
// Make newform 63.2.i.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_63_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_63_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_63_2_i_b();".
// 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_63_2_i_b();".
// 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 := [9, -21, 64, -1, 64, -19, 34, -4, 7, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [29709, -15165, 44484, -75410, 36059, -19883, 7882, -4381, 1348, -339], [98232, 130790, 30472, 47450, 51907, 25351, 10326, 4352, 2394, 658], [-398583, 81750, -414508, -2560, -118353, -85319, -20279, -21678, -796, -4192], [-336546, 31515, 82264, 361225, -150966, 233822, -28438, 54399, -9272, 8236], [-54156, 166780, 72109, 175205, -42836, 104352, -8823, 21759, -2962, 3301], [11514, -51320, -30640, -51488, -3072, -27923, -998, -5659, 248, -840], [17856, 106440, 124546, 111910, 5736, 84554, 1181, 17808, -1373, 3085], [178101, -668685, -222374, -1019855, 295851, -601717, 65033, -128469, 18997, -18476]];
    Rf_basisdens := [1, 1, 72795, 72795, 218385, 218385, 72795, 14559, 43677, 218385];
    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_63_i();"
function MakeCharacter_63_i()
    N := 63;
    order := 6;
    char_gens := [29, 10];
    v := [5, 5];
    // 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_63_i_Hecke();"
function MakeCharacter_63_i_Hecke(Kf)
    N := 63;
    order := 6;
    char_gens := [29, 10];
    char_values := [[1, 0, 0, 0, 0, 0, 1, 0, 0, 0], [1, 0, 0, 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, 0, 0, -1, 0, -1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, -1, 0, 0], [0, 1, -1, 0, 1, -1, 0, 0, 0, -1], [-1, -1, 1, 0, -1, 1, 0, 1, 0, 1], [-1, -1, 2, 1, 1, 1, 1, 1, 1, 0], [0, 1, 0, -1, 0, 0, 0, -1, 0, 0], [0, 0, 1, -2, -2, 0, -3, -1, 0, 1], [0, 0, 0, 3, 4, 2, 0, 3, 2, -2], [-2, -1, -1, 0, 1, 1, -1, 2, 0, -1], [-2, 1, -2, 0, -3, -4, -1, -4, 1, 1], [0, -4, 2, 1, 0, 1, 0, 0, -1, 1], [2, 0, 2, 0, 2, 0, 2, 0, 0, 2], [0, 5, -4, 0, 0, -4, 0, 0, 4, -4], [0, 6, -4, 2, 4, 1, -2, 1, 3, -3], [3, 0, 4, 0, 1, 1, 0, 0, 1, 1], [2, -2, -2, 0, -1, 0, 1, -1, -1, 0], [-3, 0, 2, 0, 2, 2, 0, 0, -1, -1], [-1, 0, 0, -4, 0, -4, -2, 0, 1, -1], [2, 0, 2, -1, -1, 0, 0, -2, -3, -1], [0, -2, 1, 5, 2, 3, 0, 0, -2, 2], [0, 1, 1, 0, -3, 0, 0, -3, -3, 0], [-4, 0, -1, -4, -10, -6, 0, -8, -3, 5], [0, 2, -4, 0, -2, 0, -3, 0, -2, 2], [-6, 3, 0, -2, -1, 1, -6, 2, -1, 0], [0, -3, 2, 0, 1, 6, 0, 2, -5, -1], [6, -8, 1, -1, -2, 3, 6, 1, -3, 1], [-8, 1, -7, 0, -6, -4, -4, -2, -2, -4], [0, 1, 1, 1, 3, 2, 0, 1, 2, -1], [0, -3, 2, -2, -4, 0, -5, -1, -2, 3], [-1, -5, 4, 4, 0, 2, 1, 4, 2, 2], [-1, 0, -3, -1, -2, -1, 0, -2, -2, 0], [0, -2, 4, -2, 1, -2, 0, -1, 1, -2], [-3, 5, -4, -2, 0, 2, 3, -2, 2, 2], [1, -3, 5, -1, -5, 2, -1, -1, 2, 7], [-6, 5, -1, 0, 1, -5, -3, 2, 6, -1], [-4, 0, 0, -5, 0, 0, -4, 