Embedded newform 17.5.e.a.5.1

• Label: 17.5.e.a.5.1
• Level: $17$
• Weight: $5$
• Character: 17.5
• 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			5.1
Character	\(\chi\)	\(=\)	17.5
Dual form			17.5.e.a.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.55538 - 6.16923i) q^{2} +(-1.90204 - 2.84660i) q^{3} +(-20.2157 + 20.2157i) q^{4} +(-0.0751892 + 0.378002i) q^{5} +(-12.7009 + 19.0082i) q^{6} +(-11.8164 - 59.4051i) q^{7} +(77.6662 + 32.1704i) q^{8} +(26.5120 - 64.0055i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 8 q^{2} - 8 q^{3} - 8 q^{4} - 8 q^{5} - 8 q^{6} - 8 q^{7} - 8 q^{8} - 8 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.55538	−	6.16923i	−0.638844	−	1.54231i	−0.828221	−	0.560402i	\(-0.810647\pi\)
							0.189377	−	0.981905i	\(-0.439353\pi\)
\(3\)	−1.90204	−	2.84660i	−0.211338	−	0.316289i	0.710623	−	0.703573i	\(-0.248413\pi\)
							−0.921960	+	0.387284i	\(0.873413\pi\)
\(5\)	−0.0751892	+	0.378002i	−0.00300757	+	0.0151201i	−0.982260	−	0.187524i	\(-0.939954\pi\)
							0.979252	+	0.202645i	\(0.0649536\pi\)
\(7\)	−11.8164	−	59.4051i	−0.241151	−	1.21235i	−0.891609	−	0.452807i	\(-0.850423\pi\)
							0.650457	−	0.759543i	\(-0.274577\pi\)
\(11\)	165.044	+	110.279i	1.36400	+	0.911397i	0.999801	−	0.0199239i	\(-0.00634240\pi\)
							0.364200	+	0.931321i	\(0.381342\pi\)
\(13\)	−155.868	−	155.868i	−0.922298	−	0.922298i	0.0748939	−	0.997192i	\(-0.476138\pi\)
							−0.997192	+	0.0748939i	\(0.976138\pi\)
\(17\)	288.984	+	3.04351i	0.999945	+	0.0105312i
							\(19\)	68.0987	+	164.405i	0.188639	+	0.455415i	0.989698	−	0.143171i	\(-0.0457298\pi\)
−0.801059	+	0.598585i	\(0.795730\pi\)
\(23\)	15.8194	−	23.6754i	0.0299044	−	0.0447551i	−0.816216	−	0.577747i	\(-0.803932\pi\)
							0.846121	+	0.532991i	\(0.178932\pi\)
\(29\)	−178.312	−	35.4685i	−0.212024	−	0.0421742i	0.0879354	−	0.996126i	\(-0.471973\pi\)
							−0.299959	+	0.953952i	\(0.596973\pi\)
\(31\)	−126.571	+	84.5719i	−0.131707	+	0.0880040i	−0.619679	−	0.784856i	\(-0.712737\pi\)
							0.487971	+	0.872860i	\(0.337737\pi\)
\(37\)	−899.832	−	1346.69i	−0.657292	−	0.983706i	−0.999035	−	0.0439133i	\(-0.986017\pi\)
							0.341744	−	0.939793i	\(-0.388983\pi\)
\(41\)	−359.149	−	1805.57i	−0.213652	−	1.07410i	−0.927506	−	0.373808i	\(-0.878052\pi\)
							0.713854	−	0.700295i	\(-0.246948\pi\)
\(43\)	−882.695	+	2131.01i	−0.477391	+	1.15252i	0.483438	+	0.875379i	\(0.339388\pi\)
							−0.960828	+	0.277144i	\(0.910612\pi\)
\(47\)	1192.78	+	1192.78i	0.539964	+	0.539964i	0.923518	−	0.383555i	\(-0.125300\pi\)
							−0.383555	+	0.923518i	\(0.625300\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.5.e.a.5.1	✓	40
3.2	odd	2		153.5.p.a.73.5		40
17.7	odd	16	inner	17.5.e.a.7.1	yes	40
51.41	even	16		153.5.p.a.109.5		40

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.5.e.a.5.1	✓	40	1.1	even	1	trivial
17.5.e.a.7.1	yes	40	17.7	odd	16	inner
153.5.p.a.73.5		40	3.2	odd	2
153.5.p.a.109.5		40	51.41	even	16