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By Avella P., Boccia M., Sforza A.

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N=1 Suppose, as above, that f |WN = εN f . Then f ⊗ χ is in S2 (N l2 ), and (f ⊗ χ) |WN l2 = χ(−N )εN f ⊗ χ (special case of equation (14) in [64]). 1 to obtain the following. 2. Let f be as above. Let l be an odd prime not dividing N . If χ(−N ) = ε N then L(f ⊗ χ, 1) = 0, while if χ(−N ) = −εN , then √ χ(n)a(n, f ) exp(−2πn/l N ). 10). Let √ g(χ) be the Gauss sum attached to χ: if l ≡ 1 (mod 4) then χ(−1) = +1 and g(χ) = l, while √ if l ≡ 3 (mod√4) then χ(−1) = −1 and g(χ) = i l. If we set l∗ = χ(−1)l then in all cases we have g(χ) = l∗ .

It is slightly simpler to use a modular symbol of the form {0, α} here instead of {α, ∞}, since (for suitable α) this will be integral. 1) has more terms of the form {0, β} on the right, so this is slower in practice. Remark. 1) and the remarks following it show that the modular symbol {α, ∞} lies in the rational homology H1 (X0 (N ), Q) provided that the denominator of α is coprime to N . 3) for Γ0 (N ). Implementation. In practice we only use the first method if L(f, 1) = 0 for all the rational newforms f in S2 (N ).

In order to compute the degree of the map ϕ: X → Ef , the idea used in [69] is to compute the Petersson norm ||f || in two ways. The first way involves deg(ϕ) explicitly, while the second expresses it as a sum of terms involving periods, which can be evaluated as above. 1. Let f (z) be a cusp form of weight 2 for G as above, and ϕ: X → E f the associated modular parametrization. Then 4π 2 ||f ||2 = deg(ϕ)Vol(Ef ). Remark. In terms of the fundamental periods ω1 , ω2 of Ef , the volume is given by Vol(Ef ) = |Im (ω1 ω2 )|.

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A Branch-and-Cut Algorithm for the Median-Path Problem by Avella P., Boccia M., Sforza A.


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