
// Make newform 6030.2.a.bu in Magma, downloaded from the LMFDB on 06 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6030_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_6030_2_a_bu();".
// 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_6030_2_a_bu();".
// 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 := [2, 10, -8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-4, 0, 1, 0], [-1, -6, 1, 1], [-3, 8, 0, -1]];
    Rf_basisdens := [1, 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_6030_a();"
function MakeCharacter_6030_a()
    N := 6030;
    order := 1;
    char_gens := [4691, 1207, 3151];
    v := [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_6030_a_Hecke(Kf)
    return MakeCharacter_6030_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+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, 0, 0], [0, 0, 0, 0], [1, 0, 0, 0], [0, 1, 0, 0], [1, 1, -1, 0], [1, 0, -1, 0], [0, -1, 1, 1], [0, -1, 1, 1], [3, 1, 0, -1], [-1, 1, 0, 1], [3, 0, -1, 0], [1, 0, 0, -1], [0, -1, 1, -1], [-2, -2, 2, 0], [5, -1, -2, -1], [2, 2, 0, 0], [0, -1, 1, -1], [2, 1, 0, 0], [-1, 0, 0, 0], [2, -2, 1, 1], [-4, 0, 0, 2], [3, 0, 1, -2], [5, -2, 2, 3], [3, 0, 0, -3], [-1, -1, 1, 0], [1, -2, 2, 1], [-6, -2, 2, 0], [10, 0, -2, 0], [-2, -3, 2, 2], [3, 1, 1, 2], [-4, 0, -2, 2], [12, 1, 1, -1], [3, 0, 0, 1], [6, 1, -5, -1], [-1, 1, 0, 1], [-7, -2, 4, 1], [-2, -2, -2, -2], [-9, 0, 0, -1], [7, -1, -2, 1], [12, 0, -3, -3], [6, -1, -3, -1], [-4, 4, 0, -2], [10, -2, 2, 0], [4, 5, -3, -1], [0, 0, 2, 0], [-3, -2, 2, 3], [-6, -5, 1, -1], [-8, -4, 6, 2], [10, -6, 4, 2], [-4, -3, 2, 0], [-1, 4, 0, -3], [-4, 0, -2, 2], [-1, 3, -3, 2], [9, 0, 0, 1], [-2, -1, 3, 5], [1, -3, 2, -1], [-15, -3, 2, 1], [-7, 4, -1, -2], [1, -2, 4, -1], [2, 0, -4, 0], [5, 2, -6, -1], [-4, 2, -3, 1], [-5, 7, -5, -2], [10, 4, -2, 0], [7, 1, -5, -2], [11, 2, -1, -6], [7, 5, -7, -2], [2, -5, 7, 1], [0, 5, -3, -7], [2, -4, 4, 0], [-18, 3, -1, -1], [-2, 4, 1, -3], [11, -3, -4, -1], [-1, 4, 1, 0], [-6, 3, 5, 1], [-7, -4, 0, 1], [-21, -3, 8, 5], [-1, -6, 2, -1], [8, 0, -4, -6], [8, 0, 2, 0], [-12, 5, 1, -1], [8, 6, -2, 0], [2, 8, -3, -3], [12, 1, -5, -5], [-1, -4, 4, 3], [4, -2, 2, 2], [-15, -4, -2, -3], [-4, -3, 1, -1], [3, 5, 0, 5], [11, 1, 4, 3], [3, -2, -2, 1], [-8, -2, -3, 3], [4, -5, 8, 0], [0, -5, 7, 5], [7, 3, 1, -6], [-3, -6, 4, -3], [-9, 3, 0, 1], [-10, -5, 5, -3], [-15, -2, 10, 3], [-5, 3, -4, 5], [8, 2, 2, 6], [5, 0, -1, -8], [10, -2, 2, 0], [-17, -4, -2, 3], [0, 2, -2, -2], [-14, -3, 1, 7], [6, -5, 3, -3], [11, 2, -2, 3], [-20, 0, 6, 2], [11, 1, -5, -4], [20, -2, -4, 0], [6, -8, 0, 0], [0, -1, -5, -5], [-20, -1, 1, 1], [-17, -2, 1, 2], [6, -8, -4, 0], [-16, -5, -1, -5], [3, -8, -6, 1], [6, -2, -4, -8], [-10, 5, -5, -5], [-9, 5, -2, -1], [-12, -5, 1, 5], [-6, 0, 4, 0], [-10, -6, 2, 4], [2, 6, -2, -8], [-7, -8, 2, 1], [-6, -4, -6, -2], [5, 2, 3, 2], [9, -1, -4, -3], [-1, -6, 7, -2], [-10, 1, 