Embedded newform 2352.4.a.b.1.1

• Label: 2352.4.a.b.1.1
• Level: $2352$
• Weight: $4$
• Character: 2352.1
• Self dual: yes
• Analytic conductor: $138.772$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	$N$	$=$	$2352 = 2^{4} \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2352.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$138.772492334$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 147)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2352.1

$q$-expansion

$f(q)$

$=$

$q-3.00000 q^{3} -12.0000 q^{5} +9.00000 q^{9} -20.0000 q^{11} +84.0000 q^{13} +36.0000 q^{15} +96.0000 q^{17} +12.0000 q^{19} +176.000 q^{23} +19.0000 q^{25} -27.0000 q^{27} +58.0000 q^{29} -264.000 q^{31} +60.0000 q^{33} +258.000 q^{37} -252.000 q^{39} -156.000 q^{43} -108.000 q^{45} -408.000 q^{47} -288.000 q^{51} -722.000 q^{53} +240.000 q^{55} -36.0000 q^{57} +492.000 q^{59} +492.000 q^{61} -1008.00 q^{65} -412.000 q^{67} -528.000 q^{69} -296.000 q^{71} -240.000 q^{73} -57.0000 q^{75} -776.000 q^{79} +81.0000 q^{81} +924.000 q^{83} -1152.00 q^{85} -174.000 q^{87} +744.000 q^{89} +792.000 q^{93} -144.000 q^{95} +168.000 q^{97} -180.000 q^{99} +O(q^{100})$

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
			$3$	−3.00000	−0.577350
$5$	−12.0000	−1.07331				−0.536656	−	0.843801i	$-0.680313\pi$
			−0.536656	+	0.843801i	$0.680313\pi$
$7$	0	0
			$11$	−20.0000	−0.548202	−0.274101	−	0.961701i	$-0.588380\pi$
−0.274101	+	0.961701i				$0.588380\pi$
$13$	84.0000	1.79211	0.896054	−	0.443945i	$-0.146421\pi$
			0.896054	+	0.443945i	$0.146421\pi$
$17$	96.0000	1.36961	0.684806	−	0.728725i	$-0.259887\pi$
			0.684806	+	0.728725i	$0.259887\pi$
$19$	12.0000	0.144894	0.0724471	−	0.997372i	$-0.476919\pi$
			0.0724471	+	0.997372i	$0.476919\pi$
$23$	176.000	1.59559	0.797794	−	0.602930i	$-0.206000\pi$
			0.797794	+	0.602930i	$0.206000\pi$
$29$	58.0000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
			0.185695	+	0.982607i	$0.440546\pi$
$31$	−264.000	−1.52954	−0.764771	−	0.644302i	$-0.777148\pi$
			−0.764771	+	0.644302i	$0.777148\pi$
$37$	258.000	1.14635	0.573175	−	0.819433i	$-0.305712\pi$
			0.573175	+	0.819433i	$0.305712\pi$
$41$	0	0		−	1.00000i	$-0.5\pi$
					1.00000i	$0.5\pi$
$43$	−156.000	−0.553251	−0.276625	−	0.960978i	$-0.589216\pi$
			−0.276625	+	0.960978i	$0.589216\pi$
$47$	−408.000	−1.26623	−0.633116	−	0.774057i	$-0.718224\pi$
			−0.633116	+	0.774057i	$0.718224\pi$

(See $a_n$ instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.b.1.1		1
4.3	odd	2		147.4.a.e.1.1	yes	1
7.6	odd	2		2352.4.a.bi.1.1		1
12.11	even	2		441.4.a.h.1.1		1
28.3	even	6		147.4.e.f.79.1		2
28.11	odd	6		147.4.e.e.79.1		2
28.19	even	6		147.4.e.f.67.1		2
28.23	odd	6		147.4.e.e.67.1		2
28.27	even	2		147.4.a.d.1.1	✓	1
84.11	even	6		441.4.e.f.226.1		2
84.23	even	6		441.4.e.f.361.1		2
84.47	odd	6		441.4.e.g.361.1		2
84.59	odd	6		441.4.e.g.226.1		2
84.83	odd	2		441.4.a.g.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.4.a.d.1.1	✓	1	28.27	even	2
147.4.a.e.1.1	yes	1	4.3	odd	2
147.4.e.e.67.1		2	28.23	odd	6
147.4.e.e.79.1		2	28.11	odd	6
147.4.e.f.67.1		2	28.19	even	6
147.4.e.f.79.1		2	28.3	even	6
441.4.a.g.1.1		1	84.83	odd	2
441.4.a.h.1.1		1	12.11	even	2
441.4.e.f.226.1		2	84.11	even	6
441.4.e.f.361.1		2	84.23	even	6
441.4.e.g.226.1		2	84.59	odd	6
441.4.e.g.361.1		2	84.47	odd	6
2352.4.a.b.1.1		1	1.1	even	1	trivial
2352.4.a.bi.1.1		1	7.6	odd	2