
// Make newform 688.2.i.e in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_688_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_688_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_688_2_i_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_688_2_i_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 := [1, 1, 2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [1, 0, 0, 1], [-1, -2, 2, -1]];
    Rf_basisdens := [1, 1, 2, 2];
    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_688_i();"
function MakeCharacter_688_i()
    N := 688;
    order := 3;
    char_gens := [431, 517, 433];
    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_688_i_Hecke();"
function MakeCharacter_688_i_Hecke(Kf)
    N := 688;
    order := 3;
    char_gens := [431, 517, 433];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [0, 0, 0, 1]];
    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, -1], [2, -2, 0, 2], [0, 2, 2, 1], [2, 0, -1, 0], [0, -1, -1, 3], [0, 5, 5, -2], [2, -2, 0, 2], [6, -1, 0, 6], [0, 0, 0, 3], [0, 0, 0, 0], [0, 3, 0, 0], [3, 0, -4, 0], [7, 0, 0, 1], [-6, 0, -3, 0], [3, -1, 0, 3], [3, 0, 5, 0], [0, 3, 3, -2], [1, -3, 0, 1], [0, 11, 11, -7], [0, 3, 3, -3], [0, -1, -1, 3], [-5, 13, 0, -5], [-3, 3, 0, -3], [-6, 0, 2, 0], [0, -2, -2, 7], [0, 12, 12, -3], [-14, 0, -4, 0], [9, 3, 0, 9], [-14, 0, 1, 0], [-8, 0, -14, 0], [3, 0, 8, 0], [-6, 0, -6, 0], [-15, 6, 0, -15], [-9, 0, 0, -9], [4, 0, -3, 0], [0, 3, 3, -10], [0, 0, 0, -14], [11, 2, 0, 11], [-7, 0, -2, 0], [0, -14, -14, 13], [0, 1, 1, -21], [-21, 4, 0, -21], [-7, 0, -6, 0], [0, -8, -8, 10], [11, 0, 8, 0], [6, 0, -2, 0], [-6, 0, -2, 0], [5, -4, 0, 5], [0, 6, 6, -12], [0, 13, 13, -3], [-18, 6, 0, -18], [0, 15, 15, -20], [0, 10, 10, 13], [18, 0, 9, 0], [0, 17, 0, 0], [-3, 0, -9, 0], [0, -3, -3, -9], [6, 4, 0, 6], [-1, -4, 0, -1], [0, 2, 2, -14], [-4, 0, 5, 0], [14, -23, 0, 14], [9, -24, 0, 9], [16, -18, 0, 16], [-12, 0, -28, 0], [0, 4, 4, -1], [0, 0, 0, 28], [2, 5, 0, 2], [-11, -6, 0, -11], [0, -2, -2, -14], [0, -14, -14, 28], [3, 10, 0, 3], [0, 0, 0, 6], [-5, 0, -20, 0], [-9, 0, 18, 0], [7, 0, 26, 0], [-13, -12, 0, -13], [0, -12, -12, -6], [17, 0, 5, 0], [7, 0, -2, 0], [0, -14, -14, 15], [7, 0, -11, 0], [25, 6, 0, 25], [0, 3, 3, -6], [0, -4, -4, 4], [-6, 24, 0, -6], [-6, 0, 6, 0], [-4, 22, 0, -4], [9, -24, 0, 9], [-28, 8, 0, -28], [-30, 10, 0, -30], [0, -7, -7, 10], [0, -14, -14, 1], [0, -3, -3, -9], [0, 4, 4, 7], [0, 6, 6, 9], [1, 13, 0, 1], [0, -24, -24, 3], [0, -8, -8, -25], [21, 0, 0, 21], [-24, 0, 9, 0], [-10, 0, -7, 0], [13, 8, 0, 13], [-27, -10, 0, -27], [0, -23, -23, 9], [0, -19, -19, 15], [0, -4, -4, -24], [33, 2, 0, 33], [10, 0, -12, 0], [-19, -3, 0, -19], [23, 0, 11, 0], [0, 6, 6, 0], [0, 15, 15, 6], [0, 5, 5, 19], [27, 0, 10, 0], [-24, 0, -9, 0], [33, 0, 7, 0], [-7, 0, 16, 0], [0, 20, 20, -3], [10, 0, 33, 0], [0, 24, 24, -25], [-35, 0, -2, 0], [18, -18, 0, 18], [40, -15, 0, 40], [0, -17, -17, 13], [3, 0, -16, 0], [7, -11, 0, 7], [-16, 0, -21, 0], [-24, 0, -30, 0], [-1, 0, 30, 0], [1, 20, 0, 1], [-4, -21, 0, -4], [0, -21, -21, 3], [0, 5, 5, -7], [6, -16, 0, 6], [-20, 0, -22, 0], [0, -25, -25, 30], [5, 2, 0, 5], [-26, 0, -19, 0], [-6, 33, 0, -6], [-11, 0, -7, 0], [-3, 34, 0, -3], [-7, 10, 0, -7], [22, -31, 0, 22], [0, 0, 16, 0], [0, -19, -19, 42], [20, 10, 0, 20], [30, 0, -12, 0], [-18, 17, 0, -18], [0, -14, -14, 15], [37, 0, 2, 0], [-16, 18, 0, -16], [-15, 0, -42, 0], [12, 0, -6, 0], [9, 0, -21, 0], [24, 0, 33, 0], [0, 2, 2, 32], [0, -7, -7, -44], [23, 2, 0, 23], [7, 0, 20, 0], [0, -16, -16, -13], [25, 0, 39, 0], [0, 20, 20, -15], [-38, -2, 0, -38], [-4, -4, 0, -4], [27, 0, 10, 0], [21, 0, 25, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_688_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_688_2_i_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_688_2_i_e(:prec:=4)
    chi := MakeCharacter_688_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(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_688_2_i_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_688_2_i_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_688_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 3, 8, 3, 1]>,<5,R![16, 8, 8, -2, 1]>],Snew);
    return Vf;
end function;