
// Make newform 900.4.d.b in Magma, downloaded from the LMFDB on 19 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_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_900_4_d_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, 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, 16], [-36, 0], [0, 5], [0, 39], [-140, 0], [0, -96], [6, 0], [-16, 0], [0, -17], [390, 0], [0, 26], [0, -204], [0, -57], [516, 0], [-58, 0], [0, -446], [120, 0], [0, 323], [1168, 0], [0, -366], [-1590, 0], [0, 97], [-798, 0], [0, -136], [0, -78], [-1622, 0], [0, 537], [0, -764], [-2412, 0], [0, -1053], [556, 0], [-2418, 0], [2840, 0], [0, 1027], [0, 230], [0, -1008], [0, -309], [-2964, 0], [-370, 0], [-1104, 0], [0, 1199], [0, -639], [-4472, 0], [1340, 0], [0, -1180], [0, -690], [-1694, 0], [0, -2595], [2352, 0], [-3502, 0], [-4788, 0], [0, -753], [0, -216], [54, 0], [-6496, 0], [0, -233], [4854, 0], [0, 2258], [0, 4287], [0, 1738], [-2424, 0], [0, 779], [0, 4269], [-988, 0], [0, 1273], [0, -4278], [3706, 0], [0, 5697], [-264, 0], [0, 5116], [0, 281], [7228, 0], [0, -2868], [-9186, 0], [0, -197], [1614, 0], [-1034, 0], [3708, 0], [-4930, 0], [2592, 0], [0, -1081], [-1352, 0], [0, 2766], [-3198, 0], [0, -755], [-16086, 0], [0, -2692], [0, 1302], [11136, 0], [0, 7312], [-11844, 0], [11284, 0], [0, 2016], [-17562, 0], [-3162, 0], [0, -3382], [17798, 0], [0, -9998], [0, -5547], [0, 450], [7914, 0], [-2380, 0], [0, -12863], [0, -1806], [0, 1449], [2664, 0], [-502, 0], [0, 3988], [0, -10207], [0, 3171], [-22676, 0], [-7048, 0], [20286, 0], [0, 8054], [0, -13728], [0, -6261], [-16308, 0], [32078, 0], [0, -2305], [0, -5391], [0, 1446], [-29572, 0], [-5766, 0], [-3326, 0], [-7728, 0], [0, -10808], [0, -5059], [-10460, 0], [0, 8616], [26912, 0], [0, 6919], [17238, 0], [-21698, 0], [0, 9183], [0, -15158], [0, -3747], [-11526, 0], [-33820, 0], [-19566, 0], [0, 20048], [0, -15942], [24442, 0], [-31944, 0], [0, -8743], [0, -21717], [-10820, 0], [0, 14988], [0, -20261], [-15570, 0], [0, 542], [0, 4104], [0, 17038], [15072, 0], [-24392, 0], [13602, 0], [0, -23987], [48330, 0], [0, 4662], [0, -7419], [0, 5680], [6972, 0], [0, 20583], [0, 11232], [-10192, 0], [0, -161]];
    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_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_900_4_d_b(: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_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_900_4_d_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_900_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<7,R![1024, 0, 1]>,<11,R![36, 1]>],Snew);
    return Vf;
end function;