Embedded newform 201.4.i.b.22.8

• Label: 201.4.i.b.22.8
• 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			22.8
Character	\(\chi\)	\(=\)	201.22
Dual form			201.4.i.b.64.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.497304 + 0.146022i) q^{2} +(0.426945 + 2.96946i) q^{3} +(-6.50404 + 4.17989i) q^{4} +(3.14929 - 6.89598i) q^{5} +(-0.645927 - 1.41438i) q^{6} +(18.8432 - 5.53285i) q^{7} +(5.33944 - 6.16204i) q^{8} +(-8.63544 + 2.53559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 170 q + 2 q^{2} + 51 q^{3} - 74 q^{4} - 4 q^{5} + 27 q^{6} + 12 q^{7} - 237 q^{8} - 153 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.497304	+	0.146022i	−0.175823	+	0.0516264i	−0.368459	−	0.929644i	\(-0.620115\pi\)
							0.192636	+	0.981270i	\(0.438296\pi\)
\(3\)	0.426945	+	2.96946i	0.0821655	+	0.571474i
							\(5\)	3.14929	−	6.89598i	0.281681	−	0.616796i	−0.714917	−	0.699209i	\(-0.753536\pi\)
0.996598	+	0.0824136i	\(0.0262628\pi\)
\(7\)	18.8432	−	5.53285i	1.01743	−	0.298746i	0.269843	−	0.962904i	\(-0.413028\pi\)
							0.747592	+	0.664159i	\(0.231210\pi\)
\(11\)	5.95100	−	13.0309i	0.163118	−	0.357178i	−0.810370	−	0.585919i	\(-0.800734\pi\)
							0.973487	+	0.228741i	\(0.0734610\pi\)
\(13\)	19.0995	+	22.0420i	0.407480	+	0.470257i	0.921982	−	0.387232i	\(-0.126569\pi\)
							−0.514502	+	0.857489i	\(0.672023\pi\)
\(17\)	−40.8984	−	26.2838i	−0.583490	−	0.374986i	0.215344	−	0.976538i	\(-0.430913\pi\)
							−0.798834	+	0.601552i	\(0.794549\pi\)
\(19\)	100.688	+	29.5645i	1.21575	+	0.356977i	0.825856	−	0.563881i	\(-0.190692\pi\)
							0.389897	+	0.920858i	\(0.372511\pi\)
\(23\)	19.4343	+	135.168i	0.176188	+	1.22541i	0.865485	+	0.500935i	\(0.167011\pi\)
							−0.689297	+	0.724479i	\(0.742080\pi\)
\(29\)	257.430			1.64840			0.824199	−	0.566300i	\(-0.191626\pi\)
							0.824199	+	0.566300i	\(0.191626\pi\)
\(31\)	−189.460	+	218.648i	−1.09768	+	1.26679i	−0.136564	+	0.990631i	\(0.543606\pi\)
							−0.961113	+	0.276156i	\(0.910939\pi\)
\(37\)	395.078			1.75542			0.877708	−	0.479196i	\(-0.159072\pi\)
							0.877708	+	0.479196i	\(0.159072\pi\)
\(41\)	−276.852	−	177.922i	−1.05456	−	0.677725i	−0.106015	−	0.994365i	\(-0.533809\pi\)
							−0.948546	+	0.316639i	\(0.897446\pi\)
\(43\)	−101.659	−	65.3320i	−0.360530	−	0.231698i	0.347818	−	0.937562i	\(-0.386923\pi\)
							−0.708348	+	0.705864i	\(0.750559\pi\)
\(47\)	39.1215	+	272.096i	0.121414	+	0.844453i	0.955956	+	0.293509i	\(0.0948231\pi\)
							−0.834542	+	0.550944i	\(0.814268\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.i.b.22.8	✓	170
67.64	even	11	inner	201.4.i.b.64.8	yes	170

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.i.b.22.8	✓	170	1.1	even	1	trivial
201.4.i.b.64.8	yes	170	67.64	even	11	inner