Embedded newform 164.2.o.b.95.6

• Label: 164.2.o.b.95.6
• Level: $164$
• Weight: $2$
• Character: 164.95
• 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			95.6
Character	\(\chi\)	\(=\)	164.95
Dual form			164.2.o.b.19.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.936526 + 1.05968i) q^{2} +(2.78111 + 1.15197i) q^{3} +(-0.245837 - 1.98483i) q^{4} +(-0.0971231 - 0.0153828i) q^{5} +(-3.82530 + 1.86823i) q^{6} +(-0.232482 + 0.272202i) q^{7} +(2.33352 + 1.59834i) q^{8} +(4.28620 + 4.28620i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)	\(129\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{31}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.936526	+	1.05968i	−0.662224	+	0.749306i
							\(3\)	2.78111	+	1.15197i	1.60567	+	0.665092i	0.992204	−	0.124626i	\(-0.0397732\pi\)
0.613470	+	0.789718i	\(0.289773\pi\)
\(5\)	−0.0971231	−	0.0153828i	−0.0434348	−	0.00687939i	0.134679	−	0.990889i	\(-0.457000\pi\)
							−0.178114	+	0.984010i	\(0.557000\pi\)
\(7\)	−0.232482	+	0.272202i	−0.0878701	+	0.102883i	−0.802599	−	0.596519i	\(-0.796550\pi\)
							0.714729	+	0.699402i	\(0.246550\pi\)
\(11\)	2.00934	−	1.23132i	0.605839	−	0.371258i	−0.185493	−	0.982646i	\(-0.559388\pi\)
							0.791332	+	0.611387i	\(0.209388\pi\)
\(13\)	−6.04983	−	0.476132i	−1.67792	−	0.132055i	−0.796579	−	0.604534i	\(-0.793359\pi\)
							−0.881342	+	0.472479i	\(0.843359\pi\)
\(17\)	1.01376	−	4.22260i	0.245872	−	1.02413i	−0.703679	−	0.710518i	\(-0.748461\pi\)
							0.949551	−	0.313612i	\(-0.101539\pi\)
\(19\)	0.209528	+	2.66231i	0.0480691	+	0.610776i	0.973025	+	0.230701i	\(0.0741019\pi\)
							−0.924956	+	0.380075i	\(0.875898\pi\)
\(23\)	1.23543	+	3.80227i	0.257605	+	0.792827i	0.993305	+	0.115520i	\(0.0368533\pi\)
							−0.735700	+	0.677307i	\(0.763147\pi\)
\(29\)	−0.721610	−	3.00572i	−0.134000	−	0.558149i	−0.998461	−	0.0554502i	\(-0.982341\pi\)
							0.864462	−	0.502698i	\(-0.167659\pi\)
\(31\)	−3.69803	−	2.68678i	−0.664186	−	0.482560i	0.203888	−	0.978994i	\(-0.434642\pi\)
							−0.868074	+	0.496435i	\(0.834642\pi\)
\(37\)	0.807960	−	0.587017i	0.132828	−	0.0965051i	−0.519387	−	0.854539i	\(-0.673840\pi\)
							0.652215	+	0.758034i	\(0.273840\pi\)
\(41\)	−3.91906	−	5.06369i	−0.612054	−	0.790816i
							\(43\)	5.54362	−	10.8800i	0.845394	−	1.65918i	0.0976220	−	0.995224i	\(-0.468876\pi\)
0.747772	−	0.663956i	\(-0.231124\pi\)
\(47\)	6.76166	+	7.91689i	0.986289	+	1.15480i	0.987946	+	0.154798i	\(0.0494726\pi\)
							−0.00165682	+	0.999999i	\(0.500527\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.95.6	yes	288
4.3	odd	2	inner	164.2.o.b.95.12	yes	288
41.19	odd	40	inner	164.2.o.b.19.12	yes	288
164.19	even	40	inner	164.2.o.b.19.6	✓	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.19.6	✓	288	164.19	even	40	inner
164.2.o.b.19.12	yes	288	41.19	odd	40	inner
164.2.o.b.95.6	yes	288	1.1	even	1	trivial
164.2.o.b.95.12	yes	288	4.3	odd	2	inner