Embedded newform 128.3.l.a.3.5

• Label: 128.3.l.a.3.5
• 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			3.5
Character	\(\chi\)	\(=\)	128.3
Dual form			128.3.l.a.43.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.78050 + 0.910937i) q^{2} +(3.84622 - 3.15651i) q^{3} +(2.34039 - 3.24385i) q^{4} +(4.46942 + 1.35578i) q^{5} +(-3.97282 + 9.12384i) q^{6} +(1.40191 + 7.04787i) q^{7} +(-1.21212 + 7.90764i) q^{8} +(3.07403 - 15.4542i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.78050	+	0.910937i	−0.890252	+	0.455469i
							\(3\)	3.84622	−	3.15651i	1.28207	−	1.05217i	0.286509	−	0.958077i	\(-0.407505\pi\)
0.995563	−	0.0940927i	\(-0.0299950\pi\)
\(5\)	4.46942	+	1.35578i	0.893883	+	0.271156i	0.703620	−	0.710577i	\(-0.251566\pi\)
							0.190263	+	0.981733i	\(0.439066\pi\)
\(7\)	1.40191	+	7.04787i	0.200273	+	1.00684i	0.941866	+	0.335989i	\(0.109070\pi\)
							−0.741593	+	0.670850i	\(0.765930\pi\)
\(11\)	−0.993647	−	10.0887i	−0.0903315	−	0.917152i	−0.928399	−	0.371585i	\(-0.878814\pi\)
							0.838067	−	0.545567i	\(-0.183686\pi\)
\(13\)	0.359505	+	1.18513i	0.0276543	+	0.0911639i	0.969681	−	0.244373i	\(-0.0785822\pi\)
							−0.942027	+	0.335537i	\(0.891082\pi\)
\(17\)	−10.7657	+	25.9907i	−0.633275	+	1.52886i	0.202206	+	0.979343i	\(0.435189\pi\)
							−0.835481	+	0.549519i	\(0.814811\pi\)
\(19\)	11.7984	−	22.0732i	0.620968	−	1.16175i	−0.353120	−	0.935578i	\(-0.614879\pi\)
							0.974088	−	0.226171i	\(-0.0726207\pi\)
\(23\)	−9.93298	+	6.63701i	−0.431869	+	0.288566i	−0.752442	−	0.658658i	\(-0.771124\pi\)
							0.320573	+	0.947224i	\(0.396124\pi\)
\(29\)	1.85032	−	18.7866i	0.0638040	−	0.647813i	−0.909096	−	0.416587i	\(-0.863226\pi\)
							0.972900	−	0.231226i	\(-0.0742737\pi\)
\(31\)	16.5389	+	16.5389i	0.533512	+	0.533512i	0.921616	−	0.388104i	\(-0.126870\pi\)
							−0.388104	+	0.921616i	\(0.626870\pi\)
\(37\)	−31.5883	−	59.0975i	−0.853738	−	1.59723i	−0.801033	−	0.598621i	\(-0.795716\pi\)
							−0.0527048	−	0.998610i	\(-0.516784\pi\)
\(41\)	−63.5173	+	42.4409i	−1.54920	+	1.03514i	−0.572667	+	0.819788i	\(0.694091\pi\)
							−0.976536	+	0.215356i	\(0.930909\pi\)
\(43\)	15.7372	+	12.9152i	0.365982	+	0.300354i	0.799353	−	0.600862i	\(-0.205176\pi\)
							−0.433371	+	0.901216i	\(0.642676\pi\)
\(47\)	−5.16834	+	12.4775i	−0.109965	+	0.265478i	−0.969276	−	0.245974i	\(-0.920892\pi\)
							0.859312	+	0.511452i	\(0.170892\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.l.a.3.5	✓	496
128.43	odd	32	inner	128.3.l.a.43.5	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.5	✓	496	1.1	even	1	trivial
128.3.l.a.43.5	yes	496	128.43	odd	32	inner