Embedded newform 19.10.e.a.4.4

• Label: 19.10.e.a.4.4
• Level: $19$
• Weight: $10$
• Character: 19.4
• 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			4.4
Character	\(\chi\)	\(=\)	19.4
Dual form			19.10.e.a.5.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-24.2695 - 8.83336i) q^{2} +(-19.4747 - 110.447i) q^{3} +(118.763 + 99.6543i) q^{4} +(1548.70 - 1299.52i) q^{5} +(-502.974 + 2852.50i) q^{6} +(4850.18 - 8400.77i) q^{7} +(4609.68 + 7984.19i) q^{8} +(6676.79 - 2430.15i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 6 q^{2} + 213 q^{3} - 1032 q^{4} - 6 q^{5} + 4668 q^{6} + 5715 q^{7} + 36861 q^{8} - 62421 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−24.2695	−	8.83336i	−1.07257	−	0.390383i	−0.255432	−	0.966827i	\(-0.582218\pi\)
							−0.817137	+	0.576444i	\(0.804440\pi\)
\(3\)	−19.4747	−	110.447i	−0.138811	−	0.787239i	−0.972129	−	0.234445i	\(-0.924673\pi\)
							0.833318	−	0.552794i	\(-0.186438\pi\)
\(5\)	1548.70	−	1299.52i	1.10816	−	0.929858i	0.110215	−	0.993908i	\(-0.464846\pi\)
							0.997947	+	0.0640494i	\(0.0204015\pi\)
\(7\)	4850.18	−	8400.77i	0.763514	−	1.32245i	−0.177514	−	0.984118i	\(-0.556806\pi\)
							0.941029	−	0.338327i	\(-0.109861\pi\)
\(11\)	−24669.4	−	42728.7i	−0.508033	−	0.879939i	−0.999957	−	0.00930079i	\(-0.997039\pi\)
							0.491924	−	0.870638i	\(-0.336294\pi\)
\(13\)	−5515.23	+	31278.4i	−0.0535573	+	0.303739i	−0.999806	−	0.0197017i	\(-0.993728\pi\)
							0.946249	+	0.323440i	\(0.104839\pi\)
\(17\)	−393693.	−	143292.i	−1.14324	−	0.416105i	−0.300157	−	0.953890i	\(-0.597039\pi\)
							−0.843082	+	0.537785i	\(0.819261\pi\)
\(19\)	469983.	+	319067.i	0.827353	+	0.561682i
							\(23\)	1.95064e6	+	1.63678e6i	1.45345	+	1.21959i	0.930012	+	0.367530i	\(0.119797\pi\)
0.523441	+	0.852062i	\(0.324648\pi\)
\(29\)	2.02149e6	−	735762.i	0.530738	−	0.193173i	−0.0627297	−	0.998031i	\(-0.519981\pi\)
							0.593468	+	0.804858i	\(0.297758\pi\)
\(31\)	−2.67192e6	+	4.62789e6i	−0.519631	+	0.900027i	0.480109	+	0.877209i	\(0.340597\pi\)
							−0.999740	+	0.0228183i	\(0.992736\pi\)
\(37\)	−8.23837e6			−0.722659			−0.361329	−	0.932438i	\(-0.617677\pi\)
							−0.361329	+	0.932438i	\(0.617677\pi\)
\(41\)	−349249.	−	1.98069e6i	−0.0193023	−	0.109469i	0.973634	−	0.228114i	\(-0.0732559\pi\)
							−0.992937	+	0.118646i	\(0.962145\pi\)
\(43\)	−2.50797e7	+	2.10444e7i	−1.11870	+	0.938702i	−0.998538	−	0.0540525i	\(-0.982786\pi\)
							−0.120163	+	0.992754i	\(0.538342\pi\)
\(47\)	1.88864e7	−	6.87408e6i	0.564558	−	0.205482i	−0.0439450	−	0.999034i	\(-0.513993\pi\)
							0.608503	+	0.793552i	\(0.291770\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.10.e.a.4.4	✓	84
19.5	even	9	inner	19.10.e.a.5.4	yes	84
19.9	even	9		361.10.a.i.1.33		42
19.10	odd	18		361.10.a.j.1.10		42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.10.e.a.4.4	✓	84	1.1	even	1	trivial
19.10.e.a.5.4	yes	84	19.5	even	9	inner
361.10.a.i.1.33		42	19.9	even	9
361.10.a.j.1.10		42	19.10	odd	18