LMFDB - Newform orbit 4004.2.a.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(1,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, 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("4004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	4004.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:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{9} - 3x^{8} - 19x^{7} + 51x^{6} + 116x^{5} - 247x^{4} - 249x^{3} + 288x^{2} + 189x - 14$ x^9 - 3x^8 - 19x^7 + 51x^6 + 116x^5 - 247x^4 - 249x^3 + 288x^2 + 189x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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 $1,\beta_1,\ldots,\beta_{8}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_1 q^{3} - \beta_{4} q^{5} + q^{7} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots + 2) q^{9} + q^{11} - q^{13} + (\beta_{7} - \beta_{6} - \beta_{4} + \cdots - 2) q^{15} + (\beta_{8} - \beta_{3} - \beta_{2}) q^{17}+ \cdots + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots + 2) q^{99}+O(q^{100})$ q + b1 * q^3 - b4 * q^5 + q^7 + (-b4 + b3 - b2 + b1 + 2) * q^9 + q^11 - q^13 + (b7 - b6 - b4 + b1 - 2) * q^15 + (b8 - b3 - b2) * q^17 + (b4 - b2 + 1) * q^19 + b1 * q^21 + (b8 + b5 - b1 + 1) * q^23 + (b6 + b3 + 1) * q^25 + (b7 + b6 + 2b3 + 2b1 + 3) * q^27 + (-b7 - b6 - b4 + b1 + 1) * q^29 + (-2b8 - b7 - b5 + b4 + b2 + 2) q^31 + b1 * q^33 - b4 * q^35 + (-b8 - b4 - b3) * q^37 - b1 * q^39 + (-b7 - b5 - b4 - b3 + b2 - b1 + 2) * q^41 + (b8 + b7 - b5 - b3 + b2 + b1) * q^43 + (b8 + b7 - 2b6 + 2b5 - 4b4 - b2 - 1) q^45 + (-b8 + b5 - 2b4 + b1 + 1) q^47 + q^49 + (-b8 - 2b5 + 2b4 - 2b3 + 2b2 + b1 - 2) * q^51 + (-b8 - b7 + b6 - 3b4 + b3 - b2 + b1 + 3) q^53 - b4 * q^55 + (-b8 - b7 + 2b6 - b5 + 3b4 - b3 + b2 + b1) * q^57 + (-2b7 - b6 + b5 + b2 - b1 + 3) q^59 + (b8 - b7 + b6 + b4 + b2 - 2b1 + 5) q^61 + (-b4 + b3 - b2 + b1 + 2) * q^63 + b4 * q^65 + (b7 + b6 + 2b4 - b3 - b2 + 1) q^67 + (b7 - b6 - b5 + b3 + 2b2 - 2) q^69 + (b8 + b5 + b4 + b2 + b1) * q^71 + (-b8 + 3b7 - b5 + b3 - b2 + 2b1) * q^73 + (b6 + 2b4 + 3b3 - b2 + b1 + 4) * q^75 + q^77 + (-b8 - 2b7 - b6 + b4 - b3 + b2 + b1 - 4) q^79 + (b8 + b6 + 2b5 + 4b3 - b2 + 2b1 + 8) q^81 + (b7 + 2b5 + b4 + b3 - b1 + 2) q^83 + (-b8 - 2b7 - b6 + b5 - b2 - b1 + 1) q^85 + (b8 + b7 - b4 + b2 + 3b1 + 3) q^87 + (b8 - 2b5 + b4 + 2b2) * q^89 - q^91 + (-2b7 + 2b6 + b5 + 3b4 - 2b3 - 2b2 + 1) q^93 + (-b7 - 2b6 + b5 - b4 - 2b3 - b2 - b1 - 8) * q^95 + (2b4 - b3 + 2b2 - b1 + 2) * q^97 + (-b4 + b3 - b2 + b1 + 2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$9 q + 3 q^{3} + 9 q^{7} + 20 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + 