
// Make newform 384.4.d.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_384_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_384_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_384_4_d_f();".
// 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_384_4_d_f();".
// 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, 210, 0, 107, 0, 18, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, -87, 0, -33, 0, -3, 0, 0], [-3, 0, -2, 0, 6, 0, 1, 0], [0, 328, 0, 104, 0, 8, 0, 0], [-48, 0, 272, 0, 96, 0, 8, 0], [72, 0, 864, 0, 288, 0, 24, 0], [0, 253, 0, 155, 0, 31, 0, 2], [0, 377, 0, 487, 0, 131, 0, 10]];
    Rf_basisdens := [1, 2, 1, 1, 1, 1, 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_384_d();"
function MakeCharacter_384_d()
    N := 384;
    order := 2;
    char_gens := [127, 133, 257];
    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_384_d_Hecke();"
function MakeCharacter_384_d_Hecke(Kf)
    N := 384;
    order := 2;
    char_gens := [127, 133, 257];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, 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+1)]];
    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, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, -1, 0, 0, 0, 0, 0], [0, -4, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, -1], [30, 0, 0, 0, 0, -1, 0, 0], [0, 4, 0, -3, 0, 0, 0, 0], [0, 0, 2, 0, -4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 2], [0, 0, 5, 0, 4, 0, 0, 0], [0, 0, 0, 0, 0, 0, 17, -1], [-102, 0, 0, 0, 0, -5, 0, 0], [0, -44, 0, -3, 0, 0, 0, 0], [0, 0, -14, 0, -8, 0, 0, 0], [0, 0, 0, 0, 0, 0, -13, -2], [0, -156, 0, 16, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -49, 5], [0, 196, 0, -12, 0, 0, 0, 0], [0, 0, 30, 0, 12, 0, 0, 0], [-242, 0, 0, 0, 0, -8, 0, 0], [0, 0, -11, 0, -20, 0, 0, 0], [0, 284, 0, 17, 0, 0, 0, 0], [918, 0, 0, 0, 0, -6, 0, 0], [-370, 0, 0, 0, 0, 2, 0, 0], [0, 0, 0, 0, 0, 0, -9, 14], [0, 0, 1, 0, -8, 0, 0, 0], [0, -236, 0, 10, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 35, -1], [-990, 0, 0, 0, 0, 6, 0, 0], [0, 0, 27, 0, 28, 0, 0, 0], [0, 244, 0, 28, 0, 0, 0, 0], [474, 0, 0, 0, 0, 25, 0, 0], [0, -92, 0, -84, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 11, -32], [0, 0, 51, 0, -40, 0, 0, 0], [0, 0, 0, 0, 0, 0, -49, -31], [0, 132, 0, -81, 0, 0, 0, 0], [0, 0, 36, 0, -36, 0, 0, 0], [0, 0, 0, 0, 0, 0, -191, 32], [0, 628, 0, 100, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -179, 1], [0, 0, -36, 0, -72, 0, 0, 0], [1702, 0, 0, 0, 0, 16, 0, 0], [0, 0, 0, 0, 0, 0, 51, 10], [0, 0, -95, 0, 104, 0, 0, 0], [0, -12, 0, -144, 0, 0, 0, 0], [0, 0, -1, 0, -100, 0, 0, 0], [0, -852, 0, 47, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -79, 65], [1590, 0, 0, 0, 0, -14, 0, 0], [0, 0, -56, 0, -32, 0, 0, 0], [2950, 0, 0, 0, 0, 6, 0, 0], [0, -564, 0, -133, 0, 0, 0, 0], [-1950, 0, 0, 0, 0, -16, 0, 0], [0, 0, 224, 0, 20, 0, 0, 0], [0, 0, 0, 0, 0, 0, -271, -22], [0, 0, 239, 0, 148, 0, 0, 0], [0, 0, 0, 0, 0, 0, -239, -59], [3654, 0, 0, 0, 0, -24, 0, 0], [0, -732, 0, -108, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -333, -90], [0, 4, 0, 132, 0, 0, 0, 0], [0, 0, -28, 0, 92, 0, 0, 0], [-1214, 0, 0, 0, 0, -104, 0, 0], [0, 0, 0, 0, 0, 0, -115, 94], [0, 2868, 0, 18, 0, 0, 0, 0], [1258, 0, 0, 0, 0, -48, 0, 0], [0, -676, 0, 251, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -557, 1], [-4974, 0, 0, 0, 0, -82, 0, 0], [0, 0, -106, 0, -4, 0, 0, 0], [0, 0, -11, 0, -228, 0, 0, 0], [0, 0, 0, 0, 0, 0, 365, 23], [0, -572, 0, 375, 0, 0, 0, 0], [0, 0, 100, 0, -56, 0, 0, 0], [0, 0, 0, 0, 0, 0, 67, -24], [0, 0, 0, 0, 0, 0, 231, -75], [-10050, 