
// Make newform 855.2.k.h in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_855_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_855_k_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_855_2_k_h();".
// 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_855_2_k_h();".
// 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, 12, 31, -14, 26, -3, 6, -1, 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], [-1215, 14317, -3618, 3051, -911, 729, -189, -26], [1947, 483, 989, 3314, 276, 529, 20, 115], [-11694, -567, -1161, -2599, -324, -621, 161, -135], [6390, -17152, 19028, -16046, 3534, -3834, 994, -649], [6573, 16249, 338, 12064, 193, 2845, -109, 434], [-5355, 16585, -15946, 13447, -3858, 3213, -833, 514]];
    Rf_basisdens := [1, 1, 4243, 4243, 4243, 12729, 4243, 4243];
    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_855_k();"
function MakeCharacter_855_k()
    N := 855;
    order := 3;
    char_gens := [191, 172, 496];
    v := [3, 3, 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_855_k_Hecke();"
function MakeCharacter_855_k_Hecke(Kf)
    N := 855;
    order := 3;
    char_gens := [191, 172, 496];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 1, 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 := [[0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [-1, 0, 1, 0, 0, 0, -1, 0], [0, 0, -1, -2, 0, 0, 1, 0], [-2, -1, 0, -1, 0, 2, 1, 0], [0, -2, 0, 0, 0, 0, 0, -1], [1, -1, 2, 0, 1, 0, 0, 0], [0, -1, 0, -1, -1, 0, -1, 1], [0, -1, 0, -1, 2, 0, -2, -2], [-1, 0, 1, -1, -3, 0, -1, 0], [-1, 0, -1, 0, -2, 0, 1, 0], [0, 2, 3, 0, 0, -3, 0, -2], [0, -4, -1, 0, 0, 1, 0, 1], [-3, -2, 0, -2, 2, 3, 3, -2], [0, 2, 0, 2, 3, 0, -2, -3], [0, -5, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 4, -1, 2, -4], [-2, -2, 0, -2, -2, 2, 0, 2], [0, -1, 0, 0, 0, 6, 0, 3], [0, 3, -1, 0, 0, 4, 0, -1], [0, 1, 1, 0, 0, -5, 0, -2], [0, 0, -2, 2, -1, 0, 2, 0], [3, 3, 0, 3, -2, -3, -3, 2], [0, -5, -4, 0, 0, 1, 0, 0], [-3, 1, 0, 1, -1, 3, -3, 1], [2, 0, -1, -3, 1, 0, 1, 0], [-6, 0, 1, 1, -5, 0, -1, 0], [0, -1, -2, 0, 0, -8, 0, 1], [3, 0, -2, -3, 2, 0, 2, 0], [-8, 2, 0, 2, 2, 8, 2, -2], [0, 0, -1, 0, 0, -3, 0, -8], [-3, -4, 0, -4, 1, 3, 7, -1], [4, -5, 0, -5, -4, -4, -1, 4], [0, 0, -4, 0, 0, -3, 0, 7], [5, 0, -4, -4, -4, 0, 4, 0], [0, -1, -2, 0, 0, 1, 0, 1], [2, 0, -6, 4, -1, 0, 6, 0], [-3, -6, 0, -6, 0, 3, 3, 0], [0, 6, -7, 0, 0, -3, 0, -4], [3, 0, 4, -4, 5, 0, -4, 0], [10, 7, 0, 7, -2, -10, -1, 2], [-3, 0, -5, -1, 4, 0, 5, 0], [0, -4, 2, 0, 0, -8, 0, 3], [15, 0, 3, -2, 6, 0, -3, 0], [-2, 3, 0, 3, 3, 2, 4, -3], [0, 1, 7, 0, 0, -11, 0, -2], [0, 1, -6, 0, 0, -11, 0, -6], [-18, 0, -2, -2, 4, 0, 2, 0], [-4, 0, 1, -4, 2, 0, -1, 0], [0, -5, -6, 0, 0, 9, 0, -1], [9, 0, -1, 9, -3, 0, 1, 0], [-5, -3, 0, -3, -4, 5, -7, 4], [0, -4, 0, -4, -8, 0, -1, 8], [-3, 1, 0, 1, -2, 3, 6, 2], [0, -2, 4, 0, 0, 0, 0, 0], [0, 6, -4, 0, 0, -3, 0, -6], [0, 0, 6, 0, 0, 13, 0, 5], [8, 0, -6, -2, -2, 0, 6, 0], [-9, 7, 0, 7, -3, 9, 6, 3], [0, 9, -4, 0, 0, -5, 0, -5], [12, 0, -5, -5, -3, 0, 5, 0], [0, 3, 4, 0, 0, -11, 0, -4], [9, 0, 1, -2, 2, 0, -1, 0], [-14, -1, 0, -1, -7, 14, -1, 7], [-3, 2, 0, 2, -3, 3, -9, 3], [5, 0, -2, 13, 0, 0, 2, 0], [0, 14, 4, 0, 0, 4, 0, 2], [0, -2, 3, 0, 0, -3, 0, 5], [5, 0, 6, -3, 9, 0, -6, 0], [0, 0, -6, 4, 5, 0, 6, 0], [0, 4, 6, 0, 0, 0, 0, 5], [-8, 3, 0, 3, -6, 8, 7, 6], [11, 0, 5, 6, -9, 0, -5, 0], [8, 