LMFDB - Newform orbit 8800.2.a.cd

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8800,2,Mod(1,8800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8800, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$8800 = 2^{5} \cdot 5^{2} \cdot 11$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8800.a (trivial)

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:	yes
Analytic conductor:	$70.2683537787$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{7} - 8x^{5} - 2x^{4} + 16x^{3} + 5x^{2} - 6x - 2$ x^7 - 8x^5 - 2x^4 + 16x^3 + 5x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{4}$
Twist minimal:	no (minimal twist has level 1760)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -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 $1,\beta_1,\ldots,\beta_{6}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_1 q^{3} + (\beta_{3} - 1) q^{7} + (\beta_{5} - \beta_1 + 1) q^{9} + q^{11} + (\beta_{6} - 1) q^{13} + (\beta_{5} + \beta_{4} - \beta_{2} + \cdots + 1) q^{17} + ( - \beta_{6} - \beta_{5} + \cdots + \beta_1) q^{19}+ \cdots + (\beta_{5} - \beta_1 + 1) q^{99}+O(q^{100})$ q + b1 * q^3 + (b3 - 1) * q^7 + (b5 - b1 + 1) * q^9 + q^11 + (b6 - 1) * q^13 + (b5 + b4 - b2 - b1 + 1) * q^17 + (-b6 - b5 - b4 + b2 + b1) * q^19 + (b5 + b4 - b3 - 2b1 + 2) q^21 + (-b5 + b4 - b3 + b1 - 2) * q^23 + (-b5 - b4 + b3 - b2 - 2) * q^27 + (-b5 - b4 - b3 - 2) * q^29 + (-b5 - 2b4 + b1 - 2) q^31 + b1 * q^33 + (-b6 + b5 + b4 - b1 - 2) * q^37 + (-b6 + 2b4 - b3 + b2 - b1) q^39 + (b6 - 2b5 - 2b4 - b3 + b2 + b1) * q^41 + (-b6 - b5 + b4 - b3 + b2 + b1 - 1) * q^43 + (b6 - b5 - 3b4 + 2b2 - 4) * q^47 + (-2b6 - b5 + b4 - b3 - 2b1 + 1) * q^49 + (b5 + b4 + b3 - 2b2) q^51 + (-b5 + b4 + b3 - 2b1) q^53 + (b6 - b5 - 3b4 + b2 + b1) q^57 + (b6 + 2b5 - b4 + 2b3 + b2 - 2b1 + 4) q^59 + (b6 + b5 - b4 - b2 - b1) * q^61 + (b6 - 3b5 - b4 - b3 - b2 + 5b1 - 5) * q^63 + (-b6 + b5 + b4 - b2 - 2b1 - 2) q^67 + (b6 + b4 + b2 - 4b1) q^69 + (-2b6 - b5 + 2b4 - 2b2 - b1) q^71 + (-b6 - 2b5 - 2b4 - 2b3 + 4b1 - 3) * q^73 + (b3 - 1) * q^77 + (b6 - b3 + b2 + 3b1 - 4) q^79 + (-2b6 + 2b4 - 2b3 - 2b1 - 1) * q^81 + (b6 + b5 + 3b4 - b3 - b2 - b1 - 1) q^83 + (-b6 - b5 - b4 + b2 - b1 - 4) * q^87 + (b6 + b5 - b3 - b2 - 4b1 + 2) q^89 + (2b5 + 2b4 - 4b3 + 2) q^91 + (-2b6 + b5 - b4 - b3 + b2 - 2b1 + 2) * q^93 + (-b6 - 2b5 - 3b3 + 2b2 + 3b1) * q^97 + (b5 - b1 + 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$7 q - q^{3} - 7 q^{7} + 6 q^{9} + 7 q^{11} - 10 q^{13} + 3 q^{17} + 7 q^{19} + 13 q^{21} - 14 q^{23} - 13 q^{27} - 11 q^{29} - 11 q^{31} - q^{33} - 13 q^{37} + 4 q^{39} + 4 q^{41} - 2 q^{43} - 22 q^{47}+ \cdots + 6 q^{99}+O(q^{100})$ 7 * q - q^3 - 7 * q^7 + 6 * q^9 + 7 * q^11 - 10 * q^13 + 3 * q^17 + 7 * q^19 + 13 * q^21 - 14 * q^23 - 13 * q^27 - 11 * q^29 - 11 * q^31 - q^33 - 13 * q^37 + 4 * q^39 + 4 * q^41 - 2 * q^43 - 22 * q^47 + 16 * q^49 - 7 * q^51 + 3 * q^53 + 3 * q^57 + 26 * q^59 - 5 * q^61 - 38 * q^63 - 14 * q^67 + 2 * q^69 + 3 * q^71 - 16 * q^73 - 7 * q^77 - 32 * q^79 - q^81 - 16 * q^83 - 19 * q^87 + 11 * q^89 + 8 * q^91 + 23 * q^93 + 8 * q^97 + 6 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{7} - 8x^{5} - 2x^{4} + 16x^{3} + 5x^{2} - 6x - 2$ x^7 - 8*x^5 - 2*x^4 + 16*x^3 + 5*x^2 - 6*x - 2 :

