
// Make newform 1850.2.b.o in Magma, downloaded from the LMFDB on 20 February 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1850_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1850_b_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_1850_2_b_o();".
// 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_1850_2_b_o();".
// 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, -10, 25, -2, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [362, 0, -1, 5, 25, 1], [50, 0, 5, -25, -1, -5], [-50, 248, -5, 25, 1, 5], [126, -620, 25, -1, -5, -25], [378, -1860, 199, -3, -15, -75]];
    Rf_basisdens := [1, 124, 124, 124, 124, 124];
    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_1850_b();"
function MakeCharacter_1850_b()
    N := 1850;
    order := 2;
    char_gens := [1777, 1001];
    v := [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_1850_b_Hecke();"
function MakeCharacter_1850_b_Hecke( : Kf := false)
    N := 1850;
    order := 2;
    char_gens := [1777, 1001];
    char_values := [[-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 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, 0, 0, 0, -1, 0], [0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1], [4, 1, -1, 0, 0, 0], [0, 0, 0, 2, 0, 0], [0, 0, 0, 1, 0, 1], [0, 0, -1, 0, 0, 0], [0, 0, 0, 2, 0, 2], [2, 1, 1, 0, 0, 0], [0, -3, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0], [-2, 3, -1, 0, 0, 0], [0, 0, 0, 1, -4, -1], [0, 0, 0, 1, 0, 2], [0, 0, 0, -1, 6, -3], [-4, 2, -1, 0, 0, 0], [2, -1, -1, 0, 0, 0], [0, 0, 0, 1, 0, 2], [8, 2, 2, 0, 0, 0], [0, 0, 0, -2, -6, -2], [8, -2, -1, 0, 0, 0], [0, 0, 0, 3, 0, -2], [-6, 0, 0, 0, 0, 0], [0, 0, 0, -1, 6, -3], [-4, -2, 4, 0, 0, 0], [0, 0, 0, 2, -8, 0], [0, 0, 0, -1, -4, -4], [-6, 1, -3, 0, 0, 0], [0, 0, 0, 1, 0, -3], [0, 0, 0, 3, 4, 2], [8, 2, -1, 0, 0, 0], [0, 0, 0, 2, -2, 6], [-4, -5, 3, 0, 0, 0], [-4, 2, -4, 0, 0, 0], [0, 0, 2, 0, 0, 0], [0, 0, 0, 1, -2, -1], [0, 0, 0, 3, -4, 7], [0, 0, 0, -2, 0, 4], [0, 0, 0, 3, -2, 9], [-8, -6, 3, 0, 0, 0], [8, -2, 0, 0, 0, 0], [-8, -5, 8, 0, 0, 0], [0, 0, 0, -4, -10, -4], [0, 0, 0, 4, -14, 0], [-4, -6, 3, 0, 0, 0], [12, 3, -5, 0, 0, 0], [0, 0, 0, 2, -12, -1], [0, 0, 0, -5, -12, -1], [-2, 0, -8, 0, 0, 0], [0, 0, 0, -2, 10, 6], [-4, -3, -4, 0, 0, 0], [10, 2, 6, 0, 0, 0], [0, 0, -3, 0, 0, 0], [0, 0, 0, 4, -8, -6], [0, 0, 0, 4, -4, -5], [-4, 2, -4, 0, 0, 0], [-24, -2, -2, 0, 0, 0], [0, 0, 0, -2, 16, 0], [-6, -2, -6, 0, 0, 0], [0, 0, 0, -6, -4, -4], [0, 0, 0, 1, -6, 3], [0, 0, 0, 1, 12, 4], [-16, -5, -2, 0, 0, 0], [0, 0, 0, -4, 2, -4], [0, 0, 0, -3, 14, -1], [-8, 0, 3, 0, 0, 0], [0, 0, 0, -2, 6, -6], [0, 0, 0, 4, -4, 0], [4, -6, -4, 0, 0, 0], [0, 0, 0, 11, 14, 9], [8, -4, -2, 0, 0, 0], [0, 0, 0, -2, 12, 3], [0, 0, 0, -8, 2, 4], [-12, -8, 10, 0, 0, 0], [0, 0, 0, -4, 0, 0], [10, 7, 3, 0, 0, 0], [0, 0, 0, -4, 26, 