
// Make newform 8036.2.a.f in Magma, downloaded from the LMFDB on 24 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_f();".
// 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_f();".
// 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], [1], [0], [3], [-2], [-1], [3], [-5], [-2], [-5], [7], [-1], [4], [3], [-3], [-5], [-3], [-13], [0], [1], [-11], [-4], [-5], [-2], [3], [-7], [-1], [-7], [-2], [0], [-7], [5], [12], [21], [3], [3], [-1], [8], [-11], [3], [22], [-7], [-7], [6], [-19], [4], [12], [-27], [-1], [-27], [-16], [-3], [8], [-1], [-25], [9], [-27], [-17], [-22], [-9], [26], [16], [13], [-1], [5], [21], [-18], [7], [30], [13], [-25], [27], [19], [-4], [-9], [-25], [15], [-13], [9], [24], [6], [-19], [-6], [35], [-9], [6], [-3], [-26], [32], [33], [-19], [-11], [36], [-7], [24], [11], [-13], [-11], [-1], [4], [-3], [21], [-45], [-5], [-17], [12], [-9], [17], [-42], [-23], [-29], [22], [-1], [40], [-35], [20], [-9], [-43], [-24], [45], [-34], [-27], [-5], [11], [-14], [5], [-45], [-32], [-7], [-23], [36], [-23], [-34], [13], [-2], [3], [11], [34], [41], [-36], [31], [-13], [40], [-7], [28], [46], [9], [25], [27], [-5], [-38], [-44], [15], [13], [48], [-11], [39], [58], [49], [35], [-26], [-8], [-57], [57], [-45], [-57], [-13], [-34], [3], [-25], [34], [33], [-9], [-47], [-6], [28], [-9], [-36], [-29], [47], [12], [-10], [9], [25], [37], [-29], [19], [45], [-43], [65], [4], [-9], [-55], [53], [-67], [-3], [-47], [-42], [3], [31], [8], [39], [7], [-28], [-5], [37], [57], [6], [23], [5], [54], [56], [37], [-3], [13], [35], [33], [61], [34], [49], [-8], [33], [-31], [48], [-58], [-19], [57], [39], [23], [-21], [-31], [-20], [-9], [-72], [-3], [-29], [57], [4], [48], [-41], [59], [-39], [-25], [-14], [-3], [28], [49], [21], [48], [30], [25], [37], [-58], [51], [-21], [17], [-31], [-42], [-5], [-43], [-12], [-5], [-54], [-15], [41], [30], [-70], [20], [17], [31], [57], [-31], [-1], [50], [69], [-3], [17], [-55], [55], [-11], [-33], [-48], [-38], [3], [-7], [-3], [66], [-11], [14], [-35], [-77], [-11], [36], [6], [31], [23], [-74], [67], [-12], [13], [57], [-7], [-36], [-53], [-74], [5], [-14], [-7], [-83], [-41], [9], [3], [49], [-80], [53], [76], [45], [-50], [18], [71], [-9], [-26], [8], [43], [-49], [-19], [-1], [73], [10], [13], [-61], [-52], [73], [-35], [-12], [-58], [-7], [-82], [-47], [21], [-38], [50], [68], [55], [24], [17], [3], [64], [-15], [-49], [-69], [-14], [-29], [-55], [86], [-15], [-85], [-73], [12], [-46], [-49], [51], [19], [-47], [-11], [42], [9], [94], [-25], [75], [-21], [22], [47], [-63], [-45], [-48], [-47], [-33], [62], [15], [78], [76], [-53], [24], [-17], [99], [-77], [15], [72], [-2], [-81], [87], [73], [-45], [-43], [49], [-54], [24], [69], [77], [-51], [-1], [87], [65], [-61], [-25], [-50], [-91], [-15], [-19], [19], [-60], [-35], [98], [-3], [71], [53], [62], [-31], [39], [75], [36], [-20], [-86], [-63], [-35], [-62], [25], [81]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : 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![-1, 1]>,<11,R![-3, 1]>],Snew);
    return Vf;
end function;