$\beta_{1}$	$=$	$\nu^{4} - \nu^{3} - 5\nu^{2} + 3\nu + 3$ v^4 - v^3 - 5v^2 + 3v + 3
$\beta_{2}$	$=$	$-\nu^{6} + \nu^{5} + 6\nu^{4} - 2\nu^{3} - 8\nu^{2} - 3\nu + 1$ -v^6 + v^5 + 6v^4 - 2v^3 - 8v^2 - 3v + 1
$\beta_{3}$	$=$	$\nu^{6} - \nu^{5} - 7\nu^{4} + 5\nu^{3} + 11\nu^{2} - 6\nu - 1$ v^6 - v^5 - 7v^4 + 5v^3 + 11v^2 - 6v - 1
$\beta_{4}$	$=$	$-\nu^{6} + 8\nu^{4} + 2\nu^{3} - 16\nu^{2} - 5\nu + 5$ -v^6 + 8v^4 + 2v^3 - 16v^2 - 5v + 5
$\beta_{5}$	$=$	$-\nu^{6} + 2\nu^{5} + 6\nu^{4} - 10\nu^{3} - 10\nu^{2} + 11\nu + 4$ -v^6 + 2v^5 + 6v^4 - 10v^3 - 10v^2 + 11*v + 4
$\beta_{6}$	$=$	$-2\nu^{6} + 16\nu^{4} + 2\nu^{3} - 30\nu^{2} + 7$ -2v^6 + 16v^4 + 2v^3 - 30v^2 + 7

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$( \beta_{6} - 2\beta_{4} + \beta_{3} + \beta_{2} + \beta_1 ) / 4$ (b6 - 2*b4 + b3 + b2 + b1) / 4
$\nu^{2}$	$=$	$( -\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 4 ) / 2$ (-b5 - b4 - b3 + b2 + b1 + 4) / 2
$\nu^{3}$	$=$	$( 3\beta_{6} - 2\beta_{5} - 8\beta_{4} + 3\beta_{3} + 7\beta_{2} + 7\beta _1 + 2 ) / 4$ (3b6 - 2b5 - 8b4 + 3b3 + 7b2 + 7b1 + 2) / 4
$\nu^{4}$	$=$	$( -6\beta_{5} - 6\beta_{4} - 5\beta_{3} + 7\beta_{2} + 9\beta _1 + 15 ) / 2$ (-6b5 - 6b4 - 5b3 + 7b2 + 9*b1 + 15) / 2
$\nu^{5}$	$=$	$( 5\beta_{6} - 8\beta_{5} - 20\beta_{4} + 3\beta_{3} + 21\beta_{2} + 23\beta _1 + 10 ) / 2$ (5b6 - 8b5 - 20b4 + 3b3 + 21b2 + 23b1 + 10) / 2
$\nu^{6}$	$=$	$( \beta_{6} - 68\beta_{5} - 74\beta_{4} - 47\beta_{3} + 89\beta_{2} + 121\beta _1 + 136 ) / 4$ (b6 - 68b5 - 74b4 - 47b3 + 89b2 + 121*b1 + 136) / 4

Display $\beta_i$ in terms of $\nu^j$

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}

1.1

−1.71085
1.51372
−0.687115
−1.92480
−0.340436
2.44914
0.700339

−3.19254

−4.81353

7.19231

1.2

−2.13376

−1.20437

1.55295

1.3

−0.874674

4.39214

−2.23495

1.4

−0.441641

−4.16263

−2.80495

1.5

1.45210

1.03231

−0.891413

1.6

1.64481

−1.42015

−0.294593

1.7

2.54571

−0.823764

3.48064

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$5$	$-1$
$11$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8800.2.a.cd		7
4.b	odd	2	1		8800.2.a.ci		7
5.b	even	2	1		8800.2.a.cj		7
5.c	odd	4	2		1760.2.b.f	yes	14
20.d	odd	2	1		8800.2.a.cc		7
20.e	even	4	2		1760.2.b.e	✓	14

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1760.2.b.e	✓	14	20.e	even	4	2
1760.2.b.f	yes	14	5.c	odd	4	2
8800.2.a.cc		7	20.d	odd	2	1
8800.2.a.cd		7	1.a	even	1	1	trivial
8800.2.a.ci		7	4.b	odd	2	1
8800.2.a.cj		7	5.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(8800))$ :

