Embedded newform 128.3.l.a.3.13

• Label: 128.3.l.a.3.13
• 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			3.13
Character	\(\chi\)	\(=\)	128.3
Dual form			128.3.l.a.43.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.630182 + 1.89812i) q^{2} +(-2.02704 + 1.66355i) q^{3} +(-3.20574 - 2.39233i) q^{4} +(-2.42484 - 0.735568i) q^{5} +(-1.88021 - 4.89591i) q^{6} +(-0.539873 - 2.71413i) q^{7} +(6.56113 - 4.57729i) q^{8} +(-0.414318 + 2.08292i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.630182	+	1.89812i	−0.315091	+	0.949061i
							\(3\)	−2.02704	+	1.66355i	−0.675680	+	0.554516i	−0.908438	−	0.418019i	\(-0.862725\pi\)
0.232759	+	0.972534i	\(0.425225\pi\)
\(5\)	−2.42484	−	0.735568i	−0.484969	−	0.147114i	0.0383263	−	0.999265i	\(-0.487797\pi\)
							−0.523295	+	0.852152i	\(0.675297\pi\)
\(7\)	−0.539873	−	2.71413i	−0.0771247	−	0.387732i	−0.999997	−	0.00262495i	\(-0.999164\pi\)
							0.922872	−	0.385107i	\(-0.125836\pi\)
\(11\)	−1.71554	−	17.4182i	−0.155958	−	1.58347i	−0.680928	−	0.732351i	\(-0.738423\pi\)
							0.524969	−	0.851121i	\(-0.324077\pi\)
\(13\)	−0.950147	−	3.13221i	−0.0730882	−	0.240940i	0.912783	−	0.408445i	\(-0.133929\pi\)
							−0.985871	+	0.167505i	\(0.946429\pi\)
\(17\)	−2.88229	+	6.95847i	−0.169547	+	0.409321i	−0.985699	−	0.168515i	\(-0.946103\pi\)
							0.816153	+	0.577836i	\(0.196103\pi\)
\(19\)	12.6094	−	23.5904i	0.663650	−	1.24160i	−0.294586	−	0.955625i	\(-0.595182\pi\)
							0.958236	−	0.285978i	\(-0.0923183\pi\)
\(23\)	−22.5962	+	15.0983i	−0.982444	+	0.656448i	−0.939471	−	0.342628i	\(-0.888683\pi\)
							−0.0429732	+	0.999076i	\(0.513683\pi\)
\(29\)	−0.0540521	+	0.548800i	−0.00186386	+	0.0189241i	−0.996072	−	0.0885506i	\(-0.971776\pi\)
							0.994208	+	0.107475i	\(0.0342765\pi\)
\(31\)	−29.1550	−	29.1550i	−0.940484	−	0.940484i	0.0578418	−	0.998326i	\(-0.481578\pi\)
							−0.998326	+	0.0578418i	\(0.981578\pi\)
\(37\)	−0.0272610	−	0.0510018i	−0.000736784	−	0.00137843i	0.881553	−	0.472086i	\(-0.156499\pi\)
							−0.882289	+	0.470707i	\(0.843999\pi\)
\(41\)	−31.0621	+	20.7550i	−0.757612	+	0.506220i	−0.873370	−	0.487057i	\(-0.838070\pi\)
							0.115758	+	0.993277i	\(0.463070\pi\)
\(43\)	−30.8908	−	25.3514i	−0.718391	−	0.589568i	0.202549	−	0.979272i	\(-0.435077\pi\)
							−0.920940	+	0.389704i	\(0.872577\pi\)
\(47\)	−19.1031	+	46.1190i	−0.406450	+	0.981256i	0.579615	+	0.814891i	\(0.303203\pi\)
							−0.986064	+	0.166365i	\(0.946797\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.l.a.3.13	✓	496
128.43	odd	32	inner	128.3.l.a.43.13	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.13	✓	496	1.1	even	1	trivial
128.3.l.a.43.13	yes	496	128.43	odd	32	inner