
// Make newform 27.3.d.a in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_27_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_27_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_27_3_d_a();".
// 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_27_3_d_a();".
// 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_27_d();"
function MakeCharacter_27_d()
    N := 27;
    order := 6;
    char_gens := [2];
    v := [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_27_d_Hecke();"
function MakeCharacter_27_d_Hecke(Kf)
    N := 27;
    order := 6;
    char_gens := [2];
    char_values := [[0, 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 := [[1, 1], [0, 0], [-4, 2], [-2, 2], [1, 1], [0, 4], [-9, 18], [11, 0], [32, -16], [-26, -26], [0, -32], [-34, 0], [14, -7], [61, -61], [28, 28], [0, 0], [-58, 29], [-56, 56], [0, 31], [18, -36], [65, 0], [-38, 38], [28, 28], [-72, 144], [115, -115], [-26, -26], [0, 40], [81, -162], [-52, 0], [104, -52], [-16, 0], [-184, 92], [109, 109], [0, -5], [176, -88], [-20, 20], [0, 40], [-106, 0], [-220, 110], [-134, -134], [-36, 72], [-232, 0], [-134, -134], [0, 265], [-72, 144], [290, 0], [0, 94], [52, -52], [109, 109], [0, -266], [-117, 234], [464, -232], [-119, 119], [225, -450], [-202, 101], [-26, -26], [108, -216], [-268, 0], [-56, 56], [28, 28], [0, -374], [-292, 146], [533, 0], [284, -142], [-155, 155], [28, 28], [-2, 2], [0, -77], [-130, 65], [-416, 416], [1, 1], [-342, 684], [358, -358], [0, 580], [83, 0], [644, -322], [298, 298], [362, 0], [-454, 227], [0, -221], [-904, 452], [682, -682], [162, -324], [-295, 0], [-812, 812], [-53, -53], [369, -738], [-65, 65], [460, 460], [0, -734], [117, -234], [-350, -350], [-106, 0], [266, -133], [0, 787], [-360, 720], [248, -124], [117, -234], [-250, 0], [650, 0], [-623, 623], [-306, 612], [-130, 65], [-377, -377], [0, -545], [-871, 0], [1, 1], [-108, 216], [-652, 326], [-461, 461], [0, 112], [902, 0], [410, -205], [799, -799], [830, 0], [217, 217], [0, 13], [-270, 540], [-436, 218], [568, 568], [0, 382], [-578, 578], [-404, -404], [603, -1206], [-182, 182], [0, 0], [700, -700], [342, -684], [664, -664], [0, 670], [317, 0], [716, -358], [0, -1310], [218, 0], [-760, 380], [0, -1022], [-684, 1368], [0, 130], [-364, 182], [81, -162], [299, 0], [-350, -350], [0, 814], [828, -1656], [-718, 0], [460, 460], [-1424, 1424], [-404, -404], [0, -311], [594, -1188], [0, -104], [-36, 72], [119, 0], [-1372, 686], [-695, 695], [1000, 1000], [56, 0], [460, 460], [470, 0], [464, -232], [1, 1], [-477, 954], [0, -1202], [108, -216], [482, -241], [-674, -674], [-1420, 0], [-524, 524]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_27_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_27_3_d_a();".
// 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_27_3_d_a(:prec:=2)
    chi := MakeCharacter_27_d();
    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_27_3_d_a();".
// 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_27_3_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_27_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;