Embedded newform 17.10.d.a.2.7

• Label: 17.10.d.a.2.7
• Level: $17$
• Weight: $10$
• Character: 17.2
• Analytic conductor: $8.756$
• Analytic rank: $0$
• Dimension: $52$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 17 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	17.d (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.75560921479\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Relative dimension:	\(13\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			2.7
Character	\(\chi\)	\(=\)	17.2
Dual form			17.10.d.a.9.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.23497 - 2.23497i) q^{2} +(-45.0140 + 18.6454i) q^{3} -502.010i q^{4} +(265.226 + 640.313i) q^{5} +(142.277 + 58.9331i) q^{6} +(-537.194 + 1296.90i) q^{7} +(-2266.28 + 2266.28i) q^{8} +(-12239.4 + 12239.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q - 4 q^{2} - 4 q^{3} - 1140 q^{5} + 12308 q^{6} - 4 q^{7} + 2044 q^{8} - 83224 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.23497	−	2.23497i	−0.0987728	−	0.0987728i	0.655994	−	0.754766i	\(-0.272250\pi\)
							−0.754766	+	0.655994i	\(0.772250\pi\)
\(3\)	−45.0140	+	18.6454i	−0.320850	+	0.132900i	−0.537295	−	0.843394i	\(-0.680554\pi\)
							0.216445	+	0.976295i	\(0.430554\pi\)
\(5\)	265.226	+	640.313i	0.189781	+	0.458171i	0.989917	−	0.141647i	\(-0.0452398\pi\)
							−0.800137	+	0.599818i	\(0.795240\pi\)
\(7\)	−537.194	+	1296.90i	−0.0845649	+	0.204158i	−0.960505	−	0.278261i	\(-0.910242\pi\)
							0.875940	+	0.482419i	\(0.160242\pi\)
\(11\)	−65465.7	−	27116.8i	−1.34818	−	0.558433i	−0.412392	−	0.911007i	\(-0.635306\pi\)
							−0.935784	+	0.352574i	\(0.885306\pi\)
\(13\)			180297.i			1.75082i	0.483378	+	0.875412i	\(0.339410\pi\)
							−0.483378	+	0.875412i	\(0.660590\pi\)
\(17\)	41203.0	+	341892.i	0.119649	+	0.992816i
							\(19\)	−599782.	−	599782.i	−1.05585	−	1.05585i	−0.998345	−	0.0575042i	\(-0.981686\pi\)
−0.0575042	−	0.998345i	\(-0.518314\pi\)
\(23\)	1.45714e6	+	603568.i	1.08574	+	0.449729i	0.852520	−	0.522695i	\(-0.175073\pi\)
							0.233222	+	0.972424i	\(0.425073\pi\)
\(29\)	−1.18611e6	−	2.86351e6i	−0.311410	−	0.751810i	−0.999653	−	0.0263300i	\(-0.991618\pi\)
							0.688244	−	0.725480i	\(-0.258382\pi\)
\(31\)	−3.10406e6	+	1.28574e6i	−0.603674	+	0.250050i	−0.663521	−	0.748158i	\(-0.730939\pi\)
							0.0598469	+	0.998208i	\(0.480939\pi\)
\(37\)	−1.75560e7	+	7.27194e6i	−1.53999	+	0.637885i	−0.981472	−	0.191607i	\(-0.938630\pi\)
							−0.558519	+	0.829492i	\(0.688630\pi\)
\(41\)	3.30907e6	−	7.98880e6i	0.182885	−	0.441524i	−0.805674	−	0.592360i	\(-0.798196\pi\)
							0.988559	+	0.150836i	\(0.0481964\pi\)
\(43\)	−7.27178e6	+	7.27178e6i	−0.324364	+	0.324364i	−0.850439	−	0.526074i	\(-0.823663\pi\)
							0.526074	+	0.850439i	\(0.323663\pi\)
\(47\)		−	5.71205e6i		−	0.170746i	−0.996349	−	0.0853732i	\(-0.972792\pi\)
							0.996349	−	0.0853732i	\(-0.0272082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	17.10.d.a.2.7	✓	52
17.3	odd	16		289.10.a.i.1.25		52
17.9	even	8	inner	17.10.d.a.9.7	yes	52
17.14	odd	16		289.10.a.i.1.26		52

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.10.d.a.2.7	✓	52	1.1	even	1	trivial
17.10.d.a.9.7	yes	52	17.9	even	8	inner
289.10.a.i.1.25		52	17.3	odd	16
289.10.a.i.1.26		52	17.14	odd	16