
// Make newform 1617.2.a.z in Magma, downloaded from the LMFDB on 18 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1617_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1617_2_a_z();".
// 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_1617_2_a_z();".
// 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 := [3, 4, -4, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -1, 1, 0], [2, -3, -2, 1]];
    Rf_basisdens := [1, 1, 1, 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_1617_a();"
function MakeCharacter_1617_a()
    N := 1617;
    order := 1;
    char_gens := [1079, 199, 442];
    v := [1, 1, 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;

function MakeCharacter_1617_a_Hecke( : Kf := Kf)
    return MakeCharacter_1617_a();
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, 1, 0, 0], [1, 0, 0, 0], [0, 1, -1, 0], [0, 0, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 1], [0, -1, -2, 0], [-2, 1, -2, -1], [1, -2, 0, -2], [2, -1, 0, -1], [-2, -1, -1, 2], [-1, -1, -1, 0], [2, -2, 2, -1], [2, 3, -2, 0], [3, 1, 1, 0], [-2, 3, 1, 4], [5, -2, 0, 2], [0, 2, 0, 1], [10, -3, 1, 2], [5, -1, 1, 2], [4, 0, 6, -1], [2, -4, 2, -3], [-2, 2, 0, 4], [6, -2, 6, 0], [-11, 1, -1, 2], [8, 1, 2, -1], [-2, -2, 2, -4], [6, -6, 0, 3], [6, -4, 0, 1], [0, -7, 1, 0], [8, 1, -2, 3], [8, 4, 0, 5], [2, -2, 6, -4], [-2, 5, -2, 2], [-2, -1, -4, -2], [16, -5, 4, 2], [-9, 4, -4, -2], [-2, 3, 1, -4], [12, -4, -2, 0], [4, -2, -2, 1], [-1, 3, -5, 2], [-2, -6, 4, -4], [-4, 4, -4, 2], [4, -8, -2, -4], [6, -3, 8, -6], [-14, -4, 0, 2], [2, -6, 8, -3], [-16, 5, -3, -6], [-4, 0, 4, -4], [14, -4, 10, 0], [4, -1, 2, 1], [6, -2, -2, -3], [-10, 6, -2, 0], [-11, 9, -7, -2], [6, -8, 0, 6], [4, 4, -4, -2], [12, -1, 5, 6], [6, -4, -2, 1], [-4, 2, 2, 0], [-4, 1, -8, 5], [-6, 6, 8, 1], [-4, 9, 0, -1], [14, -10, 4, 3], [-5, -4, 6, -2], [-11, 3, 3, 2], [-12, -5, -1, 0], [-4, 2, -4, 6], [4, 4, 6, 2], [16, -4, 2, 1], [-16, -4, -2, 3], [-4, 1, 1, 2], [0, 2, 0, -9], [-6, 3, 7, -2], [-6, 4, -6, -5], [-20, -1, 1, -4], [-13, 6, -8, -4], [-18, 14, 6, 0], [-7, -3, -9, 0], [-8, -5, 1, -2], [0, -12, 0, -11], [7, -3, -1, -2], [1, -6, -8, -4], [8, 8, 2, 5], [-17, -1, -3, 2], [-10, -7, 0, -6], [-1, -4, -6, -10], [-4, 4, 8, -4], [6, 4, 0, -5], [-8, 3, -4, -2], [-4, 6, -6, -6], [-9, 14, 2, -6], [6, -12, 0, 0], [12, 13, -9, 4], [-8, 8, -8, 1], [-4, 17, -7, 0], [0, 0, 14, -3], [16, 0, -2, 4], [4, 6, -8, -2], [4, -14, 4, -1], [8, -6, 8, 3], [2, -13, 10, -3], [4, -5, 0, 4], [-8, 8, -4, 7], [2, 7, -6, 2], [14, -3, 0, 3], [18, -18, 8, 4], [12, 0, 10, -6], [-8, 1, 0, -5], [-12, 8, 2, 0], [-10, 2, -8, -6], [-26, 0, -10, -1], [0, -2, -4, 4], [4, -9, -9, 4], [-10, 9, -7, 6], [-4, 7, 7, 0], [12, -9, 3, -6], [10, -2, -4, -4], [-4, -8, 4, -4], [-6, 14, 4, 0], [-2, 14, 0, 10], [3, -10, -2, 4], [8, 0, 4, 15], [14, -9, 10, -4], [11, -3, 3, -4], [4, -2, 0, 8], [24, -3, -2, 0], [-3, -5, 5, -2], [15, -11, 7, 6], [16, -10, 6, 2], [16, 6, -8, -1], [-20, 16, 6, -1], [36, -8, -2, 