Embedded newform 162.2.a.b.1.1

• Label: 162.2.a.b.1.1
• Level: $162$
• Weight: $2$
• Character: 162.1
• Self dual: yes
• Analytic conductor: $1.294$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	162.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.29357651274\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 18)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	162.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} -1.00000 q^{8} +3.00000 q^{11} +2.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -1.00000 q^{19} -3.00000 q^{22} +6.00000 q^{23} -5.00000 q^{25} -2.00000 q^{26} +2.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{34} -4.00000 q^{37} +1.00000 q^{38} -9.00000 q^{41} -1.00000 q^{43} +3.00000 q^{44} -6.00000 q^{46} +6.00000 q^{47} -3.00000 q^{49} +5.00000 q^{50} +2.00000 q^{52} -12.0000 q^{53} -2.00000 q^{56} +6.00000 q^{58} -3.00000 q^{59} +8.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +5.00000 q^{67} +3.00000 q^{68} +12.0000 q^{71} +11.0000 q^{73} +4.00000 q^{74} -1.00000 q^{76} +6.00000 q^{77} -4.00000 q^{79} +9.00000 q^{82} -12.0000 q^{83} +1.00000 q^{86} -3.00000 q^{88} -6.00000 q^{89} +4.00000 q^{91} +6.00000 q^{92} -6.00000 q^{94} +5.00000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−0.707107
			\(3\)	0	0
\(5\)	0	0					−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(7\)	2.00000	0.755929	0.377964	−	0.925820i	\(-0.376624\pi\)
			0.377964	+	0.925820i	\(0.376624\pi\)
\(11\)	3.00000	0.904534	0.452267	−	0.891883i	\(-0.350615\pi\)
			0.452267	+	0.891883i	\(0.350615\pi\)
\(13\)	2.00000	0.554700	0.277350	−	0.960769i	\(-0.410544\pi\)
			0.277350	+	0.960769i	\(0.410544\pi\)
\(17\)	3.00000	0.727607	0.363803	−	0.931476i	\(-0.381478\pi\)
			0.363803	+	0.931476i	\(0.381478\pi\)
\(19\)	−1.00000	−0.229416	−0.114708	−	0.993399i	\(-0.536593\pi\)
			−0.114708	+	0.993399i	\(0.536593\pi\)
\(23\)	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	\(-0.616954\pi\)
			−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	−4.00000	−0.657596	−0.328798	−	0.944400i	\(-0.606644\pi\)
			−0.328798	+	0.944400i	\(0.606644\pi\)
\(41\)	−9.00000	−1.40556	−0.702782	−	0.711405i	\(-0.748059\pi\)
			−0.702782	+	0.711405i	\(0.748059\pi\)
\(43\)	−1.00000	−0.152499	−0.0762493	−	0.997089i	\(-0.524294\pi\)
			−0.0762493	+	0.997089i	\(0.524294\pi\)
\(47\)	6.00000	0.875190	0.437595	−	0.899172i	\(-0.355830\pi\)
			0.437595	+	0.899172i	\(0.355830\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.2.a.b.1.1		1
3.2	odd	2		162.2.a.c.1.1		1
4.3	odd	2		1296.2.a.f.1.1		1
5.2	odd	4		4050.2.c.r.649.1		2
5.3	odd	4		4050.2.c.r.649.2		2
5.4	even	2		4050.2.a.v.1.1		1
7.6	odd	2		7938.2.a.i.1.1		1
8.3	odd	2		5184.2.a.p.1.1		1
8.5	even	2		5184.2.a.q.1.1		1
9.2	odd	6		18.2.c.a.13.1	yes	2
9.4	even	3		54.2.c.a.19.1		2
9.5	odd	6		18.2.c.a.7.1	✓	2
9.7	even	3		54.2.c.a.37.1		2
12.11	even	2		1296.2.a.g.1.1		1
15.2	even	4		4050.2.c.c.649.2		2
15.8	even	4		4050.2.c.c.649.1		2
15.14	odd	2		4050.2.a.c.1.1		1
21.20	even	2		7938.2.a.x.1.1		1
24.5	odd	2		5184.2.a.r.1.1		1
24.11	even	2		5184.2.a.o.1.1		1
36.7	odd	6		432.2.i.b.145.1		2
36.11	even	6		144.2.i.c.49.1		2
36.23	even	6		144.2.i.c.97.1		2
36.31	odd	6		432.2.i.b.289.1		2
45.2	even	12		450.2.j.e.49.2		4
45.4	even	6		1350.2.e.c.451.1		2
45.7	odd	12		1350.2.j.a.199.1		4
