
// Make newform 462.2.k.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_462_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_462_k_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_462_2_k_f();".
// 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_462_2_k_f();".
// 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, 0, 0, 0, -1, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_462_k();"
function MakeCharacter_462_k()
    N := 462;
    order := 6;
    char_gens := [155, 199, 211];
    v := [3, 5, 6];
    // 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_462_k_Hecke();"
function MakeCharacter_462_k_Hecke(Kf)
    N := 462;
    order := 6;
    char_gens := [155, 199, 211];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 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, 0, 1, 0, 0, 0, -1, 0], [1, 0, 0, -1, 0, 0, -1, 0], [0, 0, 0, -1, 1, -1, 0, 0], [0, 0, 0, 2, 0, 0, 0, -3], [0, 0, 1, 0, 0, 0, 0, 0], [0, 1, 0, -1, 0, 1, -3, 2], [-5, 0, -1, -1, 5, 1, 2, 1], [-2, 1, 2, -1, 1, 1, -2, -1], [2, -1, 2, -1, -1, 1, -2, 1], [0, -1, 0, 1, 0, -3, -1, -4], [1, 4, -3, 3, 1, -1, 0, 1], [0, 3, 1, 1, 0, 1, 1, 3], [-5, -1, -4, -1, 0, 4, 2, -3], [-1, 1, -2, 1, 0, -1, 1, 0], [0, 2, 0, 0, -6, 0, 0, 2], [5, 1, 2, 2, 5, 1, 0, -1], [-1, -3, -1, 2, 1, 1, 2, 1], [-6, 3, 4, 2, 3, -2, -4, -3], [3, 1, -3, 0, -3, -1, 6, -1], [0, -3, 0, 3, 0, 2, -1, -1], [4, 1, 3, -3, 4, -4, 0, 4], [0, 6, 2, 1, -2, 1, 2, 6], [9, 3, -2, 3, 0, -3, 1, 0], [0, 6, 1, -4, 1, -4, 1, 6], [1, 10, 0, -10, -2, -12, 2, -2], [0, -4, -3, 1, 0, 3, 6, 3], [-12, 4, -3, -1, 6, 1, 3, -4], [2, -1, 0, 5, -1, -5, 0, 1], [7, 2, 1, 0, -7, -2, -2, -2], [1, 12, 0, -12, -2, -6, -2, 6], [-8, -3, 0, -3, 0, -6, 0, 9], [0, 6, 3, -9, 2, -9, 3, 6], [-2, 2, 7, 2, -2, 0, 0, 0], [0, -6, 0, 6, 0, 12, 2, 6], [2, -4, -5, -8, -1, 8, 5, 4], [-4, -5, 2, -7, 4, 12, -4, 12], [5, -6, -3, -9, 5, -3, 0, 3], [0, 2, -1, -5, -5, -5, -1, 2], [-4, -1, -12, -1, 0, -2, 6, 3], [0, -11, 2, 4, -1, 4, 2, -11], [1, -1, 7, -7, 1, -6, 0, 6], [-3, -11, 0, 11, 6, 3, -1, -8], [-20, 4, 2, 2, 10, -2, -2, -4], [0, -1, -7, 3, 0, -2, 14, -2], [1, 13, 0, -13, -2, -3, -6, 10], [-2, -9, -9, -2, -2, 7, 0, -7], [2, -5, 14, -5, 0, 4, -7, 1], [5, -8, 0, 8, -10, 3, -8, -5], [-8, -12, -2, 12, 8, 0, 4, 0], [-4, -4, -2, -8, 2, 8, 2, 4], [6, -6, 6, -9, -3, 9, -6, 6], [-2, -4, 0, 4, 4, 3, -14, -1], [9, -1, -8, -6, 9, -5, 0, 5], [-4, 14, -4, 14, 0, -10, 2, -4], [0, 0, -1, -8, 2, -8, -1, 0], [4, 1, 4, -11, 4, -12, 0, 12], [-14, -2, 4, 2, 14, 0, -8, 0], [-12, -11, 16, -7, 6, 7, -16, 11], [-5, -4, 2, -5, 5, 9, -4, 9], [10, 2, 0, -2, -20, -7, 3, -5], [0, -1, 3, 6, 0, 7, 0, -7], [1, 8, -10, 8, 0, 10, 5, -18], [3, -10, 0, 10, -6, 1, 2, -9], [0, 1, -1, 0, 0, -1, 2, -1], [10, -12, 2, 0, -5, 0, -2, 12], [6, 9, -1, 18, -3, -18, 1, -9], [0, 0, -1, 1, -9, 1, -1, 0], [-12, -2, -14, -2, 0, 1, 7, 1], [-3, 7, 14, 3, -3, -4, 0, 4], [1, 0, 0, 0, -2, 12, 5, 12], [2, 12, -2, -4, -2, -8, 4, -8], [0, -6, -10, -3, 0, 3, 10, 6], [3, 3, 8, 13, 3, 10, 0, -10], [0, 7, 11, -6, 4, -6, 11, 7], [7, -14, -14, -14, 0, 9, 7, 5], [0, -11, 6, 7, 1, 7, 6, -11], [3, -3, -12, 10, 3, 13, 0, -13], [-10, -1, 3, 12, 5, -12, -3, 1], [26, 6, -4, 4, -13, -4, 4, -6], [0, 9, -3, 19, 0, 10, 0, -10], [2, -12, -8, -12, 0, 6, 4, 6], [-3, -1, -6, -1, 0, 7, 3, -6], [2, -3, 18, 5, 2, 8, 0, -8], [-3, 4, 0, -4, 6, -4, -11, 0], [-8, 