
// Make newform 29.2.d.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_29_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_29_d_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_29_2_d_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_29_2_d_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, -1, 1, -1, 1, -1, 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_29_d();"
function MakeCharacter_29_d()
    N := 29;
    order := 7;
    char_gens := [2];
    v := [3];
    // 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_29_d_Hecke();"
function MakeCharacter_29_d_Hecke(Kf)
    N := 29;
    order := 7;
    char_gens := [2];
    char_values := [[0, -1, 0, 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 := [[-1, 1, 0, 1, -1, 1], [-1, 0, 0, 0, 0, 1], [1, -1, 0, -2, 0, -2], [1, -2, 0, 0, 2, -1], [-1, -2, 2, -2, -1, 0], [-3, 4, -2, 4, -3, 0], [0, 0, -2, 2, -2, 2], [1, 0, 1, 0, 2, -2], [0, -1, 3, 1, 3, -1], [3, 0, 2, -2, 6, -2], [-1, 1, 0, 2, -6, 2], [3, -2, 2, -3, 0, 0], [4, 0, 2, 0, 0, -2], [0, 3, -3, 1, -3, 3], [1, 0, -6, 0, 1, 0], [-1, 1, 0, 2, -4, 2], [-8, 0, 2, -2, 2, -2], [3, -4, 2, -2, 4, -3], [7, -4, 4, -7, 0, -8], [3, 0, -6, 0, 3, 0], [0, -7, 3, -5, 3, -7], [-3, 0, 0, 3, 0, 6], [1, 4, 1, 0, -4, 4], [-1, 1, 0, 6, 0, 6], [5, -8, 5, 0, 8, -8], [0, -9, 5, -1, 5, -9], [-3, 4, -10, 4, -3, 0], [-5, 10, -10, 5, 0, 4], [-5, 2, -6, 6, -2, 5], [3, 0, 8, -8, 0, -3], [-1, 6, 6, 6, -1, 0], [0, -5, 1, -9, 1, -5], [-9, 14, -14, 9, 0, 12], [0, -1, -9, 5, -9, -1], [13, -6, 13, 0, 2, -2], [0, 5, 1, 11, 1, 5], [-2, 0, -4, -10, 10, 4], [3, -8, 4, -8, 3, 0], [3, 6, 4, -4, -6, -3], [-10, 0, -10, -2, 2, 10], [-3, 3, 0, 0, -2, 0], [-5, 2, -10, 10, -2, 5], [-8, 0, 6, 0, 0, -6], [7, -14, 7, 0, 0, 0], [0, -3, -9, 7, -9, -3], [7, 6, 7, 0, 2, -2], [-3, -2, 2, 3, 0, 16], [-13, -2, 2, 13, 0, 4], [-7, 7, 0, 10, -14, 10], [-3, 4, 6, -6, -4, 3], [10, 0, 10, -8, 8, -10], [17, -16, 22, -22, 16, -17], [-3, -2, 2, 3, 0, -12], [-3, -12, -3, 0, 10, -10], [-1, -14, -1, 0, -2, 2], [-9, 9, 0, 2, 10, 2], [-3, 10, -10, 3, 0, -6], [-9, 8, -9, 0, -10, 10], [3, 0, 12, 0, 3, 0], [15, -2, 2, -15, 0, 6], [3, -4, 4, -4, 4, -3], [-1, 18, 0, 0, -18, 1], [18, 0, 6, -6, 6, -6], [13, -16, 16, -13, 0, -22], [0, 9, 5, -9, 5, 9], [7, -7, 0, -6, 8, -6], [22, 0, 10, -8, 8, -10], [9, -4, 4, -4, 9, 0], [-6, 0, -2, 14, -14, 2], [-20, 0, -10, 20, -20, 10], [-5, 5, 0, -4, 4, -4], [3, -12, 10, -12, 3, 0], [-1, 32, -1, 0, -2, 2], [1, -6, 1, 0, 18, -18], [13, -13, 0, 6, 4, 6], [0, 17, -5, 21, -5, 17], [14, 0, 8, 8, -8, -8], [-9, 0, 0, 9, 0, 8], [13, -13, 0, 6, 6, 6], [-25, 14, -14, 14, -14, 25], [-9, 2, -28, 2, -9, 0], [0, -3, -1, 15, -1, -3], [-13, 