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Newforms and basis of new subspace [explanation]

asked 2014-12-04 16:12:37 -0500

anonymous user

Anonymous

updated 2015-01-14 04:43:29 -0500

FrédéricC gravatar image

Hi,

I am new to Sage and modular forms. I have some conceptual questions.

When I write

sage: S = CuspForms(Gamma0(55),2,prec=14)
sage: S.new_subspace().basis()

[
q + 2*q^3 - 2*q^5 - 4*q^6 - 3*q^7 + 5*q^8 + 9*q^9 - q^10 + 2*q^11 - 10*q^12 - 9*q^13 + O(q^14),
q^2 - 2*q^3 + 2*q^5 + 2*q^6 + 2*q^7 - 5*q^8 - 8*q^9 + q^10 - 2*q^11 + 6*q^12 + 8*q^13 + O(q^14),
q^4 - q^5 - 2*q^6 - q^7 + 3*q^8 + 4*q^9 - q^10 + q^11 - 4*q^12 - 3*q^13 + O(q^14)
]


sage: CuspForms(Gamma0(55),2).newforms(names='a')

[q + q^2 - q^4 + q^5 + O(q^6),
 q + a1*q^2 + (-2*a1 + 2)*q^3 + (2*a1 - 1)*q^4 - q^5 + O(q^6)]

What is the difference between those 2 and how to find a coefficient of a1?

Can someone tell me how to find newforms for some S_k if the second function is not correct.

Thank you.

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Comments

kcrisman gravatar imagekcrisman ( 2014-12-04 20:25:19 -0500 )edit

In fact, a1 is the generator of a quadratic number field here.

sage: CuspForms(Gamma0(55),2).newforms(names='a')[1].parent()
Cuspidal subspace of dimension 5 of Modular Forms space of dimension 8 for Congruence Subgroup Gamma0(55) of weight 2 over Number Field in a1 with defining polynomial x^2 - 2*x - 1
FrédéricC gravatar imageFrédéricC ( 2014-12-05 02:59:16 -0500 )edit

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answered 2014-12-07 06:18:04 -0500

Lior gravatar image

Thank you. How do we get a polynomial defining a newform?

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Asked: 2014-12-04 16:12:37 -0500

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Last updated: Dec 07 '14