
// Make newform 1170.2.i.o in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1170_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1170_i_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_1170_2_i_o();".
// 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_1170_2_i_o();".
// 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 := [16, 4, 5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [16, -5, 5, -1], [4, 0, 0, 1]];
    Rf_basisdens := [1, 1, 20, 5];
    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_1170_i();"
function MakeCharacter_1170_i()
    N := 1170;
    order := 3;
    char_gens := [911, 937, 1081];
    v := [3, 3, 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_1170_i_Hecke();"
function MakeCharacter_1170_i_Hecke(Kf)
    N := 1170;
    order := 3;
    char_gens := [911, 937, 1081];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [0, 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 := 2;
    raw_aps := [[1, 0, -1, 0], [0, 0, 0, 0], [-1, 0, 0, 0], [0, -1, -1, 0], [1, 2, -1, 2], [0, -1, 1, 1], [0, 2, 0, 0], [0, 1, 1, 0], [0, 3, 0, 3], [4, -1, -4, -1], [-2, 0, 0, -3], [1, 2, -1, 2], [6, 4, -6, 4], [0, -1, -2, 0], [-7, 0, 0, 0], [-7, 0, 0, -1], [0, 1, 8, 0], [0, 0, -6, 0], [-4, 4, 4, 4], [0, 2, -10, 0], [0, 0, 0, 6], [10, 0, 0, 1], [4, 0, 0, 2], [-11, -5, 11, -5], [0, -2, 4, 0], [-12, 2, 12, 2], [3, 0, 0, 1], [2, 0, -2, 0], [-10, 0, 0, 4], [0, -3, 4, 0], [7, 1, -7, 1], [-1, 0, 0, 2], [0, -5, 4, 0], [0, 1, -11, 0], [0, -5, -10, 0], [-6, 0, 0, -6], [17, 0, 0, -2], [0, 1, 16, 0], [-5, 6, 5, 6], [0, -11, 3, 0], [-8, -3, 8, -3], [6, 0, 0, -2], [0, 2, 14, 0], [0, 0, 0, 0], [5, 5, -5, 5], [0, -2, -6, 0], [11, 7, -11, 7], [-7, -13, 7, -13], [0, 0, -20, 0], [-18, 0, 0, -4], [10, 0, 0, -3], [2, 0, 0, 6], [0, -2, 25, 0], [0, 2, -5, 0], [-4, -1, 4, -1], [-13, 0, 13, 0], [0, -6, 12, 0], [-4, 11, 4, 11], [0, 0, 1, 0], [-10, 0, 0, -4], [-14, -1, 14, -1], [0, -7, -7, 0], [10, 0, 0, 10], [-12, 0, 0, -12], [22, 0, 0, 6], [17, 0, 0, 5], [0, 12, -12, 0], [-6, 0, 0, 0], [0, 2, 4, 0], [4, -8, -4, -8], [-4, -14, 4, -14], [-10, 0, 0, -2], [0, -8, 0, -8], [0, -5, 18, 0], [13, 3, -13, 3], [0, 5, 8, 0], [4, 0, 0, 9], [0, -2, 17, 0], [5, -3, -5, -3], [0, -5, 11, 0], [20, 8, -20, 8], [-20, 0, 0, -8], [-4, -12, 4, -12], [0, -12, 0, 0], [-22, -2, 22, -2], [10, 0, 0, 2], [0, 11, -3, 0], [-14, -4, 14, -4], [0, 9, -16, 0], [8, 0, 0, -10], [-4, 0, 0, 14], [-32, -4, 32, -4], [0, 9, 1, 0], [-19, 1, 19, 1], [4, 0, 0, 0], [0, 9, -29, 0], [14, 17, -14, 17], [11, 0, 0, -3], [22, 5, -22, 5], [-18, 0, 0, -6], [10, 0, 0, -10], [-7, -1, 7, -1], [0, 16, -8, 0], [9, 11, -9, 11], [19, 0, 0, 3], [-22, 0, 0, -12], [-34, -4, 34, -4], [-16, 0, 0, -11], [14, 0, 0, 0], [-5, -16, 5, -16], [0, 15, -9, 0], [-23, 2, 23, 2], [0, -3, -34, 0], [-19, 0, 0, -1], [0, 4, 20, 0], [0, -1, -13, 0], [0, -4, -28, 0], [21, 9, -21, 9], [-43, -1, 43, -1], [0, 9, 8, 0], [-14, -10, 14, -10], [-32, 4, 32, 4], [34, 0, 0, 2], [0, -2, 14, 0], [19, 1, -19, 1], [16, 0, 0, 13], [0, 4, -28, 0], [0, -16, 20, 0], [15, 0, 0, 15], [31, 0, 0, -7], [1, 17, -1, 17], [-4, 11, 4, 11], [16, 5, -16, 5], [5, 15, -5, 15], [0, 7, -7, 0], [38, -3, -38, -3], [0, -11, 7, 0], [0, -3, -36, 0], [0, -2, 10, 0], [-2, -8, 2, -8], [21, 0, 0, 17], [-10, 1, 10, 1], [0, 9, 9, 0], [22, 0, 0, 