Embedded newform 471.2.q.b.37.5

• Label: 471.2.q.b.37.5
• Level: $471$
• Weight: $2$
• Character: 471.37
• Analytic conductor: $3.761$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	471.q (of order \(39\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.76095393520\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(14\) over \(\Q(\zeta_{39})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{39}]$

Embedding invariants

Embedding label			37.5
Character	\(\chi\)	\(=\)	471.37
Dual form			471.2.q.b.331.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33614 + 0.701259i) q^{2} +(-0.996757 + 0.0804666i) q^{3} +(0.157371 - 0.227991i) q^{4} +(2.79334 - 2.90817i) q^{5} +(1.27538 - 0.806500i) q^{6} +(1.85954 + 2.69400i) q^{7} +(0.313387 - 2.58098i) q^{8} +(0.987050 - 0.160411i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 2 q^{2} + 14 q^{3} - 26 q^{4} + 2 q^{5} + q^{6} - 4 q^{7} + 14 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/471\mathbb{Z}\right)^\times\).

\(n\)	\(158\)	\(319\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{29}{39}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33614	+	0.701259i	−0.944792	+	0.495865i	−0.865457	−	0.500983i	\(-0.832972\pi\)
							−0.0793351	+	0.996848i	\(0.525280\pi\)
\(3\)	−0.996757	+	0.0804666i	−0.575478	+	0.0464574i
							\(5\)	2.79334	−	2.90817i	1.24922	−	1.30057i	0.312745	−	0.949837i	\(-0.398751\pi\)
0.936474	−	0.350738i	\(-0.114069\pi\)
\(7\)	1.85954	+	2.69400i	0.702839	+	1.01824i	0.998026	+	0.0627983i	\(0.0200025\pi\)
							−0.295188	+	0.955439i	\(0.595382\pi\)
\(11\)	−1.33544	+	0.844479i	−0.402649	+	0.254620i	−0.720406	−	0.693552i	\(-0.756045\pi\)
							0.317757	+	0.948172i	\(0.397070\pi\)
\(13\)	−2.44018	−	4.22652i	−0.676784	−	1.17222i	−0.975944	−	0.218022i	\(-0.930040\pi\)
							0.299160	−	0.954203i	\(-0.403294\pi\)
\(17\)	3.18873	−	3.31982i	0.773380	−	0.805174i	−0.211689	−	0.977337i	\(-0.567896\pi\)
							0.985068	+	0.172163i	\(0.0550757\pi\)
\(19\)	−5.22489	−	3.92625i	−1.19867	−	0.900744i	−0.201773	−	0.979432i	\(-0.564670\pi\)
							−0.996899	+	0.0786887i	\(0.974927\pi\)
\(23\)	0.941139	+	2.48158i	0.196241	+	0.517445i	0.996629	−	0.0820370i	\(-0.0261426\pi\)
							−0.800388	+	0.599482i	\(0.795373\pi\)
\(29\)	3.81850	−	0.941177i	0.709079	−	0.174772i	0.131749	−	0.991283i	\(-0.457941\pi\)
							0.577330	+	0.816511i	\(0.304095\pi\)
\(31\)	1.74433	+	0.582342i	0.313290	+	0.104592i	0.468965	−	0.883217i	\(-0.344627\pi\)
							−0.155675	+	0.987808i	\(0.549755\pi\)
\(37\)	1.45474	+	1.51455i	0.239158	+	0.248990i	0.829955	−	0.557831i	\(-0.188366\pi\)
							−0.590797	+	0.806820i	\(0.701187\pi\)
\(41\)	2.74067	−	7.22656i	0.428021	−	1.12860i	−0.531524	−	0.847044i	\(-0.678380\pi\)
							0.959545	−	0.281556i	\(-0.0908505\pi\)
\(43\)	6.35772	+	0.513248i	0.969543	+	0.0782696i	0.555080	−	0.831797i	\(-0.312688\pi\)
							0.414463	+	0.910066i	\(0.363970\pi\)
\(47\)	1.49139	+	5.14888i	0.217542	+	0.751042i	0.992841	+	0.119447i	\(0.0381121\pi\)
							−0.775299	+	0.631595i	\(0.782401\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.q.b.37.5	✓	336
157.17	even	39	inner	471.2.q.b.331.5	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.q.b.37.5	✓	336	1.1	even	1	trivial
471.2.q.b.331.5	yes	336	157.17	even	39	inner