
// Make newform 4368.2.a.n in Magma, downloaded from the LMFDB on 25 April 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_n();".
// 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_n();".
// 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], [-3], [-1], [5], [-1], [3], [1], [-1], [-5], [8], [1], [-12], [-11], [-8], [10], [8], [13], [-2], [-8], [11], [-2], [18], [18], [10], [-16], [-5], [10], [3], [18], [2], [-11], [-3], [12], [12], [3], [17], [-10], [15], [2], [-4], [14], [-25], [8], [16], [17], [15], [10], [10], [14], [-8], [-20], [10], [-15], [-14], [-12], [-26], [22], [22], [-22], [4], [-6], [16], [-30], [-12], [-28], [-26], [-13], [16], [-6], [-10], [-8], [16], [0], [-12], [15], [18], [18], [34], [-7], [15], [14], [-2], [16], [-1], [-18], [9], [0], [-17], [-5], [-31], [-27], [8], [-6], [36], [30], [7], [-39], [20], [3], [-20], [8], [21], [-10], [40], [18], [-18], [22], [9], [-40], [-1], [19], [15], [17], [-9], [-10], [-49], [-32], [45], [2], [10], [31], [14], [39], [-4], [22], [-18], [-10], [-49], [44], [10], [10], [-32], [-6], [12], [49], [-25], [17], [48], [-6], [-51], [-24], [-4], [45], [53], [-20], [4], [-18], [20], [10], [14], [3], [-19], [-52], [-4], [-15], [18], [44], [-34], [-18], [-39], [48], [33], [-24], [19], [-51], [-30], [14], [-10], [-8], [-23], [-58], [0], [-30], [-23], [-30], [-10], [60], [24], [-5], [1], [16], [-58], [2], [24], [52], [-8], [17], [-58], [3], [-12], [13], [-52], [25], [-42], [-49], [-12], [-6], [22], [36], [16], [-45], [14], [-38], [-40], [70], [56], [-24], [15], [-8], [-7], [0], [50], [-26], [3], [9], [-18], [19], [-60], [-68], [-12], [56], [-11], [20], [-57], [-10], [-41], [40], [-29], [-29], [18], [-30], [-23], [-20], [22], [3], [-59], [36], [58], [12], [-36], [-20], [-33], [-34], [-54], [-28], [-54], [0], [-66], [-15], [-23], [72], [-36], [38], [36], [11], [-72], [52], [-54], [7], [-56], [-60], [-46], [-7], [-40], [-26], [38], [-30], [-28], [30], [-14], [38], [7], [60], [-8], [-72], [75], [-7], [60], [20], [-60], [29], [28], [16], [21], [-12], [7], [-18], [40], [41], [45], [-25], [-75], [60], [34], [-24], [32], [74], [75], [-49], [-36], [-12], [86], [32], [-13], [-76], [48], [39], [6], [-2], [36], [-36], [-28], [-59], [-54], [36], [40], [-76], [69], [-26], [-62], [48], [-8], [67], [-39], [-30], [-52], [69], [31], [-43], [-63], [1], [-40], [-22], [-20], [-86], [-7], [20], [-38], [-37], [-46], [48], [81], [-46], [-58], [-42], [31], [82], [72], [-42], [0], [-32], [20], [-20], [-22], [63], [73], [18], [38], [4], [23], [-1], [23], [-60], [-10], [-50], [51], [85], [85], [54], [72], [-52], [39], [-35], [-80], [-60], [-63], [-55], [27], [86], [78], [-29], [-55], [38], [-76], [3], [9], [-3], [-24], [78], [-34], [18], [40], [52], [-12], [98], [-14], [-33], [-23], [-30], [72], [2], [3], [72], [-6], [39], [98], [-91], [5], [76], [-30], [83], [-76], [27], [-39], [-8], [46], [9], [-38], [24], [5], [-50], [-12], [-36], [-33], [-22], [102], [-70], [-95], [84]];
    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_n();".
// 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_n(: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_n();".
// 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_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4368_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3, 1]>,<11,R![-5, 1]>,<17,R![-3, 1]>],Snew);
    return Vf;
end function;