
// Make newform 1050.2.b.a in Magma, downloaded from the LMFDB on 10 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1050_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1050_b_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_1050_2_b_a();".
// 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_1050_2_b_a();".
// 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, 1, 1, 0], [0, 2, 0, 1], [-1, 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_1050_b();"
function MakeCharacter_1050_b()
    N := 1050;
    order := 2;
    char_gens := [701, 127, 451];
    v := [1, 2, 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_1050_b_Hecke();"
function MakeCharacter_1050_b_Hecke( : Kf := false)
    N := 1050;
    order := 2;
    char_gens := [701, 127, 451];
    char_values := [[-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, 1, 0], [-1, 0, 0, 1], [0, 0, 0, 0], [-2, 0, -1, 1], [0, -2, -2, -2], [0, 1, 0, 1], [4, 1, 0, -1], [0, 2, 6, 2], [0, 0, -4, 0], [0, 3, 4, 3], [0, 1, -4, 1], [-2, 1, 0, -1], [0, 2, 0, -2], [0, 4, 0, -4], [-4, -2, 0, 2], [0, 2, -2, 2], [-2, -2, 0, 2], [0, 1, 6, 1], [12, 0, 0, 0], [0, 1, 6, 1], [0, 1, 10, 1], [-4, -4, 0, 4], [6, 7, 0, -7], [8, -2, 0, 2], [0, 1, -2, 1], [10, 2, 0, -2], [0, 7, 6, 7], [0, 6, 4, 6], [2, 2, 0, -2], [0, -4, 2, -4], [4, -3, 0, 3], [2, -6, 0, 6], [0, -2, 6, -2], [0, 2, -14, 2], [0, -3, 8, -3], [16, 4, 0, -4], [0, 1, 8, 1], [8, 2, 0, -2], [4, -4, 0, 4], [-2, -6, 0, 6], [0, 4, 10, 4], [0, -5, -10, -5], [0, -1, -6, -1], [6, 0, 0, 0], [0, -2, 6, -2], [0, -5, -16, -5], [-8, 0, 0, 0], [0, -3, -14, -3], [2, -1, 0, 1], [0, 1, -10, 1], [0, -2, 14, -2], [0, -5, 6, -5], [0, 8, 8, 8], [10, 2, 0, -2], [8, 5, 0, -5], [0, -4, -8, -4], [-2, -2, 0, 2], [0, 3, 28, 3], [6, 9, 0, -9], [0, 0, -20, 0], [0, -1, -12, -1], [-10, 6, 0, -6], [0, 5, 12, 5], [-20, 2, 0, -2], [0, 3, 2, 3], [0, -4, -6, -4], [8, -4, 0, 4], [-14, -6, 0, 6], [0, 2, 0, 2], [0, 5, 14, 5], [-24, -3, 0, 3], [0, 5, 6, 5], [0, 5, 2, 5], [26, 5, 0, -5], [-8, -8, 0, 8], [-4, -8, 0, 8], [0, -5, -24, -5], [0, -9, -12, -9], [0, -2, -12, -2], [0, 6, 0, 6], [-22, -12, 0, 12], [-6, 2, 0, -2], [0, 11, 26, 11], [0, -7, -18, -7], [0, 5, 4, 5], [0, -2, -16, -2], [0, 2, 8, 2], [-22, -4, 0, 4], [2, 4, 0, -4], [16, 5, 0, -5], [-30, -3, 0, 3], [-28, -8, 0, 8], [-4, -11, 0, 11], [0, -6, -6, -6], [8, 8, 0, -8], [-16, -10, 0, 10], [-18, 2, 0, -2], [32, 4, 0, -4], [0, 15, 24, 15], [26, 4, 0, -4], [28, 6, 0, -6], [0, 2, -10, 2], [34, 5, 0, -5], [0, 2, -12, 2], [4, 12, 0, -12], [0, 5, 2, 5], [-14, 3, 0, -3], [-8, -17, 0, 17], [0, 1, -38, 1], [0, -2, 8, -2], [0, 1, 18, 1], [-22, -13, 0, 13], [0, 0, -2, 0], [0, 0, 14, 0], [4, -8, 0, 8], [0, -6, -16, -6], [0, -3, 16, -3], [20, 12, 0, -12], [0, 8, 14, 8], [0, -2, 34, -2], [0, -3, 22, -3], [-22, 2, 0, -2], [18, 0, 0, 0], [0, 0, 36, 0], [0, 4, -26, 4], [0, 13, 28, 13], [-18, 2, 0, -2], [32, 2, 0, -2], [0, 3, -30, 3], [0, -13, -4, -13], [4, 4, 0, -4], [0, 2, 8, 2], [16, -6, 0, 6], [-6, 7, 0, -7], [-4, 8, 0, -8], [0, 0, -36, 0], [18, -6, 0, 6], [0, 3, -40, 3], [14, 6, 0, -6], [0, -10, -24, -10], [0, 12, 22, 12], [0, 3, 28, 3], [16, 5, 0, -5], [0, -12, -24, -12], [0, -5, 14, -5], [-8, 2, 0, -2], [0, -3, -4, -3], [-20, 7, 0, -7], [0, 