4], [-28, 10, 28, 10], [0, -12, 18, 0], [-18, 0, 0, 13], [14, 0, 0, -5], [7, 0, 0, 21], [-4, 0, 0, 1], [0, -1, -10, 0], [-23, -1, 23, -1], [-2, 0, 0, -11], [-5, -18, 5, -18], [18, 23, -18, 23], [28, 0, 0, -6], [0, -6, -26, 0], [0, 2, -22, 0], [22, 0, 0, 20], [16, 0, 0, 6], [-22, -14, 22, -14], [0, -11, 34, 0], [11, 0, 0, 3], [0, -3, -9, 0], [-28, -13, 28, -13], [-29, 0, 0, -2], [-26, -15, 26, -15], [0, -17, 11, 0], [-25, 0, 0, -2], [42, 0, 0, -4], [-12, 0, 0, 8], [0, -6, 30, 0], [0, 10, 6, 0], [0, 0, -10, 0], [2, 0, 0, -15], [0, 3, -11, 0], [-40, 2, 40, 2], [20, 0, 0, 10], [0, -20, 0, -20], [38, -8, -38, -8], [4, 0, 0, -4], [21, 0, 0, -2], [-17, 0, 0, -4], [12, 0, 0, 21], [19, -2, -19, -2], [0, -3, 10, 0], [33, 0, 0, -8], [44, 0, 0, 5], [-21, -10, 21, -10], [0, -18, -2, 0], [0, 0, 22, 0], [0, -18, 8, 0], [-31, 0, 0, 1], [34, -1, -34, -1], [0, -6, -14, 0], [38, 4, -38, 4], [22, 0, 0, 1], [0, 12, -37, 0], [24, 0, 0, 9], [-4, 0, 0, 13], [0, 0, 2, 0], [0, -17, 6, 0], [-30, 8, 30, 8], [9, 0, 0, 7], [-1, -9, 1, -9], [-34, 6, 34, 6], [-24, 0, 0, -8], [0, 10, -52, 0], [-11, 15, 11, 15], [0, -13, 27, 0], [36, -8, -36, -8], [-2, 0, 0, -3], [-9, 21, 9, 21], [0, -10, 50, 0], [0, 16, -14, 0], [7, 0, 0, 8], [-48, 0, 0, -6], [0, -3, 25, 0], [23, 23, -23, 23], [-17, 0, 0, 9], [0, 20, 0, 20], [22, 0, 0, 14], [1, 0, 0, -25], [0, 8, -60, 0], [-6, -12, 6, -12], [52, 0, 0, 2], [-14, 1, 14, 1], [0, -26, 30, 0], [0, -6, 24, 0], [13, 0, 0, 2], [19, 24, -19, 24], [7, -15, -7, -15], [44, -4, -44, -4], [55, 0, 0, 7], [-20, 0, 0, -7], [5, 0, 0, -7], [0, 3, -50, 0], [-6, -20, 6, -20], [0, -13, 45, 0], [34, 2, -34, 2], [-26, 18, 26, 18], [28, 18, -28, 18], [0, 5, 47, 0], [-2, 8, 2, 8], [0, 16, 14, 0], [-36, 0, 0, 10], [0, -8, 8, 0], [29, 13, -29, 13], [0, 29, -5, 0], [5, 26, -5, 26], [-14, 5, 14, 5], [-13, -33, 13, -33], [21, 0, 0, -22], [-10, 19, 10, 19], [-39, 0, 0, -15], [0, 5, -25, 0], [0, -10, -30, 0], [-18, -18, 18, -18], [-19, 0, 0, -9], [0, 12, -26, 0], [11, 0, 0, 13], [12, 18, -12, 18], [-60, 0, 0, -4], [-30, -16, 30, -16], [0, 1, 6, 0], [0, -18, -8, 0], [0, 8, 18, 0], [5, 0, 0, 15], [0, 23, -28, 0], [0, 14, -2, 0], [18, 0, 0, 2], [-20, 0, 0, -25], [40, 16, -40, 16], [0, 1, 36, 0], [0, 8, 32, 0], [-72, 0, 72, 0], [0, 4, -6, 0], [-66, 0, 0, -8], [0, 1, 66, 0], [0, -7, -56, 0], [43, -1, -43, -1], [-16, -17, 16, -17], [16, 0, 0, -9], [12, 32, -12, 32], [42, 0, 0, -13], [18, 0, 0, -10], [2, 0, 0, -18], [-17, 0, 0, -9], [0, -5, -42, 0], [0, -7, 23, 0], [-28, 13, 28, 13], [0, 30, -24, 0], [20, -7, -20, -7], [0, 10, -7, 0], [0, 20, -2, 0], [-38, 0, 0, 3], [-46, 0, 0, -5], [77, 3, -77, 3], [16, -21, -16, -21], [20, 5, -20, 5], [0, -2, -52, 0], [34, 0, 0, -18], [16, -7, -16, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1170_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1170_2_i_o();".
// 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_1170_2_i_o(:prec:=4)
    chi := MakeCharacter_1170_i();
    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_1170_2_i_o();".
// 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_1170_2_i_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1170_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, -6, 11, 3, 1]>,<11,R![289, 0, 17, 0, 1]>,<29,R![256, -144, 65, -9, 1]>],Snew);
    return Vf;
end function;