
// Make newform 7938.2.a.bk in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7938_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_7938_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_7938_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 := [-2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 3]];
    Rf_basisdens := [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_7938_a();"
function MakeCharacter_7938_a()
    N := 7938;
    order := 1;
    char_gens := [6077, 3727];
    v := [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_7938_a_Hecke(Kf)
    return MakeCharacter_7938_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], [0, 1], [2, -1], [0, 0], [2, -1], [-3, 1], [0, -1], [5, 1], [-4, 0], [-3, -2], [2, 2], [-9, 1], [0, 1], [0, 0], [2, -1], [-4, 1], [3, -1], [-7, 0], [5, -1], [-12, -1], [-3, -2], [-4, 2], [-12, -1], [5, 1], [-6, -2], [2, 1], [12, -2], [11, 1], [12, 1], [3, -4], [8, 3], [-12, 1], [-7, -1], [14, 0], [-16, -1], [-3, 5], [-6, -4], [12, -1], [-16, 1], [0, 2], [-13, -2], [0, 4], [-19, -1], [2, 2], [-1, -3], [-18, 2], [8, -4], [-12, 2], [9, 3], [17, 2], [-6, 3], [3, 2], [-6, 4], [-6, -4], [-4, 2], [2, 5], [3, 6], [8, -4], [-6, -4], [-28, 0], [-6, -4], [-1, 4], [0, -4], [-10, 2], [-4, -2], [-6, -3], [8, -4], [-21, 0], [-15, -1], [-1, -3], [14, 5], [14, 3], [3, -5], [-6, 1], [-16, 1], [-21, 2], [35, 0], [12, 1], [-10, 1], [33, -1], [-13, -4], [-1, -3], [-18, -2], [-12, -4], [20, 2], [-30, 1], [-13, -1], [0, 0], [9, -1], [-7, -1], [6, 3], [2, 3], [3, -5], [-12, -1], [9, 6], [-16, 1], [2, 4], [2, 2], [24, -4], [-6, 3], [9, 8], [-4, 4], [2, -8], [-18, -5], [-3, -2], [-6, 4], [-1, 4], [-22, 4], [-4, 8], [3, -6], [-10, -2], [-16, 2], [-21, -6], [-34, 3], [-15, -3], [-24, 0], [18, -5], [-4, 9], [-19, 2], [-30, 1], [-12, 7], [2, -1], [-18, -1], [-4, -7], [15, 3], [-13, 3], [-22, 4], [2, -2], [-39, -1], [-13, 1], [-4, -4], [-21, 0], [-4, 2], [24, -5], [-10, 5], [0, 0], [9, 0], [2, 6], [-6, -9], [11, -7], [0, 4], [-16, 8], [-24, -2], [-10, -9], [-3, -2], [14, -7], [-21, -5], [-4, -7], [45, 2], [20, -7], [-27, 3], [-4, 8], [27, 3], [8, -6], [27, 4], [5, -6], [-6, -11], [30, 2], [15, 2], [23, 7], [18, -2], [-3, -4], [24, 2], [29, -5], [14, 0], [-16, 4], [0, 0], [-24, -4], [-4, 9], [36, 6], [-19, -2], [-34, -4], [-36, 4], [26, -5], [-30, -5], [8, -4], [-34, -4], [-28, -2], [12, 8], [2, 9], [33, -6], [-9, -5], [12, 1], [-22, -4], [-10, -2], [11, -4], [-3, 5], [-25, 2], [-6, 7], [20, 8], [6, 4], [6, 2], [-15, 4], [-13, 2], [32, -4], [-36, 4], [-39, -5], [18, -4], [-4, 2], [2, -1], [-25, 2], [18, 5], [30, -8], [-1, 11], [-6, -2], [-39, 6], [-40, -1], [59, 0], [-36, 4], [11, -1], [-6, 3], [33, 1], [53, -2], [41, 3], [27, -10], [0, -12], [42, -3], [-16, -1], [-16, -6], [-12, 