
// Make newform 3822.2.a.i in Magma, downloaded from the LMFDB on 18 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_i();".
// 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_i();".
// 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], [2], [0], [3], [1], [-5], [-1], [0], [-1], [-4], [-2], [-10], [-10], [1], [-3], [-3], [5], [5], [-1], [-12], [-6], [-16], [14], [-4], [-2], [2], [12], [-16], [15], [0], [4], [18], [-2], [8], [19], [-21], [19], [-21], [-3], [2], [-3], [20], [-14], [-24], [20], [-16], [-11], [-8], [-6], [11], [21], [-4], [18], [-6], [-6], [9], [-11], [-27], [16], [-4], [12], [27], [-14], [-34], [-6], [-20], [-17], [2], [-36], [-14], [0], [-26], [13], [28], [32], [9], [-36], [-36], [30], [-10], [8], [-8], [19], [-40], [-18], [-8], [-16], [-20], [8], [24], [33], [11], [22], [-28], [12], [28], [-42], [-6], [-34], [28], [18], [-24], [19], [36], [-38], [-31], [-14], [-38], [19], [12], [-16], [34], [4], [48], [30], [19], [38], [18], [6], [26], [14], [3], [-12], [17], [-50], [20], [18], [-12], [40], [40], [-3], [-12], [-37], [-12], [44], [-26], [5], [-2], [-5], [28], [42], [-30], [-35], [19], [9], [40], [-3], [40], [-44], [22], [26], [-38], [14], [28], [26], [-2], [46], [5], [24], [-25], [19], [-31], [14], [62], [59], [-48], [-19], [16], [-27], [-17], [-42], [-40], [-10], [14], [-49], [-23], [-30], [-14], [19], [21], [0], [31], [38], [-39], [2], [-9], [12], [42], [3], [-29], [29], [-50], [-34], [-18], [-11], [-30], [-38], [0], [48], [44], [-8], [-34], [-26], [12], [42], [11], [-42], [-46], [58], [29], [-1], [44], [-9], [-24], [28], [-10], [-45], [-57], [10], [55], [-16], [0], [-24], [20], [-19], [38], [-36], [-22], [-11], [59], [16], [21], [-19], [-42], [-20], [30], [0], [-70], [-28], [37], [-76], [-20], [30], [-32], [50], [5], [-15], [-43], [4], [22], [-56], [55], [53], [21], [-4], [33], [-23], [-37], [-44], [-76], [76], [38], [6], [-64], [60], [44], [-8], [76], [-81], [26], [-69], [28], [72], [-50], [-29], [27], [64], [28], [-58], [-66], [68], [-32], [-10], [25], [-64], [-65], [-54], [17], [-63], [-51], [50], [-36], [65], [-34], [-61], [-20], [63], [24], [36], [83], [64], [-62], [-52], [-18], [70], [-36], [-7], [31], [59], [-36], [30], [70], [2], [83], [-38], [36], [-47], [-58], [61], [-26], [14], [-33], [-91], [8], [-43], [52], [-13], [-22], [15], [-18], [-6], [-2], [19], [-55], [7], [-40], [36], [-6], [54], [-2], [19], [22], [-8], [-6], [-36], [-25], [49], [81], [-30], [-36], [36], [-2], [20], [-15], [38], [45], [-15], [-33], [21], [-66], [-29], [0], [-33], [84], [25], [22], [51], [3], [24], [44], [19], [14], [-80], [34], [61], [-48], [-16], [54], [-67], [96], [0], [-71], [-54], [55], [-52], [37], [-21], [-29], [52], [-60], [-84], [56], [-62], [3], [-17], [29], [103], [-20], [-68], [76], [-22], [61], [-24], [4], [56], [22], [-20], [59], [-59], [-29], [54], [-76], [-44], [85], [55], [16], [84], [-102], [-84], [106], [56], [-8], [68], [50], [72], [18], [78], [-28], [-28]];
    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_i();".
// 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_i(: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_i();".
// 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_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3822_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<11,R![-3, 1]>,<17,R![5, 1]>,<29,R![1, 1]>],Snew);
    return Vf;
end function;