
// Make newform 5472.2.a.bs in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5472_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_5472_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_5472_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, 5, -4, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, -3, -1, 1], [-3, -1, 1, 0], [1, -5, -1, 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_5472_a();"
function MakeCharacter_5472_a()
    N := 5472;
    order := 1;
    char_gens := [4447, 2053, 1217, 3745];
    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_5472_a_Hecke(Kf)
    return MakeCharacter_5472_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, 0, 0, 0], [0, -1, 1, 0], [0, 0, 0, 1], [-2, 1, -1, 0], [2, -1, -1, 1], [-1, -1, 0, 2], [1, 0, 0, 0], [2, 1, -1, 1], [-2, 1, 1, -1], [-2, 0, -2, 2], [4, 0, 2, -2], [0, -2, 0, 0], [0, -1, -1, 0], [2, 1, 3, -2], [-2, -1, -1, -1], [-5, -2, 1, 0], [-2, 1, -3, 0], [1, -2, -1, 0], [8, 0, 0, 2], [3, 1, 2, -2], [4, 0, 4, -2], [0, -2, 2, 2], [6, 2, 2, 2], [-2, 2, -2, 0], [6, 0, -4, 0], [6, 2, 0, 0], [1, -2, -1, 4], [2, 3, 3, -1], [2, 2, -2, 0], [0, 2, -6, -2], [4, 3, -1, -4], [3, 3, 0, -2], [4, -1, 3, 0], [2, -1, -5, 2], [-2, -2, -4, 0], [4, -2, -4, 2], [-6, -2, 4, -2], [4, 0, 4, 0], [0, 2, -4, 2], [-8, 0, 4, 0], [4, 2, -4, -2], [8, 0, 0, 3], [4, 2, 4, -2], [-4, 0, 2, -4], [4, 4, 0, -3], [-7, 2, 3, -4], [6, -4, 2, 4], [-5, -4, -1, 0], [6, -5, 7, -4], [-4, 5, -1, -2], [8, -4, 4, -1], [4, 2, -4, 4], [-12, 1, -3, 2], [2, 4, 0, -2], [-10, 1, -9, 2], [-12, 2, -8, 2], [-2, -1, 9, -1], [8, 3, -7, 0], [12, 0, -2, 4], [-10, 7, 5, -4], [6, -1, 7, 1], [-12, 2, -6, 0], [-8, -2, -10, -1], [1, 4, -1, 0], [-10, 5, -3, -3], [15, 2, 1, -4], [18, -4, 4, -2], [20, 1, 5, -2], [14, 3, -1, 0], [11, -4, 5, -4], [12, 6, -6, 1], [4, 4, -4, -4], [10, 5, -3, -5], [17, 2, 3, 0], [-2, -10, 4, 0], [-4, 1, -1, 6], [-4, 3, 5, 0], [20, 0, 6, -4], [-10, -6, -2, 6], [6, -6, 4, 2], [-2, 5, 1, -7], [20, -2, 2, 2], [4, 0, 10, 2], [-8, -4, 4, 4], [-14, 3, -3, -8], [0, -4, 6, 6], [19, 3, -4, -2], [6, -1, 7, 2], [-2, -5, 1, -2], [-4, -7, 5, -2], [-4, 0, 0, -4], [-2, -6, 8, -4], [-12, 8, 0, 0], [-20, 1, -11, 8], [2, 5, -5, 5], [-4, -4, 2, -2], [-8, 0, -14, 4], [19, 2, -3, 0], [-4, -1, -15, 2], [0, -4, 0, -4], [16, -3, 7, 0], [-12, 4, -12, 0], [-12, 6, -4, -4], [-20, 4, -4, 0], [5, -1, -6, 10], [14, 9, -9, -6], [-10, -4, 4, -8], [16, 2, 2, 0], [-2, -6, 14, 2], [24, -2, 6, -2], [0, 1, -1, 6], [12, -5, 13, -2], [14, 0, 2, 8], [-10, 5, -13, 2], [20, 0, -2, 2], [0, -3, -3, 6], [-12, 0, -4, -5], [-16, 5, -1, 0], [-19, -2, -5, 8], [-2, -11, 1, 11], [12, 4, -2, 8], [-6, -1, -5, -7], [-24, 0, 4, -4], [8, 3, 11, -6], [0, 4, -6, -4], [4, 6, -4, -2], [24, 0, 0, -1], [-20, 6, 2, -11], [6, -6, 10, -10], [10, -3, -9, 2], [-10, 0, 6, 2], [12, 8, -4, -4], [-14, -7, 1, 0], [-13, -1, -4, 2], [-25, -1, -12, -2], [-2, -3, -11, 