Newform orbit 63.6.a.g

• Label: 63.6.a.g
• Level: $63$
• Weight: $6$
• Character orbit: 63.a
• Self dual: yes
• Analytic conductor: $10.104$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	63.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(10.1041806482\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{7}) \)
Defining polynomial:	\( x^{2} - 7 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	yes
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta q^{2} - 4 q^{4} - 7 \beta q^{5} - 49 q^{7} - 36 \beta q^{8} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} - 98 q^{7}+O(q^{10}) \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−2.64575
2.64575

−5.29150

0

−4.00000

37.0405

0

−49.0000

190.494

0

−196.000

1.2

5.29150

0

−4.00000

−37.0405

0

−49.0000

−190.494

0

−196.000

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(1\)
\(7\)	\(1\)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	63.6.a.g	✓	2
3.b	odd	2	1	inner	63.6.a.g	✓	2
4.b	odd	2	1		1008.6.a.bl		2
7.b	odd	2	1		441.6.a.p		2
12.b	even	2	1		1008.6.a.bl		2
21.c	even	2	1		441.6.a.p		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.6.a.g	✓	2	1.a	even	1	1	trivial
63.6.a.g	✓	2	3.b	odd	2	1	inner
441.6.a.p		2	7.b	odd	2	1
441.6.a.p		2	21.c	even	2	1
1008.6.a.bl		2	4.b	odd	2	1
1008.6.a.bl		2	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 28 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(63))\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 28 \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 1372 \)
$7$	\( (T + 49)^{2} \)
$11$	\( T^{2} - 51772 \)