Space of modular forms of level 693, weight 2, and trivial character

• Label: 693.2.a
• Level: $693$
• Weight: $2$
• Character orbit: 693.a
• Rep. character: $\chi_{693}(1,\cdot)$
• Character field: $\Q$
• Dimension: $24$
• Newform subspaces: $13$
• Sturm bound: $192$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(192\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(693))\).

	Total	New	Old
Modular forms	104	24	80
Cusp forms	89	24	65
Eisenstein series	15	0	15

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
\(+\)	\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(6\)
Plus space			\(+\)	\(10\)
Minus space			\(-\)	\(14\)

Trace form

24 * q - 2 * q^2 + 30 * q^4 - 6 * q^5 + 6 * q^8 + 4 * q^10 + 4 * q^11 - 8 * q^13 + 4 * q^14 + 42 * q^16 - 20 * q^17 + 12 * q^19 + 16 * q^20 - 2 * q^22 + 10 * q^23 - 6 * q^25 - 4 * q^26 - 24 * q^29 + 6 * q^31 + 30 * q^32 - 20 * q^34 - 4 * q^35 - 22 * q^37 + 16 * q^38 - 12 * q^40 - 20 * q^41 + 10 * q^44 - 8 * q^46 + 16 * q^47 + 24 * q^49 + 66 * q^50 - 56 * q^52 + 16 * q^53 - 6 * q^55 + 12 * q^56 - 28 * q^58 + 22 * q^59 + 4 * q^61 - 4 * q^62 - 6 * q^64 - 16 * q^65 - 6 * q^67 - 24 * q^68 + 8 * q^70 + 6 * q^71 - 20 * q^73 - 28 * q^74 - 24 * q^76 + 4 * q^77 + 36 * q^79 - 20 * q^80 - 92 * q^82 - 24 * q^83 - 4 * q^85 - 24 * q^86 + 6 * q^88 - 42 * q^89 - 4 * q^92 + 12 * q^94 + 12 * q^95 - 50 * q^97 - 2 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(693))\) 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
693.2.a.a	$693$	$2$	693.a	1.a	$1$	$1$	$1$	$5.534$	\(\Q\)	None	✓				77.2.a.c	$1$	$0$	\(-1\)	\(0\)	\(2\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+2q^{5}-q^{7}+3q^{8}-2q^{10}+\cdots\)
693.2.a.b	$693$	$2$	693.a	1.a	$1$	$1$	$1$	$5.534$	\(\Q\)	None	✓				77.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-3q^{5}+q^{7}+q^{11}-4q^{13}+\cdots\)
693.2.a.c	$693$	$2$	693.a	1.a	$1$	$1$	$1$	$5.534$	\(\Q\)	None	✓				77.2.a.a	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{5}-q^{7}+q^{11}-4q^{13}+\cdots\)
693.2.a.d	$693$	$2$	693.a	1.a	$1$	$1$	$1$	$5.534$	\(\Q\)	None	✓				231.2.a.a	$1$	$0$	\(1\)	\(0\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+2q^{5}+q^{7}-3q^{8}+2q^{10}+\cdots\)
693.2.a.e	$693$	$2$	693.a	1.a	$1$	$2$	$2$	$5.534$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	693.2.a.e	$1$	$1$	\(-3\)	\(0\)	\(-2\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+3\beta q^{4}-q^{5}+q^{7}+\cdots\)
693.2.a.f	$693$	$2$	693.a	1.a	$1$	$2$	$2$	$5.534$	\(\Q(\sqrt{5}) \)	None	✓		✓	✓	231.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}-q^{5}+q^{7}+(-1+\cdots)q^{8}+\cdots\)
693.2.a.g	$693$	$2$	693.a	1.a	$1$	$2$	$2$	$5.534$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓	✓	693.2.a.g	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1+\beta )q^{4}-q^{5}-q^{7}-3q^{8}+\cdots\)
693.2.a.h	$693$	$2$	693.a	1.a	$1$	$2$	$2$	$5.534$	\(\Q(\sqrt{5}) \)	None	✓				77.2.a.d	$1$	$0$	\(0\)	\(0\)	\(4\)	\(2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+3q^{4}+2q^{5}+q^{7}-\beta q^{8}+\cdots\)
693.2.a.i	$693$	$2$	693.a	1.a	$1$	$2$	$2$	$5.534$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓	✓	693.2.a.g	$1$	$0$	\(1\)	\(0\)	\(2\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1+\beta )q^{4}+q^{5}-q^{7}+3q^{8}+\cdots\)
693.2.a.j	$693$	$2$	693.a	1.a	$1$	$2$	$2$	$5.534$	\(\Q(\sqrt{21}) \)	None	✓				231.2.a.b	$1$	$0$	\(1\)	\(0\)	\(-6\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(3+\beta )q^{4}-3q^{5}+q^{7}+(5+\cdots)q^{8}+\cdots\)
693.2.a.k	$693$	$2$	693.a	1.a	$1$	$2$	$2$	$5.534$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	693.2.a.e	$1$	$0$	\(3\)	\(0\)	\(2\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+3\beta q^{4}+q^{5}+q^{7}+(1+\cdots)q^{8}+\cdots\)
693.2.a.l	$693$	$2$	693.a	1.a	$1$	$3$	$3$	$5.534$	3.3.229.1	None	✓		✓		231.2.a.e	$1$	$1$	\(-2\)	\(0\)	\(-4\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+(2+\beta _{1})q^{4}+(-1+\cdots)q^{5}+\cdots\)
693.2.a.m	$693$	$2$	693.a	1.a	$1$	$3$	$3$	$5.534$	3.3.837.1	None	✓		✓		231.2.a.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{1}+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(693))\) into lower level spaces

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