
// Make newform 605.2.a.j in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_605_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_605_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_605_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 := [1, 1, -3, -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, -1, 1, 0], [1, -2, -1, 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_605_a();"
function MakeCharacter_605_a()
    N := 605;
    order := 1;
    char_gens := [122, 486];
    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_605_a_Hecke(Kf)
    return MakeCharacter_605_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+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, 0, 1, 1], [-1, 0, 0, 0], [0, -1, -2, 1], [0, 0, 0, 0], [0, 3, 0, -2], [1, 1, 1, -3], [-5, -2, 1, 0], [2, -1, -1, 0], [-5, 2, 1, 2], [0, -5, 0, 0], [3, 3, 3, -7], [-1, 1, -3, -1], [-6, 1, 2, 0], [3, -1, -3, 0], [-6, 1, 5, 1], [2, -1, -4, 5], [-2, 2, -2, -1], [-5, -3, -4, 6], [3, -3, 0, -2], [-1, -3, -2, 0], [-9, -4, -3, 6], [1, 1, -3, 6], [1, 0, -6, 0], [5, 0, -2, 8], [-2, 3, 7, -1], [-1, 0, 4, 6], [2, -8, -3, 7], [-5, 1, 4, -6], [1, -1, 1, -2], [-5, -2, 3, 3], [-6, 1, 6, -2], [-8, 9, 4, -3], [-8, 5, 5, -7], [-1, -6, -5, 7], [-12, 2, 9, 5], [6, 3, 6, -10], [4, 2, -9, -4], [7, 5, 6, -7], [11, 1, 2, -3], [-5, 12, 2, 0], [0, 3, -8, 4], [-1, 5, 6, 0], [-7, -5, 9, 2], [-2, -5, 1, 7], [-4, -2, 3, -2], [-17, -1, -2, 12], [-6, -2, 2, 10], [-10, -2, 7, 7], [1, 1, 13, 2], [12, -13, -6, -3], [4, 7, -7, -16], [-2, 10, 0, -9], [9, 2, -1, -9], [7, -1, -3, -14], [1, 7, 0, 4], [-4, 5, 6, 4], [-7, 7, 0, -5], [-6, -2, 0, 1], [6, 8, -3, -16], [-17, 9, 10, -3], [10, -1, 2, 7], [0, -5, -7, -7], [-9, -13, -7, 16], [-6, 8, -5, -2], [6, -13, -7, 4], [-2, 20, 4, -12], [-11, -6, 6, -5], [-5, 3, 0, -1], [-10, -7, 0, 16], [-4, -17, 2, 7], [5, -6, -8, 0], [-3, -2, -11, 8], [-1, 6, 5, 11], [2, -2, 4, 9], [-16, -5, -5, 7], [11, 22, 6, -18], [-11, -2, 0, -3], [-15, -11, -3, 13], [9, 8, 2, -3], [-11, -9, 9, 5], [5, 15, 0, -17], [-2, -11, -13, 15], [8, 18, -7, -17], [0, 0, -5, 8], [-3, -17, 2, -3], [16, 0, -13, -7], [-9, -8, 6, 5], [8, -5, -3, 2], [19, -2, -9, -17], [18, 4, -6, 9], [8, -13, -1, 2], [-12, 12, -3, -1], [-15, -5, -4, 0], [10, -12, -11, -5], [38, -11, -5, 0], [3, 1, 6, -9], [-5, -13, 0, -3], [12, -2, -5, -12], [4, 2, 4, -1], [-2, 2, -1, -14], [5, 20, -5, -22], [9, 2, -9, 2], [-3, -5, 13, 20], [-6, -1, 0, -16], [26, -13, -5, -8], [-11, -10, 15, 16], [36, -2, -1, -6], [-5, 3, 3, -12], [24, -9, 0, -16], [2, 11, -13, 6], [11, -15, -11, 4], [0, 5, 7, -15], [21, -10, -17, -5], [-15, -16, -12, 20], [-21, -12, 6, 12], [9, -8, -17, -4], [1, 4, 2, -29], [9, -25, 3, 6], [-10, 7, 2, -9], [-18, -14, 9, 8], [-4, -20, 13, 7], [-19, 4, 22, -6], [-15, 11, -1, -8], [10, -2, -5, 15], [-21, 23, 9, -1], [-15, 10, 7, 2], [-19, 9, 15, -6], [19, 0, -18, -6], [18, -4, -14, 11], [-4, 0, -17, -8], [-2, -4, 6, 30], [-5, -4, 12, 13], [-24, 0, 6, 6], [-7, 7, 9, -16], [5, -18, 10, 20], [9, 19, -12, -7], [-24, 6, 8, 11], [-17, -6, 5, 7], [-3, 2, 10, -8], [-4, 17, -3, 5], [-30, -7, 0, 31], [-4, 24, -4, 1], [-26, -1, 3, 35], [-7, -5, 5, 5], [12, -20, -3, 8], [11, 10, 17, -12], [-29, 20, 7, 7], [7, -3, -11, -2], [5, 2, 8, -13], [-13, -14, -4, 14], [14, 10, 17, -15], [0, 13, -8, 19], [-11, 16, 7, 17], [2, 23, -2, -8], [8, 8, -2, -20], [24, -8, -6, -22], [-8, 14, -11, 13], [-12, 7, 13, -12], [-38, -8, 5, 4], [9, -11, -6, 19], [17, 10, -9, -13], [-12, -11, 4, 7], [-17, -7, 2, 3], [-6, 15, 17, 1], [23, 8, -14, 17], [15, 7, 1, 0], [8, 19, 2, -17]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_605_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_605_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_605_2_a_j(:prec:=4)
    chi := MakeCharacter_605_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_605_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_605_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_605_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -1, -3, 1, 1]>,<3,R![-1, 5, -6, 0, 1]>],Snew);
    return Vf;
end function;