0], [6, -10, -2, 10], [-7, -8, -12, -6], [-6, 12, 8, -3], [2, 6, -8, -6], [10, -4, 4, -2], [-14, -3, -4, -4], [12, -6, -8, -6], [-22, -8, -4, -1], [6, -16, 2, -5], [6, 0, -8, 3], [12, -7, 7, 4], [-26, 2, -10, -8], [19, -3, 3, -4], [-4, -2, 0, -16], [-8, 12, 14, 8], [-12, 11, 0, 3], [-10, 4, 4, 6], [4, 12, -14, 4], [14, -6, 2, -3], [-16, 20, -4, -4], [10, -10, -6, -4], [2, 4, 10, 11], [6, -13, 19, -8], [11, -13, 1, 2], [28, -11, -4, -7], [-8, 13, 9, 10], [-8, 8, -14, 8], [18, -7, 6, 3], [25, 11, 1, 4], [2, 6, 2, 17], [24, 7, 4, -5], [-20, -10, -4, -8], [14, -1, -1, 14], [25, 2, 2, 4], [24, 2, 4, -2], [2, 16, -10, 0], [-32, 0, 2, 5], [18, -14, -8, -6], [-20, -4, -2, 4], [-18, 0, -12, -2], [0, -4, 10, 3], [8, 2, 12, 0], [20, -10, -2, -6], [26, -13, -1, 8], [-16, -7, -4, -12], [-26, 15, -13, -2], [38, -4, 14, 3], [-30, 26, 4, -2], [-4, 5, 15, 0], [16, -2, 12, -5], [-19, 18, 2, 6], [-4, -5, -12, -1], [23, 1, -5, -4], [4, 16, 4, 16], [-2, -6, 6, -4], [-2, -14, 6, -16], [16, 6, -2, 2], [-6, -2, 8, -15], [-9, -10, 4, -8], [-2, 10, -10, 1], [-10, 4, -12, 0], [24, -10, -2, -12], [12, 2, 4, 0], [-10, -17, -7, -22], [-44, 12, -18, -13], [-29, -10, -6, -10], [-2, -14, 10, -5], [-18, -12, 10, -6], [-4, -10, -6, -13], [38, -11, 14, 6], [-10, -14, 0, -18], [-26, 6, -4, -4], [43, 7, -3, -2], [26, 1, 9, 14], [-24, 9, -1, 4], [14, 14, 8, 14], [-14, -6, 10, -5], [-24, 19, 7, 10], [8, -14, 8, 12], [16, -2, 6, 10], [-2, 2, 10, 2], [16, -8, 14, -2], [16, -4, 12, -2], [9, 11, 3, 4], [8, -2, -8, 3], [26, 0, 12, 11], [24, -22, 12, 0], [-22, 28, -6, -10], [-8, -10, -4, 14], [10, 15, 0, -4], [-28, -4, -10, -4], [40, 5, 5, -4], [4, -16, 4, 7], [-10, 16, 12, 6], [-4, 9, -3, 20], [-11, 6, -6, -8], [-20, -5, -6, -19], [28, 0, 6, -8], [-1, 3, -11, 18], [-2, -11, -10, -7], [-2, 27, -12, 2], [-8, -15, -6, -17], [24, -8, -6, -2], [-24, 14, -2, 9], [-21, 2, 16, -2], [-38, 28, -8, -11], [20, -2, 22, -4], [-16, -6, -8, 8], [-15, -24, 14, -6], [14, -8, -2, -9], [16, 2, 22, 2], [-33, 4, 4, 4], [12, -21, -12, 0], [-6, -14, 6, -15], [-10, 11, -11, 12], [9, 13, -5, 0], [30, -5, 6, -2], [12, 20, -12, 15], [22, -6, -16, -4], [24, -3, -20, 3], [7, -17, -5, -8], [-22, 8, -28, 0], [-6, 8, -8, -9], [-22, -4, -18, 8], [-18, -20, 8, 4], [-6, -3, -6, -11], [-2, -18, 2, -8], [-4, -18, 0, 9], [2, 8, -2, 9], [0, -4, -24, 0], [24, -4, -2, -11], [-14, -10, -6, 2], [-60, 9, -23, -12], [6, 15, 1, 10], [30, 5, -1, 18], [-6, -3, 19, -6], [-22, 13, -14, 3], [26, -2, 14, 11], [-3, 29, -5, 4], [2, -22, 14, -18], [-2, 32, -12, 8], [24, 5, 8, -14], [-10, 2, -16, 0], [2, -23, 11, -6], [-31, 3, -9, -22], [-26, 12, -6, -10], [-42, 2, -16, 10], [-34, 10, -4, -7], [26, 6, 18, 11], [-6, -13, -3, 22], [-6, 10, -12, 6], [-58, 4, 10, 0], [30, 13, 8, 22], [27, 0, 10, -6], [-6, -18, -10, -8], [2, -11, 14, 5], [2, 17, 1, -12], [-54, -5, -8, -6], [-34, 8, -10, 2], [-29, -2, -24, 4], [28, 0, 12, -17], [-2, -8, -6, -23], [20, 14, -8, 0], [0, -20, 8, -21], [-26, 21, -1, 8], [20, -22, 2, 8], [-48, 2, -14, -24], [26, -15, 