
// Make newform 3888.2.a.bd in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3888_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_3888_2_a_bd();".
// 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_3888_2_a_bd();".
// 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, -3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-2, 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_3888_a();"
function MakeCharacter_3888_a()
    N := 3888;
    order := 1;
    char_gens := [2431, 2917, 1217];
    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_3888_a_Hecke(Kf)
    return MakeCharacter_3888_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], [0, 0, 0], [-2, -1, 0], [1, -1, 2], [1, 2, -3], [-1, 1, 1], [-3, 0, 0], [1, 0, -3], [2, -2, 3], [-4, 1, -3], [4, -1, 2], [-1, -3, 0], [1, 5, -3], [4, 2, -1], [-2, 5, 0], [-6, 0, 3], [-7, 1, 0], [2, 1, -5], [-2, 5, -1], [-3, -9, 3], [2, 6, -3], [-2, 5, -4], [-2, -1, -3], [0, -9, 6], [5, -8, 4], [-4, 4, 3], [-5, -4, -4], [3, 0, 3], [-10, -3, 0], [-5, -1, 6], [1, -3, 3], [-4, 7, 6], [-7, 1, -6], [-5, -7, 5], [-5, -4, 9], [-8, -4, 5], [-7, -11, 10], [1, -3, 12], [-1, 1, -3], [-7, 7, -3], [-3, 0, -6], [-1, -6, -3], [-11, -1, -3], [-10, 1, 4], [-12, 0, -6], [1, 6, -6], [7, -1, -4], [1, -7, -4], [-11, 2, 3], [-10, 10, -8], [-3, -9, 9], [-16, -2, -6], [2, 10, 1], [-15, -9, 6], [1, 5, 6], [13, 5, -3], [-9, 0, -9], [7, 3, -6], [-1, 13, -17], [2, 16, -12], [1, -7, -7], [-8, -16, 9], [-11, 12, 3], [2, 4, 6], [5, -2, 1], [-25, 4, 0], [1, 20, -4], [8, -8, 10], [11, 7, -6], [20, 4, 4], [17, -8, 9], [15, 9, -9], [19, -4, 5], [11, -5, -11], [1, -18, 15], [5, 10, 3], [-4, -11, 9], [8, -9, -6], [4, 2, -6], [-4, 1, 7], [14, -14, 0], [-4, -5, 16], [3, -9, -15], [-19, -15, 6], [-2, 14, -16], [16, -7, 12], [18, -9, 21], [-7, -2, 7], [-13, 10, 6], [-5, 5, 17], [21, -9, -3], [-14, -13, 15], [-8, -9, -3], [8, -8, 6], [-2, -7, 5], [24, 0, 3], [4, 11, -15], [27, 0, -3], [-2, 12, -15], [17, -3, 18], [16, 5, -7], [-15, 9, 6], [14, -11, 15], [2, -8, -15], [-23, 2, -13], [2, 15, -12], [-17, -22, 12], [-6, 9, 12], [-1, 1, 6], [-19, -2, -5], [10, 20, -16], [11, -3, -3], [7, 11, -12], [13, -7, 23], [-8, -27, 15], [5, -14, 12], [13, -10, 17], [3, 9, -6], [-23, -19, 15], [1, 11, 6], [2, 10, -20], [-16, 7, -11], [-1, 1, 18], [-9, 9, 3], [13, -19, -13], [-15, 0, -24], [-1, -2, 13], [-21, 0, 12], [-14, 8, 2], [-13, -2, -14], [19, 15, -15], [23, -14, 0], [25, -10, -13], [-25, 15, 6], [25, -10, -9], [-10, 16, -11], [0, 27, -33], [-8, 20, -19], [16, -4, -15], [18, -9, 9], [-17, 0, -3], [-13, 16, -9], [-17, 5, -4], [-9, 18, -3], [8, 24, -18], [4, -10, 21], [17, 7, -11], [17, 4, -3], [-29, -10, 14], [3, 9, 12], [-10, -5, -5], [-3, 0, -18], [1, -24, 0], [-1, -11, 9], [4, 8, -4], [-2, 5, 9], [-11, -6, -6], [23, -11, 0], [8, -15, 6], [22, -16, 21], [10, -19, 9], [3, 18, 0], [-23, -13, -4], [24, 18, -3], [16, -1, -18], [-14, 2, 0], [-2, 0, 0], [5, -5, 28], [2, -3, -3], [-2, -22, 12], [-11, -4, 18], [23, 4, -5], [47, -8, 0], [17, -8, 16], [-17, 14, -4], [4, 20, 3], [-20, -19, 8], [-12, -9, -15], [28, -12, -6], [23, -11, 28], [1, -10, 11], [43, 11, -6], [2, -11, 31], [27, -9, 21], [-1, 16, 0], [-4, -17, 33], [-16, -9, -18], [13, 29, -19], [-25, -17, 4], [18, -9, 6], [35, 15, -18], [4, -19, -6], [-26, 21, 0], [11, -5, 27], [-9, -18, 0], [-11, -10, -9], [-28, -5, -14], [11, 25, -11], [-31, -26, 15], [21, -18, 6], [4, 14, -18], [-14, 5, 11], [2, 1, 4], [-16, 19, -5], [-24, 18, -9], [-6, 18, -9], [-8, 3, -18], [38, 13, -3], [-28, -14, 15], [7, 14, 2], [35, 12, -3], [-8, 17, 3], [7, -25, -1], [-14, -16, -9], [17, -11, 36], [32, -8, -11], [4, -10, 11], [-10, 16, -42], [-18, -9, -21], [-8, -16, 