Space of modular forms of level 1760, weight 2, and trivial character

• Label: 1760.2.a
• Level: $1760$
• Weight: $2$
• Character orbit: 1760.a
• Rep. character: $\chi_{1760}(1,\cdot)$
• Character field: $\Q$
• Dimension: $40$
• Newform subspaces: $22$
• Sturm bound: $576$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1760 = 2^{5} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1760.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(576\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(7\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1760))\).

	Total	New	Old
Modular forms	304	40	264
Cusp forms	273	40	233
Eisenstein series	31	0	31

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

\(2\)	\(5\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(32\)	\(4\)	\(28\)		\(29\)	\(4\)	\(25\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(42\)	\(6\)	\(36\)		\(38\)	\(6\)	\(32\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(44\)	\(7\)	\(37\)		\(40\)	\(7\)	\(33\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(34\)	\(3\)	\(31\)		\(30\)	\(3\)	\(27\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(40\)	\(6\)	\(34\)		\(36\)	\(6\)	\(30\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(38\)	\(4\)	\(34\)		\(34\)	\(4\)	\(30\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(36\)	\(3\)	\(33\)		\(32\)	\(3\)	\(29\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(38\)	\(7\)	\(31\)		\(34\)	\(7\)	\(27\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(140\)	\(14\)	\(126\)		\(125\)	\(14\)	\(111\)		\(15\)	\(0\)	\(15\)
Minus space			\(-\)		\(164\)	\(26\)	\(138\)		\(148\)	\(26\)	\(122\)		\(16\)	\(0\)	\(16\)