0, 0, 0, 0, 5, 0, 0], [-10054, 0, 0, 0, 0, 50, 0, 0], [0, -2628, 0, 119, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 177, 69], [0, 0, -250, 0, -328, 0, 0, 0], [2366, 0, 0, 0, 0, -52, 0, 0], [0, 0, 357, 0, 16, 0, 0, 0], [0, -2580, 0, -277, 0, 0, 0, 0], [6702, 0, 0, 0, 0, 43, 0, 0], [1034, 0, 0, 0, 0, 38, 0, 0], [0, 0, 0, 0, 0, 0, -731, -192], [0, 0, -561, 0, 68, 0, 0, 0], [0, 1532, 0, -469, 0, 0, 0, 0], [0, 0, 346, 0, -152, 0, 0, 0], [0, 0, -165, 0, 240, 0, 0, 0], [0, 3700, 0, -170, 0, 0, 0, 0], [0, 3140, 0, 300, 0, 0, 0, 0], [0, 0, -2, 0, -140, 0, 0, 0], [0, 0, 0, 0, 0, 0, 657, 22], [-2550, 0, 0, 0, 0, 271, 0, 0], [0, -588, 0, 315, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 619, 79], [0, 532, 0, 789, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -71, 108], [0, 3932, 0, 89, 0, 0, 0, 0], [3882, 0, 0, 0, 0, 59, 0, 0], [0, -4060, 0, 426, 0, 0, 0, 0], [-4214, 0, 0, 0, 0, 168, 0, 0], [0, -1916, 0, 334, 0, 0, 0, 0], [-3582, 0, 0, 0, 0, -66, 0, 0], [0, 0, 654, 0, -84, 0, 0, 0], [-15218, 0, 0, 0, 0, 172, 0, 0], [0, 0, 243, 0, 108, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1219, -49], [726, 0, 0, 0, 0, 52, 0, 0], [0, 1796, 0, 228, 0, 0, 0, 0], [0, 0, -525, 0, -200, 0, 0, 0], [17502, 0, 0, 0, 0, 175, 0, 0], [0, 4948, 0, 645, 0, 0, 0, 0], [0, 0, 18, 0, 324, 0, 0, 0], [0, 0, 0, 0, 0, 0, 477, 220], [0, 3268, 0, 478, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1223, -89], [-8434, 0, 0, 0, 0, -256, 0, 0], [0, 0, 0, 0, 0, 0, 1291, -164], [0, 2028, 0, -619, 0, 0, 0, 0], [0, 2724, 0, -369, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -639, -50], [0, 0, 0, 0, 0, 0, -539, 325], [0, 0, 470, 0, -112, 0, 0, 0], [0, 0, -281, 0, 440, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1287, 351], [0, 1316, 0, -330, 0, 0, 0, 0], [0, 0, 664, 0, 76, 0, 0, 0], [0, 0, -183, 0, -148, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1533, 93], [8730, 0, 0, 0, 0, 81, 0, 0], [-7226, 0, 0, 0, 0, -308, 0, 0], [0, 0, 0, 0, 0, 0, 911, -46], [0, -2028, 0, -1197, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 333, 292], [-34662, 0, 0, 0, 0, -71, 0, 0], [0, 8372, 0, -267, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1227, -6], [0, 0, 183, 0, 488, 0, 0, 0], [0, 468, 0, -542, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 915, 447], [0, 0, -1246, 0, -532, 0, 0, 0], [0, 0, 0, 0, 0, 0, -539, 163], [2538, 0, 0, 0, 0, 99, 0, 0], [0, 628, 0, -993, 0, 0, 0, 0], [0, 0, -820, 0, -808, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2689, -115], [4914, 0, 0, 0, 0, 60, 0, 0], [0, 692, 0, 255, 0, 0, 0, 0], [0, 0, -1632, 0, -948, 0, 0, 0], [0, -8908, 0, -429, 0, 0, 0, 0], [0, 0, 1528, 0, -32, 0, 0, 0], [0, 0, 1039, 0, -168, 0, 0, 0], [43758, 0, 0, 0, 0, 81, 0, 0], [17990, 0, 0, 0, 0, 200, 0, 0], [0, 0, 0, 0, 0, 0, 2453, -200], [0, 2212, 0, 748, 0, 0, 0, 0], [-870, 0, 0, 0, 0, -529, 0, 0], [0, 0, -571, 0, -680, 0, 0, 0], [0, -1012, 0, -811, 0, 0, 0, 0], [-8034, 0, 0, 0, 0, 473, 0, 0], [0, 0, 416, 0, 284, 0, 0, 0], [0, 0, -1007, 0, -700, 0, 0, 0], [0, 0, 0, 0, 0, 0, 201, -69]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_384_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_384_4_d_f();".
// 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_384_4_d_f(:prec:=8)
    chi := MakeCharacter_384_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_384_4_d_f();".
// 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_384_4_d_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_384_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![9792, 0, 336, 0, 1]>,<7,R![9792, 0, -1104, 0, 1]>],Snew);
    return Vf;
end function;