5 q^{17} + 10 q^{19} + 3 q^{21} + 8 q^{23} + 3 q^{25} + 27 q^{27} + 14 q^{29} + 11 q^{31} + 3 q^{33} - 3 q^{39} + 14 q^{41} + 8 q^{43}+ \cdots + 20 q^{99}+O(q^{100})$ 9 * q + 3 * q^3 + 9 * q^7 + 20 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + 5 * q^17 + 10 * q^19 + 3 * q^21 + 8 * q^23 + 3 * q^25 + 27 * q^27 + 14 * q^29 + 11 * q^31 + 3 * q^33 - 3 * q^39 + 14 * q^41 + 8 * q^43 + 4 * q^45 + 10 * q^47 + 9 * q^49 - 15 * q^51 + 21 * q^53 - 8 * q^57 + 23 * q^59 + 34 * q^61 + 20 * q^63 + 10 * q^67 - 16 * q^69 + 4 * q^71 + 9 * q^73 + 30 * q^75 + 9 * q^77 - 34 * q^79 + 69 * q^81 + 15 * q^83 + 5 * q^85 + 39 * q^87 - 9 * q^91 + 3 * q^93 - 64 * q^95 + 15 * q^97 + 20 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{9} - 3x^{8} - 19x^{7} + 51x^{6} + 116x^{5} - 247x^{4} - 249x^{3} + 288x^{2} + 189x - 14$ x^9 - 3*x^8 - 19*x^7 + 51*x^6 + 116*x^5 - 247*x^4 - 249*x^3 + 288*x^2 + 189*x - 14 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( - 11 \nu^{8} - 1526 \nu^{7} + 1071 \nu^{6} + 27349 \nu^{5} - 11387 \nu^{4} - 132924 \nu^{3} + \cdots + 36543 ) / 15311$ (-11v^8 - 1526v^7 + 1071v^6 + 27349v^5 - 11387v^4 - 132924v^3 + 26935v^2 + 141008v + 36543) / 15311
$\beta_{3}$	$=$	$( 38 \nu^{8} - 296 \nu^{7} - 916 \nu^{6} + 7131 \nu^{5} + 7323 \nu^{4} - 46071 \nu^{3} - 19277 \nu^{2} + \cdots - 10711 ) / 15311$ (38v^8 - 296v^7 - 916v^6 + 7131v^5 + 7323v^4 - 46071v^3 - 19277v^2 + 55726v - 10711) / 15311
$\beta_{4}$	$=$	$( 49 \nu^{8} + 1230 \nu^{7} - 1987 \nu^{6} - 20218 \nu^{5} + 18710 \nu^{4} + 86853 \nu^{3} + \cdots + 29301 ) / 15311$ (49v^8 + 1230v^7 - 1987v^6 - 20218v^5 + 18710v^4 + 86853v^3 - 61523v^2 - 69971v + 29301) / 15311
$\beta_{5}$	$=$	$( 95 \nu^{8} - 740 \nu^{7} - 2290 \nu^{6} + 10172 \nu^{5} + 25963 \nu^{4} - 30967 \nu^{3} + \cdots + 103366 ) / 15311$ (95v^8 - 740v^7 - 2290v^6 + 10172v^5 + 25963v^4 - 30967v^3 - 117092v^2 + 16827v + 103366) / 15311
$\beta_{6}$	$=$	$( 626 \nu^{8} - 847 \nu^{7} - 9449 \nu^{6} + 9491 \nu^{5} + 32800 \nu^{4} - 14361 \nu^{3} + \cdots - 41874 ) / 15311$ (626v^8 - 847v^7 - 9449v^6 + 9491v^5 + 32800v^4 - 14361v^3 + 7997v^2 - 64309v - 41874) / 15311
$\beta_{7}$	$=$	$( - 702 \nu^{8} + 1439 \nu^{7} + 11281 \nu^{6} - 23753 \nu^{5} - 47446 \nu^{4} + 121814 \nu^{3} + \cdots + 17363 ) / 15311$ (-702v^8 + 1439v^7 + 11281v^6 - 23753v^5 - 47446v^4 + 121814v^3 + 30557v^2 - 169631v + 17363) / 15311
$\beta_{8}$	$=$	$( - 979 \nu^{8} + 1985 \nu^{7} + 18764 \nu^{6} - 31010 \nu^{5} - 110094 \nu^{4} + 127655 \nu^{3} + \cdots - 70160 ) / 15311$ (-979v^8 + 1985v^7 + 18764v^6 - 31010v^5 - 110094v^4 + 127655v^3 + 192431v^2 - 81863v - 70160) / 15311

