
// Make newform 384.3.g.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_384_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_384_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_384_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_384_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 := [1, 0, 0, 0, -1, 0, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 2, 0, 0, 0], [0, 4, 0, -4, 0, -4, 0, 0], [0, -2, -4, -2, 0, 2, 2, 0], [0, -2, 0, 2, 0, -2, 2, -4], [0, 4, 0, 4, 0, 4, 0, -8], [0, 4, -8, 4, 0, -4, 4, 0], [0, -2, 0, 2, 0, -2, -6, -4]];
    Rf_basisdens := [1, 1, 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_g();"
function MakeCharacter_384_g()
    N := 384;
    order := 2;
    char_gens := [127, 133, 257];
    v := [1, 2, 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_g_Hecke();"
function MakeCharacter_384_g_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 := 3;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [2, 0, 0, -1, 0, 0, 0, 0], [0, 0, -1, 0, -1, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, -1], [-6, 0, 0, 0, 0, -1, 1, 0], [2, 0, 0, 0, 0, 2, 1, 0], [0, -4, 2, 0, -3, 0, 0, 1], [0, 0, 2, 0, -2, 0, 0, 0], [10, 0, 0, 5, 0, -2, 0, 0], [0, -8, -1, 0, -3, 0, 0, -2], [2, 0, 0, -6, 0, -3, -1, 0], [10, 0, 0, 0, 0, 2, -5, 0], [0, 12, 2, 0, -1, 0, 0, -5], [0, 8, 2, 0, -6, 0, 0, 0], [-22, 0, 0, -3, 0, -2, -4, 0], [0, -12, -4, 0, -8, 0, 0, 8], [34, 0, 0, -2, 0, 1, 7, 0], [0, -4, 12, 0, 0, 0, 0, 0], [0, -24, 2, 0, 6, 0, 0, -8], [-2, 0, 0, -20, 0, -2, 2, 0], [0, 8, -9, 0, 7, 0, 0, 4], [0, 0, 4, 0, 19, 0, 0, -5], [-30, 0, 0, -8, 0, -4, -2, 0], [50, 0, 0, -24, 0, 0, 10, 0], [66, 0, 0, 5, 0, 2, -10, 0], [0, -64, -11, 0, 3, 0, 0, 10], [0, 20, 0, 0, 2, 0, 0, 10], [-70, 0, 0, -4, 0, 3, 9, 0], [42, 0, 0, 8, 0, -8, -6, 0], [0, -8, -19, 0, 11, 0, 0, -10], [0, -12, -24, 0, 8, 0, 0, -8], [-22, 0, 0, 8, 0, -2, 1, 0], [0, -44, 12, 0, 12, 0, 0, 4], [-78, 0, 0, 15, 0, 8, 0, 0], [0, 64, 3, 0, -7, 0, 0, 18], [42, 0, 0, -6, 0, -3, -5, 0], [0, 36, 10, 0, 11, 0, 0, 15], [0, -128, 12, 0, 8, 0, 0, 0], [74, 0, 0, -9, 0, 4, -10, 0], [0, -20, 20, 0, 16, 0, 0, -16], [34, 0, 0, 28, 0, -13, -3, 0], [0, -72, 16, 0, -20, 0, 0, 12], [-18, 0, 0, -44, 0, 2, -2, 0], [74, 0, 0, -7, 0, -6, 20, 0], [0, -112, 1, 0, -17, 0, 0, -6], [0, 28, -4, 0, -24, 0, 0, 0], [0, -72, 13, 0, 17, 0, 0, 0], [0, 8, 12, 0, -7, 0, 0, -15], [-94, 0, 0, 16, 0, -9, 9, 0], [-38, 0, 0, 16, 0, -8, 18, 0], [0, 120, -20, 0, -36, 0, 0, 0], [70, 0, 0, 12, 0, 26, -12, 0], [0, 16, 32, 0, -33, 0, 0, 15], [-62, 0, 0, -8, 0, 8, -16, 0], [0, -104, -12, 0, 16, 0, 0, -40], [98, 0, 0, -27, 0, -22, 2, 0], [0, 8, 21, 0, -47, 0, 0, 0], [210, 0, 0, 12, 0, -5, 5, 0], [-118, 0, 0, 16, 0, 12, 4, 0], [0, 12, -40, 0, -8, 0, 0, 0], [218, 0, 0, 37, 0, -2, -8, 0], [0, -84, -4, 0, 0, 0, 0, 0], [0, 88, 56, 0, -4, 0, 0, -24], [142, 0, 0, 36, 0, 2, -10, 0], [-222, 0, 0, -23, 0, -2, -2, 0], [0, -44, -48, 0, 10, 0, 0, -22], [-114, 0, 0, -4, 0, 6, -6, 0], [0, -120, 20, 0, -9, 0, 0, 15], [-30, 0, 0, 46, 0, 1, -21, 0], [74, 0, 0, 16, 0, -16, -6, 0], [0, -152, -54, 0, 18, 0, 0, 4], [0, 56, 47, 0, 33, 0, 0, -30], [106, 0, 0, 22, 0, 41, -1, 0], [0, 36, -38, 0, -7, 0, 0, 21], [0, 56, 16, 0, 56, 0, 0, 0], [298, 0, 0, -25, 0, 4, 10, 0], [154, 0, 0, -26, 0, 9, -13, 0], [-46, 0, 0, 80, 0, -10, -5, 0], [-14, 0, 0, 24, 0, -16, -34, 0], [0, 248, -4, 