
// Make newform 5824.2.a.bs in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5824_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_5824_2_a_bs();".
// 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_5824_2_a_bs();".
// 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, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-3, 0, 1]];
    Rf_basisdens := [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_5824_a();"
function MakeCharacter_5824_a()
    N := 5824;
    order := 1;
    char_gens := [2367, 1093, 4161, 4929];
    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_5824_a_Hecke(Kf)
    return MakeCharacter_5824_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], [-1, 1, -1], [-1, 1, 0], [1, 0, 0], [1, -1, 1], [-1, 0, 0], [1, 1, 1], [-1, -1, 0], [-4, 2, 1], [-8, 0, -1], [1, 1, -2], [1, -3, -1], [0, 2, -2], [4, -2, -3], [3, -1, 4], [-2, -2, 3], [-2, 2, 4], [2, 0, 0], [-2, -6, 4], [3, -3, 1], [-3, -1, -4], [6, -4, 1], [-1, -9, 4], [-1, 5, -2], [-3, -1, 0], [-11, 1, -1], [4, 4, -4], [0, 0, 2], [-1, 3, 3], [4, 0, 5], [2, -6, 0], [6, -2, -2], [-3, 1, -5], [-7, 3, 3], [-7, 1, 1], [-1, -7, 1], [1, -7, -3], [-2, 6, -2], [5, 1, -4], [-5, 7, -9], [8, 10, -3], [-7, -3, 3], [0, 4, -2], [-8, 2, -2], [2, -8, 8], [-5, 9, -1], [-2, 8, -3], [7, -1, 6], [-10, -2, -2], [6, -8, 2], [10, 2, -1], [10, -2, 2], [3, -3, 0], [-11, 7, -1], [-1, -9, 1], [4, -6, 1], [-4, 10, -4], [-20, -4, 0], [0, 4, -3], [3, -9, -1], [2, -6, -6], [7, -3, 6], [19, 3, 0], [-5, -3, -3], [-13, 3, 1], [-10, 0, 2], [0, 8, -10], [0, -12, 1], [12, -8, -4], [1, -1, 10], [-22, 4, 2], [-3, 3, -9], [-9, 9, -7], [-6, 4, -10], [3, -11, 11], [18, -2, -2], [0, -10, 12], [15, -7, 0], [-2, 0, -2], [3, -7, -2], [-5, 1, -11], [7, -9, 9], [-24, 4, 2], [2, 4, 6], [-18, 6, -12], [14, 0, 5], [-12, 2, 4], [-23, 5, -9], [-24, -6, 2], [-7, 3, 1], [-21, 9, 1], [3, 7, -8], [-22, -6, 0], [6, 2, -8], [-5, -7, 9], [16, 4, 2], [-27, 7, 0], [18, -4, -10], [4, 12, -10], [3, 3, -11], [-12, -2, -7], [10, 12, -10], [-32, -4, 0], [-4, 8, -7], [-20, 10, -13], [10, 16, -4], [-3, 9, 2], [-9, -3, 2], [14, 12, 1], [18, 0, 12], [-13, -19, 5], [10, -4, 10], [-10, 4, 12], [-4, 8, -4], [24, 4, -2], [-10, 10, -3], [-6, 18, -2], [-1, -7, 1], [-14, 16, -8], [-2, -4, 3], [7, -11, 2], [12, -8, 5], [-7, -11, -1], [-18, 6, 2], [11, -1, -2], [-20, 16, -7], [-9, 19, 3], [15, -15, 1], [20, 8, -6], [-17, 13, -18], [-15, -1, -3], [-12, -4, 0], [-10, 12, -15], [8, 4, 1], [7, -11, 10], [5, -9, 8], [8, 10, -10], [27, -9, 10], [-18, 20, 4], [-24, 16, -5], [32, -4, -4], [-8, -6, -2], [-12, 8, 10], [-10, 22, 6], [15, 15, -3], [10, -2, 14], [-3, -5, -2], [-8, 2, 2], [0, -24, 6], [12, 4, 4], [15, 7, 9], [16, -14, 10], [-30, 14, -12], [30, 10, -6], [22, -8, 5], [-24, -10, 1], [-12, 4, 12], [-23, 11, -8], [-1, -29, 7], [-31, -5, 4], [18, 14, -12], [12, 20, -5], [-15, 11, 5], [-10, -6, 4], [1, -7, 5], [-25, 15, 10], [-14, -6, -8], [32, -10, 12], [-25, 3, 7], [-11, 13, 1], [-7, 7, 1], [16, 10, -10], [6, 12, -11], [9, 5, 7], [-6, -18, 10], [22, 8, 2], [-18, 14, 6], [-20, 14, -2], [11, -3, -3], [15, -1, -3], [22, -6, 10], [39, -11, -7], [-34, 8, 6], [-11, 19, -2], [1, 3, -11], [-5, 3, -3], [-4, 0, -7], [-25, -13, 2], [32, -6, 2], [-53, 3, -2], [9, 25, -15], [-10, 10, -12], [18, 4, 1], [3, -3, -8], [8, 2, 9], [9, 1, -7], [23, 11, -5], [14, 6, -5], [15, -15, 12], [-2, -26, 4], [13, 1, 15], [33, -13, 5], [11, 13, -16], [-38, 4, 10], [3, -9, -14], [28, -14, 16], [1, 1, 14], [24, 2, 17], [9, -3, -21], [-8, 0, -2], [-18, 14, -1], [-9, 3, 3], [-16, 8, -16], [10, 10, -2], [-21, 3, 2], [-5, -23, 10], [-2, -20, 5], [-16, -2, 4], [30, 10, 2], [29, -15, 