Embedded newform 471.2.l.b.50.3

• Label: 471.2.l.b.50.3
• Level: $471$
• Weight: $2$
• Character: 471.50
• Analytic conductor: $3.761$
• Analytic rank: $0$
• Dimension: $200$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	471.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.76095393520\)
Analytic rank:	\(0\)
Dimension:	\(200\)
Relative dimension:	\(50\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			50.3
Character	\(\chi\)	\(=\)	471.50
Dual form			471.2.l.b.179.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.82955 + 1.82955i) q^{2} +(-1.72845 - 0.111624i) q^{3} -4.69449i q^{4} +(0.00122739 + 0.00458067i) q^{5} +(3.36650 - 2.95806i) q^{6} +(3.09988 - 3.09988i) q^{7} +(4.92970 + 4.92970i) q^{8} +(2.97508 + 0.385872i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 200 q - 6 q^{3} - 10 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(158\)	\(319\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.82955	+	1.82955i	−1.29369	+	1.29369i	−0.361196	+	0.932490i	\(0.617631\pi\)
							−0.932490	+	0.361196i	\(0.882369\pi\)
\(3\)	−1.72845	−	0.111624i	−0.997921	−	0.0644459i
							\(5\)	0.00122739	+	0.00458067i	0.000548904	+	0.00204854i	0.966200	−	0.257795i	\(-0.0829958\pi\)
−0.965651	+	0.259843i	\(0.916329\pi\)
\(7\)	3.09988	−	3.09988i	1.17164	−	1.17164i	0.189827	−	0.981818i	\(-0.439207\pi\)
							0.981818	−	0.189827i	\(-0.0607928\pi\)
\(11\)	−4.85159	−	2.80106i	−1.46281	−	0.844553i	−0.463668	−	0.886009i	\(-0.653467\pi\)
							−0.999140	+	0.0414562i	\(0.986800\pi\)
\(13\)	−2.70430	+	1.56133i	−0.750039	+	0.433035i	−0.825708	−	0.564098i	\(-0.809224\pi\)
							0.0756693	+	0.997133i	\(0.475891\pi\)
\(17\)	0.672268	+	0.388134i	0.163049	+	0.0941364i	0.579304	−	0.815111i	\(-0.303324\pi\)
							−0.416255	+	0.909248i	\(0.636658\pi\)
\(19\)	−3.63715	+	6.29973i	−0.834419	+	1.44526i	0.0600835	+	0.998193i	\(0.480863\pi\)
							−0.894503	+	0.447063i	\(0.852470\pi\)
\(23\)	−1.75519	−	1.75519i	−0.365982	−	0.365982i	0.500027	−	0.866010i	\(-0.333323\pi\)
							−0.866010	+	0.500027i	\(0.833323\pi\)
\(29\)	3.66506	−	3.66506i	0.680584	−	0.680584i	−0.279548	−	0.960132i	\(-0.590185\pi\)
							0.960132	+	0.279548i	\(0.0901845\pi\)
\(31\)	−7.50177	+	4.33115i	−1.34736	+	0.777897i	−0.987875	−	0.155254i	\(-0.950380\pi\)
							−0.359483	+	0.933152i	\(0.617047\pi\)
\(37\)	0.433267	+	0.750440i	0.0712286	+	0.123372i	0.899440	−	0.437044i	\(-0.143975\pi\)
							−0.828211	+	0.560416i	\(0.810641\pi\)
\(41\)	−7.65319	+	7.65319i	−1.19523	+	1.19523i	−0.219649	+	0.975579i	\(0.570491\pi\)
							−0.975579	+	0.219649i	\(0.929509\pi\)
\(43\)	1.60036	−	5.97263i	0.244053	−	0.910817i	−0.729805	−	0.683656i	\(-0.760389\pi\)
							0.973857	−	0.227161i	\(-0.0729444\pi\)
\(47\)	−1.56757	+	0.905036i	−0.228653	+	0.132013i	−0.609951	−	0.792439i	\(-0.708811\pi\)
							0.381297	+	0.924452i	\(0.375477\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.2.l.b.50.3	✓	200
3.2	odd	2	inner	471.2.l.b.50.48	yes	200
157.22	odd	12	inner	471.2.l.b.179.48	yes	200
471.179	even	12	inner	471.2.l.b.179.3	yes	200

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.2.l.b.50.3	✓	200	1.1	even	1	trivial
471.2.l.b.50.48	yes	200	3.2	odd	2	inner
471.2.l.b.179.3	yes	200	471.179	even	12	inner
471.2.l.b.179.48	yes	200	157.22	odd	12	inner