Embedded newform 201.2.i.a.40.2

• Label: 201.2.i.a.40.2
• Level: $201$
• Weight: $2$
• Character: 201.40
• Analytic conductor: $1.605$
• Analytic rank: $0$
• Dimension: $50$
• 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:	\(50\)
Relative dimension:	\(5\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			40.2
Character	\(\chi\)	\(=\)	201.40
Dual form			201.2.i.a.196.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.371960 - 0.429265i) q^{2} +(-0.415415 + 0.909632i) q^{3} +(0.238716 - 1.66030i) q^{4} +(-2.09548 - 0.615288i) q^{5} +(0.544991 - 0.160024i) q^{6} +(-1.13239 - 1.30685i) q^{7} +(-1.75717 + 1.12926i) q^{8} +(-0.654861 - 0.755750i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 2 q^{2} + 5 q^{3} - 6 q^{4} + 2 q^{5} - 9 q^{6} + 2 q^{7} + 27 q^{8} - 5 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(68\)	\(136\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.371960	−	0.429265i	−0.263016	−	0.303536i	0.608846	−	0.793288i	\(-0.291633\pi\)
							−0.871862	+	0.489752i	\(0.837087\pi\)
\(3\)	−0.415415	+	0.909632i	−0.239840	+	0.525176i
							\(5\)	−2.09548	−	0.615288i	−0.937127	−	0.275165i	−0.222709	−	0.974885i	\(-0.571490\pi\)
−0.714417	+	0.699720i	\(0.753308\pi\)
\(7\)	−1.13239	−	1.30685i	−0.428003	−	0.493941i	0.500255	−	0.865878i	\(-0.333239\pi\)
							−0.928258	+	0.371936i	\(0.878694\pi\)
\(11\)	−2.41415	−	0.708860i	−0.727895	−	0.213729i	−0.103269	−	0.994653i	\(-0.532930\pi\)
							−0.624626	+	0.780924i	\(0.714748\pi\)
\(13\)	−1.05926	−	0.680747i	−0.293786	−	0.188805i	0.385442	−	0.922732i	\(-0.374049\pi\)
							−0.679228	+	0.733927i	\(0.737685\pi\)
\(17\)	−0.337536	−	2.34761i	−0.0818645	−	0.569380i	−0.988929	−	0.148389i	\(-0.952591\pi\)
							0.907065	−	0.420991i	\(-0.138318\pi\)
\(19\)	3.76078	−	4.34017i	0.862782	−	0.995703i	−0.137205	−	0.990543i	\(-0.543812\pi\)
							0.999987	−	0.00516064i	\(-0.00164269\pi\)
\(23\)	−0.890516	+	1.94996i	−0.185685	+	0.406594i	−0.979466	−	0.201610i	\(-0.935383\pi\)
							0.793780	+	0.608204i	\(0.208110\pi\)
\(29\)	1.81679			0.337370			0.168685	−	0.985670i	\(-0.446048\pi\)
							0.168685	+	0.985670i	\(0.446048\pi\)
\(31\)	−0.514437	+	0.330609i	−0.0923956	+	0.0593790i	−0.586023	−	0.810295i	\(-0.699307\pi\)
							0.493627	+	0.869674i	\(0.335671\pi\)
\(37\)	7.08491			1.16475			0.582376	−	0.812919i	\(-0.302123\pi\)
							0.582376	+	0.812919i	\(0.302123\pi\)
\(41\)	−0.379091	−	2.63664i	−0.0592041	−	0.411774i	−0.997774	−	0.0666847i	\(-0.978758\pi\)
							0.938570	−	0.345089i	\(-0.112151\pi\)
\(43\)	1.21683	+	8.46324i	0.185565	+	1.29063i	0.843325	+	0.537404i	\(0.180595\pi\)
							−0.657760	+	0.753228i	\(0.728496\pi\)
\(47\)	−2.70803	+	5.92977i	−0.395007	+	0.864946i	0.602745	+	0.797934i	\(0.294074\pi\)
							−0.997752	+	0.0670115i	\(0.978654\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	201.2.i.a.40.2	✓	50
3.2	odd	2		603.2.u.d.442.4		50
67.62	even	11	inner	201.2.i.a.196.2	yes	50
201.62	odd	22		603.2.u.d.397.4		50

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
201.2.i.a.40.2	✓	50	1.1	even	1	trivial
201.2.i.a.196.2	yes	50	67.62	even	11	inner
603.2.u.d.397.4		50	201.62	odd	22
603.2.u.d.442.4		50	3.2	odd	2