
// Make newform 4560.2.a.o in Magma, downloaded from the LMFDB on 18 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_o();".
// 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_o();".
// 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], [-2], [4], [-2], [1], [-4], [0], [8], [8], [-8], [6], [12], [-6], [0], [2], [-8], [8], [14], [0], [-4], [0], [-12], [2], [-4], [12], [10], [-6], [-8], [18], [-22], [0], [10], [8], [18], [-14], [-8], [14], [0], [2], [18], [24], [18], [20], [-12], [-4], [12], [10], [-6], [-10], [2], [-2], [18], [-24], [0], [-12], [18], [12], [-14], [14], [32], [18], [-6], [18], [-12], [28], [12], [-10], [-6], [10], [-18], [-36], [20], [-24], [30], [-22], [12], [30], [30], [22], [-12], [-16], [40], [-24], [0], [18], [2], [6], [-8], [10], [-28], [-22], [40], [16], [20], [-28], [-4], [-18], [32], [-42], [-4], [0], [-32], [-22], [-8], [-46], [-20], [-18], [12], [-6], [-42], [40], [-32], [32], [-14], [32], [-46], [-20], [42], [4], [18], [36], [8], [2], [10], [-50], [-18], [-26], [-20], [-24], [8], [-2], [-18], [50], [34], [-28], [-2], [-30], [-12], [-38], [-34], [-28], [10], [40], [14], [-42], [-20], [16], [-12], [-18], [26], [-48], [-28], [-32], [0], [-30], [-2], [-8], [52], [34], [-58], [-12], [38], [-24], [8], [-62], [50], [-26], [60], [22], [-22], [-6], [40], [-30], [8], [42], [-44], [-50], [42], [28], [-36], [-22], [-44], [-30], [-32], [-4], [30], [38], [-16], [-64], [-12], [-8], [-8], [34], [62], [-26], [-42], [-44], [0], [-32], [-12], [50], [50], [18], [0], [-44], [-50], [-52], [-22], [-58], [-34], [12], [20], [22], [62], [52], [72], [34], [-58], [0], [0], [-54], [12], [-10], [-46], [-60], [-48], [-22], [-26], [-60], [-32], [32], [46], [12], [-50], [54], [-70], [-12], [-4], [68], [-14], [-70], [14], [30], [42], [-72], [-20], [16], [38], [2], [-8], [-50], [34], [30], [42], [-28], [18], [78], [-44], [52], [70], [44], [-42], [60], [40], [42], [76], [-6], [-58], [32], [14], [20], [-52], [6], [12], [50], [22], [-62], [56], [8], [52], [62], [-58], [18], [-6], [58], [-80], [-40], [42], [52], [-26], [-52], [-76], [-10], [48], [34], [0], [-58], [-26], [18], [0], [-84], [-72], [68], [52], [-50], [-30], [14], [76], [70], [-68], [16], [-48], [50], [-70], [88], [-86], [-10], [-52], [-42], [-88], [16], [-26], [-78], [-20], [-4], [72], [-6], [-18], [-62], [60], [36], [-32], [52], [-90], [-26], [-18], [2], [-36], [18], [-60], [8], [-6], [60], [2], [-8], [-12], [-42], [68], [-32], [-18], [-4], [10], [-26], [30], [-42], [78], [56], [18], [-58], [-48], [60], [2], [64], [38], [76], [-58], [-22], [-20], [-24], [60], [48], [-22], [-80], [-42], [74], [90], [28], [52], [-66], [22], [18], [20], [-24], [68], [58], [-14], [-88], [-50], [34], [-50], [2], [-12], [64], [-40], [-20], [28], [62], [-50], [-26], [-48], [18], [-40], [8], [-22], [72], [-64], [-10], [4], [18], [76], [48], [-2], [-38], [-20], [-8], [18], [16], [60], [-12], [92], [-40], [-96], [78], [36], [-30], [-72], [-10]];
    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_o();".
// 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_o(: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_o();".
// 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_o( : 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![2, 1]>,<13,R![-4, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;