
// Make newform 605.2.g.j in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_605_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_605_g_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_605_2_g_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_g_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 := [25, 25, 10, -5, 21, -11, 6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [375, 43855, -84434, 40069, -18164, 7489, -2159, 555], [35160, -68120, 27673, -14943, 9063, -6608, -1002, -970], [-13875, -13500, 38305, -81659, 28414, -12059, 4159, -1604], [-90925, -19395, 35785, -82042, 21007, -19912, 2252, -2052], [-12035, -4035, 72554, -82441, 50856, -17466, 6691, -2667], [66675, 54640, 22635, 59219, -26434, 21519, -1464, 4024]];
    Rf_basisdens := [1, 1, 94655, 94655, 94655, 94655, 94655, 94655];
    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_g();"
function MakeCharacter_605_g()
    N := 605;
    order := 5;
    char_gens := [122, 486];
    v := [5, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_605_g_Hecke();"
function MakeCharacter_605_g_Hecke(Kf)
    N := 605;
    order := 5;
    char_gens := [122, 486];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 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 := [[-1, 0, 1, 1, 0, 1, 1, 0], [0, -1, 0, 0, 0, 1, 0, 1], [0, 0, 0, -1, 0, 0, 0, 0], [-1, -1, 0, 1, 0, 1, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 1, 1, 0, 1, 3, 0], [0, 0, 0, -1, 1, 0, 0, 1], [0, -1, -4, -1, 0, 1, -1, 1], [-5, -1, 3, 0, -3, 3, 3, 1], [-3, 0, 0, 3, -1, 0, 3, 0], [-3, 0, 1, 1, 0, 1, 3, -2], [-2, -3, 0, 2, 1, 3, 3, 0], [0, 4, 3, 0, -3, -1, 0, -1], [5, 1, -2, 0, -2, 2, -2, -1], [0, -2, -1, -1, -1, 3, -1, 3], [2, 0, -2, -2, 0, 1, -2, -1], [3, -3, 0, -3, 2, 3, 2, 0], [-1, 4, 1, -2, -2, 0, 0, -2], [0, 3, -2, 0, 0, 0, -2, -3], [-8, 5, 8, 6, -3, 0, 0, -3], [6, 1, 0, -6, -2, -1, -2, 0], [-3, 0, -2, -2, 0, 2, 3, 3], [3, 0, -3, -5, -1, 0, 0, -1], [-3, 4, 6, 0, 2, -2, 6, -4], [-1, 0, -1, -1, 0, -2, 1, 2], [2, 0, -8, -8, 0, -1, -2, -1], [2, 2, 0, -2, 4, -2, 3, 0], [0, 5, 1, 6, 2, -7, 6, -7], [2, -3, 0, 0, -2, 2, 0, 3], [0, -6, 1, 5, 7, -1, 5, -1], [12, -3, -12, -12, 2, 0, 0, 2], [1, 1, -6, 0, 4, -4, -6, -1], [-9, 3, 9, 9, -1, 0, 0, -1], [-2, -3, 0, 2, 9, 3, 4, 0], [0, -5, 0, 10, 0, 0, 0, 0], [0, -1, -1, 4, 0, 1, 4, 