
// Make newform 4928.2.a.bp in Magma, downloaded from the LMFDB on 27 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4928_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_4928_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_4928_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 := [-6, 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_4928_a();"
function MakeCharacter_4928_a()
    N := 4928;
    order := 1;
    char_gens := [4159, 1541, 2817, 3137];
    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_4928_a_Hecke(Kf)
    return MakeCharacter_4928_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], [-2, 0], [-1, 0], [1, 0], [-2, -1], [2, 1], [0, -2], [4, 2], [2, 2], [4, -1], [-4, 0], [-2, 3], [2, 2], [-4, 1], [0, 4], [0, 3], [2, 3], [0, -6], [-8, 2], [10, -1], [-6, 2], [12, -2], [6, 0], [10, 2], [6, -1], [12, -1], [4, 0], [-10, 2], [-10, 0], [8, 4], [0, 4], [0, -4], [-8, 6], [-2, 4], [-6, -6], [-10, 0], [-4, 2], [4, -4], [2, 5], [-4, -8], [-14, -2], [8, 4], [2, -8], [6, -8], [4, -9], [20, 0], [4, 7], [8, 4], [10, 2], [-6, -2], [16, 0], [6, 3], [0, 1], [6, 4], [0, -8], [18, 2], [16, 6], [-6, 4], [-10, -2], [24, 0], [18, 3], [-8, -4], [-4, -1], [-14, -8], [-6, -8], [32, 0], [6, -6], [-2, 10], [-14, -5], [6, -8], [-10, 6], [4, -7], [-14, -4], [8, 2], [20, -3], [-6, -8], [6, -8], [-18, 4], [-14, -1], [-16, 1], [-4, -4], [-14, -2], [26, 2], [-20, 2], [-24, 0], [0, -8], [-10, -12], [30, -3], [-20, 2], [0, 3], [-4, -2], [32, -2], [-18, 6], [28, -6], [0, 10], [10, -6], [-14, -2], [4, 12], [-2, -8], [12, 0], [34, -4], [-8, -14], [6, 6], [20, 4], [2, 10], [-24, -1], [-18, -7], [-12, -2], [14, 3], [-4, -8], [2, -6], [0, 12], [-8, -1], [-12, -4], [12, 8], [8, 5], [-12, -5], [14, 12], [-10, -2], [10, -8], [22, 6], [-2, 11], [36, 4], [24, -7], [6, -6], [6, 4], [28, -7], [-4, -1], [-38, 3], [12, -12], [0, 8], [-8, 2], [22, 0], [18, -13], [22, -7], [-18, -6], [4, 6], [22, -4], [-6, 8], [4, -8], [6, -4], [-4, -16], [-38, 6], [14, 0], [12, 1], [-26, 7], [26, 7], [8, 13], [-32, 0], [-22, 6], [-6, 6], [20, -4], [4, 14], [-8, -2], [12, -10], [24, 8], [-18, 14], [6, -11], [-46, -1], [-20, -8], [-34, 4], [-34, 2], [16, -9], [14, 4], [12, -3], [16, -18], [-6, -11], [-2, -20], [-26, 10], [36, 0], [-14, -12], [10, -6], [46, -2], [20, -13], [-2, -10], [-14, 6], [28, 0], [36, 6], [-10, -1], [0, 8], [-8, 6], [14, 0], [22, -1], [4, -12], [-30, 10], [2, -20], [-48, -3], [-46, 6], [-32, 12], [-26, -2], [20, 16], [-32, -2], [-22, -12], [14, -10], [-18, 8], [2, -6], [4, 0], [-18, 9], [-24, 10], [62, 0], [16, -12], [-10, -18], [-46, 3], [24, 13], [-6, -24], [-4, 13], [12, 20], [-6, 24], [-8, -5], [38, 2], [38, -10], [44, -4], [40, -5], [24, 2], [34, 0], [38, 10], [30, 5], [12, -6], [-50, 