Embedded newform 128.3.l.a.3.15

• Label: 128.3.l.a.3.15
• Level: $128$
• Weight: $3$
• Character: 128.3
• 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			3.15
Character	\(\chi\)	\(=\)	128.3
Dual form			128.3.l.a.43.15

$q$-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 \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.15	✓	496	1.1	even	1	trivial
128.3.l.a.43.15	yes	496	128.43	odd	32	inner