
// Make newform 667.2.a.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_667_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_667_2_a_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_667_2_a_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 := [-5, 37, 9, -215, 13, 342, -77, -188, 54, 41, -13, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-20718, 158295, -31183, -433403, 194815, 277030, -131017, -62480, 30343, 4126, -2268, 69], [-29932, 117880, 49707, -328362, 24036, 209382, -30082, -51132, 7810, 4875, -607, -127], [-25638, 84019, 135717, -276488, -84627, 210856, 13625, -63527, 773, 8049, -210, -353], [34852, -75949, -216607, 177916, 255406, -143208, -114560, 52179, 21760, -8798, -1451, 549], [23075, -19087, 34877, 86000, -218601, 1464, 154428, -32306, -37175, 10271, 2852, -874], [10438, -43159, -153652, 156809, 282647, -182412, -163016, 84275, 36674, -16651, -2741, 1153], [-21970, 64932, 177063, -190488, -303228, 212320, 168053, -95833, -36402, 18320, 2642, -1227], [-10451, -43380, -89336, 89329, 291927, -130539, -190180, 73554, 44255, -16487, -3336, 1231], [8869, 63686, -238564, -88693, 605079, -103569, -383278, 106697, 88739, -28263, -6684, 2257], [-20106, -161659, 391735, 441457, -926609, -126167, 578643, -63186, -134015, 28520, 10130, -2687]];
    Rf_basisdens := [1, 1, 6469, 6469, 6469, 6469, 6469, 6469, 6469, 6469, 6469, 6469];
    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_667_a();"
function MakeCharacter_667_a()
    N := 667;
    order := 1;
    char_gens := [465, 553];
    v := [1, 1];
    // 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;

function MakeCharacter_667_a_Hecke(Kf)
    return MakeCharacter_667_a();
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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [-1, 1, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0], [-1, 0, -1, 0, 0, 1, 0, 0, 0, 0, 1, 0], [-2, 1, 0, 1, 0, 0, 1, 0, 0, 2, -1, -1], [-2, 0, 0, -1, 0, 1, -1, -1, 0, 0, 0, 0], [-1, 1, 1, 1, 2, 2, 1, 1, 0, 0, 1, 1], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [2, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, -1, 0, -1, 0, -2, -1, 0, 0, -2, 0], [1, -1, 1, -1, 0, 0, -1, 1, 2, -2, 3, 2], [0, -2, 0, -1, -3, -3, -4, -1, 0, -1, -1, -2], [-5, 1, -1, 0, 2, 0, 0, 1, 0, 1, 0, -2], [-5, 0, -1, 0, 1, 1, 2, 2, 0, 0, 0, 0], [1, -4, -1, 2, 0, 0, -2, -1, 0, -1, 2, -1], [1, 