Embedded newform 392.2.z.a.205.2

• Label: 392.2.z.a.205.2
• Level: $392$
• Weight: $2$
• Character: 392.205
• 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			205.2
Character	\(\chi\)	\(=\)	392.205
Dual form			392.2.z.a.109.2

$q$-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 \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.z.a.205.2	yes	648
8.5	even	2	inner	392.2.z.a.205.25	yes	648
49.11	even	21	inner	392.2.z.a.109.25	yes	648
392.109	even	42	inner	392.2.z.a.109.2	✓	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.z.a.109.2	✓	648	392.109	even	42	inner
392.2.z.a.109.25	yes	648	49.11	even	21	inner
392.2.z.a.205.2	yes	648	1.1	even	1	trivial
392.2.z.a.205.25	yes	648	8.5	even	2	inner