
// Make newform 2790.2.a.bk in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2790_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_2790_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_2790_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 := [6, 4, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, 0], [-1, -4, 0, 1], [-7, 0, 2, 0]];
    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_2790_a();"
function MakeCharacter_2790_a()
    N := 2790;
    order := 1;
    char_gens := [2171, 1117, 1801];
    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_2790_a_Hecke(Kf)
    return MakeCharacter_2790_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], [1, 0, -1, 0], [1, 0, 1, 0], [2, -1, 0, 0], [1, -1, 0, 1], [1, 1, -1, 0], [0, 0, 1, -1], [0, 2, 0, 0], [1, 0, 0, 0], [0, 1, 0, -2], [0, 2, 0, 0], [5, -1, 1, 2], [-2, 0, 0, -2], [-1, -1, -1, 0], [4, -1, 2, 2], [-1, 1, 4, -1], [2, 1, -2, 2], [1, -2, 1, 0], [5, -1, 3, 0], [1, -1, 1, -2], [0, 0, -2, 2], [-5, 2, -3, -2], [4, -2, 2, 0], [-1, 1, 1, -2], [4, 1, -2, -2], [-3, -1, 1, 2], [6, 0, 2, 2], [-1, 1, -3, 2], [2, -3, 0, 0], [2, 1, 4, -2], [-5, 1, -6, 1], [5, -1, 2, 1], [-3, -1, -1, -2], [-6, 2, 0, -2], [1, 1, -3, 2], [12, -1, 0, -2], [-6, 2, 5, 1], [-6, -2, 2, -2], [-4, -1, 4, -2], [2, 2, -1, 1], [4, -1, -4, 2], [6, 0, 4, 0], [-18, 2, 0, 0], [3, 3, 1, 0], [-1, -3, -1, -2], [14, 1, -4, 4], [-11, 3, -3, -2], [-4, 4, -1, -5], [-5, 1, 3, 4], [-4, 0, -6, 2], [-8, -2, 0, 2], [4, 1, -4, 2], [-11, -5, 1, 0], [-9, 3, -4, 1], [-10, -2, 2, 0], [1, -5, -1, 0], [-2, -1, -4, 0], [0, -4, 0, 0], [16, -1, 0, 2], [-6, -2, -2, 2], [-4, 1, 8, -2], [8, 3, 4, -2], [-4, -4, -2, 2], [-10, 0, -4, 0], [-9, 5, 2, -1], [-12, 0, 2, 2], [-6, -2, 4, 2], [10, -4, 8, 0], [-7, 1, 8, -3], [-7, 0, -7, 0], [-10, -3, 4, -4], [5, 3, 1, 2], [7, -3, 5, 0], [-1, -5, 4, 1], [-12, 0, 0, 0], [1, 3, 1, -2], [3, 0, -3, -6], [6, -4, -8, 0], [-4, 5, -2, -2], [-16, 2, 2, 4], [20, -1, 0, 2], [-1, 1, 5, 4], [-6, 4, -2, 0], [-9, 1, 7, 2], [-28, 1, 2, 0], [-20, 0, -6, 2], [0, 0, -4, 4], [6, -1, -4, -4], [-4, 2, 4, 4], [-3, 2, -3, 0], [-10, 7, 0, 0], [21, 2, -3, 0], [-13, 1, 0, -3], [12, -6, 0, 0], [-6, 2, -6, 0], [8, -2, -4, 0], [3, -1, -3, 4], [2, 4, 2, -2], [-10, 1, -2, 2], [-15, 1, -1, -2], [4, 0, -8, 0], [-7, 0, -7, 0], [-11, 3, -10, -3], [-12, -2, -6, 4], [12, -6, 0, 0], [0, 6, 0, -6], [15, -6, 3, 0], [-28, 2, 0, -6], [-23, -4, -5, 4], [-8, -7, -2, 4], [-3, -1, 1, -4], [-19, -1, 6, -3], [7, -5, 7, -2], [14, 3, 2, -6], [-7, 5, 7, -4], [2, 2, -9, -1], [4, 0, 0, -2], [8, -7, 2, 6], [-4, -6, 2, 4], [-16, -2, 2, -2], [-1, 11, -1, 0], [9, -1, -11, 2], [-1, -7, -5, 2], [-1, 1, 9, 2], [-22, 4, -3, 3], [-10, 6, 0, 2], [-19, 6, -9, 0], [10, -4, 2, 0], [5, 3, 0, 3], [20, -4, -5, 1], [-8, -2, 4, 4], [-4, -1, -2, -4], [19, 0, 1, 6], [17, 7, -5, 2], [15, -5, 5, 