Embedded newform 17.12.d.a.15.1

• Label: 17.12.d.a.15.1
• 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			15.1
Character	\(\chi\)	\(=\)	17.15
Dual form			17.12.d.a.8.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-60.6361 - 60.6361i) q^{2} +(-3.37027 - 8.13656i) q^{3} +5305.48i q^{4} +(4847.42 - 2007.87i) q^{5} +(-289.009 + 697.730i) q^{6} +(-22293.6 - 9234.30i) q^{7} +(197521. - 197521. i) q^{8} +(125207. - 125207. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 4 q^{2} - 4 q^{3} + 10120 q^{5} - 44716 q^{6} - 4 q^{7} + 8188 q^{8} + 621728 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−60.6361	−	60.6361i	−1.33988	−	1.33988i	−0.896169	−	0.443712i	\(-0.853661\pi\)
							−0.443712	−	0.896169i	\(-0.646339\pi\)
\(3\)	−3.37027	−	8.13656i	−0.00800752	−	0.0193319i	0.919825	−	0.392328i	\(-0.128330\pi\)
							−0.927833	+	0.372997i	\(0.878330\pi\)
\(5\)	4847.42	−	2007.87i	0.693707	−	0.287343i	−0.00783681	−	0.999969i	\(-0.502495\pi\)
							0.701544	+	0.712627i	\(0.252495\pi\)
\(7\)	−22293.6	−	9234.30i	−0.501349	−	0.207666i	0.117653	−	0.993055i	\(-0.462463\pi\)
							−0.619002	+	0.785389i	\(0.712463\pi\)
\(11\)	174707.	−	421781.i	0.327078	−	0.789637i	−0.671728	−	0.740798i	\(-0.734448\pi\)
							0.998807	−	0.0488394i	\(-0.0155522\pi\)
\(13\)			1.87634e6i			1.40160i	0.713359	+	0.700799i	\(0.247173\pi\)
							−0.713359	+	0.700799i	\(0.752827\pi\)
\(17\)	−120920.	−	5.85297e6i	−0.0206552	−	0.999787i
							\(19\)	−958538.	−	958538.i	−0.0888105	−	0.0888105i	0.661306	−	0.750116i	\(-0.270003\pi\)
−0.750116	+	0.661306i	\(0.770003\pi\)
\(23\)	9.58472e6	−	2.31396e7i	0.310510	−	0.749639i	−0.689176	−	0.724594i	\(-0.742027\pi\)
							0.999686	−	0.0250444i	\(-0.00797272\pi\)
\(29\)	−1.02587e8	+	4.24928e7i	−0.928757	+	0.384704i	−0.795207	−	0.606339i	\(-0.792638\pi\)
							−0.133550	+	0.991042i	\(0.542638\pi\)
\(31\)	−6.03904e7	−	1.45795e8i	−0.378860	−	0.914648i	−0.992180	−	0.124814i	\(-0.960167\pi\)
							0.613320	−	0.789834i	\(-0.289833\pi\)
\(37\)	−1.87086e8	−	4.51666e8i	−0.443539	−	1.07080i	−0.974698	−	0.223526i	\(-0.928243\pi\)
							0.531159	−	0.847272i	\(-0.321757\pi\)
\(41\)	−1.31306e9	−	5.43886e8i	−1.77000	−	0.733156i	−0.994848	−	0.101376i	\(-0.967675\pi\)
							−0.775148	−	0.631780i	\(-0.782325\pi\)
\(43\)	8.10092e8	−	8.10092e8i	0.840345	−	0.840345i	−0.148558	−	0.988904i	\(-0.547463\pi\)
							0.988904	+	0.148558i	\(0.0474632\pi\)
\(47\)			1.86419e9i			1.18564i	0.805335	+	0.592820i	\(0.201985\pi\)
							−0.805335	+	0.592820i	\(0.798015\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.12.d.a.15.1	yes	60
17.8	even	8	inner	17.12.d.a.8.1	✓	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.12.d.a.8.1	✓	60	17.8	even	8	inner
17.12.d.a.15.1	yes	60	1.1	even	1	trivial