
// Make newform 4928.2.a.a in Magma, downloaded from the LMFDB on 25 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_a();".
// 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_a();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
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], [-3], [1], [1], [-1], [4], [2], [-6], [5], [-10], [-1], [5], [-2], [-8], [-8], [6], [3], [2], [-3], [-1], [10], [-6], [12], [-15], [-5], [12], [12], [-10], [-4], [-19], [-2], [18], [-3], [-10], [22], [-6], [-7], [4], [2], [-16], [1], [-5], [-5], [14], [-18], [8], [-2], [-1], [4], [7], [6], [-4], [-12], [-21], [-6], [-18], [18], [-16], [-24], [-4], [0], [6], [-28], [-8], [-23], [-9], [-17], [-18], [14], [34], [9], [-8], [11], [4], [-29], [-17], [-9], [-18], [-6], [-26], [16], [-22], [20], [-25], [14], [-39], [15], [8], [-18], [-13], [3], [28], [13], [-30], [44], [0], [31], [7], [32], [-32], [-24], [-14], [20], [-18], [-20], [-25], [36], [30], [48], [8], [10], [-16], [-30], [17], [-27], [15], [-29], [21], [17], [-2], [-35], [4], [-38], [12], [15], [0], [-39], [11], [19], [4], [-18], [24], [23], [-38], [-48], [40], [6], [-22], [-23], [30], [-22], [-18], [25], [20], [29], [-45], [34], [-28], [55], [-52], [38], [27], [-44], [2], [-40], [0], [-48], [-30], [36], [-58], [5], [44], [34], [29], [-31], [27], [32], [12], [52], [-29], [24], [-2], [50], [-8], [25], [7], [26], [-47], [18], [2], [60], [-52], [-13], [-32], [-25], [18], [-16], [-60], [-16], [-46], [-29], [34], [-19], [62], [-26], [45], [-64], [1], [-18], [6], [14], [-40], [2], [-34], [45], [-15], [1], [42], [36], [-28], [-4], [59], [1], [-52], [-24], [1], [-44], [-66], [-36], [-23], [48], [26], [-3], [-61], [0], [-10], [22], [48], [-40], [-40], [-29], [14], [20], [12], [-45], [54], [51], [42], [15], [-59], [39], [58], [-36], [-29], [-42], [-18], [-40], [-20], [-54], [18], [48], [-16], [-25], [-26], [76], [54], [34], [-62], [25], [20], [-76], [-20], [-6], [16], [30], [-60], [57], [-5], [-74], [4], [17], [7], [78], [-56], [10], [19], [51], [22], [10], [60], [-24], [-5], [66], [-58], [-74], [15], [58], [-58], [-75], [72], [-33], [-52], [60], [44], [-34], [28], [-27], [-61], [-34], [12], [-88], [32], [10], [36], [-1], [23], [-17], [5], [24], [48], [90], [-26], [50], [-51], [-42], [18], [-59], [42], [-66], [-66], [80], [-67], [26], [-25], [-6], [21], [71], [-15], [54], [72], [60], [82], [10], [-10], [22], [51], [11], [42], [-15], [36], [55], [63], [40], [53], [57], [-36], [91], [19], [36], [15], [12], [47], [86], [-26], [60], [72], [53], [-26], [52], [-9], [-62], [34], [17], [24], [53], [45], [68], [80], [8], [-51], [-76], [-4], [-44], [-20], [39], [60], [22], [-58], [20], [96], [46], [31], [22], [38], [4], [-1], [-32], [-3], [10], [-77], [25], [-61], [13], [-7], [81], [54], [-56], [26], [-76], [100], [-78], [-51], [-72], [-30], [72], [-34], [-82], [60], [68], [1], [12], [-48], [-60], [-80], [10], [-51], [-8], [-17], [-38], [-14], [-19], [0], [-28], [-86], [-10], [-69], [-69], [0], [65], [38]];
    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_a();".
// 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_a(:prec:=1)
    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_a();".
// 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_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4928_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 1]>,<5,R![-1, 1]>,<13,R![-4, 1]>,<17,R![-2, 1]>,<19,R![6, 1]>,<23,R![-5, 1]>],Snew);
    return Vf;
end function;