2, 0, 0, -2, -2, 1, 1, 0, 0, -1, 1], [-3, 0, -1, 1, 5, 5, 2, 0, -2, -1, 2, 2], [3, -2, 1, -2, -3, 0, -4, 0, 1, -2, -1, 0], [2, -3, 0, -2, -4, -4, -4, -2, 1, 1, -3, -3], [1, 1, 0, -2, -1, -1, 0, -2, -2, 0, -3, -2], [1, -2, 2, -1, -1, -2, 1, -3, -1, 2, -1, 2], [-3, 3, 1, 0, 3, 0, 4, 0, 0, 1, 2, 2], [-2, 0, -1, 0, 2, 1, 1, 1, 2, -3, 1, 1], [-5, 1, 0, 2, 2, 0, 4, 1, -1, 1, 0, 1], [-2, 1, 2, -2, 2, 1, 2, 1, -1, -1, 1, 5], [0, -1, -1, -1, 2, 0, -3, 4, 2, -4, 4, 2], [-2, 1, 1, 1, 2, 0, 2, 2, -1, -1, 0, 1], [-4, 2, -1, 0, 2, 0, 0, 0, 2, -1, -1, 0], [0, 3, 0, 1, -2, 2, 1, 1, -2, 2, -2, -2], [4, 1, 1, 2, 2, 1, -1, 1, 0, 0, 2, 2], [-1, 1, -3, 3, 0, 1, 1, 2, -2, -2, -1, -3], [3, 1, 3, -3, 2, -1, 0, -2, -1, 0, 1, 2], [-1, 2, 1, 0, 0, -2, -4, 2, 2, -5, 3, 2], [5, -4, 0, -1, -3, 0, -3, 0, 1, -3, 2, 2], [-3, -2, 0, -1, 2, 4, 1, 0, -3, 1, 2, 1], [4, 1, 2, -2, -4, -5, -2, 0, 2, 1, 1, 1], [-6, 1, -1, -1, 2, 1, 1, -1, 0, 4, -2, -1], [-4, -1, 4, -2, -5, -4, -1, -1, 2, 0, 0, 1], [2, 4, 1, -1, -3, -4, -4, 1, 5, 2, 2, -1], [-3, -3, 2, -3, -7, -7, -7, -4, 0, 2, -2, -3], [-1, 1, 0, -1, -3, 2, 0, -3, 1, 1, -2, 0], [1, 2, -2, 2, 4, 3, 7, 0, -4, 2, 1, 1], [-5, 3, -1, 2, 3, 3, 3, 3, 4, 1, 2, 5], [1, -2, 2, 0, -5, -2, -2, -4, -5, 3, -6, -5], [2, -4, -6, 3, 3, 4, -2, 2, -1, -1, 3, -2], [0, -4, -2, -4, 0, -4, -7, -1, 3, -3, -1, -1], [-10, 3, -4, 0, 1, 2, -2, 4, 7, 1, 3, -1], [-3, 0, 2, 3, 2, 2, 1, -3, -2, 2, 1, -1], [-6, 1, -5, 2, 6, 2, 1, 1, 1, 0, 3, 1], [0, 1, 0, -3, -4, -2, 0, -1, 0, 0, 0, -3], [-5, -1, -5, 0, 0, 1, -2, -3, -3, -2, 0, -5], [-7, -1, -4, 2, -1, 3, 1, 3, 3, -1, 2, -3], [-4, -5, -1, 5, 3, 2, 5, 1, -1, -2, 3, 0], [6, -1, -4, -1, -4, -2, -1, 0, -2, -2, -2, 0], [-5, 1, 6, 2, 0, 0, 7, 5, 2, 5, 0, 3], [-3, 4, -1, 1, -8, -5, -1, 0, 3, 4, -7, -4], [1, -6, -1, 4, 2, 0, 2, -1, -3, 1, -2, -2], [1, -5, -3, -1, 1, 0, -2, -1, -2, -1, 1, -2], [-3, -7, -4, 1, 1, 1, -4, -6, -7, -3, 0, -5], [-6, 3, -4, -1, -5, -5, -5, 2, 0, -2, -2, -6], [-3, 4, 0, 6, 7, 1, 9, 2, -1, 5, 0, 1], [-7, 6, 3, 0, 7, 6, 7, 7, 4, 3, 4, 8], [2, -3, 2, -3, 3, 4, -3, 0, -1, 0, 3, 2], [-9, 3, -3, 1, 1, 2, -6, 0, -2, -1, 0, -3], [12, -1, 2, 3, 0, 4, 2, -2, -4, -6, 0, 4], [3, -1, 1, -3, -2, 2, -3, -6, -1, 1, -2, -5], [-7, 2, -3, 5, 6, 4, 9, 6, -1, 2, 1, 6], [2, -8, 0, -4, -4, -4, -6, -3, -3, -6, -1, 0], [-4, 6, 1, -2, -7, -4, -2, -5, 3, 6, -8, -2], [8, 4, 3, -4, -4, -2, -2, 0, 1, 0, 0, 1], [-6, -1, 5, -3, -3, 2, 