Coverage for local/lib/python2.7/site-packages/sage/modular/modform/theta.py : 79%

""" 

q-expansions of Theta Series 

AUTHOR: 

William Stein 

""" 

from six.moves import range 

from sage.rings.all import Integer, ZZ, PowerSeriesRing 

from math import sqrt 

def theta2_qexp(prec=10, var='q', K=ZZ, sparse=False): 

r""" 

Return the `q`-expansion of the series 

` \theta_2 = \sum_{n odd} q^{n^2}. ` 

INPUT: 

- prec -- integer; the absolute precision of the output 

- var -- (default: 'q') variable name 

- K -- (default: ZZ) base ring of answer 

OUTPUT: 

a power series over K 

EXAMPLES:: 

sage: theta2_qexp(18) 

q + q^9 + O(q^18) 

sage: theta2_qexp(49) 

q + q^9 + q^25 + O(q^49) 

sage: theta2_qexp(100, 'q', QQ) 

q + q^9 + q^25 + q^49 + q^81 + O(q^100) 

sage: f = theta2_qexp(100, 't', GF(3)); f 

t + t^9 + t^25 + t^49 + t^81 + O(t^100) 

sage: parent(f) 

Power Series Ring in t over Finite Field of size 3 

sage: theta2_qexp(200) 

q + q^9 + q^25 + q^49 + q^81 + q^121 + q^169 + O(q^200) 

sage: f = theta2_qexp(20,sparse=True); f 

q + q^9 + O(q^20) 

sage: parent(f) 

Sparse Power Series Ring in q over Integer Ring 

""" 

prec = Integer(prec) 

if prec <= 0: 

raise ValueError("prec must be positive") 

if sparse: 

v = {} 

else: 

v = [Integer(0)] * prec 

one = Integer(1) 

n = int(sqrt(prec)) 

if n*n < prec: 

n += 1 

for m in range(1, n, 2): 

v[m*m] = one 

R = PowerSeriesRing(K, sparse=sparse, names=var) 

return R(v, prec=prec) 

def theta_qexp(prec=10, var='q', K=ZZ, sparse=False): 

r""" 

Return the `q`-expansion of the standard `\theta` series 

` \theta = 1 + 2\sum_{n=1}{^\infty} q^{n^2}. ` 

INPUT: 

- prec -- integer; the absolute precision of the output 

- var -- (default: 'q') variable name 

- K -- (default: ZZ) base ring of answer 

OUTPUT: 

a power series over K 

EXAMPLES:: 

sage: theta_qexp(25) 

1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + O(q^25) 

sage: theta_qexp(10) 

1 + 2*q + 2*q^4 + 2*q^9 + O(q^10) 

sage: theta_qexp(100) 

1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + 2*q^25 + 2*q^36 + 2*q^49 + 2*q^64 + 2*q^81 + O(q^100) 

sage: theta_qexp(100, 't') 

1 + 2*t + 2*t^4 + 2*t^9 + 2*t^16 + 2*t^25 + 2*t^36 + 2*t^49 + 2*t^64 + 2*t^81 + O(t^100) 

sage: theta_qexp(100, 't', GF(2)) 

1 + O(t^100) 

sage: f = theta_qexp(20,sparse=True); f 

1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + O(q^20) 

sage: parent(f) 

Sparse Power Series Ring in q over Integer Ring 

""" 

prec = Integer(prec) 

if prec <= 0: 

raise ValueError("prec must be positive") 

if sparse: 

v = {} 

else: 

v = [Integer(0)] * prec 

v[0] = Integer(1) 

two = Integer(2) 

n = int(sqrt(prec)) 

if n*n != prec: 

n += 1 

for m in range(1, n): 

v[m*m] = two 

R = PowerSeriesRing(K, sparse=sparse, names=var) 

return R(v, prec=prec)