
// Make newform 8036.2.a.a 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_a();".
// 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_a();".
// 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], [-5], [-2], [-3], [-3], [-1], [-10], [-11], [7], [-1], [8], [-7], [-11], [-7], [-1], [7], [-12], [-5], [9], [0], [-3], [2], [-11], [-1], [-13], [17], [6], [4], [3], [9], [-20], [-7], [-5], [-3], [-17], [16], [-21], [7], [-22], [1], [17], [26], [-21], [-12], [8], [-9], [-27], [13], [-16], [19], [24], [-7], [19], [11], [-25], [23], [-14], [17], [-18], [12], [-1], [-11], [17], [-3], [-14], [-9], [-2], [-9], [-29], [-15], [19], [-20], [-27], [3], [29], [-13], [35], [12], [-18], [-35], [14], [-7], [39], [22], [5], [2], [8], [-33], [-17], [-35], [-20], [-3], [40], [5], [-27], [39], [11], [-4], [33], [-25], [15], [15], [-39], [36], [5], [9], [-22], [35], [27], [-6], [-11], [0], [-15], [-36], [-19], [9], [-4], [51], [34], [15], [31], [9], [50], [-19], [-11], [32], [3], [-15], [-48], [-7], [-38], [-49], [-10], [-19], [-47], [2], [-31], [-12], [-17], [-5], [-12], [51], [8], [-14], [-13], [11], [-45], [11], [-50], [-32], [-43], [17], [-28], [5], [17], [6], [27], [15], [30], [4], [-11], [-7], [-11], [-13], [37], [-14], [-39], [-33], [34], [-3], [-21], [-41], [2], [12], [43], [-8], [-27], [-25], [-16], [-42], [-45], [-27], [47], [23], [17], [-43], [-21], [-25], [20], [-53], [-9], [-15], [-25], [13], [21], [-18], [-3], [-9], [-56], [-23], [-3], [-36], [61], [-9], [-11], [30], [-71], [-39], [-42], [-8], [-33], [39], [55], [19], [-33], [-23], [-6], [-71], [48], [21], [-59], [52], [22], [-9], [-67], [-71], [15], [-41], [55], [-48], [-45], [-44], [47], [49], [-11], [20], [8], [39], [37], [31], [-13], [-2], [-13], [-16], [7], [-15], [0], [-22], [7], [-55], [54], [-51], [-17], [21], [-77], [54], [13], [49], [-48], [37], [-10], [11], [3], [54], [22], [-20], [-51], [5], [17], [15], [31], [-62], [-37], [-39], [5], [-23], [-75], [-37], [-65], [-24], [2], [61], [-47], [-11], [-6], [-45], [46], [-79], [-3], [-51], [-28], [14], [-83], [61], [-78], [21], [20], [39], [9], [-7], [24], [-49], [-42], [-63], [-22], [-27], [-19], [-63], [-3], [63], [-39], [-44], [79], [-84], [45], [46], [-18], [45], [-69], [10], [72], [23], [-63], [1], [21], [69], [50], [5], [-9], [-16], [55], [-3], [-8], [-30], [-69], [-38], [31], [41], [14], [-6], [-80], [-13], [-20], [-17], [15], [-56], [-81], [77], [59], [30], [65], [-23], [-30], [-21], [49], [11], [16], [-26], [69], [-81], [3], [75], [49], [30], [-19], [-10], [79], [-19], [67], [-66], [-67], [-43], [-19], [-84], [55], [-35], [-38], [9], [-42], [-32], [-9], [4], [9], [71], [-55], [99], [24], [78], [69], [99], [3], [67], [57], [-45], [-42], [100], [-27], [-45], [69], [-37], [93], [35], [-35], [39], [-62], [7], [63], [-29], [-9], [8], [85], [102], [27], [-81], [-25], [34], [103], [-33], [77], [48], [0], [-58], [83], [81], [-46], [-89], [39]];
    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_a();".
// 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_a(: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_a();".
// 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_a( : 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![5, 1]>],Snew);
    return Vf;
end function;