
// Make newform 1950.2.i.bh in Magma, downloaded from the LMFDB on 18 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_bh();".
// 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_bh();".
// 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;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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], [0, 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 := [[1, 0, -1, 0], [1, 0, -1, 0], [0, 0, 0, 0], [0, -1, 1, -1], [1, 1, -1, -2], [3, 2, -3, 0], [0, -2, 1, -2], [0, -1, -3, -1], [1, -3, -1, 6], [3, -2, -3, 4], [4, 0, 0, 0], [1, -4, -1, 8], [4, -3, -4, 6], [0, -1, -5, -1], [1, 10, 0, -5], [-6, 2, 0, -1], [0, -6, 2, -6], [0, -7, -2, -7], [-1, -5, 1, 10], [0, -1, -3, -1], [-4, -10, 0, 5], [12, 0, 0, 0], [-7, -2, 0, 1], [-2, -4, 2, 8], [0, 0, -10, 0], [-7, 4, 7, -8], [-11, -2, 0, 1], [3, 1, -3, -2], [10, 0, 0, 0], [0, 0, 9, 0], [4, 0, -4, 0], [0, 16, 0, -8], [0, -2, 15, -2], [0, 4, 8, 4], [0, 4, -13, 4], [7, 14, 0, -7], [1, 12, 0, -6], [0, 2, -10, 2], [0, 8, 0, -16], [0, 12, 2, 12], [-13, 3, 13, -6], [-4, 14, 0, -7], [0, -2, 6, -2], [-2, 5, 2, -10], [-4, -8, 4, 16], [0, -11, 9, -11], [18, -2, -18, 4], [10, 6, -10, -12], [0, 3, 1, 3], [8, -24, 0, 12], [-2, 12, 0, -6], [11, 18, 0, -9], [0, -6, -1, -6], [0, -2, 18, -2], [-3, 0, 3, 0], [-5, 7, 5, -14], [0, 6, 6, 6], [8, 8, -8, -16], [0, -10, -5, -10], [0, -2, 0, 1], [-1, -5, 1, 10], [0, -1, 12, -1], [21, 2, 0, -1], [-9, -14, 0, 7], [14, 0, 0, 0], [-28, 2, 0, -1], [0, -12, 8, -12], [8, 22, 0, -11], [0, 5, 3, 5], [26, 4, -26, -8], [-1, 14, 1, -28], [21, 14, 0, -7], [7, -15, -7, 30], [0, 2, 27, 2], [-2, 14, 2, -28], [0, -4, 20, -4], [-31, -8, 0, 4], [0, 10, -2, 10], [-10, -11, 10, 22], [0, -4, -23, -4], [10, 2, -10, -4], [-2, -22, 0, 11], [13, 3, -13, -6], [0, -3, -20, -3], [-15, -3, 15, 6], [0, 40, 0, -20], [0, -8, -6, -8], [-14, -3, 14, 6], [0, -4, -33, -4], [15, 10, 0, -5], [3, -22, 0, 11], [16, 0, -16, 0], [0, -17, 9, -17], [-27, -3, 27, 6], [30, -12, 0, 6], [0, 3, 27, 3], [3, -16, -3, 32], [10, 34, 0, -17], [-7, 17, 7, -34], [4, 6, 0, -3], [-31, -6, 0, 3], [-6, 5, 6, -10], [0, 0, 8, 0], [-32, -8, 32, 16], [-17, 10, 0, -5], [2, 30, 0, -15], [24, 8, -24, -16], [-9, 12, 0, -6], [-12, -24, 0, 12], [-17, 0, 17, 0], [0, -6, 38, -6], [1, -12, -1, 24], [0, 16, -5, 16], [2, 44, 0, -22], [0, 8, 4, 8], [0, 11, 10, 11], [0, 12, 12, 12], [-6, 10, 6, -20], [-2, -20, 2, 40], [0, -2, 38, -2], [16, -13, -16, 26], [12, 1, -12, -2], [14, -40, 0, 20], [0, 4, -20, 4], [-25, 9, 25, -18], [-10, 4, 0, -2], [0, 11, -18, 11], [0, -6, 22, -6], [15, 2, 0, -1], [-35, 12, 0, -6], [-16, -8, 16, 16], [-18, 2, 18, -4], [-17, 11, 17, -22], [-2, 22, 2, -44], [0, -4, -30, -4], [-42, 2, 42, -4], [0, -20, 6, -20], [0, 10, -2, 10], [0, -4, -8, -4], [24, 15, -24, -30], [-20, 16, 0, -8], [14, 14, -14, -28], [0, 6, -30, 6], [-28, 24, 0, -12], [-30, -7, 30, 14], [0, 22, -10, 22], [19, 8, 0, -4], [3, 8, 0, -4], [-1, 50, 0, -25], [-9, 14, 0, -7], [0, -2, -13, -2], [-30, -7, 30, 14], [-6, -28, 0, 14], [36, 4, -36, -8], [0, -4, 0, 8], [6, -44, 0, 22], [0, -2, 54, -2], [0, 