
// Make newform 69.2.e.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_69_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_69_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_69_2_e_b();".
// 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_69_2_e_b();".
// 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, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_69_e();"
function MakeCharacter_69_e()
    N := 69;
    order := 11;
    char_gens := [47, 28];
    v := [11, 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_69_e_Hecke();"
function MakeCharacter_69_e_Hecke(Kf)
    N := 69;
    order := 11;
    char_gens := [47, 28];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 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 := [[0, 0, 0, 0, -1, 1, 0, 0, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [1, -1, 0, -2, 2, -2, 0, -2, 2, -2], [0, 2, 0, 1, 0, 1, 0, 2, 0, 0], [-1, -1, 2, 0, 0, 0, 0, -2, 1, 1], [1, -1, 1, -1, 0, 0, 0, 1, 0, 0], [0, 0, 0, -1, 1, 0, 0, 0, 0, 3], [-2, 3, -1, 3, -2, 0, 0, 1, -1, 0], [4, -1, 2, -2, 4, -4, 2, 0, 2, -2], [-1, -1, -2, 1, -1, 2, 1, 1, 0, -3], [-3, -1, 0, 0, 0, -1, -3, 0, -3, 3], [0, -1, 1, 0, -1, 5, -1, 0, 1, -1], [-3, 3, 0, -1, -2, 5, 0, 5, -2, -1], [2, 1, -1, 1, -1, -2, 0, -2, -4, -2], [-3, 0, -7, 4, -2, 4, -4, 2, -4, 7], [2, 0, -3, -1, -7, -1, -3, 0, 2, 0], [-8, 4, -4, 8, 0, 3, -1, 2, -1, 3], [-5, 1, 0, 1, 0, 1, -5, 0, -6, 6], [4, -3, 4, 0, 4, -5, 7, -7, 5, -4], [-4, 7, -4, 0, -3, 4, -6, 6, -4, 3], [-3, -1, 2, -1, -3, 0, 3, -5, 5, -3], [4, -1, 1, -4, 0, -1, 7, 0, 7, -1], [0, 1, -5, 2, 2, 6, 2, 2, -5, 1], [6, -5, 3, -3, 5, -6, 0, -4, 5, -4], [2, -2, 0, -2, -2, 1, -1, 1, -2, -2], [0, -6, 2, -2, 2, -6, 2, -2, 2, -6], [1, 7, -4, 1, -1, 1, -1, 4, -7, -1], [3, -5, 4, -1, 4, -5, 3, 0, 5, -5], [4, -4, 3, -6, 6, -3, 4, -4, 0, 1], [0, 9, 0, 0, -7, 3, -4, 4, -3, 7], [-1, -1, 1, 0, -2, 2, 0, -1, 1, 1], [2, -7, 4, -2, 5, -2, 4, -7, 2, 0], [3, 0, 7, -2, 10, 0, 0, -10, 2, -7], [5, 0, 10, -11, 7, -11, 11, -7, 11, -10], [3, 2, -1, 3, 0, 0, -3, 1, -2, -3], [-2, 0, 0, 2, 0, 0, 2, -6, 2, 0], [-2, 1, -5, 1, -2, 0, -11, 0, 0, 11], [6, -10, 6, 0, 7, -4, -2, 2, 4, -7], [9, 6, -2, 6, -6, 2, -6, -9, 0, 3], [-2, 0, -5, 0, 0, 0, 0, 5, 0, 2], [9, -9, 0, -6, 7, 2, -1, 2, 7, -6], [-9, 9, 0, 0, -9, 9, 0, -1, -3, -1], [3, 1, 6, -7, 1, -7, 6, 1, 3, 0], [0, -1, -8, -7, 2, -2, 2, -7, -8, -1], [2, -8, 8, -2, 0, -3, 6, -1, 6, -3], [4, -10, 5, -11, 5, -10, 4, 0, 5, -5], [1, 1, -10, 1, 1, 0, 3, 4, -4, -3], [-10, 11, -8, 1, -13, 1, -8, 11, -10, 0], [5, -1, 6, -6, 1, -5, 0, 4, 7, 4], [3, 0, 3, -3, 6, 3, -3, -6, 3, -3], [3, -2, -7, 7, 2, -3, 0, 4, -2, 