Embedded newform 89.6.f.a.81.18

• Label: 89.6.f.a.81.18
• Level: $89$
• Weight: $6$
• Character: 89.81
• Analytic conductor: $14.274$
• Analytic rank: $0$
• Dimension: $360$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 89 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	89.f (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.2741599634\)
Analytic rank:	\(0\)
Dimension:	\(360\)
Relative dimension:	\(36\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			81.18
Character	\(\chi\)	\(=\)	89.81
Dual form			89.6.f.a.11.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.194195 - 0.224113i) q^{2} +(19.9890 + 2.87398i) q^{3} +(4.54156 + 31.5872i) q^{4} +(-59.2305 - 38.0651i) q^{5} +(4.52585 - 3.92167i) q^{6} +(-59.5362 + 92.6401i) q^{7} +(15.9440 + 10.2466i) q^{8} +(158.143 + 46.4349i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 360 q - q^{2} - 553 q^{4} - 17 q^{5} - 11 q^{6} - 11 q^{7} + 439 q^{8} + 1858 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{22}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.194195	−	0.224113i	0.0343291	−	0.0396179i	−0.738325	−	0.674445i	\(-0.764383\pi\)
							0.772654	+	0.634827i	\(0.218929\pi\)
\(3\)	19.9890	+	2.87398i	1.28229	+	0.184366i	0.749571	−	0.661924i	\(-0.230260\pi\)
							0.532723	+	0.846290i	\(0.321169\pi\)
\(5\)	−59.2305	−	38.0651i	−1.05955	−	0.680930i	−0.109802	−	0.993953i	\(-0.535022\pi\)
							−0.949745	+	0.313024i	\(0.898658\pi\)
\(7\)	−59.5362	+	92.6401i	−0.459236	+	0.714585i	−0.991228	−	0.132164i	\(-0.957807\pi\)
							0.531992	+	0.846749i	\(0.321444\pi\)
\(11\)	−626.151	+	402.403i	−1.56026	+	1.00272i	−0.577815	+	0.816168i	\(0.696094\pi\)
							−0.982445	+	0.186550i	\(0.940269\pi\)
\(13\)	86.4982	+	12.4366i	0.141954	+	0.0204099i	0.212926	−	0.977068i	\(-0.431701\pi\)
							−0.0709713	+	0.997478i	\(0.522610\pi\)
\(17\)	915.647	+	1056.71i	0.768432	+	0.886818i	0.996218	−	0.0868933i	\(-0.0276939\pi\)
							−0.227785	+	0.973711i	\(0.573148\pi\)
\(19\)	382.417	−	1302.39i	0.243026	−	0.827671i	−0.744147	−	0.668016i	\(-0.767144\pi\)
							0.987173	−	0.159655i	\(-0.0510381\pi\)
\(23\)	369.874	−	1259.67i	0.145792	−	0.496522i	−0.853923	−	0.520400i	\(-0.825783\pi\)
							0.999715	+	0.0238774i	\(0.00760113\pi\)
\(29\)	−877.462	+	1365.36i	−0.193746	+	0.301475i	−0.924512	−	0.381154i	\(-0.875527\pi\)
							0.730766	+	0.682629i	\(0.239163\pi\)
\(31\)	2400.57	+	8175.61i	0.448653	+	1.52797i	0.804802	+	0.593544i	\(0.202272\pi\)
							−0.356148	+	0.934429i	\(0.615910\pi\)
\(37\)			14957.0i			1.79614i	0.439849	+	0.898072i	\(0.355032\pi\)
							−0.439849	+	0.898072i	\(0.644968\pi\)
\(41\)	4443.65	−	638.901i	0.412839	−	0.0593573i	0.0672336	−	0.997737i	\(-0.478583\pi\)
							0.345605	+	0.938380i	\(0.387674\pi\)
\(43\)	−3714.65	−	5780.11i	−0.306370	−	0.476722i	0.653593	−	0.756846i	\(-0.273261\pi\)
							−0.959964	+	0.280124i	\(0.909624\pi\)
\(47\)	−1170.29	−	8139.57i	−0.0772770	−	0.537473i	−0.991281	−	0.131767i	\(-0.957935\pi\)
							0.914004	−	0.405706i	\(-0.132974\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	89.6.f.a.81.18	yes	360
89.11	even	22	inner	89.6.f.a.11.18	✓	360

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
89.6.f.a.11.18	✓	360	89.11	even	22	inner
89.6.f.a.81.18	yes	360	1.1	even	1	trivial