LMFDB - Newform orbit 416.4.a.j

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,4,Mod(1,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("416.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$416 = 2^{5} \cdot 13$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	416.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:	$24.5447945624$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 3x^{5} - 101x^{4} + 207x^{3} + 751x^{2} - 855x + 112$ x^6 - 3x^5 - 101x^4 + 207x^3 + 751x^2 - 855*x + 112
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{8}$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

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

0.152104
3.56475
−9.25891
10.2589
−2.56475
0.847896

−10.3340

9.49534

8.94239

79.7910

1.2

−4.53772

−20.8058

16.7967

−6.40912

1.3

−3.82336

3.31043

−35.2123

−12.3819

1.4

3.82336

3.31043

35.2123

−12.3819

1.5

4.53772

−20.8058

−16.7967

−6.40912

1.6

10.3340

9.49534

−8.94239

79.7910

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$13$	$+1$

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	416.4.a.j	✓	6
4.b	odd	2	1	inner	416.4.a.j	✓	6
8.b	even	2	1		832.4.a.bi		6
8.d	odd	2	1		832.4.a.bi		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
416.4.a.j	✓	6	1.a	even	1	1	trivial
416.4.a.j	✓	6	4.b	odd	2	1	inner
832.4.a.bi		6	8.b	even	2	1
832.4.a.bi		6	8.d	odd	2	1

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{6} - 142T_{3}^{4} + 4061T_{3}^{2} - 32144$ T3^6 - 142*T3^4 + 4061*T3^2 - 32144 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(416))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{6}$ T^6
$3$	$T^{6} - 142 T^{4} + \cdots - 32144$ T^6 - 142T^4 + 4061T^2 - 32144
$5$	$(T^{3} + 8 T^{2} + \cdots + 654)^{2}$ (T^3 + 8T^2 - 235T + 654)^2
$7$	$T^{6} - 1602 T^{4} + \cdots - 27973316$ T^6 - 1602T^4 + 471525T^2 - 27973316
$11$	$T^{6} - 5484 T^{4} + \cdots - 68016704$ T^6 - 5484T^4 + 1453104T^2 - 68016704
$13$	$(T + 13)^{6}$ (T + 13)^6
$17$	$(T^{3} - 48 T^{2} + \cdots - 99414)^{2}$ (T^3 - 48T^2 - 6155T - 99414)^2
$19$	$T^{6} + \cdots - 3134758976$ T^6 - 19532T^4 + 67823920T^2 - 3134758976
$23$	$T^{6} + \cdots - 306970886144$ T^6 - 33680T^4 + 265987072T^2 - 306970886144
$29$	$(T^{3} - 218 T^{2} + \cdots + 1440232)^{2}$ (T^3 - 218T^2 - 6468T + 1440232)^2
$31$	$T^{6} + \cdots - 3716023320576$ T^6 - 54564T^4 + 844502848T^2 - 3716023320576
$37$	$(T^{3} - 372 T^{2} + \cdots + 10002726)^{2}$ (T^3 - 372T^2 - 21647T + 10002726)^2
$41$	$(T^{3} - 718 T^{2} + \cdots - 12682752)^{2}$ (T^3 - 718T^2 + 167912T - 12682752)^2
$43$	$T^{6} + \cdots - 6973565239296$ T^6 - 192102T^4 + 7087804645T^2 - 6973565239296
$47$	$T^{6} + \cdots - 112256330631044$ T^6 - 206570T^4 + 11376708781T^2 - 112256330631044
$53$	$(T^{3} + 142 T^{2} + \cdots + 30322624)^{2}$ (T^3 + 142T^2 - 218136T + 30322624)^2
$59$	$T^{6} + \cdots - 16\!\cdots\!04$ T^6 - 795884T^4 + 199610345008T^2 - 16069600736995904
$61$	$(T^{3} + 62 T^{2} + \cdots - 4259328)^{2}$ (T^3 + 62T^2 - 56488T - 4259328)^2
$67$	$T^{6} + \cdots - 97\!\cdots\!16$ T^6 - 1397996T^4 + 643170666544T^2 - 97041222482596416
$71$	$T^{6} + \cdots - 11\!\cdots\!96$ T^6 - 1782962T^4 + 791594394325T^2 - 11671148432914596
$73$	$(T^{3} - 602 T^{2} + \cdots - 5585048)^{2}$ (T^3 - 602T^2 - 361988T - 5585048)^2
$79$	$T^{6} + \cdots - 17\!\cdots\!96$ T^6 - 2299056T^4 + 1469902529280T^2 - 179786365510455296
$83$	$T^{6} + \cdots - 45\!\cdots\!36$ T^6 - 1432788T^4 + 221484834048T^2 - 4545716263387136
$89$	$(T^{3} - 2198 T^{2} + \cdots + 909803736)^{2}$ (T^3 - 2198T^2 + 349500T + 909803736)^2
$97$	$(T^{3} - 2362 T^{2} + \cdots - 183907864)^{2}$ (T^3 - 2362T^2 + 1339516T - 183907864)^2
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 $\F_{q}$

