Embedded newform 128.2.k.a.5.11

• Label: 128.2.k.a.5.11
• Level: $128$
• Weight: $2$
• Character: 128.5
• Analytic conductor: $1.022$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(1.02208514587\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(15\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			5.11
Character	\(\chi\)	\(=\)	128.5
Dual form			128.2.k.a.77.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.674225 - 1.24315i) q^{2} +(0.865940 + 0.262680i) q^{3} +(-1.09084 - 1.67632i) q^{4} +(1.24951 + 0.123066i) q^{5} +(0.910389 - 0.899388i) q^{6} +(0.832309 + 1.24564i) q^{7} +(-2.81939 + 0.225861i) q^{8} +(-1.81356 - 1.21178i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 240 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.674225	−	1.24315i	0.476749	−	0.879040i
							\(3\)	0.865940	+	0.262680i	0.499951	+	0.151658i	0.530181	−	0.847885i	\(-0.322124\pi\)
−0.0302301	+	0.999543i	\(0.509624\pi\)
\(5\)	1.24951	+	0.123066i	0.558797	+	0.0550367i	0.373473	−	0.927641i	\(-0.378167\pi\)
							0.185323	+	0.982678i	\(0.440667\pi\)
\(7\)	0.832309	+	1.24564i	0.314583	+	0.470807i	0.954742	−	0.297436i	\(-0.0961314\pi\)
							−0.640159	+	0.768243i	\(0.721131\pi\)
\(11\)	−0.865009	+	1.61832i	−0.260810	+	0.487941i	−0.977909	−	0.209030i	\(-0.932969\pi\)
							0.717099	+	0.696971i	\(0.245469\pi\)
\(13\)	−0.0973539	−	0.988451i	−0.0270011	−	0.274147i	−0.999200	−	0.0399906i	\(-0.987267\pi\)
							0.972199	−	0.234156i	\(-0.0752328\pi\)
\(17\)	0.0244568	−	0.0101304i	0.00593166	−	0.00245697i	−0.379716	−	0.925103i	\(-0.623978\pi\)
							0.385647	+	0.922646i	\(0.373978\pi\)
\(19\)	−0.837843	+	1.02091i	−0.192214	+	0.234214i	−0.860196	−	0.509964i	\(-0.829659\pi\)
							0.667981	+	0.744178i	\(0.267159\pi\)
\(23\)	1.65899	+	8.34031i	0.345924	+	1.73907i	0.626662	+	0.779291i	\(0.284420\pi\)
							−0.280739	+	0.959784i	\(0.590580\pi\)
\(29\)	4.72421	−	2.52514i	0.877264	−	0.468907i	0.0296741	−	0.999560i	\(-0.490553\pi\)
							0.847589	+	0.530652i	\(0.178053\pi\)
\(31\)	0.299761	+	0.299761i	0.0538386	+	0.0538386i	0.733514	−	0.679675i	\(-0.237879\pi\)
							−0.679675	+	0.733514i	\(0.737879\pi\)
\(37\)	5.59401	−	4.59089i	0.919650	−	0.754737i	−0.0501236	−	0.998743i	\(-0.515962\pi\)
							0.969774	+	0.244006i	\(0.0784615\pi\)
\(41\)	−8.72362	+	1.73524i	−1.36240	+	0.270998i	−0.821558	−	0.570125i	\(-0.806895\pi\)
							−0.540843	+	0.841124i	\(0.681895\pi\)
\(43\)	−1.65222	+	0.501196i	−0.251962	+	0.0764317i	−0.413739	−	0.910396i	\(-0.635777\pi\)
							0.161777	+	0.986827i	\(0.448277\pi\)
\(47\)	1.47462	+	3.56006i	0.215096	+	0.519288i	0.994192	−	0.107616i	\(-0.0343218\pi\)
							−0.779096	+	0.626904i	\(0.784322\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.2.k.a.5.11	✓	240
4.3	odd	2		512.2.k.a.113.6		240
128.51	odd	32		512.2.k.a.145.6		240
128.77	even	32	inner	128.2.k.a.77.11	yes	240

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.2.k.a.5.11	✓	240	1.1	even	1	trivial
128.2.k.a.77.11	yes	240	128.77	even	32	inner
512.2.k.a.113.6		240	4.3	odd	2
512.2.k.a.145.6		240	128.51	odd	32