Embedded newform 19.10.e.a.5.2

• Label: 19.10.e.a.5.2
• Level: $19$
• Weight: $10$
• Character: 19.5
• Analytic conductor: $9.786$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	19.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.78568088711\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(14\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			5.2
Character	\(\chi\)	\(=\)	19.5
Dual form			19.10.e.a.4.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-36.8078 + 13.3970i) q^{2} +(-43.1817 + 244.896i) q^{3} +(783.124 - 657.119i) q^{4} +(-962.522 - 807.652i) q^{5} +(-1691.43 - 9592.59i) q^{6} +(83.0798 + 143.899i) q^{7} +(-9994.16 + 17310.4i) q^{8} +(-39613.3 - 14418.1i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 6 q^{2} + 213 q^{3} - 1032 q^{4} - 6 q^{5} + 4668 q^{6} + 5715 q^{7} + 36861 q^{8} - 62421 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−36.8078	+	13.3970i	−1.62669	+	0.592067i	−0.984640	−	0.174594i	\(-0.944139\pi\)
							−0.642051	+	0.766662i	\(0.721916\pi\)
\(3\)	−43.1817	+	244.896i	−0.307790	+	1.74556i	0.302283	+	0.953218i	\(0.402251\pi\)
							−0.610073	+	0.792345i	\(0.708860\pi\)
\(5\)	−962.522	−	807.652i	−0.688724	−	0.577908i	0.229817	−	0.973234i	\(-0.426187\pi\)
							−0.918541	+	0.395326i	\(0.870632\pi\)
\(7\)	83.0798	+	143.899i	0.0130784	+	0.0226524i	0.872491	−	0.488631i	\(-0.162504\pi\)
							−0.859412	+	0.511283i	\(0.829170\pi\)
\(11\)	42900.3	−	74305.5i	0.883473	−	1.53022i	0.0360185	−	0.999351i	\(-0.488532\pi\)
							0.847454	−	0.530868i	\(-0.178134\pi\)
\(13\)	13157.2	+	74618.2i	0.127767	+	0.724602i	0.979626	+	0.200831i	\(0.0643642\pi\)
							−0.851859	+	0.523771i	\(0.824525\pi\)
\(17\)	−372235.	+	135482.i	−1.08093	+	0.393426i	−0.820253	−	0.572001i	\(-0.806167\pi\)
							−0.260675	+	0.965427i	\(0.583945\pi\)
\(19\)	−212410.	+	526849.i	−0.373925	+	0.927459i
							\(23\)	1.69346e6	−	1.42098e6i	1.26183	−	1.05880i	0.266343	−	0.963878i	\(-0.414185\pi\)
0.995485	−	0.0949208i	\(-0.0302598\pi\)
\(29\)	−1.61532e6	−	587928.i	−0.424099	−	0.154360i	0.121149	−	0.992634i	\(-0.461342\pi\)
							−0.545248	+	0.838275i	\(0.683564\pi\)
\(31\)	76323.5	+	132196.i	0.0148433	+	0.0257094i	0.873352	−	0.487090i	\(-0.161942\pi\)
							−0.858508	+	0.512800i	\(0.828608\pi\)
\(37\)	7.71043e6			0.676349			0.338174	−	0.941084i	\(-0.390191\pi\)
							0.338174	+	0.941084i	\(0.390191\pi\)
\(41\)	1.44124e6	−	8.17367e6i	0.0796542	−	0.451742i	−0.918728	−	0.394890i	\(-0.870783\pi\)
							0.998383	−	0.0568516i	\(-0.0181062\pi\)
\(43\)	7.37764e6	+	6.19058e6i	0.329086	+	0.276136i	0.792327	−	0.610096i	\(-0.208869\pi\)
							−0.463241	+	0.886232i	\(0.653314\pi\)
\(47\)	4.02536e7	+	1.46511e7i	1.20327	+	0.437955i	0.864365	−	0.502864i	\(-0.167720\pi\)
							0.338907	+	0.940820i	\(0.389943\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.10.e.a.5.2	yes	84
19.2	odd	18		361.10.a.j.1.4		42
19.4	even	9	inner	19.10.e.a.4.2	✓	84
19.17	even	9		361.10.a.i.1.39		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.10.e.a.4.2	✓	84	19.4	even	9	inner
19.10.e.a.5.2	yes	84	1.1	even	1	trivial
361.10.a.i.1.39		42	19.17	even	9
361.10.a.j.1.4		42	19.2	odd	18