Space of modular forms of level 392, weight 8, and trivial character

• Label: 392.8.a
• Level: $392$
• Weight: $8$
• Character orbit: 392.a
• Rep. character: $\chi_{392}(1,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $15$
• Sturm bound: $448$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(448\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(392))\).

	Total	New	Old
Modular forms	408	72	336
Cusp forms	376	72	304
Eisenstein series	32	0	32

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

\(2\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(19\)
\(+\)	\(-\)	$-$	\(17\)
\(-\)	\(+\)	$-$	\(17\)
\(-\)	\(-\)	$+$	\(19\)
Plus space		\(+\)	\(38\)
Minus space		\(-\)	\(34\)

Trace form

\( 72 q + 14 q^{3} + 98 q^{5} + 54140 q^{9} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(392))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
392.8.a.a	$392$	$8$	392.a	1.a	$1$	$1$	$1$	$122.455$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-46\)	\(160\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-46q^{3}+160q^{5}-71q^{9}-6840q^{11}+\cdots\)
392.8.a.b	$392$	$8$	392.a	1.a	$1$	$1$	$1$	$122.455$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-44\)	\(-430\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-44q^{3}-430q^{5}-251q^{9}-3164q^{11}+\cdots\)
392.8.a.c	$392$	$8$	392.a	1.a	$1$	$1$	$1$	$122.455$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(18\)	\(-160\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+18q^{3}-160q^{5}-1863q^{9}+5704q^{11}+\cdots\)
392.8.a.d	$392$	$8$	392.a	1.a	$1$	$1$	$1$	$122.455$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(84\)	\(82\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+84q^{3}+82q^{5}+4869q^{9}-2524q^{11}+\cdots\)
392.8.a.e	$392$	$8$	392.a	1.a	$1$	$2$	$2$	$122.455$	\(\Q(\sqrt{249}) \)	None	✓	$1$	$1$	\(0\)	\(42\)	\(-14\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(21-3\beta )q^{3}+(-7+11\beta )q^{5}+(495+\cdots)q^{9}+\cdots\)
392.8.a.f	$392$	$8$	392.a	1.a	$1$	$3$	$3$	$122.455$	3.3.3109313.1	None	✓	$1$	$0$	\(0\)	\(-28\)	\(-138\)	\(0\)		$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-9+\beta _{1})q^{3}+(-46+\beta _{1}+\beta _{2})q^{5}+\cdots\)
392.8.a.g	$392$	$8$	392.a	1.a	$1$	$3$	$3$	$122.455$	3.3.294792.1	None	✓	$1$	$1$	\(0\)	\(-12\)	\(598\)	\(0\)		$+$	$-$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{1})q^{3}+(199+\beta _{1}+\beta _{2})q^{5}+\cdots\)
392.8.a.h	$392$	$8$	392.a	1.a	$1$	$4$	$4$	$122.455$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{10}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(437+\beta _{3})q^{9}+\cdots\)
392.8.a.i	$392$	$8$	392.a	1.a	$1$	$6$	$6$	$122.455$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{19}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-2\beta _{1}+\beta _{2})q^{5}+(1240+\cdots)q^{9}+\cdots\)
392.8.a.j	$392$	$8$	392.a	1.a	$1$	$7$	$7$	$122.455$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-29\)	\(237\)	\(0\)		$+$	$+$	$+$	$2^{18}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+(-4+\beta _{1})q^{3}+(34+\beta _{1}-\beta _{3})q^{5}+\cdots\)
392.8.a.k	$392$	$8$	392.a	1.a	$1$	$7$	$7$	$122.455$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-25\)	\(13\)	\(0\)		$-$	$+$	$-$	$2^{18}\cdot 3\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{1})q^{3}+(2+\beta _{1}-\beta _{2})q^{5}+\cdots\)
392.8.a.l	$392$	$8$	392.a	1.a	$1$	$7$	$7$	$122.455$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(25\)	\(-13\)	\(0\)		$-$	$-$	$+$	$2^{18}\cdot 3\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+(4+\beta _{1})q^{3}+(-2-\beta _{1}+\beta _{2})q^{5}+\cdots\)
392.8.a.m	$392$	$8$	392.a	1.a	$1$	$7$	$7$	$122.455$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(29\)	\(-237\)	\(0\)		$+$	$-$	$-$	$2^{18}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+(4-\beta _{1})q^{3}+(-34-\beta _{1}+\beta _{3})q^{5}+\cdots\)
392.8.a.n	$392$	$8$	392.a	1.a	$1$	$10$	$10$	$122.455$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{27}\cdot 7^{6}$	$\mathrm{SU}(2)$	\(q-\beta _{6}q^{3}+(-4\beta _{5}-\beta _{6}+\beta _{7})q^{5}+(413+\cdots)q^{9}+\cdots\)
392.8.a.o	$392$	$8$	392.a	1.a	$1$	$12$	$12$	$122.455$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{35}\cdot 7^{10}$	$\mathrm{SU}(2)$	\(q+\beta _{8}q^{3}-\beta _{7}q^{5}+(1236-\beta _{1})q^{9}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(392))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(392)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)