7], [34, -18, 13], [22, -26, 0], [20, -22, 0], [4, -8, -18], [-2, 2, 16], [16, -12, -19], [2, -20, 14], [24, -10, 1], [13, 3, 7], [13, 11, -7], [0, 14, 4], [3, 5, -1], [-8, 0, -12], [8, -10, 24], [6, -22, 20], [-17, -5, 4], [48, -14, -6], [-26, 16, -18], [62, -4, -1], [-2, 22, -4], [28, -16, 20], [22, -6, 12], [-44, 12, 0], [8, -14, 6], [3, 5, -20], [7, -3, -5], [-45, 15, -4], [3, 23, -10], [-15, 3, -9], [-24, -22, 7], [-14, -4, 10], [-12, -2, 22], [-22, 2, -4], [-32, 18, -26], [-18, -4, 8], [34, -14, -6], [2, 22, 1], [24, -12, -12], [-12, -6, 12], [17, -25, -14], [8, 14, -19], [-12, 10, 3], [9, -33, 18], [7, -1, 1], [-30, 8, -20], [-28, 28, 2], [-36, 14, -22], [-19, -9, 0], [9, -1, -3], [-18, 16, 10], [-15, 21, -1], [29, -1, 19], [-10, -12, 12], [28, 14, -1], [7, -5, -5], [-16, 16, 4], [13, -1, 7], [-7, -27, 19], [-10, -8, 2], [-12, -8, -12], [6, -2, -24], [44, -8, -2], [0, 0, -4], [-21, 13, -8], [-14, -10, 20], [8, 18, -13], [-18, 6, -29], [30, 16, -6], [22, -6, 6], [57, -7, 9], [-18, -2, 7], [22, -10, 18], [-34, 12, 4], [-27, 25, 14], [-6, 22, -7], [30, -8, -22], [-2, 30, 4], [10, -28, 14], [-37, 21, -1], [-47, 17, 4], [-19, 25, 7], [-31, 5, 15], [-10, 24, -5], [48, -18, 15], [18, -32, -1], [-51, 5, 5], [20, 4, -14], [23, 3, 15], [0, -28, 18], [-20, 0, 21], [11, 9, 11], [-23, -23, 5], [4, -14, 4], [6, -40, 1], [-7, 7, -5], [-6, -20, 18], [-19, 5, 24], [25, 3, -7], [23, -3, -6], [-32, -8, 15], [-34, 6, -2], [52, -14, 16], [9, 5, 11], [26, 6, -18], [38, -8, 10], [15, 3, 11], [-48, 12, 0], [-13, -5, 0], [10, 10, -4], [32, 6, 4], [-5, 23, -1], [-28, 8, -23], [25, 23, -9], [7, -5, -18], [-51, 3, -3], [-19, 9, -14], [-3, -31, -1], [40, 10, -14], [-14, 4, 2], [-50, 14, 12], [43, -13, -3], [-22, -4, -6], [-42, 8, -14], [6, -18, 20], [3, -21, 1], [58, -2, 0], [15, -33, 25], [25, -29, 27], [15, 7, 14], [-21, -13, 13], [-65, 15, 0], [27, 23, -13], [-1, -17, -3], [16, 4, 14], [-10, 0, -18], [-23, -11, 23], [7, -17, 30], [-1, 15, -4], [-22, 12, -26], [-43, 7, -23], [-20, 22, -22], [40, -14, -22], [-38, 8, -3], [-17, -3, 21], [-17, -19, 1], [54, -18, 7], [-7, -33, 14], [21, -9, 9], [-24, 24, -1], [4, -12, -10], [31, 5, -7], [-21, 5, 21], [38, -6, 9], [-39, 3, -19], [70, -6, 13], [10, -14, -18], [0, 4, 30], [-14, -4, 26], [-8, 14, 0], [-38, 0, 16], [-51, 11, 2], [-48, 10, -30], [-3, 27, 3], [-51, -15, 15], [-21, -1, 6], [-43, -15, 12], [-25, 9, 9], [-26, 24, 10], [23, -11, -19], [49, -5, 1], [20, 6, -14], [-5, -31, -10], [34, -6, 10], [26, 18, -18], [28, -28, -8], [16, 12, 9], [0, -8, -4], [-18, 4, -8], [8, 22, -5], [-24, -2, -12], [39, -11, -10], [14, 2, -3], [-23, 19, -15], [-13, 21, 8], [9, 35, 0], [-41, 5, -9], [-64, 6, -8], [-13, -29, 5], [-45, -9, 6], [41, -31, 24], [-10, 0, 10], [-8, -8, 15], [-14, -8, 3], [52, -12, 30], [22, 34, -3], [-59, -3, -3], [-14, 46, -10], [18, -26, 6], [-38, -16, 12], [-21, 1, 3], [-12, 24, -7], [-18, 8, 30], [-9, 1, -11], [-3, -13, 13], [26, -16, 0], [27, -23, -10], [60, 6, 1], [-44, -6, -4], [-13, 7, 10], [-20, -32, 10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5824_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5824_2_a_bs();".
// 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_5824_2_a_bs(:prec:=3)
    chi := MakeCharacter_5824_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_5824_2_a_bs();".
// 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_5824_2_a_bs( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5824_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-8, -6, 2, 1]>,<5,R![-2, -3, 2, 1]>,<11,R![8, -6, -2, 1]>,<17,R![-4, -10, -4, 1]>,<19,R![-4, 1, 4, 1]>],Snew);
    return Vf;
end function;