LMFDB - Newform orbit 5733.2.a.br

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5733,2,Mod(1,5733)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5733, 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("5733.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$5733 = 3^{2} \cdot 7^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5733.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:	$45.7782354788$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.4507648.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1$ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 637)
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.

$f(q)$	$=$	$q + \beta_{2} q^{2} + (\beta_{4} - \beta_{3} - \beta_1 + 1) q^{4} + ( - \beta_{4} - 1) q^{5} + ( - \beta_{5} - \beta_{3} - \beta_1) q^{8} + ( - 2 \beta_{2} - \beta_1 + 1) q^{10} + (2 \beta_{5} + \beta_{4}) q^{11}+ \cdots + ( - 2 \beta_{5} - \beta_{4} - 5 \beta_{3} + \cdots - 4) q^{97}+O(q^{100})$ q + b2 * q^2 + (b4 - b3 - b1 + 1) * q^4 + (-b4 - 1) * q^5 + (-b5 - b3 - b1) * q^8 + (-2b2 - b1 + 1) q^10 + (2b5 + b4) q^11 + q^13 + (-b4 - b3 + b2) * q^16 + (b5 + b4 + b3 + b2 + 2b1 - 3) q^17 + (-b5 - b4 + b3 - b2) * q^19 + (b3 + b2 + b1 - 3) * q^20 + (-2b5 - 2b4 + 4b3 - b2 + b1 - 3) q^22 + (-2b5 - b4 + b3 + b1) q^23 + (b4 + b3 + b1 - 1) * q^25 + b2 * q^26 + (2b4 - 2b3 + 1) * q^29 + (b5 - 3b4 - 2b2 + b1 + 1) * q^31 + (b5 + b4 + b3 - b2 - b1 + 4) * q^32 + (3b3 - 3b2 + 3b1 - 1) q^34 + (2b5 + b4 - 2b3 + 2b2 + 2b1) * q^37 + (2b5 - b3 + b1 - 1) q^38 + (b5 + b4 + b2 + 3b1) q^40 + (-b5 - 2b3 - 2b2 + 3b1) q^41 + (-2b5 + 2b4 - 4b1 + 1) q^43 + (2b5 - b4 - 2b3 - 3b2 + 4b1) * q^44 + (3b5 + 2b4 - 3b3 + b2 + b1 + 2) q^46 + (-3b5 - b4 + 3b3 - 3b2 - 6) q^47 + (b5 + b3 + 3b1 - 2) q^50 + (b4 - b3 - b1 + 1) * q^52 + (b4 - b3 + 4b2 + b1 + 2) q^53 + (-2b5 + 3b3 - 3b1 - 1) q^55 + (-2b5 + 3b2 - 2) * q^58 + (-2b5 - 2b4 + 3b3 - b2 + b1 - 5) q^59 + (3b5 - b4 - 2b3 + b2 - 3b1 + 2) q^61 + (-b5 - 3b4 + 5b3 - 3b2 - 5) q^62 + (4b3 + 2b2 + 2b1 - 4) q^64 + (-b4 - 1) * q^65 + (3b5 + b4 + b3 + b2 - b1 + 4) q^67 + (b5 - 5b4 + 4b3 - 3b2 + 5b1 - 6) * q^68 + (-3b5 - 5b4 + 5b3 - 4b2 - b1 - 2) * q^71 + (-5b5 - b4 + b3 - b2 - 2b1 - 2) * q^73 + (-4b5 + 4b3 - b2 - b1 + 1) * q^74 + (-b5 + 3b3 - b2 - 3) q^76 + (3b4 + 3b3 - 3b2 - 2b1 - 3) * q^79 + (-b5 + 2b3 - 2b2 + b1 + 4) * q^80 + (-b5 - b4 + 3b3 + b2 + 3b1 - 8) * q^82 + (2b5 - b4 - 2b1 - 7) * q^83 + (-2b5 + b4 - 2b3 - 4b2 - 6b1) * q^85 + (2b5 + 2b4 - 8b3 + 5b2 - 2b1 + 4) q^86 + (-b4 + 3b3 - b2 + 2b1 - 8) * q^88 + (2b5 - 3b4 + b3 - b2 + b1 - 4) * q^89 + (-2b5 + 4b3 + b2 - 3b1 - 3) q^92 + (6b5 - 3b3 - 4b2 + 5b1 - 5) * q^94 + (2b5 - 2b3 + 2b2 + 2b1 + 1) * q^95 + (-2b5 - b4 - 5b3 + 3b2 + 3b1 - 4) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$6 q + 4 q^{4} - 6 q^{5} + 4 q^{10} - 4 q^{11} + 6 q^{13} - 16 q^{17} + 2 q^{19} - 16 q^{20} - 12 q^{22} + 6 q^{23} - 4 q^{25} + 6 q^{29} + 6 q^{31} + 20 q^{32} - 8 q^{38} + 4 q^{40} + 8 q^{41} + 2 q^{43}+ \cdots - 14 q^{97}+O(q^{100})$ 6 * q + 4 * q^4 - 6 * q^5 + 4 * q^10 - 4 * q^11 + 6 * q^13 - 16 * q^17 + 2 * q^19 - 16 * q^20 - 12 * q^22 + 6 * q^23 - 4 * q^25 + 6 * q^29 + 6 * q^31 + 20 * q^32 - 8 * q^38 + 4 * q^40 + 8 * q^41 + 2 * q^43 + 4 * q^44 + 8 * q^46 - 30 * q^47 - 8 * q^50 + 4 * q^52 + 14 * q^53 - 8 * q^55 - 8 * q^58 - 24 * q^59 - 28 * q^62 - 20 * q^64 - 6 * q^65 + 16 * q^67 - 28 * q^68 - 8 * q^71 - 6 * q^73 + 12 * q^74 - 16 * q^76 - 22 * q^79 + 28 * q^80 - 40 * q^82 - 50 * q^83 - 8 * q^85 + 16 * q^86 - 44 * q^88 - 26 * q^89 - 20 * q^92 - 32 * q^94 + 6 * q^95 - 14 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1$ x^6 - 2*x^5 - 5*x^4 + 8*x^3 + 7*x^2 - 6*x - 1 :

