
// Make newform 4368.2.a.f 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_f();".
// 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_f();".
// 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], [0], [-1], [2], [-1], [-6], [8], [4], [-6], [-4], [-2], [0], [-4], [6], [6], [-10], [10], [4], [6], [6], [0], [-6], [-12], [-2], [-14], [-8], [0], [-6], [-10], [0], [0], [8], [-20], [12], [4], [-18], [-8], [-2], [-22], [-4], [6], [8], [10], [0], [16], [-4], [-4], [-18], [-6], [-26], [6], [22], [-16], [-18], [-4], [-30], [0], [-18], [-8], [-28], [-12], [-20], [-4], [-10], [8], [-4], [-2], [12], [30], [-16], [-10], [24], [-6], [32], [-14], [-18], [-18], [-36], [30], [16], [10], [-6], [14], [16], [24], [-12], [-34], [-8], [8], [-24], [2], [-4], [32], [28], [-36], [-24], [22], [36], [10], [-44], [36], [-12], [22], [12], [18], [-14], [-20], [-12], [38], [-24], [22], [-32], [-44], [12], [2], [-44], [32], [10], [-36], [-34], [-14], [22], [-14], [28], [-50], [50], [36], [48], [34], [-4], [38], [48], [-22], [48], [-38], [20], [28], [-18], [30], [-8], [36], [48], [-22], [34], [-30], [14], [-18], [-4], [26], [-34], [38], [-20], [0], [44], [8], [40], [-16], [-2], [-12], [-42], [-18], [4], [36], [-52], [-46], [48], [-26], [-10], [-6], [-42], [30], [52], [62], [-8], [-6], [52], [12], [8], [-62], [48], [-12], [58], [-36], [22], [6], [-18], [0], [-42], [46], [-6], [-22], [20], [-28], [-28], [6], [-14], [-46], [36], [-24], [-16], [16], [2], [26], [-10], [70], [-28], [48], [20], [28], [-2], [30], [-16], [-38], [-6], [-62], [64], [-30], [62], [28], [-10], [-16], [20], [12], [8], [-10], [42], [20], [-64], [-34], [14], [44], [-32], [-6], [-36], [-66], [-58], [60], [-20], [-24], [26], [-76], [-56], [-42], [-44], [44], [-32], [-18], [16], [-20], [-22], [32], [-70], [70], [74], [18], [-54], [32], [6], [-14], [-16], [60], [-14], [-66], [-36], [-44], [72], [48], [4], [42], [-62], [-60], [14], [-48], [30], [80], [6], [-74], [2], [8], [16], [8], [72], [-10], [-68], [-24], [-2], [-36], [44], [-50], [54], [-4], [12], [-2], [-2], [-42], [64], [6], [60], [60], [26], [-8], [-32], [-44], [-68], [62], [36], [-34], [-54], [82], [-68], [16], [62], [-20], [-50], [38], [54], [-30], [70], [-38], [44], [58], [-82], [44], [4], [-34], [56], [-4], [24], [26], [-78], [-78], [-40], [-86], [80], [-14], [22], [-72], [14], [48], [-50], [-50], [-44], [-32], [72], [-84], [90], [-64], [70], [22], [68], [-42], [4], [-16], [-6], [14], [-34], [-34], [-34], [-30], [6], [26], [20], [-26], [-60], [26], [16], [36], [-6], [88], [44], [6], [78], [8], [46], [98], [-32], [-46], [34], [10], [-20], [-48], [8], [68], [-8], [-70], [-40], [42], [24], [76], [-42], [36], [42], [4], [-50], [-26], [40], [-86], [-52], [-88], [-10], [90], [-56], [-12], [-80], [-32], [-46], [60], [-92], [-30], [50], [-18], [44], [-92], [98], [30], [62], [-96], [-68], [36], [26], [70], [-34], [32], [-14], [-100], [-8], [-36], [64], [0]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4368_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![-2, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;