
// Make newform 605.2.g.e in Magma, downloaded from the LMFDB on 25 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_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_605_2_g_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, 3, 5, 3, 4, -3, 5, -3, 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], [-7, -12, -7, -4, 0, -3, 2, -1], [9, 6, 19, -16, 20, -7, 0, 1], [-1, -10, 13, -16, 12, -9, 4, -1], [-5, -8, -13, -4, 8, -17, 10, -3], [-1, 6, 1, 16, -20, 23, -12, 3], [-7, -12, -11, -28, 32, -35, 18, -5]];
    Rf_basisdens := [1, 1, 8, 8, 8, 8, 8, 8];
    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], [-1, 0, 0, 1, -1, 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, -1, 0, 0, 0, 0, 0], [1, 0, -1, 0, 0, 1, 0, 1], [0, 0, 0, 1, 0, 0, 0, 0], [0, -1, 1, 0, 1, 1, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, -2, 2, 0, 0, 2], [-3, 0, 0, 1, -3, 1, 1, 0], [0, 0, 1, 0, 5, -1, 2, 0], [2, -1, -1, 0, 0, 0, 0, 0], [-5, -1, -2, 5, -3, 1, 2, -3], [0, 0, 5, 0, 0, 0, 0, 0], [-3, 0, 3, 3, 4, 0, -3, 4], [1, 0, -3, 0, 2, 3, -1, 1], [6, -2, -2, 0, 0, 1, 0, 0], [0, 0, 3, 0, 3, -3, -1, 0], [0, -5, -1, -1, 1, 0, 5, -5], [-2, 3, -1, 2, -7, -3, 1, -7], [-1, 4, 0, -2, -1, -2, -2, 0], [-5, -4, -4, 6, 0, 1, 0, -6], [2, 3, 0, -3, 2, 0, 0, 0], [-1, 1, 3, 1, -1, -1, -3, -1], [0, -3, -4, 6, -6, 0, 3, 3], [6, 4, 0, 1, 6, -3, -3, 0], [1, -6, -6, 0, 0, 6, 0, 0], [0, 2, 0, -8, 8, 0, -2, 13], [0, 7, 3, -1, 1, 0, -7, 3], [1, -4, 0, -1, -5, 4, 0, -5], [-7, 0, -3, 0, -9, 3, 8, -7], [5, -4, -4, 6, 0, 3, 0, -6], [-2, 0, -1, 0, -1, 1, -1, -2], [3, -5, 0, -5, 3, 3, 3, 0], [6, -6, -6, 2, 0, 5, 0, -2], [3, -5, 0, 8, 3, -4, -4, 0], [-8, 0, -5, 8, -15, 0, 5, -15], [7, -1, 0, -1, 7, -5, -5, 0], [-5, 0, 9, 0, 7, -9, -2, -5], [-10, 0, -6, 0, -4, 6, 3, -10], [0, 9, -2, 4, -4, 0, -9, 0], [0, 6, 5, -7, 7, 0, -6, 0], [3, 0, 2, 0, -8, -2, -1, 3], [0, 0, -2, 0, -5, 2, 12, 0], [-4, -11, 0, 0, -4, 8, 8, 0], [1, -1, 5, -1, 1, 1, -5, 1], [2, -14, 0, -7, 2, 9, 9, 0], [2, -1, -1, -7, 0, 6, 0, 7], [-4, 3, 3, -2, 0, -5, 0, 2], [0, -2, -1, 12, -12, 0, 2, 5], [10, 0, -2, 0, 4, 2, -2, 10], [-10, 9, 2, 10, -3, -9, -2, -3], [0, -13, -1, -2, 2, 0, 13, 3], [0, -6, -13, -3, 3, 0, 6, -9], [16, 0, -7, 0, 12, 7, -7, 16], [2, 0, 0, 9, 0, -10, 0, -9], [0, 1, -2, 9, -9, 0, -1, 0], [-7, 2, -1, 7, 7, -2, 1, 7], [-1, 0, 0, -4, 0, -7, 0, 4], [-4, 1, 0, 4, -4, -6, -6, 0], [-7, -7, -7, 7, -12, 7, 7, -12], [0, 0, -2, 1, -1, 0, 0, 5], [-16, 11, 0, 6, -16, -3, -3, 0], [3, 0, 10, 0, 20, -10, -9, 3], [10, 3, 1, -10, 17, -3, -1, 17], [0, 7, 7, 7, 0, -2, 0, -7], [16, 0, 7, 0, 7, -7, -13, 16], [2, -13, 0, 6, 2, 5, 5, 0], [0, 7, 13, -4, 4, 0, -7, 10], [-2, 4, 4, -12, 0, 16, 0, 12], [-11, 12, 6, 11, -16, -12, -6, -16], [-1, 3, 0, -5, -1, 0, 0, 0], [-16, 0, 0, 0, -6, 0, 7, -16], [-4, 2, 2, 7, 0, -19, 0, -7], [5, -2, 6, -5, 5, 2, -6, 5], [3, 9, -2, -3, -5, -9, 2, -5], [1, -5, -5, -11, 0, -1, 0, 11], [-9, 6, 0, -2, -9, -4, -4, 0], [0, 5, 5, -7, 7, 0, -5, -9], [-11, 16, 22, 11, 7, -16, -22, 7], [-11, 0, 0, -3, 0, -2, 0, 3], [-13, 8, 0, 15, -13, 3, 3, 0], [0, 2, 8, -3, 3, 0, -2, -6], [-11, 9, 9, 5, 0, -18, 0, -5], [-17, 0, 0, 0, -12, 0, 15, -17], [0, -13, -11, 