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

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} + \beta _1 + 2$ b2 + b1 + 2
$\nu^{3}$	$=$	$\beta_{3} + \beta_{2} + 4\beta _1 + 1$ b3 + b2 + 4*b1 + 1
$\nu^{4}$	$=$	$\beta_{4} + \beta_{3} + 5\beta_{2} + 6\beta _1 + 7$ b4 + b3 + 5b2 + 6b1 + 7
$\nu^{5}$	$=$	$\beta_{5} + 2\beta_{4} + 6\beta_{3} + 8\beta_{2} + 18\beta _1 + 8$ b5 + 2b4 + 6b3 + 8b2 + 18b1 + 8

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

0.758419
−0.146243
1.90903
−1.20475
2.35100
−1.66745

−2.18322

2.76645

−2.11065

−1.67333

4.60802

1.2

−1.83237

1.35758

−2.62555

1.17715

4.81098

1.3

−0.264627

−1.92997

1.43515

1.03998

−0.379780

1.4

0.656184

−1.56942

1.35996

−2.34220

0.892385

1.5

1.17619

−0.616586

−3.14862

−3.07759

−3.70337

1.6

2.44785

3.99195

−0.910286

4.87599

−2.22824

Atkin-Lehner signs

$p$	Sign
$3$	$-1$
$7$	$+1$
$13$	$-1$

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	5733.2.a.br		6
3.b	odd	2	1		637.2.a.n	yes	6
7.b	odd	2	1		5733.2.a.bu		6
21.c	even	2	1		637.2.a.m	✓	6
21.g	even	6	2		637.2.e.o		12
21.h	odd	6	2		637.2.e.n		12
39.d	odd	2	1		8281.2.a.cd		6
273.g	even	2	1		8281.2.a.cc		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
637.2.a.m	✓	6	21.c	even	2	1
637.2.a.n	yes	6	3.b	odd	2	1
637.2.e.n		12	21.h	odd	6	2
637.2.e.o		12	21.g	even	6	2
5733.2.a.br		6	1.a	even	1	1	trivial
5733.2.a.bu		6	7.b	odd	2	1
8281.2.a.cc		6	273.g	even	2	1
8281.2.a.cd		6	39.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 $S_{2}^{\mathrm{new}}(\Gamma_0(5733))$ :

