
// Make newform 4560.2.a.z in Magma, downloaded from the LMFDB on 17 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_z();".
// 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_z();".
// 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], [-2], [0], [-2], [4], [-1], [-4], [-6], [-10], [-10], [0], [-4], [-4], [10], [6], [10], [4], [-8], [-2], [6], [-6], [4], [2], [-6], [0], [-8], [-8], [6], [-8], [-16], [16], [-20], [-6], [10], [8], [-16], [-8], [2], [-6], [8], [0], [14], [26], [-4], [8], [-4], [-4], [-26], [-28], [-16], [18], [4], [-6], [-12], [-22], [20], [0], [8], [12], [-18], [20], [-8], [-34], [14], [28], [-18], [14], [-2], [24], [-24], [18], [34], [24], [-24], [-30], [-4], [4], [-22], [-12], [-24], [40], [14], [30], [22], [0], [10], [-18], [6], [-30], [16], [-24], [16], [36], [0], [-6], [-12], [28], [30], [-20], [30], [0], [-16], [-4], [-30], [30], [-24], [-36], [-14], [-8], [20], [-8], [20], [-8], [24], [44], [0], [6], [-6], [32], [-14], [38], [36], [-20], [-18], [46], [-48], [26], [44], [-28], [24], [-10], [-20], [14], [-2], [-6], [4], [-50], [-18], [24], [-54], [30], [8], [-16], [32], [16], [2], [20], [-48], [18], [-22], [-44], [-16], [20], [20], [-16], [-14], [2], [-18], [26], [2], [-18], [-38], [10], [48], [-38], [40], [42], [-26], [-30], [-24], [56], [6], [50], [6], [-20], [22], [20], [-42], [-10], [14], [-10], [-14], [36], [18], [26], [28], [-46], [-24], [-14], [-6], [16], [54], [10], [46], [22], [-32], [48], [-12], [14], [10], [30], [-58], [48], [-62], [-44], [0], [-46], [-40], [-2], [-42], [-14], [-12], [-52], [6], [-58], [-60], [32], [-34], [60], [-58], [0], [-74], [20], [-26], [-22], [-24], [32], [72], [40], [4], [22], [-52], [64], [-4], [-2], [6], [-36], [8], [4], [-68], [10], [-44], [6], [-48], [2], [-54], [8], [-12], [-4], [42], [-16], [46], [18], [-28], [58], [68], [62], [-58], [-36], [0], [50], [10], [-12], [20], [-34], [10], [36], [6], [16], [36], [-10], [-20], [-34], [-38], [22], [-44], [-22], [56], [40], [16], [0], [-8], [16], [-32], [26], [66], [8], [-36], [46], [34], [-50], [-46], [-62], [-66], [-50], [-86], [-38], [52], [-26], [-30], [32], [0], [-52], [2], [72], [20], [0], [68], [64], [-34], [60], [-76], [76], [-10], [-44], [84], [62], [22], [80], [-58], [10], [-80], [-44], [-62], [-24], [-60], [-16], [6], [18], [2], [-56], [-6], [-44], [26], [68], [-14], [-62], [-34], [-22], [-92], [10], [26], [-90], [46], [-22], [-52], [40], [10], [-60], [-82], [-46], [40], [-8], [74], [-48], [-24], [-48], [-80], [-16], [-66], [-48], [10], [-52], [-38], [-14], [56], [70], [-48], [-4], [88], [98], [60], [-32], [-74], [16], [-38], [-6], [-86], [-10], [36], [-78], [84], [8], [48], [36], [-32], [-8], [0], [-74], [-10], [60], [32], [-60], [30], [10], [-36], [-32], [-54], [32], [76], [-36], [-6], [-54], [20], [-16], [-32], [-4], [32], [34], [78], [-100], [-60], [-26], [82], [-76], [52], [88], [-72], [-50], [74], [-28], [-38], [-94], [-42], [24], [82], [-20], [-72]];
    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_z();".
// 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_z(: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_z();".
// 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_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4560_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![0, 1]>,<13,R![2, 1]>,<17,R![-4, 1]>],Snew);
    return Vf;
end function;