
// Make newform 1800.2.k.j in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1800_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1800_k_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_1800_2_k_j();".
// 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_1800_2_k_j();".
// 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 := [1, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 1, 0], [0, 0, 0, 1], [0, 1, -1, 0]];
    Rf_basisdens := [1, 1, 1, 1];
    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_1800_k();"
function MakeCharacter_1800_k()
    N := 1800;
    order := 2;
    char_gens := [1351, 901, 1001, 577];
    v := [2, 1, 2, 2];
    // 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_1800_k_Hecke();"
function MakeCharacter_1800_k_Hecke(Kf)
    N := 1800;
    order := 2;
    char_gens := [1351, 901, 1001, 577];
    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, -1, 1, 0], [0, 0, 0, 0], [0, 0, 0, 0], [1, -1, 1, -1], [0, 0, 2, 0], [-2, 2, 0, -2], [0, 2, -2, 2], [2, -2, 4, 2], [-1, -3, 3, -3], [4, -4, 0, 4], [-2, -2, 2, -2], [0, 0, 2, 0], [2, -2, 2, -2], [1, -1, 7, 1], [5, -1, 1, -1], [-2, 2, 8, -2], [-2, 2, 4, -2], [4, -4, -2, 4], [1, -1, -9, 1], [-2, -2, 2, -2], [-4, -2, 2, -2], [-8, 4, -4, 4], [1, -1, 3, 1], [-2, -4, 4, -4], [4, 6, -6, 6], [4, -4, 4, 4], [-7, -5, 5, -5], [-1, 1, 1, -1], [-4, 4, 10, -4], [-6, -4, 4, -4], [-15, -1, 1, -1], [-8, 8, 6, -8], [-2, 4, -4, 4], [-2, 2, -4, -2], [-8, 8, -6, -8], [-2, -6, 6, -6], [4, -4, 10, 4], [-3, 3, -5, -3], [-15, 3, -3, 3], [0, 0, 2, 0], [2, -2, 12, 2], [0, 0, 16, 0], [2, -10, 10, -10], [-4, 2, -2, 2], [2, -2, -16, 2], [12, -8, 8, -8], [-12, 12, 6, -12], [11, -3, 3, -3], [-11, 11, -9, -11], [0, 0, -4, 0], [12, -10, 10, -10], [20, 0, 0, 0], [6, 6, -6, 6], [4, -4, -18, 4], [-2, 0, 0, 0], [-3, -5, 5, -5], [-4, 4, 2, -4], [-2, -10, 10, -10], [0, 0, -2, 0], [-14, 2, -2, 2], [-5, 5, 1, -5], [8, -8, -2, 8], [-15, 15, -1, -15], [-14, -10, 10, -10], [18, -8, 8, -8], [2, -2, 12, 2], [0, 0, -14, 0], [6, 8, -8, 8], [5, -5, 7, 5], [0, 0, -28, 0], [6, 4, -4, 4], [-12, -4, 4, -4], [-15, 7, -7, 7], [-16, 16, -2, -16], [14, -14, -12, 14], [9, 7, -7, 7], [12, -12, 14, 12], [14, -14, -8, 14], [-6, 8, -8, 8], [6, -10, 10, -10], [6, -6, -8, 6], [8, -8, 14, 8], [18, 2, -2, 2], [16, 2, -2, 2], [20, 12, -12, 12], [-15, 15, -5, -15], [6, 10, -10, 10], [6, 12, -12, 12], [-4, 4, 4, -4], [13, -1, 1, -1], [5, -5, -17, 5], [8, -8, 8, -8], [9, -9, 9, -9], [-4, 4, 10, -4], [-2, 2, 28, -2], [-9, 5, -5, 5], [-8, 8, -12, -8], [30, 8, -8, 8], [3, -3, 17, 3], [4, -4, -20, 4], [-1, 1, -35, -1], [-12, 12, -6, -12], [11, -11, -3, 11], [10, 2, -2, 2], [-12, 12, -14, -12], [2, -24, 24, -24], [-3, 3, -9, -3], [-2, 20, -20, 20], [0, 20, -20, 20], [22, -2, 2, -2], [-5, -15, 15, -15], [6, -6, -16, 6], [-24, 2, -2, 2], [-10, 10, -16, -10], [-6, -10, 10, -10], [10, -6, 6, -6], [7, -7, -27, 7], [15, 1, -1, 1], [-14, 14, -12, -14], [-10, 10, 0, -10], [-8, 8, -22, -8], [-16, 2, -2, 2], [6, -6, 28, 6], [15, -15, 9, 15], [0, 0, 18, 0], [-4, 4, -26, -4], [12, -12, 8, 12], [-12, 8, -8, 8], [5, 11, -11, 11], [4, -4, 42, 4], [-10, 10, 12, -10], [-15, 15, -15, 15], [2, 14, -14, 14], [4, -4, 26, 4], [-22, 16, -16, 16], [6, 4, -4, 4], [6, -6, -12, 6], [9, -9, -1, 9], [-22, 22, 12, -22], [-10, -8, 8, -8], [4, -4, -22, 4], [12, -12, -14, 12], [-3, 7, -7, 7], [-7, 7, 11, -7], [-4, 4, 22, -4], [-4, -12, 12, -12], [6, -6, 32, 