Space of modular forms of level 693, weight 4, and trivial character

• Label: 693.4.a
• Level: $693$
• Weight: $4$
• Character orbit: 693.a
• Rep. character: $\chi_{693}(1,\cdot)$
• Character field: $\Q$
• Dimension: $76$
• Newform subspaces: $21$
• Sturm bound: $384$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(384\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(693))\).

	Total	New	Old
Modular forms	296	76	220
Cusp forms	280	76	204
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.

\(3\)	\(7\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(8\)
\(+\)	\(+\)	\(-\)	\(-\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)	\(8\)
\(+\)	\(-\)	\(-\)	\(+\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)	\(11\)
\(-\)	\(+\)	\(-\)	\(+\)	\(11\)
\(-\)	\(-\)	\(+\)	\(+\)	\(13\)
\(-\)	\(-\)	\(-\)	\(-\)	\(9\)
Plus space			\(+\)	\(40\)
Minus space			\(-\)	\(36\)

Trace form

\( 76 q - 4 q^{2} + 288 q^{4} - 8 q^{5} - 36 q^{8} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(693))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	7	11
693.4.a.a	$693$	$4$	693.a	1.a	$1$	$1$	$1$	$40.888$	\(\Q\)	None	✓				231.4.a.e	$1$	$1$	\(-5\)	\(0\)	\(6\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}+17q^{4}+6q^{5}+7q^{7}-45q^{8}+\cdots\)
693.4.a.b	$693$	$4$	693.a	1.a	$1$	$1$	$1$	$40.888$	\(\Q\)	None	✓				77.4.a.a	$1$	$0$	\(-3\)	\(0\)	\(-12\)	\(7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}-12q^{5}+7q^{7}+21q^{8}+\cdots\)
693.4.a.c	$693$	$4$	693.a	1.a	$1$	$1$	$1$	$40.888$	\(\Q\)	None	✓				231.4.a.d	$1$	$0$	\(-3\)	\(0\)	\(14\)	\(-7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+q^{4}+14q^{5}-7q^{7}+21q^{8}+\cdots\)
693.4.a.d	$693$	$4$	693.a	1.a	$1$	$1$	$1$	$40.888$	\(\Q\)	None	✓				231.4.a.c	$1$	$1$	\(-2\)	\(0\)	\(-11\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-4q^{4}-11q^{5}-7q^{7}+24q^{8}+\cdots\)
693.4.a.e	$693$	$4$	693.a	1.a	$1$	$1$	$1$	$40.888$	\(\Q\)	None	✓				231.4.a.b	$1$	$0$	\(2\)	\(0\)	\(-1\)	\(-7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-4q^{4}-q^{5}-7q^{7}-24q^{8}+\cdots\)
693.4.a.f	$693$	$4$	693.a	1.a	$1$	$1$	$1$	$40.888$	\(\Q\)	None	✓				231.4.a.a	$1$	$1$	\(3\)	\(0\)	\(4\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}+4q^{5}-7q^{7}-21q^{8}+\cdots\)
693.4.a.g	$693$	$4$	693.a	1.a	$1$	$2$	$2$	$40.888$	\(\Q(\sqrt{17}) \)	None	✓				231.4.a.h	$1$	$1$	\(-3\)	\(0\)	\(-1\)	\(14\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(-3+3\beta )q^{4}+(-2+\cdots)q^{5}+\cdots\)
693.4.a.h	$693$	$4$	693.a	1.a	$1$	$2$	$2$	$40.888$	\(\Q(\sqrt{37}) \)	None	✓				231.4.a.i	$1$	$1$	\(-3\)	\(0\)	\(-2\)	\(14\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(2+3\beta )q^{4}+(1-4\beta )q^{5}+\cdots\)
693.4.a.i	$693$	$4$	693.a	1.a	$1$	$2$	$2$	$40.888$	\(\Q(\sqrt{2}) \)	None	✓				77.4.a.b	$1$	$1$	\(2\)	\(0\)	\(4\)	\(14\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+(2-3\beta )q^{5}+\cdots\)
