
// Make newform 8050.2.a.n in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8050_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_8050_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_8050_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_8050_a();"
function MakeCharacter_8050_a()
    N := 8050;
    order := 1;
    char_gens := [2577, 1151, 3501];
    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_8050_a_Hecke(Kf)
    return MakeCharacter_8050_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], [-2], [0], [-1], [4], [0], [-6], [-6], [1], [10], [4], [2], [-10], [4], [-12], [6], [-2], [0], [0], [-8], [6], [-8], [14], [-14], [2], [-4], [4], [-16], [-10], [2], [-16], [6], [-10], [10], [18], [16], [12], [0], [-12], [16], [12], [-12], [-8], [-14], [-6], [4], [12], [-8], [14], [-20], [-26], [0], [-18], [-2], [22], [-24], [16], [0], [-30], [10], [-26], [-8], [2], [8], [-14], [-30], [-32], [-22], [-24], [-16], [14], [-24], [-24], [14], [-20], [-20], [-26], [-24], [-18], [6], [-26], [-22], [24], [-30], [8], [20], [-30], [10], [-24], [-8], [30], [12], [32], [28], [20], [-16], [0], [22], [22], [-18], [32], [-18], [-6], [-10], [28], [38], [14], [-18], [24], [-30], [-16], [42], [-30], [10], [40], [46], [-22], [36], [-10], [-44], [-32], [-10], [36], [28], [6], [6], [-2], [-28], [-44], [24], [0], [-32], [0], [10], [-10], [14], [16], [22], [-20], [6], [38], [6], [-40], [48], [-20], [-36], [16], [-54], [34], [-24], [38], [26], [12], [28], [40], [32], [-24], [54], [38], [24], [-24], [-2], [8], [-22], [-50], [-44], [32], [20], [-2], [-48], [24], [40], [-8], [-42], [20], [10], [20], [-22], [32], [-24], [-16], [30], [-26], [6], [48], [-12], [-62], [46], [-46], [-36], [-26], [-4], [-16], [16], [-40], [-2], [-10], [10], [30], [24], [-54], [44], [20], [-46], [66], [-60], [20], [-24], [-34], [-22], [34], [-52], [32], [-42], [24], [-10], [64], [6], [0], [54], [50], [72], [-30], [-16], [-46], [34], [30], [40], [-64], [16], [46], [-50], [-8], [10], [14], [16], [-50], [-42], [4], [-64], [20], [-50], [-32], [-34], [-26], [-20], [48], [42], [44], [-24], [14], [18], [-8], [-38], [12], [76], [14], [42], [44], [26], [32], [36], [36], [0], [2], [14], [18], [10], [56], [-10], [-16], [-44], [-50], [8], [30], [64], [16], [-26], [-10], [-54], [-24], [-32], [-60], [0], [2], [-56], [-26], [-42], [84], [26], [-30], [-18], [74], [66], [58], [60], [48], [-24], [-78], [-10], [50], [-38], [32], [-24], [52], [22], [-8], [-8], [-24], [26], [-72], [78], [-46], [40], [-32], [-46], [-14], [-40], [-42], [-30], [-50], [22], [-28], [40], [90], [14], [-52], [42], [8], [2], [-2], [-10], [-4], [-18], [24], [66], [46], [-74], [90], [-4], [-30], [-6], [40], [-64], [58], [-24], [72], [-48], [-40], [24], [82], [38], [22], [76], [-22], [-66], [60], [-30], [-2], [64], [-58], [42], [96], [-24], [90], [-86], [-8], [-64], [86], [-84], [34], [56], [-54], [-40], [-86], [-74], [-24], [38], [-66], [-6], [-12], [42], [-32], [-98], [30], [-88], [0], [-4], [-8], [-20], [38], [60], [12], [30], [-24], [-74], [-36], [46], [-88], [-22], [100], [-18], [48], [90], [0], [-76], [86], [-2], [-62], [-14], [-34], [-66], [-28], [48], [-58], [-40], [48], [-52], [50], [0], [-10], [8], [48], [-30], [2], [84], [-12], [74], [-30], [-52]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8050_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8050_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_8050_2_a_n(:prec:=1)
    chi := MakeCharacter_8050_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(2999) - 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_8050_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_8050_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8050_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<11,R![-4, 1]>,<13,R![0, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;