Embedded newform 201.4.p.a.2.15

• Label: 201.4.p.a.2.15
• 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			2.15
Character	\(\chi\)	\(=\)	201.2
Dual form			201.4.p.a.101.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.168996 + 3.54767i) q^{2} +(-2.66059 - 4.46332i) q^{3} +(-4.59361 - 0.438637i) q^{4} +(8.27468 - 9.54949i) q^{5} +(16.2840 - 8.68460i) q^{6} +(-5.63397 - 10.9284i) q^{7} +(-1.71122 + 11.9018i) q^{8} +(-12.8425 + 23.7501i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1320 q - 22 q^{3} + 214 q^{4} + q^{6} + 22 q^{7} + 48 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.168996	+	3.54767i	−0.0597492	+	1.25429i	0.747573	+	0.664179i	\(0.231219\pi\)
							−0.807322	+	0.590111i	\(0.799084\pi\)
\(3\)	−2.66059	−	4.46332i	−0.512031	−	0.858967i
							\(5\)	8.27468	−	9.54949i	0.740110	−	0.854132i	−0.253461	−	0.967346i	\(-0.581569\pi\)
0.993571	+	0.113214i	\(0.0361144\pi\)
\(7\)	−5.63397	−	10.9284i	−0.304206	−	0.590077i	0.686337	−	0.727284i	\(-0.259218\pi\)
							−0.990542	+	0.137207i	\(0.956187\pi\)
\(11\)	2.20976	+	6.38469i	0.0605699	+	0.175005i	0.971226	−	0.238160i	\(-0.0765443\pi\)
							−0.910656	+	0.413165i	\(0.864423\pi\)
\(13\)	−36.1104	−	45.9181i	−0.770402	−	0.979645i	−0.999980	−	0.00635501i	\(-0.997977\pi\)
							0.229578	−	0.973290i	\(-0.426265\pi\)
\(17\)	−0.371259	−	3.88800i	−0.00529667	−	0.0554693i	0.992437	−	0.122754i	\(-0.0391725\pi\)
							−0.997734	+	0.0672845i	\(0.978566\pi\)
\(19\)	−101.108	−	52.1247i	−1.22083	−	0.629381i	−0.277335	−	0.960773i	\(-0.589451\pi\)
							−0.943493	+	0.331393i	\(0.892481\pi\)
\(23\)	−29.0096	+	7.03765i	−0.262996	+	0.0638022i	−0.365089	−	0.930973i	\(-0.618961\pi\)
							0.102092	+	0.994775i	\(0.467446\pi\)
\(29\)	−172.565	−	99.6306i	−1.10498	−	0.637963i	−0.167459	−	0.985879i	\(-0.553556\pi\)
							−0.937526	+	0.347916i	\(0.886890\pi\)
\(31\)	105.450	−	134.090i	0.610946	−	0.776881i	−0.377896	−	0.925848i	\(-0.623352\pi\)
							0.988842	+	0.148967i	\(0.0475948\pi\)
\(37\)	17.8006	+	30.8316i	0.0790920	+	0.136991i	0.902858	−	0.429938i	\(-0.141465\pi\)
							−0.823766	+	0.566929i	\(0.808131\pi\)
\(41\)	29.2761	+	41.1125i	0.111516	+	0.156602i	0.866562	−	0.499070i	\(-0.166325\pi\)
							−0.755046	+	0.655672i	\(0.772385\pi\)
\(43\)	−20.9560	−	9.57030i	−0.0743201	−	0.0339408i	0.377908	−	0.925843i	\(-0.376643\pi\)
							−0.452228	+	0.891902i	\(0.649371\pi\)
\(47\)	−255.829	−	268.306i	−0.793968	−	0.832690i	0.195002	−	0.980803i	\(-0.437529\pi\)
							−0.988970	+	0.148113i	\(0.952680\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.p.a.2.15	✓	1320
3.2	odd	2	inner	201.4.p.a.2.52	yes	1320
67.34	odd	66	inner	201.4.p.a.101.52	yes	1320
201.101	even	66	inner	201.4.p.a.101.15	yes	1320

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.p.a.2.15	✓	1320	1.1	even	1	trivial
201.4.p.a.2.52	yes	1320	3.2	odd	2	inner
201.4.p.a.101.15	yes	1320	201.101	even	66	inner
201.4.p.a.101.52	yes	1320	67.34	odd	66	inner