Embedded newform 1960.2.a.g.1.1

• Label: 1960.2.a.g.1.1
• Level: $1960$
• Weight: $2$
• Character: 1960.1
• Self dual: yes
• Analytic conductor: $15.651$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1960.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(15.6506787962\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1960.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{5} -3.00000 q^{9} +4.00000 q^{11} +2.00000 q^{13} -2.00000 q^{17} -4.00000 q^{19} +4.00000 q^{23} +1.00000 q^{25} -2.00000 q^{29} +8.00000 q^{31} +6.00000 q^{37} +6.00000 q^{41} -8.00000 q^{43} +3.00000 q^{45} -4.00000 q^{47} +6.00000 q^{53} -4.00000 q^{55} +4.00000 q^{59} +2.00000 q^{61} -2.00000 q^{65} +8.00000 q^{67} +6.00000 q^{73} +9.00000 q^{81} +16.0000 q^{83} +2.00000 q^{85} +6.00000 q^{89} +4.00000 q^{95} +14.0000 q^{97} -12.0000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(5\)	−1.00000	−0.447214
			\(7\)	0	0
\(11\)	4.00000	1.20605				0.603023	−	0.797724i	\(-0.293963\pi\)
			0.603023	+	0.797724i	\(0.293963\pi\)
\(13\)	2.00000	0.554700	0.277350	−	0.960769i	\(-0.410544\pi\)
			0.277350	+	0.960769i	\(0.410544\pi\)
\(17\)	−2.00000	−0.485071	−0.242536	−	0.970143i	\(-0.577979\pi\)
			−0.242536	+	0.970143i	\(0.577979\pi\)
\(19\)	−4.00000	−0.917663	−0.458831	−	0.888523i	\(-0.651732\pi\)
			−0.458831	+	0.888523i	\(0.651732\pi\)
\(23\)	4.00000	0.834058	0.417029	−	0.908893i	\(-0.363071\pi\)
			0.417029	+	0.908893i	\(0.363071\pi\)
\(29\)	−2.00000	−0.371391	−0.185695	−	0.982607i	\(-0.559454\pi\)
			−0.185695	+	0.982607i	\(0.559454\pi\)
\(31\)	8.00000	1.43684	0.718421	−	0.695608i	\(-0.244865\pi\)
			0.718421	+	0.695608i	\(0.244865\pi\)
\(37\)	6.00000	0.986394	0.493197	−	0.869918i	\(-0.335828\pi\)
			0.493197	+	0.869918i	\(0.335828\pi\)
\(41\)	6.00000	0.937043	0.468521	−	0.883452i	\(-0.344787\pi\)
			0.468521	+	0.883452i	\(0.344787\pi\)
\(43\)	−8.00000	−1.21999	−0.609994	−	0.792406i	\(-0.708828\pi\)
			−0.609994	+	0.792406i	\(0.708828\pi\)
\(47\)	−4.00000	−0.583460	−0.291730	−	0.956501i	\(-0.594231\pi\)
			−0.291730	+	0.956501i	\(0.594231\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1960.2.a.g.1.1		1
4.3	odd	2		3920.2.a.s.1.1		1
5.4	even	2		9800.2.a.x.1.1		1
7.2	even	3		1960.2.q.i.361.1		2
7.3	odd	6		1960.2.q.h.961.1		2
7.4	even	3		1960.2.q.i.961.1		2
7.5	odd	6		1960.2.q.h.361.1		2
7.6	odd	2		40.2.a.a.1.1	✓	1
21.20	even	2		360.2.a.a.1.1		1
28.27	even	2		80.2.a.a.1.1		1
35.13	even	4		200.2.c.b.49.2		2
35.27	even	4		200.2.c.b.49.1		2
35.34	odd	2		200.2.a.c.1.1		1
