Embedded newform 7220.2.a.f.1.1

• Label: 7220.2.a.f.1.1
• Level: $7220$
• Weight: $2$
• Character: 7220.1
• Self dual: yes
• Analytic conductor: $57.652$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7220 = 2^{2} \cdot 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7220.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(57.6519902594\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 20)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	7220.1

$q$-expansion

\(f(q)\)

\(=\)

\(q+2.00000 q^{3} -1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} -2.00000 q^{13} -2.00000 q^{15} -6.00000 q^{17} +4.00000 q^{21} +6.00000 q^{23} +1.00000 q^{25} -4.00000 q^{27} -6.00000 q^{29} +4.00000 q^{31} -2.00000 q^{35} -2.00000 q^{37} -4.00000 q^{39} -6.00000 q^{41} -10.0000 q^{43} -1.00000 q^{45} -6.00000 q^{47} -3.00000 q^{49} -12.0000 q^{51} +6.00000 q^{53} -12.0000 q^{59} +2.00000 q^{61} +2.00000 q^{63} +2.00000 q^{65} -2.00000 q^{67} +12.0000 q^{69} +12.0000 q^{71} +2.00000 q^{73} +2.00000 q^{75} -8.00000 q^{79} -11.0000 q^{81} +6.00000 q^{83} +6.00000 q^{85} -12.0000 q^{87} +6.00000 q^{89} -4.00000 q^{91} +8.00000 q^{93} -2.00000 q^{97} +O(q^{100})\)

