Embedded newform 128.5.l.a.3.6

• Label: 128.5.l.a.3.6
• 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			3.6
Character	\(\chi\)	\(=\)	128.3
Dual form			128.5.l.a.43.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.87913 - 0.975872i) q^{2} +(13.1338 - 10.7787i) q^{3} +(14.0953 + 7.57107i) q^{4} +(-12.6494 - 3.83717i) q^{5} +(-61.4665 + 28.9949i) q^{6} +(15.2716 + 76.7754i) q^{7} +(-47.2893 - 43.1245i) q^{8} +(40.5159 - 203.687i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1008 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.87913	−	0.975872i	−0.969783	−	0.243968i
							\(3\)	13.1338	−	10.7787i	1.45932	−	1.19763i	0.520193	−	0.854049i	\(-0.325860\pi\)
0.939123	−	0.343581i	\(-0.111640\pi\)
\(5\)	−12.6494	−	3.83717i	−0.505978	−	0.153487i	0.0269678	−	0.999636i	\(-0.491415\pi\)
							−0.532945	+	0.846150i	\(0.678915\pi\)
\(7\)	15.2716	+	76.7754i	0.311665	+	1.56684i	0.745908	+	0.666049i	\(0.232016\pi\)
							−0.434243	+	0.900796i	\(0.642984\pi\)
\(11\)	−8.68399	−	88.1700i	−0.0717685	−	0.728678i	−0.961978	−	0.273127i	\(-0.911942\pi\)
							0.890210	−	0.455551i	\(-0.150558\pi\)
\(13\)	−75.8004	−	249.880i	−0.448523	−	1.47858i	−0.831866	−	0.554977i	\(-0.812727\pi\)
							0.383343	−	0.923606i	\(-0.374773\pi\)
\(17\)	120.896	−	291.868i	0.418324	−	1.00992i	−0.564509	−	0.825427i	\(-0.690934\pi\)
							0.982833	−	0.184497i	\(-0.0590655\pi\)
\(19\)	−62.8229	+	117.533i	−0.174025	+	0.325577i	−0.953779	−	0.300508i	\(-0.902844\pi\)
							0.779755	+	0.626085i	\(0.215344\pi\)
\(23\)	382.039	−	255.270i	0.722190	−	0.482552i	−0.139349	−	0.990243i	\(-0.544501\pi\)
							0.861539	+	0.507691i	\(0.169501\pi\)
\(29\)	23.9176	−	242.840i	0.0284395	−	0.288751i	−0.970437	−	0.241356i	\(-0.922408\pi\)
							0.998876	−	0.0473955i	\(-0.0150921\pi\)
\(31\)	−27.0958	−	27.0958i	−0.0281955	−	0.0281955i	0.692869	−	0.721064i	\(-0.256347\pi\)
							−0.721064	+	0.692869i	\(0.756347\pi\)
\(37\)	459.523	+	859.706i	0.335663	+	0.627981i	0.991964	−	0.126518i	\(-0.0403800\pi\)
							−0.656301	+	0.754499i	\(0.727880\pi\)
\(41\)	1443.46	−	964.491i	0.858693	−	0.573760i	−0.0464248	−	0.998922i	\(-0.514783\pi\)
							0.905117	+	0.425162i	\(0.139783\pi\)
\(43\)	213.229	+	174.993i	0.115321	+	0.0946418i	0.690287	−	0.723536i	\(-0.257484\pi\)
							−0.574965	+	0.818178i	\(0.694984\pi\)
\(47\)	−1073.26	+	2591.08i	−0.485859	+	1.17297i	0.470927	+	0.882172i	\(0.343920\pi\)
							−0.956786	+	0.290794i	\(0.906080\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.l.a.3.6	✓	1008
128.43	odd	32	inner	128.5.l.a.43.6	yes	1008

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.l.a.3.6	✓	1008	1.1	even	1	trivial
128.5.l.a.43.6	yes	1008	128.43	odd	32	inner