Embedded newform 17.10.d.a.15.12

• Label: 17.10.d.a.15.12
• 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			15.12
Character	\(\chi\)	\(=\)	17.15
Dual form			17.10.d.a.8.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(26.4264 + 26.4264i) q^{2} +(-94.0407 - 227.034i) q^{3} +884.705i q^{4} +(-1431.34 + 592.881i) q^{5} +(3514.54 - 8484.85i) q^{6} +(-8665.10 - 3589.20i) q^{7} +(-9849.24 + 9849.24i) q^{8} +(-28783.0 + 28783.0i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	26.4264	+	26.4264i	1.16789	+	1.16789i	0.982703	+	0.185188i	\(0.0592895\pi\)
							0.185188	+	0.982703i	\(0.440711\pi\)
\(3\)	−94.0407	−	227.034i	−0.670302	−	1.61825i	−0.781098	−	0.624409i	\(-0.785340\pi\)
							0.110796	−	0.993843i	\(-0.464660\pi\)
\(5\)	−1431.34	+	592.881i	−1.02418	+	0.424231i	−0.830610	−	0.556855i	\(-0.812008\pi\)
							−0.193574	+	0.981086i	\(0.562008\pi\)
\(7\)	−8665.10	−	3589.20i	−1.36406	−	0.565011i	−0.423886	−	0.905716i	\(-0.639334\pi\)
							−0.940170	+	0.340705i	\(0.889334\pi\)
\(11\)	12187.7	−	29423.6i	0.250988	−	0.605939i	−0.747296	−	0.664491i	\(-0.768648\pi\)
							0.998284	+	0.0585519i	\(0.0186483\pi\)
\(13\)			3836.87i			0.0372590i	0.999826	+	0.0186295i	\(0.00593030\pi\)
							−0.999826	+	0.0186295i	\(0.994070\pi\)
\(17\)	271096.	+	212355.i	0.787233	+	0.616656i
							\(19\)	−498424.	−	498424.i	−0.877420	−	0.877420i	0.115847	−	0.993267i	\(-0.463042\pi\)
−0.993267	+	0.115847i	\(0.963042\pi\)
\(23\)	−54124.7	+	130669.i	−0.0403293	+	0.0973634i	−0.942761	−	0.333470i	\(-0.891780\pi\)
							0.902431	+	0.430834i	\(0.141780\pi\)
\(29\)	−4.99241e6	+	2.06793e6i	−1.31075	+	0.542930i	−0.925103	−	0.379717i	\(-0.876022\pi\)
							−0.385646	+	0.922647i	\(0.626022\pi\)
\(31\)	−3.36298e6	−	8.11894e6i	−0.654028	−	1.57896i	−0.806881	−	0.590713i	\(-0.798846\pi\)
							0.152854	−	0.988249i	\(-0.451154\pi\)
\(37\)	−112891.	−	272543.i	−0.00990264	−	0.0239071i	0.918851	−	0.394605i	\(-0.129119\pi\)
							−0.928754	+	0.370698i	\(0.879119\pi\)
\(41\)	1.30639e7	+	5.41125e6i	0.722015	+	0.299068i	0.713266	−	0.700893i	\(-0.247215\pi\)
							0.00874870	+	0.999962i	\(0.497215\pi\)
\(43\)	−1.32798e7	+	1.32798e7i	−0.592355	+	0.592355i	−0.938267	−	0.345912i	\(-0.887570\pi\)
							0.345912	+	0.938267i	\(0.387570\pi\)
\(47\)		−	2.19574e7i		−	0.656357i	−0.944616	−	0.328179i	\(-0.893565\pi\)
							0.944616	−	0.328179i	\(-0.106435\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.15.12	yes	52
17.5	odd	16		289.10.a.i.1.6		52
17.8	even	8	inner	17.10.d.a.8.12	✓	52
17.12	odd	16		289.10.a.i.1.5		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.8.12	✓	52	17.8	even	8	inner
17.10.d.a.15.12	yes	52	1.1	even	1	trivial
289.10.a.i.1.5		52	17.12	odd	16
289.10.a.i.1.6		52	17.5	odd	16