Embedded newform 201.4.i.a.22.15

• Label: 201.4.i.a.22.15
• 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			22.15
Character	\(\chi\)	\(=\)	201.22
Dual form			201.4.i.a.64.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.28771 - 1.25899i) q^{2} +(-0.426945 - 2.96946i) q^{3} +(10.0694 - 6.47123i) q^{4} +(4.82712 - 10.5699i) q^{5} +(-5.56913 - 12.1947i) q^{6} +(18.6881 - 5.48732i) q^{7} +(11.6165 - 13.4061i) q^{8} +(-8.63544 + 2.53559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 170 q + 2 q^{2} - 51 q^{3} - 82 q^{4} + 4 q^{5} - 27 q^{6} - 16 q^{7} - 177 q^{8} - 153 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.28771	−	1.25899i	1.51594	−	0.445119i	0.585224	−	0.810872i	\(-0.301007\pi\)
							0.930712	+	0.365753i	\(0.119188\pi\)
\(3\)	−0.426945	−	2.96946i	−0.0821655	−	0.571474i
							\(5\)	4.82712	−	10.5699i	0.431750	−	0.945401i	−0.561289	−	0.827620i	\(-0.689694\pi\)
0.993040	−	0.117782i	\(-0.0375783\pi\)
\(7\)	18.6881	−	5.48732i	1.00906	−	0.296288i	0.264893	−	0.964278i	\(-0.414663\pi\)
							0.744170	+	0.667990i	\(0.232845\pi\)
\(11\)	−12.3314	+	27.0021i	−0.338006	+	0.740131i	−0.999956	−	0.00941800i	\(-0.997002\pi\)
							0.661949	+	0.749549i	\(0.269729\pi\)
\(13\)	−26.7688	−	30.8929i	−0.571103	−	0.659088i	0.394565	−	0.918868i	\(-0.370895\pi\)
							−0.965668	+	0.259780i	\(0.916350\pi\)
\(17\)	60.6199	+	38.9581i	0.864853	+	0.555807i	0.896174	−	0.443702i	\(-0.146335\pi\)
							−0.0313216	+	0.999509i	\(0.509972\pi\)
\(19\)	−26.9333	−	7.90832i	−0.325206	−	0.0954891i	0.115055	−	0.993359i	\(-0.463296\pi\)
							−0.440260	+	0.897870i	\(0.645114\pi\)
\(23\)	−17.2465	−	119.952i	−0.156354	−	1.08747i	−0.905280	−	0.424815i	\(-0.860339\pi\)
							0.748926	−	0.662654i	\(-0.230570\pi\)
\(29\)	57.5562			0.368549			0.184275	−	0.982875i	\(-0.441006\pi\)
							0.184275	+	0.982875i	\(0.441006\pi\)
\(31\)	28.0961	−	32.4246i	0.162781	−	0.187859i	−0.668499	−	0.743713i	\(-0.733063\pi\)
							0.831280	+	0.555854i	\(0.187608\pi\)
\(37\)	−72.2926			−0.321212			−0.160606	−	0.987019i	\(-0.551345\pi\)
							−0.160606	+	0.987019i	\(0.551345\pi\)
\(41\)	114.003	+	73.2650i	0.434249	+	0.279075i	0.739448	−	0.673214i	\(-0.235087\pi\)
							−0.305199	+	0.952289i	\(0.598723\pi\)
\(43\)	307.847	+	197.841i	1.09177	+	0.701639i	0.957247	−	0.289271i	\(-0.0934128\pi\)
							0.134525	+	0.990910i	\(0.457049\pi\)
\(47\)	21.6414	+	150.520i	0.0671644	+	0.467139i	0.995451	+	0.0952736i	\(0.0303726\pi\)
							−0.928287	+	0.371865i	\(0.878718\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.i.a.22.15	✓	170
67.64	even	11	inner	201.4.i.a.64.15	yes	170

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.i.a.22.15	✓	170	1.1	even	1	trivial
201.4.i.a.64.15	yes	170	67.64	even	11	inner