
// Make newform 165.3.g.b in Magma, downloaded from the LMFDB on 04 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_165_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_165_g_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_165_3_g_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_165_3_g_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 := [3, -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_165_g();"
function MakeCharacter_165_g()
    N := 165;
    order := 2;
    char_gens := [56, 67, 46];
    v := [1, 1, 2];
    // 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_165_g_Hecke();"
function MakeCharacter_165_g_Hecke(Kf)
    N := 165;
    order := 2;
    char_gens := [56, 67, 46];
    char_values := [[-1, 0], [-1, 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 := [[1, 0], [3, -1], [5, 0], [3, -6], [1, -2], [-6, 12], [19, 0], [11, 0], [-32, 0], [-3, 6], [-31, 0], [-15, 30], [-6, 12], [-18, 36], [-14, 0], [13, 0], [0, 0], [47, 0], [-12, 24], [27, -54], [-24, 48], [-100, 0], [22, 0], [51, -102], [48, -96], [12, -24], [-6, 12], [-92, 0], [110, 0], [-8, 0], [12, -24], [27, -54], [-38, 0], [134, 0], [-27, 54], [-40, 0], [-27, 54], [-75, 150], [43, 0], [220, 0], [-60, 120], [-340, 0], [36, -72], [33, -66], [-110, 0], [-43, 0], [131, 0], [78, -156], [-200, 0], [-310, 0], [-263, 0], [30, -60], [-280, 0], [108, -216], [268, 0], [247, 0], [-42, 84], [152, 0], [-60, 120], [60, -120], [30, -60], [-464, 0], [-96, 192], [-9, 18], [48, -96], [-431, 0], [122, 0], [-153, 306], [190, 0], [-334, 0], [670, 0], [-48, 96], [-42, 84], [-48, 96], [50, 0], [-620, 0], [6, -12], [-84, 168], [129, -258], [650, 0], [-60, 120], [-190, 0], [-246, 492], [-12, 24], [-28, 0], [490, 0], [156, -312], [123, -246], [-159, 318], [-66, 132], [445, 0], [42, -84], [54, -108], [273, -546], [278, 0], [-458, 0], [198, -396], [24, -48], [-42, 84], [605, 0], [0, 0], [376, 0], [-638, 0], [96, -192], [599, 0], [-102, 204], [631, 0], [286, 0], [-171, 342], [-418, 0], [315, -630], [204, -408], [262, 0], [302, 0], [125, 0], [297, -594], [-315, 630], [706, 0], [487, 0], [-99, 198], [-46, 0], [261, -522], [220, 0], [-1109, 0], [728, 0], [57, -114], [-148, 0], [-387, 774], [342, -684], [246, -492], [-1066, 0], [415, 0], [-481, 0], [-108, 216], [282, -564], [1436, 0], [1477, 0], [12, -24], [298, 0], [-444, 888], [-997, 0], [192, -384], [-246, 492], [484, 0], [290, 0], [-312, 624], [96, -192], [-1271, 0], [1142, 0], [1132, 0], [84, -168], [156, -312], [-159, 318], [154, 0], [249, -498], [153, -306], [1298, 0], [393, -786], [60, -120], [-111, 222], [901, 0], [-209, 0], [-285, 570], [-402, 804], [-944, 0], [-1676, 0], [-562, 0], [-558, 1116]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_165_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_165_3_g_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_165_3_g_b(:prec:=2)
    chi := MakeCharacter_165_g();
    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_165_3_g_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_165_3_g_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_165_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![-1, 1]>],Snew);
    return Vf;
end function;