
// Make newform 162.3.b.a in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_162_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_162_b_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_b_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_b_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, 0, 0, 1], [-3, 0, 3, 0], [0, 12, 0, -3]];
    Rf_basisdens := [1, 2, 1, 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_b();"
function MakeCharacter_162_b()
    N := 162;
    order := 2;
    char_gens := [83];
    v := [1];
    // 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_b_Hecke();"
function MakeCharacter_162_b_Hecke(Kf)
    N := 162;
    order := 2;
    char_gens := [83];
    char_values := [[-1, 0, 0, 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, 0, 1, 0], [-1, 0, 0, 1], [0, 3, 1, 0], [5, 0, 0, 2], [0, -6, 2, 0], [-10, 0, 0, 2], [0, -3, -1, 0], [0, -6, 1, 0], [-19, 0, 0, -3], [32, 0, 0, -2], [0, 18, 7, 0], [23, 0, 0, -3], [0, -21, 3, 0], [0, -30, 10, 0], [0, -39, -7, 0], [-31, 0, 0, -6], [53, 0, 0, -3], [0, -24, -10, 0], [-52, 0, 0, -6], [-7, 0, 0, 5], [0, 15, -21, 0], [0, 66, -10, 0], [-7, 0, 0, 14], [0, -108, -1, 0], [-37, 0, 0, 9], [0, -48, -20, 0], [-16, 0, 0, 18], [0, -132, -3, 0], [-34, 0, 0, 6], [0, 81, -23, 0], [0, 30, 37, 0], [11, 0, 0, 13], [0, 78, -1, 0], [-175, 0, 0, -15], [-109, 0, 0, -12], [-58, 0, 0, -26], [0, -87, 33, 0], [0, 60, 3, 0], [0, -84, -32, 0], [140, 0, 0, -14], [0, -9, -1, 0], [125, 0, 0, -30], [0, 84, 8, 0], [80, 0, 0, -10], [-103, 0, 0, -7], [203, 0, 0, -15], [0, 87, 9, 0], [143, 0, 0, 36], [0, -48, 16, 0], [0, -51, 3, 0], [143, 0, 0, -8], [0, 96, 16, 0], [0, -24, 9, 0], [0, -45, 77, 0], [0, 6, 10, 0], [-34, 0, 0, 42], [-49, 0, 0, 0], [0, -114, -35, 0], [-145, 0, 0, -27], [0, 96, 29, 0], [110, 0, 0, 6], [0, -63, 31, 0], [-397, 0, 0, 4], [0, 126, -55, 0], [245, 0, 0, -31], [335, 0, 0, 4], [0, -159, -69, 0], [107, 0, 0, 28], [0, 102, -35, 0], [0, -252, 12, 0], [77, 0, 0, -15], [113, 0, 0, -46], [-328, 0, 0, 22], [0, 297, 61, 0], [0, -72, -13, 0], [-124, 0, 0, -18], [0, 54, -65, 0], [-385, 0, 0, -8], [0, -195, 51, 0], [269, 0, 0, -2], [0, 156, 92, 0], [344, 0, 0, -18], [-139, 0, 0, 57], [0, -489, 27, 0], [0, 450, -18, 0], [401, 0, 0, -42], [0, -78, -23, 0], [-283, 0, 0, -31], [0, -516, -20, 0], [0, 123, 93, 0], [-190, 0, 0, -22], [0, 249, -115, 0], [35, 0, 0, 81], [0, -216, 114, 0], [0, -378, -65, 0], [0, -60, 56, 0], [488, 0, 0, 54], [-274, 0, 0, -32], [41, 0, 0, 69], [0, 102, -106, 0], [0, 21, 111, 0], [0, -180, -1, 0], [-7, 0, 0, -11], [-88, 0, 0, -6], [0, -45, -97, 0], [0, -150, 26, 0], [0, 33, 161, 0], [-751, 0, 0, -6], [-745, 0, 0, 61], [56, 0, 0, -78], [0, 162, 7, 0], [-175, 0, 0, -3], [-58, 0, 0, -14], [0, -24, -53, 0], [593, 0, 0, -27], [0, 252, 12, 0], [0, -84, 125, 0], [0, 279, -7, 0], [395, 0, 0, -84], [137, 0, 0, -58], [0, 288, 11, 0], [0, 576, 60, 0], [707, 0, 0, 45], [0, -102, -122, 0], [-349, 0, 0, -112], [0, 144, 60, 0], [-643, 0, 0, -27], [-55, 0, 0, 90], [272, 0, 0, -90], [0, -375, -99, 0], [-241, 0, 0, -143], [866, 0, 0, 24], [0, -222, 123, 0], [-157, 0, 0, 68], [0, 90, 78, 0], [359, 0, 0, -63], [0, 858, -21, 0], [0, 174, 2, 0], [74, 0, 0, 90], [0, 774, 29, 0], [-1579, 0, 0, 5], [0, -132, -148, 0], [980, 0, 0, 14], [0, 771, -9, 0], [77, 0, 0, 54], [0, 42, 165, 0], [245, 0, 0, 97], [0, -276, 176, 0], [317, 0, 0, 94], [0, -462, -170, 0], [-838, 0, 0, 74], [0, 513, -29, 0], [335, 0, 0, -115], [0, -777, 33, 0], [38, 0, 0, -174], [0, -102, 93, 0], [896, 0, 0, 58], [0, 456, -55, 0], [0, -273, -75, 0], [0, -90, 114, 0], [629, 0, 0, 81], [0, 576, -30, 0], [0, -1092, 5, 0], [0, 567, 5, 0], [650, 0, 0, -30], [275, 0, 0, -132]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_162_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_162_3_b_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_b_a(:prec:=4)
    chi := MakeCharacter_162_b();
    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_b_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_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_162_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![27, 0, 1]>],Snew);
    return Vf;
end function;