
// Make newform 273.2.p.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_273_p();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_273_p_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_273_2_p_f();".
// 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_273_2_p_f();".
// 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 := [16, -32, 32, 160, 320, 80, 0, -8, 60, 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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-956416, -156757872, -6794816, -4189816, 1884316, -11267844, 479716, -745616, 28834, -140579, 5381, -12309], [13274488, 262576728, 93750596, 58720980, -23444608, 74667404, -5541268, 10325938, -400085, 1270592, -60738, 170557], [9887728, -69115624, -288671368, -223768168, -46962688, 11897532, -30830234, -33375518, -24618, 119552, -583343, -556438], [8903008, -7918288, -51309608, -199641288, -45708008, -2447648, 11897532, -35281738, 10762, -24618, 119552, -583343], [-22588976, 159724320, -256163344, 487489968, 152924728, 146207356, -197893628, 144220708, -400372, 2016198, -3033606, 2412741], [62241040, -227943584, 255016704, 952913392, 427286336, -74231220, -67932876, 205726644, 2509892, -1302018, -676828, 3446659], [26709024, -23754864, -153928824, -571700956, -137124024, -7342944, 35692596, -105845214, 32286, -73854, 358656, -1750029], [-316463600, -399110384, 443625424, 1688398144, 469375944, -61660468, -111309124, 334136424, 239104, -1166686, -1112876, 5569761], [58170888, -52760848, -328183716, -937941424, -274186516, -13524756, 67292332, -196850238, -134391, -135251, 676169, -3299703], [119102928, -834676912, -1734207216, -2667267136, -619540376, -61660468, -2417492, -482550816, 239104, -1166686, -1112876, -8041693]];
    Rf_basisdens := [1, 1, 54445816, 54445816, 54445816, 54445816, 108891632, 108891632, 54445816, 108891632, 54445816, 108891632];
    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_273_p();"
function MakeCharacter_273_p()
    N := 273;
    order := 4;
    char_gens := [92, 157, 106];
    v := [4, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_273_p_Hecke();"
function MakeCharacter_273_p_Hecke(Kf)
    N := 273;
    order := 4;
