
// Make newform 5850.2.e.bi in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5850_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_5850_e_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_5850_2_e_bi();".
// 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_5850_2_e_bi();".
// 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 := [100, 0, 21, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 11, 0, 1], [11, 0, 1, 0]];
    Rf_basisdens := [1, 1, 10, 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_5850_e();"
function MakeCharacter_5850_e()
    N := 5850;
    order := 2;
    char_gens := [3251, 3277, 2251];
    v := [2, 1, 2];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_5850_e_Hecke();"
function MakeCharacter_5850_e_Hecke(Kf)
    N := 5850;
    order := 2;
    char_gens := [3251, 3277, 2251];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, 1, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 1, -1, 0], [-2, 0, 0, 1], [0, 0, -1, 0], [0, 1, 3, 0], [-1, 0, 0, -1], [0, 0, 0, 0], [-1, 0, 0, 2], [-2, 0, 0, 3], [0, -1, -2, 0], [1, 0, 0, -1], [0, 2, -4, 0], [0, 0, 7, 0], [0, 2, 5, 0], [0, 0, 0, -1], [6, 0, 0, -3], [0, -2, -1, 0], [-1, 0, 0, -1], [0, 0, -12, 0], [3, 0, 0, -1], [0, -1, 7, 0], [6, 0, 0, -2], [0, 2, 10, 0], [-4, 0, 0, 1], [0, -2, -6, 0], [0, -3, 8, 0], [-7, 0, 0, -1], [0, 0, 14, 0], [0, 1, 10, 0], [-9, 0, 0, 5], [0, -1, 10, 0], [-4, 0, 0, 2], [12, 0, 0, 2], [-6, 0, 0, 3], [0, -1, 19, 0], [0, 4, -4, 0], [0, 3, 10, 0], [0, 4, 15, 0], [-8, 0, 0, 6], [14, 0, 0, 1], [2, 0, 0, -4], [0, -6, -6, 0], [0, 2, 2, 0], [-11, 0, 0, -1], [-4, 0, 0, -4], [0, -2, 4, 0], [0, 3, -5, 0], [-13, 0, 0, 1], [0, -6, -2, 0], [-2, 0, 0, -3], [-8, 0, 0, -2], [-3, 0, 0, -1], [0, -1, -19, 0], [0, 0, -2, 0], [15, 0, 0, -6], [6, 0, 0, -5], [0, 0, 22, 0], [3, 0, 0, -7], [0, -2, 14, 0], [0, 2, -14, 0], [0, -1, 2, 0], [-30, 0, 0, 0], [0, 6, 1, 0], [0, 4, 16, 0], [-12, 0, 0, 0], [0, 3, 3, 0], [0, -5, -8, 0], [6, 0, 0, -2], [0, 5, 4, 0], [-13, 0, 0, 8], [0, 1, 6, 0], [0, 1, 9, 0], [22, 0, 0, 3], [0, 5, -14, 0], [-19, 0, 0, 3], [0, -7, -4, 0], [-2, 0, 0, 6], [-24, 0, 0, 2], [-9, 0, 0, -7], [-6, 0, 0, -2], [-11, 0, 0, 9], [0, 3, 0, 0], [13, 0, 0, 1], [0, 5, -14, 0], [5, 0, 0, 1], [0, 2, 10, 0], [-20, 0, 0, 6], [0, -1, -25, 0], [0, 1, 34, 0], [-19, 0, 0, 8], [0, 3, 19, 0], [-28, 0, 0, 2], [-1, 0, 0, 6], [0, -6, -10, 0], [0, 0, 0, 2], [20, 0, 0, -6], [0, 4, -10, 0], [-18, 0, 0, -6], [0, -8, 2, 0], [0, -4, 2, 0], [0, -5, 14, 0], [-14, 0, 0, -1], [40, 0, 0, 0], [0, 6, -2, 0], [0, 5, 11, 0], [0, 7, 8, 0], [-8, 0, 0, -6], [-11, 0, 0, -8], [0, -3, 24, 0], [0, 4, 2, 0], [0, 3, -16, 0], [24, 0, 0, 4], [12, 0, 0, -8], [8, 0, 0, 5], [0, -1, -14, 0], [0, 0, -24, 0], [0, 1, 25, 0], [5, 0, 0, 9], [-5, 0, 0, 13], [0, -10, -1, 0], [0, -8, 2, 0], [0, 