1, -2, 3, 0, -4, 3], [-3, 2, 2, 5, 5, 8, 8, 0, 0, -2, 4, 5], [5, 2, -3, -1, -5, -4, -2, -2, 3, 0, -3, -5], [0, 2, 4, -7, -8, -4, 2, 0, 4, 2, -4, -1], [-11, 1, 3, -3, -5, -2, 0, -7, -6, 4, -9, -1], [0, -2, -5, 2, 5, 1, -6, 2, -1, -3, 6, -3], [3, -5, 7, 0, -6, -6, -1, 2, 4, 0, -2, -1], [4, 10, -2, 1, 3, 3, 5, 0, -5, 1, -3, 1], [0, -1, 4, -3, -2, -4, 1, -4, -5, 5, -5, -4], [-1, 0, -2, -2, 0, 1, 1, 1, -2, -1, -2, -1], [2, 1, -5, 7, 9, 3, 6, 7, 3, 2, 8, 1], [0, 0, -4, 4, 3, 6, 3, 1, 0, -3, 1, 3], [0, 3, 0, 6, 1, 6, 0, -1, -2, 2, 0, -3], [12, -8, 3, 0, -6, -7, -8, -4, 1, -2, -2, -6], [6, 2, -2, 5, -3, -7, -1, 0, -4, -1, 1, -4], [5, -6, 1, 1, -1, -1, -1, 2, 5, -3, 8, 4], [-1, -4, -1, -5, 2, 5, -2, 1, 3, -2, 3, 5], [1, 1, 3, -3, 2, -4, 3, -4, 5, 3, -4, 0], [-3, 5, 1, 1, 3, 2, -2, 2, -5, 2, 3, 1], [17, -5, 3, 5, -1, 0, 1, -2, 2, 2, -3, -4], [5, -2, 1, 3, 7, -1, 7, 3, -8, 6, 1, -3], [7, -1, 8, 0, 3, -5, 2, 4, -4, -2, 5, 7], [4, 1, -7, -4, -2, 1, -4, -3, -1, -2, 0, -3], [-1, -5, 5, -5, -11, -8, -8, -1, 1, -3, -2, 2], [6, -10, 4, -5, -6, -7, -8, -6, -5, -2, -3, -1], [-11, 5, -8, -3, 0, 2, 2, 1, -4, -4, -4, -3], [2, -4, 3, -2, -4, -6, -4, 7, 7, -4, 2, 4], [9, -6, -5, 3, 2, -2, -5, 7, -4, -7, 6, -4], [-1, -2, -1, 1, -4, 0, 3, 3, 7, 1, 2, -3], [-6, -2, 8, -1, -11, -2, 0, -6, 5, 8, -6, 1], [-8, 4, -1, 1, 1, -4, -4, -2, 3, 4, -4, -6], [-1, -3, 5, -6, -4, -3, -2, 3, 9, 1, -2, 5], [4, -3, 2, 2, 2, 2, 2, -7, -8, 6, -6, -3], [-3, -3, 6, 2, -1, 2, 1, 2, 8, 2, 5, 6], [-5, 6, -4, 2, 1, 0, 0, 7, -3, -2, 3, -3], [-8, 12, 4, -4, -1, -1, 1, -3, -8, 1, -4, -5], [4, 1, 2, 0, -5, 1, 8, -4, 0, 3, -5, -3], [11, -8, 3, -3, -7, -7, -2, -2, -1, 9, -6, -3], [9, -7, -2, 1, -2, 0, -5, -3, -4, 0, 1, -8], [-10, 7, -4, 4, 3, 6, 9, -3, -3, 5, -6, -4], [6, -7, 3, 0, 5, 13, 5, -1, -2, 2, 0, 4], [-7, -15, 1, 0, -2, 1, -4, -4, 1, -1, 2, 5], [1, 3, -4, 1, 3, -2, 0, -2, -1, 4, -1, 1], [-6, 4, -2, 3, 2, 3, 8, 0, 1, 5, -5, -1], [1, 0, -4, 7, 7, -1, 5, 3, -3, -3, -4, -3], [11, -9, 3, 3, 0, 8, 5, -1, -2, -2, -1, 3], [3, -1, 4, -2, 3, -1, 1, 3, 5, -4, -1, 4], [5, 9, -5, 2, 6, -3, 5, -4, -5, -1, -2, -4], [5, -8, -7, 0, -1, -6, -1, 2, 8, -4, 5, -2], [-2, 1, 3, 10, 0, 3, 2, -3, -7, 11, -4, -5], [-10, -6, -8, 7, 8, 9, 5, 0, -4, 1, 3, -2], [-1, 11, -4, -4, 2, 5, 0, 0, -9, -3, 0, 2], [-3, -1, 7, -4, -8, 0, -8, -5, 2, 4, -3, 0], [-9, 6, -4, 1, 2, -4, 6, 0, -3, -4, -2, -2], [-10, 1, -1, -3, 3, 1, 5, 10, 6, -2, 