
// Make newform 6630.2.a.bk in Magma, downloaded from the LMFDB on 13 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6630_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_6630_2_a_bk();".
// 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_6630_2_a_bk();".
// 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 := [3, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4, 0, 1]];
    Rf_basisdens := [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_6630_a();"
function MakeCharacter_6630_a()
    N := 6630;
    order := 1;
    char_gens := [4421, 1327, 4591, 5851];
    v := [1, 1, 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_6630_a_Hecke(Kf)
    return MakeCharacter_6630_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]];
    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, 0], [1, 0, 0], [-1, 0, 0], [1, -1, 0], [3, 0, 0], [1, 0, 0], [1, 0, 0], [-1, 3, -1], [1, 1, 0], [4, -2, 0], [2, 0, 3], [0, -2, -1], [5, -1, 0], [-5, 2, -2], [0, 3, -3], [3, -3, 2], [5, -1, 2], [-3, 1, -1], [0, 1, 3], [0, 6, -2], [6, -5, -1], [-2, -1, -3], [3, -6, 0], [2, -4, -1], [-8, -1, 1], [9, 0, 0], [2, 0, 2], [1, 1, 1], [10, -4, 4], [6, 0, 4], [-6, 4, 2], [-6, 3, 1], [3, -3, -6], [-5, 2, -6], [-4, 5, -3], [8, 6, -2], [-3, 1, 1], [-6, 4, 2], [1, 4, 5], [9, -6, -2], [2, -1, 6], [6, -2, 2], [15, -3, 2], [4, -4, 6], [-2, 4, 0], [-8, 2, 2], [0, -2, -7], [2, 0, -5], [5, 5, -4], [5, 6, -5], [-2, -5, -5], [-2, 1, -2], [10, -1, 8], [8, 2, 2], [2, -1, 8], [-3, -6, 0], [1, 7, -2], [14, -6, -4], [-4, 9, -3], [15, 6, 1], [-2, 8, -8], [17, -1, 1], [18, -2, 8], [-2, 10, 1], [-7, 0, 6], [10, 7, 1], [8, -12, 4], [-16, 6, -7], [0, -3, -6], [-6, -2, 3], [-6, 6, 0], [12, 0, 0], [6, -11, 0], [6, -11, 4], [-6, 1, -1], [-2, 13, -5], [14, -4, 9], [-7, 6, -1], [10, -2, -2], [-8, -7, 1], [6, -12, 0], [-2, -4, -5], [24, 3, -1], [-7, 3, -12], [-12, -5, -3], [-12, 3, 1], [10, -2, -2], [-24, 4, -6], [0, 0, 0], [19, 8, -2], [-12, -6, -2], [8, 5, -1], [14, 0, 6], [-2, -8, -6], [-8, -10, 0], [-6, 12, -4], [-2, 1, 3], [8, -4, -6], [-15, -2, 5], [12, -2, -4], [-9, -11, 4], [-6, -9, 8], [14, -1, 13], [-8, 10, 2], [17, 0, 3], [8, -18, 4], [14, 2, 5], [-1, -1, -4], [-1, -1, -4], [23, -9, 6], [-41, -1, 1], [20, 0, 2], [-5, -8, 4], [24, 1, 5], [16, -10, -4], [8, 17, -4], [-4, 3, -4], [-10, -4, -13], [-10, 8, -3], [-13, 17, 5], [2, 12, -5], [19, -4, 11], [-3, 0, 2], [8, -4, 6], [14, 15, -3], [-5, -2, 1], [-13, -3, -2], [-5, -20, 1], [6, -5, 9], [-7, 3, -2], [13, -4, 13], [-6, -3, -4], [14, 12, 0], [4, 17, -6], [5, 8, -10], [-18, 4, 4], [7, -5, 5], [-5, -4, -5], [6, -6, 6], [-24, 9, -16], [-4, -18, 4], [36, -9, -4], [-3, 4, -7], [-8, 10, 0], [16, -7, -5], [10, -5, -9], [1, -4, -8], [-22, 17, 11], [-13, 3, -2], [-4, -7, -1], [-5, 14, 6], [-1, 5, 5], [-7, 18, -12], [-12, 12, 6], [-9, 19, 8], [7, 10, 10], [4, 17, -8], [29, -13, -2], [-16, 6, -6], [27, 3, 9], [-38, 4, -2], [12, -3, 8], [-2, -10, 13], [13, -20, 9], [-19, 5, -10], [10, -2, 8], [10, 5, 13], [20, -6, -4], [-3, 7, -5], [38, -10, -5], [20, -16, 12], [-31, 6, -8], [-5, -2, -13], [-37, 9, -3], [30, 13, -6], [3, -3, 13], [-2, -7, -5], [-9, -9, 13], [16, -7, -5], [4, 2, -14], [1, 2, -4], [-20, 10, -18], [-22, -18, 6], [2, 14, -12], [-24, 18, 4], [-24, 0, -5], [-32, -10, 4], [20, 21, -5], [8, 24, -6], [-10, 14, 2], [-4, -6, 4], [-5, -17, 6], [-6, -2, 16], [-2, -2, -8], [-41, 7, -7], [36, -6, 0], [24, 10, -8], [-12, -2, 0], [-6, -6, -5], [-11, -8, -13], [15, -9, 13], [-24, -2, 6], [-23, -1, 6], [-16, -12, 5], [-25, 5, 10], [-14, -2, -8], [1, 5, 4], [-8, -11, -9], [-6, 12, -6], [14, 6, -4], [-16, 12, -3], [-9, 0, 15], [27, -8, 23], [-36, 0, -13], [6, -24, 2], [-31, -6, -3], [-34, 0, 0], [14, 