
// Make newform 900.3.g.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_900_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_900_g_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_g_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_3_g_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 := [18, -16, 17, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [12, -25, 3, -2], [8, -1, 1, 0], [-78, 160, -12, 8]];
    Rf_basisdens := [1, 15, 1, 15];
    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_g();"
function MakeCharacter_900_g()
    N := 900;
    order := 2;
    char_gens := [451, 101, 577];
    v := [2, 1, 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_g_Hecke();"
function MakeCharacter_900_g_Hecke(Kf)
    N := 900;
    order := 2;
    char_gens := [451, 101, 577];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [1, 0, 0, 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, 0, -1, 0], [0, -7, 0, 0], [0, 0, 3, 0], [0, 0, 0, -2], [-12, 0, 0, 0], [0, 0, 0, 1], [0, -6, 0, 0], [-38, 0, 0, 0], [0, 0, -1, 0], [0, -49, 0, 0], [0, 0, -10, 0], [0, 0, 0, 8], [0, 0, 0, 0], [0, -59, 0, 0], [-70, 0, 0, 0], [0, 0, 16, 0], [0, -84, 0, 0], [0, 0, 2, 0], [-30, 0, 0, 0], [0, 0, 0, -14], [0, 23, 0, 0], [0, 0, -14, 0], [0, -56, 0, 0], [0, 0, 13, 0], [0, 0, 0, -6], [-74, 0, 0, 0], [0, 0, 0, 18], [0, 0, -25, 0], [0, -117, 0, 0], [0, 0, 0, 12], [-62, 0, 0, 0], [0, 112, 0, 0], [70, 0, 0, 0], [0, 0, 7, 0], [0, 0, -14, 0], [0, 0, 0, -21], [0, 0, 0, -28], [0, -95, 0, 0], [-22, 0, 0, 0], [0, 138, 0, 0], [0, 0, -40, 0], [0, 0, 0, 17], [-294, 0, 0, 0], [-42, 0, 0, 0], [0, 0, 29, 0], [0, 0, 0, 28], [422, 0, 0, 0], [0, 0, 0, -22], [0, 50, 0, 0], [280, 0, 0, 0], [0, 57, 0, 0], [0, 0, 0, 0], [0, 0, 0, 7], [0, 186, 0, 0], [322, 0, 0, 0], [0, 0, 5, 0], [0, 231, 0, 0], [0, 0, 16, 0], [0, 0, 0, 21], [0, 0, -24, 0], [0, 126, 0, 0], [0, 0, 8, 0], [0, 0, 0, -29], [40, 0, 0, 0], [0, 0, -46, 0], [0, 0, 0, 42], [406, 0, 0, 0], [0, 0, 0, 56], [0, -114, 0, 0], [0, 0, -37, 0], [0, 0, 15, 0], [-538, 0, 0, 0], [0, 0, 0, -28], [0, 350, 0, 0], [0, 0, 41, 0], [0, 175, 0, 0], [242, 0, 0, 0], [0, -559, 0, 0], [358, 0, 0, 0], [0, -10, 0, 0], [0, 0, 38, 0], [690, 0, 0, 0], [0, 0, 0, -42], [0, 57, 0, 0], [0, 0, 50, 0], [0, -456, 0, 0], [0, 0, -39, 0], [0, 0, 0, 48], [0, 56, 0, 0], [0, 0, 117, 0], [0, -399, 0, 0], [72, 0, 0, 0], [0, 0, 0, 24], [0, -348, 0, 0], [0, 49, 0, 0], [0, 0, -50, 0], [-590, 0, 0, 0], [0, 0, -34, 0], [0, 0, 0, 72], [0, 0, 0, 2], [0, 77, 0, 0], [-704, 0, 0, 0], [0, 0, -166, 0], [0, 0, 0, -102], [0, 0, 0, 0], [0, -210, 0, 0], [20, 0, 0, 0], [0, 0, 1, 0], [0, 0, -7, 0], [0, 0, 0, 4], [-178, 0, 0, 0], [-154, 0, 0, 0], [0, -571, 0, 0], [0, 0, 150, 0], [0, 0, 0, -84], [0, 0, 0, 7], [0, -371, 0, 0], [74, 0, 0, 0], [0, 0, 102, 0], [0, 0, 0, -29], [0, 0, 0, -44], [-68, 0, 0, 0], [0, 92, 0, 0], [210, 0, 0, 0], [0, 86, 0, 0], [0, 0, 3, 0], [0, 0, -209, 0], [-416, 0, 0, 0], [0, 0, 0, -84], [86, 0, 0, 0], [0, 0, -7, 0], [0, -261, 0, 0], [-384, 0, 0, 0], [0, 0, 0, -21], [0, 0, -70, 0], [0, 0, 0, 52], [0, -1047, 0, 0], [-602, 0, 0, 0], [0, 994, 0, 0], [0, 0, -189, 0], [0, 0, 0, -122], [-1074, 0, 0, 0], [0, 526, 0, 0], [0, 0, 25, 0], [0, 0, 0, -12], [-154, 0, 0, 0], [0, 0, 0, 111], [0, 0, 205, 0], [0, -609, 0, 0], [0, 0, 150, 0], [0, 0, 0, 63], [0, 0, -138, 0], [0, 540, 0, 0], [54, 0, 0, 0], [0, 105, 0, 0], [0, 0, -116, 0], [0, 242, 0, 0], [0, 0, 0, 42], [0, 0, 0, -112], [0, 0, -59, 0], [0, 483, 0, 0], [0, 0, 0, -168], [0, 0, 0, -100], [1054, 0, 0, 0], [0, 0, -177, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_900_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_900_3_g_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_3_g_c(:prec:=4)
    chi := MakeCharacter_900_g();
    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_g_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_3_g_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_900_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![-46, 0, 1]>],Snew);
    return Vf;
end function;