Embedded newform 201.3.g.b.29.15

• Label: 201.3.g.b.29.15
• 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			29.15
Character	\(\chi\)	\(=\)	201.29
Dual form			201.3.g.b.104.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29938 - 0.750199i) q^{2} +(2.00560 + 2.23105i) q^{3} +(-0.874403 - 1.51451i) q^{4} -2.91605i q^{5} +(-0.932314 - 4.40358i) q^{6} +(-6.24309 - 10.8134i) q^{7} +8.62550i q^{8} +(-0.955134 + 8.94917i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 6 q^{3} + 82 q^{4} + 3 q^{6} - 10 q^{7} + 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29938	−	0.750199i	−0.649691	−	0.375099i	0.138647	−	0.990342i	\(-0.455725\pi\)
							−0.788338	+	0.615242i	\(0.789058\pi\)
\(3\)	2.00560	+	2.23105i	0.668533	+	0.743682i
							\(5\)		−	2.91605i		−	0.583209i	−0.956539	−	0.291605i	\(-0.905811\pi\)
0.956539	−	0.291605i	\(-0.0941891\pi\)
\(7\)	−6.24309	−	10.8134i	−0.891870	−	1.54476i	−0.837631	−	0.546237i	\(-0.816060\pi\)
							−0.0542394	−	0.998528i	\(-0.517273\pi\)
\(11\)	−5.22494	+	3.01662i	−0.474994	+	0.274238i	−0.718328	−	0.695705i	\(-0.755092\pi\)
							0.243334	+	0.969943i	\(0.421759\pi\)
\(13\)	−8.25323	+	14.2950i	−0.634864	+	1.09962i	0.351680	+	0.936120i	\(0.385610\pi\)
							−0.986544	+	0.163496i	\(0.947723\pi\)
\(17\)	−22.0606	−	12.7367i	−1.29768	−	0.749219i	−0.317681	−	0.948198i	\(-0.602904\pi\)
							−0.980004	+	0.198979i	\(0.936237\pi\)
\(19\)	4.82969	−	8.36527i	0.254194	−	0.440278i	−0.710482	−	0.703715i	\(-0.751523\pi\)
							0.964676	+	0.263438i	\(0.0848564\pi\)
\(23\)	−26.3817	−	15.2315i	−1.14703	−	0.662239i	−0.198870	−	0.980026i	\(-0.563727\pi\)
							−0.948162	+	0.317787i	\(0.897060\pi\)
\(29\)	4.04518	−	2.33549i	0.139489	−	0.0805340i	−0.428631	−	0.903479i	\(-0.641004\pi\)
							0.568120	+	0.822945i	\(0.307671\pi\)
\(31\)	−20.3967	−	35.3281i	−0.657957	−	1.13962i	−0.981144	−	0.193280i	\(-0.938087\pi\)
							0.323187	−	0.946335i	\(-0.395246\pi\)
\(37\)	−5.99853	+	10.3898i	−0.162122	+	0.280804i	−0.935630	−	0.352983i	\(-0.885167\pi\)
							0.773507	+	0.633788i	\(0.218501\pi\)
\(41\)	−10.4648	+	6.04185i	−0.255239	+	0.147362i	−0.622161	−	0.782890i	\(-0.713745\pi\)
							0.366922	+	0.930252i	\(0.380412\pi\)
\(43\)	56.0143			1.30266			0.651329	−	0.758795i	\(-0.274212\pi\)
							0.651329	+	0.758795i	\(0.274212\pi\)
\(47\)	−17.7786	+	10.2645i	−0.378268	+	0.218393i	−0.677064	−	0.735924i	\(-0.736748\pi\)
							0.298796	+	0.954317i	\(0.403415\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.g.b.29.15	✓	84
3.2	odd	2	inner	201.3.g.b.29.28	yes	84
67.37	even	3	inner	201.3.g.b.104.28	yes	84
201.104	odd	6	inner	201.3.g.b.104.15	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.g.b.29.15	✓	84	1.1	even	1	trivial
201.3.g.b.29.28	yes	84	3.2	odd	2	inner
201.3.g.b.104.15	yes	84	201.104	odd	6	inner
201.3.g.b.104.28	yes	84	67.37	even	3	inner