Embedded newform 392.2.x.a.253.10

• Label: 392.2.x.a.253.10
• Level: $392$
• Weight: $2$
• Character: 392.253
• Analytic conductor: $3.130$
• Analytic rank: $0$
• Dimension: $324$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.x (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.13013575923\)
Analytic rank:	\(0\)
Dimension:	\(324\)
Relative dimension:	\(54\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			253.10
Character	\(\chi\)	\(=\)	392.253
Dual form			392.2.x.a.141.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.20676 - 0.737382i) q^{2} +(-1.25303 - 2.60195i) q^{3} +(0.912535 + 1.77969i) q^{4} +(1.55709 + 3.23334i) q^{5} +(-0.406522 + 4.06389i) q^{6} +(-2.52866 + 0.778385i) q^{7} +(0.211097 - 2.82054i) q^{8} +(-3.32958 + 4.17517i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q - 5 q^{2} - 5 q^{4} - 9 q^{6} - 12 q^{7} + q^{8} + 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{6}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.20676	−	0.737382i	−0.853308	−	0.521408i
							\(3\)	−1.25303	−	2.60195i	−0.723439	−	1.50224i	−0.859280	−	0.511506i	\(-0.829088\pi\)
0.135841	−	0.990731i	\(-0.456626\pi\)
\(5\)	1.55709	+	3.23334i	0.696354	+	1.44599i	0.885783	+	0.464099i	\(0.153622\pi\)
							−0.189430	+	0.981894i	\(0.560664\pi\)
\(7\)	−2.52866	+	0.778385i	−0.955743	+	0.294202i
							\(11\)	2.55723	−	2.03932i	0.771033	−	0.614878i	−0.156905	−	0.987614i	\(-0.550152\pi\)
0.927938	+	0.372736i	\(0.121580\pi\)
\(13\)	4.25521	−	3.39342i	1.18018	−	0.941165i	0.181078	−	0.983469i	\(-0.442041\pi\)
							0.999105	+	0.0423042i	\(0.0134699\pi\)
\(17\)	−0.846538	−	3.70892i	−0.205316	−	0.899546i	−0.967637	−	0.252347i	\(-0.918797\pi\)
							0.762321	−	0.647199i	\(-0.224060\pi\)
\(19\)		−	2.92074i		−	0.670064i	−0.942207	−	0.335032i	\(-0.891253\pi\)
							0.942207	−	0.335032i	\(-0.108747\pi\)
\(23\)	0.516983	−	2.26505i	0.107798	−	0.472295i	−0.891997	−	0.452042i	\(-0.850696\pi\)
							0.999795	−	0.0202532i	\(-0.00644724\pi\)
\(29\)	1.62989	−	0.372012i	0.302663	−	0.0690808i	−0.0684924	−	0.997652i	\(-0.521819\pi\)
							0.371155	+	0.928571i	\(0.378962\pi\)
\(31\)	5.20799			0.935382			0.467691	−	0.883892i	\(-0.345086\pi\)
							0.467691	+	0.883892i	\(0.345086\pi\)
\(37\)	9.64017	−	2.20031i	1.58483	−	0.361728i	0.662787	−	0.748808i	\(-0.269373\pi\)
							0.922046	+	0.387080i	\(0.126516\pi\)
\(41\)	−2.84788	+	1.37146i	−0.444763	+	0.214187i	−0.642841	−	0.766000i	\(-0.722244\pi\)
							0.198078	+	0.980186i	\(0.436530\pi\)
\(43\)	1.83949	−	3.81974i	0.280519	−	0.582504i	−0.712335	−	0.701840i	\(-0.752362\pi\)
							0.992854	+	0.119336i	\(0.0380764\pi\)
\(47\)	0.0234141	+	0.0293603i	0.00341529	+	0.00428264i	0.783536	−	0.621346i	\(-0.213414\pi\)
							−0.780121	+	0.625629i	\(0.784842\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.x.a.253.10	yes	324
8.5	even	2	inner	392.2.x.a.253.22	yes	324
49.43	even	7	inner	392.2.x.a.141.22	yes	324
392.141	even	14	inner	392.2.x.a.141.10	✓	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.x.a.141.10	✓	324	392.141	even	14	inner
392.2.x.a.141.22	yes	324	49.43	even	7	inner
392.2.x.a.253.10	yes	324	1.1	even	1	trivial
392.2.x.a.253.22	yes	324	8.5	even	2	inner