
// Make newform 403.2.a.b in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_403_a();"
// 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_a_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_a_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, -4, 8, 4, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-2, 0, 1, 0, 0, 0], [2, -3, -1, 1, 0, 0], [-2, 4, 3, -5, -1, 1], [2, 6, -2, -6, 0, 1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1];
    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_403_a();"
function MakeCharacter_403_a()
    N := 403;
    order := 1;
    char_gens := [249, 313];
    v := [1, 1];
    // 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;

function MakeCharacter_403_a_Hecke(Kf)
    return MakeCharacter_403_a();
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, 0, 1, 0], [-1, 1, 0, 0, 0, 0], [-2, 0, 0, 0, -1, 1], [0, 0, 1, -1, -1, 0], [0, 0, -2, 0, 1, -1], [1, 0, 0, 0, 0, 0], [-4, 0, 0, 0, -1, -1], [-1, -2, 2, 3, -2, 0], [-3, 0, 1, -1, 0, 2], [-2, 1, -1, -1, 2, 1], [1, 0, 0, 0, 0, 0], [-1, 4, -2, -1, 2, -2], [-2, -1, 4, 1, 0, 1], [-1, -3, 1, -1, -1, 2], [-2, 0, 0, 2, 2, 1], [-6, 2, 2, -1, -1, 2], [1, 2, -4, -1, 1, 0], [-3, 5, -1, -2, -1, 3], [-8, 1, 2, 3, -3, 2], [4, 1, -1, -5, 3, -2], [-4, 1, 0, 2, 1, -1], [1, 0, -4, -2, 4, -3], [-1, 3, 0, 0, -1, 2], [1, 1, -1, 0, 1, -5], [2, 1, -6, 0, 4, -4], [-5, -4, 3, 3, -5, 2], [5, -2, 2, 1, 1, 3], [5, -1, -2, 0, 3, -4], [4, -2, 0, -4, -1, 4], [-3, 5, 0, 1, 4, -2], [1, 1, -2, -5, 1, 0], [-2, 3, 1, 4, 0, -3], [-3, -3, -2, 2, -6, -4], [1, 1, -5, 1, -2, 0], [1, -4, 1, 4, -4, 1], [-3, -10, 3, 1, -3, 2], [-10, -5, 0, 7, -4, -1], [7, -3, 1, -3, 6, 1], [0, 4, -2, 1, 4, -2], [-14, 5, -2, 1, -3, 1], [-10, -4, 5, 6, -8, 1], [-6, -1, 2, 0, -3, 3], [-8, -3, 3, 2, -7, -2], [4, -1, 6, 1, 3, 0], [-1, -12, 0, 1, -2, -1], [6, -3, 0, -5, 4, -8], [8, 10, -6, -6, 8, -1], [-2, -8, 0, 5, 1, -5], [3, -2, -1, -3, -6, 6], [-4, -1, -2, -1, 8, 0], [-3, 1, 3, -3, -4, 2], [7, 7, -4, 2, 0, 1], [5, 7, -5, 1, -3, 1], [-4, 0, 3, -2, 3, -1], [3, -4, -3, 9, 0, -5], [5, 1, 6, 0, 1, 1], [5, 0, -1, -8, 3, -3], [-1, 8, 5, 2, -7, 11], [1, 5, -6, -1, 0, 0], [12, -1, -11, -3, 8, -9], [-3, -9, -1, 6, 0, -1], [-13, -2, -1, 0, -3, -1], [-1, 4, -2, 3, 9, 1], [7, -2, 3, -3, -11, 6], [-14, -7, 1, 6, -4, -1], [-10, 8, 0, -2, -6, 9], [7, 3, 8, -5, -4, 0], [2, -1, -4, 4, -5, -1], [-11, -2, 7, 2, 5, -4], [1, -2, -8, -5, 13, -6], [2, -3, 0, -2, -1, -4], [9, -8, 8, 2, -7, 0], [8, -6, 5, 3, 4, -8], [12, -6, -4, -1, -3, -4], [14, 4, -7, -7, 10, 0], [-4, 9, -3, -8, -1, 2], [-17, -10, 15, 0, -6, 5], [-1, 5, 9, 3, -12, 16], [11, 2, -1, -4, 1, 1], [-1, -1, 4, -4, 