Embedded newform 28.6.f.a.19.2

• Label: 28.6.f.a.19.2
• Level: $28$
• Weight: $6$
• Character: 28.19
• Analytic conductor: $4.491$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	28.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.49074695476\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			19.2
Character	\(\chi\)	\(=\)	28.19
Dual form			28.6.f.a.3.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.50906 - 1.28461i) q^{2} +(13.0818 + 22.6584i) q^{3} +(28.6996 + 14.1540i) q^{4} +(-60.0746 - 34.6841i) q^{5} +(-42.9616 - 141.632i) q^{6} +(33.9161 + 125.127i) q^{7} +(-139.926 - 114.843i) q^{8} +(-220.769 + 382.383i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 12 q^{4} - 6 q^{5} - 72 q^{8} - 1136 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/28\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(17\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.50906	−	1.28461i	−0.973874	−	0.227089i
							\(3\)	13.0818	+	22.6584i	0.839201	+	1.45354i	0.890564	+	0.454858i	\(0.150310\pi\)
−0.0513635	+	0.998680i	\(0.516357\pi\)
\(5\)	−60.0746	−	34.6841i	−1.07465	−	0.620448i	−0.145199	−	0.989402i	\(-0.546382\pi\)
							−0.929447	+	0.368955i	\(0.879716\pi\)
\(7\)	33.9161	+	125.127i	0.261614	+	0.965173i
							\(11\)	−444.727	+	256.763i	−1.10818	+	0.639810i	−0.938358	−	0.345664i	\(-0.887654\pi\)
−0.169826	+	0.985474i	\(0.554320\pi\)
\(13\)			23.3349i			0.0382954i	0.999817	+	0.0191477i	\(0.00609528\pi\)
							−0.999817	+	0.0191477i	\(0.993905\pi\)
\(17\)	−26.5430	+	15.3246i	−0.0222755	+	0.0128608i	−0.511096	−	0.859523i	\(-0.670760\pi\)
							0.488821	+	0.872384i	\(0.337427\pi\)
\(19\)	20.4078	−	35.3473i	0.0129692	−	0.0224632i	−0.859468	−	0.511190i	\(-0.829205\pi\)
							0.872437	+	0.488726i	\(0.162538\pi\)
\(23\)	2131.04	+	1230.35i	0.839985	+	0.484965i	0.857259	−	0.514885i	\(-0.172165\pi\)
							−0.0172742	+	0.999851i	\(0.505499\pi\)
\(29\)	6823.51			1.50665			0.753326	−	0.657647i	\(-0.228448\pi\)
							0.753326	+	0.657647i	\(0.228448\pi\)
\(31\)	3004.28	+	5203.57i	0.561483	+	0.972517i	0.997367	+	0.0725145i	\(0.0231024\pi\)
							−0.435884	+	0.900003i	\(0.643564\pi\)
\(37\)	5623.56	−	9740.29i	0.675316	−	1.16968i	−0.301060	−	0.953605i	\(-0.597341\pi\)
							0.976376	−	0.216077i	\(-0.0693261\pi\)
\(41\)			10626.3i			0.987235i	0.869679	+	0.493618i	\(0.164326\pi\)
							−0.869679	+	0.493618i	\(0.835674\pi\)
\(43\)			7944.77i			0.655254i	0.944807	+	0.327627i	\(0.106249\pi\)
							−0.944807	+	0.327627i	\(0.893751\pi\)
\(47\)	−3922.20	+	6793.44i	−0.258991	+	0.448586i	−0.965972	−	0.258647i	\(-0.916723\pi\)
							0.706981	+	0.707233i	\(0.250057\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.6.f.a.19.2	yes	36
4.3	odd	2	inner	28.6.f.a.19.14	yes	36
7.2	even	3		196.6.d.b.195.23		36
7.3	odd	6	inner	28.6.f.a.3.14	yes	36
7.5	odd	6		196.6.d.b.195.24		36
28.3	even	6	inner	28.6.f.a.3.2	✓	36
28.19	even	6		196.6.d.b.195.21		36
28.23	odd	6		196.6.d.b.195.22		36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.f.a.3.2	✓	36	28.3	even	6	inner
28.6.f.a.3.14	yes	36	7.3	odd	6	inner
28.6.f.a.19.2	yes	36	1.1	even	1	trivial
28.6.f.a.19.14	yes	36	4.3	odd	2	inner
196.6.d.b.195.21		36	28.19	even	6
196.6.d.b.195.22		36	28.23	odd	6
196.6.d.b.195.23		36	7.2	even	3
196.6.d.b.195.24		36	7.5	odd	6