Embedded newform 28.6.f.a.3.14

• Label: 28.6.f.a.3.14
• Level: $28$
• Weight: $6$
• Character: 28.3
• Analytic conductor: $4.491$
• Analytic rank: $0$
• Dimension: $36$
• 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			3.14
Character	\(\chi\)	\(=\)	28.3
Dual form			28.6.f.a.19.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.86703 - 4.12869i) q^{2} +(-13.0818 + 22.6584i) q^{3} +(-2.09210 - 31.9315i) q^{4} +(-60.0746 + 34.6841i) q^{5} +(42.9616 + 141.632i) q^{6} +(-33.9161 + 125.127i) q^{7} +(-139.926 - 114.843i) q^{8} +(-220.769 - 382.383i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 12 q^{4} - 6 q^{5} - 72 q^{8} - 1136 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.86703	−	4.12869i	0.683601	−	0.729855i
							\(3\)	−13.0818	+	22.6584i	−0.839201	+	1.45354i	0.0513635	+	0.998680i	\(0.483643\pi\)
−0.890564	+	0.454858i	\(0.849690\pi\)
\(5\)	−60.0746	+	34.6841i	−1.07465	+	0.620448i	−0.929447	−	0.368955i	\(-0.879716\pi\)
							−0.145199	+	0.989402i	\(0.546382\pi\)
\(7\)	−33.9161	+	125.127i	−0.261614	+	0.965173i
							\(11\)	444.727	+	256.763i	1.10818	+	0.639810i	0.938358	−	0.345664i	\(-0.112346\pi\)
0.169826	+	0.985474i	\(0.445680\pi\)
\(13\)		−	23.3349i		−	0.0382954i	−0.999817	−	0.0191477i	\(-0.993905\pi\)
							0.999817	−	0.0191477i	\(-0.00609528\pi\)
\(17\)	−26.5430	−	15.3246i	−0.0222755	−	0.0128608i	0.488821	−	0.872384i	\(-0.337427\pi\)
							−0.511096	+	0.859523i	\(0.670760\pi\)
\(19\)	−20.4078	−	35.3473i	−0.0129692	−	0.0224632i	0.859468	−	0.511190i	\(-0.170795\pi\)
							−0.872437	+	0.488726i	\(0.837462\pi\)
\(23\)	−2131.04	+	1230.35i	−0.839985	+	0.484965i	−0.857259	−	0.514885i	\(-0.827835\pi\)
							0.0172742	+	0.999851i	\(0.494501\pi\)
\(29\)	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	0.435884	+	0.900003i	\(0.356436\pi\)
							−0.997367	+	0.0725145i	\(0.976898\pi\)
\(37\)	5623.56	+	9740.29i	0.675316	+	1.16968i	0.976376	+	0.216077i	\(0.0693261\pi\)
							−0.301060	+	0.953605i	\(0.597341\pi\)
\(41\)		−	10626.3i		−	0.987235i	−0.869679	−	0.493618i	\(-0.835674\pi\)
							0.869679	−	0.493618i	\(-0.164326\pi\)
\(43\)			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	\(-0.0832767\pi\)
							−0.706981	+	0.707233i	\(0.749943\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.f.a.3.2	✓	36	4.3	odd	2	inner
28.6.f.a.3.14	yes	36	1.1	even	1	trivial
28.6.f.a.19.2	yes	36	7.5	odd	6	inner
28.6.f.a.19.14	yes	36	28.19	even	6	inner
196.6.d.b.195.21		36	28.11	odd	6
196.6.d.b.195.22		36	28.3	even	6
196.6.d.b.195.23		36	7.3	odd	6
196.6.d.b.195.24		36	7.4	even	3