1, -41, 0], [-33, 0, 0, -4], [-12, 0, 0, 7], [6, 0, 0, 8], [16, 0, 0, 10], [0, -6, -30, 0], [0, -7, 22, 0], [17, 0, 0, 8], [0, -8, 1, 0], [-5, 0, 0, -5], [0, 3, 29, 0], [-3, 0, 0, -11], [8, 0, 0, 0], [0, -2, -34, 0], [0, -3, -45, 0], [0, -7, -23, 0], [18, 0, 0, 4], [46, 0, 0, -1], [26, 0, 0, 2], [0, -3, 6, 0], [0, -1, 17, 0], [-8, 0, 0, -5], [0, 0, 0, 0], [0, -5, -18, 0], [0, 8, 22, 0], [-10, 0, 0, -6], [0, 16, 5, 0], [0, -11, 2, 0], [10, 0, 0, -11], [0, -8, 16, 0], [0, 0, -20, 0], [0, -10, -12, 0], [-26, 0, 0, 6], [13, 0, 0, 3], [55, 0, 0, 1], [0, 7, 11, 0], [4, 0, 0, 12], [0, -5, 29, 0], [0, 1, 35, 0], [0, -3, -41, 0], [45, 0, 0, 3], [0, 2, -16, 0], [0, -5, -11, 0], [-1, 0, 0, 3], [0, 5, 29, 0], [-10, 0, 0, 2], [0, 5, 21, 0], [2, 0, 0, -2], [15, 0, 0, 3], [2, 0, 0, -8], [0, 0, -8, 0], [-47, 0, 0, 3], [-8, 0, 0, 15], [-18, 0, 0, 7], [-2, 0, 0, 16], [0, 11, 0, 0], [-20, 0, 0, 7], [0, -2, -52, 0], [-4, 0, 0, -8], [0, 8, 22, 0], [0, 15, 18, 0], [0, 5, -1, 0], [1, 0, 0, 3], [0, -13, -27, 0], [0, 8, 35, 0], [16, 0, 0, 8], [-22, 0, 0, 3], [0, 3, -4, 0], [0, -3, 31, 0], [4, 0, 0, 16], [10, 0, 0, 10], [0, -3, -32, 0], [0, 5, 31, 0], [36, 0, 0, 4], [0, 3, 31, 0], [0, 0, -18, 0], [0, -8, -34, 0], [-4, 0, 0, 0], [17, 0, 0, 1], [0, 8, -18, 0], [-11, 0, 0, 14], [-10, 0, 0, -9], [0, 9, -4, 0], [-8, 0, 0, 5], [0, 13, -8, 0], [-2, 0, 0, -6], [4, 0, 0, -10], [0, -2, -57, 0], [-5, 0, 0, 9], [0, -12, -36, 0], [0, -11, 23, 0], [-1, 0, 0, 8], [-22, 0, 0, 14], [0, 18, 6, 0], [8, 0, 0, 9], [0, -7, 23, 0], [0, 6, 4, 0], [2, 0, 0, 4], [-8, 0, 0, -8], [-71, 0, 0, 1], [0, 5, -33, 0], [0, 3, -54, 0], [16, 0, 0, -5], [0, -4, 50, 0], [28, 0, 0, -10], [0, -3, 12, 0], [-28, 0, 0, -7], [0, -6, -22, 0], [-2, 0, 0, -6], [42, 0, 0, -5], [-18, 0, 0, -8], [0, -4, 52, 0], [0, -8, -32, 0], [22, 0, 0, 0], [0, -10, -48, 0], [-49, 0, 0, 7], [-38, 0, 0, -2], [0, 6, -16, 0], [-34, 0, 0, -6], [0, -13, 28, 0], [-19, 0, 0, 5], [0, 10, 28, 0], [-2, 0, 0, -16], [0, -16, -28, 0], [20, 0, 0, -4], [-3, 0, 0, 10], [0, -8, -14, 0], [0, -19, -14, 0], [22, 0, 0, -6], [0, 1, -42, 0], [43, 0, 0, -2], [0, 9, -12, 0], [-26, 0, 0, 1], [52, 0, 0, 2], [0, -18, -21, 0], [0, -6, 31, 0], [0, 6, 46, 0], [0, 19, -4, 0], [0, 9, 6, 0], [-1, 0, 0, -5], [0, -2, 18, 0], [0, 14, 24, 0], [40, 0, 0, 0], [14, 0, 0, -8], [23, 0, 0, -9], [0, 5, -41, 0], [0, -7, 17, 0], [10, 0, 0, 8], [0, 4, 17, 0], [0, -4, 14, 0], [-29, 0, 0, 17], [0, -2, 59, 0], [0, 15, 21, 0], [0, 14, -24, 0], [25, 0, 0, -7], [-4, 0, 0, 4], [3, 0, 0, 5], [0, -14, -34, 0], [38, 0, 0, -10], [0, -18, -28, 0], [14, 0, 0, -8], [0, 10, -33, 0], [70, 0, 0, 2], [0, -15, 5, 0], [0, -18, 2, 0], [-14, 0, 0, 23], [-38, 0, 0, -5], [8, 0, 0, -1], [0, -3, 0, 0], [0, -5, 40, 0], [-48, 0, 0, -5], [0, 17, 29, 0], [-16, 0, 0, 21], [-8, 0, 0, -8], [0, 12, -2, 0], [52, 