
// Make newform 4368.2.a.x 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_x();".
// 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_x();".
// 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], [2], [-1], [4], [-1], [2], [-4], [4], [10], [-8], [2], [10], [-4], [8], [-6], [8], [2], [8], [-8], [6], [-8], [-8], [2], [14], [6], [4], [-12], [-6], [10], [0], [20], [-2], [-20], [10], [-8], [-6], [-8], [8], [14], [12], [-6], [-12], [26], [-6], [4], [-20], [-16], [8], [-6], [18], [24], [14], [28], [-30], [-4], [6], [0], [-26], [-18], [4], [-30], [12], [0], [10], [2], [-32], [34], [-4], [18], [-30], [-8], [-36], [-18], [16], [24], [-6], [2], [-18], [6], [12], [2], [-32], [-14], [20], [36], [-10], [10], [-6], [16], [12], [-8], [16], [12], [-8], [0], [34], [-38], [12], [-14], [28], [18], [12], [-6], [-12], [-42], [0], [18], [-44], [-14], [-4], [-22], [-18], [28], [-48], [10], [44], [0], [-46], [-36], [-22], [34], [14], [4], [-28], [-6], [-38], [24], [-44], [2], [40], [-32], [-32], [-10], [42], [-50], [-30], [-28], [22], [10], [44], [42], [-40], [12], [-38], [24], [-22], [-54], [-28], [48], [2], [-30], [-4], [-8], [28], [-36], [-40], [18], [-22], [-30], [-20], [-38], [-32], [-28], [-18], [24], [-24], [-38], [50], [6], [-4], [-30], [-36], [-38], [-52], [-38], [32], [-6], [-28], [2], [48], [-36], [-58], [-22], [8], [62], [38], [12], [-14], [-64], [42], [36], [-52], [66], [-52], [10], [-54], [46], [42], [0], [-30], [-28], [-22], [-22], [40], [38], [56], [36], [30], [-52], [34], [-10], [-16], [56], [48], [-26], [-16], [18], [8], [66], [-58], [16], [-26], [24], [12], [22], [26], [-20], [-4], [12], [-2], [28], [24], [-22], [-12], [8], [-18], [-54], [12], [56], [-36], [-44], [-44], [2], [-6], [-48], [8], [0], [-16], [-12], [-38], [66], [-24], [-70], [14], [-20], [-18], [-28], [46], [-34], [8], [-44], [26], [18], [-38], [60], [34], [-22], [64], [-6], [26], [-24], [-10], [-40], [42], [16], [-36], [-46], [-22], [36], [56], [64], [-16], [58], [-4], [20], [-62], [-54], [40], [82], [-6], [12], [6], [0], [86], [-10], [-28], [54], [60], [36], [26], [18], [8], [84], [60], [26], [-36], [58], [40], [22], [40], [74], [18], [-4], [-48], [-14], [0], [40], [22], [-30], [-20], [-86], [38], [56], [30], [-38], [32], [-68], [52], [10], [90], [18], [76], [-48], [56], [-48], [74], [-6], [30], [-12], [-14], [90], [66], [-64], [66], [-36], [42], [8], [8], [46], [-36], [90], [-78], [-44], [-66], [-54], [-64], [32], [-46], [-24], [-46], [42], [-76], [44], [12], [58], [18], [56], [90], [-52], [52], [56], [-38], [52], [-10], [48], [44], [22], [-14], [-14], [-46], [6], [64], [38], [44], [-24], [-80], [-30], [-96], [72], [66], [-54], [-28], [4], [-32], [-38], [40], [-54], [52], [42], [34], [-86], [88], [-6], [-6], [20], [-62], [-58], [4], [80], [-22], [42], [-12], [-28], [106], [46], [48], [-28], [10], [48], [18], [58], [48], [-36], [-2], [-22], [-44], [54], [-76], [40]];
    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_x();".
// 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_x(: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_x();".
// 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_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4368_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<11,R![-4, 1]>,<17,R![-2, 1]>],Snew);
    return Vf;
end function;