
// Make newform 224.3.o.c in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_224_o();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_224_o_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_224_3_o_c();".
// 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_224_3_o_c();".
// 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 := [4096, 0, 512, -768, 192, -48, 148, -12, 12, -12, 2, 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], [-49152, 57856, -18432, 30912, -4656, 7012, -1980, 1212, 420, 298, 48, -11], [52224, 28800, 2112, 1008, -3516, 444, -12, 828, -14, 36, -51, 6], [99328, 21888, 13952, -10416, -3604, 7268, 748, 804, -826, -60, -97, -10], [-512, -28032, -192, 4144, 11372, 1548, -396, -340, -98, 96, 319, 16], [46080, -3072, 3968, -9584, 2064, -324, 956, -132, 84, -86, 20, 3], [11904, 8192, -2816, -2436, -2592, 136, 16, 170, -42, 43, -23, -8], [-3072, 1024, 384, 480, -48, -52, -84, 4, 12, 6, 0, -1], [-115200, 64256, -43200, 23856, -4224, 7596, -3924, 668, -756, 306, -24, 51], [-115200, -28928, -43200, 23856, -4224, 7596, -3924, 668, -756, 306, -24, 51], [512, 86272, 192, 19152, -14648, 7188, -1788, 1796, -448, 450, -46, 75], [-192512, 71296, -72192, -2688, -35708, -3264, -2568, 1744, -266, -486, -267, -81]];
    Rf_basisdens := [1, 93184, 46592, 46592, 46592, 6656, 11648, 1024, 46592, 46592, 46592, 46592];
    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_224_o();"
function MakeCharacter_224_o()
    N := 224;
    order := 6;
    char_gens := [127, 197, 129];
    v := [3, 3, 2];
    // 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_224_o_Hecke();"
function MakeCharacter_224_o_Hecke(Kf)
    N := 224;
    order := 6;
    char_gens := [127, 197, 129];
    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, -1, 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 := 3;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 1, 0, 0, -1, 1, 0, 0, 0, 1, 0, 0], [0, -1, 0, 0, 0, 1, 1, 0, 0, 0, -1, 0], [2, 0, -2, 0, 0, 0, 0, 0, 2, 0, 0, -1], [0, 1, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0], [-14, 0, 14, 0, 0, 0, 0, 0, 3, 0, 0, -1], [0, 0, 16, 1, 0, 0, 0, 2, -2, 0, 0, 0], [0, 5, 0, 0, 1, -1, 3, 0, 0, -1, -3, 0], [0, -9, 0, 0, 0, 4, -5, 0, 0, 9, 0, 0], [0, 0, 0, 0, -5, 0, 0, 0, 0, 1, -2, 0], [0, -6, 0, 0, -1, 1, 5, 0, 0, 1, -5, 0], [11, 0, 0, 3, 0, 0, 0, 3, 0, 0, 0, 3], [-2, 0, 0, 4, 0, 0, 0, -6, 0, 0, 0, 4], [0, 12, 0, 0, -1, 1, -6, 0, 0, 1, 6, 0], [0, 0, 0, 0, -5, 0, 0, 0, 0, -9, 9, 