4], [0, -5, -3, 9, 3, 2, 3, 9, -3, -5], [-9, 10, -15, 11, -13, 13, -11, 15, -10, 9], [2, -11, 2, 0, 7, -7, 10, -10, 7, -7], [-8, 8, -1, 8, -8, 0, -1, -1, 1, 1], [-4, -5, 5, 4, 0, 11, -1, 1, -1, 11], [-10, 0, -5, 0, -7, 0, -5, 0, -10, 0], [-2, 2, 0, -5, -1, -2, 0, -2, -1, -5], [4, 0, -4, 3, -10, 3, -3, 10, -3, 4], [-7, 7, 0, 8, -14, 4, -18, 4, -14, 8], [-13, 4, -3, 14, -9, 14, -3, 4, -13, 0], [-1, 7, 1, -6, 6, -1, -7, 1, 0, 7], [10, 3, -7, -6, -7, 3, 10, 0, 0, 0], [2, -2, 1, -2, 4, -4, 2, -1, 2, -2], [0, -19, 17, -10, 15, -8, 15, -10, 17, -19], [3, -3, 0, -14, 7, -16, 11, -16, 7, -14], [0, -10, 12, -1, 3, -4, 3, -1, 12, -10], [14, -11, 12, 2, 12, -11, 14, 0, 3, -3], [14, -14, 14, 0, 11, -8, 14, -14, 8, -11], [-6, -2, 1, -2, -6, 0, -6, -5, 5, 6], [12, 3, 3, -3, 3, 3, 12, 0, 0, 0], [0, 10, -1, -1, -9, 18, -9, -1, -1, 10], [14, 0, -6, 2, -3, -3, 3, 3, -2, 6], [-1, 1, 0, 7, -1, 4, -16, 4, -1, 7], [0, 1, -5, 1, 6, -6, -1, 5, -1, 0], [4, 4, 5, 8, 5, 4, 4, 0, -1, 1], [-11, 18, -11, 0, -2, -4, -4, 4, 4, 2], [11, 4, 8, 11, -11, -8, -4, -11, 0, -1], [3, 4, -4, -3, 0, -1, -4, 4, -4, -1], [-13, 13, 0, 13, -11, 10, 1, 10, -11, 13], [-21, 21, 0, 7, -7, 2, 3, 2, -7, 7], [8, -3, 4, 5, -3, 5, 4, -3, 8, 0], [-9, -4, -8, 9, -9, 8, 4, 9, 0, 14], [4, 3, -20, 3, 4, 0, -1, 2, -2, 1], [-18, 4, -18, 0, -5, 7, -13, 13, -7, 5], [2, -9, 8, -1, 1, -8, 9, -2, 0, 8], [21, -21, 21, -23, 13, -13, 23, -21, 21, -21], [3, -7, 7, -7, 7, -3, 0, -9, -7, -9], [6, 0, 21, -4, 1, 4, -4, -1, 4, -21], [-4, -7, 13, -13, 7, 4, 0, 1, 5, 1], [-2, 11, -16, 9, -7, 9, -16, 11, -2, 0], [7, 10, -9, 10, 7, 0, -6, 5, -5, 6], [3, -7, 22, -7, 3, 0, 0, -2, 2, 0], [-2, -11, -2, -1, -2, -11, -2, 0, 7, -7], [-11, 0, 6, -3, -4, -3, 6, 0, -11, 0], [9, -8, 0, 0, 8, -9, 0, -13, -6, -13], [-6, -7, 5, -5, 7, 6, 0, 11, 1, 11], [-9, 1, -11, 2, -11, 1, -9, 0, -1, 1], [6, -2, 2, 8, -8, -2, 2, -6, 0, -9], [9, -9, 8, 6, 7, -7, -6, -8, 9, -9], [0, 0, 0, 9, 6, 7, 8, 7, 6, 9], [2, -2, 0, 3, 14, 5, -1, 5, 14, 3], [-12, 11, -6, 5, -19, 19, -5, 6, -11, 12], [3, -24, 17, -7, 7, -17, 24, -3, 0, 1], [10, -8, 1, -8, 10, 0, -5, -4, 4, 5], [-18, 28, -18, 0, -5, 11, -5, 5, -11, 5], [-25, 29, -16, 20, -13, 13, -20, 16, -29, 25], [5, -5, 0, 4, -6, 4, -3, 4, -6, 4], [-3, 0, -4, 11, 6, -7, 7, -6, -11, 4], [1, -1, -20, 20, 1, -1, 0, 9, 5, 9], [0, 10, -4, 3, -2, 31, -2, 3, -4, 10], [14, 0, -6, -4, -6, 0, 14, 0, 12, -12], [7, 10, 11, 10, 7, 0, -11, -1, 1, 11], [12, 10, 12, 0, 1, 6, -9, 9, -6, -1], [-17, 15, -15, 17, 0, 23, -5, -7, -5, 23], [-6, 2, -7, 7, -2, 6, 0, -3, -7, -3], [1, 0, 12, -11, 9, -9, 9, -9, 11, -12], [-18, 10, -1, 1, -10, 18, 0, -1, 13, -1], [0, 11, -12, -6, -5, -14, -5, -6, -12, 