Embedded newform 2304.3.g.f.1279.1

• Label: 2304.3.g.f.1279.1
• Level: $2304$
• Weight: $3$
• Character: 2304.1279
• Self dual: yes
• Analytic conductor: $62.779$
• Analytic rank: $0$
• Dimension: $1$
• CM discriminant: -4
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$2304 = 2^{8} \cdot 3^{2}$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	2304.g (of order $2$, degree $1$, not minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	$62.7794529086$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1279.1
Character	$\chi$	$=$	2304.1279

$q$-expansion

$f(q)$

$=$

$q+8.00000 q^{5} +24.0000 q^{13} -30.0000 q^{17} +39.0000 q^{25} +40.0000 q^{29} +24.0000 q^{37} +18.0000 q^{41} +49.0000 q^{49} -56.0000 q^{53} +120.000 q^{61} +192.000 q^{65} -110.000 q^{73} -240.000 q^{85} +78.0000 q^{89} -130.000 q^{97} +O(q^{100})$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	8.00000	1.60000				0.800000	−	0.600000i	$-0.204833\pi$
			0.800000	+	0.600000i	$0.204833\pi$
$7$	0	0	1.00000			$0$
			−1.00000			$\pi$
$11$	0	0	1.00000			$0$
			−1.00000			$\pi$
$13$	24.0000	1.84615	0.923077	−	0.384615i	$-0.125666\pi$
			0.923077	+	0.384615i	$0.125666\pi$
$17$	−30.0000	−1.76471	−0.882353	−	0.470588i	$-0.844042\pi$
			−0.882353	+	0.470588i	$0.844042\pi$
$19$	0	0	1.00000			$0$
			−1.00000			$\pi$
$23$	0	0	1.00000			$0$
			−1.00000			$\pi$
$29$	40.0000	1.37931	0.689655	−	0.724138i	$-0.257762\pi$
			0.689655	+	0.724138i	$0.257762\pi$
$31$	0	0	1.00000			$0$
			−1.00000			$\pi$
$37$	24.0000	0.648649	0.324324	−	0.945946i	$-0.394863\pi$
			0.324324	+	0.945946i	$0.394863\pi$
$41$	18.0000	0.439024	0.219512	−	0.975610i	$-0.429553\pi$
			0.219512	+	0.975610i	$0.429553\pi$
$43$	0	0	1.00000			$0$
			−1.00000			$\pi$
$47$	0	0	1.00000			$0$
			−1.00000			$\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.g.f.1279.1		1
3.2	odd	2		256.3.c.a.255.1		1
4.3	odd	2	CM	2304.3.g.f.1279.1		1
8.3	odd	2		2304.3.g.a.1279.1		1
8.5	even	2		2304.3.g.a.1279.1		1
12.11	even	2		256.3.c.a.255.1		1
16.3	odd	4		1152.3.b.a.703.1		2
16.5	even	4		1152.3.b.a.703.2		2
16.11	odd	4		1152.3.b.a.703.2		2
16.13	even	4		1152.3.b.a.703.1		2
24.5	odd	2		256.3.c.b.255.1		1
24.11	even	2		256.3.c.b.255.1		1
48.5	odd	4		128.3.d.a.63.1	✓	2
48.11	even	4		128.3.d.a.63.1	✓	2
48.29	odd	4		128.3.d.a.63.2	yes	2
48.35	even	4		128.3.d.a.63.2	yes	2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.d.a.63.1	✓	2	48.5	odd	4
128.3.d.a.63.1	✓	2	48.11	even	4
128.3.d.a.63.2	yes	2	48.29	odd	4
128.3.d.a.63.2	yes	2	48.35	even	4
256.3.c.a.255.1		1	3.2	odd	2
256.3.c.a.255.1		1	12.11	even	2
256.3.c.b.255.1		1	24.5	odd	2
256.3.c.b.255.1		1	24.11	even	2
1152.3.b.a.703.1		2	16.3	odd	4
1152.3.b.a.703.1		2	16.13	even	4
1152.3.b.a.703.2		2	16.5	even	4
1152.3.b.a.703.2		2	16.11	odd	4
2304.3.g.a.1279.1		1	8.3	odd	2
2304.3.g.a.1279.1		1	8.5	even	2
2304.3.g.f.1279.1		1	1.1	even	1	trivial
2304.3.g.f.1279.1		1	4.3	odd	2	CM