
// Make newform 5440.2.a.bv in Magma, downloaded from the LMFDB on 20 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_5440_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_5440_2_a_bv();".
// 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_5440_2_a_bv();".
// 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, 4, -3, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-1, 2, 0, 0], [-2, -1, 1, 0], [3, -5, -3, 2]];
    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_5440_a();"
function MakeCharacter_5440_a()
    N := 5440;
    order := 1;
    char_gens := [511, 4421, 2177, 1601];
    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_5440_a_Hecke( : Kf := Kf)
    return MakeCharacter_5440_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], [-1, 0, 1, 0], [1, 0, 0, 0], [0, 0, 0, -1], [0, -1, 1, 0], [-1, 0, 1, 1], [-1, 0, 0, 0], [-2, 0, 2, 0], [-2, -2, 0, 1], [1, 0, 1, -1], [3, -1, 2, 1], [-1, 2, 1, -1], [3, 0, -1, 1], [-1, 1, 0, -2], [-1, 1, 4, 0], [0, 2, 0, 0], [-2, 2, 0, -2], [6, 0, 0, 0], [-5, -1, -2, 2], [5, 1, 2, -1], [-1, -2, 1, -1], [5, -1, 0, 1], [3, -1, -2, 0], [4, 2, -2, 0], [0, 4, 2, -2], [8, 0, 0, -2], [7, -1, -2, -2], [-3, 2, 1, 0], [2, -4, 0, 0], [3, 2, 5, -1], [-1, 1, 4, 2], [-8, 3, 1, 0], [3, 0, -3, 1], [-8, -1, 1, 2], [0, 2, -4, 0], [10, -2, 0, 2], [2, 6, 2, -2], [3, -2, -5, 2], [12, 2, 2, -1], [5, -2, 3, 1], [4, -2, -2, 0], [7, 4, -5, 1], [-4, 2, 2, -2], [-1, -2, 1, -1], [-9, -2, 1, 3], [9, 1, -6, 1], [-4, 1, -1, 0], [-1, -1, -2, 0], [-3, 4, 3, -2], [8, 0, -4, -2], [2, 0, 4, 4], [8, -2, 2, -2], [9, 0, 9, -1], [0, 0, -8, 0], [-1, 0, -3, -3], [-15, -5, 0, 2], [10, 4, -8, 0], [2, -4, 2, 6], [-2, 0, -12, 0], [14, -2, 10, 2], [7, 2, 3, -6], [-2, -2, -6, -2], [-11, -5, 0, 0], [9, 1, -10, -1], [-5, 2, 1, 7], [8, -4, 6, 6], [-4, -4, 0, 2], [12, -4, -6, 2], [7, -4, 9, 4], [4, -4, 6, 2], [-8, 0, 2, -2], [-2, 0, -14, 0], [14, -4, -2, 1], [-17, 0, 9, 1], [2, 5, -3, 0], [-3, 1, -10, -4], [2, 2, 4, 2], [-8, -4, 2, -2], [14, 4, -8, 0], [-4, 2, 4, 0], [6, -1, -9, -4], [0, -6, 2, 0], [1, 9, 2, -5], [1, 0, 3, 7], [19, 7, -10, 3], [1, -1, 8, 0], [17, 4, 1, -5], [-9, -4, 1, 1], [12, 0, 2, 6], [-17, 3, -6, -2], [15, -3, 4, -2], [9, -7, -6, 1], [16, -4, -4, -1], [18, 4, 2, -4], [4, 7, 1, -2], [0, -2, -14, 1], [-10, 4, 16, 0], [10, -8, 0, 0], [5, -3, 6, 2], [11, -4, -1, 5], [-15, 6, 1, 0], [-3, 4, -9, 3], [15, 3, -2, 4], [-18, -2, 14, -2], [-10, 5, -7, -2], [-11, 4, -1, -1], [-5, -1, -10, 4], [6, -4, 4, -4], [-6, 2, 12, 0], [9, 4, -11, 3], [-4, 2, 14, -3], [0, 6, -8, 0], [-11, 6, 7, -3], [10, -7, 5, 8], [-10, 8, 2, 0], [17, -4, 1, 3], [-11, -4, -17, 6], [-9, -3, 8, -4], [9, 2, 3, 1], [-4, -6, 2, -2], [-4, 2, 0, -4], [-9, -2, -11, 7], [-12, 0, -10, 6], [13, 4, -9, 2], [-22, -5, 3, -2], [16, -2, 6, -4], [-1, -8, 11, 5], [-27, -1, 4, 1], [23, -1, -2, -2], [8, -10, -8, 0], [14, 4, 6, 4], [2, 4, 6, 1], [3, 5, 