
// Make newform 441.4.p.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_441_p();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_441_p_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_p_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_441_4_p_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 := [130321, 0, 12996, 0, 935, 0, 36, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [467856, 0, 33660, 0, 935, 0, 36, 0], [10227852, 0, 536690, 0, 33660, 0, -148, 0], [0, 3261996, 0, 234685, 0, 15895, 0, 251], [0, -2899, 0, 0, 0, 0, 0, 1], [12996, 0, 1148, 0, 72, 0, 2, 0], [0, -4107458, 0, -178090, 0, -6050, 0, -148], [0, 13410428, 0, -3027530, 0, -102850, 0, -4682]];
    Rf_basisdens := [1, 337535, 337535, 6413165, 17765, 361, 377245, 6413165];
    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_441_p();"
function MakeCharacter_441_p()
    N := 441;
    order := 6;
    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_p_Hecke();"
function MakeCharacter_441_p_Hecke(Kf)
    N := 441;
    order := 6;
    char_gens := [344, 199];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 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 := 4;
    raw_aps := [[0, 0, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -11, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -3, -3], [0, 0, 0, 0, 55, 0, 0, 0], [0, 0, 0, 89, 89, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 8], [0, 184, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -5, 0, 0], [-190, 0, 0, 0, 0, 0, 0, 0], [0, 0, 2, 0, 0, -2, 0, 0], [0, 0, 0, 253, 0, 0, 0, 0], [0, 0, -4, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -22, -22], [-296, 296, 0, 0, 0, 0, 0, 0], [0, 0, 0, 233, 233, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -27], [0, -836, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -58, 0, 0], [0, 0, 33, 0, 0, -33, 0, 0], [0, 0, 0, 0, 0, 0, 19, 0], [0, 0, -35, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 22, 22], [0, 0, 0, 0, -1403, 0, 0, 0], [844, -844, 0, 0, 0, 0, 0, 0], [0, 0, 0, 407, 407, 0, 0, 0], [-220, 0, 0, 0, 0, 0, 0, 0], [0, 0, -62, 0, 0, 62, 0, 0], [0, 0, 0, -1109, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -22, 0], [0, 0, 0, 0, -1037, 0, 0, 0], [-1970, 1970, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -88], [0, 1336, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -110, 0, 0], [0, 0, 11, 0, 0, -11, 0, 0], [0, 0, 0, -77, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 87, 0], [0, 0, 0, 0, 1573, 0, 0, 0], [-3740, 3740, 0, 0, 0, 0, 0, 0], [0, 0, 0, -847, -847, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 27], [3590, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -101, 0], [0, 0, -176, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -11, -11], [0, 0, 0, 0, -455, 0, 0, 0], [0, 0, 0, -457, -457, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 13], [0, 0, 0, 0, 0, -294, 0, 0], [0, 0, 169, 0, 0, -169, 0, 0], [0, 0, 0, -425, 0, 0, 0, 0], [0, 0, -231, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -60, -60], [1126, -1126, 0, 0, 0, 0, 0, 0], [0, 0, 0, -3589, -3589, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 239], [0, 0, 0, 0, 0, -11, 0, 0], [0, 0, 0, 0, 0, 0, -297, 0], [0, 0, 308, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -364, -364], [0, 0, 0, 0, -1259, 0, 0, 0], [0, 9526, 0, 0, 0, 0, 0, 0], [-8272, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 6919, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -212, 0], [0, 0, 259, 0, 0, 0, 0, 0], [0, 0, 0, 0, -7667, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -125], [0, -4202, 0, 0, 0, 0, 0, 0], [-2506, 0, 0, 0, 0, 0, 0, 0], [0, 0, 622, 0, 0, -622, 0, 0], [0, 0, 0, -3275, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 