
// Make newform 546.2.s.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_546_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_546_s_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_546_2_s_b();".
// 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_546_2_s_b();".
// 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, -1, 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_546_s();"
function MakeCharacter_546_s()
    N := 546;
    order := 6;
    char_gens := [365, 157, 379];
    v := [6, 6, 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_546_s_Hecke();"
function MakeCharacter_546_s_Hecke(Kf)
    N := 546;
    order := 6;
    char_gens := [365, 157, 379];
    char_values := [[1, 0, 0, 0], [1, 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 := [[0, 1, 0, 0], [-1, 0, 1, 0], [1, 0, -2, 1], [0, 1, 0, -1], [3, 0, 3, 0], [0, 1, 0, 3], [0, 1, -4, 1], [-4, 3, 2, -3], [-4, -2, 4, 4], [7, -2, -7, 4], [1, 0, -2, 5], [1, 5, 1, 0], [-2, -5, -2, 0], [0, -1, -7, -1], [4, 0, -8, 3], [-3, -8, 0, 4], [8, 2, -4, -2], [0, -1, -2, -1], [0, 10, 0, 0], [-2, 9, 1, -9], [-2, 0, 4, -2], [9, 0, 0, 0], [-1, 0, 2, -11], [-8, -3, -8, 0], [-6, -9, 3, 9], [-2, 4, 2, -8], [11, 6, 0, -3], [13, -4, -13, 8], [-1, 0, 2, -13], [0, -5, 1, -5], [4, 6, -4, -12], [-8, -8, 0, 4], [-4, -16, 2, 16], [0, -11, 2, -11], [0, -8, 0, 8], [-3, 0, 6, -4], [-8, -4, 0, 2], [-14, 5, 7, -5], [4, -12, 4, 0], [0, 3, -3, 3], [0, -2, 0, 4], [16, -2, 0, 1], [0, 3, 3, 3], [9, -8, 9, 0], [-1, 2, -1, 0], [0, 9, 9, 9], [-18, -4, 18, 8], [-2, -2, -2, 0], [-20, 4, 10, -4], [6, 0, -12, -7], [3, 18, 0, -9], [9, 0, -18, -11], [8, 4, -4, -4], [0, 13, -5, 13], [-6, -11, 6, 22], [9, -7, -9, 14], [0, 0, 4, 0], [5, -1, 5, 0], [0, -6, -4, -6], [1, 0, -2, -11], [-4, -8, 4, 16], [-8, 8, 4, -8], [0, 0, 0, 5], [20, -6, 0, 3], [19, -14, 0, 7], [12, 0, -24, 6], [24, -14, -12, 14], [-11, 0, 0, 0], [0, 2, 3, 2], [-8, 20, -8, 0], [10, 14, 10, 0], [-17, 0, 34, 5], [4, -10, -4, 20], [0, 7, 3, 7], [-11, -17, -11, 0], [0, 31, 0, -31], [18, 4, 0, -2], [-12, -11, 6, 11], [4, -18, 4, 0], [-18, -1, 9, 1], [-1, 9, 1, -18], [10, 0, -20, -8], [7, 1, 7, 0], [0, -8, 26, -8], [6, 10, -6, -20], [-25, 0, 0, 0], [30, 3, -15, -3], [-16, -2, -16, 0], [8, 0, -4, 0], [-5, 0, 10, -24], [12, 8, 0, -4], [8, -9, 8, 0], [-34, 0, 17, 0], [4, -6, -4, 12], [9, 0, -18, -5], [0, 4, 24, 4], [-5, -21, -5, 0], [-6, -18, 0, 9], [20, 3, -20, -6], [11, 0, -22, 11], [1, -6, 0, 3], [1, 18, 1, 0], [0, -5, -5, -5], [-7, 5, 7, -10], [17, 26, 0, -13], [-9, 0, 18, 13], [-7, 7, -7, 0], [0, 0, 0, 37], [-6, -48, 0, 24], [5, 25, -5, -50], [0, 5, 17, 5], [-16, -20, -16, 0], [-6, 19, 3, -19], [6, 0, -12, -1], [-14, 14, 7, -14], [0, -10, 0, -10], [20, -15, -10, 15], [-22, -1, 22, 2], [-13, 16, 13, -32], [0, 14, -11, 14], [-2, 26, -2, 0], [-9, -20, 9, 40], [45, -2, 0, 1], [-20, 0, 10, 0], [-4, 44, -4, 0], [-5, -24, 0, 12], [42, 15, -21, -15], [0, 17, 10, 17], [14, -12, 0, 6], [-12, 0, 24, 5], [-18, -8, -18, 0], [11, -5, 11, 0], [19, -8, -19, 16], [-11, -3, 11, 6], [-28, 6, 14, -6], [1, -45, 1, 0], [36, -12, -18, 12], [16, -5, -8, 5], [0, 21, -1, 21], [11, -1, -11, 2], [-22, 0, 44, -16], [3, -24, 3, 0], [0, 16, -6, 16], [-6, 0, 12, -6], [24, -13, -24, 26], [-20, -20, 10, 20], [-14, 0, 28, 1], [-16, -4, 0, 2], [22, -6, 0, 3], [-8, 0, 16, 2], [-18, -35, 9, 35], [-12, -16, 12, 32], [-3, -26, 0, 13], [12, 5, -12, -10], [20, 6, -20, -12], [27, 14, 0, -7], [0, 16, -15, 16], [8, -3, -4, 3], [16, 32, 0, -16], [-10, 0, 20, 10], [-19, -10, -19, 0], [0, 5, -9, 5], [-12, 0, 24, -8], [0, -22, 0, -22], [-6, -30, -6, 0], [0, 0, 0, -18], [-13, -14, 13, 28], [0, -19, -24, -19]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_546_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_546_2_s_b();".
// 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_546_2_s_b(:prec:=4)
    chi := MakeCharacter_546_s();
    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_546_2_s_b();".
// 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_546_2_s_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_546_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, 0, 8, 0, 1]>,<11,R![27, -9, 1]>],Snew);
    return Vf;
end function;