
// Make newform 288.4.d.a in Magma, downloaded from the LMFDB on 04 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_288_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_288_d_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_288_4_d_a();".
// 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_288_4_d_a();".
// 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, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [-2, 4]];
    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_288_d();"
function MakeCharacter_288_d()
    N := 288;
    order := 2;
    char_gens := [127, 37, 65];
    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_288_d_Hecke();"
function MakeCharacter_288_d_Hecke( : Kf := false)
    N := 288;
    order := 2;
    char_gens := [127, 37, 65];
    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 := 4;
    raw_aps := [[0, 0], [0, 0], [0, -2], [8, 0], [0, 3], [0, -10], [14, 0], [0, -7], [-152, 0], [0, -30], [-224, 0], [0, -46], [70, 0], [0, -83], [336, 0], [0, 6], [0, -101], [0, -18], [0, 33], [-72, 0], [-294, 0], [464, 0], [0, 103], [-266, 0], [994, 0], [0, 142], [-1176, 0], [0, 51], [0, 358], [1710, 0], [1664, 0], [0, 127], [1062, 0], [0, 509], [0, 150], [-744, 0], [0, -34], [0, -335], [-1960, 0], [0, 378], [0, -617], [0, 442], [3904, 0], [3330, 0], [0, -226], [1736, 0], [0, -551], [-1568, 0], [0, 239], [0, -974], [838, 0], [6288, 0], [-2926, 0], [0, -1029], [-2562, 0], [-5896, 0], [0, -1014], [1680, 0], [0, 298], [2742, 0], [0, 565], [0, -1746], [0, 489], [-2744, 0], [2282, 0], [0, 1810], [0, -803], [6130, 0], [0, -469], [0, 62], [10206, 0], [-3176, 0], [11760, 0], [0, 2074], [0, 581], [2688, 0], [0, 1982], [0, 1078], [-12402, 0], [-12278, 0], [0, 1551], [0, 274], [7632, 0], [3794, 0], [1848, 0], [0, 2331], [3582, 0], [2714, 0], [0, 1578], [-2224, 0], [0, -1945], [17696, 0], [-1304, 0], [0, -3149], [0, 585], [-19320, 0], [0, -846], [2982, 0], [0, 381], [0, -2914], [0, -2383], [0, -1350], [0, 1431], [-15594, 0], [0, -3163], [6594, 0], [0, 4387], [-14322, 0], [-16088, 0], [-21238, 0], [13664, 0], [0, -3854], [3782, 0], [0, 1101], [-2056, 0], [-11842, 0], [0, -3071], [19320, 0], [0, -438], [0, -5241], [0, -4246], [-10078, 0], [0, 3054], [0, 3091], [0, -5527], [0, 466], [0, 5986], [-9296, 0], [-21672, 0], [0, 1790], [0, 1297], [17432, 0], [11632, 0], [0, 3162], [-39466, 0], [35266, 0], [0, 3070], [0, 6585], [0, 482], [24390, 0], [0, -1811], [0, -1658], [21688, 0], [0, 3675], [0, 3694], [-18760, 0], [0, -5430], [-8778, 0], [0, -1067], [-9312, 0], [0, -26], [31150, 0], [0, 2377], [37128, 0], [0, 6653], [-10512, 0], [46104, 0], [5726, 0], [1274, 0], [0, -4998], [0, -4521], [-38250, 0], [-4664, 0], [0, -5853], [4814, 0], [-12376, 0], [-45344, 0], [0, 4914]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_288_d_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_288_4_d_a();".
// 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_288_4_d_a(:prec:=2)
    chi := MakeCharacter_288_d();
    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_288_4_d_a();".
// 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_288_4_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_288_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![112, 0, 1]>],Snew);
    return Vf;
end function;