
// Make newform 550.4.b.l in Magma, downloaded from the LMFDB on 22 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_550_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_550_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_550_4_b_l();".
// 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_550_4_b_l();".
// 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;
    Rf_num := [[1, 0], [0, 2]];
    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_550_b();"
function MakeCharacter_550_b()
    N := 550;
    order := 2;
    char_gens := [177, 101];
    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_550_b_Hecke();"
function MakeCharacter_550_b_Hecke( : Kf := false)
    N := 550;
    order := 2;
    char_gens := [177, 101];
    char_values := [[-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 := 4;
    raw_aps := [[0, -1], [0, 4], [0, 0], [0, 6], [-11, 0], [0, -17], [0, 43], [4, 0], [0, 74], [-134, 0], [-280, 0], [0, -215], [-6, 0], [0, -68], [0, 14], [0, -329], [-4, 0], [-90, 0], [0, -48], [816, 0], [0, -215], [-1296, 0], [0, -304], [-810, 0], [0, -353], [-18, 0], [0, 866], [0, -208], [-1638, 0], [0, 97], [0, 746], [-732, 0], [0, 1163], [-1444, 0], [-574, 0], [1072, 0], [0, 925], [0, 1180], [0, -1466], [0, -2001], [2244, 0], [-3026, 0], [312, 0], [0, 1837], [0, 2293], [-3000, 0], [5652, 0], [0, 1630], [0, 1932], [-1982, 0], [0, -3423], [2384, 0], [6354, 0], [2812, 0], [0, 1247], [0, 2454], [-4982, 0], [-7800, 0], [0, 3037], [-8182, 0], [0, -3148], [0, 1035], [0, -1332], [-5168, 0], [0, -939], [0, 1269], [-5748, 0], [0, 4003], [0, 232], [-3078, 0], [0, -5615], [-1576, 0], [0, 3574], [0, 91], [-180, 0], [0, 1110], [-10814, 0], [0, 4093], [3378, 0], [3142, 0], [-748, 0], [-1378, 0], [11160, 0], [0, 6601], [-13928, 0], [0, 7288], [6254, 0], [0, -4409], [9510, 0], [0, -4394], [0, 8880], [-5280, 0], [0, 3362], [17436, 0], [-4348, 0], [0, -8946], [7354, 0], [6490, 0], [0, -6252], [6422, 0], [0, -4564], [0, -3055], [0, -5176], [-1562, 0], [-5956, 0], [0, 5559], [0, -5292], [0, 3293], [23832, 0], [-19990, 0], [0, -8390], [0, 13019], [0, 2691], [-3316, 0], [-27456, 0], [13266, 0], [0, -11860], [0, 9946], [0, 3579], [-2812, 0], [3566, 0], [0, 14109], [0, -6035], [0, 6304], [11780, 0], [-26938, 0], [3746, 0], [-17040, 0], [0, -4246], [0, 16199], [-6636, 0], [0, 2654], [-3832, 0], [0, -1447], [-5910, 0], [-21394, 0], [0, 815], [0, -524], [0, 6885], [39318, 0], [6172, 0], [-8882, 0], [0, -16110], [0, -5112], [33146, 0], [-264, 0], [0, -7613], [0, 10023], [-12724, 0], [0, -994], [0, -15871], [-34142, 0], [0, 10700], [0, -20378], [0, 1732], [14488, 0], [-10712, 0], [-626, 0], [0, -6033], [44102, 0], [0, 3352], [0, -12959], [0, 19198], [33516, 0], [0, -5897], [0, 13682], [-36440, 0], [0, -8347]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_550_b_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_550_4_b_l();".
// 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_550_4_b_l(:prec:=2)
    chi := MakeCharacter_550_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_550_4_b_l();".
// 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_550_4_b_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_550_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![64, 0, 1]>,<7,R![144, 0, 1]>],Snew);
    return Vf;
end function;