693.4.a.j	$693$	$4$	693.a	1.a	$1$	$2$	$2$	$40.888$	\(\Q(\sqrt{17}) \)	None	✓				231.4.a.g	$1$	$1$	\(3\)	\(0\)	\(-25\)	\(14\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(-3+3\beta )q^{4}+(-14+\cdots)q^{5}+\cdots\)
693.4.a.k	$693$	$4$	693.a	1.a	$1$	$2$	$2$	$40.888$	\(\Q(\sqrt{17}) \)	None	✓				231.4.a.f	$1$	$0$	\(3\)	\(0\)	\(19\)	\(14\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(-3+3\beta )q^{4}+(8+3\beta )q^{5}+\cdots\)
693.4.a.l	$693$	$4$	693.a	1.a	$1$	$4$	$4$	$40.888$	4.4.522072.1	None	✓				77.4.a.d	$1$	$0$	\(2\)	\(0\)	\(-10\)	\(-28\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(6-\beta _{2}-2\beta _{3})q^{4}+(-3+\cdots)q^{5}+\cdots\)
693.4.a.m	$693$	$4$	693.a	1.a	$1$	$4$	$4$	$40.888$	4.4.509800.1	None	✓				77.4.a.c	$1$	$1$	\(4\)	\(0\)	\(18\)	\(-28\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{2}+(6-\beta _{1}-2\beta _{2}-\beta _{3})q^{4}+\cdots\)
693.4.a.n	$693$	$4$	693.a	1.a	$1$	$5$	$5$	$40.888$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		231.4.a.l	$1$	$1$	\(-5\)	\(0\)	\(-7\)	\(-35\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2})q^{4}+\cdots\)
693.4.a.o	$693$	$4$	693.a	1.a	$1$	$5$	$5$	$40.888$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				77.4.a.e	$1$	$0$	\(-1\)	\(0\)	\(24\)	\(35\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(9+\beta _{1}+\beta _{2})q^{4}+(4-\beta _{3}+\cdots)q^{5}+\cdots\)
693.4.a.p	$693$	$4$	693.a	1.a	$1$	$5$	$5$	$40.888$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				231.4.a.k	$1$	$0$	\(1\)	\(0\)	\(-21\)	\(35\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(4+\beta _{1}+\beta _{2})q^{4}+(-4+\beta _{4})q^{5}+\cdots\)
693.4.a.q	$693$	$4$	693.a	1.a	$1$	$5$	$5$	$40.888$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		231.4.a.j	$1$	$0$	\(1\)	\(0\)	\(-7\)	\(-35\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}+(-2+2\beta _{1}+\cdots)q^{5}+\cdots\)
693.4.a.r	$693$	$4$	693.a	1.a	$1$	$8$	$8$	$40.888$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	693.4.a.r	$1$	$1$	\(-6\)	\(0\)	\(-10\)	\(56\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2})q^{4}+\cdots\)
693.4.a.s	$693$	$4$	693.a	1.a	$1$	$8$	$8$	$40.888$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	693.4.a.s	$1$	$1$	\(-2\)	\(0\)	\(-10\)	\(-56\)		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(4+\beta _{2})q^{4}+(-1-\beta _{5})q^{5}+\cdots\)
693.4.a.t	$693$	$4$	693.a	1.a	$1$	$8$	$8$	$40.888$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	693.4.a.s	$1$	$0$	\(2\)	\(0\)	\(10\)	\(-56\)		$+$	$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(4+\beta _{2})q^{4}+(1+\beta _{5})q^{5}+\cdots\)
693.4.a.u	$693$	$4$	693.a	1.a	$1$	$8$	$8$	$40.888$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	693.4.a.r	$1$	$0$	\(6\)	\(0\)	\(10\)	\(56\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(4-\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(693))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(693)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(231))\)\(^{\oplus 2}\)