
// Make newform 2016.3.d.c in Magma, downloaded from the LMFDB on 21 April 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_2016_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2016_d_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_2016_3_d_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_2016_3_d_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 := [9, 0, 8, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 5, 0, 1], [0, 11, 0, 1], [4, 0, 1, 0]];
    Rf_basisdens := [1, 3, 3, 1];
    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_2016_d();"
function MakeCharacter_2016_d()
    N := 2016;
    order := 2;
    char_gens := [127, 1765, 1793, 577];
    v := [2, 2, 1, 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_2016_d_Hecke();"
function MakeCharacter_2016_d_Hecke( : Kf := false)
    N := 2016;
    order := 2;
    char_gens := [127, 1765, 1793, 577];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 0, 0], [1, 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 := 3;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, 2, 0, 0], [0, 0, 0, 1], [0, 0, -3, 0], [14, 0, 0, 0], [0, -4, 0, 0], [0, 0, 0, 8], [0, 0, -3, 0], [0, 15, 0, 0], [0, 0, 0, -12], [-60, 0, 0, 0], [0, -6, 0, 0], [0, 0, 0, -8], [0, 0, -14, 0], [0, -45, 0, 0], [0, 0, -18, 0], [10, 0, 0, 0], [0, 0, 0, 26], [0, 0, 23, 0], [42, 0, 0, 0], [0, 0, 0, 34], [0, 0, -24, 0], [0, -26, 0, 0], [-126, 0, 0, 0], [0, 32, 0, 0], [0, 0, 0, 32], [0, 0, -43, 0], [0, 0, 0, 0], [0, 41, 0, 0], [0, 0, 0, 94], [0, 0, -12, 0], [0, 81, 0, 0], [0, 0, 0, 88], [0, -19, 0, 0], [0, 0, 0, 32], [190, 0, 0, 0], [0, 0, 0, -38], [0, 0, 50, 0], [0, -152, 0, 0], [0, 0, -27, 0], [-98, 0, 0, 0], [0, 0, 21, 0], [188, 0, 0, 0], [0, 153, 0, 0], [0, 0, 0, 76], [0, 0, 0, 24], [0, 0, 0, 20], [0, 0, -70, 0], [-266, 0, 0, 0], [0, -173, 0, 0], [0, 0, -101, 0], [-98, 0, 0, 0], [0, 0, -82, 0], [0, 306, 0, 0], [0, 0, -3, 0], [0, 130, 0, 0], [0, 0, 0, 104], [70, 0, 0, 0], [0, 111, 0, 0], [0, 0, 0, 32], [0, -204, 0, 0], [0, 0, 0, -140], [0, 0, 110, 0], [-410, 0, 0, 0], [0, 19, 0, 0], [0, 0, 0, -64], [196, 0, 0, 0], [0, 0, -131, 0], [-58, 0, 0, 0], [0, 200, 0, 0], [0, 0, -15, 0], [0, 0, 0, 24], [214, 0, 0, 0], [0, 0, 0, -72], [0, 0, 148, 0], [0, -199, 0, 0], [-270, 0, 0, 0], [0, -467, 0, 0], [-602, 0, 0, 0], [0, 0, -94, 0], [98, 0, 0, 0], [0, 0, 21, 0], [-166, 0, 0, 0], [0, 0, 0, 24], [0, 0, 233, 0], [0, 69, 0, 0], [616, 0, 0, 0], [0, 104, 0, 0], [0, 0, 0, -214], [0, 0, -122, 0], [0, 0, 166, 0], [0, 0, 0, -200], [0, 0, -65, 0], [0, 0, 0, 40], [0, 0, -60, 0], [0, 306, 0, 0], [0, -144, 0, 0], [0, 0, 0, 128], [238, 0, 0, 0], [0, 0, 0, -322], [0, -423, 0, 0], [0, 0, -42, 0], [0, 51, 0, 0], [0, 0, 0, -206], [986, 0, 0, 0], [0, 0, 206, 0], [0, -44, 0, 0], [0, 0, 289, 0], [534, 0, 0, 0], [0, 0, 0, -160], [722, 0, 0, 0], [0, 475, 0, 0], [0, 0, 0, -380], [0, 0, 0, 14], [0, -577, 0, 0], [0, 0, 0, -208], [0, 0, 14, 0], [0, 775, 0, 0], [0, 0, 111, 0], [1170, 0, 0, 0], [-1160, 0, 0, 0], [0, 744, 0, 0], [0, 0, -9, 0], [0, 0, 0, -260], [0, -125, 0, 0], [-672, 0, 0, 0], [0, 0, -156, 0], [0, 0, 0, 152], [-546, 0, 0, 0], [0, 0, 0, -162], [0, 0, 285, 0], [0, 0, 0, -536], [-1232, 0, 0, 0], [0, 674, 0, 0], [-170, 0, 0, 0], [0, -4, 0, 0], [0, 0, 0, 76], [0, -160, 0, 