
// Make newform 441.4.e.n in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_441_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_441_e_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_441_4_e_n();".
// 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_441_4_e_n();".
// 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_441_e();"
function MakeCharacter_441_e()
    N := 441;
    order := 3;
    char_gens := [344, 199];
    v := [3, 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_441_e_Hecke();"
function MakeCharacter_441_e_Hecke(Kf)
    N := 441;
    order := 3;
    char_gens := [344, 199];
    char_values := [[1, 0], [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 := 4;
    raw_aps := [[0, 4], [0, 0], [0, 4], [0, 0], [62, -62], [62, 0], [-84, 84], [0, 100], [0, -42], [10, 0], [-48, 48], [0, 246], [-248, 0], [68, 0], [0, -324], [258, -258], [-120, 120], [0, 622], [-904, 904], [678, 0], [-642, 642], [0, -740], [468, 0], [0, -200], [1266, 0], [-232, 232], [0, -1792], [0, -1906], [90, -90], [-458, 0], [804, 0], [0, -812], [414, -414], [1620, 0], [0, 2370], [568, -568], [-266, 266], [0, 272], [-1876, 0], [0, 152], [610, -610], [-1042, 0], [0, -2038], [2602, -2602], [-2354, 0], [1680, -1680], [-668, 0], [1832, 0], [-4944, 4944], [0, -5470], [0, -2802], [1170, 0], [-2338, 2338], [2792, 0], [0, -7024], [2438, -2438], [6780, -6780], [0, -1928], [-5554, 5554], [-1942, 0], [4828, -4828], [-6152, 0], [-5884, 0], [-9132, 9132], [0, -9382], [0, 3114], [0, -1532], [-4166, 0], [-11366, 11366], [-9310, 0], [8572, -8572], [0, -4790], [5424, -5424], [0, -1838], [-4260, 0], [0, -9048], [-11490, 11490], [0, -1866], [0, 13662], [-13210, 13210], [6960, 0], [8162, 0], [16602, -16602], [-7738, 0], [0, -840], [0, 6618], [-3090, 0], [0, -5914], [-15968, 0], [-1172, 0], [0, -5304], [-5740, 5740], [-8944, 8944], [5558, 0], [0, 19820], [1848, 0], [0, -340], [-10212, 10212], [0, -9332], [0, 8998], [-3416, 0], [-526, 526], [6712, -6712], [0, 4190], [-3032, 3032], [5434, -5434], [464, 0], [0, -11748], [7650, -7650], [22878, 0], [0, 704], [-24958, 24958], [8826, 0], [21220, -21220], [-3268, 0], [13062, -13062], [28012, 0], [-3844, 3844], [0, -28482], [9330, 0], [8782, -8782], [-10562, 0], [0, 26016], [8898, -8898], [0, 30572], [30618, 0], [0, 8130], [0, 27840], [-14624, 0], [0, -20862], [13920, -13920], [-25578, 0], [0, -33472], [25934, 0], [0, -26952], [-23450, 0], [-39568, 39568], [-12356, 12356], [-21736, 0], [-38310, 38310], [-2132, 0], [0, 5002], [3612, -3612], [27666, 0], [12890, -12890], [-9340, 0], [33082, 0], [-7544, 7544], [0, 8180], [0, 10518], [0, -14134], [6492, 0], [38228, 0], [0, 43076], [32236, -32236], [46518, 0], [0, -17840], [0, -7000], [-36114, 0], [4748, -4748], [0, 42694], [16742, 0], [-9956, 0], [0, 26388], [-786, 786], [-51888, 51888], [51928, -51928], [-386, 386]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_441_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_441_4_e_n();".
// 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_441_4_e_n(:prec:=2)
    chi := MakeCharacter_441_e();
    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_441_4_e_n();".
// 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_441_4_e_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_441_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![16, -4, 1]>,<5,R![16, -4, 1]>,<13,R![-62, 1]>],Snew);
    return Vf;
end function;