Space of modular forms of level 1682, weight 2, and trivial character

• Label: 1682.2.a
• Level: $1682$
• Weight: $2$
• Character orbit: 1682.a
• Rep. character: $\chi_{1682}(1,\cdot)$
• Character field: $\Q$
• Dimension: $68$
• Newform subspaces: $22$
• Sturm bound: $435$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1682 = 2 \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1682.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(435\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1682))\).

	Total	New	Old
Modular forms	247	68	179
Cusp forms	188	68	120
Eisenstein series	59	0	59

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

\(2\)	\(29\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(56\)	\(15\)	\(41\)		\(42\)	\(15\)	\(27\)		\(14\)	\(0\)	\(14\)
\(+\)	\(-\)	\(-\)		\(67\)	\(19\)	\(48\)		\(52\)	\(19\)	\(33\)		\(15\)	\(0\)	\(15\)
\(-\)	\(+\)	\(-\)		\(64\)	\(22\)	\(42\)		\(49\)	\(22\)	\(27\)		\(15\)	\(0\)	\(15\)
\(-\)	\(-\)	\(+\)		\(60\)	\(12\)	\(48\)		\(45\)	\(12\)	\(33\)		\(15\)	\(0\)	\(15\)
Plus space		\(+\)		\(116\)	\(27\)	\(89\)		\(87\)	\(27\)	\(60\)		\(29\)	\(0\)	\(29\)
Minus space		\(-\)		\(131\)	\(41\)	\(90\)		\(101\)	\(41\)	\(60\)		\(30\)	\(0\)	\(30\)

