Space of modular forms of level 1860, weight 2, and trivial character

• Label: 1860.2.a
• Level: $1860$
• Weight: $2$
• Character orbit: 1860.a
• Rep. character: $\chi_{1860}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $9$
• Sturm bound: $768$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1860.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(768\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1860))\).

	Total	New	Old
Modular forms	396	20	376
Cusp forms	373	20	353
Eisenstein series	23	0	23

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

\(2\)	\(3\)	\(5\)	\(31\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(1\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(2\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(1\)
Plus space				\(+\)	\(6\)
Minus space				\(-\)	\(14\)

Trace form

20 * q + 20 * q^9 - 8 * q^11 + 8 * q^17 + 8 * q^23 + 20 * q^25 + 8 * q^29 - 8 * q^33 - 8 * q^35 - 8 * q^37 + 8 * q^43 + 32 * q^47 + 20 * q^49 + 8 * q^51 + 24 * q^53 + 8 * q^55 + 24 * q^59 + 24 * q^61 + 16 * q^67 + 8 * q^71 + 40 * q^73 + 64 * q^77 + 16 * q^79 + 20 * q^81 + 8 * q^83 - 8 * q^87 + 64 * q^89 - 4 * q^93 - 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1860))\) 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	3	5	31
1860.2.a.a	$1860$	$2$	1860.a	1.a	$1$	$1$	$1$	$14.852$	\(\Q\)	None	✓	✓	✓	✓	1860.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}-2q^{13}-q^{15}-4q^{17}+\cdots\)
1860.2.a.b	$1860$	$2$	1860.a	1.a	$1$	$1$	$1$	$14.852$	\(\Q\)	None	✓	✓	✓	✓	1860.2.a.b	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{7}+q^{9}-4q^{11}-4q^{13}+\cdots\)
1860.2.a.c	$1860$	$2$	1860.a	1.a	$1$	$1$	$1$	$14.852$	\(\Q\)	None	✓	✓			1860.2.a.c	$1$	$0$	\(0\)	\(1\)	\(1\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+2q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots\)
1860.2.a.d	$1860$	$2$	1860.a	1.a	$1$	$2$	$2$	$14.852$	\(\Q(\sqrt{6}) \)	None	✓	✓	✓	✓	1860.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+\beta q^{7}+q^{9}+(-2-\beta )q^{13}+\cdots\)
1860.2.a.e	$1860$	$2$	1860.a	1.a	$1$	$2$	$2$	$14.852$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	1860.2.a.e	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(1+\beta )q^{7}+q^{9}-4q^{11}+\cdots\)
1860.2.a.f	$1860$	$2$	1860.a	1.a	$1$	$3$	$3$	$14.852$	3.3.564.1	None	✓	✓	✓	✓	1860.2.a.f	$1$	$0$	\(0\)	\(-3\)	\(-3\)	\(4\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+(2-\beta _{1}+\beta _{2})q^{7}+q^{9}+\cdots\)
1860.2.a.g	$1860$	$2$	1860.a	1.a	$1$	$3$	$3$	$14.852$	3.3.404.1	None	✓	✓	✓	✓	1860.2.a.g	$1$	$0$	\(0\)	\(3\)	\(-3\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(-1+\beta _{2})q^{7}+q^{9}+2\beta _{2}q^{11}+\cdots\)
1860.2.a.h	$1860$	$2$	1860.a	1.a	$1$	$3$	$3$	$14.852$	3.3.7636.1	None	✓	✓	✓		1860.2.a.h	$1$	$0$	\(0\)	\(3\)	\(3\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-\beta _{1}q^{7}+q^{9}+2q^{11}+\cdots\)
1860.2.a.i	$1860$	$2$	1860.a	1.a	$1$	$4$	$4$	$14.852$	4.4.224148.1	None	✓	✓	✓	✓	1860.2.a.i	$1$	$0$	\(0\)	\(-4\)	\(4\)	\(-4\)		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+(-1+\beta _{1})q^{7}+q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1860))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1860)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(93))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(124))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(186))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(310))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(372))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(465))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(620))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(930))\)\(^{\oplus 2}\)