8, -7], [3, 4, 5, -7], [-10, 10, 4, 2], [0, -6, -6, 4], [-1, 0, -19, 1], [-5, -6, -1, -8], [-6, 2, -14, -2], [-28, 4, -10, -2], [-32, 1, -1, 4], [4, 0, -2, -2], [-14, 2, 0, 5], [7, 4, -3, 10], [-8, 2, -8, -12], [21, 5, 6, 1], [-5, 2, -19, -5], [-13, -2, 5, -5], [-4, 0, -4, -2], [7, -3, -8, 0], [-4, 12, -10, -6], [11, -4, -5, 5], [13, 7, -4, 8], [0, 10, 2, 1], [-11, 0, -9, -2], [17, 7, -8, 3], [22, 8, 2, -2], [5, 12, 1, -9], [-20, 0, -6, 6], [9, -4, -3, -5], [11, -6, 7, 0], [-3, 0, 3, -13], [-19, 3, -4, -10], [18, -2, 0, 10], [-2, -8, 8, 0], [4, 4, 8, 5], [-17, -3, -4, -7], [-1, -10, -15, 3], [4, 8, 2, 2], [-13, -2, 1, -9], [-2, -8, 10, 6], [0, 12, 4, -6], [-3, 1, 6, -5], [-39, 4, 3, 3], [8, 4, -4, -6], [18, -4, -12, 4], [-22, 5, -7, -6], [5, 8, 1, 3], [27, -5, -2, 0], [-26, 6, -4, -6], [13, 13, -6, -2], [-32, -7, -1, 4], [-14, 0, -4, 12], [2, 0, 8, 8], [-15, 3, 0, 2], [-10, -4, -10, 0], [4, 4, -18, 2], [-9, -1, -6, 2], [-15, 0, -11, -1], [17, 3, 24, -5], [-14, 4, 12, 0], [15, 2, -25, 0], [18, 0, 14, 4], [24, 2, -2, 8], [-9, 6, -9, -4], [1, 2, -13, 5], [27, -4, -1, -3], [-22, 0, 16, 4], [-3, -2, -5, -11], [-21, 1, 8, 4], [-7, -4, 1, -13], [23, 1, 12, 1], [-18, -2, -22, -2], [-12, -2, 14, -4], [-6, 4, -22, 2], [15, -12, 1, 5], [-16, 12, 4, 0], [-3, -7, 2, -4], [0, 0, -6, -2], [16, -10, -14, -6], [-16, 4, 18, 6], [16, -2, 10, -4], [12, -2, 18, 3], [11, 13, 8, -6], [17, -9, 4, -3], [-7, -8, 17, -1], [29, 1, -2, -2], [22, 4, 8, 0], [-28, 4, -20, -7], [-7, 8, -1, 3], [-32, -6, -8, 4], [13, -1, -4, 5], [-6, 0, -20, 4], [2, -4, 10, -1], [11, 5, -8, 4], [-2, 2, 20, 6], [44, 4, -6, 2], [21, 5, -10, -3], [11, -5, 6, -8], [-16, 9, -13, -12], [22, -6, -18, -6], [-12, 13, 3, -2], [4, 10, -6, -4], [20, 4, 20, -6], [-39, 1, 6, 2], [-31, -1, -12, 6], [17, -8, 9, -1], [-2, -4, -4, -4], [-10, -3, 9, -8], [2, 8, -10, 4], [-11, 6, -11, 2], [-20, 2, 2, 14], [29, -1, 8, 8], [12, 8, 0, -6], [10, -4, 8, 0], [5, -9, 4, -3], [14, -2, 0, 11], [-24, 1, -13, 4], [8, 0, 24, -6], [-9, 3, -6, 4], [-26, -2, 6, -6], [-20, 0, 10, 2], [7, -5, -6, 12], [7, 0, -29, -3], [-5, 4, 13, 1], [0, 4, -12, 4], [11, -4, -5, 5], [-25, 2, -1, -10], [11, 6, 13, -9], [1, 0, -21, -9], [32, 0, 8, -7], [17, -11, 10, 6], [31, 4, -17, 1], [-4, -4, -22, 2], [-48, -4, -14, 6], [0, -4, -12, 4], [16, -4, -26, 2], [22, -8, -8, 8], [-17, -6, -5, 2], [10, -6, -10, 2], [13, -8, 5, 11], [-35, -1, -4, 0], [-4, 4, 14, 2], [-22, 4, 10, 0], [26, 4, -24, 0], [-33, -1, -2, 4], [21, -9, -12, 2], [-14, 4, 2, 12], [-6, 12, 10, 0], [16, -12, -4, 10], [45, 1, -10, 2], [-3, 13, 6, 7], [20, 12, -20, 5], [-8, -8, -10, 10], [-33, -14, -1, 2], [-16, 8, 8, 8], [21, -6, -5, -7], [19, -2, -7, -1], [0, -4, 12, -2], [18, -12, -2, 8], [16, -8, -6, 10], [15, 6, -5, 10], [31, 4, 1, 1], [-10, 7, -29, -2], [4, 0, 26, -2], [-12, 8, 2, -2], [-8, -4, 20, 4], [27, 4, 1, 9], [-6, 13, 25, -4], [33, -3, 2, 0], [-41, 4, 5, -7], [-3, 