Space of modular forms of level 392, weight 4, and trivial character

• Label: 392.4.a
• Level: $392$
• Weight: $4$
• Character orbit: 392.a
• Rep. character: $\chi_{392}(1,\cdot)$
• Character field: $\Q$
• Dimension: $31$
• Newform subspaces: $14$
• Sturm bound: $224$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	392.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(224\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(392))\).

	Total	New	Old
Modular forms	184	31	153
Cusp forms	152	31	121
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
\(+\)	\(+\)	$+$	\(9\)
\(+\)	\(-\)	$-$	\(7\)
\(-\)	\(+\)	$-$	\(7\)
\(-\)	\(-\)	$+$	\(8\)
Plus space		\(+\)	\(17\)
Minus space		\(-\)	\(14\)

Trace form

\( 31 q + 2 q^{3} - 12 q^{5} + 287 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(392))\) 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
392.4.a.a	$392$	$4$	392.a	1.a	$1$	$1$	$1$	$23.129$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(-8\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{3}-8q^{5}+9q^{9}+56q^{11}+28q^{13}+\cdots\)
392.4.a.b	$392$	$4$	392.a	1.a	$1$	$1$	$1$	$23.129$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-4\)	\(12\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+12q^{5}-11q^{9}+12q^{11}+\cdots\)
392.4.a.c	$392$	$4$	392.a	1.a	$1$	$1$	$1$	$23.129$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(16\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2^{4}q^{5}-23q^{9}+24q^{11}+\cdots\)
392.4.a.d	$392$	$4$	392.a	1.a	$1$	$1$	$1$	$23.129$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(4\)	\(-12\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}-12q^{5}-11q^{9}+12q^{11}+\cdots\)
392.4.a.e	$392$	$4$	392.a	1.a	$1$	$1$	$1$	$23.129$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(4\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}+2q^{5}-11q^{9}-44q^{11}-22q^{13}+\cdots\)
392.4.a.f	$392$	$4$	392.a	1.a	$1$	$2$	$2$	$23.129$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-10\beta q^{5}-5^{2}q^{9}+54q^{11}+\cdots\)
392.4.a.g	$392$	$4$	392.a	1.a	$1$	$2$	$2$	$23.129$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{5}+5q^{9}-4q^{11}-\beta q^{13}+\cdots\)
392.4.a.h	$392$	$4$	392.a	1.a	$1$	$2$	$2$	$23.129$	\(\Q(\sqrt{57}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-22\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-11-\beta )q^{5}+(31+2\beta )q^{9}+\cdots\)
392.4.a.i	$392$	$4$	392.a	1.a	$1$	$3$	$3$	$23.129$	3.3.1929.1	None	✓	$1$	$1$	\(0\)	\(-7\)	\(-3\)	\(0\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+(-1+\beta _{1}-\beta _{2})q^{5}+\cdots\)
392.4.a.j	$392$	$4$	392.a	1.a	$1$	$3$	$3$	$23.129$	3.3.1929.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-13\)	\(0\)		$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-4-\beta _{1})q^{5}+(15+3\beta _{1}+\cdots)q^{9}+\cdots\)
392.4.a.k	$392$	$4$	392.a	1.a	$1$	$3$	$3$	$23.129$	3.3.1929.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(13\)	\(0\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(4+\beta _{1})q^{5}+(15+3\beta _{1}-2\beta _{2})q^{9}+\cdots\)
392.4.a.l	$392$	$4$	392.a	1.a	$1$	$3$	$3$	$23.129$	3.3.1929.1	None	✓	$1$	$0$	\(0\)	\(7\)	\(3\)	\(0\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+(4+\cdots)q^{9}+\cdots\)
392.4.a.m	$392$	$4$	392.a	1.a	$1$	$4$	$4$	$23.129$	\(\Q(\sqrt{2}, \sqrt{65})\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(2\beta _{1}+\beta _{3})q^{3}+(-3\beta _{1}-\beta _{3})q^{5}+\cdots\)
392.4.a.n	$392$	$4$	392.a	1.a	$1$	$4$	$4$	$23.129$	\(\Q(\sqrt{2}, \sqrt{113})\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2\beta _{1}+\beta _{3})q^{3}+(-9\beta _{1}+\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(392))\) into lower level spaces

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