
// Make newform 1840.2.a.u in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1840_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_1840_2_a_u();".
// 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_1840_2_a_u();".
// 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 := [2, 5, -4, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, -3, -1, 1], [-3, -1, 1, 0], [1, -5, -1, 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_1840_a();"
function MakeCharacter_1840_a()
    N := 1840;
    order := 1;
    char_gens := [1151, 1381, 737, 1201];
    v := [1, 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_1840_a_Hecke(Kf)
    return MakeCharacter_1840_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+1)]];
    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], [1, 0, 1, 0], [1, 0, 0, 0], [1, 0, 0, 1], [-1, -1, 0, 1], [0, 1, -1, 1], [0, 1, 0, 0], [1, 1, 0, -1], [1, 0, 0, 0], [5, -1, 1, 0], [0, -2, 1, -1], [1, 0, 4, -1], [3, 1, -1, 0], [1, 1, -2, 1], [-1, 2, 1, -2], [4, 3, -2, -2], [-6, 1, -2, 2], [-1, -1, -2, 1], [1, -2, 2, -1], [2, 2, 1, 1], [-8, -1, 1, -1], [-1, -3, 0, 1], [4, -1, -2, 0], [-1, 1, -4, 3], [-5, -1, 0, -1], [4, -3, -2, 2], [2, 2, -4, 2], [-3, 2, -2, -1], [-3, 1, -4, -3], [3, 0, 2, -1], [5, -2, 3, 2], [2, -3, -5, 1], [0, -4, 2, 4], [-10, 0, -5, 3], [8, 0, 0, 2], [0, -1, 3, -1], [1, 0, 2, 1], [5, 0, -1, -2], [-4, 0, 4, 0], [2, -2, -4, 4], [0, -3, -3, 5], [-3, 1, -2, -5], [10, 0, 2, 0], [-6, -1, -7, 1], [12, 3, 1, -1], [-9, 3, 0, -3], [0, 3, 2, 0], [4, 2, -8, 0], [-5, 7, 2, -5], [-5, 1, 0, -1], [-6, 1, 3, -5], [-10, 0, -7, 1], [8, -2, 0, 2], [12, -2, 8, -4], [-16, 3, -1, 1], [-11, 0, 2, -5], [-11, -3, 1, -4], [4, -3, 6, 0], [10, -3, -3, 5], [19, -1, 0, -1], [6, 1, 8, -4], [14, -3, 8, -6], [2, 4, 0, -2], [10, 1, 3, -3], [-16, -1, 0, 4], [6, -2, 8, -4], [-6, -2, -5, 3], [-7, 1, 0, 1], [-8, -8, 4, 0], [-3, 7, -3, -4], [-4, 5, -7, -1], [1, -5, 8, -1], [-2, 3, -6, 2], [4, -6, 4, -2], [8, 4, 4, -8], [-7, -2, -4, 1], [0, -4, 2, 8], [-8, -1, 3, 1], [-8, -2, -4, 6], [7, -1, 5, 2], [5, -1, 6, 1], [-3, -3, 0, 9], [-1, 3, -6, -5], [-14, 1, -10, 4], [-12, 5, -7, 1], [7, -2, 9, 2], [-4, -3, -2, -2], [-16, 3, 0, -4], [5, 4, -9, -4], [-10, 0, -4, -6], [2, 7, -12, 0], [16, -4, 8, 0], [-17, 0, -9, 4], [12, 2, -9, 1], [10, 6, 1, -3], [0, -1, -4, -6], [5, -4, -5, 8], [-12, 0, 2, 0], [3, -1, 2, -5], [9, 0, 7, -8], [-5, 0, -1, 4], [25, 6, -6, -3], [0, -7, 6, 8], [12, -2, 12, 2], [12, 0, 8, 0], [-6, 3, -11, -3], [3, 0, 1, -6], [-4, 4, 0, 2], [-20, 0, -4, -8], [15, -3, 5, 6], [-2, -6, 4, -2], [10, -6, 2, -2], [1, -2, 4, -9], [18, -2, 8, 2], [7, 3, -10, -1], [9, -7, 4, -3], [-16, -7, 4, 2], [-3, -2, 1, -4], [2, 3, -9, -1], [12, 2, 8, -4], [22, 0, 0, 0], [-2, 1, 11, -1], [0, -7, 10, -6], [13, 2, -5, 6], [-18, 2, 4, 2], [5, 3, -4, -7], [1, -1, 16, 1], [0, 7, 2, 4], [-4, 7, 4, -2], [-12, -7, 4, 4], [0, 0, -1, 1], [2, 2, -4, 