Embedded newform 17.14.d.a.2.17

• Label: 17.14.d.a.2.17
• Level: $17$
• Weight: $14$
• Character: 17.2
• 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			2.17
Character	\(\chi\)	\(=\)	17.2
Dual form			17.14.d.a.9.17

$q$-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 \(\left(\mathbb{Z}/17\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.14.d.a.2.17	✓	76	1.1	even	1	trivial
17.14.d.a.9.17	yes	76	17.9	even	8	inner