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