
// Make newform 1296.3.o.o in Magma, downloaded from the LMFDB on 06 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1296_o();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1296_o_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_1296_3_o_o();".
// 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_1296_3_o_o();".
// 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;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_1296_o();"
function MakeCharacter_1296_o()
    N := 1296;
    order := 6;
    char_gens := [1135, 325, 1217];
    v := [3, 6, 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_1296_o_Hecke();"
function MakeCharacter_1296_o_Hecke(Kf)
    N := 1296;
    order := 6;
    char_gens := [1135, 325, 1217];
    char_values := [[-1, 0], [1, 0], [-1, 1]];
    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], [6, -6], [0, 0], [0, 0], [-10, 10], [-30, 0], [0, 0], [0, 0], [0, -42], [0, 0], [-70, 0], [-18, 18], [0, 0], [0, 0], [90, 0], [0, 0], [0, 22], [0, 0], [0, 0], [-110, 0], [0, 0], [0, 0], [-78, 0], [0, -130], [0, 198], [0, 0], [0, 0], [-182, 0], [30, -30], [0, 0], [0, 0], [0, -210], [0, 0], [102, -102], [0, 0], [-170, 170], [0, 0], [0, 0], [0, -330], [0, 0], [-38, 0], [0, 0], [190, -190], [-390, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-442, 442], [210, 0], [0, 0], [0, -418], [0, 0], [510, -510], [0, 0], [138, 0], [0, 0], [0, 230], [0, 462], [0, 0], [-570, 570], [0, 0], [0, 0], [0, -50], [0, 150], [0, 0], [350, -350], [0, 0], [0, 598], [0, -450], [0, 0], [0, 0], [550, -550], [0, 0], [0, 0], [0, -378], [650, 0], [798, -798], [782, -782], [0, 0], [0, -58], [0, 0], [290, 0], [0, 0], [0, 0], [-702, 0], [0, -850], [0, -522], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [918, -918], [-558, 0], [0, 0], [682, 0], [0, 0], [330, 0], [0, 0], [0, 462], [0, 0], [-1150, 0], [0, 0], [930, 0], [0, 0], [0, 1102], [0, 0], [-70, 0], [-210, 210], [0, 0], [0, 0], [0, -1218], [0, 0], [0, 0], [630, -630], [0, 0], [-1178, 1178], [0, -770], [0, 1350], [0, 0], [0, 0], [-1302, 0], [0, 518], [0, 0], [0, 0], [-1450, 1450], [0, 0], [0, 0], [0, 0], [-1190, 0], [78, -78], [-962, 962], [-390, 0], [0, 0], [1110, -1110], [-1518, 0], [0, 0], [0, -858], [0, 0], [0, 0], [1258, 0], [0, 0], [0, -410], [0, -1650], [0, 0], [0, 0], [-1610, 1610], [738, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -258], [-430, 0], [-1482, 1482], [0, 0], [-1230, 0], [0, 0], [0, 0], [-1890, 1890], [0, 0], [0, 0], [0, -1850], [-1118, 0], [-90, 90], [0, 0], [1558, -1558], [0, 0], [0, 2030], [0, 0], [1998, -1998], [0, 0], [1722, 0], [0, 0], [0, 1462], [0, 0], [0, 0], [-2170, 2170], [1170, 0], [0, 0], [0, -282], [-470, 0], [0, 0], [-658, 658], [0, 0], [2050, 0], [0, 0], [0, 0], [0, 2262], [0, 0], [1710, -1710], [-98, 98], [0, -490], [0, -1410], [0, 0], [-2442, 2442], [0, 0], [2150, -2150], [0, 1598], [0, 0], [-2070, 0], [0, 0], [0, 0], [0, -2322], [0, 0], [-2590, 0], [102, -102], [0, 0], [0, 0], [0, 0], [0, 2542], [0, 0], [0, -1122], [0, 0], [-2730, 2730], [1862, -1862], [0, 0], [318, -318], [0, 0], [0, 0], [0, 742], [0, -2610], [0, 0], [0, 0], [0, 0], [2870, -2870], [0, 0], [0, 0], [-1938, 1938], [0, 0], [0, 0], [-1378, 1378], [-2790, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [1802, 0], [990, -990], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [1430, -1430], [-3198, 0], [0, 0], [0, 3182], [0, 2550], [0, 0], [2842, 0], [0, 0], [570, 0], [-1870, 0], [0, 0], [0, 0], [2438, -2438], [2090, 0], [-3330, 3330], [0, 0], [1482, 0], [0, 2958], [0, 0], [2310, -2310], [118, -118], [0, 0], [0, 590], [0, 0], [-2530, 2530], [0, 0], [0, 0], [0, 3478], [1298, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -122], [0, 0], [0, 0], [610, 0], [-2970, 2970], [0, 0], [-2622, 0], [0, -1098], [0, 0], [-3570, 3570], [0, 0], [0, 3190], [-3498, 3498], [0, 0], [0, 1830], [0, 0], [0, 0], [-3410, 3410], [-630, 0], [0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1296_o_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1296_3_o_o();".
// 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_1296_3_o_o(:prec:=2)
    chi := MakeCharacter_1296_o();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1296_3_o_o();".
// 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_1296_3_o_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1296_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![36, -6, 1]>,<7,R![0, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;