Embedded newform 201.4.p.a.2.17

• Label: 201.4.p.a.2.17
• Level: $201$
• Weight: $4$
• Character: 201.2
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $1320$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.p (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(1320\)
Relative dimension:	\(66\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			2.17
Character	\(\chi\)	\(=\)	201.2
Dual form			201.4.p.a.101.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.147428 + 3.09489i) q^{2} +(-3.36537 + 3.95908i) q^{3} +(-1.59284 - 0.152098i) q^{4} +(11.9344 - 13.7731i) q^{5} +(-11.7568 - 10.9991i) q^{6} +(-6.80969 - 13.2090i) q^{7} +(-2.82203 + 19.6277i) q^{8} +(-4.34859 - 26.6475i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 22 q^{3} + 214 q^{4} + q^{6} + 22 q^{7} + 48 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{66}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.147428	+	3.09489i	−0.0521236	+	1.09421i	0.809649	+	0.586914i	\(0.199658\pi\)
							−0.861773	+	0.507295i	\(0.830646\pi\)
\(3\)	−3.36537	+	3.95908i	−0.647665	+	0.761925i
							\(5\)	11.9344	−	13.7731i	1.06745	−	1.23190i	0.0958178	−	0.995399i	\(-0.469453\pi\)
0.971631	−	0.236503i	\(-0.0760012\pi\)
\(7\)	−6.80969	−	13.2090i	−0.367689	−	0.713217i	0.630317	−	0.776338i	\(-0.282925\pi\)
							−0.998006	+	0.0631211i	\(0.979895\pi\)
\(11\)	10.7424	+	31.0382i	0.294452	+	0.850762i	0.991073	+	0.133323i	\(0.0425647\pi\)
							−0.696621	+	0.717439i	\(0.745314\pi\)
\(13\)	51.9401	+	66.0472i	1.10812	+	1.40909i	0.902622	+	0.430435i	\(0.141640\pi\)
							0.205501	+	0.978657i	\(0.434118\pi\)
\(17\)	11.8013	+	123.589i	0.168367	+	1.76322i	0.546493	+	0.837464i	\(0.315963\pi\)
							−0.378126	+	0.925754i	\(0.623431\pi\)
\(19\)	−88.9317	−	45.8475i	−1.07381	−	0.553586i	−0.171723	−	0.985145i	\(-0.554933\pi\)
							−0.902085	+	0.431559i	\(0.857964\pi\)
\(23\)	177.032	−	42.9475i	1.60495	−	0.389356i	0.669496	−	0.742815i	\(-0.266510\pi\)
							0.935450	+	0.353460i	\(0.114995\pi\)
\(29\)	91.4352	+	52.7902i	0.585486	+	0.338031i	0.763311	−	0.646032i	\(-0.223573\pi\)
							−0.177825	+	0.984062i	\(0.556906\pi\)
\(31\)	−92.0453	+	117.045i	−0.533285	+	0.678127i	−0.975808	−	0.218630i	\(-0.929841\pi\)
							0.442523	+	0.896757i	\(0.354084\pi\)
\(37\)	86.3587	+	149.578i	0.383711	+	0.664606i	0.991589	−	0.129424i	\(-0.0413127\pi\)
							−0.607879	+	0.794030i	\(0.707979\pi\)
\(41\)	106.339	+	149.333i	0.405059	+	0.568825i	0.965840	−	0.259139i	\(-0.0834386\pi\)
							−0.560781	+	0.827964i	\(0.689499\pi\)
\(43\)	81.2834	+	37.1209i	0.288270	+	0.131648i	0.554302	−	0.832316i	\(-0.312985\pi\)
							−0.266032	+	0.963964i	\(0.585713\pi\)
\(47\)	−116.053	−	121.713i	−0.360173	−	0.377738i	0.518569	−	0.855036i	\(-0.326465\pi\)
							−0.878742	+	0.477298i	\(0.841616\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.p.a.2.17	✓	1320
3.2	odd	2	inner	201.4.p.a.2.50	yes	1320
67.34	odd	66	inner	201.4.p.a.101.50	yes	1320
201.101	even	66	inner	201.4.p.a.101.17	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.p.a.2.17	✓	1320	1.1	even	1	trivial
201.4.p.a.2.50	yes	1320	3.2	odd	2	inner
201.4.p.a.101.17	yes	1320	201.101	even	66	inner
201.4.p.a.101.50	yes	1320	67.34	odd	66	inner