Embedded newform 201.4.j.b.5.17

• Label: 201.4.j.b.5.17
• Level: $201$
• Weight: $4$
• Character: 201.5
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(64\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.17
Character	\(\chi\)	\(=\)	201.5
Dual form			201.4.j.b.161.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.00708 + 2.31629i) q^{2} +(0.0979400 - 5.19523i) q^{3} +(-0.198329 - 1.37941i) q^{4} +(4.53381 - 1.33125i) q^{5} +(11.8371 + 10.6541i) q^{6} +(-2.62239 - 2.27231i) q^{7} +(-17.0337 - 10.9469i) q^{8} +(-26.9808 - 1.01764i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 11 q^{3} - 290 q^{4} + 35 q^{6} - 22 q^{7} - 33 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.00708	+	2.31629i	−0.709610	+	0.818934i	−0.990017	−	0.140947i	\(-0.954985\pi\)
							0.280407	+	0.959881i	\(0.409531\pi\)
\(3\)	0.0979400	−	5.19523i	0.0188486	−	0.999822i
							\(5\)	4.53381	−	1.33125i	0.405517	−	0.119070i	−0.0726120	−	0.997360i	\(-0.523133\pi\)
0.478129	+	0.878290i	\(0.341315\pi\)
\(7\)	−2.62239	−	2.27231i	−0.141596	−	0.122693i	0.581183	−	0.813773i	\(-0.302590\pi\)
							−0.722779	+	0.691079i	\(0.757135\pi\)
\(11\)	−21.0844	+	6.19093i	−0.577925	+	0.169694i	−0.557615	−	0.830100i	\(-0.688284\pi\)
							−0.0203101	+	0.999794i	\(0.506465\pi\)
\(13\)	20.1534	+	31.3593i	0.429965	+	0.669039i	0.986866	−	0.161540i	\(-0.0516462\pi\)
							−0.556901	+	0.830579i	\(0.688010\pi\)
\(17\)	25.9206	+	3.72682i	0.369804	+	0.0531698i	0.324712	−	0.945813i	\(-0.394733\pi\)
							0.0450924	+	0.998983i	\(0.485642\pi\)
\(19\)	57.2207	+	66.0362i	0.690912	+	0.797354i	0.987495	−	0.157653i	\(-0.0503926\pi\)
							−0.296583	+	0.955007i	\(0.595847\pi\)
\(23\)	−116.354	+	53.1370i	−1.05484	+	0.481731i	−0.865880	−	0.500252i	\(-0.833241\pi\)
							−0.188965	+	0.981984i	\(0.560513\pi\)
\(29\)			75.8605i			0.485756i	0.970057	+	0.242878i	\(0.0780915\pi\)
							−0.970057	+	0.242878i	\(0.921908\pi\)
\(31\)	−28.3326	+	44.0863i	−0.164151	+	0.255424i	−0.913578	−	0.406663i	\(-0.866692\pi\)
							0.749427	+	0.662086i	\(0.230329\pi\)
\(37\)	−209.134			−0.929227			−0.464613	−	0.885514i	\(-0.653807\pi\)
							−0.464613	+	0.885514i	\(0.653807\pi\)
\(41\)	−57.0562	+	396.834i	−0.217334	+	1.51159i	0.530490	+	0.847691i	\(0.322008\pi\)
							−0.747824	+	0.663897i	\(0.768901\pi\)
\(43\)	−152.689	−	21.9533i	−0.541507	−	0.0778570i	−0.133867	−	0.990999i	\(-0.542739\pi\)
							−0.407641	+	0.913142i	\(0.633648\pi\)
\(47\)	−75.2176	+	34.3507i	−0.233438	+	0.106608i	−0.528700	−	0.848809i	\(-0.677320\pi\)
							0.295262	+	0.955416i	\(0.404593\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.j.b.5.17	✓	640
3.2	odd	2	inner	201.4.j.b.5.48	yes	640
67.27	odd	22	inner	201.4.j.b.161.48	yes	640
201.161	even	22	inner	201.4.j.b.161.17	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.j.b.5.17	✓	640	1.1	even	1	trivial
201.4.j.b.5.48	yes	640	3.2	odd	2	inner
201.4.j.b.161.17	yes	640	201.161	even	22	inner
201.4.j.b.161.48	yes	640	67.27	odd	22	inner