Embedded newform 19.8.e.a.5.5

• Label: 19.8.e.a.5.5
• Level: $19$
• Weight: $8$
• Character: 19.5
• 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			5.5
Character	\(\chi\)	\(=\)	19.5
Dual form			19.8.e.a.4.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.08875 + 2.94406i) q^{2} +(-14.2479 + 80.8039i) q^{3} +(-41.2933 + 34.6492i) q^{4} +(63.0220 + 52.8818i) q^{5} +(-122.644 - 695.549i) q^{6} +(204.570 + 354.325i) q^{7} +(782.905 - 1356.03i) q^{8} +(-4271.15 - 1554.57i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 6 q^{2} + 33 q^{3} + 72 q^{4} - 6 q^{5} - 876 q^{6} + 588 q^{7} - 4611 q^{8} + 5025 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.08875	+	2.94406i	−0.714951	+	0.260221i	−0.673781	−	0.738931i	\(-0.735331\pi\)
							−0.0411704	+	0.999152i	\(0.513109\pi\)
\(3\)	−14.2479	+	80.8039i	−0.304668	+	1.72786i	0.320397	+	0.947283i	\(0.396184\pi\)
							−0.625064	+	0.780573i	\(0.714927\pi\)
\(5\)	63.0220	+	52.8818i	0.225474	+	0.189196i	0.748526	−	0.663106i	\(-0.230762\pi\)
							−0.523051	+	0.852301i	\(0.675206\pi\)
\(7\)	204.570	+	354.325i	0.225423	+	0.390444i	0.956446	−	0.291908i	\(-0.0942903\pi\)
							−0.731023	+	0.682353i	\(0.760957\pi\)
\(11\)	−1041.40	+	1803.76i	−0.235908	+	0.408605i	−0.959536	−	0.281585i	\(-0.909140\pi\)
							0.723628	+	0.690190i	\(0.242473\pi\)
\(13\)	−371.554	−	2107.19i	−0.0469051	−	0.266012i	0.952332	−	0.305063i	\(-0.0986776\pi\)
							−0.999237	+	0.0390510i	\(0.987567\pi\)
\(17\)	25825.9	−	9399.85i	1.27492	−	0.464034i	0.386173	−	0.922426i	\(-0.373797\pi\)
							0.888750	+	0.458392i	\(0.151575\pi\)
\(19\)	714.131	+	29889.2i	0.0238858	+	0.999715i
							\(23\)	−78005.1	+	65454.1i	−1.33683	+	1.12173i	−0.354399	+	0.935094i	\(0.615315\pi\)
−0.982429	+	0.186638i	\(0.940241\pi\)
\(29\)	−89607.8	−	32614.6i	−0.682265	−	0.248324i	−0.0224451	−	0.999748i	\(-0.507145\pi\)
							−0.659820	+	0.751424i	\(0.729367\pi\)
\(31\)	−4877.25	−	8447.65i	−0.0294042	−	0.0509296i	0.850949	−	0.525249i	\(-0.176028\pi\)
							−0.880353	+	0.474319i	\(0.842694\pi\)
\(37\)	−399236.			−1.29576			−0.647879	−	0.761743i	\(-0.724344\pi\)
							−0.647879	+	0.761743i	\(0.724344\pi\)
\(41\)	−126624.	+	718121.i	−0.286928	+	1.62725i	0.411388	+	0.911461i	\(0.365044\pi\)
							−0.698316	+	0.715790i	\(0.746067\pi\)
\(43\)	−307492.	−	258017.i	−0.589786	−	0.494889i	0.298358	−	0.954454i	\(-0.403561\pi\)
							−0.888144	+	0.459565i	\(0.848005\pi\)
\(47\)	1.09355e6	+	398021.i	1.53638	+	0.559195i	0.965173	−	0.261613i	\(-0.0842545\pi\)
							0.571203	+	0.820809i	\(0.306477\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.8.e.a.5.5	yes	66
19.2	odd	18		361.8.a.i.1.13		33
19.4	even	9	inner	19.8.e.a.4.5	✓	66
19.17	even	9		361.8.a.j.1.21		33

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.8.e.a.4.5	✓	66	19.4	even	9	inner
19.8.e.a.5.5	yes	66	1.1	even	1	trivial
361.8.a.i.1.13		33	19.2	odd	18
361.8.a.j.1.21		33	19.17	even	9