
// Make newform 5328.2.a.bp in Magma, downloaded from the LMFDB on 01 May 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_5328_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_5328_2_a_bp();".
// 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_5328_2_a_bp();".
// 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 := [10, 3, -8, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-4, -2, 1, 0], [5, -4, -3, 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_5328_a();"
function MakeCharacter_5328_a()
    N := 5328;
    order := 1;
    char_gens := [1999, 1333, 2369, 1297];
    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_5328_a_Hecke( : Kf := Kf)
    return MakeCharacter_5328_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, 0, 0, 0], [-1, 0, 0, 1], [0, 0, 1, -1], [1, 0, 1, 0], [1, 1, -1, 1], [-2, 0, 0, 0], [0, -2, 0, -2], [-3, 1, -1, -1], [-1, -1, -1, 1], [1, 0, -2, 3], [1, 0, 0, 0], [-1, 1, -2, 0], [-2, 0, 0, -2], [-8, 2, 1, 1], [0, 2, 1, -1], [-2, -2, -2, -2], [-1, 0, 0, -3], [1, -2, 0, -1], [-2, -4, 1, 1], [3, 0, 1, 4], [-1, -5, 1, 1], [2, 2, 3, -3], [6, -2, -2, 0], [-2, 4, 2, -2], [-4, 2, -1, -3], [-7, 0, 2, -1], [1, -1, -1, 1], [-2, 4, 0, 4], [-6, 4, 0, 0], [-2, 4, -1, 3], [2, 2, 0, 4], [7, 2, -4, 3], [-11, -1, -1, 5], [-2, -2, 3, -1], [-6, 4, 0, -2], [8, -6, -3, -1], [-4, -2, 2, 0], [-17, 0, 0, -1], [-12, 6, 1, -1], [-2, 0, 2, -4], [4, -4, -3, 5], [3, -4, -2, -7], [2, 2, -4, 6], [-4, 6, -1, 1], [-16, 0, -2, -2], [5, 4, -1, 2], [0, -2, 3, 3], [-2, 4, 0, -6], [2, 4, 3, -3], [-9, -3, 5, -3], [-1, -4, 0, 3], [6, 2, -2, 8], [4, -4, -4, 4], [-2, -4, -2, -2], [-4, -6, 5, -7], [2, -2, 2, -4], [6, 2, 5, 3], [1, 0, 0, -1], [6, -2, 0, -2], [-2, 8, -4, 2], [-4, -6, 2, 2], [5, 3, 4, -4], [-3, -3, -1, 3], [-4, 4, -6, 4], [-6, -6, -2, 4], [8, 0, -6, 6], [3, 0, -3, 0], [-10, 0, -2, 0], [-8, -2, 6, -6], [-16, 10, 2, 2], [-6, 0, 1, 1], [-2, 2, -6, 6], [-2, 8, 1, -1], [-3, -1, 4, 0], [-12, 4, -4, 0], [-9, 3, 9, -5], [-14, -4, -1, 1], [8, -4, 2, -4], [20, -6, -4, 0], [-3, 7, 2, 10], [7, 7, -1, -3], [8, -4, 0, 0], [-5, -3, 2, -8], [-9, 3, 3, -1], [23, 4, -1, 4], [-6, 0, 0, -4], [-20, 8, 2, 0], [-10, 0, -6, -6], [15, -1, 5, 1], [20, -4, -2, 2], [7, -1, -3, 9], [-4, -4, 2, -6], [-25, 1, 1, -1], [2, 2, 2, 2], [-15, -4, 0, 5], [-6, 10, 5, -3], [4, -2, 3, 1], [2, -6, 2, -6], [-17, 0, -8, 5], [2, 4, -4, -6], [5, 0, -2, 9], [0, -2, 2, -4], [-10, -6, 4, -10], [27, 1, 2, -6], [-16, 8, 0, -2], [-4, -2, 0, 10], [9, -8, -1, 8], [2, 0, -7, 1], [13, -2, 8, -3], [-11, -3, -1, -5], [2, 0, -1, -3], [23, -7, 0, -2], [-9, -4, -5, 8], [-1, -5, 5, -3], [7, -3, 2, 8], [18, 2, 0, 4], [-13, 3, 1, 5], [-5, 11, 1, 7], [19, -3, -6, -2], [-5, -5, -1, 5], [-3, -5, 6, -4], [-22, 2, -11, -3], [8, 4, 8, -4], [-12, -4, 4, 0], [-17, 8, -4, 5], [-5, 0, 6, -5], [2, -2, 5, -5], [15, -8, 4, 3], [-8, -10, 1, 7], [-5, 4, -5, 12], [-8, 2, 11, -9], [-14, 4, 7, -9], [5, -4, -2, 