
// Make newform 403.2.l.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_403_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_403_l_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_l_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_403_2_l_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 := [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_403_l();"
function MakeCharacter_403_l()
    N := 403;
    order := 6;
    char_gens := [249, 313];
    v := [3, 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_l_Hecke();"
function MakeCharacter_403_l_Hecke(Kf)
    N := 403;
    order := 6;
    char_gens := [249, 313];
    char_values := [[-1, 0], [-1, 1]];
    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, 2], [0, 1], [1, 1], [2, -1], [3, 3], [1, -4], [0, -3], [-6, 3], [0, 0], [-6, 0], [1, -6], [-6, 3], [-3, -3], [0, 1], [2, -4], [-3, 3], [2, -1], [10, 0], [-5, -5], [-5, -5], [5, 5], [0, 5], [7, 7], [-4, 8], [-4, 8], [18, 0], [19, -19], [0, 9], [-8, 16], [9, -9], [0, -7], [0, 9], [-3, -3], [-4, 0], [-22, 11], [-2, 4], [-22, 0], [6, -12], [10, -5], [9, -9], [0, -3], [-23, 23], [15, -15], [26, -13], [5, 5], [-25, 25], [0, 5], [2, -1], [-5, -5], [2, -1], [-18, 0], [-9, -9], [2, -1], [0, 21], [-15, 15], [0, 0], [21, -21], [-6, 12], [26, 0], [-8, 16], [4, 0], [-30, 15], [2, -1], [24, 0], [0, -19], [18, -9], [7, 7], [-14, 0], [0, 33], [0, 0], [34, -17], [-5, -5], [31, -31], [-26, 0], [-14, 7], [15, 15], [0, 33], [18, -9], [-4, 8], [-30, 15], [12, 0], [-19, -19], [26, -13], [-26, 0], [19, -19], [15, -15], [-8, 16], [-4, 8], [8, -16], [10, -20], [-36, 0], [2, -1], [-13, -13], [-9, 9], [42, -21], [0, 21], [5, 5], [0, 21], [20, 0], [10, -5], [19, -19], [0, 0], [3, -3], [21, -21], [-37, 37], [-46, 23], [-6, 12], [0, 0], [-33, 33], [0, 29], [7, -7], [-54, 27], [-14, 7], [-6, 12], [-13, -13], [-15, 15], [-10, 20], [-12, 0], [-6, 0], [-36, 0], [-3, -3], [1, -1], [9, -9], [22, -44], [3, 3], [0, 9], [28, -56], [0, 9], [0, 5], [-7, -7], [-13, -13], [10, -20], [0, -47], [17, -17], [10, -5], [58, -29], [20, -40], [-30, 15], [-3, 3], [0, -15], [-29, -29], [24, -48], [0, 49], [3, 3], [-14, 0], [10, -20], [8, -16], [21, -21], [11, -11], [7, 7], [-11, -11], [-15, 15], [-56, 0], [0, -27], [28, 0], [0, 21], [11, -11], [-12, 24], [0, -7], [-15, -15], [50, -25], [54, 0], [50, -25], [27, -27], [-20, 40], [27, 27], [-52, 0], [25, -25]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_403_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_403_2_l_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_403_2_l_b(:prec:=2)
    chi := MakeCharacter_403_l();
    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_l_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_403_2_l_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_403_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, 0, 1]>,<5,R![3, -3, 1]>],Snew);
    return Vf;
end function;