
// Make newform 3840.2.a.bq in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3840_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_3840_2_a_bq();".
// 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_3840_2_a_bq();".
// 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, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-1, 2, 0], [-6, 0, 2]];
    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_3840_a();"
function MakeCharacter_3840_a()
    N := 3840;
    order := 1;
    char_gens := [511, 2821, 2561, 1537];
    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_3840_a_Hecke(Kf)
    return MakeCharacter_3840_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 := [[0, 0, 0], [1, 0, 0], [-1, 0, 0], [-1, -1, 0], [1, -1, 1], [0, 0, 1], [2, 0, -1], [1, -1, 0], [1, -1, 0], [-2, 0, 0], [-2, 0, -1], [2, 2, -1], [4, 2, 0], [2, 2, 0], [1, -1, 2], [-2, 0, 0], [3, 1, -1], [-2, 2, -2], [-4, 0, 0], [2, 2, -2], [6, 0, 0], [6, 0, -1], [6, 2, 0], [4, -2, 0], [6, 0, -2], [-2, 0, 0], [1, 1, 2], [6, -2, -2], [2, 2, 0], [10, 4, -1], [-7, 1, 2], [-1, -3, 1], [10, 0, 1], [-9, 1, 2], [2, 0, 0], [-4, 2, 1], [10, 2, 1], [-6, 2, 0], [-5, 1, -2], [6, 4, -2], [5, -1, -1], [-14, 2, 0], [-8, 0, 0], [6, -4, 2], [14, 0, 2], [12, -2, -1], [-13, -3, 2], [-13, -1, 2], [0, 0, -2], [6, -2, -2], [14, 0, 1], [10, -2, 0], [6, 4, -2], [5, 3, 3], [6, 0, 3], [13, -1, 2], [10, 4, 0], [20, -2, 3], [2, -2, -3], [6, 4, -2], [-20, 0, 0], [-2, 0, -4], [-14, 2, -4], [-10, -6, 4], [-6, 0, -2], [14, 8, -2], [-11, 3, -4], [2, -4, 4], [-4, 4, -2], [6, -6, 2], [2, 4, 1], [2, 2, -2], [1, 5, -4], [-10, -2, 1], [-5, -3, -2], [3, 1, 0], [10, 0, 4], [0, -4, 1], [-4, 2, -2], [6, 0, 4], [1, -1, 3], [-22, 2, 2], [-2, -2, 2], [-2, 0, -4], [0, 2, -1], [4, -4, -2], [-18, 0, -4], [-14, -4, -2], [-10, 4, 0], [19, -1, 0], [18, -2, 4], [-14, 2, 2], [11, 3, -2], [13, -1, 1], [-7, -1, 4], [13, 3, -2], [6, -8, -4], [-10, 0, 0], [-4, -8, 4], [-10, -2, -4], [2, -6, 4], [14, -4, 2], [-12, -4, -2], [-16, -2, 2], [-15, -1, 0], [2, -4, 2], [10, 2, 2], [-2, 0, -1], [-14, 2, -6], [6, 8, -4], [-15, 1, -4], [-4, 0, 1], [-18, 0, -3], [-17, 1, 2], [18, 4, -1], [-6, 4, -6], [-2, -2, -4], [-5, -11, 6], [2, -8, 2], [13, 7, -3], [-18, -6, 2], [-18, 0, 0], [10, 8, -6], [26, -2, 0], [-5, 5, 0], [-14, 4, -4], [18, 2, -2], [16, 4, -2], [-9, -9, 6], [14, 10, -3], [-15, 7, -8], [21, -1, -4], [14, 0, -5], [22, 2, 1], [-6, -4, 8], [2, 0, 6], [-2, 0, -2], [-8, 4, -8], [10, 0, 0], [-14, 4, 6], [13, 3, 6], [-22, 4, 4], [1, -3, 6], [20, 0, 0], [34, 2, 0], [6, -10, 2], [12, 8, -7], [-10, -4, 3], [11, 5, 2], [3, -7, 6], [-6, -6, 1], [-4, -2, 8], [-6, 2, 8], [15, 5, -6], [-16, -4, -4], [6, 2, -8], [-30, -4, 3], [-14, -4, 4], [-2, 12, 0], [-18, -4, 0], [-4, 0, 4], [-26, -4, -3], [19, 7, 2], [13, -5, -5], [-10, -4, -1], [9, 11, 0], [-8, 6, 1], [4, 0, 7], [-18, 8, -2], [10, -8, 0], [11, 1, 7], [6, -6, 12], [-8, 8, 8], [14, 4, -2], [26, -4, -1], [16, -6, 0], [1, 7, 2], [-34, 4, 0], [-11, -3, 0], [-34, 2, -4], [17, 5, 2], [-9, -7, 3], [16, 8, -5], [6, 0, 7], [47, 5, 0], [22, -4, -4], [38, -6, -1], [6, 6, -4], [-2, -4, -8], [-4, 4, 0], [2, 8, -6], [24, 12, -4], [-17, -7, 2], [-38, 0, 0], [10, 10, -2], [2, 4, -5], [14, -8, 0], [20, -4, -7], [22, -8, 3], [45, -1, 4], [-2, 12, -8], [22, 12, -3], [-4, -4, 5], [2, -12, 2], [-21, 5, 1], [14, -8, 12], [2, 8, -11], [4, 12, -10], [0, 10, -8], [-7, -9, 0], [14, 4, 6], [2, -8, -4], [-55, 1, 2], [-24, -4, 0], [-16, 4, 6], [2, 12, 2], [1, -3, 4], [0, 6, 2], [-5, 9, 0], [-14, 8, 