Trace form

\( 40 q + 40 q^{9} + 32 q^{13} + 16 q^{21} + 40 q^{25} - 16 q^{29} + 32 q^{37} - 16 q^{45} + 72 q^{49} + 32 q^{57} + 32 q^{61} - 16 q^{69} + 32 q^{73} + 88 q^{81} + 32 q^{89} + 32 q^{93} + 16 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1760))\) 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	5	11
1760.2.a.a	$1760$	$2$	1760.a	1.a	$1$	$1$	$1$	$14.054$	\(\Q\)	None	✓	✓			1760.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+2q^{7}+q^{9}+q^{11}-2q^{13}+\cdots\)
1760.2.a.b	$1760$	$2$	1760.a	1.a	$1$	$1$	$1$	$14.054$	\(\Q\)	None	✓	✓			1760.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}-4q^{7}+q^{9}+q^{11}+4q^{13}+\cdots\)
1760.2.a.c	$1760$	$2$	1760.a	1.a	$1$	$1$	$1$	$14.054$	\(\Q\)	None	✓	✓			1760.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+q^{9}-q^{11}+4q^{13}+\cdots\)
1760.2.a.d	$1760$	$2$	1760.a	1.a	$1$	$1$	$1$	$14.054$	\(\Q\)	None	✓	✓			1760.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+3q^{7}-2q^{9}-q^{11}-2q^{13}+\cdots\)
1760.2.a.e	$1760$	$2$	1760.a	1.a	$1$	$1$	$1$	$14.054$	\(\Q\)	None	✓	✓			1760.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{7}-2q^{9}-q^{11}-2q^{13}+\cdots\)
1760.2.a.f	$1760$	$2$	1760.a	1.a	$1$	$1$	$1$	$14.054$	\(\Q\)	None	✓	✓			1760.2.a.f	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+3q^{7}-2q^{9}+q^{11}-2q^{13}+\cdots\)
1760.2.a.g	$1760$	$2$	1760.a	1.a	$1$	$1$	$1$	$14.054$	\(\Q\)	None	✓	✓			1760.2.a.g	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{9}-q^{11}-2q^{13}-2q^{17}+\cdots\)
1760.2.a.h	$1760$	$2$	1760.a	1.a	$1$	$1$	$1$	$14.054$	\(\Q\)	None	✓	✓			1760.2.a.g	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{9}+q^{11}-2q^{13}-2q^{17}+\cdots\)
1760.2.a.i	$1760$	$2$	1760.a	1.a	$1$	$1$	$1$	$14.054$	\(\Q\)	None	✓	✓			1760.2.a.d	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-3q^{7}-2q^{9}+q^{11}-2q^{13}+\cdots\)
1760.2.a.j	$1760$	$2$	1760.a	1.a	$1$	$1$	$1$	$14.054$	\(\Q\)	None	✓	✓			1760.2.a.f	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-3q^{7}-2q^{9}-q^{11}-2q^{13}+\cdots\)
1760.2.a.k	$1760$	$2$	1760.a	1.a	$1$	$1$	$1$	$14.054$	\(\Q\)	None	✓	✓			1760.2.a.e	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-q^{7}-2q^{9}+q^{11}-2q^{13}+\cdots\)
1760.2.a.l	$1760$	$2$	1760.a	1.a	$1$	$1$	$1$	$14.054$	\(\Q\)	None	✓	✓			1760.2.a.a	$1$	$1$	\(0\)	\(2\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}-2q^{7}+q^{9}-q^{11}-2q^{13}+\cdots\)
1760.2.a.m	$1760$	$2$	1760.a	1.a	$1$	$1$	$1$	$14.054$	\(\Q\)	None	✓	✓			1760.2.a.c	$1$	$0$	\(0\)	\(2\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+q^{9}+q^{11}+4q^{13}+\cdots\)
1760.2.a.n	$1760$	$2$	1760.a	1.a	$1$	$1$	$1$	$14.054$	\(\Q\)	None	✓	✓			1760.2.a.b	$1$	$0$	\(0\)	\(2\)	\(1\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+4q^{7}+q^{9}-q^{11}+4q^{13}+\cdots\)
1760.2.a.o	$1760$	$2$	1760.a	1.a	$1$	$2$	$2$	$14.054$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		1760.2.a.o	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}-q^{5}-2q^{7}+(3+2\beta )q^{9}+\cdots\)
1760.2.a.p	$1760$	$2$	1760.a	1.a	$1$	$2$	$2$	$14.054$	\(\Q(\sqrt{5}) \)	None	✓	✓			1760.2.a.o	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(4\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-q^{5}+2q^{7}+(3+2\beta )q^{9}+\cdots\)
1760.2.a.q	$1760$	$2$	1760.a	1.a	$1$	$3$	$3$	$14.054$	3.3.229.1	None	✓	✓	✓		1760.2.a.q	$1$	$0$	\(0\)	\(-2\)	\(-3\)	\(4\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}-q^{5}+(1-\beta _{1})q^{7}+\cdots\)
1760.2.a.r	$1760$	$2$	1760.a	1.a	$1$	$3$	$3$	$14.054$	3.3.229.1	None	✓	✓			1760.2.a.r	$1$	$0$	\(0\)	\(-1\)	\(-3\)	\(7\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-q^{5}+(2-\beta _{1})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
1760.2.a.s	$1760$	$2$	1760.a	1.a	$1$	$3$	$3$	$14.054$	3.3.229.1	None	✓	✓	✓		1760.2.a.r	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(-7\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}-q^{5}+(-2+\beta _{1})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
1760.2.a.t	$1760$	$2$	1760.a	1.a	$1$	$3$	$3$	$14.054$	3.3.229.1	None	✓	✓			1760.2.a.q	$1$	$0$	\(0\)	\(2\)	\(-3\)	\(-4\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}-q^{5}+(-1+\beta _{1})q^{7}+\cdots\)
1760.2.a.u	$1760$	$2$	1760.a	1.a	$1$	$5$	$5$	$14.054$	5.5.792644.1	None	✓	✓	✓		1760.2.a.u	$1$	$0$	\(0\)	\(0\)	\(5\)	\(-2\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+q^{5}-\beta _{4}q^{7}+(2-\beta _{1})q^{9}+\cdots\)
1760.2.a.v	$1760$	$2$	1760.a	1.a	$1$	$5$	$5$	$14.054$	5.5.792644.1	None	✓	✓	✓		1760.2.a.u	$1$	$0$	\(0\)	\(0\)	\(5\)	\(2\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+q^{5}+\beta _{4}q^{7}+(2-\beta _{1})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1760))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1760)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(160))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(220))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(352))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(440))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(880))\)\(^{\oplus 2}\)