Embedded newform 19.10.e.a.4.2

• Label: 19.10.e.a.4.2
• Level: $19$
• Weight: $10$
• Character: 19.4
• Analytic conductor: $9.786$
• Analytic rank: $0$
• Dimension: $84$
• 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			4.2
Character	\(\chi\)	\(=\)	19.4
Dual form			19.10.e.a.5.2

$q$-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 \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

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

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.10.e.a.4.2	✓	84	1.1	even	1	trivial
19.10.e.a.5.2	yes	84	19.5	even	9	inner
361.10.a.i.1.39		42	19.9	even	9
361.10.a.j.1.4		42	19.10	odd	18