Embedded newform 128.5.l.a.3.5

• Label: 128.5.l.a.3.5
• Level: $128$
• Weight: $5$
• Character: 128.3
• Analytic conductor: $13.231$
• Analytic rank: $0$
• Dimension: $1008$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	128.l (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.2313552747\)
Analytic rank:	\(0\)
Dimension:	\(1008\)
Relative dimension:	\(63\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			3.5
Character	\(\chi\)	\(=\)	128.3
Dual form			128.5.l.a.43.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.89182 + 0.923973i) q^{2} +(-7.47268 + 6.13267i) q^{3} +(14.2925 - 7.19188i) q^{4} +(-34.4748 - 10.4578i) q^{5} +(23.4159 - 30.7718i) q^{6} +(-4.53604 - 22.8042i) q^{7} +(-48.9789 + 41.1954i) q^{8} +(2.42901 - 12.2115i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1008 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.89182	+	0.923973i	−0.972955	+	0.230993i
							\(3\)	−7.47268	+	6.13267i	−0.830298	+	0.681408i	−0.950290	−	0.311367i	\(-0.899213\pi\)
0.119992	+	0.992775i	\(0.461713\pi\)
\(5\)	−34.4748	−	10.4578i	−1.37899	−	0.418312i	−0.488202	−	0.872730i	\(-0.662347\pi\)
							−0.890788	+	0.454418i	\(0.849847\pi\)
\(7\)	−4.53604	−	22.8042i	−0.0925723	−	0.465392i	−0.999067	−	0.0431762i	\(-0.986252\pi\)
							0.906495	−	0.422216i	\(-0.138748\pi\)
\(11\)	−15.5512	−	157.894i	−0.128522	−	1.30491i	−0.815725	−	0.578441i	\(-0.803661\pi\)
							0.687202	−	0.726466i	\(-0.258839\pi\)
\(13\)	−30.2246	−	99.6372i	−0.178844	−	0.589569i	−0.999827	−	0.0186162i	\(-0.994074\pi\)
							0.820983	−	0.570953i	\(-0.193426\pi\)
\(17\)	26.2758	−	63.4353i	0.0909196	−	0.219499i	−0.871878	−	0.489723i	\(-0.837098\pi\)
							0.962797	+	0.270224i	\(0.0870978\pi\)
\(19\)	−129.558	+	242.385i	−0.358885	+	0.671427i	−0.994968	−	0.100194i	\(-0.968054\pi\)
							0.636083	+	0.771621i	\(0.280554\pi\)
\(23\)	−496.261	+	331.591i	−0.938112	+	0.626826i	−0.927783	−	0.373120i	\(-0.878288\pi\)
							−0.0103290	+	0.999947i	\(0.503288\pi\)
\(29\)	−52.0724	+	528.700i	−0.0619172	+	0.628656i	0.913282	+	0.407327i	\(0.133539\pi\)
							−0.975199	+	0.221328i	\(0.928961\pi\)
\(31\)	844.449	+	844.449i	0.878720	+	0.878720i	0.993402	−	0.114683i	\(-0.0365851\pi\)
							−0.114683	+	0.993402i	\(0.536585\pi\)
\(37\)	875.288	+	1637.55i	0.639363	+	1.19616i	0.967820	+	0.251645i	\(0.0809716\pi\)
							−0.328456	+	0.944519i	\(0.606528\pi\)
\(41\)	452.714	−	302.494i	0.269312	−	0.179949i	−0.413580	−	0.910468i	\(-0.635722\pi\)
							0.682892	+	0.730519i	\(0.260722\pi\)
\(43\)	−364.268	−	298.947i	−0.197008	−	0.161680i	0.530717	−	0.847549i	\(-0.321923\pi\)
							−0.727726	+	0.685868i	\(0.759423\pi\)
\(47\)	−1167.26	+	2818.02i	−0.528411	+	1.27570i	0.404152	+	0.914692i	\(0.367567\pi\)
							−0.932563	+	0.361006i	\(0.882433\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.l.a.3.5	✓	1008
128.43	odd	32	inner	128.5.l.a.43.5	yes	1008

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.l.a.3.5	✓	1008	1.1	even	1	trivial
128.5.l.a.43.5	yes	1008	128.43	odd	32	inner