2, -8], [-32, -17, 4], [-4, -19, 8], [-8, -31, 11], [29, 0, 3], [-22, 8, 1], [-32, 14, -14], [-12, -9, 5], [-8, 5, 1], [14, 14, 8], [-25, 14, -10], [-30, 16, 2], [19, -20, -7], [-15, 13, -1], [-30, 16, 9], [-3, -11, 12], [-46, 8, 2], [-68, 2, 1], [8, -10, 8], [-3, 10, 9], [-18, 15, -7], [-50, 10, 4], [-9, -3, -2], [-3, 3, 0], [8, 0, -8], [-8, 5, 13], [5, -24, 6], [15, 12, 4], [23, -22, 14], [-35, -1, -4], [17, 8, 11], [2, 3, -12], [-22, 14, -7], [43, -7, 14], [2, 14, -2], [40, 10, -2], [-3, 7, 6], [-46, 2, -10], [16, 10, -2], [37, -1, 3], [-31, -9, -9], [-33, -18, 6], [-9, -14, -1], [8, -18, 10], [-6, -6, 12], [10, -28, 18], [4, -4, -8], [-31, 2, -5], [39, -11, 12], [34, 7, 3], [1, -20, 10], [-40, 9, 6], [22, 10, -8], [4, -1, -10], [-38, 11, -21], [22, -19, 16], [-5, 5, 9], [-13, 21, 0], [-17, 26, 3], [-26, -2, -17], [-22, 24, -12], [-32, -7, -7], [1, -5, -13], [-13, 2, 0], [-36, -2, -10], [-9, -27, 8], [-9, 4, -1], [52, -7, -8], [-21, -33, 12], [-39, 4, -19], [6, -21, -5], [-31, -15, 2], [-52, -1, 1], [26, 2, 12], [-24, 3, -4], [20, -4, -10], [-7, 2, -7], [-24, 10, -4], [-22, 20, 2], [-64, 9, -2], [49, -5, 16], [-10, 26, 0], [38, 6, -2], [9, 1, 6], [-27, 9, 13], [30, -2, 14], [-33, 15, 3], [-15, 28, 8], [-34, 21, 7], [-24, -15, 3], [19, 17, 3], [-5, -17, -3], [-30, 10, -10], [-15, 15, 16], [19, 1, -7], [-40, 2, -20], [0, -2, -10], [-52, 2, 8], [14, -12, 9], [-16, 2, -6], [18, -9, 13], [-8, 2, -10], [28, -5, -5], [-29, 8, 5], [-10, 9, 3], [-22, 20, -20], [1, -4, -17], [-12, -6, -21], [-3, 4, -18], [26, -18, 6], [35, -12, -9], [-34, 8, 0], [17, 14, -1], [-2, 8, 14], [3, -6, -3], [-48, 19, -13], [-19, 8, 18], [36, -14, 16], [-9, 3, 18], [1, 8, -20], [-24, 18, -9], [-18, 16, 12], [-12, -5, 17], [-10, -12, -20], [3, -39, 9], [30, -27, 2], [43, 11, -5], [-22, -13, 0], [-26, -14, 4], [-30, 7, 2], [-28, -15, 15], [30, -30, -10], [-33, 6, -3], [4, -10, -2], [0, 25, -19], [12, 6, -14], [12, -5, 20], [70, -7, -7], [0, 0, 2], [-10, 20, 6], [23, -16, -14], [-16, 17, -13], [18, 12, 2], [47, -6, 1], [38, 11, -6], [-16, -16, -4], [-26, 22, -3], [30, -2, -11], [-34, 30, -2], [-18, 9, 1], [-15, -8, -5], [-25, 9, -8], [22, 1, 16], [-9, -14, -1], [16, 10, 18], [10, 10, -1], [-37, 9, -27], [20, 21, -7], [24, -12, -21], [-28, 8, -4], [30, -11, 14], [-68, 4, 2], [33, 13, -6], [-24, 27, -8], [-7, 29, -4], [-20, 20, 16], [-26, 4, 1], [-6, -17, 19], [-14, 7, -6], [9, -35, 16], [45, 10, -11], [21, -29, 21], [4, -26, 19], [38, -9, -4], [-4, 20, -2], [-38, -14, 12], [55, -10, 23], [8, -22, 14], [-29, 26, -2], [6, -14, 2], [50, -4, -10], [-6, 10, -21], [-36, 30, 0], [-51, 19, -4], [48, 3, 8], [38, 6, -1], [1, 4, 10], [-48, 3, 3], [30, -20, -20], [-25, -9, -2], [-47, 4, -8], [56, -18, -6], [76, 4, 5], [-10, 12, 5], [5, -7, 2], [-40, 17, -1], [-1, -12, 10], [62, 0, -6], [38, -10, -4], [14, -1, 17], [19, -19, -7], [54, -12, -11], [5, -34, 16], [34, 10, -5], [28, -4, -21], [-14, -17, 4], [-11, -11, 5], [60, -5, 1], [53, -1, 5], [-16, 35, -6], [-23, 25, 6], [-49, 9, -9], [4, 4, -23], [0, -11, 13], [-9, 33, 2], [27, -14, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6630_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6630_2_a_bk();".
// 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_6630_2_a_bk(:prec:=3)
    chi := MakeCharacter_6630_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(3019) - 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_6630_2_a_bk();".
// 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_6630_2_a_bk( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6630_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, -4, -2, 1]>,<11,R![-3, 1]>,<19,R![141, -47, -1, 1]>,<23,R![6, 0, -4, 1]>],Snew);
    return Vf;
end function;