
// Make newform 4560.2.a.l in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4560_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_4560_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_4560_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_4560_a();"
function MakeCharacter_4560_a()
    N := 4560;
    order := 1;
    char_gens := [1711, 1141, 3041, 2737, 1921];
    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_4560_a_Hecke(Kf)
    return MakeCharacter_4560_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], [1], [-4], [4], [2], [-6], [-1], [-4], [6], [-8], [10], [2], [0], [4], [2], [12], [-2], [-4], [0], [-14], [-8], [0], [2], [-18], [6], [16], [-4], [14], [14], [8], [-12], [2], [12], [22], [8], [22], [-8], [-8], [-22], [20], [6], [8], [14], [6], [24], [12], [8], [20], [-10], [-6], [0], [10], [12], [22], [28], [-18], [-16], [30], [-30], [24], [-30], [4], [8], [-6], [18], [36], [-2], [-16], [14], [-6], [0], [-12], [-14], [4], [0], [6], [22], [-30], [-6], [12], [-2], [0], [30], [16], [-24], [-14], [42], [14], [4], [-24], [-24], [8], [-36], [20], [-20], [-26], [10], [-4], [30], [-28], [6], [36], [10], [-12], [-46], [16], [-14], [32], [-30], [0], [-18], [18], [-4], [40], [-22], [16], [20], [6], [36], [-18], [-26], [-14], [12], [-20], [-18], [-10], [-24], [28], [-34], [12], [-24], [-32], [-26], [10], [-46], [10], [-44], [42], [10], [12], [6], [-28], [4], [14], [-16], [38], [-18], [20], [16], [-6], [-30], [8], [-48], [-52], [-48], [56], [50], [-30], [30], [24], [-50], [4], [-52], [6], [56], [16], [46], [-38], [-50], [-12], [-34], [24], [42], [0], [-54], [28], [6], [-56], [46], [60], [-44], [58], [-34], [-12], [-26], [26], [20], [-6], [24], [-10], [-56], [-12], [-10], [0], [54], [-46], [46], [50], [-28], [6], [-16], [58], [26], [-36], [-42], [-24], [-20], [58], [44], [-46], [54], [-20], [20], [-56], [2], [-28], [18], [32], [-66], [-34], [40], [-70], [-60], [20], [54], [-18], [-48], [0], [60], [46], [60], [-16], [-14], [40], [52], [-14], [-26], [-20], [-24], [-60], [44], [20], [38], [30], [72], [-60], [-52], [4], [60], [14], [2], [-60], [74], [30], [12], [-42], [-12], [18], [2], [-56], [28], [38], [58], [2], [-28], [54], [42], [68], [54], [14], [44], [34], [-40], [-2], [36], [-72], [38], [-46], [-4], [-16], [40], [4], [-10], [16], [-48], [-30], [-22], [8], [-30], [-2], [-16], [-26], [-28], [10], [-66], [8], [38], [36], [-80], [-14], [10], [72], [12], [-4], [-82], [-12], [-58], [-56], [-74], [-84], [86], [34], [4], [-60], [-46], [20], [-32], [34], [34], [-76], [50], [70], [64], [50], [42], [4], [76], [24], [30], [-18], [-6], [72], [-88], [20], [72], [-2], [38], [58], [-68], [18], [-62], [-10], [-32], [90], [-60], [70], [52], [-72], [14], [-60], [-2], [14], [32], [-50], [-66], [48], [92], [42], [8], [46], [58], [-16], [-60], [40], [-26], [-90], [8], [50], [12], [28], [-4], [-90], [-48], [70], [60], [72], [-14], [66], [6], [14], [-6], [-20], [66], [-44], [80], [-80], [46], [64], [32], [50], [-94], [20], [-8], [-40], [46], [56], [-30], [-68], [-74], [14], [2], [0], [14], [38], [-80], [38], [-30], [4], [-84], [-106], [-10], [20], [28], [50], [-18], [24], [0], [-38], [8], [30], [66], [84], [84], [6], [10], [-76], [26], [60], [-32]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4560_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4560_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_4560_2_a_l(:prec:=1)
    chi := MakeCharacter_4560_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_4560_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_4560_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4560_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![-4, 1]>,<13,R![-2, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;