Embedded newform 164.2.o.b.63.10

• Label: 164.2.o.b.63.10
• Level: $164$
• Weight: $2$
• Character: 164.63
• 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			63.10
Character	\(\chi\)	\(=\)	164.63
Dual form			164.2.o.b.151.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0656449 + 1.41269i) q^{2} +(0.934786 + 2.25677i) q^{3} +(-1.99138 - 0.185472i) q^{4} +(3.26707 - 0.517454i) q^{5} +(-3.24948 + 1.17242i) q^{6} +(-2.24222 + 1.91504i) q^{7} +(0.392738 - 2.80103i) q^{8} +(-2.09788 + 2.09788i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)	\(129\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{29}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0656449	+	1.41269i	−0.0464180	+	0.998922i
							\(3\)	0.934786	+	2.25677i	0.539699	+	1.30295i	0.924933	+	0.380129i	\(0.124121\pi\)
−0.385235	+	0.922819i	\(0.625879\pi\)
\(5\)	3.26707	−	0.517454i	1.46108	−	0.231412i	0.625260	−	0.780416i	\(-0.284993\pi\)
							0.835820	+	0.549004i	\(0.184993\pi\)
\(7\)	−2.24222	+	1.91504i	−0.847481	+	0.723817i	−0.963004	−	0.269488i	\(-0.913146\pi\)
							0.115523	+	0.993305i	\(0.463146\pi\)
\(11\)	2.22901	−	3.63741i	0.672071	−	1.09672i	−0.316878	−	0.948466i	\(-0.602635\pi\)
							0.988949	−	0.148254i	\(-0.0473655\pi\)
\(13\)	−0.268081	−	3.40630i	−0.0743524	−	0.944736i	−0.915078	−	0.403276i	\(-0.867872\pi\)
							0.840726	−	0.541461i	\(-0.182128\pi\)
\(17\)	−5.36900	+	1.28898i	−1.30217	+	0.312624i	−0.824475	−	0.565899i	\(-0.808529\pi\)
							−0.477700	+	0.878523i	\(0.658529\pi\)
\(19\)	−3.38628	−	0.266506i	−0.776867	−	0.0611407i	−0.316175	−	0.948701i	\(-0.602399\pi\)
							−0.460692	+	0.887560i	\(0.652399\pi\)
\(23\)	0.260126	−	0.800585i	0.0542400	−	0.166934i	−0.920267	−	0.391291i	\(-0.872028\pi\)
							0.974507	+	0.224358i	\(0.0720284\pi\)
\(29\)	5.60558	+	1.34578i	1.04093	+	0.249905i	0.717654	−	0.696400i	\(-0.245216\pi\)
							0.323276	+	0.946305i	\(0.395216\pi\)
\(31\)	1.06891	−	0.776609i	0.191982	−	0.139483i	−0.487642	−	0.873044i	\(-0.662143\pi\)
							0.679624	+	0.733560i	\(0.262143\pi\)
\(37\)	4.45624	+	3.23765i	0.732601	+	0.532266i	0.890385	−	0.455208i	\(-0.150435\pi\)
							−0.157784	+	0.987474i	\(0.550435\pi\)
\(41\)	−6.09190	−	1.97201i	−0.951394	−	0.307976i
							\(43\)	−3.42830	−	6.72843i	−0.522811	−	1.02608i	−0.989887	−	0.141858i	\(-0.954692\pi\)
0.467076	−	0.884217i	\(-0.345308\pi\)
\(47\)	7.24631	+	6.18894i	1.05698	+	0.902749i	0.995443	−	0.0953553i	\(-0.0303987\pi\)
							0.0615404	+	0.998105i	\(0.480399\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.63.10	yes	288
4.3	odd	2	inner	164.2.o.b.63.4	✓	288
41.28	odd	40	inner	164.2.o.b.151.4	yes	288
164.151	even	40	inner	164.2.o.b.151.10	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.63.4	✓	288	4.3	odd	2	inner
164.2.o.b.63.10	yes	288	1.1	even	1	trivial
164.2.o.b.151.4	yes	288	41.28	odd	40	inner
164.2.o.b.151.10	yes	288	164.151	even	40	inner