8], [-10, -10, -2], [20, -2, -1], [-4, 2, 8], [-5, -9, 8], [-18, -6, -4], [10, -10, 4], [-2, -16, -1], [-28, 8, -2], [11, -1, 0], [13, 7, -5], [-44, 4, 1], [25, 7, -2], [12, -2, 7], [2, 8, -6], [48, -4, 8], [1, 7, -4], [-14, 4, -10], [18, 8, 2], [3, -11, -3], [-14, 10, 6], [-8, 8, 2], [-31, -9, 8], [-27, 1, -10], [-14, -10, 2], [30, -8, 1], [-44, 0, -6], [1, -3, -4], [-37, 1, -1], [-29, -3, 4], [23, 5, -6], [24, -8, -9], [-6, 0, -4], [-5, -3, -8], [14, -12, 10], [2, -4, 2], [45, -1, -3], [-34, -10, 0], [6, -2, -4], [18, -12, 2], [42, 12, -4], [-5, -17, 10], [-12, -8, 0], [30, -2, -2], [50, 2, 3], [46, 4, -3], [-31, -1, 4], [14, -4, -4], [40, -6, 2], [0, 4, -8], [-30, 0, -4], [6, 2, -8], [22, -10, 8], [14, 4, -2], [22, 8, -1], [34, 8, -2], [1, 9, 6], [-16, 0, -2], [14, 2, -2], [-6, -4, -6], [-21, 9, 5], [11, 5, -4], [12, -10, 11], [23, 1, 6], [30, -6, 0], [36, 0, -8], [-28, -12, 4], [14, 4, 8], [26, 0, -2], [-4, 2, -3], [24, 2, 2], [26, -8, -4], [-10, -10, -2], [6, 0, 13], [-19, -1, 11], [-26, -18, 3], [10, 0, -12], [6, -4, 5], [-46, -12, 6], [63, 1, -3], [50, 2, -4], [18, 8, -10], [10, 0, 12], [20, 2, -3], [-42, -2, -2], [17, 15, -4], [-10, 12, 4], [-14, 6, -12], [-34, -10, 2], [32, 20, -6], [46, 6, 1], [47, -3, 6], [-18, 0, 0], [4, -14, 14], [8, 4, -8], [-33, 5, -4], [42, -8, 2], [-11, 7, -13], [-12, 16, 6], [34, -4, 0], [8, 10, 2], [-5, -11, 0], [2, 8, -10], [-22, -12, 8], [7, 15, -2], [2, -16, 7], [-10, -8, 12], [17, -9, -6], [-10, -2, -4], [-43, -1, -4], [-6, 8, -8], [-34, -2, 6], [-26, -4, 6], [38, -4, 9], [-26, -10, 2], [29, 19, -6], [-2, 2, 8], [30, 18, -4], [10, -12, -7], [6, 0, 0], [-51, -3, -2], [6, -10, 1], [-2, 12, -9], [-6, 12, 4], [-8, -6, -5], [2, 4, -10], [21, 19, -3], [2, 18, -14], [-4, 8, 0], [-52, 8, -6], [-42, -8, 8], [19, 13, -10], [26, 4, -10], [10, -4, 12], [-45, 7, -10], [-46, 6, -6], [22, 0, 5], [44, 4, 4], [49, 19, -5], [-70, -8, 1], [57, -5, -2], [8, -12, -5], [-4, -2, 10], [-15, -5, 8], [63, 1, 1], [20, 0, -4], [6, 0, 14], [-10, 16, -8], [-3, -15, 16], [-30, -12, 0], [-29, -11, 5], [-49, 9, -2], [-25, 5, 4], [-30, -20, 4], [60, 10, -7], [-44, 4, -1], [-31, -9, 9], [-50, 2, -4], [10, -8, 12], [34, -12, 8], [34, -8, 4], [14, -12, -4], [-50, 4, -7], [-1, -1, 8], [30, 0, -3], [9, 7, 10], [9, -5, 10], [-34, 0, 3], [-14, 14, 1], [-18, -18, 8], [9, -9, -12], [-22, -16, 2], [30, -4, 14], [-17, -19, 3], [20, 8, 0], [-56, -8, 8], [10, -12, 6], [-24, -14, 3], [-6, 4, -10], [57, -1, -4], [34, 16, 0], [6, 2, -4], [-54, -8, 5], [-51, -3, 2], [-46, -8, 9], [18, 4, -12], [-18, 8, -9], [-24, 8, -13], [0, 14, -2], [-18, -10, 0], [-35, 3, -9], [-54, 4, -6], [-46, 0, 4], [-22, 14, -20], [19, -3, 16], [-34, 8, 12], [46, 8, 0], [24, -16, -8], [23, -1, -8], [-14, 8, 9], [57, 7, -4], [-22, -8, 12], [4, -8, -1], [-59, -9, 4], [11, 21, -13], [34, -8, 6], [-26, 16, 4], [-38, 6, 12], [6, -12, -8], [9, -9, -4], [74, 2, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3840_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3840_2_a_bq();".
// 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_3840_2_a_bq(:prec:=3)
    chi := MakeCharacter_3840_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_3840_2_a_bq();".
// 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_3840_2_a_bq( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3840_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-16, -16, 2, 1]>,<11,R![64, -24, -4, 1]>,<13,R![16, -28, 0, 1]>,<17,R![32, -16, -6, 1]>,<19,R![16, -12, -4, 1]>],Snew);
    return Vf;
end function;