2], [-7, -9], [0, 0], [14, 8], [-21, 0], [6, -12], [-37, 1], [42, 2], [-4, 4], [56, 0], [8, -10], [-51, 2], [2, 6], [-12, 6], [-4, 2], [54, 4], [-12, -4], [39, -9], [48, -5], [2, 6], [-40, 6], [32, -4], [51, 6], [3, -5], [35, -7], [66, 2], [14, 6], [3, 1], [2, -12], [-3, 12], [27, 9], [-1, 4], [12, 15], [6, 7], [-34, -5], [-28, 7], [48, 4], [56, 0], [8, 2], [-48, -7], [14, 7], [14, 0], [-12, 6], [26, 1], [-48, 0], [0, 0], [50, -3], [30, 0], [-4, -5], [20, -4], [44, -8], [-40, 4], [5, -6], [5, -13], [24, 11], [-10, 4], [41, 2], [-12, -8], [36, 10], [32, 4], [33, 1], [62, 4], [8, -4], [3, 11], [-28, 2], [-18, 13], [-64, 4], [39, -2], [42, 1], [-30, -6], [6, -8], [54, -6], [-22, 0], [-6, -11], [41, 11], [6, -3], [-30, -6], [14, -7], [47, 8], [12, 11], [-16, 0], [24, -8], [-28, -3], [-7, 10], [-18, 10], [14, 0], [18, 8], [20, 8], [45, 9], [-18, -4], [60, -6], [-16, -7], [30, -4], [41, -10], [30, 6], [-27, -3], [-25, 2], [9, 12], [62, -3], [41, 8], [-12, -8], [-31, -1], [3, -14], [23, 6], [-10, 3], [-22, -3], [-15, -1], [-36, -4], [-10, -11], [-24, -8], [-28, -14], [-12, -8], [-10, -3], [-12, 13], [14, -13], [30, -8], [26, 8], [-10, 12], [-22, -1], [9, -4], [36, 3], [53, 1], [-18, 11], [12, 0], [-10, -9], [-10, 9], [9, 13], [-30, 8], [32, 5], [-1, -4], [60, -3], [-1, -3], [-4, -8], [-3, -2], [-60, 2], [-36, -3], [-33, -6], [21, -3], [-22, -16], [63, 0], [33, -5], [42, -11], [44, -1], [44, -4], [-18, -5], [17, 11], [-37, 2], [0, 1], [32, 12], [24, -16], [-60, -1], [-49, -7], [8, -7], [18, 5], [15, -7], [26, 8], [-36, -10], [44, 6], [-36, 4], [15, 0], [29, -1], [63, -6], [-16, 8], [-51, 1], [-40, 12], [50, -4], [-34, -7], [15, 17], [-61, 2], [-12, -8], [18, -15], [2, 6], [-48, 12], [-1, -4], [-34, -4], [-9, -6], [14, 10], [-12, 0], [14, 0], [63, 2], [-13, 15], [-84, 0], [18, 5], [-49, -7], [50, 5], [45, 8], [-4, 16], [-18, 2], [-19, 6], [-72, -3], [66, 6], [-16, -16], [-19, -12], [42, 7], [-15, 7], [-19, -1], [-42, 0], [-27, 3], [-12, 0], [68, 8], [0, 4], [-36, -3], [20, 18], [-12, -1], [24, 5], [51, -4], [-52, 12], [45, 9], [14, 0], [48, -4], [50, -6], [-63, 7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7938_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7938_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_7938_2_a_bk(:prec:=2)
    chi := MakeCharacter_7938_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(3023) - 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_7938_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_7938_2_a_bk( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7938_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![-18, 0, 1]>,<13,R![-14, -4, 1]>,<17,R![0, 1]>,<23,R![-9, 6, 1]>],Snew);
    return Vf;
end function;