Embedded newform 201.4.p.a.2.5

• Label: 201.4.p.a.2.5
• 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			2.5
Character	\(\chi\)	\(=\)	201.2
Dual form			201.4.p.a.101.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.240289 + 5.04428i) q^{2} +(5.03306 - 1.29161i) q^{3} +(-17.4232 - 1.66372i) q^{4} +(-12.5554 + 14.4897i) q^{5} +(5.30585 + 25.6985i) q^{6} +(-10.2423 - 19.8673i) q^{7} +(6.82934 - 47.4991i) q^{8} +(23.6635 - 13.0015i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 22 q^{3} + 214 q^{4} + q^{6} + 22 q^{7} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.240289	+	5.04428i	−0.0849549	+	1.78342i	0.407620	+	0.913152i	\(0.366359\pi\)
							−0.492575	+	0.870270i	\(0.663944\pi\)
\(3\)	5.03306	−	1.29161i	0.968614	−	0.248571i
							\(5\)	−12.5554	+	14.4897i	−1.12299	+	1.29600i	−0.172583	+	0.984995i	\(0.555211\pi\)
−0.950408	+	0.311005i	\(0.899334\pi\)
\(7\)	−10.2423	−	19.8673i	−0.553033	−	1.07273i	−0.985275	−	0.170976i	\(-0.945308\pi\)
							0.432243	−	0.901757i	\(-0.357722\pi\)
\(11\)	−3.99314	−	11.5374i	−0.109452	−	0.316242i	0.876952	−	0.480578i	\(-0.159573\pi\)
							−0.986404	+	0.164336i	\(0.947452\pi\)
\(13\)	−30.5476	−	38.8444i	−0.651722	−	0.828732i	0.342107	−	0.939661i	\(-0.388860\pi\)
							−0.993828	+	0.110930i	\(0.964617\pi\)
\(17\)	4.34210	+	45.4725i	0.0619478	+	0.648747i	0.972115	+	0.234505i	\(0.0753470\pi\)
							−0.910167	+	0.414241i	\(0.864047\pi\)
\(19\)	76.8760	+	39.6324i	0.928241	+	0.478541i	0.854916	−	0.518767i	\(-0.173609\pi\)
							0.0733249	+	0.997308i	\(0.476639\pi\)
\(23\)	−202.710	+	49.1769i	−1.83774	+	0.445831i	−0.995622	−	0.0934658i	\(-0.970205\pi\)
							−0.842116	+	0.539296i	\(0.818690\pi\)
\(29\)	−138.472	−	79.9466i	−0.886673	−	0.511921i	−0.0138202	−	0.999904i	\(-0.504399\pi\)
							−0.872853	+	0.487984i	\(0.837733\pi\)
\(31\)	9.00759	−	11.4541i	0.0521874	−	0.0663617i	−0.759259	−	0.650789i	\(-0.774438\pi\)
							0.811446	+	0.584427i	\(0.198681\pi\)
\(37\)	70.6719	+	122.407i	0.314011	+	0.543882i	0.979227	−	0.202769i	\(-0.0649939\pi\)
							−0.665216	+	0.746651i	\(0.731661\pi\)
\(41\)	−28.0426	−	39.3803i	−0.106817	−	0.150004i	0.757707	−	0.652595i	\(-0.226320\pi\)
							−0.864524	+	0.502591i	\(0.832380\pi\)
\(43\)	−289.379	−	132.155i	−1.02628	−	0.468685i	−0.170133	−	0.985421i	\(-0.554420\pi\)
							−0.856145	+	0.516736i	\(0.827147\pi\)
\(47\)	−149.458	−	156.747i	−0.463844	−	0.486466i	0.449715	−	0.893172i	\(-0.351526\pi\)
							−0.913559	+	0.406707i	\(0.866677\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.p.a.2.5	✓	1320
3.2	odd	2	inner	201.4.p.a.2.62	yes	1320
67.34	odd	66	inner	201.4.p.a.101.62	yes	1320
201.101	even	66	inner	201.4.p.a.101.5	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.p.a.2.5	✓	1320	1.1	even	1	trivial
201.4.p.a.2.62	yes	1320	3.2	odd	2	inner
201.4.p.a.101.5	yes	1320	201.101	even	66	inner
201.4.p.a.101.62	yes	1320	67.34	odd	66	inner