
// Make newform 5850.2.a.e in Magma, downloaded from the LMFDB on 13 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5850_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_5850_2_a_e();".
// 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_5850_2_a_e();".
// 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_5850_a();"
function MakeCharacter_5850_a()
    N := 5850;
    order := 1;
    char_gens := [3251, 3277, 2251];
    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_5850_a_Hecke(Kf)
    return MakeCharacter_5850_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], [6], [1], [4], [2], [6], [10], [4], [-6], [-10], [0], [-8], [6], [6], [-6], [-12], [0], [2], [-8], [4], [-14], [14], [-10], [-2], [0], [4], [4], [-10], [8], [14], [16], [-16], [16], [6], [-4], [16], [22], [24], [6], [0], [-2], [6], [-8], [4], [8], [4], [8], [12], [4], [18], [20], [-28], [10], [-10], [-8], [-14], [18], [-12], [-18], [-4], [8], [-8], [-2], [2], [16], [-16], [-4], [26], [20], [-18], [-18], [10], [-20], [-30], [-10], [-30], [-34], [-8], [4], [16], [16], [-32], [0], [-38], [14], [4], [4], [-8], [20], [32], [0], [-18], [14], [-16], [-6], [16], [-32], [36], [18], [-24], [42], [12], [2], [28], [-14], [32], [22], [42], [-38], [18], [-22], [40], [18], [-20], [6], [10], [32], [-32], [44], [-6], [20], [18], [6], [-8], [-8], [38], [-18], [-2], [36], [-48], [-22], [-30], [10], [18], [-12], [30], [2], [-50], [28], [-14], [44], [26], [-40], [-26], [0], [28], [-24], [18], [30], [40], [6], [40], [48], [-56], [-46], [48], [8], [-12], [24], [-4], [36], [-58], [28], [-32], [-10], [-14], [-34], [-30], [-48], [48], [-34], [-16], [30], [54], [48], [-18], [-30], [0], [0], [-14], [42], [-4], [-10], [-26], [36], [-10], [12], [36], [4], [-20], [0], [-36], [20], [30], [6], [26], [14], [12], [-8], [-6], [18], [-66], [-54], [44], [24], [10], [0], [-56], [-66], [26], [-60], [60], [-14], [-46], [-14], [4], [18], [-54], [20], [46], [72], [12], [50], [-24], [-16], [-50], [-18], [-2], [28], [-8], [-66], [4], [-48], [-18], [42], [0], [0], [60], [56], [-50], [-16], [22], [-16], [-72], [2], [2], [-6], [22], [6], [-36], [30], [14], [-26], [-22], [28], [-2], [26], [74], [20], [-40], [34], [-54], [-28], [20], [22], [-44], [50], [50], [68], [62], [-40], [-4], [44], [36], [-40], [-2], [-60], [-46], [52], [38], [-56], [44], [-24], [36], [10], [36], [30], [-58], [72], [-66], [-38], [2], [-14], [-8], [-54], [-20], [-76], [24], [-78], [40], [-84], [-40], [18], [40], [50], [-60], [62], [6], [8], [38], [-52], [-72], [22], [-66], [-48], [-62], [10], [20], [18], [-58], [-60], [-82], [82], [10], [28], [-82], [-66], [-20], [18], [80], [44], [70], [-12], [-12], [58], [22], [-26], [50], [40], [-76], [8], [-34], [36], [-74], [-20], [80], [66], [74], [-86], [4], [-86], [-74], [36], [-64], [38], [56], [44], [34], [-10], [-10], [-70], [-4], [-52], [30], [64], [-38], [-20], [-64], [-60], [82], [56], [10], [74], [-40], [-22], [-78], [-28], [-24], [6], [-48], [94], [-30], [-84], [24], [82], [-92], [-62], [-38], [-102], [30], [-24], [12], [12], [24], [-10], [8], [68], [52], [-36], [8], [-102], [-24], [32], [-78], [42], [-76], [78], [-44], [-66], [36], [-68], [-92], [2], [0], [54], [-86], [18], [94], [-6], [24], [-12], [-94], [-46], [-40], [6], [-106], [-80]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5850_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5850_2_a_e();".
// 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_5850_2_a_e(:prec:=1)
    chi := MakeCharacter_5850_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_5850_2_a_e();".
// 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_5850_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5850_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![-6, 1]>,<17,R![-4, 1]>,<23,R![-6, 1]>,<31,R![-4, 1]>],Snew);
    return Vf;
end function;