
// Make newform 4410.2.a.s 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_s();".
// 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_s();".
// 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], [8], [6], [4], [6], [-4], [-10], [4], [4], [4], [-10], [-14], [10], [-4], [-12], [-4], [4], [-2], [8], [0], [-2], [-4], [12], [10], [2], [-12], [18], [2], [10], [10], [0], [-2], [-4], [12], [18], [-4], [-26], [-12], [-18], [-18], [-4], [20], [8], [-18], [14], [-22], [8], [-4], [22], [12], [0], [26], [-16], [2], [10], [26], [6], [2], [16], [8], [-18], [-28], [-14], [4], [-10], [-12], [8], [-8], [-6], [-4], [20], [-18], [-26], [14], [20], [6], [-34], [0], [40], [32], [-4], [-18], [6], [-6], [16], [-10], [4], [-44], [12], [28], [-24], [-6], [-12], [34], [-34], [28], [6], [18], [38], [-36], [20], [2], [-20], [28], [-40], [16], [-2], [6], [6], [4], [-18], [-18], [28], [-14], [4], [2], [22], [-46], [12], [46], [38], [42], [36], [-20], [-30], [12], [12], [40], [-2], [-12], [-28], [-18], [-6], [2], [10], [-2], [18], [-40], [-44], [-14], [-36], [-50], [-24], [-38], [24], [-42], [-40], [-20], [4], [4], [-40], [-4], [-12], [-12], [-10], [12], [-2], [-52], [-62], [-54], [20], [-36], [42], [-2], [-14], [36], [-26], [36], [-42], [-4], [48], [20], [-26], [-8], [50], [28], [-18], [22], [12], [-44], [-30], [26], [26], [6], [-36], [-36], [36], [36], [42], [-52], [12], [10], [46], [-68], [-48], [66], [-44], [-14], [8], [42], [6], [20], [-60], [2], [2], [-14], [-18], [4], [38], [32], [-28], [32], [20], [48], [34], [18], [16], [30], [-20], [22], [-38], [-36], [-48], [-8], [-36], [30], [-38], [20], [-26], [34], [-32], [-4], [30], [-36], [-40], [36], [14], [-72], [14], [-44], [20], [48], [-46], [-20], [48], [-6], [24], [56], [-64], [-26], [-76], [-54], [78], [46], [-24], [60], [-28], [-62], [-54], [-8], [-30], [-70], [0], [36], [66], [-50], [52], [28], [56], [-28], [-72], [4], [46], [-2], [34], [-24], [12], [52], [58], [-38], [-16], [-38], [14], [68], [8], [-46], [30], [38], [34], [-38], [-6], [48], [-6], [22], [70], [32], [58], [-52], [-36], [-4], [-10], [-60], [-54], [-48], [-22], [40], [-10], [82], [-60], [24], [-8], [-18], [-68], [4], [-46], [50], [62], [78], [56], [26], [-4], [-4], [74], [44], [2], [-82], [34], [4], [-30], [20], [54], [-6], [-84], [-16], [-52], [82], [-42], [58], [-36], [-38], [12], [-26], [-28], [-24], [70], [-30], [-62], [-30], [-52], [-26], [36], [20], [66], [-42], [68], [42], [-48], [12], [-44], [-38], [26], [42], [52], [22], [-84], [-54], [32], [-6], [-64], [-54], [34], [40], [-4], [-68], [-8], [-22], [74], [-16], [-2], [82], [-48], [20], [-70], [92], [-12], [10], [-26], [4], [62], [-48], [-18], [44], [28], [-100], [-50], [-54], [-42], [80], [32], [10], [36], [-54], [-62], [-50], [54], [4], [-62], [-36], [28], [-58], [58], [-20], [60], [-88], [-40], [6], [98], [16], [-42], [24], [-10], [28], [-38], [-70], [36]];
    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_s();".
// 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_s(: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_s();".
// 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_s( : 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![-8, 1]>,<19,R![-6, 1]>,<29,R![-6, 1]>,<31,R![4, 1]>],Snew);
    return Vf;
end function;