
// Make newform 1920.2.m.t in Magma, downloaded from the LMFDB on 06 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1920_m();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1920_m_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_1920_2_m_t();".
// 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_1920_2_m_t();".
// 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, 3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [1, 3, 1, 1], [0, -4, 0, -2], [2, -3, 1, -1]];
    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_1920_m();"
function MakeCharacter_1920_m()
    N := 1920;
    order := 2;
    char_gens := [511, 901, 641, 1537];
    v := [1, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1920_m_Hecke();"
function MakeCharacter_1920_m_Hecke( : Kf := false)
    N := 1920;
    order := 2;
    char_gens := [511, 901, 641, 1537];
    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 := ISA(Type(Kf), Fld), 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]];
    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], [1, 0, 0, -1], [1, 0, 1, 0], [-1, 1, 0, 1], [0, 0, -2, 0], [2, -2, 0, -2], [2, -2, 0, -2], [-2, -2, 0, -2], [-1, -1, 0, 1], [0, 2, 0, 2], [2, 2, 0, -2], [-6, -2, 0, -2], [0, 0, -2, 0], [-1, -1, -4, 1], [1, 1, 4, -1], [4, 4, 2, -4], [4, 4, 2, -4], [4, 4, 0, -4], [3, 3, 4, -3], [0, 0, 0, 0], [-4, -4, 0, 4], [-2, -2, -4, 2], [-3, -5, 0, -5], [0, 0, 4, 0], [-4, -4, -4, 4], [-8, -2, 0, -2], [-5, -3, 0, -3], [-3, 3, 0, 3], [-4, -4, -4, 4], [10, -2, 0, -2], [9, -1, 0, -1], [-4, -4, 2, 4], [-6, -2, 0, -2], [-10, 6, 0, 6], [6, 8, 0, 8], [-2, -2, -4, 2], [-6, -2, 0, -2], [1, 1, -8, -1], [1, 1, 0, -1], [0, 0, 2, 0], [-4, -4, 2, 4], [-4, -4, 2, 4], [-4, -4, 0, -4], [4, 4, 8, -4], [-4, -4, -2, 4], [2, 2, -8, -2], [-6, -6, 0, -6], [-1, 9, 0, 9], [-15, -1, 0, -1], [-4, -4, -10, 4], [-14, -2, 0, -2], [20, 4, 0, 4], [-8, 2, 0, 2], [8, 8, 6, -8], [18, 6, 0, 6], [5, 5, 4, -5], [-2, 8, 0, 8], [-2, -2, -12, 2], [2, 6, 0, 6], [0, 0, -6, 0], [3, 3, 0, -3], [-8, -8, -2, 8], [5, 5, -4, -5], [16, 0, 0, 0], [0, 0, -12, 0], [-4, -4, -6, 4], [-2, -2, 0, -2], [8, 8, 4, -8], [-7, -1, 0, -1], [-12, -12, -2, 12], [2, 6, 0, 6], [4, -12, 0, -12], [-1, -7, 0, -7], [26, -2, 0, -2], [-22, -6, 0, -6], [-5, -5, -8, 5], [18, 0, 0, 0], [10, 6, 0, 6], [0, 0, 0, 0], [0, 2, 0, 2], [0, 0, -2, 0], [-4, -4, 0, 4], [0, 16, 0, 16], [4, 4, -4, -4], [-6, -6, 0, 6], [-5, 5, 0, 5], [-8, -8, 6, 8], [8, 8, 16, -8], [-16, 2, 0, 2], [29, 3, 0, 3], [-17, 1, 0, 1], [-12, -12, 0, -12], [9, 7, 0, 7], [-4, -4, 2, 4], [-6, 10, 0, 10], [17, 17, 8, -17], [8, 6, 0, 6], [0, 0, -4, 0], [-3, -3, 0, 3], [4, 4, -6, -4], [-1, -1, 8, 1], [-8, -8, 2, 8], [-9, 9, 0, 9], [-16, -16, -6, 16], [14, -2, 0, -2], [0, 0, 4, 0], [-39, -1, 0, -1], [26, -2, 0, -2], [4, 4, 0, 4], [0, -2, 0, -2], [-37, 5, 0, 5], [-6, 6, 0, 6], [-22, 6, 0, 6], [30, -2, 0, -2], [2, 2, -8, -2], [8, 8, 2, -8], [-11, -11, -4, 11], [-3, -3, 12, 3], [4, 4, 14, -4], [4, 4, -6, -4], [4, 4, -4, -4], [-4, -4, -12, 4], [4, 4, 10, -4], [9, -1, 0, -1], [14, 14, 0, 14], [10, 0, 0, 0], [-4, -4, -14, 4], [-8, 8, 0, 8], [-15, -1, 0, -1], [-6, -2, 0, -2], [-14, 2, 0, 2], [19, 19, 12, -19], [-14, -14, 0, 14], [-14, 6, 0, 6], [8, 8, 0, -8], [10, -8, 0, -8], [20, 20, 14, -20], [5, 5, 4, -5], [4, 4, -14, -4], [0, 0, -20, 0], [-22, -6, 0, -6], [22, 8, 0, 8], [-21, -3, 0, -3], [-9, -15, 0, -15], [4, 4, 