Embedded newform 19.8.e.a.4.1

• Label: 19.8.e.a.4.1
• Level: $19$
• Weight: $8$
• Character: 19.4
• Analytic conductor: $5.935$
• Analytic rank: $0$
• Dimension: $66$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	19.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.93531548420\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			4.1
Character	\(\chi\)	\(=\)	19.4
Dual form			19.8.e.a.5.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-20.4728 - 7.45148i) q^{2} +(-4.63945 - 26.3116i) q^{3} +(265.556 + 222.828i) q^{4} +(384.427 - 322.572i) q^{5} +(-101.078 + 573.243i) q^{6} +(-238.816 + 413.641i) q^{7} +(-2381.93 - 4125.62i) q^{8} +(1384.33 - 503.855i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q - 6 q^{2} + 33 q^{3} + 72 q^{4} - 6 q^{5} - 876 q^{6} + 588 q^{7} - 4611 q^{8} + 5025 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−20.4728	−	7.45148i	−1.80956	−	0.658624i	−0.997143	−	0.0755408i	\(-0.975932\pi\)
							−0.812412	−	0.583083i	\(-0.801846\pi\)
\(3\)	−4.63945	−	26.3116i	−0.0992069	−	0.562630i	−0.993377	−	0.114902i	\(-0.963344\pi\)
							0.894170	−	0.447728i	\(-0.147767\pi\)
\(5\)	384.427	−	322.572i	1.37537	−	1.15407i	0.404476	−	0.914549i	\(-0.367454\pi\)
							0.970891	−	0.239521i	\(-0.0769905\pi\)
\(7\)	−238.816	+	413.641i	−0.263160	+	0.455806i	−0.967080	−	0.254473i	\(-0.918098\pi\)
							0.703920	+	0.710279i	\(0.251431\pi\)
\(11\)	48.1985	+	83.4822i	0.0109184	+	0.0189112i	0.871433	−	0.490515i	\(-0.163191\pi\)
							−0.860515	+	0.509426i	\(0.829858\pi\)
\(13\)	1328.14	−	7532.25i	0.167665	−	0.950874i	−0.778609	−	0.627509i	\(-0.784075\pi\)
							0.946274	−	0.323365i	\(-0.104814\pi\)
\(17\)	2361.69	+	859.586i	0.116588	+	0.0424344i	0.399655	−	0.916666i	\(-0.369130\pi\)
							−0.283068	+	0.959100i	\(0.591352\pi\)
\(19\)	−29839.3	−	1868.00i	−0.998046	−	0.0624796i
							\(23\)	−29067.3	−	24390.4i	−0.498147	−	0.417995i	0.358789	−	0.933419i	\(-0.383190\pi\)
−0.856935	+	0.515424i	\(0.827634\pi\)
\(29\)	47958.4	−	17455.4i	0.365150	−	0.132904i	−0.152927	−	0.988237i	\(-0.548870\pi\)
							0.518078	+	0.855334i	\(0.326648\pi\)
\(31\)	−30374.3	+	52609.8i	−0.183122	+	0.317176i	−0.942942	−	0.332957i	\(-0.891954\pi\)
							0.759820	+	0.650133i	\(0.225287\pi\)
\(37\)	41889.9			0.135958			0.0679788	−	0.997687i	\(-0.478345\pi\)
							0.0679788	+	0.997687i	\(0.478345\pi\)
\(41\)	44349.3	+	251518.i	0.100495	+	0.569934i	0.992924	+	0.118748i	\(0.0378880\pi\)
							−0.892430	+	0.451187i	\(0.851001\pi\)
\(43\)	−65611.8	+	55054.9i	−0.125847	+	0.105598i	−0.703539	−	0.710657i	\(-0.748398\pi\)
							0.577692	+	0.816255i	\(0.303953\pi\)
\(47\)	756311.	−	275275.i	1.06257	−	0.386744i	0.249177	−	0.968458i	\(-0.419840\pi\)
							0.813394	+	0.581714i	\(0.197618\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.8.e.a.4.1	✓	66
19.5	even	9	inner	19.8.e.a.5.1	yes	66
19.9	even	9		361.8.a.j.1.32		33
19.10	odd	18		361.8.a.i.1.2		33

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.8.e.a.4.1	✓	66	1.1	even	1	trivial
19.8.e.a.5.1	yes	66	19.5	even	9	inner
361.8.a.i.1.2		33	19.10	odd	18
361.8.a.j.1.32		33	19.9	even	9