1], [-15, 7, 6, -8], [2, -8, 4, 4], [-4, -12, 5, -7], [18, -6, -5, 5], [17, 1, 3, 9], [16, 8, 0, -6], [-1, -8, -9, 8], [20, -2, 5, -5], [8, 0, 0, -8], [-6, 10, 8, -4], [3, -5, 1, -13], [10, 0, -4, -2], [21, 4, 2, 5], [-26, 4, 8, -4], [-4, 6, 0, 10], [0, 4, 4, -8], [10, 4, 8, 8], [-7, 10, 0, 9], [8, 0, 0, 0], [-10, 0, -7, 13], [4, 10, 2, 20], [4, 8, -8, 8], [-4, 0, -8, 0], [5, 6, 0, 1], [-3, 15, -8, 6], [0, -2, 6, -2], [16, -4, -10, 6], [-3, -4, 5, -14], [-3, 8, -4, -7], [21, -6, -4, -5], [26, -4, 6, 2], [2, -10, 3, 5], [-5, 3, -1, 15], [22, 0, -4, 4], [11, -4, 3, -2], [-9, 8, -2, 3], [2, 2, 10, 2], [47, -5, 1, -1], [5, -8, 2, -5], [7, -11, -4, 10], [0, -4, -1, -19], [-18, -6, -4, 6], [-2, -4, -2, 4], [34, 2, 2, -8], [0, -2, -1, 7], [-36, 4, 0, -6], [0, -8, -12, 8], [0, -6, 0, 2], [-21, 4, 4, 1], [2, -4, 2, 6], [-26, 4, 4, -2], [18, -14, -3, -3], [8, 6, 11, -3], [26, -2, 12, 4], [-6, -10, -8, -2], [-26, -6, -3, 11], [-10, -10, 4, 6], [24, 0, -8, 12], [18, -16, -6, 8], [-6, -2, -15, 5], [7, 12, -3, 6], [24, -2, 9, -1], [36, -4, 4, 2], [-26, -8, -4, 4], [13, -5, 8, -2], [7, -12, 4, -1], [15, -8, -6, 7], [-14, -4, 3, -5], [8, -8, 8, -6], [7, 9, 0, -14], [-3, -5, 11, -19], [-21, -4, 4, -3], [2, -6, 2, 2], [7, -2, 8, -3], [-10, 10, 10, 0], [-14, 10, 3, -11], [22, -8, 6, -10], [14, 0, 6, -2], [-5, 7, -3, 5], [37, -14, -4, 3], [-9, 8, 0, -9], [21, 0, -11, 10], [-13, -4, 2, 5], [2, 16, -2, 6], [-27, 8, -8, -3], [6, -14, -2, -8], [8, -10, 1, -17], [-6, -10, 7, -7], [36, 10, -5, 13], [7, -13, 3, -1], [15, -4, 7, -12], [-13, 15, -1, 1], [-3, -1, -13, 15], [8, 0, -9, 9], [-4, 16, -3, 7], [8, -10, 6, 0], [14, 8, -5, -3], [-3, 4, -13, 6], [10, -12, -11, 1], [19, -8, -11, -4], [-33, 8, -11, 2], [-16, 8, 9, -9], [25, 7, -5, 15], [-19, -7, 5, 3], [-12, -8, -8, 12], [-9, 0, -2, -15], [-12, 24, 0, -4], [10, -4, 6, -8], [0, -4, -3, -17], [17, 0, 4, 3], [-7, 15, 3, 19], [-17, 7, -3, 1], [15, -9, -7, 5], [12, -16, -6, 2], [5, 15, 4, -16], [20, 8, 4, 4], [-1, 0, -2, -17], [29, -14, 8, -3], [12, -14, -2, 0], [18, 4, -4, 0], [-29, 15, 5, -1], [-19, 15, -2, -2], [34, -4, -3, 11], [0, 0, 4, 8], [26, -8, 11, -3], [-20, -6, -2, -10], [-3, -3, 1, -7], [-16, -6, -4, -10], [10, 2, 6, 0], [-4, -10, -2, 14], [0, -2, -8, 0], [-7, -5, 2, -14], [8, 8, 15, 7], [-38, 6, -2, 12], [55, 0, -1, 8], [13, 9, -1, -7], [33, 0, -6, -3], [-20, 0, 2, -6], [2, -16, 4, 4], [27, -1, 15, 11], [15, 5, 5, -15], [-24, 16, -8, 0], [-7, -1, 11, -11], [1, 0, 10, 13], [-33, 1, -4, -2], [-46, 18, 8, 4], [-10, -4, -13, -1], [8, 4, -6, -2], [-46, 0, -8, 2], [-16, -6, 7, -11], [-8, 14, 0, 14], [16, 12, -1, -3], [-34, -6, 6, -16], [14, -10, -2, -4], [47, 4, 6, -3], [28, 8, 2, 12], [14, -8, -10, 14], [12, -2, -10, -4], [-28, 14, 5, -1], [-50, 6, -1, 9], [12, 8, 13, -15], [-16, 8, -11, -7], [-22, 0, -5, 7], [-32, 0, 8, -10], [2, -6, 12, -8], [-39, 11, 0, 4], [-22, 14, 12, -14], [-6, 0, 3, -11], [-12, -12, -3, -13], [-8, 