Embedded newform 128.2.k.a.5.1

• Label: 128.2.k.a.5.1
• 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			5.1
Character	\(\chi\)	\(=\)	128.5
Dual form			128.2.k.a.77.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41035 - 0.104463i) q^{2} +(1.66739 + 0.505799i) q^{3} +(1.97817 + 0.294660i) q^{4} +(0.127039 + 0.0125123i) q^{5} +(-2.29877 - 0.887535i) q^{6} +(1.18113 + 1.76769i) q^{7} +(-2.75914 - 0.622220i) q^{8} +(0.0299647 + 0.0200218i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41035	−	0.104463i	−0.997268	−	0.0738667i
							\(3\)	1.66739	+	0.505799i	0.962671	+	0.292023i	0.732226	−	0.681062i	\(-0.238482\pi\)
0.230445	+	0.973085i	\(0.425982\pi\)
\(5\)	0.127039	+	0.0125123i	0.0568137	+	0.00559567i	0.126384	−	0.991981i	\(-0.459663\pi\)
							−0.0695703	+	0.997577i	\(0.522163\pi\)
\(7\)	1.18113	+	1.76769i	0.446427	+	0.668125i	0.984623	−	0.174695i	\(-0.0558940\pi\)
							−0.538196	+	0.842820i	\(0.680894\pi\)
\(11\)	1.19515	−	2.23597i	0.360351	−	0.674170i	−0.634784	−	0.772690i	\(-0.718911\pi\)
							0.995135	+	0.0985202i	\(0.0314109\pi\)
\(13\)	0.393262	+	3.99285i	0.109071	+	1.10742i	0.881457	+	0.472264i	\(0.156563\pi\)
							−0.772386	+	0.635154i	\(0.780937\pi\)
\(17\)	1.68215	−	0.696769i	0.407981	−	0.168991i	−0.169248	−	0.985573i	\(-0.554134\pi\)
							0.577229	+	0.816582i	\(0.304134\pi\)
\(19\)	1.73148	−	2.10981i	0.397228	−	0.484024i	−0.535535	−	0.844513i	\(-0.679890\pi\)
							0.932763	+	0.360489i	\(0.117390\pi\)
\(23\)	−0.387809	−	1.94965i	−0.0808638	−	0.406530i	−0.999923	−	0.0124102i	\(-0.996050\pi\)
							0.919059	−	0.394119i	\(-0.128950\pi\)
\(29\)	−5.51926	+	2.95011i	−1.02490	+	0.547821i	−0.896101	−	0.443850i	\(-0.853612\pi\)
							−0.128800	+	0.991671i	\(0.541112\pi\)
\(31\)	−3.26985	−	3.26985i	−0.587282	−	0.587282i	0.349612	−	0.936894i	\(-0.386313\pi\)
							−0.936894	+	0.349612i	\(0.886313\pi\)
\(37\)	−4.27534	+	3.50868i	−0.702862	+	0.576824i	−0.916474	−	0.400095i	\(-0.868977\pi\)
							0.213612	+	0.976919i	\(0.431477\pi\)
\(41\)	−8.09161	+	1.60952i	−1.26370	+	0.251365i	−0.781060	−	0.624456i	\(-0.785321\pi\)
							−0.482637	+	0.875821i	\(0.660321\pi\)
\(43\)	9.33199	−	2.83083i	1.42311	−	0.431697i	0.517458	−	0.855709i	\(-0.326878\pi\)
							0.905657	+	0.424011i	\(0.139378\pi\)
\(47\)	−3.95511	−	9.54848i	−0.576912	−	1.39279i	−0.895571	−	0.444919i	\(-0.853232\pi\)
							0.318659	−	0.947870i	\(-0.396768\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.2.k.a.5.1	✓	240
4.3	odd	2		512.2.k.a.113.4		240
128.51	odd	32		512.2.k.a.145.4		240
128.77	even	32	inner	128.2.k.a.77.1	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.2.k.a.5.1	✓	240	1.1	even	1	trivial
128.2.k.a.77.1	yes	240	128.77	even	32	inner
512.2.k.a.113.4		240	4.3	odd	2
512.2.k.a.145.4		240	128.51	odd	32