Embedded newform 201.4.i.b.22.15

• Label: 201.4.i.b.22.15
• Level: $201$
• Weight: $4$
• Character: 201.22
• Analytic conductor: $11.859$
• Analytic rank: $0$
• Dimension: $170$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 201 = 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	201.i (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8593839112\)
Analytic rank:	\(0\)
Dimension:	\(170\)
Relative dimension:	\(17\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			22.15
Character	\(\chi\)	\(=\)	201.22
Dual form			201.4.i.b.64.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.03324 - 1.18427i) q^{2} +(0.426945 + 2.96946i) q^{3} +(8.13449 - 5.22772i) q^{4} +(8.01749 - 17.5559i) q^{5} +(5.23860 + 11.4709i) q^{6} +(5.21241 - 1.53050i) q^{7} +(4.59562 - 5.30363i) q^{8} +(-8.63544 + 2.53559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 170 q + 2 q^{2} + 51 q^{3} - 74 q^{4} - 4 q^{5} + 27 q^{6} + 12 q^{7} - 237 q^{8} - 153 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.03324	−	1.18427i	1.42596	−	0.418701i	0.524447	−	0.851443i	\(-0.324272\pi\)
							0.901518	+	0.432742i	\(0.142454\pi\)
\(3\)	0.426945	+	2.96946i	0.0821655	+	0.571474i
							\(5\)	8.01749	−	17.5559i	0.717107	−	1.57024i	−0.100809	−	0.994906i	\(-0.532143\pi\)
0.817915	−	0.575338i	\(-0.195130\pi\)
\(7\)	5.21241	−	1.53050i	0.281443	−	0.0826393i	−0.137964	−	0.990437i	\(-0.544056\pi\)
							0.419408	+	0.907798i	\(0.362238\pi\)
\(11\)	13.6256	−	29.8358i	0.373478	−	0.817803i	−0.625806	−	0.779979i	\(-0.715230\pi\)
							0.999284	−	0.0378246i	\(-0.0120428\pi\)
\(13\)	−27.2604	−	31.4602i	−0.581591	−	0.671192i	0.386355	−	0.922350i	\(-0.373734\pi\)
							−0.967946	+	0.251158i	\(0.919189\pi\)
\(17\)	9.07640	+	5.83305i	0.129491	+	0.0832189i	0.603783	−	0.797149i	\(-0.293659\pi\)
							−0.474292	+	0.880368i	\(0.657296\pi\)
\(19\)	137.240	+	40.2974i	1.65711	+	0.486571i	0.970630	−	0.240577i	\(-0.0773367\pi\)
							0.686480	+	0.727149i	\(0.259155\pi\)
\(23\)	25.4797	+	177.215i	0.230995	+	1.60661i	0.693814	+	0.720154i	\(0.255929\pi\)
							−0.462819	+	0.886453i	\(0.653162\pi\)
\(29\)	−111.216			−0.712145			−0.356073	−	0.934458i	\(-0.615884\pi\)
							−0.356073	+	0.934458i	\(0.615884\pi\)
\(31\)	94.3256	−	108.858i	0.546496	−	0.630690i	−0.413567	−	0.910474i	\(-0.635717\pi\)
							0.960063	+	0.279784i	\(0.0902628\pi\)
\(37\)	−147.113			−0.653655			−0.326828	−	0.945084i	\(-0.605980\pi\)
							−0.326828	+	0.945084i	\(0.605980\pi\)
\(41\)	129.886	+	83.4730i	0.494753	+	0.317958i	0.764114	−	0.645081i	\(-0.223176\pi\)
							−0.269362	+	0.963039i	\(0.586813\pi\)
\(43\)	38.9217	+	25.0135i	0.138035	+	0.0887098i	0.607836	−	0.794063i	\(-0.292038\pi\)
							−0.469801	+	0.882772i	\(0.655674\pi\)
\(47\)	56.6118	+	393.744i	0.175695	+	1.22199i	0.866586	+	0.499028i	\(0.166310\pi\)
							−0.690890	+	0.722960i	\(0.742781\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.4.i.b.22.15	✓	170
67.64	even	11	inner	201.4.i.b.64.15	yes	170

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.4.i.b.22.15	✓	170	1.1	even	1	trivial
201.4.i.b.64.15	yes	170	67.64	even	11	inner