
// Make newform 4017.2.a.c in Magma, downloaded from the LMFDB on 19 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_c();".
// 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_c();".
// 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 := [[0], [1], [1], [2], [0], [-1], [-3], [-4], [-8], [5], [-1], [1], [-6], [-8], [3], [0], [0], [6], [-12], [-1], [-7], [13], [-6], [2], [6], [2], [1], [-5], [11], [-12], [16], [-1], [8], [-11], [18], [0], [-14], [-20], [-20], [6], [25], [10], [-20], [-2], [-27], [-2], [-16], [6], [-25], [14], [0], [-6], [-26], [28], [0], [4], [-5], [7], [-2], [11], [-24], [-7], [-23], [16], [-7], [12], [-17], [-5], [9], [14], [17], [32], [23], [-23], [-16], [-11], [-12], [-29], [-22], [0], [-15], [16], [4], [14], [16], [14], [30], [9], [24], [13], [15], [-1], [28], [-19], [7], [35], [-6], [6], [1], [-2], [-32], [-6], [38], [24], [16], [10], [-10], [18], [24], [26], [-39], [4], [6], [26], [-32], [-39], [14], [39], [24], [-12], [-17], [-19], [5], [-9], [25], [3], [-20], [28], [-40], [-26], [-4], [21], [-31], [-3], [7], [-37], [10], [-52], [-15], [36], [47], [40], [8], [9], [-24], [-10], [-14], [-43], [-18], [-1], [-43], [-10], [-47], [35], [17], [-58], [-18], [-10], [30], [36], [40], [17], [-11], [-40], [32], [24], [-40], [58], [-20], [10], [56], [61], [-15], [22], [-21], [-6], [17], [-62], [-8], [40], [63], [15], [55], [62], [1], [42], [34], [-46], [25], [-38], [-44], [-42], [-9], [-31], [30], [7], [22], [4], [7], [-14], [-41], [-40], [42], [33], [18], [19], [29], [12], [-55], [-3], [-17], [-14], [16], [-3], [-4], [1], [23], [-66], [-8], [72], [46], [-50], [-39], [-49], [14], [0], [6], [-40], [-44], [66], [26], [-47], [62], [66], [16], [3], [15], [-11], [-56], [-52], [42], [-18], [4], [-10], [4], [-39], [0], [-46], [54], [-9], [44], [50], [16], [-35], [-63], [3], [-29], [-56], [66], [-46], [-48], [-51], [-58], [16], [8], [62], [-14], [51], [-1], [-16], [7], [-65], [20], [56], [-22], [82], [-78], [54], [-2], [-53], [0], [15], [-42], [-44], [-48], [-72], [-22], [72], [44], [-32], [30], [-68], [-70], [-84], [-37], [-10], [37], [-22], [36], [-74], [-13], [-15], [-60], [-51], [-32], [8], [-2], [3], [-30], [20], [-12], [72], [-36], [-59], [-40], [-61], [45], [47], [-62], [76], [-4], [57], [-87], [-8], [40], [28], [30], [36], [-70], [26], [70], [-48], [-24], [92], [15], [28], [35], [-74], [-24], [38], [-37], [-68], [15], [-62], [27], [-36], [-70], [62], [5], [66], [-42], [28], [63], [-11], [-66], [-48], [84], [-45], [-2], [-1], [-6], [-47], [-2], [-36], [59], [25], [1], [71], [-54], [-31], [-34], [-56], [34], [80], [42], [52], [49], [-32], [66], [20], [-72], [-18], [-32], [-6], [39], [56], [32], [-70], [-79], [-25], [-23], [66], [27], [-53], [-72], [10], [50], [30], [-44], [30], [-34], [-90], [17], [-78], [66], [-62], [-8], [-27], [-92], [32], [-2], [87], [-26], [58], [42], [54], [-60], [4], [-18], [31], [-18], [0], [-63], [-84], [35], [26], [-49], [45], [-28], [-20]];
    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_c();".
// 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_c(: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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4017_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<23,R![8, 1]>],Snew);
    return Vf;
end function;