LMFDB - Newform orbit 8800.2.a.bw

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:	$5$
Coefficient field:	5.5.5060976.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{5} - 2x^{4} - 7x^{3} + 12x^{2} - x - 2$ x^5 - 2x^4 - 7x^3 + 12*x^2 - x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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 $1,\beta_1,\beta_2,\beta_3,\beta_4$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_1 q^{3} + ( - \beta_{3} - 1) q^{7} + (\beta_{2} + 1) q^{9} - q^{11} + ( - \beta_{4} + \beta_{3}) q^{13} + (\beta_{4} + \beta_{3} - \beta_1 + 1) q^{17} + (\beta_{4} - \beta_{2} - 1) q^{19} + ( - \beta_{3} - \beta_1 + 1) q^{21}+ \cdots + ( - \beta_{2} - 1) q^{99}+O(q^{100})$ q + b1 * q^3 + (-b3 - 1) * q^7 + (b2 + 1) * q^9 - q^11 + (-b4 + b3) * q^13 + (b4 + b3 - b1 + 1) * q^17 + (b4 - b2 - 1) * q^19 + (-b3 - b1 + 1) * q^21 + (-b3 - b2 - b1) * q^23 + (b4 + b3 + b2 + b1 + 1) * q^27 + (-b3 - 2b1) q^29 + (-2b4 + b3 - b2 + 2b1 - 4) * q^31 - b1 * q^33 + (-b4 - b1) * q^37 + (b3 - b2 + b1 - 3) * q^39 + (3b3 + 3b1 + 1) * q^41 + (-b4 - 2b3 - b1 + 1) q^43 + (b4 - b3 + b2 - b1 - 1) * q^47 + (-b4 - b2 + b1 - 3) * q^49 + (b3 - 3) * q^51 + (2b4 + 2b3 + b2 - 2b1 + 4) q^53 + (-b4 - b3 - 5b1 + 1) q^57 + (b4 - b3 - b2 - b1 - 1) * q^59 + (-b4 + 2b3 + 4b1 - 2) * q^61 + (2b3 - b2 + b1) q^63 + (-2b4 - 2) q^67 + (-b4 - 2b3 - 2b2 - 3b1 - 4) q^69 + (b4 - 2b3 - 3b1 - 1) * q^71 + (b4 + 2b3 + 2b1) * q^73 + (b3 + 1) * q^77 + (-2b4 - 3b3 + b2 - 3b1 - 1) q^79 + (b4 + 2b3 + 3b1 + 3) * q^81 + (3b4 + b3 + b2 + 6) q^83 + (-b3 - 2b2 - 7) q^87 + (2b4 + b2 - 2b1 + 3) * q^89 + (3b4 - b3 + b2 - 2b1 - 3) * q^91 + (-b4 - b2 - 5b1 + 2) q^93 + (b4 + 2b3 - b2 - 3b1 - 1) * q^97 + (-b2 - 1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$5 q + 2 q^{3} - 4 q^{7} + 3 q^{9} - 5 q^{11} + q^{13} - 5 q^{19} + 4 q^{21} + q^{23} + 2 q^{27} - 3 q^{29} - 11 q^{31} - 2 q^{33} - 12 q^{39} + 8 q^{41} + 7 q^{43} - 10 q^{47} - 9 q^{49} - 16 q^{51} + 8 q^{53}+ \cdots - 3 q^{99}+O(q^{100})$ 5 * q + 2 * q^3 - 4 * q^7 + 3 * q^9 - 5 * q^11 + q^13 - 5 * q^19 + 4 * q^21 + q^23 + 2 * q^27 - 3 * q^29 - 11 * q^31 - 2 * q^33 - 12 * q^39 + 8 * q^41 + 7 * q^43 - 10 * q^47 - 9 * q^49 - 16 * q^51 + 8 * q^53 - 2 * q^57 - 6 * q^59 - 2 * q^61 + 2 * q^63 - 6 * q^67 - 18 * q^69 - 11 * q^71 + 4 * q^77 - 6 * q^79 + 17 * q^81 + 21 * q^83 - 30 * q^87 + 5 * q^89 - 26 * q^91 + 4 * q^93 - 13 * q^97 - 3 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{5} - 2x^{4} - 7x^{3} + 12x^{2} - x - 2$ x^5 - 2*x^4 - 7*x^3 + 12*x^2 - x - 2 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} - 4$ v^2 - 4
$\beta_{3}$	$=$	$\nu^{4} - \nu^{3} - 8\nu^{2} + 4\nu + 3$ v^4 - v^3 - 8v^2 + 4v + 3
$\beta_{4}$	$=$	$-\nu^{4} + 2\nu^{3} + 7\nu^{2} - 11\nu$ -v^4 + 2v^3 + 7v^2 - 11*v

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + 4$ b2 + 4
$\nu^{3}$	$=$	$\beta_{4} + \beta_{3} + \beta_{2} + 7\beta _1 + 1$ b4 + b3 + b2 + 7*b1 + 1
$\nu^{4}$	$=$	$\beta_{4} + 2\beta_{3} + 9\beta_{2} + 3\beta _1 + 30$ b4 + 2b3 + 9b2 + 3*b1 + 30

