\(q\)-expansions of theta series

AUTHOR:

• William Stein

sage.modular.modform.theta.theta2_qexp(prec=10, var='q', K=Integer Ring, sparse=False)

Return the \(q\)-expansion of the series \(\theta_2 = \sum_{n \text{ odd}} q^{n^2}\).

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

sage.modular.modform.theta.theta_qexp(prec=10, var='q', K=Integer Ring, sparse=False)

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