Embedded newform 19.4.e.a.4.4

• Label: 19.4.e.a.4.4
• Level: $19$
• Weight: $4$
• Character: 19.4
• Analytic conductor: $1.121$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	19.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.12103629011\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) 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			19.4.e.a.5.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.98397 + 1.08608i) q^{2} +(-0.341646 - 1.93757i) q^{3} +(1.59618 + 1.33936i) q^{4} +(-6.63193 + 5.56485i) q^{5} +(1.08489 - 6.15271i) q^{6} +(-4.36312 + 7.55715i) q^{7} +(-9.39359 - 16.2702i) q^{8} +(21.7342 - 7.91062i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{2} - 3 q^{3} - 24 q^{4} - 6 q^{5} + 42 q^{6} + 3 q^{7} - 75 q^{8} - 51 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.98397	+	1.08608i	1.05499	+	0.383986i	0.810545	−	0.585676i	\(-0.199171\pi\)
							0.244449	+	0.969662i	\(0.421393\pi\)
\(3\)	−0.341646	−	1.93757i	−0.0657497	−	0.372885i	−0.999873	−	0.0159348i	\(-0.994928\pi\)
							0.934123	−	0.356951i	\(-0.116184\pi\)
\(5\)	−6.63193	+	5.56485i	−0.593178	+	0.497735i	−0.889245	−	0.457432i	\(-0.848769\pi\)
							0.296067	+	0.955167i	\(0.404325\pi\)
\(7\)	−4.36312	+	7.55715i	−0.235587	+	0.408048i	−0.959443	−	0.281903i	\(-0.909034\pi\)
							0.723856	+	0.689951i	\(0.242368\pi\)
\(11\)	−3.07408	−	5.32447i	−0.0842609	−	0.145944i	0.820815	−	0.571194i	\(-0.193520\pi\)
							−0.905076	+	0.425250i	\(0.860186\pi\)
\(13\)	−4.94890	+	28.0666i	−0.105583	+	0.598790i	0.885403	+	0.464824i	\(0.153882\pi\)
							−0.990986	+	0.133966i	\(0.957229\pi\)
\(17\)	69.5906	+	25.3289i	0.992836	+	0.361363i	0.786818	−	0.617186i	\(-0.211727\pi\)
							0.206018	+	0.978548i	\(0.433949\pi\)
\(19\)	67.3348	+	48.2186i	0.813034	+	0.582216i
							\(23\)	−132.929	−	111.541i	−1.20512	−	1.01121i	−0.999469	−	0.0325854i	\(-0.989626\pi\)
−0.205646	−	0.978626i	\(-0.565930\pi\)
\(29\)	−276.350	+	100.583i	−1.76955	+	0.644064i	−0.769567	+	0.638566i	\(0.779528\pi\)
							−0.999985	+	0.00549778i	\(0.998250\pi\)
\(31\)	129.528	−	224.349i	0.750448	−	1.29981i	−0.197158	−	0.980372i	\(-0.563171\pi\)
							0.947606	−	0.319442i	\(-0.103495\pi\)
\(37\)	81.0926			0.360312			0.180156	−	0.983638i	\(-0.442340\pi\)
							0.180156	+	0.983638i	\(0.442340\pi\)
\(41\)	−16.2062	−	91.9101i	−0.0617314	−	0.350096i	−0.999991	−	0.00416513i	\(-0.998674\pi\)
							0.938260	−	0.345931i	\(-0.112437\pi\)
\(43\)	109.072	−	91.5224i	0.386822	−	0.324582i	−0.428552	−	0.903517i	\(-0.640976\pi\)
							0.815374	+	0.578935i	\(0.196532\pi\)
\(47\)	−295.906	+	107.701i	−0.918346	+	0.334251i	−0.757580	−	0.652742i	\(-0.773619\pi\)
							−0.160766	+	0.986993i	\(0.551396\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.4.e.a.4.4	✓	24
3.2	odd	2		171.4.u.b.118.1		24
19.5	even	9	inner	19.4.e.a.5.4	yes	24
19.9	even	9		361.4.a.n.1.2		12
19.10	odd	18		361.4.a.m.1.11		12
57.5	odd	18		171.4.u.b.100.1		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.e.a.4.4	✓	24	1.1	even	1	trivial
19.4.e.a.5.4	yes	24	19.5	even	9	inner
171.4.u.b.100.1		24	57.5	odd	18
171.4.u.b.118.1		24	3.2	odd	2
361.4.a.m.1.11		12	19.10	odd	18
361.4.a.n.1.2		12	19.9	even	9