
// Make newform 5850.2.a.d in Magma, downloaded from the LMFDB on 07 September 2025.
// 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_d();".
// 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_d();".
// 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 := 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]];
    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], [4], [-1], [2], [-8], [0], [-6], [-4], [2], [10], [-4], [8], [-10], [-4], [-2], [16], [8], [-2], [8], [12], [-14], [-10], [2], [-16], [12], [-2], [-6], [0], [-4], [-10], [12], [6], [12], [-14], [16], [0], [-10], [12], [-10], [8], [14], [18], [-8], [12], [4], [20], [22], [18], [0], [10], [-4], [-6], [8], [26], [-4], [-22], [26], [4], [26], [8], [0], [6], [-6], [8], [-18], [-12], [6], [14], [0], [-16], [-6], [0], [-24], [26], [-6], [-6], [2], [-4], [22], [8], [30], [16], [-4], [-6], [30], [6], [20], [-4], [16], [-4], [-36], [0], [40], [-42], [14], [20], [-34], [28], [18], [4], [30], [-20], [-18], [4], [-42], [24], [26], [16], [2], [6], [32], [-36], [-2], [16], [24], [-10], [36], [-2], [14], [38], [44], [-32], [50], [6], [24], [40], [2], [40], [-24], [-40], [-54], [26], [2], [-54], [-40], [-2], [-2], [0], [-42], [16], [28], [-2], [40], [2], [18], [-44], [40], [-22], [-26], [-4], [0], [-28], [-40], [-16], [-46], [-26], [46], [-4], [-30], [-4], [28], [6], [24], [-48], [26], [58], [-34], [-60], [-42], [-16], [-18], [-56], [-42], [-16], [-26], [0], [-2], [-28], [28], [-38], [-18], [24], [26], [2], [-40], [-14], [-24], [-34], [36], [20], [-50], [52], [66], [-30], [66], [-42], [-64], [54], [48], [-22], [2], [44], [14], [-52], [-28], [18], [44], [62], [-6], [-16], [12], [-32], [42], [24], [-18], [-8], [-46], [6], [-44], [-70], [52], [-36], [-26], [10], [-48], [8], [60], [18], [-12], [-28], [54], [-28], [-48], [58], [42], [-28], [0], [44], [-20], [48], [-26], [38], [48], [-12], [36], [48], [-24], [34], [-14], [-8], [74], [-42], [60], [54], [16], [54], [-26], [-64], [-36], [-74], [-62], [6], [60], [54], [-6], [64], [-18], [-2], [48], [14], [72], [-34], [36], [-36], [22], [-70], [-12], [-24], [68], [-32], [46], [-32], [-48], [30], [58], [-4], [-42], [2], [-20], [-50], [12], [34], [-30], [64], [-42], [60], [-8], [22], [-6], [8], [60], [-4], [70], [-36], [14], [-24], [-38], [-64], [-2], [-42], [-44], [48], [-38], [-36], [-72], [-26], [-82], [-68], [-42], [-30], [28], [78], [34], [-32], [-8], [-64], [78], [-18], [22], [40], [-44], [64], [-28], [86], [-34], [-62], [-40], [2], [-30], [6], [-64], [-6], [-20], [38], [56], [-88], [-58], [48], [62], [30], [64], [6], [66], [-24], [36], [-54], [0], [-46], [22], [0], [68], [-28], [38], [-6], [-44], [10], [-52], [-12], [40], [-70], [32], [-62], [-12], [48], [-82], [30], [-10], [30], [70], [92], [-26], [48], [88], [-12], [42], [-48], [-32], [10], [42], [-28], [20], [32], [-74], [-76], [-26], [-76], [-2], [6], [34], [-12], [46], [-18], [0], [-78], [90], [88], [-4], [78], [54], [-76], [12], [6], [50], [-16], [0], [-74], [0], [-22], [-14], [-64], [100], [6], [2], [-52], [-78], [56], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5850_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5850_2_a_d();".
// 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_d(: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_d();".
// 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_d( : 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![-4, 1]>,<17,R![-2, 1]>,<23,R![0, 1]>,<31,R![4, 1]>],Snew);
    return Vf;
end function;