Embedded newform 164.2.o.b.111.10

• Label: 164.2.o.b.111.10
• Level: $164$
• Weight: $2$
• Character: 164.111
• Analytic conductor: $1.310$
• Analytic rank: $0$
• Dimension: $288$
• 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			111.10
Character	\(\chi\)	\(=\)	164.111
Dual form			164.2.o.b.99.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0891741 + 1.41140i) q^{2} +(3.05939 + 1.26724i) q^{3} +(-1.98410 + 0.251721i) q^{4} +(0.290054 - 0.147790i) q^{5} +(-1.51576 + 4.43102i) q^{6} +(-1.49074 - 2.43266i) q^{7} +(-0.532208 - 2.77790i) q^{8} +(5.63263 + 5.63263i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(83\)	\(129\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{7}{40}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0891741	+	1.41140i	0.0630556	+	0.998010i
							\(3\)	3.05939	+	1.26724i	1.76634	+	0.731641i	0.995516	+	0.0945895i	\(0.0301538\pi\)
0.770821	+	0.637052i	\(0.219846\pi\)
\(5\)	0.290054	−	0.147790i	0.129716	−	0.0660936i	−0.387928	−	0.921690i	\(-0.626809\pi\)
							0.517644	+	0.855596i	\(0.326809\pi\)
\(7\)	−1.49074	−	2.43266i	−0.563446	−	0.919460i	−0.999806	−	0.0196880i	\(-0.993733\pi\)
							0.436360	−	0.899772i	\(-0.356267\pi\)
\(11\)	−3.50558	−	0.275895i	−1.05697	−	0.0831854i	−0.461958	−	0.886902i	\(-0.652853\pi\)
							−0.595013	+	0.803716i	\(0.702853\pi\)
\(13\)	−0.883038	−	3.67812i	−0.244911	−	1.02013i	−0.950307	−	0.311314i	\(-0.899231\pi\)
							0.705396	−	0.708813i	\(-0.250769\pi\)
\(17\)	1.39866	−	1.19457i	0.339225	−	0.289725i	−0.463473	−	0.886111i	\(-0.653397\pi\)
							0.802698	+	0.596386i	\(0.203397\pi\)
\(19\)	−1.69124	−	0.406031i	−0.387997	−	0.0931498i	0.0347536	−	0.999396i	\(-0.488935\pi\)
							−0.422751	+	0.906246i	\(0.638935\pi\)
\(23\)	−3.85416	+	2.80021i	−0.803649	+	0.583885i	−0.911982	−	0.410229i	\(-0.865449\pi\)
							0.108333	+	0.994115i	\(0.465449\pi\)
\(29\)	6.55512	+	5.59860i	1.21725	+	1.03963i	0.997951	+	0.0639768i	\(0.0203783\pi\)
							0.219303	+	0.975657i	\(0.429622\pi\)
\(31\)	−1.45882	−	4.48978i	−0.262012	−	0.806389i	−0.992367	−	0.123321i	\(-0.960645\pi\)
							0.730355	−	0.683068i	\(-0.239355\pi\)
\(37\)	−1.02582	+	3.15716i	−0.168644	+	0.519034i	−0.999286	−	0.0377730i	\(-0.987974\pi\)
							0.830642	+	0.556807i	\(0.187974\pi\)
\(41\)	4.73492	−	4.31052i	0.739470	−	0.673190i
							\(43\)	0.501753	+	0.0794698i	0.0765166	+	0.0121190i	0.194575	−	0.980888i	\(-0.437667\pi\)
−0.118059	+	0.993007i	\(0.537667\pi\)
\(47\)	0.324411	−	0.529390i	0.0473201	−	0.0772195i	−0.828105	−	0.560572i	\(-0.810581\pi\)
							0.875426	+	0.483353i	\(0.160581\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.111.10	yes	288
4.3	odd	2	inner	164.2.o.b.111.7	yes	288
41.17	odd	40	inner	164.2.o.b.99.7	✓	288
164.99	even	40	inner	164.2.o.b.99.10	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.99.7	✓	288	41.17	odd	40	inner
164.2.o.b.99.10	yes	288	164.99	even	40	inner
164.2.o.b.111.7	yes	288	4.3	odd	2	inner
164.2.o.b.111.10	yes	288	1.1	even	1	trivial