Embedded newform 19.3.f.a.15.2

• Label: 19.3.f.a.15.2
• Level: $19$
• Weight: $3$
• Character: 19.15
• 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			15.2
Root			\(-1.89323i\) of defining polynomial
Character	\(\chi\)	\(=\)	19.15
Dual form			19.3.f.a.14.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.647524 - 1.77906i) q^{2} +(-1.91357 + 0.337414i) q^{3} +(0.318417 + 0.267183i) q^{4} +(-2.13959 + 1.79533i) q^{5} +(-0.638803 + 3.62283i) q^{6} +(-1.20796 + 2.09224i) q^{7} +(7.23987 - 4.17994i) q^{8} +(-4.90934 + 1.78685i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 6 q^{2} - 6 q^{5} - 36 q^{6} + 6 q^{7} - 9 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.647524	−	1.77906i	0.323762	−	0.889529i	−0.665891	−	0.746049i	\(-0.731948\pi\)
							0.989653	−	0.143480i	\(-0.0458293\pi\)
\(3\)	−1.91357	+	0.337414i	−0.637856	+	0.112471i	−0.483218	−	0.875500i	\(-0.660532\pi\)
							−0.154638	+	0.987971i	\(0.549421\pi\)
\(5\)	−2.13959	+	1.79533i	−0.427918	+	0.359066i	−0.831165	−	0.556025i	\(-0.812326\pi\)
							0.403248	+	0.915091i	\(0.367881\pi\)
\(7\)	−1.20796	+	2.09224i	−0.172565	+	0.298892i	−0.939316	−	0.343053i	\(-0.888539\pi\)
							0.766751	+	0.641945i	\(0.221872\pi\)
\(11\)	−9.60360	−	16.6339i	−0.873055	−	1.51218i	−0.858821	−	0.512277i	\(-0.828802\pi\)
							−0.0142343	−	0.999899i	\(-0.504531\pi\)
\(13\)	14.2961	+	2.52079i	1.09970	+	0.193907i	0.693911	−	0.720061i	\(-0.255886\pi\)
							0.405788	+	0.913967i	\(0.366997\pi\)
\(17\)	16.1226	+	5.86814i	0.948387	+	0.345184i	0.769472	−	0.638680i	\(-0.220519\pi\)
							0.178914	+	0.983865i	\(0.442742\pi\)
\(19\)	−13.2208	+	13.6459i	−0.695834	+	0.718203i
							\(23\)	−5.84413	−	4.90381i	−0.254092	−	0.213209i	0.506840	−	0.862040i	\(-0.330814\pi\)
−0.760932	+	0.648831i	\(0.775258\pi\)
\(29\)	2.81743	+	7.74083i	0.0971528	+	0.266925i	0.978743	−	0.205090i	\(-0.0657489\pi\)
							−0.881590	+	0.472015i	\(0.843527\pi\)
\(31\)	5.19837	+	3.00128i	0.167689	+	0.0968155i	0.581496	−	0.813549i	\(-0.302468\pi\)
							−0.413806	+	0.910365i	\(0.635801\pi\)
\(37\)		−	59.5153i		−	1.60852i	−0.594277	−	0.804260i	\(-0.702562\pi\)
							0.594277	−	0.804260i	\(-0.297438\pi\)
\(41\)	−30.4205	+	5.36396i	−0.741964	+	0.130828i	−0.531838	−	0.846846i	\(-0.678498\pi\)
							−0.210126	+	0.977674i	\(0.567387\pi\)
\(43\)	18.8243	−	15.7955i	0.437775	−	0.367337i	−0.397101	−	0.917775i	\(-0.629984\pi\)
							0.834876	+	0.550438i	\(0.185539\pi\)
\(47\)	−68.7602	+	25.0267i	−1.46298	+	0.532483i	−0.946186	−	0.323625i	\(-0.895099\pi\)
							−0.516798	+	0.856107i	\(0.672876\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	19.3.f.a.15.2	yes	12
3.2	odd	2		171.3.ba.b.91.1		12
4.3	odd	2		304.3.z.a.129.2		12
19.2	odd	18		361.3.f.e.307.2		12
19.3	odd	18		361.3.f.f.116.1		12
19.4	even	9		361.3.d.f.69.5		12
19.5	even	9		361.3.f.g.299.1		12
19.6	even	9		361.3.d.d.293.2		12
19.7	even	3		361.3.f.e.127.2		12
19.8	odd	6		361.3.f.b.333.2		12
19.9	even	9		361.3.b.c.360.9		12
19.10	odd	18		361.3.b.c.360.4		12
19.11	even	3		361.3.f.f.333.1		12
19.12	odd	6		361.3.f.c.127.1		12
19.13	odd	18		361.3.d.f.293.5		12
19.14	odd	18	inner	19.3.f.a.14.2	✓	12
19.15	odd	18		361.3.d.d.69.2		12
19.16	even	9		361.3.f.b.116.2		12
19.17	even	9		361.3.f.c.307.1		12
19.18	odd	2		361.3.f.g.262.1		12
57.14	even	18		171.3.ba.b.109.1		12
76.71	even	18		304.3.z.a.33.2		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.3.f.a.14.2	✓	12	19.14	odd	18	inner
19.3.f.a.15.2	yes	12	1.1	even	1	trivial
171.3.ba.b.91.1		12	3.2	odd	2
171.3.ba.b.109.1		12	57.14	even	18
304.3.z.a.33.2		12	76.71	even	18
304.3.z.a.129.2		12	4.3	odd	2
361.3.b.c.360.4		12	19.10	odd	18
361.3.b.c.360.9		12	19.9	even	9
361.3.d.d.69.2		12	19.15	odd	18
361.3.d.d.293.2		12	19.6	even	9
361.3.d.f.69.5		12	19.4	even	9
361.3.d.f.293.5		12	19.13	odd	18
361.3.f.b.116.2		12	19.16	even	9
361.3.f.b.333.2		12	19.8	odd	6
361.3.f.c.127.1		12	19.12	odd	6
361.3.f.c.307.1		12	19.17	even	9
361.3.f.e.127.2		12	19.7	even	3
361.3.f.e.307.2		12	19.2	odd	18
361.3.f.f.116.1		12	19.3	odd	18
361.3.f.f.333.1		12	19.11	even	3
361.3.f.g.262.1		12	19.18	odd	2
361.3.f.g.299.1		12	19.5	even	9