79\)	\(14\)	\(65\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(79\)	\(12\)	\(67\)		\(75\)	\(12\)	\(63\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(69\)	\(10\)	\(59\)		\(65\)	\(10\)	\(55\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(73\)	\(12\)	\(61\)		\(69\)	\(12\)	\(57\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(75\)	\(10\)	\(65\)		\(71\)	\(10\)	\(61\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(79\)	\(12\)	\(67\)		\(75\)	\(12\)	\(63\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(69\)	\(14\)	\(55\)		\(65\)	\(14\)	\(51\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(288\)	\(44\)	\(244\)		\(273\)	\(44\)	\(229\)		\(15\)	\(0\)	\(15\)
Minus space			\(-\)		\(304\)	\(52\)	\(252\)		\(288\)	\(52\)	\(236\)		\(16\)	\(0\)	\(16\)

Trace form

\( 96 q + 96 q^{9} + 96 q^{25} + 96 q^{45} + 96 q^{49} + 96 q^{53} - 32 q^{57} - 32 q^{65} + 96 q^{69} + 64 q^{81} - 32 q^{89} + 96 q^{93} - 32 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3808))\) 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	7	17
3808.2.a.a	$3808$	$2$	3808.a	1.a	$1$	$1$	$1$	$30.407$	\(\Q\)	None	✓	✓			3808.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{7}+q^{9}-6q^{11}+6q^{13}+\cdots\)
3808.2.a.b	$3808$	$2$	3808.a	1.a	$1$	$1$	$1$	$30.407$	\(\Q\)	None	✓	✓			3808.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{7}+q^{9}+6q^{11}+6q^{13}+\cdots\)
3808.2.a.c	$3808$	$2$	3808.a	1.a	$1$	$4$	$4$	$30.407$	\(\Q(\zeta_{20})^+\)	None	✓	✓			3808.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{1})q^{5}-q^{7}+\beta _{2}q^{9}+\cdots\)
3808.2.a.d	$3808$	$2$	3808.a	1.a	$1$	$4$	$4$	$30.407$	\(\Q(\zeta_{20})^+\)	None	✓	✓			3808.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1-\beta _{1})q^{5}+q^{7}+\beta _{2}q^{9}+\cdots\)
3808.2.a.e	$3808$	$2$	3808.a	1.a	$1$	$5$	$5$	$30.407$	5.5.804272.1	None	✓	✓			3808.2.a.e	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots\)
3808.2.a.f	$3808$	$2$	3808.a	1.a	$1$	$5$	$5$	$30.407$	5.5.804272.1	None	✓	✓			3808.2.a.e	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(5\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{1}-\beta _{2}+\beta _{3})q^{5}+\cdots\)
3808.2.a.g	$3808$	$2$	3808.a	1.a	$1$	$6$	$6$	$30.407$	6.6.147697840.1	None	✓	✓	✓		3808.2.a.g	$1$	$1$	\(0\)	\(-4\)	\(-4\)	\(6\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1-\beta _{5})q^{5}+q^{7}+\cdots\)
3808.2.a.h	$3808$	$2$	3808.a	1.a	$1$	$6$	$6$	$30.407$	6.6.109859312.1	None	✓	✓			3808.2.a.h	$1$	$1$	\(0\)	\(-4\)	\(4\)	\(-6\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{5})q^{3}+(1-\beta _{1})q^{5}-q^{7}+\cdots\)
3808.2.a.i	$3808$	$2$	3808.a	1.a	$1$	$6$	$6$	$30.407$	6.6.80686992.1	None	✓	✓	✓		3808.2.a.i	$1$	$1$	\(0\)	\(-2\)	\(-6\)	\(6\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{4})q^{5}+q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
3808.2.a.j	$3808$	$2$	3808.a	1.a	$1$	$6$	$6$	$30.407$	6.6.4022000.1	None	✓	✓	✓		3808.2.a.j	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(-6\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}-\beta _{1}q^{5}-q^{7}+(1-\beta _{5})q^{9}+\cdots\)
3808.2.a.k	$3808$	$2$	3808.a	1.a	$1$	$6$	$6$	$30.407$	6.6.93059344.1	None	✓	✓	✓		3808.2.a.k	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-6\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{5})q^{5}-q^{7}+(2-\beta _{2}+\cdots)q^{9}+\cdots\)
3808.2.a.l	$3808$	$2$	3808.a	1.a	$1$	$6$	$6$	$30.407$	6.6.93059344.1	None	✓	✓			3808.2.a.k	$1$	$0$	\(0\)	\(0\)	\(4\)	\(6\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{5})q^{5}+q^{7}+(2-\beta _{2}+\cdots)q^{9}+\cdots\)
3808.2.a.m	$3808$	$2$	3808.a	1.a	$1$	$6$	$6$	$30.407$	6.6.80686992.1	None	✓	✓			3808.2.a.i	$1$	$1$	\(0\)	\(2\)	\(-6\)	\(-6\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{4})q^{5}-q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
3808.2.a.n	$3808$	$2$	3808.a	1.a	$1$	$6$	$6$	$30.407$	6.6.4022000.1	None	✓	✓			3808.2.a.j	$1$	$0$	\(0\)	\(2\)	\(2\)	\(6\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}-\beta _{1}q^{5}+q^{7}+(1-\beta _{5})q^{9}+\cdots\)
3808.2.a.o	$3808$	$2$	3808.a	1.a	$1$	$6$	$6$	$30.407$	6.6.147697840.1	None	✓	✓			3808.2.a.g	$1$	$0$	\(0\)	\(4\)	\(-4\)	\(-6\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1-\beta _{5})q^{5}-q^{7}+\cdots\)
3808.2.a.p	$3808$	$2$	3808.a	1.a	$1$	$6$	$6$	$30.407$	6.6.109859312.1	None	✓	✓			3808.2.a.h	$1$	$0$	\(0\)	\(4\)	\(4\)	\(6\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{5})q^{3}+(1-\beta _{1})q^{5}+q^{7}+(2+\cdots)q^{9}+\cdots\)
3808.2.a.q	$3808$	$2$	3808.a	1.a	$1$	$8$	$8$	$30.407$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		3808.2.a.q	$1$	$0$	\(0\)	\(-2\)	\(6\)	\(-8\)		$+$	$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{3}+(1+\beta _{5})q^{5}-q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)
3808.2.a.r	$3808$	$2$	3808.a	1.a	$1$	$8$	$8$	$30.407$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		3808.2.a.q	$1$	$0$	\(0\)	\(2\)	\(6\)	\(8\)		$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{3}+(1+\beta _{5})q^{5}+q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3808))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3808)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(119))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(238))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(272))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(476))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(544))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(952))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1904))\)\(^{\oplus 2}\)