0], [-9, 0, 9, 0, 0, 0, 0, 0, 11, 0, 0, 4], [0, 32, 0, 0, 3, -3, 11, 0, 0, -3, -11, 0], [33, 0, -33, 0, 0, 0, 0, 0, -9, 0, 0, 10], [0, -22, 0, 0, 0, -8, 8, 0, 0, 22, 0, 0], [11, 0, -11, 0, 0, 0, 0, 0, 8, 0, 0, 6], [0, -17, 0, 0, 5, -5, -3, 0, 0, -5, 3, 0], [28, 0, 0, -14, 0, 0, 0, -6, 0, 0, 0, -14], [0, 0, -7, -8, 0, 0, 0, 4, -4, 0, 0, 0], [-11, 0, 0, 13, 0, 0, 0, -1, 0, 0, 0, 13], [0, 0, 0, 0, -13, 0, 0, 0, 0, -12, -34, 0], [0, 25, 0, 0, 9, -9, 9, 0, 0, -9, -9, 0], [0, 0, -4, 1, 0, 0, 0, -18, 18, 0, 0, 0], [0, 0, 0, 0, 23, 0, 0, 0, 0, 10, 14, 0], [19, 0, 0, -5, 0, 0, 0, -43, 0, 0, 0, -5], [0, -2, 0, 0, 0, 20, 16, 0, 0, 2, 0, 0], [0, 0, 5, 18, 0, 0, 0, 31, -31, 0, 0, 0], [-106, 0, 106, 0, 0, 0, 0, 0, -37, 0, 0, 3], [-26, 0, 0, 16, 0, 0, 0, -24, 0, 0, 0, 16], [0, -29, 0, 0, 7, -7, 14, 0, 0, -7, -14, 0], [0, 0, 0, 0, 3, 0, 0, 0, 0, 25, -30, 0], [0, 0, 0, 0, 15, 0, 0, 0, 0, -39, 9, 0], [0, 0, -71, 6, 0, 0, 0, 29, -29, 0, 0, 0], [0, -31, 0, 0, 0, 6, -53, 0, 0, 31, 0, 0], [0, 44, 0, 0, -5, 5, -43, 0, 0, 5, 43, 0], [-196, 0, 196, 0, 0, 0, 0, 0, -36, 0, 0, -3], [0, 14, 0, 0, 0, -20, 28, 0, 0, -14, 0, 0], [0, -22, 0, 0, 19, -19, 44, 0, 0, -19, -44, 0], [-60, 0, 60, 0, 0, 0, 0, 0, 47, 0, 0, 25], [0, -43, 0, 0, 0, 0, -55, 0, 0, 43, 0, 0], [0, 0, 0, 0, 3, 0, 0, 0, 0, -85, 18, 0], [100, 0, 0, 2, 0, 0, 0, -26, 0, 0, 0, 2], [0, -68, 0, 0, 0, -4, 54, 0, 0, 68, 0, 0], [44, 0, -44, 0, 0, 0, 0, 0, -22, 0, 0, 13], [0, -23, 0, 0, -1, 1, 38, 0, 0, 1, -38, 0], [0, 0, -230, -13, 0, 0, 0, -15, 15, 0, 0, 0], [0, 51, 0, 0, 0, 38, 5, 0, 0, -51, 0, 0], [-119, 0, 119, 0, 0, 0, 0, 0, 8, 0, 0, 40], [15, 0, 0, 21, 0, 0, 0, 39, 0, 0, 0, 21], [0, 0, 118, -13, 0, 0, 0, 47, -47, 0, 0, 0], [0, 0, 0, 0, -11, 0, 0, 0, 0, -36, 91, 0], [0, 0, 0, 0, 27, 0, 0, 0, 0, 43, 19, 0], [0, 91, 0, 0, 29, -29, 27, 0, 0, -29, -27, 0], [0, 0, 0, 0, 35, 0, 0, 0, 0, -53, 5, 0], [31, 0, 0, 19, 0, 0, 0, 97, 0, 0, 0, 19], [-132, 0, 132, 0, 0, 0, 0, 0, 4, 0, 0, 17], [0, 0, 0, 0, 0, -16, -28, 0, 0, 0, 0, 0], [-180, 0, 0, 30, 0, 0, 0, 60, 0, 0, 0, 30], [0, 0, 0, 0, 45, 0, 0, 0, 0, -4, -55, 0], [0, 0, -69, -6, 0, 0, 0, -52, 52, 0, 0, 0], [0, 24, 0, 0, -17, 17, 45, 0, 0, 17, -45, 0], [0, 0, 123, 16, 0, 0, 0, 57, -57, 0, 0, 0], [-143, 0, 0, 17, 0, 0, 0, -51, 0, 0, 0, 17], [213, 0, -213, 0, 0, 0, 0, 0, -19, 0, 0, -34], [0, 33, 0, 0, 0, -36, -19, 0, 0, -33, 0, 0], [-44, 0, 44, 0, 0, 0, 0, 0, -125, 0, 0, -27], [0, -12, 0, 0, -29, 29, 20, 0, 0, 29, -20, 0], [0, 0, 0, 0, 25, 0, 0, 0, 0, 40, -45, 0], [0, 