
// Make newform 1040.2.f.d in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1040_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1040_f_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1040_2_f_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_1040_2_f_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
        poly := [9, -3, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-6, 1, 2, 1], [-3, 1, -1, 1], [9, 4, -1, -2]];
    Rf_basisdens := [1, 3, 3, 3];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1040_f();"
function MakeCharacter_1040_f()
    N := 1040;
    order := 2;
    char_gens := [911, 261, 417, 561];
    v := [2, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1040_f_Hecke();"
function MakeCharacter_1040_f_Hecke(Kf)
    N := 1040;
    order := 2;
    char_gens := [911, 261, 417, 561];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 0, 0], [-1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 0, 0], [0, 0, -1, 0], [1, -1, 0, 0], [0, -1, 0, -1], [0, -1, -1, 1], [1, -1, 1, 1], [0, 1, -2, -1], [0, 1, -1, -1], [0, 1, 3, -1], [-2, -2, 0, -2], [0, -1, 1, 1], [2, -3, 0, -3], [0, 0, -4, 0], [0, -2, -1, 2], [-8, 1, 0, 1], [0, 0, 2, 0], [0, 1, 3, -1], [-2, 2, 0, 2], [4, 0, 0, 0], [0, 0, -5, 0], [-10, 0, 0, 0], [0, 2, 0, 2], [4, -2, 0, -2], [0, 2, 0, -2], [-2, 2, 0, 2], [-10, 2, 0, 2], [0, -3, 3, 3], [0, -1, 5, 1], [0, -3, 0, 3], [0, -4, 0, 4], [0, 1, -1, -1], [4, 1, 0, 1], [6, 0, 0, 0], [-4, 1, 0, 1], [0, 2, -6, -2], [0, 0, 3, 0], [0, 0, 6, 0], [20, -2, 0, -2], [8, -4, 0, -4], [0, -4, 6, 4], [-12, -3, 0, -3], [6, 4, 0, 4], [0, 0, 0, 0], [14, 0, 0, 0], [2, -1, 0, -1], [-8, 0, 0, 0], [4, 3, 0, 3], [-8, 3, 0, 3], [4, 4, 0, 4], [0, -5, 8, 5], [0, -3, 2, 3], [0, -2, 3, 2], [0, 2, 4, -2], [4, -8, 0, -8], [0, -1, -6, 1], [0, -5, -9, 5], [-2, -2, 0, -2], [0, -4, 7, 4], [0, -2, -10, 2], [0, 2, 0, -2], [0, -5, 5, 5], [-14, -5, 0, -5], [4, -6, 0, -6], [-8, -2, 0, -2], [0, -3, -6, 3], [2, 8, 0, 8], [0, -3, 3, 3], [0, -7, -2, 7], [0, -2, -9, 2], [0, 3, 12, -3], [-2, 4, 0, 4], [0, -5, -3, 5], [0, 3, 3, -3], [0, 6, 6, -6], [0, 5, -5, -5], [-24, -3, 0, -3], [6, 0, 0, 0], [18, 4, 0, 4], [0, 2, -12, -2], [0, -6, 0, 6], [-12, 3, 0, 3], [0, 1, -4, -1], [0, 4, -9, -4], [0, 1, 14, -1], [-8, 6, 0, 6], [0, 2, -1, -2], [0, 8, 0, -8], [6, -8, 0, -8], [0, -9, -4, 9], [16, 0, 0, 0], [0, -5, 9, 5], [0, 6, -5, -6], [8, 4, 0, 4], [28, 1, 0, 1], [0, -9, 3, 9], [0, 1, 3, -1], [0, -2, -2, 2], [2, -7, 0, -7], [0, 3, -3, -3], [0, -11, 8, 11], [0, -4, 7, 4], [-6, -3, 0, -3], [0, 10, 3, -10], [-14, 1, 0, 1], [-20, -3, 0, -3], [-2, 8, 0, 8], [-4, -10, 0, -10], [-18, 6, 0, 6], [-8, 10, 0, 10], [-30, 1, 0, 1], [0, 1, -1, -1], [10, -4, 0, -4], [-10, 2, 0, 2], [0, 7, -1, -7], [0, 8, 19, -8], [2, 8, 0, 8], [-28, 4, 0, 4], [0, 9, -5, -9], [0, 2, -6, -2], [12, -12, 0, -12], [0, -6, -6, 6], [0, 5, -2, -5], [0, 4, -14, -4], [-4, 2, 0, 2], [0, 7, -13, -7], [-2, -8, 0, -8], [0, 6, -6, -6], [48, 0, 0, 0], [0, -5, -1, 5], [10, 11, 0, 11], [0, -5, -13, 5], [0, 3, 0, 3], [-16, 4, 0, 4], [0, 0, 18, 0], [0, 2, 0, -2], [0, 4, 8, -4], [-14, 7, 0, 7], [-44, 0, 0, 0], [0, 0, -10, 0], [18, -9, 0, -9], [0, -1, -17, 1], [0, 5, 12, -5], [0, -7, 19, 7], [-20, -2, 0, -2], [-34, 