3, -1], [5, -1, -10, -1], [0, 4, -4, 0], [15, 6, -9, -6], [1, 0, -2, 5], [-2, 0, -6, 2], [14, 8, -3, 7], [-8, -4, 10, -2], [-4, 8, -7, 1], [-26, 0, 0, 0], [-3, -5, -7, 4], [-5, -7, 0, 5], [18, 4, -4, 2], [21, 4, -2, 3], [-10, 4, -2, -2], [9, 8, -7, -8], [-12, 8, 2, -4], [-13, -1, 7, 8], [-22, 1, -5, -3], [21, 1, -4, 1], [-8, -10, 8, -2], [11, 4, -4, -7], [-22, -4, 2, 0], [-15, -3, 6, -1], [-11, 13, -7, -2], [10, 2, -3, 5], [-13, 4, -3, 2], [2, -9, 1, -3], [-13, -5, 3, 10], [14, -10, 2, 2], [9, 4, -8, -3], [-4, 11, 1, -3], [4, 4, 2, 10], [10, -11, 11, 3], [12, -7, -9, -5], [8, 10, -6, -6], [15, -2, -5, 0], [3, -2, -4, 1], [-3, -11, 3, 2], [2, -6, 10, -2], [-30, -1, 7, 7], [6, 6, -8, -8], [-1, -4, -1, 8], [17, 5, 5, -6], [-11, 8, -7, 0], [-2, -3, -1, -1], [-13, 1, 11, 0], [-13, 1, 10, 3], [-28, 9, -6, -2], [-24, -4, 4, 6], [6, 4, 4, 10], [4, -7, -1, -3], [-10, 0, 0, 0], [6, 9, 1, -9], [-23, 10, -8, -7], [-1, -7, -4, -3], [9, 0, 1, -10], [2, 2, 6, 4], [-26, 6, 4, 0], [-4, 8, -2, -2], [-2, 3, 3, -1], [-5, 4, 6, 5], [-1, -3, 9, -6], [-17, -12, 8, -3], [-30, 1, -3, 3], [-6, 9, -11, -5], [5, -5, 9, 0], [-3, 0, 7, 4], [-6, -4, 4, 12], [5, 1, 10, -1], [-1, 3, -6, -1], [17, 2, -4, 5], [19, 8, -1, -2], [-34, -10, 6, -2], [-16, -2, 2, -2], [19, 4, -11, -12], [-27, 4, 0, 1], [5, -6, 8, 1], [-3, -2, -6, -7], [-3, 5, 7, 2], [2, -3, -3, -3], [-3, -8, -4, -9], [29, 7, -8, 1], [24, 11, -13, -1], [-8, -4, 8, -4], [16, 2, 0, -6], [26, -6, 2, 0], [-3, 2, -8, 3], [-9, 2, 10, 9], [-6, -10, 13, 11], [1, 9, -4, -5], [-16, 8, 0, -4], [-8, 5, 11, 5], [4, -4, -4, -4], [-4, 17, -11, -3], [-14, 14, -12, -8], [10, -1, -1, -7], [-6, -10, -2, -4], [8, 4, -6, 2], [0, -4, 4, 4], [7, 12, -7, 0], [7, -3, -1, -4], [9, -6, 8, 1], [-28, 11, -7, -1], [9, -9, 3, 4], [-18, 14, 0, -6], [-39, 3, 5, 0], [-8, -4, -7, -3], [8, -9, 3, 9], [-3, -6, -1, -10], [-3, -4, 14, 3], [-9, -3, 9, 6], [-2, 15, -2, -4], [-34, -3, -4, 8], [-15, -2, 6, 9], [-20, -8, 12, 12], [-12, -8, 4, 0], [-37, 1, 3, 6], [-10, 1, -1, 7], [7, -14, 4, 11], [5, 2, 2, -3], [-9, 6, -2, -5], [-11, -8, -4, -3], [25, 6, -6, 1], [-8, 8, -8, 2], [23, 2, -10, -11], [34, -7, 8, 8], [33, 2, 6, -1], [-2, 4, -11, -1], [-52, 0, 4, 6], [10, -8, -6, 0], [31, 2, 6, -7], [40, -5, 6, 0], [23, -14, 7, 10], [-2, -15, 15, 1], [-20, -8, 0, 4], [41, -8, 2, 1], [-10, 4, -4, 0], [10, 0, -10, -8], [-28, -4, -4, 2], [-16, -5, 0, -10], [48, -2, 0, 0], [23, -1, 3, -6], [6, -10, 4, 2], [-16, 4, -4, 2], [0, -11, 20, 0], [8, 15, -9, -5], [24, 10, 2, -4], [25, 6, -18, -3], [12, -8, 8, 4], [-31, 3, 4, -3], [21, 10, -10, 3], [-29, 2, 8, 3], [36, 7, -6, 0], [-28, -8, 4, 0], [-52, -4, -3, -1], [26, -3, -3, 1], [6, -10, -1, -3], [-15, -10, 1, -12], [-30, -5, 5, -7], [-35, -7, 14, 9], [-20, -4, 0, 8], [-8, 3, 7, -1], [22, 11, 3, -1], [7, -2, -1, 10], [-13, -1, -4, -7], [-11, -8, 13, 4], [-34, 4, -4, -4], [38, 9, -5, -1], [30, 10, 4, 2], [9, 7, -15, -6], [6, -6, 11, 5], [15, -10, 6, 1], [-15, 6, -4, -7], [32, 