
// Make newform 1350.3.d.d in Magma, downloaded from the LMFDB on 15 May 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_d();".
// 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_d();".
// 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, 6], [1, 0], [0, -18], [29, 0], [0, 6], [0, -12], [-10, 0], [25, 0], [0, 12], [-14, 0], [0, -6], [0, 36], [0, 66], [23, 0], [19, 0], [0, 72], [97, 0], [77, 0], [0, 84], [0, -54], [49, 0], [0, 96], [163, 0], [0, -18], [2, 0], [0, 78], [178, 0], [0, 84], [0, 66], [-163, 0], [0, -24], [149, 0], [-242, 0], [-173, 0], [0, -138], [0, 228], [0, -216], [263, 0], [0, -102], [-71, 0], [0, 54], [173, 0], [341, 0], [58, 0], [0, -96], [-262, 0], [0, -216], [0, 120], [-121, 0], [0, 252], [0, 60], [0, 120], [0, -198], [29, 0], [382, 0], [0, -156], [-62, 0], [0, 222], [34, 0], [0, -42], [-239, 0], [0, 12], [-499, 0], [145, 0], [0, 312], [-121, 0], [0, -312], [0, -306], [-149, 0], [361, 0], [173, 0], [0, -480], [0, 294], [286, 0], [0, -240], [215, 0], [0, 426], [455, 0], [0, -234], [-218, 0], [374, 0], [0, 12], [0, -342], [22, 0], [0, -282], [883, 0], [0, -90], [0, 276], [-317, 0], [0, -258], [710, 0], [0, -198], [0, -186], [0, 486], [763, 0], [-313, 0], [139, 0], [0, 198], [0, 420], [0, 438], [-163, 0], [-1127, 0], [0, 714], [0, -252], [0, -636], [-166, 0], [523, 0], [-335, 0], [0, -354], [5, 0], [245, 0], [0, 456], [82, 0], [0, -324], [0, -228], [0, -732], [143, 0], [-671, 0], [0, -6], [0, -72], [-850, 0], [0, -198], [-1201, 0], [0, 684], [-950, 0], [-98, 0], [1262, 0], [0, 546], [197, 0], [241, 0], [0, 354], [431, 0], [0, 180], [-125, 0], [0, 600], [0, 792], [-322, 0], [0, 618], [-269, 0], [0, -18], [-1105, 0], [0, -84], [-1391, 0], [0, -384], [845, 0], [0, 882], [-1151, 0], [0, 918], [-677, 0], [0, -732], [-5, 0], [0, 708], [1190, 0], [0, -336], [-1775, 0], [0, -114], [0, -402], [0, 1278], [-701, 0], [0, 270], [0, -228], [0, 30], [-475, 0], [1534, 0], [-1033, 0], [0, -174], [0, 132], [1106, 0], [0, 1398], [265, 0], [917, 0], [0, 696], [-1426, 0], [0, -954], [43, 0], [554, 0], [1075, 0], [0, -516], [-386, 0], [0, -738], [0, -1272], [0, -1236], [-1127, 0], [1003, 0], [-1441, 0], [0, 234], [457, 0], [0, -564], [-835, 0], [0, 582], [0, -738], [0, 258], [-1081, 0], [-1967, 0], [0, -114], [0, 1080], [0, -1626], [-643, 0], [1009, 0], [1103, 0], [0, 864], [0, 648], [1685, 0], [0, 150], [0, -390], [845, 0], [529, 0], [0, -906], [2515, 0], [0, -78], [0, 120], [-1993, 0], [-2150, 0], [0, -1578], [0, 1728], [0, -696], [791, 0], [566, 0], [0, -600], [-38, 0], [0, -1038], [-409, 0], [0, -1482], [0, 198], [-293, 0], [0, -786], [1153, 0], [-538, 0], [-370, 0], [0, -1266], [-2309, 0], [0, -906], [-505, 0], [0, -342], [0, -708], [0, -1116], [0, -1752], [1469, 0], [-773, 0], [-1897, 0], [0, -1326], [0, -1650], [1195, 0], [0, 372], [-586, 0], [0, -252], [2926, 0], [0, -1386], [0, 618], [2255, 0], [0, -762], [0, 2250], [1271, 0], [-590, 0], [0, 720], [-2234, 0], [859, 0], [0, 210], [-433, 0], [-263, 0], [0, -408], [-1858, 0], [0, -54], [0, 2004], [2026, 0], [0, 726], [935, 0], [1195, 0], [238, 0], [-955, 0], [-1322, 0], [-893, 0], [0, 762], [-2974, 0], [-70, 0], [0, 708], [0, 1554], [-1378, 0], [0, -1140], [1895, 0], [-2285, 0], [0, 558], [745, 0], [0, -408], [-3091, 0], [0, 918], [0, -2334], [0, -972], [0, -804], [0, 1452], [310, 0], [0, 444], [-2227, 0], [0, 1614], [0, -1440], [-5, 0], [-1799, 0], [0, -1080], [3350, 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_d();".
// 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_d(: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_d();".
// 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_d( : 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![72, 0, 1]>],Snew);
    return Vf;
end function;