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$-\beta_{4} + \beta_{3} - \beta_{2} + \beta _1 + 5$ -b4 + b3 - b2 + b1 + 5
$\nu^{3}$	$=$	$\beta_{7} + \beta_{6} + 2\beta_{3} + 8\beta _1 + 3$ b7 + b6 + 2b3 + 8b1 + 3
$\nu^{4}$	$=$	$\beta_{8} + \beta_{6} + 2\beta_{5} - 9\beta_{4} + 13\beta_{3} - 10\beta_{2} + 11\beta _1 + 44$ b8 + b6 + 2b5 - 9b4 + 13b3 - 10b2 + 11*b1 + 44
$\nu^{5}$	$=$	$\beta_{8} + 11\beta_{7} + 12\beta_{6} + 31\beta_{3} - \beta_{2} + 74\beta _1 + 49$ b8 + 11b7 + 12b6 + 31b3 - b2 + 74b1 + 49
$\nu^{6}$	$=$	$19\beta_{8} + 21\beta_{6} + 29\beta_{5} - 81\beta_{4} + 148\beta_{3} - 95\beta_{2} + 124\beta _1 + 434$ 19b8 + 21b6 + 29b5 - 81b4 + 148b3 - 95b2 + 124*b1 + 434
$\nu^{7}$	$=$	$22\beta_{8} + 110\beta_{7} + 133\beta_{6} + 3\beta_{5} - 2\beta_{4} + 394\beta_{3} - 31\beta_{2} + 734\beta _1 + 673$ 22b8 + 110b7 + 133b6 + 3b5 - 2b4 + 394b3 - 31b2 + 734b1 + 673
$\nu^{8}$	$=$	$249 \beta_{8} + 5 \beta_{7} + 310 \beta_{6} + 337 \beta_{5} - 741 \beta_{4} + 1649 \beta_{3} + \cdots + 4485$ 249b8 + 5b7 + 310b6 + 337b5 - 741b4 + 1649b3 - 924b2 + 1440b1 + 4485

