LMFDB - Newform orbit 1800.4.f.ba

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,4,Mod(649,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("1800.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1800.f (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$106.203438010$
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} + 175x^{4} + 8109x^{2} + 72900 $ x^6 + 175x^4 + 8109x^2 + 72900
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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_{4} - 3 \beta_1) q^{7}+O(q^{10}) $ q + (-b4 - 3b1) q^7	\( q + ( - \beta_{4} - 3 \beta_1) q^{7} + ( - \beta_{3} + \beta_{2} - 3) q^{11} + (2 \beta_{5} + 5 \beta_1) q^{13} + (3 \beta_{5} + \beta_{4} + 15 \beta_1) q^{17} + ( - \beta_{3} + 2 \beta_{2} + 3) q^{19} + ( - \beta_{5} - 7 \beta_{4} - 13 \beta_1) q^{23} + ( - 7 \beta_{3} + 3 \beta_{2} - 1) q^{29} + (7 \beta_{3} - 4 \beta_{2} + 21) q^{31} + (4 \beta_{5} + 4 \beta_{4} + 2 \beta_1) q^{37} + ( - 11 \beta_{3} - \beta_{2} - 133) q^{41} + ( - 6 \beta_{5} - 7 \beta_{4} - 51 \beta_1) q^{43} + ( - 10 \beta_{5} + 10 \beta_{4} + 94 \beta_1) q^{47} + (4 \beta_{3} - 8 \beta_{2} - 34) q^{49} + (\beta_{5} - 13 \beta_{4} + 85 \beta_1) q^{53} + ( - 6 \beta_{3} - 10 \beta_{2} + 62) q^{59} + ( - 28 \beta_{3} + 6 \beta_{2} - 127) q^{61} + ( - 10 \beta_{5} - 13 \beta_{4} + 51 \beta_1) q^{67} + (5 \beta_{3} - 13 \beta_{2} - 425) q^{71} + ( - 4 \beta_{5} + 28 \beta_{4} + 134 \beta_1) q^{73} + ( - 5 \beta_{5} - 15 \beta_{4} + 247 \beta_1) q^{77} + ( - 12 \beta_{3} + 4 \beta_{2} + 96) q^{79} + (14 \beta_{5} + 18 \beta_{4} + 470 \beta_1) q^{83} + (24 \beta_{3} - 8 \beta_{2} + 312) q^{89} + (27 \beta_{3} + 26 \beta_{2} + 275) q^{91} + ( - 16 \beta_{5} + 36 \beta_{4} - 227 \beta_1) q^{97}+O(q^{100}) \) q + (-b4 - 3b1) q^7 + (-b3 + b2 - 3) * q^11 + (2b5 + 5b1) * q^13 + (3b5 + b4 + 15b1) * q^17 + (-b3 + 2b2 + 3) q^19 + (-b5 - 7b4 - 13b1) * q^23 + (-7b3 + 3b2 - 1) * q^29 + (7b3 - 4b2 + 21) * q^31 + (4b5 + 4b4 + 2b1) q^37 + (-11b3 - b2 - 133) q^41 + (-6b5 - 7b4 - 51b1) q^43 + (-10b5 + 10b4 + 94b1) q^47 + (4b3 - 8b2 - 34) * q^49 + (b5 - 13b4 + 85b1) * q^53 + (-6b3 - 10b2 + 62) * q^59 + (-28b3 + 6b2 - 127) * q^61 + (-10b5 - 13b4 + 51b1) q^67 + (5b3 - 13b2 - 425) * q^71 + (-4b5 + 28b4 + 134b1) q^73 + (-5b5 - 15b4 + 247b1) q^77 + (-12b3 + 4b2 + 96) * q^79 + (14b5 + 18b4 + 470b1) q^83 + (24b3 - 8b2 + 312) * q^89 + (27b3 + 26b2 + 275) * q^91 + (-16b5 + 36b4 - 227b1) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q - 16 q^{11} + 22 q^{19} + 118 q^{31} - 800 q^{41} - 220 q^{49} + 352 q^{59} - 750 q^{61} - 2576 q^{71} + 584 q^{79} + 1856 q^{89} + 1702 q^{91}+O(q^{100}) $ 6 * q - 16 * q^11 + 22 * q^19 + 118 * q^31 - 800 * q^41 - 220 * q^49 + 352 * q^59 - 750 * q^61 - 2576 * q^71 + 584 * q^79 + 1856 * q^89 + 1702 * q^91