4], [2, -4, 0, 0, 0, 0], [18, 2, 6, 0, 0, 0], [-4, 12, -2, 0, 0, 0], [22, 0, 0, 0, 0, 0], [16, 3, 2, 0, 0, 0], [0, 0, 0, 4, 22, 4], [4, -3, 2, 0, 0, 0], [0, 0, 0, 1, 12, -4], [-6, -4, -4, 0, 0, 0], [0, 0, 0, -1, -18, 5], [-14, 5, -3, 0, 0, 0], [0, 0, 0, 4, 8, 4], [0, 0, 0, -3, -4, 9], [12, -2, 5, 0, 0, 0], [0, 0, 0, -12, 0, -2], [-4, 8, 0, 0, 0, 0], [-4, 4, -5, 0, 0, 0], [0, 0, 0, 4, -32, 0], [2, 6, -2, 0, 0, 0], [16, 3, 1, 0, 0, 0], [0, 0, 0, 8, -4, 0], [6, 4, -4, 0, 0, 0], [0, 0, 0, -1, 4, -7], [0, 0, 0, 2, 0, 4], [0, 0, 0, 7, -4, 13], [-10, -8, 8, 0, 0, 0], [4, -1, 11, 0, 0, 0], [0, 0, 0, 8, 8, 6], [0, 0, 0, 3, -12, -11], [0, 0, 0, 10, -2, 2], [-16, -8, -10, 0, 0, 0], [28, 3, -3, 0, 0, 0], [0, 0, 0, -6, 0, -10], [0, 0, 0, 5, 18, 11], [0, 0, 0, 2, -18, 2], [-12, -5, 3, 0, 0, 0], [-12, -7, 2, 0, 0, 0], [-28, 1, 3, 0, 0, 0], [0, 0, 0, -1, 12, 7], [0, 0, 0, -4, 24, -4], [0, 0, 0, -8, 2, 8], [4, 0, 4, 0, 0, 0], [10, 1, 1, 0, 0, 0], [0, 0, 0, 0, 18, 4], [0, 0, 0, -8, -2, 0], [0, 0, 0, -9, 12, -7], [4, -9, 5, 0, 0, 0], [-2, -8, 8, 0, 0, 0], [6, -3, -7, 0, 0, 0], [-16, -10, 12, 0, 0, 0], [0, 0, 0, 8, -8, 0], [0, 0, 0, 5, -14, -9], [-4, 11, 3, 0, 0, 0], [0, 0, 0, 2, -24, 7], [0, 2, -8, 0, 0, 0], [0, 0, 0, -4, 6, -12], [-38, -1, -5, 0, 0, 0], [-2, 6, -6, 0, 0, 0], [0, 0, 0, -3, -2, -5], [0, 0, 0, -1, 12, -8], [0, 0, 0, 12, 0, 10], [30, 4, -4, 0, 0, 0], [-4, -12, -4, 0, 0, 0], [-18, -10, 14, 0, 0, 0], [0, 0, 0, -3, -24, 6], [0, 0, 0, 7, 4, -1], [-18, 1, -3, 0, 0, 0], [-16, 0, 4, 0, 0, 0], [0, 0, 0, -4, 30, 4], [0, 0, 0, 7, 14, 13], [4, -6, -5, 0, 0, 0], [0, 0, 0, 0, -4, -11], [0, 0, 0, 1, -26, -1], [8, 5, 3, 0, 0, 0], [0, 0, 0, -11, 12, -7], [0, 0, 0, 12, 0, 1], [0, 0, 0, -6, -20, 4], [0, -10, 3, 0, 0, 0], [-24, 2, 1, 0, 0, 0], [-6, -9, -5, 0, 0, 0], [0, 0, 0, 8, -6, 8], [0, 6, -8, 0, 0, 0], [0, 0, 0, -1, 44, -5], [0, 0, 0, -2, 2, 2], [0, 0, 0, -14, 16, 8], [28, -1, 3, 0, 0, 0], [0, 0, 0, 1, -34, 3], [0, 0, 0, -10, -4, -19], [-12, 3, 0, 0, 0, 0], [0, 0, 0, -2, 6, 2], [-24, -11, 15, 0, 0, 0], [0, 0, 0, 0, -34, 0], [4, -4, -13, 0, 0, 0], [6, 3, 11, 0, 0, 0], [20, -1, 0, 0, 0, 0], [0, 0, 0, -10, -2, 2], [32, 2, 2, 0, 0, 0], [-2, 0, -4, 0, 0, 0], [-12, 0, 11, 0, 0, 0], [12, -2, -12, 0, 0, 0], [0, 0, 0, 4, -8, -3], [-26, -6, 10, 0, 0, 0], [0, 0, 0, 6, 32, 2], [-40, -6, -1, 0, 0, 0], [0, 0, 0, -16, 4, -2], [0, 0, 0, -9, -18, -3], [0, 0, 0, 8, 24, 9], [22, 8, -8, 0, 0, 0], [0, 0, 0, -5, -22, -3], [0, 0, 0, -7, 4, 3], [-34, -10, 14, 0, 0, 0], [32, 4, -2, 0, 0, 0], [0, 0, 0, -16, -36, -10], [0, 0, 0, -13, -16, -2], [20, 0, 3, 0, 0, 0], [30, 10, 2, 0, 0, 0], [0, 0, 0, 5, 4, 16], [0, 0, 0, 4, -26, 12], [-26, -4, 8, 0, 0, 0], [0, 0, 0, -2, 20, 0], [0, 0, 0, -8, -30, -12], [0, 0, 0, 8, -8, 10], [10, 20, 0, 0, 0, 0], [16, -8, -4, 0, 0, 0], [0, 0, 0, 5, 38, -1], [-12, 1, 7, 0, 0, 0], [-4, -23, 11, 0, 0, 0], [0, 