\(L\)-series#

\(L\)-series of \(\Delta\)#

Thanks to wrapping work of Jennifer Balakrishnan of M.I.T., we can compute explicitly with the \(L\)-series of the modular form \(\Delta\). Like for elliptic curves, behind these scenes this uses Dokchitsers \(L\)-functions calculation Pari program.

Sage

sage: L = delta_lseries(); L
L-series of conductor 1 and weight 12
sage: L(1)
0.0374412812685155

Python

>>> from sage.all import *
>>> L = delta_lseries(); L
L-series of conductor 1 and weight 12
>>> L(Integer(1))
0.0374412812685155

\(L\)-series of a Cusp Form#

In some cases we can also compute with \(L\)-series attached to a cusp form.

Sage

sage: f = CuspForms(2,8).newforms()[0]
sage: L = f.lseries()
sage: L(1)
0.0884317737041015
sage: L(0.5)
0.0296568512531983

Python

>>> from sage.all import *
>>> f = CuspForms(Integer(2),Integer(8)).newforms()[Integer(0)]
>>> L = f.lseries()
>>> L(Integer(1))
0.0884317737041015
>>> L(RealNumber('0.5'))
0.0296568512531983

\(L\)-series of a General Newform is Not Implemented#

Unfortunately, computing with the \(L\)-series of a general newform is not yet implemented.

Sage

sage: S = CuspForms(23,2); S
Cuspidal subspace of dimension 2 of Modular Forms space of
dimension 3 for Congruence Subgroup Gamma0(23) of weight
2 over Rational Field
sage: f = S.newforms('a')[0]; f
q + a0*q^2 + (-2*a0 - 1)*q^3 + (-a0 - 1)*q^4 + 2*a0*q^5 + O(q^6)

Python

>>> from sage.all import *
>>> S = CuspForms(Integer(23),Integer(2)); S
Cuspidal subspace of dimension 2 of Modular Forms space of
dimension 3 for Congruence Subgroup Gamma0(23) of weight
2 over Rational Field
>>> f = S.newforms('a')[Integer(0)]; f
q + a0*q^2 + (-2*a0 - 1)*q^3 + (-a0 - 1)*q^4 + 2*a0*q^5 + O(q^6)

Computing with \(L(f,s)\) totally not implemented yet, though should be easy via Dokchitser.