
// Make newform 4410.2.a.t in Magma, downloaded from the LMFDB on 23 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_t();".
// 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_t();".
// 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], [4], [2], [2], [-4], [8], [2], [0], [6], [-6], [-4], [0], [10], [12], [-14], [-12], [8], [-10], [16], [-12], [10], [-2], [6], [8], [-12], [14], [-18], [-16], [20], [-10], [4], [10], [-8], [18], [20], [-8], [-18], [-4], [-6], [16], [2], [-6], [24], [-12], [16], [4], [10], [22], [0], [14], [12], [-14], [-8], [-18], [-16], [22], [6], [-28], [6], [-20], [-8], [6], [2], [12], [18], [4], [-14], [-14], [-24], [32], [-10], [28], [16], [26], [-30], [14], [38], [20], [-26], [0], [14], [8], [36], [-2], [10], [14], [0], [-12], [32], [-8], [36], [-12], [40], [30], [-6], [20], [-34], [-12], [18], [20], [-26], [28], [-34], [12], [-30], [24], [38], [-16], [6], [22], [4], [-40], [-34], [28], [24], [-14], [-36], [-38], [-30], [-26], [-12], [-36], [-30], [-26], [-16], [-8], [-46], [-28], [24], [16], [22], [10], [-34], [-26], [-20], [30], [22], [-28], [-22], [-24], [-28], [18], [8], [26], [-22], [20], [-32], [14], [2], [-28], [8], [28], [-32], [-8], [-14], [-10], [-18], [-52], [-26], [-8], [28], [-18], [-24], [-32], [-22], [-14], [54], [20], [-46], [8], [10], [16], [42], [28], [-6], [-24], [-62], [48], [-12], [-26], [-38], [-48], [-42], [-34], [60], [-22], [0], [-2], [52], [-12], [30], [-4], [-6], [50], [-34], [-14], [24], [18], [-16], [-38], [62], [-36], [-34], [32], [-36], [-10], [4], [-46], [-42], [-56], [28], [8], [-10], [-32], [66], [-56], [-30], [-58], [-24], [-66], [64], [20], [54], [-54], [-48], [8], [4], [-50], [-4], [64], [-10], [-12], [32], [-18], [42], [20], [8], [12], [20], [-24], [14], [66], [24], [-48], [-44], [44], [16], [-2], [-46], [-8], [-42], [46], [-4], [-74], [-28], [-58], [70], [16], [-36], [10], [-14], [18], [44], [50], [-54], [-52], [-6], [2], [4], [-26], [32], [14], [-40], [68], [62], [10], [4], [16], [8], [-24], [42], [52], [32], [-46], [-22], [40], [-46], [50], [52], [6], [-68], [78], [30], [0], [22], [-20], [28], [-10], [50], [-48], [76], [60], [34], [-12], [-30], [24], [6], [0], [74], [-66], [-76], [-8], [-74], [-60], [48], [-2], [78], [-4], [-38], [62], [-16], [-74], [78], [4], [-28], [0], [26], [46], [2], [64], [36], [-52], [-4], [-2], [18], [-10], [-32], [54], [-58], [-26], [-56], [-30], [-52], [-54], [-36], [-48], [86], [12], [10], [18], [64], [-42], [-86], [0], [60], [-18], [-56], [-58], [-6], [-48], [-12], [28], [42], [14], [-40], [90], [-84], [68], [48], [42], [72], [-2], [-60], [48], [-66], [34], [-26], [62], [22], [8], [-2], [44], [24], [48], [-70], [-52], [16], [-62], [6], [-60], [-20], [-40], [26], [-64], [90], [-20], [42], [-14], [30], [32], [-22], [70], [24], [46], [14], [76], [-76], [14], [42], [36], [68], [-86], [-50], [80], [8], [-62], [-24], [-62], [-86], [-48], [76], [-10], [-82], [44], [6], [20], [56]];
    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_t();".
// 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_t(: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_t();".
// 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_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4410_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-4, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>,<19,R![4, 1]>,<29,R![-2, 1]>,<31,R![0, 1]>],Snew);
    return Vf;
end function;