LMFDB - Newform orbit 4000.2.a.n

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4000,2,Mod(1,4000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4000, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4000 = 2^{5} \cdot 5^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4000.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:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.30040000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 11x^{4} - 4x^{3} + 29x^{2} + 22x + 4$ x^6 - 11x^4 - 4x^3 + 29x^2 + 22x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - \beta_1 q^{3} + ( - \beta_{5} + 1) q^{7} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{9} + (\beta_{4} + \beta_{2} + \beta_1 + 2) q^{11} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 2) q^{13} + ( - \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{17}+ \cdots + (\beta_{4} + 6 \beta_{3} + 2 \beta_{2} + \cdots + 10) q^{99}+O(q^{100})$ q - b1 * q^3 + (-b5 + 1) * q^7 + (b4 + b3 + b1 + 1) * q^9 + (b4 + b2 + b1 + 2) * q^11 + (-b5 + b4 + b3 - b2 - b1 + 2) * q^13 + (-b5 - b4 + b2 - b1) * q^17 + (-b5 + 2b3 - b2 + b1 + 3) q^19 + (-b5 - b3 + b2) * q^21 + (b4 + 2b3 + b1 + 2) q^23 + (-b2 - 2) * q^27 + (b5 - b4 + b2 + b1) * q^29 + (2b5 - 2b3 - b2 - b1) * q^31 + (-b5 - 3b3 - b2 - 4b1 - 2) * q^33 + (b5 + 3b2 + 3) q^37 + (2b4 + 4b3 + b2 - b1 + 6) * q^39 + (b4 - 2b2 - b1 - 1) q^41 + (-2b4 - 2b3 - b2 - 2) * q^43 + (2b5 + b4 + b2 + 2) q^47 + (-b5 - 2b3 + b2 - 2b1 - 1) * q^49 + (-2b5 - 4b3 + 3b1 + 2) q^51 + (-b5 - b4 + b3 - 3b2 - 3b1 + 3) * q^53 + (-b4 + b3 - 3b1 - 4) q^57 + (b5 - 2b4 + 2b3 - 2b2 - b1 + 5) q^59 + (3b5 + b3 + b2 + 2b1 - 2) * q^61 + (b5 - 4b3 + b2 + b1 - 3) q^63 + (2b5 + 2b3 + 3b2 + b1) q^67 + (-2b2 - 4b1 - 2) * q^69 + (-2b4 - 4b3 - b2 - 4b1 + 4) q^71 + (2b5 + b4 + b3 - 2b2 - b1 - 4) * q^73 + (-2b5 + 2b4 - 4b3 + 2b2 + 1) * q^77 + (2b5 - 2b4 + 4b3 + 3b2 + 4) * q^79 + (b5 - 3b4 + b2 - b1 - 3) q^81 + (-2b5 + 2b4 + 2b1) q^83 + (-2b4 - 4b3 - 2b2 - b1 - 6) q^87 + (-3b4 - 4b2 - b1) * q^89 + (-b5 - b4 - 8b3 + 3b2 + 1) * q^91 + (3b5 + b4 + 6b3 + b2 - b1 + 4) * q^93 + (2b5 + b4 + 5b3 + 2b2 + b1 - 2) q^97 + (b4 + 6b3 + 2b2 + 4b1 + 10) q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 3 q^{7} + 4 q^{9} + 13 q^{11} + 7 q^{13} - 4 q^{17} + 9 q^{19} + 7 q^{23} - 12 q^{27} + 2 q^{29} + 12 q^{31} - 6 q^{33} + 21 q^{37} + 26 q^{39} - 5 q^{41} - 8 q^{43} + 19 q^{47} - 3 q^{49} + 18 q^{51}+ \cdots + 43 q^{99}+O(q^{100})$ 6 * q + 3 * q^7 + 4 * q^9 + 13 * q^11 + 7 * q^13 - 4 * q^17 + 9 * q^19 + 7 * q^23 - 12 * q^27 + 2 * q^29 + 12 * q^31 - 6 * q^33 + 21 * q^37 + 26 * q^39 - 5 * q^41 - 8 * q^43 + 19 * q^47 - 3 * q^49 + 18 * q^51 + 11 * q^53 - 28 * q^57 + 25 * q^59 - 6 * q^61 - 3 * q^63 - 12 * q^69 + 34 * q^71 - 20 * q^73 + 14 * q^77 + 16 * q^79 - 18 * q^81 - 4 * q^83 - 26 * q^87 - 3 * q^89 + 26 * q^91 + 16 * q^93 - 20 * q^97 + 43 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{6} - 11x^{4} - 4x^{3} + 29x^{2} + 22x + 4$ x^6 - 11*x^4 - 4*x^3 + 29*x^2 + 22*x + 4 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{3} - 6\nu - 2$ v^3 - 6*v - 2
$\beta_{3}$	$=$	$( \nu^{5} - 11\nu^{3} - 2\nu^{2} + 29\nu + 10 ) / 2$ (v^5 - 11v^3 - 2v^2 + 29*v + 10) / 2
$\beta_{4}$	$=$	$( -\nu^{5} + 11\nu^{3} + 4\nu^{2} - 31\nu - 18 ) / 2$ (-v^5 + 11v^3 + 4v^2 - 31*v - 18) / 2
$\beta_{5}$	$=$	$( -3\nu^{5} + 2\nu^{4} + 31\nu^{3} - 6\nu^{2} - 79\nu - 26 ) / 2$ (-3v^5 + 2v^4 + 31v^3 - 6v^2 - 79*v - 26) / 2

