Embedded newform 89.10.f.a.11.20

• Label: 89.10.f.a.11.20
• 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			11.20
Character	\(\chi\)	\(=\)	89.11
Dual form			89.10.f.a.81.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.7990 - 15.9249i) q^{2} +(202.658 - 29.1378i) q^{3} +(9.67508 - 67.2917i) q^{4} +(2267.64 - 1457.33i) q^{5} +(-3260.50 - 2825.24i) q^{6} +(6234.30 + 9700.76i) q^{7} +(-10281.2 + 6607.30i) q^{8} +(21335.5 - 6264.68i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 660 q - 9 q^{2} - 825 q^{3} - 16393 q^{4} - 5127 q^{5} - 11 q^{6} - 11 q^{7} + 1015 q^{8} + 489535 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.7990	−	15.9249i	−0.609837	−	0.703789i	0.363907	−	0.931435i	\(-0.381442\pi\)
							−0.973744	+	0.227646i	\(0.926897\pi\)
\(3\)	202.658	−	29.1378i	1.44450	−	0.207688i	0.624937	−	0.780675i	\(-0.285125\pi\)
							0.819564	+	0.572987i	\(0.194215\pi\)
\(5\)	2267.64	−	1457.33i	1.62259	−	1.04278i	0.668306	−	0.743887i	\(-0.267020\pi\)
							0.954287	−	0.298890i	\(-0.0966165\pi\)
\(7\)	6234.30	+	9700.76i	0.981401	+	1.52709i	0.843839	+	0.536596i	\(0.180290\pi\)
							0.137562	+	0.990493i	\(0.456074\pi\)
\(11\)	−6514.61	−	4186.68i	−0.134159	−	0.0862190i	0.471840	−	0.881684i	\(-0.343590\pi\)
							−0.605999	+	0.795465i	\(0.707226\pi\)
\(13\)	39028.9	−	5611.51i	0.379002	−	0.0544922i	0.0498185	−	0.998758i	\(-0.484136\pi\)
							0.329183	+	0.944266i	\(0.393227\pi\)
\(17\)	−33000.2	+	38084.3i	−0.0958290	+	0.110593i	−0.801638	−	0.597809i	\(-0.796038\pi\)
							0.705809	+	0.708402i	\(0.250583\pi\)
\(19\)	−70006.7	−	238421.i	−0.123239	−	0.419714i	0.874643	−	0.484768i	\(-0.161096\pi\)
							−0.997882	+	0.0650546i	\(0.979278\pi\)
\(23\)	−532141.	−	1.81231e6i	−0.396508	−	1.35038i	−0.879976	−	0.475019i	\(-0.842441\pi\)
							0.483468	−	0.875362i	\(-0.339377\pi\)
\(29\)	1.70389e6	+	2.65131e6i	0.447354	+	0.696097i	0.989556	−	0.144152i	\(-0.0460453\pi\)
							−0.542201	+	0.840249i	\(0.682409\pi\)
\(31\)	−1.09426e6	+	3.72672e6i	−0.212811	+	0.724768i	0.782023	+	0.623250i	\(0.214188\pi\)
							−0.994834	+	0.101518i	\(0.967630\pi\)
\(37\)		−	3.70136e6i		−	0.324679i	−0.986735	−	0.162339i	\(-0.948096\pi\)
							0.986735	−	0.162339i	\(-0.0519040\pi\)
\(41\)	2.37905e7	+	3.42056e6i	1.31485	+	0.189047i	0.763816	−	0.645434i	\(-0.223323\pi\)
							0.551036	+	0.834481i	\(0.314233\pi\)
\(43\)	1.60516e7	−	2.49768e7i	0.715997	−	1.11411i	−0.272395	−	0.962186i	\(-0.587816\pi\)
							0.988392	−	0.151928i	\(-0.0485480\pi\)
\(47\)	−30504.4	+	212162.i	−0.000911846	+	0.00634203i	−0.990272	−	0.139143i	\(-0.955565\pi\)
							0.989360	+	0.145485i	\(0.0464743\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	89.10.f.a.11.20	✓	660
89.81	even	22	inner	89.10.f.a.81.20	yes	660

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
89.10.f.a.11.20	✓	660	1.1	even	1	trivial
89.10.f.a.81.20	yes	660	89.81	even	22	inner