Embedded newform 201.4.i.b.22.7

• Label: 201.4.i.b.22.7
• Level: $201$
• Weight: $4$
• Character: 201.22
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $170$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(170\)
Relative dimension:	\(17\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			22.7
Character	\(\chi\)	\(=\)	201.22
Dual form			201.4.i.b.64.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26486 + 0.371395i) q^{2} +(0.426945 + 2.96946i) q^{3} +(-5.26810 + 3.38560i) q^{4} +(-4.40899 + 9.65433i) q^{5} +(-1.64287 - 3.59738i) q^{6} +(-28.6135 + 8.40167i) q^{7} +(12.3122 - 14.2090i) q^{8} +(-8.63544 + 2.53559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 170 q + 2 q^{2} + 51 q^{3} - 74 q^{4} - 4 q^{5} + 27 q^{6} + 12 q^{7} - 237 q^{8} - 153 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{10}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26486	+	0.371395i	−0.447194	+	0.131308i	−0.497573	−	0.867422i	\(-0.665775\pi\)
							0.0503787	+	0.998730i	\(0.483957\pi\)
\(3\)	0.426945	+	2.96946i	0.0821655	+	0.571474i
							\(5\)	−4.40899	+	9.65433i	−0.394352	+	0.863510i	0.603460	+	0.797393i	\(0.293788\pi\)
−0.997812	+	0.0661168i	\(0.978939\pi\)
\(7\)	−28.6135	+	8.40167i	−1.54498	+	0.453647i	−0.939596	−	0.342285i	\(-0.888799\pi\)
							−0.605385	+	0.795933i	\(0.706981\pi\)
\(11\)	−3.35429	+	7.34488i	−0.0919416	+	0.201324i	−0.950016	−	0.312202i	\(-0.898934\pi\)
							0.858074	+	0.513526i	\(0.171661\pi\)
\(13\)	−11.1627	−	12.8825i	−0.238153	−	0.274843i	0.624074	−	0.781365i	\(-0.285476\pi\)
							−0.862227	+	0.506522i	\(0.830931\pi\)
\(17\)	43.4785	+	27.9419i	0.620299	+	0.398642i	0.812707	−	0.582673i	\(-0.197993\pi\)
							−0.192408	+	0.981315i	\(0.561630\pi\)
\(19\)	87.9929	+	25.8371i	1.06247	+	0.311970i	0.765846	−	0.643024i	\(-0.222320\pi\)
							0.296626	+	0.954994i	\(0.404139\pi\)
\(23\)	−5.99273	−	41.6804i	−0.0543292	−	0.377868i	−0.998787	−	0.0492357i	\(-0.984321\pi\)
							0.944458	−	0.328632i	\(-0.106588\pi\)
\(29\)	−132.340			−0.847412			−0.423706	−	0.905800i	\(-0.639271\pi\)
							−0.423706	+	0.905800i	\(0.639271\pi\)
\(31\)	−11.2199	+	12.9484i	−0.0650047	+	0.0750195i	−0.787321	−	0.616544i	\(-0.788532\pi\)
							0.722316	+	0.691563i	\(0.243078\pi\)
\(37\)	274.554			1.21990			0.609952	−	0.792439i	\(-0.291189\pi\)
							0.609952	+	0.792439i	\(0.291189\pi\)
\(41\)	−322.120	−	207.014i	−1.22699	−	0.788541i	−0.243573	−	0.969882i	\(-0.578320\pi\)
							−0.983420	+	0.181342i	\(0.941956\pi\)
\(43\)	−305.278	−	196.190i	−1.08266	−	0.695785i	−0.127492	−	0.991840i	\(-0.540693\pi\)
							−0.955171	+	0.296055i	\(0.904329\pi\)
\(47\)	−52.7326	−	366.763i	−0.163656	−	1.13825i	−0.891668	−	0.452689i	\(-0.850465\pi\)
							0.728012	−	0.685564i	\(-0.240444\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.i.b.22.7	✓	170
67.64	even	11	inner	201.4.i.b.64.7	yes	170

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.i.b.22.7	✓	170	1.1	even	1	trivial
201.4.i.b.64.7	yes	170	67.64	even	11	inner