Embedded newform 128.2.k.a.5.5

• Label: 128.2.k.a.5.5
• Level: $128$
• Weight: $2$
• Character: 128.5
• Analytic conductor: $1.022$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.02208514587\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(15\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			5.5
Character	\(\chi\)	\(=\)	128.5
Dual form			128.2.k.a.77.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.919458 + 1.07452i) q^{2} +(-2.80857 - 0.851969i) q^{3} +(-0.309193 - 1.97596i) q^{4} +(1.35768 + 0.133720i) q^{5} +(3.49782 - 2.23451i) q^{6} +(1.67452 + 2.50610i) q^{7} +(2.40750 + 1.48457i) q^{8} +(4.66778 + 3.11891i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.919458	+	1.07452i	−0.650155	+	0.759802i
							\(3\)	−2.80857	−	0.851969i	−1.62153	−	0.491885i	−0.656563	−	0.754271i	\(-0.727991\pi\)
−0.964963	+	0.262386i	\(0.915491\pi\)
\(5\)	1.35768	+	0.133720i	0.607173	+	0.0598013i	0.396932	−	0.917848i	\(-0.370075\pi\)
							0.210241	+	0.977650i	\(0.432575\pi\)
\(7\)	1.67452	+	2.50610i	0.632911	+	0.947218i	0.999856	+	0.0169753i	\(0.00540366\pi\)
							−0.366945	+	0.930243i	\(0.619596\pi\)
\(11\)	2.23846	−	4.18786i	0.674921	−	1.26269i	−0.278323	−	0.960487i	\(-0.589779\pi\)
							0.953244	−	0.302201i	\(-0.0977213\pi\)
\(13\)	0.297700	+	3.02259i	0.0825670	+	0.838317i	0.943890	+	0.330259i	\(0.107136\pi\)
							−0.861323	+	0.508057i	\(0.830364\pi\)
\(17\)	6.57785	−	2.72463i	1.59536	−	0.660820i	0.604612	−	0.796520i	\(-0.293328\pi\)
							0.990750	+	0.135700i	\(0.0433283\pi\)
\(19\)	−0.530432	+	0.646334i	−0.121690	+	0.148279i	−0.830292	−	0.557329i	\(-0.811826\pi\)
							0.708602	+	0.705608i	\(0.249326\pi\)
\(23\)	0.595434	+	2.99345i	0.124157	+	0.624177i	0.991885	+	0.127138i	\(0.0405793\pi\)
							−0.867728	+	0.497039i	\(0.834421\pi\)
\(29\)	4.38039	−	2.34137i	0.813418	−	0.434781i	−0.0115685	−	0.999933i	\(-0.503682\pi\)
							0.824987	+	0.565152i	\(0.191182\pi\)
\(31\)	2.41787	+	2.41787i	0.434262	+	0.434262i	0.890075	−	0.455813i	\(-0.150652\pi\)
							−0.455813	+	0.890075i	\(0.650652\pi\)
\(37\)	−1.25366	+	1.02885i	−0.206100	+	0.169142i	−0.731781	−	0.681540i	\(-0.761311\pi\)
							0.525681	+	0.850682i	\(0.323811\pi\)
\(41\)	−7.68881	+	1.52940i	−1.20079	+	0.238852i	−0.754660	−	0.656116i	\(-0.772198\pi\)
							−0.446131	+	0.894968i	\(0.647198\pi\)
\(43\)	2.94295	−	0.892733i	0.448795	−	0.136140i	−0.0577933	−	0.998329i	\(-0.518406\pi\)
							0.506588	+	0.862188i	\(0.330906\pi\)
\(47\)	1.75089	+	4.22702i	0.255394	+	0.616575i	0.998623	−	0.0524621i	\(-0.0167069\pi\)
							−0.743229	+	0.669037i	\(0.766707\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.2.k.a.5.5	✓	240
4.3	odd	2		512.2.k.a.113.14		240
128.51	odd	32		512.2.k.a.145.14		240
128.77	even	32	inner	128.2.k.a.77.5	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.2.k.a.5.5	✓	240	1.1	even	1	trivial
128.2.k.a.77.5	yes	240	128.77	even	32	inner
512.2.k.a.113.14		240	4.3	odd	2
512.2.k.a.145.14		240	128.51	odd	32