68, 0, 0, -9, 9, -83, 0, 0, 9, 83, 0], [123, 0, 0, 9, 0, 0, 0, 127, 0, 0, 0, 9], [0, -130, 0, 0, -69, 69, 10, 0, 0, 69, -10, 0], [0, 0, 0, 0, -41, 0, 0, 0, 0, 20, 2, 0], [0, -41, 0, 0, -17, 17, 56, 0, 0, 17, -56, 0], [0, 0, 8, 77, 0, 0, 0, -31, 31, 0, 0, 0], [12, 0, -12, 0, 0, 0, 0, 0, 101, 0, 0, -19], [184, 0, 0, -10, 0, 0, 0, -104, 0, 0, 0, -10], [0, 169, 0, 0, 0, 72, 27, 0, 0, -169, 0, 0], [0, 0, 0, 0, 27, 0, 0, 0, 0, -119, 4, 0], [51, 0, 0, -17, 0, 0, 0, 125, 0, 0, 0, -17], [0, -16, 0, 0, 43, -43, -48, 0, 0, -43, 48, 0], [0, 0, -96, -31, 0, 0, 0, 20, -20, 0, 0, 0], [195, 0, 0, 7, 0, 0, 0, -9, 0, 0, 0, 7], [0, 0, 77, 20, 0, 0, 0, 62, -62, 0, 0, 0], [0, -61, 0, 0, 0, 40, -9, 0, 0, 61, 0, 0], [0, 132, 0, 0, 0, -28, -104, 0, 0, -132, 0, 0], [0, 0, 311, 52, 0, 0, 0, 23, -23, 0, 0, 0], [0, 0, 0, 0, -23, 0, 0, 0, 0, 186, -45, 0], [0, 0, 0, 0, -63, 0, 0, 0, 0, 56, -9, 0], [647, 0, 0, -3, 0, 0, 0, -61, 0, 0, 0, -3], [0, 0, -551, 18, 0, 0, 0, 41, -41, 0, 0, 0], [0, 152, 0, 0, 0, -76, 86, 0, 0, -152, 0, 0], [0, -105, 0, 0, -1, 1, 190, 0, 0, 1, -190, 0], [235, 0, -235, 0, 0, 0, 0, 0, -158, 0, 0, 14], [0, 0, -586, -37, 0, 0, 0, -4, 4, 0, 0, 0], [0, 13, 0, 0, 51, -51, 0, 0, 0, -51, 0, 0], [-41, 0, 0, -11, 0, 0, 0, 155, 0, 0, 0, -11], [0, 0, 0, 0, -17, 0, 0, 0, 0, -76, -28, 0], [117, 0, -117, 0, 0, 0, 0, 0, 55, 0, 0, 82], [0, 0, -341, 46, 0, 0, 0, 8, -8, 0, 0, 0], [-222, 0, 222, 0, 0, 0, 0, 0, 58, 0, 0, 83], [-197, 0, 197, 0, 0, 0, 0, 0, 150, 0, 0, -10], [-400, 0, 0, 10, 0, 0, 0, 32, 0, 0, 0, 10], [0, 0, 22, -97, 0, 0, 0, -143, 143, 0, 0, 0], [0, 0, 0, 0, 37, 0, 0, 0, 0, 32, -3, 0], [11, 0, 0, 43, 0, 0, 0, -211, 0, 0, 0, 43], [0, -176, 0, 0, -49, 49, 84, 0, 0, 49, -84, 0], [0, 0, 0, 0, -49, 0, 0, 0, 0, 216, 204, 0], [409, 0, 0, 65, 0, 0, 0, 49, 0, 0, 0, 65], [25, 0, -25, 0, 0, 0, 0, 0, -1, 0, 0, -46], [0, 210, 0, 0, 0, 24, -168, 0, 0, -210, 0, 0], [-35, 0, 35, 0, 0, 0, 0, 0, 100, 0, 0, 56], [-538, 0, 0, -56, 0, 0, 0, -150, 0, 0, 0, -56], [0, 0, 0, 0, -131, 0, 0, 0, 0, 314, -53, 0], [0, -176, 0, 0, -101, 101, 77, 0, 0, 101, -77, 0], [-298, 0, 0, -84, 0, 0, 0, 36, 0, 0, 0, -84], [0, 0, 0, 0, 39, 0, 0, 0, 0, 111, -153, 0], [65, 0, 0, -123, 0, 0, 0, 13, 0, 0, 0, -123], [0, -105, 0, 0, 139, -139, -48, 0, 0, -139, 48, 0], [58, 0, -58, 0, 0, 0, 0, 0, -16, 0, 0, -17], [0, 0, 771, -4, 0, 0, 0, 45, -45, 0, 0, 0], [0, -44, 0, 0, 0, 84, 100, 0, 0, 44, 0, 0], [0, 86, 0, 0, -49, 49, -59, 0, 0, 49, 59, 0], [0, 175, 0, 0, 165, -165, -1, 0, 0, -165, 1, 0], [0, -218, 