
// Make newform 4368.2.a.o in Magma, downloaded from the LMFDB on 17 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_o();".
// 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_o();".
// 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], [0], [-1], [4], [0], [6], [-6], [-8], [-2], [-10], [12], [-6], [-14], [10], [14], [4], [12], [-6], [-8], [-6], [-14], [-18], [4], [-8], [-2], [-14], [6], [-8], [0], [12], [-4], [0], [-16], [-2], [-20], [10], [-20], [6], [-14], [-18], [-14], [-4], [-24], [-8], [4], [10], [-26], [10], [8], [18], [-20], [4], [14], [24], [-12], [26], [-12], [4], [-18], [12], [-8], [-14], [12], [4], [-2], [22], [2], [18], [-28], [0], [34], [12], [26], [22], [-2], [24], [22], [12], [-6], [16], [-38], [-40], [18], [20], [-26], [10], [32], [12], [6], [24], [2], [20], [-4], [-18], [-32], [12], [-38], [16], [-16], [-24], [18], [28], [18], [-18], [-10], [-6], [10], [8], [-10], [-20], [-32], [16], [30], [44], [-12], [-6], [-30], [26], [14], [40], [-28], [-8], [30], [-14], [4], [-24], [-26], [-12], [-4], [32], [-38], [-2], [14], [6], [-16], [-8], [-26], [12], [24], [-44], [-48], [26], [-26], [22], [28], [-4], [44], [-18], [56], [56], [-36], [32], [38], [44], [-50], [-22], [18], [-36], [6], [-24], [-52], [-40], [-6], [-48], [38], [-10], [12], [52], [30], [-62], [46], [-16], [44], [-20], [36], [-8], [62], [16], [4], [6], [-42], [0], [-8], [54], [-44], [22], [38], [-10], [24], [44], [-42], [18], [-32], [-14], [-34], [2], [-36], [60], [24], [-2], [-34], [-50], [-56], [-60], [-66], [36], [60], [-26], [-12], [-20], [42], [6], [-54], [60], [-28], [32], [-12], [6], [-68], [12], [16], [8], [-54], [-56], [-2], [48], [64], [-2], [-60], [8], [-30], [28], [54], [34], [-24], [22], [72], [68], [20], [-16], [6], [-22], [-8], [12], [10], [-12], [-26], [-62], [18], [-72], [-26], [-44], [24], [-18], [24], [72], [-34], [-48], [46], [-70], [-34], [-26], [20], [-60], [-30], [-4], [34], [6], [4], [26], [0], [-10], [-4], [-20], [30], [26], [-44], [72], [72], [-48], [-26], [8], [38], [30], [-16], [56], [-84], [74], [12], [-40], [42], [14], [56], [-80], [-12], [-88], [-40], [34], [-40], [68], [-70], [32], [22], [-10], [-46], [72], [-42], [36], [48], [74], [24], [8], [-26], [-34], [60], [-2], [-26], [20], [-22], [-24], [-88], [-84], [30], [20], [-28], [-2], [-30], [10], [74], [72], [-66], [20], [-34], [-74], [-90], [22], [-48], [46], [42], [-6], [64], [48], [-6], [2], [-68], [62], [-32], [-4], [-58], [72], [-40], [-30], [-56], [46], [46], [-66], [56], [46], [-24], [-14], [-84], [-28], [-2], [38], [-16], [-2], [18], [-12], [-60], [26], [40], [-62], [-46], [2], [34], [84], [-98], [30], [-56], [-48], [-28], [44], [-54], [40], [-46], [-76], [-48], [-50], [-62], [72], [-4], [36], [-74], [16], [-100], [4], [8], [30], [-30], [-72], [42], [-90], [8], [-14], [36], [40], [-42], [18], [-66], [-18], [4], [22], [84], [-24], [80], [-2], [-16], [-78], [-102], [60], [-16], [30], [54], [-30], [64], [100], [12]];
    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_o();".
// 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_o(: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_o();".
// 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_o( : 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![0, 1]>,<17,R![-4, 1]>],Snew);
    return Vf;
end function;