
// Make newform 187.2.g.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_187_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_187_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_187_2_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_187_2_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, -1, 1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_187_g();"
function MakeCharacter_187_g()
    N := 187;
    order := 5;
    char_gens := [35, 122];
    v := [4, 5];
    // 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_187_g_Hecke();"
function MakeCharacter_187_g_Hecke(Kf)
    N := 187;
    order := 5;
    char_gens := [35, 122];
    char_values := [[0, 0, 0, -1], [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 := 2;
    raw_aps := [[0, 1, -1, 0], [0, -1, 2, -1], [-1, 2, -1, 0], [-3, 3, 0, 1], [-3, 1, 1, 1], [3, 1, -1, -3], [0, -1, 0, 0], [0, -3, 4, -3], [3, 0, -2, 2], [4, -4, 0, 3], [0, 4, -4, 0], [3, -3, 0, -7], [0, -1, 5, -1], [-6, 0, 0, 0], [0, 1, -5, 1], [-10, 8, -8, 10], [4, -4, 0, -7], [-3, 7, -3, 0], [12, 0, 3, -3], [-3, -3, -3, 0], [-5, 5, 0, 11], [8, -2, 2, -8], [-1, -1, -1, 0], [4, 0, 8, -8], [4, 0, 0, -4], [-7, -3, 3, 7], [-6, 6, 0, 9], [0, 12, -3, 12], [-6, 0, 3, -3], [0, 8, -10, 8], [-10, 7, -10, 0], [-9, 0, -12, 12], [2, 1, 2, 0], [4, -4, 0, 3], [-9, -3, -9, 0], [0, 0, -3, 0], [0, 2, 12, 2], [-8, 0, 0, 8], [-15, 11, -11, 15], [0, -6, -3, -6], [0, -11, -2, -11], [-16, 1, -16, 0], [9, -9, 0, 6], [0, -4, 0, 0], [-12, 0, 5, -5], [-12, 0, -19, 19], [13, -18, 18, -13], [0, -4, 16, -4], [4, -4, 0, 10], [8, -2, 2, -8], [1, -11, 11, -1], [0, 13, -9, 13], [26, 0, 3, -3], [-3, -9, 9, 3], [-1, 1, 0, 10], [16, 0, 4, -4], [19, -12, 19, 0], [-6, 6, 0, -9], [-8, 3, -3, 8], [10, 8, 10, 0], [0, -3, -2, -3], [-11, 11, 0, 19], [5, 0, 14, -14], [0, 17, -24, 17], [2, 0, 2, 0], [19, -20, 20, -19], [-3, 0, 0, 0], [-12, 12, 0, -12], [-12, 3, -12, 0], [0, -21, 13, -21], [21, 0, -6, 6], [15, -15, 0, -15], [-12, 12, 0, 3], [18, 0, -12, 12], [-12, 1, -12, 0], [-12, 26, -26, 12], [-14, 14, 0, 7], [-4, 0, -24, 24], [0, -27, 0, 0], [23, -17, 17, -23], [30, 0, 15, -15], [0, 6, 9, 6], [8, -3, 3, -8], [-14, 14, 0, 18], [9, 0, 18, -18], [0, -19, -4, -19], [3, 23, -23, -3], [-12, 28, -12, 0], [-3, 0, -10, 10], [-7, 0, 8, -8], [-21, 15, -21, 0], [24, 3, 24, 0], [0, -8, -3, -8], [-8, 8, 0, 12], [12, -12, 0, -21], [0, -14, 21, -14], [0, 12, -21, 12], [-18, 18, 0, 27], [9, 14, 9, 0], [-38, 26, -26, 38], [0, -6, 16, -6], [-1, 1, 0, -15], [-28, 15, -15, 28], [0, 14, 3, 14], [11, 0, -12, 12], [31, -16, 31, 0], [3, -3, 0, 18], [22, 0, 1, -1], [-23, 24, -23, 0], [12, -12, 0, 17], [-12, -6, 6, 12], [0, 2, -32, 2], [19, 0, 22, -22], [0, 16, -33, 16], [15, -15, 0, -17], [0, -14, 29, -14], [-11, 39, -11, 0], [14, -10, 10, -14], [-8, 8, 0, -15], [-18, 0, -6, 6], [11, 0, 3, -3], [36, -21, 21, -36], [-9, -36, -9, 0], [-18, 0, -24, 24], [-43, 36, -36, 43], [0, 13, -17, 13], [-18, -21, -18, 0], [-10, 10, 0, 5], [27, 0, 18, -18], [11, -11, 0, -2], [24, -16, 16, -24], [16, -32, 16, 0], [0, 3, 23, 3], [-2, -21, 21, 2], [18, -18, 18, -18], [12, 0, -6, 6], [0, 23, -20, 23], [-5, 12, -5, 0], [0, -3, 0, 0], [9, 3, 9, 0], [0, 18, -17, 18], [-14, 14, 0, -25], [-31, 37, -37, 31], [0, -14, 14, 0], [-3, 3, 0, 19], [0, -23, 9, -23], [-12, -3, -12, 0], [7, 0, 13, -13], [13, 0, 6, -6], [4, -51, 4, 0], [0, 12, -13, 12], [7, 0, -20, 20], [0, 11, 31, 11], [14, -14, 0, -40], [0, 47, 0, 0], [-18, -9, 9, 18], [0, 10, 0, 0], [1, 42, 1, 0], [-33, 8, -8, 33], [-5, -32, -5, 0], [-13, 0, -2, 2], [-14, 14, 0, 13], [19, 0, 22, -22], [0, 14, 30, 14], [10, -39, 39, -10], [42, -42, 0, -30], [17, 0, -25, 25], [12, -12, 0, -14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_187_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_187_2_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_187_2_g_b(:prec:=4)
    chi := MakeCharacter_187_g();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_187_2_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_187_2_g_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_187_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -3, 4, -2, 1]>,<3,R![1, -1, 6, 4, 1]>],Snew);
    return Vf;
end function;