Embedded newform 19.3.f.a.3.1

• Label: 19.3.f.a.3.1
• Level: $19$
• Weight: $3$
• Character: 19.3
• Analytic conductor: $0.518$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	19.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.517712502285\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{18})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial:	\( x^{12} + 24x^{10} + 216x^{8} + 905x^{6} + 1770x^{4} + 1395x^{2} + 361 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.1
Root			\(-2.01431i\) of defining polynomial
Character	\(\chi\)	\(=\)	19.3
Dual form			19.3.f.a.13.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29478 + 1.54305i) q^{2} +(0.621128 + 1.70654i) q^{3} +(-0.00997859 - 0.0565914i) q^{4} +(1.24445 - 7.05761i) q^{5} +(-3.43750 - 1.25115i) q^{6} +(0.422527 + 0.731838i) q^{7} +(-6.87755 - 3.97075i) q^{8} +(4.36793 - 3.66513i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} - 6 q^{5} - 36 q^{6} + 6 q^{7} - 9 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29478	+	1.54305i	−0.647388	+	0.771527i	−0.985518	−	0.169572i	\(-0.945761\pi\)
							0.338129	+	0.941100i	\(0.390206\pi\)
\(3\)	0.621128	+	1.70654i	0.207043	+	0.568845i	0.999136	−	0.0415544i	\(-0.0132310\pi\)
							−0.792093	+	0.610400i	\(0.791009\pi\)
\(5\)	1.24445	−	7.05761i	0.248890	−	1.41152i	−0.562394	−	0.826870i	\(-0.690119\pi\)
							0.811283	−	0.584653i	\(-0.198769\pi\)
\(7\)	0.422527	+	0.731838i	0.0603610	+	0.104548i	0.894627	−	0.446814i	\(-0.147441\pi\)
							−0.834266	+	0.551362i	\(0.814108\pi\)
\(11\)	−3.11282	+	5.39155i	−0.282983	+	0.490141i	−0.972118	−	0.234492i	\(-0.924657\pi\)
							0.689135	+	0.724633i	\(0.257991\pi\)
\(13\)	−5.42246	+	14.8981i	−0.417112	+	1.14601i	0.536218	+	0.844079i	\(0.319852\pi\)
							−0.953331	+	0.301928i	\(0.902370\pi\)
\(17\)	−19.3149	−	16.2071i	−1.13617	−	0.953359i	−0.136862	−	0.990590i	\(-0.543702\pi\)
							−0.999307	+	0.0372316i	\(0.988146\pi\)
\(19\)	−3.87853	−	18.5999i	−0.204133	−	0.978943i
							\(23\)	6.55112	+	37.1532i	0.284831	+	1.61536i	0.705888	+	0.708324i	\(0.250548\pi\)
−0.421056	+	0.907035i	\(0.638341\pi\)
\(29\)	12.5198	+	14.9205i	0.431718	+	0.514501i	0.937417	−	0.348208i	\(-0.113210\pi\)
							−0.505699	+	0.862710i	\(0.668765\pi\)
\(31\)	8.08793	−	4.66957i	0.260901	−	0.150631i	−0.363845	−	0.931460i	\(-0.618536\pi\)
							0.624745	+	0.780828i	\(0.285203\pi\)
\(37\)			1.20580i			0.0325891i	0.999867	+	0.0162945i	\(0.00518694\pi\)
							−0.999867	+	0.0162945i	\(0.994813\pi\)
\(41\)	−4.05309	−	11.1358i	−0.0988558	−	0.271604i	0.880400	−	0.474232i	\(-0.157274\pi\)
							−0.979256	+	0.202628i	\(0.935052\pi\)
\(43\)	−4.58077	+	25.9788i	−0.106530	+	0.604159i	0.884069	+	0.467357i	\(0.154794\pi\)
							−0.990598	+	0.136802i	\(0.956317\pi\)
\(47\)	20.0697	−	16.8405i	0.427014	−	0.358308i	−0.403809	−	0.914843i	\(-0.632314\pi\)
							0.830824	+	0.556535i	\(0.187870\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.3.f.a.3.1	✓	12
3.2	odd	2		171.3.ba.b.136.2		12
4.3	odd	2		304.3.z.a.193.1		12
19.2	odd	18		361.3.d.f.69.2		12
19.3	odd	18		361.3.d.d.293.5		12
19.4	even	9		361.3.f.c.333.1		12
19.5	even	9		361.3.b.c.360.10		12
19.6	even	9		361.3.f.g.127.2		12
19.7	even	3		361.3.f.f.299.1		12
19.8	odd	6		361.3.f.c.116.1		12
19.9	even	9		361.3.f.b.262.2		12
19.10	odd	18		361.3.f.f.262.1		12
19.11	even	3		361.3.f.e.116.2		12
19.12	odd	6		361.3.f.b.299.2		12
19.13	odd	18	inner	19.3.f.a.13.1	yes	12
19.14	odd	18		361.3.b.c.360.3		12
19.15	odd	18		361.3.f.e.333.2		12
19.16	even	9		361.3.d.f.293.2		12
19.17	even	9		361.3.d.d.69.5		12
19.18	odd	2		361.3.f.g.307.2		12
57.32	even	18		171.3.ba.b.127.2		12
76.51	even	18		304.3.z.a.241.1		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.3.f.a.3.1	✓	12	1.1	even	1	trivial
19.3.f.a.13.1	yes	12	19.13	odd	18	inner
171.3.ba.b.127.2		12	57.32	even	18
171.3.ba.b.136.2		12	3.2	odd	2
304.3.z.a.193.1		12	4.3	odd	2
304.3.z.a.241.1		12	76.51	even	18
361.3.b.c.360.3		12	19.14	odd	18
361.3.b.c.360.10		12	19.5	even	9
361.3.d.d.69.5		12	19.17	even	9
361.3.d.d.293.5		12	19.3	odd	18
361.3.d.f.69.2		12	19.2	odd	18
361.3.d.f.293.2		12	19.16	even	9
361.3.f.b.262.2		12	19.9	even	9
361.3.f.b.299.2		12	19.12	odd	6
361.3.f.c.116.1		12	19.8	odd	6
361.3.f.c.333.1		12	19.4	even	9
361.3.f.e.116.2		12	19.11	even	3
361.3.f.e.333.2		12	19.15	odd	18
361.3.f.f.262.1		12	19.10	odd	18
361.3.f.f.299.1		12	19.7	even	3
361.3.f.g.127.2		12	19.6	even	9
361.3.f.g.307.2		12	19.18	odd	2