
// Make newform 1350.3.d.g in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1350_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1350_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_1350_3_d_g();".
// 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_1350_3_d_g();".
// 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 := [2, 0, 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_1350_d();"
function MakeCharacter_1350_d()
    N := 1350;
    order := 2;
    char_gens := [1001, 1027];
    v := [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_1350_d_Hecke();"
function MakeCharacter_1350_d_Hecke(Kf)
    N := 1350;
    order := 2;
    char_gens := [1001, 1027];
    char_values := [[-1, 0], [1, 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, 1], [0, 0], [0, 0], [5, 0], [0, 1], [-9, 0], [0, -8], [-21, 0], [0, 1], [0, -27], [40, 0], [-25, 0], [0, 37], [-64, 0], [0, -16], [0, 51], [0, -64], [-97, 0], [-131, 0], [0, -63], [17, 0], [117, 0], [0, -41], [0, -104], [-41, 0], [0, -64], [13, 0], [0, 87], [-8, 0], [0, -27], [8, 0], [0, -96], [0, -189], [37, 0], [0, -184], [109, 0], [118, 0], [-203, 0], [0, 72], [0, 8], [0, 89], [-127, 0], [0, -72], [-271, 0], [0, 224], [-147, 0], [141, 0], [8, 0], [0, 49], [8, 0], [0, 224], [0, 145], [79, 0], [0, -33], [0, -120], [0, -200], [0, 72], [-221, 0], [-88, 0], [0, 144], [-32, 0], [0, -123], [-486, 0], [0, 48], [281, 0], [0, -368], [-59, 0], [-55, 0], [0, -423], [439, 0], [0, 368], [0, 39], [-589, 0], [-9, 0], [-157, 0], [0, 200], [0, 424], [296, 0], [0, 275], [-145, 0], [0, 496], [505, 0], [0, 31], [32, 0], [504, 0], [0, -168], [0, 48], [752, 0], [0, -432], [-597, 0], [0, 600], [0, -119], [-507, 0], [0, -303], [-870, 0], [0, 327], [0, 579], [0, 611], [163, 0], [697, 0], [389, 0], [0, 768], [0, 440], [0, 328], [-923, 0], [-247, 0], [0, -321], [0, 3], [0, 159], [-736, 0], [-437, 0], [335, 0], [0, 181], [965, 0], [275, 0], [0, -344], [1152, 0], [0, -489], [0, -248], [0, -352], [-577, 0], [489, 0], [0, 424], [0, -167], [640, 0], [0, -773], [489, 0], [0, 439], [-1080, 0], [-248, 0], [-848, 0], [0, -24], [-133, 0], [1271, 0], [0, -381], [81, 0], [0, -315], [395, 0], [0, 760], [0, -888], [-752, 0], [0, -352], [531, 0], [0, -248], [705, 0], [0, -1144], [1639, 0], [0, -699], [-355, 0], [0, 1032], [1129, 0], [0, -117], [-1227, 0], [0, 168], [-1005, 0], [0, -552], [-600, 0], [0, -891], [1465, 0], [0, 181], [0, 288], [0, -352], [-491, 0], [0, 360], [0, -333], [0, 425], [-755, 0], [24, 0], [-793, 0], [0, -1104], [0, 537], [-984, 0], [0, 568], [1015, 0], [1027, 0], [0, -1029], [1184, 0], [0, 141], [533, 0], [-616, 0], [-325, 0], [0, -976], [-296, 0], [0, -1128], [0, -352], [0, -141], [-1327, 0], [1123, 0], [-631, 0], [0, -1336], [1617, 0], [0, 856], [-1235, 0], [0, 192], [0, -288], [0, 963], [9, 0], [-1857, 0], [0, 651], [0, -895], [0, -456], [-963, 0], [369, 0], [-407, 0], [0, -761], [0, -957], [-2355, 0], [0, -1385], [0, 485], [1645, 0], [79, 0], [0, -1176], [-965, 0], [0, 1137], [0, 425], [-863, 0], [-1970, 0], [0, 797], [0, -1447], [0, 259], [1791, 0], [-674, 0], [0, 240], [712, 0], [0, 1017], [351, 0], [0, -152], [0, 1793], [2557, 0], [0, -1976], [583, 0], [-608, 0], [2600, 0], [0, 429], [-19, 0], [0, 1864], [-1065, 0], [0, -112], [0, 897], [0, -456], [0, 903], [1019, 0], [-1363, 0], [-377, 0], [0, -776], [0, 975], [-645, 0], [0, 312], [1904, 0], [0, -1247], [936, 0], [0, 64], [0, -352], [745, 0], [0, -1267], [0, 1560], [1711, 0], [-1610, 0], [0, -1195], [736, 0], [-1701, 0], [0, -1865], [697, 0], [97, 0], [0, 312], [1712, 0], [0, -2139], [0, 1549], [-784, 0], [0, 2376], [-415, 0], [-995, 0], [2098, 0], [-155, 0], [-832, 0], [307, 0], [0, -1033], [-2344, 0], [2960, 0], [0, -1047], [0, -1041], [-1858, 0], [0, -175], [1945, 0], [355, 0], [0, -272], [1425, 0], [0, -688], [-811, 0], [0, 763], [0, 1051], [0, 2073], [0, 1971], [0, 1512], [-3640, 0], [0, 1944], [1563, 0], [0, 264], [0, -2615], [-165, 0], [-1799, 0], [0, 555], [-910, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1350_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1350_3_d_g();".
// 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_1350_3_d_g(:prec:=2)
    chi := MakeCharacter_1350_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(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_1350_3_d_g();".
// 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_1350_3_d_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1350_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![-5, 1]>,<11,R![2, 0, 1]>],Snew);
    return Vf;
end function;