9], [5, 0, -3, 0], [-18, -1, 7, -1], [-15, 1, -4, 2], [-17, 0, -17, 4], [4, 1, -9, -4], [-16, 6, -2, -1], [7, 6, -3, -8], [-10, 1, -3, 3], [-2, -8, 14, 6], [-2, 8, -8, -4], [4, 4, 2, 0], [-2, -9, -3, 0], [4, 2, -2, 10], [22, 7, -5, 3], [-20, 3, -3, 2], [-12, 9, 5, -2], [-24, -4, -4, -10], [-9, 6, 5, -4], [18, -8, 2, 2], [-18, -3, -1, 5], [-9, -8, -3, 4], [15, -7, 6, -2], [-10, 11, 11, -9], [2, -6, 0, 2], [-24, 2, -12, 6], [32, 8, -4, 4], [28, -8, 16, -8], [-14, 6, -14, 6], [4, 10, 6, -2], [-2, -10, -4, 10], [8, -9, -3, 8], [-34, 2, 6, -6], [16, 1, 3, 0], [-3, 8, 5, -12], [-10, -3, -7, 9], [-8, -12, 8, 4], [-9, 4, -3, 0], [-16, 4, -4, 0], [-24, 5, -5, -2], [-6, -6, 0, 10], [4, 15, -11, -6], [18, -6, 4, -6], [-6, -9, -1, 2], [0, 8, 8, -9], [5, -4, -7, 4], [16, 6, -4, 2], [-6, 2, 6, -6], [8, 8, 0, -1], [20, 7, -3, 6], [26, -1, 7, -9], [4, -6, -6, 4], [10, 2, 2, -4], [14, 5, 15, -15], [12, -8, 2, 6], [10, 15, -7, -10], [-12, -2, -10, -4], [42, -1, 3, 1], [-26, 2, -12, 2], [-6, 8, -12, -10], [-1, 5, 10, -6], [-28, -4, -14, 8], [5, 0, -5, -4], [-20, 0, 4, -3], [-4, 4, -6, -2], [14, 2, 8, -2], [-2, 5, -15, 5], [8, -10, 12, 4], [12, -10, 6, 6], [-10, 7, -17, -2], [-4, 6, -14, 5], [15, -4, 3, 4], [5, 3, 10, -14], [-35, -6, -1, -4], [1, 0, -13, -4], [2, 6, 6, 2], [-6, -7, 11, 2], [17, -4, -11, 0], [-6, 2, 12, -2], [-44, -4, -2, 2], [-10, 3, -11, -5], [24, 4, 6, 6], [-24, -4, -20, 2], [-24, -7, -5, 2], [-8, -2, 12, -2], [-2, 4, 10, 0], [2, -2, 2, -12], [32, 2, 2, 5], [-7, 2, -5, -12], [0, -5, 9, 2], [22, 4, 20, 0], [2, 2, -12, -8], [-4, -4, -8, -2], [-16, 7, -9, 2], [30, -3, 25, -8], [29, 2, -1, 4], [8, -10, 6, 8], [10, 8, -4, -12], [8, -5, 3, -10], [-4, -2, 10, -3], [8, -7, 19, -6], [-6, 5, 9, -6], [0, -6, -10, 2], [-16, 10, 2, -20], [-20, -2, -2, -8], [20, 7, 11, -14], [10, -7, 23, -7], [2, 5, -3, -3], [-4, -10, 4, 8], [-14, -7, -13, 18], [-22, 3, 5, -10], [-20, 6, 14, 0], [20, -12, 4, 0], [20, -2, 2, -5], [2, -4, 24, 4], [15, 1, 26, -6], [6, -7, 19, -11], [-24, 6, -4, -14], [46, 5, 5, -6], [-12, 6, -2, -10], [-10, 11, -9, -4], [-4, -8, -8, 8], [-48, 4, -2, 2], [2, -12, -4, 12], [-6, 14, 8, -2], [8, 10, -2, 4], [-22, 15, -1, -6], [14, -3, 17, -7], [-12, 1, -21, 14], [29, 10, -1, -12], [-46, 1, 5, 1], [11, 1, 14, 2], [3, 2, -11, -4], [16, -3, -1, 10], [14, -7, -3, 5], [7, 4, -13, -8], [-9, -11, 10, 6], [20, -2, 10, -13], [22, 8, -4, -8], [-50, -3, -9, 5], [-14, -3, 23, 0], [-14, -3, 5, 1], [22, 2, -2, 2], [-8, -7, 9, 0], [-16, 4, -12, -2], [-42, -5, 5, -5], [4, 2, 14, -9], [18, -11, 21, 1], [20, 15, -5, -8], [38, 3, 17, -6], [-1, 7, -12, 6], [12, -4, 2, 2], [-12, -6, -18, -5], [-12, 4, 10, -6], [32, -6, 0, -6], [-6, -1, -15, 6], [8, 4, 6, 2], [16, -2, 10, 4], [40, -2, 12, 10], [-10, 14, 10, -14], [-20, -14, 14, 10], [26, 7, 11, -16], [9, -2, -1, -12], [0, 7, -1, -10], [16, 9, -13, -6], [-10, 13, 5, -3], [-24, 