
// Make newform 4017.2.a.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4017_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_4017_2_a_a();".
// 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_4017_2_a_a();".
// 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_4017_a();"
function MakeCharacter_4017_a()
    N := 4017;
    order := 1;
    char_gens := [1340, 1237, 1756];
    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_4017_a_Hecke(Kf)
    return MakeCharacter_4017_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], [-1], [2], [-2], [4], [-1], [-6], [-2], [6], [6], [0], [-8], [6], [-12], [0], [0], [8], [-2], [-4], [0], [-4], [0], [4], [6], [-14], [8], [-1], [2], [16], [4], [0], [-18], [14], [20], [6], [-20], [10], [-10], [-12], [12], [-26], [-6], [-8], [-4], [-22], [-8], [4], [-26], [4], [-22], [4], [-20], [8], [0], [-8], [4], [10], [-8], [-2], [-30], [-4], [-18], [-16], [14], [10], [2], [-20], [2], [-22], [16], [-6], [0], [-8], [10], [-24], [24], [-4], [-4], [-6], [-18], [10], [-2], [8], [-34], [16], [0], [-18], [28], [-14], [4], [-22], [16], [-16], [-34], [-20], [6], [-18], [4], [-36], [30], [-40], [22], [-24], [24], [0], [24], [0], [22], [12], [-26], [20], [-2], [2], [14], [-42], [10], [-34], [42], [-8], [-18], [4], [34], [-22], [44], [-12], [46], [-38], [52], [-48], [28], [-34], [-24], [-4], [26], [26], [12], [-6], [-6], [-22], [-32], [16], [38], [-56], [-12], [14], [4], [26], [30], [-12], [16], [-24], [36], [16], [34], [-24], [-12], [0], [-10], [10], [18], [-4], [6], [48], [-4], [10], [60], [4], [30], [6], [-32], [-52], [8], [-26], [4], [-20], [-2], [-8], [-42], [-52], [42], [40], [2], [14], [54], [0], [-18], [58], [58], [-60], [12], [50], [-4], [12], [14], [-28], [-54], [2], [2], [30], [48], [-6], [8], [-42], [-26], [68], [-42], [-64], [-36], [70], [16], [46], [-60], [0], [28], [24], [0], [-40], [8], [-48], [-34], [-38], [-10], [-18], [-60], [-28], [-46], [36], [42], [-40], [20], [-46], [56], [2], [-52], [16], [0], [0], [-26], [24], [40], [-24], [-28], [-56], [70], [-34], [-34], [-12], [48], [-50], [-30], [-18], [30], [40], [-58], [30], [12], [-42], [10], [-58], [-62], [44], [62], [60], [70], [58], [60], [18], [-30], [-76], [-48], [62], [4], [-2], [60], [-42], [22], [16], [42], [14], [54], [6], [-24], [16], [-62], [2], [-30], [46], [2], [46], [-84], [-16], [24], [54], [76], [34], [-24], [-72], [-66], [-8], [-2], [22], [10], [-16], [50], [-60], [62], [-16], [-58], [-48], [-18], [56], [70], [12], [-56], [-72], [-10], [-60], [32], [-72], [44], [-68], [-24], [42], [6], [66], [38], [-38], [-72], [-64], [26], [-10], [72], [48], [-52], [88], [56], [44], [50], [-6], [-32], [52], [-48], [46], [8], [66], [-24], [-74], [38], [72], [-50], [46], [-38], [-66], [24], [10], [-28], [-96], [28], [-36], [-96], [2], [66], [-72], [92], [-84], [-38], [-14], [48], [38], [82], [12], [32], [92], [100], [10], [68], [-54], [-26], [-94], [-34], [30], [38], [-50], [42], [-60], [24], [68], [-58], [76], [-18], [-92], [102], [20], [-76], [-60], [38], [66], [44], [52], [6], [-4], [100], [-56], [-18], [-6], [16], [74], [-10], [88], [0], [-18], [54], [68], [60], [14], [0], [-28], [64], [2], [54], [-58], [94], [48], [-42], [20], [14], [102], [46], [92], [8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4017_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4017_2_a_a();".
// 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_4017_2_a_a(:prec:=1)
    chi := MakeCharacter_4017_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_4017_2_a_a();".
// 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_4017_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4017_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<23,R![-6, 1]>],Snew);
    return Vf;
end function;