Space of modular forms of level 693, weight 6, and trivial character

• Label: 693.6.a
• Level: $693$
• Weight: $6$
• Character orbit: 693.a
• Rep. character: $\chi_{693}(1,\cdot)$
• Character field: $\Q$
• Dimension: $124$
• Newform subspaces: $18$
• Sturm bound: $576$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(576\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(693))\).

	Total	New	Old
Modular forms	488	124	364
Cusp forms	472	124	348
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
\(+\)	\(+\)	\(+\)	\(+\)	\(12\)
\(+\)	\(+\)	\(-\)	\(-\)	\(12\)
\(+\)	\(-\)	\(+\)	\(-\)	\(12\)
\(+\)	\(-\)	\(-\)	\(+\)	\(12\)
\(-\)	\(+\)	\(+\)	\(-\)	\(19\)
\(-\)	\(+\)	\(-\)	\(+\)	\(19\)
\(-\)	\(-\)	\(+\)	\(+\)	\(17\)
\(-\)	\(-\)	\(-\)	\(-\)	\(21\)
Plus space			\(+\)	\(60\)
Minus space			\(-\)	\(64\)

Trace form

\( 124 q + 8 q^{2} + 1992 q^{4} - 26 q^{5} - 468 q^{8} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(693))\) 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
693.6.a.a	$693$	$6$	693.a	1.a	$1$	$1$	$1$	$111.146$	\(\Q\)	None	✓				77.6.a.a	$1$	$0$	\(2\)	\(0\)	\(74\)	\(-49\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-28q^{4}+74q^{5}-7^{2}q^{7}-120q^{8}+\cdots\)
693.6.a.b	$693$	$6$	693.a	1.a	$1$	$1$	$1$	$111.146$	\(\Q\)	None	✓				231.6.a.a	$1$	$1$	\(2\)	\(0\)	\(76\)	\(49\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-28q^{4}+76q^{5}+7^{2}q^{7}-120q^{8}+\cdots\)
693.6.a.c	$693$	$6$	693.a	1.a	$1$	$4$	$4$	$111.146$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				231.6.a.b	$1$	$1$	\(-7\)	\(0\)	\(38\)	\(196\)		$-$	$-$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(24+\beta _{1}-\beta _{2}+\beta _{3})q^{4}+\cdots\)
693.6.a.d	$693$	$6$	693.a	1.a	$1$	$4$	$4$	$111.146$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				77.6.a.b	$1$	$1$	\(8\)	\(0\)	\(-57\)	\(196\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+(-2-3\beta _{1}+2\beta _{2})q^{4}+\cdots\)
693.6.a.e	$693$	$6$	693.a	1.a	$1$	$5$	$5$	$111.146$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				77.6.a.c	$1$	$0$	\(-10\)	\(0\)	\(-119\)	\(-245\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(23-3\beta _{1}+\beta _{2}-2\beta _{3}+\cdots)q^{4}+\cdots\)
693.6.a.f	$693$	$6$	693.a	1.a	$1$	$5$	$5$	$111.146$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				231.6.a.e	$1$	$1$	\(-5\)	\(0\)	\(2\)	\(-245\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(14-\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
693.6.a.g	$693$	$6$	693.a	1.a	$1$	$5$	$5$	$111.146$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				231.6.a.d	$1$	$0$	\(13\)	\(0\)	\(2\)	\(-245\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{2}+(15-4\beta _{1}+\beta _{3})q^{4}+\cdots\)
693.6.a.h	$693$	$6$	693.a	1.a	$1$	$5$	$5$	$111.146$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				231.6.a.c	$1$	$0$	\(13\)	\(0\)	\(114\)	\(245\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{2}+(15-3\beta _{1}+\beta _{2})q^{4}+\cdots\)
693.6.a.i	$693$	$6$	693.a	1.a	$1$	$6$	$6$	$111.146$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				77.6.a.d	$1$	$1$	\(-4\)	\(0\)	\(62\)	\(-294\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(12-\beta _{1}+\beta _{2})q^{4}+\cdots\)
693.6.a.j	$693$	$6$	693.a	1.a	$1$	$8$	$8$	$111.146$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		231.6.a.i	$1$	$0$	\(-5\)	\(0\)	\(2\)	\(-392\)		$-$	$+$	$+$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(21+\beta _{2})q^{4}+(1-2\beta _{1}+\cdots)q^{5}+\cdots\)
693.6.a.k	$693$	$6$	693.a	1.a	$1$	$8$	$8$	$111.146$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				77.6.a.e	$1$	$0$	\(-4\)	\(0\)	\(-50\)	\(392\)		$-$	$-$	$-$	$-$	$2^{5}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(21-\beta _{1}+\beta _{2})q^{4}+\cdots\)
693.6.a.l	$693$	$6$	693.a	1.a	$1$	$8$	$8$	$111.146$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		231.6.a.h	$1$	$1$	\(-3\)	\(0\)	\(-86\)	\(392\)		$-$	$-$	$+$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(20+\beta _{1}+\beta _{2})q^{4}+(-11+\cdots)q^{5}+\cdots\)
693.6.a.m	$693$	$6$	693.a	1.a	$1$	$8$	$8$	$111.146$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				231.6.a.g	$1$	$0$	\(3\)	\(0\)	\(14\)	\(392\)		$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(20+\beta _{1}+\beta _{2})q^{4}+(4\beta _{1}+\cdots)q^{5}+\cdots\)
693.6.a.n	$693$	$6$	693.a	1.a	$1$	$8$	$8$	$111.146$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		231.6.a.f	$1$	$1$	\(5\)	\(0\)	\(-98\)	\(-392\)		$-$	$+$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(21+\beta _{2})q^{4}+(-12+\cdots)q^{5}+\cdots\)
693.6.a.o	$693$	$6$	693.a	1.a	$1$	$12$	$12$	$111.146$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓	✓	693.6.a.o	$1$	$1$	\(-12\)	\(0\)	\(-50\)	\(588\)		$+$	$-$	$-$	$+$	$2^{6}\cdot 3^{5}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(2^{4}-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
693.6.a.p	$693$	$6$	693.a	1.a	$1$	$12$	$12$	$111.146$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓	✓	693.6.a.p	$1$	$1$	\(-4\)	\(0\)	\(-50\)	\(-588\)		$+$	$+$	$+$	$+$	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2^{4}+\beta _{2})q^{4}+(-4+\beta _{6}+\cdots)q^{5}+\cdots\)
693.6.a.q	$693$	$6$	693.a	1.a	$1$	$12$	$12$	$111.146$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓	✓	693.6.a.p	$1$	$0$	\(4\)	\(0\)	\(50\)	\(-588\)		$+$	$+$	$-$	$-$	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2^{4}+\beta _{2})q^{4}+(4-\beta _{6})q^{5}+\cdots\)
693.6.a.r	$693$	$6$	693.a	1.a	$1$	$12$	$12$	$111.146$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓	✓	693.6.a.o	$1$	$0$	\(12\)	\(0\)	\(50\)	\(588\)		$+$	$-$	$+$	$-$	$2^{6}\cdot 3^{5}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2^{4}-2\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(693))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(693)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(231))\)\(^{\oplus 2}\)