Embedded newform 17.12.d.a.15.2

• Label: 17.12.d.a.15.2
• 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			15.2
Character	\(\chi\)	\(=\)	17.15
Dual form			17.12.d.a.8.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-47.6155 - 47.6155i) q^{2} +(95.9964 + 231.756i) q^{3} +2486.47i q^{4} +(-10717.4 + 4439.28i) q^{5} +(6464.25 - 15606.1i) q^{6} +(36174.9 + 14984.1i) q^{7} +(20877.8 - 20877.8i) q^{8} +(80766.4 - 80766.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 4 q^{2} - 4 q^{3} + 10120 q^{5} - 44716 q^{6} - 4 q^{7} + 8188 q^{8} + 621728 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−47.6155	−	47.6155i	−1.05216	−	1.05216i	−0.998562	−	0.0536009i	\(-0.982930\pi\)
							−0.0536009	−	0.998562i	\(-0.517070\pi\)
\(3\)	95.9964	+	231.756i	0.228081	+	0.550635i	0.995944	−	0.0899788i	\(-0.0286799\pi\)
							−0.767863	+	0.640614i	\(0.778680\pi\)
\(5\)	−10717.4	+	4439.28i	−1.53374	+	0.635298i	−0.980287	−	0.197579i	\(-0.936692\pi\)
							−0.553458	+	0.832877i	\(0.686692\pi\)
\(7\)	36174.9	+	14984.1i	0.813519	+	0.336971i	0.750357	−	0.661033i	\(-0.229881\pi\)
							0.0631621	+	0.998003i	\(0.479881\pi\)
\(11\)	−138064.	+	333316.i	−0.258476	+	0.624017i	−0.998838	−	0.0481917i	\(-0.984654\pi\)
							0.740362	+	0.672209i	\(0.234654\pi\)
\(13\)		−	1.41087e6i		−	1.05390i	−0.849898	−	0.526948i	\(-0.823336\pi\)
							0.849898	−	0.526948i	\(-0.176664\pi\)
\(17\)	−5.59505e6	−	1.72258e6i	−0.955730	−	0.294246i
							\(19\)	−5.45909e6	−	5.45909e6i	−0.505797	−	0.505797i	0.407437	−	0.913233i	\(-0.366423\pi\)
−0.913233	+	0.407437i	\(0.866423\pi\)
\(23\)	−1.54905e6	+	3.73973e6i	−0.0501835	+	0.121154i	−0.946983	−	0.321283i	\(-0.895886\pi\)
							0.896800	+	0.442437i	\(0.145886\pi\)
\(29\)	1.55963e8	−	6.46021e7i	1.41200	−	0.584868i	0.459160	−	0.888354i	\(-0.348150\pi\)
							0.952836	+	0.303486i	\(0.0981504\pi\)
\(31\)	−7.27888e7	−	1.75728e8i	−0.456642	−	1.10243i	−0.969749	−	0.244106i	\(-0.921506\pi\)
							0.513107	−	0.858325i	\(-0.328494\pi\)
\(37\)	−3.66189e7	−	8.84058e7i	−0.0868152	−	0.209590i	0.874509	−	0.485009i	\(-0.161184\pi\)
							−0.961324	+	0.275418i	\(0.911184\pi\)
\(41\)	6.26718e8	+	2.59595e8i	0.844814	+	0.349933i	0.762750	−	0.646694i	\(-0.223849\pi\)
							0.0820639	+	0.996627i	\(0.473849\pi\)
\(43\)	1.04908e9	−	1.04908e9i	1.08826	−	1.08826i	0.0925536	−	0.995708i	\(-0.470497\pi\)
							0.995708	−	0.0925536i	\(-0.0295029\pi\)
\(47\)		−	1.18218e9i		−	0.751878i	−0.926644	−	0.375939i	\(-0.877320\pi\)
							0.926644	−	0.375939i	\(-0.122680\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.12.d.a.15.2	yes	60
17.8	even	8	inner	17.12.d.a.8.2	✓	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.12.d.a.8.2	✓	60	17.8	even	8	inner
17.12.d.a.15.2	yes	60	1.1	even	1	trivial