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{4} + \beta_{3} + \beta _1 + 4$ b4 + b3 + b1 + 4
$\nu^{3}$	$=$	$\beta_{2} + 6\beta _1 + 2$ b2 + 6*b1 + 2
$\nu^{4}$	$=$	$\beta_{5} + 6\beta_{4} + 9\beta_{3} + \beta_{2} + 8\beta _1 + 24$ b5 + 6b4 + 9b3 + b2 + 8*b1 + 24
$\nu^{5}$	$=$	$2\beta_{4} + 4\beta_{3} + 11\beta_{2} + 39\beta _1 + 20$ 2b4 + 4b3 + 11b2 + 39b1 + 20

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

2.71210
2.29226
−0.306558
−0.481965
−1.81030
−2.40554

−2.71210

−0.0156607

4.35547

1.2

−2.29226

0.938845

2.25447

1.3

0.306558

2.60656

−2.90602

1.4

0.481965

−2.69841

−2.76771

1.5

1.81030

4.37760

0.277175

1.6

2.40554

−2.20893

2.78662

Atkin-Lehner signs

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

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	4000.2.a.n	yes	6
4.b	odd	2	1		4000.2.a.k	✓	6
5.b	even	2	1		4000.2.a.l	yes	6
5.c	odd	4	2		4000.2.c.g		12
8.b	even	2	1		8000.2.a.bw		6
8.d	odd	2	1		8000.2.a.bv		6
20.d	odd	2	1		4000.2.a.m	yes	6
20.e	even	4	2		4000.2.c.f		12
40.e	odd	2	1		8000.2.a.bx		6
40.f	even	2	1		8000.2.a.bu		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4000.2.a.k	✓	6	4.b	odd	2	1
4000.2.a.l	yes	6	5.b	even	2	1
4000.2.a.m	yes	6	20.d	odd	2	1
4000.2.a.n	yes	6	1.a	even	1	1	trivial
4000.2.c.f		12	20.e	even	4	2
4000.2.c.g		12	5.c	odd	4	2
8000.2.a.bu		6	40.f	even	2	1
8000.2.a.bv		6	8.d	odd	2	1
8000.2.a.bw		6	8.b	even	2	1
8000.2.a.bx		6	40.e	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4000))$ :

