Embedded newform 128.3.l.a.3.19

• Label: 128.3.l.a.3.19
• 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			3.19
Character	\(\chi\)	\(=\)	128.3
Dual form			128.3.l.a.43.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.359443 - 1.96744i) q^{2} +(1.06123 - 0.870929i) q^{3} +(-3.74160 - 1.41436i) q^{4} +(-6.51407 - 1.97602i) q^{5} +(-1.33205 - 2.40095i) q^{6} +(-0.702367 - 3.53104i) q^{7} +(-4.12755 + 6.85298i) q^{8} +(-1.38812 + 6.97855i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.359443	−	1.96744i	0.179721	−	0.983718i
							\(3\)	1.06123	−	0.870929i	0.353743	−	0.290310i	−0.440692	−	0.897658i	\(-0.645267\pi\)
0.794435	+	0.607349i	\(0.207767\pi\)
\(5\)	−6.51407	−	1.97602i	−1.30281	−	0.395204i	−0.438760	−	0.898604i	\(-0.644582\pi\)
							−0.864053	+	0.503400i	\(0.832082\pi\)
\(7\)	−0.702367	−	3.53104i	−0.100338	−	0.504434i	−0.997970	−	0.0636876i	\(-0.979714\pi\)
							0.897632	−	0.440746i	\(-0.145286\pi\)
\(11\)	−0.787497	−	7.99559i	−0.0715906	−	0.726872i	−0.962244	−	0.272189i	\(-0.912253\pi\)
							0.890653	−	0.454683i	\(-0.150247\pi\)
\(13\)	−2.30539	−	7.59984i	−0.177337	−	0.584603i	−0.999874	−	0.0158904i	\(-0.994942\pi\)
							0.822536	−	0.568712i	\(-0.192558\pi\)
\(17\)	7.94744	−	19.1868i	0.467496	−	1.12864i	−0.497756	−	0.867317i	\(-0.665842\pi\)
							0.965252	−	0.261319i	\(-0.0841575\pi\)
\(19\)	7.19499	−	13.4609i	0.378684	−	0.708467i	−0.618311	−	0.785933i	\(-0.712183\pi\)
							0.996995	+	0.0774661i	\(0.0246829\pi\)
\(23\)	−20.1780	+	13.4825i	−0.877306	+	0.586197i	−0.910619	−	0.413248i	\(-0.864394\pi\)
							0.0333123	+	0.999445i	\(0.489394\pi\)
\(29\)	3.83729	−	38.9607i	0.132320	−	1.34347i	−0.668117	−	0.744056i	\(-0.732899\pi\)
							0.800437	−	0.599416i	\(-0.204601\pi\)
\(31\)	6.89353	+	6.89353i	0.222372	+	0.222372i	0.809497	−	0.587125i	\(-0.199740\pi\)
							−0.587125	+	0.809497i	\(0.699740\pi\)
\(37\)	−18.2260	−	34.0985i	−0.492595	−	0.921580i	−0.998267	−	0.0588541i	\(-0.981255\pi\)
							0.505672	−	0.862726i	\(-0.331245\pi\)
\(41\)	−0.962894	+	0.643385i	−0.0234852	+	0.0156923i	−0.567257	−	0.823541i	\(-0.691996\pi\)
							0.543772	+	0.839233i	\(0.316996\pi\)
\(43\)	59.6302	+	48.9372i	1.38675	+	1.13808i	0.973822	+	0.227311i	\(0.0729935\pi\)
							0.412927	+	0.910764i	\(0.364507\pi\)
\(47\)	8.29527	−	20.0266i	0.176495	−	0.426097i	−0.810732	−	0.585418i	\(-0.800930\pi\)
							0.987227	+	0.159321i	\(0.0509305\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.l.a.3.19	✓	496
128.43	odd	32	inner	128.3.l.a.43.19	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.19	✓	496	1.1	even	1	trivial
128.3.l.a.43.19	yes	496	128.43	odd	32	inner