
// Make newform 2106.2.e.bd in Magma, downloaded from the LMFDB on 11 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2106_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2106_e_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_2106_2_e_bd();".
// 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_2106_2_e_bd();".
// 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, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1], [0, 2, 0, -1]];
    Rf_basisdens := [1, 1, 1, 1];
    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_2106_e();"
function MakeCharacter_2106_e()
    N := 2106;
    order := 3;
    char_gens := [1379, 1783];
    v := [2, 3];
    // 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_2106_e_Hecke();"
function MakeCharacter_2106_e_Hecke(Kf)
    N := 2106;
    order := 3;
    char_gens := [1379, 1783];
    char_values := [[0, -1, 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 := [[-1, 1, 0, 0], [0, 0, 0, 0], [0, 1, -1, 0], [-1, 1, 2, -2], [-4, 4, -1, 1], [0, 1, 0, 0], [0, 0, 0, 4], [0, 0, 0, -1], [0, 1, 3, 0], [-8, 8, -1, 1], [0, -4, 2, 0], [-6, 0, 0, -2], [0, -3, -1, 0], [2, -2, -4, 4], [-4, 4, 2, -2], [0, 0, 0, -5], [0, -2, 7, 0], [2, -2, -3, 3], [0, 2, -6, 0], [7, 0, 0, -2], [-10, 0, 0, 2], [-11, 11, 3, -3], [-6, 6, -5, 5], [-9, 0, 0, 3], [-11, 11, 1, -1], [-4, 4, 7, -7], [0, -4, 2, 0], [5, 0, 0, -1], [-5, 0, 0, -9], [0, 1, -4, 0], [-8, 0, 0, -2], [0, -2, 4, 0], [6, -6, -4, 4], [0, 5, -3, 0], [0, -21, 1, 0], [3, -3, 6, -6], [0, 4, -1, 0], [-10, 0, 0, -2], [0, -3, -6, 0], [-2, 2, -3, 3], [6, 0, 0, 2], [6, 0, 0, 9], [2, -2, 4, -4], [0, 1, 3, 0], [-6, 0, 0, 12], [-4, 0, 0, 0], [0, 7, -7, 0], [9, -9, -12, 12], [-6, 6, -5, 5], [0, 6, -4, 0], [7, 0, 0, 6], [0, 9, 4, 0], [9, -9, -1, 1], [-3, 0, 0, -3], [0, 3, 2, 0], [21, -21, -1, 1], [-18, 0, 0, -4], [-5, 0, 0, -6], [0, 0, -9, 9], [-10, 10, -12, 12], [0, -4, 2, 0], [0, -4, 0, 0], [-16, 0, 0, -4], [0, 18, 8, 0], [-23, 23, 0, 0], [-3, 3, 7, -7], [-2, 2, -2, 2], [0, 4, -8, 0], [0, 21, 1, 0], [20, -20, 2, -2], [5, -5, 9, -9], [6, 0, 0, -8], [-11, 11, 9, -9], [0, -14, -7, 0], [16, 0, 0, -11], [0, 3, 6, 0], [0, 0, 8, -8], [20, 0, 0, -8], [0, 9, -3, 0], [0, 20, -2, 0], [0, 9, -5, 0], [-13, 13, -13, 13], [33, 0, 0, -2], [13, 0, 0, 6], [28, -28, 0, 0], [0, 0, 16, -16], [-8, 0, 0, 0], [-31, 31, 5, -5], [-33, 33, -3, 3], [0, 17, 10, 0], [-13, 0, 0, -7], [-3, 3, -14, 14], [11, 0, 0, 6], [0, 14, -10, 0], [0, 2, 3, 0], [-5, 0, 0, -15], [0, 14, 2, 0], [3, 0, 0, 14], [-3, 0, 0, -13], [23, 0, 0, 7], [-34, 34, -4, 4], [-6, 0, 0, -14], [0, 37, -3, 0], [-28, 28, 0, 0], [0, 33, -1, 0], [6, 0, 0, 2], [4, -4, -6, 6], [-1, 0, 0, 5], [0, -7, -17, 0], [11, -11, 18, -18], [0, -18, 6, 0], [-4, 0, 0, 4], [0, -7, 1, 0], [2, -2, -17, 17], [25, 0, 0, 12], [29, -29, -4, 4], [0, 34, -1, 0], [-29, 0, 0, -5], [0, -12, 14, 0], [-6, 6, 2, -2], [0, -2, -2, 0], [29, -29, 10, -10], [0, 0, -5, 5], [28, 0, 0, 10], [8, -8, 21, -21], [-6, 0, 0, 11], [10, -10, -2, 2], [35, 0, 0, 1], [-40, 40, -4, 4], [0, -17, 17, 0], [-24, 0, 0, 11], [0, 11, 10, 0], [0, 6, 0, 0], [6, 0, 0, -5], [0, -24, 12, 0], [0, 12, 16, 0], [20, 0, 0, -8], [0, -44, 3, 0], [0, 6, 15, 0], [9, 0, 0, -12], [-10, 0, 0, 5], [-26, 26, -6, 6], [0, -36, -2, 0], [0, 0, 0, 12], [10, 0, 0, 1], [-8, 8, 18, -18], [-37, 37, 