
// Make newform 464.2.u.h in Magma, downloaded from the LMFDB on 02 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_464_u();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_464_u_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_464_2_u_h();".
// 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_464_2_u_h();".
// 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, 3072, 3328, 576, -80, -140, 197, 35, -5, -9, 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], [293952, 163104, -16252, -53096, 50792, 1185, -1945, -233, 1067, -291, 97, 1], [-838912, -244288, 592832, 6384, -70768, 4093, 7123, 443, -3473, 1469, -435, 53], [324672, 274224, 39372, -66491, 5981, 25423, -8945, -307, 1137, -79, 40, 30], [-54272, -250432, -105168, 140576, 2656, -15837, -1587, 1317, 177, -749, 195, -69], [1298688, 1540352, 490080, -241200, -81487, 118748, 30394, -28698, 12256, 750, -250, 325], [1024, -72720, -39944, 4207, 13254, -12733, -247, 495, 57, -269, 76, -25], [-3739648, 4524800, 289408, 419888, -880016, 265585, 31327, 46567, -72085, 37297, -9535, 1961], [-122880, 232512, 174384, 22092, -64091, 10181, 19513, -9995, -157, 1407, -469, 130], [-1331200, 300288, 458752, 302880, -215200, -35987, 54723, 19019, -27073, 15181, -3475, 725], [2008064, 2440960, 500352, 210032, -144192, 151369, 30183, 9327, -14093, 13609, -2951, 913]];
    Rf_basisdens := [1, 1, 443456, 1773824, 443456, 443456, 1773824, 110864, 7095296, 443456, 1773824, 1773824];
    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_464_u();"
function MakeCharacter_464_u()
    N := 464;
    order := 7;
    char_gens := [175, 117, 321];
    v := [7, 7, 6];
    // 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_464_u_Hecke();"
function MakeCharacter_464_u_Hecke(Kf)
    N := 464;
    order := 7;
    char_gens := [175, 117, 321];
    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, 0, 0, 1, 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]];
    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, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 1, 0, 1, -1, 1, 0, -1, 0, -1, 1, -1], [0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0], [0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0], [-1, 1, -1, -1, 1, 1, 0, -1, -1, 0, 0, -1], [-1, 1, 0, 3, 1, 0, -1, -1, 3, -1, 1, -1], [0, 0, 0, 0, 0, 1, -4, 0, 0, -1, 1, -1], [-4, 0, 0, 0, 4, -1, 0, 0, 0, 1, 0, 1], [0, -1, 0, 1, 0, -1, 0, 1, 4, 1, 0, 2], [0, -1, -4, 0, 0, 1, 0, -1, 0, 1, 0, 0], [0, 0, -3, -4, 3, 0, -4, -1, 0, 