Embedded newform 89.6.f.a.11.18

• Label: 89.6.f.a.11.18
• Level: $89$
• Weight: $6$
• Character: 89.11
• 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			11.18
Character	\(\chi\)	\(=\)	89.11
Dual form			89.6.f.a.81.18

$q$-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 \(\left(\mathbb{Z}/89\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
89.6.f.a.11.18	✓	360	1.1	even	1	trivial
89.6.f.a.81.18	yes	360	89.81	even	22	inner