10, 4, 14], [24, -8, -6, -2], [-2, -2, 10, 0], [2, -8, 16, -14], [37, -7, -1, -3], [-26, -10, 5, -9], [-17, 3, 5, -5], [-32, -2, 11, -13], [18, 4, 7, 21], [41, -12, 1, 2], [53, -4, -5, 6], [41, 9, -3, 13], [-40, -4, -8, 10], [6, 6, -5, 21], [21, 9, -11, 9], [-43, 11, 4, 6], [-46, 6, -12, -10], [-18, 12, 4, -2], [5, 0, -3, 18], [-21, -12, -2, -21], [-12, 14, 2, 8], [-27, 4, -7, 14], [10, 8, 0, 16], [1, 7, -17, 9], [14, -8, 20, -6], [25, 4, 12, 9], [10, -20, 0, -4], [-26, -6, -17, 3], [-31, 24, 6, 5], [-39, 0, 6, -5], [28, -20, 10, -2], [20, -16, -2, -10], [7, 6, 16, -19], [-52, -12, 9, -15], [-32, 6, -5, 5], [14, 8, 8, 4], [-26, 8, -4, 10], [-58, 22, 7, -1], [-17, 13, -6, -16], [-29, 15, -11, 7], [-9, -8, -14, 5], [25, -3, 19, -3], [-36, -10, 2, 0], [-4, 8, -15, 5], [-18, 6, 17, -17], [25, 13, 3, 3], [-10, -8, -11, 15], [-7, 15, -1, 17], [1, -16, 1, 10], [-30, -20, 16, -20], [-32, -8, -2, 14], [10, 12, -8, 24], [-11, -5, 7, -17], [1, 17, 9, -7], [16, -10, 8, -10], [7, -16, -5, 10], [5, 8, -10, -9], [13, -3, -15, 15], [15, 8, -7, 4], [-31, 8, 2, -9], [-18, 22, 2, 2], [-3, -17, -4, -24], [-4, -16, 20, -10], [-17, -13, -1, 5], [31, 4, 0, -13], [22, -10, 0, -14], [-28, -20, 8, 0], [-38, -2, 2, -10], [0, 24, 0, 8], [4, -12, -8, -18], [1, -23, 1, 1], [-24, -16, 9, -7], [20, -8, 8, -8], [14, -4, -5, 15], [11, -19, -7, -15], [-28, -4, -2, 12], [-13, 5, 6, -4], [-3, 1, -1, 9], [-32, 12, -2, 16], [29, -4, 0, -15], [-27, 26, 8, -3], [20, 6, 16, -14], [-38, 2, -9, -11], [-20, 4, 5, 3], [-43, 8, 0, -5], [-18, -4, 4, -10], [-27, -16, 18, -13], [12, -2, -3, -13], [29, 8, -10, 25], [10, -20, 6, 0], [2, 8, -10, 12], [-8, 16, 5, -13], [-6, -4, -8, -12], [31, 0, 4, -13], [-11, -4, -19, 18], [-33, 12, 3, -8], [44, -24, 4, -2], [0, -12, -11, -23], [28, -2, -10, 14], [47, 15, 1, -3], [-18, -12, 0, -4], [-23, 1, 5, -9], [-10, -2, 7, -11], [1, 16, 10, -19], [-7, 2, 12, -11], [-39, 11, -5, 1], [-45, -7, 4, -8], [-12, -6, 5, -1], [44, 4, -7, 21], [18, 12, -12, 2], [16, -26, -2, -16], [30, 18, -4, 26], [-10, 16, 0, 4], [-22, -4, 8, -10], [10, 22, -3, 27], [2, 4, 20, -12], [7, 3, 3, -17], [-69, 12, 2, -1], [5, -8, -6, 1], [-52, -4, 0, -8], [-18, -2, 3, -19], [18, 8, 12, -12], [-8, -18, -5, 1], [-17, 22, 12, 9], [-1, 4, 5, -28], [7, 26, -4, 21], [-8, 0, 8, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5328_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5328_2_a_bp();".
// 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_5328_2_a_bp(:prec:=4)
    chi := MakeCharacter_5328_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_5328_2_a_bp();".
// 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_5328_2_a_bp( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5328_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-16, -26, -1, 5, 1]>,<7,R![64, 4, -18, -1, 1]>,<11,R![-52, 63, -12, -4, 1]>,<13,R![-160, 122, -17, -5, 1]>],Snew);
    return Vf;
end function;