
// Make newform 1950.2.i.ba in Magma, downloaded from the LMFDB on 27 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1950_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1950_i_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_1950_2_i_ba();".
// 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_1950_2_i_ba();".
// 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 := [100, 0, 10, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 10, 10];
    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_1950_i();"
function MakeCharacter_1950_i()
    N := 1950;
    order := 3;
    char_gens := [1301, 1327, 301];
    v := [3, 3, 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_1950_i_Hecke();"
function MakeCharacter_1950_i_Hecke(Kf)
    N := 1950;
    order := 3;
    char_gens := [1301, 1327, 301];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, -1, 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, 1, 0], [0, 0, -1, 0], [0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 3, 0], [-1, -1, 1, 0], [-4, -1, -4, 0], [0, -1, 0, 0], [0, -1, -1, -1], [0, -2, -2, -2], [6, 0, 0, -1], [0, 3, 1, 3], [0, -3, 0, -3], [-2, -1, -2, 0], [6, 0, 0, 0], [4, 0, 0, -1], [6, 0, 6, 0], [-5, -3, -5, 0], [0, 0, 4, 0], [11, -1, 11, 0], [5, 0, 0, 2], [6, 0, 0, -1], [-1, 0, 0, 4], [0, -1, -4, -1], [-11, 2, -11, 0], [0, -4, -4, -4], [4, 0, 0, -3], [0, 0, -6, 0], [-1, 0, 0, -3], [4, -2, 4, 0], [0, -2, 14, -2], [10, 0, 0, 2], [-12, 0, -12, 0], [12, 2, 12, 0], [10, 1, 10, 0], [-6, 0, 0, 5], [-13, 0, 0, -1], [0, -5, 0, 0], [0, 1, -5, 1], [-8, -2, -8, 0], [0, -4, 11, -4], [-3, 0, 0, 5], [-19, -1, -19, 0], [0, -2, 11, -2], [0, 0, 6, 0], [-12, -1, -12, 0], [0, -1, 10, -1], [0, 7, -4, 7], [3, -6, 3, 0], [-7, 0, 0, -3], [-20, 0, 0, 2], [9, 0, 0, 3], [-8, 0, -8, 0], [-15, 0, -15, 0], [0, -5, 10, -5], [0, -3, -9, -3], [16, 1, 16, 0], [0, -2, -6, -2], [7, 5, 7, 0], [-12, 0, 0, -3], [0, -3, 10, -3], [12, -3, 12, 0], [-4, 0, 0, -4], [3, 0, 0, -3], [5, 0, 0, -4], [16, 0, 0, 5], [12, -1, 12, 0], [4, 0, 0, 0], [-5, -2, -5, 0], [0, 9, -7, 9], [0, -5, 4, -5], [-12, 0, 0, 0], [0, -4, 6, -4], [-1, 3, -1, 0], [0, 0, 2, 0], [1, 1, 1, 0], [-22, 0, 0, -2], [-18, -2, -18, 0], [0, -2, 10, -2], [-24, 2, -24, 0], [0, 0, 9, 0], [25, 0, 0, 1], [0, 3, -15, 3], [-3, 10, -3, 0], [0, 4, -24, 4], [9, 0, 0, 0], [-12, -3, -12, 0], [0, 4, 23, 4], [14, -1, 14, 0], [-14, 0, 0, 3], [3, 0, 0, 0], [0, -2, -20, -2], [0, 1, 0, 0], [0, 6, -3, 6], [24, 0, 0, 5], [-1, 5, -1, 0], [0, 11, 8, 11], [-8, 0, 0, -4], [0, 6, -14, 6], [15, 0, 0, 5], [8, 0, 0, -7], [0, 6, 0, 6], [15, -6, 15, 0], [0, 6, 12, 6], [-26, 0, 0, -2], [43, 0, 0, 0], [0, 0, 21, 0], [2, 0, 0, 1], [17, 0, 0, -5], [0, -8, 18, -8], [30, -2, 30, 0], [0, 2, -36, 2], [-22, 8, -22, 0], [8, 0, 0, 0], [-10, -8, -10, 0], [12, 12, 12, 0], [-18, -2, -18, 0], [0, 3, 33, 3], [0, -3, 0, -3], [15, 0, 15, 0], [0, 0, 26, 0], [0, -4, -9, -4], [20, 0, 0, 4], [9, 12, 9, 0], [0, 12, 14, 12], [6, 0, 0, -9], [-11, -3, -11, 0], [31, -5, 31, 0], [-6, 0, 0, -2], [-29, 0, 0, 3], [0, -8, -6, -8], [0, -4, 32, -4], [0, -2, -24, -2], [0, -6, -8, -6], [18, -9, 18, 0], [0, -6, -4, -6], [-22, -4, -22, 0], [-4, -3, -4, 0], [4, 16, 4, 0], [0, -1, -16, -1], [6, 0, 0, -4], [0, -2, -2, -2], [12, 1, 12, 