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} + 175x^{4} + 8109x^{2} + 72900 $ x^6 + 175*x^4 + 8109*x^2 + 72900 :

$\beta_{1}$	$=$	$ ( -\nu^{5} + 95\nu^{3} + 15651\nu ) / 49410 $ (-v^5 + 95v^3 + 15651v) / 49410
$\beta_{2}$	$=$	$ ( -2\nu^{4} + 190\nu^{2} + 20871 ) / 549 $ (-2v^4 + 190v^2 + 20871) / 549
$\beta_{3}$	$=$	$ ( 10\nu^{4} + 1246\nu^{2} + 24660 ) / 549 $ (10v^4 + 1246v^2 + 24660) / 549
$\beta_{4}$	$=$	$ ( 14\nu^{5} + 1964\nu^{3} + 50994\nu ) / 4941 $ (14v^5 + 1964v^3 + 50994*v) / 4941
$\beta_{5}$	$=$	$ ( 47\nu^{5} + 6515\nu^{3} + 230643\nu ) / 16470 $ (47v^5 + 6515v^3 + 230643*v) / 16470

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

$\nu$	$=$	$ ( \beta_{5} - \beta_{4} + \beta_1 ) / 4 $ (b5 - b4 + b1) / 4
$\nu^{2}$	$=$	$ ( \beta_{3} + 5\beta_{2} - 235 ) / 4 $ (b3 + 5*b2 - 235) / 4
$\nu^{3}$	$=$	$ ( -41\beta_{5} + 44\beta_{4} + 379\beta_1 ) / 2 $ (-41b5 + 44b4 + 379*b1) / 2
$\nu^{4}$	$=$	$ ( 95\beta_{3} - 623\beta_{2} + 19417 ) / 4 $ (95b3 - 623b2 + 19417) / 4
$\nu^{5}$	$=$	$ ( 7861\beta_{5} - 7291\beta_{4} - 109979\beta_1 ) / 4 $ (7861b5 - 7291b4 - 109979*b1) / 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1001$	$1351$
$\chi(n)$	$-1$	$1$	$1$	$1$

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} $

649.1

		3.43305i
	−	7.73485i
	−	10.1679i
		10.1679i
		7.73485i
	−	3.43305i

−

23.6993i

649.2

−

22.6681i

649.3

−

7.96885i

649.4

7.96885i

649.5

22.6681i

649.6

23.6993i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1800.4.f.ba		6
3.b	odd	2	1		1800.4.f.bb		6
5.b	even	2	1	inner	1800.4.f.ba		6
5.c	odd	4	1		1800.4.a.br	✓	3
5.c	odd	4	1		1800.4.a.bt	yes	3
15.d	odd	2	1		1800.4.f.bb		6
15.e	even	4	1		1800.4.a.bs	yes	3
15.e	even	4	1		1800.4.a.bu	yes	3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1800.4.a.br	✓	3	5.c	odd	4	1
1800.4.a.bs	yes	3	15.e	even	4	1
1800.4.a.bt	yes	3	5.c	odd	4	1
1800.4.a.bu	yes	3	15.e	even	4	1
1800.4.f.ba		6	1.a	even	1	1	trivial
1800.4.f.ba		6	5.b	even	2	1	inner
1800.4.f.bb		6	3.b	odd	2	1
1800.4.f.bb		6	15.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 $S_{4}^{\mathrm{new}}(1800, [\chi])$:

