Space of modular forms of level 1755, weight 2, and trivial character

• Label: 1755.2.a
• Level: $1755$
• Weight: $2$
• Character orbit: 1755.a
• Rep. character: $\chi_{1755}(1,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $22$
• Sturm bound: $504$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1755.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(504\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1755))\).

	Total	New	Old
Modular forms	264	64	200
Cusp forms	241	64	177
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.

\(3\)	\(5\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(7\)
\(+\)	\(+\)	\(-\)	$-$	\(11\)
\(+\)	\(-\)	\(+\)	$-$	\(9\)
\(+\)	\(-\)	\(-\)	$+$	\(5\)
\(-\)	\(+\)	\(+\)	$-$	\(9\)
\(-\)	\(+\)	\(-\)	$+$	\(5\)
\(-\)	\(-\)	\(+\)	$+$	\(7\)
\(-\)	\(-\)	\(-\)	$-$	\(11\)
Plus space			\(+\)	\(24\)
Minus space			\(-\)	\(40\)

Trace form

\( 64 q + 60 q^{4} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1755))\) 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	5	13
1755.2.a.a	$1755$	$2$	1755.a	1.a	$1$	$1$	$1$	$14.014$	\(\Q\)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-q^{5}+2q^{10}-5q^{11}+\cdots\)
1755.2.a.b	$1755$	$2$	1755.a	1.a	$1$	$1$	$1$	$14.014$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(1\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+q^{5}-q^{7}-2q^{10}+\cdots\)
1755.2.a.c	$1755$	$2$	1755.a	1.a	$1$	$1$	$1$	$14.014$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-q^{5}-q^{7}+q^{13}+4q^{16}+\cdots\)
1755.2.a.d	$1755$	$2$	1755.a	1.a	$1$	$1$	$1$	$14.014$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}+q^{5}-q^{7}+q^{13}+4q^{16}+\cdots\)
1755.2.a.e	$1755$	$2$	1755.a	1.a	$1$	$1$	$1$	$14.014$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(0\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}-q^{5}-q^{7}-2q^{10}+\cdots\)
1755.2.a.f	$1755$	$2$	1755.a	1.a	$1$	$1$	$1$	$14.014$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(1\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+q^{5}+2q^{10}+5q^{11}+\cdots\)
1755.2.a.g	$1755$	$2$	1755.a	1.a	$1$	$2$	$2$	$14.014$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(2\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}+q^{5}+(-1+\cdots)q^{7}+\cdots\)
1755.2.a.h	$1755$	$2$	1755.a	1.a	$1$	$2$	$2$	$14.014$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(-2\)	\(0\)	\(-2\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(2-2\beta )q^{4}-q^{5}+(2+\cdots)q^{7}+\cdots\)
1755.2.a.i	$1755$	$2$	1755.a	1.a	$1$	$2$	$2$	$14.014$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}-q^{5}-2\beta q^{7}+\cdots\)
1755.2.a.j	$1755$	$2$	1755.a	1.a	$1$	$2$	$2$	$14.014$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(1\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+q^{5}-2\beta q^{7}+\cdots\)
1755.2.a.k	$1755$	$2$	1755.a	1.a	$1$	$2$	$2$	$14.014$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1+2\beta )q^{4}-q^{5}+(-1+\cdots)q^{7}+\cdots\)
1755.2.a.l	$1755$	$2$	1755.a	1.a	$1$	$2$	$2$	$14.014$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(2\)	\(0\)	\(2\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(2+2\beta )q^{4}+q^{5}+(2-\beta )q^{7}+\cdots\)
1755.2.a.m	$1755$	$2$	1755.a	1.a	$1$	$4$	$4$	$14.014$	4.4.1957.1	None	✓	$1$	$1$	\(-3\)	\(0\)	\(4\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+(1-\beta _{1}-\beta _{3})q^{4}+\cdots\)
1755.2.a.n	$1755$	$2$	1755.a	1.a	$1$	$4$	$4$	$14.014$	4.4.3981.1	None	✓	$1$	$1$	\(-3\)	\(0\)	\(4\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1755.2.a.o	$1755$	$2$	1755.a	1.a	$1$	$4$	$4$	$14.014$	4.4.12357.1	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-4\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}-q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)
1755.2.a.p	$1755$	$2$	1755.a	1.a	$1$	$4$	$4$	$14.014$	4.4.8957.1	None	✓	$1$	$1$	\(-1\)	\(0\)	\(-4\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}-q^{5}+(1+\cdots)q^{7}+\cdots\)
1755.2.a.q	$1755$	$2$	1755.a	1.a	$1$	$4$	$4$	$14.014$	4.4.12357.1	None	✓	$1$	$0$	\(1\)	\(0\)	\(4\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)
1755.2.a.r	$1755$	$2$	1755.a	1.a	$1$	$4$	$4$	$14.014$	4.4.8957.1	None	✓	$1$	$0$	\(1\)	\(0\)	\(4\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}+(-\beta _{2}+\beta _{3})q^{4}+q^{5}+(1+\cdots)q^{7}+\cdots\)
1755.2.a.s	$1755$	$2$	1755.a	1.a	$1$	$4$	$4$	$14.014$	4.4.1957.1	None	✓	$1$	$0$	\(3\)	\(0\)	\(-4\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+(1-\beta _{1}-\beta _{3})q^{4}-q^{5}+\cdots\)
1755.2.a.t	$1755$	$2$	1755.a	1.a	$1$	$4$	$4$	$14.014$	4.4.3981.1	None	✓	$1$	$0$	\(3\)	\(0\)	\(-4\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}-q^{5}+\cdots\)
1755.2.a.u	$1755$	$2$	1755.a	1.a	$1$	$7$	$7$	$14.014$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(-7\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}-q^{5}-\beta _{4}q^{7}+\cdots\)
1755.2.a.v	$1755$	$2$	1755.a	1.a	$1$	$7$	$7$	$14.014$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(0\)	\(7\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+q^{5}-\beta _{4}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1755))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1755)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(351))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(585))\)\(^{\oplus 2}\)