Embedded newform 392.2.z.a.317.54

• Label: 392.2.z.a.317.54
• Level: $392$
• Weight: $2$
• Character: 392.317
• Analytic conductor: $3.130$
• Analytic rank: $0$
• Dimension: $648$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.z (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13013575923\)
Analytic rank:	\(0\)
Dimension:	\(648\)
Relative dimension:	\(54\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			317.54
Character	\(\chi\)	\(=\)	392.317
Dual form			392.2.z.a.277.54

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.40339 + 0.174600i) q^{2} +(0.428172 + 1.38810i) q^{3} +(1.93903 + 0.490066i) q^{4} +(-1.89578 + 2.04316i) q^{5} +(0.358531 + 2.02281i) q^{6} +(2.04312 - 1.68097i) q^{7} +(2.63566 + 1.02631i) q^{8} +(0.735231 - 0.501272i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{19}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.40339	+	0.174600i	0.992349	+	0.123461i
							\(3\)	0.428172	+	1.38810i	0.247205	+	0.801419i	0.991100	+	0.133122i	\(0.0425002\pi\)
−0.743895	+	0.668297i	\(0.767024\pi\)
\(5\)	−1.89578	+	2.04316i	−0.847817	+	0.913730i	−0.997180	−	0.0750528i	\(-0.976087\pi\)
							0.149362	+	0.988783i	\(0.452278\pi\)
\(7\)	2.04312	−	1.68097i	0.772227	−	0.635346i
							\(11\)	−1.25708	+	1.84380i	−0.379024	+	0.555926i	−0.967591	−	0.252523i	\(-0.918740\pi\)
0.588567	+	0.808448i	\(0.299692\pi\)
\(13\)	−1.37674	−	2.85883i	−0.381839	−	0.792897i	−0.999977	−	0.00682699i	\(-0.997827\pi\)
							0.618138	−	0.786070i	\(-0.287887\pi\)
\(17\)	−1.80893	+	4.60907i	−0.438730	+	1.11786i	0.525028	+	0.851085i	\(0.324055\pi\)
							−0.963758	+	0.266780i	\(0.914040\pi\)
\(19\)	−3.05266	−	1.76245i	−0.700328	−	0.404334i	0.107142	−	0.994244i	\(-0.465830\pi\)
							−0.807469	+	0.589909i	\(0.799163\pi\)
\(23\)	−0.243450	−	0.620301i	−0.0507629	−	0.129342i	0.903243	−	0.429129i	\(-0.141179\pi\)
							−0.954006	+	0.299787i	\(0.903084\pi\)
\(29\)	1.79670	−	1.43282i	0.333639	−	0.266068i	−0.442312	−	0.896861i	\(-0.645842\pi\)
							0.775951	+	0.630793i	\(0.217270\pi\)
\(31\)	−2.45168	−	4.24643i	−0.440334	−	0.762682i	0.557380	−	0.830258i	\(-0.311807\pi\)
							−0.997714	+	0.0675761i	\(0.978473\pi\)
\(37\)	−1.04306	−	6.92023i	−0.171477	−	1.13768i	−0.894570	−	0.446927i	\(-0.852518\pi\)
							0.723093	−	0.690751i	\(-0.242720\pi\)
\(41\)	2.32176	−	10.1723i	0.362598	−	1.58865i	−0.383976	−	0.923343i	\(-0.625445\pi\)
							0.746574	−	0.665303i	\(-0.231698\pi\)
\(43\)	4.45981	−	1.01792i	0.680115	−	0.155232i	0.131512	−	0.991315i	\(-0.458017\pi\)
							0.548603	+	0.836083i	\(0.315160\pi\)
\(47\)	0.0584488	−	0.779945i	0.00852564	−	0.113767i	−0.991317	−	0.131492i	\(-0.958023\pi\)
							0.999843	+	0.0177251i	\(0.00564236\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.317.54	yes	648
8.5	even	2	inner	392.2.z.a.317.2	yes	648
49.32	even	21	inner	392.2.z.a.277.2	✓	648
392.277	even	42	inner	392.2.z.a.277.54	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.277.2	✓	648	49.32	even	21	inner
392.2.z.a.277.54	yes	648	392.277	even	42	inner
392.2.z.a.317.2	yes	648	8.5	even	2	inner
392.2.z.a.317.54	yes	648	1.1	even	1	trivial