
// Make newform 8036.2.a.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8036_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_8036_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_8036_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_8036_a();"
function MakeCharacter_8036_a()
    N := 8036;
    order := 1;
    char_gens := [4019, 493, 785];
    v := [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_8036_a_Hecke(Kf)
    return MakeCharacter_8036_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], [1], [-3], [0], [3], [-4], [0], [-7], [6], [6], [-10], [2], [-1], [-4], [12], [-6], [6], [-13], [-4], [-9], [14], [-1], [12], [-12], [2], [0], [-4], [6], [8], [-3], [14], [18], [-12], [14], [18], [-4], [14], [14], [12], [-21], [3], [-4], [0], [14], [3], [-16], [23], [8], [21], [-4], [6], [-27], [-22], [12], [-6], [12], [-15], [8], [-13], [6], [26], [12], [20], [12], [20], [-18], [-28], [23], [-15], [14], [-15], [36], [-28], [29], [-16], [-27], [15], [14], [27], [-31], [-6], [8], [18], [23], [-1], [18], [9], [-16], [27], [-1], [30], [21], [8], [-42], [-4], [9], [-18], [30], [38], [14], [20], [-12], [-15], [30], [5], [20], [-33], [-6], [-30], [-34], [-28], [-7], [-27], [44], [20], [12], [5], [-12], [36], [-48], [-31], [-22], [-15], [12], [-1], [15], [-10], [3], [-40], [-22], [-16], [6], [-43], [-40], [18], [35], [36], [-28], [-30], [-12], [38], [-9], [5], [12], [38], [-3], [-49], [-3], [8], [24], [17], [45], [32], [3], [-10], [0], [59], [0], [-34], [-45], [-18], [-6], [23], [0], [-42], [12], [-43], [-28], [38], [-6], [51], [-22], [-27], [-19], [5], [-30], [53], [-45], [17], [-10], [-34], [30], [50], [30], [12], [-9], [-58], [62], [-22], [-33], [50], [51], [-28], [-15], [-12], [54], [14], [-10], [-60], [-21], [-21], [8], [2], [8], [-12], [12], [-16], [-3], [15], [44], [38], [6], [-34], [18], [-27], [17], [-49], [18], [24], [66], [14], [-58], [60], [29], [6], [-52], [-21], [-36], [53], [0], [59], [20], [-4], [-45], [-19], [-3], [44], [-54], [-30], [-24], [-36], [-67], [-13], [14], [54], [-6], [2], [21], [14], [-42], [-37], [57], [48], [-19], [30], [30], [38], [68], [42], [-34], [-4], [-3], [-40], [-52], [51], [-28], [54], [27], [14], [60], [32], [62], [53], [-4], [-58], [-82], [-60], [-4], [-46], [51], [-48], [-37], [6], [-37], [56], [3], [-52], [-75], [8], [48], [48], [-36], [6], [-12], [-16], [72], [-55], [51], [39], [17], [-31], [36], [-16], [-36], [32], [-70], [-54], [-79], [69], [32], [-24], [-54], [-57], [32], [-18], [-73], [18], [-84], [14], [48], [-82], [5], [-9], [11], [54], [56], [71], [-1], [-45], [27], [62], [-54], [-88], [-51], [38], [-36], [20], [30], [56], [68], [62], [75], [-42], [-13], [6], [6], [35], [-10], [24], [30], [20], [-46], [48], [26], [86], [30], [66], [36], [6], [90], [-31], [66], [-39], [-60], [20], [-16], [-72], [-52], [-31], [-63], [65], [36], [48], [20], [38], [12], [-54], [-49], [54], [-4], [0], [-63], [-58], [-57], [-79], [-18], [65], [-82], [2], [81], [98], [0], [69], [62], [12], [-22], [47], [-51], [8], [0], [-31], [-42], [50], [30], [-18], [-28], [53], [12], [-85], [-48], [-1], [-3], [-99], [-49], [68], [42], [66], [-43], [48], [66], [30], [-4], [-48], [-51], [29], [27], [-3], [-36], [-79], [9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8036_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8036_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_8036_2_a_d(:prec:=1)
    chi := MakeCharacter_8036_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_8036_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_8036_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8036_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 1]>,<5,R![3, 1]>,<11,R![-3, 1]>],Snew);
    return Vf;
end function;