Embedded newform 19.4.e.a.6.1

• Label: 19.4.e.a.6.1
• Level: $19$
• Weight: $4$
• Character: 19.6
• 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			6.1
Character	\(\chi\)	\(=\)	19.6
Dual form			19.4.e.a.16.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.26193 + 1.89799i) q^{2} +(-8.82799 - 3.21312i) q^{3} +(0.124797 - 0.707761i) q^{4} +(-0.0925462 - 0.524856i) q^{5} +(26.0668 - 9.48752i) q^{6} +(-11.8949 + 20.6025i) q^{7} +(-10.7499 - 18.6194i) q^{8} +(46.9260 + 39.3756i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{2} - 3 q^{3} - 24 q^{4} - 6 q^{5} + 42 q^{6} + 3 q^{7} - 75 q^{8} - 51 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.26193	+	1.89799i	−0.799713	+	0.671039i	−0.948129	−	0.317886i	\(-0.897027\pi\)
							0.148416	+	0.988925i	\(0.452583\pi\)
\(3\)	−8.82799	−	3.21312i	−1.69895	−	0.618366i	−0.703242	−	0.710950i	\(-0.748265\pi\)
							−0.995705	+	0.0925844i	\(0.970487\pi\)
\(5\)	−0.0925462	−	0.524856i	−0.00827759	−	0.0469445i	0.980389	−	0.197071i	\(-0.0631430\pi\)
							−0.988667	+	0.150127i	\(0.952032\pi\)
\(7\)	−11.8949	+	20.6025i	−0.642263	+	1.11243i	0.342664	+	0.939458i	\(0.388671\pi\)
							−0.984926	+	0.172974i	\(0.944662\pi\)
\(11\)	−18.5849	−	32.1900i	−0.509414	−	0.882331i	−0.999941	−	0.0109047i	\(-0.996529\pi\)
							0.490527	−	0.871426i	\(-0.336804\pi\)
\(13\)	−14.9925	+	5.45682i	−0.319859	+	0.116419i	−0.496960	−	0.867774i	\(-0.665550\pi\)
							0.177101	+	0.984193i	\(0.443328\pi\)
\(17\)	−47.7006	+	40.0256i	−0.680535	+	0.571037i	−0.916163	−	0.400807i	\(-0.868730\pi\)
							0.235627	+	0.971843i	\(0.424285\pi\)
\(19\)	−57.8848	+	59.2314i	−0.698930	+	0.715190i
							\(23\)	−3.18128	+	18.0420i	−0.0288410	+	0.163566i	−0.995827	−	0.0912656i	\(-0.970909\pi\)
0.966986	+	0.254831i	\(0.0820199\pi\)
\(29\)	13.6165	+	11.4256i	0.0871906	+	0.0731616i	0.685341	−	0.728222i	\(-0.259653\pi\)
							−0.598151	+	0.801384i	\(0.704098\pi\)
\(31\)	−76.5514	+	132.591i	−0.443517	+	0.768195i	−0.997948	−	0.0640356i	\(-0.979603\pi\)
							0.554430	+	0.832230i	\(0.312936\pi\)
\(37\)	−41.5560			−0.184642			−0.0923212	−	0.995729i	\(-0.529429\pi\)
							−0.0923212	+	0.995729i	\(0.529429\pi\)
\(41\)	−314.270	−	114.385i	−1.19709	−	0.435706i	−0.334883	−	0.942260i	\(-0.608697\pi\)
							−0.862208	+	0.506554i	\(0.830919\pi\)
\(43\)	−53.2608	−	302.057i	−0.188888	−	1.07124i	−0.920857	−	0.389901i	\(-0.872509\pi\)
							0.731969	−	0.681338i	\(-0.238602\pi\)
\(47\)	37.2512	+	31.2575i	0.115610	+	0.0970080i	0.698760	−	0.715356i	\(-0.253736\pi\)
							−0.583150	+	0.812364i	\(0.698180\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.4.e.a.6.1	✓	24
3.2	odd	2		171.4.u.b.82.4		24
19.4	even	9		361.4.a.n.1.3		12
19.15	odd	18		361.4.a.m.1.10		12
19.16	even	9	inner	19.4.e.a.16.1	yes	24
57.35	odd	18		171.4.u.b.73.4		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.e.a.6.1	✓	24	1.1	even	1	trivial
19.4.e.a.16.1	yes	24	19.16	even	9	inner
171.4.u.b.73.4		24	57.35	odd	18
171.4.u.b.82.4		24	3.2	odd	2
361.4.a.m.1.10		12	19.15	odd	18
361.4.a.n.1.3		12	19.4	even	9