Embedded newform 162.2.a.d.1.1

• Label: 162.2.a.d.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:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\)

\(=\)

\(q+1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} +3.00000 q^{10} -1.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -4.00000 q^{19} +3.00000 q^{20} +4.00000 q^{25} -1.00000 q^{26} -4.00000 q^{28} -9.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} +3.00000 q^{34} -12.0000 q^{35} -1.00000 q^{37} -4.00000 q^{38} +3.00000 q^{40} -6.00000 q^{41} +8.00000 q^{43} +12.0000 q^{47} +9.00000 q^{49} +4.00000 q^{50} -1.00000 q^{52} +6.00000 q^{53} -4.00000 q^{56} -9.00000 q^{58} -1.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -3.00000 q^{65} -4.00000 q^{67} +3.00000 q^{68} -12.0000 q^{70} +12.0000 q^{71} +11.0000 q^{73} -1.00000 q^{74} -4.00000 q^{76} -16.0000 q^{79} +3.00000 q^{80} -6.00000 q^{82} +12.0000 q^{83} +9.00000 q^{85} +8.00000 q^{86} +3.00000 q^{89} +4.00000 q^{91} +12.0000 q^{94} -12.0000 q^{95} +2.00000 q^{97} +9.00000 q^{98} +O(q^{100})\)

Coefficient data

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

Currently showing only \(a_p\); display all \(a_n\)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	0.707107
			\(3\)	0	0
\(5\)	3.00000	1.34164				0.670820	−	0.741620i	\(-0.265942\pi\)
			0.670820	+	0.741620i	\(0.265942\pi\)
\(7\)	−4.00000	−1.51186	−0.755929	−	0.654654i	\(-0.772814\pi\)
			−0.755929	+	0.654654i	\(0.772814\pi\)
\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(13\)	−1.00000	−0.277350	−0.138675	−	0.990338i	\(-0.544284\pi\)
			−0.138675	+	0.990338i	\(0.544284\pi\)
\(17\)	3.00000	0.727607	0.363803	−	0.931476i	\(-0.381478\pi\)
			0.363803	+	0.931476i	\(0.381478\pi\)
\(19\)	−4.00000	−0.917663	−0.458831	−	0.888523i	\(-0.651732\pi\)
			−0.458831	+	0.888523i	\(0.651732\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	−9.00000	−1.67126	−0.835629	−	0.549294i	\(-0.814897\pi\)
			−0.835629	+	0.549294i	\(0.814897\pi\)
\(31\)	−4.00000	−0.718421	−0.359211	−	0.933257i	\(-0.616954\pi\)
			−0.359211	+	0.933257i	\(0.616954\pi\)
\(37\)	−1.00000	−0.164399	−0.0821995	−	0.996616i	\(-0.526194\pi\)
			−0.0821995	+	0.996616i	\(0.526194\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	8.00000	1.21999	0.609994	−	0.792406i	\(-0.291172\pi\)
			0.609994	+	0.792406i	\(0.291172\pi\)
\(47\)	12.0000	1.75038	0.875190	−	0.483779i	\(-0.160736\pi\)
			0.875190	+	0.483779i	\(0.160736\pi\)
\(53\)	6.00000	0.824163	0.412082	−	0.911147i	\(-0.364802\pi\)
			0.412082	+	0.911147i	\(0.364802\pi\)
\(59\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(61\)	−1.00000	−0.128037	−0.0640184	−	0.997949i	\(-0.520392\pi\)
			−0.0640184	+	0.997949i	\(0.520392\pi\)
\(67\)	−4.00000	−0.488678	−0.244339	−	0.969690i	\(-0.578571\pi\)
			−0.244339	+	0.969690i	\(0.578571\pi\)
\(71\)	12.0000	1.42414	0.712069	−	0.702109i	\(-0.247758\pi\)
			0.712069	+	0.702109i	\(0.247758\pi\)
\(73\)	11.0000	1.28745	0.643726	−	0.765256i	\(-0.277388\pi\)
			0.643726	+	0.765256i	\(0.277388\pi\)
\(79\)	−16.0000	−1.80014	−0.900070	−	0.435745i	\(-0.856485\pi\)
			−0.900070	+	0.435745i	\(0.856485\pi\)
\(83\)	12.0000	1.31717	0.658586	−	0.752506i	\(-0.271155\pi\)
			0.658586	+	0.752506i	\(0.271155\pi\)
\(89\)	3.00000	0.317999	0.159000	−	0.987279i	\(-0.449173\pi\)
			0.159000	+	0.987279i	\(0.449173\pi\)
\(97\)	2.00000	0.203069	0.101535	−	0.994832i	\(-0.467625\pi\)
			0.101535	+	0.994832i	\(0.467625\pi\)

Currently showing only \(a_p\); display all \(a_n\)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.2.a.d.1.1	yes	1
3.2	odd	2		162.2.a.a.1.1	✓	1
4.3	odd	2		1296.2.a.l.1.1		1
5.2	odd	4		4050.2.c.n.649.2		2
5.3	odd	4		4050.2.c.n.649.1		2
5.4	even	2		4050.2.a.r.1.1		1
7.6	odd	2		7938.2.a.s.1.1		1
8.3	odd	2		5184.2.a.h.1.1		1
8.5	even	2		5184.2.a.c.1.1		1
9.2	odd	6		162.2.c.d.109.1		2
9.4	even	3		162.2.c.a.55.1		2
9.5	odd	6		162.2.c.d.55.1		2
9.7	even	3		162.2.c.a.109.1		2
12.11	even	2		1296.2.a.c.1.1		1
15.2	even	4		4050.2.c.g.649.1		2
15.8	even	4		4050.2.c.g.649.2		2
15.14	odd	2		4050.2.a.bh.1.1		1
21.20	even	2		7938.2.a.n.1.1		1
24.5	odd	2		5184.2.a.y.1.1		1
24.11	even	2		5184.2.a.bd.1.1		1
36.7	odd	6		1296.2.i.b.433.1		2
36.11	even	6		1296.2.i.n.433.1		2
36.23	even	6		1296.2.i.n.865.1		2
36.31	odd	6		1296.2.i.b.865.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.2.a.a.1.1	✓	1	3.2	odd	2
162.2.a.d.1.1	yes	1	1.1	even	1	trivial
162.2.c.a.55.1		2	9.4	even	3
162.2.c.a.109.1		2	9.7	even	3
162.2.c.d.55.1		2	9.5	odd	6
162.2.c.d.109.1		2	9.2	odd	6
1296.2.a.c.1.1		1	12.11	even	2
1296.2.a.l.1.1		1	4.3	odd	2
1296.2.i.b.433.1		2	36.7	odd	6
1296.2.i.b.865.1		2	36.31	odd	6
1296.2.i.n.433.1		2	36.11	even	6
1296.2.i.n.865.1		2	36.23	even	6
4050.2.a.r.1.1		1	5.4	even	2
4050.2.a.bh.1.1		1	15.14	odd	2
4050.2.c.g.649.1		2	15.2	even	4
4050.2.c.g.649.2		2	15.8	even	4
4050.2.c.n.649.1		2	5.3	odd	4
4050.2.c.n.649.2		2	5.2	odd	4
5184.2.a.c.1.1		1	8.5	even	2
5184.2.a.h.1.1		1	8.3	odd	2
5184.2.a.y.1.1		1	24.5	odd	2
5184.2.a.bd.1.1		1	24.11	even	2
7938.2.a.n.1.1		1	21.20	even	2
7938.2.a.s.1.1		1	7.6	odd	2