Space of modular forms of level 714, weight 4, and trivial character

• Label: 714.4.a
• Level: $714$
• Weight: $4$
• Character orbit: 714.a
• Rep. character: $\chi_{714}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $19$
• Sturm bound: $576$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	714.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(576\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(714))\).

	Total	New	Old
Modular forms	440	48	392
Cusp forms	424	48	376
Eisenstein series	16	0	16

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

\(2\)	\(3\)	\(7\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(3\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(3\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(4\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(2\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(4\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(4\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(2\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(5\)
Plus space				\(+\)	\(30\)
Minus space				\(-\)	\(18\)

Trace form

\( 48 q + 192 q^{4} + 432 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(714))\) 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	3	7	17
714.4.a.a	$714$	$4$	714.a	1.a	$1$	$1$	$1$	$42.127$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(3\)	\(2\)	\(-7\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}+2q^{5}-6q^{6}+\cdots\)
714.4.a.b	$714$	$4$	714.a	1.a	$1$	$1$	$1$	$42.127$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(3\)	\(7\)	\(-7\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}+7q^{5}-6q^{6}+\cdots\)
714.4.a.c	$714$	$4$	714.a	1.a	$1$	$1$	$1$	$42.127$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(-3\)	\(-17\)	\(7\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-17q^{5}-6q^{6}+\cdots\)
714.4.a.d	$714$	$4$	714.a	1.a	$1$	$1$	$1$	$42.127$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(-3\)	\(-2\)	\(-7\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-2q^{5}-6q^{6}+\cdots\)
714.4.a.e	$714$	$4$	714.a	1.a	$1$	$1$	$1$	$42.127$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(-3\)	\(-2\)	\(7\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-2q^{5}-6q^{6}+\cdots\)
714.4.a.f	$714$	$4$	714.a	1.a	$1$	$1$	$1$	$42.127$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(3\)	\(-5\)	\(7\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}-5q^{5}+6q^{6}+\cdots\)
714.4.a.g	$714$	$4$	714.a	1.a	$1$	$2$	$2$	$42.127$	\(\Q(\sqrt{10}) \)	None	✓	$1$	$1$	\(-4\)	\(6\)	\(-2\)	\(14\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}+(-1+3\beta )q^{5}+\cdots\)
714.4.a.h	$714$	$4$	714.a	1.a	$1$	$2$	$2$	$42.127$	\(\Q(\sqrt{137}) \)	None	✓	$1$	$0$	\(4\)	\(-6\)	\(12\)	\(-14\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}+(7-2\beta )q^{5}+\cdots\)
714.4.a.i	$714$	$4$	714.a	1.a	$1$	$2$	$2$	$42.127$	\(\Q(\sqrt{22}) \)	None	✓	$1$	$1$	\(4\)	\(6\)	\(-18\)	\(-14\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+(-9+\beta )q^{5}+\cdots\)
714.4.a.j	$714$	$4$	714.a	1.a	$1$	$3$	$3$	$42.127$	3.3.356300.1	None	✓	$1$	$1$	\(-6\)	\(-9\)	\(-5\)	\(-21\)		$+$	$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+(-2-\beta _{2})q^{5}+\cdots\)
714.4.a.k	$714$	$4$	714.a	1.a	$1$	$3$	$3$	$42.127$	3.3.75201.1	None	✓	$1$	$0$	\(-6\)	\(-9\)	\(-4\)	\(21\)		$+$	$+$	$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+(-1+\beta _{2})q^{5}+\cdots\)
714.4.a.l	$714$	$4$	714.a	1.a	$1$	$3$	$3$	$42.127$	3.3.2140348.1	None	✓	$1$	$1$	\(-6\)	\(-9\)	\(-2\)	\(21\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+(-1+\beta _{1})q^{5}+\cdots\)
714.4.a.m	$714$	$4$	714.a	1.a	$1$	$3$	$3$	$42.127$	3.3.515564.1	None	✓	$1$	$0$	\(-6\)	\(-9\)	\(7\)	\(-21\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}+(2+\beta _{1})q^{5}+\cdots\)
714.4.a.n	$714$	$4$	714.a	1.a	$1$	$3$	$3$	$42.127$	3.3.80300.1	None	✓	$1$	$1$	\(6\)	\(-9\)	\(2\)	\(-21\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}+(1-\beta _{1}+\beta _{2})q^{5}+\cdots\)
714.4.a.o	$714$	$4$	714.a	1.a	$1$	$4$	$4$	$42.127$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-8\)	\(12\)	\(-3\)	\(-28\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}+(-1+\beta _{1})q^{5}+\cdots\)
714.4.a.p	$714$	$4$	714.a	1.a	$1$	$4$	$4$	$42.127$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-8\)	\(12\)	\(0\)	\(28\)		$+$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-\beta _{1}q^{5}-6q^{6}+\cdots\)
714.4.a.q	$714$	$4$	714.a	1.a	$1$	$4$	$4$	$42.127$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(8\)	\(-12\)	\(3\)	\(28\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}+(1-\beta _{1})q^{5}+\cdots\)
714.4.a.r	$714$	$4$	714.a	1.a	$1$	$4$	$4$	$42.127$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(8\)	\(12\)	\(14\)	\(-28\)		$-$	$-$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+(3-\beta _{3})q^{5}+\cdots\)
714.4.a.s	$714$	$4$	714.a	1.a	$1$	$5$	$5$	$42.127$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(10\)	\(15\)	\(13\)	\(35\)		$-$	$-$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+(3-\beta _{1})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(714))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(714)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(119))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(238))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(357))\)\(^{\oplus 2}\)