
// Make newform 5184.2.a.bz 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_bz();".
// 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_bz();".
// 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 := [-3, 0, 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], [2, 1], [0, -2], [-2, 0], [-1, 2], [-4, 1], [-4, -2], [2, -4], [6, -1], [4, 4], [-3, 2], [0, -4], [-8, 2], [0, 4], [-4, -4], [-8, 0], [-7, -2], [-4, 2], [-2, 0], [1, 0], [-4, 2], [-4, 4], [0, 3], [2, -8], [-12, 0], [-4, -4], [12, -4], [-1, -2], [0, 1], [-4, 2], [14, -4], [-12, -5], [-12, 4], [-14, -1], [-8, 4], [5, -6], [0, 0], [18, 0], [-6, -1], [0, 4], [6, -8], [6, 8], [11, 4], [6, -5], [-4, 0], [-8, -10], [-4, 2], [-6, 12], [11, -2], [-8, -1], [4, 4], [-15, 4], [-10, 0], [8, 11], [8, 12], [-2, 13], [-20, 2], [14, 0], [8, -7], [-4, -8], [6, -7], [-12, -4], [-12, 0], [3, 12], [6, 1], [-4, 14], [-6, 8], [-18, -4], [6, 0], [-16, 8], [6, -12], [-16, -4], [10, 0], [-4, -12], [18, 0], [20, 4], [-11, 6], [12, 3], [9, 4], [-12, -8], [7, -6], [-20, -4], [7, 12], [12, 0], [8, -4], [-8, -12], [-31, -4], [-4, -12], [40, 0], [-12, -16], [-2, 8], [-28, -4], [30, 8], [0, 14], [12, 8], [-12, -12], [8, 12], [-32, -2], [3, -6], [-4, -8], [2, -9], [-8, 4], [28, 9], [0, -8], [11, -8], [-26, 0], [16, -9], [-12, -4], [7, -8], [-8, -18], [10, -8], [0, -17], [16, -12], [8, -12], [4, 15], [12, 12], [10, 16], [-12, 12], [18, -12], [13, 10], [19, 4], [-4, 20], [-8, 4], [-28, 6], [6, 17], [-5, -10], [2, 4], [0, 18], [-26, -8], [-28, 0], [-16, -4], [0, 10], [22, 8], [44, 3], [11, -12], [30, -1], [16, 2], [26, -13], [-28, -1], [-8, 24], [-18, 5], [-20, 0], [2, -8], [-10, -8], [30, -4], [-14, 0], [-12, -9], [-12, -8], [26, 4], [-3, -10], [-16, -20], [44, -4], [10, 20], [16, 12], [26, -12], [-4, 18], [20, 9], [-9, 24], [-6, 7], [6, -4], [12, 15], [-8, -10], [6, 4], [-24, -4], [36, 0], [8, -22], [17, -18], [17, -20], [44, -4], [0, -20], [-3, 30], [4, 28], [-35, 0], [-20, -6], [-16, 20], [32, 10], [-30, 5], [8, -12], [-35, -2], [-8, 12], [-10, -12], [3, 2], [-24, 20], [0, 0], [42, 5], [47, -2], [0, -6], [-26, 16], [-42, -8], [13, 8], [36, 8], [-8, -18], [2, 19], [-54, -4], [32, 8], [-17, -16], [38, 8], [8, -23], [-42, 4], [34, 17], [-44, 10], [34, 0], [1, -8], [6, 28], [-12, 12], [-8, 10], [36, -8], [8, -20], [28, 18], [6, -8], [-6, 11], [32, -14], [2, 16], [4, 8], [14, -24], [4, -6], [-20, 5], [-60, -4], [18, -31], [43, 2], [24, -6], [36, 5], [40, -4], [24, 8], [-35, -10], [24, -12], [30, -16], [-24, 16], [4, -28], [-21, 14], [-44, -6], [4, 18], [-8, -9], [-12, -28], [16, -8], [17, -12], [-4, -8], [-28, 8], [38, 0], [-18, -16], [-16, 2], [32, 12], [2, -8], [0, 12], [-2, -20], [-40, -14], [6, 16], [-56, -4], [4, 0], [39, -6], [24, 9], [40, -8], [1, 0], [6, -9], [-18, -4], [-29, -18], [4, 6], [6, -19], [-2, -40], [8, 32], [12, 8], [-25, -6], [6, 0], [-8, -24], [-28, 20], [-14, -17], [-64, -1], [8, 20], [10, 13], [-67, -6], [8, 18], [58, 0], [20, -8], [25, -8], [52, -12], [0, 24], [-11, -22], [-17, -16], [-20, 32], [32, -24], [-16, -28], [32, -4], [-11, 10], [8, -4], [-4, -20], [-19, 12], [-36, 12], [-56, -6], [-40, 5], [20, -28], [-42, -20], [16, -31], [12, 28], [-63, -10], [50, -17], [40, 0], [30, -1], [30, 8], [-64, 12], [23, 4], [28, -16], [-8, -14], [10, -16], [-72, -4], [26, 16], [30, -8], [22, 32], [16, -4], [1, 42], [-18, 8], [-12, -4], [-4, 13], [4, -30], [20, 20], [-22, -8], [-12, 0], [12, -24], [5, -16], [48, 20], [-56, -20], [-26, 0], [-2, 21], [8, 4], [-8, 12], [33, -20], [24, 10], [24, 14], [-70, 8], [-84, -4], [-10, -16], [-30, -25], [24, 16], [30, 8], [12, -10], [48, 0], [17, 14], [0, 5], [26, 24], [-24, -12], [-67, 6], [0, 20], [6, -29], [-36, 28], [62, 13], [34, -12], [15, -34], [16, -20], [36, -4], [-26, -11], [-72, 6], [-43, 4], [-10, 25], [-36, 6], [-71, 14], [-40, -11], [-52, 12], [-46, -8], [88, 4], [-30, 32], [25, 26], [24, -13], [-66, -12], [-40, 16], [-8, 32], [-9, -24], [-36, 28], [-4, 8], [58, 16], [-58, -24], [-48, 2], [-20, -32], [-82, 9], [-40, -24], [-5, -2], [54, -8], [-12, -16], [31, 0], [24, 0], [5, -36], [18, -15], [-4, 29], [12, 36], [-24, 1], [-56, 14], [54, -4], [-60, 24], [31, 30], [28, -6], [-12, 0], [-11, -16], [2, 21], [-30, 16], [24, 34], [-24, 8], [61, -4], [24, -42], [-84, 9], [56, 2], [2, -5], [37, -22], [96, -4], [-40, -2], [-4, 33], [38, 23], [-28, 2], [-13, -30], [36, -19], [32, 26], [24, 28], [-1, 24], [-22, -37], [2, 12], [40, -16], [-3, -56], [-10, -37], [-36, -28], [-32, -12], [-36, 19], [-32, -24], [-68, 8], [-50, -32], [-44, 8], [8, -40], [19, -28], [-2, 51], [4, 20], [0, -19], [0, -4], [-28, 0]];
    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_bz();".
// 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_bz(: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_bz();".
// 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_bz( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5184_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, -4, 1]>,<7,R![-12, 0, 1]>,<11,R![2, 1]>],Snew);
    return Vf;
end function;