6, 0, 6], [0, -1, 5, 1], [2, 3, 0, 3], [0, 8, 12, -8], [4, -12, 0, -12], [8, 5, 0, 5], [26, -9, 0, -9], [-38, 7, 0, 7], [0, 6, -21, -6], [0, 3, -17, -3], [0, -2, -1, 2], [-16, 8, 0, 8], [-24, 8, 0, 8], [0, -8, 16, 8], [0, 0, -12, 0], [0, 11, -12, -11], [20, -4, 0, -4], [0, -1, 6, 1], [16, 9, 0, 9], [-4, -7, 0, -7], [-2, 10, 0, 10], [-16, 11, 0, 11], [16, 6, 0, 6], [0, -10, -2, 10], [0, 2, 4, -2], [0, 10, -2, -10], [0, 7, -9, -7], [0, 3, 12, -3], [-40, -4, 0, -4], [6, -14, 0, -14], [16, 12, 0, 12], [-14, -5, 0, -5], [0, -9, -9, 9], [0, -6, 2, 6], [0, 11, 7, -11], [14, 12, 0, 12], [32, 4, 0, 4], [-4, 8, 0, 8], [0, 6, -18, -6], [6, 0, 0, 0], [16, -8, 0, -8], [-26, 4, 0, 4], [0, -8, -10, 8], [-12, 2, 0, 2], [0, 8, -20, -8], [0, 7, -15, -7], [0, 15, -6, -15], [44, -4, 0, -4], [20, 13, 0, 13], [0, 10, 22, -10], [0, 4, -9, -4], [0, 0, 20, 0], [0, -12, 0, 12], [0, 0, 6, 0], [30, -12, 0, -12], [0, 1, 27, -1], [0, -13, -12, 13], [0, 2, 0, 2], [-46, 3, 0, 3], [10, -1, 0, -1], [0, -9, -17, 9], [0, 12, 14, -12], [0, -2, 11, 2], [0, -5, -15, 5], [0, 8, 12, -8], [12, -13, 0, -13], [0, -8, 8, 8], [22, 10, 0, 10], [0, -7, -29, 7], [12, 6, 0, 6], [0, 10, -9, -10], [0, -2, -16, 2], [0, 7, 11, -7], [-6, -3, 0, -3], [-40, 8, 0, 8], [26, 5, 0, 5], [6, -2, 0, -2], [0, -14, -1, 14], [0, 8, 0, -8], [40, 3, 0, 3], [0, -4, -1, 4], [6, 16, 0, 16], [0, -9, -10, 9], [-48, 6, 0, 6], [0, 9, -21, -9], [0, 7, 15, -7], [0, 12, 6, -12], [12, 8, 0, 8], [0, -5, -7, 5], [-30, 3, 0, 3], [0, 12, 3, -12], [24, -15, 0, -15], [0, -10, -20, 10], [2, -13, 0, -13], [36, 0, 0, 0], [56, -4, 0, -4], [4, 10, 0, 10], [44, -11, 0, -11], [0, 3, -9, -3], [0, 13, -16, -13], [-18, -18, 0, -18], [24, 6, 0, 6], [-8, -21, 0, -21], [0, -5, 3, 5], [0, -9, -33, 9], [0, -11, 15, 11], [26, 3, 0, 3], [0, -4, -24, 4], [16, 1, 0, 1], [34, -13, 0, -13], [0, -4, 6, 4], [0, -17, 3, 17], [38, -12, 0, -12], [44, -14, 0, -14], [0, -4, -30, 4], [38, -12, 0, -12], [0, 7, -13, -7], [0, -16, -1, 16], [0, -11, -28, 11], [0, -14, 2, 14], [-50, -2, 0, -2], [-44, -9, 0, -9], [0, -3, -16, 3], [0, -2, -8, 2], [-36, -4, 0, -4], [0, 2, 6, -2], [14, -24, 0, -24], [-28, -14, 0, -14], [-58, -6, 0, -6], [16, -18, 0, -18], [0, 12, 0, -12], [-8, 0, 0, 0], [12, 18, 0, 18], [0, -2, 14, 2], [0, -4, 4, 4], [-44, 13, 0, 13], [0, -3, 7, 3], [0, -10, -17, 10], [0, 7, -9, -7], [0, 2, -14, -2], [28, -6, 0, -6], [32, -10, 0, -10], [0, 17, -2, -17], [34, -11, 0, -11], [0, 6, -9, -6], [50, -4, 0, -4], [14, -10, 0, -10], [0, 10, 3, -10], [6, 6, 0, 6], [0, -17, 15, 17], [0, -18, 18, 18], [14, 14, 0, 14], [-32, 0, 0, 0], [0, 2, -18, -2], [44, 5, 0, 5], [-60, 8, 0, 8], [0, -9, 30, 9], [-22, 5, 0, 5], [16, -12, 0, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1040_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1040_2_f_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_1040_2_f_d(:prec:=4)
    chi := MakeCharacter_1040_f();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1040_2_f_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_1040_2_f_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1040_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 0, 7, 0, 1]>,<7,R![-8, 1, 1]>,<37,R![-74, -1, 1]>],Snew);
    return Vf;
end function;