0, 0], [-2, -1, 7, 5, 0, 4], [11, 9, -1, -6, 11, -2], [5, 10, -8, -1, 10, 0], [0, 0, 4, -5, 6, -3], [-12, 8, 0, -3, -5, 8], [14, 8, -2, 0, 3, -1], [9, 1, 4, -5, 7, 4], [6, 10, 2, -5, -2, -2], [-2, -7, 7, -1, -3, -5], [12, 8, 0, -3, 1, -1], [1, 4, 6, 5, -13, 0], [2, 2, -6, 9, 4, 0], [-1, 4, -10, 3, 5, 0], [-5, 15, -10, 1, -9, 2], [-9, 1, 0, -1, 2, -6], [-3, 6, -11, -6, 8, -10], [-14, 7, 14, -1, -4, 13], [-1, 6, 8, -12, -6, 15], [-8, 17, -2, -3, 1, 5], [5, 9, 3, -9, 5, -3], [-10, -11, -2, 2, -3, 1], [-2, -19, -2, 9, 2, -2], [-3, -17, 4, 8, -3, -5], [9, 1, -12, 4, 14, -16], [-9, -6, 1, 0, -15, 3], [-14, -9, 5, 11, -10, 0], [-3, 1, 13, -3, -9, 4], [-6, -17, 15, 10, -8, 2], [19, -2, 0, -9, 11, 0], [-12, -3, 3, 7, -1, -5], [-8, 5, -8, 2, 8, 5], [7, -4, -11, 5, 2, -1], [16, -7, 2, 2, 3, -12], [5, -7, -7, 0, -8, -10], [-3, 3, 14, -7, 0, 11], [-4, 10, -2, -3, -6, 3], [-14, -14, -4, 8, -5, -7], [-17, -6, 3, 0, -7, 1], [-9, -15, 3, -2, -11, 0], [0, -11, 3, -3, 2, 5], [8, 5, -7, 1, 2, -4], [1, 8, -8, -1, 20, -1], [-9, -6, -4, -2, 12, -2], [10, -15, -8, -1, -7, -12], [-15, 6, -2, 6, -7, 0], [-1, -2, 4, -4, -10, 6], [-6, 4, 3, -11, 0, 9], [-14, -2, -4, 13, -4, 4], [-7, 7, -3, 2, -7, 8], [9, -6, -3, -3, -3, -5], [7, 2, 7, -3, -4, 2], [-18, 6, 8, -1, -11, 7], [8, -3, -1, 14, 1, 7], [-11, -22, -1, 11, 1, -2], [14, -3, -4, -12, 2, 6], [4, -5, -6, 1, -2, -2], [-22, 10, -8, -3, 1, 12], [9, -3, -1, -1, 14, -7], [-21, 1, 2, 1, 5, -3], [-24, 17, 7, -2, 3, 12], [16, 2, 7, -12, 4, -1], [4, -2, 9, -2, 1, 9], [1, -2, 5, -3, 3, -7], [4, 13, -13, 1, 2, -5], [6, -6, 10, -4, -7, 0], [-13, 0, -11, 8, -3, -11], [-26, -15, 22, 5, -18, 7], [9, 6, -5, -7, 2, -1], [8, 4, -9, 4, 5, -1], [-11, -12, 16, 8, -12, 0], [-3, -11, -7, 10, 2, -11], [4, 11, -2, -5, 2, 4], [-9, -1, 10, -6, -11, 10], [11, 5, -10, -1, 9, -17], [-3, 7, 4, -4, 1, -4], [10, -4, 5, -2, -13, 12], [5, 4, -9, -4, 21, -7], [-11, 1, -12, 5, 3, 9], [-5, -2, 7, 6, -9, 3], [9, -11, -5, 3, -5, 4], [-10, 4, -2, 1, -16, 3], [5, -10, -7, -6, 10, 1], [-17, 8, -11, 1, 13, 2], [16, -21, -7, 1, -1, -4], [2, -5, 11, 9, -4, 14], [9, -7, 13, -5, 9, 3], [-9, -4, -2, -3, -4, 1], [3, 0, 4, -17, -1, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_403_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_403_2_a_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_a_b(:prec:=6)
    chi := MakeCharacter_403_a();
    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_a_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_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_403_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-3, 7, 6, -13, -7, 2, 1]>],Snew);
    return Vf;
end function;