9, -24, 9], [-2, -22, 0, 11], [2, -12, 0, 6], [0, -16, 0, 32], [0, 16, -10, 16], [1, -18, 0, 9], [0, -4, 48, -4], [-3, -14, 3, 28], [4, -32, 0, 16], [11, 11, -11, -22], [0, 4, -9, 4], [-34, -12, 0, 6], [12, 46, 0, -23], [9, 2, 0, -1], [0, -17, -18, -17], [0, 4, -8, 4], [0, 5, -40, 5], [-26, 12, 0, -6], [0, -11, 30, -11], [31, -11, -31, 22], [21, -16, 0, 8], [16, 0, -16, 0], [-2, -20, 2, 40], [-31, 6, 0, -3], [4, 24, 0, -12], [13, -36, 0, 18], [-42, -16, 0, 8], [23, 19, -23, -38], [0, -14, -39, -14], [26, -44, 0, 22], [-20, -32, 0, 16], [1, 6, -1, -12], [0, 1, 47, 1], [0, 1, 6, 1], [0, -1, 1, -1], [13, -26, 0, 13], [6, 8, -6, -16], [0, -11, 39, -11], [-6, -6, 6, 12], [9, 4, 0, -2], [0, -2, -27, -2], [-21, -16, 0, 8], [-2, 20, 0, -10], [0, 6, -51, 6], [0, 6, 6, 6], [-39, -6, 39, 12], [19, 12, 0, -6], [13, 9, -13, -18], [-6, 7, 6, -14], [-9, 30, 0, -15], [0, 0, -44, 0], [-4, -5, 4, 10], [0, -14, -10, -14], [32, -13, -32, 26], [-54, -20, 0, 10], [-19, -5, 19, 10], [0, -31, 15, -31], [0, -4, -28, -4], [7, 0, 0, 0], [-32, 16, 0, -8], [0, -3, 50, -3], [38, 18, -38, -36], [-10, -8, 0, 4], [-12, 21, 12, -42], [4, 0, 0, 0], [30, -18, 0, 9], [0, -4, 8, -4], [9, 23, -9, -46], [-24, -14, 0, 7], [1, 4, -1, -8], [0, -10, 38, -10], [0, 10, -46, 10], [-28, 0, 0, 0], [-7, 24, 7, -48], [26, -22, -26, 44], [-2, -30, 2, 60], [-18, -30, 0, 15], [-4, -40, 0, 20], [44, 32, 0, -16], [0, 18, 6, 18], [14, -17, -14, 34], [0, 22, -18, 22], [15, 21, -15, -42], [-15, -13, 15, 26], [-17, -15, 17, 30], [0, 23, 13, 23], [14, 21, -14, -42], [0, -6, 43, -6], [30, -20, 0, 10], [0, -36, 0, -36], [12, -16, -12, 32], [0, 1, -29, 1], [-23, 9, 23, -18], [-6, -10, 6, 20], [-8, 40, 8, -80], [46, 12, 0, -6], [-37, -4, 37, 8], [64, -8, 0, 4], [0, -31, 9, -31], [0, -32, -10, -32], [-55, -11, 55, 22], [46, 16, 0, -8], [0, -11, -4, -11], [18, -12, 0, 6], [22, -18, -22, 36], [22, -26, 0, 13], [-19, 2, 19, -4], [0, 0, -21, 0], [0, -19, 19, -19], [0, 0, 6, 0], [20, 46, 0, -23], [0, -3, -23, -3], [0, -5, 18, -5], [62, -22, 0, 11], [46, 20, 0, -10], [14, 3, -14, -6], [0, 12, 8, 12], [0, -36, -2, -36], [-1, -39, 1, 78], [0, -8, 44, -8], [-52, 6, 0, -3], [0, 18, 43, 18], [0, -19, -3, -19], [21, 9, -21, -18], [-38, 10, 38, -20], [52, 0, 0, 0], [32, -11, -32, 22], [-47, 34, 0, -17], [47, -6, 0, 3], [0, 58, 0, -29], [-26, -38, 0, 19], [0, -31, -23, -31], [0, 1, 10, 1], [22, -30, -22, 60], [0, -37, 9, -37], [-33, -4, 33, 8], [0, -20, -12, -20], [0, 18, -26, 18], [-15, 20, 0, -10], [24, 32, 0, -16], [-18, 1, 18, -2], [-26, -22, 26, 44], [21, 21, -21, -42], [0, 15, 20, 15], [-30, -2, 0, 1], [37, -15, -37, 30]];
    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_bh();".
// 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_bh(: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_bh();".
// 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_bh( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1950_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 4, 6, -2, 1]>,<11,R![4, 4, 6, -2, 1]>,<17,R![121, 22, 15, -2, 1]>,<19,R![36, 36, 30, 6, 1]>],Snew);
    return Vf;
end function;