Newform orbit 63.6.e.a

• Label: 63.6.e.a
• Level: $63$
• Weight: $6$
• Character orbit: 63.e
• Analytic conductor: $10.104$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	63.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1041806482\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 21)
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.

\(f(q)\)	\(=\)	\( q + (2 \zeta_{6} - 2) q^{2} + 28 \zeta_{6} q^{4} + ( - 11 \zeta_{6} + 11) q^{5} + (7 \zeta_{6} + 126) q^{7} - 120 q^{8}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 28 q^{4} + 11 q^{5} + 259 q^{7} - 240 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\)	\(10\)	\(29\)
\(\chi(n)\)	\(-1 + \zeta_{6}\)	\(1\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

37.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−1.00000

+

1.73205i

0

14.0000

+

24.2487i

5.50000

−

9.52628i

0

129.500

+

6.06218i

−120.000

0

11.0000

+

19.0526i

46.1

−1.00000

−

1.73205i

0

14.0000

−

24.2487i

5.50000

+

9.52628i

0

129.500

−

6.06218i

−120.000

0

11.0000

−

19.0526i

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	63.6.e.a		2
3.b	odd	2	1		21.6.e.a	✓	2
7.c	even	3	1	inner	63.6.e.a		2
7.c	even	3	1		441.6.a.g		1
7.d	odd	6	1		441.6.a.h		1
12.b	even	2	1		336.6.q.b		2
21.c	even	2	1		147.6.e.g		2
21.g	even	6	1		147.6.a.d		1
21.g	even	6	1		147.6.e.g		2
21.h	odd	6	1		21.6.e.a	✓	2
21.h	odd	6	1		147.6.a.c		1
84.n	even	6	1		336.6.q.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.6.e.a	✓	2	3.b	odd	2	1
21.6.e.a	✓	2	21.h	odd	6	1
63.6.e.a		2	1.a	even	1	1	trivial
63.6.e.a		2	7.c	even	3	1	inner
147.6.a.c		1	21.h	odd	6	1
147.6.a.d		1	21.g	even	6	1
147.6.e.g		2	21.c	even	2	1
147.6.e.g		2	21.g	even	6	1
336.6.q.b		2	12.b	even	2	1
336.6.q.b		2	84.n	even	6	1
441.6.a.g		1	7.c	even	3	1
441.6.a.h		1	7.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} + 4 \) acting on \(S_{6}^{\mathrm{new}}(63, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T + 4 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 11T + 121 \)
$7$	\( T^{2} - 259T + 16807 \)
$11$	\( T^{2} - 269T + 72361 \)