
// Make newform 1080.2.f.e in Magma, downloaded from the LMFDB on 27 December 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1080_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1080_f_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_1080_2_f_e();".
// 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_1080_2_f_e();".
// 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 := [9, 0, 0, 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, 3, 3];
    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_1080_f();"
function MakeCharacter_1080_f()
    N := 1080;
    order := 2;
    char_gens := [271, 541, 1001, 217];
    v := [2, 2, 2, 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_1080_f_Hecke();"
function MakeCharacter_1080_f_Hecke(Kf)
    N := 1080;
    order := 2;
    char_gens := [271, 541, 1001, 217];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 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]];
    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 := [[0, 0, 0, 0], [0, 0, 0, 0], [-1, 0, -1, 1], [0, 0, -1, 0], [2, -1, 0, 1], [0, 2, -1, 2], [0, -2, 0, -2], [1, 2, 0, -2], [0, 1, 2, 1], [-2, 1, 0, -1], [6, 0, 0, 0], [0, 2, -5, 2], [2, 3, 0, -3], [0, 0, -6, 0], [0, 2, 0, 2], [0, -1, -2, -1], [0, -2, 0, 2], [-1, 4, 0, -4], [0, 4, 5, 4], [14, -1, 0, 1], [0, -2, 1, -2], [-1, 2, 0, -2], [0, -1, 10, -1], [-12, 0, 0, 0], [0, -2, -11, -2], [12, -2, 0, 2], [0, 4, 1, 4], [0, 5, 2, 5], [0, 0, 0, 0], [0, -3, -10, -3], [0, -4, 6, -4], [12, -2, 0, 2], [0, -3, -10, -3], [5, -6, 0, 6], [0, 4, 0, -4], [-11, 2, 0, -2], [0, 0, 2, 0], [0, 0, 1, 0], [0, -4, 12, -4], [0, 2, 12, 2], [-2, 3, 0, -3], [-5, 0, 0, 0], [12, 4, 0, -4], [0, 2, 19, 2], [0, 0, 12, 0], [-7, -6, 0, 6], [7, -6, 0, 6], [0, -8, -6, -8], [0, -7, -2, -7], [0, -4, 0, 4], [0, -6, -12, -6], [-14, -3, 0, 3], [5, -8, 0, 8], [22, 3, 0, -3], [0, 2, -12, 2], [0, 2, 12, 2], [0, -10, 0, 10], [-1, -6, 0, 6], [0, -4, -12, -4], [12, 6, 0, -6], [0, -8, 6, -8], [0, 11, 2, 11], [0, 0, 16, 0], [0, -10, 0, 10], [0, -2, 19, -2], [0, 8, 0, 8], [-17, 2, 0, -2], [0, 2, -25, 2], [0, 5, -2, 5], [1, -4, 0, 4], [0, -4, 24, -4], [2, 7, 0, -7], [0, 4, 13, 4], [0, -6, -7, -6], [13, 2, 0, -2], [0, -12, 0, -12], [12, -8, 0, 8], [0, -8, -12, -8], [10, 3, 0, -3], [7, -4, 0, 4], [-12, 8, 0, -8], [19, 4, 0, -4], [-2, 9, 0, -9], [0, 8, 12, 8], [18, -4, 0, 4], [0, -2, 36, -2], [-12, 8, 0, -8], [0, 8, -12, 8], [12, -4, 0, 4], [0, 8, -13, 8], [0, 0, -12, 0], [-2, 11, 0, -11], [0, 12, 11, 12], [2, 11, 0, -11], [-8, 12, 0, -12], [0, 3, -26, 3], [2, -7, 0, 7], [-2, -9, 0, 9], [0, 4, -7, 4], [19, -8, 0, 8], [0, 0, 11, 0], [0, 2, 24, 2], [0, 0, -24, 0], [0, 2, 0, -2], [25, 6, 0, -6], [0, 2, -19, 2], [0, -11, 2, -11], [0, 3, -26, 3], [-10, -13, 0, 13], [24, 8, 0, -8], [0, -8, -11, -8], [0, -2, -19, -2], [0, -9, 10, -9], [11, -10, 0, 10], [11, 6, 0, -6], [-24, -4, 0, 4], [0, -4, -30, -4], [0, 5, 14, 5], [0, -4, 0, -4], [-24, -4, 0, 4], [-37, -4, 0, 4], [0, 6, 31, 6], [0, -6, 0, -6], [0, 3, -22, 3], [6, -8, 0, 8], [-2, -1, 0, 1], [-19, 8, 0, -8], [2, 1, 0, -1], [0, 4, 30, 4], [0, 4, -24, 4], [18, -8, 0, 8], [0, -2, 24, -2], [11, 2, 0, -2], [0, 6, 1, 6], [-26, 3, 0, -3], [-13, 12, 0, -12], [0, 3, -34, 3], [0, -4, 31, -4], [0, 10, -12, 10], [-12, 10, 0, -10], [6, -8, 0, 8], [-24, 6, 