27], [-31, 16, 13], [22, 17, -28], [-5, 35, -3], [19, -12, 3], [-1, -11, 9], [-25, 10, -26], [11, -17, -6], [-6, -9, 6], [-32, 8, 5], [16, 23, 9], [-7, 25, 10], [28, 0, 24], [-26, -1, 17], [19, -25, 45], [43, -13, 5], [-17, 26, -36], [14, 19, -29], [-27, -18, 15], [-37, -14, 3], [39, 9, -15], [-11, 20, -33], [-11, 12, 12], [-26, -13, -7], [20, -9, 18], [4, -4, -18], [1, 14, -24], [1, 12, -36], [11, -26, 36], [19, -13, 32], [-18, 9, -3], [-19, 22, -29], [24, -27, -15], [-4, -23, -9], [-37, 1, -8], [-28, 4, -12], [-21, 0, -21], [-16, 6, -21], [-35, -7, 14], [-15, -27, 33], [17, 12, -3], [22, 5, 29], [-2, 5, 27], [-4, -2, -2], [-28, 6, -9], [-29, 11, -39], [-20, -28, 35], [0, 18, -12], [14, 4, 3], [7, 41, -25], [-2, 8, 0], [-10, -44, 19], [46, -21, 3], [-34, -35, 19], [-17, -10, -13], [-25, 28, -23], [7, 21, -15], [14, 46, -24], [-10, -14, 1], [26, -12, 36], [13, 26, -15], [-31, 22, -36], [-20, 5, 11], [10, 29, -39], [23, -32, 19], [1, 23, 2], [-6, 9, -3], [-1, -3, 9], [-17, 5, -3], [-8, 20, -7], [6, -36, 15], [35, 7, 3], [3, -27, 36], [-5, 11, 3], [23, -5, 18], [-1, 10, 10], [13, -37, 24], [1, -16, 23], [-13, 37, -30], [-30, 9, 18], [40, 50, -34], [38, 4, -11], [-30, -45, 24], [-35, 21, -27], [8, -8, 15], [-17, -25, 2], [-46, 9, 3], [-35, -55, 36], [-1, -26, 13], [-67, -5, 9], [-10, 21, -21], [-65, -16, 12], [39, 0, -3], [-1, 19, -9], [-23, 14, -37], [-9, 27, -24], [17, 0, -15], [19, 23, -6], [-55, -8, 15], [8, -14, -26], [-26, -25, 18], [22, -16, 23], [14, 16, -32], [18, -36, 42], [-26, -42, 27], [11, 13, -9], [-10, 12, -12], [-38, 21, -33], [1, 2, -43], [1, -49, 12], [-12, -36, 15], [20, 1, 13], [-35, 29, 0], [-2, 32, -22], [-2, 44, -12], [-2, -9, -18], [0, -27, 18], [20, 9, -27], [22, 17, 6], [-10, 22, 22], [25, 39, -6], [29, 16, -20], [-52, 1, 3], [-11, 14, 3], [40, -13, 2], [-1, 25, -6], [9, 27, -21], [20, 18, 9], [43, -10, -10], [16, -19, 15], [-6, -9, 9], [-2, -31, 47], [-55, -12, 18], [46, -1, -6], [7, -34, 44], [47, 31, -26], [-33, 9, 15], [26, 34, -18], [-42, -9, -3], [-47, -13, 12], [-47, 23, -18], [8, 4, 46], [11, 7, 33], [-42, 36, 9], [1, -19, -9], [46, -15, 0], [14, 37, -5], [-13, -32, 21], [-8, 12, 9], [35, -33, -12], [-5, 41, -21], [-8, 0, 6], [41, 37, -15], [-13, -29, 9], [37, -4, -4], [53, 12, -3], [-1, 4, 12], [-39, 9, -12], [-10, -42, 42], [-57, 9, -18], [-31, -23, 1], [32, 1, -12], [-35, -4, -21], [19, 3, -27], [-19, 16, 21], [-35, -25, 29], [0, 18, -21], [22, 20, 5], [23, 4, 1], [19, 15, 30], [-13, 13, -63], [-13, 19, -32], [8, 4, -39], [-36, 18, 18], [-62, -1, 20], [37, 29, -30], [20, -5, -8], [1, 36, 15], [-31, -8, -21], [-5, 29, -22], [19, -4, -12], [-43, -17, 22], [-24, 9, 3], [-53, 2, -7], [13, 11, -6], [-6, 18, 0], [-17, -33, 0], [32, -29, 10], [-19, 40, 24], [13, 35, -19], [37, -16, -21], [-16, 1, -44], [8, -35, -27], [-6, 9, -30], [10, 35, -37], [47, 49, -38], [33, 0, 15], [60, 0, 12], [-11, 14, -40], [-57, 27, 21], [-10, -5, 3], [44, -32, 24], [-1, 54, -6], [-53, 2, 24], [-49, -2, -3], [-28, 0, -6], [52, 5, -15], [13, -4, 33], [-36, 36, -45], [16, 21, 21], [1, 53, -36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3888_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3888_2_a_bd();".
// 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_3888_2_a_bd(:prec:=3)
    chi := MakeCharacter_3888_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_3888_2_a_bd();".
// 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_3888_2_a_bd( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3888_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3, 9, 6, 1]>,<7,R![17, -6, -3, 1]>,<11,R![3, -18, -3, 1]>,<13,R![-17, -6, 3, 1]>],Snew);
    return Vf;
end function;