
// Make newform 405.4.e.j in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_405_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_405_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_405_4_e_j();".
// 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_405_4_e_j();".
// 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_405_e();"
function MakeCharacter_405_e()
    N := 405;
    order := 3;
    char_gens := [326, 82];
    v := [2, 3];
    // 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_405_e_Hecke();"
function MakeCharacter_405_e_Hecke(Kf)
    N := 405;
    order := 3;
    char_gens := [326, 82];
    char_values := [[0, -1], [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 := 4;
    raw_aps := [[2, -2], [0, 0], [0, -5], [0, 0], [-10, 10], [0, 80], [7, 0], [-113, 0], [0, 81], [220, -220], [0, 189], [170, 0], [0, 130], [-10, 10], [-160, 160], [631, 0], [0, 560], [-229, 229], [0, -750], [890, 0], [-890, 0], [27, -27], [-429, 429], [-750, 0], [1480, -1480], [1500, -1500], [0, 460], [-420, 0], [-607, 0], [0, -2170], [-1610, 0], [0, -2370], [-1797, 1797], [0, 124], [0, 70], [-2248, 2248], [0, -1010], [590, 0], [0, -2403], [-801, 801], [-2360, 0], [1241, 0], [4990, -4990], [0, 2260], [-2247, 0], [4564, 0], [0, -4949], [-3890, 3890], [-2453, 2453], [0, 6213], [-3450, 0], [0, -6490], [3401, -3401], [-4980, 0], [0, -3357], [4540, -4540], [8410, 0], [259, 0], [4170, -4170], [1740, -1740], [0, 5070], [0, 159], [6490, 0], [0, 8220], [4660, -4660], [6817, -6817], [-192, 192], [0, -4840], [0, 860], [-5377, 5377], [8010, -8010], [12930, 0], [6000, -6000], [0, 140], [6217, 0], [0, -4551], [2310, -2310], [-2900, 0], [0, -2250], [0, 11263], [0, 6910], [5249, -5249], [-11880, 0], [-4280, 0], [-6463, 6463], [-11721, 11721], [-2180, 0], [17840, -17840], [2250, -2250], [0, -1230], [5813, 0], [6750, -6750], [-6610, 0], [0, 4990], [0, -1483], [-11641, 0], [0, -2620], [-13690, 0], [-10220, 0], [-2778, 0], [12830, -12830], [-4950, 0], [0, -6540], [15240, -15240], [0, 5281], [-10510, 0], [-4107, 4107], [26129, 0], [0, 4360], [16639, -16639], [0, -490], [18400, 0], [0, -7827], [19756, -19756], [9829, 0], [6000, -6000], [0, -8280], [-16637, 0], [0, -19751], [-14260, 14260], [0, -22318], [-20040, 20040], [2310, -2310], [-26739, 0], [-5101, 5101], [26030, 0], [3854, -3854], [-870, 0], [-35780, 35780], [0, 3400], [-683, 0], [0, 13400], [0, 23219], [19630, 0], [0, -2940], [0, 13987], [19839, 0], [0, 38390], [0, 28027], [8630, 0], [-1932, 0], [18090, -18090], [0, -12890], [14887, 0], [12666, 0], [-43820, 43820], [-19320, 19320], [3653, -3653], [0, -24373], [-17629, 0], [0, 21210], [39340, 0], [-4240, 0], [0, 15933], [6780, -6780], [24740, -24740], [-48344, 0], [29650, -29650], [10260, 0], [0, 16270], [23103, -23103], [-32090, 0], [0, 42010], [17490, 0], [0, 22130], [-40959, 40959], [61169, 0], [-26190, 26190]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_405_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_405_4_e_j();".
// 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_405_4_e_j(:prec:=2)
    chi := MakeCharacter_405_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_405_4_e_j();".
// 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_405_4_e_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_405_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![4, -2, 1]>,<7,R![0, 1]>],Snew);
    return Vf;
end function;