Embedded newform 128.5.l.a.3.12

• Label: 128.5.l.a.3.12
• Level: $128$
• Weight: $5$
• Character: 128.3
• Analytic conductor: $13.231$
• Analytic rank: $0$
• Dimension: $1008$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	128.l (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.2313552747\)
Analytic rank:	\(0\)
Dimension:	\(1008\)
Relative dimension:	\(63\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			3.12
Character	\(\chi\)	\(=\)	128.3
Dual form			128.5.l.a.43.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.26315 + 2.31341i) q^{2} +(9.84830 - 8.08229i) q^{3} +(5.29629 - 15.0980i) q^{4} +(-14.5438 - 4.41182i) q^{5} +(-13.4388 + 49.1569i) q^{6} +(-8.09349 - 40.6887i) q^{7} +(17.6452 + 61.5195i) q^{8} +(15.8633 - 79.7501i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1008 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{3}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.26315	+	2.31341i	−0.815787	+	0.578352i
							\(3\)	9.84830	−	8.08229i	1.09426	−	0.898033i	0.0989770	−	0.995090i	\(-0.468443\pi\)
0.995279	+	0.0970572i	\(0.0309430\pi\)
\(5\)	−14.5438	−	4.41182i	−0.581753	−	0.176473i	−0.0143271	−	0.999897i	\(-0.504561\pi\)
							−0.567426	+	0.823425i	\(0.692061\pi\)
\(7\)	−8.09349	−	40.6887i	−0.165173	−	0.830382i	−0.971156	−	0.238443i	\(-0.923363\pi\)
							0.805983	−	0.591938i	\(-0.201637\pi\)
\(11\)	−6.45886	−	65.5779i	−0.0533790	−	0.541966i	−0.984270	−	0.176670i	\(-0.943468\pi\)
							0.930891	−	0.365297i	\(-0.119032\pi\)
\(13\)	−23.2592	−	76.6754i	−0.137629	−	0.453701i	0.860849	−	0.508861i	\(-0.169933\pi\)
							−0.998478	+	0.0551599i	\(0.982433\pi\)
\(17\)	−33.3348	+	80.4774i	−0.115345	+	0.278468i	−0.971000	−	0.239078i	\(-0.923155\pi\)
							0.855655	+	0.517547i	\(0.173155\pi\)
\(19\)	−218.707	+	409.173i	−0.605838	+	1.13344i	0.372812	+	0.927907i	\(0.378394\pi\)
							−0.978650	+	0.205536i	\(0.934106\pi\)
\(23\)	−247.473	+	165.356i	−0.467812	+	0.312582i	−0.767040	−	0.641600i	\(-0.778271\pi\)
							0.299227	+	0.954182i	\(0.403271\pi\)
\(29\)	43.7002	−	443.695i	0.0519621	−	0.527580i	−0.933606	−	0.358301i	\(-0.883356\pi\)
							0.985568	−	0.169279i	\(-0.0541439\pi\)
\(31\)	−805.336	−	805.336i	−0.838019	−	0.838019i	0.150579	−	0.988598i	\(-0.451886\pi\)
							−0.988598	+	0.150579i	\(0.951886\pi\)
\(37\)	−704.762	−	1318.52i	−0.514800	−	0.963124i	−0.996239	−	0.0866441i	\(-0.972386\pi\)
							0.481439	−	0.876480i	\(-0.340114\pi\)
\(41\)	1136.33	−	759.268i	0.675982	−	0.451676i	−0.169607	−	0.985512i	\(-0.554250\pi\)
							0.845588	+	0.533835i	\(0.179250\pi\)
\(43\)	−811.369	−	665.873i	−0.438815	−	0.360126i	0.388895	−	0.921282i	\(-0.372857\pi\)
							−0.827710	+	0.561156i	\(0.810357\pi\)
\(47\)	−200.100	+	483.083i	−0.0905838	+	0.218689i	−0.962678	−	0.270649i	\(-0.912762\pi\)
							0.872094	+	0.489338i	\(0.162762\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.l.a.3.12	✓	1008
128.43	odd	32	inner	128.5.l.a.43.12	yes	1008

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.l.a.3.12	✓	1008	1.1	even	1	trivial
128.5.l.a.43.12	yes	1008	128.43	odd	32	inner