Embedded newform 17.10.d.a.2.12

• Label: 17.10.d.a.2.12
• Level: $17$
• Weight: $10$
• Character: 17.2
• 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			2.12
Character	\(\chi\)	\(=\)	17.2
Dual form			17.10.d.a.9.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(27.1947 + 27.1947i) q^{2} +(246.648 - 102.165i) q^{3} +967.098i q^{4} +(-306.747 - 740.553i) q^{5} +(9485.84 + 3929.16i) q^{6} +(-1300.23 + 3139.03i) q^{7} +(-12376.2 + 12376.2i) q^{8} +(36479.5 - 36479.5i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	27.1947	+	27.1947i	1.20185	+	1.20185i	0.973605	+	0.228240i	\(0.0732972\pi\)
							0.228240	+	0.973605i	\(0.426703\pi\)
\(3\)	246.648	−	102.165i	1.75805	−	0.728209i	0.761236	−	0.648475i	\(-0.224593\pi\)
							0.996816	−	0.0797336i	\(-0.0254070\pi\)
\(5\)	−306.747	−	740.553i	−0.219490	−	0.529897i	0.775329	−	0.631558i	\(-0.217584\pi\)
							−0.994819	+	0.101661i	\(0.967584\pi\)
\(7\)	−1300.23	+	3139.03i	−0.204682	+	0.494145i	−0.992570	−	0.121673i	\(-0.961174\pi\)
							0.787889	+	0.615818i	\(0.211174\pi\)
\(11\)	−62784.5	−	26006.2i	−1.29296	−	0.535562i	−0.373094	−	0.927794i	\(-0.621703\pi\)
							−0.919866	+	0.392232i	\(0.871703\pi\)
\(13\)			130769.i			1.26987i	0.772564	+	0.634936i	\(0.218974\pi\)
							−0.772564	+	0.634936i	\(0.781026\pi\)
\(17\)	−335376.	+	78171.5i	−0.973894	+	0.227001i
							\(19\)	−192865.	−	192865.i	−0.339518	−	0.339518i	0.516668	−	0.856186i	\(-0.327172\pi\)
−0.856186	+	0.516668i	\(0.827172\pi\)
\(23\)	−809452.	−	335286.i	−0.603137	−	0.249827i	0.0601542	−	0.998189i	\(-0.480841\pi\)
							−0.663291	+	0.748362i	\(0.730841\pi\)
\(29\)	862956.	+	2.08336e6i	0.226567	+	0.546982i	0.995755	−	0.0920405i	\(-0.0293389\pi\)
							−0.769188	+	0.639023i	\(0.779339\pi\)
\(31\)	2.65070e6	−	1.09796e6i	0.515505	−	0.213529i	−0.109736	−	0.993961i	\(-0.535001\pi\)
							0.625241	+	0.780432i	\(0.285001\pi\)
\(37\)	1.25250e7	−	5.18801e6i	1.09867	−	0.455085i	0.241650	−	0.970363i	\(-0.422311\pi\)
							0.857023	+	0.515278i	\(0.172311\pi\)
\(41\)	5.44928e6	−	1.31557e7i	0.301170	−	0.727089i	−0.698761	−	0.715355i	\(-0.746265\pi\)
							0.999931	−	0.0117336i	\(-0.00373502\pi\)
\(43\)	−1.40119e7	+	1.40119e7i	−0.625014	+	0.625014i	−0.946809	−	0.321796i	\(-0.895714\pi\)
							0.321796	+	0.946809i	\(0.395714\pi\)
\(47\)			1.09462e7i			0.327206i	0.986526	+	0.163603i	\(0.0523116\pi\)
							−0.986526	+	0.163603i	\(0.947688\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.2.12	✓	52
17.3	odd	16		289.10.a.i.1.48		52
17.9	even	8	inner	17.10.d.a.9.12	yes	52
17.14	odd	16		289.10.a.i.1.47		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.2.12	✓	52	1.1	even	1	trivial
17.10.d.a.9.12	yes	52	17.9	even	8	inner
289.10.a.i.1.47		52	17.14	odd	16
289.10.a.i.1.48		52	17.3	odd	16