Newform orbit 21.6.g.a

• Label: 21.6.g.a
• Level: $21$
• Weight: $6$
• Character orbit: 21.g
• Analytic conductor: $3.368$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.36806021607\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (9 \zeta_{6} + 9) q^{3} + (32 \zeta_{6} - 32) q^{4} + (87 \zeta_{6} + 62) q^{7} + 243 \zeta_{6} q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 27 q^{3} - 32 q^{4} + 211 q^{7} + 243 q^{9}+O(q^{10}) \)

Character values

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

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

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

5.1

0.500000	−	0.866025i
0.500000	+	0.866025i

0

13.5000

−

7.79423i

−16.0000

−

27.7128i

0

0

105.500

−

75.3442i

0

121.500

−

210.444i

0

17.1

0

13.5000

+

7.79423i

−16.0000

+

27.7128i

0

0

105.500

+

75.3442i

0

121.500

+

210.444i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
7.d	odd	6	1	inner
21.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	21.6.g.a	✓	2
3.b	odd	2	1	CM	21.6.g.a	✓	2
7.c	even	3	1		147.6.c.a		2
7.d	odd	6	1	inner	21.6.g.a	✓	2
7.d	odd	6	1		147.6.c.a		2
21.g	even	6	1	inner	21.6.g.a	✓	2
21.g	even	6	1		147.6.c.a		2
21.h	odd	6	1		147.6.c.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.6.g.a	✓	2	1.a	even	1	1	trivial
21.6.g.a	✓	2	3.b	odd	2	1	CM
21.6.g.a	✓	2	7.d	odd	6	1	inner
21.6.g.a	✓	2	21.g	even	6	1	inner
147.6.c.a		2	7.c	even	3	1
147.6.c.a		2	7.d	odd	6	1
147.6.c.a		2	21.g	even	6	1
147.6.c.a		2	21.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{6}^{\mathrm{new}}(21, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 27T + 243 \)
$5$	\( T^{2} \)
$7$	\( T^{2} - 211T + 16807 \)
$11$	\( T^{2} \)