Embedded newform 201.3.g.b.29.10

• Label: 201.3.g.b.29.10
• 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			29.10
Character	\(\chi\)	\(=\)	201.29
Dual form			201.3.g.b.104.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.22999 - 1.28749i) q^{2} +(-2.28852 + 1.93976i) q^{3} +(1.31525 + 2.27807i) q^{4} -0.0457250i q^{5} +(7.60080 - 1.37922i) q^{6} +(-4.62513 - 8.01096i) q^{7} +3.52645i q^{8} +(1.47465 - 8.87837i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 6 q^{3} + 82 q^{4} + 3 q^{6} - 10 q^{7} + 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.22999	−	1.28749i	−1.11500	−	0.643744i	−0.174878	−	0.984590i	\(-0.555953\pi\)
							−0.940119	+	0.340847i	\(0.889286\pi\)
\(3\)	−2.28852	+	1.93976i	−0.762840	+	0.646587i
							\(5\)		−	0.0457250i		−	0.00914500i	−0.999990	−	0.00457250i	\(-0.998545\pi\)
0.999990	−	0.00457250i	\(-0.00145548\pi\)
\(7\)	−4.62513	−	8.01096i	−0.660733	−	1.14442i	−0.980423	−	0.196901i	\(-0.936912\pi\)
							0.319691	−	0.947522i	\(-0.396421\pi\)
\(11\)	−15.6453	+	9.03281i	−1.42230	+	0.821164i	−0.996495	−	0.0836519i	\(-0.973342\pi\)
							−0.425803	+	0.904816i	\(0.640008\pi\)
\(13\)	3.86109	−	6.68760i	0.297007	−	0.514431i	−0.678443	−	0.734653i	\(-0.737345\pi\)
							0.975450	+	0.220222i	\(0.0706783\pi\)
\(17\)	24.1115	+	13.9208i	1.41832	+	0.818870i	0.996152	−	0.0876462i	\(-0.0279345\pi\)
							0.422172	+	0.906516i	\(0.361268\pi\)
\(19\)	−8.96629	+	15.5301i	−0.471910	+	0.817372i	−0.999483	−	0.0321373i	\(-0.989769\pi\)
							0.527573	+	0.849509i	\(0.323102\pi\)
\(23\)	−2.77975	−	1.60489i	−0.120859	−	0.0697779i	0.438352	−	0.898803i	\(-0.355562\pi\)
							−0.559211	+	0.829026i	\(0.688896\pi\)
\(29\)	36.3339	−	20.9774i	1.25289	−	0.723359i	0.281211	−	0.959646i	\(-0.409264\pi\)
							0.971683	+	0.236287i	\(0.0759306\pi\)
\(31\)	15.9484	+	27.6234i	0.514464	+	0.891077i	0.999859	+	0.0167824i	\(0.00534224\pi\)
							−0.485396	+	0.874295i	\(0.661324\pi\)
\(37\)	−14.0862	+	24.3980i	−0.380708	+	0.659405i	−0.991164	−	0.132645i	\(-0.957653\pi\)
							0.610456	+	0.792050i	\(0.290986\pi\)
\(41\)	−32.4444	+	18.7318i	−0.791326	+	0.456873i	−0.840429	−	0.541921i	\(-0.817697\pi\)
							0.0491028	+	0.998794i	\(0.484364\pi\)
\(43\)	70.6486			1.64299			0.821495	−	0.570216i	\(-0.193140\pi\)
							0.821495	+	0.570216i	\(0.193140\pi\)
\(47\)	12.5830	−	7.26477i	0.267722	−	0.154570i	−0.360130	−	0.932902i	\(-0.617268\pi\)
							0.627852	+	0.778333i	\(0.283934\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.g.b.29.10	✓	84
3.2	odd	2	inner	201.3.g.b.29.33	yes	84
67.37	even	3	inner	201.3.g.b.104.33	yes	84
201.104	odd	6	inner	201.3.g.b.104.10	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.g.b.29.10	✓	84	1.1	even	1	trivial
201.3.g.b.29.33	yes	84	3.2	odd	2	inner
201.3.g.b.104.10	yes	84	201.104	odd	6	inner
201.3.g.b.104.33	yes	84	67.37	even	3	inner