45.13	odd	12		1350.2.j.a.1099.1		4
45.14	odd	6		450.2.e.i.151.1		2
45.22	odd	12		1350.2.j.a.1099.2		4
45.23	even	12		450.2.j.e.349.2		4
45.29	odd	6		450.2.e.i.301.1		2
45.32	even	12		450.2.j.e.349.1		4
45.34	even	6		1350.2.e.c.901.1		2
45.38	even	12		450.2.j.e.49.1		4
45.43	odd	12		1350.2.j.a.199.2		4
63.2	odd	6		882.2.h.c.67.1		2
63.4	even	3		2646.2.h.h.667.1		2
63.5	even	6		882.2.e.g.655.1		2
63.11	odd	6		882.2.e.i.373.1		2
63.13	odd	6		2646.2.f.g.883.1		2
63.16	even	3		2646.2.h.h.361.1		2
63.20	even	6		882.2.f.d.589.1		2
63.23	odd	6		882.2.e.i.655.1		2
63.25	even	3		2646.2.e.b.1549.1		2
63.31	odd	6		2646.2.h.i.667.1		2
63.32	odd	6		882.2.h.c.79.1		2
63.34	odd	6		2646.2.f.g.1765.1		2
63.38	even	6		882.2.e.g.373.1		2
63.40	odd	6		2646.2.e.c.2125.1		2
63.41	even	6		882.2.f.d.295.1		2
63.47	even	6		882.2.h.b.67.1		2
63.52	odd	6		2646.2.e.c.1549.1		2
63.58	even	3		2646.2.e.b.2125.1		2
63.59	even	6		882.2.h.b.79.1		2
63.61	odd	6		2646.2.h.i.361.1		2
72.5	odd	6		576.2.i.g.385.1		2
72.11	even	6		576.2.i.a.193.1		2
72.13	even	6		1728.2.i.e.1153.1		2
72.29	odd	6		576.2.i.g.193.1		2
72.43	odd	6		1728.2.i.f.577.1		2
72.59	even	6		576.2.i.a.385.1		2
72.61	even	6		1728.2.i.e.577.1		2
72.67	odd	6		1728.2.i.f.1153.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.2.c.a.7.1	✓	2	9.5	odd	6
18.2.c.a.13.1	yes	2	9.2	odd	6
54.2.c.a.19.1		2	9.4	even	3
54.2.c.a.37.1		2	9.7	even	3
144.2.i.c.49.1		2	36.11	even	6
144.2.i.c.97.1		2	36.23	even	6
162.2.a.b.1.1		1	1.1	even	1	trivial
162.2.a.c.1.1		1	3.2	odd	2
432.2.i.b.145.1		2	36.7	odd	6
432.2.i.b.289.1		2	36.31	odd	6
450.2.e.i.151.1		2	45.14	odd	6
450.2.e.i.301.1		2	45.29	odd	6
450.2.j.e.49.1		4	45.38	even	12
450.2.j.e.49.2		4	45.2	even	12
450.2.j.e.349.1		4	45.32	even	12
450.2.j.e.349.2		4	45.23	even	12
576.2.i.a.193.1		2	72.11	even	6
576.2.i.a.385.1		2	72.59	even	6
576.2.i.g.193.1		2	72.29	odd	6
576.2.i.g.385.1		2	72.5	odd	6
882.2.e.g.373.1		2	63.38	even	6
882.2.e.g.655.1		2	63.5	even	6
882.2.e.i.373.1		2	63.11	odd	6
882.2.e.i.655.1		2	63.23	odd	6
882.2.f.d.295.1		2	63.41	even	6
882.2.f.d.589.1		2	63.20	even	6
882.2.h.b.67.1		2	63.47	even	6
882.2.h.b.79.1		2	63.59	even	6
882.2.h.c.67.1		2	63.2	odd	6
882.2.h.c.79.1		2	63.32	odd	6
1296.2.a.f.1.1		1	4.3	odd	2
1296.2.a.g.1.1		1	12.11	even	2
1350.2.e.c.451.1		2	45.4	even	6
1350.2.e.c.901.1		2	45.34	even	6
1350.2.j.a.199.1		4	45.7	odd	12
1350.2.j.a.199.2		4	45.43	odd	12
1350.2.j.a.1099.1		4	45.13	odd	12
1350.2.j.a.1099.2		4	45.22	odd	12
1728.2.i.e.577.1		2	72.61	even	6
1728.2.i.e.1153.1		2	72.13	even	6
1728.2.i.f.577.1		2	72.43	odd	6
1728.2.i.f.1153.1		2	72.67	odd	6
2646.2.e.b.1549.1		2	63.25	even	3
2646.2.e.b.2125.1		2	63.58	even	3
2646.2.e.c.1549.1		2	63.52	odd	6
2646.2.e.c.2125.1		2	63.40	odd	6
2646.2.f.g.883.1		2	63.13	odd	6
2646.2.f.g.1765.1		2	63.34	odd	6
2646.2.h.h.361.1		2	63.16	even	3
2646.2.h.h.667.1		2	63.4	even	3
2646.2.h.i.361.1		2	63.61	odd	6
2646.2.h.i.667.1		2	63.31	odd	6
4050.2.a.c.1.1		1	15.14	odd	2
4050.2.a.v.1.1		1	5.4	even	2
4050.2.c.c.649.1		2	15.8	even	4
4050.2.c.c.649.2		2	15.2	even	4
4050.2.c.r.649.1		2	5.2	odd	4
4050.2.c.r.649.2		2	5.3	odd	4
5184.2.a.o.1.1		1	24.11	even	2
5184.2.a.p.1.1		1	8.3	odd	2
5184.2.a.q.1.1		1	8.5	even	2
5184.2.a.r.1.1		1	24.5	odd	2
7938.2.a.i.1.1		1	7.6	odd	2
7938.2.a.x.1.1		1	21.20	even	2