270, 270], [0, 0, 0, 0, 9583, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 341], [0, 0, 0, 0, 0, 424, 0, 0], [5606, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 625, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 66, 0], [0, 0, 0, 0, 0, 0, -13, -13], [0, 0, 0, 0, 1855, 0, 0, 0], [0, 0, 0, 3635, 3635, 0, 0, 0], [0, -3608, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -73, 0, 0], [1772, 0, 0, 0, 0, 0, 0, 0], [0, 0, -606, 0, 0, 606, 0, 0], [0, 0, 220, 0, 0, 0, 0, 0], [-1958, 1958, 0, 0, 0, 0, 0, 0], [0, 0, 0, 5039, 5039, 0, 0, 0], [0, -10310, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -6, 0, 0], [0, 0, 337, 0, 0, -337, 0, 0], [0, 0, 213, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 312, 312], [0, 15646, 0, 0, 0, 0, 0, 0], [1880, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 4951, 0, 0, 0, 0], [0, 0, 398, 0, 0, 0, 0, 0], [0, 0, 0, 0, 12661, 0, 0, 0], [24736, -24736, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 582], [0, 0, 0, 0, 0, 158, 0, 0], [0, 0, -1071, 0, 0, 1071, 0, 0], [0, 0, 0, -17831, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -446, 0], [0, 0, 0, 0, 0, 0, -617, -617], [26554, -26554, 0, 0, 0, 0, 0, 0], [0, 0, 0, -18721, -18721, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 550], [24860, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3733, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 89, 0], [0, 0, 1092, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4147, 0, 0, 0], [0, 0, 0, -1951, -1951, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 704], [-13636, 0, 0, 0, 0, 0, 0, 0], [0, 0, 473, 0, 0, -473, 0, 0], [0, 0, 0, -143, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 674, 674], [0, 0, 0, 16181, 16181, 0, 0, 0], [0, -12212, 0, 0, 0, 0, 0, 0], [0, 0, 1368, 0, 0, -1368, 0, 0], [0, 0, 0, 0, 0, 0, -110, 0], [0, 0, 0, 0, 0, 0, -407, -407], [-1760, 1760, 0, 0, 0, 0, 0, 0], [0, 0, 0, -15865, -15865, 0, 0, 0], [0, 5122, 0, 0, 0, 0, 0, 0], [-18772, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1375, 0, 0, -1375, 0, 0], [0, 0, 0, 0, 0, 0, -394, 0], [0, 0, -603, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 884], [0, 0, 0, 0, 0, -923, 0, 0], [0, 0, 0, -24461, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 960, 0], [0, 0, 0, 0, 20131, 0, 0, 0], [-23468, 23468, 0, 0, 0, 0, 0, 0], [0, 0, 0, -26479, -26479, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -363], [0, 0, 0, 0, 0, -2026, 0, 0], [0, 0, 0, 0, 0, 0, 154, 0], [0, 0, -227, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1243, 1243], [0, 0, 0, 0, -845, 0, 0, 0], [0, -34298, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1029, 0, 0], [16034, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2222, 0, 0, 2222, 0, 0], [-51722, 51722, 0, 0, 0, 0, 0, 0], [0, 0, 0, 27269, 27269, 0, 0, 0], [0, -30470, 0, 0, 0, 0, 0, 0], [0, 0, -1255, 0, 0, 1255, 0, 0], [0, 0, 0, 0, 0, 0, 660, 0], [0, 0, -187, 0, 0, 0, 0, 0], [0, 0, 0, 0, -36827, 0, 0, 0], [0, 0, 0, 11309, 11309, 0, 0, 0], [-24772, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2616, 0, 0, 2616, 0, 0], [0, 0, 0, 31933, 0, 0, 0, 0], [0, 0, 1606, 0, 0, 0, 0, 0], [18382, -18382, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1298]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_441_p_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_441_4_p_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_441_4_p_a(:prec:=8)
    chi := MakeCharacter_441_p();
    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_p_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_441_4_p_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_441_p();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![64, 0, -8, 0, 1]>],Snew);
    return Vf;
end function;