Embedded newform 128.3.l.a.3.3

• Label: 128.3.l.a.3.3
• 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			3.3
Character	\(\chi\)	\(=\)	128.3
Dual form			128.3.l.a.43.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.90117 + 0.620914i) q^{2} +(0.779996 - 0.640126i) q^{3} +(3.22893 - 2.36093i) q^{4} +(0.179045 + 0.0543127i) q^{5} +(-1.08545 + 1.70130i) q^{6} +(-2.44195 - 12.2765i) q^{7} +(-4.67283 + 6.49343i) q^{8} +(-1.55718 + 7.82847i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.90117	+	0.620914i	−0.950587	+	0.310457i
							\(3\)	0.779996	−	0.640126i	0.259999	−	0.213375i	−0.495359	−	0.868689i	\(-0.664963\pi\)
0.755357	+	0.655313i	\(0.227463\pi\)
\(5\)	0.179045	+	0.0543127i	0.0358090	+	0.0108625i	0.308138	−	0.951342i	\(-0.400294\pi\)
							−0.272329	+	0.962204i	\(0.587794\pi\)
\(7\)	−2.44195	−	12.2765i	−0.348850	−	1.75379i	−0.613739	−	0.789509i	\(-0.710335\pi\)
							0.264889	−	0.964279i	\(-0.414665\pi\)
\(11\)	−0.360362	−	3.65882i	−0.0327602	−	0.332620i	−0.997570	−	0.0696713i	\(-0.977805\pi\)
							0.964810	−	0.262949i	\(-0.0846950\pi\)
\(13\)	−4.42408	−	14.5842i	−0.340314	−	1.12187i	−0.945457	−	0.325748i	\(-0.894384\pi\)
							0.605143	−	0.796117i	\(-0.293116\pi\)
\(17\)	5.83214	−	14.0800i	0.343067	−	0.828237i	−0.654335	−	0.756205i	\(-0.727051\pi\)
							0.997402	−	0.0720325i	\(-0.0229485\pi\)
\(19\)	1.99470	−	3.73182i	0.104984	−	0.196411i	−0.823944	−	0.566671i	\(-0.808231\pi\)
							0.928928	+	0.370259i	\(0.120731\pi\)
\(23\)	28.6157	−	19.1204i	1.24416	−	0.831322i	0.253456	−	0.967347i	\(-0.418433\pi\)
							0.990705	+	0.136025i	\(0.0434328\pi\)
\(29\)	0.216869	−	2.20191i	0.00747825	−	0.0759279i	−0.990721	−	0.135910i	\(-0.956604\pi\)
							0.998199	+	0.0599818i	\(0.0191043\pi\)
\(31\)	7.33155	+	7.33155i	0.236502	+	0.236502i	0.815400	−	0.578898i	\(-0.196517\pi\)
							−0.578898	+	0.815400i	\(0.696517\pi\)
\(37\)	28.2321	+	52.8185i	0.763029	+	1.42753i	0.899517	+	0.436885i	\(0.143918\pi\)
							−0.136488	+	0.990642i	\(0.543582\pi\)
\(41\)	−46.7256	+	31.2210i	−1.13965	+	0.761488i	−0.974413	−	0.224766i	\(-0.927838\pi\)
							−0.165235	+	0.986254i	\(0.552838\pi\)
\(43\)	−25.1836	−	20.6676i	−0.585664	−	0.480642i	0.294280	−	0.955719i	\(-0.404920\pi\)
							−0.879944	+	0.475077i	\(0.842420\pi\)
\(47\)	5.14627	−	12.4242i	0.109495	−	0.264345i	−0.859629	−	0.510919i	\(-0.829305\pi\)
							0.969124	+	0.246575i	\(0.0793051\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.l.a.3.3	✓	496
128.43	odd	32	inner	128.3.l.a.43.3	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.3	✓	496	1.1	even	1	trivial
128.3.l.a.43.3	yes	496	128.43	odd	32	inner