
// Make newform 4896.2.a.l in Magma, downloaded from the LMFDB on 10 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4896_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_4896_2_a_l();".
// 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_4896_2_a_l();".
// 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_4896_a();"
function MakeCharacter_4896_a()
    N := 4896;
    order := 1;
    char_gens := [2143, 613, 3809, 4321];
    v := [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_4896_a_Hecke(Kf)
    return MakeCharacter_4896_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], [0], [0], [2], [0], [-6], [1], [-4], [-6], [8], [-2], [12], [2], [12], [4], [14], [12], [-4], [4], [6], [-6], [14], [12], [-6], [-10], [-2], [-8], [16], [-8], [6], [12], [0], [14], [-16], [-22], [20], [-2], [-4], [-6], [0], [20], [20], [4], [-2], [12], [10], [16], [-16], [-20], [14], [-10], [20], [-10], [-28], [-2], [-4], [0], [20], [0], [-22], [-24], [26], [12], [-6], [-18], [-12], [-20], [-14], [20], [26], [10], [8], [14], [10], [12], [20], [-30], [4], [-30], [6], [-4], [18], [18], [14], [34], [4], [18], [22], [-18], [0], [-28], [22], [-18], [-20], [8], [-22], [18], [10], [-12], [-20], [-32], [-26], [-12], [14], [12], [2], [-44], [-18], [-8], [-30], [-10], [-26], [22], [44], [16], [-22], [20], [28], [-24], [12], [-38], [-26], [36], [-12], [24], [30], [-8], [6], [4], [-46], [4], [34], [18], [-2], [30], [-38], [-30], [40], [38], [42], [-28], [36], [22], [48], [6], [-18], [32], [18], [20], [0], [8], [54], [52], [-58], [48], [30], [-20], [-30], [-42], [28], [-56], [26], [-24], [60], [-10], [30], [-30], [28], [-50], [28], [-28], [26], [6], [-10], [20], [30], [-44], [28], [-16], [54], [-16], [24], [-4], [6], [-36], [-10], [-44], [-4], [-18], [-46], [-14], [-12], [-28], [-22], [-60], [-50], [50], [-20], [6], [48], [28], [22], [22], [34], [-20], [-38], [24], [36], [-66], [-44], [-2], [-46], [42], [12], [-18], [-2], [-40], [-2], [-18], [6], [22], [6], [-30], [-42], [44], [-54], [46], [10], [28], [-20], [-30], [-68], [-8], [-30], [12], [-16], [-10], [60], [-48], [-40], [44], [28], [-36], [-18], [-26], [-46], [-58], [68], [-12], [32], [-26], [14], [12], [2], [-22], [36], [28], [-28], [12], [-6], [62], [-12], [4], [-24], [-6], [76], [-34], [50], [-4], [10], [-8], [-68], [58], [-64], [62], [-64], [12], [-30], [22], [-12], [4], [-50], [-42], [26], [-24], [-24], [-50], [68], [20], [-6], [-8], [20], [38], [72], [-44], [56], [-16], [6], [-12], [4], [-58], [18], [42], [20], [36], [-82], [-44], [-4], [16], [6], [-14], [-72], [66], [52], [48], [70], [36], [-70], [-14], [-54], [4], [-2], [-46], [-16], [-78], [-38], [-48], [0], [22], [60], [32], [36], [2], [-12], [52], [24], [10], [62], [50], [-24], [50], [-2], [60], [-56], [-34], [44], [-12], [84], [90], [52], [-20], [-2], [42], [86], [46], [54], [0], [84], [-62], [76], [-44], [54], [60], [28], [-92], [-26], [-72], [-68], [-2], [-36], [-56], [-30], [-34], [-4], [40], [36], [-74], [18], [-14], [-38], [56], [-6], [26], [70], [52], [-66], [-4], [-98], [100], [0], [50], [-12], [-36], [-92], [-76], [-2], [36], [42], [40], [-90], [-84], [-74], [8], [-30], [18], [-74], [-2], [38], [-4], [76], [-62], [-54], [-44], [-40], [-50], [12], [10], [22], [-102], [28], [-26], [32], [22], [-44], [-34], [-14], [-40], [-10], [28], [94]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4896_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4896_2_a_l();".
// 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_4896_2_a_l(:prec:=1)
    chi := MakeCharacter_4896_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_4896_2_a_l();".
// 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_4896_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4896_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![-2, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;