
// Make newform 1700.2.a.e in Magma, downloaded from the LMFDB on 06 April 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1700_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_1700_2_a_e();".
// 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_1700_2_a_e();".
// 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 := [-1, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-4, 0, 1], [-3, -2, 1]];
    Rf_basisdens := [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_1700_a();"
function MakeCharacter_1700_a()
    N := 1700;
    order := 1;
    char_gens := [851, 477, 1601];
    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_1700_a_Hecke( : Kf := Kf)
    return MakeCharacter_1700_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, 0, 0], [0, 0, 1], [0, 0, 0], [0, 0, 1], [1, 1, 0], [-1, -1, -1], [-1, 0, 0], [0, 0, 2], [-2, 2, -1], [0, 2, 0], [3, -1, 2], [2, 0, 2], [6, 0, -2], [3, -1, -1], [1, 1, -1], [-4, 2, -2], [-6, -2, 0], [-2, 0, -4], [9, 1, -1], [5, 1, 0], [4, 2, -4], [-5, -1, 0], [5, -3, 1], [9, 1, 3], [2, 4, 0], [7, -1, 1], [-1, -1, -3], [2, 2, -5], [6, 0, -4], [-2, 0, -2], [9, -3, 1], [5, -3, 2], [-9, -1, 3], [1, 1, 2], [10, 0, -4], [2, -2, 0], [-8, -2, -2], [6, 2, 5], [-10, -2, 1], [-8, 2, 0], [-2, 2, -4], [-6, 0, -2], [10, 2, -6], [12, 2, -8], [0, 2, 0], [5, 1, 4], [7, -1, -6], [-13, -1, 3], [12, 0, 3], [5, 1, 7], [2, -4, 0], [-6, -2, -6], [-10, 0, -2], [4, 4, 0], [-3, 1, 9], [-9, 3, -5], [2, -4, 4], [-20, 0, 0], [8, -6, 2], [-6, -4, 4], [6, 2, 9], [-8, -2, -2], [-15, 1, -1], [-11, 1, 0], [8, -2, 0], [-4, -2, 2], [-12, 0, 6], [4, 2, 6], [0, -4, -5], [14, 0, -8], [-2, 0, 4], [-8, 4, 2], [-12, -4, 1], [3, 3, 3], [-5, -5, -4], [-15, 1, -7], [-15, 1, 3], [14, 4, -12], [-22, 0, 0], [10, 4, -12], [-15, 1, -8], [3, -9, 1], [5, 1, 10], [5, 1, 5], [-5, 7, 2], [9, 5, -9], [8, 6, 8], [-11, -7, 1], [2, 4, -4], [-1, -9, 1], [27, 3, -1], [-11, -7, 4], [-4, 4, 5], [-8, -4, -2], [25, 1, -8], [-2, -2, 3], [22, 0, -8], [22, -4, 4], [-5, -5, 1], [-6, 0, -6], [-14, -6, -5], [11, 3, -1], [5, -3, 9], [6, 8, -4], [19, -5, 2], [19, -5, 3], [9, -3, 1], [18, 4, 4], [-10, 2, 2], [10, -4, 2], [-14, -2, -7], [4, 2, -2], [-2, 0, -6], [23, -1, 2], [-12, -4, -6], [0, -2, -4], [12, 4, 1], [9, -7, 3], [12, 6, 8], [-10, -2, -10], [14, 4, 8], [6, 0, -6], [-12, -2, -6], [8, 0, 1], [-3, 5, -4], [-1, 3, 13], [16, 2, 8], [-21, 7, -2], [7, 3, -7], [-8, -2, 2], [8, 0, -2], [24, 4, -1], [-23, -3, 4], [25, 5, 9], [5, 5, 3], [-3, 5, -1], [-23, -3, 9], [-18, 2, -1], [-8, -2, -2], [-20, -2, 10], [23, 7, 0], [8, -2, 2], [-10, -6, 3], [4, 4, 7], [10, 8, 0], [33, -3, 6], [38, -4, 2], [24, 6, 8], [32, 0, -2], [-3, 1, 11], [-14, -8, 4], [14, 0, -6], [29, 1, 3], [-26, -6, 1], [0, -4, -1], [11, 7, 0], [-4, -12, 4], [-2, -8, -6], [2, -4, 0], [16, 2, -8], [34, -2, -5], [19, -5, -5], [-29, -5, 11], [14, 6, 12], [-22, -4, -4], [-16, 0, -1], [-15, 5, 0], [0, 2, 4], [8, 10, -2], [-18, 4, 6], [-4, 8, -8], [31, -1, -7], [-19, -7, 18], [-9, -1, -1], [16, -4, 2], [18, 8, -8], [-5, 3, 4], [-14, 0, -2], [15, 3, -5], [-7, 1, -5], [17, 1, -5], [31, -1, -4], [10, 8, -8], [22, -8, 0], [7, -5, 7], [-9, -9, 5], [14, 4, -4], [-3, -7, 19], [-12, -6, -8], [-1, -5, 6], [28, 2, 2], [-14, -6, 13], [-4, 8, 10], [25, 1, 7], [-10, -6, -9], [24, 6, 12], [8, 6, -4], [-24, 10, -2], [4, 2, -8], [-27, -7, -1], [14, -12, 10], [-27, -7, 2], [44, 2, 2], [3, -1, -11], [-20, -8, 0], [5, -7, -3], [-24, 0, 4], [15, -13, 3], [-26, 4, -8], [-2, -6, 14], [18, 4, -20], [-43, -3, 3], [42, 6, -5], [-21, -5, 3], [3, -1, -12], [10, 