Space of modular forms of level 1859, weight 2, and trivial character

• Label: 1859.2.a
• Level: $1859$
• Weight: $2$
• Character orbit: 1859.a
• Rep. character: $\chi_{1859}(1,\cdot)$
• Character field: $\Q$
• Dimension: $128$
• Newform subspaces: $20$
• Sturm bound: $364$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 1859 = 11 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1859.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(364\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1859))\).

	Total	New	Old
Modular forms	196	128	68
Cusp forms	169	128	41
Eisenstein series	27	0	27

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

\(11\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(26\)
\(+\)	\(-\)	$-$	\(37\)
\(-\)	\(+\)	$-$	\(40\)
\(-\)	\(-\)	$+$	\(25\)
Plus space		\(+\)	\(51\)
Minus space		\(-\)	\(77\)

Trace form

\( 128 q - q^{2} - q^{3} + 129 q^{4} - q^{5} + 2 q^{6} - 6 q^{7} - 3 q^{8} + 129 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1859))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		11	13
1859.2.a.a	$1859$	$2$	1859.a	1.a	$1$	$1$	$1$	$14.844$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}+q^{5}+2q^{7}-2q^{9}+q^{11}+\cdots\)
1859.2.a.b	$1859$	$2$	1859.a	1.a	$1$	$1$	$1$	$14.844$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(-1\)	\(-1\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}-q^{5}-2q^{6}+2q^{7}+\cdots\)
1859.2.a.c	$1859$	$2$	1859.a	1.a	$1$	$2$	$2$	$14.844$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(-6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1-\beta )q^{3}+q^{4}-\beta q^{5}+\cdots\)
1859.2.a.d	$1859$	$2$	1859.a	1.a	$1$	$2$	$2$	$14.844$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{3}+q^{4}-\beta q^{5}+\cdots\)
1859.2.a.e	$1859$	$2$	1859.a	1.a	$1$	$3$	$3$	$14.844$	3.3.756.1	None	✓	$1$	$1$	\(-3\)	\(0\)	\(-3\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta _{1}q^{3}-q^{4}+(-1-\beta _{2})q^{5}+\cdots\)
1859.2.a.f	$1859$	$2$	1859.a	1.a	$1$	$3$	$3$	$14.844$	3.3.361.1	None	✓	$1$	$0$	\(-1\)	\(1\)	\(-1\)	\(5\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{2}q^{3}+(2+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1859.2.a.g	$1859$	$2$	1859.a	1.a	$1$	$3$	$3$	$14.844$	3.3.361.1	None	✓	$1$	$1$	\(1\)	\(1\)	\(1\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}-\beta _{2})q^{2}+\beta _{1}q^{3}+(2+\beta _{1}+\cdots)q^{4}+\cdots\)
1859.2.a.h	$1859$	$2$	1859.a	1.a	$1$	$3$	$3$	$14.844$	3.3.756.1	None	✓	$1$	$0$	\(3\)	\(0\)	\(3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{1}q^{3}-q^{4}+(1+\beta _{2})q^{5}+\beta _{1}q^{6}+\cdots\)
1859.2.a.i	$1859$	$2$	1859.a	1.a	$1$	$4$	$4$	$14.844$	4.4.1957.1	None	✓	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1}-\beta _{2})q^{2}+(-\beta _{2}-\beta _{3})q^{3}+\cdots\)
1859.2.a.j	$1859$	$2$	1859.a	1.a	$1$	$6$	$6$	$14.844$	6.6.84512328.1	None	✓	$1$	$1$	\(-1\)	\(-2\)	\(-4\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1859.2.a.k	$1859$	$2$	1859.a	1.a	$1$	$6$	$6$	$14.844$	6.6.28561300.1	None	✓	$1$	$1$	\(0\)	\(1\)	\(-6\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(-\beta _{2}-\beta _{4})q^{3}+(1+\beta _{2}+\cdots)q^{4}+\cdots\)
1859.2.a.l	$1859$	$2$	1859.a	1.a	$1$	$6$	$6$	$14.844$	6.6.28561300.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(6\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(-\beta _{2}-\beta _{4})q^{3}+(1+\beta _{2}+\cdots)q^{4}+\cdots\)
1859.2.a.m	$1859$	$2$	1859.a	1.a	$1$	$6$	$6$	$14.844$	6.6.194616205.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(-1\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{3})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1859.2.a.n	$1859$	$2$	1859.a	1.a	$1$	$6$	$6$	$14.844$	6.6.84512328.1	None	✓	$1$	$0$	\(1\)	\(-2\)	\(4\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\cdots\)
1859.2.a.o	$1859$	$2$	1859.a	1.a	$1$	$8$	$8$	$14.844$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-8\)	\(-14\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{6}q^{3}+(\beta _{1}+\beta _{2}+\beta _{4}-\beta _{5}+\cdots)q^{4}+\cdots\)
1859.2.a.p	$1859$	$2$	1859.a	1.a	$1$	$8$	$8$	$14.844$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(8\)	\(14\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{6}q^{3}+(\beta _{1}+\beta _{2}+\beta _{4}-\beta _{5}+\cdots)q^{4}+\cdots\)
1859.2.a.q	$1859$	$2$	1859.a	1.a	$1$	$9$	$9$	$14.844$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-5\)	\(-4\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1-\beta _{1}+\beta _{3})q^{3}+(1-\beta _{5}+\cdots)q^{4}+\cdots\)
1859.2.a.r	$1859$	$2$	1859.a	1.a	$1$	$9$	$9$	$14.844$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-5\)	\(4\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{1}+\beta _{3})q^{3}+(1-\beta _{5}+\cdots)q^{4}+\cdots\)
1859.2.a.s	$1859$	$2$	1859.a	1.a	$1$	$21$	$21$	$14.844$		None	✓	$1$	$0$	\(0\)	\(6\)	\(-7\)	\(-1\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
1859.2.a.t	$1859$	$2$	1859.a	1.a	$1$	$21$	$21$	$14.844$		None	✓	$1$	$0$	\(0\)	\(6\)	\(7\)	\(1\)		$-$	$+$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1859))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1859)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(143))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)