Embedded newform 19.3.f.a.13.2

• Label: 19.3.f.a.13.2
• Level: $19$
• Weight: $3$
• Character: 19.13
• 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			13.2
Root			\(-0.728740i\) of defining polynomial
Character	\(\chi\)	\(=\)	19.13
Dual form			19.3.f.a.3.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.468425 + 0.558247i) q^{2} +(-1.14207 + 3.13782i) q^{3} +(0.602375 - 3.41624i) q^{4} +(-1.13111 - 6.41483i) q^{5} +(-2.28665 + 0.832274i) q^{6} +(-5.47943 + 9.49065i) q^{7} +(4.71370 - 2.72146i) q^{8} +(-1.64718 - 1.38215i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} - 6 q^{5} - 36 q^{6} + 6 q^{7} - 9 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.468425	+	0.558247i	0.234212	+	0.279123i	0.870330	−	0.492468i	\(-0.163905\pi\)
							−0.636118	+	0.771592i	\(0.719461\pi\)
\(3\)	−1.14207	+	3.13782i	−0.380691	+	1.04594i	0.590375	+	0.807129i	\(0.298980\pi\)
							−0.971066	+	0.238811i	\(0.923242\pi\)
\(5\)	−1.13111	−	6.41483i	−0.226221	−	1.28297i	−0.860336	−	0.509727i	\(-0.829747\pi\)
							0.634115	−	0.773239i	\(-0.281365\pi\)
\(7\)	−5.47943	+	9.49065i	−0.782776	+	1.35581i	0.147543	+	0.989056i	\(0.452864\pi\)
							−0.930319	+	0.366752i	\(0.880470\pi\)
\(11\)	−2.46116	−	4.26286i	−0.223742	−	0.387533i	0.732199	−	0.681090i	\(-0.238494\pi\)
							−0.955941	+	0.293558i	\(0.905161\pi\)
\(13\)	2.41607	+	6.63809i	0.185851	+	0.510622i	0.997270	−	0.0738434i	\(-0.0235265\pi\)
							−0.811419	+	0.584465i	\(0.801304\pi\)
\(17\)	−4.19721	+	3.52188i	−0.246895	+	0.207169i	−0.757834	−	0.652448i	\(-0.773742\pi\)
							0.510939	+	0.859617i	\(0.329298\pi\)
\(19\)	18.4580	+	4.50568i	0.971475	+	0.237141i
							\(23\)	3.59777	−	20.4040i	0.156425	−	0.887129i	−0.801047	−	0.598602i	\(-0.795723\pi\)
0.957472	−	0.288527i	\(-0.0931656\pi\)
\(29\)	16.3721	−	19.5115i	0.564554	−	0.672809i	−0.405950	−	0.913895i	\(-0.633059\pi\)
							0.970504	+	0.241087i	\(0.0775038\pi\)
\(31\)	−4.26339	−	2.46147i	−0.137529	−	0.0794022i	0.429657	−	0.902992i	\(-0.358634\pi\)
							−0.567186	+	0.823590i	\(0.691968\pi\)
\(37\)		−	25.2454i		−	0.682309i	−0.940007	−	0.341155i	\(-0.889182\pi\)
							0.940007	−	0.341155i	\(-0.110818\pi\)
\(41\)	−24.5133	+	67.3497i	−0.597885	+	1.64268i	0.157595	+	0.987504i	\(0.449626\pi\)
							−0.755480	+	0.655172i	\(0.772596\pi\)
\(43\)	4.59562	+	26.0631i	0.106875	+	0.606118i	0.990455	+	0.137838i	\(0.0440152\pi\)
							−0.883580	+	0.468280i	\(0.844874\pi\)
\(47\)	−35.0584	−	29.4175i	−0.745923	−	0.625904i	0.188498	−	0.982073i	\(-0.439638\pi\)
							−0.934421	+	0.356170i	\(0.884082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.3.f.a.13.2	yes	12
3.2	odd	2		171.3.ba.b.127.1		12
4.3	odd	2		304.3.z.a.241.2		12
19.2	odd	18		361.3.f.f.299.2		12
19.3	odd	18	inner	19.3.f.a.3.2	✓	12
19.4	even	9		361.3.b.c.360.7		12
19.5	even	9		361.3.f.c.116.2		12
19.6	even	9		361.3.d.f.69.4		12
19.7	even	3		361.3.f.e.333.1		12
19.8	odd	6		361.3.f.b.262.1		12
19.9	even	9		361.3.d.d.293.3		12
19.10	odd	18		361.3.d.f.293.4		12
19.11	even	3		361.3.f.f.262.2		12
19.12	odd	6		361.3.f.c.333.2		12
19.13	odd	18		361.3.d.d.69.3		12
19.14	odd	18		361.3.f.e.116.1		12
19.15	odd	18		361.3.b.c.360.6		12
19.16	even	9		361.3.f.g.307.1		12
19.17	even	9		361.3.f.b.299.1		12
19.18	odd	2		361.3.f.g.127.1		12
57.41	even	18		171.3.ba.b.136.1		12
76.3	even	18		304.3.z.a.193.2		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.3.f.a.3.2	✓	12	19.3	odd	18	inner
19.3.f.a.13.2	yes	12	1.1	even	1	trivial
171.3.ba.b.127.1		12	3.2	odd	2
171.3.ba.b.136.1		12	57.41	even	18
304.3.z.a.193.2		12	76.3	even	18
304.3.z.a.241.2		12	4.3	odd	2
361.3.b.c.360.6		12	19.15	odd	18
361.3.b.c.360.7		12	19.4	even	9
361.3.d.d.69.3		12	19.13	odd	18
361.3.d.d.293.3		12	19.9	even	9
361.3.d.f.69.4		12	19.6	even	9
361.3.d.f.293.4		12	19.10	odd	18
361.3.f.b.262.1		12	19.8	odd	6
361.3.f.b.299.1		12	19.17	even	9
361.3.f.c.116.2		12	19.5	even	9
361.3.f.c.333.2		12	19.12	odd	6
361.3.f.e.116.1		12	19.14	odd	18
361.3.f.e.333.1		12	19.7	even	3
361.3.f.f.262.2		12	19.11	even	3
361.3.f.f.299.2		12	19.2	odd	18
361.3.f.g.127.1		12	19.18	odd	2
361.3.f.g.307.1		12	19.16	even	9