Embedded newform 392.2.bc.a.59.42

• Label: 392.2.bc.a.59.42
• Level: $392$
• Weight: $2$
• Character: 392.59
• 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			59.42
Character	\(\chi\)	\(=\)	392.59
Dual form			392.2.bc.a.299.42

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.967776 - 1.03122i) q^{2} +(-0.513626 - 0.753351i) q^{3} +(-0.126818 - 1.99598i) q^{4} +(0.131428 + 1.75378i) q^{5} +(-1.27394 - 0.199415i) q^{6} +(2.59747 + 0.503146i) q^{7} +(-2.18102 - 1.80088i) q^{8} +(0.792297 - 2.01874i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 22 q^{3} - 13 q^{4} - 14 q^{6} - 4 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{13}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.967776	−	1.03122i	0.684321	−	0.729181i
							\(3\)	−0.513626	−	0.753351i	−0.296542	−	0.434948i	0.648833	−	0.760931i	\(-0.275257\pi\)
−0.945376	+	0.325983i	\(0.894305\pi\)
\(5\)	0.131428	+	1.75378i	0.0587763	+	0.784316i	0.946246	+	0.323449i	\(0.104842\pi\)
							−0.887469	+	0.460867i	\(0.847539\pi\)
\(7\)	2.59747	+	0.503146i	0.981751	+	0.190171i
							\(11\)	−1.93199	−	4.92263i	−0.582517	−	1.48423i	−0.853213	−	0.521563i	\(-0.825349\pi\)
0.270696	−	0.962665i	\(-0.412746\pi\)
\(13\)	1.33852	−	1.67845i	0.371239	−	0.465519i	−0.560761	−	0.827978i	\(-0.689491\pi\)
							0.932000	+	0.362459i	\(0.118063\pi\)
\(17\)	−4.42994	+	4.77433i	−1.07442	+	1.15795i	−0.0875919	+	0.996156i	\(0.527917\pi\)
							−0.986825	+	0.161790i	\(0.948273\pi\)
\(19\)	5.47732	−	3.16233i	1.25658	−	0.725488i	0.284174	−	0.958773i	\(-0.408281\pi\)
							0.972409	+	0.233284i	\(0.0749473\pi\)
\(23\)	3.24772	+	3.50021i	0.677196	+	0.729844i	0.974363	−	0.224981i	\(-0.0722321\pi\)
							−0.297167	+	0.954826i	\(0.596042\pi\)
\(29\)	−5.75422	−	1.31336i	−1.06853	−	0.243886i	−0.348133	−	0.937445i	\(-0.613184\pi\)
							−0.720399	+	0.693560i	\(0.756041\pi\)
\(31\)	0.509359	−	0.882235i	0.0914835	−	0.158454i	−0.816652	−	0.577130i	\(-0.804172\pi\)
							0.908136	+	0.418676i	\(0.137506\pi\)
\(37\)	1.61008	+	5.21976i	0.264696	+	0.858123i	0.986178	+	0.165690i	\(0.0529850\pi\)
							−0.721482	+	0.692433i	\(0.756539\pi\)
\(41\)	−0.546709	+	1.13525i	−0.0853816	+	0.177297i	−0.939288	−	0.343130i	\(-0.888513\pi\)
							0.853906	+	0.520427i	\(0.174227\pi\)
\(43\)	−4.10880	+	1.97870i	−0.626587	+	0.301748i	−0.720101	−	0.693870i	\(-0.755904\pi\)
							0.0935138	+	0.995618i	\(0.470190\pi\)
\(47\)	6.05368	+	0.912446i	0.883020	+	0.133094i	0.574871	−	0.818244i	\(-0.305052\pi\)
							0.308149	+	0.951338i	\(0.400290\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.bc.a.59.42	yes	648
8.3	odd	2	inner	392.2.bc.a.59.36	✓	648
49.5	odd	42	inner	392.2.bc.a.299.36	yes	648
392.299	even	42	inner	392.2.bc.a.299.42	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.bc.a.59.36	✓	648	8.3	odd	2	inner
392.2.bc.a.59.42	yes	648	1.1	even	1	trivial
392.2.bc.a.299.36	yes	648	49.5	odd	42	inner
392.2.bc.a.299.42	yes	648	392.299	even	42	inner