Embedded newform 392.2.bc.a.243.16

• Label: 392.2.bc.a.243.16
• Level: $392$
• Weight: $2$
• Character: 392.243
• Analytic conductor: $3.130$
• Analytic rank: $0$
• Dimension: $648$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.bc (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			243.16
Character	\(\chi\)	\(=\)	392.243
Dual form			392.2.bc.a.171.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.841391 + 1.13669i) q^{2} +(1.33594 + 0.524316i) q^{3} +(-0.584122 - 1.91280i) q^{4} +(-1.33135 + 0.200669i) q^{5} +(-1.72003 + 1.07739i) q^{6} +(0.0948942 - 2.64405i) q^{7} +(2.66573 + 0.945447i) q^{8} +(-0.689338 - 0.639612i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 22 q^{3} - 13 q^{4} - 14 q^{6} - 4 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{5}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.841391	+	1.13669i	−0.594953	+	0.803760i
							\(3\)	1.33594	+	0.524316i	0.771303	+	0.302714i	0.718165	−	0.695873i	\(-0.244982\pi\)
0.0531382	+	0.998587i	\(0.483078\pi\)
\(5\)	−1.33135	+	0.200669i	−0.595399	+	0.0897420i	−0.439827	−	0.898083i	\(-0.644960\pi\)
							−0.155572	+	0.987825i	\(0.549722\pi\)
\(7\)	0.0948942	−	2.64405i	0.0358666	−	0.999357i
							\(11\)	4.38419	−	4.06793i	1.32188	−	1.22653i	0.366761	−	0.930315i	\(-0.380467\pi\)
0.955122	−	0.296213i	\(-0.0957239\pi\)
\(13\)	0.410890	+	1.80023i	0.113960	+	0.499293i	0.999403	+	0.0345447i	\(0.0109981\pi\)
							−0.885443	+	0.464749i	\(0.846145\pi\)
\(17\)	4.98676	−	0.373706i	1.20947	−	0.0906370i	0.545313	−	0.838233i	\(-0.316411\pi\)
							0.664154	+	0.747596i	\(0.268792\pi\)
\(19\)	3.75682	+	2.16900i	0.861874	+	0.497603i	0.864639	−	0.502393i	\(-0.167547\pi\)
							−0.00276565	+	0.999996i	\(0.500880\pi\)
\(23\)	−2.60604	−	0.195295i	−0.543396	−	0.0407219i	−0.199795	−	0.979838i	\(-0.564028\pi\)
							−0.343601	+	0.939116i	\(0.611647\pi\)
\(29\)	4.05444	−	8.41914i	0.752891	−	1.56339i	−0.0715440	−	0.997437i	\(-0.522793\pi\)
							0.824435	−	0.565957i	\(-0.191493\pi\)
\(31\)	1.75079	+	3.03246i	0.314451	+	0.544645i	0.979321	−	0.202314i	\(-0.0648462\pi\)
							−0.664870	+	0.746959i	\(0.731513\pi\)
\(37\)	0.211647	+	0.310429i	0.0347946	+	0.0510343i	0.843242	−	0.537534i	\(-0.180644\pi\)
							−0.808447	+	0.588568i	\(0.799692\pi\)
\(41\)	−1.39696	+	1.11404i	−0.218169	+	0.173984i	−0.726476	−	0.687192i	\(-0.758843\pi\)
							0.508307	+	0.861176i	\(0.330272\pi\)
\(43\)	−4.91778	+	6.16670i	−0.749955	+	0.940413i	−0.999610	−	0.0279215i	\(-0.991111\pi\)
							0.249656	+	0.968335i	\(0.419683\pi\)
\(47\)	4.83384	+	1.49104i	0.705088	+	0.217491i	0.626506	−	0.779417i	\(-0.284485\pi\)
							0.0785822	+	0.996908i	\(0.474961\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.bc.a.243.16	yes	648
8.3	odd	2	inner	392.2.bc.a.243.28	yes	648
49.24	odd	42	inner	392.2.bc.a.171.28	yes	648
392.171	even	42	inner	392.2.bc.a.171.16	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.bc.a.171.16	✓	648	392.171	even	42	inner
392.2.bc.a.171.28	yes	648	49.24	odd	42	inner
392.2.bc.a.243.16	yes	648	1.1	even	1	trivial
392.2.bc.a.243.28	yes	648	8.3	odd	2	inner