
// Make newform 5550.2.a.bj in Magma, downloaded from the LMFDB on 04 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5550_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_5550_2_a_bj();".
// 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_5550_2_a_bj();".
// 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_5550_a();"
function MakeCharacter_5550_a()
    N := 5550;
    order := 1;
    char_gens := [3701, 1777, 2851];
    v := [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_5550_a_Hecke(Kf)
    return MakeCharacter_5550_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 := [[1], [1], [0], [-2], [-2], [-1], [1], [-4], [-6], [2], [-5], [1], [0], [8], [-7], [-3], [7], [-14], [-3], [-7], [1], [-13], [-14], [-6], [18], [3], [-9], [10], [5], [-6], [-6], [1], [12], [1], [1], [4], [16], [-22], [10], [-13], [3], [18], [-18], [6], [-6], [-8], [13], [2], [3], [14], [-22], [-20], [4], [12], [21], [8], [-9], [-14], [-11], [-15], [2], [1], [9], [-4], [14], [33], [8], [14], [21], [2], [9], [-15], [-2], [-2], [4], [-4], [18], [12], [-15], [38], [-30], [-35], [20], [-9], [-8], [0], [-9], [-14], [-28], [36], [-5], [16], [33], [-28], [28], [18], [-18], [-18], [8], [-22], [-20], [14], [-32], [-5], [29], [20], [-44], [-4], [-1], [-26], [-1], [36], [28], [21], [-12], [-20], [26], [20], [26], [-30], [-17], [26], [-17], [-51], [20], [0], [-26], [1], [24], [-16], [-23], [-9], [12], [45], [40], [20], [18], [-11], [-30], [34], [-13], [13], [-2], [5], [6], [-9], [22], [-17], [14], [-20], [-2], [50], [26], [-19], [-26], [-10], [48], [-34], [-55], [-13], [3], [-12], [-61], [-24], [-7], [-32], [-11], [-3], [-3], [-42], [36], [-21], [0], [6], [-24], [-9], [-22], [-2], [20], [46], [49], [-9], [-22], [54], [-35], [-46], [-10], [36], [10], [47], [34], [-44], [-4], [3], [38], [-14], [-22], [38], [4], [24], [42], [46], [-14], [29], [36], [-62], [14], [18], [57], [67], [48], [-66], [-1], [-30], [-50], [25], [-47], [-3], [-46], [30], [32], [40], [-42], [-44], [8], [-57], [-18], [-41], [4], [20], [-34], [35], [10], [16], [-35], [-44], [5], [30], [-15], [50], [19], [-20], [36], [-45], [-70], [-20], [17], [3], [-44], [-36], [7], [-18], [-8], [34], [-30], [-54], [-12], [-23], [-63], [-22], [-60], [-69], [-14], [52], [13], [25], [63], [30], [-21], [26], [-5], [52], [-44], [-11], [23], [-10], [-48], [-5], [22], [-67], [-80], [73], [-28], [-8], [74], [-73], [2], [-27], [-35], [-63], [-82], [-79], [24], [14], [40], [77], [56], [-22], [12], [41], [86], [59], [-30], [25], [80], [-16], [12], [-5], [-23], [-14], [23], [70], [-38], [-65], [42], [-24], [8], [6], [51], [19], [4], [-78], [-60], [-52], [-72], [-32], [71], [32], [-50], [5], [-84], [50], [-48], [-61], [2], [74], [36], [0], [50], [24], [-38], [-56], [38], [-49], [14], [-60], [-16], [-32], [-68], [66], [20], [75], [-34], [-60], [-52], [60], [-72], [11], [-48], [-44], [-58], [-36], [-72], [23], [39], [46], [-30], [-72], [0], [-84], [-28], [55], [71], [-95], [8], [-78], [27], [-19], [27], [-43], [4], [-30], [-16], [-8], [22], [81], [6], [26], [-70], [62], [-83], [18], [93], [16], [-75], [54], [-65], [72], [28], [-22], [-4], [-35], [-48], [-33], [70], [-52], [-63], [-10], [-84], [42], [-73], [102], [-44], [-42], [12], [6], [48], [-36], [68], [30], [16], [27], [-96], [-104], [-67], [-69], [-30], [-42], [-99], [-18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5550_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5550_2_a_bj();".
// 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_5550_2_a_bj(:prec:=1)
    chi := MakeCharacter_5550_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_5550_2_a_bj();".
// 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_5550_2_a_bj( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5550_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![2, 1]>,<13,R![1, 1]>,<17,R![-1, 1]>],Snew);
    return Vf;
end function;