LMFDB - Newform orbit 7056.2.k.g

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7056,2,Mod(881,7056)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7056, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7056.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7056.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$56.3424436662$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 441)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{4} q^{5}+O(q^{10}) $ q + b4 * q^5	$ q + \beta_{4} q^{5} + (2 \beta_{2} + \beta_1) q^{11} - \beta_{5} q^{13} + ( - \beta_{7} - \beta_{4}) q^{17} + 2 \beta_{5} q^{19} - \beta_1 q^{23} + ( - \beta_{3} + 5) q^{25} + (2 \beta_{2} - \beta_1) q^{29} - 2 \beta_{5} q^{31} + ( - \beta_{3} - 4) q^{37} + ( - \beta_{7} + \beta_{4}) q^{41} + ( - 6 \beta_{3} + 4) q^{43} - 2 \beta_{4} q^{47} + 5 \beta_{2} q^{53} + (2 \beta_{6} - 2 \beta_{5}) q^{55} - 2 \beta_{4} q^{59} + (\beta_{6} + \beta_{5}) q^{61} + ( - \beta_{2} + 5 \beta_1) q^{65} + 6 \beta_{3} q^{67} + (2 \beta_{2} - \beta_1) q^{71} + (\beta_{6} - \beta_{5}) q^{73} + ( - 4 \beta_{3} + 4) q^{79} + 4 \beta_{4} q^{83} + ( - 9 \beta_{3} - 8) q^{85} + ( - 2 \beta_{7} - \beta_{4}) q^{89} + (2 \beta_{2} - 10 \beta_1) q^{95} + ( - \beta_{6} + \beta_{5}) q^{97}+O(q^{100}) $ q + b4 * q^5 + (2b2 + b1) q^11 - b5 * q^13 + (-b7 - b4) * q^17 + 2b5 q^19 - b1 * q^23 + (-b3 + 5) * q^25 + (2b2 - b1) q^29 - 2b5 q^31 + (-b3 - 4) * q^37 + (-b7 + b4) * q^41 + (-6b3 + 4) q^43 - 2b4 q^47 + 5b2 q^53 + (2b6 - 2b5) * q^55 - 2b4 q^59 + (b6 + b5) * q^61 + (-b2 + 5b1) q^65 + 6b3 q^67 + (2b2 - b1) q^71 + (b6 - b5) * q^73 + (-4b3 + 4) q^79 + 4b4 q^83 + (-9b3 - 8) q^85 + (-2b7 - b4) q^89 + (2b2 - 10b1) * q^95 + (-b6 + b5) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 40 q^{25} - 32 q^{37} + 32 q^{43} + 32 q^{79} - 64 q^{85}+O(q^{100}) $ 8 * q + 40 * q^25 - 32 * q^37 + 32 * q^43 + 32 * q^79 - 64 * q^85

Display 100 coefficients 10 coefficients

Basis of coefficient ring

$\beta_{1}$	$=$	$ 2\zeta_{16}^{4} $ 2*v^4
$\beta_{2}$	$=$	$ \zeta_{16}^{6} + \zeta_{16}^{2} $ v^6 + v^2
$\beta_{3}$	$=$	$ -\zeta_{16}^{6} + \zeta_{16}^{2} $ -v^6 + v^2
$\beta_{4}$	$=$	$ -2\zeta_{16}^{7} + \zeta_{16}^{5} - \zeta_{16}^{3} + 2\zeta_{16} $ -2v^7 + v^5 - v^3 + 2v
$\beta_{5}$	$=$	$ \zeta_{16}^{7} - 2\zeta_{16}^{5} - 2\zeta_{16}^{3} + \zeta_{16} $ v^7 - 2v^5 - 2v^3 + v
$\beta_{6}$	$=$	$ 3\zeta_{16}^{7} + \zeta_{16}^{5} + \zeta_{16}^{3} + 3\zeta_{16} $ 3v^7 + v^5 + v^3 + 3v
$\beta_{7}$	$=$	$ -\zeta_{16}^{7} - 3\zeta_{16}^{5} + 3\zeta_{16}^{3} + \zeta_{16} $ -v^7 - 3v^5 + 3v^3 + v

