Embedded newform 19.8.e.a.4.9

• Label: 19.8.e.a.4.9
• Level: $19$
• Weight: $8$
• Character: 19.4
• Analytic conductor: $5.935$
• Analytic rank: $0$
• Dimension: $66$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	19.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.93531548420\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			4.9
Character	\(\chi\)	\(=\)	19.4
Dual form			19.8.e.a.5.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(12.6181 + 4.59260i) q^{2} +(-9.09979 - 51.6075i) q^{3} +(40.0701 + 33.6228i) q^{4} +(154.381 - 129.541i) q^{5} +(122.191 - 692.979i) q^{6} +(257.216 - 445.511i) q^{7} +(-508.193 - 880.216i) q^{8} +(-525.417 + 191.236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 6 q^{2} + 33 q^{3} + 72 q^{4} - 6 q^{5} - 876 q^{6} + 588 q^{7} - 4611 q^{8} + 5025 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	12.6181	+	4.59260i	1.11529	+	0.405933i	0.832931	−	0.553376i	\(-0.186661\pi\)
							0.282359	+	0.959309i	\(0.408883\pi\)
\(3\)	−9.09979	−	51.6075i	−0.194584	−	1.10354i	−0.913010	−	0.407937i	\(-0.866248\pi\)
							0.718426	−	0.695603i	\(-0.244863\pi\)
\(5\)	154.381	−	129.541i	0.552330	−	0.463460i	−0.323399	−	0.946263i	\(-0.604826\pi\)
							0.875729	+	0.482803i	\(0.160381\pi\)
\(7\)	257.216	−	445.511i	0.283436	−	0.490925i	−0.688793	−	0.724958i	\(-0.741859\pi\)
							0.972229	+	0.234033i	\(0.0751924\pi\)
\(11\)	2620.72	+	4539.22i	0.593672	+	1.02827i	0.993733	+	0.111781i	\(0.0356556\pi\)
							−0.400061	+	0.916488i	\(0.631011\pi\)
\(13\)	−780.644	+	4427.25i	−0.0985488	+	0.558898i	0.895053	+	0.445959i	\(0.147137\pi\)
							−0.993602	+	0.112939i	\(0.963974\pi\)
\(17\)	869.267	+	316.387i	0.0429123	+	0.0156188i	0.363387	−	0.931638i	\(-0.381620\pi\)
							−0.320475	+	0.947257i	\(0.603842\pi\)
\(19\)	−6512.66	+	29179.7i	−0.217832	+	0.975986i
							\(23\)	24707.8	+	20732.3i	0.423435	+	0.355304i	0.829468	−	0.558554i	\(-0.188644\pi\)
−0.406033	+	0.913858i	\(0.633088\pi\)
\(29\)	65245.6	−	23747.5i	0.496773	−	0.180811i	−0.0814688	−	0.996676i	\(-0.525961\pi\)
							0.578242	+	0.815865i	\(0.303739\pi\)
\(31\)	10292.4	−	17826.9i	0.0620511	−	0.107476i	−0.833331	−	0.552774i	\(-0.813569\pi\)
							0.895382	+	0.445299i	\(0.146902\pi\)
\(37\)	548328.			1.77965			0.889825	−	0.456303i	\(-0.150827\pi\)
							0.889825	+	0.456303i	\(0.150827\pi\)
\(41\)	−52113.9	−	295553.i	−0.118089	−	0.669717i	−0.985174	−	0.171557i	\(-0.945120\pi\)
							0.867085	−	0.498160i	\(-0.165991\pi\)
\(43\)	−605067.	+	507712.i	−1.16055	+	0.973818i	−0.999913	−	0.0131958i	\(-0.995800\pi\)
							−0.160638	+	0.987013i	\(0.551355\pi\)
\(47\)	663151.	−	241367.i	0.931687	−	0.339106i	0.168809	−	0.985649i	\(-0.446008\pi\)
							0.762878	+	0.646542i	\(0.223786\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.8.e.a.4.9	✓	66
19.5	even	9	inner	19.8.e.a.5.9	yes	66
19.9	even	9		361.8.a.j.1.8		33
19.10	odd	18		361.8.a.i.1.26		33

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.8.e.a.4.9	✓	66	1.1	even	1	trivial
19.8.e.a.5.9	yes	66	19.5	even	9	inner
361.8.a.i.1.26		33	19.10	odd	18
361.8.a.j.1.8		33	19.9	even	9