    char_gens := [92, 157, 106];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 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, 1, 0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 0, -1, -1, 0, 1, 0, -1, -1, 0], [0, 1, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0], [0, 1, -1, 1, 1, -1, 0, 1, 0, 1, 0, 1], [-2, -2, 0, -1, 0, -2, 0, 2, 0, -2, -1, 0], [0, 0, -2, 0, 0, 0, -1, 1, 0, 1, 0, 1], [0, 0, -1, 0, -2, 0, 0, 0, -1, 0, 0, 2], [-2, -1, 0, -2, 0, -1, 0, 2, 0, -2, -2, 0], [2, 2, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0], [1, 2, 0, 4, -1, 0, 1, -1, 0, 3, 0, 3], [-3, -1, 0, 0, 3, 0, -2, 2, 0, -2, 0, -2], [0, -2, 0, -2, 0, 2, 0, 0, 0, 0, 2, 0], [3, 0, 0, 0, 3, 1, 1, 1, 4, 2, 0, -2], [-2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0], [1, 0, 0, 0, 1, 3, 1, 1, 0, 2, 0, -2], [0, 2, -3, 3, 0, -2, 0, 0, -3, 0, -3, 4], [-2, 0, 0, 0, -2, 0, -2, -2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 5, -1, -1, 0, 2, 4, -2], [-3, 0, 0, 0, -3, 4, 0, 0, 0, -1, 2, 1], [4, 3, 1, 1, 0, 3, 0, -2, -1, 4, 1, 0], [1, -1, 0, 2, -1, 0, 3, -3, 0, 0, 0, 0], [-1, 0, 0, 0, -1, -5, 0, 0, -4, -4, 0, 4], [-1, -2, 4, -4, 1, 0, -1, 1, 0, -2, 0, -2], [4, 3, 0, 1, 0, 3, 0, -2, 0, 6, 1, 0], [-8, 0, 1, 1, 0, 0, 0, -2, -1, 2, 1, 0], [-2, 0, 1, 0, 0, 0, 0, -4, -1, 2, 0, 0], [1, 0, 0, 0, 1, 2, 2, 2, 4, 0, 4, 0], [-2, -3, 0, 0, 0, -3, 0, 6, 0, -4, 0, 0], [0, -3, -1, -1, -4, 3, -2, 0, -1, 0, 1, -4], [0, 1, 4, 2, -2, -1, 0, 0, 4, 0, -2, 0], [0, 1, 2, -4, 0, 0, -3, 3, 0, -4, 0, -4], [0, -4, 3, 1, -4, 4, 0, 0, 3, 0, -1, -4], [-2, 0, 0, 0, -2, -3, 3, 3, 0, -2, -4, 2], [-2, 0, -2, -2, 2, 0, -2, 2, 0, 0, 0, 0], [0, -2, -2, -4, -2, 2, -4, 0, -2, 0, 4, -6], [-2, 0, 0, 4, 2, 0, -1, 1, 0, 1, 0, 1], [3, 0, 0, 0, 3, -3, 1, 1, 4, -2, 2, 2], [-8, -5, 4, -5, 0, -5, 0, 0, -4, -10, -5, 0], [0, -1, -3, -2, 6, 1, -6, 0, -3, 0, 2, 2], [8, 6, 0, 0, 0, 6, 0, -2, 0, 6, 0, 0], [14, -1, -1, 0, 0, -1, 0, -2, 1, 4, 0, 0], [-1, -4, 6, -6, 1, 0, 1, -1, 0, -7, 0, -7], [6, 0, 0, 0, 6, 9, -3, -3, 2, 8, 8, -8], [6, 2, 0, 2, 0, 2, 0, -6, 0, 0, 2, 0], [-4, -1, -5, 1, 0, -1, 0, 2, 5, 4, 1, 0], [2, 0, 0, -2, -2, 0, 1, -1, 0, 1, 0, 1], [-3, -1, -4, 2, 3, 0, -1, 1, 0, -2, 0, -2], [3, 0, 0, 0, 3, 0, -2, -2, 4, 7, 2, -7], [0, -3, -4, 2, 2, 3, -6, 0, -4, 0, -2, 0], [-8, 0, 0, 0, -8, 1, -1, -1, 4, 2, -4, -2], [3, 0, 0, 0, 3, -8, 2, 2, 0, -1, -2, 1], [-2, -3, 0, -4, 0, -3, 0, 8, 0, -4, -4, 0], [-4, -1, 0, -5, 0, -1, 0, -2, 0, -6, -5, 0], [-2, 2, -1, -2, 0, 2, 0, 0, 1, 0, -2, 0], [0, -5, -2, -3, -4, 5, -4, 0, -2, 0, 3, 2], [2, 0, 0, 0, 2, -4, -2, -2, -2, 2, 2, -2], [0, -2, 0, -4, 6, 2, 2, 0, 0, 0, 4, -6], [-4, -1, -6, 4, 4, 0, -3, 3, 0, 4, 0, 4], [2, -4, 0, -4, 0, -4, 0, 4, 0, -2, -4, 0], [-1, 0, 0, 0, -1, -3, -4, -4, 0, -4, 2, 4], [0, 0, 0, 0, 0, 10, 1, 1, 2, 1, 4, -1], [6, 3, 0, 0, 0, 3, 0, -6, 0, 8, 0, 0], [0, -8, -3, -7, 10, 8, -8, 0, -3, 0, 7, -6], [2, 0, 0, 0, 2, -3, 5, 5, 6, 4, 