
// Make newform 8550.2.a.l in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8550_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_8550_2_a_l();".
// 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_8550_2_a_l();".
// 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_8550_a();"
function MakeCharacter_8550_a()
    N := 8550;
    order := 1;
    char_gens := [1901, 1027, 1351];
    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_8550_a_Hecke(Kf)
    return MakeCharacter_8550_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], [0], [0], [1], [0], [3], [-7], [-1], [-5], [5], [10], [-2], [-2], [-6], [0], [9], [7], [-4], [-7], [0], [9], [-10], [-2], [10], [18], [4], [0], [9], [13], [8], [6], [20], [-3], [12], [-4], [-6], [-18], [18], [14], [6], [-24], [-10], [7], [2], [-10], [17], [5], [22], [-25], [-18], [26], [27], [-28], [-4], [-6], [-8], [10], [5], [-28], [26], [26], [31], [-4], [-33], [29], [3], [-17], [14], [-18], [10], [27], [-27], [-8], [-3], [-7], [0], [-24], [20], [-18], [26], [26], [31], [-12], [-12], [10], [24], [-14], [41], [10], [-16], [-8], [24], [2], [-14], [42], [-17], [42], [0], [19], [-28], [20], [8], [-44], [8], [10], [11], [30], [-34], [-30], [-10], [-22], [16], [-2], [-8], [0], [12], [2], [-9], [-12], [-15], [17], [28], [19], [36], [10], [-24], [-34], [-29], [-11], [-24], [50], [34], [46], [-22], [45], [49], [15], [-11], [5], [39], [11], [56], [31], [-9], [-51], [48], [46], [24], [56], [24], [5], [-2], [-26], [52], [-5], [32], [11], [-21], [13], [37], [-28], [44], [-12], [-36], [-18], [-18], [-46], [-14], [-48], [-62], [-15], [27], [24], [23], [-38], [-30], [-10], [18], [-54], [-34], [-25], [-7], [-10], [16], [3], [26], [41], [-12], [36], [63], [-8], [0], [-52], [-25], [-24], [38], [25], [44], [-7], [-27], [2], [16], [-41], [-2], [-30], [-36], [69], [-65], [-49], [-27], [33], [12], [-16], [29], [46], [-12], [65], [-38], [-34], [6], [62], [-26], [-36], [65], [-23], [-10], [-34], [-40], [-28], [22], [32], [-35], [-44], [-42], [-46], [-3], [-14], [-22], [-42], [-66], [-12], [-8], [11], [11], [18], [-72], [72], [32], [-40], [11], [56], [-15], [-73], [20], [-22], [-20], [10], [68], [6], [-58], [-58], [36], [-30], [47], [14], [-31], [59], [29], [31], [-24], [55], [-13], [21], [-5], [-12], [-49], [-48], [51], [-60], [-60], [-2], [-39], [-21], [-65], [52], [36], [25], [10], [-61], [2], [80], [-8], [12], [-4], [-46], [46], [-30], [22], [5], [12], [-34], [23], [-51], [71], [-66], [26], [47], [37], [-72], [-12], [-8], [0], [-12], [76], [-76], [22], [-30], [24], [-37], [18], [55], [43], [-39], [-32], [-43], [0], [-7], [-45], [-22], [54], [24], [75], [29], [30], [50], [-62], [41], [-72], [39], [-73], [-27], [33], [13], [30], [17], [-24], [10], [31], [60], [-40], [4], [-16], [44], [-52], [10], [-40], [-33], [8], [-79], [38], [-54], [-42], [28], [-57], [-56], [78], [-22], [12], [30], [34], [-37], [-73], [-55], [32], [-76], [21], [-88], [-58], [-9], [58], [18], [-50], [43], [18], [-27], [-90], [-21], [46], [-58], [-12], [-2], [-25], [-66], [-28], [26], [6], [-66], [88], [52], [30], [34], [-66], [46], [-18], [18], [53], [-32], [40], [-65], [78], [-26], [-26], [-37], [-8], [57], [2], [52], [-102], [29], [-72], [-34], [-63], [13], [25], [-44], [27], [-81], [41], [-40], [-81], [-66], [-22], [-24], [62], [-52], [15], [-58], [54], [-48], [36], [60], [7], [-22], [10], [91], [90], [-52], [-94], [4], [38], [1], [-62], [-24], [-66], [50], [-54], [98], [-21], [12], [-14], [-71], [-106], [-43], [77], [54], [-39], [-68], [66], [53], [-52], [-44], [73], [62], [107], [84], [-30], [-96], [-1], [3], [-39], [24], [70], [96], [-22], [103], [64], [70], [-69], [-36], [-38], [78], [52], [-104], [-18], [-60], [-62], [-92], [16], [69], [5], [-70], [25], [-76]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8550_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8550_2_a_l();".
// 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_8550_2_a_l(:prec:=1)
    chi := MakeCharacter_8550_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(3593) - 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_8550_2_a_l();".
// 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_8550_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8550_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-1, 1]>,<11,R![0, 1]>,<13,R![-3, 1]>,<17,R![7, 1]>,<23,R![5, 1]>,<53,R![-9, 1]>],Snew);
    return Vf;
end function;