
// Make newform 162.3.d.c in Magma, downloaded from the LMFDB on 09 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_c();".
// 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_c();".
// 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, 0, 0, 0, -1, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 3, 0, 0, 0, 3, 0], [0, 1, 0, 0, 0, 0, 0, -1], [0, 0, 6, 0, 0, 0, -3, 0], [0, 0, 0, 1, 0, 1, 0, -1], [0, 3, 0, 2, 0, -1, 0, 1], [0, 1, 0, -1, 0, 2, 0, 3]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1];
    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, 1, 0, 0, 0, 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, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 0, -1, 1], [-2, 2, 2, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, -2, 4, 0], [0, 16, -1, 0, 0, 0, 0, 0], [0, 0, 0, 5, 0, -5, 0, -5], [14, 0, 0, 0, -2, 0, 0, 0], [0, 0, 0, -2, 0, 0, 4, -4], [0, 0, 0, 0, 0, -23, 1, 0], [0, -8, 0, 0, 0, 0, 0, 0], [-19, 0, 0, 0, 8, 0, 0, 0], [0, 0, 0, -7, 0, 0, 8, -8], [22, -22, -6, 0, 6, 0, 0, 0], [0, 0, 0, 0, 0, 12, 0, 0], [0, 0, 0, -35, 0, 35, 0, 8], [0, 0, 0, 52, 0, 0, 4, -4], [13, -13, 0, 0, 0, 0, 0, 0], [0, 10, 6, 0, 0, 0, 0, 0], [0, 0, 0, 38, 0, -38, 0, 16], [56, 0, 0, 0, 3, 0, 0, 0], [-26, 26, -14, 0, 14, 0, 0, 0], [0, 0, 0, 0, 0, -48, 12, 0], [0, 0, 0, -34, 0, 34, 0, -29], [-8, 8, 16, 0, -16, 0, 0, 0], [0, 0, 0, 0, 0, -13, -16, 0], [0, 28, 24, 0, 0, 0, 0, 0], [0, 0, 0, -92, 0, 92, 0, -16], [-16, 0, 0, 0, -15, 0, 0, 0], [0, 0, 0, 21, 0, 0, -27, 27], [110, 0, 0, 0, -6, 0, 0, 0], [0, 0, 0, -46, 0, 0, -4, 4], [0, 0, 0, 0, 0, -62, -29, 0], [0, 40, 4, 0, 0, 0, 0, 0], [0, 0, 0, 31, 0, 0, -17, 17], [-32, 32, 0, 0, 0, 0, 0, 0], [0, -125, -24, 0, 0, 0, 0, 0], [8, 0, 0, 0, -4, 0, 0, 0], [0, 0, 0, 90, 0, 0, 24, -24], [0, 0, 0, 0, 0, 87, -33, 0], [0, 0, 0, 184, 0, -184, 0, -4], [8, 0, 0, 0, 32, 0, 0, 0], [0, 0, 0, 0, 0, -58, -16, 0], [0, 55, 0, 0, 0, 0, 0, 0], [0, 0, 0, 23, 0, -23, 0, -23], [-292, 0, 0, 0, 16, 0, 0, 0], [0, -110, 38, 0, 0, 0, 0, 0], [58, -58, 54, 0, -54, 0, 0, 0], [0, 0, 0, 0, 0, -138, -24, 0], [0, -152, -15, 0, 0, 0, 0, 0], [0, 0, 0, 82, 0, -82, 0, 53], [0, 0, 0, 144, 0, 0, -84, 84], [232, -232, 29, 0, -29, 0, 0, 0], [0, 0, 0, 82, 0, -82, 0, -64], [0, 0, 0, 6, 0, 0, -51, 51], [0, 0, 0, 0, 0, 200, 8, 0], [0, 0, 0, 70, 0, -70, 0, -25], [254, 0, 0, 0, 30, 0, 0, 0], [-248, 248, 48, 0, -48, 0, 0, 0], [0, 0, 0, 0, 0, 19, 43, 0], [0, 172, 24, 0, 0, 0, 0, 0], [0, 0, 0, 238, 0, 0, 1, -1], [-52, 0, 0, 0, -12, 0, 0, 0], [0, 0, 0, -52, 0, 0, -28, 28], [313, -313, 32, 0, -32, 0, 0, 0], [0, 0, 0, 0, 0, 62, -31, 0], [-98, 98, -38, 0, 38, 0, 0, 0], [0, -32, 88, 0, 0, 0, 0, 0], [0, 0, 0, -270, 0, 0, 96, -96], [70, -70, -112, 0, 112, 0, 0, 0], [0, 0, 0, 0, 0, 247, -80, 0], [0, 0, 0, -378, 0, 378, 0, 0], [-140, 140, 24, 0, -24, 0, 0, 0], [0, -362, 56, 0, 0, 0, 0, 0], [-676, 0, 0, 0, -4, 0, 0, 0], [0, 0, 0, 2, 0, 0, -100, 100], [0, 0, 0, 0, 0, -211, 56, 0], [-61, 0, 0, 0, 24, 0, 0, 0], [0, 0, 0, 206, 0, 0, 77, -77], [0, -104, -83, 0, 0, 0, 0, 0], [0, 0, 0, -384, 0, 0, 36, -36], [376, -376, -73, 0, 73, 0, 