Embedded newform 6480.2.a.z.1.1

• Label: 6480.2.a.z.1.1
• Level: $6480$
• Weight: $2$
• Character: 6480.1
• Self dual: yes
• Analytic conductor: $51.743$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$6480 = 2^{4} \cdot 3^{4} \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	6480.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$51.7430605098$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 90)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6480.1

$q$-expansion

$f(q)$

$=$

$q+1.00000 q^{5} +4.00000 q^{7} -3.00000 q^{11} -4.00000 q^{13} +3.00000 q^{17} -5.00000 q^{19} -6.00000 q^{23} +1.00000 q^{25} +6.00000 q^{29} -2.00000 q^{31} +4.00000 q^{35} -4.00000 q^{37} -3.00000 q^{41} -11.0000 q^{43} +9.00000 q^{49} +6.00000 q^{53} -3.00000 q^{55} +3.00000 q^{59} -10.0000 q^{61} -4.00000 q^{65} -5.00000 q^{67} -6.00000 q^{71} -7.00000 q^{73} -12.0000 q^{77} -14.0000 q^{79} -12.0000 q^{83} +3.00000 q^{85} +6.00000 q^{89} -16.0000 q^{91} -5.00000 q^{95} +11.0000 q^{97} +O(q^{100})$

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
			$3$	0	0
$5$	1.00000	0.447214
			$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
0.755929	+	0.654654i				$0.227186\pi$
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
			−0.452267	+	0.891883i	$0.649385\pi$
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
			−0.554700	+	0.832050i	$0.687167\pi$
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
			0.363803	+	0.931476i	$0.381478\pi$
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
			−0.573539	+	0.819178i	$0.694430\pi$
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
			−0.625543	+	0.780189i	$0.715123\pi$
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
			0.557086	+	0.830455i	$0.311919\pi$
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
			−0.179605	+	0.983739i	$0.557482\pi$
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
			−0.328798	+	0.944400i	$0.606644\pi$
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
			−0.234261	+	0.972174i	$0.575267\pi$
$43$	−11.0000	−1.67748	−0.838742	−	0.544529i	$-0.816708\pi$
			−0.838742	+	0.544529i	$0.816708\pi$
$47$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.z.1.1		1
3.2	odd	2		6480.2.a.l.1.1		1
4.3	odd	2		810.2.a.a.1.1		1
9.2	odd	6		2160.2.q.d.1441.1		2
9.4	even	3		720.2.q.c.241.1		2
9.5	odd	6		2160.2.q.d.721.1		2
9.7	even	3		720.2.q.c.481.1		2
12.11	even	2		810.2.a.e.1.1		1
20.3	even	4		4050.2.c.p.649.2		2
20.7	even	4		4050.2.c.p.649.1		2
20.19	odd	2		4050.2.a.bi.1.1		1
36.7	odd	6		90.2.e.b.31.1	✓	2
36.11	even	6		270.2.e.a.91.1		2
36.23	even	6		270.2.e.a.181.1		2
36.31	odd	6		90.2.e.b.61.1	yes	2
60.23	odd	4		4050.2.c.d.649.1		2
60.47	odd	4		4050.2.c.d.649.2		2
60.59	even	2		4050.2.a.q.1.1		1
180.7	even	12		450.2.j.a.49.1		4
180.23	odd	12		1350.2.j.c.1099.2		4
180.43	even	12		450.2.j.a.49.2		4
180.47	odd	12		1350.2.j.c.199.2		4
180.59	even	6		1350.2.e.g.451.1		2
180.67	even	12		450.2.j.a.349.2		4
180.79	odd	6		450.2.e.d.301.1		2
180.83	odd	12		1350.2.j.c.199.1		4
180.103	even	12		450.2.j.a.349.1		4
180.119	even	6		1350.2.e.g.901.1		2
180.139	odd	6		450.2.e.d.151.1		2
180.167	odd	12		1350.2.j.c.1099.1		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.e.b.31.1	✓	2	36.7	odd	6
90.2.e.b.61.1	yes	2	36.31	odd	6
270.2.e.a.91.1		2	36.11	even	6
270.2.e.a.181.1		2	36.23	even	6
450.2.e.d.151.1		2	180.139	odd	6
450.2.e.d.301.1		2	180.79	odd	6
450.2.j.a.49.1		4	180.7	even	12
450.2.j.a.49.2		4	180.43	even	12
450.2.j.a.349.1		4	180.103	even	12
450.2.j.a.349.2		4	180.67	even	12
720.2.q.c.241.1		2	9.4	even	3
720.2.q.c.481.1		2	9.7	even	3
810.2.a.a.1.1		1	4.3	odd	2
810.2.a.e.1.1		1	12.11	even	2
1350.2.e.g.451.1		2	180.59	even	6
1350.2.e.g.901.1		2	180.119	even	6
1350.2.j.c.199.1		4	180.83	odd	12
1350.2.j.c.199.2		4	180.47	odd	12
1350.2.j.c.1099.1		4	180.167	odd	12
1350.2.j.c.1099.2		4	180.23	odd	12
2160.2.q.d.721.1		2	9.5	odd	6
2160.2.q.d.1441.1		2	9.2	odd	6
4050.2.a.q.1.1		1	60.59	even	2
4050.2.a.bi.1.1		1	20.19	odd	2
4050.2.c.d.649.1		2	60.23	odd	4
4050.2.c.d.649.2		2	60.47	odd	4
4050.2.c.p.649.1		2	20.7	even	4
4050.2.c.p.649.2		2	20.3	even	4
6480.2.a.l.1.1		1	3.2	odd	2
6480.2.a.z.1.1		1	1.1	even	1	trivial