Newform orbit 784.2.a.m

• Label: 784.2.a.m
• Level: $784$
• Weight: $2$
• Character orbit: 784.a
• Self dual: yes
• Analytic conductor: $6.260$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$784 = 2^{4} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	784.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$6.26027151847$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2})$
Defining polynomial:	$x^{2} - 2$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 196)
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^3 + b * q^5 + 5 * q^9 - 4 * q^11 + 3*b * q^13 + 4 * q^15 + b * q^17 - 2*b * q^19 + 4 * q^23 - 3 * q^25 + 4*b * q^27 + 8 * q^29 - 8*b * q^33 - 8 * q^37 + 12 * q^39 - 5*b * q^41 + 4 * q^43 + 5*b * q^45 - 4*b * q^47 + 4 * q^51 + 10 * q^53 - 4*b * q^55 - 8 * q^57 - 10*b * q^59 - 5*b * q^61 + 6 * q^65 + 8*b * q^69 - 5*b * q^73 - 6*b * q^75 - 8 * q^79 + q^81 + 10*b * q^83 + 2 * q^85 + 16*b * q^87 + 5*b * q^89 - 4 * q^95 - b * q^97 - 20 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 10 q^{9} - 8 q^{11} + 8 q^{15} + 8 q^{23} - 6 q^{25} + 16 q^{29} - 16 q^{37} + 24 q^{39} + 8 q^{43} + 8 q^{51} + 20 q^{53} - 16 q^{57} + 12 q^{65} - 16 q^{79} + 2 q^{81} + 4 q^{85} - 8 q^{95} - 40 q^{99}+O(q^{100})$

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

−2.82843

0

−1.41421

0

0

0

5.00000

0

1.2

0

2.82843

0

1.41421

0

0

0

5.00000

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$7$	$+1$

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	784.2.a.m		2
3.b	odd	2	1		7056.2.a.cr		2
4.b	odd	2	1		196.2.a.c	✓	2
7.b	odd	2	1	inner	784.2.a.m		2
7.c	even	3	2		784.2.i.l		4
7.d	odd	6	2		784.2.i.l		4
8.b	even	2	1		3136.2.a.bs		2
8.d	odd	2	1		3136.2.a.br		2
12.b	even	2	1		1764.2.a.l		2
20.d	odd	2	1		4900.2.a.y		2
20.e	even	4	2		4900.2.e.p		4
21.c	even	2	1		7056.2.a.cr		2
28.d	even	2	1		196.2.a.c	✓	2
28.f	even	6	2		196.2.e.b		4
28.g	odd	6	2		196.2.e.b		4
56.e	even	2	1		3136.2.a.br		2
56.h	odd	2	1		3136.2.a.bs		2
84.h	odd	2	1		1764.2.a.l		2
84.j	odd	6	2		1764.2.k.l		4
84.n	even	6	2		1764.2.k.l		4
140.c	even	2	1		4900.2.a.y		2
140.j	odd	4	2		4900.2.e.p		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
196.2.a.c	✓	2	4.b	odd	2	1
196.2.a.c	✓	2	28.d	even	2	1
196.2.e.b		4	28.f	even	6	2
196.2.e.b		4	28.g	odd	6	2
784.2.a.m		2	1.a	even	1	1	trivial
784.2.a.m		2	7.b	odd	2	1	inner
784.2.i.l		4	7.c	even	3	2
784.2.i.l		4	7.d	odd	6	2
1764.2.a.l		2	12.b	even	2	1
1764.2.a.l		2	84.h	odd	2	1
1764.2.k.l		4	84.j	odd	6	2
1764.2.k.l		4	84.n	even	6	2
3136.2.a.br		2	8.d	odd	2	1
3136.2.a.br		2	56.e	even	2	1
3136.2.a.bs		2	8.b	even	2	1
3136.2.a.bs		2	56.h	odd	2	1
4900.2.a.y		2	20.d	odd	2	1
4900.2.a.y		2	140.c	even	2	1
4900.2.e.p		4	20.e	even	4	2
4900.2.e.p		4	140.j	odd	4	2
7056.2.a.cr		2	3.b	odd	2	1
7056.2.a.cr		2	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(784))$:

$T_{3}^{2} - 8$

$T_{5}^{2} - 2$

$T_{11} + 4$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2} - 8$
$5$	$T^{2} - 2$
$7$	$T^{2}$
$11$	$(T + 4)^{2}$