Embedded newform 89.10.f.a.11.15

• Label: 89.10.f.a.11.15
• Level: $89$
• Weight: $10$
• Character: 89.11
• Analytic conductor: $45.838$
• Analytic rank: $0$
• Dimension: $660$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 89 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	89.f (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(45.8381894186\)
Analytic rank:	\(0\)
Dimension:	\(660\)
Relative dimension:	\(66\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			11.15
Character	\(\chi\)	\(=\)	89.11
Dual form			89.10.f.a.81.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-18.6156 - 21.4835i) q^{2} +(-248.677 + 35.7544i) q^{3} +(-42.1371 + 293.070i) q^{4} +(754.308 - 484.765i) q^{5} +(5397.41 + 4676.88i) q^{6} +(5204.71 + 8098.69i) q^{7} +(-5163.46 + 3318.35i) q^{8} +(41676.3 - 12237.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 660 q - 9 q^{2} - 825 q^{3} - 16393 q^{4} - 5127 q^{5} - 11 q^{6} - 11 q^{7} + 1015 q^{8} + 489535 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/89\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{21}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−18.6156	−	21.4835i	−0.822701	−	0.949448i	0.176693	−	0.984266i	\(-0.443460\pi\)
							−0.999394	+	0.0348184i	\(0.988915\pi\)
\(3\)	−248.677	+	35.7544i	−1.77252	+	0.254849i	−0.949634	−	0.313362i	\(-0.898545\pi\)
							−0.822883	+	0.568211i	\(0.807636\pi\)
\(5\)	754.308	−	484.765i	0.539739	−	0.346869i	−0.242198	−	0.970227i	\(-0.577868\pi\)
							0.781937	+	0.623358i	\(0.214232\pi\)
\(7\)	5204.71	+	8098.69i	0.819324	+	1.27489i	0.958633	+	0.284644i	\(0.0918753\pi\)
							−0.139310	+	0.990249i	\(0.544488\pi\)
\(11\)	62867.0	+	40402.2i	1.29466	+	0.832028i	0.992620	−	0.121268i	\(-0.0386959\pi\)
							0.302040	+	0.953295i	\(0.402332\pi\)
\(13\)	123742.	−	17791.5i	1.20164	−	0.172769i	0.487721	−	0.873000i	\(-0.337828\pi\)
							0.713917	+	0.700230i	\(0.246919\pi\)
\(17\)	−352691.	+	407027.i	−1.02417	+	1.18196i	−0.0410248	+	0.999158i	\(0.513062\pi\)
							−0.983150	+	0.182802i	\(0.941483\pi\)
\(19\)	156343.	+	532455.i	0.275225	+	0.937329i	0.974858	+	0.222827i	\(0.0715285\pi\)
							−0.699633	+	0.714502i	\(0.746653\pi\)
\(23\)	−201470.	−	686145.i	−0.150119	−	0.511259i	0.849754	−	0.527179i	\(-0.176750\pi\)
							−0.999873	+	0.0159204i	\(0.994932\pi\)
\(29\)	986243.	+	1.53462e6i	0.258936	+	0.402913i	0.946244	−	0.323454i	\(-0.104844\pi\)
							−0.687308	+	0.726366i	\(0.741208\pi\)
\(31\)	1.53103e6	−	5.21421e6i	0.297753	−	1.01405i	−0.665710	−	0.746211i	\(-0.731871\pi\)
							0.963463	−	0.267842i	\(-0.0863106\pi\)
\(37\)			1.39853e7i			1.22678i	0.789782	+	0.613388i	\(0.210194\pi\)
							−0.789782	+	0.613388i	\(0.789806\pi\)
\(41\)	−194747.	−	28000.4i	−0.0107633	−	0.00154752i	0.136931	−	0.990581i	\(-0.456276\pi\)
							−0.147694	+	0.989033i	\(0.547185\pi\)
\(43\)	1.60697e7	−	2.50049e7i	0.716803	−	1.11537i	−0.271437	−	0.962456i	\(-0.587499\pi\)
							0.988240	−	0.152911i	\(-0.0488648\pi\)
\(47\)	−9.23250e6	+	6.42135e7i	−0.275981	+	1.91949i	0.104158	+	0.994561i	\(0.466785\pi\)
							−0.380139	+	0.924929i	\(0.624124\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	89.10.f.a.11.15	✓	660
89.81	even	22	inner	89.10.f.a.81.15	yes	660

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
89.10.f.a.11.15	✓	660	1.1	even	1	trivial
89.10.f.a.81.15	yes	660	89.81	even	22	inner