T_{3}^{6} - 11T_{3}^{4} + 4T_{3}^{3} + 29T_{3}^{2} - 22T_{3} + 4

T_{7}^{6} - 3T_{7}^{5} - 15T_{7}^{4} + 30T_{7}^{3} + 55T_{7}^{2} - 63T_{7} - 1

T_{11}^{6} - 13T_{11}^{5} + 41T_{11}^{4} + 50T_{11}^{3} - 325T_{11}^{2} + 375T_{11} - 125

T_{13}^{6} - 7T_{13}^{5} - 34T_{13}^{4} + 245T_{13}^{3} + 150T_{13}^{2} - 1375T_{13} + 625

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6}$ T^6
$3$	$T^{6} - 11 T^{4} + \cdots + 4$ T^6 - 11T^4 + 4T^3 + 29T^2 - 22T + 4
$5$	$T^{6}$ T^6
$7$	$T^{6} - 3 T^{5} + \cdots - 1$ T^6 - 3T^5 - 15T^4 + 30T^3 + 55T^2 - 63*T - 1
$11$	$T^{6} - 13 T^{5} + \cdots - 125$ T^6 - 13T^5 + 41T^4 + 50T^3 - 325T^2 + 375*T - 125
$13$	$T^{6} - 7 T^{5} + \cdots + 625$ T^6 - 7T^5 - 34T^4 + 245T^3 + 150T^2 - 1375*T + 625
$17$	$T^{6} + 4 T^{5} + \cdots + 2000$ T^6 + 4T^5 - 66T^4 - 240T^3 + 1025T^2 + 3500*T + 2000
$19$	$T^{6} - 9 T^{5} + \cdots + 125$ T^6 - 9T^5 - 16T^4 + 285T^3 - 300T^2 - 1125*T + 125
$23$	$T^{6} - 7 T^{5} + \cdots - 64$ T^6 - 7T^5 - 11T^4 + 104T^3 + 24T^2 - 192*T - 64
$29$	$T^{6} - 2 T^{5} + \cdots - 2636$ T^6 - 2T^5 - 54T^4 + 64T^3 + 841T^2 - 310*T - 2636
$31$	$T^{6} - 12 T^{5} + \cdots - 29500$ T^6 - 12T^5 - 59T^4 + 700T^3 + 2125T^2 - 9750*T - 29500
$37$	$T^{6} - 21 T^{5} + \cdots - 8000$ T^6 - 21T^5 + 69T^4 + 1120T^3 - 8600T^2 + 16000*T - 8000
$41$	$T^{6} + 5 T^{5} + \cdots + 10475$ T^6 + 5T^5 - 75T^4 - 450T^3 + 695T^2 + 7525*T + 10475
$43$	$T^{6} + 8 T^{5} + \cdots - 24256$ T^6 + 8T^5 - 74T^4 - 456T^3 + 2181T^2 + 6280*T - 24256
$47$	$T^{6} - 19 T^{5} + \cdots + 2549$ T^6 - 19T^5 + 56T^4 + 627T^3 - 2856T^2 - 899*T + 2549
$53$	$T^{6} - 11 T^{5} + \cdots - 9875$ T^6 - 11T^5 - 106T^4 + 1325T^3 - 650T^2 - 10875*T - 9875
$59$	$T^{6} - 25 T^{5} + \cdots - 59375$ T^6 - 25T^5 + 80T^4 + 2425T^3 - 24500T^2 + 76875*T - 59375
$61$	$T^{6} + 6 T^{5} + \cdots - 32220$ T^6 + 6T^5 - 181T^4 - 390T^3 + 7205T^2 + 5220*T - 32220
$67$	$T^{6} - 159 T^{4} + \cdots - 5364$ T^6 - 159T^4 + 68T^3 + 3509T^2 + 4074T - 5364
$71$	$T^{6} - 34 T^{5} + \cdots - 372500$ T^6 - 34T^5 + 274T^4 + 2280T^3 - 45275T^2 + 228250*T - 372500
$73$	$T^{6} + 20 T^{5} + \cdots + 2500$ T^6 + 20T^5 - 45T^4 - 2800T^3 - 14375T^2 - 8750*T + 2500
$79$	$T^{6} - 16 T^{5} + \cdots + 174500$ T^6 - 16T^5 - 206T^4 + 4100T^3 - 2275T^2 - 139500*T + 174500
$83$	$T^{6} + 4 T^{5} + \cdots - 46080$ T^6 + 4T^5 - 144T^4 - 256T^3 + 5056T^2 + 3840*T - 46080
$89$	$T^{6} + 3 T^{5} + \cdots - 142144$ T^6 + 3T^5 - 361T^4 - 136T^3 + 27344T^2 + 768*T - 142144
$97$	$T^{6} + 20 T^{5} + \cdots + 177500$ T^6 + 20T^5 - 45T^4 - 2400T^3 - 5375T^2 + 58750*T + 177500
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Elliptic curves over $\Q$
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