Newform orbit 7056.2.a.cl

• Label: 7056.2.a.cl
• Level: $7056$
• Weight: $2$
• Character orbit: 7056.a
• Self dual: yes
• Analytic conductor: $56.342$
• Analytic rank: $1$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7056.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(56.3424436662\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 98)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

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

\(f(q)\)	\(=\)	q + 2*b * q^5 - 2 * q^11 - b * q^17 - 5*b * q^19 - 4 * q^23 + 3 * q^25 - 2 * q^29 + 6*b * q^31 + 10 * q^37 - 7*b * q^41 - 2 * q^43 - 2*b * q^47 + 2 * q^53 - 4*b * q^55 + b * q^59 - 2*b * q^61 - 12 * q^67 - 12 * q^71 + b * q^73 + 4 * q^79 - 7*b * q^83 - 4 * q^85 - 5*b * q^89 - 20 * q^95 - 7*b * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{11} - 8 q^{23} + 6 q^{25} - 4 q^{29} + 20 q^{37} - 4 q^{43} + 4 q^{53} - 24 q^{67} - 24 q^{71} + 8 q^{79} - 8 q^{85} - 40 q^{95}+O(q^{100}) \)

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

−1.41421
1.41421

0

0

0

−2.82843

0

0

0

0

0

1.2

0

0

0

2.82843

0

0

0

0

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( -1 \)
\(3\)	\( -1 \)
\(7\)	\( +1 \)

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	7056.2.a.cl		2
3.b	odd	2	1		784.2.a.l		2
4.b	odd	2	1		882.2.a.n		2
7.b	odd	2	1	inner	7056.2.a.cl		2
12.b	even	2	1		98.2.a.b	✓	2
21.c	even	2	1		784.2.a.l		2
21.g	even	6	2		784.2.i.m		4
21.h	odd	6	2		784.2.i.m		4
24.f	even	2	1		3136.2.a.bn		2
24.h	odd	2	1		3136.2.a.bm		2
28.d	even	2	1		882.2.a.n		2
28.f	even	6	2		882.2.g.l		4
28.g	odd	6	2		882.2.g.l		4
60.h	even	2	1		2450.2.a.bj		2
60.l	odd	4	2		2450.2.c.v		4
84.h	odd	2	1		98.2.a.b	✓	2
84.j	odd	6	2		98.2.c.c		4
84.n	even	6	2		98.2.c.c		4
168.e	odd	2	1		3136.2.a.bn		2
168.i	even	2	1		3136.2.a.bm		2
420.o	odd	2	1		2450.2.a.bj		2
420.w	even	4	2		2450.2.c.v		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
98.2.a.b	✓	2	12.b	even	2	1
98.2.a.b	✓	2	84.h	odd	2	1
98.2.c.c		4	84.j	odd	6	2
98.2.c.c		4	84.n	even	6	2
784.2.a.l		2	3.b	odd	2	1
784.2.a.l		2	21.c	even	2	1
784.2.i.m		4	21.g	even	6	2
784.2.i.m		4	21.h	odd	6	2
882.2.a.n		2	4.b	odd	2	1
882.2.a.n		2	28.d	even	2	1
882.2.g.l		4	28.f	even	6	2
882.2.g.l		4	28.g	odd	6	2
2450.2.a.bj		2	60.h	even	2	1
2450.2.a.bj		2	420.o	odd	2	1
2450.2.c.v		4	60.l	odd	4	2
2450.2.c.v		4	420.w	even	4	2
3136.2.a.bm		2	24.h	odd	2	1
3136.2.a.bm		2	168.i	even	2	1
3136.2.a.bn		2	24.f	even	2	1
3136.2.a.bn		2	168.e	odd	2	1
7056.2.a.cl		2	1.a	even	1	1	trivial
7056.2.a.cl		2	7.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7056))\):

\( T_{5}^{2} - 8 \)

\( T_{11} + 2 \)

\( T_{13} \)

\( T_{17}^{2} - 2 \)

\( T_{23} + 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} - 8 \)
$7$	\( T^{2} \)
$11$	\( (T + 2)^{2} \)