4, 19, -5], [9, -1, 4, 9], [9, -4, -33, -4], [18, 3, 15, 2], [32, -8, -2, -6], [25, -1, 0, 4], [-9, -4, -5, 17], [-6, -2, -24, -4], [-32, 2, 0, 0], [38, -2, -8, -3], [13, 4, 21, -9], [18, -20, -8, 4], [-23, -7, -2, -8], [-19, 9, 22, 10], [-6, -4, 0, 0], [-6, -20, 2, 8], [3, -3, 8, -7], [-42, 0, -8, 0], [-8, 2, 8, 4], [-32, -1, 17, -4], [10, 8, -16, -4], [12, 14, -4, -8], [7, 3, -6, 6], [12, 16, 22, -10], [10, -14, 14, 6], [22, 7, 35, -8], [11, 6, -5, -10], [-18, -10, 20, -3], [0, -16, -14, 10], [2, 16, 8, -12], [29, -2, -9, 1], [33, -1, -8, 5], [3, -9, -10, -2], [-2, 9, -3, -6], [33, -8, -17, 4], [-32, -10, -4, 0], [3, 6, -31, 3], [5, 0, 9, -13], [9, -1, -4, 2], [-50, -2, 2, -2], [56, 8, 10, -10], [-2, 4, 20, 4], [-2, 2, -16, -12], [-11, -12, 15, -5], [24, -5, 9, -6], [-14, 8, 16, 4], [11, -7, 4, 8], [-39, 3, -16, -5], [-1, 2, 17, 15], [-2, 2, 8, -6], [-16, -8, -26, 10], [0, -10, 4, 8], [-6, 4, 2, -13], [20, -4, -4, -10], [39, -12, -11, 1], [10, 4, -14, -8], [24, -7, 7, 8], [-7, -2, 15, 9], [21, -5, -12, -6], [31, 6, 13, -13], [-16, -8, 30, -2], [-41, -1, 6, 2], [0, 15, 5, -2], [-13, -5, 2, -12], [-8, -4, -34, 10], [18, -8, -16, -4], [-23, 5, 14, 6], [25, 4, 9, -5], [48, 0, 8, 6], [4, -11, 7, 6], [24, 6, -2, 7], [18, 2, 32, 2], [28, 10, 2, 6], [-23, 2, 19, -7], [0, -1, 1, -14], [3, -5, -6, -3], [-7, -4, -9, -5], [22, 2, -2, 6], [-30, 4, -4, -4], [25, -4, 9, 11], [16, -12, 26, 6], [-34, -6, -12, 7], [-14, -16, 4, 4], [18, -13, 15, 16], [-30, 6, 4, 5], [-52, -4, -16, 6], [-19, 0, 15, -1], [-41, 12, 1, 2], [-51, -9, 16, -4], [57, -8, -19, -1], [56, -4, 6, 6], [-40, 2, -2, 10], [-3, 9, -6, -6], [14, 2, 32, -6], [-31, 6, -25, 1], [-8, 6, 10, 6], [-6, -12, 8, 8], [-2, 9, 29, -2], [0, 10, -14, 0], [2, 4, 32, -8], [-23, 0, 7, -9], [5, -13, 4, -12], [64, -8, -6, 2], [32, -2, -8, -4], [-23, -9, -4, -9], [13, 4, 3, 19], [18, -6, -8, -18], [31, 9, -32, 6], [-48, 7, -7, -10], [38, -4, 0, 12], [7, 0, 21, 1], [-5, 25, 4, -8], [-2, -1, -9, 6], [32, -4, 10, 14], [33, 12, 17, -5], [-13, 17, 4, -13], [26, 10, 16, -9], [26, 8, 12, -4], [37, 7, 16, -4], [12, -10, 0, 12], [14, 4, -16, 4], [-30, 6, -24, 1], [-40, 6, 10, 4], [46, 12, 8, -12], [28, -10, -32, 0], [-3, -16, -1, 0], [51, -16, -17, 1], [38, 10, -8, 8], [27, -3, 0, -17]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5440_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5440_2_a_bv();".
// 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_5440_2_a_bv(:prec:=4)
    chi := MakeCharacter_5440_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_5440_2_a_bv();".
// 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_5440_2_a_bv( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5440_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 2, 1]>,<7,R![132, 0, -24, 0, 1]>,<11,R![-12, -48, -24, 0, 1]>,<13,R![64, -32, -24, 4, 1]>,<19,R![-8, 4, 1]>],Snew);
    return Vf;
end function;