6], [-6, -8, 8, -8], [-14, 14, 4, -14], [39, 5, -5, 5], [-16, 16, -26, -16], [-6, -2, 2, -2], [-19, 19, -5, -19], [19, -19, 19, -19], [-7, 7, -17, -7], [18, 18, -18, 18], [-32, -4, 4, -4], [14, 2, -2, 2], [20, -14, 14, -14], [0, 0, 32, 0], [7, -7, 9, 7], [-38, 12, -12, 12], [-25, 5, -5, 5], [4, -4, -30, 4], [-28, 2, -2, 2], [-47, -1, 1, -1], [38, 2, -2, 2], [-18, 18, -20, -18], [-26, -12, 12, -12], [-22, 22, 0, -22], [-30, 30, -4, -30], [16, -16, -8, 16], [-42, -6, 6, -6], [-32, 10, -10, 10], [4, -12, 12, -12], [-14, 12, -12, 12], [0, 0, -2, 0], [0, 0, -48, 0], [-3, 15, -15, 15], [-16, 16, 22, -16], [25, 15, -15, 15], [28, -28, 2, 28], [-2, 2, -44, -2], [-8, -10, 10, -10], [35, 9, -9, 9], [-4, 4, 40, -4], [-8, 8, 42, -8], [3, -3, 33, 3], [-2, -10, 10, -10], [46, -6, 6, -6], [-26, -4, 4, -4], [-11, 11, -9, -11], [20, -20, -2, 20], [-24, 24, -6, -24], [7, -7, -39, 7], [36, -10, 10, -10], [14, 2, -2, 2], [-22, 22, 8, -22], [6, -4, 4, -4], [-49, -3, 3, -3], [12, -12, 0, 12], [2, -2, 2, -2], [-18, 18, 28, -18], [14, 16, -16, 16], [2, -2, -32, 2], [24, -24, 18, 24], [20, -16, 16, -16], [-3, 3, 39, -3], [18, 28, -28, 28], [16, -16, 42, 16], [12, -30, 30, -30], [8, -8, -6, 8], [-25, -15, 15, -15], [5, -5, 23, 5], [48, 4, -4, 4], [-34, -14, 14, -14], [29, 11, -11, 11], [-30, 8, -8, 8], [45, -9, 9, -9], [-14, 14, 0, -14], [-24, 24, 2, -24], [-20, -4, 4, -4], [6, -36, 36, -36], [15, 25, -25, 25], [-13, 13, 45, -13], [-8, 8, -30, -8], [-24, 6, -6, 6], [0, -4, 4, -4], [3, 41, -41, 41], [0, 0, 30, 0], [22, -22, -16, 22], [-30, 30, 0, -30], [34, -2, 2, -2], [10, -30, 30, -30], [-19, 19, 11, -19], [-45, -3, 3, -3], [26, -2, 2, -2], [6, -6, 0, 6], [26, -26, 24, 26], [-10, 18, -18, 18], [9, -9, -5, 9], [4, -4, 42, 4], [11, -11, 11, -11], [8, -8, 12, 8], [6, -36, 36, -36], [12, -36, 36, -36], [5, -13, 13, -13], [8, -8, -30, 8], [-22, 22, -4, -22], [35, -7, 7, -7], [10, -10, -12, 10], [10, 0, 0, 0], [35, 13, -13, 13], [-6, 38, -38, 38], [8, -8, -6, 8], [-2, 2, 4, -2], [4, -4, -28, 4], [35, -35, -15, 35], [4, -4, -26, 4], [-46, 0, 0, 0], [29, -9, 9, -9], [13, -13, 23, 13], [0, 0, 20, 0], [8, -8, 10, 8], [20, -10, 10, -10], [-18, 18, -44, -18], [-16, 16, 52, -16], [-18, -16, 16, -16], [19, -19, -15, 19], [-20, 20, -22, -20], [0, 0, 16, 0], [11, -11, 1, 11], [0, -10, 10, -10], [48, -12, 12, -12], [-58, -12, 12, -12], [-17, -23, 23, -23], [-5, 5, -59, -5], [20, -20, -32, 20], [-26, -2, 2, -2], [-28, 28, 6, -28], [63, 5, -5, 5], [-6, 2, -2, 2], [33, 11, -11, 11], [-8, 8, 24, -8], [-1, 1, 21, -1], [18, -2, 2, -2], [12, 10, -10, 10], [-38, 38, 16, -38], [-4, -16, 16, -16], [42, 2, -2, 2], [0, 0, -10, 0], [-21, 21, -19, -21], [-60, -6, 6, -6], [4, -4, 58, 4], [-4, 4, 2, -4], [-32, 32, -18, -32], [-10, 10, -10, 10], [36, -36, 18, 36], [-34, 34, -4, -34], [-29, 29, 1, -29], [14, 8, -8, 8], [-28, 28, 30, -28], [-16, 16, -16, 16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1800_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1800_2_k_j();".
// 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_1800_2_k_j(:prec:=4)
    chi := MakeCharacter_1800_k();
    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_1800_2_k_j();".
// 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_1800_2_k_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1800_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-2, -2, 1]>,<11,R![4, 0, 1]>,<17,R![-12, 0, 1]>],Snew);
    return Vf;
end function;