10], [-23, -1, -3, 2], [-14, -2, 8, -4], [-1, -6, 1, -2], [21, -1, 3, 4], [-10, 2, 2, 0], [0, 9, 2, -4], [2, -4, 2, 0], [-8, 6, -5, -5], [-11, 0, -3, -8], [9, -7, 1, 6], [14, -8, 8, 0], [-7, 3, 16, -1], [28, 0, 3, 1], [10, -4, 10, 4], [6, 3, -10, -2], [13, -5, 11, 0], [24, -2, 6, 6], [6, 3, 6, -2], [-14, -2, 6, -8], [12, -4, 8, -4], [-5, 8, -9, 4], [18, -4, 4, 0], [-6, 2, -6, 0], [8, -2, -6, 2], [-7, 9, -6, -1], [-12, 7, -6, -8], [18, -3, 8, -2], [-8, -2, -4, 2], [1, -3, -8, 3], [1, 3, 5, -6], [6, 0, 2, -4], [18, -4, -2, -6], [13, 1, -5, -6], [-4, 3, 16, -2], [44, -2, 5, 1], [24, 1, -8, 2], [-30, 8, -8, 0], [14, 4, -2, -4], [25, -10, 3, -2], [-36, 8, -8, 0], [7, -3, 1, 6], [35, 2, -3, 0], [8, 6, -11, 5], [0, -9, -2, 6], [-21, 0, 3, 0], [22, -4, -2, 4], [-15, -5, -8, 5], [-2, 6, 10, 0], [-2, -8, -2, 0], [25, -7, 9, 2], [16, 3, 8, -6], [-2, 0, 6, 2], [39, 0, 3, 0], [1, 1, -5, 4], [-9, -5, 11, -2], [-1, 5, -10, -5], [-16, -2, 8, 0], [-12, -10, 0, 0], [5, 3, -6, -1], [20, -6, 6, 12], [-11, 3, 5, -6], [15, -7, 7, 8], [4, 0, -14, -6], [11, 9, -3, 2], [-5, 7, -5, -2], [-3, -3, 9, -2], [5, -1, 3, -6], [18, 1, -4, -2], [-22, 2, 0, -4], [28, 12, 4, -8], [-12, 6, -6, 6], [-2, -4, 10, 0], [4, 0, -12, -4], [-5, 1, 1, 4], [-2, 2, 2, -4], [-4, 1, 6, -6], [-25, 5, -5, -2], [-6, 4, -10, 4], [-3, -3, 17, -4], [16, 5, 2, 6], [-14, 2, 18, -8], [15, 3, 4, -7], [-38, 0, -8, 0], [24, 0, 1, -3], [42, 2, -6, 4], [32, -2, 0, -4], [17, -6, 3, 12], [-23, 13, -15, -2], [-32, 10, -1, 3], [18, -8, 0, 0], [-4, 8, -4, 0], [8, -3, 6, 0], [-22, -1, 12, 2], [-8, -9, 4, -2], [16, -2, 0, 8], [10, 4, 14, 0], [39, -2, -3, -6], [20, -6, 4, -4], [32, -12, 7, 1], [55, -1, -1, 0], [-18, 10, 2, -2], [27, -2, -1, 4], [2, 2, -8, -2], [8, -6, 12, 8], [-19, 3, -2, 5], [-6, 3, 4, 0], [1, 1, 0, 5], [9, -3, -10, 7], [-25, 4, -5, -8], [30, 5, -4, -4], [31, 2, 3, 4], [33, -5, 2, 5], [12, -2, -4, 4], [13, 3, 1, -2], [26, -10, 10, 4], [36, -8, 11, 1], [7, 1, -11, -2], [-14, 6, -10, 2], [-12, -1, 2, -14], [38, -4, 8, 4], [-25, 3, 1, -4], [4, -10, 6, 4], [-16, -6, -2, -4], [23, 6, -3, 0], [-18, 12, -12, -6], [-4, 0, -1, -5], [-11, -3, 9, 2], [13, 1, -16, 5], [9, -9, -3, -2], [-1, 5, -11, -2], [0, 0, -16, 4], [4, -7, -8, 10], [-18, 6, -2, 2], [16, 10, 2, -12], [22, 0, -4, -6], [-11, -3, -1, 12], [7, -9, 5, 0], [-40, 2, -6, 0], [24, 5, 10, -6], [27, 5, 3, -6], [-1, -11, -4, 7], [-16, 6, -6, -12], [-15, -2, -19, 4], [-10, 4, 0, -10], [-12, -8, 4, 0], [-8, 12, 0, -8], [46, 0, 4, 4], [16, 3, -16, 6], [18, -8, -4, -2], [-43, -5, -1, 4], [4, -12, 4, -6], [-22, 4, -18, 0], [20, 1, -2, 4], [-15, 3, -5, 2], [21, 1, -7, 10], [-25, 3, -22, -3], [-18, 9, -16, -2], [0, 5, 10, 0], [12, -8, 0, 0], [-2, 4, 16, -2], [4, 2, 18, -8], [11, 2, -13, 0], [-7, 3, 23, -8], [24, -2, -2, 0], [9, -3, 5, -8], [22, 0, 0, -10], [13, -9, -7, 2], [-23, 7, -4, 3], [-4, 