4, -6], [-23, -7, -5], [18, 0, 0], [24, 4, -19], [-21, 3, -9], [-38, 0, 8], [19, 7, 6], [10, -8, 12], [16, 0, -1], [-1, 11, -7], [-17, -5, 25], [-44, -6, 2], [17, -3, -10], [3, -5, 3], [31, 7, 14], [-8, -10, -10], [-37, -5, -8], [-10, 2, 8], [-45, 3, 1], [35, 3, -7], [-21, -5, -1], [22, -8, 14], [26, -8, 8], [3, -5, 12], [16, -8, -2], [-14, -6, -11], [-30, -10, -6], [-37, -5, 5], [41, 5, -5], [-14, 4, 16], [3, 7, -2], [14, 2, 21], [-9, 15, -2], [24, 8, 2], [11, 15, -5], [16, 14, 6], [4, 6, 10], [-33, -1, 11], [-10, 0, -18], [25, -11, 1], [20, 8, 4], [-4, 6, -12], [-38, 2, 3], [-20, -2, -4], [-7, 5, 1], [-12, -4, -13], [3, 7, -3], [-52, -2, -4], [12, 6, -10], [-30, 4, 4], [4, -8, 16], [22, -8, 4], [18, -8, 12], [-26, 10, -3], [-28, -6, 18], [16, -6, 16], [25, 1, 3], [-50, 0, -4], [16, 4, -10], [-10, -8, 0], [-33, -5, 7], [21, -7, 1], [-21, -5, 9], [13, -7, -9], [36, -16, 6], [-11, 1, -5], [17, -11, 4], [20, 0, 5], [10, 4, 20], [10, 2, -13], [-20, -8, 8], [-2, -8, -6], [-6, 0, 6], [-20, -8, -14], [3, 3, -27], [0, -10, -6], [-38, 2, 11], [-19, -7, 17], [-47, 1, -8], [-42, -4, 8], [-2, 16, -12], [0, 4, -24], [19, 11, 19], [-21, 3, 6], [15, 7, 7], [-13, -13, 7], [13, 17, 1], [-13, -1, -24], [16, 4, -11], [5, -3, -18], [30, -8, 4], [13, -3, -17], [2, -8, 6], [34, -2, -14], [20, 10, 6], [-10, -14, 11], [-14, -16, 6], [-12, -10, -22], [-21, -9, -9], [5, -11, 11], [-34, -4, -8], [-12, -4, -14], [17, 5, -8], [6, 0, 16], [30, 8, -24], [-31, 9, -8], [20, -6, 10], [-2, -12, -8], [-21, -5, 9], [-4, 10, -22], [46, 0, 12], [-7, -15, -6], [50, 6, -15], [18, 2, 9], [0, 2, 10], [32, -10, 14], [46, -4, 2], [-13, -1, -2], [-3, 17, -13], [15, -9, -10], [48, 0, -9], [14, 12, 4], [-14, 4, -18], [-24, -10, -12], [7, -9, 19], [-24, 6, -18], [26, 12, 0], [-22, 12, -16], [-14, 18, -10], [3, -5, 7], [-27, -11, -20], [28, -6, 2], [3, -1, 5], [7, 3, -26], [-34, -4, 2], [-6, 2, 12], [14, 0, -20], [18, 0, 0], [-4, 4, 15], [9, 13, -21], [55, -1, -5], [16, -4, 2], [-25, 15, -12], [-40, -2, 8], [-25, 3, 11], [38, 4, 14], [-28, -2, 2], [15, 11, -27], [33, 1, -8], [-15, -3, -11], [2, -12, 16], [20, 6, 14], [1, 5, -9], [16, -10, -12], [16, 4, 6], [-45, 3, 2], [14, -6, 3], [-9, -13, 1], [-2, -2, -30], [-4, -18, 16], [-31, 1, -10], [7, 11, -8], [-11, 17, -11], [34, 4, -20], [34, 12, -12], [22, -4, 10], [38, 0, -4], [-18, -14, 1], [54, 0, 4], [13, 13, -8], [-6, -18, -9], [32, 8, 14], [21, 1, -7], [8, -4, -27], [-41, 3, 13], [-40, 6, 4], [-12, -2, 6], [-22, -14, 34], [11, -1, 9], [30, 2, 0], [24, -10, -4], [-18, -6, 14], [-26, 4, -8], [-9, -1, 14], [-41, 3, -3], [-18, 0, -4], [31, -1, -25], [-33, 3, 17], [-20, 10, -14], [-2, -20, 16], [43, 7, 18], [-31, -3, 21], [49, -15, 7], [-41, 11, -9], [-39, -7, 34], [24, 6, -22], [25, 13, -15], [-5, -5, 1], [25, -7, -14], [-34, 8, -12], [-34, -12, 6], [5, -7, -18], [22, 10, -7], [10, -12, 8], [-45, -5, -3], [-50, 12, -4], [-16, 2, -10], [6, -10, 5], [-18, -6, -8], [-44, 10, -10], [16, -10, 18], [36, 8, 9], [-2, -8, -2], [-26, -6, 10], [-23, -11, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1700_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1700_2_a_e();".
// 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_1700_2_a_e(:prec:=3)
    chi := MakeCharacter_1700_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_1700_2_a_e();".
// 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_1700_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1700_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, -8, 0, 1]>],Snew);
    return Vf;
end function;