Embedded newform 128.3.l.a.43.17

• Label: 128.3.l.a.43.17
• Level: $128$
• Weight: $3$
• Character: 128.43
• 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			43.17
Character	\(\chi\)	\(=\)	128.43
Dual form			128.3.l.a.3.17

$q$-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 \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{29}{32}\right)\)	\(-1\)

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

(See \(a_n\) instead)

Twists

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

    

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