10, -2, -4], [-38, -6, -2, 0], [32, -6, 6, 0], [34, -2, -10, 8], [2, -6, -14, -4], [19, -5, 12, 13], [-28, 14, -12, -4], [30, -4, -2, -4], [43, -3, 5, 0], [12, -2, -10, 10], [-24, 6, -6, -8], [-32, 7, -4, 2], [-26, -6, 6, 4], [2, 12, -4, 4], [29, 0, -9, 2], [-3, 11, -6, 1], [-13, 7, -15, -12], [-31, -5, 15, -10], [9, 4, 3, 4], [9, -9, -3, 12], [20, 6, -8, 14], [7, 7, -11, 4], [24, 1, -16, 2], [27, -5, 4, 5], [-11, 5, -9, -2], [-20, -2, 14, -12], [-22, -2, 6, 14], [10, 6, -10, 0], [-22, -6, -10, 0], [-45, -7, -1, 2], [23, 1, -5, -2], [-11, 11, -8, 1], [-9, -3, 15, -12], [-11, -9, 11, -6], [-36, -1, -2, 0], [-54, -3, 8, -4], [35, 5, -4, 3], [-14, 10, 2, -4], [1, -1, 5, -6], [-40, 12, -12, -10], [-28, -5, -6, 2], [0, 14, 4, -6], [13, -17, 7, 4], [-30, -6, 0, 6], [-8, -12, -4, 2], [-43, 11, -14, -7], [23, 7, -11, 8], [-25, -3, 9, 2], [4, -5, -6, -4], [2, 4, 8, -8], [-7, -15, -2, 7], [-24, -6, 22, -10], [41, 4, 9, -4], [39, -1, -6, 9], [-8, -8, -12, 4], [-45, 3, 7, -6], [-23, 12, -15, -8], [46, -6, 9, 1], [42, -5, -8, 2], [-18, 7, 10, 6], [-35, 5, -25, 0], [12, -4, 4, 14], [-36, -5, 2, -4], [-61, -7, -3, 6], [8, 13, -10, 6], [25, -1, -13, 0], [-26, -4, 0, -2], [-10, 8, 0, -4], [-44, 10, -20, -8], [6, -15, -2, 6], [23, -2, -1, -12], [-3, 14, -15, 0], [22, -4, 16, 4], [-22, 2, -18, -4], [-11, 13, 7, 4], [-20, 10, -4, -16], [-7, 5, -2, -5], [-56, -1, -2, -8], [-11, 3, 18, -5], [-3, -5, -4, -1], [50, -6, -4, -6], [-16, 0, 8, -8], [42, -7, 6, 10], [3, -11, 13, 0], [14, -2, 5, -3], [-14, 0, -8, 4], [-23, -7, -9, -2], [-18, 5, -4, -2], [-38, -7, -2, 10], [-49, -4, -1, 0], [0, 6, -22, 8], [-1, -7, -7, -8], [20, 0, -12, -4], [11, -3, -12, 9], [-2, -12, 6, 16], [-4, 2, 13, -7], [39, 1, -17, 8], [-9, -8, 21, -8], [-34, 0, -4, 0], [-32, -4, 8, -4], [4, -10, -8, 4], [58, 2, 2, 0], [-30, 3, -2, 2], [-33, 5, 7, -6], [3, -4, 3, 12], [-12, 2, 10, 6], [14, -12, 12, -4], [-28, 2, -2, -14], [-43, -9, -8, 11], [4, 2, 18, 8], [-15, 7, 19, -10], [-16, -4, -2, 10], [43, -8, 1, 16], [-25, 9, 9, -10], [20, -8, 0, -4], [18, 0, -18, 12], [12, -5, 16, 6], [-10, -10, -13, 15], [10, -5, 4, 18], [52, -6, 6, -4], [-21, -5, -7, -2], [-3, 1, -19, -2], [-27, 0, 5, -8], [-29, 1, -6, -13], [-10, 6, 0, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2790_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2790_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_2790_2_a_bk(:prec:=4)
    chi := MakeCharacter_2790_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_2790_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_2790_2_a_bk( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2790_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![16, 24, -4, -5, 1]>,<11,R![24, 20, -10, -3, 1]>,<13,R![64, 56, -12, -6, 1]>,<17,R![48, -80, -40, 0, 1]>,<19,R![64, 48, -12, -7, 1]>],Snew);
    return Vf;
end function;