T_{3}^{7} + T_{3}^{6} - 13T_{3}^{5} - 7T_{3}^{4} + 46T_{3}^{3} + 12T_{3}^{2} - 40T_{3} - 16

T_{7}^{7} + 7T_{7}^{6} - 8T_{7}^{5} - 136T_{7}^{4} - 232T_{7}^{3} + 16T_{7}^{2} + 256T_{7} + 128

T_{13}^{7} + 10T_{13}^{6} - 240T_{13}^{4} - 448T_{13}^{3} + 1120T_{13}^{2} + 2624T_{13} + 1024

T_{17}^{7} - 3T_{17}^{6} - 34T_{17}^{5} + 172T_{17}^{4} - 168T_{17}^{3} - 240T_{17}^{2} + 384T_{17} - 128

T_{19}^{7} - 7T_{19}^{6} - 52T_{19}^{5} + 340T_{19}^{4} + 608T_{19}^{3} - 3584T_{19}^{2} + 3584T_{19} - 1024

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{7}$ T^7
$3$	$T^{7} + T^{6} + \cdots - 16$ T^7 + T^6 - 13T^5 - 7T^4 + 46T^3 + 12T^2 - 40*T - 16
$5$	$T^{7}$ T^7
$7$	$T^{7} + 7 T^{6} + \cdots + 128$ T^7 + 7T^6 - 8T^5 - 136T^4 - 232T^3 + 16T^2 + 256T + 128
$11$	$(T - 1)^{7}$ (T - 1)^7
$13$	$T^{7} + 10 T^{6} + \cdots + 1024$ T^7 + 10T^6 - 240T^4 - 448T^3 + 1120T^2 + 2624*T + 1024
$17$	$T^{7} - 3 T^{6} + \cdots - 128$ T^7 - 3T^6 - 34T^5 + 172T^4 - 168T^3 - 240T^2 + 384T - 128
$19$	$T^{7} - 7 T^{6} + \cdots - 1024$ T^7 - 7T^6 - 52T^5 + 340T^4 + 608T^3 - 3584T^2 + 3584T - 1024
$23$	$T^{7} + 14 T^{6} + \cdots + 10816$ T^7 + 14T^6 + 21T^5 - 398T^4 - 1556T^3 + 760T^2 + 10000T + 10816
$29$	$T^{7} + 11 T^{6} + \cdots + 256$ T^7 + 11T^6 - 12T^5 - 368T^4 - 576T^3 + 1488T^2 + 2432T + 256
$31$	$T^{7} + 11 T^{6} + \cdots + 7424$ T^7 + 11T^6 - 41T^5 - 387T^4 + 1456T^3 + 992T^2 - 8192T + 7424
$37$	$T^{7} + 13 T^{6} + \cdots - 256$ T^7 + 13T^6 - 23T^5 - 871T^4 - 3220T^3 - 848T^2 + 6592T - 256
$41$	$T^{7} - 4 T^{6} + \cdots - 652544$ T^7 - 4T^6 - 180T^5 + 272T^4 + 11504T^3 + 9536T^2 - 252096T - 652544
$43$	$T^{7} + 2 T^{6} + \cdots - 20864$ T^7 + 2T^6 - 116T^5 + 224T^4 + 3456T^3 - 18816T^2 + 34432T - 20864
$47$	$T^{7} + 22 T^{6} + \cdots + 1262336$ T^7 + 22T^6 - 16T^5 - 3232T^4 - 17216T^3 + 77536T^2 + 749376T + 1262336
$53$	$T^{7} - 3 T^{6} + \cdots + 8192$ T^7 - 3T^6 - 212T^5 + 784T^4 + 5056T^3 - 25856T^2 + 24576T + 8192
$59$	$T^{7} - 26 T^{6} + \cdots + 708608$ T^7 - 26T^6 + 85T^5 + 3672T^4 - 50784T^3 + 280640T^2 - 721664T + 708608
$61$	$T^{7} + 5 T^{6} + \cdots + 20224$ T^7 + 5T^6 - 132T^5 - 656T^4 + 2976T^3 + 22576T^2 + 40832T + 20224
$67$	$T^{7} + 14 T^{6} + \cdots - 5312$ T^7 + 14T^6 - 27T^5 - 774T^4 - 1292T^3 + 4712T^2 + 2704T - 5312
$71$	$T^{7} - 3 T^{6} + \cdots + 256$ T^7 - 3T^6 - 357T^5 + 535T^4 + 27760T^3 + 41536T^2 + 14016T + 256
$73$	$T^{7} + 16 T^{6} + \cdots - 5921408$ T^7 + 16T^6 - 204T^5 - 3616T^4 + 12320T^3 + 258080T^2 - 216064T - 5921408
$79$	$T^{7} + 32 T^{6} + \cdots + 483328$ T^7 + 32T^6 + 276T^5 - 496T^4 - 15680T^3 - 42752T^2 + 125952T + 483328
$83$	$T^{7} + 16 T^{6} + \cdots - 14848$ T^7 + 16T^6 - 100T^5 - 2520T^4 - 6736T^3 + 24608T^2 - 3008T - 14848
$89$	$T^{7} - 11 T^{6} + \cdots - 859696$ T^7 - 11T^6 - 253T^5 + 1563T^4 + 22892T^3 - 16048T^2 - 492704T - 859696
$97$	$T^{7} - 8 T^{6} + \cdots - 842752$ T^7 - 8T^6 - 251T^5 + 1304T^4 + 20544T^3 - 39424T^2 - 565760T - 842752
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
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$n$ :		e.g. 2-40 or 990-1000
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