8, -4], [-4, 12, 0, 12], [-46, -2, 0, -2], [18, -10, 0, -10], [26, -6, 0, -6], [1, 1, -20, -1], [-38, -2, 0, -2], [8, 8, 10, -8], [1, 1, 0, -1], [-11, -11, -4, 11], [19, 19, 16, -19], [40, -8, 0, -8], [-2, -2, -20, 2], [-8, -8, -22, 8], [-12, -12, -12, 12], [-40, 6, 0, 6], [-21, 13, 0, 13], [-6, -2, 0, -2], [-19, 11, 0, 11], [0, 0, 14, 0], [26, 6, 0, 6], [3, 3, 4, -3], [22, 22, 12, -22], [2, -18, 0, -18], [-6, 8, 0, 8], [4, 4, -10, -4], [4, 4, 10, -4], [-12, -12, -6, 12], [44, -4, 0, -4], [-4, -4, -20, 4], [6, 6, 20, -6], [8, 8, 24, -8], [38, -10, 0, -10], [16, 10, 0, 10], [21, -13, 0, -13], [12, 12, 4, -12], [1, -9, 0, -9], [-8, -8, -2, 8], [34, -2, 0, -2], [-6, -10, 0, -10], [-9, -9, -4, 9], [-8, -18, 0, -18], [2, -10, 0, -10], [11, 11, 12, -11], [16, -14, 0, -14], [12, 12, 0, 12], [-8, -8, 0, 8], [21, -5, 0, -5], [-30, 2, 0, 2], [-2, 8, 0, 8], [-35, -5, 0, -5], [-14, -18, 0, -18], [-32, -6, 0, -6], [-6, -10, 0, -10], [-38, -2, 0, -2], [-1, -1, -8, 1], [-40, 6, 0, 6], [-6, -6, 8, 6], [-38, 6, 0, 6], [-42, -8, 0, -8], [8, 8, 6, -8], [16, 16, 2, -16], [-18, -18, -12, 18], [-21, -19, 0, -19], [8, 8, 0, -8], [42, 10, 0, 10], [4, 4, 0, -4], [22, -8, 0, -8], [-49, 1, 0, 1], [-17, -23, 0, -23], [-32, 0, 0, 0], [-8, 6, 0, 6], [23, -15, 0, -15], [0, 0, -8, 0], [7, 7, 8, -7], [-4, -4, -22, 4], [4, 4, -4, -4], [2, 2, 16, -2], [16, 16, 0, -16], [-7, -1, 0, -1], [-45, 5, 0, 5], [4, 4, -4, -4], [34, -2, 0, -2], [-24, -8, 0, -8], [3, 5, 0, 5], [12, 12, 18, -12], [-6, 22, 0, 22], [-26, -10, 0, -10], [-2, -2, 4, 2], [-16, -16, -22, 16], [-23, -23, -20, 23], [3, 3, 8, -3], [-8, 18, 0, 18], [-12, -12, 10, 12], [0, 0, -2, 0], [-56, -8, 0, -8], [27, -19, 0, -19], [-10, 6, 0, 6], [-25, -23, 0, -23], [4, 4, 30, -4], [2, 6, 0, 6], [40, -8, 0, -8], [23, 1, 0, 1], [-20, -20, -14, 20], [6, 6, 0, 6], [-23, -23, -20, 23], [2, 14, 0, 14], [-8, -8, -20, 8], [-5, -5, 20, 5], [-24, 10, 0, 10], [8, 8, 22, -8], [-12, -12, -14, 12], [-4, -4, 10, 4], [-9, -9, 20, 9], [-16, -16, -22, 16], [8, 8, 8, -8], [-13, -3, 0, -3], [15, 1, 0, 1], [-4, -4, -18, 4], [2, 22, 0, 22], [-6, -10, 0, -10], [50, 2, 0, 2], [-8, -26, 0, -26], [-24, -24, 0, 24], [21, 21, 0, -21], [-16, -16, 2, 16], [-12, -12, -10, 12], [9, 9, 16, -9], [-4, -4, -12, 4], [6, 6, 4, -6], [0, 0, -24, 0], [-9, 9, 0, 9], [13, -13, 0, -13], [4, 4, 22, -4], [-24, -6, 0, -6], [-4, -4, 10, 4], [-9, -9, 20, 9], [6, 6, 12, -6], [9, 9, 32, -9], [-4, -4, -10, 4], [23, 23, 20, -23], [20, -12, 0, -12], [12, 12, 8, -12], [-20, -20, -14, 20], [10, 10, 16, -10], [24, 24, 6, -24], [26, -16, 0, -16], [-19, -5, 0, -5], [2, 14, 0, 14], [4, 4, -22, -4], [-14, 6, 0, 6], [14, 8, 0, 8], [2, 2, 16, -2], [0, 0, -30, 0], [-8, -8, -34, 8], [-1, -1, -8, 1], [-16, -16, -12, 16], [16, 16, 30, -16], [18, 18, -8, -18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1920_m_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1920_2_m_t();".
// 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_1920_2_m_t(:prec:=4)
    chi := MakeCharacter_1920_m();
    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_1920_2_m_t();".
// 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_1920_2_m_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1920_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, 2, 1]>,<13,R![-16, -4, 1]>,<17,R![-16, -4, 1]>,<19,R![-16, 4, 1]>,<29,R![-20, 0, 1]>,<83,R![-116, 6, 1]>],Snew);
    return Vf;
end function;