
// Make newform 165.2.m.a in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_165_m();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_165_m_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_2_m_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_165_2_m_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 := [1, 3, 5, 3, 4, -3, 5, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-7, -12, -7, -4, 0, -3, 2, -1], [9, 6, 19, -16, 20, -7, 0, 1], [-1, -10, 13, -16, 12, -9, 4, -1], [-5, -8, -13, -4, 8, -17, 10, -3], [-1, 6, 1, 16, -20, 23, -12, 3], [-7, -12, -11, -28, 32, -35, 18, -5]];
    Rf_basisdens := [1, 1, 8, 8, 8, 8, 8, 8];
    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_165_m();"
function MakeCharacter_165_m()
    N := 165;
    order := 5;
    char_gens := [56, 67, 46];
    v := [5, 5, 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_m_Hecke();"
function MakeCharacter_165_m_Hecke(Kf)
    N := 165;
    order := 5;
    char_gens := [56, 67, 46];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 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 := 2;
    raw_aps := [[1, 0, 0, -1, 1, -1, -1, 0], [-1, 0, 0, 1, -1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, -1], [1, 0, -1, 0, 0, 1, -2, 1], [0, 2, 3, -1, 2, -3, -2, 1], [2, -2, 0, -2, 2, 1, 1, 0], [0, 0, 0, 0, 0, 0, 0, 5], [-1, 5, 3, 1, -1, -5, -3, -1], [-1, 1, 1, -2, 0, 0, 0, 2], [1, 0, 4, 0, 4, -4, -1, 1], [-1, 3, 0, 0, -1, 0, 0, 0], [-4, 0, 1, 0, -3, -1, -1, -4], [-7, -3, -4, 7, -6, 3, 4, -6], [-4, 1, 1, 6, 0, -1, 0, -6], [5, 5, 4, -5, 6, -5, -4, 6], [-1, 3, 0, -4, -1, 3, 3, 0], [7, 0, 1, 0, 3, -1, -1, 7], [0, -6, -3, 5, -5, 0, 6, -8], [0, -5, -5, -4, 0, 10, 0, 4], [0, -2, 3, 1, -1, 0, 2, -7], [1, 0, 5, 0, 8, -5, 3, 1], [5, 0, 0, -4, 5, 2, 2, 0], [0, 2, 5, 3, -3, 0, -2, 9], [2, 0, 0, 5, 0, -2, 0, -5], [-1, 1, 0, -2, -1, 3, 3, 0], [-5, 4, 0, 2, -5, 1, 1, 0], [10, 0, 0, 0, 3, 0, -5, 10], [-7, -1, -7, 7, -9, 1, 7, -9], [3, -7, -7, 4, 0, 1, 0, -4], [2, -1, -5, -2, -9, 1, 5, -9], [0, 3, -4, 0, 0, 0, -3, -5], [-2, 6, 6, -6, 0, 5, 0, 6], [0, -1, -3, 2, -2, 0, 1, 6], [-2, 0, -3, 0, -2, 3, -4, -2], [0, 1, -3, -1, 1, 0, -1, 2], [3, -4, 0, -3, 6, 4, 0, 6], [9, 5, 11, -9, 10, -5, -11, 10], [-7, 0, 0, -6, -7, 8, 8, 0], [-2, 0, 0, -2, -2, -6, -6, 0], [-5, -8, -7, 5, -6, 8, 7, -6], [3, 3, 3, -3, 1, -3, -3, 1], [0, 6, -3, -2, 2, 0, -6, 14], [-1, 0, 6, 0, 12, -6, -11, -1], [0, -10, -2, -4, 4, 0, 10, 3], [14, 3, 3, -2, 0, 1, 0, 2], [-4, 0, 0, 2, 0, -9, 0, -2], [-17, 3, 0, 11, -17, 3, 3, 0], [9, -2, -6, -9, -8, 2, 6, -8], [3, 0, 5, 0, 15, -5, -1, 3], [8, -13, 0, 7, 8, 3, 3, 0], [10, -10, 0, -7, 10, 5, 5, 0], [-7, 9, 6, 7, -4, -9, -6, -4], [-3, 4, 4, 2, 0, -8, 0, -2], [12, -12, 0, 3, 12, 6, 6, 0], [-4, 0, -6, 0, 1, 6, 7, -4], [2, -9, -9, 6, 0, 13, 0, -6], [0, 7, 0, -6, 6, 0, -7, 15], [1, 0, 1, 0, 13, -1, -1, 1], [7, -11, 0, 4, 7, 4, 4, 0], [0, -8, -8, 6, -6, 0, 8, -9], [-8, 6, 3, 8, 5, -6, -3, 5], [14, 0, 0, 0, 6, 0, -11, 14], [0, 9, 9, -7, 0, -7, 0, 7], [-2, -13, 2, 2, -1, 13, -2, -1], [0, 6, 10, -3, 3, 0, -6, 1], [6, -10, 0, -12, 6, 6, 6, 0], [-9, 3, 3, -3, 0, 7, 0, 3], [18, 0, -6, 0, 17, 6, -11, 18], [0, -1, 5, 6, -6, 0, 1, 11], [-2, -7, 9, 2, 9, 7, -9, 9], [-18, 6, 6, -1, 0, -10, 0, 1], [11, 0, 11, 0, 14, -11, -12, 11], [3, 0, -11, 0, -6, 11, -2, 3], [-6, 4, 4, -7, 0, 1, 0, 7], [0, -11, -2, 0, 0, 0, 11, -10], [17, -1, 0, -10, 17, -11, -11, 0], [-11, 0, -13, 0, -23, 13, 6, -11], [-8, 13, 13, 