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

−2.98175
−2.56319
−1.21819
−0.648104
0.0675528
1.32285
2.29815
3.32951
3.39317

−2.98175

2.28847

1.00000

5.89080

1.2

−2.56319

−1.31026

1.00000

3.56996

1.3

−1.21819

3.23173

1.00000

−1.51602

1.4

−0.648104

−1.99671

1.00000

−2.57996

1.5

0.0675528

−1.58844

1.00000

−2.99544

1.6

1.32285

−0.265081

1.00000

−1.25008

1.7

2.29815

0.952431

1.00000

2.28151

1.8

3.32951

2.67963

1.00000

8.08561

1.9

3.39317

−3.99177

1.00000

8.51361

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$7$	$-1$
$11$	$-1$
$13$	$+1$

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	4004.2.a.h	✓	9

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
4004.2.a.h	✓	9	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{9} - 3T_{3}^{8} - 19T_{3}^{7} + 51T_{3}^{6} + 116T_{3}^{5} - 247T_{3}^{4} - 249T_{3}^{3} + 288T_{3}^{2} + 189T_{3} - 14$ T3^9 - 3*T3^8 - 19*T3^7 + 51*T3^6 + 116*T3^5 - 247*T3^4 - 249*T3^3 + 288*T3^2 + 189*T3 - 14 acting on $S_{2}^{\mathrm{new}}(\Gamma_0(4004))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{9}$ T^9
$3$	$T^{9} - 3 T^{8} + \cdots - 14$ T^9 - 3T^8 - 19T^7 + 51T^6 + 116T^5 - 247T^4 - 249T^3 + 288T^2 + 189T - 14
$5$	$T^{9} - 24 T^{7} + \cdots + 83$ T^9 - 24T^7 + 4T^6 + 171T^5 + 5T^4 - 452T^3 - 125T^2 + 311*T + 83
$7$	$(T - 1)^{9}$ (T - 1)^9
$11$	$(T - 1)^{9}$ (T - 1)^9
$13$	$(T + 1)^{9}$ (T + 1)^9
$17$	$T^{9} - 5 T^{8} + \cdots + 61022$ T^9 - 5T^8 - 95T^7 + 411T^6 + 2752T^5 - 8325T^4 - 34129T^3 + 44498T^2 + 157069T + 61022
$19$	$T^{9} - 10 T^{8} + \cdots - 19801$ T^9 - 10T^8 - 42T^7 + 662T^6 - 611T^5 - 11075T^4 + 31154T^3 + 3857T^2 - 51339T - 19801
$23$	$T^{9} - 8 T^{8} + \cdots - 11536$ T^9 - 8T^8 - 61T^7 + 576T^6 + 733T^5 - 12613T^4 + 10176T^3 + 76964T^2 - 122080T - 11536
$29$	$T^{9} - 14 T^{8} + \cdots - 147280$ T^9 - 14T^8 - 47T^7 + 1438T^6 - 4831T^5 - 19321T^4 + 158216T^3 - 384444T^2 + 400064T - 147280
$31$	$T^{9} - 11 T^{8} + \cdots - 15554608$ T^9 - 11T^8 - 186T^7 + 2254T^6 + 9793T^5 - 148923T^4 - 71368T^3 + 3315032T^2 - 2692592T - 15554608
$37$	$T^{9} - 188 T^{7} + \cdots + 118400$ T^9 - 188T^7 - 175T^6 + 9023T^5 + 12410T^4 - 134392T^3 - 148032T^2 + 586240*T + 118400
$41$	$T^{9} - 14 T^{8} + \cdots - 1040000$ T^9 - 14T^8 - 86T^7 + 1445T^6 + 2309T^5 - 42230T^4 - 36824T^3 + 432224T^2 + 320512T - 1040000
$43$	$T^{9} - 8 T^{8} + \cdots + 1770695$ T^9 - 8T^8 - 211T^7 + 1806T^6 + 10221T^5 - 107699T^4 - 15742T^3 + 1661021T^2 - 3285048T + 1770695
$47$	$T^{9} - 10 T^{8} + \cdots - 11326672$ T^9 - 10T^8 - 241T^7 + 2584T^6 + 16373T^5 - 204293T^4 - 226384T^3 + 5301524T^2 - 6222048T - 11326672
$53$	$T^{9} - 21 T^{8} + \cdots - 199204765$ T^9 - 21T^8 - 200T^7 + 5936T^6 + 4023T^5 - 524141T^4 + 568286T^3 + 18083584T^2 - 17592009T - 199204765
$59$	$T^{9} - 23 T^{8} + \cdots - 775936$ T^9 - 23T^8 - 13T^7 + 2693T^6 - 4421T^5 - 100504T^4 + 78804T^3 + 1094656T^2 + 125552T - 775936
$61$	$T^{9} - 34 T^{8} + \cdots + 262558$ T^9 - 34T^8 + 311T^7 + 1211T^6 - 33072T^5 + 151099T^4 - 88711T^3 - 379519T^2 + 157753T + 262558
$67$	$T^{9} - 10 T^{8} + \cdots + 235526$ T^9 - 10T^8 - 253T^7 + 1936T^6 + 13904T^5 - 55626T^4 - 263163T^3 + 4606T^2 + 435639T + 235526
$71$	$T^{9} - 4 T^{8} + \cdots - 19670560$ T^9 - 4T^8 - 253T^7 + 1260T^6 + 18585T^5 - 104722T^4 - 449208T^3 + 2827664T^2 + 2442384T - 19670560
$73$	$T^{9} - 9 T^{8} + \cdots - 41371456$ T^9 - 9T^8 - 464T^7 + 4593T^6 + 57870T^5 - 634711T^4 - 1016792T^3 + 14138360T^2 + 12753440T - 41371456
$79$	$T^{9} + 34 T^{8} + \cdots + 3373045$ T^9 + 34T^8 + 243T^7 - 2746T^6 - 35595T^5 + 7905T^4 + 1082440T^3 + 1443135T^2 - 6722294T + 3373045
$83$	$T^{9} - 15 T^{8} + \cdots + 743353$ T^9 - 15T^8 - 261T^7 + 2506T^6 + 27639T^5 - 33263T^4 - 708834T^3 - 1291590T^2 + 470694T + 743353
$89$	$T^{9} - 538 T^{7} + \cdots - 318322999$ T^9 - 538T^7 - 80T^6 + 92901T^5 - 44233T^4 - 5905196T^3 + 10882147T^2 + 113607021*T - 318322999
$97$	$T^{9} - 15 T^{8} + \cdots + 336832$ T^9 - 15T^8 - 90T^7 + 1903T^6 - 492T^5 - 51883T^4 - 11368T^3 + 529720T^2 + 930720T + 336832
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1.9
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Embeddings

Atkin-Lehner signs

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	4004.2.a.h

Level	$4004$
Weight	$2$
Character orbit	4004.a
Self dual	yes
Analytic conductor	$31.972$
Analytic rank	$0$
Dimension	$9$
CM	no
Inner twists	$1$