Embedded newform 164.2.o.b.111.12

• Label: 164.2.o.b.111.12
• Level: $164$
• Weight: $2$
• Character: 164.111
• 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			111.12
Character	\(\chi\)	\(=\)	164.111
Dual form			164.2.o.b.99.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.651149 - 1.25539i) q^{2} +(-1.44458 - 0.598363i) q^{3} +(-1.15201 - 1.63489i) q^{4} +(2.07896 - 1.05928i) q^{5} +(-1.69181 + 1.42388i) q^{6} +(-0.213546 - 0.348475i) q^{7} +(-2.80256 + 0.381665i) q^{8} +(-0.392561 - 0.392561i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 12 q^{2} - 20 q^{4} - 32 q^{5} - 28 q^{6} - 24 q^{8} - 40 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.651149	−	1.25539i	0.460432	−	0.887695i
							\(3\)	−1.44458	−	0.598363i	−0.834026	−	0.345465i	−0.0755307	−	0.997143i	\(-0.524065\pi\)
−0.758495	+	0.651679i	\(0.774065\pi\)
\(5\)	2.07896	−	1.05928i	0.929738	−	0.473725i	0.0775665	−	0.996987i	\(-0.475285\pi\)
							0.852172	+	0.523262i	\(0.175285\pi\)
\(7\)	−0.213546	−	0.348475i	−0.0807128	−	0.131711i	0.809737	−	0.586793i	\(-0.199610\pi\)
							−0.890450	+	0.455082i	\(0.849610\pi\)
\(11\)	−1.39182	−	0.109539i	−0.419649	−	0.0330271i	−0.133123	−	0.991100i	\(-0.542501\pi\)
							−0.286526	+	0.958072i	\(0.592501\pi\)
\(13\)	−0.0467934	−	0.194908i	−0.0129781	−	0.0540579i	0.965505	−	0.260384i	\(-0.0838492\pi\)
							−0.978483	+	0.206326i	\(0.933849\pi\)
\(17\)	1.14996	−	0.982159i	0.278906	−	0.238208i	−0.498980	−	0.866613i	\(-0.666292\pi\)
							0.777886	+	0.628405i	\(0.216292\pi\)
\(19\)	5.19518	+	1.24725i	1.19186	+	0.286139i	0.780394	−	0.625288i	\(-0.215018\pi\)
							0.411462	+	0.911427i	\(0.365018\pi\)
\(23\)	7.30867	−	5.31006i	1.52396	−	1.10722i	0.564484	−	0.825444i	\(-0.309075\pi\)
							0.959479	−	0.281781i	\(-0.0909250\pi\)
\(29\)	5.29830	+	4.52518i	0.983870	+	0.840304i	0.987167	−	0.159690i	\(-0.0510496\pi\)
							−0.00329739	+	0.999995i	\(0.501050\pi\)
\(31\)	−0.798881	−	2.45870i	−0.143483	−	0.441596i	0.853330	−	0.521372i	\(-0.174580\pi\)
							−0.996813	+	0.0797759i	\(0.974580\pi\)
\(37\)	−0.0154177	+	0.0474509i	−0.00253466	+	0.00780088i	−0.952316	−	0.305114i	\(-0.901305\pi\)
							0.949781	+	0.312915i	\(0.101305\pi\)
\(41\)	−3.12320	+	5.58978i	−0.487762	+	0.872977i
							\(43\)	−6.26523	−	0.992315i	−0.955439	−	0.151327i	−0.340800	−	0.940136i	\(-0.610698\pi\)
−0.614638	+	0.788809i	\(0.710698\pi\)
\(47\)	2.42806	−	3.96224i	0.354169	−	0.577952i	−0.625092	−	0.780551i	\(-0.714939\pi\)
							0.979262	+	0.202599i	\(0.0649387\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.o.b.111.12	yes	288
4.3	odd	2	inner	164.2.o.b.111.9	yes	288
41.17	odd	40	inner	164.2.o.b.99.9	✓	288
164.99	even	40	inner	164.2.o.b.99.12	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.o.b.99.9	✓	288	41.17	odd	40	inner
164.2.o.b.99.12	yes	288	164.99	even	40	inner
164.2.o.b.111.9	yes	288	4.3	odd	2	inner
164.2.o.b.111.12	yes	288	1.1	even	1	trivial