
// Make newform 4560.2.a.q in Magma, downloaded from the LMFDB on 19 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_q();".
// 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_q();".
// 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], [-6], [-6], [1], [-4], [6], [8], [2], [10], [8], [-12], [2], [4], [-2], [12], [16], [-14], [-8], [0], [-6], [14], [6], [-8], [12], [6], [-10], [16], [-12], [2], [-12], [-10], [-8], [22], [16], [0], [-6], [12], [-2], [8], [-2], [6], [16], [20], [16], [4], [6], [10], [-16], [-22], [-4], [-18], [-20], [14], [8], [-2], [-22], [16], [2], [20], [8], [-6], [-14], [-4], [14], [-32], [-2], [26], [16], [-4], [10], [-4], [-24], [-26], [22], [-22], [-6], [-20], [-26], [32], [14], [-32], [-8], [10], [10], [-18], [-20], [8], [-24], [16], [12], [28], [12], [6], [-14], [-20], [-34], [36], [-26], [20], [34], [-4], [18], [-32], [34], [0], [2], [-24], [46], [-14], [20], [32], [-14], [40], [-12], [6], [12], [-10], [22], [-30], [12], [20], [-18], [6], [-24], [-28], [-50], [-44], [-16], [32], [-10], [-6], [18], [10], [52], [-38], [42], [-28], [6], [-4], [-12], [-26], [0], [-10], [6], [44], [-24], [18], [-14], [16], [8], [12], [-16], [0], [-14], [-30], [46], [24], [-42], [-52], [-28], [-2], [0], [-16], [46], [-38], [14], [12], [22], [-8], [-54], [-32], [10], [20], [-26], [0], [-50], [-12], [-52], [-30], [22], [-60], [6], [-62], [52], [26], [-24], [38], [-40], [12], [6], [0], [-34], [18], [62], [66], [-12], [22], [-48], [-14], [10], [60], [-10], [-16], [-20], [-54], [20], [-46], [-42], [36], [-12], [-24], [-14], [-36], [-70], [-24], [-34], [54], [-24], [-30], [-20], [20], [-10], [-26], [-32], [8], [-4], [-34], [52], [-48], [42], [-32], [20], [-14], [6], [-36], [8], [36], [-28], [-4], [-66], [-42], [-8], [-20], [-60], [-4], [44], [-2], [66], [52], [26], [-66], [-4], [-58], [-28], [50], [-62], [-48], [-36], [38], [34], [50], [-52], [-26], [-22], [36], [54], [-58], [-52], [34], [16], [-50], [-20], [24], [-66], [18], [-20], [72], [-32], [68], [-18], [56], [16], [34], [26], [0], [-54], [-34], [48], [30], [12], [-14], [30], [8], [70], [-52], [8], [-14], [-6], [64], [-84], [-12], [46], [-28], [-34], [-24], [-58], [12], [22], [42], [-60], [4], [-30], [36], [0], [2], [50], [60], [82], [6], [8], [-78], [42], [28], [-20], [-64], [-34], [-82], [-22], [0], [-88], [-52], [-24], [38], [14], [58], [-76], [-86], [-62], [86], [16], [-6], [-4], [38], [-12], [56], [78], [60], [94], [-82], [-56], [-74], [-26], [0], [-4], [-22], [-32], [-82], [-54], [56], [12], [-16], [-26], [38], [-16], [-30], [92], [-44], [12], [6], [-40], [-58], [-60], [-72], [-46], [34], [-58], [30], [42], [100], [50], [44], [-24], [8], [-82], [72], [8], [-62], [82], [4], [-32], [72], [78], [40], [90], [-28], [22], [70], [-46], [8], [-74], [6], [72], [70], [90], [-28], [60], [86], [-10], [68], [20], [82], [78], [24], [88], [58], [96], [-18], [-70], [20], [-84], [-10], [42], [52], [-6], [36], [-96]];
    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_q();".
// 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_q(: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_q();".
// 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_q( : 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![6, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;