Space of modular forms of level 1694, weight 2, and trivial character

• Label: 1694.2.a
• Level: $1694$
• Weight: $2$
• Character orbit: 1694.a
• Rep. character: $\chi_{1694}(1,\cdot)$
• Character field: $\Q$
• Dimension: $54$
• Newform subspaces: $27$
• Sturm bound: $528$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1694 = 2 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1694.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 27 \)
Sturm bound:			\(528\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1694))\).

	Total	New	Old
Modular forms	288	54	234
Cusp forms	241	54	187
Eisenstein series	47	0	47

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

\(2\)	\(7\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(30\)	\(4\)	\(26\)		\(25\)	\(4\)	\(21\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(41\)	\(9\)	\(32\)		\(35\)	\(9\)	\(26\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(42\)	\(10\)	\(32\)		\(36\)	\(10\)	\(26\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(31\)	\(4\)	\(27\)		\(25\)	\(4\)	\(21\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(36\)	\(8\)	\(28\)		\(30\)	\(8\)	\(22\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(35\)	\(6\)	\(29\)		\(29\)	\(6\)	\(23\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(36\)	\(2\)	\(34\)		\(30\)	\(2\)	\(28\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(37\)	\(11\)	\(26\)		\(31\)	\(11\)	\(20\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(132\)	\(16\)	\(116\)		\(109\)	\(16\)	\(93\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(156\)	\(38\)	\(118\)		\(132\)	\(38\)	\(94\)		\(24\)	\(0\)	\(24\)

