
// Make newform 4560.2.a.bp in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4560_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_4560_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_4560_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 := [-7, 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_4560_a();"
function MakeCharacter_4560_a()
    N := 4560;
    order := 1;
    char_gens := [1711, 1141, 3041, 2737, 1921];
    v := [1, 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_4560_a_Hecke(Kf)
    return MakeCharacter_4560_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], [1, 0], [1, 0], [1, 1], [3, 1], [1, -1], [0, 0], [1, 0], [-2, 0], [7, -1], [2, 0], [1, 3], [-1, -1], [-3, -3], [-6, 0], [4, 0], [2, 2], [6, -2], [-4, 0], [-2, -2], [-2, 0], [8, 0], [0, -6], [1, 5], [5, -5], [0, -6], [-4, 4], [-4, 4], [0, -2], [2, -6], [-8, -4], [-5, 1], [10, 4], [2, -6], [6, 0], [-6, 0], [0, -6], [-7, 1], [-4, -2], [14, 2], [2, -2], [0, 2], [-3, 3], [-5, -3], [24, 0], [18, -2], [0, -4], [12, 4], [-20, 2], [2, -2], [22, 0], [21, 3], [6, 4], [-7, -5], [6, 6], [-4, -2], [-3, -3], [-18, -2], [-14, 4], [3, 3], [17, 5], [-2, 6], [-18, -2], [7, 1], [-24, -2], [12, -4], [-6, 4], [-9, 5], [-4, 2], [-6, 8], [-14, 8], [-3, -5], [13, 5], [-11, -1], [-10, 0], [16, 4], [-6, 0], [-14, 0], [9, 1], [-12, -2], [31, -3], [10, 4], [6, -10], [-13, 5], [16, 8], [2, 4], [-5, -1], [4, -2], [-6, 0], [21, 5], [-10, -4], [-5, -7], [2, 2], [9, -9], [-14, 2], [-4, -2], [37, 1], [3, -9], [-22, -6], [-26, 0], [14, -10], [-10, 0], [-12, -6], [19, 7], [2, 6], [12, -6], [-26, 4], [-26, -4], [16, 4], [0, 6], [-14, -10], [12, 10], [0, 0], [22, -6], [-16, 8], [-7, 5], [-37, 3], [20, -2], [-4, 0], [8, 8], [0, 2], [-9, -7], [-4, 0], [-8, 0], [-26, 2], [-10, 0], [-6, 10], [5, -1], [-5, -17], [-22, 4], [0, -4], [0, -12], [6, 16], [-4, 6], [10, 8], [-10, -12], [0, 0], [-6, 2], [34, -2], [2, 4], [-16, -4], [20, 2], [-7, -19], [-28, -2], [-12, 6], [34, 2], [-40, 2], [8, 12], [-24, -8], [20, 8], [7, 9], [-44, -2], [-23, 1], [-12, 12], [-38, 6], [8, 12], [-8, 4], [0, 6], [12, -2], [-3, 5], [-32, -2], [-34, -2], [-33, 7], [-20, -12], [-8, 16], [-28, 2], [-28, -12], [-48, 2], [30, 0], [18, -12], [6, -6], [10, -4], [33, 11], [14, 12], [20, -12], [40, -2], [-6, 14], [-36, -6], [-10, 10], [-6, -16], [31, -9], [-16, -8], [-25, 5], [14, 2], [2, 8], [-10, -4], [1, -9], [-34, -2], [8, 6], [27, -3], [17, -1], [-42, 4], [0, 4], [33, 5], [18, 0], [-26, -2], [6, -2], [-30, 0], [-32, -4], [-6, 12], [37, -7], [-20, -12], [3, -7], [-14, 16], [-49, 5], [-24, 0], [24, 8], [-16, -14], [20, 18], [-24, 12], [36, 6], [-44, 10], [-19, -19], [-36, 10], [22, -6], [20, 10], [17, 9], [-17, -5], [-16, 2], [4, 8], [6, 4], [-2, -8], [17, -3], [-25, -13], [-36, 2], [-22, 0], [-34, -6], [-10, -18], [20, 12], [-17, -19], [-28, 2], [26, 8], [-28, -12], [57, -3], [3, -13], [-14, 16], [54, -6], [-18, -20], [-9, -3], [42, -10], [16, -16], [28, 0], [5, 9], [-8, -6], [-52, -8], [7, -19], [43, -1], [68, -4], [-30, -8], [-12, 14], [-16, 18], [46, 4], [44, 2], [12, 2], [-32, 4], [-11, 11], [38, -6], [52, 4], [-60, 4], [10, -8], [-24, 8], [-8, -4], [-38, 8], [27, 5], [24, 12], [-2, -12], [37, -7], [-60, -6], [0, 20], [24, -4], [22, -8], [-4, 8], [20, -2], [14, -6], [11, 25], [-21, 19], [32, 2], [-12, -22], [52, -10], [13, -17], [0, -28], [-64, 4], [-22, 4], [-34, 8], [27, -1], [-29, 13], [-6, -16], [0, 12], [8, -16], [-37, 15], [-4, 10], [8, -22], [36, 12], [26, 6], [41, 3], [-4, 26], [40, 8], [-8, -16], [48, -12], [63, 7], [22, 0], [20, 0], [-24, 12], [24, 18], [46, -10], [-1, 21], [24, -20], [-50, 12], [23, -7], [-36, -18], [-2, -16], [10, 4], [-59, 5], [2, 18], [-12, -2], [18, -16], [31, 13], [-12, 12], [4, -18], [-10, -20], [70, 4], [-20, 18], [-35, -7], [46, -6], [40, 8], [10, -16], [16, 20], [86, -2], [36, 18], [28, -4], [34, 8], [-36, 0], [38, 6], [20, -6], [-2, 30], [-12, 22], [74, 0], [-56, 8], [14, -2], [9, 13], [55, 9], [-18, -12], [-19, 13], [-26, -4], [20, 16], [40, 12], [10, 18], [-24, -16], [32, -4], [-6, 0], [22, 0], [51, 9], [64, -2], [30, 2], [0, -22], [-4, -12], [-1, -3], [15, -7], [-4, -16], [-4, 0], [22, -12], [38, -8], [-16, -28], [-22, -6], [9, -31], [25, -13], [-6, -4], [-82, -2], [4, -22], [-45, 21], [4, 16], [16, 10], [-7, 17], [-56, 12], [56, -10], [8, 28], [-69, -3], [-34, -4], [52, 10], [-61, -15], [45, 21], [24, -12], [83, -1], [-2, 8], [-6, 0], [20, -18], [58, 14], [-38, 4], [-9, -29], [0, -6], [4, -30], [16, -28], [55, 13], [-13, -17], [-6, -14], [57, 15], [22, -4], [-33, 15], [-30, -28], [58, -8], [-22, 8], [-32, 4], [34, -14], [18, -6], [51, -9], [-36, 16], [0, 34], [-41, 11], [-6, -22], [15, 13], [17, 3], [4, 20], [-8, -28], [74, -2], [58, -4], [42, -4], [-32, 0], [20, 4], [-42, 12], [8, 20], [75, -9], [-37, -11], [30, 0], [-4, -32], [11, 5], [-10, -22], [-20, 20], [34, 16], [-34, -22], [-29, 21]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4560_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4560_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_4560_2_a_bp(:prec:=2)
    chi := MakeCharacter_4560_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_4560_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_4560_2_a_bp( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4560_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-6, -2, 1]>,<11,R![2, -6, 1]>,<13,R![-6, -2, 1]>,<17,R![0, 1]>],Snew);
    return Vf;
end function;