3, 0, -13, 4, 13, 0, -3], [-4, 9, 0, 8, 2, -8, 0, -9], [1, -10, 0, 10, -2, 8, 15, -2], [0, 9, -9, -12, -13, -12, -9, 9], [-3, 7, -16, 7, 0, 3, 8, -10], [-4, 6, -12, 6, 0, -2, 6, -4], [0, -7, -2, 16, 16, 16, -2, -7], [10, 1, 0, -4, -10, 3, 0, 3], [-15, -4, -3, 7, 15, -3, 6, -3], [-4, -8, 0, 8, 8, 4, 6, -4], [0, 16, -2, 6, 8, 6, -2, 16], [12, 6, 0, 6, 0, 3, 0, -9], [0, -5, -11, -2, 15, -2, -11, -5], [11, 6, 17, -8, -11, 2, -34, 2], [4, 0, -3, 17, -2, -17, 3, 0], [0, -5, 10, -4, -7, -4, 10, -5], [13, -2, 8, -2, 0, 1, -4, 1], [-7, 6, -15, -4, -7, -10, 0, 10], [4, 23, 1, -12, -4, -11, -2, -11], [-34, -3, -3, 0, 17, 0, 3, 3], [0, 3, -1, -8, 0, 5, 2, 5], [-4, -6, 9, -16, -4, -10, 0, 10], [-3, -13, 6, -13, 0, 10, -3, 3], [0, -4, 10, -7, -25, -7, 10, -4], [-1, -6, 9, -21, -1, -15, 0, 15], [13, -5, 0, 5, -26, 9, 7, 4], [0, 5, 4, 6, 0, -6, -4, -5], [-10, 4, 12, 12, 10, -16, -24, -16], [16, -2, 0, 2, -32, -4, -1, -6], [4, -8, 8, 4, 4, 12, 0, -12], [21, -10, 0, -10, 0, -5, 0, 15], [4, -10, -7, -6, 4, 4, 0, -4], [-9, 0, 0, 0, 18, 7, -21, 7], [0, 15, 9, 4, 0, -19, -18, -19], [-2, -14, 22, -15, 1, 15, -22, 14], [-14, 0, 0, 0, 28, 8, -10, 8], [1, 8, 29, 5, 1, -3, 0, 3], [8, -2, -22, -2, 0, 11, 11, -9], [0, -2, -7, 0, 19, 0, -7, -2], [-5, 1, 13, -8, -5, -9, 0, 9], [-6, 19, -5, 9, 3, -9, 5, -19], [5, -13, 0, 13, -10, -1, -16, -14], [0, -7, -11, 11, -18, 11, -11, -7], [0, 7, 11, -18, -5, -18, 11, 7], [1, 19, 0, -19, -2, -11, 9, 8], [-4, -5, -29, 8, 2, -8, 29, 5], [-5, 2, -5, 11, 5, -13, 10, -13], [2, 15, 0, -15, -4, 11, -6, 26], [0, -5, -5, 12, 7, 12, -5, -5], [-3, 3, -22, 3, 0, -3, 11, 0], [0, 9, -9, -18, 1, -18, -9, 9], [-4, 0, 0, 0, 8, 8, -18, 8], [-26, 4, 10, 4, 26, -8, -20, -8], [-11, 11, -24, -7, -11, -18, 0, 18], [-11, 5, -6, 5, 0, -1, 3, -4], [12, 0, -4, -12, 12, -12, 0, 12], [8, -2, 0, 2, -16, -4, 2, -6], [18, 3, 12, 14, -9, -14, -12, -3], [-11, 12, 11, -3, 11, -9, -22, -9], [-1, 0, 0, 0, 2, -7, 2, -7], [-8, 3, 9, -7, -8, -10, 0, 10], [-7, 21, 14, 21, 0, -13, -7, -8], [2, -8, 0, 8, -4, 24, -8, 16], [-25, -4, 1, 11, 25, -7, -2, -7], [14, 6, 1, 1, -7, -1, -1, -6], [-18, 15, -16, -5, 9, 5, 16, -15], [0, 1, -1, 6, 10, 6, -1, 1], [11, 4, 4, 4, 0, -14, -2, 10], [-1, 20, 26, 20, 0, -17, -13, -3], [0, 9, -4, -13, -20, -13, -4, 9], [-24, 0, 2, -21, 24, 21, -4, 21], [-9, -19, 0, 19, 18, 17, 13, -2], [0, -10, 4, -4, -2, -4, 4, -10], [0, -14, -15, 20, -10, 20, -15, -14], [4, 18, 0, -18, -8, -25, 1, -7], [0, -22, -7, 5, 0, 17, 14, 17], [28, 0, -20, 0, -14, 0, 20, 0], [12, 8, 0, -8, -24, 1, 13, 9], [6, 12, 28, 12, 0, -10, -14, -2], [0, -24, -9, 31, -13, 31, -9, -24], [15, 14, -5, 20, 15, 6, 0, -6], [6, 12, 8, -14, -6, 2, -16, 2], [7, 5, -10, -15, -7, 10, 20, 10], [11, -2, 1, -20, 11, -18, 0, 18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_462_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_462_2_k_f();".
// 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_462_2_k_f(:prec:=8)
    chi := MakeCharacter_462_k();
    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_462_2_k_f();".
// 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_462_2_k_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_462_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, -8, 20, -8, 22, -16, 14, -4, 1]>,<17,R![141376, 150400, 107360, 40960, 11224, 2000, 260, 20, 1]>],Snew);
    return Vf;
end function;