
// Make newform 4368.2.a.u in Magma, downloaded from the LMFDB on 20 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_u();".
// 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_u();".
// 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], [4], [-4], [-2], [6], [0], [10], [4], [4], [0], [10], [-4], [-2], [0], [6], [14], [4], [-16], [12], [6], [0], [-8], [-18], [-10], [-14], [20], [-12], [-6], [20], [22], [8], [14], [8], [8], [-16], [10], [10], [-6], [-2], [-2], [0], [20], [16], [-12], [-6], [6], [10], [-18], [0], [-8], [-2], [-4], [0], [10], [18], [-20], [-12], [-4], [4], [18], [-10], [-4], [-6], [34], [-10], [32], [6], [-8], [6], [-20], [-20], [-26], [22], [34], [14], [-8], [-22], [-2], [-14], [16], [-38], [-26], [2], [4], [4], [0], [-16], [-16], [-38], [20], [0], [-24], [-20], [-44], [34], [-40], [-42], [24], [-10], [-40], [34], [-36], [-24], [10], [-38], [8], [38], [18], [-4], [4], [6], [-12], [36], [14], [6], [38], [-2], [-32], [-26], [-44], [-18], [-34], [24], [32], [-22], [40], [-10], [16], [-10], [-44], [26], [44], [4], [-24], [34], [-12], [-2], [20], [18], [10], [40], [38], [-32], [-28], [-54], [-18], [-44], [4], [-36], [36], [-6], [32], [-12], [-22], [-32], [2], [18], [-28], [-36], [-46], [52], [8], [-42], [14], [-48], [2], [6], [26], [6], [0], [-20], [8], [-30], [-16], [-58], [52], [-36], [10], [-24], [2], [-48], [38], [20], [22], [36], [-34], [-30], [-20], [-36], [54], [36], [-42], [-14], [28], [-44], [70], [-16], [-14], [-50], [-56], [-24], [0], [-26], [30], [4], [2], [-28], [16], [-24], [32], [26], [-24], [-20], [-30], [-6], [-10], [-24], [30], [64], [-60], [-30], [40], [54], [-32], [30], [-66], [36], [-72], [26], [12], [-52], [-18], [-2], [26], [-72], [-2], [-12], [0], [-54], [76], [-28], [-8], [20], [48], [-50], [30], [0], [38], [-42], [-64], [2], [74], [28], [16], [34], [4], [-62], [-30], [-42], [-32], [-20], [-18], [32], [40], [78], [-2], [44], [46], [8], [-2], [40], [2], [10], [38], [8], [-8], [68], [80], [14], [20], [10], [34], [78], [48], [8], [82], [60], [-6], [-44], [-66], [-24], [-48], [68], [-24], [44], [10], [-30], [-20], [86], [40], [34], [42], [-14], [-18], [74], [16], [-14], [38], [-60], [18], [38], [32], [2], [-34], [6], [44], [-66], [0], [-40], [6], [46], [16], [92], [2], [30], [-34], [-18], [88], [12], [72], [36], [94], [-4], [-66], [16], [54], [78], [16], [-48], [-46], [6], [-70], [-28], [-28], [-40], [-20], [-58], [-18], [88], [-26], [-44], [-84], [32], [-78], [54], [-58], [16], [-78], [-14], [-92], [38], [-96], [-92], [-34], [42], [28], [78], [-30], [-80], [66], [-72], [62], [-30], [36], [62], [-36], [6], [48], [-34], [68], [52], [-16], [58], [88], [20], [30], [-72], [82], [92], [-26], [-94], [56], [-36], [-32], [-78], [30], [-6], [60], [28], [16], [-40], [-62], [38], [-100], [12], [82], [6], [-38], [-52], [38], [-84], [2], [-8], [0], [84], [-26], [38], [54], [12], [-54], [36], [-70], [42], [28], [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_u();".
// 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_u(: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_u();".
// 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_u( : 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![-4, 1]>],Snew);
    return Vf;
end function;