
// Make newform 187.2.e.a in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_187_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_187_e_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_e_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_187_2_e_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 := [1, 0, 0, 0, 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_e();"
function MakeCharacter_187_e()
    N := 187;
    order := 4;
    char_gens := [35, 122];
    v := [4, 3];
    // 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_e_Hecke();"
function MakeCharacter_187_e_Hecke(Kf)
    N := 187;
    order := 4;
    char_gens := [35, 122];
    char_values := [[1, 0, 0, 0], [0, 0, 1, 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, 0, 2, 0], [-1, 2, -1, 0], [2, 0, 2, 0], [0, 0, 0, -1], [0, 0, 0, -1], [-2, -1, 0, 1], [0, -4, 0, -1], [0, -1, 4, -1], [3, 0, -3, 0], [2, 1, 2, 0], [-2, -4, -2, 0], [1, 8, 1, 0], [0, 0, 0, 5], [0, -1, -4, -1], [3, 6, 0, -6], [0, 6, -5, 6], [0, -4, 1, -4], [-2, 0, 2, 4], [1, 0, 0, 0], [-4, -10, -4, 0], [2, -11, 2, 0], [2, 0, -2, 6], [0, 2, 2, 2], [1, -2, 0, 2], [-4, -2, -4, 0], [2, -9, 0, 9], [11, 4, 0, -4], [-4, 5, -4, 0], [-6, 0, 6, 7], [-4, 0, 4, 2], [0, -3, 10, -3], [2, -5, 2, 0], [7, 2, 0, -2], [-2, 9, -2, 0], [16, 3, 0, -3], [0, -6, 2, -6], [-9, -8, 0, 8], [-5, 0, 5, 0], [10, -10, 10, 0], [-2, 3, -2, 0], [0, 16, 3, 16], [-6, 0, 6, 10], [10, -2, 0, 2], [12, 0, -12, 4], [8, 0, -8, -8], [-14, -2, -14, 0], [0, 0, 0, -9], [0, 0, -6, 0], [-2, 0, 2, 11], [0, 4, -1, 4], [-12, -5, -12, 0], [18, 5, 0, -5], [-4, 19, -4, 0], [6, 0, 0, 0], [0, -6, 19, -6], [0, -11, 10, -11], [9, 10, 9, 0], [-18, -8, 0, 8], [-14, 5, -14, 0], [0, 6, 8, 6], [6, 0, -6, 10], [18, -5, 0, 5], [-18, -3, 0, 3], [-4, 8, -4, 0], [-9, 0, 9, 2], [10, 0, -10, 2], [0, -4, -6, -4], [6, 23, 6, 0], [2, 0, -2, -19], [0, -21, -6, -21], [-15, 2, 0, -2], [0, -10, -18, -10], [-10, 0, 10, 4], [16, 9, 0, -9], [-9, -22, -9, 0], [0, 18, 6, 18], [0, 0, -9, 0], [-13, 0, 13, -18], [0, 0, 0, 12], [-4, 10, 0, -10], [14, 0, -14, 8], [-9, -8, 0, 8], [-8, 0, 8, -21], [0, -6, -3, -6], [6, 11, 6, 0], [-4, 12, 0, -12], [0, 0, 0, 16], [0, 6, 28, 6], [0, -21, -4, -21], [9, 22, 0, -22], [0, 22, -3, 22], [14, 7, 14, 0], [3, 0, -3, -4], [0, 13, 20, 13], [-2, 0, 2, 10], [4, 0, -4, -17], [29, 8, 0, -8], [-1, 0, 1, -6], [-12, 4, 0, -4], [-8, 19, -8, 0], [-16, 14, -16, 0], [-4, 20, 0, -20], [0, -19, 10, -19], [0, 6, -28, 6], [4, 0, -4, 23], [33, -8, 0, 8], [0, 22, 6, 22], [0, 8, 14, 8], [11, -12, 0, 12], [-8, 0, 8, 16], [20, 18, 20, 0], [24, -9, 0, 9], [0, 24, 0, 0], [-14, 0, 14, 10], [0, 4, -15, 4], [3, 2, 3, 0], [-7, 18, -7, 0], [-3, -18, 0, 18], [-8, 0, 8, -14], [10, 23, 0, -23], [0, 0, -21, 0], [16, 0, -16, -7], [-6, -31, -6, 0], [-15, -30, -15, 0], [23, 0, -23, -6], [-24, 10, 0, -10], [5, 10, 5, 0], [-1, -22, -1, 0], [9, -10, 0, 10], [0, -14, -14, -14], [0, -8, -24, -8], [8, -19, 8, 0], [-1, -26, -1, 0], [0, 2, -20, 2], [12, -5, 0, 5], [10, -17, 0, 17], [0, 28, 2, 28], [-2, -9, -2, 0], [0, -6, 28, -6], [-22, 0, 22, 15], [-16, -10, -16, 0], [-20, 4, -20, 0], [-2, 0, 2, -32], [-10, 0, 10, -35], [16, 12, 0, -12], [-7, 0, 7, 26], [-6, 31, -6, 0], [14, 0, -14, -36], [0, -8, 1, -8], [-42, 8, 0, -8], [4, 0, -4, 22], [-14, -2, -14, 0], [15, 4, 0, -4], [-12, 12, -12, 0], [-2, 0, 2, -18], [1, 0, -1, -6], [-6, 17, 0, -17], [-21, 0, 21, -4], [0, -8, 30, -8], [-8, 0, 8, -7], [18, -14, 18, 0], [-36, -4, 0, 4], [0, -15, -2, -15], [0, 30, 14, 30], [0, 10, -45, 10], [14, -6, 14, 0], [13, 36, 13, 0], [-12, 0, 12, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_187_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_187_2_e_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_187_2_e_a(:prec:=4)
    chi := MakeCharacter_187_e();
    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_e_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_187_2_e_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_187_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, 0, 1]>],Snew);
    return Vf;
end function;