4, -22, 4], [0, 2, 28, 2], [-30, -7, 0, 7], [-8, -6, 0, 6], [4, 8, 0, -8], [-20, 2, 0, -2], [20, -2, 0, 2], [0, -7, -42, -7], [-20, -20, 0, 20], [-28, -8, 0, 8], [0, -13, -26, -13], [-22, 0, 0, 0], [0, 10, 48, 10], [0, -6, 30, -6], [16, 19, 0, -19], [58, 0, 0, 0], [0, 6, -6, 6], [-20, 6, 0, -6], [20, -2, 0, 2], [0, -1, -24, -1], [22, 12, 0, -12], [-6, 10, 0, -10], [0, -4, -18, -4], [0, 11, 6, 11], [0, 7, -18, 7], [26, 0, 0, 0], [0, -1, 28, -1], [-12, -2, 0, 2], [8, 16, 0, -16], [0, -11, -16, -11], [0, -5, -26, -5], [0, -23, -14, -23], [0, -7, 0, 7], [-2, 10, 0, -10], [-42, -3, 0, 3], [-32, -15, 0, 15], [0, -10, -24, -10], [58, -2, 0, 2], [-22, -19, 0, 19], [0, 7, 16, 7], [38, -2, 0, 2], [4, -4, 0, 4], [0, -13, 6, -13], [0, -16, 0, -16], [12, 20, 0, -20], [-2, 20, 0, -20], [0, -16, -28, -16], [0, 11, 0, -11], [-18, -20, 0, 20], [22, 11, 0, -11], [-8, -11, 0, 11], [-16, 10, 0, -10], [0, 15, -4, 15], [0, 7, -28, 7], [0, -19, 8, -19], [0, -10, -36, -10], [-22, -2, 0, 2], [-38, -6, 0, 6], [0, 11, -20, 11], [0, -12, -16, -12], [0, 12, 48, 12], [0, -2, 38, -2], [-22, -4, 0, 4], [18, -10, 0, 10], [-32, -23, 0, 23], [22, -11, 0, 11], [-4, 6, 0, -6], [0, -8, -8, -8], [16, 9, 0, -9], [28, 20, 0, -20], [0, 12, 0, 12], [0, 22, 18, 22], [-42, -14, 0, 14], [0, 11, 20, 11], [0, 2, 28, 2], [40, -1, 0, 1], [26, 3, 0, -3], [-18, 2, 0, -2], [-16, 5, 0, -5], [0, 19, 10, 19], [0, 7, -6, 7], [0, -6, 34, -6], [-22, -3, 0, 3], [0, 10, 14, 10], [-12, 6, 0, -6], [0, 0, 20, 0], [0, 13, 32, 13], [48, 0, 0, 0], [0, 2, 16, 2], [0, 30, 26, 30], [0, -4, -38, -4], [-28, -26, 0, 26], [0, 4, -40, 4], [0, -22, -22, -22], [0, 5, 14, 5], [-2, -12, 0, 12], [16, -3, 0, 3], [20, 20, 0, -20], [0, -1, -34, -1], [22, 4, 0, -4], [8, 8, 0, -8], [0, 10, 36, 10], [10, -17, 0, 17], [0, -8, 0, 8], [0, 6, 4, 6], [0, 20, 4, 20], [-14, 12, 0, -12], [0, 10, 46, 10], [-18, -10, 0, 10], [0, 7, 44, 7], [50, 12, 0, -12], [0, -1, 26, -1], [44, -7, 0, 7], [0, 22, 0, 22], [0, -5, 34, -5], [0, 17, -4, 17], [-4, -7, 0, 7], [0, 10, 34, 10], [0, 7, 28, 7], [-12, -20, 0, 20], [-68, -4, 0, 4], [0, -16, -30, -16], [0, -13, -18, -13], [0, 8, 0, -8], [0, -19, -30, -19], [-24, 6, 0, -6], [0, -13, 14, -13], [0, -5, -66, -5], [0, -12, 16, -12], [2, 2, 0, -2], [22, -10, 0, 10], [-2, 20, 0, -20], [-48, -12, 0, 12], [20, -2, 0, 2], [28, 10, 0, -10], [0, 23, 38, 23], [0, 23, 20, 23], [0, 3, -2, 3], [22, 6, 0, -6], [0, -20, -42, -20], [0, 9, 8, 9], [24, 24, 0, -24], [0, 19, 4, 19], [22, -1, 0, 1], [0, -24, -18, -24], [-34, 8, 0, -8], [-54, -5, 0, 5], [-46, -16, 0, 16], [0, 1, 16, 1], [-22, 4, 0, -4], [66, 16, 0, -16], [0, 3, -20, 3], [0, -27, -18, -27], [0, 0, -62, 0], [12, 2, 0, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1050_b_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1050_2_b_a();".
// 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_1050_2_b_a(:prec:=4)
    chi := MakeCharacter_1050_b();
    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_1050_2_b_a();".
// 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_1050_2_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1050_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![20, 0, 1]>,<17,R![4, -6, 1]>,<37,R![4, 6, 1]>],Snew);
    return Vf;
end function;