Embedded newform 17.6.d.a.2.7

• Label: 17.6.d.a.2.7
• Level: $17$
• Weight: $6$
• Character: 17.2
• Analytic conductor: $2.727$
• Analytic rank: $0$
• Dimension: $28$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.72652493682\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			2.7
Character	\(\chi\)	\(=\)	17.2
Dual form			17.6.d.a.9.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.23579 + 7.23579i) q^{2} +(8.11856 - 3.36282i) q^{3} +72.7132i q^{4} +(-35.2492 - 85.0992i) q^{5} +(83.0768 + 34.4115i) q^{6} +(7.61170 - 18.3763i) q^{7} +(-294.592 + 294.592i) q^{8} +(-117.224 + 117.224i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 4 q^{2} - 4 q^{3} + 40 q^{5} - 364 q^{6} - 4 q^{7} + 124 q^{8} - 136 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.23579	+	7.23579i	1.27912	+	1.27912i	0.941162	+	0.337957i	\(0.109736\pi\)
							0.337957	+	0.941162i	\(0.390264\pi\)
\(3\)	8.11856	−	3.36282i	0.520806	−	0.215725i	−0.106765	−	0.994284i	\(-0.534049\pi\)
							0.627571	+	0.778559i	\(0.284049\pi\)
\(5\)	−35.2492	−	85.0992i	−0.630558	−	1.52230i	−0.838923	−	0.544250i	\(-0.816814\pi\)
							0.208366	−	0.978051i	\(-0.433186\pi\)
\(7\)	7.61170	−	18.3763i	0.0587133	−	0.141747i	−0.891800	−	0.452429i	\(-0.850558\pi\)
							0.950514	+	0.310682i	\(0.100558\pi\)
\(11\)	348.122	+	144.197i	0.867461	+	0.359314i	0.771621	−	0.636083i	\(-0.219446\pi\)
							0.0958402	+	0.995397i	\(0.469446\pi\)
\(13\)		−	728.530i		−	1.19561i	−0.801642	−	0.597805i	\(-0.796040\pi\)
							0.801642	−	0.597805i	\(-0.203960\pi\)
\(17\)	−1066.41	+	531.625i	−0.894957	+	0.446152i
							\(19\)	1472.30	+	1472.30i	0.935650	+	0.935650i	0.998051	−	0.0624010i	\(-0.0198758\pi\)
−0.0624010	+	0.998051i	\(0.519876\pi\)
\(23\)	−1934.83	−	801.433i	−0.762646	−	0.315898i	−0.0327567	−	0.999463i	\(-0.510429\pi\)
							−0.729890	+	0.683565i	\(0.760429\pi\)
\(29\)	−212.349	−	512.656i	−0.0468873	−	0.113196i	0.898701	−	0.438563i	\(-0.144512\pi\)
							−0.945588	+	0.325367i	\(0.894512\pi\)
\(31\)	480.883	−	199.188i	0.0898743	−	0.0372272i	−0.337293	−	0.941400i	\(-0.609511\pi\)
							0.427168	+	0.904172i	\(0.359511\pi\)
\(37\)	2695.88	−	1116.67i	0.323740	−	0.134098i	−0.214894	−	0.976637i	\(-0.568941\pi\)
							0.538634	+	0.842540i	\(0.318941\pi\)
\(41\)	6060.26	−	14630.8i	0.563030	−	1.35928i	−0.344301	−	0.938859i	\(-0.611884\pi\)
							0.907331	−	0.420416i	\(-0.138116\pi\)
\(43\)	−1403.28	+	1403.28i	−0.115738	+	0.115738i	−0.762604	−	0.646866i	\(-0.776079\pi\)
							0.646866	+	0.762604i	\(0.276079\pi\)
\(47\)		−	13740.1i		−	0.907286i	−0.891183	−	0.453643i	\(-0.850124\pi\)
							0.891183	−	0.453643i	\(-0.149876\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.6.d.a.2.7	✓	28
17.3	odd	16		289.6.a.j.1.26		28
17.9	even	8	inner	17.6.d.a.9.7	yes	28
17.14	odd	16		289.6.a.j.1.25		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.6.d.a.2.7	✓	28	1.1	even	1	trivial
17.6.d.a.9.7	yes	28	17.9	even	8	inner
289.6.a.j.1.25		28	17.14	odd	16
289.6.a.j.1.26		28	17.3	odd	16