Display $\zeta_{16}^j$ in terms of $\beta_i$

$\zeta_{16}$	$=$	$ ( \beta_{7} + 2\beta_{6} + \beta_{5} + 3\beta_{4} ) / 14 $ (b7 + 2b6 + b5 + 3b4) / 14
$\zeta_{16}^{2}$	$=$	$ ( \beta_{3} + \beta_{2} ) / 2 $ (b3 + b2) / 2
$\zeta_{16}^{3}$	$=$	$ ( 2\beta_{7} + \beta_{6} - 3\beta_{5} - \beta_{4} ) / 14 $ (2b7 + b6 - 3b5 - b4) / 14
$\zeta_{16}^{4}$	$=$	$ ( \beta_1 ) / 2 $ (b1) / 2
$\zeta_{16}^{5}$	$=$	$ ( -2\beta_{7} + \beta_{6} - 3\beta_{5} + \beta_{4} ) / 14 $ (-2b7 + b6 - 3b5 + b4) / 14
$\zeta_{16}^{6}$	$=$	$ ( -\beta_{3} + \beta_{2} ) / 2 $ (-b3 + b2) / 2
$\zeta_{16}^{7}$	$=$	$ ( -\beta_{7} + 2\beta_{6} + \beta_{5} - 3\beta_{4} ) / 14 $ (-b7 + 2b6 + b5 - 3b4) / 14

Display $\beta_i$ in terms of $\zeta_{16}^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times$.

$n$	$785$	$1765$	$4609$	$6175$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

881.1

−0.382683	+	0.923880i
−0.382683	−	0.923880i
−0.923880	+	0.382683i
−0.923880	−	0.382683i
0.923880	−	0.382683i
0.923880	+	0.382683i
0.382683	−	0.923880i
0.382683	+	0.923880i

−3.37849

−3.37849

−2.93015

−2.93015

2.93015

2.93015

3.37849

3.37849

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	7056.2.k.g		8
3.b	odd	2	1	inner	7056.2.k.g		8
4.b	odd	2	1		441.2.c.b	✓	8
7.b	odd	2	1	inner	7056.2.k.g		8
12.b	even	2	1		441.2.c.b	✓	8
21.c	even	2	1	inner	7056.2.k.g		8
28.d	even	2	1		441.2.c.b	✓	8
28.f	even	6	2		441.2.p.c		16
28.g	odd	6	2		441.2.p.c		16
84.h	odd	2	1		441.2.c.b	✓	8
84.j	odd	6	2		441.2.p.c		16
84.n	even	6	2		441.2.p.c		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
441.2.c.b	✓	8	4.b	odd	2	1
441.2.c.b	✓	8	12.b	even	2	1
441.2.c.b	✓	8	28.d	even	2	1
441.2.c.b	✓	8	84.h	odd	2	1
441.2.p.c		16	28.f	even	6	2
441.2.p.c		16	28.g	odd	6	2
441.2.p.c		16	84.j	odd	6	2
441.2.p.c		16	84.n	even	6	2
7056.2.k.g		8	1.a	even	1	1	trivial
7056.2.k.g		8	3.b	odd	2	1	inner
7056.2.k.g		8	7.b	odd	2	1	inner
7056.2.k.g		8	21.c	even	2	1	inner