2, -6, -2], [0, 3, 5, -7], [2, -8, 2, 8], [12, 4, -6, 0], [7, 8, -1, 0], [27, -8, 5, 4], [-1, 8, -2, 5], [5, 17, -6, -3], [-13, 10, -20, 5], [45, 8, -16, -3], [-3, 5, -6, -5], [-42, 12, 0, 0], [20, 11, -11, -9], [-6, 2, -4, 2], [-18, 3, -5, -5], [10, 7, -7, -11], [12, -19, 11, 11], [-16, 0, 12, 2], [9, 9, -6, 9], [22, -15, -8, -2], [-38, 13, 3, -3], [-25, -10, 14, -1], [-32, -3, -5, 3], [-5, -9, -1, -6], [49, 4, -8, -7], [52, 4, -8, -2], [36, 8, -18, -4], [14, -10, 15, 5], [-5, -10, -9, -8], [22, 7, -9, 1], [4, 6, -2, 2], [9, 6, -16, -11], [-33, -11, 2, -5], [-8, 3, -5, 3], [-44, -14, 6, -4], [14, 6, -2, -4], [44, 6, 0, -6], [34, -3, -5, 1], [-37, 8, -2, -1], [-7, 0, 4, -3], [66, -4, 5, -1], [-50, -1, 13, 11], [-27, -1, 10, 9], [24, -12, 10, -2], [1, -2, 7, 18], [6, -2, -4, 0], [-34, 7, -3, -1], [-37, 7, -11, 0], [69, 3, -2, -7], [55, -1, 2, 5], [-23, -7, -8, -1], [-31, -3, 17, 8], [56, 2, -11, -5], [13, -2, 10, 7], [1, -9, -1, -2], [33, -10, 10, 3], [-11, -4, 2, 1], [9, 8, -4, -5], [5, -1, 8, -3], [0, -15, 5, -1], [-27, 12, -16, 5], [51, 7, 3, 10], [18, -4, -13, 1], [10, 0, 8, -2], [12, 8, -2, 4], [13, 18, -14, 3], [-38, 4, 0, 2], [-37, 9, 4, -5], [2, -6, 14, -6], [-3, -6, 9, -8], [-5, 14, -11, -4], [20, 5, -7, -1], [6, -4, 4, -2], [12, -17, -1, 9], [50, 8, -16, -4], [38, 5, -19, -11], [61, -2, -2, -7], [29, -5, 15, -6], [-25, -9, 18, 9], [3, -16, 20, 5], [39, -3, -1, -8], [25, -4, 4, 13], [-27, -12, -5, 6], [5, 0, 8, -5], [-58, 0, 10, 8], [-9, 14, -4, -5], [2, 16, 0, 4], [-34, 2, -2, -2], [24, 17, -3, 7], [-14, 10, 10, 8], [16, 8, 2, -2], [-34, 10, -2, -2], [-5, -12, 2, -7], [-16, 12, -18, 4], [9, -17, -5, -2], [2, -4, 0, -12], [5, -15, -4, 7], [12, 3, 13, 1], [22, 3, 2, 0], [51, 4, 0, 9], [-25, -10, 22, 11], [-23, 8, -9, -2], [-7, -10, -7, 6], [-16, -18, 20, 6], [15, 16, -8, -9], [-1, 12, 0, -5], [68, 0, -8, -6], [48, -12, 11, 7], [27, -2, 12, -1], [56, 0, 8, 0], [-13, -9, 3, -10], [-5, 9, 8, 5], [-1, 16, -17, 0], [46, 0, -18, -4], [0, 25, -25, -1], [-13, 17, -8, -1], [-25, 16, -18, 3], [-35, 22, -12, -5], [-14, -6, 20, 10], [1, 18, 2, -1], [-1, 11, 1, 4], [23, -24, 9, 8], [-9, 12, 8, -1], [-6, -2, 2, 14], [0, -12, 0, -8], [34, -8, 0, -6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6030_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6030_2_a_bu();".
// 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_6030_2_a_bu(:prec:=4)
    chi := MakeCharacter_6030_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(2999) - 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_6030_2_a_bu();".
// 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_6030_2_a_bu( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6030_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![52, 12, -20, -1, 1]>,<11,R![112, 32, -20, -3, 1]>,<13,R![-8, 28, -20, -2, 1]>,<17,R![32, 80, -32, -2, 1]>,<23,R![-448, 204, 8, -12, 1]>,<29,R![-112, 148, -52, 2, 1]>],Snew);
    return Vf;
end function;