
// Make newform 4410.2.a.bq 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_bq();".
// 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_bq();".
// 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
        poly := [-7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
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], [0, 0], [1, 0], [0, 0], [-2, 1], [-2, -1], [2, 2], [1, 2], [-3, 0], [0, 0], [2, 0], [2, 3], [-4, -1], [6, -2], [-1, -4], [-5, -2], [4, -2], [6, -2], [10, 2], [8, 2], [2, 0], [6, -2], [-10, 2], [4, 4], [-4, 0], [12, 0], [12, -2], [0, -6], [10, -4], [2, -4], [12, 1], [2, -1], [0, 6], [0, 4], [-6, 6], [14, 0], [-18, 1], [8, 0], [-9, 0], [-13, 2], [18, -3], [-6, -2], [6, 0], [8, 6], [-1, 2], [2, -6], [3, -2], [0, -2], [2, 2], [12, -2], [12, 6], [-2, 4], [-15, 4], [-6, 3], [-22, 2], [-4, -4], [6, -6], [4, 8], [16, -4], [8, -7], [8, 6], [-17, -2], [-18, -2], [6, -6], [14, 6], [22, 4], [3, 10], [6, -2], [4, -2], [12, 4], [12, 0], [8, -4], [-24, 1], [-4, 0], [7, 2], [-21, 0], [20, 2], [-12, 4], [4, -5], [22, -4], [30, 3], [-16, -6], [18, 6], [8, 0], [12, -8], [34, -2], [0, -9], [-12, -2], [4, -8], [8, 3], [14, 2], [6, 12], [8, 6], [24, 6], [4, -4], [-36, 0], [4, -14], [-4, 5], [-6, -8], [16, -10], [-14, 2], [1, -14], [-10, 8], [-12, -9], [-20, -4], [-6, -8], [2, -4], [-18, -6], [30, 0], [10, 8], [-24, -5], [-14, -1], [-10, 2], [11, 6], [2, 12], [-16, 11], [12, -2], [7, -8], [11, -10], [-8, -2], [12, 4], [-12, 4], [-33, 6], [8, 8], [-24, 4], [4, 10], [0, -8], [36, -6], [28, -1], [30, 7], [-27, -2], [-3, -12], [20, 6], [12, -8], [-36, 3], [9, -8], [-3, -6], [-12, 10], [-6, 12], [12, 9], [-1, 6], [0, -6], [-24, -2], [38, -4], [26, 6], [14, -10], [26, -9], [0, 6], [-20, 8], [1, -8], [-10, 15], [-40, -1], [22, -4], [20, 8], [-10, 12], [0, -18], [8, 0], [16, -11], [-10, -18], [14, -4], [8, 14], [0, 0], [-24, -2], [-30, -9], [-42, 0], [-7, -16], [0, -2], [-16, -12], [-15, -8], [-29, -2], [56, 2], [14, 0], [-24, 6], [-44, -4], [-10, 6], [40, -8], [-57, -2], [-18, -12], [-4, 15], [-12, -2], [-28, -9], [-18, -9], [2, -9], [32, 2], [-33, 0], [6, -6], [-30, -5], [-4, -12], [3, 16], [-36, 6], [10, -16], [4, 10], [9, -2], [-10, -4], [18, 6], [18, 6], [-15, 4], [-24, 4], [18, 0], [20, 8], [-20, -8], [14, 0], [-32, -4], [-29, -4], [24, 6], [11, 2], [-34, -6], [6, 18], [-12, -3], [11, -18], [22, -10], [0, 18], [0, 10], [20, -10], [0, -12], [23, 0], [24, -11], [-12, 3], [-35, -8], [-15, -6], [20, 0], [-28, 6], [-36, 3], [16, -1], [-4, 8], [-16, 6], [-26, -8], [38, 2], [12, -17], [20, -10], [18, -17], [-17, 2], [0, -14], [-48, 3], [-48, -2], [-12, 12], [30, 4], [5, 2], [-22, -19], [-20, -14], [10, -14], [-49, -6], [4, 2], [-12, -14], [-6, -24], [12, 6], [44, 9], [6, -15], [8, 12], [29, -4], [12, 16], [-60, -3], [-15, 0], [43, -4], [-6, 0], [22, -5], [32, 6], [-18, -8], [39, -6], [-14, 14], [-20, -7], [-8, -2], [-18, 4], [-40, -12], [-12, 12], [-1, -6], [-30, 6], [44, 11], [36, 10], [-33, 18], [4, 20], [58, 8], [-10, 12], [4, -4], [-16, 6], [20, 21], [38, -10], [20, -12], [25, -16], [10, 19], [-5, 16], [44, -12], [3, 12], [4, -16], [2, 0], [0, -24], [48, -8], [-57, 6], [8, 12], [36, -9], [18, 6], [-46, 8], [-32, 4], [26, -7], [-2, -13], [12, 0], [0, 10], [10, 16], [6, -27], [44, -6], [28, -16], [-13, 2], [12, -2], [18, 0], [20, 0], [28, -22], [44, 8], [-2, -4], [-12, 18], [-10, 15], [33, 0], [-6, 0], [40, 4], [38, 0], [9, 0], [-1, 0], [42, -3], [-36, -12], [-32, 20], [-52, -4], [-36, 16], [60, -2], [26, -16], [40, -1], [36, -12], [-7, -24], [23, 6], [70, -4], [-8, -8], [-30, -12], [-34, -12], [-6, 0], [-36, -2], [0, -12], [5, -30], [60, 6], [24, 22], [30, 6], [-63, 4], [32, -3], [-12, 16], [28, -2], [12, -6], [-28, 18], [18, 12], [16, -14], [-14, -10], [-48, -14], [-30, 6], [33, 18], [-31, -6], [-14, -28], [60, 6], [-4, -3], [-32, -22], [42, -12], [12, -12], [-8, 22], [-6, 6], [-36, 0], [-22, -9], [-16, 8], [8, 8], [34, -11], [66, -8], [-24, -14], [-5, -26], [24, 13], [9, 16], [18, -3], [-27, -2], [-44, -8], [-6, -24], [-46, 12], [60, 4], [-10, -19], [-6, -6], [14, 6], [0, 12], [-62, -4], [40, -2], [-20, 4], [-56, -7], [-6, -18], [-59, 14], [-9, -12], [40, 20], [62, -3], [-40, 12], [-8, 16], [-34, -12], [-2, 16], [-56, 10], [28, 8], [-44, 10], [6, -8], [-54, -2], [-32, 19], [13, -22], [-6, 0], [-26, -10], [-72, -6], [12, -23], [-24, 24], [-14, 16], [48, -8], [-10, 3], [24, 24], [-14, 14], [-6, -21], [14, -30], [-6, 24], [6, 6], [51, -2], [32, 12], [-32, 22], [-30, -18], [4, 17], [36, 0], [71, -4], [4, -2], [-34, 21], [-60, 12], [-64, 2], [-8, -16], [-41, -2], [50, 2], [0, 27], [39, -2], [20, 2]];
    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_bq();".
// 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_bq(:prec:=2)
    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_bq();".
// 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_bq( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4410_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![-3, 4, 1]>,<13,R![-3, 4, 1]>,<17,R![-24, -4, 1]>,<19,R![-27, -2, 1]>,<29,R![0, 1]>,<31,R![-2, 1]>],Snew);
    return Vf;
end function;