56.13	odd	2		320.2.a.c.1.1		1
56.27	even	2		320.2.a.d.1.1		1
63.13	odd	6		3240.2.q.k.2161.1		2
63.20	even	6		3240.2.q.x.1081.1		2
63.34	odd	6		3240.2.q.k.1081.1		2
63.41	even	6		3240.2.q.x.2161.1		2
77.76	even	2		4840.2.a.f.1.1		1
84.83	odd	2		720.2.a.e.1.1		1
91.90	odd	2		6760.2.a.i.1.1		1
105.62	odd	4		1800.2.f.a.649.1		2
105.83	odd	4		1800.2.f.a.649.2		2
105.104	even	2		1800.2.a.v.1.1		1
112.13	odd	4		1280.2.d.j.641.1		2
112.27	even	4		1280.2.d.a.641.2		2
112.69	odd	4		1280.2.d.j.641.2		2
112.83	even	4		1280.2.d.a.641.1		2
140.27	odd	4		400.2.c.d.49.2		2
140.83	odd	4		400.2.c.d.49.1		2
140.139	even	2		400.2.a.e.1.1		1
168.83	odd	2		2880.2.a.bg.1.1		1
168.125	even	2		2880.2.a.t.1.1		1
280.13	even	4		1600.2.c.k.449.2		2
280.27	odd	4		1600.2.c.m.449.2		2
280.69	odd	2		1600.2.a.o.1.1		1
280.83	odd	4		1600.2.c.m.449.1		2
280.139	even	2		1600.2.a.k.1.1		1
280.237	even	4		1600.2.c.k.449.1		2
308.307	odd	2		9680.2.a.q.1.1		1
420.83	even	4		3600.2.f.t.2449.1		2
420.167	even	4		3600.2.f.t.2449.2		2
420.419	odd	2		3600.2.a.h.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.a.a.1.1	✓	1	7.6	odd	2
80.2.a.a.1.1		1	28.27	even	2
200.2.a.c.1.1		1	35.34	odd	2
200.2.c.b.49.1		2	35.27	even	4
200.2.c.b.49.2		2	35.13	even	4
320.2.a.c.1.1		1	56.13	odd	2
320.2.a.d.1.1		1	56.27	even	2
360.2.a.a.1.1		1	21.20	even	2
400.2.a.e.1.1		1	140.139	even	2
400.2.c.d.49.1		2	140.83	odd	4
400.2.c.d.49.2		2	140.27	odd	4
720.2.a.e.1.1		1	84.83	odd	2
1280.2.d.a.641.1		2	112.83	even	4
1280.2.d.a.641.2		2	112.27	even	4
1280.2.d.j.641.1		2	112.13	odd	4
1280.2.d.j.641.2		2	112.69	odd	4
1600.2.a.k.1.1		1	280.139	even	2
1600.2.a.o.1.1		1	280.69	odd	2
1600.2.c.k.449.1		2	280.237	even	4
1600.2.c.k.449.2		2	280.13	even	4
1600.2.c.m.449.1		2	280.83	odd	4
1600.2.c.m.449.2		2	280.27	odd	4
1800.2.a.v.1.1		1	105.104	even	2
1800.2.f.a.649.1		2	105.62	odd	4
1800.2.f.a.649.2		2	105.83	odd	4
1960.2.a.g.1.1		1	1.1	even	1	trivial
1960.2.q.h.361.1		2	7.5	odd	6
1960.2.q.h.961.1		2	7.3	odd	6
1960.2.q.i.361.1		2	7.2	even	3
1960.2.q.i.961.1		2	7.4	even	3
2880.2.a.t.1.1		1	168.125	even	2
2880.2.a.bg.1.1		1	168.83	odd	2
3240.2.q.k.1081.1		2	63.34	odd	6
3240.2.q.k.2161.1		2	63.13	odd	6
3240.2.q.x.1081.1		2	63.20	even	6
3240.2.q.x.2161.1		2	63.41	even	6
3600.2.a.h.1.1		1	420.419	odd	2
3600.2.f.t.2449.1		2	420.83	even	4
3600.2.f.t.2449.2		2	420.167	even	4
3920.2.a.s.1.1		1	4.3	odd	2
4840.2.a.f.1.1		1	77.76	even	2
6760.2.a.i.1.1		1	91.90	odd	2
9680.2.a.q.1.1		1	308.307	odd	2
9800.2.a.x.1.1		1	5.4	even	2