
// Make newform 6017.2.a.a in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6017_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_6017_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_6017_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_6017_a();"
function MakeCharacter_6017_a()
    N := 6017;
    order := 1;
    char_gens := [3830, 2190];
    v := [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_6017_a_Hecke(Kf)
    return MakeCharacter_6017_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], [0], [0], [-2], [-1], [-5], [-4], [-7], [2], [-1], [4], [-6], [2], [-8], [-9], [2], [-6], [-14], [-5], [10], [-6], [-6], [4], [8], [15], [10], [-6], [-8], [-4], [9], [15], [-3], [7], [-15], [-15], [-10], [-5], [-10], [-5], [-8], [-13], [2], [3], [11], [-4], [8], [-22], [-20], [3], [-14], [9], [12], [8], [16], [10], [9], [-9], [-28], [-6], [-2], [4], [-11], [0], [21], [-5], [29], [8], [-12], [5], [5], [35], [8], [-16], [-26], [-11], [16], [-12], [4], [-9], [-13], [19], [-28], [-26], [-16], [17], [-7], [-6], [-4], [-28], [-26], [15], [35], [-8], [-8], [-23], [4], [30], [-18], [6], [28], [1], [23], [36], [0], [-3], [-14], [-39], [-38], [-41], [25], [8], [-5], [-6], [-2], [-19], [0], [1], [48], [-18], [4], [-10], [29], [-37], [-9], [-24], [-15], [-20], [-44], [-28], [0], [-20], [-24], [39], [-53], [27], [-44], [-30], [-29], [-2], [10], [20], [-42], [-13], [12], [7], [35], [-10], [-38], [37], [34], [-2], [-12], [-36], [-16], [28], [-48], [8], [-42], [20], [-41], [24], [23], [-54], [-20], [-12], [26], [-13], [-46], [-6], [6], [-57], [4], [24], [14], [35], [3], [-16], [-19], [-37], [44], [-61], [48], [-58], [-42], [63], [-33], [38], [4], [-11], [-14], [-36], [-7], [36], [-18], [3], [37], [-46], [1], [42], [-24], [-21], [-23], [17], [18], [21], [67], [-20], [9], [-69], [-50], [-2], [-20], [21], [12], [16], [25], [61], [50], [39], [-32], [-13], [45], [-71], [36], [10], [-40], [54], [66], [4], [-12], [-48], [16], [13], [20], [-44], [-60], [-30], [-54], [32], [-20], [-66], [-74], [-13], [-6], [49], [-48], [46], [20], [64], [-69], [-12], [48], [36], [45], [-3], [51], [31], [-42], [-44], [-51], [60], [-70], [70], [-15], [-25], [40], [6], [22], [52], [66], [46], [-16], [34], [-32], [-7], [-43], [8], [26], [-21], [-60], [16], [-65], [-72], [-10], [58], [20], [2], [78], [-32], [9], [44], [-21], [-56], [65], [13], [52], [2], [-30], [36], [53], [-16], [-8], [-2], [28], [68], [40], [26], [-37], [-88], [16], [-63], [0], [-30], [84], [9], [-86], [9], [9], [15], [6], [-47], [-24], [-36], [57], [-72], [1], [0], [33], [-39], [26], [-40], [-62], [17], [-52], [-82], [56], [46], [-69], [23], [11], [-14], [-94], [-6], [4], [54], [62], [-14], [22], [-44], [30], [-25], [86], [93], [28], [-38], [-51], [-20], [24], [-42], [-4], [42], [18], [-92], [-90], [14], [-54], [30], [69], [0], [-10], [10], [76], [10], [32], [6], [41], [-2], [-42], [96], [-44], [-60], [5], [95], [96], [44], [24], [100], [18], [-70], [49], [-38], [81], [4], [-68], [-12], [51], [-56], [-68], [12], [15], [31], [12], [32], [-76], [-87], [61], [37], [-75], [-92], [-89], [28], [88], [6], [44], [-78], [65], [-59], [56], [8], [-48], [61], [-14], [12], [67], [23], [96], [-28], [22], [-14], [-39]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6017_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6017_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_6017_2_a_a(:prec:=1)
    chi := MakeCharacter_6017_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_6017_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_6017_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6017_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<3,R![0, 1]>],Snew);
    return Vf;
end function;