4], [-42, 10], [28, 2], [12, -24], [-8, -8], [12, -12], [-22, -18], [-26, -14], [24, 8], [-44, 2], [-38, -2], [-8, 12], [-44, 4], [16, -4], [-6, -16], [0, 5], [-36, 10], [42, -6], [42, 8], [0, -6], [-4, 1], [-36, -6], [-16, -22], [36, 3], [28, -4], [30, -9], [36, 4], [28, -5], [-40, -3], [12, -24], [0, 4], [18, 0], [26, 7], [52, -6], [-2, 0], [10, -21], [60, -4], [16, 4], [-4, 2], [-14, -4], [-10, -13], [-26, 10], [44, -4], [54, -7], [-38, -7], [-18, -24], [-24, -1], [-24, 12], [-2, -22], [26, 10], [-26, -14], [14, 24], [20, 2], [-6, 10], [8, -24], [10, -17], [20, -9], [24, -4], [10, 24], [-50, 12], [36, 10], [0, -12], [-16, 6], [-48, -10], [26, 1], [44, -14], [0, -18], [28, -8], [-10, -8], [-12, 3], [-6, 21], [-36, 4], [40, 15], [-42, 14], [16, -2], [-58, 12], [46, 13], [-36, 11], [-22, -10], [4, 6], [36, -6], [14, 1], [38, 6], [-22, -2], [-12, -6], [-56, 2], [20, 8], [-52, 0], [-46, 0], [24, 20], [6, 0], [-36, 8], [-2, 16], [2, -2], [0, 16], [30, -22], [6, 5], [8, -13], [48, -4], [-30, -6], [-6, -18], [-32, 16], [-10, -12], [26, -1], [28, -4], [30, 4], [6, 4], [64, -2], [48, 7], [-6, -14], [-18, 6], [-6, -20], [-8, 4], [8, 20], [40, 2], [-46, -2], [-48, 1], [-40, 20], [30, 19], [30, -16], [0, -30], [-2, 8], [-36, 8], [10, 21], [4, -12], [-78, 4], [12, -20], [22, 22], [20, -22], [-24, -8], [-18, -2], [-24, -6], [-40, -8], [14, 32], [8, 6], [18, -4], [54, -5], [4, -1], [20, 12], [-14, -8], [0, 16], [34, -4], [22, -13], [64, 4], [42, -18], [-8, 5], [-24, 20], [26, -3], [40, 0], [14, 28], [-48, -2], [16, -9], [40, 12], [-2, -24], [32, 4], [-60, -4], [24, -17], [-22, 22], [-70, 1], [50, -15], [-30, -23], [-38, 20], [-32, 8], [-10, 2], [14, 4], [56, -2], [76, 3], [-40, 20], [6, -12], [-4, 8], [44, 11], [-80, 4], [2, 2], [-40, 18], [16, 7], [-48, -12], [6, 34], [-16, 14], [-10, -12], [4, -2], [10, 2], [54, 9], [18, 32], [72, 0], [-66, -12], [-42, -17], [-20, -8], [-70, -8], [18, -28], [-8, 25], [-40, 5], [-18, 1], [-6, 10], [-72, -12], [-46, 2], [10, -22], [-18, -4], [32, 16], [44, -3], [22, -4], [8, -8], [-46, 20], [42, 3], [-68, 10], [-52, -12], [-10, -4], [-30, -2], [24, 10], [30, 26], [16, -5], [16, -22]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4928_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4928_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_4928_2_a_bp(:prec:=2)
    chi := MakeCharacter_4928_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_4928_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_4928_2_a_bp( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4928_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-6, 0, 1]>,<5,R![2, 1]>,<13,R![-2, 4, 1]>,<17,R![-2, -4, 1]>,<19,R![-24, 0, 1]>,<23,R![-8, -8, 1]>],Snew);
    return Vf;
end function;