0, 0, 0, 64, -244, 0, 0, 218, 0, 0], [0, -6, 0, 0, 91, -91, -49, 0, 0, -91, 49, 0], [-788, 0, 788, 0, 0, 0, 0, 0, 58, 0, 0, -55], [0, 144, 0, 0, 0, -16, 10, 0, 0, -144, 0, 0], [0, 288, 0, 0, -17, 17, -70, 0, 0, 17, 70, 0], [0, -89, 0, 0, 0, -36, -65, 0, 0, 89, 0, 0], [0, 0, 26, 27, 0, 0, 0, 19, -19, 0, 0, 0], [305, 0, 0, -95, 0, 0, 0, -151, 0, 0, 0, -95], [0, 0, 0, 0, 79, 0, 0, 0, 0, -140, 14, 0], [417, 0, -417, 0, 0, 0, 0, 0, 165, 0, 0, -54], [0, 83, 0, 0, 0, 20, 127, 0, 0, -83, 0, 0], [389, 0, -389, 0, 0, 0, 0, 0, -2, 0, 0, 48], [530, 0, 0, 80, 0, 0, 0, -258, 0, 0, 0, 80], [0, -281, 0, 0, -25, 25, -196, 0, 0, 25, 196, 0], [0, 0, 0, 0, -75, 0, 0, 0, 0, 8, 169, 0], [330, 0, 0, -36, 0, 0, 0, -182, 0, 0, 0, -36], [0, 0, 0, 0, 75, 0, 0, 0, 0, -254, -90, 0], [0, 12, 0, 0, 0, -88, 184, 0, 0, -12, 0, 0], [0, -109, 0, 0, 0, 172, -3, 0, 0, 109, 0, 0], [-347, 0, 347, 0, 0, 0, 0, 0, -118, 0, 0, 60], [0, 0, -98, -157, 0, 0, 0, 142, -142, 0, 0, 0], [0, -305, 0, 0, -199, 199, -79, 0, 0, 199, 79, 0], [0, 345, 0, 0, 23, -23, 88, 0, 0, -23, -88, 0], [156, 0, 0, -66, 0, 0, 0, -204, 0, 0, 0, -66], [-540, 0, 0, 30, 0, 0, 0, -154, 0, 0, 0, 30], [0, 206, 0, 0, 83, -83, 108, 0, 0, -83, -108, 0], [1121, 0, -1121, 0, 0, 0, 0, 0, -71, 0, 0, -106], [0, -126, 0, 0, 0, -96, 14, 0, 0, 126, 0, 0], [0, -8, 0, 0, 139, -139, 44, 0, 0, -139, -44, 0], [0, 0, -141, -62, 0, 0, 0, 12, -12, 0, 0, 0], [-1254, 0, 0, -28, 0, 0, 0, -4, 0, 0, 0, -28], [0, 0, 0, 0, -69, 0, 0, 0, 0, 133, 83, 0], [0, 0, -5, -156, 0, 0, 0, -279, 279, 0, 0, 0], [-447, 0, 0, 53, 0, 0, 0, -149, 0, 0, 0, 53], [0, -246, 0, 0, 0, 204, 8, 0, 0, 246, 0, 0], [0, 0, -629, -64, 0, 0, 0, 177, -177, 0, 0, 0], [276, 0, -276, 0, 0, 0, 0, 0, -93, 0, 0, 25], [0, 0, 0, 0, 3, 0, 0, 0, 0, 51, 364, 0], [0, 0, 0, 0, -19, 0, 0, 0, 0, -102, 255, 0], [0, 0, 0, 0, -49, 0, 0, 0, 0, -439, 49, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_224_o_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_224_3_o_c();".
// 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_224_3_o_c(:prec:=12)
    chi := MakeCharacter_224_o();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_224_3_o_c();".
// 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_224_3_o_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_224_o();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![361, 209, 178, 5, 20, 3, 1]>,<5,R![678446209, 0, -121144597, 0, 18167550, 0, -566489, 0, 13038, 0, -133, 0, 1]>],Snew);
    return Vf;
end function;