Embedded newform 17.13.e.a.6.7

• Label: 17.13.e.a.6.7
• Level: $17$
• Weight: $13$
• Character: 17.6
• 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			6.7
Character	\(\chi\)	\(=\)	17.6
Dual form			17.13.e.a.3.7

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

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.13.e.a.3.7	✓	136	17.3	odd	16	inner
17.13.e.a.6.7	yes	136	1.1	even	1	trivial