2], [9, -5, -6, 5], [-21, 0, -8, 5], [-9, -5, 1, -2], [-4, 4, -4, 6], [3, 3, 8, 3], [8, -9, 2, 0], [-3, -6, -4, -1], [0, -7, 2, -2], [-8, 6, 5, -1], [26, 0, 0, 4], [23, -2, 5, 2], [8, 1, 2, -4], [-18, -3, 6, -4], [-11, -5, -6, 9], [-6, 8, 0, -12], [-8, -7, 11, 1], [-14, -12, 9, 1], [-5, -2, -9, 4], [0, 2, -4, -6], [8, 2, 14, -12], [4, 0, 4, -12], [-3, 2, 5, 0], [41, 2, 6, 3], [29, 3, 4, 7], [9, -3, -10, 5], [11, -1, -5, 0], [21, -1, 10, -1], [2, -4, 10, 10], [7, -2, 1, 2], [12, 2, 16, -10], [3, 4, 3, 4], [34, 4, -2, 0], [-11, 6, -12, 3], [26, -5, 0, 8], [-38, -3, -2, -2], [8, 8, 8, -6], [11, 7, 6, 1], [-30, 1, -5, -1], [1, 7, 0, -13], [-9, 11, 1, -6], [36, 2, 4, -4], [-14, 3, -8, -10], [6, -1, -3, -5], [-3, -9, -12, 5], [-2, 3, -3, 7], [13, -3, 3, 2], [-5, -10, 2, -3], [8, 4, 2, -8], [15, -10, 19, -4], [-5, -13, 2, 3], [-4, -1, -11, 3], [28, 6, 4, -6], [-2, 5, 12, -4], [-2, -2, 10, 2], [0, 10, 8, -2], [-5, -10, 6, 5], [5, 1, -7, 0], [10, -1, -6, 6], [8, 9, 3, -15], [-2, -8, 4, 6], [-2, -6, 6, 8], [12, 11, 2, -6], [0, -2, 14, -6], [-25, -2, -2, -5], [-5, 1, -6, 1], [20, 8, -6, 4], [-10, -2, -14, 6], [21, 0, 3, -2], [-4, 6, -14, 0], [-20, -4, -4, 8], [14, 1, 1, -3], [17, -13, -2, 7], [14, -10, 4, 2], [12, -1, 21, -5], [23, 1, 12, -11], [-11, 10, 5, -12], [-3, 3, 3, -6], [2, 3, 9, -1], [18, -13, -1, 3], [21, 2, 13, 2], [-11, 1, -6, -11], [-16, 6, -6, -10], [4, 7, 2, 0], [-3, -9, -4, 13], [7, 16, -11, -2], [3, -1, 13, -2], [16, -1, 16, 6], [20, -3, 11, 1], [-21, 1, -5, 0], [17, 9, -12, -1], [25, 11, -3, -4], [-27, 2, 4, -3], [3, -8, 5, -2], [-3, -3, -17, 6], [-13, -3, 2, 1], [-18, -2, -17, 7], [-16, 10, 6, -18], [2, 3, 10, 2], [-22, -4, -12, 16], [-9, 5, -2, 7], [-20, 0, 2, -14], [-36, -3, -10, -2], [-6, 13, -2, -6], [5, -3, 10, 5], [30, 0, 6, -2], [25, -5, -6, -1], [-48, -2, -3, -1], [-12, -14, 3, 9], [-6, 13, -14, -2], [18, -8, 17, 1], [1, -6, 25, 0], [15, -20, 1, 4], [-8, -16, 4, 2], [2, 1, 7, 1], [17, 4, -7, -4], [6, -6, 18, 4], [24, 0, 6, -10], [6, -5, 4, -8], [4, 4, 22, 0], [-8, -2, 8, -6], [-16, -9, 16, 6], [25, 5, 16, 1], [16, -2, -12, 10], [-16, -8, 15, 1], [-13, -9, 2, -7], [28, 6, 10, -14], [-28, -1, 1, -17], [22, -7, 17, 1], [-18, 7, -18, 6], [39, 8, -12, -1], [-14, -13, 10, 16], [3, -6, 18, -9], [16, -4, 16, 6], [-19, -1, -4, 11], [-5, -13, -6, 9], [17, 13, -12, 5], [35, -6, 2, 1], [30, -1, 13, -3], [-5, 11, 17, -2], [20, -3, 24, -6], [-17, -8, 14, -9], [33, 7, -10, -7], [-32, 1, -3, -9], [-13, -8, -7, 14], [-28, -4, 8, 0], [2, -7, 14, 0], [58, 2, 4, 0], [0, 0, 12, -8], [-1, 2, -1, 8], [0, 2, -2, 6], [23, -11, 8, -7], [0, 12, 8, -6], [0, -2, 0, -8], [22, -1, -6, 2], [-24, 7, -14, 10], [37, 15, -14, -5], [-18, 6, -12, -2], [24, -2, 12, 6], [-10, 4, -28, -4], [-10, -1, -22, 6], [18, 10, 4, -8], [-17, -3, -16, -3], [-4, -7, -23, 11], [-36, -4, -12, 2], [-26, -4, -4, 2], [50, -5, -5, 3], [-38, -4, -1, -5], [-23, 8, -5, 6], [-9, 2, 