Embedded newform 17.6.d.a.2.3

• Label: 17.6.d.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	17.2
Dual form			17.6.d.a.9.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.63520 - 1.63520i) q^{2} +(3.76372 - 1.55898i) q^{3} -26.6522i q^{4} +(-7.23148 - 17.4583i) q^{5} +(-8.70370 - 3.60519i) q^{6} +(93.1665 - 224.924i) q^{7} +(-95.9083 + 95.9083i) q^{8} +(-160.092 + 160.092i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 4 q^{2} - 4 q^{3} + 40 q^{5} - 364 q^{6} - 4 q^{7} + 124 q^{8} - 136 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.63520	−	1.63520i	−0.289066	−	0.289066i	0.547645	−	0.836711i	\(-0.315524\pi\)
							−0.836711	+	0.547645i	\(0.815524\pi\)
\(3\)	3.76372	−	1.55898i	0.241443	−	0.100009i	−0.258681	−	0.965963i	\(-0.583288\pi\)
							0.500124	+	0.865954i	\(0.333288\pi\)
\(5\)	−7.23148	−	17.4583i	−0.129361	−	0.312304i	0.845907	−	0.533330i	\(-0.179060\pi\)
							−0.975268	+	0.221026i	\(0.929060\pi\)
\(7\)	93.1665	−	224.924i	0.718646	−	1.73496i	0.0414740	−	0.999140i	\(-0.486795\pi\)
							0.677172	−	0.735825i	\(-0.263205\pi\)
\(11\)	522.229	+	216.314i	1.30130	+	0.539018i	0.922335	−	0.386391i	\(-0.126279\pi\)
							0.378969	+	0.925409i	\(0.376279\pi\)
\(13\)			679.500i			1.11515i	0.830128	+	0.557573i	\(0.188267\pi\)
							−0.830128	+	0.557573i	\(0.811733\pi\)
\(17\)	755.598	−	921.373i	0.634116	−	0.773238i
							\(19\)	−185.984	−	185.984i	−0.118193	−	0.118193i	0.645536	−	0.763730i	\(-0.276634\pi\)
−0.763730	+	0.645536i	\(0.776634\pi\)
\(23\)	831.458	+	344.401i	0.327733	+	0.135752i	0.540483	−	0.841355i	\(-0.318242\pi\)
							−0.212749	+	0.977107i	\(0.568242\pi\)
\(29\)	1807.49	+	4363.67i	0.399100	+	0.963512i	0.987880	+	0.155219i	\(0.0496084\pi\)
							−0.588780	+	0.808293i	\(0.700392\pi\)
\(31\)	429.659	−	177.971i	0.0803008	−	0.0332617i	−0.342171	−	0.939638i	\(-0.611162\pi\)
							0.422472	+	0.906376i	\(0.361162\pi\)
\(37\)	−8362.22	+	3463.74i	−1.00419	+	0.415950i	−0.823333	−	0.567558i	\(-0.807888\pi\)
							−0.180860	+	0.983509i	\(0.557888\pi\)
\(41\)	3414.99	−	8244.51i	0.317271	−	0.765959i	−0.682126	−	0.731234i	\(-0.738945\pi\)
							0.999397	−	0.0347245i	\(-0.0110554\pi\)
\(43\)	6161.24	−	6161.24i	0.508156	−	0.508156i	−0.405804	−	0.913960i	\(-0.633008\pi\)
							0.913960	+	0.405804i	\(0.133008\pi\)
\(47\)			16695.6i			1.10245i	0.834358	+	0.551223i	\(0.185839\pi\)
							−0.834358	+	0.551223i	\(0.814161\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.6.d.a.2.3	✓	28
17.3	odd	16		289.6.a.j.1.12		28
17.9	even	8	inner	17.6.d.a.9.3	yes	28
17.14	odd	16		289.6.a.j.1.11		28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.6.d.a.2.3	✓	28	1.1	even	1	trivial
17.6.d.a.9.3	yes	28	17.9	even	8	inner
289.6.a.j.1.11		28	17.14	odd	16
289.6.a.j.1.12		28	17.3	odd	16