
// Make newform 162.3.d.a in Magma, downloaded from the LMFDB on 10 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_162_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_162_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_162_3_d_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_162_3_d_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 := [4, 0, -2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 2, 2];
    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_162_d();"
function MakeCharacter_162_d()
    N := 162;
    order := 6;
    char_gens := [83];
    v := [5];
    // 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_162_d_Hecke();"
function MakeCharacter_162_d_Hecke(Kf)
    N := 162;
    order := 6;
    char_gens := [83];
    char_values := [[0, 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, 0, 0], [0, -6, 0, 6], [-5, 0, 5, 0], [0, -6, 0, 0], [0, 0, 1, 0], [0, 0, 0, 18], [29, 0, 0, 0], [0, -6, 0, 6], [0, 12, 0, 0], [0, 0, 10, 0], [-25, 0, 0, 0], [0, 12, 0, -12], [-14, 0, 14, 0], [0, -6, 0, 0], [0, 0, 0, -36], [0, 66, 0, -66], [-23, 0, 23, 0], [0, 0, 19, 0], [0, 0, 0, 72], [-97, 0, 0, 0], [-77, 0, 77, 0], [0, 84, 0, 0], [0, 0, 0, -54], [49, 0, -49, 0], [0, -96, 0, 0], [0, 0, 163, 0], [0, 0, 0, 18], [2, 0, 0, 0], [0, -78, 0, 78], [-178, 0, 0, 0], [0, 84, 0, -84], [0, 66, 0, 0], [0, 0, 163, 0], [0, -24, 0, 24], [-149, 0, 149, 0], [0, 0, -242, 0], [173, 0, 0, 0], [0, 138, 0, -138], [0, 228, 0, 0], [0, 0, 0, -216], [263, 0, 0, 0], [0, 102, 0, 0], [0, 0, -71, 0], [0, 0, 0, -54], [173, 0, 0, 0], [0, 0, -341, 0], [58, 0, -58, 0], [0, -96, 0, 0], [0, 0, 262, 0], [0, 0, 0, 216], [0, 120, 0, -120], [121, 0, -121, 0], [0, 0, 0, 252], [0, -60, 0, 60], [0, 120, 0, 0], [0, 0, 0, -198], [29, 0, 0, 0], [382, 0, -382, 0], [0, 156, 0, 0], [0, 0, -62, 0], [0, -222, 0, 222], [-34, 0, 0, 0], [0, -42, 0, 42], [-239, 0, 239, 0], [0, 12, 0, 0], [499, 0, -499, 0], [0, 0, 145, 0], [0, -312, 0, 312], [121, 0, -121, 0], [0, -312, 0, 0], [0, 0, 0, -306], [-149, 0, 149, 0], [0, 0, 361, 0], [173, 0, 0, 0], [0, 480, 0, -480], [0, -294, 0, 0], [-286, 0, 0, 0], [0, -240, 0, 240], [0, 0, -215, 0], [0, 426, 0, -426], [-455, 0, 455, 0], [0, 0, 0, -234], [218, 0, 0, 0], [-374, 0, 374, 0], [0, 12, 0, 0], [0, 0, 0, -342], [22, 0, -22, 0], [0, 282, 0, 0], [0, 0, 883, 0], [0, 0, 0, 90], [0, -276, 0, 0], [317, 0, 0, 0], [0, -258, 0, 258], [0, 0, -710, 0], [0, 0, 0, 198], [0, -186, 0, 186], [0, 0, 0, 486], [-763, 0, 0, 0], [-313, 0, 0, 0], [139, 0, -139, 0], [0, 0, 0, -198], [0, -420, 0, 420], [0, -438, 0, 0], [0, 0, 163, 0], [1127, 0, 0, 0], [0, 714, 0, 0], [0, 0, 0, 252], [0, -636, 0, 636], [166, 0, -166, 0], [0, 0, 523, 0], [335, 0, 0, 0], [0, 354, 0, -354], [-5, 0, 5, 0], [245, 0, 0, 0], [0, -456, 0, 0], [0, 0, 82, 0], [0, 0, 0, 324], [0, 228, 0, -228], [0, 732, 0, 0], [0, 0, -143, 0], [-671, 0, 671, 0], [0, -6, 0, 0], [0, 0, 0, 72], [850, 0, -850, 0], [0, 0, 0, -198], [1201, 0, -1201, 0], [0, 0, 0, 684], [-950, 0, 950, 0], [0, 0, -98, 0], [1262, 0, 0, 0], [0, -546, 0, 546], [0, 0, -197, 0], [-241, 0, 0, 0], [0, 354, 0, -354], [0, 0, -431, 0], [0, 0, 0, -180], [0, 0, -125, 0], [0, -600, 0, 600], [0, 0, 0, 792], [-322, 0, 0, 0], [0, -618, 0, 0], [0, 0, -269, 0], [0, 0, 0, 18], [-1105, 0, 0, 0], [0, 84, 0, 0], [-1391, 0, 1391, 0], [0, -384, 0, 0], [0, 0, -845, 0], [0, 0, 0, -882], [0, 0, -1151, 0], [0, 0, 0, 918], [677, 0, 0, 0], [0, 732, 0, -732], [-5, 0, 5, 0], [0, -708, 0, 0], [1190, 0, 0, 0], [0, 336, 0, 0], [1775, 0, 0, 0], [0, -114, 0, 114], [0, -402, 0, 0], [0, 0, 0, -1278], [0, 0, -701, 0], [0, 0, 0, 270], [0, 228, 0, -228], [0, 30, 0, 0], [-475, 0, 0, 0], [1534, 0, -1534, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_162_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_162_3_d_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_162_3_d_a(:prec:=4)
    chi := MakeCharacter_162_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(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_162_3_d_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_162_3_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_162_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![5184, 0, -72, 0, 1]>],Snew);
    return Vf;
end function;