Embedded newform 4356.2.a.x.1.2

• Label: 4356.2.a.x.1.2
• Level: $4356$
• Weight: $2$
• Character: 4356.1
• Self dual: yes
• Analytic conductor: $34.783$
• Analytic rank: $1$
• Dimension: $4$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$4356 = 2^{2} \cdot 3^{2} \cdot 11^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4356.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$34.7828351205$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.22000.1
Defining polynomial:	$x^{4} - 15x^{2} + 55$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 396)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.52626$ of defining polynomial
Character	$\chi$	$=$	4356.1

$q$-expansion

$f(q)$	$=$	$q-2.52626 q^{5} -1.00000 q^{7} +0.236068 q^{13} +6.61382 q^{17} -3.85410 q^{19} +0.964944 q^{23} +1.38197 q^{25} -3.12262 q^{29} +6.85410 q^{31} +2.52626 q^{35} -6.70820 q^{37} +9.14008 q^{41} -3.00000 q^{43} +9.73645 q^{47} -6.00000 q^{49} -1.56131 q^{53} -10.7014 q^{59} -0.909830 q^{61} -0.596368 q^{65} +4.32624 q^{67} +10.7014 q^{71} -14.4721 q^{73} +0.708204 q^{79} -6.01745 q^{83} -16.7082 q^{85} +7.21019 q^{89} -0.236068 q^{91} +9.73645 q^{95} +11.5623 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{7} - 8 q^{13} - 2 q^{19} + 10 q^{25} + 14 q^{31} - 12 q^{43} - 24 q^{49} - 26 q^{61} - 14 q^{67} - 40 q^{73} - 24 q^{79} - 40 q^{85} + 8 q^{91} + 6 q^{97}+O(q^{100})$

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
$5$	−2.52626	−1.12978				−0.564888	−	0.825168i	$-0.691081\pi$
			−0.564888	+	0.825168i	$0.691081\pi$
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
			−0.188982	+	0.981981i	$0.560519\pi$
$11$	0	0
			$13$	0.236068	0.0654735	0.0327367	−	0.999464i	$-0.489578\pi$
0.0327367	+	0.999464i				$0.489578\pi$
$17$	6.61382	1.60409	0.802044	−	0.597265i	$-0.203746\pi$
			0.802044	+	0.597265i	$0.203746\pi$
$19$	−3.85410	−0.884192	−0.442096	−	0.896968i	$-0.645765\pi$
			−0.442096	+	0.896968i	$0.645765\pi$
$23$	0.964944	0.201205	0.100602	−	0.994927i	$-0.467923\pi$
			0.100602	+	0.994927i	$0.467923\pi$
$29$	−3.12262	−0.579857	−0.289928	−	0.957048i	$-0.593631\pi$
			−0.289928	+	0.957048i	$0.593631\pi$
$31$	6.85410	1.23103	0.615517	−	0.788124i	$-0.288947\pi$
			0.615517	+	0.788124i	$0.288947\pi$
$37$	−6.70820	−1.10282	−0.551411	−	0.834234i	$-0.685910\pi$
			−0.551411	+	0.834234i	$0.685910\pi$
$41$	9.14008	1.42744	0.713720	−	0.700431i	$-0.247009\pi$
			0.713720	+	0.700431i	$0.247009\pi$
$43$	−3.00000	−0.457496	−0.228748	−	0.973486i	$-0.573463\pi$
			−0.228748	+	0.973486i	$0.573463\pi$
$47$	9.73645	1.42021	0.710103	−	0.704098i	$-0.248648\pi$
			0.710103	+	0.704098i	$0.248648\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4356.2.a.x.1.2		4
3.2	odd	2	inner	4356.2.a.x.1.3		4
11.7	odd	10		396.2.j.d.181.1	✓	8
11.8	odd	10		396.2.j.d.361.1	yes	8
11.10	odd	2		4356.2.a.z.1.2		4
33.8	even	10		396.2.j.d.361.2	yes	8
33.29	even	10		396.2.j.d.181.2	yes	8
33.32	even	2		4356.2.a.z.1.3		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
396.2.j.d.181.1	✓	8	11.7	odd	10
396.2.j.d.181.2	yes	8	33.29	even	10
396.2.j.d.361.1	yes	8	11.8	odd	10
396.2.j.d.361.2	yes	8	33.8	even	10
4356.2.a.x.1.2		4	1.1	even	1	trivial
4356.2.a.x.1.3		4	3.2	odd	2	inner
4356.2.a.z.1.2		4	11.10	odd	2
4356.2.a.z.1.3		4	33.32	even	2