
// Make newform 4176.2.a.g in Magma, downloaded from the LMFDB on 03 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4176_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_4176_2_a_g();".
// 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_4176_2_a_g();".
// 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_4176_a();"
function MakeCharacter_4176_a()
    N := 4176;
    order := 1;
    char_gens := [1567, 1045, 929, 4033];
    v := [1, 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_4176_a_Hecke(Kf)
    return MakeCharacter_4176_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], [0], [-3], [5], [-4], [-6], [1], [5], [6], [1], [0], [1], [-7], [-1], [-13], [2], [-13], [-2], [4], [10], [-12], [-8], [12], [-6], [-12], [-8], [3], [-5], [4], [-9], [2], [-6], [14], [0], [-21], [5], [-23], [-1], [2], [11], [-8], [-4], [9], [-26], [-11], [-16], [1], [-8], [15], [-1], [-14], [-22], [-19], [-16], [-12], [-5], [-2], [14], [8], [10], [-18], [16], [-28], [3], [19], [12], [27], [-20], [-15], [-4], [-22], [15], [-20], [14], [16], [2], [20], [10], [-34], [22], [1], [2], [4], [-28], [-13], [-24], [29], [-25], [30], [16], [-34], [28], [-17], [8], [-40], [17], [27], [18], [14], [37], [2], [-3], [6], [-21], [36], [34], [-39], [26], [4], [-34], [16], [6], [-9], [35], [-1], [27], [46], [16], [-4], [6], [-20], [-17], [0], [-1], [14], [41], [-28], [18], [16], [30], [-16], [41], [-8], [23], [20], [-30], [0], [38], [-30], [-54], [-30], [-2], [-56], [2], [-43], [3], [-21], [38], [-9], [12], [28], [18], [-30], [8], [28], [0], [9], [-54], [23], [18], [0], [54], [-38], [-28], [20], [48], [55], [-5], [9], [-28], [32], [10], [36], [-26], [-13], [18], [-2], [-30], [-30], [60], [0], [-42], [28], [-30], [30], [-6], [-55], [37], [2], [-56], [55], [54], [-52], [-2], [-54], [-4], [44], [28], [-24], [-22], [-6], [43], [-41], [-4], [56], [-3], [14], [9], [-24], [-68], [2], [-47], [-8], [26], [-3], [14], [-49], [66], [-44], [-64], [38], [16], [33], [36], [28], [-51], [57], [56], [46], [-12], [-46], [8], [2], [-39], [8], [-3], [-4], [-66], [-25], [21], [68], [-26], [40], [41], [-54], [-42], [44], [-68], [14], [-48], [46], [12], [51], [-66], [-70], [15], [-33], [-21], [-14], [54], [-34], [-22], [-50], [-66], [-21], [-33], [48], [17], [5], [45], [30], [48], [-25], [50], [44], [-50], [-18], [-10], [-38], [31], [-18], [75], [-2], [-56], [23], [-3], [2], [20], [1], [-18], [-14], [-72], [-62], [-32], [2], [19], [-78], [-6], [31], [-41], [16], [61], [-26], [-52], [4], [-63], [-36], [-50], [40], [46], [56], [30], [-78], [-62], [-12], [17], [10], [-24], [55], [45], [47], [51], [-35], [-50], [-54], [28], [74], [64], [88], [9], [-39], [83], [-39], [-46], [16], [77], [-34], [74], [-30], [23], [1], [-70], [8], [19], [93], [34], [21], [77], [-64], [62], [40], [46], [-36], [-4], [-37], [-55], [24], [9], [74], [-20], [-64], [30], [93], [-24], [23], [4], [-30], [-77], [86], [48], [-24], [0], [-42], [-25], [34], [-82], [75], [44], [-28], [-3], [-94], [-48], [-68], [33], [20], [-42], [-26], [8], [5], [-8], [-14], [77], [-50], [33], [-23], [-39], [75], [-57], [52], [0], [-34], [-18], [34], [6], [-27], [-64], [34], [-57], [-36], [-9], [58], [-39], [61], [-34], [-18], [-90], [-71], [-53], [46], [-103], [73], [2], [49], [0], [-14], [-66], [97], [-13], [-12], [-43]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4176_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4176_2_a_g();".
// 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_4176_2_a_g(:prec:=1)
    chi := MakeCharacter_4176_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_4176_2_a_g();".
// 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_4176_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4176_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![3, 1]>,<7,R![-5, 1]>,<11,R![4, 1]>,<17,R![-1, 1]>],Snew);
    return Vf;
end function;