Embedded newform 201.2.i.b.22.7

• Label: 201.2.i.b.22.7
• Level: $201$
• Weight: $2$
• Character: 201.22
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $70$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	201.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.60499308063\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(7\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			22.7
Character	\(\chi\)	\(=\)	201.22
Dual form			201.2.i.b.64.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.62393 - 0.770455i) q^{2} +(-0.142315 - 0.989821i) q^{3} +(4.60890 - 2.96196i) q^{4} +(-1.23993 + 2.71506i) q^{5} +(-1.13604 - 2.48757i) q^{6} +(-2.41757 + 0.709862i) q^{7} +(6.22967 - 7.18942i) q^{8} +(-0.959493 + 0.281733i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 2 q^{2} - 7 q^{3} - 12 q^{4} - 2 q^{5} + 9 q^{6} - 10 q^{7} + 15 q^{8} - 7 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.62393	−	0.770455i	1.85540	−	0.544794i	0.855775	−	0.517348i	\(-0.173081\pi\)
							0.999623	−	0.0274461i	\(-0.00873746\pi\)
\(3\)	−0.142315	−	0.989821i	−0.0821655	−	0.571474i
							\(5\)	−1.23993	+	2.71506i	−0.554512	+	1.21421i	0.400131	+	0.916458i	\(0.368965\pi\)
−0.954643	+	0.297753i	\(0.903763\pi\)
\(7\)	−2.41757	+	0.709862i	−0.913754	+	0.268302i	−0.704620	−	0.709585i	\(-0.748883\pi\)
							−0.209134	+	0.977887i	\(0.567064\pi\)
\(11\)	−1.71376	+	3.75260i	−0.516717	+	1.13145i	0.453951	+	0.891027i	\(0.350014\pi\)
							−0.970668	+	0.240425i	\(0.922713\pi\)
\(13\)	−3.09359	−	3.57019i	−0.858007	−	0.990193i	−1.00000		0.000325632i	\(-0.999896\pi\)
							0.141993	−	0.989868i	\(-0.454649\pi\)
\(17\)	2.41623	+	1.55282i	0.586023	+	0.376614i	0.799798	−	0.600269i	\(-0.204940\pi\)
							−0.213776	+	0.976883i	\(0.568576\pi\)
\(19\)	0.794411	+	0.233260i	0.182250	+	0.0535136i	0.371584	−	0.928399i	\(-0.378815\pi\)
							−0.189334	+	0.981913i	\(0.560633\pi\)
\(23\)	−1.02122	−	7.10277i	−0.212940	−	1.48103i	−0.763268	−	0.646082i	\(-0.776406\pi\)
							0.550328	−	0.834949i	\(-0.314503\pi\)
\(29\)	8.03680			1.49240			0.746198	−	0.665724i	\(-0.231877\pi\)
							0.746198	+	0.665724i	\(0.231877\pi\)
\(31\)	−2.77973	+	3.20798i	−0.499254	+	0.576169i	−0.948315	−	0.317332i	\(-0.897213\pi\)
							0.449061	+	0.893501i	\(0.351759\pi\)
\(37\)	1.44672			0.237839			0.118919	−	0.992904i	\(-0.462057\pi\)
							0.118919	+	0.992904i	\(0.462057\pi\)
\(41\)	1.80468	+	1.15980i	0.281843	+	0.181130i	0.673923	−	0.738801i	\(-0.264608\pi\)
							−0.392080	+	0.919931i	\(0.628244\pi\)
\(43\)	0.730201	+	0.469272i	0.111355	+	0.0715633i	0.595133	−	0.803627i	\(-0.297099\pi\)
							−0.483778	+	0.875191i	\(0.660736\pi\)
\(47\)	−0.280490	−	1.95085i	−0.0409137	−	0.284561i	−0.999999	−	0.00144367i	\(-0.999540\pi\)
							0.959085	−	0.283117i	\(-0.0913686\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.i.b.22.7	✓	70
3.2	odd	2		603.2.u.e.424.1		70
67.64	even	11	inner	201.2.i.b.64.7	yes	70
201.131	odd	22		603.2.u.e.64.1		70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.i.b.22.7	✓	70	1.1	even	1	trivial
201.2.i.b.64.7	yes	70	67.64	even	11	inner
603.2.u.e.64.1		70	201.131	odd	22
603.2.u.e.424.1		70	3.2	odd	2