Trace form

54 * q + 6 * q^3 + 54 * q^4 + 2 * q^5 + 2 * q^6 + 62 * q^9 - 6 * q^10 + 6 * q^12 + 6 * q^13 - 2 * q^14 + 16 * q^15 + 54 * q^16 - 4 * q^18 + 10 * q^19 + 2 * q^20 + 6 * q^21 + 2 * q^24 + 66 * q^25 - 2 * q^26 + 36 * q^27 + 8 * q^29 + 16 * q^30 + 36 * q^31 + 4 * q^34 - 2 * q^35 + 62 * q^36 + 12 * q^37 + 14 * q^38 + 8 * q^39 - 6 * q^40 - 24 * q^41 - 2 * q^42 + 12 * q^43 - 14 * q^45 + 8 * q^46 + 4 * q^47 + 6 * q^48 + 54 * q^49 + 4 * q^50 - 12 * q^51 + 6 * q^52 - 8 * q^53 + 20 * q^54 - 2 * q^56 - 4 * q^57 + 2 * q^59 + 16 * q^60 + 10 * q^61 - 12 * q^62 - 12 * q^63 + 54 * q^64 + 4 * q^65 + 36 * q^67 + 24 * q^69 + 2 * q^70 + 16 * q^71 - 4 * q^72 - 4 * q^73 + 12 * q^74 + 42 * q^75 + 10 * q^76 + 16 * q^78 - 24 * q^79 + 2 * q^80 + 82 * q^81 - 4 * q^82 + 10 * q^83 + 6 * q^84 + 4 * q^85 + 16 * q^86 - 36 * q^87 + 12 * q^89 - 6 * q^90 + 6 * q^91 + 32 * q^93 - 4 * q^94 - 32 * q^95 + 2 * q^96 + 8 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1694))\) 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	7	11
1694.2.a.a	$1694$	$2$	1694.a	1.a	$1$	$1$	$1$	$13.527$	\(\Q\)	None	✓	✓			1694.2.a.a	$1$	$0$	\(-1\)	\(-1\)	\(2\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}-q^{7}+\cdots\)
1694.2.a.b	$1694$	$2$	1694.a	1.a	$1$	$1$	$1$	$13.527$	\(\Q\)	None	✓	✓			1694.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(-1\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
1694.2.a.c	$1694$	$2$	1694.a	1.a	$1$	$1$	$1$	$13.527$	\(\Q\)	None	✓				154.2.a.c	$1$	$1$	\(-1\)	\(0\)	\(2\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
1694.2.a.d	$1694$	$2$	1694.a	1.a	$1$	$1$	$1$	$13.527$	\(\Q\)	None	✓	✓			1694.2.a.d	$1$	$0$	\(-1\)	\(3\)	\(2\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+3q^{3}+q^{4}+2q^{5}-3q^{6}-q^{7}+\cdots\)
1694.2.a.e	$1694$	$2$	1694.a	1.a	$1$	$1$	$1$	$13.527$	\(\Q\)	None	✓				14.2.a.a	$1$	$1$	\(1\)	\(-2\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}+q^{4}-2q^{6}-q^{7}+q^{8}+\cdots\)
1694.2.a.f	$1694$	$2$	1694.a	1.a	$1$	$1$	$1$	$13.527$	\(\Q\)	None	✓	✓			1694.2.a.a	$1$	$0$	\(1\)	\(-1\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+q^{7}+\cdots\)
1694.2.a.g	$1694$	$2$	1694.a	1.a	$1$	$1$	$1$	$13.527$	\(\Q\)	None	✓				154.2.a.a	$1$	$0$	\(1\)	\(0\)	\(-4\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-4q^{5}+q^{7}+q^{8}-3q^{9}+\cdots\)
1694.2.a.h	$1694$	$2$	1694.a	1.a	$1$	$1$	$1$	$13.527$	\(\Q\)	None	✓	✓			1694.2.a.b	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-q^{7}+q^{8}-3q^{9}+\cdots\)
1694.2.a.i	$1694$	$2$	1694.a	1.a	$1$	$1$	$1$	$13.527$	\(\Q\)	None	✓				154.2.a.b	$1$	$0$	\(1\)	\(2\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}+2q^{5}+2q^{6}+q^{7}+\cdots\)
1694.2.a.j	$1694$	$2$	1694.a	1.a	$1$	$1$	$1$	$13.527$	\(\Q\)	None	✓	✓			1694.2.a.d	$1$	$0$	\(1\)	\(3\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}+q^{4}+2q^{5}+3q^{6}+q^{7}+\cdots\)
1694.2.a.k	$1694$	$2$	1694.a	1.a	$1$	$2$	$2$	$13.527$	\(\Q(\sqrt{5}) \)	None	✓				154.2.f.a	$1$	$0$	\(-2\)	\(-2\)	\(-3\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}+q^{6}+\cdots\)
1694.2.a.l	$1694$	$2$	1694.a	1.a	$1$	$2$	$2$	$13.527$	\(\Q(\sqrt{5}) \)	None	✓				154.2.a.d	$1$	$0$	\(-2\)	\(-2\)	\(2\)	\(-2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1-\beta )q^{3}+q^{4}+(1+\beta )q^{5}+\cdots\)