$f(q)$	$=$	$q - \beta_{3} q^{3} + ( - \beta_1 - 3) q^{5} + \beta_{4} q^{7} + (\beta_{2} - 2 \beta_1 + 20) q^{9} + ( - \beta_{5} - 3 \beta_{3}) q^{11} - 13 q^{13} + (2 \beta_{5} - 5 \beta_{3}) q^{15} + ( - 2 \beta_{2} - \beta_1 + 15) q^{17}+ \cdots + (12 \beta_{5} - 7 \beta_{4} - 27 \beta_{3}) q^{99}+O(q^{100})$ q - b3 * q^3 + (-b1 - 3) * q^5 + b4 * q^7 + (b2 - 2b1 + 20) q^9 + (-b5 - 3b3) q^11 - 13 * q^13 + (2b5 - 5b3) * q^15 + (-2b2 - b1 + 15) q^17 + (-b4 - 11b3) q^19 + (-3b2 - 2b1 - 13) * q^21 + (-3b5 + b4 + 2b3) * q^23 + (2b2 + 13b1 + 58) * q^25 + (7b5 - 7b4 - 31b3) q^27 + (b2 - 9b1 + 70) q^29 + (-3b5 - 2b4 + 7b3) q^31 + (b2 + 9b1 + 158) q^33 + (-6b5 + 2b4 - 7b3) q^35 + (5b2 + 10b1 + 129) * q^37 + 13b3 q^39 + (-b2 + 3b1 + 240) q^41 + (-3b5 + 11b4 + 5b3) q^43 + (9b2 - 13b1 + 282) * q^45 + (b5 - 6b4 + 34b3) * q^47 + (-15b2 - 6b1 + 184) * q^49 + (-4b5 + 14b4 + 21b3) q^51 + (-9b2 + 19b1 - 44) * q^53 + (21b5 + 7b4 + 34b3) q^55 + (14b2 - 20b1 + 530) * q^57 + (13b5 - 14b4 - 19b3) q^59 + (3b2 - 13b1 - 24) * q^61 + (-5b5 - 6b4 + 63b3) q^63 + (13b1 + 39) q^65 + (2b5 + 25b4 + 41b3) q^67 + (-11b2 + 47b1 - 56) * q^69 + (-20b5 - b4 + 40b3) * q^71 + (-17b2 - 3b1 + 194) * q^73 + (-20b5 - 14b4 + 2b3) q^75 + (-11b2 - 47b1 - 274) * q^77 + (-20b5 - 6b4 + 66b3) q^79 + (39b2 - 99b1 + 889) * q^81 + (6b5 - 31b4 + 3b3) q^83 + (-8b2 + 21b1 + 141) * q^85 + (21b5 - 7b4 - 164b3) q^87 + (2b2 - 70b1 + 710) * q^89 - 13b4 q^91 + (-7b2 + 63b1 - 252) * q^93 + (28b5 - 2b4 - 48b3) q^95 + (-b2 + 45b1 + 802) q^97 + (12b5 - 7b4 - 27b3) q^99
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 16 q^{5} + 122 q^{9} - 78 q^{13} + 96 q^{17} - 68 q^{21} + 318 q^{25} + 436 q^{29} + 928 q^{33} + 744 q^{37} + 1436 q^{41} + 1700 q^{45} + 1146 q^{49} - 284 q^{53} + 3192 q^{57} - 124 q^{61} + 208 q^{65}+ \cdots + 4724 q^{97}+O(q^{100})$ 6 * q - 16 * q^5 + 122 * q^9 - 78 * q^13 + 96 * q^17 - 68 * q^21 + 318 * q^25 + 436 * q^29 + 928 * q^33 + 744 * q^37 + 1436 * q^41 + 1700 * q^45 + 1146 * q^49 - 284 * q^53 + 3192 * q^57 - 124 * q^61 + 208 * q^65 - 408 * q^69 + 1204 * q^73 - 1528 * q^77 + 5454 * q^81 + 820 * q^85 + 4396 * q^89 - 1624 * q^93 + 4724 * q^97

$\beta_{1}$	$=$	$( -5\nu^{4} + 10\nu^{3} + 478\nu^{2} - 483\nu - 1612 ) / 134$ (-5v^4 + 10v^3 + 478v^2 - 483v - 1612) / 134
$\beta_{2}$	$=$	$( 3\nu^{4} - 6\nu^{3} - 394\nu^{2} + 397\nu + 4612 ) / 134$ (3v^4 - 6v^3 - 394v^2 + 397v + 4612) / 134
$\beta_{3}$	$=$	$( -62\nu^{5} + 155\nu^{4} + 6294\nu^{3} - 9596\nu^{2} - 47707\nu + 25458 ) / 1742$ (-62v^5 + 155v^4 + 6294v^3 - 9596v^2 - 47707*v + 25458) / 1742
$\beta_{4}$	$=$	$( -62\nu^{5} + 155\nu^{4} + 6294\nu^{3} - 9596\nu^{2} - 40739\nu + 21974 ) / 1742$ (-62v^5 + 155v^4 + 6294v^3 - 9596v^2 - 40739*v + 21974) / 1742
$\beta_{5}$	$=$	$( 90\nu^{5} - 225\nu^{4} - 9586\nu^{3} + 14604\nu^{2} + 110161\nu - 57522 ) / 1742$ (90v^5 - 225v^4 - 9586v^3 + 14604v^2 + 110161*v - 57522) / 1742

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more