Embedded newform 392.2.bc.a.283.52

• Label: 392.2.bc.a.283.52
• Level: $392$
• Weight: $2$
• Character: 392.283
• 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			283.52
Character	\(\chi\)	\(=\)	392.283
Dual form			392.2.bc.a.187.52

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.39914 - 0.205943i) q^{2} +(-0.0159293 - 0.105684i) q^{3} +(1.91518 - 0.576284i) q^{4} +(0.474246 - 1.20836i) q^{5} +(-0.0440522 - 0.144586i) q^{6} +(0.874709 + 2.49697i) q^{7} +(2.56091 - 1.20072i) q^{8} +(2.85580 - 0.880899i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 648 q - 13 q^{2} - 22 q^{3} - 13 q^{4} - 14 q^{6} - 4 q^{8} - 76 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(197\)	\(295\)	\(297\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(e\left(\frac{19}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.39914	−	0.205943i	0.989340	−	0.145623i
							\(3\)	−0.0159293	−	0.105684i	−0.00919681	−	0.0610168i	0.983762	−	0.179480i	\(-0.0574414\pi\)
−0.992958	+	0.118463i	\(0.962203\pi\)
\(5\)	0.474246	−	1.20836i	0.212089	−	0.540395i	−0.784752	−	0.619809i	\(-0.787210\pi\)
							0.996842	+	0.0794147i	\(0.0253051\pi\)
\(7\)	0.874709	+	2.49697i	0.330609	+	0.943768i
							\(11\)	−5.31040	−	1.63804i	−1.60114	−	0.493888i	−0.639640	−	0.768675i	\(-0.720916\pi\)
−0.961505	+	0.274787i	\(0.911393\pi\)
\(13\)	0.995053	+	4.35961i	0.275978	+	1.20914i	0.902829	+	0.430000i	\(0.141486\pi\)
							−0.626851	+	0.779139i	\(0.715657\pi\)
\(17\)	−2.89123	−	4.24065i	−0.701225	−	1.02851i	−0.997307	−	0.0733457i	\(-0.976632\pi\)
							0.296081	−	0.955163i	\(-0.404320\pi\)
\(19\)	−1.79329	+	1.03535i	−0.411408	+	0.237526i	−0.691394	−	0.722477i	\(-0.743003\pi\)
							0.279987	+	0.960004i	\(0.409670\pi\)
\(23\)	3.08354	−	4.52272i	0.642963	−	0.943053i	−0.356997	−	0.934105i	\(-0.616199\pi\)
							0.999960	−	0.00894771i	\(-0.00284818\pi\)
\(29\)	0.00224034	−	0.00465212i	0.000416021	−	0.000863877i	−0.900761	−	0.434316i	\(-0.856990\pi\)
							0.901177	+	0.433452i	\(0.142705\pi\)
\(31\)	−3.36177	+	5.82275i	−0.603791	+	1.04580i	0.388450	+	0.921470i	\(0.373011\pi\)
							−0.992241	+	0.124327i	\(0.960323\pi\)
\(37\)	−4.79579	+	0.359395i	−0.788423	+	0.0590841i	−0.462854	−	0.886434i	\(-0.653175\pi\)
							−0.325569	+	0.945518i	\(0.605556\pi\)
\(41\)	1.08487	−	0.865153i	0.169428	−	0.135114i	−0.535105	−	0.844786i	\(-0.679728\pi\)
							0.704533	+	0.709671i	\(0.251157\pi\)
\(43\)	−0.0907032	+	0.113738i	−0.0138321	+	0.0173449i	−0.788699	−	0.614779i	\(-0.789245\pi\)
							0.774867	+	0.632124i	\(0.217817\pi\)
\(47\)	−8.08777	+	7.50435i	−1.17972	+	1.09462i	−0.186022	+	0.982545i	\(0.559560\pi\)
							−0.993700	+	0.112077i	\(0.964250\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.bc.a.283.52	yes	648
8.3	odd	2	inner	392.2.bc.a.283.33	yes	648
49.40	odd	42	inner	392.2.bc.a.187.33	✓	648
392.187	even	42	inner	392.2.bc.a.187.52	yes	648

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.2.bc.a.187.33	✓	648	49.40	odd	42	inner
392.2.bc.a.187.52	yes	648	392.187	even	42	inner
392.2.bc.a.283.33	yes	648	8.3	odd	2	inner
392.2.bc.a.283.52	yes	648	1.1	even	1	trivial