1], [0, -3, -1, 4, 1, 2, 4, 2], [6, 0, -7, -7, 0, 2, -6, -1], [-11, 0, 8, 8, 0, 3, 11, 0], [0, 5, -8, -13, -5, 0, -13, 0], [0, 6, -5, -2, 0, -6, -2, -6], [-2, -1, 2, -9, 3, 0, 0, 3], [-12, -1, 0, 12, -4, 1, 8, 0], [-1, 2, 1, -11, 1, 0, 0, 1], [-2, -7, -2, 0, -7, 7, -2, 7], [6, -4, -7, 0, 3, -3, -7, 4], [-14, 0, 14, 14, 0, 1, 14, -4], [0, 4, -6, 0, -2, -2, 0, -2], [8, 0, 0, -8, -7, 0, -19, 0], [14, 0, -1, -1, 0, 5, -14, 5], [-2, 0, -5, -5, 0, -1, 2, 0], [0, -2, 8, 17, 3, -1, 17, -1], [-9, 6, 13, 0, -4, 4, 13, -6], [-4, 0, 0, 0, 0, -8, 4, 1], [13, 7, 0, -13, 1, -7, -2, 0], [6, 1, -4, 0, 8, -8, -4, -1], [0, -1, 0, 3, 1, 0, 0, 1], [1, 1, 0, -1, -10, -1, -3, 0], [11, 0, -4, -4, 0, -12, -11, 2], [13, -5, -13, 3, 2, 0, 0, 2], [0, 5, -16, -9, -5, 0, -9, 0], [-11, 7, 0, 11, 0, -7, -2, 0], [6, 1, 8, 0, 5, -5, 8, -1], [0, 2, 5, -3, 3, -5, -3, -5], [5, 13, -5, -13, -8, 0, 0, -8], [1, 0, 0, 0, 0, 5, -1, 1], [-2, -4, -8, 0, 0, 0, -8, 4], [-7, -2, 0, 7, -4, 2, 10, 0], [1, -1, -1, -4, 3, 0, 0, 3], [0, -9, -5, -10, 3, 6, -10, 6], [-12, -3, -5, 0, 0, 0, -5, 3], [-4, 2, 0, 4, 8, -2, 9, 0], [-5, 4, 0, 5, 1, -4, -19, 0], [8, -2, -10, 0, -7, 7, -10, 2], [7, -6, -7, 4, 2, 0, 0, 2], [-12, 0, 4, 4, 0, 3, 12, 5], [4, 2, 0, -4, -10, -2, -3, 0], [4, -2, 7, 0, 6, -6, 7, 2], [-6, -12, 6, 1, -1, 0, 0, -1], [-14, 0, 19, 19, 0, 8, 14, -10], [-9, 5, 2, 0, 7, -7, 2, -5], [0, -7, -2, -7, 1, 6, -7, 6], [12, 0, -6, -6, 0, 1, -12, -1], [-2, -12, 0, 2, -5, 12, 3, 0], [-2, -14, 3, 0, -5, 5, 3, 14], [0, -5, -24, 3, -3, 8, 3, 8], [5, 0, -11, -11, 0, -6, -5, 5], [-11, -4, 11, 19, 0, 0, 0, 0], [-17, -3, 1, 0, -5, 5, 1, 3], [8, 2, -8, 0, -9, 9, -8, -2], [5, -6, -5, 0, 6, 0, 0, 6], [23, 4, -23, 2, 5, 0, 0, 5], [0, -3, 11, -2, 6, -3, -2, -3], [6, -3, 0, -6, 6, 3, 0, 0], [-10, 6, 0, 10, -1, -6, 1, 0], [0, -4, -4, 5, 3, 1, 5, 1], [0, 13, 20, 3, -9, -4, 3, -4], [-17, -3, 0, 17, -6, 3, -3, 0], [-1, 19, 1, -17, -6, 0, 0, -6], [-1, 0, 6, 6, 0, 2, 1, 12], [0, -15, -19, -9, 2, 13, -9, 13], [-5, 6, 0, 5, 11, -6, 25, 0], [7, 0, 14, 14, 0, 4, -7, -5], [0, -6, 7, 1, -5, 11, 1, 11], [5, -1, 2, 0, 6, -6, 2, 1], [3, 2, -3, -2, -9, 0, 0, -9], [1, -12, 0, -1, 11, 12, 21, 0], [-8, 2, 5, 0, 9, -9, 5, -2], [1, -4, -1, 23, 1, 0, 0, 1], [6, 13, 0, -6, 14, -13, 5, 0], [-1, 0, -10, -10, 0, 9, 1, 7], [0, 12, 9, -9, 1, -13, -9, -13], [-20, 11, 14, 0, 7, -7, 