Prove
Lemma 39 ([A fixed-progression moment]). There are absolute constants η>0 and…Proposition 38 ([Bounded-factor completion]). There is an absolute integer A_0…Proposition
Use the Shimura packet of E^ (-σ) . Its positive index D tests the twist by χ_ -D , which is E^ (σD) . The nonvanishing at D_0 selects a globally occurring member admitting that Fourier functional. Take its spherical vector outside S and its holomorphic vector of weight 3/2 at infinity. Here is the support-local vector selection. Start with a vector on which the D_0 Fourier functional is nonzero. Projection under a sufficiently large compact group of upper unipotents to additive frequency D_0 restricts the potentially nonzero unit translates to arbitrarily small neighborhoods of the two solutions of u^2=1 . Choose a unit character agreeing with the forced action at -1 . It can have 2 -power order: the image of -1 is detected in the 2 -primary quotient of the unit group. The neighborhoods of 1 and -1 then contribute with the same sign, so this projection preserves the nonzero functional. Upper integral translations act trivially, and smoothness gives sufficiently deep lower-unipotent invariance. The vector therefore defines a classical form at a level supported on S , allowing arbitrary depth at 2 and at the bad primes. The unit-squareclass and coprimality sieves commute with good T(p^2) , since their conditions are unchanged by multiplication of the index by p^2 . The classical Hecke formula gives , with η=ψχ_ -σ . At the level just chosen, let V be the full space of weight- 3/2 cusp forms orthogonal to unary theta series, with scalar character ψ and good T(p^2) eigenvalues η(p)a_p(E) . It contains the constructed form, whose coefficient at D_0 is nonzero. For D/D_0 a unit square at every support prime, Waldspurger's comparison gives, for every f∈V , a_f(D)^2= ψ(D/D_0)(L_*(σD))/(L_*(σD_0))a_f(D_0)^2. Its hypothesis (H1) follows from the nonzero eigenspace ; the final paragraph of the comparison proof removes (H2) for these local squareclasses. Factoring the displayed quadratic identity in the two coefficient functionals shows that they are proportional on all of V , with one scalar independent of f . This includes oldvectors and their linear combinations; if the central value at D vanishes, the first functional is simply zero. Choose an algebraic form g in the filtered subspace of V with a(D_0)≠0 . To justify this choice, multiplication by ordinary theta embeds the space into an integral-weight space, and divisibility by theta is specified by algebraic cusp conditions . The character, good-Hecke and filter conditions are algebraic linear conditions as well. Since evaluation at D_0 is nonzero after extension to ℂ , it is nonzero on some algebraic form. Let u be an automorphism of ℚ acting on the character values by an odd power e , and put ν=η^ (e-1)/2 . The forms g^u and g⊗ν have the same scalar character ψ^e=ψν^2 and the same good eigenvalues η(p)^e a_p(E)=ν(p)^2η(p)a_p(E) ; here ψ/η is quadratic. They belong to one good-Hecke eigenspace at a common level still supported on S . Applying the preceding proportionality in that nonzero space and normalizing at D_0 gives u(ξ)=ν(D/D_0)ξ, ξ=a(D)/a(D_0). Local square roots of D/D_0 at the conductor of η give a value z of η with η(D/D_0)=z^2 . Then u(z)=z^e and ν(D/D_0)=z^ e-1 , so ξ/z is fixed by every algebraic automorphism and is rational. Moreover the same local-square condition gives ψ(D/D_0)=η(D/D_0)=z^2 . The comparison therefore reads (ξ/z)^2=(L_*(σD))/(L_*(σD_0)), which proves since v(z)=0 . There is no product over the new prime factors: the unramified squarefree formula includes both local valuations zero and one. No exceptional local representation at 2 is excluded by the same-local-class comparison. The vanishing assertion and follow.