
// Make newform 4410.2.a.bj in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4410_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_4410_2_a_bj();".
// 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_4410_2_a_bj();".
// 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_4410_a();"
function MakeCharacter_4410_a()
    N := 4410;
    order := 1;
    char_gens := [3431, 2647, 1081];
    v := [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_4410_a_Hecke(Kf)
    return MakeCharacter_4410_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 := [[1], [0], [1], [0], [1], [1], [0], [-3], [-7], [8], [-2], [11], [11], [8], [5], [11], [-4], [0], [0], [6], [-6], [-8], [-8], [10], [-16], [0], [-16], [10], [6], [-6], [-17], [5], [18], [20], [0], [6], [-7], [16], [3], [15], [19], [-24], [6], [22], [1], [-24], [5], [12], [-8], [14], [-18], [18], [7], [-13], [20], [-24], [-20], [32], [-22], [1], [-14], [-27], [20], [-12], [-12], [18], [-13], [-12], [-14], [12], [-24], [-4], [25], [6], [-19], [-35], [-10], [14], [5], [2], [5], [22], [0], [-32], [32], [-16], [-11], [-18], [12], [-13], [0], [-22], [-16], [-20], [-32], [-36], [34], [33], [2], [-10], [16], [33], [38], [9], [-32], [-14], [18], [-16], [-2], [-26], [-37], [-41], [-28], [29], [-26], [3], [-10], [17], [-7], [-36], [-24], [28], [13], [-4], [-12], [18], [44], [-2], [11], [11], [-19], [-49], [26], [2], [-27], [-5], [-33], [22], [18], [39], [9], [-42], [-24], [10], [-8], [40], [-41], [14], [12], [53], [17], [53], [-42], [-44], [-10], [2], [-36], [21], [12], [-54], [-22], [20], [-20], [45], [-54], [-49], [22], [14], [11], [-45], [36], [-2], [10], [-44], [20], [-14], [-7], [-10], [11], [22], [57], [-23], [-9], [10], [9], [24], [-23], [24], [25], [42], [-30], [6], [35], [-22], [-20], [16], [29], [46], [-14], [-4], [-32], [-10], [62], [13], [32], [-27], [-20], [-44], [-15], [-1], [16], [-34], [32], [-18], [40], [7], [-17], [69], [3], [-31], [44], [38], [-25], [-49], [-48], [46], [-38], [16], [17], [4], [-19], [37], [10], [-55], [-58], [56], [-22], [39], [43], [-26], [-24], [-49], [-8], [-46], [18], [-18], [55], [25], [60], [-3], [-14], [27], [-27], [-17], [-66], [55], [38], [-2], [-5], [8], [15], [18], [-34], [-54], [-16], [-41], [72], [-25], [14], [65], [-56], [-66], [14], [-48], [-66], [-39], [-72], [20], [-31], [27], [-1], [-76], [33], [64], [-22], [-60], [-16], [69], [76], [-61], [68], [-6], [36], [-35], [7], [0], [-26], [6], [15], [-42], [28], [-3], [-46], [42], [-44], [38], [44], [-10], [-6], [-31], [45], [10], [-18], [-2], [-57], [-91], [89], [12], [-80], [-74], [60], [-22], [6], [37], [24], [59], [9], [-74], [-48], [-54], [54], [38], [30], [-12], [3], [-74], [-74], [-36], [-11], [-17], [14], [-30], [-90], [70], [58], [-12], [76], [-58], [12], [-47], [7], [14], [-42], [17], [66], [54], [36], [-24], [-24], [-20], [29], [22], [-48], [25], [-42], [22], [-37], [45], [-93], [57], [63], [-36], [90], [-10], [-34], [15], [44], [-48], [14], [-34], [-6], [12], [33], [12], [-89], [-9], [20], [63], [32], [56], [22], [-54], [-36], [-20], [46], [94], [-8], [-69], [81], [42], [40], [-6], [15], [84], [-90], [36], [103], [66], [-84], [81], [48], [-14], [-84], [-49], [36], [-10], [80], [-9], [92], [-1], [-106], [77], [24], [72], [64], [77], [-32], [-69], [19], [-90]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4410_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4410_2_a_bj();".
// 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_4410_2_a_bj(:prec:=1)
    chi := MakeCharacter_4410_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_4410_2_a_bj();".
// 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_4410_2_a_bj( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4410_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-1, 1]>,<13,R![-1, 1]>,<17,R![0, 1]>,<19,R![3, 1]>,<29,R![-8, 1]>,<31,R![2, 1]>],Snew);
    return Vf;
end function;