
// Make newform 675.2.k.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_675_k();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_675_k_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_675_2_k_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_675_2_k_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 := [36, 144, 288, 384, 304, 96, 0, -24, -16, 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, 0, 0, 0, 0], [-194196, 54216, 96048, 113100, 66640, -2470, -5676, -4564, -3014, 1118, -516, 473], [98004, -80298, -141012, -164352, -93902, 7532, 8976, 6311, 4162, -1768, 816, -748], [-15984, -49284, 132080, -98118, -57708, -11380, -14442, 13990, 2220, -2887, 2392, -962], [-15984, -49284, 97432, -98118, -57708, -11380, -14442, 13990, 2220, -2887, 2392, -962], [-38664, -119214, 157488, -236286, -137484, -24718, -32710, 34543, 5370, -7159, 5669, -2327], [5172, 12468, 22776, 27758, 10408, 1364, -1416, -1866, -32, 53, -90, 118], [-105120, -282372, -400856, -420176, -151968, -12648, 24587, 26378, 684, -840, 1054, -1688], [1015848, 2709180, 4346412, 3952074, 1356228, 85550, -282250, -226010, -84, 77, -3520, 14137], [-737076, -1783248, -3266544, -3992950, -1528612, -219272, 206064, 280340, -3016, -4217, 12498, -17172], [708480, 1902042, 2696676, 2820480, 1007136, 68632, -152420, -179857, -4254, 6136, -7163, 11258], [1273908, 3074676, 5621880, 6858478, 2589800, 350356, -351240, -467946, -2248, 11125, -21978, 29270]];
    Rf_basisdens := [1, 51972, 51972, 51972, 34648, 51972, 1704, 12993, 103944, 103944, 51972, 103944];
    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_675_k();"
function MakeCharacter_675_k()
    N := 675;
    order := 6;
    char_gens := [326, 352];
    v := [2, 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_675_k_Hecke();"
function MakeCharacter_675_k_Hecke(Kf)
    N := 675;
    order := 6;
    char_gens := [326, 352];
    char_values := [[0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 1, 1, 0, 0, 0], [-1, 0, -1, 0, 0, 0, 1, 0, 0, -1, 0, 0], [0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [-1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 2, 0, 0, 0, 0, 0, 0], [2, -1, -2, 0, 0, 0, -2, 0, 0, -2, 0, -1], [0, 0, 0, 0, 0, 0, -3, 0, 0, -1, 0, -2], [0, 0, 0, 0, 2, 0, 0, -1, -2, 0, 1, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, 1, 0, -1], [0, 0, 0, 1, 0, 1, 0, -1, 2, 0, 1, 0], [0, 0, 0, 3, 0, -1, 0, -3, -3, 0, -1, 0], [0, 0, 0, 0, 2, 0, 0, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1], [-1, 1, -1, 0, 0, 0, 1, 0, 0, -1, 0, 1], [0, 0, 0, 1, -1, 3, 0, 0, 0, 0, 0, 0], [2, 1, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 0, -2, -4, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2], [0, 0, 0, 3, 0, 0, 0, -3, -3, 0, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 4, 0, -2, -2, 0, 4, 0], [3, -1, 2, 0, 0, 0, -3, 0, 0, 2, 0, -1], [0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 3, -1, 0, 2, 0], [-2, -2, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 6, -4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, 0, -1, 5, 0, 1, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0], [0, 0, 0, -3, 0, 2, 0, 3, 2, 0, 2, 0], [0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 