
// Make newform 507.4.b.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_507_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_507_b_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_507_4_b_d();".
// 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_507_4_b_d();".
// 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];
        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_507_b();"
function MakeCharacter_507_b()
    N := 507;
    order := 2;
    char_gens := [170, 340];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_507_b_Hecke();"
function MakeCharacter_507_b_Hecke(Kf)
    N := 507;
    order := 2;
    char_gens := [170, 340];
    char_values := [[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 := 4;
    raw_aps := [[0, 1], [3, 0], [0, -7], [0, -10], [0, -22], [0, 0], [-37, 0], [0, -30], [162, 0], [-113, 0], [0, -196], [0, 13], [0, -285], [246, 0], [0, -462], [-537, 0], [0, 576], [-635, 0], [0, -202], [0, 1086], [0, -805], [884, 0], [0, -518], [0, 194], [0, 1202], [429, 0], [1302, 0], [-1338, 0], [0, 1034], [1077, 0], [988, 0], [560, 0], [0, -519], [-348, 0], [0, 645], [0, 2914], [-2079, 0], [0, 1700], [0, 3680], [-4146, 0], [-3674, 0], [3283, 0], [-596, 0], [0, -393], [0, 3522], [-2018, 0], [160, 0], [0, -4072], [0, 5794], [0, 6482], [-6890, 0], [0, -2466], [0, -3617], [-4860, 0], [-565, 0], [-498, 0], [5546, 0], [0, -2256], [-2309, 0], [0, 5833], [-1650, 0], [0, 2991], [0, -2422], [3402, 0], [2310, 0], [0, 257], [0, -1028], [-2487, 0], [-2850, 0], [0, -2018], [0, 5287], [0, -7278], [-4202, 0], [-1583, 0], [0, 2052], [0, 6872], [11653, 0], [0, 6134], [0, -10795], [0, 8489], [1496, 0], [0, 11695], [0, -10590], [13949, 0], [10726, 0], [16228, 0], [0, 7538], [0, -15539], [0, -4811], [0, 562], [-4914, 0], [0, -3600], [0, 17130], [11838, 0], [0, -8976], [1682, 0], [0, -15167], [-6783, 0], [-13918, 0], [0, -1335], [-3806, 0], [0, -1905], [4800, 0], [-14678, 0], [586, 0], [0, -8939], [0, 13792], [0, 9569], [-5192, 0], [-3677, 0], [-10960, 0], [0, 26027], [0, -17681], [0, 3192], [0, 7580], [27707, 0], [0, -11216], [2536, 0], [17730, 0], [18920, 0], [0, 5241], [-20467, 0], [-70, 0], [0, 6432], [0, 6666], [14054, 0], [0, -71], [-3936, 0], [-34202, 0], [0, 27363], [0, 21776], [0, -2484], [-32906, 0], [-3914, 0], [0, -33038], [0, 17586], [0, -18314], [0, -42068], [4282, 0], [40221, 0], [0, 7084], [0, -17338], [-35496, 0], [0, -14992], [20659, 0], [0, 28716], [0, 13377], [27419, 0], [2422, 0], [0, 34522], [0, -13733], [22759, 0], [2168, 0], [15888, 0], [11628, 0], [-12584, 0], [17184, 0], [0, -12777], [9191, 0], [0, -50498], [0, 1560], [21498, 0], [0, 418], [18132, 0], [0, -12501], [0, -43708], [-39614, 0], [-36503, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_507_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_507_4_b_d();".
// 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_507_4_b_d(:prec:=2)
    chi := MakeCharacter_507_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_507_4_b_d();".
// 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_507_4_b_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_507_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![1, 0, 1]>,<5,R![49, 0, 1]>],Snew);
    return Vf;
end function;