
// Make newform 8550.2.a.n in Magma, downloaded from the LMFDB on 12 May 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_n();".
// 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_n();".
// 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], [2], [-4], [6], [4], [1], [4], [-6], [-6], [-10], [-4], [-12], [4], [-10], [-10], [2], [-12], [-8], [2], [10], [2], [8], [-2], [10], [12], [8], [-16], [-14], [20], [-12], [0], [-4], [-14], [-2], [-12], [16], [24], [-18], [2], [0], [8], [-14], [22], [-4], [-8], [-24], [20], [-26], [-12], [0], [10], [0], [-18], [12], [-14], [20], [-28], [-4], [-28], [2], [12], [-16], [-6], [18], [-12], [34], [-2], [22], [0], [0], [22], [-14], [8], [-16], [10], [24], [24], [-6], [0], [-8], [40], [-30], [-38], [6], [-12], [22], [-18], [18], [-22], [24], [-20], [-12], [-4], [-8], [-38], [-40], [-4], [38], [4], [10], [24], [-12], [-28], [-10], [22], [-8], [28], [-6], [28], [16], [-48], [4], [8], [20], [20], [-24], [-6], [-22], [-8], [-26], [-6], [-12], [4], [-10], [14], [-40], [-26], [8], [-20], [48], [-38], [-8], [34], [-18], [6], [-4], [2], [42], [-8], [42], [26], [48], [-40], [0], [36], [30], [4], [32], [34], [-18], [-36], [48], [20], [-12], [16], [-42], [-26], [-10], [-38], [6], [18], [42], [-42], [24], [-2], [52], [34], [18], [-6], [40], [0], [2], [-50], [42], [12], [-26], [24], [22], [-46], [-58], [-58], [-10], [-52], [42], [10], [20], [10], [40], [54], [-6], [-32], [-66], [-30], [-54], [-26], [44], [-64], [-36], [30], [-26], [54], [-2], [12], [30], [-4], [-24], [-42], [56], [34], [22], [-42], [-52], [44], [54], [-38], [-72], [-8], [-22], [-4], [-62], [-4], [-14], [20], [-58], [-66], [-56], [36], [-12], [52], [28], [2], [-48], [48], [52], [14], [42], [32], [8], [-68], [52], [-10], [12], [58], [64], [-26], [-30], [-40], [12], [8], [6], [24], [-2], [38], [0], [-30], [28], [-78], [-62], [-40], [-72], [66], [-46], [-68], [44], [46], [6], [-60], [-54], [-16], [12], [-46], [4], [-62], [-2], [22], [44], [74], [-52], [-72], [8], [-24], [-24], [32], [0], [46], [62], [-32], [-8], [-18], [-22], [-46], [-6], [26], [30], [-14], [-34], [2], [-20], [26], [62], [72], [16], [52], [-34], [88], [-28], [24], [56], [8], [70], [0], [-76], [36], [46], [40], [-12], [26], [-62], [-16], [50], [-6], [36], [52], [-38], [-56], [-4], [-24], [-30], [-62], [-2], [72], [18], [36], [18], [-12], [-50], [66], [50], [-22], [-76], [-30], [-66], [74], [-18], [-38], [28], [-24], [46], [-68], [74], [-78], [84], [-80], [6], [0], [-12], [-32], [48], [92], [-6], [24], [82], [52], [-42], [-42], [-44], [78], [44], [20], [64], [-70], [60], [-60], [-10], [80], [46], [-34], [54], [-42], [76], [-42], [-28], [-40], [48], [44], [-60], [56], [48], [-66], [-6], [48], [64], [36], [26], [-34], [-40], [80], [18], [-64], [-76], [8], [14], [-46], [-52], [36], [-60], [-20], [-28], [46], [-14], [12], [-4], [-78], [98], [36], [-32], [-88], [-40], [-18], [74], [28], [74], [38], [-6], [-24], [-50], [-44], [24], [-62], [-12], [-52], [-16], [-4], [42], [-58], [20], [-44], [68], [-54], [30], [-56], [-84], [82], [-54], [-112], [32], [-98], [-52], [28], [-48], [-86], [22], [-54], [54], [-76], [-70], [-24], [78], [56], [-14], [30], [-60], [44], [-78], [58], [-54], [102], [12], [-16], [12], [-52], [-50], [90], [-14], [-14], [4], [12], [14], [22], [-8], [34], [-30], [106], [70], [42], [30], [-40], [98], [-34], [64], [10], [106], [100], [30], [-76], [-90], [-24], [56], [-90], [-94], [-42]];
    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_n();".
// 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_n(: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_n();".
// 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_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8550_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-2, 1]>,<11,R![4, 1]>,<13,R![-6, 1]>,<17,R![-4, 1]>,<23,R![-4, 1]>,<53,R![10, 1]>],Snew);
    return Vf;
end function;