Embedded newform 2736.2.a.s.1.1

• Label: 2736.2.a.s.1.1
• Level: $2736$
• Weight: $2$
• Character: 2736.1
• Self dual: yes
• Analytic conductor: $21.847$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$2736 = 2^{4} \cdot 3^{2} \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2736.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 57)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2736.1

$q$-expansion

$f(q)$

$=$

$q+2.00000 q^{5} +6.00000 q^{13} +6.00000 q^{17} +1.00000 q^{19} +4.00000 q^{23} -1.00000 q^{25} -2.00000 q^{29} -8.00000 q^{31} -10.0000 q^{37} +2.00000 q^{41} +4.00000 q^{43} +12.0000 q^{47} -7.00000 q^{49} +6.00000 q^{53} -12.0000 q^{59} -2.00000 q^{61} +12.0000 q^{65} +4.00000 q^{67} +10.0000 q^{73} +16.0000 q^{83} +12.0000 q^{85} +2.00000 q^{89} +2.00000 q^{95} +10.0000 q^{97} +O(q^{100})$

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
$5$	2.00000	0.894427				0.447214	−	0.894427i	$-0.352416\pi$
			0.447214	+	0.894427i	$0.352416\pi$
$7$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$11$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
			0.832050	+	0.554700i	$0.187167\pi$
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
			0.727607	+	0.685994i	$0.240633\pi$
$19$	1.00000	0.229416
			$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
0.417029	+	0.908893i				$0.363071\pi$
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
			−0.185695	+	0.982607i	$0.559454\pi$
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
			−0.718421	+	0.695608i	$0.755135\pi$
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
			−0.821995	+	0.569495i	$0.807139\pi$
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
			0.156174	+	0.987730i	$0.450084\pi$
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
			0.304997	+	0.952353i	$0.401344\pi$
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
			0.875190	+	0.483779i	$0.160736\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.a.s.1.1		1
3.2	odd	2		912.2.a.b.1.1		1
4.3	odd	2		171.2.a.a.1.1		1
12.11	even	2		57.2.a.c.1.1	✓	1
20.19	odd	2		4275.2.a.m.1.1		1
24.5	odd	2		3648.2.a.bf.1.1		1
24.11	even	2		3648.2.a.o.1.1		1
28.27	even	2		8379.2.a.e.1.1		1
60.23	odd	4		1425.2.c.g.799.1		2
60.47	odd	4		1425.2.c.g.799.2		2
60.59	even	2		1425.2.a.a.1.1		1
76.75	even	2		3249.2.a.g.1.1		1
84.83	odd	2		2793.2.a.i.1.1		1
132.131	odd	2		6897.2.a.a.1.1		1
156.155	even	2		9633.2.a.h.1.1		1
228.227	odd	2		1083.2.a.a.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.a.c.1.1	✓	1	12.11	even	2
171.2.a.a.1.1		1	4.3	odd	2
912.2.a.b.1.1		1	3.2	odd	2
1083.2.a.a.1.1		1	228.227	odd	2
1425.2.a.a.1.1		1	60.59	even	2
1425.2.c.g.799.1		2	60.23	odd	4
1425.2.c.g.799.2		2	60.47	odd	4
2736.2.a.s.1.1		1	1.1	even	1	trivial
2793.2.a.i.1.1		1	84.83	odd	2
3249.2.a.g.1.1		1	76.75	even	2
3648.2.a.o.1.1		1	24.11	even	2
3648.2.a.bf.1.1		1	24.5	odd	2
4275.2.a.m.1.1		1	20.19	odd	2
6897.2.a.a.1.1		1	132.131	odd	2
8379.2.a.e.1.1		1	28.27	even	2
9633.2.a.h.1.1		1	156.155	even	2