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
			\(3\)	2.00000	1.15470	0.577350	−	0.816497i	\(-0.304087\pi\)
0.577350	+	0.816497i				\(0.304087\pi\)
\(5\)	−1.00000	−0.447214
			\(7\)	2.00000	0.755929	0.377964	−	0.925820i	\(-0.376624\pi\)
0.377964	+	0.925820i				\(0.376624\pi\)
\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
			−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	−6.00000	−1.45521	−0.727607	−	0.685994i	\(-0.759367\pi\)
			−0.727607	+	0.685994i	\(0.759367\pi\)
\(19\)	0	0
			\(23\)	6.00000	1.25109	0.625543	−	0.780189i	\(-0.284877\pi\)
0.625543	+	0.780189i				\(0.284877\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	4.00000	0.718421	0.359211	−	0.933257i	\(-0.383046\pi\)
			0.359211	+	0.933257i	\(0.383046\pi\)
\(37\)	−2.00000	−0.328798	−0.164399	−	0.986394i	\(-0.552568\pi\)
			−0.164399	+	0.986394i	\(0.552568\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	−10.0000	−1.52499	−0.762493	−	0.646997i	\(-0.776025\pi\)
			−0.762493	+	0.646997i	\(0.776025\pi\)
\(47\)	−6.00000	−0.875190	−0.437595	−	0.899172i	\(-0.644170\pi\)
			−0.437595	+	0.899172i	\(0.644170\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7220.2.a.f.1.1		1
19.18	odd	2		20.2.a.a.1.1	✓	1
57.56	even	2		180.2.a.a.1.1		1
76.75	even	2		80.2.a.b.1.1		1
95.18	even	4		100.2.c.a.49.1		2
95.37	even	4		100.2.c.a.49.2		2
95.94	odd	2		100.2.a.a.1.1		1
133.18	odd	6		980.2.i.i.961.1		2
133.37	odd	6		980.2.i.i.361.1		2
133.75	even	6		980.2.i.c.361.1		2
133.94	even	6		980.2.i.c.961.1		2
133.132	even	2		980.2.a.h.1.1		1
152.37	odd	2		320.2.a.f.1.1		1
152.75	even	2		320.2.a.a.1.1		1
171.56	even	6		1620.2.i.b.1081.1		2
171.94	odd	6		1620.2.i.h.541.1		2
171.113	even	6		1620.2.i.b.541.1		2
171.151	odd	6		1620.2.i.h.1081.1		2
209.208	even	2		2420.2.a.a.1.1		1
228.227	odd	2		720.2.a.h.1.1		1
247.18	even	4		3380.2.f.b.3041.2		2
247.151	even	4		3380.2.f.b.3041.1		2
247.246	odd	2		3380.2.a.c.1.1		1
285.113	odd	4		900.2.d.c.649.1		2
285.227	odd	4		900.2.d.c.649.2		2
285.284	even	2		900.2.a.b.1.1		1
304.37	odd	4		1280.2.d.c.641.2		2
304.75	even	4		1280.2.d.g.641.1		2
304.189	odd	4		1280.2.d.c.641.1		2
304.227	even	4		1280.2.d.g.641.2		2
323.132	odd	4		5780.2.c.a.5201.1		2
323.208	odd	4		5780.2.c.a.5201.2		2
323.322	odd	2		5780.2.a.f.1.1		1
380.227	odd	4		400.2.c.b.49.1		2
380.303	odd	4		400.2.c.b.49.2		2
380.379	even	2		400.2.a.c.1.1		1
399.398	odd	2		8820.2.a.g.1.1		1
456.227	odd	2		2880.2.a.f.1.1		1
456.341	even	2		2880.2.a.m.1.1		1
532.531	odd	2		3920.2.a.h.1.1		1
665.132	odd	4		4900.2.e.f.2549.1		2
665.398	odd	4		4900.2.e.f.2549.2		2
665.664	even	2		4900.2.a.e.1.1		1
760.37	even	4		1600.2.c.d.449.1		2
760.189	odd	2		1600.2.a.c.1.1		1
760.227	odd	4		1600.2.c.e.449.2		2
760.379	even	2		1600.2.a.w.1.1		1
760.493	even	4		1600.2.c.d.449.2		2
760.683	odd	4		1600.2.c.e.449.1		2
836.835	odd	2		9680.2.a.ba.1.1		1
1140.227	even	4		3600.2.f.j.2449.1		2
1140.683	even	4		3600.2.f.j.2449.2		2
1140.1139	odd	2		3600.2.a.be.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.2.a.a.1.1	✓	1	19.18	odd	2
80.2.a.b.1.1		1	76.75	even	2
100.2.a.a.1.1		1	95.94	odd	2
100.2.c.a.49.1		2	95.18	even	4
100.2.c.a.49.2		2	95.37	even	4
180.2.a.a.1.1		1	57.56	even	2
320.2.a.a.1.1		1	152.75	even	2
320.2.a.f.1.1		1	152.37	odd	2
400.2.a.c.1.1		1	380.379	even	2
400.2.c.b.49.1		2	380.227	odd	4
400.2.c.b.49.2		2	380.303	odd	4
720.2.a.h.1.1		1	228.227	odd	2
900.2.a.b.1.1		1	285.284	even	2
900.2.d.c.649.1		2	285.113	odd	4
900.2.d.c.649.2		2	285.227	odd	4
980.2.a.h.1.1		1	133.132	even	2
980.2.i.c.361.1		2	133.75	even	6
980.2.i.c.961.1		2	133.94	even	6
980.2.i.i.361.1		2	133.37	odd	6
980.2.i.i.961.1		2	133.18	odd	6
1280.2.d.c.641.1		2	304.189	odd	4
1280.2.d.c.641.2		2	304.37	odd	4
1280.2.d.g.641.1		2	304.75	even	4
1280.2.d.g.641.2		2	304.227	even	4
1600.2.a.c.1.1		1	760.189	odd	2
1600.2.a.w.1.1		1	760.379	even	2
1600.2.c.d.449.1		2	760.37	even	4
1600.2.c.d.449.2		2	760.493	even	4
1600.2.c.e.449.1		2	760.683	odd	4
1600.2.c.e.449.2		2	760.227	odd	4
1620.2.i.b.541.1		2	171.113	even	6
1620.2.i.b.1081.1		2	171.56	even	6
1620.2.i.h.541.1		2	171.94	odd	6
1620.2.i.h.1081.1		2	171.151	odd	6
2420.2.a.a.1.1		1	209.208	even	2
2880.2.a.f.1.1		1	456.227	odd	2
2880.2.a.m.1.1		1	456.341	even	2
3380.2.a.c.1.1		1	247.246	odd	2
3380.2.f.b.3041.1		2	247.151	even	4
3380.2.f.b.3041.2		2	247.18	even	4
3600.2.a.be.1.1		1	1140.1139	odd	2
3600.2.f.j.2449.1		2	1140.227	even	4
3600.2.f.j.2449.2		2	1140.683	even	4
3920.2.a.h.1.1		1	532.531	odd	2
4900.2.a.e.1.1		1	665.664	even	2
4900.2.e.f.2549.1		2	665.132	odd	4
4900.2.e.f.2549.2		2	665.398	odd	4
5780.2.a.f.1.1		1	323.322	odd	2
5780.2.c.a.5201.1		2	323.132	odd	4
5780.2.c.a.5201.2		2	323.208	odd	4
7220.2.a.f.1.1		1	1.1	even	1	trivial
8820.2.a.g.1.1		1	399.398	odd	2
9680.2.a.ba.1.1		1	836.835	odd	2