11, -11], [17, -17, -12, 12], [0, 13, 13, 0], [7, 0, 0, 28], [0, 15, 9, 0], [18, 0, 0, 14], [-29, 0, 0, 3], [0, 23, -3, 0], [6, -6, 4, -4], [0, 0, 20, -20], [-6, 0, 0, -22], [-12, 12, 14, -14], [30, 0, 0, 6], [0, -4, -22, 0], [-18, 18, 18, -18], [2, 0, 0, -32], [0, -11, -6, 0], [-3, 0, 0, 13], [0, -27, -1, 0], [-23, 23, 18, -18], [-3, 0, 0, -9], [-20, 20, -11, 11], [20, 0, 0, 14], [0, -32, -14, 0], [-18, 18, 6, -6], [0, -6, -14, 0], [0, 29, -7, 0], [-16, 16, -4, 4], [0, -27, -5, 0], [0, 29, 16, 0], [-8, 8, 26, -26], [-14, 0, 0, 20], [-23, 0, 0, -9], [16, -16, -6, 6], [-15, 15, -12, 12], [38, -38, 0, 0], [0, 2, -33, 0], [-34, 0, 0, 8], [0, 29, 10, 0], [50, -50, -4, 4], [-32, 0, 0, -4], [12, -12, -7, 7], [0, 43, -1, 0], [12, 0, 0, -18], [-14, 0, 0, 6], [18, -18, 5, -5], [-11, 0, 0, -5], [-35, 35, -1, 1], [-17, 0, 0, -9], [0, -15, -12, 0], [0, -20, -24, 0], [8, -8, 13, -13], [-20, 20, -24, 24], [-2, 0, 0, -20], [0, 24, -26, 0], [19, -19, 17, -17], [0, 40, -14, 0], [-34, 34, 14, -14], [-2, 0, 0, 9], [-54, 0, 0, -5], [10, 0, 0, 16], [0, -45, 1, 0], [6, -6, 12, -12], [0, -18, 6, 0], [23, 0, 0, -14], [0, -60, 1, 0], [19, -19, 1, -1], [-16, 16, -29, 29], [0, 27, -16, 0], [2, -2, -6, 6], [0, -7, 23, 0], [-11, 11, 24, -24], [34, 0, 0, 8], [0, 24, -20, 0], [0, 32, 6, 0], [0, 19, 6, 0], [0, -30, 10, 0], [13, 0, 0, -4], [0, -9, -15, 0], [-20, 20, 1, -1], [-56, 56, 4, -4], [3, 0, 0, 19], [11, -11, 1, -1], [0, 0, 4, -4], [0, 32, -14, 0], [4, 0, 0, 38], [0, -3, 14, 0], [0, 23, 26, 0], [1, -1, 1, -1], [25, -25, -18, 18], [0, -16, 18, 0], [-2, 0, 0, 14], [0, 27, 23, 0], [-4, 0, 0, -2], [-12, 12, -23, 23], [25, 0, 0, -15], [0, -45, 5, 0], [-7, 0, 0, 3], [0, -56, -10, 0], [-45, 45, 13, -13], [2, 0, 0, 36], [0, 50, -5, 0], [0, 16, -7, 0], [0, 0, 0, 0], [0, 1, -19, 0], [38, 0, 0, -2], [0, 23, 12, 0], [1, -1, 4, -4], [-10, 10, -35, 35], [-8, 0, 0, -39], [-24, 0, 0, 16], [16, -16, -24, 24], [18, 0, 0, -5], [-23, 0, 0, 21], [38, -38, 12, -12], [27, -27, 9, -9], [0, -20, 0, 0], [-2, 0, 0, 3], [0, 21, 1, 0], [41, -41, -13, 13], [7, 0, 0, 33], [-11, 11, 37, -37], [0, -36, 19, 0], [0, 12, 12, 0], [0, -10, -31, 0], [-44, 0, 0, 5], [45, -45, -5, 5], [0, 1, -9, 0], [0, 43, -10, 0], [18, 0, 0, 20], [0, -8, 29, 0], [-3, 3, 21, -21], [-29, 0, 0, -3], [9, -9, 13, -13], [0, -32, 14, 0], [0, 9, 6, 0], [-32, 32, -12, 12], [19, -19, -23, 23], [0, 54, -10, 0], [10, 0, 0, 8], [35, 0, 0, -12], [0, 4, 18, 0], [31, -31, 4, -4], [11, 0, 0, 8], [18, -18, 25, -25], [-73, 0, 0, -1], [0, -3, -1, 0], [0, -20, 26, 0], [20, -20, -26, 26], [0, 30, -19, 0], [12, -12, 12, -12], [28, -28, -3, 3], [-41, 0, 0, 15], [-56, 56, 8, -8], [0, 61, -10, 0], [-13, 0, 0, 15], [18, 0, 0, -24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2106_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2106_2_e_bd();".
// 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_2106_2_e_bd(:prec:=4)
    chi := MakeCharacter_2106_e();
    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_2106_2_e_bd();".
// 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_2106_2_e_bd( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2106_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, 4, 6, -2, 1]>,<7,R![121, -22, 15, 2, 1]>,<11,R![169, 104, 51, 8, 1]>,<19,R![-3, 0, 1]>],Snew);
    return Vf;
end function;