
// Make newform 5850.2.e.j in Magma, downloaded from the LMFDB on 06 September 2025.
// 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_j();".
// 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_j();".
// 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, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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 := false)
    N := 5850;
    order := 2;
    char_gens := [3251, 3277, 2251];
    char_values := [[1, 0], [-1, 0], [1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, -1], [0, 0], [0, 0], [0, 1], [-3, 0], [0, 1], [0, 3], [4, 0], [0, 6], [-3, 0], [-1, 0], [0, -2], [0, 0], [0, -10], [0, -3], [0, -3], [-15, 0], [-13, 0], [0, 13], [0, 0], [0, -10], [4, 0], [0, -15], [6, 0], [0, 4], [-9, 0], [0, 8], [0, -6], [-14, 0], [0, -18], [0, 16], [6, 0], [0, 18], [-2, 0], [12, 0], [11, 0], [0, -5], [0, -16], [0, 0], [0, 15], [-24, 0], [-25, 0], [0, 0], [0, -4], [0, -24], [-20, 0], [-22, 0], [0, -4], [0, 9], [-20, 0], [0, -6], [-21, 0], [8, 0], [0, 0], [0, -15], [0, -18], [21, 0], [23, 0], [0, 22], [-24, 0], [0, 26], [0, 12], [0, 4], [-18, 0], [0, -19], [0, -12], [-4, 0], [0, 13], [0, -6], [-32, 0], [0, -6], [9, 0], [0, 10], [0, -31], [-11, 0], [0, 12], [-30, 0], [0, -2], [-24, 0], [-14, 0], [6, 0], [20, 0], [12, 0], [0, -22], [-8, 0], [0, 6], [-6, 0], [0, 10], [36, 0], [0, 29], [0, 30], [27, 0], [0, -11], [30, 0], [7, 0], [0, -24], [-36, 0], [6, 0], [0, -4], [-28, 0], [0, 22], [0, 12], [0, 36], [21, 0], [8, 0], [0, -32], [0, 33], [0, -42], [-42, 0], [5, 0], [0, -14], [0, 44], [0, -6], [-44, 0], [20, 0], [-3, 0], [0, -4], [0, 18], [0, -21], [-12, 0], [-10, 0], [0, 23], [0, 6], [0, -39], [17, 0], [-3, 0], [-8, 0], [18, 0], [0, -44], [0, -22], [43, 0], [0, 3], [14, 0], [0, 13], [24, 0], [40, 0], [0, -24], [0, 7], [0, -33], [6, 0], [-43, 0], [0, 0], [0, -4], [0, 15], [-17, 0], [-24, 0], [0, 26], [0, -6], [10, 0], [0, 39], [0, 46], [39, 0], [0, -28], [0, -12], [0, -44], [-36, 0], [-44, 0], [-6, 0], [0, 13], [-30, 0], [0, 3], [0, 3], [0, 13], [30, 0], [0, -30], [0, 9], [50, 0], [0, -35], [-2, 0], [0, -57], [-36, 0], [56, 0], [-48, 0], [0, -40], [22, 0], [27, 0], [29, 0], [42, 0], [0, 2], [55, 0], [0, 64], [12, 0], [0, 26], [0, -48], [0, -9], [-54, 0], [0, -47], [0, 17], [34, 0], [-51, 0], [0, 2], [0, -27], [-16, 0], [18, 0], [0, -12], [0, 63], [-46, 0], [0, 47], [0, -42], [0, -36], [6, 0], [2, 0], [0, -44], [-5, 0], [39, 0], [0, 42], [55, 0], [0, 0], [-24, 0], [-58, 0], [0, 43], [-42, 0], [0, 26], [0, 21], [9, 0], [62, 0], [0, -44], [-3, 0], [0, 3], [0, 54], [-58, 0], [64, 0], [0, 0], [0, 29], [0, 24], [-17, 0], [0, -6], [48, 0], [0, 10], [63, 0], [0, 2], [-74, 0], [47, 0], [-54, 0], [0, 2], [0, -24], [10, 0], [0, -36], [-18, 0], [30, 0], [0, 60], [56, 0], [0, 44], [-2, 0], [0, 54], [48, 0], [0, -20], [36, 0], [49, 0], [0, 66], [0, 4], [12, 0], [0, -24], [-71, 0], [0, -6], [-15, 0], [-58, 0], [0, -35], [0, -45], [0, 34], [0, -16], [0, -42], [52, 0], [0, -70], [0, 24], [10, 0], [60, 0], [-72, 0], [0, -13], [0, -33], [-58, 0], [0, -17], [0, -28], [4, 0], [0, 19], [0, -13], [0, 12], [40, 0], [2, 0], [-24, 0], [0, 72], [-40, 0], [0, 30], [44, 0], [0, 49], [30, 0], [0, -61], [0, -6], [31, 0], [-15, 0], [-27, 0], [0, 24], [0, 54], [21, 0], [0, -79], [-45, 0], [-34, 0], [0, 42], [-42, 0], [0, 49], [0, -19], [0, 66], [10, 0], [0, 84], [44, 0], [0, -20], [0, 36], [-26, 0], [12, 0], [0, -25], [0, -12], [60, 0], [-69, 0], [0, -16], [0, 87], [22, 0], [3, 0], [-24, 0], [0, 44], [51, 0], [38, 0], [0, -32], [30, 0], [0, -49], [0, -66], [-1, 0], [-11, 0], [0, 77], [0, 12], [0, -21], [80, 0], [0, 66], [-62, 0], [0, -9], [-61, 0], [0, 39], [-62, 0], [0, 0], [-70, 0], [0, 52], [0, -16], [0, -6], [-48, 0], [68, 0], [0, -12], [6, 0], [-79, 0], [0, 85], [9, 0], [0, -21], [-25, 0], [0, -92], [-42, 0], [0, 38], [-86, 0], [0, 27], [-3, 0], [21, 0], [0, -63], [0, 57], [0, 34], [-6, 0], [0, 84], [36, 0], [0, 28], [0, -10], [0, 54], [0, 35], [-61, 0], [-24, 0], [-8, 0], [0, 72], [15, 0], [-94, 0], [0, -23], [-87, 0], [84, 0], [0, 2], [3, 0], [0, 10], [72, 0], [0, -24], [0, 43], [0, -72], [-44, 0], [0, -48], [59, 0], [0, 7], [0, -4], [0, 78], [-86, 0], [0, -42], [9, 0], [0, 64], [99, 0], [0, -97], [25, 0], [99, 0], [38, 0], [-30, 0], [-86, 0], [0, -51], [0, 67], [0, -12], [42, 0], [-52, 0], [0, 82], [48, 0], [0, 59], [-12, 0], [0, -19], [0, 63], [0, -48], [-28, 0], [0, 70], [-99, 0], [-48, 0], [0, 28], [0, 18], [0, 66], [-27, 0], [0, 70], [0, 3], [-84, 0], [0, 38], [0, -6], [0, -30], [-78, 0], [23, 0], [-24, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5850_e_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5850_2_e_j();".
// 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_j(:prec:=2)
    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_j();".
// 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_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5850_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 0, 1]>,<11,R![3, 1]>,<17,R![9, 0, 1]>,<19,R![-4, 1]>,<29,R![3, 1]>,<31,R![1, 1]>],Snew);
    return Vf;
end function;