
// Make newform 792.2.q.f in Magma, downloaded from the LMFDB on 08 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_792_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_792_q_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_792_2_q_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_792_2_q_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 := [4, 4, 24, -16, 88, -18, 67, -9, 39, -2, 7, 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], [-6600, -742314, 1603488, -923458, 1443341, -468778, 993670, -106013, 181033, -6521, 27283, -227], [740800, -2646058, 3877124, -3776456, 2968745, -2265832, 1665582, -264513, 277191, -15881, 42545, 6647], [380300, -931250, -466670, -503041, -680639, -429749, -657714, 26763, -132954, 1841, -19652, 3664], [653222, 11500, 5440, 366719, -79328, 274222, -167199, 175701, -32247, 30185, -4391, 4540], [114850, -952402, 1416528, -1085500, 1179117, -674471, 778577, -272146, 142775, -40674, 19879, -5314], [49112, 773962, -3493784, 3586466, -2679077, 2635708, -1630536, 1567515, -310833, 270815, -34235, 39985], [-439952, 75878, -1948864, -766968, -1648883, -840050, -961076, -532631, -170621, -106231, -27867, -14427], [201370, -254154, 1046126, -1073249, 866583, -787162, 487779, -463938, 92862, -80210, 9948, -11815], [156984, 1021494, -1066300, 3869790, -1114611, 3260036, -605628, 1711377, -119847, 305421, -5685, 41411], [70592, 289238, -1599004, 3746686, -1819103, 2970940, -982652, 1937689, -203471, 347805, -13317, 52023], [-971548, -2578064, -791960, -7960921, -139940, -5884328, -7725, -3372969, 33177, -607687, -17441, -86060]];
    Rf_basisdens := [1, 240564, 721692, 360846, 360846, 120282, 721692, 240564, 120282, 240564, 240564, 360846];
    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_792_q();"
function MakeCharacter_792_q()
    N := 792;
    order := 3;
    char_gens := [199, 397, 353, 145];
    v := [3, 3, 1, 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_792_q_Hecke();"
function MakeCharacter_792_q_Hecke( : Kf := false)
    N := 792;
    order := 3;
    char_gens := [199, 397, 353, 145];
    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], [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 := ISA(Type(Kf), Fld), 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, -1, 0], [0, 0, 0, 0, -1, 0, -1, 0, 1, 0, 0, 0], [-1, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 1, 1, 0, 1, -1], [1, 1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0], [1, 0, -1, 0, 1, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, -1, -1, 0, -1, 0, 1, 0, 1, -1], [-2, 0, 0, 1, 0, 2, 2, 1, 1, 0, 1, 0], [0, 0, 0, 0, -1, 0, 0, 1, 1, 1, 1, -1], [0, -2, 0, 2, 0, 2, 2, 1, 0, 0, -1, -2], [0, -1, 0, -1, 0, 0, -1, -2, 0, -2, 0, 1], [-1, 0, 1, -1, 0, 0, 1, 1, 0, 0, 1, 0], [-2, -2, -1, 0, 0, -1, 2, -3, 0, 0, -3, 0], [1, 2, 1, -2, -1, 2, 2, 1, 0, 1, 3, -2], [0, -1, 0, -3, 1, 0, 1, -1, -1, -1, 3, -2], [-1, -3, 0, -2, 0, 1, 1, 0, 0, 0, 0, 0], [0, -2, 0, 0, 0, 0, 0, -1, 0, -3, 1, 1], [2, -4, 1, 4, 0, 1, 1, -2, 0, 1, -1, -1], [-2, -2, 2, 2, -1, 0, 0, 2, 0, 2, -4, 0], [1, 1, 2, -2, 0, -3, -1, 0, -1, 0, 0, 0], [-2, 0, 0, 0, 0, -3, 2, -3, -1, 0, -3, 0], [4, -4, -1, 4, -2, 0, 0, -3, 0, -1, -1, 0], [1, -2, 0, -4, 0, 0, -1, 2, -2, 0, 2, 0], [3, 4, -1, 0, 0, -4, -3, 0, 3, 0, 0, 0], [0, 3, 0, -3, 1, 0, 0, -1, -1, 4, -1, -2], [7, -1, -3, 1, 4, -2, -2, -5, 0, -3, 2, 2], [-3, -3, 3, 3, -2, 5, 5, 3, 0, 3, -1, -5], [0, 2, 0, -3, 3, 0, -2, -1, -3, 4, 0, -2], [-1, -2, -4, 2, 2, -1, -1, -1, 0, -4, -2, 1], [0, -4, 0, -1, -2, 0, -7, -6, 2, 0, -1, 5], [-3, 0, -2, 3, 0, 0, 3, -3, -2, 0, -3, 0], [0, 0, 0, 3, -1, 0, -8, 2, 1, -1, -1, 1], [0, 3, 0, 3, 2, 0, 0, 4, -2, 1, -2, -1], [3, -1, 0, 4, 0, 0, -3, -5, 0, 0, -5, 0], [0, 0, 0, -2, 4, 0, 4, 0, -4, -1, 2, -2], [-1, -3, 3, 3, -3, 0, 0, 2, 0, 3, -5, 0], [0, -1, 0, -3, 3, 0, -7, -6, -3, 0, -2, 3], [-5, -1, 4, -1, 0, 0, 5, 0, -2, 0, 0, 0], [3, -3, 1, 3, 0, -3, -3, -4, 0, 1, -2, 3], [2, 1, -7, -1, 1, -3, -3, -3, 0, -7, 1, 3], [-7, -3, 5, -5, 0, -1, 7, 1, -1, 0, 1, 0], [0, -4, 0, 0, 6, 0, 6, -2, -6, 2, 2, 2], [4, 6, -1, -6, -1, 0, 0, 1, 0, -1, 5, 0], [4, -3, -2, 3, -3, 2, 2, -4, 0, -2, 3, -2], [0, -2, 0, 2, 5, 0, -1, -2, -5, -1, -2, 4], [-7, 3, -2, 3, 0, -4, 7, -4, -1, 0, -4, 0], [-2, 3, -1, 3, 0, 1, 2, 1, 0, 0, 1, 0], [0, -4, 0, 3, -3, 0, 7, 4, 3, -2, 5, -1], [5, 4, -5, -4, 8, -6, -6, -2, 0, -5, 0, 6], [0, -6, 0, 0, -6, 0, -11, -2, 6, -1, 4, 2], [13, 0, 3, 0, 0, -2, -13, -2, 3, 0, -2, 0], [2, 0, -1, 0, -5, -1, -1, -1, 0, -1, 0, 1], [0, -3, 0, 1, 5, 0, -3, -6, -5, -2, -4, 7], [-6, -7, 5, -2, 0, 6, 6, 1, -1, 0, 1, 0], [9, -1, -2, 1, 0, 1, 1, 1, 0, -2, -1, -1], [9, -2, 4, 2, -4, 1, 1, -3, 0, 4, 2, -1], [1, 4, 1, -6, 0, -7, -1, 3, 4, 0, 3, 0], [-1, 10, 1, 2, 0, -3, 1, 5, -2, 0, 5, 0], [0, 8, 0, -4, 1, 0, 2, 3, -1, 0, -1, -7], [0, 3, 0, -3, 0, 0, 1, -1, 0, 2, -1, -2], [5, -3, -2, 3, -7, 1, 1, 0, 0, -2, -2, -1], [0, -4, 0, -1, 4, 0, 1, 2, -4, -3, 7, -3], [3, 4, 1, 3, 0, 0, -3, 1, -3, 0, 1, 0], [0, 7, 1, 2, 0, 3, 0, 8, 9, 0, 8, 0], [-12, 1, -6, 