Embedded newform 9216.2.a.y.1.3

• Label: 9216.2.a.y.1.3
• Level: $9216$
• Weight: $2$
• Character: 9216.1
• Self dual: yes
• Analytic conductor: $73.590$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$9216 = 2^{10} \cdot 3^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	9216.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$73.5901305028$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4352.1
Defining polynomial:	$x^{4} - 6x^{2} - 4x + 2$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 48)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.334904$ of defining polynomial
Character	$\chi$	$=$	9216.1

$q$-expansion

$f(q)$	$=$	$q+0.473626 q^{5} +4.55765 q^{7} -3.49824 q^{11} -0.0840215 q^{13} +3.61706 q^{17} +3.61706 q^{19} -2.82843 q^{23} -4.77568 q^{25} -7.30205 q^{29} -0.557647 q^{31} +2.15862 q^{35} +6.20285 q^{37} +9.27391 q^{41} +2.27744 q^{43} +2.82843 q^{47} +13.7721 q^{49} -0.697947 q^{53} -1.65685 q^{55} -5.65685 q^{59} -3.85970 q^{61} -0.0397948 q^{65} -5.33962 q^{67} +9.11529 q^{71} -0.541560 q^{73} -15.9437 q^{77} +10.9937 q^{79} +15.0496 q^{83} +1.71313 q^{85} +14.6533 q^{89} -0.382941 q^{91} +1.71313 q^{95} +4.31724 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	4 * q - 4 * q^5 + 4 * q^7 + 8 * q^13 + 4 * q^25 - 12 * q^29 + 12 * q^31 + 16 * q^37 + 4 * q^49 - 20 * q^53 + 16 * q^55 + 16 * q^61 + 8 * q^65 - 16 * q^67 + 8 * q^71 - 8 * q^73 - 24 * q^77 + 12 * q^79 + 24 * q^85 + 8 * q^89 - 16 * q^91 + 24 * q^95

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
$5$	0.473626	0.211812				0.105906	−	0.994376i	$-0.466226\pi$
			0.105906	+	0.994376i	$0.466226\pi$
$7$	4.55765	1.72263	0.861314	−	0.508072i	$-0.169642\pi$
			0.861314	+	0.508072i	$0.169642\pi$
$11$	−3.49824	−1.05476	−0.527379	−	0.849630i	$-0.676825\pi$
			−0.527379	+	0.849630i	$0.676825\pi$
$13$	−0.0840215	−0.0233034	−0.0116517	−	0.999932i	$-0.503709\pi$
			−0.0116517	+	0.999932i	$0.503709\pi$
$17$	3.61706	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
			0.438633	+	0.898666i	$0.355463\pi$
$19$	3.61706	0.829810	0.414905	−	0.909865i	$-0.363815\pi$
			0.414905	+	0.909865i	$0.363815\pi$
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
			−0.294884	+	0.955533i	$0.595281\pi$
$29$	−7.30205	−1.35596	−0.677979	−	0.735082i	$-0.737144\pi$
			−0.677979	+	0.735082i	$0.737144\pi$
$31$	−0.557647	−0.100156	−0.0500782	−	0.998745i	$-0.515947\pi$
			−0.0500782	+	0.998745i	$0.515947\pi$
$37$	6.20285	1.01974	0.509871	−	0.860251i	$-0.329693\pi$
			0.509871	+	0.860251i	$0.329693\pi$
$41$	9.27391	1.44834	0.724171	−	0.689620i	$-0.242223\pi$
			0.724171	+	0.689620i	$0.242223\pi$
$43$	2.27744	0.347307	0.173653	−	0.984807i	$-0.444443\pi$
			0.173653	+	0.984807i	$0.444443\pi$
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
			0.206284	+	0.978492i	$0.433863\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9216.2.a.y.1.3		4
3.2	odd	2		3072.2.a.t.1.2		4
4.3	odd	2		9216.2.a.x.1.3		4
8.3	odd	2		9216.2.a.bn.1.2		4
8.5	even	2		9216.2.a.bo.1.2		4
12.11	even	2		3072.2.a.n.1.2		4
24.5	odd	2		3072.2.a.i.1.3		4
24.11	even	2		3072.2.a.o.1.3		4
32.3	odd	8		1152.2.k.f.289.2		8
32.5	even	8		144.2.k.b.37.1		8
32.11	odd	8		1152.2.k.f.865.2		8
32.13	even	8		144.2.k.b.109.1		8
32.19	odd	8		576.2.k.b.145.3		8
32.21	even	8		1152.2.k.c.865.2		8
32.27	odd	8		576.2.k.b.433.3		8
32.29	even	8		1152.2.k.c.289.2		8
48.5	odd	4		3072.2.d.f.1537.2		8
48.11	even	4		3072.2.d.i.1537.6		8
48.29	odd	4		3072.2.d.f.1537.7		8
48.35	even	4		3072.2.d.i.1537.3		8
96.5	odd	8		48.2.j.a.37.4	yes	8
96.11	even	8		384.2.j.a.97.2		8
96.29	odd	8		384.2.j.b.289.4		8
96.35	even	8		384.2.j.a.289.2		8
96.53	odd	8		384.2.j.b.97.4		8
96.59	even	8		192.2.j.a.49.3		8
96.77	odd	8		48.2.j.a.13.4	✓	8
96.83	even	8		192.2.j.a.145.3		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.2.j.a.13.4	✓	8	96.77	odd	8
48.2.j.a.37.4	yes	8	96.5	odd	8
144.2.k.b.37.1		8	32.5	even	8
144.2.k.b.109.1		8	32.13	even	8
192.2.j.a.49.3		8	96.59	even	8
192.2.j.a.145.3		8	96.83	even	8
384.2.j.a.97.2		8	96.11	even	8
384.2.j.a.289.2		8	96.35	even	8
384.2.j.b.97.4		8	96.53	odd	8
384.2.j.b.289.4		8	96.29	odd	8
576.2.k.b.145.3		8	32.19	odd	8
576.2.k.b.433.3		8	32.27	odd	8
1152.2.k.c.289.2		8	32.29	even	8
1152.2.k.c.865.2		8	32.21	even	8
1152.2.k.f.289.2		8	32.3	odd	8
1152.2.k.f.865.2		8	32.11	odd	8
3072.2.a.i.1.3		4	24.5	odd	2
3072.2.a.n.1.2		4	12.11	even	2
3072.2.a.o.1.3		4	24.11	even	2
3072.2.a.t.1.2		4	3.2	odd	2
3072.2.d.f.1537.2		8	48.5	odd	4
3072.2.d.f.1537.7		8	48.29	odd	4
3072.2.d.i.1537.3		8	48.35	even	4
3072.2.d.i.1537.6		8	48.11	even	4
9216.2.a.x.1.3		4	4.3	odd	2
9216.2.a.y.1.3		4	1.1	even	1	trivial
9216.2.a.bn.1.2		4	8.3	odd	2
9216.2.a.bo.1.2		4	8.5	even	2