0, 53, 0, 0, -19], [-206, 0, 0, -52, 0, 3, -23, 0], [0, 8, -74, 0, -6, 0, 0, -12], [66, 0, 0, 8, 0, -44, 8, 0], [0, 64, 13, 0, -25, 0, 0, 22], [0, 56, -20, 0, 87, 0, 0, 15], [66, 0, 0, -88, 0, -14, 53, 0], [-142, 0, 0, 24, 0, -32, -6, 0], [74, 0, 0, 71, 0, 4, 34, 0], [0, 40, -31, 0, 47, 0, 0, 30], [0, 72, 44, 0, -59, 0, 0, 45], [0, -112, 6, 0, -2, 0, 0, 0], [0, -80, 31, 0, 31, 0, 0, 0], [0, 172, -36, 0, 22, 0, 0, -34], [0, -140, -40, 0, 40, 0, 0, 0], [0, 72, -38, 0, 74, 0, 0, 20], [-102, 0, 0, 105, 0, -22, -40, 0], [-158, 0, 0, -72, 0, 22, -13, 0], [0, -284, 10, 0, -23, 0, 0, 45], [98, 0, 0, 36, 0, 19, 49, 0], [0, 276, -10, 0, -43, 0, 0, 25], [50, 0, 0, -53, 0, -60, 12, 0], [0, -40, -96, 0, 15, 0, 0, 15], [58, 0, 0, 112, 0, -38, -37, 0], [0, -36, -12, 0, -138, 0, 0, 6], [-90, 0, 0, 52, 0, 50, 6, 0], [0, -196, -36, 0, 10, 0, 0, -30], [-374, 0, 0, 24, 0, -24, 26, 0], [0, -80, 30, 0, -90, 0, 0, -44], [-26, 0, 0, -108, 0, 26, -22, 0], [0, 8, -19, 0, -73, 0, 0, 10], [-54, 0, 0, -114, 0, 41, 43, 0], [154, 0, 0, 104, 0, 4, -48, 0], [0, 188, -24, 0, -60, 0, 0, 20], [0, -176, -57, 0, -1, 0, 0, -64], [106, 0, 0, 56, 0, 38, -11, 0], [0, -332, 70, 0, 65, 0, 0, -35], [0, 128, -62, 0, -110, 0, 0, 20], [90, 0, 0, 123, 0, 40, -54, 0], [0, 132, 4, 0, -54, 0, 0, 18], [362, 0, 0, -150, 0, -67, -5, 0], [-270, 0, 0, -8, 0, -60, 44, 0], [450, 0, 0, 107, 0, -60, 16, 0], [0, 256, 76, 0, 17, 0, 0, 25], [0, -188, 114, 0, -101, 0, 0, -1], [-126, 0, 0, -47, 0, -34, 62, 0], [-414, 0, 0, 112, 0, -25, 13, 0], [0, -16, -62, 0, 18, 0, 0, -64], [0, -64, -77, 0, -51, 0, 0, -50], [386, 0, 0, -120, 0, -9, 21, 0], [0, 148, 36, 0, 38, 0, 0, -90], [0, 224, 120, 0, 36, 0, 0, 80], [0, -280, -25, 0, -41, 0, 0, 52], [290, 0, 0, 0, 0, 15, 81, 0], [258, 0, 0, 56, 0, 66, -51, 0], [-274, 0, 0, 36, 0, -22, -46, 0], [1090, 0, 0, 9, 0, 30, 10, 0], [0, 92, 22, 0, -73, 0, 0, -5], [-62, 0, 0, -29, 0, 28, -64, 0], [42, 0, 0, 80, 0, -10, 105, 0], [0, 532, -26, 0, -109, 0, 0, 55], [-334, 0, 0, 41, 0, -2, -66, 0], [0, -256, -117, 0, 79, 0, 0, 20], [0, 20, 32, 0, 78, 0, 0, -90], [138, 0, 0, 12, 0, 3, -47, 0], [0, 232, 54, 0, 18, 0, 0, 64], [202, 0, 0, -78, 0, 77, 59, 0], [-550, 0, 0, 216, 0, 50, 27, 0], [0, -20, -130, 0, -79, 0, 0, -3], [0, 328, -64, 0, 156, 0, 0, -20], [-670, 0, 0, -58, 0, 65, -5, 0], [-30, 0, 0, 8, 0, -32, 52, 0], [0, -380, 50, 0, 111, 0, 0, -5], [0, -424, 124, 0, -128, 0, 0, 40], [0, -500, -82, 0, -11, 0, 0, -95], [0, -8, 44, 0, 136, 0, 0, -92], [0, -304, 27, 0, 191, 0, 0, -12], [-1230, 0, 0, -8, 0, -10, -17, 0], [242, 0, 0, 96, 0, -48, -8, 0], [42, 0, 0, 127, 0, -44, -2, 0], [0, 12, 20, 0, 128, 0, 0, 80], [386, 0, 0, -168, 0, -74, 43, 0], [0, 208, -31, 0, 13, 0, 0, 60], [0, -320, 180, 0, 61, 0, 0, -43], [-302, 0, 0, -56, 0, -42, -113, 0], [0, -264, 100, 0, 152, 0, 0, -40], [0, 184, -17, 0, -93, 0, 0, 136], [-350, 0, 0, 54, 0, -55, 7, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_384_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_384_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_384_3_g_b(:prec:=8)
    chi := MakeCharacter_384_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_384_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_384_3_g_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_384_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-128, 128, -16, -8, 1]>],Snew);
    return Vf;
end function;