5, 0, 0], [2, 8, -4, 1, 1, 0, 4, 0, -2, 2], [1, -12, 0, -1, -5, 5, 1, 1, -5, 0], [12, -4, 5, 0, 0, 5, 12, 0, -5, 5], [9, 0, -6, 1, -2, -3, 0, 2, -1, -3], [10, -2, -5, 0, 4, 1, 5, 8, 3, -4], [-3, 2, -1, 0, 5, -5, -6, 0, 1, -1], [0, 6, -3, -1, 3, -4, 0, 0, 3, -3], [1, 0, 7, -5, -8, -3, 0, -10, -1, 9], [-1, 4, -2, -4, -3, -1, -2, 0, 3, -3], [-4, 5, -4, 0, -1, -8, -2, 0, 7, -1], [-1, 0, 2, -4, 3, -1, -1, 4, 1, 2], [4, 2, -5, 0, -10, -7, 2, -10, -3, 0], [-3, 4, -1, 1, 7, -8, -3, -1, 8, -1], [6, -3, 1, 0, 3, 5, 3, 4, -2, -1], [4, -2, 1, 4, 6, -1, -4, 4, -1, -7], [-1, 7, -14, 4, 9, -1, 1, 4, -1, -10], [-1, -5, 11, -2, -4, 1, 1, -2, 1, 5], [-9, 0, 2, -2, -3, -1, 0, -4, 4, 8], [0, 1, -1, 12, 10, 8, 9, 6, 6, -4], [-4, 2, -4, 4, 10, -2, 4, 4, -2, -12], [0, -3, -1, -12, -17, -5, -3, -6, -10, 11], [2, 1, -14, -3, -12, -6, -2, -3, -6, 6], [1, -12, 7, 6, -1, 8, 1, -6, -8, 7], [6, -5, 4, 0, 0, 8, 3, -1, -8, 1], [1, -4, 2, 13, 5, 8, 2, 0, 3, -3], [0, -7, 1, -5, 3, -2, 0, 5, 2, 1], [-2, 2, -1, 3, 10, -7, -4, 0, 5, -5], [12, 0, 4, 4, 14, 10, 0, 8, 3, -5], [-2, 8, -4, -13, -6, -7, -4, 0, 1, -1], [4, 0, 0, 14, 9, 5, 8, 0, 3, -3], [13, 0, 16, 1, 13, 12, 0, 2, 5, 3], [-7, -3, 2, -7, -3, 5, -7, 7, -5, 2], [-6, 0, 0, -1, -6, 5, -12, 0, 3, -3], [-4, 4, 6, 0, -3, -2, -2, -7, -1, 4], [-9, -2, 0, 3, -8, 8, -9, -3, -8, 0], [2, 1, -2, 4, 1, 7, -2, 4, 7, 6], [-2, 0, -2, 7, 14, -2, 2, 7, -2, -16], [0, 3, 4, 0, 5, 2, -1, 0, 7, -5], [-2, 0, 11, 0, 10, 10, 0, 0, -2, -2], [0, 1, -1, -18, -23, -4, 6, -9, -9, 14], [4, -1, 14, -5, -3, 5, -4, -5, 5, 8], [-1, 7, -2, -9, -3, -3, 1, -9, -3, 0], [-4, -1, -1, 0, -7, -6, -2, -13, -1, 6], [-8, 2, -1, 7, 5, 2, -16, 0, 6, -6], [-3, -1, -5, 7, -6, 1, -3, -7, -1, -5], [0, 6, -6, 4, 8, -2, -13, 2, 2, -6], [-2, 10, 1, 0, -2, -5, -1, 2, 3, -4], [8, -4, 2, -12, -3, -9, 16, 0, -3, 3], [-5, -6, 3, 8, -8, 16, -10, 0, -3, 3], [1, 18, -9, -10, 0, -10, 2, 0, 6, -6], [3, 10, -5, 2, 8, -6, 6, 0, 1, -1], [1, 0, -5, 3, 2, -1, 0, 6, 3, -3], [0, 2, -6, -10, -8, -11, 6, -5, -9, 3], [7, 3, -6, -5, 8, -14, 7, 5, 14, -6], [-6, -6, -3, 7, 5, -8, -6, -7, 8, -3], [-12, -8, 3, -6, 4, -1, -12, 6, 1, 3], [0, -9, 5, 0, -9, 5, -5, 0, -4, 9], [1, -5, 10, -5, -10, 0, -1, -5, 0, 10], [0, -9, 6, -10, -9, -9, -5, -5, -8, 4], [-12, 0, 9, 0, -6, -6, 0, 0, 3, 3], [0, -2, -4, 6, 4, -1, 18, 3, -3, -1], [0, 15, -10, -6, 10, -14, 9, -3, 2, -13], [1, -8, 16, 9, 10, 8, -1, 9, 8, -2], [-11, 21, -15, -4, 1, -16, -11, 4, 16, -15], [0, -3, 6, -2, -4, 4, 20, -1, 2, 3], [2, -5, 19, 0, 20, 15, 1, 11, 5, 9], [15, 0, -7, -9, -19, -10, 0, -18, -7, 11], [11, -12, 6, -16, -20, 4, 22, 0, -19, 19], [-4, 0, 2, 12, 25, 13, 0, 24, 11, -13], [2, 6, -6, 0, -6, -9, 1, -3, 3, -3], [0, 14, -5, 16, 28, 5, -18, 8, 17, -20], [-15, 24, -6, -5, 11, -17, -15, 5, 17, -6], [6, 6, -3, -1, -3, 2, 12, 0, -6, 6], [8, -6, -19, 0, -7, -7, 4, -1, 0, -6], [26, 2, 3, 0, -6, -4, 13, -11, -2, 