1694.2.a.m	$1694$	$2$	1694.a	1.a	$1$	$2$	$2$	$13.527$	\(\Q(\sqrt{5}) \)	None	✓				154.2.f.c	$1$	$1$	\(-2\)	\(-1\)	\(4\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta q^{3}+q^{4}+2q^{5}+\beta q^{6}-q^{7}+\cdots\)
1694.2.a.n	$1694$	$2$	1694.a	1.a	$1$	$2$	$2$	$13.527$	\(\Q(\sqrt{5}) \)	None	✓				154.2.f.d	$1$	$1$	\(-2\)	\(0\)	\(-3\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1-2\beta )q^{3}+q^{4}+(-1-\beta )q^{5}+\cdots\)
1694.2.a.o	$1694$	$2$	1694.a	1.a	$1$	$2$	$2$	$13.527$	\(\Q(\sqrt{5}) \)	None	✓		✓		154.2.f.b	$1$	$1$	\(-2\)	\(0\)	\(-1\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1-2\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
1694.2.a.p	$1694$	$2$	1694.a	1.a	$1$	$2$	$2$	$13.527$	\(\Q(\sqrt{3}) \)	None	✓	✓			1694.2.a.p	$1$	$0$	\(-2\)	\(2\)	\(2\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{5}+\cdots\)
1694.2.a.q	$1694$	$2$	1694.a	1.a	$1$	$2$	$2$	$13.527$	\(\Q(\sqrt{5}) \)	None	✓		✓	✓	154.2.f.a	$1$	$1$	\(2\)	\(-2\)	\(-3\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+(-1-\beta )q^{5}-q^{6}+\cdots\)
1694.2.a.r	$1694$	$2$	1694.a	1.a	$1$	$2$	$2$	$13.527$	\(\Q(\sqrt{5}) \)	None	✓				154.2.f.c	$1$	$0$	\(2\)	\(-1\)	\(4\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta q^{3}+q^{4}+2q^{5}-\beta q^{6}+q^{7}+\cdots\)
1694.2.a.s	$1694$	$2$	1694.a	1.a	$1$	$2$	$2$	$13.527$	\(\Q(\sqrt{5}) \)	None	✓				154.2.f.d	$1$	$0$	\(2\)	\(0\)	\(-3\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-2\beta )q^{3}+q^{4}+(-1-\beta )q^{5}+\cdots\)
1694.2.a.t	$1694$	$2$	1694.a	1.a	$1$	$2$	$2$	$13.527$	\(\Q(\sqrt{5}) \)	None	✓				154.2.f.b	$1$	$0$	\(2\)	\(0\)	\(-1\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-2\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots\)
1694.2.a.u	$1694$	$2$	1694.a	1.a	$1$	$2$	$2$	$13.527$	\(\Q(\sqrt{3}) \)	None	✓	✓			1694.2.a.p	$1$	$0$	\(2\)	\(2\)	\(2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{5}+\cdots\)
1694.2.a.v	$1694$	$2$	1694.a	1.a	$1$	$3$	$3$	$13.527$	3.3.2024.1	None	✓	✓	✓		1694.2.a.v	$1$	$0$	\(-3\)	\(2\)	\(-1\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1-\beta _{1})q^{3}+q^{4}-\beta _{2}q^{5}+(-1+\cdots)q^{6}+\cdots\)
1694.2.a.w	$1694$	$2$	1694.a	1.a	$1$	$3$	$3$	$13.527$	3.3.2024.1	None	✓	✓	✓		1694.2.a.v	$1$	$0$	\(3\)	\(2\)	\(-1\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}-\beta _{2}q^{5}+(1+\cdots)q^{6}+\cdots\)
1694.2.a.x	$1694$	$2$	1694.a	1.a	$1$	$4$	$4$	$13.527$	\(\Q(\sqrt{70 +2 \sqrt{5}})\)	None	✓				154.2.f.e	$1$	$0$	\(-4\)	\(-3\)	\(-5\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1694.2.a.y	$1694$	$2$	1694.a	1.a	$1$	$4$	$4$	$13.527$	\(\Q(\sqrt{3 + \sqrt{3}})\)	None	✓	✓			1694.2.a.y	$1$	$0$	\(-4\)	\(4\)	\(4\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta _{1})q^{3}+q^{4}+(1+\beta _{2}-\beta _{3})q^{5}+\cdots\)
1694.2.a.z	$1694$	$2$	1694.a	1.a	$1$	$4$	$4$	$13.527$	\(\Q(\sqrt{70 +2 \sqrt{5}})\)	None	✓		✓		154.2.f.e	$1$	$1$	\(4\)	\(-3\)	\(-5\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1694.2.a.ba	$1694$	$2$	1694.a	1.a	$1$	$4$	$4$	$13.527$	\(\Q(\sqrt{3 + \sqrt{3}})\)	None	✓	✓	✓		1694.2.a.y	$1$	$0$	\(4\)	\(4\)	\(4\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta _{1})q^{3}+q^{4}+(1+\beta _{2}-\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1694))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1694)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(847))\)\(^{\oplus 2}\)