4, 0, 5, 12, 2, 3, 0, 2, 0], [0, 0, 0, -1, 1, 0, 6, 5, -1, -1, 6, -6], [0, 9, 0, -4, 0, 0, -1, 4, 0, -1, -1, -8], [6, 2, 4, 3, 0, 3, -6, 2, 4, 0, 2, 0], [-13, -2, -3, 0, 0, 1, 13, -1, -8, 0, -1, 0], [4, -2, -1, 2, -9, 3, 3, 3, 0, -1, -2, -3], [4, 2, 1, 5, 0, -1, -4, -4, 8, 0, -4, 0], [0, 7, 0, -3, -1, 0, 3, 2, 1, 8, -2, -5], [11, 8, 0, -8, -4, 3, 3, 3, 0, 0, 8, -3], [0, -1, 0, 10, -1, 0, 2, 9, 1, 3, 0, 1], [2, -4, -3, -1, 0, 3, -2, 0, 5, 0, 0, 0], [-13, -3, 7, 3, -9, 6, 6, 6, 0, 7, -3, -6], [0, 2, 0, 2, -6, 0, -9, 6, 6, 2, 2, -4], [0, 9, 0, -5, 2, 0, 9, 0, -2, 2, -4, -5], [0, 3, 0, 7, -9, 0, -9, 4, 9, 5, -6, 3], [8, 9, -2, 4, 0, -5, -8, 0, 5, 0, 0, 0], [7, 2, 7, -2, 6, -4, -4, 4, 0, 7, -6, 4], [-7, -2, -1, 2, 0, -1, -1, -4, 0, -1, 1, 1], [-10, 0, 0, 8, 0, 8, 10, 0, -2, 0, 0, 0], [7, 5, -3, -5, 0, -9, -7, 1, 6, 0, 1, 0], [0, 8, 2, -8, -1, 6, 6, 13, 0, 2, 1, -6], [12, -9, 5, -2, 0, -1, -12, -8, -1, 0, -8, 0], [-10, 7, -5, -5, 0, -2, 10, 10, 1, 0, 10, 0], [0, -2, 0, -10, -2, 0, 0, -7, 2, -4, 5, -3], [2, -7, 1, 7, -9, 3, 3, -3, 0, 1, -1, -3], [-2, 6, -4, -4, 0, 0, 2, 10, 10, 0, 10, 0], [10, 2, 0, -2, 3, -2, -2, -9, 0, 0, 9, 2], [0, 5, 0, 8, 7, 0, 1, 3, -7, -5, -10, 5], [0, 3, 0, 4, -1, 0, 7, 4, 1, -6, -3, 0], [8, -7, 1, 7, 2, -2, -2, 0, 0, 1, -9, 2], [0, -10, 0, 0, 2, 0, -4, -4, -2, -2, 6, 4], [4, -1, -5, 1, -2, -7, -7, -7, 0, -5, -1, 7], [10, 3, -2, -3, 7, -4, -4, 7, 0, -2, -8, 4], [-3, 5, 5, -5, 10, 4, 4, 3, 0, 5, 6, -4], [-14, 1, 1, 2, 0, -1, 14, -2, -9, 0, -2, 0], [-4, 6, 1, -6, 3, 3, 3, 5, 0, 1, 4, -3], [0, 4, 0, 6, -1, 0, -12, 5, 1, -2, -5, 1], [-16, -7, -1, -4, 0, 3, 16, 0, -10, 0, 0, 0], [0, 1, 0, 5, 1, 0, -8, 7, -1, -3, 1, -2], [0, -4, 1, 4, -1, 4, 4, -6, 0, 1, 6, -4], [-2, -1, -2, 5, 0, 0, 2, -6, 10, 0, -6, 0], [16, -5, 4, 5, 2, -3, -3, -1, 0, 4, -7, 3], [0, -7, 0, -3, 5, 0, 5, -5, -5, 2, 5, 2], [-2, 6, -4, 0, 0, -1, 2, 5, 5, 0, 5, 0], [0, -4, 0, -4, 9, 0, 8, -8, -9, 0, 0, 4], [-14, 9, -4, -9, -1, -3, -3, -3, 0, -4, 9, 3], [0, -5, 0, 1, -10, 0, -19, -3, 10, -1, 1, 4], [-4, 7, -2, 8, 0, -1, 4, -2, -1, 0, -2, 0], [6, 5, 1, -5, 2, -9, -9, -9, 0, 1, 5, 9], [2, -9, 7, -7, 0, -3, -2, -5, 3, 0, -5, 0], [0, 6, 0, 9, 0, 0, -14, 10, 0, 5, -5, -1], [-2, 1, -8, -1, -4, -4, -4, -1, 0, -8, -2, 4], [0, -8, 0, 1, 5, 0, -3, -3, -5, 1, 4, 4], [8, -6, 4, 6, 0, 6, -8, -6, -6, 0, -6, 0], [0, 9, 0, 8, -5, 0, 1, 10, 5, 6, -7, -2], [22, 5, -3, 2, 0, -7, -22, -4, 1, 0, -4, 0], [-7, 5, -2, 10, 0, 7, 7, 2, -6, 0, 2, 0], [-11, 16, 1, -16, 6, 1, 1, 11, 0, 1, 6, -1], [12, -6, -3, -6, 