5, 1], [-3, 9, -2, 5, 10, 10, 8, 5, -3, 3, 4, 2], [12, -4, -4, 4, -5, -4, -1, -1, -5, 1, -9, -2], [-5, 0, -3, 2, -5, -1, 1, -3, 2, 6, -7, -12], [7, -4, 2, 0, 8, 7, 3, 8, 7, -5, 10, 11], [5, 2, 1, -2, -7, -5, 5, -5, -3, 4, -3, -8], [-13, -2, 4, 3, 5, 5, 3, -1, -6, 5, -2, 4], [5, 1, 9, -8, -4, -6, 2, -1, 8, 0, 1, 11], [-10, 0, -6, -2, 5, 1, -1, -1, -2, -6, 6, 0], [5, 1, 4, 3, 15, 4, 11, 3, -6, 5, 0, 5], [-12, 4, 7, 9, 12, 13, 18, 3, -4, -1, 5, 5], [-8, 9, 2, -5, 1, -5, -2, -8, -6, -2, -8, -1], [-11, -6, -2, -2, 1, -9, -6, 1, 5, -5, 4, 1], [3, 1, -5, 5, -5, 1, 2, 1, 2, -1, -6, -3], [0, 1, 0, -1, -2, -5, 2, -1, -3, -5, 3, 4], [-7, 5, 0, 3, 11, 0, 2, -2, -6, -1, 3, -5], [-8, 2, -5, 2, -4, 4, 1, -1, 5, 6, -8, -3], [1, -3, 5, -4, 2, 2, 5, -9, -3, 11, -2, 3], [3, -4, -3, 7, 8, 11, -4, -1, 4, 2, 9, 2], [-10, -6, 1, -6, -4, -3, -8, -5, -3, -2, 2, -1], [-4, 1, 4, 2, 10, 2, 1, 9, 5, -8, 10, 13], [-2, 2, 3, 8, 8, 13, 12, 3, 4, -2, 5, 11], [-7, 6, -13, 2, 9, 1, 8, 4, -3, 2, 3, -4], [11, 1, 7, -4, -15, -13, -6, -8, 5, 0, -7, 3], [6, -3, 6, 2, -3, 0, -11, 2, 4, 8, 5, 1], [-8, 3, -7, 7, 4, 4, 7, 2, -6, 6, 0, -4], [-2, 9, -2, 5, -6, -11, 7, 9, 2, 5, -5, 0], [2, 7, 4, -2, -14, -18, 0, -2, 5, 0, -12, -5], [1, -2, -3, 0, 2, 1, 1, -5, -5, -6, 3, 6], [10, -2, -2, -10, 9, -1, -3, -3, 4, -5, 5, 2], [-23, 7, -8, -3, 3, 0, 3, 9, 2, -3, 2, 1], [5, -4, 10, -2, -12, -11, 3, 1, 0, 3, -10, -3], [-11, 1, -5, 2, -4, -2, 3, -5, -6, 8, -9, -14], [1, 2, 5, -1, 2, -7, -5, 2, 10, -2, -2, 2], [-7, 0, -2, -4, -4, -1, -5, -1, 13, -2, -5, 4], [2, 5, 0, 0, 1, -1, -1, 2, 8, -1, 6, -3], [-9, 5, -3, -1, -5, -3, -8, 5, 5, -2, 5, 3], [-12, -2, -5, 0, 13, 10, -2, 5, -1, -6, 12, 4], [3, -3, 5, -10, -13, -9, -7, 0, 3, 7, -5, 0], [4, -2, -2, 0, 7, -7, 6, 6, -2, -6, 4, 5], [-13, 7, 0, 0, 3, 9, 8, -3, -6, 4, -5, -6], [-5, 12, 8, -7, -1, 0, 4, 8, 9, -1, 2, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_667_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_667_2_a_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_667_2_a_b(:prec:=12)
    chi := MakeCharacter_667_a();
    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_667_2_a_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_667_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_667_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-5, -37, 9, 215, 13, -342, -77, 188, 54, -41, -13, 3, 1]>],Snew);
    return Vf;
end function;