Space of modular forms of level 722, weight 2, and trivial character

• Label: 722.2.a
• Level: $722$
• Weight: $2$
• Character orbit: 722.a
• Rep. character: $\chi_{722}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $14$
• Sturm bound: $190$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 722 = 2 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	722.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(190\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(722))\).

	Total	New	Old
Modular forms	115	28	87
Cusp forms	76	28	48
Eisenstein series	39	0	39

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

\(2\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(6\)
\(+\)	\(-\)	\(-\)	\(8\)
\(-\)	\(+\)	\(-\)	\(10\)
\(-\)	\(-\)	\(+\)	\(4\)
Plus space		\(+\)	\(10\)
Minus space		\(-\)	\(18\)

Trace form

\( 28 q + 28 q^{4} + 2 q^{5} + 2 q^{6} - 4 q^{7} + 34 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(722))\) 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	19
722.2.a.a	$722$	$2$	722.a	1.a	$1$	$1$	$1$	$5.765$	\(\Q\)	None	✓	✓			722.2.a.a	$1$	$1$	\(-1\)	\(-3\)	\(2\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-3q^{3}+q^{4}+2q^{5}+3q^{6}-3q^{7}+\cdots\)
722.2.a.b	$722$	$2$	722.a	1.a	$1$	$1$	$1$	$5.765$	\(\Q\)	None	✓				38.2.a.b	$1$	$0$	\(-1\)	\(1\)	\(-4\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-4q^{5}-q^{6}+3q^{7}+\cdots\)
722.2.a.c	$722$	$2$	722.a	1.a	$1$	$1$	$1$	$5.765$	\(\Q\)	None	✓				38.2.c.a	$1$	$1$	\(-1\)	\(1\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}-4q^{7}-q^{8}+\cdots\)
722.2.a.d	$722$	$2$	722.a	1.a	$1$	$1$	$1$	$5.765$	\(\Q\)	None	✓				38.2.c.a	$1$	$1$	\(1\)	\(-1\)	\(0\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}-4q^{7}+q^{8}+\cdots\)
722.2.a.e	$722$	$2$	722.a	1.a	$1$	$1$	$1$	$5.765$	\(\Q\)	None	✓				38.2.a.a	$1$	$1$	\(1\)	\(-1\)	\(0\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}-q^{7}+q^{8}+\cdots\)
722.2.a.f	$722$	$2$	722.a	1.a	$1$	$1$	$1$	$5.765$	\(\Q\)	None	✓	✓			722.2.a.a	$1$	$0$	\(1\)	\(3\)	\(2\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}+q^{4}+2q^{5}+3q^{6}-3q^{7}+\cdots\)
722.2.a.g	$722$	$2$	722.a	1.a	$1$	$2$	$2$	$5.765$	\(\Q(\sqrt{7}) \)	None	✓				38.2.c.b	$1$	$0$	\(-2\)	\(0\)	\(2\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}+(1+\beta )q^{5}-\beta q^{6}+\cdots\)
722.2.a.h	$722$	$2$	722.a	1.a	$1$	$2$	$2$	$5.765$	\(\Q(\sqrt{5}) \)	None	✓	✓			722.2.a.h	$1$	$0$	\(-2\)	\(2\)	\(-5\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2\beta q^{3}+q^{4}+(-3+\beta )q^{5}+\cdots\)
722.2.a.i	$722$	$2$	722.a	1.a	$1$	$2$	$2$	$5.765$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		722.2.a.h	$1$	$1$	\(2\)	\(-2\)	\(-5\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2\beta q^{3}+q^{4}+(-3+\beta )q^{5}+\cdots\)
722.2.a.j	$722$	$2$	722.a	1.a	$1$	$2$	$2$	$5.765$	\(\Q(\sqrt{7}) \)	None	✓				38.2.c.b	$1$	$0$	\(2\)	\(0\)	\(2\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}+(1-\beta )q^{5}+\beta q^{6}+\cdots\)
722.2.a.k	$722$	$2$	722.a	1.a	$1$	$3$	$3$	$5.765$	\(\Q(\zeta_{18})^+\)	None	✓		✓		38.2.e.a	$1$	$0$	\(-3\)	\(0\)	\(6\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta _{1}q^{3}+q^{4}+2q^{5}-\beta _{1}q^{6}+\cdots\)
722.2.a.l	$722$	$2$	722.a	1.a	$1$	$3$	$3$	$5.765$	\(\Q(\zeta_{18})^+\)	None	✓				38.2.e.a	$1$	$0$	\(3\)	\(0\)	\(6\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta _{1}q^{3}+q^{4}+2q^{5}-\beta _{1}q^{6}+\cdots\)
722.2.a.m	$722$	$2$	722.a	1.a	$1$	$4$	$4$	$5.765$	\(\Q(\zeta_{20})^+\)	None	✓	✓	✓		722.2.a.m	$1$	$1$	\(-4\)	\(-2\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1-\beta _{2}-\beta _{3})q^{3}+q^{4}+(-1+\cdots)q^{5}+\cdots\)
722.2.a.n	$722$	$2$	722.a	1.a	$1$	$4$	$4$	$5.765$	\(\Q(\zeta_{20})^+\)	None	✓	✓	✓		722.2.a.m	$1$	$0$	\(4\)	\(2\)	\(-2\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta _{2}+\beta _{3})q^{3}+q^{4}+(-1+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(722))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(722)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(361))\)\(^{\oplus 2}\)