
// Make newform 3822.2.a.f in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3822_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_3822_2_a_f();".
// 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_3822_2_a_f();".
// 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_3822_a();"
function MakeCharacter_3822_a()
    N := 3822;
    order := 1;
    char_gens := [2549, 3433, 1471];
    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_3822_a_Hecke(Kf)
    return MakeCharacter_3822_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], [-1], [1], [0], [-3], [-1], [-2], [0], [6], [9], [5], [-8], [-4], [0], [0], [1], [-7], [-4], [-4], [6], [6], [-13], [-3], [-8], [-15], [-2], [-16], [17], [16], [12], [5], [15], [-12], [4], [-6], [-11], [-14], [-4], [-4], [-14], [12], [-8], [-6], [19], [-22], [-24], [20], [-1], [-13], [-14], [-14], [22], [-27], [23], [-26], [-26], [-3], [9], [-22], [0], [14], [15], [-18], [2], [13], [3], [8], [1], [-28], [-34], [6], [-2], [-3], [-26], [-24], [-26], [34], [34], [6], [-11], [0], [-12], [26], [-38], [-29], [39], [8], [1], [-26], [-8], [12], [30], [-9], [-29], [16], [-26], [-21], [28], [-12], [-32], [8], [17], [3], [0], [20], [-3], [-9], [-40], [18], [19], [1], [-28], [4], [32], [-13], [12], [-32], [36], [39], [-12], [26], [17], [31], [-41], [8], [-39], [-26], [2], [-9], [22], [-18], [-20], [13], [22], [-32], [-13], [18], [28], [7], [4], [36], [3], [20], [-27], [38], [44], [58], [-50], [40], [40], [-18], [50], [10], [24], [-8], [-50], [-56], [28], [-37], [-39], [12], [12], [21], [13], [-22], [6], [15], [8], [-55], [27], [35], [48], [2], [19], [61], [14], [32], [-27], [32], [-10], [-43], [-19], [18], [-42], [-60], [19], [24], [2], [13], [64], [5], [-33], [-32], [6], [3], [-44], [58], [50], [-58], [-54], [21], [48], [52], [-22], [20], [-59], [48], [-27], [12], [-66], [34], [-21], [-61], [3], [12], [-66], [33], [-28], [-26], [6], [-18], [20], [54], [53], [37], [48], [34], [2], [-45], [8], [-74], [60], [13], [72], [62], [-48], [49], [-39], [-40], [30], [28], [40], [25], [-42], [14], [26], [-27], [-40], [12], [-46], [-20], [-66], [-36], [58], [-59], [21], [24], [-18], [9], [-73], [36], [35], [-52], [-22], [-8], [22], [-39], [12], [-56], [79], [36], [44], [-73], [9], [-7], [-8], [-49], [48], [-22], [44], [36], [-3], [-54], [28], [-86], [16], [75], [65], [-5], [-12], [14], [-27], [32], [-60], [-10], [83], [80], [-42], [4], [10], [-74], [41], [-11], [-29], [58], [46], [67], [26], [6], [-66], [0], [-78], [-10], [54], [-68], [58], [85], [-60], [-17], [34], [-80], [38], [-39], [-20], [30], [3], [58], [4], [-28], [-21], [-80], [71], [36], [-60], [-8], [-44], [-68], [18], [43], [-87], [64], [60], [-21], [5], [51], [9], [-70], [-82], [-68], [73], [-56], [41], [-18], [-83], [0], [76], [-40], [5], [-42], [4], [58], [-66], [-58], [-36], [58], [-17], [14], [-80], [-23], [69], [54], [-54], [89], [-40], [22], [52], [4], [25], [24], [-2], [78], [-62], [-12], [26], [94], [24], [-52], [-4], [-8], [-84], [-29], [-50], [-75], [58], [96], [-17], [-25], [-4], [-22], [-19], [-18], [36], [72], [-18], [-45], [97], [-86], [6], [-94], [45], [51], [63], [45], [74], [22], [69], [-54], [76], [22], [76], [63], [-18], [48], [-65], [30], [18], [9], [82], [-22], [42]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3822_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3822_2_a_f();".
// 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_3822_2_a_f(:prec:=1)
    chi := MakeCharacter_3822_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_3822_2_a_f();".
// 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_3822_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3822_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-1, 1]>,<11,R![3, 1]>,<17,R![2, 1]>,<29,R![-9, 1]>],Snew);
    return Vf;
end function;