https://github.com/haochenuw/zero-modform

Computations of zero of modular forms
https://github.com/haochenuw/zero-modform

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Computations of zero of modular forms

• Host: GitHub
• URL: https://github.com/haochenuw/zero-modform
• Owner: haochenuw
• Created: 2014-06-27T02:11:03.000Z (over 10 years ago)
• Default Branch: master
• Last Pushed: 2015-11-06T20:49:22.000Z (about 9 years ago)
• Last Synced: 2023-06-03T19:50:16.694Z (over 1 year ago)
• Language: Python
• Homepage:
• Size: 2.53 MB
• Stars: 1
• Watchers: 2
• Forks: 1
• Open Issues: 0
• Metadata Files:
  • Readme: README.md

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README

        
Computing zero polynomials of modular forms.

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### 1. zero-modform.sage:

#### 1.0. What does it do:

It computes the critical j-polynomial of atkin-lehner eigenform of _squarefree_ conductor N. (**Norm** method).

#### 1.1. Usage:

    sage: load('zero-modform.sage')

    sage: f = EllipticCurve('57a').modular_form();

    sage: zero_poly_comp(f).factor()

    (x - 54000)^2 * x^2 * (x^4 - 399605224650084576000*x^3 - 7985216535621460489954944000000*x^2 + 58827548670433207062445836288000000000*x + 120020259495560805847424176128000000000000)

exact-point.sage: Take as input the j-polynomial of a modular form, computes upper half plane representatives [z] for its zeros.

exceptional-point.sage: computes the set of points on X_0(p) where the map z \mapsto (j(z), j(pz)) is not injective, i.e., when

[z] maps to a singular point on the planar model of X_0(p).

916c/916c.sage: compute a polynomial relation between two modular functions on X0(916) and use them to obtain a critical polynomial

for E = 916c.

atkin-lehner.sage: Computes set of fixed points of any atkin_lehner operator $w_d$ on $X_0(N)$.

multimod.sage: Computes an u-polynomial of atkin-lehner eigenforms of any conductor, with a chosen u satisfying some integrality assumptions( which u = j always satisfy).(**Multimodular** method). N does not have to be square free.

polyrel-ZZ.sage: Computes u-polynomial of atkin-lehner eigenforms with r and u having concentrated poles. N does not have to be square free. (**Yang-product** method)

recognize-hilbert.sage: Given an irreducible polynomial f(x) \in ZZ[x], determine if there exists a negative discriminant D

such that $f = H_D(x)$. In the first case it finds such a $D$.

results/:

critpolys.json: contains an incomplete list critical polynomials of elliptic curves with conductor \<1000. (Now it contains

all rank 2 curves of conductor <= 1000)

crit-poly.txt: contains all critical polynomials of elliptic curves with prime conductor p \< 1000 such that genus(X_0(p)) > 1.

389-zero-pol.txt: contains zero polynomials for traces of the 5 newforms of weight 2 and level 389.