
// Make newform 900.3.c.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_900_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_900_c_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_c_f();".
// 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_c_f();".
// 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, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-1, 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_c();"
function MakeCharacter_900_c()
    N := 900;
    order := 2;
    char_gens := [451, 101, 577];
    v := [1, 2, 2];
    // 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_c_Hecke();"
function MakeCharacter_900_c_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 := 3;
    raw_aps := [[-1, -1], [0, 0], [0, 0], [0, -6], [0, 6], [18, 0], [-10, 0], [0, -8], [0, 4], [36, 0], [0, 4], [54, 0], [-18, 0], [0, -12], [0, 0], [26, 0], [0, -18], [-74, 0], [0, -24], [0, -60], [36, 0], [0, -52], [0, -52], [18, 0], [-72, 0], [-36, 0], [0, -6], [0, -108], [-26, 0], [10, 0], [0, -126], [0, -78], [-110, 0], [0, 108], [-288, 0], [0, 108], [234, 0], [0, -72], [0, 76], [-146, 0], [0, -42], [262, 0], [0, 108], [180, 0], [154, 0], [0, 108], [0, 140], [0, -54], [0, -124], [338, 0], [182, 0], [0, -204], [-106, 0], [0, 186], [-14, 0], [0, 108], [108, 0], [0, -188], [270, 0], [234, 0], [0, 48], [58, 0], [0, 156], [0, 156], [468, 0], [-250, 0], [0, 216], [-468, 0], [0, -324], [-434, 0], [158, 0], [0, 264], [0, -126], [-270, 0], [0, 188], [0, -32], [288, 0], [-306, 0], [450, 0], [-50, 0], [0, 426], [-286, 0], [0, -72], [-36, 0], [0, -452], [0, -124], [-54, 0], [288, 0], [288, 0], [0, -234], [0, 332], [0, -84], [0, 150], [0, 42], [0, 256], [0, 64], [252, 0], [-54, 0], [0, 360], [-650, 0], [0, 396], [-574, 0], [0, 324], [-198, 0], [0, -104], [-504, 0], [0, -236], [-998, 0], [0, -312], [-614, 0], [0, -378], [414, 0], [-58, 0], [0, -108], [0, 476], [-810, 0], [0, 240], [0, 568], [950, 0], [0, -654], [-242, 0], [324, 0], [806, 0], [0, 332], [0, -448], [-756, 0], [-310, 0], [0, 48], [0, 630], [1206, 0], [0, -280], [0, -648], [0, 140], [846, 0], [1458, 0], [1282, 0], [422, 0], [0, 696], [94, 0], [270, 0], [0, 108], [-1188, 0], [0, 222], [0, -260], [718, 0], [0, 528], [-666, 0], [182, 0], [0, 572], [0, 676], [-774, 0], [1602, 0], [0, -240], [0, -212], [0, -828], [0, 624], [0, 108], [-54, 0], [936, 0], [0, 0], [0, 596], [-1550, 0], [0, -702], [0, -1062], [206, 0], [0, -416], [0, 764], [-198, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_900_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_900_3_c_f();".
// 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_c_f(:prec:=2)
    chi := MakeCharacter_900_c();
    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_c_f();".
// 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_c_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_900_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![108, 0, 1]>,<13,R![-18, 1]>,<17,R![10, 1]>],Snew);
    return Vf;
end function;