5], [0, 3, -6, 8, 10, -1, -1, 4, 1, -6], [10, -8, 10, -8, -16, -6, -10, -8, -6, 10], [5, 8, -3, -11, -6, -3, -5, -11, -3, 3], [-1, 0, -12, 3, -3, -6, 0, 6, -3, -9], [12, 2, -10, -23, -30, -22, -12, -23, -22, 8], [7, 9, -4, -1, 7, 5, -7, -1, 5, -2], [0, -12, 11, 12, 10, 7, 24, 6, 5, -4], [-12, -1, 14, 0, -2, 13, -6, -4, -15, 2], [0, 7, -5, 0, -3, -13, 0, -4, 10, 1], [2, -6, 3, -8, -10, 2, 4, 0, -6, 6], [0, 6, -7, 20, 16, 13, -5, 10, 9, -6], [-3, 0, 0, 10, 13, 3, 0, 20, 17, -3], [-18, 4, -2, 0, -14, -11, -9, -19, -3, 5], [6, 18, -9, 8, 26, -18, 12, 0, 14, -14], [9, -20, 10, -4, -10, 6, 18, 0, -14, 14], [1, 0, 4, 6, 5, -1, 0, 12, 12, 0], [12, -24, 11, 2, -10, 21, 12, -2, -21, 11], [-12, -6, 5, 0, 7, 12, -6, 8, -5, -1], [-32, 1, -1, 0, 6, 5, -16, 13, 1, -7], [0, 6, 0, 2, 10, -1, -1, 1, 7, -9], [0, -2, -2, 4, -1, 3, 0, 4, 3, 4], [0, -6, 6, -2, -8, 5, -14, -1, -1, 7], [-23, 0, 14, -2, 0, 2, 0, -4, -3, 1], [0, -17, 7, 16, 4, 10, 18, 8, -2, 4], [-7, 3, 8, -22, -21, -9, 7, -22, -9, 12], [0, -9, 8, 12, 1, 16, 15, 6, 5, 5], [-5, 10, -5, 1, 7, -6, -10, 0, 1, -1], [-15, -1, 11, 0, -6, 17, -15, 0, -17, 11], [-24, -8, 1, 0, -2, 10, -12, -1, -12, -1], [18, -2, 1, 6, 1, 5, 36, 0, 2, -2], [5, -12, 6, 26, 13, 13, 10, 0, -4, 4], [2, 0, -19, 3, 1, -2, 0, 6, 2, -4], [-7, 10, -5, -4, 0, -4, -14, 0, 12, -12], [-18, -4, -13, 0, -2, 5, -9, 12, -7, -14], [0, -10, 6, -4, -8, -2, 6, -2, -6, 6], [-2, 12, 14, 0, 0, 1, -1, -1, -1, 1], [3, 1, 14, -5, 10, 4, 3, 5, -4, 14], [16, 3, 1, 0, -4, 0, 8, -2, -4, -2], [11, -2, 10, -8, -11, 1, -11, -8, 1, 12], [0, 12, -8, 6, 2, 11, -4, 3, 7, 1], [6, 0, -14, -1, -15, -14, 0, -2, -7, -5], [22, 0, -10, -12, -23, -11, 0, -24, -17, 7], [0, 4, -2, -20, -14, -14, -15, -10, -8, 4], [-4, 12, -6, -2, 12, -18, -4, 2, 18, -6], [10, -5, 13, 0, 9, 5, 5, -4, 4, 13], [11, -3, 5, 12, 8, -3, 11, -12, 3, 5], [18, 0, 1, -3, -11, -8, 0, -6, -11, -5], [-19, 6, -3, -9, -8, -1, -38, 0, -1, 1], [-3, 0, -10, 15, 20, 5, 0, 30, 5, -25], [1, 6, -3, -13, 3, -16, 2, 0, 12, -12], [10, -8, 4, 5, 5, 0, 20, 0, -2, 2], [1, -18, 9, -1, -5, 14, 1, 1, -14, 9], [3, -9, 1, -15, 8, -7, 3, 15, 7, 1], [4, -18, 9, -7, -9, 2, 8, 0, -3, 3], [-27, 1, 2, -8, -2, 4, -27, 8, -4, 2], [0, 27, -10, -6, 3, 5, 7, -3, 14, -6], [6, 6, -2, 25, 42, 18, -6, 25, 18, -24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_63_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_63_2_i_b();".
// 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_63_2_i_b(:prec:=10)
    chi := MakeCharacter_63_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(997) - 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_63_2_i_b();".
// 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_63_2_i_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_63_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, 0, 43, 0, 101, 0, 63, 0, 14, 0, 1]>],Snew);
    return Vf;
end function;