
// Make newform 678.2.a.j in Magma, downloaded from the LMFDB on 29 November 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_678_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_678_2_a_j();".
// 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_678_2_a_j();".
// 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 := [4, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4, 0, 1]];
    Rf_basisdens := [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_678_a();"
function MakeCharacter_678_a()
    N := 678;
    order := 1;
    char_gens := [227, 229];
    v := [1, 1];
    // 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;

function MakeCharacter_678_a_Hecke( : Kf := Kf)
    return MakeCharacter_678_a();
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, 0, 0], [-1, 0, 0], [1, -1, 1], [1, 0, 1], [2, 0, -1], [-1, 1, -1], [-1, 2, -1], [2, -2, -1], [5, 1, 0], [-2, 4, 0], [-3, 3, 2], [0, 0, -1], [4, -2, 2], [2, -1, 2], [4, -1, 0], [2, -3, -2], [-1, 1, -1], [-9, 1, 1], [6, -5, -2], [5, 2, -1], [-4, 0, 0], [-4, 2, -2], [-4, -3, -2], [-2, -2, -5], [-2, 1, -6], [-5, 1, 3], [-4, 4, -6], [-1, 5, -1], [-11, 5, 3], [-1, 0, 0], [4, -2, -3], [6, -1, 0], [11, -3, 3], [-7, -1, -1], [-10, -4, -1], [0, 0, 4], [6, -4, 6], [-7, 3, 5], [7, 3, 1], [-6, -8, 3], [3, 3, -1], [-14, -1, -4], [5, -3, 4], [-2, -4, 2], [10, 4, 0], [4, -6, 6], [-5, 5, 1], [-2, 8, 4], [10, -9, 0], [-4, -10, 2], [10, 2, 4], [-6, 0, 4], [2, 0, -1], [8, 2, 1], [-6, 8, -6], [-1, -1, -7], [-13, -1, 3], [2, 2, -8], [13, 1, 7], [3, -9, 0], [-9, 5, -1], [-9, -1, 7], [1, 3, -7], [12, -4, 2], [-5, 5, 6], [-2, 4, 3], [13, -5, -5], [3, -6, -3], [-8, 4, 0], [-8, 12, 1], [0, -2, 2], [12, -9, 8], [-1, -3, -4], [-8, -2, 2], [-4, 0, -3], [-8, 6, 10], [-6, 8, -8], [16, 5, 6], [2, -8, -2], [16, 0, 0], [-15, 7, 5], [21, 1, 1], [15, 8, 1], [24, -2, 6], [-4, -1, -6], [4, -5, -10], [17, -3, -6], [6, 8, 8], [-20, 5, -8], [15, 2, 3], [-2, -6, 10], [-4, 8, -3], [16, -10, 2], [9, 1, -7], [14, 5, 2], [16, -6, 10], [12, -7, -8], [-14, -2, -4], [-2, -10, -5], [-4, 13, 6], [8, 2, 10], [16, -2, -7], [-22, -4, 5], [-12, -8, 6], [-18, -14, 5], [16, 4, -4], [12, 1, -2], [-26, 2, -4], [-4, -12, 4], [5, -9, -4], [12, -8, -10], [12, -7, -2], [16, -2, 12], [6, -9, 6], [-4, -12, 2], [5, -2, 9], [-18, 6, 1], [4, 4, -6], [-20, 5, -12], [20, -6, 8], [-30, 6, -1], [26, 6, 2], [-10, -12, 8], [5, -13, 11], [-31, -9, 3], [10, 0, 2], [-5, -1, -11], [24, -10, -4], [19, -5, 17], [-8, -2, 10], [4, 0, 12], [-12, 0, -9], [-22, -6, 2], [14, -2, -1], [-14, 4, 4], [-6, 8, -13], [-18, 8, 1], [-12, 0, -10], [-2, -14, 4], [10, 6, 8], [0, 15, -8], [6, 3, -10], [-28, 6, -19], [10, -1, 12], [-22, 9, -12], [31, -9, 1], [14, 2, -12], [-11, -2, 9], [-20, 13, -12], [-10, 14, 2], [-10, 1, -8], [38, -6, -5], [32, 8, -5], [-7, 21, 2], [20, 4, 12], [10, 2, 12], [24, -9, 10], [8, 2, 0], [12, -14, -12], [-13, 7, 7], [-12, -4, -4], [28, 0, 6], [-15, 13, -9], [-5, 15, 7], [2, 0, 4], [9, -6, -1], [-5, -20, 7], [14, 5, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_678_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_678_2_a_j();".
// 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_678_2_a_j(:prec:=3)
    chi := MakeCharacter_678_a();
    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_678_2_a_j();".
// 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_678_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_678_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, -12, -1, 1]>,<7,R![11, -6, -2, 1]>],Snew);
    return Vf;
end function;