
// Make newform 8550.2.a.v 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_v();".
// 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_v();".
// 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], [-6], [4], [-6], [1], [0], [0], [8], [-8], [12], [-2], [0], [-6], [12], [2], [16], [0], [10], [8], [0], [12], [-8], [-6], [16], [-12], [2], [-6], [-20], [-18], [6], [8], [18], [8], [10], [10], [24], [6], [12], [2], [-6], [-20], [6], [20], [-4], [16], [12], [2], [6], [6], [2], [-6], [-6], [-12], [-24], [20], [10], [-24], [-14], [-18], [16], [18], [-26], [-30], [-28], [-8], [24], [14], [30], [-6], [34], [4], [-4], [24], [-18], [-14], [0], [14], [-30], [-10], [-12], [4], [8], [-36], [-12], [22], [42], [-14], [36], [-30], [16], [6], [32], [12], [36], [24], [4], [-10], [16], [-30], [12], [36], [-16], [-2], [-12], [6], [12], [-34], [-8], [-38], [-30], [-16], [8], [-12], [-14], [-12], [30], [0], [50], [-8], [42], [12], [8], [-30], [-46], [-18], [10], [22], [44], [24], [-16], [22], [-6], [50], [-30], [28], [-42], [30], [20], [-30], [-14], [12], [2], [-24], [22], [6], [-52], [24], [-20], [-30], [34], [0], [-44], [-48], [56], [-18], [10], [24], [0], [18], [34], [24], [30], [0], [32], [10], [2], [18], [24], [-58], [-30], [-50], [-40], [30], [-16], [42], [16], [14], [-26], [48], [4], [-54], [24], [-6], [16], [4], [-58], [30], [-20], [-12], [-28], [48], [-36], [-54], [14], [46], [18], [0], [24], [-64], [4], [2], [6], [30], [-16], [-36], [-30], [20], [-26], [-18], [-14], [-12], [-36], [14], [-14], [-24], [24], [54], [-58], [32], [36], [46], [-12], [-10], [18], [-12], [28], [30], [-26], [44], [-40], [12], [-26], [0], [14], [-6], [30], [-36], [12], [-52], [46], [-46], [30], [-66], [70], [36], [44], [36], [-50], [-66], [60], [62], [6], [66], [26], [-68], [18], [10], [-8], [12], [38], [76], [-30], [-76], [-48], [-18], [-44], [78], [-10], [16], [-62], [-4], [-56], [-62], [0], [-22], [62], [54], [-48], [32], [-48], [-58], [-50], [66], [70], [42], [56], [36], [-6], [-24], [-66], [-72], [-44], [30], [8], [54], [12], [-26], [-14], [-6], [32], [-36], [-64], [-8], [36], [-82], [-30], [-2], [-24], [30], [48], [-32], [-60], [2], [-66], [0], [-74], [-6], [20], [58], [-72], [-68], [66], [50], [-52], [28], [-72], [6], [-34], [18], [-28], [48], [-16], [-24], [-10], [-18], [-82], [70], [-44], [-42], [-36], [-64], [90], [-24], [74], [-8], [60], [18], [20], [52], [-66], [-80], [-22], [-18], [54], [42], [18], [0], [-62], [42], [-48], [72], [-38], [28], [-30], [16], [38], [54], [-4], [-36], [-60], [32], [-38], [60], [-42], [70], [-90], [-32], [84], [90], [-38], [-18], [68], [0], [-28], [-14], [34], [72], [38], [18], [90], [58], [-12], [4], [80], [0], [20], [30], [38], [18], [4], [-78], [0], [-40], [10], [60], [40], [90], [16], [-42], [60], [-16], [58], [90], [60], [28], [78], [-72], [-84], [76], [-48], [12], [100], [30], [-36], [30], [-16], [102], [-10], [30], [-76], [96], [22], [-42], [74], [-70], [-80], [-4], [24], [-30], [-10], [72], [-82], [102], [-2], [-48], [62], [86], [-68], [84], [24], [6], [-2], [-84], [-22], [96], [34], [-6], [-52], [56], [-12], [-10], [-14], [34], [80], [72], [-6], [-76], [4], [12], [-12], [-10], [24], [-80], [18], [68], [84], [-30], [76], [60], [52], [-96], [10], [-48], [-94], [-24], [92], [-40], [-8], [0], [86], [-66], [-30], [-10], [-32], [-30], [-112], [92], [-66], [106], [30]];
    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_v();".
// 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_v(: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_v();".
// 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_v( : 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![6, 1]>,<13,R![-4, 1]>,<17,R![6, 1]>,<23,R![0, 1]>,<53,R![6, 1]>],Snew);
    return Vf;
end function;