Embedded newform 201.2.f.b.38.19

• Label: 201.2.f.b.38.19
• Level: $201$
• Weight: $2$
• Character: 201.38
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			38.19
Character	\(\chi\)	\(=\)	201.38
Dual form			201.2.f.b.164.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30392 - 2.25846i) q^{2} +(1.69795 + 0.341983i) q^{3} +(-2.40042 - 4.15766i) q^{4} -3.98713 q^{5} +(2.98635 - 3.38884i) q^{6} +(2.31524 - 1.33671i) q^{7} -7.30417 q^{8} +(2.76609 + 1.16134i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 24 q^{4} - 2 q^{6} - 6 q^{7} + 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.30392	−	2.25846i	0.922012	−	1.59697i	0.125714	−	0.992067i	\(-0.459878\pi\)
							0.796298	−	0.604905i	\(-0.206789\pi\)
\(3\)	1.69795	+	0.341983i	0.980314	+	0.197444i
							\(5\)	−3.98713			−1.78310			−0.891549	−	0.452924i	\(-0.850381\pi\)
−0.891549	+	0.452924i	\(0.850381\pi\)
\(7\)	2.31524	−	1.33671i	0.875079	−	0.505227i	0.00604636	−	0.999982i	\(-0.498075\pi\)
							0.869033	+	0.494755i	\(0.164742\pi\)
\(11\)	2.06265	+	3.57261i	0.621912	+	1.07718i	0.989129	+	0.147048i	\(0.0469772\pi\)
							−0.367217	+	0.930135i	\(0.619690\pi\)
\(13\)	1.09982	+	0.634982i	0.305035	+	0.176112i	0.644703	−	0.764433i	\(-0.276981\pi\)
							−0.339667	+	0.940546i	\(0.610314\pi\)
\(17\)	1.45732	+	0.841382i	0.353451	+	0.204065i	0.666204	−	0.745769i	\(-0.267918\pi\)
							−0.312753	+	0.949834i	\(0.601251\pi\)
\(19\)	0.716358	−	1.24077i	0.164344	−	0.284652i	−0.772078	−	0.635528i	\(-0.780783\pi\)
							0.936422	+	0.350876i	\(0.114116\pi\)
\(23\)	−4.35792	−	2.51605i	−0.908690	−	0.524632i	−0.0286803	−	0.999589i	\(-0.509130\pi\)
							−0.880009	+	0.474956i	\(0.842464\pi\)
\(29\)	−1.28781	+	0.743520i	−0.239141	+	0.138068i	−0.614782	−	0.788697i	\(-0.710756\pi\)
							0.375641	+	0.926765i	\(0.377423\pi\)
\(31\)	−7.70707	+	4.44968i	−1.38423	+	0.799186i	−0.992657	−	0.120961i	\(-0.961403\pi\)
							−0.391574	+	0.920147i	\(0.628069\pi\)
\(37\)	−0.0952630	+	0.165000i	−0.0156611	+	0.0271259i	−0.873750	−	0.486376i	\(-0.838319\pi\)
							0.858089	+	0.513502i	\(0.171652\pi\)
\(41\)	−0.545494	−	0.944823i	−0.0851918	−	0.147557i	0.820281	−	0.571961i	\(-0.193817\pi\)
							−0.905473	+	0.424404i	\(0.860484\pi\)
\(43\)			2.84850i			0.434392i	0.976128	+	0.217196i	\(0.0696911\pi\)
							−0.976128	+	0.217196i	\(0.930309\pi\)
\(47\)	−5.34630	+	3.08669i	−0.779838	+	0.450239i	−0.836373	−	0.548161i	\(-0.815328\pi\)
							0.0565351	+	0.998401i	\(0.481995\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.f.b.38.19	yes	40
3.2	odd	2	inner	201.2.f.b.38.2	✓	40
67.30	odd	6	inner	201.2.f.b.164.2	yes	40
201.164	even	6	inner	201.2.f.b.164.19	yes	40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.f.b.38.2	✓	40	3.2	odd	2	inner
201.2.f.b.38.19	yes	40	1.1	even	1	trivial
201.2.f.b.164.2	yes	40	67.30	odd	6	inner
201.2.f.b.164.19	yes	40	201.164	even	6	inner