
// Make newform 900.3.l.g in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_900_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_900_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_900_3_l_g();".
// 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_3_l_g();".
// 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 := [9, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 3, 3];
    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_l();"
function MakeCharacter_900_l()
    N := 900;
    order := 4;
    char_gens := [451, 101, 577];
    v := [4, 4, 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_l_Hecke();"
function MakeCharacter_900_l_Hecke(Kf)
    N := 900;
    order := 4;
    char_gens := [451, 101, 577];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [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 := 3;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 4, 0, 0], [15, 0, 0, 0], [0, 0, 0, -2], [0, -3, 0, 0], [0, 0, 17, 0], [0, 0, 0, 6], [0, 0, -42, 0], [-14, 0, 0, 0], [0, 24, 0, 0], [39, 0, 0, 0], [0, 0, 0, 20], [0, 42, 0, 0], [0, 0, 0, 54], [0, 0, 6, 0], [92, 0, 0, 0], [0, 29, 0, 0], [102, 0, 0, 0], [0, 0, 0, 3], [0, 0, -104, 0], [0, 0, 0, -39], [0, 0, 87, 0], [0, -76, 0, 0], [24, 0, 0, 0], [0, 0, 0, 6], [0, 87, 0, 0], [0, 0, 64, 0], [0, 0, 0, 69], [0, 54, 0, 0], [30, 0, 0, 0], [0, 39, 0, 0], [0, 0, -163, 0], [0, 0, -18, 0], [-20, 0, 0, 0], [0, 86, 0, 0], [0, 0, 0, 5], [0, -66, 0, 0], [0, 0, 0, 144], [0, 0, -231, 0], [-170, 0, 0, 0], [42, 0, 0, 0], [0, 0, 0, -161], [0, -186, 0, 0], [0, 0, 136, 0], [-211, 0, 0, 0], [0, 0, 0, -20], [0, -168, 0, 0], [0, 0, 68, 0], [0, 0, 0, -156], [0, 0, -162, 0], [29, 0, 0, 0], [-225, 0, 0, 0], [0, -108, 0, 0], [0, 0, 0, -78], [0, 0, 24, 0], [110, 0, 0, 0], [0, -38, 0, 0], [-126, 0, 0, 0], [0, 0, 0, -239], [0, 0, 0, 60], [0, -181, 0, 0], [-552, 0, 0, 0], [0, 0, 0, 332], [0, -348, 0, 0], [-157, 0, 0, 0], [0, -191, 0, 0], [0, 75, 0, 0], [0, 0, 500, 0], [0, 0, 0, -12], [0, 0, 12, 0], [0, -44, 0, 0], [0, 0, 0, -226], [0, 0, -181, 0], [0, 0, 0, -204], [0, 0, 162, 0], [0, -296, 0, 0], [405, 0, 0, 0], [0, 0, -643, 0], [0, 0, -207, 0], [-104, 0, 0, 0], [444, 0, 0, 0], [0, 0, 0, 433], [0, 0, 826, 0], [0, 0, 0, -219], [0, 0, 483, 0], [0, 27, 0, 0], [660, 0, 0, 0], [0, 0, 0, -204], [0, -300, 0, 0], [0, 0, -822, 0], [0, 122, 0, 0], [-150, 0, 0, 0], [0, 0, -518, 0], [0, 0, 0, 312], [0, 0, -342, 0], [-9, 0, 0, 0], [0, 0, 0, -169], [-856, 0, 0, 0], [0, -393, 0, 0], [0, 30, 0, 0], [0, 0, 0, 252], [0, 0, -435, 0], [-562, 0, 0, 0], [0, 191, 0, 0], [0, -33, 0, 0], [0, 0, 0, 513], [0, 0, -60, 0], [745, 0, 0, 0], [0, 456, 0, 0], [0, 0, 0, 78], [0, 312, 0, 0], [0, 0, 250, 0], [962, 0, 0, 0], [-582, 0, 0, 0], [0, 0, 0, -324], [0, 696, 0, 0], [0, 0, 0, -288], [0, 0, 153, 0], [658, 0, 0, 0], [0, 0, 0, 336], [0, 144, 0, 0], [0, 0, 0, -447], [103, 0, 0, 0], [870, 0, 0, 0], [0, 0, 92, 0], [0, 0, -192, 0], [0, -502, 0, 0], [0, 0, 0, -558], [0, 0, 470, 0], [0, 0, 0, 246], [-314, 0, 0, 0], [0, -164, 0, 0], [-1227, 0, 0, 0], [0, 0, -1121, 0], [0, 0, 0, 150], [0, -188, 0, 0], [0, 126, 0, 0], [0, 0, -618, 0], [-1286, 0, 0, 0], [-186, 0, 0, 0], [0, 0, 0, -628], [0, -555, 0, 0], [0, 0, 434, 0], [0, 0, 1176, 0], [0, 0, 0, 730], [0, -411, 0, 0], [0, 0, -683, 0], [0, 0, 0, -474], [0, 532, 0, 0], [78, 0, 0, 0], [0, 0, 0, 461], [0, 384, 0, 0], [0, -732, 0, 0], [588, 0, 0, 0], [0, 0, 98, 0], [0, 0, -1626, 0], [0, -299, 0, 0], [-1242, 0, 0, 0], [0, -636, 0, 0], [0, 0, 0, -831], [0, 996, 0, 0], [-1191, 0, 0, 0], [0, -951, 0, 0], [0, 0, 0, 432], [-64, 0, 0, 0], [0, -288, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_900_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_900_3_l_g();".
// 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_3_l_g(:prec:=4)
    chi := MakeCharacter_900_l();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_3_l_g();".
// 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_3_l_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_900_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![2304, 0, 0, 0, 1]>,<11,R![-15, 1]>],Snew);
    return Vf;
end function;