5, 0, -4, 0, -5], [0, -19, -11, 2, -2, 0, 19, -16], [16, 4, 0, -18, 16, 0, 0, 0], [1, -9, -9, -13, 0, 5, 0, 13], [-10, 13, -8, 10, -19, -13, 8, -19], [-13, 14, 0, 12, -13, -16, -16, 0], [9, 0, -4, 0, -5, 4, -9, 9], [-16, 3, 3, 8, 0, -8, 0, -8], [-14, -2, -8, 14, -15, 2, 8, -15], [0, 1, 0, -10, 0, 6, 6, 0], [0, 16, 21, -9, 9, 0, -16, 21], [-15, 6, 6, 5, 0, 10, 0, -5], [14, -8, -8, -9, 0, -6, 0, 9], [0, -5, -17, 21, -21, 0, 5, -14], [0, -8, 1, 26, -26, 0, 8, -21], [-4, 8, 8, 4, -9, -8, -8, -9], [-14, 0, 5, 0, -4, -5, 2, -14], [-21, 0, -14, 0, -6, 14, 11, -21], [-7, 6, 6, 7, 5, -6, -6, 5], [2, -10, -3, -2, 14, 10, 3, 14], [7, 0, -17, 0, -18, 17, 11, 7], [0, 4, -3, 5, -5, 0, -4, -22], [6, 10, 0, 17, 6, -17, -17, 0], [-3, 14, -4, 3, 4, -14, 4, 4], [-3, 0, -5, 0, -4, 5, -4, -3], [-5, -9, 0, -22, -5, 9, 9, 0], [17, -17, -3, -17, 7, 17, 3, 7], [-17, 10, 10, -12, 0, -17, 0, 12], [0, -2, 3, -6, 6, 0, 2, 11], [-9, 0, -13, 0, -12, 13, 6, -9], [3, 8, 8, 13, 0, -20, 0, -13], [0, 8, -10, -5, 5, 0, -8, 10], [19, 0, -11, 0, 4, 11, 3, 19], [-13, -2, 0, 26, -13, 7, 7, 0], [3, -4, -5, -3, -9, 4, 5, -9], [12, -8, -8, -5, 0, 31, 0, 5], [-19, -22, -28, 19, -21, 22, 28, -21], [21, 0, 5, 0, 13, -5, -17, 21], [-5, 3, -4, 5, -7, -3, 4, -7], [0, -6, -6, 7, -7, 0, 6, -12], [2, -3, 0, -18, 2, 2, 2, 0], [-5, 0, -10, 0, -8, 10, -4, -5], [-25, 6, 6, 20, 0, -10, 0, -20], [-13, 0, 0, 14, 0, 0, 0, -14], [-14, 27, 0, 8, -14, -4, -4, 0], [0, -3, 22, -18, 18, 0, 3, 26], [11, 6, 6, -19, 0, 8, 0, 19], [-4, -16, 0, 5, -4, 23, 23, 0], [18, 20, 13, -18, 27, -20, -13, 27], [0, -6, -2, -23, 23, 0, 6, 25], [-8, 0, -3, 0, 7, 3, 22, -8], [-20, 10, 10, 6, 0, -20, 0, -6], [1, 0, 6, 0, 30, -6, 7, 1], [15, -19, 0, -8, 15, 15, 15, 0], [0, -16, -11, 8, -8, 0, 16, -20], [-2, -20, -19, 2, -9, 20, 19, -9], [10, 10, 0, 12, 10, -28, -28, 0], [-3, -3, 0, -24, -3, 8, 8, 0], [21, 0, 0, 4, 0, 6, 0, -4], [14, -13, -5, -14, 20, 13, 5, 20], [0, -4, -1, -2, 2, 0, 4, -18], [0, 6, 3, 6, -6, 0, -6, -6], [0, -2, 13, -2, 2, 0, 2, 5], [17, -18, -3, -17, 0, 18, 3, 0], [4, 0, -19, 0, -28, 19, 13, 4], [9, 8, 0, 13, 9, -25, -25, 0], [2, -11, 0, 10, 2, -2, -2, 0], [-33, 0, 6, 0, -25, -6, 9, -33], [15, 14, 4, -15, 5, -14, -4, 5], [0, -3, -7, -3, 3, 0, 3, -4], [27, -15, -15, 1, 0, 9, 0, -1], [31, -3, -3, 9, 0, -18, 0, -9], [0, 11, -11, 26, -26, 0, -11, -13], [-9, -13, -13, 9, -10, 13, 13, -10], [8, -8, -8, 5, 0, -9, 0, -5], [-2, 46, 34, 2, 25, -46, -34, 25], [-15, 0, 3, 0, -19, -3, 29, -15], [0, 0, 6, -30, 30, 0, 0, 21], [-11, 20, 0, 23, -11, -16, -16, 0], [0, 6, 15, -28, 28, 0, -6, 15], [0, 2, 7, 18, -18, 0, -2, -8], [-9, 4, 0, -16, -9, -9, -9, 0], [0, 4, 18, -22, 22, 0, -4, 9], [-30, 15, 15, 0, 0, -3, 0, 0], [6, 0, 23, 0, 18, -23, -15, 6], [-2, -1, -1, -14, 0, -8, 0, 14], [-18, 20, 22, 18, 12, -20, -22, 12], [-10, 5, 0, -9, -10, -11, -11, 0], [3, 0, 5, 0, 28, -5, 10, 3], [-34, 4, 4, 10, 0, 7, 0, -10], [-6, 0, 6, 0, 18, -6, 0, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_165_m_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_165_2_m_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_165_2_m_a(:prec:=8)
    chi := MakeCharacter_165_m();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_2_m_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_165_2_m_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_165_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![121, -165, 177, -85, 24, 5, 3, 0, 1]>],Snew);
    return Vf;
end function;