Embedded newform 128.3.l.a.3.17

• Label: 128.3.l.a.3.17
• 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			3.17
Character	\(\chi\)	\(=\)	128.3
Dual form			128.3.l.a.43.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0173622 - 1.99992i) q^{2} +(3.90317 - 3.20325i) q^{3} +(-3.99940 - 0.0694462i) q^{4} +(4.26052 + 1.29242i) q^{5} +(-6.33848 - 7.86165i) q^{6} +(-0.713727 - 3.58815i) q^{7} +(-0.208325 + 7.99729i) q^{8} +(3.21811 - 16.1785i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{2} - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{6} - 16 q^{7} - 16 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0173622	−	1.99992i	0.00868110	−	0.999962i
							\(3\)	3.90317	−	3.20325i	1.30106	−	1.06775i	0.307843	−	0.951437i	\(-0.400393\pi\)
0.993213	−	0.116311i	\(-0.0371071\pi\)
\(5\)	4.26052	+	1.29242i	0.852104	+	0.258483i	0.685979	−	0.727622i	\(-0.259374\pi\)
							0.166126	+	0.986105i	\(0.446874\pi\)
\(7\)	−0.713727	−	3.58815i	−0.101961	−	0.512592i	−0.997686	−	0.0679847i	\(-0.978343\pi\)
							0.895725	−	0.444608i	\(-0.146657\pi\)
\(11\)	1.43644	+	14.5844i	0.130585	+	1.32585i	0.807533	+	0.589822i	\(0.200802\pi\)
							−0.676948	+	0.736031i	\(0.736698\pi\)
\(13\)	−3.73328	−	12.3070i	−0.287175	−	0.946689i	−0.975260	−	0.221059i	\(-0.929049\pi\)
							0.688085	−	0.725630i	\(-0.258451\pi\)
\(17\)	−7.23134	+	17.4580i	−0.425373	+	1.02694i	0.555364	+	0.831607i	\(0.312579\pi\)
							−0.980737	+	0.195334i	\(0.937421\pi\)
\(19\)	−15.3176	+	28.6572i	−0.806190	+	1.50827i	0.0541382	+	0.998533i	\(0.482759\pi\)
							−0.860328	+	0.509741i	\(0.829741\pi\)
\(23\)	12.3054	−	8.22221i	0.535018	−	0.357488i	−0.258522	−	0.966005i	\(-0.583236\pi\)
							0.793540	+	0.608518i	\(0.208236\pi\)
\(29\)	4.17878	−	42.4279i	0.144096	−	1.46303i	−0.602695	−	0.797972i	\(-0.705906\pi\)
							0.746791	−	0.665059i	\(-0.231594\pi\)
\(31\)	10.9719	+	10.9719i	0.353932	+	0.353932i	0.861570	−	0.507639i	\(-0.169481\pi\)
							−0.507639	+	0.861570i	\(0.669481\pi\)
\(37\)	20.5444	+	38.4360i	0.555255	+	1.03881i	0.990371	+	0.138435i	\(0.0442073\pi\)
							−0.435116	+	0.900374i	\(0.643293\pi\)
\(41\)	28.2572	−	18.8808i	0.689199	−	0.460508i	−0.161010	−	0.986953i	\(-0.551475\pi\)
							0.850210	+	0.526444i	\(0.176475\pi\)
\(43\)	−17.4937	−	14.3567i	−0.406830	−	0.333877i	0.408620	−	0.912705i	\(-0.366010\pi\)
							−0.815450	+	0.578828i	\(0.803510\pi\)
\(47\)	−17.1221	+	41.3363i	−0.364299	+	0.879496i	0.630362	+	0.776301i	\(0.282907\pi\)
							−0.994661	+	0.103195i	\(0.967093\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.3.l.a.3.17	✓	496
128.43	odd	32	inner	128.3.l.a.43.17	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.3.l.a.3.17	✓	496	1.1	even	1	trivial
128.3.l.a.43.17	yes	496	128.43	odd	32	inner