
// Make newform 162.3.d.b in Magma, downloaded from the LMFDB on 15 November 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_b();".
// 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_b();".
// 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]];
    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, 3, 0, -3], [4, 0, -4, 0], [0, 12, 0, 0], [0, 0, -8, 0], [0, 0, 0, 9], [-16, 0, 0, 0], [0, 12, 0, -12], [0, 3, 0, 0], [0, 0, -44, 0], [-34, 0, 0, 0], [0, -33, 0, 33], [40, 0, -40, 0], [0, -60, 0, 0], [0, 0, 0, -27], [0, -24, 0, 24], [-50, 0, 50, 0], [0, 0, -8, 0], [0, 0, 0, 36], [-16, 0, 0, 0], [76, 0, -76, 0], [0, 84, 0, 0], [0, 0, 0, -9], [-176, 0, 176, 0], [0, 21, 0, 0], [0, 0, 28, 0], [0, 0, 0, 0], [56, 0, 0, 0], [0, 111, 0, -111], [92, 0, 0, 0], [0, 120, 0, -120], [0, 111, 0, 0], [0, 0, -152, 0], [0, -195, 0, 195], [148, 0, -148, 0], [0, 0, 82, 0], [56, 0, 0, 0], [0, -24, 0, 24], [0, -123, 0, 0], [0, 0, 0, 144], [-232, 0, 0, 0], [0, -24, 0, 0], [0, 0, -206, 0], [0, 0, 0, -117], [20, 0, 0, 0], [0, 0, -296, 0], [436, 0, -436, 0], [0, 12, 0, 0], [0, 0, -8, 0], [0, 0, 0, 9], [0, -96, 0, 96], [-32, 0, 32, 0], [0, 0, 0, -36], [0, 129, 0, -129], [0, 264, 0, 0], [0, 0, 0, 243], [380, 0, 0, 0], [328, 0, -328, 0], [0, 201, 0, 0], [0, 0, 208, 0], [0, 309, 0, -309], [-520, 0, 0, 0], [0, 264, 0, -264], [94, 0, -94, 0], [0, 237, 0, 0], [-536, 0, 536, 0], [0, 0, 208, 0], [0, -204, 0, 204], [238, 0, -238, 0], [0, -159, 0, 0], [0, 0, 0, 396], [-284, 0, 284, 0], [0, 0, 190, 0], [-160, 0, 0, 0], [0, 192, 0, -192], [0, -285, 0, 0], [146, 0, 0, 0], [0, -231, 0, 231], [0, 0, -368, 0], [0, -276, 0, 276], [40, 0, -40, 0], [0, 0, 0, 108], [542, 0, 0, 0], [4, 0, -4, 0], [0, 228, 0, 0], [0, 0, 0, -153], [400, 0, -400, 0], [0, -213, 0, 0], [0, 0, 604, 0], [0, 0, 0, -252], [0, 372, 0, 0], [596, 0, 0, 0], [0, 192, 0, -192], [0, 0, -224, 0], [0, 0, 0, -612], [0, -339, 0, 339], [0, 0, 0, 369], [-736, 0, 0, 0], [-808, 0, 0, 0], [-536, 0, 536, 0], [0, 0, 0, 117], [0, 228, 0, -228], [0, 111, 0, 0], [0, 0, -368, 0], [-142, 0, 0, 0], [0, 264, 0, 0], [0, 0, 0, -783], [0, -564, 0, 564], [-158, 0, 158, 0], [0, 0, -332, 0], [578, 0, 0, 0], [0, 39, 0, -39], [-896, 0, 896, 0], [20, 0, 0, 0], [0, 183, 0, 0], [0, 0, -728, 0], [0, 0, 0, 324], [0, -213, 0, 213], [0, -744, 0, 0], [0, 0, -62, 0], [670, 0, -670, 0], [0, -915, 0, 0], [0, 0, 0, 396], [40, 0, -40, 0], [0, 0, 0, -675], [-968, 0, 968, 0], [0, 0, 0, -828], [508, 0, -508, 0], [0, 0, 1144, 0], [-304, 0, 0, 0], [0, -600, 0, 600], [0, 0, -188, 0], [-1240, 0, 0, 0], [0, 111, 0, -111], [0, 0, 910, 0], [0, 0, 0, 981], [0, 0, 1360, 0], [0, 75, 0, -75], [0, 0, 0, 783], [-160, 0, 0, 0], [0, 309, 0, 0], [0, 0, -332, 0], [0, 0, 0, -72], [632, 0, 0, 0], [0, 516, 0, 0], [-446, 0, 446, 0], [0, -303, 0, 0], [0, 0, -728, 0], [0, 0, 0, 648], [0, 0, 910, 0], [0, 0, 0, 657], [1064, 0, 0, 0], [0, 984, 0, -984], [1768, 0, -1768, 0], [0, -168, 0, 0], [380, 0, 0, 0], [0, 471, 0, 0], [-178, 0, 0, 0], [0, 309, 0, -309], [0, 1272, 0, 0], [0, 0, 0, -927], [0, 0, -1700, 0], [0, 0, 0, -324], [0, -537, 0, 537], [0, -744, 0, 0], [-772, 0, 0, 0], [-194, 0, 194, 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_b();".
// 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_b(: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_b();".
// 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_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_162_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![324, 0, -18, 0, 1]>],Snew);
    return Vf;
end function;