
// Make newform 8550.2.a.p in Magma, downloaded from the LMFDB on 25 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_p();".
// 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_p();".
// 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], [2], [4], [6], [1], [8], [6], [-8], [8], [12], [0], [0], [10], [6], [-6], [-12], [-12], [10], [-8], [-8], [8], [14], [-2], [14], [4], [-2], [-18], [-18], [-2], [-6], [-20], [2], [20], [-24], [-4], [-16], [2], [-2], [14], [-12], [2], [-26], [-24], [4], [14], [16], [30], [-10], [28], [-18], [-18], [-10], [16], [-10], [0], [-4], [0], [20], [30], [-16], [-20], [-14], [-18], [-28], [-6], [-12], [-22], [-10], [-8], [-14], [-28], [12], [16], [26], [-20], [-24], [10], [-10], [10], [0], [34], [-28], [12], [8], [-10], [-30], [-10], [32], [12], [26], [-18], [-12], [-24], [14], [24], [36], [-38], [-8], [18], [16], [24], [12], [-2], [0], [18], [-8], [30], [-22], [20], [22], [-12], [20], [-28], [4], [24], [-2], [-30], [-18], [2], [6], [-4], [-36], [-42], [14], [-24], [22], [-36], [-28], [24], [16], [36], [-28], [-2], [6], [12], [2], [16], [4], [38], [-26], [48], [-10], [-16], [12], [-38], [20], [40], [-12], [52], [36], [-8], [-8], [0], [-20], [-36], [-2], [18], [-12], [-18], [2], [10], [26], [24], [-60], [56], [14], [-14], [30], [-30], [56], [10], [0], [12], [20], [18], [10], [38], [-34], [-54], [40], [-26], [-40], [18], [20], [44], [-50], [12], [34], [20], [52], [-38], [24], [-26], [-30], [56], [-30], [-8], [-46], [0], [-52], [-50], [34], [46], [4], [32], [4], [28], [50], [42], [-30], [-8], [-60], [10], [10], [40], [-8], [-46], [-38], [-44], [-44], [-26], [-36], [54], [-34], [24], [26], [10], [-40], [-20], [20], [16], [-36], [-48], [34], [-10], [42], [68], [16], [-44], [-18], [30], [-6], [60], [2], [-10], [-68], [24], [-52], [-64], [48], [10], [-78], [46], [58], [-52], [42], [-14], [-26], [-8], [-50], [56], [-30], [28], [-6], [40], [20], [-74], [2], [-48], [-54], [-52], [-22], [-46], [-76], [6], [-54], [-10], [-40], [72], [-56], [42], [-8], [44], [-34], [22], [-16], [-28], [-30], [32], [-54], [-50], [-48], [-38], [-88], [26], [-10], [-56], [66], [78], [-24], [-20], [68], [-2], [-60], [-6], [60], [56], [8], [-42], [-24], [56], [-80], [38], [-14], [12], [-86], [0], [-20], [50], [30], [2], [-74], [14], [-36], [48], [88], [66], [-10], [-22], [28], [0], [-4], [-24], [50], [26], [58], [82], [-20], [22], [54], [-4], [-30], [-66], [22], [32], [-20], [42], [-84], [62], [6], [-74], [14], [-14], [-96], [-30], [50], [32], [88], [-56], [16], [-66], [36], [-34], [18], [-26], [-58], [42], [20], [8], [-34], [-80], [36], [78], [24], [-34], [-52], [14], [2], [-6], [-78], [6], [-20], [-56], [12], [60], [48], [48], [14], [-14], [-30], [-76], [-64], [-58], [-100], [-4], [-76], [-54], [50], [-54], [-2], [22], [86], [-20], [-84], [48], [28], [14], [22], [22], [84], [44], [-38], [-2], [-56], [-26], [26], [-24], [-70], [12], [-88], [42], [50], [-10], [52], [48], [-28], [48], [-50], [98], [-68], [96], [76], [56], [38], [-34], [28], [100], [-36], [68], [10], [84], [26], [90], [-64], [72], [18], [58], [-44], [84], [84], [-4], [-46], [50], [-10], [18], [-56], [30], [-68], [-40], [110], [-30], [104], [-46], [24], [-44], [40], [-20], [46], [96], [8], [50], [90], [8], [-46], [-60], [16], [-74], [-34], [-12], [78], [46], [-106], [60], [-42], [66], [-4], [32], [16], [64], [90], [102], [-2], [58], [-52], [42], [64], [-4], [-114], [74], [58]];
    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_p();".
// 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_p(: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_p();".
// 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_p( : 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![-2, 1]>,<13,R![-4, 1]>,<17,R![-6, 1]>,<23,R![-8, 1]>,<53,R![-10, 1]>],Snew);
    return Vf;
end function;