0, 0, -12], [0, -1, 53, 0], [0, -2, -23, 0], [0, -12, 26, 0], [20, 0, 0, 2], [0, -1, 50, 0], [-4, 0, 0, -10], [0, -4, 0, 0], [0, 19, 6, 0], [-26, 0, 0, -8], [-65, 0, 0, -1], [0, -7, 27, 0], [0, 4, 36, 0], [-14, 0, 0, 6], [16, 0, 0, 7], [0, -6, -4, 0], [0, -6, -3, 0], [-29, 0, 0, -3], [36, 0, 0, 5], [-24, 0, 0, -8], [0, -10, -36, 0], [-54, 0, 0, 1], [-10, 0, 0, 12], [0, -6, -18, 0], [25, 0, 0, -5], [0, 3, -27, 0], [0, -5, 18, 0], [-5, 0, 0, 24], [31, 0, 0, -4], [0, -2, -53, 0], [0, -2, -66, 0], [0, -8, -57, 0], [11, 0, 0, 1], [0, 16, -2, 0], [-72, 0, 0, 2], [0, -3, -19, 0], [15, 0, 0, 14], [0, -11, -27, 0], [-26, 0, 0, 20], [0, -9, -14, 0], [54, 0, 0, 4], [0, 5, 40, 0], [0, 3, -8, 0], [0, -10, 10, 0], [16, 0, 0, -2], [-80, 0, 0, -4], [0, -12, -16, 0], [16, 0, 0, 4], [-30, 0, 0, 3], [0, 11, -31, 0], [-33, 0, 0, 0], [0, 10, 23, 0], [55, 0, 0, -2], [0, 6, 68, 0], [-74, 0, 0, 4], [0, 11, 14, 0], [-24, 0, 0, 18], [0, -5, -9, 0], [-17, 0, 0, -4], [-8, 0, 0, 19], [0, 5, 69, 0], [0, 2, -5, 0], [0, -5, -34, 0], [4, 0, 0, -2], [0, 4, 20, 0], [14, 0, 0, -2], [0, 12, -2, 0], [0, 0, -22, 0], [0, 10, 6, 0], [0, -5, -79, 0], [53, 0, 0, 2], [3, 0, 0, 5], [-20, 0, 0, -18], [0, -4, -40, 0], [-55, 0, 0, 10], [-11, 0, 0, 23], [0, -1, -3, 0], [56, 0, 0, -9], [-48, 0, 0, 6], [0, 14, -12, 0], [-16, 0, 0, -9], [0, -3, -60, 0], [-10, 0, 0, -20], [0, 21, 30, 0], [0, 3, -47, 0], [0, 8, 6, 0], [-47, 0, 0, -11], [0, 11, -42, 0], [-50, 0, 0, 3], [0, -3, 3, 0], [0, -11, 62, 0], [0, 4, -38, 0], [26, 0, 0, -26], [0, 18, -8, 0], [36, 0, 0, 7], [0, 6, 70, 0], [-10, 0, 0, -3], [0, -4, 11, 0], [-10, 0, 0, 1], [-20, 0, 0, -9], [-30, 0, 0, 4], [12, 0, 0, -14], [-9, 0, 0, 3], [0, 11, -11, 0], [0, 5, -3, 0], [0, -13, -16, 0], [30, 0, 0, 8], [-28, 0, 0, 18], [0, 12, -22, 0], [-22, 0, 0, -8], [0, -6, -27, 0], [-2, 0, 0, 26], [0, -29, -17, 0], [0, -11, 41, 0], [0, -18, -26, 0], [14, 0, 0, -12], [0, -7, -56, 0], [55, 0, 0, -10], [-23, 0, 0, 29], [0, 17, -10, 0], [0, 16, -26, 0], [0, -2, -10, 0], [-23, 0, 0, 20], [0, 5, -26, 0], [0, -2, -71, 0], [-5, 0, 0, -5], [0, 8, -4, 0], [0, 6, 68, 0], [0, -16, 28, 0], [15, 0, 0, -7], [-46, 0, 0, -3], [-28, 0, 0, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5850_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5850_2_e_bi();".
// 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_5850_2_e_bi(:prec:=4)
    chi := MakeCharacter_5850_e();
    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_5850_2_e_bi();".
// 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_5850_2_e_bi( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5850_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![64, 0, 25, 0, 1]>,<11,R![-8, 3, 1]>,<17,R![16, 0, 33, 0, 1]>,<19,R![-8, 3, 1]>,<29,R![-41, 0, 1]>,<31,R![-92, 1, 1]>],Snew);
    return Vf;
end function;