Embedded newform 128.3.l.a.3.1

• Label: 128.3.l.a.3.1
• 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			3.1
Character	\(\chi\)	\(=\)	128.3
Dual form			128.3.l.a.43.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99185 - 0.180370i) q^{2} +(1.05127 - 0.862751i) q^{3} +(3.93493 + 0.718539i) q^{4} +(-7.90123 - 2.39681i) q^{5} +(-2.24958 + 1.52885i) q^{6} +(1.29801 + 6.52556i) q^{7} +(-7.70819 - 2.14097i) q^{8} +(-1.39499 + 7.01311i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99185	−	0.180370i	−0.995925	−	0.0901849i
							\(3\)	1.05127	−	0.862751i	0.350422	−	0.287584i	−0.442672	−	0.896684i	\(-0.645969\pi\)
0.793093	+	0.609100i	\(0.208469\pi\)
\(5\)	−7.90123	−	2.39681i	−1.58025	−	0.479362i	−0.626360	−	0.779534i	\(-0.715456\pi\)
							−0.953885	+	0.300172i	\(0.902956\pi\)
\(7\)	1.29801	+	6.52556i	0.185431	+	0.932222i	0.955664	+	0.294460i	\(0.0951398\pi\)
							−0.770233	+	0.637762i	\(0.779860\pi\)
\(11\)	0.0388787	+	0.394742i	0.00353443	+	0.0358856i	0.996790	−	0.0800584i	\(-0.0255107\pi\)
							−0.993256	+	0.115944i	\(0.963011\pi\)
\(13\)	−1.94755	−	6.42020i	−0.149811	−	0.493862i	0.849627	−	0.527385i	\(-0.176827\pi\)
							−0.999438	+	0.0335229i	\(0.989327\pi\)
\(17\)	−11.7599	+	28.3908i	−0.691756	+	1.67005i	0.0494563	+	0.998776i	\(0.484251\pi\)
							−0.741212	+	0.671271i	\(0.765749\pi\)
\(19\)	−7.55037	+	14.1257i	−0.397388	+	0.743460i	−0.998445	−	0.0557458i	\(-0.982246\pi\)
							0.601057	+	0.799206i	\(0.294746\pi\)
\(23\)	−19.3042	+	12.8987i	−0.839314	+	0.560812i	−0.899272	−	0.437389i	\(-0.855903\pi\)
							0.0599585	+	0.998201i	\(0.480903\pi\)
\(29\)	2.58223	−	26.2178i	0.0890423	−	0.904062i	−0.842080	−	0.539353i	\(-0.818669\pi\)
							0.931122	−	0.364708i	\(-0.118831\pi\)
\(31\)	−40.1489	−	40.1489i	−1.29513	−	1.29513i	−0.931573	−	0.363553i	\(-0.881563\pi\)
							−0.363553	−	0.931573i	\(-0.618437\pi\)
\(37\)	19.9517	+	37.3269i	0.539234	+	1.00884i	0.993045	+	0.117738i	\(0.0375641\pi\)
							−0.453811	+	0.891098i	\(0.649936\pi\)
\(41\)	24.1411	−	16.1305i	0.588806	−	0.393428i	−0.225175	−	0.974318i	\(-0.572295\pi\)
							0.813982	+	0.580890i	\(0.197295\pi\)
\(43\)	−38.9814	−	31.9912i	−0.906544	−	0.743981i	0.0606133	−	0.998161i	\(-0.480694\pi\)
							−0.967157	+	0.254180i	\(0.918194\pi\)
\(47\)	18.1618	−	43.8464i	0.386421	−	0.932902i	−0.604271	−	0.796779i	\(-0.706536\pi\)
							0.990692	−	0.136123i	\(-0.0434643\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.l.a.3.1	✓	496
128.43	odd	32	inner	128.3.l.a.43.1	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.1	✓	496	1.1	even	1	trivial
128.3.l.a.43.1	yes	496	128.43	odd	32	inner