Space of modular forms of level 861, weight 2, and trivial character

• Label: 861.2.a
• Level: $861$
• Weight: $2$
• Character orbit: 861.a
• Rep. character: $\chi_{861}(1,\cdot)$
• Character field: $\Q$
• Dimension: $39$
• Newform subspaces: $13$
• Sturm bound: $224$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 861 = 3 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	861.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(224\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(861))\).

	Total	New	Old
Modular forms	116	39	77
Cusp forms	109	39	70
Eisenstein series	7	0	7

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

\(3\)	\(7\)	\(41\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(4\)
\(+\)	\(+\)	\(-\)	$-$	\(5\)
\(+\)	\(-\)	\(+\)	$-$	\(8\)
\(+\)	\(-\)	\(-\)	$+$	\(3\)
\(-\)	\(+\)	\(+\)	$-$	\(5\)
\(-\)	\(+\)	\(-\)	$+$	\(4\)
\(-\)	\(-\)	\(+\)	$+$	\(3\)
\(-\)	\(-\)	\(-\)	$-$	\(7\)
Plus space			\(+\)	\(14\)
Minus space			\(-\)	\(25\)

Trace form

\( 39 q + q^{2} - q^{3} + 45 q^{4} - 6 q^{5} + 5 q^{6} + 3 q^{7} - 3 q^{8} + 39 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(861))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	7	41
861.2.a.a	$861$	$2$	861.a	1.a	$1$	$1$	$1$	$6.875$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-1\)	\(2\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+2q^{5}+q^{6}+q^{7}+\cdots\)
861.2.a.b	$861$	$2$	861.a	1.a	$1$	$1$	$1$	$6.875$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(-3\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-3q^{5}-q^{6}+q^{7}+\cdots\)
861.2.a.c	$861$	$2$	861.a	1.a	$1$	$1$	$1$	$6.875$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(3\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}+3q^{5}-q^{6}-q^{7}+\cdots\)
861.2.a.d	$861$	$2$	861.a	1.a	$1$	$1$	$1$	$6.875$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(1\)	\(-1\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
861.2.a.e	$861$	$2$	861.a	1.a	$1$	$2$	$2$	$6.875$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-2\)	\(2\)	\(-2\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}-q^{5}+\cdots\)
861.2.a.f	$861$	$2$	861.a	1.a	$1$	$2$	$2$	$6.875$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(-1\)	\(-2\)	\(4\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(2+\beta )q^{4}+2q^{5}+\beta q^{6}+\cdots\)
861.2.a.g	$861$	$2$	861.a	1.a	$1$	$2$	$2$	$6.875$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(-1\)	\(2\)	\(-3\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+(2+\beta )q^{4}+(-2+\beta )q^{5}+\cdots\)
861.2.a.h	$861$	$2$	861.a	1.a	$1$	$3$	$3$	$6.875$	3.3.148.1	None	✓	$1$	$1$	\(-1\)	\(-3\)	\(1\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+(\beta _{1}+\beta _{2})q^{5}+\cdots\)
861.2.a.i	$861$	$2$	861.a	1.a	$1$	$4$	$4$	$6.875$	4.4.8468.1	None	✓	$1$	$1$	\(1\)	\(-4\)	\(-3\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1-\beta _{2}+\cdots)q^{5}+\cdots\)
861.2.a.j	$861$	$2$	861.a	1.a	$1$	$5$	$5$	$6.875$	5.5.981328.1	None	✓	$1$	$0$	\(-3\)	\(-5\)	\(1\)	\(-5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
861.2.a.k	$861$	$2$	861.a	1.a	$1$	$5$	$5$	$6.875$	5.5.1197392.1	None	✓	$1$	$0$	\(3\)	\(-5\)	\(-9\)	\(5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-q^{3}+(2+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
861.2.a.l	$861$	$2$	861.a	1.a	$1$	$5$	$5$	$6.875$	5.5.626512.1	None	✓	$1$	$0$	\(3\)	\(5\)	\(3\)	\(-5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+q^{3}+(2-\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
861.2.a.m	$861$	$2$	861.a	1.a	$1$	$7$	$7$	$6.875$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(7\)	\(1\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(861))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(861)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(41))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(123))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(287))\)\(^{\oplus 2}\)