Embedded newform 201.3.g.b.29.7

• Label: 201.3.g.b.29.7
• 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			29.7
Character	\(\chi\)	\(=\)	201.29
Dual form			201.3.g.b.104.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.37828 - 1.37310i) q^{2} +(0.338320 + 2.98086i) q^{3} +(1.77082 + 3.06716i) q^{4} +5.91841i q^{5} +(3.28841 - 7.55389i) q^{6} +(1.61899 + 2.80418i) q^{7} +1.25873i q^{8} +(-8.77108 + 2.01697i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 6 q^{3} + 82 q^{4} + 3 q^{6} - 10 q^{7} + 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.37828	−	1.37310i	−1.18914	−	0.686551i	−0.231030	−	0.972947i	\(-0.574210\pi\)
							−0.958112	+	0.286395i	\(0.907543\pi\)
\(3\)	0.338320	+	2.98086i	0.112773	+	0.993621i
							\(5\)			5.91841i			1.18368i	0.806055	+	0.591841i	\(0.201599\pi\)
−0.806055	+	0.591841i	\(0.798401\pi\)
\(7\)	1.61899	+	2.80418i	0.231285	+	0.400597i	0.958186	−	0.286145i	\(-0.0923737\pi\)
							−0.726902	+	0.686742i	\(0.759040\pi\)
\(11\)	0.716568	−	0.413711i	0.0651425	−	0.0376101i	−0.467075	−	0.884218i	\(-0.654692\pi\)
							0.532218	+	0.846608i	\(0.321359\pi\)
\(13\)	3.02795	−	5.24456i	0.232919	−	0.403427i	−0.725747	−	0.687962i	\(-0.758506\pi\)
							0.958666	+	0.284534i	\(0.0918390\pi\)
\(17\)	−19.4654	−	11.2384i	−1.14502	−	0.661080i	−0.197354	−	0.980332i	\(-0.563235\pi\)
							−0.947670	+	0.319252i	\(0.896568\pi\)
\(19\)	−8.74373	+	15.1446i	−0.460196	+	0.797083i	−0.998970	−	0.0453669i	\(-0.985554\pi\)
							0.538774	+	0.842450i	\(0.318888\pi\)
\(23\)	−8.43772	−	4.87152i	−0.366857	−	0.211805i	0.305227	−	0.952280i	\(-0.401268\pi\)
							−0.672085	+	0.740474i	\(0.734601\pi\)
\(29\)	−16.9909	+	9.80972i	−0.585894	+	0.338266i	−0.763472	−	0.645841i	\(-0.776507\pi\)
							0.177578	+	0.984107i	\(0.443174\pi\)
\(31\)	15.5885	+	27.0001i	0.502855	+	0.870970i	0.999995	+	0.00329959i	\(0.00105029\pi\)
							−0.497140	+	0.867670i	\(0.665616\pi\)
\(37\)	−27.3675	+	47.4019i	−0.739663	+	1.28113i	0.212985	+	0.977056i	\(0.431682\pi\)
							−0.952647	+	0.304078i	\(0.901652\pi\)
\(41\)	22.5982	−	13.0471i	0.551177	−	0.318222i	−0.198420	−	0.980117i	\(-0.563581\pi\)
							0.749596	+	0.661895i	\(0.230248\pi\)
\(43\)	−51.2268			−1.19132			−0.595660	−	0.803236i	\(-0.703110\pi\)
							−0.595660	+	0.803236i	\(0.703110\pi\)
\(47\)	−9.27400	+	5.35435i	−0.197319	+	0.113922i	−0.595404	−	0.803426i	\(-0.703008\pi\)
							0.398085	+	0.917348i	\(0.369675\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.3.g.b.29.7	✓	84
3.2	odd	2	inner	201.3.g.b.29.36	yes	84
67.37	even	3	inner	201.3.g.b.104.36	yes	84
201.104	odd	6	inner	201.3.g.b.104.7	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.3.g.b.29.7	✓	84	1.1	even	1	trivial
201.3.g.b.29.36	yes	84	3.2	odd	2	inner
201.3.g.b.104.7	yes	84	201.104	odd	6	inner
201.3.g.b.104.36	yes	84	67.37	even	3	inner