Embedded newform 128.3.l.a.43.15

• Label: 128.3.l.a.43.15
• Level: $128$
• Weight: $3$
• Character: 128.43
• Analytic conductor: $3.488$
• Analytic rank: $0$
• Dimension: $496$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	128.l (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.48774738381\)
Analytic rank:	\(0\)
Dimension:	\(496\)
Relative dimension:	\(31\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			43.15
Character	\(\chi\)	\(=\)	128.43
Dual form			128.3.l.a.3.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.203195 - 1.98965i) q^{2} +(2.70039 + 2.21615i) q^{3} +(-3.91742 + 0.808573i) q^{4} +(7.86468 - 2.38573i) q^{5} +(3.86067 - 5.82314i) q^{6} +(-2.00235 + 10.0665i) q^{7} +(2.40478 + 7.63001i) q^{8} +(0.624959 + 3.14188i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(5\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{29}{32}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.203195	−	1.98965i	−0.101597	−	0.994826i
							\(3\)	2.70039	+	2.21615i	0.900130	+	0.738717i	0.965845	−	0.259121i	\(-0.0834328\pi\)
−0.0657150	+	0.997838i	\(0.520933\pi\)
\(5\)	7.86468	−	2.38573i	1.57294	−	0.477145i	0.621095	−	0.783735i	\(-0.286688\pi\)
							0.951842	+	0.306590i	\(0.0991881\pi\)
\(7\)	−2.00235	+	10.0665i	−0.286051	+	1.43807i	0.524002	+	0.851717i	\(0.324438\pi\)
							−0.810053	+	0.586357i	\(0.800562\pi\)
\(11\)	−0.496976	+	5.04589i	−0.0451797	+	0.458717i	0.945804	+	0.324737i	\(0.105276\pi\)
							−0.990984	+	0.133980i	\(0.957224\pi\)
\(13\)	4.42453	−	14.5857i	0.340348	−	1.12198i	−0.605085	−	0.796161i	\(-0.706861\pi\)
							0.945433	−	0.325817i	\(-0.105639\pi\)
\(17\)	3.79620	+	9.16483i	0.223306	+	0.539107i	0.995335	−	0.0964794i	\(-0.0307582\pi\)
							−0.772029	+	0.635587i	\(0.780758\pi\)
\(19\)	−14.2242	−	26.6116i	−0.748641	−	1.40061i	−0.910379	−	0.413776i	\(-0.864210\pi\)
							0.161738	−	0.986834i	\(-0.448290\pi\)
\(23\)	−9.19133	−	6.14145i	−0.399623	−	0.267020i	0.339478	−	0.940614i	\(-0.389749\pi\)
							−0.739101	+	0.673594i	\(0.764749\pi\)
\(29\)	−1.72942	−	17.5591i	−0.0596351	−	0.605486i	−0.977836	−	0.209371i	\(-0.932858\pi\)
							0.918201	−	0.396115i	\(-0.129642\pi\)
\(31\)	−36.5215	+	36.5215i	−1.17811	+	1.17811i	−0.197888	+	0.980225i	\(0.563408\pi\)
							−0.980225	+	0.197888i	\(0.936592\pi\)
\(37\)	−29.8268	+	55.8021i	−0.806131	+	1.50816i	0.0542582	+	0.998527i	\(0.482721\pi\)
							−0.860389	+	0.509638i	\(0.829779\pi\)
\(41\)	−31.5638	−	21.0903i	−0.769849	−	0.514396i	0.107526	−	0.994202i	\(-0.465707\pi\)
							−0.877375	+	0.479806i	\(0.840707\pi\)
\(43\)	29.0695	−	23.8567i	0.676034	−	0.554807i	−0.232511	−	0.972594i	\(-0.574694\pi\)
							0.908545	+	0.417787i	\(0.137194\pi\)
\(47\)	15.5404	+	37.5178i	0.330646	+	0.798251i	0.998541	+	0.0539955i	\(0.0171956\pi\)
							−0.667895	+	0.744256i	\(0.732804\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.l.a.43.15	yes	496
128.3	odd	32	inner	128.3.l.a.3.15	✓	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.15	✓	496	128.3	odd	32	inner
128.3.l.a.43.15	yes	496	1.1	even	1	trivial