11], [-22, 4, -21, 18, -21, 4, -22, 0, -16, 16], [-15, 21, -8, 17, -17, 8, -21, 15, 0, 7], [15, -8, 6, -30, 30, -6, 8, -15, 0, -28], [-6, 9, -9, 6, 0, 2, 2, 29, 2, 2], [3, 14, 2, -2, 5, -2, 2, 14, 3, 0], [-13, 0, -17, -5, 2, 0, 0, -2, 5, 17], [19, -25, 34, -14, 37, -37, 14, -34, 25, -19], [-22, 8, -16, 8, -22, 0, -4, 7, -7, 4], [9, -9, 7, -19, 19, -7, 9, -9, 0, -4], [0, -13, 11, -23, 10, -18, 10, -23, 11, -13], [11, -8, -2, 2, 8, -11, 0, -22, 21, -22], [11, -16, 17, -17, 16, -11, 0, -13, -8, -13], [22, -3, -2, 3, 6, 3, -2, -3, 22, 0], [-5, 4, -2, 12, -2, 4, -5, 0, -18, 18], [-15, 19, -15, 0, -24, 16, -24, 24, -16, 24], [-6, 13, -6, 0, -24, 5, -11, 11, -5, 24], [21, -16, 16, -21, 0, -4, 11, -22, 11, -4], [0, 14, -20, 18, -4, 22, -4, 18, -20, 14], [-3, 3, 0, 8, 4, 4, -8, 4, 4, 8], [-15, -12, 13, -15, 13, -12, -15, 0, 6, -6], [-2, -2, 19, -2, -2, 0, 9, -19, 19, -9], [-20, 10, 3, -11, 11, -3, -10, 20, 0, 13], [-14, 4, 0, 21, -12, 21, 0, 4, -14, 0], [-3, 3, 0, -3, 2, 18, 21, 18, 2, -3], [10, 0, -4, -6, -5, -19, 19, 5, 6, 4], [-27, 0, 3, 5, -4, 16, -16, 4, -5, -3], [7, -10, 10, -20, 7, -7, 20, -10, 10, -7], [-4, 21, -4, 0, -10, -3, -6, 6, 3, 10], [-1, -30, 8, -13, 13, -8, 30, 1, 0, -4], [-27, 12, -11, 3, -11, 12, -27, 0, -9, 9], [1, -2, 3, 5, 8, -8, -5, -3, 2, -1], [-8, 18, -10, 10, -18, 8, 0, -10, -21, -10], [-21, 26, -2, 16, -4, 16, -2, 26, -21, 0], [0, 13, -5, 0, -30, -1, -30, 0, -5, 13], [15, -13, 13, -15, 0, -17, -4, 3, -4, -17], [2, 8, -8, -2, 0, 17, -13, 11, -13, 17], [0, -3, -3, 14, -15, 7, -15, 14, -3, -3], [22, 0, 15, -11, 16, -3, 3, -16, 11, -15], [0, 5, 1, 4, 8, 6, 8, 4, 1, 5], [2, -17, 15, -18, 18, -15, 17, -2, 0, -16], [-13, -1, -13, 0, -7, 22, -4, 4, -22, 7], [8, -18, 5, -18, 8, 0, 11, 13, -13, -11], [10, 6, -6, -10, 0, -6, 14, -20, 14, -6], [6, 0, 13, 7, 0, -18, 18, 0, -7, -13], [22, -22, 0, -3, 3, -29, 5, -29, 3, -3], [-4, 21, -23, 34, -17, 17, -34, 23, -21, 4], [1, 4, -6, 6, -6, 6, -4, -1, 0, -19], [-4, -24, -4, 0, -2, -8, 9, -9, 8, 2], [14, -1, 6, 19, 6, -1, 14, 0, 2, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_69_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_69_2_e_b();".
// 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_69_2_e_b(:prec:=10)
    chi := MakeCharacter_69_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(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_69_2_e_b();".
// 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_69_2_e_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_69_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 8, 31, 39, -18, -23, 47, -31, 16, -4, 1]>],Snew);
    return Vf;
end function;