
// Make newform 156.3.g.b in Magma, downloaded from the LMFDB on 09 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_156_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_156_g_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_156_3_g_b();".
// 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_156_3_g_b();".
// 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 := [9, 0, -5, 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, -2, 0, 1], [-3, 0, 1, 0]];
    Rf_basisdens := [1, 1, 3, 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_156_g();"
function MakeCharacter_156_g()
    N := 156;
    order := 2;
    char_gens := [79, 53, 145];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_156_g_Hecke();"
function MakeCharacter_156_g_Hecke( : Kf := false)
    N := 156;
    order := 2;
    char_gens := [79, 53, 145];
    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 := 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 := 3;
    raw_aps := [[0, 0, 0, 0], [-2, 0, 0, 1], [0, -2, 1, 0], [0, 0, -7, 0], [0, 12, -6, 0], [0, 0, -13, 0], [7, 0, 0, 14], [0, 0, -30, 0], [-8, 0, 0, -16], [-10, 0, 0, -20], [0, 0, -10, 0], [0, 0, 13, 0], [0, -28, 14, 0], [55, 0, 0, 0], [0, -30, 15, 0], [12, 0, 0, 24], [0, 56, -28, 0], [14, 0, 0, 0], [0, 0, -16, 0], [0, -18, 9, 0], [0, 0, 4, 0], [54, 0, 0, 0], [0, -20, 10, 0], [0, 12, -6, 0], [0, 0, 78, 0], [20, 0, 0, 40], [-70, 0, 0, 0], [-28, 0, 0, -56], [0, 0, -175, 0], [-4, 0, 0, -8], [140, 0, 0, 0], [25, 0, 0, 50], [0, -40, 20, 0], [-11, 0, 0, 0], [0, -8, 4, 0], [0, 0, 215, 0], [-140, 0, 0, 0], [0, 0, 188, 0], [0, -120, 60, 0], [66, 0, 0, 132], [75, 0, 0, 150], [94, 0, 0, 0], [30, 0, 0, 60], [0, 0, -82, 0], [0, 70, -35, 0], [252, 0, 0, 0], [-153, 0, 0, 0], [0, 0, 29, 0], [0, -20, 10, 0], [0, 0, -385, 0], [57, 0, 0, 114], [0, 262, -131, 0], [0, 0, 130, 0], [-100, 0, 0, -200], [81, 0, 0, 162], [-54, 0, 0, -108], [50, 0, 0, 100], [0, 0, -245, 0], [430, 0, 0, 0], [0, 196, -98, 0], [70, 0, 0, 0], [0, 50, -25, 0], [0, 0, -232, 0], [-70, 0, 0, -140], [-155, 0, 0, 0], [0, -120, 60, 0], [0, 0, 490, 0], [475, 0, 0, 0], [-63, 0, 0, -126], [0, 0, 85, 0], [0, -140, 70, 0], [0, 56, -28, 0], [140, 0, 0, 0], [-60, 0, 0, 0], [0, 0, -220, 0], [0, 70, -35, 0], [0, 0, 0, 0], [0, 0, 478, 0], [0, -384, 192, 0], [0, 0, 200, 0], [145, 0, 0, 290], [0, 0, 385, 0], [0, -154, 77, 0], [235, 0, 0, 0], [-336, 0, 0, 0], [197, 0, 0, 394], [0, -108, 54, 0], [0, 0, 518, 0], [0, 462, -231, 0], [0, 0, -462, 0], [-168, 0, 0, -336], [0, -18, 9, 0], [0, 0, -462, 0], [-175, 0, 0, -350], [0, 0, -350, 0], [126, 0, 0, 252], [0, 72, -36, 0], [105, 0, 0, 210], [910, 0, 0, 0], [0, 0, 755, 0], [715, 0, 0, 0], [0, 210, -105, 0], [-223, 0, 0, -446], [-155, 0, 0, -310], [479, 0, 0, 0], [0, 0, 448, 0], [0, 440, -220, 0], [0, 460, -230, 0], [-240, 0, 0, -480], [-101, 0, 0, 0], [-1100, 0, 0, 0], [0, 0, 338, 0], [0, -560, 280, 0], [0, 0, 770, 0], [0, 0, -385, 0], [-180, 0, 0, -360], [0, 0, 168, 0], [112, 0, 0, 224], [-224, 0, 0, -448], [-220, 0, 0, -440], [0, 0, 1270, 0], [-1035, 0, 0, 0], [42, 0, 0, 84], [0, -420, 210, 0], [0, 0, 100, 0], [340, 0, 0, 680], [0, 0, 330, 0], [140, 0, 0, 280], [-130, 0, 0, 0], [0, 0, -817, 0], [0, 0, -310, 0], [0, -850, 425, 0], [674, 0, 0, 0], [860, 0, 0, 0], [0, 576, -288, 0], [0, 0, -540, 0], [0, -750, 375, 0], [0, 0, -626, 0], [-338, 0, 0, -676], [-5, 0, 0, -10], [0, 0, 740, 0], [0, 582, -291, 0], [420, 0, 0, 0], [0, 60, -30, 0], [-1008, 0, 0, 0], [0, 712, -356, 0], [0, 0, -797, 0], [-28, 0, 0, -56], [34, 0, 0, 0], [0, -70, 35, 0], [0, 0, 679, 0], [-135, 0, 0, -270], [525, 0, 0, 0], [402, 0, 0, 804], [445, 0, 0, 0], [150, 0, 0, 300], [-756, 0, 0, 0], [0, 432, -216, 0], [10, 0, 0, 0], [0, -878, 439, 0], [0, -260, 130, 0], [87, 0, 0, 174], [0, 0, 109, 0], [-525, 0, 0, -1050], [0, -240, 120, 0], [0, -270, 135, 0], [-266, 0, 0, 0], [1890, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_156_g_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_156_3_g_b();".
// 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_156_3_g_b(:prec:=4)
    chi := MakeCharacter_156_g();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_156_3_g_b();".
// 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_156_3_g_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_156_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-11, 0, 1]>],Snew);
    return Vf;
end function;