Space of modular forms of level 1681, weight 2, and trivial character

• Label: 1681.2.a
• Level: $1681$
• Weight: $2$
• Character orbit: 1681.a
• Rep. character: $\chi_{1681}(1,\cdot)$
• Character field: $\Q$
• Dimension: $117$
• Newform subspaces: $13$
• Sturm bound: $287$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1681 = 41^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1681.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(287\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1681))\).

	Total	New	Old
Modular forms	164	156	8
Cusp forms	123	117	6
Eisenstein series	41	39	2

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

\(41\)	Dim
\(+\)	\(54\)
\(-\)	\(63\)

Trace form

\( 117 q + q^{2} + 95 q^{4} + 2 q^{5} + 6 q^{6} - 6 q^{7} + 9 q^{8} + 81 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1681))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		41
1681.2.a.a	$1681$	$2$	1681.a	1.a	$1$	$2$	$2$	$13.423$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$1$	\(2\)	\(0\)	\(-4\)	\(0\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}-q^{4}-2q^{5}+\beta q^{6}-\beta q^{7}+\cdots\)
1681.2.a.b	$1681$	$2$	1681.a	1.a	$1$	$3$	$3$	$13.423$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(-3\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1-\beta _{1}+2\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
1681.2.a.c	$1681$	$2$	1681.a	1.a	$1$	$3$	$3$	$13.423$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(-2\)	\(3\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{1}-2\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
1681.2.a.d	$1681$	$2$	1681.a	1.a	$1$	$3$	$3$	$13.423$	3.3.148.1	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(-6\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}-\beta _{2})q^{2}-\beta _{2}q^{3}+(1+2\beta _{1}+\cdots)q^{4}+\cdots\)
1681.2.a.e	$1681$	$2$	1681.a	1.a	$1$	$4$	$4$	$13.423$	4.4.725.1	None	✓	$1$	$1$	\(-3\)	\(-3\)	\(-1\)	\(1\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1+\beta _{1})q^{3}+(-\beta _{1}+\cdots)q^{4}+\cdots\)
1681.2.a.f	$1681$	$2$	1681.a	1.a	$1$	$4$	$4$	$13.423$	4.4.725.1	None	✓	$1$	$1$	\(-3\)	\(3\)	\(-1\)	\(-1\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-\beta _{1})q^{3}+(-\beta _{1}+\cdots)q^{4}+\cdots\)
1681.2.a.g	$1681$	$2$	1681.a	1.a	$1$	$6$	$6$	$13.423$	6.6.40716288.1	None	✓	$2$	$0$	\(2\)	\(0\)	\(8\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(\beta _{1}-\beta _{4})q^{3}+(2+\beta _{5})q^{4}+\cdots\)
1681.2.a.h	$1681$	$2$	1681.a	1.a	$1$	$8$	$8$	$13.423$	\(\Q(\zeta_{60})^+\)	None	✓	$2$	$1$	\(-2\)	\(0\)	\(-6\)	\(0\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{3}-\beta _{4})q^{2}+\beta _{1}q^{3}-\beta _{2}q^{4}+\cdots\)
1681.2.a.i	$1681$	$2$	1681.a	1.a	$1$	$12$	$12$	$13.423$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(-5\)	\(-2\)	\(0\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{8}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1681.2.a.j	$1681$	$2$	1681.a	1.a	$1$	$12$	$12$	$13.423$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(-1\)	\(5\)	\(-2\)	\(0\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{8}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1681.2.a.k	$1681$	$2$	1681.a	1.a	$1$	$18$	$18$	$13.423$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$1$	\(1\)	\(-7\)	\(2\)	\(-11\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{11}q^{3}+(1-\beta _{8}+\beta _{9})q^{4}+\cdots\)
1681.2.a.l	$1681$	$2$	1681.a	1.a	$1$	$18$	$18$	$13.423$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(7\)	\(2\)	\(11\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{11}q^{3}+(1-\beta _{8}+\beta _{9})q^{4}+\cdots\)
1681.2.a.m	$1681$	$2$	1681.a	1.a	$1$	$24$	$24$	$13.423$		None	✓	$2$	$0$	\(8\)	\(0\)	\(12\)	\(0\)		$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1681))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1681)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(41))\)\(^{\oplus 2}\)