0, -4], [-4, 0, 0, 0, -4, -8, 5, 5, 2, -3, -2, 3], [0, -7, 7, -3, 0, 7, -2, 0, 7, 0, 3, -8], [2, 1, -3, 0, 0, 1, 0, -2, 3, -6, 0, 0], [7, 0, 0, 0, 7, 0, -5, -5, 2, -2, 2, 2], [-3, 7, -4, 0, 3, 0, -4, 4, 0, -2, 0, -2], [2, -13, -2, -12, -2, 0, -3, 3, 0, -8, 0, -8], [0, 6, 5, 1, 2, -6, 6, 0, 5, 0, -1, 4], [-6, 3, -3, 3, 0, 3, 0, -2, 3, 0, 3, 0], [-6, 0, 0, 0, -6, 4, 2, 2, -6, -4, 2, 4], [9, -5, 4, -2, -9, 0, -3, 3, 0, -6, 0, -6], [0, -6, 2, -6, -6, 6, 0, 0, 2, 0, 6, -12], [-5, 0, 0, 0, -5, -18, 8, 8, 4, -9, -6, 9], [-12, -1, 8, 0, 12, 0, 1, -1, 0, -6, 0, -6], [3, -2, 0, 8, -3, 0, 3, -3, 0, -2, 0, -2], [0, 7, -2, 2, -2, -7, 4, 0, -2, 0, -2, -4], [-9, 0, 0, 0, -9, -10, 1, 1, -6, -11, -10, 11], [-6, 0, 0, 0, -6, 1, -5, -5, -6, 2, 4, -2], [-6, -2, 5, -1, 0, -2, 0, 2, -5, 0, -1, 0], [2, 4, -1, 5, 0, 4, 0, -4, 1, -2, 5, 0], [-6, 0, 3, -2, 0, 0, 0, -4, -3, -8, -2, 0], [2, -9, -2, -8, -2, 0, -3, 3, 0, 0, 0, 0], [15, 0, 0, 0, 15, 10, -6, -6, 0, 4, 4, -4], [-1, 3, -4, 8, 1, 0, -2, 2, 0, 10, 0, 10], [-2, -4, -4, -4, 2, 0, -2, 2, 0, 0, 0, 0], [-6, -1, -6, -4, 0, -1, 0, 4, 6, 4, -4, 0], [-3, 0, 0, 0, -3, 7, -5, -5, 4, 4, 4, -4], [-4, 0, 0, 0, -4, -8, -2, -2, 2, -4, -2, 4], [0, -2, 1, -2, 18, 2, 4, 0, 1, 0, 2, 2], [-6, 0, 0, 0, -6, -2, 7, 7, -8, -3, -4, 3], [0, -5, -4, 0, -6, 5, -8, 0, -4, 0, 0, -4], [3, -1, 8, 0, -3, 0, 4, -4, 0, 0, 0, 0], [0, 1, -4, 3, -4, -1, 4, 0, -4, 0, -3, 10], [0, 8, -5, 3, -8, -8, 6, 0, -5, 0, -3, 10], [9, 6, 2, -2, -9, 0, -3, 3, 0, -1, 0, -1], [-16, -2, 4, 0, 0, -2, 0, 4, -4, 0, 0, 0], [2, 3, 0, 8, -2, 0, -3, 3, 0, 0, 0, 0], [0, -2, -6, -2, 0, -2, 0, 4, 6, 0, -2, 0], [0, 7, -4, 4, -18, -7, 2, 0, -4, 0, -4, 14], [0, 7, -7, 5, 0, -7, -2, 0, -7, 0, -5, 16], [-5, -2, -8, -2, 5, 0, 2, -2, 0, 5, 0, 5], [-5, 1, -8, 6, 5, 0, -5, 5, 0, 4, 0, 4], [-1, 0, 0, 0, -1, -9, -2, -2, -12, -4, -2, 4], [6, 8, 5, 6, 0, 8, 0, -12, -5, 6, 6, 0], [0, -2, 3, 5, -6, 2, 14, 0, 3, 0, -5, 0], [0, -8, 1, -5, 10, 8, -4, 0, 1, 0, 5, -14], [9, 0, 0, 0, 9, 2, 0, 0, -2, 10, 2, -10], [-2, 0, 0, 0, -2, 13, 3, 3, 4, 6, 0, -6], [6, 0, 0, 0, 6, -4, 6, 6, -2, 2, 2, -2], [-2, -2, 6, 2, 2, 0, -1, 1, 0, -5, 0, -5], [0, -2, -4, 0, 14, 2, 4, 0, -4, 0, 0, 4], [8, 22, 0, 6, -8, 0, 7, -7, 0, 13, 0, 13], [0, 4, 2, -4, 0, 4, 0, 6, -2, -4, -4, 0], [-6, 11, -10, 8, 0, 11, 0, 2, 10, 12, 8, 0], [-6, 3, 5, 6, 0, 3, 0, -10, -5, -4, 6, 0], [-3, 0, 0, 0, -3, 4, 9, 9, 6, 6, 2, -6], [0, -6, -4, -4, -8, 6, -8, 0, -4, 0, 4, -4], [0, 8, 2, 3, -10, -8, 14, 0, 2, 0, -3, 14], [8, 5, 4, 0, -8, 0, 1, -1, 0, 4, 0, 4], [-8, 8, -4, 4, 8, 0, -2, 2, 0, 0, 0, 