Embedded newform 17.10.d.a.15.2

• Label: 17.10.d.a.15.2
• Level: $17$
• Weight: $10$
• Character: 17.15
• Analytic conductor: $8.756$
• Analytic rank: $0$
• Dimension: $52$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.75560921479\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Relative dimension:	\(13\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			15.2
Character	\(\chi\)	\(=\)	17.15
Dual form			17.10.d.a.8.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-25.4395 - 25.4395i) q^{2} +(-16.0088 - 38.6487i) q^{3} +782.338i q^{4} +(2220.20 - 919.637i) q^{5} +(-575.948 + 1390.46i) q^{6} +(8017.68 + 3321.03i) q^{7} +(6877.27 - 6877.27i) q^{8} +(12680.5 - 12680.5i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{3}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−25.4395	−	25.4395i	−1.12428	−	1.12428i	−0.991091	−	0.133188i	\(-0.957479\pi\)
							−0.133188	−	0.991091i	\(-0.542521\pi\)
\(3\)	−16.0088	−	38.6487i	−0.114107	−	0.275480i	0.856500	−	0.516146i	\(-0.172634\pi\)
							−0.970608	+	0.240667i	\(0.922634\pi\)
\(5\)	2220.20	−	919.637i	1.58865	−	0.658038i	0.598893	−	0.800829i	\(-0.295608\pi\)
							0.989753	+	0.142791i	\(0.0456076\pi\)
\(7\)	8017.68	+	3321.03i	1.26214	+	0.522795i	0.910566	−	0.413365i	\(-0.135647\pi\)
							0.351574	+	0.936160i	\(0.385647\pi\)
\(11\)	−24192.5	+	58405.8i	−0.498211	+	1.20279i	0.452235	+	0.891899i	\(0.350627\pi\)
							−0.950446	+	0.310889i	\(0.899373\pi\)
\(13\)		−	128302.i		−	1.24591i	−0.782256	−	0.622957i	\(-0.785931\pi\)
							0.782256	−	0.622957i	\(-0.214069\pi\)
\(17\)	332859.	+	88277.9i	0.966584	+	0.256349i
							\(19\)	−194628.	−	194628.i	−0.342622	−	0.342622i	0.514730	−	0.857352i	\(-0.327892\pi\)
−0.857352	+	0.514730i	\(0.827892\pi\)
\(23\)	−416449.	+	1.00540e6i	−0.310304	+	0.749139i	0.689390	+	0.724390i	\(0.257879\pi\)
							−0.999694	+	0.0247488i	\(0.992121\pi\)
\(29\)	1.32700e6	−	549662.i	0.348402	−	0.144313i	−0.201618	−	0.979464i	\(-0.564620\pi\)
							0.550020	+	0.835151i	\(0.314620\pi\)
\(31\)	−1.72229e6	−	4.15798e6i	−0.334949	−	0.808638i	−0.998185	−	0.0602278i	\(-0.980817\pi\)
							0.663236	−	0.748411i	\(-0.269183\pi\)
\(37\)	1.03267e6	+	2.49309e6i	0.0905846	+	0.218691i	0.962678	−	0.270648i	\(-0.0872381\pi\)
							−0.872094	+	0.489339i	\(0.837238\pi\)
\(41\)	−2.31653e7	−	9.59538e6i	−1.28030	−	0.530316i	−0.364218	−	0.931314i	\(-0.618664\pi\)
							−0.916079	+	0.400998i	\(0.868664\pi\)
\(43\)	−3.40152e6	+	3.40152e6i	−0.151728	+	0.151728i	−0.778889	−	0.627162i	\(-0.784217\pi\)
							0.627162	+	0.778889i	\(0.284217\pi\)
\(47\)			2.48324e7i			0.742299i	0.928573	+	0.371149i	\(0.121036\pi\)
							−0.928573	+	0.371149i	\(0.878964\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.15.2	yes	52
17.5	odd	16		289.10.a.i.1.46		52
17.8	even	8	inner	17.10.d.a.8.2	✓	52
17.12	odd	16		289.10.a.i.1.45		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.8.2	✓	52	17.8	even	8	inner
17.10.d.a.15.2	yes	52	1.1	even	1	trivial
289.10.a.i.1.45		52	17.12	odd	16
289.10.a.i.1.46		52	17.5	odd	16