0, -6], [0, -4, -19, -4], [0, 2, 48, 2], [-19, -4, 0, 4], [36, 6, 0, -6], [0, 6, -19, 6], [0, -1, -2, -1], [-31, -6, 0, 6], [0, -12, -12, -12], [0, -10, 5, -10], [-2, 11, 0, -11], [0, -4, -11, -4], [0, 8, 24, 8], [0, 8, -5, 8], [-12, 6, 0, -6], [-18, -12, 0, 12], [22, -7, 0, 7], [0, -10, -7, -10], [-46, -3, 0, 3], [0, 2, 0, 2], [0, -12, -12, -12], [0, -8, 5, -8], [-24, -14, 0, 14], [0, 5, 2, 5], [0, 7, 38, 7], [-17, 6, 0, -6], [0, -4, 48, -4], [-5, -4, 0, 4], [0, -8, -12, -8], [22, -13, 0, 13], [-36, 8, 0, -8], [0, -22, 0, 22], [0, -6, 11, -6], [-17, -10, 0, 10], [2, -13, 0, 13], [-18, 0, 0, 0], [-34, 11, 0, -11], [0, 4, -5, 4], [-12, -12, 0, 12], [0, -12, -17, -12], [-60, 2, 0, -2], [0, 0, -24, 0], [0, 0, 12, 0], [0, 0, 0, 0], [-34, 1, 0, -1], [0, 6, -1, 6], [0, -12, 23, -12], [-7, -8, 0, 8], [-12, -4, 0, 4], [0, -14, 13, -14], [0, 2, 36, 2], [-17, -14, 0, 14], [-24, 6, 0, -6], [0, -20, 12, -20], [0, 3, 10, 3], [-23, 0, 0, 0], [0, -14, -25, -14], [0, -5, 2, -5], [0, 5, 14, 5], [12, 12, 0, -12], [-35, 2, 0, -2], [0, -2, 23, -2], [-31, -8, 0, 8], [-22, -11, 0, 11], [0, -7, -22, -7], [-1, 14, 0, -14], [0, 1, -14, 1], [10, 5, 0, -5], [25, -2, 0, 2], [0, 14, 19, 14], [-12, -8, 0, 8], [0, -8, -31, -8], [0, 13, 10, 13], [10, 9, 0, -9], [37, 4, 0, -4], [0, 24, -4, 24], [-10, -5, 0, 5], [0, 11, 34, 11], [0, -21, -2, -21], [5, -12, 0, 12], [-4, 12, 0, -12], [-12, -6, 0, 6], [0, 16, -30, 16], [0, 13, -38, 13], [1, 4, 0, -4], [0, 4, 12, 4], [-14, 11, 0, -11], [0, -4, -13, -4], [0, -6, 0, 6], [0, -2, -37, -2], [18, -4, 0, 4], [42, -4, 0, 4], [-10, -17, 0, 17], [0, 8, 1, 8], [0, 2, -48, 2], [55, -4, 0, 4], [0, -16, -12, -16], [10, -11, 0, 11], [-36, -6, 0, 6], [0, -21, 10, -21], [-25, 2, 0, -2], [0, 0, 31, 0], [-11, 8, 0, -8], [0, 12, 36, 12], [26, -9, 0, 9], [0, 8, -41, 8], [12, 26, 0, -26], [30, 4, 0, -4], [0, -7, -10, -7], [0, 12, 48, 12], [24, 2, 0, -2], [0, 0, -12, 0], [-7, 20, 0, -20], [0, -3, -34, -3], [-12, -4, 0, 4], [17, 16, 0, -16], [0, 0, -40, 0], [0, 19, -2, 19], [0, -4, 12, -4], [0, 8, -1, 8], [0, -5, 14, -5], [5, -8, 0, 8], [0, -10, -41, -10], [0, 6, 36, 6], [-18, -12, 0, 12], [-50, -1, 0, 1], [38, -11, 0, 11], [0, 28, -6, 28], [0, 14, -24, 14], [-23, 20, 0, -20], [0, 8, -49, 8], [0, 0, 26, 0], [-1, -18, 0, 18], [0, 12, -12, 12], [0, -8, 23, -8], [0, 17, 26, 17], [-12, 28, 0, -28], [-12, -8, 0, 8], [-34, 11, 0, -11], [0, -11, -14, -11], [28, 12, 0, -12], [0, -25, 10, -25], [1, 16, 0, -16], [0, -4, -11, -4], [12, -4, 0, 4], [0, -6, 13, -6], [0, 10, -60, 10], [29, -2, 0, 2], [62, 3, 0, -3], [58, 5, 0, -5], [0, -25, 10, -25], [0, 7, 22, 7], [0, 8, 0, -8], [0, -8, -48, -8], [-36, -10, 0, 10], [-55, -2, 0, 2], [0, -12, 36, -12], [58, 1, 0, -1], [0, -16, 19, -16], [0, -6, 31, -6], [0, -17, -10, -17], [-44, -12, 0, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1080_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1080_2_f_e();".
// 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_1080_2_f_e(:prec:=4)
    chi := MakeCharacter_1080_f();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1080_2_f_e();".
// 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_1080_2_f_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1080_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 0, 1]>,<11,R![-2, -4, 1]>],Snew);
    return Vf;
end function;