
// Make newform 2240.2.a.p in Magma, downloaded from the LMFDB on 09 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2240_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_2240_2_a_p();".
// 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_2240_2_a_p();".
// 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_2240_a();"
function MakeCharacter_2240_a()
    N := 2240;
    order := 1;
    char_gens := [1471, 1541, 897, 1921];
    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_2240_a_Hecke(Kf)
    return MakeCharacter_2240_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], [1], [1], [-4], [-2], [6], [4], [-8], [-6], [0], [-10], [10], [-12], [0], [6], [-12], [-2], [-4], [-12], [-10], [-4], [0], [18], [-10], [-10], [8], [12], [2], [2], [-8], [-20], [10], [4], [2], [-20], [14], [20], [0], [14], [-12], [-18], [-20], [-6], [-18], [-8], [4], [-16], [24], [14], [-14], [12], [2], [-4], [22], [0], [-10], [16], [22], [-6], [-8], [6], [16], [32], [-10], [-2], [-4], [-6], [12], [-10], [-26], [-36], [24], [14], [4], [32], [10], [-18], [34], [-14], [28], [-22], [-4], [-2], [16], [4], [18], [-14], [-10], [8], [32], [24], [16], [4], [20], [24], [6], [34], [-24], [10], [-12], [6], [8], [10], [-28], [-18], [-16], [-2], [-12], [-46], [-8], [6], [-30], [44], [-28], [18], [8], [-32], [-10], [-20], [22], [2], [22], [-4], [-20], [-6], [2], [40], [32], [-34], [-20], [-16], [4], [22], [18], [-30], [6], [32], [-2], [-38], [-28], [-38], [40], [-12], [-50], [-32], [-42], [22], [-12], [0], [38], [-54], [4], [8], [-28], [-52], [-12], [10], [-18], [-2], [-52], [2], [-8], [4], [2], [48], [-60], [38], [34], [54], [12], [-10], [36], [-14], [-40], [-30], [44], [58], [16], [30], [-48], [4], [-42], [-42], [-24], [30], [38], [-24], [-6], [56], [38], [-44], [-52], [22], [12], [6], [-14], [6], [54], [-32], [-30], [-40], [22], [-6], [-36], [-2], [-64], [60], [-6], [-20], [-46], [54], [56], [56], [-64], [-22], [8], [-38], [24], [-10], [-30], [16], [50], [24], [-72], [26], [14], [-12], [8], [-44], [-74], [-12], [-60], [42], [8], [-48], [-38], [6], [-28], [0], [60], [-12], [-32], [-62], [-18], [-72], [24], [-12], [44], [16], [-50], [-30], [-48], [-70], [-18], [-52], [-54], [80], [70], [6], [-56], [12], [-26], [30], [-66], [-20], [34], [-6], [-36], [-34], [-10], [-60], [46], [20], [70], [56], [4], [-46], [-22], [52], [-16], [-44], [24], [-58], [32], [52], [-38], [-66], [16], [-30], [2], [8], [-62], [60], [62], [-10], [-8], [-26], [52], [32], [30], [6], [76], [-12], [44], [2], [-20], [-82], [44], [-34], [24], [66], [50], [4], [64], [18], [-84], [-60], [-26], [34], [28], [-86], [-50], [-56], [-78], [-62], [-36], [-8], [-64], [6], [50], [22], [0], [24], [60], [-48], [34], [-50], [18], [8], [30], [-22], [30], [44], [46], [-68], [-34], [-44], [72], [54], [4], [18], [-78], [-64], [10], [-66], [-16], [-52], [-38], [56], [62], [-6], [-72], [60], [56], [-94], [78], [32], [74], [-4], [84], [0], [-62], [80], [-66], [-36], [44], [62], [42], [-10], [70], [-78], [16], [18], [-12], [24], [76], [54], [-20], [56], [-62], [70], [-76], [-40], [-20], [26], [56], [-6], [52], [-30], [-66], [26], [8], [-18], [78], [32], [-2], [98], [-80], [-60], [-74], [-74], [-12], [-92], [-78], [6], [-20], [40], [94], [-40], [-78], [-90], [-64], [-52], [54], [-18], [36], [-6], [44], [-32]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2240_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2240_2_a_p();".
// 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_2240_2_a_p(:prec:=1)
    chi := MakeCharacter_2240_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_2240_2_a_p();".
// 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_2240_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2240_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<11,R![4, 1]>,<13,R![2, 1]>,<19,R![-4, 1]>],Snew);
    return Vf;
end function;