Embedded newform 17.5.e.a.7.1

• Label: 17.5.e.a.7.1
• 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			7.1
Character	\(\chi\)	\(=\)	17.7
Dual form			17.5.e.a.5.1

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

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.5.e.a.5.1	✓	40	17.5	odd	16	inner
17.5.e.a.7.1	yes	40	1.1	even	1	trivial
153.5.p.a.73.5		40	51.5	even	16
153.5.p.a.109.5		40	3.2	odd	2