4, -4, 5], [27, 6, -15, -4], [10, -5, 21, -2], [-34, 0, 4, 0], [-25, 12, -13, 4], [-2, 5, -7, 3], [-2, -9, -11, 18], [42, -6, 6, -2], [2, -9, 1, -2], [0, 2, -8, -2], [6, 14, 6, -8], [0, -4, 8, 4], [16, 0, 12, 4], [8, 16, 6, -8], [0, -11, -3, -6], [10, -2, -16, 8], [1, -13, -6, 6], [-36, 3, -3, 2], [7, 8, -1, -8], [-11, 1, -4, 6], [-14, -1, 11, -1], [34, -10, 24, -16], [37, 8, 11, -8], [10, 8, -8, 16], [-35, 8, 5, -4], [-1, 8, -17, 12], [-4, -8, 20, 4], [-6, -9, -21, -2], [-4, 5, 7, -20], [-2, 9, -7, 5], [-22, 3, 1, -21], [-20, -5, -1, 2], [-42, 7, -11, -4], [32, 7, -1, 0], [-38, -9, -1, 15], [34, -8, 4, 2], [33, 11, -6, 14], [-46, -3, -13, 9], [26, -9, -9, -1], [3, -5, 8, 10], [10, 5, -7, -1], [-12, -12, 16, 6], [10, 11, -17, -13], [-4, 8, -8, 16], [8, 2, 20, -2], [23, 12, -21, 0], [-6, 0, 2, 0], [38, 3, 3, -2], [8, -8, -12, 8], [16, 0, -2, -10], [-10, 9, -15, -10], [-14, 16, -22, 10], [0, -6, 16, 2], [20, 10, -8, 12], [-6, 13, 11, -11], [30, -6, 4, -2], [23, -1, 8, 18], [-6, -8, 18, -4], [-10, -7, 5, -2], [2, -4, 8, 8], [34, -12, 6, 4], [7, 10, -23, -4], [-28, -13, -1, 10], [48, 2, 8, 4], [-28, -6, -12, -6], [4, -4, 16, -6], [16, 14, 4, -14], [-24, 2, -18, 18], [7, 0, -1, 0], [-36, 2, -2, -5], [-10, -3, -3, -5], [66, 9, -5, -6], [42, 13, -11, -6], [-11, 4, 9, 4], [-12, -12, 4, 0], [52, 7, 5, 6], [25, -15, 12, 2], [4, 10, -4, -8], [0, -2, 16, 6], [-32, 1, -1, -6], [24, -12, 4, 11], [0, 5, -5, -10], [-27, 8, 5, -12], [10, 16, 2, -16], [-18, 1, -21, -3], [46, -6, 18, -14], [38, -7, 23, -8], [4, 12, 0, 4], [-16, -2, 4, -14], [10, -3, 1, 5], [-6, 4, -6, 12], [-24, 7, 7, 2], [-48, -2, -18, -12], [12, 10, 12, -2], [-30, 6, 0, -8], [2, -12, 0, 10], [36, 4, 4, 8], [8, 4, 18, -16], [64, 6, -4, -10], [8, -11, 19, -14], [8, 14, -18, -2], [-20, -4, 2, -12], [16, -6, 20, -6], [48, 2, 18, 1], [36, 13, 7, 6], [0, 10, -28, -12], [3, -24, 11, 0], [-28, -1, 3, -12], [50, 6, 6, -6], [26, -5, 15, 8], [-13, 2, 5, -16], [-16, -9, -1, 2], [-11, -8, -25, 8], [14, -14, -2, 26], [18, -20, -2, 0], [-22, -4, -10, 22], [-19, 8, 11, -16], [8, 6, 14, -16], [16, 2, -12, -2], [-18, 1, -3, 11], [-24, 0, -32, 1], [13, 0, 21, -20], [18, 10, -30, 0], [-2, -21, 3, 1], [15, -4, -21, 12], [7, -11, -18, 18], [-42, -3, -13, 14], [0, 2, -14, 15]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5472_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5472_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_5472_2_a_bs(:prec:=4)
    chi := MakeCharacter_5472_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_5472_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_5472_2_a_bs( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5472_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![32, -24, -18, 1, 1]>,<7,R![4, 19, -17, 1, 1]>,<11,R![16, -28, 0, 7, 1]>,<13,R![-344, 158, 7, -10, 1]>,<23,R![64, 132, -17, -8, 1]>],Snew);
    return Vf;
end function;