T_{2}^{6} - 8T_{2}^{4} + 14T_{2}^{2} - 4T_{2} - 2

T_{5}^{6} + 6T_{5}^{5} + 5T_{5}^{4} - 24T_{5}^{3} - 31T_{5}^{2} + 26T_{5} + 31

T_{11}^{6} + 4T_{11}^{5} - 38T_{11}^{4} - 156T_{11}^{3} + 186T_{11}^{2} + 692T_{11} - 562

T_{17}^{6} + 16T_{17}^{5} + 56T_{17}^{4} - 324T_{17}^{3} - 2792T_{17}^{2} - 6792T_{17} - 5294

T_{19}^{6} - 2T_{19}^{5} - 17T_{19}^{4} + 16T_{19}^{3} + 83T_{19}^{2} - 6T_{19} - 73

T_{31}^{6} - 6T_{31}^{5} - 115T_{31}^{4} + 508T_{31}^{3} + 4181T_{31}^{2} - 10046T_{31} - 44249

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6} - 8 T^{4} + \cdots - 2$ T^6 - 8T^4 + 14T^2 - 4*T - 2
$3$	$T^{6}$ T^6
$5$	$T^{6} + 6 T^{5} + \cdots + 31$ T^6 + 6T^5 + 5T^4 - 24T^3 - 31T^2 + 26*T + 31
$7$	$T^{6}$ T^6
$11$	$T^{6} + 4 T^{5} + \cdots - 562$ T^6 + 4T^5 - 38T^4 - 156T^3 + 186T^2 + 692*T - 562
$13$	$(T - 1)^{6}$ (T - 1)^6
$17$	$T^{6} + 16 T^{5} + \cdots - 5294$ T^6 + 16T^5 + 56T^4 - 324T^3 - 2792T^2 - 6792*T - 5294
$19$	$T^{6} - 2 T^{5} + \cdots - 73$ T^6 - 2T^5 - 17T^4 + 16T^3 + 83T^2 - 6*T - 73
$23$	$T^{6} - 6 T^{5} + \cdots + 529$ T^6 - 6T^5 - 37T^4 + 272T^3 - 169T^2 - 874*T + 529
$29$	$T^{6} - 6 T^{5} + \cdots + 529$ T^6 - 6T^5 - 33T^4 + 268T^3 - 401T^2 - 230*T + 529
$31$	$T^{6} - 6 T^{5} + \cdots - 44249$ T^6 - 6T^5 - 115T^4 + 508T^3 + 4181T^2 - 10046*T - 44249
$37$	$T^{6} - 82 T^{4} + \cdots + 254$ T^6 - 82T^4 + 236T^3 + 378T^2 - 716T + 254
$41$	$T^{6} - 8 T^{5} + \cdots + 28784$ T^6 - 8T^5 - 120T^4 + 968T^3 + 1720T^2 - 19744*T + 28784
$43$	$T^{6} - 2 T^{5} + \cdots + 35153$ T^6 - 2T^5 - 161T^4 - 188T^3 + 6575T^2 + 29214*T + 35153
$47$	$T^{6} + 30 T^{5} + \cdots - 135617$ T^6 + 30T^5 + 215T^4 - 1480T^3 - 25621T^2 - 107350*T - 135617
$53$	$T^{6} - 14 T^{5} + \cdots - 1319$ T^6 - 14T^5 - 37T^4 + 680T^3 + 1199T^2 - 3386*T - 1319
$59$	$T^{6} + 24 T^{5} + \cdots + 1532$ T^6 + 24T^5 + 146T^4 - 44T^3 - 1236T^2 - 760*T + 1532
$61$	$T^{6} - 246 T^{4} + \cdots - 216584$ T^6 - 246T^4 + 112T^3 + 16252T^2 - 20768T - 216584
$67$	$T^{6} - 16 T^{5} + \cdots + 6112$ T^6 - 16T^5 + 800T^3 - 1984T^2 - 3712T + 6112
$71$	$T^{6} + 8 T^{5} + \cdots - 1206162$ T^6 + 8T^5 - 316T^4 - 1856T^3 + 33274T^2 + 104028*T - 1206162
$73$	$T^{6} + 6 T^{5} + \cdots - 142657$ T^6 + 6T^5 - 241T^4 - 808T^3 + 12011T^2 + 22234*T - 142657
$79$	$T^{6} + 22 T^{5} + \cdots + 7913$ T^6 + 22T^5 - 65T^4 - 3172T^3 - 11129T^2 + 9486*T + 7913
$83$	$T^{6} + 50 T^{5} + \cdots - 167041$ T^6 + 50T^5 + 941T^4 + 7992T^3 + 26145T^2 - 13034*T - 167041
$89$	$T^{6} + 26 T^{5} + \cdots + 9959$ T^6 + 26T^5 + 131T^4 - 1188T^3 - 9053T^2 + 26*T + 9959
$97$	$T^{6} + 14 T^{5} + \cdots + 217287$ T^6 + 14T^5 - 261T^4 - 3620T^3 + 13171T^2 + 193902*T + 217287
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