2, -1], [37, -4, 4, 19], [0, -12, 4, 4], [-15, 6, -1, 6], [8, 12, 0, -4], [20, -17, -17, 13], [20, -2, -4, 0], [-2, 6, -6, 10], [18, 8, -14, 0], [-38, 2, -8, 0], [29, 8, -7, 0], [-5, -11, 2, -3], [-16, -2, 2, 0], [-27, -6, -9, -6], [-37, 2, -2, 9], [-15, 7, -24, 5], [-10, -2, -1, -13], [-2, 4, -27, 5], [-4, -11, -4, 8], [-33, 12, 1, -16], [-59, -7, 2, -7], [-11, -12, -18, 5], [7, 0, 17, 4], [0, -10, 16, 12], [-26, 9, -6, -4], [40, 6, 2, -8], [24, -10, 10, 6], [21, -16, -9, 14], [-19, 11, 20, -9], [8, -17, -4, 4], [-19, -13, 25, 8], [12, 13, -9, -3], [-34, 0, -5, -1], [39, -6, 15, -12], [13, 15, -12, -5], [-21, -4, -13, 0], [22, 4, -4, -12], [-13, 3, -20, 3], [11, -20, 9, -4], [5, 4, -8, -3], [32, -7, -13, 5], [-3, 9, 12, -15], [8, -14, 24, -2], [6, 8, -18, -8], [36, -7, 6, -6], [-26, -5, 2, -10], [-29, 17, -13, 0], [-26, 8, -4, -10], [-16, -4, 6, 10], [-28, 13, 2, -2], [-20, 16, -15, 3], [6, 15, -11, -7], [-4, 9, -18, -6], [-8, -9, 20, 2], [24, 2, 14, 0], [-20, 14, -20, 2], [-2, -16, 4, -6], [-64, 0, -14, 2], [12, -9, 1, -5], [10, 12, 16, -22], [52, -1, -6, 10], [25, -3, 29, -8], [1, 20, -25, 6], [10, -4, -6, -4], [3, 18, -17, 0], [2, -3, 9, 13], [22, 5, -1, -17], [-3, -6, -14, 7], [-22, -10, -20, 0], [-32, 10, -9, -3], [42, -3, 14, 10], [19, -4, -3, -6], [14, 0, -10, -6], [-8, 8, -8, 8], [6, 7, -9, -5], [16, 0, -1, 1], [-23, -11, -8, 3], [6, 5, 2, -12], [12, -6, 14, 8], [-2, 11, 11, -5], [12, 26, -16, 2], [35, 5, -6, -7], [2, 10, 20, -22], [-22, -9, 11, -9], [17, 13, 6, 9], [-47, 4, 3, 2], [-8, 0, 15, -11], [12, 16, 0, 6], [30, 11, -16, 0], [-2, 5, -10, 10], [2, -10, -10, 14], [24, 2, 0, 6], [30, 10, -3, 5], [-20, -8, -4, -12], [11, -11, -18, 15], [-28, 19, 6, -14], [39, -11, -6, 23], [-27, 7, 8, -7], [32, -20, -8, 12], [50, 9, 10, 12], [63, -1, 27, -8], [-6, 5, -15, -11], [17, -14, -16, 13], [0, 11, -4, 0], [-19, -7, -21, 18], [30, 2, 9, -11], [-17, -4, 16, -15], [-9, -2, -25, -2], [-33, 7, 6, -5], [40, -4, -8, 12], [-4, -1, 15, 5], [34, 3, 3, -5], [7, 5, 20, -15], [22, 4, 10, 16], [1, -14, -4, 7], [-27, -16, 7, 4], [-32, -16, 16, 16], [-42, -10, 4, -10], [-25, -12, -2, -1], [17, 5, 14, -11], [-11, -15, -12, 13], [11, -1, -4, -1], [16, -10, -8, 4], [16, 4, 16, 4], [-6, -9, -11, 29], [22, 13, -13, -5], [9, -5, -12, 7], [-20, -7, 18, 10], [38, -7, 11, 5], [14, -2, 29, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1840_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1840_2_a_u();".
// 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_1840_2_a_u(:prec:=4)
    chi := MakeCharacter_1840_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_1840_2_a_u();".
// 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_1840_2_a_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1840_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, -1, 1]>,<7,R![-32, 52, -14, -3, 1]>,<11,R![32, -40, -16, 4, 1]>],Snew);
    return Vf;
end function;