0], [0, -949, 0, 0], [0, 0, 0, -436], [0, -191, 0, 0], [0, 0, 0, 378], [0, 0, -105, 0], [14, 0, 0, 0], [0, 0, 282, 0], [-390, 0, 0, 0], [0, -1074, 0, 0], [0, 0, 0, -244], [0, 0, -159, 0], [1670, 0, 0, 0], [0, 1140, 0, 0], [0, 0, 0, 320], [0, 0, -22, 0], [0, 0, 0, 584], [0, 0, -85, 0], [0, 0, 0, 416], [0, 1146, 0, 0], [-726, 0, 0, 0], [0, -354, 0, 0], [0, 0, 9, 0], [0, 687, 0, 0], [0, 0, 0, 2], [0, 0, 356, 0], [0, 927, 0, 0], [0, 0, -440, 0], [0, 0, 0, -568], [1126, 0, 0, 0], [1022, 0, 0, 0], [0, 996, 0, 0], [0, 0, 523, 0], [-306, 0, 0, 0], [0, 0, 159, 0], [770, 0, 0, 0], [0, 0, 0, -676], [0, -20, 0, 0], [0, 0, 0, 342], [0, -675, 0, 0], [0, 0, 0, 60], [-1382, 0, 0, 0], [0, 0, 0, -2], [0, 0, -88, 0], [-938, 0, 0, 0], [0, 464, 0, 0], [0, 0, -179, 0], [0, 1236, 0, 0], [-196, 0, 0, 0], [0, 0, 0, 416], [1862, 0, 0, 0], [0, 0, -480, 0], [-2170, 0, 0, 0], [0, 0, 349, 0], [0, 0, 0, 352], [0, 1048, 0, 0], [0, 0, -559, 0], [0, -732, 0, 0], [2192, 0, 0, 0], [2310, 0, 0, 0], [0, -6, 0, 0], [0, 0, -298, 0], [0, -375, 0, 0], [0, 0, 0, 572], [-2114, 0, 0, 0], [98, 0, 0, 0], [0, 0, -408, 0], [0, 368, 0, 0], [0, 0, 0, -644], [0, 0, -323, 0], [0, 967, 0, 0], [0, 0, 0, 628], [-1186, 0, 0, 0], [0, -146, 0, 0], [0, 0, 0, -248], [0, 0, -92, 0], [0, 0, 410, 0], [-2194, 0, 0, 0], [0, 0, 0, 112], [0, -1200, 0, 0], [0, 0, -545, 0], [0, 1343, 0, 0], [1400, 0, 0, 0], [0, 0, 0, -656], [0, -313, 0, 0], [0, 0, 0, -320], [0, 0, 0, 0], [1188, 0, 0, 0], [0, 642, 0, 0], [0, 0, -667, 0], [0, 0, 0, 872], [0, 0, 447, 0], [-784, 0, 0, 0], [0, 0, 0, 116], [0, 0, 0, -544], [0, 1285, 0, 0], [0, 0, 0, 516], [0, 0, 316, 0], [-1862, 0, 0, 0], [0, -467, 0, 0], [0, 0, 269, 0], [0, 0, 40, 0], [0, 0, -197, 0], [0, 0, 0, 508], [0, 0, 0, -180], [1480, 0, 0, 0], [0, 482, 0, 0], [0, 0, 730, 0], [0, 0, 0, -956], [0, 0, -390, 0], [0, 0, 0, 538], [0, 0, -567, 0], [-28, 0, 0, 0], [0, 68, 0, 0], [0, 0, -93, 0], [-2282, 0, 0, 0], [0, 2104, 0, 0], [0, 0, 371, 0], [154, 0, 0, 0], [0, 0, 0, -132], [0, -456, 0, 0], [382, 0, 0, 0], [0, 0, 0, 238], [0, 0, -737, 0], [658, 0, 0, 0], [366, 0, 0, 0], [0, -1928, 0, 0], [0, 0, 0, -1020], [0, -421, 0, 0], [0, 1224, 0, 0], [0, 0, 0, 682], [0, -2019, 0, 0], [-3238, 0, 0, 0], [0, 0, 0, 856], [-546, 0, 0, 0], [0, 0, 0, -304], [-1862, 0, 0, 0], [0, 0, 0, 416], [0, 0, 579, 0], [1128, 0, 0, 0], [-3188, 0, 0, 0], [0, 0, 578, 0], [0, 0, 96, 0], [0, 0, 0, -1158], [0, 0, 160, 0], [-3082, 0, 0, 0], [0, 0, 0, 280], [0, 0, -309, 0], [2632, 0, 0, 0], [0, -461, 0, 0], [0, 0, 0, 136], [0, -146, 0, 0], [0, 409, 0, 0], [0, 0, -848, 0], [0, -1437, 0, 0], [0, 0, 206, 0], [-2662, 0, 0, 0], [0, 1116, 0, 0], [0, 0, 0, -1084], [0, -1630, 0, 0], [0, 0, -838, 0], [0, 0, 0, -968], [1054, 0, 0, 0], [0, 1143, 0, 0], [0, 0, 0, -266]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2016_d_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2016_3_d_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_2016_3_d_c(:prec:=4)
    chi := MakeCharacter_2016_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2016_3_d_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_2016_3_d_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2016_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![8, 0, 1]>,<19,R![-448, 0, 1]>],Snew);
    return Vf;
end function;