0, 0, 0], [-4, 0, 0, -4, 3, 0, -3, 1, -4, -1, -1, 0], [0, 1, 0, 0, 0, 0, 0, -1, 0, -1, 0, -1], [4, -2, 4, 0, 0, -2, 0, 1, 0, 0, -1, 1], [1, -2, 0, 0, 0, -1, 1, 1, 0, 2, 0, 0], [-4, 0, 0, -4, 0, 0, 0, 1, -4, 0, -1, 0], [-1, 0, -1, 0, -3, 1, -1, -1, -1, -1, 1, -3], [0, -1, 0, 0, 0, -1, 0, 1, -4, 1, -1, 0], [0, 3, 0, 4, -4, 3, 0, -3, 4, 0, 2, -3], [-7, 3, -4, -4, 7, 4, -8, -3, -8, -2, 0, -2], [0, -2, -8, -4, 8, -2, -4, 0, -4, 2, -2, 0], [4, 1, 0, 4, 0, 1, 0, 0, 0, -3, 3, -1], [4, -3, 0, 1, -1, 0, 4, 0, 0, 3, 0, 0], [1, 2, 4, 1, 0, 1, 8, 0, 4, -2, 2, -1], [0, -1, -4, -4, 0, -1, -3, 1, -3, -1, 0, -1], [0, 1, 0, 4, 4, 1, 0, 1, -4, 0, -3, 1], [0, 0, -4, -4, 4, -1, -4, 4, -4, 1, 0, 0], [-7, 0, -4, 0, 8, -3, -4, 4, -7, 4, -3, 0], [0, 0, 5, 0, -8, 0, 5, 3, 0, 3, 0, 4], [8, -1, 8, 0, -4, -1, 0, 4, 4, 0, -1, 4], [0, 1, -4, -4, 0, 4, 0, -1, 0, -2, 0, -2], [0, -1, -1, -5, 1, 1, -5, 2, -5, -1, -1, 0], [0, 1, 4, 4, 0, -5, 4, -1, 4, 0, 0, 0], [0, 0, 5, 0, 0, -3, 4, 0, 5, 0, 0, 3], [-4, 3, -4, -4, 4, 1, -8, -3, -8, -1, 0, -1], [8, 3, 0, 5, 0, 0, 0, 0, 5, -1, 0, -3], [-4, -1, -4, -4, 0, -1, 0, -2, 0, 0, 1, -2], [-4, 0, 0, 0, -4, 2, 0, -2, -4, -2, 2, -5], [-1, 1, 0, -1, 1, 0, -1, -3, -1, -5, 3, -1], [-4, 0, 0, 4, -4, -3, -4, 3, 0, 0, 1, -1], [4, 0, 1, 0, -4, -1, 1, 0, 4, 0, -1, 5], [-8, 1, 0, -8, 4, 0, -4, -3, -8, -4, 3, -1], [3, -6, 3, 3, 0, -4, 3, 0, 3, 4, -4, 4], [4, -2, 8, 8, -4, -3, 1, 2, 1, -3, 0, -3], [0, -1, 8, 0, 0, -1, 8, 0, 8, 2, -2, 1], [0, 1, 8, -4, -8, -1, -4, -1, 8, 1, 1, 0], [0, 1, -8, -4, 8, 3, -4, 0, 0, -3, 1, 0], [8, 0, 0, 4, -4, -3, 8, 3, 0, 0, -4, 4], [-13, 0, -5, 0, 1, -1, -5, -3, -13, -3, -1, -1], [0, 4, 0, 5, -4, 0, 4, 0, 5, -1, 0, -4], [-4, 0, -8, 0, -4, -3, -8, 4, -4, 4, -3, 1], [0, 3, 4, 1, -4, 3, 1, 1, -4, -3, 3, 0], [4, -1, 0, 4, 0, 1, 4, 0, 0, -1, 1, -1], [5, 1, 0, 5, 0, 3, -4, 0, 0, -7, 7, -3], [0, -6, 8, 8, -8, -3, 0, 3, 0, 6, -4, 4], [0, 1, -1, -1, 1, 4, -1, -6, -5, -4, 1, 0], [0, 4, -4, 0, 0, 1, -8, 0, -4, -1, 1, -1], [8, 3, 8, 0, -5, 3, 0, -2, 5, 0, 4, -2], [0, -5, 1, 8, -1, -6, 8, 8, 4, 6, -5, 0], [8, 0, 8, 0, 8, -4, 8, 0, 8, 0, -4, -1], [9, 0, -4, 0, -4, 2, -4, 0, 9, 0, 2, 4], [-4, -1, 0, 12, -4, 0, 4, 0, 12, 5, 0, 1], [0, -3, 8, -8, -8, -1, -8, 5, 0, 1, -3, 0], [-19, 2, -12, -12, 19, 3, -4, -2, -4, -1, 0, -1], [-11, -3, -8, 0, 0, -2, -11, 2, 0, 3, 0, 0], [-8, 0, 0, -8, 12, 0, -12, -3, -8, -2, 3, 0], [11, -2, 11, 7, -3, -2, 0, 5, 3, 0, -4, 5], [4, -2, 0, 8, -12, 0, 12, -2, 8, 0, 2, 2], [-9, 1, 0, -1, -3, 0, 3, -2, -1, -5, 2, -1], [-5, 1, 