$ T_{7}^{6} + 1139T_{7}^{4} + 356899T_{7}^{2} + 18326961 $

$ T_{11}^{3} + 8T_{11}^{2} - 1376T_{11} + 11712 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} $ T^6
$7$	$ T^{6} + 1139 T^{4} + \cdots + 18326961 $ T^6 + 1139T^4 + 356899T^2 + 18326961
$11$	$ (T^{3} + 8 T^{2} + \cdots + 11712)^{2} $ (T^3 + 8T^2 - 1376T + 11712)^2
$13$	$ T^{6} + \cdots + 4431964329 $ T^6 + 10027T^4 + 25970643T^2 + 4431964329
$17$	$ T^{6} + 26624 T^{4} + \cdots + 691900416 $ T^6 + 26624T^4 + 150390784T^2 + 691900416
$19$	$ (T^{3} - 11 T^{2} + \cdots + 137891)^{2} $ (T^3 - 11T^2 - 4681T + 137891)^2
$23$	$ T^{6} + \cdots + 118512193536 $ T^6 + 63104T^4 + 973225984T^2 + 118512193536
$29$	$ (T^{3} - 30016 T - 1273920)^{2} $ (T^3 - 30016*T - 1273920)^2
$31$	$ (T^{3} - 59 T^{2} + \cdots + 1145475)^{2} $ (T^3 - 59T^2 - 34745T + 1145475)^2
$37$	$ T^{6} + \cdots + 5207706561600 $ T^6 + 70348T^4 + 1188054576T^2 + 5207706561600
$41$	$ (T^{3} + 400 T^{2} + \cdots - 8631360)^{2} $ (T^3 + 400T^2 - 19584T - 8631360)^2
$43$	$ T^{6} + \cdots + 30168929390625 $ T^6 + 185891T^4 + 4702238275T^2 + 30168929390625
$47$	$ T^{6} + \cdots + 12217988939776 $ T^6 + 304128T^4 + 11340054528T^2 + 12217988939776
$53$	$ T^{6} + \cdots + 1078540406784 $ T^6 + 201856T^4 + 4114059264T^2 + 1078540406784
$59$	$ (T^{3} - 176 T^{2} + \cdots + 27431424)^{2} $ (T^3 - 176T^2 - 157824T + 27431424)^2
$61$	$ (T^{3} + 375 T^{2} + \cdots - 151917099)^{2} $ (T^3 + 375T^2 - 366517T - 151917099)^2
$67$	$ T^{6} + \cdots + 44\!\cdots\!89 $ T^6 + 547011T^4 + 88411956099T^2 + 4457696485956289
$71$	$ (T^{3} + 1288 T^{2} + \cdots - 35439552)^{2} $ (T^3 + 1288T^2 + 355296T - 35439552)^2
$73$	$ T^{6} + \cdots + 21\!\cdots\!84 $ T^6 + 874732T^4 + 244925756208T^2 + 21473406545477184
$79$	$ (T^{3} - 292 T^{2} + \cdots - 868032)^{2} $ (T^3 - 292T^2 - 52304T - 868032)^2
$83$	$ T^{6} + \cdots + 16405999386624 $ T^6 + 1724416T^4 + 11471192064T^2 + 16405999386624
$89$	$ (T^{3} - 928 T^{2} + \cdots + 132710400)^{2} $ (T^3 - 928T^2 - 35840T + 132710400)^2
$97$	$ T^{6} + \cdots + 120967376199289 $ T^6 + 1777851T^4 + 432675065139T^2 + 120967376199289
