
// Make newform 1441.2.a.a in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1441_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_1441_2_a_a();".
// 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_1441_2_a_a();".
// 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_1441_a();"
function MakeCharacter_1441_a()
    N := 1441;
    order := 1;
    char_gens := [1311, 133];
    v := [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_1441_a_Hecke(Kf)
    return MakeCharacter_1441_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], [-1], [0], [-2], [4], [1], [-6], [1], [1], [0], [0], [-8], [-6], [-11], [0], [3], [7], [4], [-8], [-14], [7], [-9], [-16], [-8], [-12], [-16], [1], [-18], [-1], [-1], [0], [18], [0], [23], [-4], [-18], [-2], [17], [17], [25], [16], [0], [-20], [-4], [5], [12], [17], [14], [20], [20], [-25], [14], [6], [-14], [10], [0], [14], [-14], [-16], [-2], [0], [17], [-3], [-11], [4], [-6], [0], [-29], [-20], [19], [10], [31], [-21], [-21], [22], [-2], [-28], [-36], [-6], [-2], [-21], [22], [-9], [-7], [14], [12], [31], [1], [-6], [-29], [0], [28], [-44], [21], [15], [10], [-20], [26], [36], [12], [-46], [14], [-43], [12], [4], [0], [44], [-10], [42], [-34], [-19], [-13], [-45], [-21], [17], [-17], [10], [43], [-8], [24], [-4], [29], [8], [21], [-25], [25], [12], [-28], [42], [-5], [-2], [-22], [34], [22], [32], [-3], [8], [-36], [2], [-7], [-36], [9], [-17], [-26], [18], [-17], [-40], [18], [-5], [-44], [48], [56], [-44], [-32], [-22], [-4], [8], [-1], [26], [18], [-43], [9], [9], [8], [32], [-12], [9], [50], [18], [-24], [62], [-29], [-15], [-34], [-9], [8], [10], [40], [-26], [37], [-42], [-3], [-22], [-14], [28], [2], [-4], [49], [64], [1], [-6], [16], [-27], [-12], [-29], [24], [6], [-2], [-48], [22], [52], [51], [7], [-19], [4], [30], [4], [-14], [-67], [-3], [-12], [-32], [43], [36], [-38], [-4], [43], [-2], [-46], [-3], [-43], [20], [-14], [-14], [32], [-22], [-14], [21], [-68], [64], [-42], [-21], [30], [-31], [18], [15], [15], [9], [30], [-64], [5], [54], [-66], [16], [44], [62], [-16], [-32], [48], [33], [-10], [46], [18], [-50], [16], [-15], [38], [-50], [-52], [50], [-56], [-2], [-4], [-11], [27], [-78], [66], [41], [-53], [-18], [60], [-4], [47], [47], [0], [56], [38], [24], [15], [14], [-26], [14], [-45], [46], [-38], [-11], [36], [15], [-32], [-58], [2], [-28], [14], [-43], [9], [-4], [-30], [54], [14], [-70], [-36], [-5], [-29], [-61], [65], [-32], [-6], [44], [-14], [-60], [-13], [-68], [-10], [-9], [-18], [34], [-58], [-10], [9], [-20], [11], [-16], [7], [9], [-33], [-84], [64], [-26], [-86], [-28], [24], [28], [-57], [7], [64], [5], [90], [75], [41], [-4], [-13], [-51], [54], [11], [57], [6], [13], [28], [29], [-10], [-88], [-94], [-26], [-11], [-6], [-60], [9], [-92], [52], [-37], [12], [-13], [-13], [-78], [-62], [-6], [-33], [44], [74], [80], [-72], [-41], [20], [-50], [88], [64], [72], [70], [91], [38], [24], [48], [-1], [-72], [-25], [44], [33], [-81], [13], [71], [61], [-65], [-102], [70], [-6], [-68], [18], [50], [-81], [-4], [18], [82], [-84], [14], [30], [4], [-63], [28], [20], [-76], [-8], [66], [39], [-36], [-57], [-18], [-24], [-57], [-22], [72], [-18], [-74], [15], [-21], [60], [-9], [-36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1441_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1441_2_a_a();".
// 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_1441_2_a_a(:prec:=1)
    chi := MakeCharacter_1441_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_1441_2_a_a();".
// 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_1441_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1441_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>],Snew);
    return Vf;
end function;