Trace form

68 * q + 4 * q^3 + 68 * q^4 + 2 * q^5 - 2 * q^6 + 4 * q^7 + 66 * q^9 - 4 * q^10 + 4 * q^11 + 4 * q^12 - 2 * q^13 - 8 * q^15 + 68 * q^16 - 4 * q^17 + 8 * q^18 + 8 * q^19 + 2 * q^20 - 8 * q^21 + 2 * q^22 - 4 * q^23 - 2 * q^24 + 70 * q^25 + 4 * q^26 + 4 * q^27 + 4 * q^28 + 4 * q^30 - 12 * q^33 - 12 * q^34 - 4 * q^35 + 66 * q^36 - 14 * q^38 + 8 * q^39 - 4 * q^40 + 4 * q^42 + 4 * q^43 + 4 * q^44 + 14 * q^45 - 4 * q^46 - 24 * q^47 + 4 * q^48 + 76 * q^49 + 8 * q^50 - 12 * q^51 - 2 * q^52 + 2 * q^53 - 20 * q^54 - 32 * q^57 + 2 * q^59 - 8 * q^60 + 4 * q^61 + 4 * q^62 + 68 * q^64 + 4 * q^65 + 14 * q^67 - 4 * q^68 + 4 * q^69 + 8 * q^70 - 8 * q^71 + 8 * q^72 + 8 * q^73 - 18 * q^74 + 8 * q^75 + 8 * q^76 - 8 * q^77 - 16 * q^78 - 8 * q^79 + 2 * q^80 + 52 * q^81 - 4 * q^82 - 6 * q^83 - 8 * q^84 - 20 * q^85 + 18 * q^86 + 2 * q^88 + 16 * q^89 - 16 * q^90 + 4 * q^91 - 4 * q^92 - 8 * q^93 - 4 * q^94 - 24 * q^95 - 2 * q^96 + 8 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1682))\) 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	29
1682.2.a.a	$1682$	$2$	1682.a	1.a	$1$	$1$	$1$	$13.431$	\(\Q\)	None	✓	✓			1682.2.a.a	$1$	$1$	\(-1\)	\(-2\)	\(-2\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}+q^{4}-2q^{5}+2q^{6}-5q^{7}+\cdots\)
1682.2.a.b	$1682$	$2$	1682.a	1.a	$1$	$1$	$1$	$13.431$	\(\Q\)	None	✓	✓			1682.2.a.b	$1$	$0$	\(-1\)	\(-2\)	\(0\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{7}-q^{8}+\cdots\)
1682.2.a.c	$1682$	$2$	1682.a	1.a	$1$	$1$	$1$	$13.431$	\(\Q\)	None	✓				58.2.b.a	$1$	$0$	\(-1\)	\(-1\)	\(-1\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-2q^{7}+\cdots\)
1682.2.a.d	$1682$	$2$	1682.a	1.a	$1$	$1$	$1$	$13.431$	\(\Q\)	None	✓				58.2.a.b	$1$	$1$	\(-1\)	\(1\)	\(1\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-2q^{7}+\cdots\)
1682.2.a.e	$1682$	$2$	1682.a	1.a	$1$	$1$	$1$	$13.431$	\(\Q\)	None	✓	✓			1682.2.a.e	$1$	$0$	\(-1\)	\(3\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+3q^{3}+q^{4}-3q^{6}+4q^{7}-q^{8}+\cdots\)
1682.2.a.f	$1682$	$2$	1682.a	1.a	$1$	$1$	$1$	$13.431$	\(\Q\)	None	✓	✓			1682.2.a.e	$1$	$0$	\(1\)	\(-3\)	\(0\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}+q^{4}-3q^{6}+4q^{7}+q^{8}+\cdots\)
1682.2.a.g	$1682$	$2$	1682.a	1.a	$1$	$1$	$1$	$13.431$	\(\Q\)	None	✓				58.2.b.a	$1$	$1$	\(1\)	\(1\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-2q^{7}+\cdots\)
1682.2.a.h	$1682$	$2$	1682.a	1.a	$1$	$1$	$1$	$13.431$	\(\Q\)	None	✓	✓			1682.2.a.a	$1$	$1$	\(1\)	\(2\)	\(-2\)	\(-5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}-2q^{5}+2q^{6}-5q^{7}+\cdots\)
1682.2.a.i	$1682$	$2$	1682.a	1.a	$1$	$1$	$1$	$13.431$	\(\Q\)	None	✓	✓			1682.2.a.b	$1$	$0$	\(1\)	\(2\)	\(0\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}+2q^{6}-q^{7}+q^{8}+\cdots\)
1682.2.a.j	$1682$	$2$	1682.a	1.a	$1$	$1$	$1$	$13.431$	\(\Q\)	None	✓				58.2.a.a	$1$	$0$	\(1\)	\(3\)	\(-3\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}+q^{4}-3q^{5}+3q^{6}-2q^{7}+\cdots\)
1682.2.a.k	$1682$	$2$	1682.a	1.a	$1$	$2$	$2$	$13.431$	\(\Q(\sqrt{5}) \)	None	✓	✓			1682.2.a.k	$1$	$0$	\(-2\)	\(-1\)	\(4\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta q^{3}+q^{4}+2q^{5}+\beta q^{6}+(4+\cdots)q^{7}+\cdots\)
1682.2.a.l	$1682$	$2$	1682.a	1.a	$1$	$2$	$2$	$13.431$	\(\Q(\sqrt{5}) \)	None	✓	✓			1682.2.a.k	$1$	$0$	\(2\)	\(1\)	\(4\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}+2q^{5}+\beta q^{6}+(4+\cdots)q^{7}+\cdots\)
1682.2.a.m	$1682$	$2$	1682.a	1.a	$1$	$3$	$3$	$13.431$	\(\Q(\zeta_{14})^+\)	None	✓				58.2.d.a	$1$	$1$	\(-3\)	\(-2\)	\(5\)	\(1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1-\beta _{2})q^{3}+q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
1682.2.a.n	$1682$	$2$	1682.a	1.a	$1$	$3$	$3$	$13.431$	\(\Q(\zeta_{14})^+\)	None	✓				58.2.d.a	$1$	$0$	\(3\)	\(2\)	\(5\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+(2+\beta _{2})q^{5}+\cdots\)
1682.2.a.o	$1682$	$2$	1682.a	1.a	$1$	$4$	$4$	$13.431$	\(\Q(\zeta_{15})^+\)	None	✓	✓			1682.2.a.o	$1$	$1$	\(-4\)	\(2\)	\(-3\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+q^{4}+(-1+\cdots)q^{5}+\cdots\)
1682.2.a.p	$1682$	$2$	1682.a	1.a	$1$	$4$	$4$	$13.431$	\(\Q(\zeta_{15})^+\)	None	✓	✓			1682.2.a.o	$1$	$1$	\(4\)	\(-2\)	\(-3\)	\(-5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+q^{4}+(-1+\cdots)q^{5}+\cdots\)
1682.2.a.q	$1682$	$2$	1682.a	1.a	$1$	$6$	$6$	$13.431$	6.6.13716913.1	None	✓		✓		58.2.d.b	$1$	$1$	\(-6\)	\(-2\)	\(0\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1682.2.a.r	$1682$	$2$	1682.a	1.a	$1$	$6$	$6$	$13.431$	\(\Q(\zeta_{28})^+\)	None	✓				58.2.e.a	$1$	$0$	\(-6\)	\(8\)	\(-6\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta _{4})q^{3}+q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1682.2.a.s	$1682$	$2$	1682.a	1.a	$1$	$6$	$6$	$13.431$	\(\Q(\zeta_{28})^+\)	None	✓		✓		58.2.e.a	$1$	$1$	\(6\)	\(-8\)	\(-6\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-2+\beta _{2}+\beta _{4})q^{3}+q^{4}+(-1+\cdots)q^{5}+\cdots\)
1682.2.a.t	$1682$	$2$	1682.a	1.a	$1$	$6$	$6$	$13.431$	6.6.13716913.1	None	✓				58.2.d.b	$1$	$0$	\(6\)	\(2\)	\(0\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{1}q^{3}+q^{4}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
1682.2.a.u	$1682$	$2$	1682.a	1.a	$1$	$8$	$8$	$13.431$	8.8.\(\cdots\).1	None	✓	✓	✓		1682.2.a.u	$1$	$0$	\(-8\)	\(-1\)	\(5\)	\(7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{1}q^{3}+q^{4}+(1-\beta _{1}-\beta _{4}+\cdots)q^{5}+\cdots\)
1682.2.a.v	$1682$	$2$	1682.a	1.a	$1$	$8$	$8$	$13.431$	8.8.\(\cdots\).1	None	✓	✓	✓		1682.2.a.u	$1$	$0$	\(8\)	\(1\)	\(5\)	\(7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{1}q^{3}+q^{4}+(1-\beta _{1}-\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1682))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1682)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(841))\)\(^{\oplus 2}\)