Embedded newform 17.14.d.a.9.17

• Label: 17.14.d.a.9.17
• Level: $17$
• Weight: $14$
• Character: 17.9
• Analytic conductor: $18.229$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(18.2292579218\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(19\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			9.17
Character	\(\chi\)	\(=\)	17.9
Dual form			17.14.d.a.2.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(114.094 - 114.094i) q^{2} +(-2071.82 - 858.176i) q^{3} -17842.8i q^{4} +(18497.9 - 44657.9i) q^{5} +(-334294. + 138469. i) q^{6} +(-142277. - 343487. i) q^{7} +(-1.10109e6 - 1.10109e6i) q^{8} +(2.42861e6 + 2.42861e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 4 q^{2} - 4 q^{3} - 62720 q^{5} - 309100 q^{6} - 4 q^{7} + 32764 q^{8} - 1963432 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{8}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	114.094	−	114.094i	1.26057	−	1.26057i	0.309753	−	0.950817i	\(-0.399754\pi\)
							0.950817	−	0.309753i	\(-0.100246\pi\)
\(3\)	−2071.82	−	858.176i	−1.64083	−	0.679654i	−0.644449	−	0.764647i	\(-0.722913\pi\)
							−0.996382	+	0.0849931i	\(0.972913\pi\)
\(5\)	18497.9	−	44657.9i	0.529441	−	1.27818i	−0.402449	−	0.915442i	\(-0.631841\pi\)
							0.931890	−	0.362741i	\(-0.118159\pi\)
\(7\)	−142277.	−	343487.i	−0.457085	−	1.10350i	−0.969572	−	0.244806i	\(-0.921276\pi\)
							0.512487	−	0.858695i	\(-0.328724\pi\)
\(11\)	6.93848e6	−	2.87401e6i	1.18090	−	0.489143i	0.296116	−	0.955152i	\(-0.404309\pi\)
							0.884780	+	0.466009i	\(0.154309\pi\)
\(13\)			8.36630e6i			0.480730i	0.970683	+	0.240365i	\(0.0772672\pi\)
							−0.970683	+	0.240365i	\(0.922733\pi\)
\(17\)	9.02064e7	+	4.20402e7i	0.906399	+	0.422423i
							\(19\)	6.00410e7	−	6.00410e7i	0.292785	−	0.292785i	−0.545394	−	0.838180i	\(-0.683620\pi\)
0.838180	+	0.545394i	\(0.183620\pi\)
\(23\)	2.97239e8	−	1.23121e8i	0.418673	−	0.173420i	−0.163394	−	0.986561i	\(-0.552244\pi\)
							0.582067	+	0.813141i	\(0.302244\pi\)
\(29\)	5.69007e8	−	1.37370e9i	0.177636	−	0.428851i	−0.809834	−	0.586659i	\(-0.800443\pi\)
							0.987470	+	0.157808i	\(0.0504428\pi\)
\(31\)	7.77882e7	+	3.22209e7i	0.0157421	+	0.00652059i	0.390541	−	0.920586i	\(-0.372288\pi\)
							−0.374798	+	0.927106i	\(0.622288\pi\)
\(37\)	5.65314e9	+	2.34161e9i	0.362225	+	0.150038i	0.556371	−	0.830934i	\(-0.312193\pi\)
							−0.194146	+	0.980973i	\(0.562193\pi\)
\(41\)	1.06195e10	+	2.56379e10i	0.349149	+	0.842921i	0.996721	+	0.0809161i	\(0.0257846\pi\)
							−0.647572	+	0.762004i	\(0.724215\pi\)
\(43\)	−5.65861e10	−	5.65861e10i	−1.36510	−	1.36510i	−0.867281	−	0.497819i	\(-0.834134\pi\)
							−0.497819	−	0.867281i	\(-0.665866\pi\)
\(47\)			8.93969e10i			1.20972i	0.796330	+	0.604862i	\(0.206772\pi\)
							−0.796330	+	0.604862i	\(0.793228\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.14.d.a.9.17	yes	76
17.2	even	8	inner	17.14.d.a.2.17	✓	76

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.14.d.a.2.17	✓	76	17.2	even	8	inner
17.14.d.a.9.17	yes	76	1.1	even	1	trivial