Embedded newform 21.6.e.c.16.3

• Label: 21.6.e.c.16.3
• Level: $21$
• Weight: $6$
• Character: 21.16
• Analytic conductor: $3.368$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	21.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.36806021607\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial:	\( x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3\cdot 7^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			16.3
Root			\(0.895402 - 1.55088i\) of defining polynomial
Character	\(\chi\)	\(=\)	21.16
Dual form			21.6.e.c.4.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.395402 - 0.684857i) q^{2} +(-4.50000 - 7.79423i) q^{3} +(15.6873 + 27.1712i) q^{4} +(52.0958 - 90.2327i) q^{5} -7.11724 q^{6} +(-7.12980 - 129.446i) q^{7} +50.1170 q^{8} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} - 36 q^{3} - 69 q^{4} + 54 q^{6} + 258 q^{7} + 246 q^{8} - 324 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/21\mathbb{Z}\right)^\times\).

\(n\)	\(8\)	\(10\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.395402	−	0.684857i	0.0698979	−	0.121067i	−0.828958	−	0.559310i	\(-0.811066\pi\)
							0.898856	+	0.438244i	\(0.144399\pi\)
\(3\)	−4.50000	−	7.79423i	−0.288675	−	0.500000i
							\(5\)	52.0958	−	90.2327i	0.931919	−	1.61413i	0.151881	−	0.988399i	\(-0.451467\pi\)
0.780038	−	0.625732i	\(-0.215200\pi\)
\(7\)	−7.12980	−	129.446i	−0.0549961	−	0.998487i
							\(11\)	248.830	+	430.986i	0.620041	+	1.07394i	0.989477	+	0.144687i	\(0.0462176\pi\)
−0.369436	+	0.929256i	\(0.620449\pi\)
\(13\)	−206.551			−0.338977			−0.169488	−	0.985532i	\(-0.554211\pi\)
							−0.169488	+	0.985532i	\(0.554211\pi\)
\(17\)	−31.5793	−	54.6969i	−0.0265021	−	0.0459030i	0.852470	−	0.522776i	\(-0.175104\pi\)
							−0.878972	+	0.476873i	\(0.841770\pi\)
\(19\)	−661.977	+	1146.58i	−0.420687	+	0.728651i	−0.996007	−	0.0892772i	\(-0.971544\pi\)
							0.575320	+	0.817929i	\(0.304878\pi\)
\(23\)	97.2187	−	168.388i	0.0383204	−	0.0663729i	−0.846229	−	0.532819i	\(-0.821133\pi\)
							0.884550	+	0.466446i	\(0.154466\pi\)
\(29\)	4323.14			0.954562			0.477281	−	0.878751i	\(-0.341622\pi\)
							0.477281	+	0.878751i	\(0.341622\pi\)
\(31\)	3762.66	+	6517.11i	0.703219	+	1.21801i	0.967331	+	0.253518i	\(0.0815879\pi\)
							−0.264112	+	0.964492i	\(0.585079\pi\)
\(37\)	−5177.82	+	8968.25i	−0.621789	+	1.07697i	0.367364	+	0.930077i	\(0.380260\pi\)
							−0.989152	+	0.146892i	\(0.953073\pi\)
\(41\)	−4180.92			−0.388429			−0.194215	−	0.980959i	\(-0.562216\pi\)
							−0.194215	+	0.980959i	\(0.562216\pi\)
\(43\)	5960.87			0.491630			0.245815	−	0.969317i	\(-0.420944\pi\)
							0.245815	+	0.969317i	\(0.420944\pi\)
\(47\)	2194.87	−	3801.62i	0.144932	−	0.251029i	−0.784416	−	0.620236i	\(-0.787037\pi\)
							0.929348	+	0.369206i	\(0.120370\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.6.e.c.16.3	yes	8
3.2	odd	2		63.6.e.e.37.2		8
4.3	odd	2		336.6.q.j.289.4		8
7.2	even	3		147.6.a.m.1.2		4
7.3	odd	6		147.6.e.o.67.3		8
7.4	even	3	inner	21.6.e.c.4.3	✓	8
7.5	odd	6		147.6.a.l.1.2		4
7.6	odd	2		147.6.e.o.79.3		8
21.2	odd	6		441.6.a.w.1.3		4
21.5	even	6		441.6.a.v.1.3		4
21.11	odd	6		63.6.e.e.46.2		8
28.11	odd	6		336.6.q.j.193.4		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.e.c.4.3	✓	8	7.4	even	3	inner
21.6.e.c.16.3	yes	8	1.1	even	1	trivial
63.6.e.e.37.2		8	3.2	odd	2
63.6.e.e.46.2		8	21.11	odd	6
147.6.a.l.1.2		4	7.5	odd	6
147.6.a.m.1.2		4	7.2	even	3
147.6.e.o.67.3		8	7.3	odd	6
147.6.e.o.79.3		8	7.6	odd	2
336.6.q.j.193.4		8	28.11	odd	6
336.6.q.j.289.4		8	4.3	odd	2
441.6.a.v.1.3		4	21.5	even	6
441.6.a.w.1.3		4	21.2	odd	6