2, 0, 0, 5, 0, -2], [-7, -2, -1, 0, 0, 0, 7, 0, 0, -1, 0, -2], [0, 0, 0, -2, -2, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, 7, -2, 0, 1, 0], [0, 0, 0, -6, 1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, -6, 0, -2, 0, 6, 4, 0, -2, 0], [-4, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-14, -1, -3, 0, 0, 0, 14, 0, 0, -3, 0, -1], [0, 0, 0, 5, -8, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, -3, -2, 0, 4, 0], [5, -2, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0, 0, -1, 0, 2], [0, 0, 0, -2, 0, -3, 0, 2, 9, 0, -3, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, -12, 0, 0, 4, 0, -1], [0, 0, 0, 0, -2, 0, 0, -3, 2, 0, 5, 0], [0, 0, 0, 0, 0, 0, 7, 0, 0, -4, 0, -3], [8, 0, -5, 0, 0, 0, -8, 0, 0, -5, 0, 0], [12, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -9, 0, -1, 0, 9, 6, 0, -1, 0], [2, -6, -7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-5, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, -5, 0, -2, -4, 0, -5, 0], [-12, -2, -5, 0, 0, 0, 12, 0, 0, -5, 0, -2], [0, 0, 0, -4, -5, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 6, -4, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 9, 0, 0, -4, -9, 0, 2, 0], [0, 0, 0, 0, 0, 0, -6, 0, 0, -5, 0, 1], [0, 0, 0, -8, 0, 5, 0, 8, -10, 0, 5, 0], [0, 0, 0, -6, 0, -1, 0, 6, -2, 0, -1, 0], [-9, 3, 2, 0, 0, 0, 9, 0, 0, 2, 0, 3], [0, 0, 0, -5, -2, 6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, -2, -7, 0, 0, 0, 0, 0, 0], [-11, -2, 0, 0, 0, 0, 11, 0, 0, 0, 0, -2], [0, 0, 0, -6, 0, 4, 0, 6, -8, 0, 4, 0], [-6, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 5, 0, 1, 0, -5, 2, 0, 1, 0], [0, 0, 0, 2, -6, 4, 0, 0, 0, 0, 0, 0], [-4, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 2, -1, 0, 0, 0, 0, 0, 0], [9, -1, 5, 0, 0, 0, -9, 0, 0, 5, 0, -1], [0, 0, 0, 0, 4, 0, 0, 5, -4, 0, 4, 0], [0, 0, 0, 0, 0, 0, -3, 0, 0, -2, 0, -5], [0, 0, 0, 0, 0, 0, -10, 0, 0, -2, 0, -2], [0, 0, 0, 0, 0, 0, 11, 0, 0, -13, 0, -2], [2, -2, 2, 0, 0, 0, -2, 0, 0, 2, 0, -2], [12, -4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, 0, 0, 4, -4, 0, 9, 0], [-5, -1, 4, 0, 0, 0, 5, 0, 0, 4, 0, -1], [0, 0, 0, 0, 0, 5, 0, 0, 5, 0, 5, 0], [23, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, -3, 0, 1, 14, 0, -3, 0], [6, 5, 8, 0, 0, 0, -6, 0, 0, 8, 0, 5], [0, 0, 0, 8, 8, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -14, 0, 0, -3, 14, 0, -5, 0], [-12, 4, -2, 0, 0, 0, 12, 0, 0, -2, 0, 4], [0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 3, 0], [0, 0, 0, 0, 0, 0, 7, 0, 0, 2, 0, 3], [0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, -1], [0, 0, 0, 0, -5, 0, 0, -6, 5, 0, -9, 0], [0, 0, 0, 0, 0, 0, -17, 0, 0, 4, 0, 2], [2, 7, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -3, 0, 0, -5, 3, 0, -11, 0], [-3, -6, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -10, 0, 6, 0, 10, -7, 0, 6, 0], [0, 0, 0, 0, 4, 0, 0, -3, -4, 0, -4, 0], [0, 0, 0, 6, 7, -5, 0, 0, 0, 0, 0, 0], [-18, -4, 8, 0, 0, 0, 18, 0, 0, 8, 0, -4], [0, 0, 0, 0, 0, 0, -20, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 0, 0, -4, -2, 0, -8, 0], [0, 0, 0, 12, 0, 2, 0, -12, -1, 0, 2, 0], [0, 0, 0, 0, 10, 0, 0, -3, -10, 0, 3, 0], [0, 0, 0, 0, 0, 0, 20, 0, 0, 2, 0, 4], [-2, -2, 10, 0, 0, 