7, -5, 5, 0, -5, 0, 0, -1, -4, 4], [-8, -2, -8, -12, 12, -2, 0, 4, -12, 0, -2, 4], [4, -3, -8, 4, 0, -2, -12, 0, -8, 2, -2, 2], [15, 2, 3, 15, 0, -3, 15, 0, 3, 6, -6, 3], [24, -1, 12, 8, -8, -3, 24, 3, 0, 1, -2, 2], [-12, 0, -8, -8, 12, -1, 4, 0, 4, 1, 0, 1], [-17, 0, 0, -13, 13, 0, -13, 0, -13, 1, 0, 0], [0, 3, -8, -11, 8, 6, -11, -4, -8, -6, 3, 0], [3, 0, 15, -5, 5, -3, 3, 3, 0, 0, -3, 3], [12, 0, 9, 0, -8, -2, 9, 4, 12, 4, -2, 1], [-12, -1, -12, -12, -8, -1, 0, 1, 8, 0, -1, 1], [2, -4, 2, 2, -2, -2, 2, 4, 2, 6, 0, 6], [0, 3, 8, 0, 0, 1, 12, 0, 8, 1, -1, -1], [12, -1, 18, 16, -16, 3, 12, -3, 0, 1, 2, -2], [-4, 1, -16, -4, 0, 4, -8, 0, -16, 1, -1, -4], [0, -4, -4, -12, 4, 7, -12, -5, -4, -7, -4, 0], [5, 0, 0, 0, -5, 0, 8, 0, 8, 4, 0, 4], [-8, 0, 5, 0, -12, -5, 5, -3, -8, -3, -5, -3], [-4, 0, -4, 0, 2, -10, -4, 5, -4, 5, -10, -1], [0, -8, 0, 0, -8, 0, 8, 4, 0, 3, -4, 8], [-4, 0, -12, 0, 12, 4, -12, 0, -4, 0, 4, 0], [4, 3, -4, -4, -4, -3, -4, -3, -4, -5, 0, -5], [12, -3, 8, 8, -12, 5, -4, 3, -4, 1, 0, 1], [8, -6, 8, 8, -8, -7, 8, 7, 0, 6, -1, 1], [4, -2, 20, 20, -4, -3, 4, 2, 4, 4, 0, 4], [8, -5, 12, 8, 0, -1, 12, 0, 12, -3, 3, 1], [-13, 0, -13, 3, 17, 0, 0, -4, -17, 0, 6, -4], [-12, -4, 0, -19, 12, 0, -12, 5, -19, 5, -5, 4], [-8, -3, 0, 8, -4, 0, 4, -7, 8, 6, 7, 3], [9, 3, 8, 9, 0, 0, 12, 0, 8, -3, 3, 0], [-12, 9, 0, -12, 0, 3, -4, 0, 0, -5, 5, -3], [4, -5, 5, 5, -4, -4, 4, 5, 4, 7, 0, 7], [-4, -2, 0, 8, 16, 0, -16, 2, 8, 3, -2, 2], [-9, 4, -9, -9, 1, 4, 0, -4, -1, 0, 4, -4], [0, -3, -8, 12, 8, -4, 12, 6, 12, 4, -3, 0], [-3, 0, 4, 0, -8, 0, 4, 3, -3, 3, 0, 3], [0, 0, 4, 0, -20, -2, 4, 0, 0, 0, -2, -4], [-12, -2, -12, 12, 3, -2, 0, 0, -3, 0, 5, 0], [-4, -1, 0, -4, 0, 1, 4, 0, 0, -4, 4, -1], [0, 6, 4, 0, -4, 8, 0, 1, 2, -8, 6, 0], [-4, -1, 8, 8, -8, -1, -4, 1, 0, 1, -4, 4], [17, 6, 20, 17, 0, 0, 24, 0, 20, -6, 6, 0], [0, -1, 7, -13, -7, -4, -13, 4, 7, 4, -1, 0], [8, -3, 12, 8, 0, 2, 4, 0, 12, -2, 2, -2], [12, 0, 8, 0, -4, 3, 8, -1, 12, -1, 3, 0], [-4, 0, -20, 0, -2, 2, -20, -11, -4, -11, 2, -5], [0, 2, 4, 4, -4, 1, 0, -1, 0, -2, 2, -2], [0, -6, 8, 16, -8, -3, 16, -7, 16, 3, -6, 0], [-8, 0, -3, -3, 8, 4, -4, 0, -4, -7, 0, -7], [0, 4, -12, -12, 12, 1, -12, 7, -12, -1, 4, 0], [-12, -8, -11, -11, 12, -8, -16, 8, -16, 0, 0, 0], [-12, 9, 8, 8, 12, 3, 5, -9, 5, 0, 0, 0], [-1, 0, -5, -1, 0, 0, 3, 0, -5, 1, -1, 0], [12, 10, 12, -12, 0, 10, 0, -1, 0, 0, 5, -1], [-4, 3, 12, -4, 4, -5, -4, 5, 0, -3, 4, -4], [-11, 15, -12, 0, 0, 11, -11, -11, 0, -15, 15, -15], [1, 4, 1, -24, -4, 