Embedded newform 17.13.e.a.3.7

• Label: 17.13.e.a.3.7
• Level: $17$
• Weight: $13$
• Character: 17.3
• Analytic conductor: $15.538$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	17.e (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.5378948937\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(17\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			3.7
Character	\(\chi\)	\(=\)	17.3
Dual form			17.13.e.a.6.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-38.9153 + 16.1192i) q^{2} +(-71.3658 - 14.1956i) q^{3} +(-1641.74 + 1641.74i) q^{4} +(-4258.30 - 6372.99i) q^{5} +(3006.04 - 597.939i) q^{6} +(-77511.3 + 116004. i) q^{7} +(103450. - 249749. i) q^{8} +(-486096. - 201348. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−38.9153	+	16.1192i	−0.608051	+	0.251863i	−0.665395	−	0.746491i	\(-0.731737\pi\)
							0.0573439	+	0.998354i	\(0.481737\pi\)
\(3\)	−71.3658	−	14.1956i	−0.0978955	−	0.0194726i	0.145899	−	0.989299i	\(-0.453392\pi\)
							−0.243795	+	0.969827i	\(0.578392\pi\)
\(5\)	−4258.30	−	6372.99i	−0.272531	−	0.407871i	0.669806	−	0.742536i	\(-0.266377\pi\)
							−0.942337	+	0.334664i	\(0.891377\pi\)
\(7\)	−77511.3	+	116004.i	−0.658835	+	0.986016i	0.340123	+	0.940381i	\(0.389531\pi\)
							−0.998958	+	0.0456354i	\(0.985469\pi\)
\(11\)	365593.	+	1.83796e6i	0.206368	+	1.03748i	0.935559	+	0.353171i	\(0.114897\pi\)
							−0.729191	+	0.684310i	\(0.760103\pi\)
\(13\)	−2.94466e6	−	2.94466e6i	−0.610063	−	0.610063i	0.332899	−	0.942962i	\(-0.391973\pi\)
							−0.942962	+	0.332899i	\(0.891973\pi\)
\(17\)	2.40286e7	−	2.29095e6i	0.995486	−	0.0949121i
							\(19\)	6.13450e7	−	2.54099e7i	1.30394	−	0.540109i	0.380829	−	0.924645i	\(-0.375639\pi\)
0.923110	+	0.384536i	\(0.125639\pi\)
\(23\)	1.90243e8	−	3.78418e7i	1.28512	−	0.255626i	0.495168	−	0.868798i	\(-0.335107\pi\)
							0.789950	+	0.613172i	\(0.210107\pi\)
\(29\)	−6.66461e8	+	4.45315e8i	−1.12043	+	0.748650i	−0.970752	−	0.240084i	\(-0.922825\pi\)
							−0.149682	+	0.988734i	\(0.547825\pi\)
\(31\)	1.22442e8	−	6.15558e8i	0.137962	−	0.693583i	−0.848447	−	0.529280i	\(-0.822462\pi\)
							0.986410	−	0.164304i	\(-0.0525377\pi\)
\(37\)	−3.97744e9	−	7.91162e8i	−1.55022	−	0.308358i	−0.655573	−	0.755132i	\(-0.727573\pi\)
							−0.894647	+	0.446774i	\(0.852573\pi\)
\(41\)	−2.95094e8	+	4.41639e8i	−0.0621237	+	0.0929747i	−0.861231	−	0.508214i	\(-0.830306\pi\)
							0.799107	+	0.601189i	\(0.205306\pi\)
\(43\)	1.10036e10	+	4.55784e9i	1.74070	+	0.721022i	0.998719	+	0.0506005i	\(0.0161135\pi\)
							0.741981	+	0.670421i	\(0.233886\pi\)
\(47\)	−1.35611e10	−	1.35611e10i	−1.25808	−	1.25808i	−0.952009	−	0.306070i	\(-0.900986\pi\)
							−0.306070	−	0.952009i	\(-0.599014\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.13.e.a.3.7	✓	136
17.6	odd	16	inner	17.13.e.a.6.7	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.13.e.a.3.7	✓	136	1.1	even	1	trivial
17.13.e.a.6.7	yes	136	17.6	odd	16	inner