15, -15, 0, 13, -13], [-8, 25, 18, 8, 9, -25, -18, 9], [0, 5, 5, -8, 0, -5, 0, 8], [-3, 0, -2, 0, -6, 2, -17, -3], [0, 13, 0, 7, -7, 0, -13, 9], [5, -14, 0, -9, 5, 6, 6, 0], [-8, 3, 3, -2, 0, 2, 0, 2], [19, -9, -9, -17, 0, 7, 0, 17], [-9, -10, 0, -18, -9, 6, 6, 0], [2, -12, 0, 8, 2, -1, -1, 0], [-1, 0, 3, 0, -13, -3, 12, -1], [-15, 1, 5, 15, -15, -1, -5, -15], [-10, -1, -12, 10, -5, 1, 12, -5], [0, 0, -5, 0, -38, 5, 11, 0], [-9, 0, -6, 0, -6, 6, 1, -9], [5, -13, -13, -5, 8, 13, 13, 8], [-12, 3, 0, 12, -12, -5, -5, 0], [0, 4, 2, -1, 1, 0, -4, -3], [14, 0, -1, 0, 16, 1, -2, 14], [5, -25, -20, -5, -17, 25, 20, -17], [0, -9, 2, 2, -2, 0, 9, -11], [-20, 0, 13, 0, -17, -13, 5, -20], [6, 0, 0, 16, 0, 1, 0, -16], [8, 8, 0, -26, 8, 5, 5, 0], [11, -25, -10, -11, -5, 25, 10, -5], [-36, 1, 1, 6, 0, 1, 0, -6], [12, 0, 0, 5, 12, -3, -3, 0], [24, 9, 9, -24, 8, -9, -9, 8], [0, -13, 11, 6, -6, 0, 13, -8], [-4, 0, -11, 0, -15, 11, 15, -4], [0, 7, 7, -15, 0, -2, 0, 15], [-5, 0, 17, 0, 16, -17, -10, -5], [15, -4, -16, -15, -5, 4, 16, -5], [12, 0, -6, 0, -9, 6, -12, 12], [4, -9, 0, -9, 4, 17, 17, 0], [0, -2, -4, 29, -29, 0, 2, -28], [-9, -28, -25, 9, -15, 28, 25, -15], [10, -2, -2, 9, 0, -5, 0, -9], [-18, 9, 9, 8, 0, -23, 0, -8], [0, 13, -20, 7, -7, 0, -13, -3], [-6, -18, 0, -19, -6, 22, 22, 0], [-15, -1, -1, -8, 0, 12, 0, 8], [0, 5, 2, -15, 15, 0, -5, 25], [1, 0, 9, 0, 22, -9, -23, 1], [-2, -3, 0, 15, -2, -7, -7, 0], [19, -6, 9, -19, 25, 6, -9, 25], [19, -18, -18, -6, 0, 18, 0, 6], [18, -10, 4, -18, 29, 10, -4, 29], [0, -17, 0, -8, 8, 0, 17, 12], [30, -10, 0, -2, 30, 6, 6, 0], [13, 0, -12, 0, 8, 12, -4, 13], [0, -6, 0, -6, 6, 0, 6, -18], [0, 9, 7, -16, 16, 0, -9, 23], [-5, -10, -10, -20, 0, 28, 0, 20], [-7, 0, 12, 0, 2, -12, 19, -7], [11, -2, 0, -24, 11, 8, 8, 0], [-7, 11, 0, 17, -7, -5, -5, 0], [-8, -8, 0, -3, -8, 10, 10, 0], [-5, 0, -3, 0, -1, 3, -17, -5], [-30, 7, 7, 30, 1, -7, -7, 1], [0, 4, -24, -1, 1, 0, -4, -3], [0, 3, -1, 35, -35, 0, -3, -9], [7, -10, -5, -7, 2, 10, 5, 2], [8, 0, 3, 0, 20, -3, -20, 8], [-12, -7, 0, 11, -12, 17, 17, 0], [29, -7, -7, -7, 0, -13, 0, 7], [7, -11, -11, -2, 0, 8, 0, 2], [13, 6, 0, -5, 13, -8, -8, 0], [-14, 0, -4, 0, -1, 4, 14, -14], [14, 17, 17, -15, 0, -7, 0, 15], [19, 0, 8, 0, 19, -8, 13, 19], [-11, -9, -16, 11, 6, 9, 16, 6], [8, -25, 0, -2, 8, 2, 2, 0], [0, 2, -8, 20, -20, 0, -2, -12], [-22, -2, 0, 24, -22, -6, -6, 0], [-13, -25, 0, 8, -13, 11, 11, 0], [0, 13, 7, -12, 12, 0, -13, 24], [4, -13, 0, -38, 4, 5, 5, 0], [9, -6, -6, 19, 0, -5, 0, -19], [17, -19, -10, -17, 4, 19, 10, 4], [12, -4, -4, -7, 0, 15, 0, 7], [3, 0, -2, 0, -14, 2, -7, 3], [0, -17, -15, -1, 1, 0, 17, -5], [-23, 22, 8, 23, -40, -22, -8, -40], [15, 1, 1, 0, 0, 6, 0, 0], [8, -17, -19, -8, -9, 17, 19, -9]];
    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_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_605_2_g_e(: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_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_605_2_g_e( : 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, -3, 5, -3, 4, 3, 5, 3, 1]>,<3,R![1, 0, 7, 15, 14, -15, 13, -5, 1]>],Snew);
    return Vf;
end function;