Embedded newform 17.5.e.a.7.5

• Label: 17.5.e.a.7.5
• Level: $17$
• Weight: $5$
• Character: 17.7
• Analytic conductor: $1.757$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	17.e (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.75728937243\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(5\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			7.5
Character	\(\chi\)	\(=\)	17.7
Dual form			17.5.e.a.5.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.44646 - 5.90629i) q^{2} +(0.660541 - 0.988570i) q^{3} +(-17.5853 - 17.5853i) q^{4} +(5.91128 + 29.7180i) q^{5} +(-4.22279 - 6.31985i) q^{6} +(9.51676 - 47.8440i) q^{7} +(-52.3853 + 21.6987i) q^{8} +(30.4564 + 73.5283i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{11}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.44646	−	5.90629i	0.611616	−	1.47657i	−0.249608	−	0.968347i	\(-0.580302\pi\)
							0.861224	−	0.508225i	\(-0.169698\pi\)
\(3\)	0.660541	−	0.988570i	0.0733935	−	0.109841i	−0.792957	−	0.609278i	\(-0.791460\pi\)
							0.866351	+	0.499436i	\(0.166460\pi\)
\(5\)	5.91128	+	29.7180i	0.236451	+	1.18872i	0.898403	+	0.439173i	\(0.144728\pi\)
							−0.661951	+	0.749547i	\(0.730272\pi\)
\(7\)	9.51676	−	47.8440i	0.194220	−	0.976408i	−0.753535	−	0.657408i	\(-0.771653\pi\)
							0.947755	−	0.319000i	\(-0.103347\pi\)
\(11\)	9.22808	−	6.16601i	0.0762651	−	0.0509587i	−0.516853	−	0.856074i	\(-0.672897\pi\)
							0.593118	+	0.805115i	\(0.297897\pi\)
\(13\)	−175.656	+	175.656i	−1.03938	+	1.03938i	−0.0401923	+	0.999192i	\(0.512797\pi\)
							−0.999192	+	0.0401923i	\(0.987203\pi\)
\(17\)	−53.0713	−	284.085i	−0.183638	−	0.982994i
							\(19\)	−174.904	+	422.254i	−0.484497	+	1.16968i	0.472954	+	0.881087i	\(0.343188\pi\)
−0.957452	+	0.288593i	\(0.906812\pi\)
\(23\)	−402.540	−	602.444i	−0.760946	−	1.13884i	−0.986365	−	0.164574i	\(-0.947375\pi\)
							0.225419	−	0.974262i	\(-0.427625\pi\)
\(29\)	−259.258	+	51.5697i	−0.308274	+	0.0613195i	−0.346803	−	0.937938i	\(-0.612733\pi\)
							0.0385290	+	0.999257i	\(0.487733\pi\)
\(31\)	1173.66	+	784.215i	1.22129	+	0.816040i	0.987710	−	0.156297i	\(-0.0499558\pi\)
							0.233580	+	0.972338i	\(0.424956\pi\)
\(37\)	753.227	−	1127.28i	0.550202	−	0.823436i	−0.447277	−	0.894396i	\(-0.647606\pi\)
							0.997479	+	0.0709595i	\(0.0226061\pi\)
\(41\)	143.026	−	719.038i	0.0850837	−	0.427744i	−0.914641	−	0.404268i	\(-0.867526\pi\)
							0.999724	−	0.0234768i	\(-0.00747358\pi\)
\(43\)	−415.416	−	1002.90i	−0.224670	−	0.542402i	0.770843	−	0.637025i	\(-0.219835\pi\)
							−0.995513	+	0.0946231i	\(0.969835\pi\)
\(47\)	1964.34	−	1964.34i	0.889242	−	0.889242i	−0.105208	−	0.994450i	\(-0.533551\pi\)
							0.994450	+	0.105208i	\(0.0335508\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.5.e.a.7.5	yes	40
3.2	odd	2		153.5.p.a.109.1		40
17.5	odd	16	inner	17.5.e.a.5.5	✓	40
51.5	even	16		153.5.p.a.73.1		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.5.e.a.5.5	✓	40	17.5	odd	16	inner
17.5.e.a.7.5	yes	40	1.1	even	1	trivial
153.5.p.a.73.1		40	51.5	even	16
153.5.p.a.109.1		40	3.2	odd	2