26, -13, 0, -8, 8], [-3, 3, 0, -10, -12, -10], [-3, 12, -3, 0, 0, 0], [-7, 12, -12, 7, 0, 18], [0, -1, -5, -19, -5, -1], [3, -2, -6, 6, 2, -3], [-3, -6, -10, 10, 6, 3], [-10, 0, 10, 4, -4, -10], [25, -18, 14, -14, 18, -25], [0, 21, -5, 7, -5, 21], [0, -11, 3, -11, 3, -11], [-19, 19, 0, 18, -4, 18], [0, 7, -23, -5, -23, 7], [5, -2, 5, 0, 0, 0], [17, -6, 8, -6, 17, 0], [10, 0, -2, -8, 8, 2], [-16, 0, -18, 10, -10, 18], [-13, 8, -13, 0, -18, 18], [-3, -26, -3, 0, -2, 2], [0, -5, 5, -19, 5, -5], [0, 0, -24, 18, -18, 24], [-15, 4, -26, 4, -15, 0], [7, 2, -2, -7, 0, -18], [5, -10, 22, -22, 10, -5], [23, -24, 18, -18, 24, -23], [-19, 14, -4, 14, -19, 0], [11, -20, 11, 0, 12, -12], [-1, 4, -4, 1, 0, 30], [-23, 23, 0, 8, 10, 8], [-5, -14, -5, 0, -10, 10], [7, -10, 10, -7, 0, -30], [-13, 12, -13, 0, -32, 32], [9, 12, -4, 4, -12, -9], [-11, 30, -16, 16, -30, 11], [21, -21, 0, -2, 16, -2], [-15, 20, -20, 15, 0, 20], [0, -13, 11, -15, 11, -13], [-1, -12, 12, 1, 0, -2], [0, -7, 7, 9, 7, -7], [0, 9, -5, -7, -5, 9], [3, 18, 3, 0, 16, -16], [-1, -4, 18, -4, -1, 0], [1, -1, 0, -20, 20, -20], [-27, 27, 0, 26, -12, 26], [-13, 6, 16, 6, -13, 0], [0, -15, 5, 23, 5, -15], [-5, 5, 0, -4, 38, -4], [5, 8, -8, -5, 0, -22], [0, -11, 27, -11, 27, -11], [13, -12, -2, -12, 13, 0], [15, -14, 22, -22, 14, -15], [19, 0, 0, 0, 0, -19], [-15, 10, 0, 0, -10, 15], [0, 19, -29, 17, -29, 19], [3, -12, 3, 0, -4, 4], [7, -2, 7, 0, -2, 2], [0, -9, 17, 7, 17, -9], [-13, 8, -12, 12, -8, 13], [38, 0, 20, -32, 32, -20], [-3, 0, 0, 3, 0, -10], [13, -14, -6, -14, 13, 0], [0, -25, 9, 3, 9, -25], [6, 0, 26, -22, 22, -26], [7, -7, 0, -4, -38, -4], [0, 0, 2, -20, 20, -2], [11, -18, 18, -18, 11, 0], [27, -34, 12, -34, 27, 0], [-7, 10, -30, 30, -10, 7], [1, 14, -8, 8, -14, -1], [-1, 8, -24, 8, -1, 0], [7, -4, -18, -4, 7, 0], [-12, 0, -24, 30, -30, 24], [13, -30, 30, -13, 0, -30], [-4, 0, 12, 0, 0, -12], [9, -8, 8, -9, 0, 0], [-4, 0, 10, 2, -2, -10], [15, -20, 20, -15, 0, 2], [-7, -10, -2, -10, -7, 0], [-5, 22, -5, 0, -24, 24], [31, -24, 31, 0, 6, -6], [-35, 16, -35, 0, -10, 10], [0, 11, -27, 27, -27, 11], [-3, 12, -12, 3, 0, -26], [-29, 20, -18, 18, -20, 29], [21, -21, 0, -4, 16, -4], [23, -18, 26, -18, 23, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_29_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_29_2_d_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_29_2_d_a(:prec:=6)
    chi := MakeCharacter_29_d();
    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_29_2_d_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_29_2_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_29_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;