
// Make newform 5184.2.a.bp in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5184_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_5184_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_5184_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 := [-8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_5184_a();"
function MakeCharacter_5184_a()
    N := 5184;
    order := 1;
    char_gens := [2431, 325, 1217];
    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_5184_a_Hecke(Kf)
    return MakeCharacter_5184_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], [0, -1], [2, -1], [1, 0], [-2, -1], [3, -1], [-3, -1], [-2, -1], [-2, 1], [4, -1], [-4, 2], [7, -2], [-3, -2], [-2, 1], [-4, -2], [7, 0], [0, 1], [-1, -2], [4, 0], [-5, 3], [-4, 1], [12, 1], [-6, 0], [-3, -2], [0, 3], [-10, 1], [-1, 5], [0, -2], [2, -1], [2, -2], [16, -1], [-13, 2], [-1, 2], [-6, 1], [-2, -1], [-4, -3], [12, 0], [12, -3], [-14, 1], [4, -8], [10, 4], [2, 5], [-3, -2], [20, 2], [-2, 6], [12, 1], [2, 7], [-15, 0], [-6, 7], [-3, 1], [2, -5], [1, -2], [17, 3], [5, 4], [14, -3], [-8, 2], [-12, 4], [-22, 3], [2, -5], [-12, 5], [22, 1], [-29, 1], [18, 3], [-3, -6], [14, -7], [20, -5], [-1, 8], [5, -10], [-24, -3], [5, -8], [-6, 6], [20, 1], [-14, 3], [13, 3], [0, 9], [-6, -7], [4, -6], [3, -6], [1, -2], [4, 7], [28, -3], [6, 2], [25, -3], [0, -3], [7, -10], [23, 3], [1, 2], [-14, -3], [20, -9], [5, -1], [-6, -7], [2, -2], [5, 2], [1, -8], [-16, -4], [-24, 3], [-5, 3], [4, -4], [12, 6], [21, 2], [-14, 0], [5, -2], [-13, 6], [17, -4], [3, -5], [-5, -6], [26, 0], [-28, 5], [-13, -8], [-4, 3], [8, -2], [-19, 8], [-7, 6], [-32, 0], [11, 6], [-3, 0], [-10, 10], [-30, -3], [12, -3], [20, -11], [2, 1], [-4, 7], [-13, 1], [-8, 3], [-26, 4], [12, -7], [-36, 4], [18, 3], [26, -5], [-35, 3], [-38, -1], [-40, -1], [-18, -4], [-2, -9], [26, -3], [10, -8], [-12, 13], [20, 7], [-25, -5], [11, 9], [22, 1], [32, 3], [12, -8], [-6, 4], [-4, 11], [-26, 3], [-30, 3], [-7, -4], [0, -8], [-16, 5], [-6, 8], [-1, 13], [0, 5], [-37, 0], [8, 3], [22, -6], [-2, 15], [30, 0], [-4, -13], [-49, 2], [15, 3], [-6, 1], [4, 4], [-5, -2], [-48, 3], [32, -8], [-28, 9], [18, 4], [8, -5], [1, 4], [-24, -3], [20, 1], [19, 0], [-10, 9], [26, 7], [-3, 14], [22, 4], [-20, 12], [8, 13], [38, 3], [35, -6], [14, -13], [-11, -11], [-6, -9], [-8, 7], [0, -14], [-24, 3], [-19, 10], [-20, 4], [-9, -1], [-43, -4], [17, 3], [12, -1], [-43, -1], [5, 4], [15, -4], [28, -7], [9, -16], [-38, -2], [-6, 13], [-2, -13], [16, 9], [-13, 4], [1, -5], [6, -12], [-8, -16], [1, -2], [-3, -4], [-37, 6], [-35, 1], [-2, -11], [-18, 11], [-12, 13], [48, -7], [-7, 6], [-4, -9], [3, 10], [14, 2], [-8, -5], [18, 11], [24, -9], [-29, 10], [-18, 10], [-1, 8], [30, -1], [30, 3], [44, -4], [-34, -1], [-24, 17], [14, 11], [44, 0], [-22, 9], [17, 0], [-32, 7], [38, 1], [2, 9], [-18, 20], [-5, -4], [26, -18], [-1, 14], [20, -8], [28, 3], [18, -20], [-15, 12], [-40, 7], [36, -4], [25, -14], [12, -23], [-26, 18], [-18, 15], [45, -3], [-4, 1], [34, -3], [-42, -3], [29, 11], [-32, 6], [44, 3], [40, -2], [-3, -7], [-40, -11], [29, 8], [-4, -3], [-30, -12], [-22, 3], [11, -20], [-44, -10], [29, 4], [-13, 10], [10, 17], [-28, -15], [4, 0], [-5, 12], [18, 5], [-25, 4], [20, -12], [-12, 3], [-30, -1], [-9, 23], [33, -16], [2, -9], [14, 1], [46, 5], [-10, -17], [29, 4], [-30, 10], [26, -12], [0, -9], [-4, 21], [-7, 13], [8, 7], [-5, 13], [-51, 4], [55, 4], [-16, -19], [18, 5], [26, 3], [-50, 1], [20, 8], [-20, 15], [57, -2], [-42, 4], [40, 8], [-36, 11], [-8, 1], [18, -20], [28, -11], [24, -7], [6, -13], [40, 6], [-40, -1], [-32, -14], [6, -13], [32, -3], [4, -4], [-21, -5], [45, 12], [36, 9], [43, -4], [34, -5], [1, 14], [13, 18], [-4, 6], [-6, 14], [-10, 15], [-71, 1], [53, -13], [-11, 10], [52, -13], [-50, 4], [16, -19], [76, -5], [-16, 17], [-31, -4], [-11, -17], [12, 0], [22, -4], [49, 4], [-1, -12], [12, 0], [32, 9], [-14, 7], [-28, -13], [-52, 5], [30, 17], [19, 21], [-28, 26], [-5, 8], [8, -13], [58, -4], [-36, -15], [-15, -11], [-6, -7], [60, 9], [12, 17], [42, -4], [-4, 9], [36, -8], [-2, 11], [2, -7], [30, 11], [25, 16], [48, -12], [-15, 16], [39, 13], [63, -12], [8, -1], [-22, -14], [5, 25], [-4, -3], [-23, 19], [-28, -11], [-92, 3], [-26, 9], [14, -8], [-29, 16], [62, -10], [-26, 0], [-21, 15], [2, 9], [42, 9], [47, -14], [0, 0], [-19, 32], [3, 22], [20, 20], [54, 3], [16, -15], [8, -12], [60, -3], [-33, -4], [-20, 3], [61, -5], [5, -10], [38, -7], [-18, 5], [-48, -12], [3, -6], [35, -2], [-24, -13], [66, -1], [-41, 19], [-12, -1], [46, 15], [26, -20], [-28, -8], [14, 15], [-29, -2], [-7, -20], [-92, 1], [-25, 12], [50, 1], [59, -3], [-93, 4], [-26, -11], [-82, 4], [86, 2], [-58, -9], [11, -25], [-32, 9], [60, 7], [-52, 10], [36, 11], [7, -16], [-54, -4], [-52, 15], [-27, 8], [22, 4], [-68, -8], [-64, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5184_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5184_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_5184_2_a_bp(:prec:=2)
    chi := MakeCharacter_5184_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_5184_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_5184_2_a_bp( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5184_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-8, 1, 1]>,<7,R![-6, -3, 1]>,<11,R![-1, 1]>],Snew);
    return Vf;
end function;