0, 0, -10, -12, -4], [-24, -6, 2, 0, 0, 0], [0, 0, 0, -17, 4, 2], [10, 8, 0, 0, 0, 0], [20, -13, -1, 0, 0, 0], [0, 0, 0, 3, -20, 11], [18, 15, -1, 0, 0, 0], [0, 0, 0, -8, -8, 0], [0, 0, 0, 1, 24, 4], [20, 11, -8, 0, 0, 0], [4, -7, -13, 0, 0, 0], [0, 0, 0, 12, 8, 12], [-26, -10, 14, 0, 0, 0], [0, 0, 0, 6, 8, 12], [0, 0, 0, -7, 10, -13], [8, -10, 0, 0, 0, 0], [-8, 4, 0, 0, 0, 0], [-4, -1, 13, 0, 0, 0], [0, 0, 0, 4, 0, 24], [0, 0, 0, -15, 0, -18], [14, 8, 16, 0, 0, 0], [0, 0, 0, 10, 30, -2], [16, 2, 4, 0, 0, 0], [0, 0, 0, 6, 40, -1], [-8, -16, 7, 0, 0, 0], [0, 0, 0, -1, 18, -11], [-12, 8, 6, 0, 0, 0], [-32, -4, 10, 0, 0, 0], [18, -12, -4, 0, 0, 0], [0, 0, 0, -7, 20, -20], [0, 0, 0, 0, -4, -11], [26, -8, 0, 0, 0, 0], [0, 0, 0, 6, -8, 4], [56, 2, 1, 0, 0, 0], [0, 10, -11, 0, 0, 0], [0, 0, 0, -3, 12, -1], [-36, 6, 5, 0, 0, 0], [0, 0, 0, -13, -4, -10], [-46, -5, 7, 0, 0, 0], [0, 0, 0, -4, 18, 4], [-8, 21, -10, 0, 0, 0], [0, 0, 0, 12, -16, -14], [12, -18, 7, 0, 0, 0], [28, -3, 1, 0, 0, 0], [0, 0, 0, -14, -8, -16], [0, 0, 0, 8, 4, 10], [-4, -18, 4, 0, 0, 0], [0, 0, 0, 22, -4, 3], [-42, 2, 6, 0, 0, 0], [0, 0, 0, -4, 18, 4], [36, 5, 5, 0, 0, 0], [2, -2, 6, 0, 0, 0], [0, 0, 0, -5, 24, 0], [0, 0, 0, 1, -6, -1], [0, 0, 0, 17, 10, 3], [0, 0, 0, -2, -16, -10], [0, 0, 0, 8, 36, 12], [26, -4, -4, 0, 0, 0], [0, 0, 0, 15, -26, 5], [0, 0, 0, -8, 4, 10], [12, -7, 7, 0, 0, 0], [-52, -4, -2, 0, 0, 0], [-10, 14, 6, 0, 0, 0], [0, 0, 0, -15, -8, -10], [0, 0, 0, -8, -44, -2], [10, 16, -20, 0, 0, 0], [0, 0, 0, -21, -4, -9], [0, 0, 0, 0, -24, 2], [16, -5, 14, 0, 0, 0], [0, 0, 0, 20, -26, -12], [0, 0, 0, 8, -8, 23], [0, 0, 0, -5, 0, 0], [14, 0, -8, 0, 0, 0], [-50, -3, -7, 0, 0, 0], [-4, 10, -9, 0, 0, 0], [0, 0, 0, 13, 8, 6], [-4, -5, -16, 0, 0, 0], [0, 0, 0, -2, -16, -3], [-10, -10, -10, 0, 0, 0], [0, 0, 0, 14, 4, 20], [16, -2, -4, 0, 0, 0], [0, 0, 0, -19, -12, -11], [0, 0, 0, -15, 14, 7], [0, -6, -3, 0, 0, 0], [-14, 20, 0, 0, 0, 0], [6, 5, -19, 0, 0, 0], [0, 0, 0, 8, 20, 16], [0, 0, 0, 2, -18, -2], [20, -19, 5, 0, 0, 0], [0, 0, 0, -5, -10, -11], [-38, -16, 16, 0, 0, 0], [-24, 26, -4, 0, 0, 0], [0, 0, 0, 0, -26, -4], [12, 0, -1, 0, 0, 0], [0, 0, 0, -3, -56, 4], [0, 0, 0, 13, 22, 15], [0, 0, 0, -8, -18, -12], [48, 4, -6, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1850_b_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1850_2_b_o();".
// 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_1850_2_b_o(:prec:=6)
    chi := MakeCharacter_1850_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1850_2_b_o();".
// 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_1850_2_b_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1850_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, 0, 100, 0, 20, 0, 1]>,<7,R![100, 0, 84, 0, 17, 0, 1]>,<13,R![1024, 0, 1600, 0, 80, 0, 1]>],Snew);
    return Vf;
end function;