
// Make newform 403.2.be.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_403_be();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_403_be_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_403_2_be_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_403_2_be_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_403_be();"
function MakeCharacter_403_be()
    N := 403;
    order := 12;
    char_gens := [249, 313];
    v := [9, 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_403_be_Hecke();"
function MakeCharacter_403_be_Hecke(Kf)
    N := 403;
    order := 12;
    char_gens := [249, 313];
    char_values := [[0, 0, 0, 1], [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, 1, 1, -1], [1, 1, 1, 0], [0, 1, -1, -1], [-2, 2, -1, -3], [1, -2, 1, 1], [0, -3, 2, 3], [0, 0, 0, 0], [4, 4, -5, 1], [-1, -6, 0, 3], [-2, 0, 4, 5], [0, 5, 0, 1], [-2, -2, -2, 4], [-6, 4, 2, 2], [1, -3, -1, 6], [3, -4, 4, -3], [-10, 2, 5, -2], [-7, 7, 3, -4], [3, 0, -6, 2], [5, 1, -6, -6], [3, -5, 2, 2], [2, 5, -7, -7], [-6, -2, -6, 0], [12, 7, -5, 5], [-5, 2, 2, -5], [-3, -4, -4, -3], [4, 0, -8, 3], [-6, -7, 3, 7], [3, 5, -3, -10], [1, -8, -8, 1], [0, -4, 0, -4], [-4, 6, 4, -12], [0, -10, 0, 20], [2, -4, 2, 2], [-10, 0, 20, 2], [-10, -10, 6, 4], [11, -3, -3, 11], [-4, 10, 0, -5], [0, 7, -7, 0], [4, 4, -3, -1], [16, 8, -8, -8], [9, -7, -9, 14], [0, 0, -19, 0], [0, 5, -3, 5], [-7, 7, -1, -8], [2, 3, 1, -1], [0, -10, 4, -10], [5, -1, -5, 2], [14, -14, -1, 13], [4, 3, -7, -7], [-3, -3, 13, -10], [4, 0, -8, 0], [3, 4, 1, -1], [5, 5, -13, 8], [-4, -14, 4, 28], [-12, 3, 6, -3], [-13, 0, 26, -3], [-24, 3, 12, -3], [-1, -4, -4, -1], [16, 12, 0, -6], [-17, -12, 12, 17], [-7, 0, 14, -5], [2, -2, 14, 16], [11, -11, -15, -4], [-4, 0, 8, -20], [-10, -5, -10, 0], [-12, -12, 12, 0], [3, 10, 7, -7], [-13, 0, 0, 0], [1, 17, 1, 0], [-15, -14, 14, 15], [3, -3, -13, -10], [15, 13, -2, 2], [16, 6, -8, -6], [20, 18, 0, -9], [3, 3, -6, 3], [4, -7, -11, 11], [-5, -6, 5, 12], [1, -1, 7, 8], [-7, -8, -8, -7], [-1, 1, -19, -20], [30, 0, 0, 0], [12, -7, -5, -5], [19, 19, 0, -19], [-12, -2, 0, 1], [4, -4, -2, 4], [0, -6, 16, -6], [4, -10, -10, 4], [12, 0, 0, -12], [2, -6, 6, -2], [7, -10, -10, 7], [6, 0, -12, -8], [-16, 16, 1, -15], [14, 5, -9, 9], [0, 3, -1, 3], [22, -22, -13, 9], [-4, -6, 4, 12], [6, 13, 7, -7], [-18, -8, 18, 16], [17, 0, -34, 7], [-14, 14, -8, -22], [0, -2, 6, -2], [-1, -4, -4, -1], [6, 27, -3, -27], [0, 7, 28, 7], [0, 13, 7, 13], [7, 7, -15, 8], [-17, -9, 9, 17], [14, 16, -16, -14], [0, -7, -27, -7], [12, 19, 12, 0], [0, 21, 9, 21], [-8, 8, 14, 6], [-16, -16, -6, 22], [1, -1, -1, 1], [12, 11, -23, -23], [0, -8, -27, -8], [-5, 12, -12, 5], [-45, -2, 0, 1], [12, 30, 0, -15], [-13, -18, 0, 9], [-12, -4, 8, -8], [0, -8, 20, -8], [6, 8, -3, -8], [-7, -27, 27, 7], [11, -16, 5, 5], [0, 8, 0, 0], [19, -26, -26, 19], [16, 4, -16, -8], [6, 6, 6, 0], [20, -21, 1, 1], [29, -25, -4, -4], [-17, 4, -4, 17], [18, 14, 18, 0], [12, 24, -6, -24], [-12, -12, -8, 20], [1, 1, -23, 22], [19, 6, -6, -19], [-27, -27, 14, 13], [0, -12, -17, -12], [2, -34, 2, 0], [-29, 16, 13, 13], [24, 36, -36, -24], [-4, 12, 4, -24], [4, -5, 1, 1], [2, -24, 0, 12], [-8, 9, -9, 8], [-23, -4, 4, 23], [-12, -19, 6, 19], [6, 21, -3, -21], [-11, -5, 6, -6], [-6, 1, 5, 5], [0, 17, -10, 17], [25, 6, 0, -3], [-15, 7, 15, -14], [7, 0, -14, 17], [0, 22, 0, 0], [0, -1, 25, -1], [-7, -20, 20, 7], [1, 0, -1, 0], [24, 11, -13, 13], [10, 10, 9, -19], [24, -10, 0, 5], [20, -20, -1, 19], [0, -26, -10, -26], [14, -2, -2, 14], [-13, 2, 11, 11], [-8, 0, 16, -10], [0, 18, 10, 18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_403_be_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_403_2_be_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_403_2_be_a(:prec:=4)
    chi := MakeCharacter_403_be();
    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_403_2_be_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_403_2_be_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_403_be();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -2, 2, 2, 1]>,<7,R![169, 104, 41, 10, 1]>],Snew);
    return Vf;
end function;