Embedded newform 201.4.j.b.5.7

• Label: 201.4.j.b.5.7
• Level: $201$
• Weight: $4$
• Character: 201.5
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.j (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(64\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			5.7
Character	\(\chi\)	\(=\)	201.5
Dual form			201.4.j.b.161.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.03287 + 3.50012i) q^{2} +(2.67965 + 4.45190i) q^{3} +(-1.91401 - 13.3122i) q^{4} +(-16.4593 + 4.83290i) q^{5} +(-23.7092 - 4.12294i) q^{6} +(-19.2826 - 16.7085i) q^{7} +(21.2303 + 13.6439i) q^{8} +(-12.6389 + 23.8591i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 11 q^{3} - 290 q^{4} + 35 q^{6} - 22 q^{7} - 33 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.03287	+	3.50012i	−1.07228	+	1.23748i	−0.102183	+	0.994766i	\(0.532583\pi\)
							−0.970098	+	0.242712i	\(0.921963\pi\)
\(3\)	2.67965	+	4.45190i	0.515700	+	0.856769i
							\(5\)	−16.4593	+	4.83290i	−1.47217	+	0.432268i	−0.916803	−	0.399339i	\(-0.869240\pi\)
−0.555365	+	0.831607i	\(0.687421\pi\)
\(7\)	−19.2826	−	16.7085i	−1.04116	−	0.902172i	−0.0458560	−	0.998948i	\(-0.514602\pi\)
							−0.995306	+	0.0967758i	\(0.969147\pi\)
\(11\)	−40.6036	+	11.9223i	−1.11295	+	0.326792i	−0.785986	−	0.618245i	\(-0.787844\pi\)
							−0.326965	+	0.945037i	\(0.606026\pi\)
\(13\)	40.0694	+	62.3492i	0.854865	+	1.33020i	0.942556	+	0.334048i	\(0.108415\pi\)
							−0.0876913	+	0.996148i	\(0.527949\pi\)
\(17\)	75.5905	+	10.8683i	1.07844	+	0.155056i	0.658561	−	0.752527i	\(-0.271165\pi\)
							0.419874	+	0.907583i	\(0.362074\pi\)
\(19\)	−3.95794	−	4.56771i	−0.0477902	−	0.0551528i	0.731351	−	0.682001i	\(-0.238890\pi\)
							−0.779141	+	0.626848i	\(0.784345\pi\)
\(23\)	−3.29763	+	1.50598i	−0.0298958	+	0.0136529i	−0.430307	−	0.902683i	\(-0.641595\pi\)
							0.400411	+	0.916336i	\(0.368867\pi\)
\(29\)		−	240.668i		−	1.54107i	−0.637398	−	0.770535i	\(-0.719989\pi\)
							0.637398	−	0.770535i	\(-0.280011\pi\)
\(31\)	−4.81600	+	7.49385i	−0.0279026	+	0.0434173i	−0.854925	−	0.518752i	\(-0.826397\pi\)
							0.827022	+	0.562169i	\(0.190033\pi\)
\(37\)	276.823			1.22999			0.614993	−	0.788533i	\(-0.289159\pi\)
							0.614993	+	0.788533i	\(0.289159\pi\)
\(41\)	17.6840	−	122.995i	0.0673605	−	0.468503i	−0.928023	−	0.372523i	\(-0.878493\pi\)
							0.995383	−	0.0959794i	\(-0.0305983\pi\)
\(43\)	−357.497	−	51.4003i	−1.26785	−	0.182290i	−0.524629	−	0.851331i	\(-0.675796\pi\)
							−0.743225	+	0.669041i	\(0.766705\pi\)
\(47\)	181.411	−	82.8478i	0.563012	−	0.257119i	−0.113516	−	0.993536i	\(-0.536211\pi\)
							0.676528	+	0.736417i	\(0.263484\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.j.b.5.7	✓	640
3.2	odd	2	inner	201.4.j.b.5.58	yes	640
67.27	odd	22	inner	201.4.j.b.161.58	yes	640
201.161	even	22	inner	201.4.j.b.161.7	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.j.b.5.7	✓	640	1.1	even	1	trivial
201.4.j.b.5.58	yes	640	3.2	odd	2	inner
201.4.j.b.161.7	yes	640	201.161	even	22	inner
201.4.j.b.161.58	yes	640	67.27	odd	22	inner