
// Make newform 192.3.g.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_192_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_192_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_192_3_g_c();".
// 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_192_3_g_c();".
// 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, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 0, 4], [-1, 0, 2, 0], [0, 8, 0, -4]];
    Rf_basisdens := [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_192_g();"
function MakeCharacter_192_g()
    N := 192;
    order := 2;
    char_gens := [127, 133, 65];
    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_192_g_Hecke();"
function MakeCharacter_192_g_Hecke(Kf)
    N := 192;
    order := 2;
    char_gens := [127, 133, 65];
    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 := 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, -1, 0], [2, 0, 0, -1], [0, -1, 4, 0], [0, -2, -4, 0], [-10, 0, 0, -2], [-6, 0, 0, -2], [0, 6, 4, 0], [0, -2, 0, 0], [10, 0, 0, 1], [0, -1, -20, 0], [-18, 0, 0, 4], [-22, 0, 0, 2], [0, 10, -20, 0], [0, -14, -8, 0], [-6, 0, 0, 1], [0, -8, -44, 0], [14, 0, 0, 0], [0, -8, 28, 0], [0, -10, 48, 0], [-30, 0, 0, 8], [0, 19, 12, 0], [0, 14, 4, 0], [-78, 0, 0, 4], [-62, 0, 0, -12], [-166, 0, 0, 1], [0, 1, 12, 0], [0, -12, 52, 0], [6, 0, 0, 2], [82, 0, 0, -12], [0, -3, -92, 0], [0, 16, -4, 0], [26, 0, 0, -26], [0, -16, -60, 0], [-62, 0, 0, -9], [0, 7, -44, 0], [-194, 0, 0, 4], [0, 18, -60, 0], [0, 56, -8, 0], [-46, 0, 0, -17], [0, -24, -52, 0], [54, 0, 0, -14], [0, -8, -32, 0], [82, 0, 0, -8], [106, 0, 0, 21], [0, -43, 60, 0], [0, -56, -20, 0], [0, 17, -44, 0], [0, 58, 4, 0], [-26, 0, 0, 30], [-302, 0, 0, -12], [0, 4, 224, 0], [-190, 0, 0, 20], [0, -18, 44, 0], [130, 0, 0, -32], [0, -64, -8, 0], [138, 0, 0, 9], [0, 81, 100, 0], [22, 0, 0, 34], [194, 0, 0, 0], [0, -64, -44, 0], [26, 0, 0, -71], [0, 24, -180, 0], [0, -68, 56, 0], [-46, 0, 0, -56], [154, 0, 0, -35], [0, 20, -108, 0], [-94, 0, 0, 8], [0, -58, 172, 0], [-146, 0, 0, -32], [242, 0, 0, -12], [0, -42, 16, 0], [0, 31, 220, 0], [62, 0, 0, -40], [0, 46, 12, 0], [0, -64, -192, 0], [98, 0, 0, 63], [174, 0, 0, 24], [74, 0, 0, -54], [98, 0, 0, 60], [0, -94, 164, 0], [278, 0, 0, 34], [0, -18, -216, 0], [210, 0, 0, -24], [0, 97, -4, 0], [0, -106, 140, 0], [-198, 0, 0, 86], [610, 0, 0, -12], [-494, 0, 0, -57], [0, -103, -60, 0], [0, 18, 212, 0], [0, 14, -136, 0], [0, -153, 68, 0], [0, 4, -140, 0], [0, 32, -212, 0], [0, -50, 288, 0], [346, 0, 0, 53], [458, 0, 0, 42], [0, 14, -228, 0], [38, 0, 0, -6], [0, 110, 52, 0], [-798, 0, 0, 11], [0, 14, -28, 0], [-182, 0, 0, -46], [0, -140, 4, 0], [834, 0, 0, 8], [0, 92, -396, 0], [18, 0, 0, -76], [0, 50, -432, 0], [-542, 0, 0, 0], [0, -203, 164, 0], [-738, 0, 0, 0], [2, 0, 0, 56], [0, 56, 452, 0], [0, 183, 180, 0], [-422, 0, 0, -98], [0, -122, -12, 0], [0, 222, -48, 0], [-862, 0, 0, -61], [0, -164, 252, 0], [238, 0, 0, 52], [-686, 0, 0, 16], [818, 0, 0, -53], [0, 42, 588, 0], [0, -254, 84, 0], [-198, 0, 0, -3], [-122, 0, 0, 30], [0, -270, 152, 0], [0, 7, 20, 0], [662, 0, 0, -106], [0, 116, 108, 0], [0, -172, -152, 0], [0, 91, -372, 0], [-122, 0, 0, -18], [698, 0, 0, 6], [242, 0, 0, 112], [-150, 0, 0, 29], [0, -46, -332, 0], [-62, 0, 0, 11], [-102, 0, 0, 118], [0, -54, -20, 0], [874, 0, 0, 5], [0, -61, -156, 0], [0, 156, 356, 0], [6, 0, 0, 50], [0, -38, -512, 0], [302, 0, 0, 156], [-118, 0, 0, 146], [0, 22, 844, 0], [0, -24, 256, 0], [-274, 0, 0, -136], [450, 0, 0, -104], [0, 42, 116, 0], [0, 112, -120, 0], [0, 190, 220, 0], [0, -148, -672, 0], [0, -189, -380, 0], [-310, 0, 0, -222], [162, 0, 0, -48], [-718, 0, 0, 63], [0, -56, -756, 0], [234, 0, 0, 26], [0, -7, -116, 0], [0, 2, -564, 0], [-662, 0, 0, 2], [0, 200, 136, 0], [0, -45, 700, 0], [-290, 0, 0, -24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_192_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_192_3_g_c();".
// 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_192_3_g_c(:prec:=4)
    chi := MakeCharacter_192_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_192_3_g_c();".
// 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_192_3_g_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_192_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![-44, -4, 1]>],Snew);
    return Vf;
end function;