0], [-9, 0, 0, -6], [0, 6, 14, 6], [1, -11, 1, 0], [6, 0, 0, 4], [2, 0, 0, -14], [28, 0, 0, -5], [9, 0, 0, -9], [-41, 5, -41, 0], [0, 9, 6, 9], [-4, 0, 0, -4], [0, -4, -4, -4], [0, -1, 54, -1], [11, 0, 0, 7], [-6, -7, -6, 0], [-28, -4, -28, 0], [17, 0, 0, -4], [-12, 0, 0, 0], [0, 4, -35, 4], [-6, 9, -6, 0], [34, 0, 0, 6], [-42, 0, -42, 0], [0, 2, 32, 2], [-4, 0, 0, -8], [0, 7, 36, 7], [26, -6, 26, 0], [45, 0, 0, 2], [42, 0, 0, -3], [36, 0, 0, 0], [7, 9, 7, 0], [16, -14, 16, 0], [-4, 6, -4, 0], [6, 0, 0, -4], [-12, 3, -12, 0], [0, 6, 32, 6], [-34, 0, 0, -2], [0, -14, 14, -14], [0, -1, -29, -1], [48, 0, 0, -2], [-7, 0, 0, 4], [-4, 0, 0, 2], [-42, 0, 0, 0], [0, -14, 4, -14], [48, -3, 48, 0], [7, 0, 0, 9], [42, 0, 0, -5], [0, -10, 1, -10], [25, -11, 25, 0], [12, -8, 12, 0], [-3, 12, -3, 0], [-10, 0, 0, -15], [0, 1, -44, 1], [7, 10, 7, 0], [0, -4, -34, -4], [32, 0, 0, 6], [11, -3, 11, 0], [-18, 0, 0, -9], [-35, 0, 0, 5], [2, 11, 2, 0], [18, 14, 18, 0], [0, -10, -30, -10], [16, 0, 0, 10], [0, 0, 12, 0], [0, 1, -26, 1], [46, 0, 0, -2], [-37, 2, -37, 0], [0, 11, -10, 11], [-20, 3, -20, 0], [0, 6, 19, 6], [-8, 0, 0, -1], [0, -3, -38, -3], [3, -12, 3, 0], [-48, 6, -48, 0], [-57, 0, 0, 2], [22, 0, 0, 15], [24, 6, 24, 0], [0, 3, 21, 3], [8, 0, 0, 13], [0, -3, 29, -3], [-2, 0, 0, 13], [-16, 0, 0, 1], [-36, -5, -36, 0], [0, -12, 3, -12], [-15, 0, 0, 5], [0, -10, 2, -10], [20, -10, 20, 0], [-52, 0, -52, 0], [46, 0, 0, 8], [0, 17, 3, 17], [0, -1, 10, -1], [0, 0, 50, 0], [18, 0, 0, 0], [14, 0, 0, 14], [-5, 0, 0, 5], [24, 8, 24, 0], [0, 12, -6, 12], [9, 6, 9, 0], [0, 15, 3, 15], [0, 12, 21, 12], [0, 2, -38, 2], [36, 4, 36, 0], [0, -5, -21, -5], [-24, -15, -24, 0], [-11, 0, 0, -1], [-12, 13, -12, 0], [0, -10, -10, -10], [6, 11, 6, 0], [0, 22, 4, 22], [0, 5, -33, 5], [0, 11, -10, 11], [33, 0, 0, 9], [0, 4, 3, 4], [36, 0, 0, 6], [-15, 12, -15, 0], [-3, -1, -3, 0], [0, -6, 22, -6], [-26, 0, 0, 8], [9, -20, 9, 0], [4, 0, 0, 9], [0, -12, -15, -12], [22, 0, 0, -8], [0, 11, -27, 11], [8, 11, 8, 0], [-52, -8, -52, 0], [0, -6, 0, 0], [-30, 0, 0, -12], [-6, 7, -6, 0], [-38, 4, -38, 0], [18, 0, 0, 14], [28, 0, 0, 3], [0, -10, 0, -10], [-22, 13, -22, 0], [68, 0, 68, 0], [0, 12, -26, 12], [22, -14, 22, 0], [10, 0, 0, -8], [37, -6, 37, 0], [1, -2, 1, 0], [0, -2, 4, -2], [0, 3, 8, 3], [23, 0, 0, -11], [0, 3, 47, 3], [12, 0, 0, -11], [27, 0, 0, 9], [-22, 0, 0, -4], [18, 0, 0, -9], [26, 7, 26, 0], [44, 5, 44, 0], [0, 0, -18, 0], [23, -10, 23, 0], [0, -3, 42, -3], [18, -6, 18, 0], [-53, 5, -53, 0], [-60, 0, 0, -3], [-28, 0, 0, 15], [0, 9, -48, 9], [0, 4, -41, 4], [0, 3, 34, 3], [38, 0, 38, 0], [-6, 0, 0, 3], [0, -3, -10, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1950_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1950_2_i_ba();".
// 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_1950_2_i_ba(:prec:=4)
    chi := MakeCharacter_1950_i();
    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_1950_2_i_ba();".
// 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_1950_2_i_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1950_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![100, 0, 10, 0, 1]>,<11,R![9, 3, 1]>,<17,R![36, 48, 58, 8, 1]>,<19,R![100, 0, 10, 0, 1]>],Snew);
    return Vf;
end function;