Embedded newform 164.2.o.b.11.1

• Label: 164.2.o.b.11.1
• Level: $164$
• Weight: $2$
• Character: 164.11
• Analytic conductor: $1.310$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 164 = 2^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	164.o (of order \(40\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.30954659315\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(18\) over \(\Q(\zeta_{40})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{40}]$

Embedding invariants

Embedding label			11.1
Character	\(\chi\)	\(=\)	164.11
Dual form			164.2.o.b.15.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38810 - 0.270514i) q^{2} +(0.869335 - 2.09876i) q^{3} +(1.85364 + 0.751000i) q^{4} +(1.95666 - 3.84017i) q^{5} +(-1.77447 + 2.67812i) q^{6} +(0.861690 + 3.58920i) q^{7} +(-2.36989 - 1.54390i) q^{8} +(-1.52773 - 1.52773i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).

\(n\)	\(83\)	\(129\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38810	−	0.270514i	−0.981535	−	0.191282i
							\(3\)	0.869335	−	2.09876i	0.501911	−	1.21172i	−0.446531	−	0.894768i	\(-0.647341\pi\)
0.948442	−	0.316952i	\(-0.102659\pi\)
\(5\)	1.95666	−	3.84017i	0.875046	−	1.71737i	0.199723	−	0.979852i	\(-0.435996\pi\)
							0.675323	−	0.737522i	\(-0.264004\pi\)
\(7\)	0.861690	+	3.58920i	0.325688	+	1.35659i	0.859565	+	0.511026i	\(0.170735\pi\)
							−0.533877	+	0.845562i	\(0.679265\pi\)
\(11\)	−1.33744	+	1.56594i	−0.403253	+	0.472149i	−0.924459	−	0.381283i	\(-0.875482\pi\)
							0.521206	+	0.853431i	\(0.325482\pi\)
\(13\)	−0.539305	−	0.880065i	−0.149576	−	0.244086i	0.769073	−	0.639162i	\(-0.220718\pi\)
							−0.918649	+	0.395075i	\(0.870718\pi\)
\(17\)	0.209993	+	2.66821i	0.0509307	+	0.647136i	0.968412	+	0.249356i	\(0.0802190\pi\)
							−0.917481	+	0.397779i	\(0.869781\pi\)
\(19\)	0.0153290	+	0.00939360i	0.00351670	+	0.00215504i	0.524256	−	0.851561i	\(-0.324344\pi\)
							−0.520739	+	0.853716i	\(0.674344\pi\)
\(23\)	−3.82860	−	2.78164i	−0.798318	−	0.580012i	0.112103	−	0.993697i	\(-0.464242\pi\)
							−0.910420	+	0.413685i	\(0.864242\pi\)
\(29\)	−0.347443	+	4.41468i	−0.0645186	+	0.819786i	0.876494	+	0.481414i	\(0.159876\pi\)
							−0.941012	+	0.338373i	\(0.890124\pi\)
\(31\)	−2.93653	+	9.03770i	−0.527416	+	1.62322i	0.232072	+	0.972699i	\(0.425449\pi\)
							−0.759488	+	0.650521i	\(0.774551\pi\)
\(37\)	1.09296	+	3.36379i	0.179682	+	0.553003i	0.999816	−	0.0191673i	\(-0.00610151\pi\)
							−0.820135	+	0.572171i	\(0.806102\pi\)
\(41\)	6.09681	+	1.95675i	0.952162	+	0.305593i
							\(43\)	−0.739649	−	4.66996i	−0.112795	−	0.712162i	−0.977666	−	0.210166i	\(-0.932599\pi\)
0.864870	−	0.501995i	\(-0.167401\pi\)
\(47\)	0.161108	−	0.671065i	0.0235001	−	0.0978849i	−0.959387	−	0.282092i	\(-0.908972\pi\)
							0.982887	+	0.184207i	\(0.0589717\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.11.1	✓	288
4.3	odd	2	inner	164.2.o.b.11.3	yes	288
41.15	odd	40	inner	164.2.o.b.15.3	yes	288
164.15	even	40	inner	164.2.o.b.15.1	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.11.1	✓	288	1.1	even	1	trivial
164.2.o.b.11.3	yes	288	4.3	odd	2	inner
164.2.o.b.15.1	yes	288	164.15	even	40	inner
164.2.o.b.15.3	yes	288	41.15	odd	40	inner