0, -2, -5, -3, 0, 2, 0], [0, -2, -1, 0, 0, -3, 0, -3], [12, -8, 0, -8, 0, -12, 5, 0], [0, -8, 3, 0, 0, 13, 0, 0], [0, -9, 6, 0, 0, 6, 0, 3], [13, 0, 0, 0, 1, -13, -2, -1], [-3, 0, 3, 1, 5, 0, -3, 0], [0, -14, -7, 0, 0, 4, 0, -2], [-18, 3, 0, 3, -4, 18, -4, 4], [7, -1, 0, -1, -4, -7, -2, 4], [0, -6, 1, 0, 0, 10, 0, 4], [-15, 4, 0, 4, 0, 15, 5, 0], [9, 0, 2, 4, -7, 0, -2, 0], [-10, 0, 4, -4, 5, 0, -4, 0], [0, -3, 2, 0, 0, -6, 0, 4], [20, 0, 3, 11, -7, 0, -3, 0], [18, 0, -9, 9, 0, 0, 9, 0], [6, -12, 0, -12, 7, -6, -3, -7], [14, 0, 0, -12, -5, 0, 0, 0], [0, 0, 4, 0, 0, 3, 0, -5], [0, 4, 9, 0, 0, 13, 0, 7], [-9, 7, 0, 7, 8, 9, -2, -8], [-18, 0, 0, 0, 1, 18, 6, -1], [12, 0, 5, 2, 1, 0, -5, 0], [-5, -7, 0, -7, 10, 5, -2, -10], [-32, -3, 0, -3, -5, 32, -6, 5], [-8, 2, 0, 2, -5, 8, 6, 5], [15, 6, 0, 6, 3, -15, -11, -3], [6, 0, 1, -13, 2, 0, -1, 0], [-3, 0, 1, -13, -2, 0, -1, 0], [-7, 0, -2, -2, 6, 0, 2, 0], [2, 0, 8, 10, -1, 0, -8, 0], [0, 9, 0, 0, 0, -9, 0, -1], [-6, 4, 0, 4, -2, 6, -10, 2], [6, 5, 0, 5, -8, -6, -1, 8], [5, 0, -1, -3, -8, 0, 1, 0], [14, 0, 7, 1, 7, 0, -7, 0], [0, 13, -3, 0, 0, -20, 0, 5], [-6, -3, 0, -3, -5, 6, -1, 5], [-19, 0, 0, -9, -2, 0, 0, 0], [-17, -9, 0, -9, -12, 17, -9, 12], [0, -3, 9, 0, 0, -9, 0, 5], [0, -4, 9, 0, 0, 7, 0, 10], [-9, 0, -1, 4, 11, 0, 1, 0], [-3, 0, 0, 1, 2, 0, 0, 0], [-18, -9, 0, -9, 11, 18, 3, -11], [10, -1, 0, -1, -2, -10, 6, 2], [23, 0, -10, 3, 10, 0, 10, 0], [3, 0, 9, 6, 5, 0, -9, 0], [-9, 0, -2, -9, -10, 0, 2, 0], [14, 0, 12, 9, 2, 0, -12, 0], [0, 3, 2, 0, 0, 3, 0, 2], [-2, 7, 0, 7, 0, 2, -14, 0], [0, -6, 0, 0, 0, -3, 0, 8], [0, 2, -5, 0, 0, 7, 0, -5], [-7, 0, -10, -5, -2, 0, 10, 0], [0, -17, 1, 0, 0, 1, 0, 1], [0, -4, 1, 0, 0, 3, 0, -1], [4, 11, 0, 11, 4, -4, 0, -4], [0, -3, -9, 0, 0, 10, 0, 0], [-9, 0, 7, -2, -1, 0, -7, 0], [1, 1, 0, 1, 23, -1, 9, -23], [21, -7, 0, -7, 11, -21, -4, -11], [2, 0, 14, -6, 7, 0, -14, 0], [15, 0, 16, 11, -4, 0, -16, 0], [-21, 0, 9, -11, -9, 0, -9, 0], [1, -12, 0, -12, 0, -1, 14, 0], [-6, 14, 0, 14, 9, 6, 1, -9], [4, 4, 0, 4, 16, -4, 12, -16], [3, -7, 0, -7, -18, -3, 0, 18], [11, 0, -3, 0, 19, 0, 3, 0], [0, 0, 8, 0, 0, -30, 0, -7], [0, -9, -1, 0, 0, 4, 0, -17], [0, -2, -15, 0, 0, 3, 0, 2], [4, 11, 0, 11, 1, -4, 6, -1], [-9, 0, -1, 14, 9, 0, 1, 0], [0, 2, -13, 0, 0, -11, 0, 8], [15, 0, 5, -1, 14, 0, -5, 0], [13, -10, 0, -10, -9, -13, -3, 9], [3, -7, 0, -7, 3, -3, 14, -3], [0, -10, -6, 0, 0, -8, 0, 4], [21, 0, -2, 12, 4, 0, 2, 0], [17, 0, 0, 1, 7, 0, 0, 0], [0, -15, 0, 0, 0, 6, 0, 13], [-11, -9, 0, -9, 9, 11, -8, -9], [-27, -11, 0, -11, -8, 27, -1, 8], [0, -9, -2, 0, 0, 3, 0, -7], [0, 7, -2, 0, 0, 21, 0, 2], [0, 4, -4, 0, 0, -14, 0, 8], [0, -7, 8, 0, 0, 3, 0, -9], [6, 0, -3, -1, 3, 0, 3, 0], [0, 6, -3, 0, 0, 15, 0, 7], [0, 4, 10, 0, 0, -14, 0, -16], [4, 13, 0, 13, 10, -4, -3, -10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_855_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_855_2_k_h();".
// 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_855_2_k_h(:prec:=8)
    chi := MakeCharacter_855_k();
    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_855_2_k_h();".
// 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_855_2_k_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_855_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![36, 6, 37, 6, 31, 4, 7, -1, 1]>,<7,R![-8, -15, -1, 4, 1]>],Snew);
    return Vf;
end function;