
// Make newform 900.4.d.c in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_900_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_900_d_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_900_4_d_c();".
// 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_900_4_d_c();".
// 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;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [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_900_d();"
function MakeCharacter_900_d()
    N := 900;
    order := 2;
    char_gens := [451, 101, 577];
    v := [2, 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_900_d_Hecke();"
function MakeCharacter_900_d_Hecke(Kf)
    N := 900;
    order := 2;
    char_gens := [451, 101, 577];
    char_values := [[1, 0], [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, 0], [0, 0], [0, 0], [0, 4], [-36, 0], [0, 5], [0, -9], [100, 0], [0, 36], [-234, 0], [-16, 0], [0, -113], [-90, 0], [0, -226], [0, -216], [0, 207], [-684, 0], [422, 0], [0, 166], [360, 0], [0, -13], [-512, 0], [0, -594], [-630, 0], [0, -527], [-558, 0], [0, -4], [0, -882], [-1622, 0], [0, -567], [0, -296], [1908, 0], [0, -477], [-2564, 0], [-738, 0], [-2440, 0], [0, -1277], [0, 410], [0, -972], [0, -621], [1116, 0], [1070, 0], [576, 0], [0, 671], [0, -711], [-872, 0], [1340, 0], [0, -2440], [0, -1350], [-254, 0], [0, 2205], [-3888, 0], [5138, 0], [-4788, 0], [0, 2943], [0, 1116], [-666, 0], [-5536, 0], [0, 1063], [2934, 0], [0, -1018], [0, 1143], [0, 622], [-1224, 0], [0, -949], [0, 4581], [-4348, 0], [0, 3577], [0, 918], [-5894, 0], [0, 5553], [13176, 0], [0, -3056], [0, 6809], [-692, 0], [0, -4032], [12654, 0], [0, -53], [4014, 0], [-3914, 0], [4428, 0], [-15490, 0], [-6768, 0], [0, -649], [2248, 0], [0, -4806], [162, 0], [0, 685], [15354, 0], [0, 6512], [0, 7218], [12096, 0], [0, 3028], [-7524, 0], [-5276, 0], [0, 2484], [10998, 0], [8838, 0], [0, -11218], [-4762, 0], [0, -3002], [0, 2637], [0, -6210], [-21366, 0], [21140, 0], [0, 1633], [0, -8514], [0, 4761], [-10296, 0], [-3382, 0], [0, -10328], [0, 11057], [0, -9981], [604, 0], [152, 0], [-4194, 0], [0, 3626], [0, 3348], [0, 14211], [-19908, 0], [14318, 0], [0, -15025], [0, -11079], [0, -1566], [-20932, 0], [21834, 0], [-12446, 0], [-12528, 0], [0, 5788], [0, 14669], [-2540, 0], [0, -9396], [4832, 0], [0, -10409], [-12042, 0], [-13058, 0], [0, -5913], [0, 5998], [0, -3483], [-40806, 0], [-17980, 0], [12834, 0], [0, 18932], [0, -21258], [-45638, 0], [17496, 0], [0, -16087], [0, 19467], [-29780, 0], [0, -24048], [0, 10651], [7470, 0], [0, -382], [0, -16164], [0, -18158], [13392, 0], [-38072, 0], [-12798, 0], [0, 17437], [-17190, 0], [0, -20142], [0, 7749], [0, 18580], [-18468, 0], [0, -5193], [0, 22068], [-28432, 0], [0, -19889]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_900_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_900_4_d_c();".
// 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_900_4_d_c(:prec:=2)
    chi := MakeCharacter_900_d();
    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_900_4_d_c();".
// 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_900_4_d_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_900_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<7,R![64, 0, 1]>,<11,R![36, 1]>],Snew);
    return Vf;
end function;