0, 2, 0, 0, 10, 0, -2], [0, 0, 0, -5, -7, -5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 13, 0], [0, 0, 0, 3, -10, 8, 0, 0, 0, 0, 0, 0], [5, 2, 5, 0, 0, 0, -5, 0, 0, 5, 0, 2], [19, 3, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -4, -7, 0, 0, 0, 0, 0, 0, -7, 0, -4], [0, 0, 0, -10, 13, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 15, 0, 0, 6, -15, 0, 1, 0], [0, 0, 0, 9, 6, 1, 0, 0, 0, 0, 0, 0], [1, 5, 0, 0, 0, 0, -1, 0, 0, 0, 0, 5], [0, 0, 0, 0, 0, 0, -4, 0, 0, 12, 0, -2], [0, 0, 0, 16, 0, -1, 0, -16, -2, 0, -1, 0], [0, 0, 0, 0, 0, -9, 0, 0, 6, 0, -9, 0], [0, 0, 0, 0, -6, 0, 0, 15, 6, 0, 3, 0], [-4, 0, 10, 0, 0, 0, 4, 0, 0, 10, 0, 0], [23, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [22, -3, 2, 0, 0, 0, -22, 0, 0, 2, 0, -3], [-3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 5, 0, -9, 0, -5, 3, 0, -9, 0], [0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0], [-13, 3, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3, 1, 12, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 14, 0, 0, 1, 0, 5], [0, 0, 0, 0, 16, 0, 0, -1, -16, 0, 13, 0], [0, 0, 0, 0, 0, 0, 6, 0, 0, -4, 0, 5], [0, 0, 0, 0, 0, 0, 20, 0, 0, -4, 0, 5], [0, 0, 0, 0, 4, 0, 0, -18, -4, 0, -4, 0], [0, 0, 0, 13, -2, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 6, 14, -5, 0, 0, 0, 0, 0, 0], [7, -1, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 6, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 7, 7, 0, 0, 0, -3, 0, 0, 7, 0, 7], [0, 0, 0, -1, -7, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 5, 0, 0, 18, -5, 0, 5, 0], [9, 5, -5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [29, 2, 1, 0, 0, 0, -29, 0, 0, 1, 0, 2], [0, 0, 0, -2, 0, 3, 0, 2, 2, 0, 3, 0], [0, 0, 0, -3, 0, 3, 0, 3, -8, 0, 3, 0], [0, 0, 0, 0, 0, 0, -2, 0, 0, -1, 0, 7], [0, 0, 0, 0, 7, 0, 0, 0, -7, 0, 9, 0], [0, 0, 0, -5, 8, -15, 0, 0, 0, 0, 0, 0], [-5, 5, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 3, 0, 0, -2, -3, 0, -3, 0], [0, 0, 0, -15, -6, 3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, -5, 0, -2, -1, 0, -5, 0], [29, -7, -2, 0, 0, 0, -29, 0, 0, -2, 0, -7], [14, -2, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, -4, -14, 0, 0, 0, 4, 0, 0, -14, 0, -4], [0, 0, 0, 0, -14, 0, 0, -7, 14, 0, 0, 0], [0, 0, 0, 0, 0, 0, 4, 0, 0, 10, 0, -1], [0, 0, 0, 3, 0, -5, 0, -3, 19, 0, -5, 0], [0, 0, 0, 0, -2, 0, 0, 6, 2, 0, -5, 0], [0, 0, 0, -2, -3, 6, 0, 0, 0, 0, 0, 0], [-27, -4, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -12, 6, 7, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 12, 0, 0, -3, 0, 12, 0], [-31, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -10, 0, 8, 0, 10, -18, 0, 8, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_675_k_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_675_2_k_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_675_2_k_b(:prec:=12)
    chi := MakeCharacter_675_k();
    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_675_2_k_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_675_2_k_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_675_k();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![81, 0, -279, 0, 862, 0, -323, 0, 90, 0, -11, 0, 1]>],Snew);
    return Vf;
end function;