Hecke kernels

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

$ T_{5}^{4} - 20T_{5}^{2} + 98 $

$ T_{11}^{4} + 24T_{11}^{2} + 16 $

$ T_{13}^{4} + 20T_{13}^{2} + 98 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} $ T^8
$5$	$ (T^{4} - 20 T^{2} + 98)^{2} $ (T^4 - 20*T^2 + 98)^2
$7$	$ T^{8} $ T^8
$11$	$ (T^{4} + 24 T^{2} + 16)^{2} $ (T^4 + 24*T^2 + 16)^2
$13$	$ (T^{4} + 20 T^{2} + 98)^{2} $ (T^4 + 20*T^2 + 98)^2
$17$	$ (T^{4} - 52 T^{2} + 98)^{2} $ (T^4 - 52*T^2 + 98)^2
$19$	$ (T^{4} + 80 T^{2} + 1568)^{2} $ (T^4 + 80*T^2 + 1568)^2
$23$	$ (T^{2} + 4)^{4} $ (T^2 + 4)^4
$29$	$ (T^{4} + 24 T^{2} + 16)^{2} $ (T^4 + 24*T^2 + 16)^2
$31$	$ (T^{4} + 80 T^{2} + 1568)^{2} $ (T^4 + 80*T^2 + 1568)^2
$37$	$ (T^{2} + 8 T + 14)^{4} $ (T^2 + 8*T + 14)^4
$41$	$ (T^{4} - 68 T^{2} + 98)^{2} $ (T^4 - 68*T^2 + 98)^2
$43$	$ (T^{2} - 8 T - 56)^{4} $ (T^2 - 8*T - 56)^4
$47$	$ (T^{4} - 80 T^{2} + 1568)^{2} $ (T^4 - 80*T^2 + 1568)^2
$53$	$ (T^{2} + 50)^{4} $ (T^2 + 50)^4
$59$	$ (T^{4} - 80 T^{2} + 1568)^{2} $ (T^4 - 80*T^2 + 1568)^2
$61$	$ (T^{4} + 68 T^{2} + 98)^{2} $ (T^4 + 68*T^2 + 98)^2
$67$	$ (T^{2} - 72)^{4} $ (T^2 - 72)^4
$71$	$ (T^{4} + 24 T^{2} + 16)^{2} $ (T^4 + 24*T^2 + 16)^2
$73$	$ (T^{4} + 52 T^{2} + 98)^{2} $ (T^4 + 52*T^2 + 98)^2
$79$	$ (T^{2} - 8 T - 16)^{4} $ (T^2 - 8*T - 16)^4
$83$	$ (T^{4} - 320 T^{2} + 25088)^{2} $ (T^4 - 320*T^2 + 25088)^2
$89$	$ (T^{4} - 164 T^{2} + 4802)^{2} $ (T^4 - 164*T^2 + 4802)^2
$97$	$ (T^{4} + 52 T^{2} + 98)^{2} $ (T^4 + 52*T^2 + 98)^2
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Level:	\( N \)	\(=\)	\( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7056.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{4} q^{5}+O(q^{10}) \) q + b4 * q^5	\( q + \beta_{4} q^{5} + (2 \beta_{2} + \beta_1) q^{11} - \beta_{5} q^{13} + ( - \beta_{7} - \beta_{4}) q^{17} + 2 \beta_{5} q^{19} - \beta_1 q^{23} + ( - \beta_{3} + 5) q^{25} + (2 \beta_{2} - \beta_1) q^{29} - 2 \beta_{5} q^{31} + ( - \beta_{3} - 4) q^{37} + ( - \beta_{7} + \beta_{4}) q^{41} + ( - 6 \beta_{3} + 4) q^{43} - 2 \beta_{4} q^{47} + 5 \beta_{2} q^{53} + (2 \beta_{6} - 2 \beta_{5}) q^{55} - 2 \beta_{4} q^{59} + (\beta_{6} + \beta_{5}) q^{61} + ( - \beta_{2} + 5 \beta_1) q^{65} + 6 \beta_{3} q^{67} + (2 \beta_{2} - \beta_1) q^{71} + (\beta_{6} - \beta_{5}) q^{73} + ( - 4 \beta_{3} + 4) q^{79} + 4 \beta_{4} q^{83} + ( - 9 \beta_{3} - 8) q^{85} + ( - 2 \beta_{7} - \beta_{4}) q^{89} + (2 \beta_{2} - 10 \beta_1) q^{95} + ( - \beta_{6} + \beta_{5}) q^{97}+O(q^{100}) \) q + b4 * q^5 + (2b2 + b1) q^11 - b5 * q^13 + (-b7 - b4) * q^17 + 2b5 q^19 - b1 * q^23 + (-b3 + 5) * q^25 + (2b2 - b1) q^29 - 2b5 q^31 + (-b3 - 4) * q^37 + (-b7 + b4) * q^41 + (-6b3 + 4) q^43 - 2b4 q^47 + 5b2 q^53 + (2b6 - 2b5) * q^55 - 2b4 q^59 + (b6 + b5) * q^61 + (-b2 + 5b1) q^65 + 6b3 q^67 + (2b2 - b1) q^71 + (b6 - b5) * q^73 + (-4b3 + 4) q^79 + 4b4 q^83 + (-9b3 - 8) q^85 + (-2b7 - b4) q^89 + (2b2 - 10b1) * q^95 + (-b6 + b5) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 40 q^{25} - 32 q^{37} + 32 q^{43} + 32 q^{79} - 64 q^{85}+O(q^{100}) \) 8 * q + 40 * q^25 - 32 * q^37 + 32 * q^43 + 32 * q^79 - 64 * q^85

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