Embedded newform 392.2.z.a.109.2

• Label: 392.2.z.a.109.2
• Level: $392$
• Weight: $2$
• Character: 392.109
• 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.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			109.2
Character	\(\chi\)	\(=\)	392.109
Dual form			392.2.z.a.205.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41279 + 0.0633787i) q^{2} +(-0.0286441 - 0.190041i) q^{3} +(1.99197 - 0.179082i) q^{4} +(-3.07037 - 1.20503i) q^{5} +(0.0525127 + 0.266673i) q^{6} +(0.384038 - 2.61773i) q^{7} +(-2.80289 + 0.379254i) q^{8} +(2.83142 - 0.873378i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 13 q^{4} - 6 q^{6} - 24 q^{7} - 16 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{20}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41279	+	0.0633787i	−0.998995	+	0.0448155i
							\(3\)	−0.0286441	−	0.190041i	−0.0165377	−	0.109720i	0.979039	−	0.203673i	\(-0.0652879\pi\)
−0.995577	+	0.0939526i	\(0.970050\pi\)
\(5\)	−3.07037	−	1.20503i	−1.37311	−	0.538906i	−0.439838	−	0.898077i	\(-0.644964\pi\)
							−0.933272	+	0.359171i	\(0.883060\pi\)
\(7\)	0.384038	−	2.61773i	0.145153	−	0.989409i
							\(11\)	−0.799625	+	2.59232i	−0.241096	+	0.781614i	0.751455	+	0.659784i	\(0.229352\pi\)
−0.992551	+	0.121829i	\(0.961124\pi\)
\(13\)	−2.39344	+	0.546288i	−0.663822	+	0.151513i	−0.541139	−	0.840933i	\(-0.682007\pi\)
							−0.122683	+	0.992446i	\(0.539150\pi\)
\(17\)	−4.97960	+	3.39504i	−1.20773	+	0.823417i	−0.988346	−	0.152221i	\(-0.951357\pi\)
							−0.219385	+	0.975638i	\(0.570405\pi\)
\(19\)	−2.12167	+	1.22495i	−0.486744	+	0.281022i	−0.723223	−	0.690615i	\(-0.757340\pi\)
							0.236479	+	0.971637i	\(0.424007\pi\)
\(23\)	−4.24195	−	2.89212i	−0.884509	−	0.603048i	0.0335742	−	0.999436i	\(-0.489311\pi\)
							−0.918083	+	0.396388i	\(0.870263\pi\)
\(29\)	2.12183	−	4.40602i	0.394013	−	0.818177i	−0.605733	−	0.795668i	\(-0.707120\pi\)
							0.999747	−	0.0225095i	\(-0.00716561\pi\)
\(31\)	−2.86186	+	4.95689i	−0.514005	+	0.890283i	0.485863	+	0.874035i	\(0.338505\pi\)
							−0.999868	+	0.0162480i	\(0.994828\pi\)
\(37\)	−3.51280	+	0.263248i	−0.577501	+	0.0432777i	−0.360279	−	0.932845i	\(-0.617319\pi\)
							−0.217222	+	0.976122i	\(0.569700\pi\)
\(41\)	−1.26219	−	1.58274i	−0.197121	−	0.247183i	0.673440	−	0.739242i	\(-0.264816\pi\)
							−0.870562	+	0.492059i	\(0.836244\pi\)
\(43\)	−8.29503	−	6.61506i	−1.26498	−	1.00879i	−0.998995	−	0.0448107i	\(-0.985732\pi\)
							−0.265985	−	0.963977i	\(-0.585697\pi\)
\(47\)	−4.91600	+	4.56138i	−0.717072	+	0.665345i	−0.951801	−	0.306715i	\(-0.900770\pi\)
							0.234730	+	0.972061i	\(0.424580\pi\)