Embedded newform 19.8.e.a.4.3

• Label: 19.8.e.a.4.3
• 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			4.3
Character	\(\chi\)	\(=\)	19.4
Dual form			19.8.e.a.5.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-14.5692 - 5.30275i) q^{2} +(15.4254 + 87.4821i) q^{3} +(86.0882 + 72.2366i) q^{4} +(40.6287 - 34.0916i) q^{5} +(239.159 - 1356.34i) q^{6} +(-203.416 + 352.328i) q^{7} +(121.087 + 209.728i) q^{8} +(-5360.06 + 1950.90i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 6 q^{2} + 33 q^{3} + 72 q^{4} - 6 q^{5} - 876 q^{6} + 588 q^{7} - 4611 q^{8} + 5025 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−14.5692	−	5.30275i	−1.28775	−	0.468701i	−0.394759	−	0.918785i	\(-0.629172\pi\)
							−0.892986	+	0.450083i	\(0.851394\pi\)
\(3\)	15.4254	+	87.4821i	0.329848	+	1.87066i	0.473153	+	0.880980i	\(0.343116\pi\)
							−0.143305	+	0.989679i	\(0.545773\pi\)
\(5\)	40.6287	−	34.0916i	0.145358	−	0.121970i	−0.567209	−	0.823574i	\(-0.691977\pi\)
							0.712567	+	0.701604i	\(0.247532\pi\)
\(7\)	−203.416	+	352.328i	−0.224152	+	0.388243i	−0.956065	−	0.293156i	\(-0.905295\pi\)
							0.731913	+	0.681399i	\(0.238628\pi\)
\(11\)	−3392.81	−	5876.53i	−0.768574	−	1.33121i	−0.938336	−	0.345724i	\(-0.887633\pi\)
							0.169762	−	0.985485i	\(-0.445700\pi\)
\(13\)	−460.717	+	2612.85i	−0.0581611	+	0.329848i	−0.999980	−	0.00627196i	\(-0.998004\pi\)
							0.941819	+	0.336120i	\(0.109115\pi\)
\(17\)	−6151.24	−	2238.87i	−0.303663	−	0.110524i	0.185695	−	0.982607i	\(-0.440546\pi\)
							−0.489358	+	0.872083i	\(0.662769\pi\)
\(19\)	−22728.7	−	19423.7i	−0.760215	−	0.649672i
							\(23\)	38283.0	+	32123.3i	0.656083	+	0.550519i	0.908910	−	0.416993i	\(-0.136916\pi\)
−0.252827	+	0.967511i	\(0.581360\pi\)
\(29\)	161665.	−	58841.2i	1.23090	−	0.448011i	0.356995	−	0.934106i	\(-0.383801\pi\)
							0.873906	+	0.486095i	\(0.161579\pi\)
\(31\)	−145968.	+	252825.i	−0.880021	+	1.52424i	−0.0287053	+	0.999588i	\(0.509138\pi\)
							−0.851316	+	0.524654i	\(0.824195\pi\)
\(37\)	−303525.			−0.985118			−0.492559	−	0.870279i	\(-0.663938\pi\)
							−0.492559	+	0.870279i	\(0.663938\pi\)
\(41\)	−18119.9	−	102763.i	−0.0410594	−	0.232860i	0.957371	−	0.288860i	\(-0.0932763\pi\)
							−0.998431	+	0.0560005i	\(0.982165\pi\)
\(43\)	−382186.	+	320692.i	−0.733053	+	0.615104i	−0.930962	−	0.365116i	\(-0.881029\pi\)
							0.197909	+	0.980220i	\(0.436585\pi\)
\(47\)	−306473.	+	111547.i	−0.430575	+	0.156717i	−0.548211	−	0.836340i	\(-0.684691\pi\)
							0.117635	+	0.993057i	\(0.462469\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.8.e.a.4.3	✓	66
19.5	even	9	inner	19.8.e.a.5.3	yes	66
19.9	even	9		361.8.a.j.1.26		33
19.10	odd	18		361.8.a.i.1.8		33

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.8.e.a.4.3	✓	66	1.1	even	1	trivial
19.8.e.a.5.3	yes	66	19.5	even	9	inner
361.8.a.i.1.8		33	19.10	odd	18
361.8.a.j.1.26		33	19.9	even	9