Embedded newform 17.5.e.a.6.5

• Label: 17.5.e.a.6.5
• Level: $17$
• Weight: $5$
• Character: 17.6
• 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			6.5
Character	\(\chi\)	\(=\)	17.6
Dual form			17.5.e.a.3.5

$q$-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 \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{15}{16}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.5.e.a.3.5	✓	40	17.3	odd	16	inner
17.5.e.a.6.5	yes	40	1.1	even	1	trivial
153.5.p.a.37.1		40	51.20	even	16
153.5.p.a.91.1		40	3.2	odd	2