
// Make newform 1078.2.e.q in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1078_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1078_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_1078_2_e_q();".
// 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_1078_2_e_q();".
// 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], [-1, -2, 2, -1], [-1, 6, -2, 1], [2, 0, 0, 1]];
    Rf_basisdens := [1, 2, 2, 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_1078_e();"
function MakeCharacter_1078_e()
    N := 1078;
    order := 3;
    char_gens := [199, 981];
    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_1078_e_Hecke();"
function MakeCharacter_1078_e_Hecke(Kf)
    N := 1078;
    order := 3;
    char_gens := [199, 981];
    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+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, 1, 0, 0], [1, 1, 1, 0], [0, 1, 1, 1], [0, 0, 0, 0], [-1, -1, 0, 0], [-1, 0, 0, -1], [2, 2, 2, 0], [0, -5, 1, 1], [0, 4, 0, 0], [0, 0, 0, 2], [-2, -2, 0, 0], [0, -2, -4, -4], [2, 0, 0, -2], [-6, 0, 0, -2], [0, -2, 0, 0], [-4, -4, 2, 0], [-5, -5, -1, 0], [0, -3, -1, -1], [2, 2, 6, 0], [2, 0, 0, 2], [-4, -4, 4, 0], [0, 0, 0, 0], [-1, 0, 0, -5], [0, 10, 0, 0], [8, 0, 0, 2], [3, 3, 5, 0], [0, -6, 4, 4], [0, -2, -2, -2], [10, 10, 0, 0], [4, 0, 0, -2], [-12, 0, 0, 0], [0, 7, 1, 1], [2, 2, -8, 0], [-15, 0, 0, -3], [0, 0, 10, 10], [-12, -12, 0, 0], [-3, -3, -7, 0], [0, -6, 6, 6], [-2, 0, 0, 6], [0, -1, 1, 1], [0, 0, -4, 0], [7, 0, 0, 1], [0, 2, 2, 2], [6, 6, -4, 0], [18, 0, 0, 0], [-10, -10, 4, 0], [-18, 0, 0, 2], [4, 0, 0, -2], [17, 17, -1, 0], [0, 15, -1, -1], [0, -6, 4, 4], [20, 0, 0, 0], [-2, -2, 6, 0], [27, 0, 0, 1], [0, -2, -4, -4], [-4, -4, -4, 0], [25, 25, 1, 0], [0, -8, -4, -4], [-8, -8, -2, 0], [2, 0, 0, 12], [1, 1, 7, 0], [-11, 0, 0, -7], [3, 0, 0, -13], [8, 8, -6, 0], [0, 24, 2, 2], [0, -22, 4, 4], [0, -8, 4, 4], [18, 0, 0, 0], [2, 2, 2, 0], [15, 0, 0, 3], [26, 26, 4, 0], [0, 0, -12, -12], [-8, -8, -6, 0], [0, -6, 0, 0], [10, 0, 0, 2], [0, -6, 8, 8], [-20, -20, 6, 0], [0, -17, 3, 3], [0, -8, -10, -10], [10, 10, 6, 0], [15, 0, 0, 1], [12, 0, 0, 2], [-32, -32, -4, 0], [24, 0, 0, 2], [0, 0, -4, -4], [0, -16, 4, 4], [10, 0, 0, 4], [0, 18, -4, -4], [27, 0, 0, -1], [-26, 0, 0, -2], [0, -7, 9, 9], [10, 10, 10, 0], [-8, -8, 4, 0], [-8, 0, 0, -4], [0, -10, 10, 10], [4, 0, 0, 0], [0, -5, -13, -13], [-12, -12, -10, 0], [0, -21, -3, -3], [0, -18, 4, 4], [8, 0, 0, -4], [2, 2, 12, 0], [-29, -29, 1, 0], [0, -10, 12, 12], [28, 28, 8, 0], [-18, -18, 4, 0], [-17, 0, 0, 5], [0, 24, 0, 0], [10, 10, -10, 0], [-8, 0, 0, -12], [0, -32, 0, 0], [-24, -24, 2, 0], [8, 0, 0, 6], [15, 15, 15, 0], [-18, 0, 0, -6], [-32, -32, -2, 0], [-21, 0, 0, 1], [2, 2, -12, 0], [0, 24, -10, -10], [30, 0, 0, -6], [-17, -17, -9, 0], [14, 0, 0, -8], [0, 3, 13, 13], [-24, -24, 4, 0], [0, -3, -7, -7], [2, 0, 0, -20], [0, 0, -2, -2], [0, -10, -12, -12], [18, 0, 0, 0], [0, 29, -9, -9], [-20, -20, 0, 0], [4, 0, 0, 8], [0, -8, 4, 4], [8, 0, 0, -14], [0, 2, 6, 6], [-30, 0, 0, -6], [-9, -9, 3, 0], [17, 17, -5, 0], [-17, 0, 0, -11], [30, 30, 4, 0], [7, 0, 0, -1], [0, 12, 6, 6], [6, 6, -18, 0], [8, 0, 0, 4], [-5, -5, 3, 0], [-10, 0, 0, 12], [-1, 0, 0, 