Embedded newform 4080.2.a.c.1.1

• Label: 4080.2.a.c.1.1
• Level: $4080$
• Weight: $2$
• Character: 4080.1
• Self dual: yes
• Analytic conductor: $32.579$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4080.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(32.5789640247\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	4080.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} -4.00000 q^{11} +2.00000 q^{13} +1.00000 q^{15} +1.00000 q^{17} +4.00000 q^{19} -4.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{29} +4.00000 q^{31} +4.00000 q^{33} -6.00000 q^{37} -2.00000 q^{39} -10.0000 q^{41} +8.00000 q^{43} -1.00000 q^{45} -7.00000 q^{49} -1.00000 q^{51} +6.00000 q^{53} +4.00000 q^{55} -4.00000 q^{57} +8.00000 q^{59} +10.0000 q^{61} -2.00000 q^{65} +8.00000 q^{67} +4.00000 q^{69} -8.00000 q^{71} -2.00000 q^{73} -1.00000 q^{75} -4.00000 q^{79} +1.00000 q^{81} -4.00000 q^{83} -1.00000 q^{85} -2.00000 q^{87} -14.0000 q^{89} -4.00000 q^{93} -4.00000 q^{95} -10.0000 q^{97} -4.00000 q^{99} +O(q^{100})\)

Coefficient data

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

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

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	−1.00000	−0.577350
\(5\)	−1.00000	−0.447214
			\(7\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(11\)	−4.00000	−1.20605	−0.603023	−	0.797724i	\(-0.706037\pi\)
			−0.603023	+	0.797724i	\(0.706037\pi\)
\(13\)	2.00000	0.554700	0.277350	−	0.960769i	\(-0.410544\pi\)
			0.277350	+	0.960769i	\(0.410544\pi\)
\(17\)	1.00000	0.242536
			\(19\)	4.00000	0.917663	0.458831	−	0.888523i	\(-0.348268\pi\)
0.458831	+	0.888523i				\(0.348268\pi\)
\(23\)	−4.00000	−0.834058	−0.417029	−	0.908893i	\(-0.636929\pi\)
			−0.417029	+	0.908893i	\(0.636929\pi\)
\(29\)	2.00000	0.371391	0.185695	−	0.982607i	\(-0.440546\pi\)
			0.185695	+	0.982607i	\(0.440546\pi\)
\(31\)	4.00000	0.718421	0.359211	−	0.933257i	\(-0.383046\pi\)
			0.359211	+	0.933257i	\(0.383046\pi\)
\(37\)	−6.00000	−0.986394	−0.493197	−	0.869918i	\(-0.664172\pi\)
			−0.493197	+	0.869918i	\(0.664172\pi\)
\(41\)	−10.0000	−1.56174	−0.780869	−	0.624695i	\(-0.785223\pi\)
			−0.780869	+	0.624695i	\(0.785223\pi\)
\(43\)	8.00000	1.21999	0.609994	−	0.792406i	\(-0.291172\pi\)
			0.609994	+	0.792406i	\(0.291172\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(53\)	6.00000	0.824163	0.412082	−	0.911147i	\(-0.364802\pi\)
			0.412082	+	0.911147i	\(0.364802\pi\)
\(59\)	8.00000	1.04151	0.520756	−	0.853706i	\(-0.325650\pi\)
			0.520756	+	0.853706i	\(0.325650\pi\)
\(61\)	10.0000	1.28037	0.640184	−	0.768221i	\(-0.278858\pi\)
			0.640184	+	0.768221i	\(0.278858\pi\)
\(67\)	8.00000	0.977356	0.488678	−	0.872464i	\(-0.337479\pi\)
			0.488678	+	0.872464i	\(0.337479\pi\)
\(71\)	−8.00000	−0.949425	−0.474713	−	0.880141i	\(-0.657448\pi\)
			−0.474713	+	0.880141i	\(0.657448\pi\)
\(73\)	−2.00000	−0.234082	−0.117041	−	0.993127i	\(-0.537341\pi\)
			−0.117041	+	0.993127i	\(0.537341\pi\)
\(79\)	−4.00000	−0.450035	−0.225018	−	0.974355i	\(-0.572244\pi\)
			−0.225018	+	0.974355i	\(0.572244\pi\)
\(83\)	−4.00000	−0.439057	−0.219529	−	0.975606i	\(-0.570452\pi\)
			−0.219529	+	0.975606i	\(0.570452\pi\)
\(89\)	−14.0000	−1.48400	−0.741999	−	0.670402i	\(-0.766122\pi\)
			−0.741999	+	0.670402i	\(0.766122\pi\)
\(97\)	−10.0000	−1.01535	−0.507673	−	0.861550i	\(-0.669494\pi\)
			−0.507673	+	0.861550i	\(0.669494\pi\)

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	4080.2.a.c.1.1		1
4.3	odd	2		510.2.a.f.1.1	✓	1
12.11	even	2		1530.2.a.f.1.1		1
20.3	even	4		2550.2.d.s.2449.1		2
20.7	even	4		2550.2.d.s.2449.2		2
20.19	odd	2		2550.2.a.d.1.1		1
60.59	even	2		7650.2.a.bw.1.1		1
68.67	odd	2		8670.2.a.r.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.a.f.1.1	✓	1	4.3	odd	2
1530.2.a.f.1.1		1	12.11	even	2
2550.2.a.d.1.1		1	20.19	odd	2
2550.2.d.s.2449.1		2	20.3	even	4
2550.2.d.s.2449.2		2	20.7	even	4
4080.2.a.c.1.1		1	1.1	even	1	trivial
7650.2.a.bw.1.1		1	60.59	even	2
8670.2.a.r.1.1		1	68.67	odd	2