Embedded newform 201.3.g.b.29.12

• Label: 201.3.g.b.29.12
• Level: $201$
• Weight: $3$
• Character: 201.29
• Analytic conductor: $5.477$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	201.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.47685331364\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(42\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			29.12
Character	\(\chi\)	\(=\)	201.29
Dual form			201.3.g.b.104.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.76138 - 1.01693i) q^{2} +(2.83847 + 0.971138i) q^{3} +(0.0683121 + 0.118320i) q^{4} -2.80344i q^{5} +(-4.01204 - 4.59708i) q^{6} +(5.21582 + 9.03406i) q^{7} +7.85760i q^{8} +(7.11378 + 5.51309i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 6 q^{3} + 82 q^{4} + 3 q^{6} - 10 q^{7} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.76138	−	1.01693i	−0.880691	−	0.508467i	−0.00980497	−	0.999952i	\(-0.503121\pi\)
							−0.870886	+	0.491485i	\(0.836454\pi\)
\(3\)	2.83847	+	0.971138i	0.946155	+	0.323713i
							\(5\)		−	2.80344i		−	0.560687i	−0.959900	−	0.280344i	\(-0.909552\pi\)
0.959900	−	0.280344i	\(-0.0904484\pi\)
\(7\)	5.21582	+	9.03406i	0.745117	+	1.29058i	0.950140	+	0.311823i	\(0.100940\pi\)
							−0.205024	+	0.978757i	\(0.565727\pi\)
\(11\)	2.46092	−	1.42081i	0.223720	−	0.129165i	−0.383952	−	0.923353i	\(-0.625437\pi\)
							0.607672	+	0.794188i	\(0.292104\pi\)
\(13\)	1.23271	−	2.13511i	0.0948238	−	0.164240i	−0.814711	−	0.579867i	\(-0.803105\pi\)
							0.909535	+	0.415627i	\(0.136438\pi\)
\(17\)	3.41478	+	1.97153i	0.200870	+	0.115972i	0.597061	−	0.802196i	\(-0.296335\pi\)
							−0.396191	+	0.918168i	\(0.629668\pi\)
\(19\)	−2.36839	+	4.10217i	−0.124652	+	0.215903i	−0.921597	−	0.388149i	\(-0.873115\pi\)
							0.796945	+	0.604052i	\(0.206448\pi\)
\(23\)	14.4304	+	8.33139i	0.627408	+	0.362234i	0.779748	−	0.626094i	\(-0.215347\pi\)
							−0.152339	+	0.988328i	\(0.548681\pi\)
\(29\)	27.5632	−	15.9136i	0.950455	−	0.548746i	0.0572328	−	0.998361i	\(-0.481772\pi\)
							0.893222	+	0.449615i	\(0.148439\pi\)
\(31\)	−24.1450	−	41.8203i	−0.778870	−	1.34904i	−0.932594	−	0.360928i	\(-0.882460\pi\)
							0.153724	−	0.988114i	\(-0.450873\pi\)
\(37\)	−15.1770	+	26.2873i	−0.410188	+	0.710467i	−0.994910	−	0.100767i	\(-0.967870\pi\)
							0.584722	+	0.811234i	\(0.301204\pi\)
\(41\)	38.8350	−	22.4214i	0.947194	−	0.546863i	0.0549861	−	0.998487i	\(-0.482489\pi\)
							0.892208	+	0.451624i	\(0.149155\pi\)
\(43\)	−21.2915			−0.495151			−0.247576	−	0.968869i	\(-0.579634\pi\)
							−0.247576	+	0.968869i	\(0.579634\pi\)
\(47\)	−12.9689	+	7.48760i	−0.275934	+	0.159311i	−0.631581	−	0.775310i	\(-0.717594\pi\)
							0.355647	+	0.934620i	\(0.384260\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.g.b.29.12	✓	84
3.2	odd	2	inner	201.3.g.b.29.31	yes	84
67.37	even	3	inner	201.3.g.b.104.31	yes	84
201.104	odd	6	inner	201.3.g.b.104.12	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.g.b.29.12	✓	84	1.1	even	1	trivial
201.3.g.b.29.31	yes	84	3.2	odd	2	inner
201.3.g.b.104.12	yes	84	201.104	odd	6	inner
201.3.g.b.104.31	yes	84	67.37	even	3	inner