0], [0, -6, -2, 4, 10, 6, 0, 0, -2, 0, -4, -6], [-5, -12, -6, -6, 5, 0, -4, 4, 0, 4, 0, 4], [10, -3, -4, 0, 0, -3, 0, 0, 4, 8, 0, 0], [-8, -10, 8, -6, 0, -10, 0, 10, -8, -18, -6, 0], [-7, -12, -2, -8, 7, 0, -8, 8, 0, -5, 0, -5], [-2, 12, -4, 8, 2, 0, 2, -2, 0, 8, 0, 8], [2, 0, 0, 0, 2, -19, 5, 5, 2, -4, -12, 4], [0, 2, 2, -8, -4, -2, -4, 0, 2, 0, 8, -8], [24, 9, -5, 9, 0, 9, 0, -14, 5, 12, 9, 0], [13, 0, 0, 0, 13, -5, 6, 6, 8, -2, 0, 2], [-13, -16, 4, -8, 13, 0, -5, 5, 0, -12, 0, -12], [5, 15, -4, 2, -5, 0, 4, -4, 0, 4, 0, 4], [-12, 0, 0, 0, -12, -16, 10, 10, -6, -8, -10, 8], [10, 10, -4, 3, 0, 10, 0, -10, 4, 10, 3, 0], [10, 3, 0, -2, 0, 3, 0, -2, 0, 12, -2, 0], [0, -14, -8, -12, 0, 0, -4, 4, 0, -2, 0, -2], [-6, 0, 0, 0, -6, 11, 1, 1, 0, 0, 8, 0], [0, -10, -8, 0, 8, 10, -8, 0, -8, 0, 0, -4], [-2, 5, -2, 4, 2, 0, -5, 5, 0, -6, 0, -6], [-2, 2, 7, 5, 0, 2, 0, -4, -7, 2, 5, 0], [5, 0, 0, 0, 5, 3, -9, -9, -8, 2, -10, -2], [11, 0, 0, 0, 11, -14, 6, 6, 6, -7, -12, 7], [-4, 7, 4, 9, 0, 7, 0, -8, -4, 2, 9, 0], [0, 6, -2, 2, -2, -6, 0, 0, -2, 0, -2, 14], [10, 0, 0, 0, 10, -5, -5, -5, 2, 0, 0, 0], [-7, 0, 0, 0, -7, 6, -8, -8, -10, -2, 6, 2], [-24, -1, -4, 3, 0, -1, 0, -2, 4, 10, 3, 0], [0, 3, 5, -1, -16, -3, 16, 0, 5, 0, 1, 2], [0, -6, -4, -4, 4, 6, -2, 0, -4, 0, 4, -4], [0, -3, -1, 1, 16, 3, -10, 0, -1, 0, -1, 8], [10, -7, -5, -6, 0, -7, 0, -2, 5, 0, -6, 0], [12, 3, -5, 5, 0, 3, 0, -12, 5, 2, 5, 0], [3, -3, 8, -6, -3, 0, -6, 6, 0, -10, 0, -10], [0, 0, -1, -5, 8, 0, 2, 0, -1, 0, 5, -2], [-27, -5, -4, 4, 27, 0, 0, 0, 0, 0, 0, 0], [-4, 11, -8, 8, 4, 0, -3, 3, 0, 4, 0, 4], [0, 1, 4, -2, -14, -1, -6, 0, 4, 0, 2, -10], [10, 0, 0, 0, 10, 8, -1, -1, 4, 7, 0, -7], [0, -12, -4, -8, -12, 12, -2, 0, -4, 0, 8, -4], [0, 0, 0, 0, 0, -7, 5, 5, 12, -4, 4, 4], [9, 0, 0, 0, 9, -13, -3, -3, -4, -6, 2, 6], [-8, -10, -2, 0, 0, -10, 0, 4, 2, 0, 0, 0], [0, -4, 12, -4, 12, 4, 16, 0, 12, 0, 4, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_273_p_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_273_2_p_f();".
// 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_273_2_p_f(:prec:=12)
    chi := MakeCharacter_273_p();
    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_273_2_p_f();".
// 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_273_2_p_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_273_p();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1369, -148, 8, 188, 1247, 52, 0, -4, 75, 0, 0, 0, 1]>,<5,R![16, -160, 800, -1600, 1760, -848, 480, -520, 480, -240, 72, -12, 1]>],Snew);
    return Vf;
end function;