14, -11], [0, -11, -12, -4, 12, -1, -4, -1], [-24, -4, 0, 24, 0, 4, 27, 0], [0, 14, -5, -2, -12, -2, -2, -2], [7, -1, -7, 20, 2, 0, 0, 2], [12, 0, 7, 7, 0, -16, -12, 0], [27, 3, 0, -27, 3, -3, -16, 0], [0, 3, -23, 0, 0, 0, -23, -3], [-14, 12, 13, 0, -11, 11, 13, -12], [5, 0, 13, 13, 0, -6, -5, 7], [8, -12, -8, 9, 2, 0, 0, 2], [2, 9, -1, 0, -3, 3, -1, -9], [-5, 0, -11, -11, 0, 20, 5, 5], [0, 2, 9, 12, 11, -13, 12, -13], [-11, 7, 11, 8, -4, 0, 0, -4], [-7, -7, 0, 7, -1, 7, 6, 0], [9, -10, -24, 0, 14, -14, -24, 10], [25, 2, 0, -25, 8, -2, -17, 0], [-22, 0, 21, 21, 0, 10, 22, 1], [-4, 2, 4, -8, 8, 0, 0, 8], [0, 0, 17, -19, -8, 8, -19, 8], [-14, 0, -2, -2, 0, -4, 14, 10], [-6, 0, -11, -11, 0, -9, 6, 5], [3, 18, 10, 0, 2, -2, 10, -18], [0, -11, -10, -27, -3, 14, -27, 14], [-19, 10, 19, -10, -12, 0, 0, -12], [6, -5, -6, -26, 10, 0, 0, 10], [-30, 8, 30, 7, -10, 0, 0, -10], [0, 8, -18, 10, -3, -5, 10, -5], [1, -13, 0, -1, 6, 13, 10, 0], [-5, 0, -12, -12, 0, 6, 5, -2], [-10, 0, 14, 14, 0, -5, 10, -19], [-22, 11, 0, 22, 5, -11, 13, 0], [0, -3, -3, -5, 10, -7, -5, -7], [13, -21, -13, -6, 8, 0, 0, 8], [18, 12, -12, 0, 15, -15, -12, -12], [-12, 3, 5, 0, 7, -7, 5, -3], [21, -8, -21, -27, 6, 0, 0, 6], [0, -20, 7, 10, 20, 0, 10, 0], [-1, -4, 8, 0, 9, -9, 8, 4], [0, 1, 9, 11, -7, 6, 11, 6], [-10, 4, 0, 10, 9, -4, 14, 0], [2, 19, -2, -21, -9, 0, 0, -9], [-20, 0, 26, 26, 0, 4, 20, 6], [4, -26, -4, 14, 20, 0, 0, 20], [-4, 25, 4, -9, -8, 0, 0, -8], [11, 0, -22, -22, 0, 9, -11, -11], [9, -13, -9, -5, 18, 0, 0, 18], [7, -3, 11, 0, 8, -8, 11, 3], [0, -10, 0, 0, 1, 10, -18, 0], [-18, -1, 37, 0, 2, -2, 37, 1], [0, -3, -36, -5, 3, 0, -5, 0], [-28, 0, 12, 12, 0, -1, 28, -5], [-1, 0, 0, 1, 8, 0, 12, 0], [0, 5, -29, 0, -9, 9, -29, -5], [5, 3, 0, -5, -12, -3, -4, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_605_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_605_2_g_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_g_j(:prec:=8)
    chi := MakeCharacter_605_g();
    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_g_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_g_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_605_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -7, 117, -45, 6, 5, 3, -1, 1]>,<3,R![25, 0, 35, 25, 16, -1, 1, -1, 1]>],Snew);
    return Vf;
end function;