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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{4} - 3 \beta_1) q^{7}+O(q^{10}) \) q + (-b4 - 3b1) q^7	\( q + ( - \beta_{4} - 3 \beta_1) q^{7} + ( - \beta_{3} + \beta_{2} - 3) q^{11} + (2 \beta_{5} + 5 \beta_1) q^{13} + (3 \beta_{5} + \beta_{4} + 15 \beta_1) q^{17} + ( - \beta_{3} + 2 \beta_{2} + 3) q^{19} + ( - \beta_{5} - 7 \beta_{4} - 13 \beta_1) q^{23} + ( - 7 \beta_{3} + 3 \beta_{2} - 1) q^{29} + (7 \beta_{3} - 4 \beta_{2} + 21) q^{31} + (4 \beta_{5} + 4 \beta_{4} + 2 \beta_1) q^{37} + ( - 11 \beta_{3} - \beta_{2} - 133) q^{41} + ( - 6 \beta_{5} - 7 \beta_{4} - 51 \beta_1) q^{43} + ( - 10 \beta_{5} + 10 \beta_{4} + 94 \beta_1) q^{47} + (4 \beta_{3} - 8 \beta_{2} - 34) q^{49} + (\beta_{5} - 13 \beta_{4} + 85 \beta_1) q^{53} + ( - 6 \beta_{3} - 10 \beta_{2} + 62) q^{59} + ( - 28 \beta_{3} + 6 \beta_{2} - 127) q^{61} + ( - 10 \beta_{5} - 13 \beta_{4} + 51 \beta_1) q^{67} + (5 \beta_{3} - 13 \beta_{2} - 425) q^{71} + ( - 4 \beta_{5} + 28 \beta_{4} + 134 \beta_1) q^{73} + ( - 5 \beta_{5} - 15 \beta_{4} + 247 \beta_1) q^{77} + ( - 12 \beta_{3} + 4 \beta_{2} + 96) q^{79} + (14 \beta_{5} + 18 \beta_{4} + 470 \beta_1) q^{83} + (24 \beta_{3} - 8 \beta_{2} + 312) q^{89} + (27 \beta_{3} + 26 \beta_{2} + 275) q^{91} + ( - 16 \beta_{5} + 36 \beta_{4} - 227 \beta_1) q^{97}+O(q^{100}) \) q + (-b4 - 3b1) q^7 + (-b3 + b2 - 3) * q^11 + (2b5 + 5b1) * q^13 + (3b5 + b4 + 15b1) * q^17 + (-b3 + 2b2 + 3) q^19 + (-b5 - 7b4 - 13b1) * q^23 + (-7b3 + 3b2 - 1) * q^29 + (7b3 - 4b2 + 21) * q^31 + (4b5 + 4b4 + 2b1) q^37 + (-11b3 - b2 - 133) q^41 + (-6b5 - 7b4 - 51b1) q^43 + (-10b5 + 10b4 + 94b1) q^47 + (4b3 - 8b2 - 34) * q^49 + (b5 - 13b4 + 85b1) * q^53 + (-6b3 - 10b2 + 62) * q^59 + (-28b3 + 6b2 - 127) * q^61 + (-10b5 - 13b4 + 51b1) q^67 + (5b3 - 13b2 - 425) * q^71 + (-4b5 + 28b4 + 134b1) q^73 + (-5b5 - 15b4 + 247b1) q^77 + (-12b3 + 4b2 + 96) * q^79 + (14b5 + 18b4 + 470b1) q^83 + (24b3 - 8b2 + 312) * q^89 + (27b3 + 26b2 + 275) * q^91 + (-16b5 + 36b4 - 227b1) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q - 16 q^{11} + 22 q^{19} + 118 q^{31} - 800 q^{41} - 220 q^{49} + 352 q^{59} - 750 q^{61} - 2576 q^{71} + 584 q^{79} + 1856 q^{89} + 1702 q^{91}+O(q^{100}) \) 6 * q - 16 * q^11 + 22 * q^19 + 118 * q^31 - 800 * q^41 - 220 * q^49 + 352 * q^59 - 750 * q^61 - 2576 * q^71 + 584 * q^79 + 1856 * q^89 + 1702 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more