Embedded newform 17.12.d.a.15.3

• Label: 17.12.d.a.15.3
• Level: $17$
• Weight: $12$
• Character: 17.15
• Analytic conductor: $13.062$
• Analytic rank: $0$
• Dimension: $60$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(13.0618340695\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(15\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			15.3
Character	\(\chi\)	\(=\)	17.15
Dual form			17.12.d.a.8.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-46.2135 - 46.2135i) q^{2} +(-297.050 - 717.141i) q^{3} +2223.37i q^{4} +(146.307 - 60.6024i) q^{5} +(-19413.9 + 46869.3i) q^{6} +(68413.6 + 28337.8i) q^{7} +(8104.28 - 8104.28i) q^{8} +(-300791. + 300791. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 4 q^{2} - 4 q^{3} + 10120 q^{5} - 44716 q^{6} - 4 q^{7} + 8188 q^{8} + 621728 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−46.2135	−	46.2135i	−1.02118	−	1.02118i	−0.999771	−	0.0214119i	\(-0.993184\pi\)
							−0.0214119	−	0.999771i	\(-0.506816\pi\)
\(3\)	−297.050	−	717.141i	−0.705768	−	1.70388i	−0.710313	−	0.703886i	\(-0.751447\pi\)
							0.00454461	−	0.999990i	\(-0.498553\pi\)
\(5\)	146.307	−	60.6024i	0.0209378	−	0.00867271i	−0.372190	−	0.928157i	\(-0.621393\pi\)
							0.393128	+	0.919484i	\(0.371393\pi\)
\(7\)	68413.6	+	28337.8i	1.53852	+	0.637276i	0.981195	−	0.193019i	\(-0.0618277\pi\)
							0.557325	+	0.830294i	\(0.311828\pi\)
\(11\)	64668.6	−	156124.i	0.121069	−	0.292287i	−0.851713	−	0.524009i	\(-0.824436\pi\)
							0.972782	+	0.231722i	\(0.0744359\pi\)
\(13\)			936544.i			0.699584i	0.936827	+	0.349792i	\(0.113748\pi\)
							−0.936827	+	0.349792i	\(0.886252\pi\)
\(17\)	−2.14755e6	+	5.44609e6i	−0.366838	+	0.930285i
							\(19\)	−135336.	−	135336.i	−0.0125391	−	0.0125391i	0.700809	−	0.713349i	\(-0.252822\pi\)
−0.713349	+	0.700809i	\(0.752822\pi\)
\(23\)	−2.86351e6	+	6.91312e6i	−0.0927674	+	0.223960i	−0.963452	−	0.267882i	\(-0.913676\pi\)
							0.870684	+	0.491842i	\(0.163676\pi\)
\(29\)	−1.97283e8	+	8.17175e7i	−1.78608	+	0.739820i	−0.794994	+	0.606617i	\(0.792526\pi\)
							−0.991089	+	0.133203i	\(0.957474\pi\)
\(31\)	7.82038e6	+	1.88801e7i	0.0490612	+	0.118444i	0.946510	−	0.322674i	\(-0.104582\pi\)
							−0.897449	+	0.441119i	\(0.854582\pi\)
\(37\)	1.21073e8	+	2.92296e8i	0.287037	+	0.692969i	0.999966	−	0.00826761i	\(-0.00263169\pi\)
							−0.712929	+	0.701237i	\(0.752632\pi\)
\(41\)	3.03749e8	+	1.25817e8i	0.409453	+	0.169601i	0.577896	−	0.816111i	\(-0.303874\pi\)
							−0.168443	+	0.985711i	\(0.553874\pi\)
\(43\)	9.11143e8	−	9.11143e8i	0.945171	−	0.945171i	−0.0534024	−	0.998573i	\(-0.517007\pi\)
							0.998573	+	0.0534024i	\(0.0170066\pi\)
\(47\)			2.02267e9i			1.28643i	0.765685	+	0.643216i	\(0.222400\pi\)
							−0.765685	+	0.643216i	\(0.777600\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.12.d.a.15.3	yes	60
17.8	even	8	inner	17.12.d.a.8.3	✓	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.12.d.a.8.3	✓	60	17.8	even	8	inner
17.12.d.a.15.3	yes	60	1.1	even	1	trivial