Newform orbit 21.2.e.a

• Label: 21.2.e.a
• Level: $21$
• Weight: $2$
• Character orbit: 21.e
• Analytic conductor: $0.168$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.167685844245\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (2 \zeta_{6} - 2) q^{2} - \zeta_{6} q^{3} - 2 \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 2) q^{5} + 2 q^{6} + (\zeta_{6} - 3) q^{7} + (\zeta_{6} - 1) q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - q^{3} - 2 q^{4} + 2 q^{5} + 4 q^{6} - 5 q^{7} - q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(8\)	\(10\)
\(\chi(n)\)	\(1\)	\(-1 + \zeta_{6}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

4.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−1.00000

−

1.73205i

−0.500000

+

0.866025i

−1.00000

+

1.73205i

1.00000

+

1.73205i

2.00000

−2.50000

−

0.866025i

0

−0.500000

−

0.866025i

2.00000

−

3.46410i

16.1

−1.00000

+

1.73205i

−0.500000

−

0.866025i

−1.00000

−

1.73205i

1.00000

−

1.73205i

2.00000

−2.50000

+

0.866025i

0

−0.500000

+

0.866025i

2.00000

+

3.46410i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	21.2.e.a	✓	2
3.b	odd	2	1		63.2.e.b		2
4.b	odd	2	1		336.2.q.f		2
5.b	even	2	1		525.2.i.e		2
5.c	odd	4	2		525.2.r.e		4
7.b	odd	2	1		147.2.e.a		2
7.c	even	3	1	inner	21.2.e.a	✓	2
7.c	even	3	1		147.2.a.c		1
7.d	odd	6	1		147.2.a.b		1
7.d	odd	6	1		147.2.e.a		2
8.b	even	2	1		1344.2.q.m		2
8.d	odd	2	1		1344.2.q.c		2
9.c	even	3	1		567.2.g.a		2
9.c	even	3	1		567.2.h.f		2
9.d	odd	6	1		567.2.g.f		2
9.d	odd	6	1		567.2.h.a		2
12.b	even	2	1		1008.2.s.d		2
21.c	even	2	1		441.2.e.e		2
21.g	even	6	1		441.2.a.a		1
21.g	even	6	1		441.2.e.e		2
21.h	odd	6	1		63.2.e.b		2
21.h	odd	6	1		441.2.a.b		1
28.d	even	2	1		2352.2.q.c		2
28.f	even	6	1		2352.2.a.w		1
28.f	even	6	1		2352.2.q.c		2
28.g	odd	6	1		336.2.q.f		2
28.g	odd	6	1		2352.2.a.d		1
35.i	odd	6	1		3675.2.a.c		1
35.j	even	6	1		525.2.i.e		2
35.j	even	6	1		3675.2.a.a		1
35.l	odd	12	2		525.2.r.e		4
56.j	odd	6	1		9408.2.a.bz		1
56.k	odd	6	1		1344.2.q.c		2
56.k	odd	6	1		9408.2.a.cv		1
56.m	even	6	1		9408.2.a.k		1
56.p	even	6	1		1344.2.q.m		2
56.p	even	6	1		9408.2.a.bg		1
63.g	even	3	1		567.2.h.f		2
63.h	even	3	1		567.2.g.a		2
63.j	odd	6	1		567.2.g.f		2
63.n	odd	6	1		567.2.h.a		2
84.j	odd	6	1		7056.2.a.m		1
84.n	even	6	1		1008.2.s.d		2
84.n	even	6	1		7056.2.a.bp		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.2.e.a	✓	2	1.a	even	1	1	trivial
21.2.e.a	✓	2	7.c	even	3	1	inner
63.2.e.b		2	3.b	odd	2	1
63.2.e.b		2	21.h	odd	6	1
147.2.a.b		1	7.d	odd	6	1
147.2.a.c		1	7.c	even	3	1
147.2.e.a		2	7.b	odd	2	1
147.2.e.a		2	7.d	odd	6	1
336.2.q.f		2	4.b	odd	2	1
336.2.q.f		2	28.g	odd	6	1
441.2.a.a		1	21.g	even	6	1
441.2.a.b		1	21.h	odd	6	1
441.2.e.e		2	21.c	even	2	1
441.2.e.e		2	21.g	even	6	1
525.2.i.e		2	5.b	even	2	1
525.2.i.e		2	35.j	even	6	1
525.2.r.e		4	5.c	odd	4	2
525.2.r.e		4	35.l	odd	12	2
567.2.g.a		2	9.c	even	3	1
567.2.g.a		2	63.h	even	3	1
567.2.g.f		2	9.d	odd	6	1
567.2.g.f		2	63.j	odd	6	1
567.2.h.a		2	9.d	odd	6	1
567.2.h.a		2	63.n	odd	6	1
567.2.h.f		2	9.c	even	3	1
567.2.h.f		2	63.g	even	3	1
1008.2.s.d		2	12.b	even	2	1
1008.2.s.d		2	84.n	even	6	1
1344.2.q.c		2	8.d	odd	2	1
1344.2.q.c		2	56.k	odd	6	1
1344.2.q.m		2	8.b	even	2	1
1344.2.q.m		2	56.p	even	6	1
2352.2.a.d		1	28.g	odd	6	1
2352.2.a.w		1	28.f	even	6	1
2352.2.q.c		2	28.d	even	2	1
2352.2.q.c		2	28.f	even	6	1
3675.2.a.a		1	35.j	even	6	1
3675.2.a.c		1	35.i	odd	6	1
7056.2.a.m		1	84.j	odd	6	1
7056.2.a.bp		1	84.n	even	6	1
9408.2.a.k		1	56.m	even	6	1
9408.2.a.bg		1	56.p	even	6	1
9408.2.a.bz		1	56.j	odd	6	1
9408.2.a.cv		1	56.k	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T + 4 \)
$3$	\( T^{2} + T + 1 \)
$5$	\( T^{2} - 2T + 4 \)
$7$	\( T^{2} + 5T + 7 \)
$11$	\( T^{2} - 2T + 4 \)