Embedded newform 17.5.e.a.3.5

• Label: 17.5.e.a.3.5
• Level: $17$
• Weight: $5$
• Character: 17.3
• Analytic conductor: $1.757$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	17.e (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.75728937243\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(5\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			3.5
Character	\(\chi\)	\(=\)	17.3
Dual form			17.5.e.a.6.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(6.53721 - 2.70780i) q^{2} +(-10.8435 - 2.15691i) q^{3} +(24.0892 - 24.0892i) q^{4} +(16.7298 + 25.0379i) q^{5} +(-76.7267 + 15.2619i) q^{6} +(-31.8191 + 47.6207i) q^{7} +(48.9226 - 118.110i) q^{8} +(38.0949 + 15.7794i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	6.53721	−	2.70780i	1.63430	−	0.676950i	0.638598	−	0.769541i	\(-0.279515\pi\)
							0.995704	+	0.0925906i	\(0.0295148\pi\)
\(3\)	−10.8435	−	2.15691i	−1.20483	−	0.239656i	−0.448464	−	0.893801i	\(-0.648029\pi\)
							−0.756369	+	0.654145i	\(0.773029\pi\)
\(5\)	16.7298	+	25.0379i	0.669191	+	1.00151i	0.998360	+	0.0572439i	\(0.0182313\pi\)
							−0.329169	+	0.944271i	\(0.606769\pi\)
\(7\)	−31.8191	+	47.6207i	−0.649370	+	0.971851i	0.350014	+	0.936745i	\(0.386177\pi\)
							−0.999384	+	0.0351063i	\(0.988823\pi\)
\(11\)	−33.5675	−	168.755i	−0.277417	−	1.39467i	−0.828392	−	0.560149i	\(-0.810744\pi\)
							0.550975	−	0.834522i	\(-0.314256\pi\)
\(13\)	−34.7421	−	34.7421i	−0.205574	−	0.205574i	0.596809	−	0.802383i	\(-0.296435\pi\)
							−0.802383	+	0.596809i	\(0.796435\pi\)
\(17\)	284.753	+	49.3615i	0.985306	+	0.170801i
							\(19\)	−117.014	+	48.4686i	−0.324137	+	0.134262i	−0.538818	−	0.842422i	\(-0.681129\pi\)
0.214681	+	0.976684i	\(0.431129\pi\)
\(23\)	−327.601	+	65.1639i	−0.619284	+	0.123183i	−0.494755	−	0.869033i	\(-0.664742\pi\)
							−0.124529	+	0.992216i	\(0.539742\pi\)
\(29\)	−24.3617	+	16.2780i	−0.0289676	+	0.0193555i	−0.569970	−	0.821666i	\(-0.693045\pi\)
							0.541002	+	0.841021i	\(0.318045\pi\)
\(31\)	200.760	−	1009.29i	0.208907	−	1.05025i	−0.723910	−	0.689895i	\(-0.757657\pi\)
							0.932817	−	0.360351i	\(-0.117343\pi\)
\(37\)	14.9014	+	2.96407i	0.0108849	+	0.00216513i	0.200530	−	0.979688i	\(-0.435734\pi\)
							−0.189645	+	0.981853i	\(0.560734\pi\)
\(41\)	−867.598	+	1298.45i	−0.516120	+	0.772428i	−0.994389	−	0.105785i	\(-0.966264\pi\)
							0.478269	+	0.878213i	\(0.341264\pi\)
\(43\)	1974.21	+	817.745i	1.06772	+	0.442263i	0.846185	−	0.532889i	\(-0.178894\pi\)
							0.221533	+	0.975153i	\(0.428894\pi\)
\(47\)	−486.981	−	486.981i	−0.220453	−	0.220453i	0.588236	−	0.808689i	\(-0.299823\pi\)
							−0.808689	+	0.588236i	\(0.799823\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.5.e.a.3.5	✓	40
3.2	odd	2		153.5.p.a.37.1		40
17.6	odd	16	inner	17.5.e.a.6.5	yes	40
51.23	even	16		153.5.p.a.91.1		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.5.e.a.3.5	✓	40	1.1	even	1	trivial
17.5.e.a.6.5	yes	40	17.6	odd	16	inner
153.5.p.a.37.1		40	3.2	odd	2
153.5.p.a.91.1		40	51.23	even	16