Embedded newform 392.2.bc.a.243.28

• Label: 392.2.bc.a.243.28
• 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			243.28
Character	\(\chi\)	\(=\)	392.243
Dual form			392.2.bc.a.171.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0548704 - 1.41315i) q^{2} +(1.33594 + 0.524316i) q^{3} +(-1.99398 + 0.155080i) q^{4} +(1.33135 - 0.200669i) q^{5} +(0.667634 - 1.91665i) q^{6} +(-0.0948942 + 2.64405i) q^{7} +(0.328562 + 2.80928i) q^{8} +(-0.689338 - 0.639612i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 22 q^{3} - 13 q^{4} - 14 q^{6} - 4 q^{8} - 76 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0548704	−	1.41315i	−0.0387992	−	0.999247i
							\(3\)	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	0.155572	−	0.987825i	\(-0.450278\pi\)
							0.439827	+	0.898083i	\(0.355040\pi\)
\(7\)	−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	\(-0.989002\pi\)
							0.885443	−	0.464749i	\(-0.153855\pi\)
\(17\)	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	0.343601	−	0.939116i	\(-0.388353\pi\)
							0.199795	+	0.979838i	\(0.435972\pi\)
\(29\)	−4.05444	+	8.41914i	−0.752891	+	1.56339i	0.0715440	+	0.997437i	\(0.477207\pi\)
							−0.824435	+	0.565957i	\(0.808507\pi\)
\(31\)	−1.75079	−	3.03246i	−0.314451	−	0.544645i	0.664870	−	0.746959i	\(-0.268487\pi\)
							−0.979321	+	0.202314i	\(0.935154\pi\)
\(37\)	−0.211647	−	0.310429i	−0.0347946	−	0.0510343i	0.808447	−	0.588568i	\(-0.200308\pi\)
							−0.843242	+	0.537534i	\(0.819356\pi\)
\(41\)	−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	−0.0785822	−	0.996908i	\(-0.525039\pi\)
							−0.626506	+	0.779417i	\(0.715515\pi\)

(See \(a_n\) instead)

Twists

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

    

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