
// Make newform 3528.2.a.d in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3528_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_3528_2_a_d();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_3528_2_a_d();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_3528_a();"
function MakeCharacter_3528_a()
    N := 3528;
    order := 1;
    char_gens := [2647, 1765, 785, 1081];
    v := [1, 1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_3528_a_Hecke(Kf)
    return MakeCharacter_3528_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0], [0], [-2], [0], [-4], [2], [2], [4], [8], [-6], [-8], [6], [-6], [4], [0], [2], [4], [2], [-4], [-8], [-10], [-8], [-4], [-6], [-2], [-18], [-16], [12], [-2], [-18], [-8], [-4], [6], [12], [-14], [-16], [2], [12], [24], [6], [-12], [-6], [0], [2], [18], [-16], [-20], [8], [12], [-22], [-10], [16], [-18], [20], [2], [8], [-10], [-8], [-26], [-26], [28], [-18], [-12], [-24], [6], [-6], [20], [18], [12], [-30], [2], [24], [8], [-10], [20], [0], [2], [-14], [30], [6], [12], [-10], [-32], [14], [0], [-20], [14], [-22], [-26], [8], [-36], [-16], [-32], [12], [12], [24], [6], [26], [-4], [-18], [44], [26], [28], [-10], [36], [-2], [-44], [-14], [-24], [38], [40], [38], [-42], [44], [16], [14], [-12], [8], [-6], [-12], [10], [34], [-2], [-4], [4], [-6], [-10], [-32], [-48], [-14], [-4], [8], [24], [38], [-22], [-2], [-18], [-28], [22], [-26], [-4], [-30], [-16], [28], [50], [-24], [10], [42], [12], [32], [-18], [50], [-4], [8], [4], [-16], [16], [50], [-42], [6], [-12], [54], [-16], [36], [30], [-24], [40], [26], [-46], [-50], [-36], [-30], [40], [10], [40], [26], [-28], [-30], [-16], [-14], [8], [-52], [54], [-54], [-48], [-18], [-50], [36], [10], [0], [62], [12], [-20], [22], [36], [-22], [-14], [46], [2], [24], [-38], [-8], [-38], [-34], [-60], [6], [-56], [-12], [22], [-4], [18], [-18], [-32], [-28], [8], [22], [-40], [-62], [24], [-54], [-10], [0], [-18], [8], [60], [-10], [-6], [-48], [-48], [-52], [-66], [-28], [-40], [-10], [60], [-64], [-50], [50], [-20], [40], [36], [28], [-64], [46], [18], [-40], [56], [60], [52], [16], [78], [-30], [-8], [-10], [54], [-12], [38], [-4], [78], [-58], [-8], [-60], [-22], [66], [-14], [4], [-38], [26], [-76], [-62], [50], [-52], [38], [40], [14], [64], [12], [62], [10], [76], [-80], [16], [-8], [58], [28], [-16], [50], [18], [16], [2], [26], [12], [38], [-12], [-18], [-74], [40], [-66], [-12], [-44], [-74], [58], [-40], [20], [84], [2], [-36], [34], [-56], [-58], [48], [-30], [46], [-20], [8], [22], [-84], [-48], [-2], [78], [20], [-38], [-58], [-56], [-58], [-18], [12], [-84], [32], [-30], [-2], [42], [-24], [-36], [-12], [84], [-2], [66], [22], [-72], [22], [54], [78], [-48], [26], [20], [42], [52], [0], [-34], [84], [-22], [-22], [24], [-10], [26], [64], [-76], [-34], [24], [38], [42], [-48], [-20], [68], [42], [-42], [-32], [-38], [52], [-36], [-80], [2], [-16], [-82], [60], [96], [-34], [66], [-26], [-42], [70], [48], [30], [20], [24], [24], [58], [4], [-48], [-62], [-98], [12], [-60], [-72], [58], [-24], [-6], [-28], [-46], [-62], [-18], [40], [-22], [46], [-64], [-2], [-66], [100], [12], [46], [-46], [76], [76], [42], [22], [0], [-32], [-30], [-40], [10], [58], [16], [-76], [-10], [-10], [-28], [6], [12], [88]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3528_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3528_2_a_d();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_3528_2_a_d(:prec:=1)
    chi := MakeCharacter_3528_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_3528_2_a_d();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_3528_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3528_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<11,R![4, 1]>,<13,R![-2, 1]>,<23,R![-8, 1]>],Snew);
    return Vf;
end function;