
// Make newform 112.2.w.a in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_112_w();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_112_w_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_112_2_w_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_112_2_w_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, -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_112_w();"
function MakeCharacter_112_w()
    N := 112;
    order := 12;
    char_gens := [15, 85, 17];
    v := [12, 3, 4];
    // 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_112_w_Hecke();"
function MakeCharacter_112_w_Hecke(Kf)
    N := 112;
    order := 12;
    char_gens := [15, 85, 17];
    char_values := [[1, 0, 0, 0], [0, 0, 0, 1], [-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, 1, -1, -1], [-1, -1, 0, 0], [-2, -2, 1, 1], [0, 2, 0, -3], [3, -1, -2, -2], [1, -2, 2, -1], [0, 2, -3, 2], [-1, 1, -2, -3], [-2, 1, -2, 0], [-3, -2, 2, 3], [0, 1, -2, 1], [6, 6, -1, -5], [4, 0, -8, -2], [5, -4, -4, 5], [-6, -1, 6, 2], [-3, -4, 7, 7], [9, -5, -4, -4], [-4, 4, 1, -3], [-1, -5, -4, 4], [2, 0, -4, -4], [-6, 2, 3, -2], [8, -5, -8, 10], [7, 4, -4, -7], [3, 0, 3, 0], [4, 8, 0, -4], [-5, 12, -7, -7], [2, -3, 2, 0], [1, 1, -2, 1], [7, 8, 1, -1], [2, 4, 0, -2], [6, -4, 0, 2], [3, -3, 4, 7], [-18, 2, 9, -2], [-9, 8, 8, -9], [4, 4, 3, -7], [-12, 1, 6, -1], [1, -12, -13, 13], [-7, 7, 2, -5], [8, 0, -16, 8], [-2, 2, 3, 1], [-7, -3, 4, -4], [9, -12, -12, 9], [0, 5, 0, -10], [0, 0, 23, 0], [-13, 10, 10, -13], [8, -5, -4, 5], [-5, 8, -8, 5], [-4, -16, 0, 8], [5, 15, 10, -10], [-8, -8, -5, 13], [3, 6, 3, 0], [-12, -4, 0, 2], [0, 4, -7, 4], [-11, 18, 18, -11], [9, -6, -9, 12], [12, 15, -6, -15], [5, 14, 9, -9], [0, 7, 0, -14], [9, 0, -9, -9], [4, 0, -8, -6], [-1, -11, 12, 12], [-5, 8, 8, -5], [9, 0, 0, -9], [-4, 3, 2, -3], [9, -6, 9, 0], [10, -10, -9, 1], [-13, -13, 14, -1], [-20, -16, 0, 8], [17, -11, -6, -6], [-13, -14, 14, 13], [0, -12, 7, -12], [2, -1, 2, 0], [0, 7, 6, 7], [2, 2, -1, -1], [-5, -14, -14, -5], [20, -3, -20, 6], [-1, 16, -15, -15], [-14, 14, 1, -13], [15, 0, -15, 0], [10, 0, -5, 0], [-19, 0, 0, 19], [5, -10, -10, 5], [0, 5, -22, 5], [-2, 36, 0, -18], [2, 11, 2, 0], [-9, -9, 8, 1], [-6, 20, 0, -10], [3, 24, 3, 0], [5, -10, 10, -5], [16, 16, 0, -8], [15, -15, -4, 11], [0, 9, 0, 9], [-20, -5, 10, 5], [3, 18, 18, 3], [-15, 15, 6, -9], [8, 0, -16, 18], [-8, 8, 11, 3], [34, 14, -17, -14], [-25, -25, 24, 1], [-2, 2, -11, -13], [7, 22, -22, -7], [-5, -2, 3, -3], [-17, -25, -8, 8], [-7, -18, -7, 0], [7, -17, 10, 10], [0, 14, 11, 14], [19, -8, -8, 19], [1, -6, -1, 12], [-28, -5, 14, 5], [0, 0, 0, -10], [2, 11, -2, -22], [-13, 2, 11, 11], [2, 0, -4, -4], [-19, 7, 12, 12], [-14, 0, 28, -8], [0, 8, 25, 8], [11, 12, -12, -11], [-40, 1, 20, -1], [22, -22, -17, 5], [-3, -16, 16, 3], [17, -10, -7, -7], [20, -24, 0, 12], [22, 22, 17, -39], [-1, 33, -32, -32], [-15, 15, 10, -5], [3, 12, -12, -3], [2, 2, -25, 23], [-16, -9, 16, 18], [10, 0, -20, 24], [12, -12, 21, 33], [5, 35, 30, -30], [-4, 0, 8, 18], [6, 11, -6, -22], [-15, 10, 10, -15], [-9, -2, -9, 0], [-4, -16, 0, 8], [11, -4, -7, -7], [-1, -11, -10, 10], [-27, -10, 10, 27], [30, -8, -15, 8], [17, -20, -20, 17], [10, 10, -3, -7], [28, 7, -14, -7], [5, 24, 24, 5], [-13, -20, -7, 7], [-16, 0, 32, 10], [-17, -14, -14, -17], [-2, 12, 1, -12], [25, 25, -14, -11], [24, -5, -24, 10], [-22, 22, 21, -1], [50, 0, 0, 0], [-5, 0, 0, 5], [4, -39, 4, 0], [-7, 17, -10, -10], [-10, -20, 0, 10], [10, -1, 10, 0], [-5, 26, 5, -52], [-4, 0, 8, 36], [17, 6, -11, 11], [-3, 3, 22, 19], [2, 0, -4, -20], [14, 0, -28, 36], [-37, -37, 38, -1], [0, 8, 31, 8], [-24, -1, 12, 1], [0, 9, 24, 9], [-3, 0, 3, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_112_w_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_112_2_w_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_112_2_w_a(:prec:=4)
    chi := MakeCharacter_112_w();
    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_112_2_w_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_112_2_w_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_112_w();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 2, 5, 4, 1]>],Snew);
    return Vf;
end function;