4, 0, 1, 4, 0, -5, 1], [-12, -4, -20, -20, 12, 0, -20, 4, -20, 3, 0, 3], [12, 10, -4, 12, -12, 12, 12, -12, 0, -10, 8, -8], [0, -6, -4, 0, 4, -4, 0, 3, 6, 4, -6, 0], [-4, 3, 8, 8, 4, -8, 8, -3, 8, 5, 0, 5], [-12, 9, -12, -4, 8, 9, 0, -11, -8, 0, 2, -11], [-20, 0, -16, 0, 20, 2, -16, -5, -20, -5, 2, 3], [-12, 0, 1, 0, -12, 11, 1, 1, -12, 1, 11, -5], [12, 0, 9, 0, -4, -1, 9, 6, 12, 6, -1, 4], [8, -6, 0, 0, -8, -12, 5, 6, 5, 4, 0, 4], [-5, 1, 3, -5, 0, 4, -5, 0, 3, 7, -7, -4], [-20, -2, -4, -20, 0, -1, -8, 0, -4, 1, -1, 1], [-12, 10, -11, -11, 12, 5, -8, -10, -8, -6, 0, -6], [5, 0, -8, 0, 0, 10, -8, -8, 5, -8, 10, -5], [-4, 11, 0, -12, 4, 0, -4, 1, -12, -6, -1, -11], [0, -9, 4, -11, -4, 2, -11, 3, -16, -2, -9, 0], [-12, 6, -12, -12, -12, 6, 0, -6, 12, 0, 11, -6], [-20, 3, -12, -12, 20, -6, 0, -3, 0, -1, 0, -1], [8, 5, 0, 4, -21, 0, 21, -9, 4, -3, 9, -5], [-4, 6, 16, -8, 8, 1, -4, -1, 0, -6, -1, 1], [4, -3, 0, 13, -20, 0, 20, -1, 13, 10, 1, 3], [0, 4, 0, 10, 0, 4, 0, -4, 0, 0, -4, -4], [8, -9, 8, -20, 4, -9, 0, 6, -4, 0, -3, 6], [-20, 0, -4, 0, 4, -5, -4, 7, -20, 7, -5, 4], [-12, 0, -12, 0, -10, 7, -12, -1, -12, -1, 7, 0], [4, -2, 4, 0, 11, -2, 0, 1, -11, 0, -8, 1], [12, -10, 12, -12, 0, -10, 0, 10, 0, 0, -2, 10], [20, -8, 0, -4, -4, 0, 4, 0, -4, 2, 0, 8], [0, 1, -4, -4, 4, -4, -4, 7, 8, 4, 1, 0], [-4, -16, 0, 0, 0, 0, 0, 3, 0, 1, -3, 16], [0, -10, 4, 8, -4, -5, 8, 2, -8, 5, -10, 0], [0, -2, 0, -8, 23, 0, -23, -2, -8, 3, 2, 2], [0, -2, 3, 7, -3, -1, 7, 7, 19, 1, -2, 0], [-8, -4, -8, -2, 8, -4, 0, 6, -8, 0, 0, 6], [-4, 2, -8, -4, 0, 6, -24, 0, -8, 3, -3, -6], [-15, 3, -12, -15, 0, 0, 20, 0, -12, -1, 1, 0], [16, 4, 8, 16, 0, -1, 0, 0, 8, -7, 7, 1], [-28, -2, -24, -24, 28, 2, -20, 2, -20, -3, 0, -3], [0, -1, 3, 3, -3, -8, 3, 7, -9, 8, -1, 0], [-12, 0, -8, 0, 28, -6, -8, 3, -12, 3, -6, -2], [4, 1, -4, -12, 12, -4, 4, 4, 0, -1, -12, 12], [13, -8, 13, 8, -16, -8, 0, 6, 16, 0, -9, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_464_u_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_464_2_u_h();".
// 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_464_2_u_h(:prec:=12)
    chi := MakeCharacter_464_u();
    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_464_2_u_h();".
// 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_464_2_u_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_464_u();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4096, 3072, 3328, 576, -80, -140, 197, 35, -5, -9, 13, -3, 1]>],Snew);
    return Vf;
end function;