
// Make newform 4368.2.a.s in Magma, downloaded from the LMFDB on 12 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4368_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_4368_2_a_s();".
// 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_4368_2_a_s();".
// 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_4368_a();"
function MakeCharacter_4368_a()
    N := 4368;
    order := 1;
    char_gens := [3823, 1093, 1457, 1249, 2017];
    v := [1, 1, 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_4368_a_Hecke(Kf)
    return MakeCharacter_4368_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], [1], [1], [1], [-1], [-7], [-3], [-3], [-8], [7], [8], [-7], [-8], [-10], [-4], [7], [-2], [-4], [-1], [-2], [6], [14], [-14], [4], [5], [18], [-11], [6], [-18], [5], [-19], [0], [0], [9], [-13], [-2], [-15], [14], [-12], [-22], [-19], [16], [-12], [-1], [3], [-14], [22], [10], [12], [-24], [10], [-7], [18], [4], [-2], [-2], [-14], [10], [-32], [14], [32], [-30], [-24], [12], [10], [-21], [12], [-18], [14], [0], [8], [-12], [-12], [9], [-2], [18], [18], [5], [-1], [34], [38], [12], [17], [10], [-39], [-12], [21], [17], [25], [-5], [-16], [30], [-12], [-34], [-3], [-35], [20], [-43], [4], [-24], [-19], [-6], [-16], [42], [38], [30], [3], [-28], [1], [37], [-33], [1], [-27], [50], [-17], [-12], [35], [-6], [22], [7], [-22], [19], [20], [10], [-6], [42], [-23], [-32], [-22], [-6], [28], [18], [16], [5], [-27], [-55], [16], [-10], [45], [-24], [-20], [41], [7], [4], [-44], [-18], [44], [-46], [34], [-49], [-31], [36], [-4], [-13], [-38], [28], [2], [-46], [13], [36], [-13], [40], [-45], [3], [-42], [-14], [22], [8], [21], [30], [8], [6], [47], [18], [46], [20], [0], [25], [59], [16], [-22], [-30], [-24], [4], [-56], [41], [-14], [45], [-56], [33], [4], [35], [50], [-45], [0], [66], [18], [-8], [36], [-3], [-14], [-10], [20], [26], [-8], [-44], [-57], [16], [-71], [60], [14], [-18], [61], [-3], [6], [-9], [-64], [44], [-60], [56], [-59], [-44], [63], [46], [-29], [0], [5], [63], [30], [6], [67], [-60], [-58], [-27], [49], [0], [-46], [-60], [-68], [-52], [57], [-38], [62], [-60], [-66], [24], [66], [-69], [71], [44], [-64], [-22], [-12], [15], [-60], [-20], [-6], [11], [44], [-12], [-50], [35], [24], [-54], [30], [42], [64], [-6], [-2], [-6], [-53], [-36], [64], [0], [39], [-41], [-52], [12], [-72], [55], [-28], [-32], [-59], [-60], [15], [22], [4], [45], [-77], [47], [-83], [-60], [-34], [-60], [56], [-6], [-21], [7], [68], [12], [-74], [-60], [55], [44], [-60], [5], [54], [-46], [76], [-64], [-48], [73], [-90], [-56], [36], [12], [17], [-74], [50], [48], [24], [1], [9], [-38], [-64], [-35], [85], [-69], [39], [-89], [-40], [-70], [-52], [-2], [23], [-76], [82], [53], [46], [-72], [-93], [82], [-54], [-86], [-19], [2], [8], [26], [-48], [28], [60], [52], [-90], [51], [-9], [-18], [30], [12], [-47], [59], [69], [-76], [-94], [-26], [-63], [-17], [13], [2], [-12], [80], [49], [-5], [8], [96], [27], [-11], [-57], [-2], [42], [-53], [-13], [22], [60], [5], [-93], [15], [-12], [2], [22], [78], [-64], [-4], [92], [2], [10], [-61], [13], [-22], [36], [-62], [-87], [-20], [-66], [-15], [30], [-3], [-35], [20], [-6], [-51], [-56], [-9], [-63], [-32], [14], [-57], [106], [72], [19], [-34], [-32], [4], [27], [14], [-62], [18], [25], [-72]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4368_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4368_2_a_s();".
// 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_4368_2_a_s(:prec:=1)
    chi := MakeCharacter_4368_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_4368_2_a_s();".
// 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_4368_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4368_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 1]>,<11,R![-1, 1]>,<17,R![1, 1]>],Snew);
    return Vf;
end function;