0, 0, -12, 0, -6, 0, 0, 0], [5, 0, -5, 0, 7, 0, 0, 0, 0, -5, 0, 0], [-19, 2, 0, 11, 0, 5, 19, -4, -6, 0, -4, 0], [8, -2, 2, 2, 2, -5, -5, -1, 0, 2, -6, 5], [7, -2, 0, -2, 0, -8, -7, -8, -8, 0, -8, 0], [0, -4, 0, 8, 5, 0, -18, 1, -5, -10, -3, 7], [-3, -8, 0, 8, -10, -1, -1, 1, 0, 0, -10, 1], [0, 4, 0, 6, 8, 0, 2, 0, -8, -2, -10, 6], [0, 5, 0, -8, 1, 0, -12, -3, -1, -1, 0, -5], [2, -8, -5, 8, 1, -5, -5, -8, 0, -5, -5, 5], [0, -5, 0, -5, 0, 0, 9, -2, 0, -2, 8, -3], [0, -4, 0, -2, -4, 0, 10, -8, 4, -4, -2, 6], [-4, -12, -9, 12, -1, -7, -7, -13, 0, -9, -6, 7], [0, -1, 0, 4, 0, 0, 0, 11, 0, 8, 8, -7], [0, -6, 0, -1, 2, 0, 0, -6, -2, -6, 1, 5], [8, 14, -4, -14, 1, -2, -2, 4, 0, -4, 8, 2], [11, -1, 0, 1, 9, -5, -5, 4, 0, 0, -10, 5], [8, -7, -1, 3, 0, -2, -8, -12, 3, 0, -12, 0], [0, -5, 0, -4, 11, 0, 19, -5, -11, 1, 4, 1], [10, 8, 3, -8, -3, -3, -3, -1, 0, 3, 6, 3], [-7, 5, 0, -5, -6, -7, -7, -5, 0, 0, 3, 7], [8, -13, -1, 1, 0, 9, -8, -5, -3, 0, -5, 0], [1, 0, 3, 2, 0, 4, -1, 2, -7, 0, 2, 0], [4, 5, -7, 2, 0, -8, -4, -5, 5, 0, -5, 0], [0, 2, 0, 1, -1, 0, -5, 3, 1, 3, 0, -2], [-10, -1, -10, 1, 12, -2, -2, -1, 0, -10, -2, 2], [0, -3, 0, -8, -1, 0, 1, -9, 1, -3, 2, 1], [5, -13, -5, 13, 7, 1, 1, -8, 0, -5, -4, -1], [2, 2, 0, -2, 11, 1, 1, -3, 0, 0, 6, -1], [0, 3, 0, 15, 3, 0, -6, 6, -3, -1, -12, 9], [2, -3, -3, 3, 0, 4, -2, -2, 12, 0, -2, 0], [14, 10, 1, -1, 0, -5, -14, 6, 0, 0, 6, 0], [-2, -1, 3, 1, 0, 6, 6, 16, 0, 3, -11, -6], [-12, -9, 2, -2, 0, 10, 12, 3, -3, 0, 3, 0], [-17, 6, 0, -6, 7, 7, 7, 7, 0, 0, 6, -7], [0, 10, 0, 12, -3, 0, -2, 7, 3, 8, -15, 5], [-4, 12, 7, 8, 0, 0, 4, 4, -4, 0, 4, 0], [-19, -7, -4, 7, -5, 2, 2, -6, 0, -4, 1, -2], [0, -13, 0, 2, -3, 0, -19, -11, 3, -1, 0, 13], [-32, -5, 5, 5, -6, 4, 4, 0, 0, 5, -1, -4], [0, 15, 0, -4, 0, 0, 8, 8, 0, 10, -3, -12], [16, 9, -4, -10, 0, -13, -16, 6, 14, 0, 6, 0], [-9, -3, 4, 3, 6, 5, 5, -4, 0, 4, 6, -5], [-16, -7, -3, -8, 0, 1, 16, 2, 5, 0, 2, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_792_q_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_792_2_q_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_792_2_q_f(:prec:=12)
    chi := MakeCharacter_792_q();
    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_792_2_q_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_792_2_q_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_792_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3721, 4453, 6061, 3638, 3211, 1564, 1127, 391, 196, 50, 22, 4, 1]>],Snew);
    return Vf;
end function;