Embedded newform 392.2.bc.a.187.38

• Label: 392.2.bc.a.187.38
• Level: $392$
• Weight: $2$
• Character: 392.187
• 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			187.38
Character	\(\chi\)	\(=\)	392.187
Dual form			392.2.bc.a.283.38

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.790032 - 1.17297i) q^{2} +(0.339641 - 2.25337i) q^{3} +(-0.751699 - 1.85336i) q^{4} +(-0.471958 - 1.20253i) q^{5} +(-2.37480 - 2.17862i) q^{6} +(-1.61261 + 2.09749i) q^{7} +(-2.76780 - 0.582497i) q^{8} +(-2.09561 - 0.646411i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 22 q^{3} - 13 q^{4} - 14 q^{6} - 4 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{23}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.790032	−	1.17297i	0.558637	−	0.829412i
							\(3\)	0.339641	−	2.25337i	0.196092	−	1.30099i	−0.646700	−	0.762745i	\(-0.723851\pi\)
0.842791	−	0.538240i	\(-0.180911\pi\)
\(5\)	−0.471958	−	1.20253i	−0.211066	−	0.537788i	0.785658	−	0.618661i	\(-0.212324\pi\)
							−0.996724	+	0.0808729i	\(0.974229\pi\)
\(7\)	−1.61261	+	2.09749i	−0.609511	+	0.792778i
							\(11\)	3.85241	−	1.18831i	1.16155	−	0.358289i	0.346705	−	0.937974i	\(-0.387301\pi\)
0.814841	+	0.579685i	\(0.196824\pi\)
\(13\)	0.375174	−	1.64374i	0.104054	−	0.455892i	−0.895878	−	0.444299i	\(-0.853453\pi\)
							0.999933	−	0.0115929i	\(-0.00369020\pi\)
\(17\)	−0.707547	+	1.03778i	−0.171605	+	0.251699i	−0.902401	−	0.430898i	\(-0.858197\pi\)
							0.730795	+	0.682597i	\(0.239149\pi\)
\(19\)	−2.10761	−	1.21683i	−0.483519	−	0.279160i	0.238363	−	0.971176i	\(-0.423389\pi\)
							−0.721882	+	0.692016i	\(0.756723\pi\)
\(23\)	−0.847421	−	1.24294i	−0.176700	−	0.259171i	0.727664	−	0.685934i	\(-0.240606\pi\)
							−0.904363	+	0.426763i	\(0.859654\pi\)
\(29\)	3.39373	+	7.04715i	0.630199	+	1.30862i	0.934472	+	0.356037i	\(0.115872\pi\)
							−0.304273	+	0.952585i	\(0.598413\pi\)
\(31\)	0.410826	+	0.711571i	0.0737865	+	0.127802i	0.900558	−	0.434736i	\(-0.143158\pi\)
							−0.826771	+	0.562538i	\(0.809825\pi\)
\(37\)	5.95446	+	0.446225i	0.978907	+	0.0733589i	0.554563	−	0.832142i	\(-0.312886\pi\)
							0.424344	+	0.905501i	\(0.360505\pi\)
\(41\)	7.01655	+	5.59551i	1.09580	+	0.873872i	0.992677	−	0.120796i	\(-0.0385448\pi\)
							0.103124	+	0.994669i	\(0.467116\pi\)
\(43\)	−5.77050	−	7.23598i	−0.879993	−	1.10348i	−0.993933	−	0.109988i	\(-0.964919\pi\)
							0.113940	−	0.993488i	\(-0.463653\pi\)
\(47\)	−8.04885	−	7.46824i	−1.17404	−	1.08935i	−0.994406	−	0.105622i	\(-0.966317\pi\)
							−0.179638	−	0.983733i	\(-0.557493\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.bc.a.187.38	yes	648
8.3	odd	2	inner	392.2.bc.a.187.12	✓	648
49.38	odd	42	inner	392.2.bc.a.283.12	yes	648
392.283	even	42	inner	392.2.bc.a.283.38	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.bc.a.187.12	✓	648	8.3	odd	2	inner
392.2.bc.a.187.38	yes	648	1.1	even	1	trivial
392.2.bc.a.283.12	yes	648	49.38	odd	42	inner
392.2.bc.a.283.38	yes	648	392.283	even	42	inner