18, -9], [-10, 6, -14, 21], [32, 3, 14, 5], [-12, 9, 14, 1], [1, -10, 0, -8], [-16, -13, 1, 0], [30, -6, -6, -14], [-12, -8, -4, 18], [-35, -2, 4, -10], [13, -14, 4, 10], [18, 20, -6, -5], [2, 18, -2, 11], [-44, 18, -20, -2], [20, -3, 18, 14], [24, 5, -15, -2], [0, 16, 14, 15], [-20, 18, 14, 11], [-17, -14, 4, -2], [-2, -10, -30, -5], [-21, -6, 6, -8], [12, -1, -2, -6], [-2, -37, 8, -10], [9, 6, 6, 4], [-28, 11, -16, 8], [54, -5, 17, -8], [-38, 1, -16, -17], [-10, -18, 4, -12], [-4, 27, -3, 24], [14, 6, -4, -12], [38, -4, -12, -4], [8, 10, -2, 5], [12, 14, 14, 1], [26, -6, 0, -10], [32, 11, -4, -6], [-6, 8, -8, -12], [20, 16, 14, 27], [4, 10, 4, 28], [21, -27, 17, 10], [4, -15, 10, -20], [-31, 16, 16, 6], [32, -1, 0, 6], [-6, -8, -12, -2], [-2, 0, 18, 4], [58, -19, 16, 19], [6, 14, -2, 16], [-14, 23, -33, 4], [-48, 12, -8, -15], [-1, -6, -4, -6], [-12, 20, -8, 28], [-22, 38, 2, 2], [2, 15, 19, 2], [24, 9, 18, 22], [-3, 26, -2, 10], [24, -13, -9, 6], [6, 13, 6, -11], [-22, 16, -20, 15], [26, -13, 16, -7], [-40, 8, -28, -4], [-28, 16, 16, 2], [-32, 25, -18, 1], [28, 14, -8, -8], [8, -16, 0, 3], [-14, 7, -4, -20], [20, -12, -10, -16], [30, -28, -6, -3], [-14, -7, 11, -10], [-7, 4, -16, 18], [22, -1, 30, 13], [0, 24, -30, 13], [34, -4, -12, 5], [-23, 18, 10, -14], [26, 7, 21, 12], [46, -6, -4, 1], [-8, 8, -20, 1], [34, 0, -10, 11], [-7, -28, 12, -6], [1, -5, -5, -2], [26, -34, 2, -4], [26, -12, -4, 15], [-20, -8, -16, 1], [10, 6, 12, -28], [10, -25, -9, -20], [4, 26, -16, -2], [-4, -4, -22, 9], [-4, 21, -4, 14], [-2, -9, -6, -1], [-16, 0, -20, 8], [-12, -9, -11, -14], [-25, 19, 11, -6], [0, 23, 18, -2], [-9, -21, -5, -6], [9, 7, -1, -14], [-10, -2, -6, 2], [17, 17, 7, 32], [16, 29, -15, 8], [25, -16, 36, -14], [34, 6, 20, 17], [10, 5, 6, 32], [62, -9, -3, -10], [-18, -6, 8, -2], [52, -21, 2, 4], [24, 0, 16, -14], [-28, 12, 6, 8], [-14, 10, -32, -7], [36, -27, 1, -12], [-6, -31, -4, 15], [-20, -12, -2, -19], [-33, 15, -9, 16], [20, 3, -4, -10], [-16, 29, 7, 6], [-17, -35, 3, -26], [10, 2, 22, 2], [48, 17, 2, 24], [5, 5, 13, 2], [-12, -17, 14, -14], [32, 4, 10, -30], [58, -10, 4, 16], [-10, -18, 24, -4], [-2, -18, -8, -4], [8, 2, 4, -16], [30, -20, -2, -3], [66, 4, 10, 12], [14, -34, 14, -3], [29, -5, 7, 20], [22, 18, 10, -11], [-13, -2, 2, -6], [-42, -7, -9, 0], [-48, -6, 0, -12], [-8, 31, 2, 1], [-42, -14, 4, -8], [-20, -10, 0, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1617_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1617_2_a_z();".
// 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_1617_2_a_z(:prec:=4)
    chi := MakeCharacter_1617_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_1617_2_a_z();".
// 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_1617_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1617_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, 4, -4, -2, 1]>,<5,R![-12, 20, -4, -4, 1]>,<13,R![-4, -16, -8, 2, 1]>,<17,R![15, 124, -40, -2, 1]>],Snew);
    return Vf;
end function;