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

−2.56529
−0.342572
0.594438
1.24959
3.06383

−2.56529

−1.28048

3.58073

1.2

−0.342572

−1.74484

−2.88264

1.3

0.594438

−3.46571

−2.64664

1.4

1.24959

3.00650

−1.43851

1.5

3.06383

−0.515465

6.38707

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.bw	yes	5
4.b	odd	2	1		8800.2.a.bt	yes	5
5.b	even	2	1		8800.2.a.bs	✓	5
20.d	odd	2	1		8800.2.a.bx	yes	5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
8800.2.a.bs	✓	5	5.b	even	2	1
8800.2.a.bt	yes	5	4.b	odd	2	1
8800.2.a.bw	yes	5	1.a	even	1	1	trivial
8800.2.a.bx	yes	5	20.d	odd	2	1

Hecke kernels

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

T_{3}^{5} - 2T_{3}^{4} - 7T_{3}^{3} + 12T_{3}^{2} - T_{3} - 2

T_{7}^{5} + 4T_{7}^{4} - 5T_{7}^{3} - 34T_{7}^{2} - 39T_{7} - 12

T_{13}^{5} - T_{13}^{4} - 36T_{13}^{3} - 28T_{13}^{2} + 247T_{13} + 361

T_{17}^{5} - 39T_{17}^{3} - 36T_{17}^{2} + 43T_{17} + 40

T_{19}^{5} + 5T_{19}^{4} - 48T_{19}^{3} - 288T_{19}^{2} - 237T_{19} + 23

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{5}$ T^5
$3$	$T^{5} - 2 T^{4} + \cdots - 2$ T^5 - 2T^4 - 7T^3 + 12*T^2 - T - 2
$5$	$T^{5}$ T^5
$7$	$T^{5} + 4 T^{4} + \cdots - 12$ T^5 + 4T^4 - 5T^3 - 34T^2 - 39T - 12
$11$	$(T + 1)^{5}$ (T + 1)^5
$13$	$T^{5} - T^{4} + \cdots + 361$ T^5 - T^4 - 36T^3 - 28T^2 + 247*T + 361
$17$	$T^{5} - 39 T^{3} + \cdots + 40$ T^5 - 39T^3 - 36T^2 + 43*T + 40
$19$	$T^{5} + 5 T^{4} + \cdots + 23$ T^5 + 5T^4 - 48T^3 - 288T^2 - 237T + 23
$23$	$T^{5} - T^{4} + \cdots - 25$ T^5 - T^4 - 51T^3 + 159T^2 - 33*T - 25
$29$	$T^{5} + 3 T^{4} + \cdots + 9$ T^5 + 3T^4 - 31T^3 - 65T^2 + 147T + 9
$31$	$T^{5} + 11 T^{4} + \cdots - 1385$ T^5 + 11T^4 - 78T^3 - 1318T^2 - 4247T - 1385
$37$	$T^{5} - 38 T^{3} + \cdots - 32$ T^5 - 38T^3 - 8T^2 + 317*T - 32
$41$	$T^{5} - 8 T^{4} + \cdots - 8356$ T^5 - 8T^4 - 104T^3 + 620T^2 + 2987T - 8356
$43$	$T^{5} - 7 T^{4} + \cdots + 464$ T^5 - 7T^4 - 48T^3 + 172T^2 + 768T + 464
$47$	$T^{5} + 10 T^{4} + \cdots + 64$ T^5 + 10T^4 - 24T^3 - 78T^2 + 39T + 64
$53$	$T^{5} - 8 T^{4} + \cdots - 15514$ T^5 - 8T^4 - 141T^3 + 1216T^2 + 889T - 15514
$59$	$T^{5} + 6 T^{4} + \cdots + 2560$ T^5 + 6T^4 - 56T^3 - 338T^2 + 311T + 2560
$61$	$T^{5} + 2 T^{4} + \cdots + 14890$ T^5 + 2T^4 - 137T^3 - 350T^2 + 4539T + 14890
$67$	$T^{5} + 6 T^{4} + \cdots - 736$ T^5 + 6T^4 - 76T^3 - 120T^2 + 1168T - 736
$71$	$T^{5} + 11 T^{4} + \cdots - 1744$ T^5 + 11T^4 - 48T^3 - 332T^2 + 1632T - 1744
$73$	$T^{5} - 89 T^{3} + \cdots - 4132$ T^5 - 89T^3 + 92T^2 + 2003*T - 4132
$79$	$T^{5} + 6 T^{4} + \cdots + 61760$ T^5 + 6T^4 - 261T^3 - 1392T^2 + 11317T + 61760
$83$	$T^{5} - 21 T^{4} + \cdots - 3643$ T^5 - 21T^4 - 59T^3 + 1835T^2 + 6317T - 3643
$89$	$T^{5} - 5 T^{4} + \cdots - 9991$ T^5 - 5T^4 - 107T^3 + 569T^2 + 1827T - 9991
$97$	$T^{5} + 13 T^{4} + \cdots - 6693$ T^5 + 13T^4 - 151T^3 - 1337T^2 + 6507T - 6693
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