
// Make newform 2240.2.a.y in Magma, downloaded from the LMFDB on 25 April 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_y();".
// 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_y();".
// 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], [3], [-1], [-1], [3], [-1], [-1], [-4], [4], [9], [6], [8], [6], [8], [-7], [8], [-4], [-10], [8], [-12], [-14], [-5], [16], [-14], [-17], [0], [1], [-10], [-5], [16], [20], [-8], [-14], [10], [6], [-5], [-14], [2], [-7], [-19], [24], [0], [9], [18], [-18], [16], [-19], [17], [3], [26], [-24], [-21], [4], [6], [6], [-18], [-6], [-2], [2], [-17], [-11], [9], [33], [-4], [21], [-10], [-20], [-14], [4], [0], [3], [-8], [-3], [-14], [8], [-20], [15], [-7], [-27], [-2], [-24], [-1], [5], [18], [26], [40], [-17], [22], [-40], [14], [5], [-40], [18], [5], [-37], [-9], [-18], [16], [-20], [1], [-42], [0], [36], [6], [28], [33], [28], [1], [-35], [-14], [-41], [-34], [18], [36], [37], [6], [19], [12], [-24], [-9], [4], [10], [19], [-30], [8], [-47], [1], [36], [-16], [35], [19], [-4], [47], [20], [-8], [-30], [-45], [35], [29], [-47], [-22], [15], [-2], [-18], [26], [-28], [6], [-22], [-36], [34], [16], [24], [-10], [-8], [4], [-8], [39], [36], [3], [-34], [22], [-6], [-50], [-18], [-36], [11], [-24], [25], [9], [30], [12], [-34], [61], [26], [56], [12], [15], [15], [-16], [-54], [30], [26], [48], [38], [2], [-8], [-16], [-45], [-3], [-42], [46], [36], [53], [-10], [0], [21], [35], [-6], [47], [-23], [46], [-10], [-66], [-48], [40], [-15], [-14], [14], [-6], [10], [34], [-10], [2], [-33], [18], [-62], [-28], [-46], [48], [-34], [-58], [-2], [-47], [30], [-49], [74], [-13], [-55], [33], [-20], [26], [42], [-48], [-39], [19], [13], [50], [44], [-71], [-12], [-16], [-12], [49], [-70], [21], [-8], [-60], [-26], [-7], [-12], [-44], [-30], [-60], [-4], [50], [-41], [18], [68], [-57], [-27], [56], [30], [-4], [-61], [42], [8], [-45], [14], [0], [40], [38], [30], [31], [-16], [-50], [25], [-52], [42], [-47], [22], [45], [-53], [40], [22], [31], [69], [-44], [16], [56], [24], [15], [49], [26], [-34], [-4], [-46], [6], [83], [-72], [7], [-49], [-66], [-11], [-24], [41], [32], [-28], [56], [-80], [40], [24], [21], [-54], [10], [-16], [70], [-6], [77], [83], [5], [-38], [-42], [82], [-26], [-58], [52], [-63], [-92], [62], [-6], [74], [-2], [18], [-31], [52], [-57], [43], [-6], [0], [-83], [-6], [12], [8], [37], [76], [-52], [-10], [22], [84], [-3], [4], [-54], [23], [-43], [-13], [86], [-66], [76], [20], [-2], [10], [28], [78], [71], [52], [68], [-67], [20], [-47], [75], [90], [-20], [29], [58], [-18], [36], [21], [-11], [6], [5], [-22], [-34], [-96], [-34], [86], [45], [89], [23], [40], [39], [29], [1], [-55], [-44], [-55], [44], [-14], [90], [31], [-42], [-102], [-64], [-20], [-57], [-20], [-58], [10], [63], [-68], [-20], [-53], [6], [-100], [-52], [40], [48], [-7], [-40], [-70], [17], [-81], [-90], [-54], [-72], [-7], [65], [24], [-6], [94], [42]];
    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_y();".
// 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_y(: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_y();".
// 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_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2240_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-3, 1]>,<11,R![-3, 1]>,<13,R![1, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;