
// Make newform 8550.2.a.i 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_i();".
// 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_i();".
// 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], [0], [2], [4], [0], [-1], [-8], [-2], [-2], [8], [-2], [-4], [4], [-2], [0], [-10], [0], [-16], [-6], [14], [6], [-18], [-10], [14], [-6], [-8], [0], [-14], [18], [6], [-16], [12], [18], [14], [14], [4], [-24], [-18], [-12], [-8], [4], [-6], [10], [20], [12], [-2], [8], [-10], [-24], [0], [10], [26], [22], [-24], [-30], [-20], [26], [10], [-12], [6], [12], [-20], [-26], [-18], [-28], [-22], [14], [-14], [12], [-12], [20], [16], [12], [8], [-30], [22], [22], [10], [-6], [-4], [0], [-14], [-22], [-34], [34], [-26], [2], [-28], [14], [-20], [14], [38], [28], [-24], [-18], [14], [40], [2], [-28], [-34], [12], [10], [-28], [-38], [-6], [-16], [0], [-14], [22], [-10], [12], [-44], [-28], [6], [-28], [-32], [6], [-16], [-8], [10], [38], [44], [36], [-34], [-18], [-20], [-16], [-18], [-4], [16], [-10], [-34], [-24], [-14], [-6], [20], [-6], [-12], [-36], [6], [4], [-20], [-24], [-16], [42], [18], [28], [-48], [8], [12], [-28], [-16], [40], [48], [20], [20], [-38], [22], [42], [-26], [-48], [-40], [10], [24], [-26], [-10], [10], [30], [36], [20], [-12], [46], [-6], [-48], [-28], [-54], [2], [-58], [44], [52], [8], [-58], [12], [42], [16], [-52], [-46], [-48], [-22], [58], [52], [30], [14], [42], [-34], [30], [-32], [12], [-30], [58], [-28], [14], [2], [58], [-40], [12], [-12], [32], [-14], [50], [-36], [48], [24], [6], [48], [-66], [32], [6], [24], [74], [38], [-68], [-44], [-42], [-2], [72], [-50], [42], [6], [-16], [-6], [-2], [-20], [-48], [-42], [46], [62], [-24], [-16], [-44], [-40], [20], [-18], [24], [-28], [20], [32], [24], [14], [-40], [-44], [38], [-38], [34], [30], [28], [-6], [62], [62], [-60], [66], [-60], [12], [76], [-30], [48], [40], [30], [60], [-28], [54], [8], [-14], [-36], [30], [-20], [-14], [54], [-72], [40], [72], [-60], [-28], [0], [18], [18], [40], [26], [-30], [6], [-42], [-52], [-44], [74], [-58], [34], [52], [12], [74], [82], [16], [64], [-60], [-2], [24], [-40], [-60], [-38], [68], [6], [70], [48], [36], [6], [-42], [40], [-26], [52], [16], [30], [-66], [46], [-24], [-30], [-20], [-28], [32], [22], [34], [30], [-28], [14], [-60], [-74], [0], [66], [-46], [56], [-88], [24], [-30], [38], [34], [64], [-30], [48], [8], [-78], [-32], [-74], [-34], [-30], [32], [26], [8], [-78], [32], [-24], [14], [68], [-48], [-76], [76], [94], [66], [90], [94], [90], [72], [-32], [-14], [-52], [-38], [8], [-16], [10], [60], [-90], [18], [-20], [68], [-72], [60], [-8], [64], [26], [-4], [80], [-10], [10], [54], [20], [24], [46], [-74], [-22], [40], [-102], [92], [44], [-66], [86], [90], [8], [-70], [-22], [56], [-10], [62], [-18], [-100], [20], [38], [-42], [-72], [-2], [48], [-40], [18], [-104], [4], [16], [26], [-18], [16], [16], [68], [-16], [74], [-70], [76], [-8], [26], [-92], [-22], [-92], [52], [-8], [-30], [-60], [20], [-80], [10], [-66], [20], [-24], [-106], [16], [-68], [24], [10], [76], [18], [42], [-44], [24], [-54], [54], [-56], [-74], [-24], [-32], [92], [46], [70], [-102], [32], [-92], [62], [32], [80], [-50], [112], [92], [-38], [-64], [-4], [-14], [-34], [-94], [78], [102], [36], [14], [26], [-84], [32], [78], [-64], [-80], [-62], [-18], [-10], [2], [52], [-38], [64], [60], [30], [-40], [-22]];
    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_i();".
// 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_i(: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_i();".
// 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_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8550_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![-2, 1]>,<13,R![-4, 1]>,<17,R![0, 1]>,<23,R![8, 1]>,<53,R![2, 1]>],Snew);
    return Vf;
end function;