0, 0], [0, 0, 0, 308, 0, -308, 0, 124], [281, 0, 0, 0, 72, 0, 0, 0], [472, -472, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -192, 0], [0, 0, 0, -345, 0, 345, 0, -24], [-464, 464, -27, 0, 27, 0, 0, 0], [0, 0, 0, 0, 0, 181, 88, 0], [0, 16, -28, 0, 0, 0, 0, 0], [0, 0, 0, -20, 0, 20, 0, -160], [0, 0, 0, 0, 0, 126, -12, 0], [-220, 0, 0, 0, -44, 0, 0, 0], [0, 0, 0, -518, 0, 0, 64, -64], [0, 154, -54, 0, 0, 0, 0, 0], [0, 0, 0, -24, 0, 24, 0, 60], [0, 0, 0, 107, 0, 0, -184, 184], [0, 0, 0, 17, 0, -17, 0, -152], [398, 0, 0, 0, 42, 0, 0, 0], [80, 0, 0, 0, -121, 0, 0, 0], [508, -508, -60, 0, 60, 0, 0, 0], [0, 0, 0, 185, 0, -185, 0, 31], [0, 0, 0, -12, 0, 0, 120, -120], [0, 0, 0, 0, 0, 53, 5, 0], [0, 184, 52, 0, 0, 0, 0, 0], [335, 0, 0, 0, -72, 0, 0, 0], [0, 0, 0, 0, 0, -286, 188, 0], [0, 0, 0, 50, 0, -50, 0, -59], [0, 0, 0, -68, 0, 0, 64, -64], [31, -31, 48, 0, -48, 0, 0, 0], [0, -374, 70, 0, 0, 0, 0, 0], [-430, 0, 0, 0, 48, 0, 0, 0], [0, 0, 0, 458, 0, 0, 59, -59], [-824, 824, -12, 0, 12, 0, 0, 0], [-64, 0, 0, 0, 128, 0, 0, 0], [0, 0, 0, 0, 0, 202, -197, 0], [0, 244, -72, 0, 0, 0, 0, 0], [0, 0, 0, -102, 0, 102, 0, 192], [0, 0, 0, -251, 0, 0, -56, 56], [0, 0, 0, 0, 0, -274, 92, 0], [0, 91, -96, 0, 0, 0, 0, 0], [433, -433, -8, 0, 8, 0, 0, 0], [0, 0, 0, 0, 0, -373, 56, 0], [0, 0, 0, -24, 0, 24, 0, 240], [166, -166, -162, 0, 162, 0, 0, 0], [0, 0, 0, -398, 0, 398, 0, -79], [-584, 584, 79, 0, -79, 0, 0, 0], [0, 0, 0, 18, 0, -18, 0, -72], [130, -130, -162, 0, 162, 0, 0, 0], [0, -728, 96, 0, 0, 0, 0, 0], [56, 0, 0, 0, -132, 0, 0, 0], [0, 0, 0, 276, 0, 0, -132, 132], [0, 1210, 34, 0, 0, 0, 0, 0], [848, 0, 0, 0, 72, 0, 0, 0], [0, 0, 0, -558, 0, 0, 51, -51], [0, -191, 152, 0, 0, 0, 0, 0], [0, 0, 0, -567, 0, 567, 0, -81], [0, 550, 78, 0, 0, 0, 0, 0], [0, 0, 0, 129, 0, 0, 9, -9], [0, 0, 0, -850, 0, 850, 0, -5], [308, 0, 0, 0, -180, 0, 0, 0], [0, 0, 0, 0, 0, -106, -55, 0], [0, -56, -52, 0, 0, 0, 0, 0], [0, 0, 0, 830, 0, -830, 0, 16], [-400, 0, 0, 0, -8, 0, 0, 0], [0, 0, 0, 0, 0, -282, -24, 0], [-446, 446, 96, 0, -96, 0, 0, 0], [0, 0, 0, 0, 0, -738, 69, 0], [0, 904, 88, 0, 0, 0, 0, 0], [0, 0, 0, 38, 0, -38, 0, 196], [0, 373, -80, 0, 0, 0, 0, 0], [0, 0, 0, -311, 0, 311, 0, -40], [-40, 0, 0, 0, -68, 0, 0, 0], [0, 0, 0, -190, 0, 0, -292, 292], [-1232, 1232, 28, 0, -28, 0, 0, 0], [0, 0, 0, 0, 0, 102, 288, 0], [1010, 0, 0, 0, -102, 0, 0, 0], [0, 0, 0, 0, 0, 861, -123, 0], [-1207, 0, 0, 0, 104, 0, 0, 0], [0, 0, 0, -497, 0, 0, -113, 113], [0, 0, 0, 0, 0, 210, 36, 0], [0, 0, 0, 666, 0, -666, 0, -27], [0, 190, 282, 0, 0, 0, 0, 0], [0, 0, 0, -750, 0, 750, 0, -60], [0, 0, 0, 97, 0, 0, 40, -40], [0, 0, 0, 0, 0, -172, 140, 0], [-358, 0, 0, 0, -186, 0, 0, 0], [-131, 131, 144, 0, -144, 0, 0, 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_c();".
// 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_c(:prec:=8)
    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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_162_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![6561, 0, -2916, 0, 1215, 0, -36, 0, 1]>],Snew);
    return Vf;
end function;