15], [42, 42, -2, 0], [0, -5, -9, -9], [0, 14, 6, 6], [0, 18, 4, 4], [2, 0, 0, 12], [24, 0, 0, -12], [0, -22, 10, 10], [2, 2, -10, 0], [-28, 0, 0, 0], [0, -40, -8, -8], [0, 0, -18, -18], [-12, 0, 0, 4], [23, 23, 5, 0], [0, 8, -4, -4], [14, 0, 0, 4], [28, 0, 0, 8], [0, 17, -15, -15], [2, 2, -12, 0], [-14, -14, 0, 0], [-22, -22, -10, 0], [-13, -13, 5, 0], [-30, 0, 0, 4], [0, 19, 3, 3], [-10, -10, -18, 0], [27, 0, 0, 9], [0, -28, -4, -4], [26, 26, 8, 0], [-10, -10, 4, 0], [-30, 0, 0, -12], [32, 0, 0, -4], [18, 18, 12, 0], [44, 0, 0, 8], [0, 5, 19, 19], [0, -12, -20, -20], [17, 0, 0, 13], [-16, 0, 0, 18], [0, -12, -8, -8], [36, 36, -4, 0], [-15, -15, 17, 0], [-18, -18, -8, 0], [31, 31, 15, 0], [0, 30, 0, 0], [-2, -2, 10, 0], [0, 14, -8, -8], [14, 0, 0, -6], [0, -18, -6, -6], [0, -13, 23, 23], [12, 12, 24, 0], [-34, -34, 0, 0], [18, 18, 4, 0], [0, -26, 4, 4], [-52, 0, 0, -4], [0, 24, -4, -4], [0, 0, 6, 0], [2, 0, 0, 10], [0, 13, 25, 25], [-20, -20, -12, 0], [-15, 0, 0, 7], [7, 7, -1, 0], [0, 50, -4, -4], [0, -16, -4, -4], [-10, 0, 0, 0], [13, 13, 5, 0], [0, -2, 4, 4], [-3, 0, 0, 15], [24, 0, 0, 0], [0, 13, -7, -7], [0, 0, -12, 0], [0, -28, -2, -2], [12, 12, 4, 0], [-2, -2, 2, 0], [0, -2, -14, -14], [-36, 0, 0, 14], [0, -48, -10, -10], [0, 0, 0, -12], [0, 10, -12, -12], [0, -36, -4, -4], [-37, 0, 0, 3], [10, 0, 0, 4], [0, -6, 20, 20], [0, 0, 8, 0], [0, -12, -24, -24], [0, 22, -2, -2], [6, 6, -8, 0], [-5, -5, -19, 0], [12, 0, 0, -20], [-2, -2, 0, 0], [-51, 0, 0, 3], [-8, -8, -4, 0], [0, 0, -2, -2], [0, 44, -10, -10], [-10, 0, 0, 6], [-48, 0, 0, -10], [-14, -14, -18, 0], [0, -33, 13, 13], [56, 56, 6, 0], [0, 0, 10, 10], [-16, 0, 0, -4], [0, 30, 10, 10], [28, 0, 0, 2], [-57, -57, -5, 0], [20, 20, 16, 0], [-36, 0, 0, 12], [18, 0, 0, -16], [0, -8, 0, 0], [2, 2, -18, 0], [-10, 0, 0, -4], [-9, -9, 13, 0], [0, -10, -10, -10], [18, 18, 20, 0], [7, 7, -11, 0], [-7, 0, 0, -1], [0, -22, 2, 2], [-44, -44, -8, 0], [8, 0, 0, 20], [-25, -25, 5, 0], [29, 0, 0, 13], [-8, -8, 22, 0], [0, -25, -21, -21], [20, 0, 0, 18], [22, 0, 0, 16], [24, 0, 0, -12], [-44, -44, -10, 0], [0, 7, 19, 19], [2, 2, -18, 0], [26, 26, -8, 0], [0, 60, 8, 8], [28, 0, 0, -12], [0, -56, -2, -2], [0, 28, -16, -16], [50, 50, 4, 0], [0, -18, -16, -16], [0, -33, 5, 5], [-4, -4, -4, 0], [-62, -62, 6, 0], [18, 0, 0, -18], [-3, 0, 0, -27], [0, -37, -7, -7], [0, -28, 16, 16], [6, 6, -20, 0], [-72, 0, 0, 6], [50, 50, 0, 0], [20, 0, 0, 24], [-32, -32, 22, 0], [27, 27, -5, 0], [0, -66, -4, -4], [27, 0, 0, -25], [4, 0, 0, -22], [5, 5, 15, 0], [0, 32, -14, -14], [-1, 0, 0, 9], [0, 45, -1, -1], [53, 0, 0, -7], [0, 14, -6, -6], [0, -62, 4, 4], [0, 0, -8, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1078_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1078_2_e_q();".
// 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_1078_2_e_q(:prec:=4)
    chi := MakeCharacter_1078_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_1078_2_e_q();".
// 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_1078_2_e_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1078_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 8, 8, -2, 1]>,<5,R![16, -8, 8, 2, 1]>,<13,R![-4, 2, 1]>],Snew);
    return Vf;
end function;