Embedded newform 201.4.j.b.5.3

• Label: 201.4.j.b.5.3
• 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			5.3
Character	\(\chi\)	\(=\)	201.5
Dual form			201.4.j.b.161.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.44200 + 3.97228i) q^{2} +(3.74600 - 3.60104i) q^{3} +(-2.79312 - 19.4266i) q^{4} +(2.64211 - 0.775793i) q^{5} +(1.41059 + 27.2749i) q^{6} +(-19.3454 - 16.7629i) q^{7} +(51.4083 + 33.0381i) q^{8} +(1.06506 - 26.9790i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 640 q - 11 q^{3} - 290 q^{4} + 35 q^{6} - 22 q^{7} - 33 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.44200	+	3.97228i	−1.21693	+	1.40441i	−0.329067	+	0.944307i	\(0.606734\pi\)
							−0.887864	+	0.460107i	\(0.847811\pi\)
\(3\)	3.74600	−	3.60104i	0.720918	−	0.693020i
							\(5\)	2.64211	−	0.775793i	0.236317	−	0.0693890i	−0.161430	−	0.986884i	\(-0.551610\pi\)
0.397747	+	0.917495i	\(0.369792\pi\)
\(7\)	−19.3454	−	16.7629i	−1.04455	−	0.905110i	−0.0489508	−	0.998801i	\(-0.515588\pi\)
							−0.995601	+	0.0936917i	\(0.970133\pi\)
\(11\)	10.2013	−	2.99536i	0.279618	−	0.0821032i	−0.138917	−	0.990304i	\(-0.544362\pi\)
							0.418535	+	0.908201i	\(0.362544\pi\)
\(13\)	39.5225	+	61.4982i	0.843198	+	1.31204i	0.948234	+	0.317573i	\(0.102868\pi\)
							−0.105036	+	0.994468i	\(0.533496\pi\)
\(17\)	−103.397	−	14.8663i	−1.47515	−	0.212095i	−0.642668	−	0.766145i	\(-0.722172\pi\)
							−0.832483	+	0.554050i	\(0.813081\pi\)
\(19\)	−45.5959	−	52.6205i	−0.550548	−	0.635366i	0.410463	−	0.911877i	\(-0.365367\pi\)
							−0.961011	+	0.276511i	\(0.910822\pi\)
\(23\)	−114.027	+	52.0746i	−1.03376	+	0.472100i	−0.858711	−	0.512460i	\(-0.828734\pi\)
							−0.175044	+	0.984561i	\(0.556007\pi\)
\(29\)			123.286i			0.789435i	0.918803	+	0.394718i	\(0.129158\pi\)
							−0.918803	+	0.394718i	\(0.870842\pi\)
\(31\)	97.0553	−	151.021i	0.562311	−	0.874974i	−0.437393	−	0.899270i	\(-0.644098\pi\)
							0.999705	+	0.0242967i	\(0.00773464\pi\)
\(37\)	−244.586			−1.08675			−0.543375	−	0.839490i	\(-0.682854\pi\)
							−0.543375	+	0.839490i	\(0.682854\pi\)
\(41\)	26.2043	−	182.255i	0.0998152	−	0.694230i	−0.877054	−	0.480392i	\(-0.840494\pi\)
							0.976869	−	0.213838i	\(-0.0685965\pi\)
\(43\)	298.046	+	42.8526i	1.05701	+	0.151976i	0.648837	−	0.760927i	\(-0.275256\pi\)
							0.408177	+	0.912903i	\(0.366165\pi\)
\(47\)	−286.877	+	131.013i	−0.890327	+	0.406599i	−0.807433	−	0.589960i	\(-0.799144\pi\)
							−0.0828942	+	0.996558i	\(0.526416\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.j.b.5.3	✓	640
3.2	odd	2	inner	201.4.j.b.5.62	yes	640
67.27	odd	22	inner	201.4.j.b.161.62	yes	640
201.161	even	22	inner	201.4.j.b.161.3	yes	640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.j.b.5.3	✓	640	1.1	even	1	trivial
201.4.j.b.5.62	yes	640	3.2	odd	2	inner
201.4.j.b.161.3	yes	640	201.161	even	22	inner
201.4.j.b.161.62	yes	640	67.27	odd	22	inner