
// Make newform 3822.2.a.d in Magma, downloaded from the LMFDB on 25 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_d();".
// 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_d();".
// 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], [1], [1], [5], [-1], [-5], [-3], [-2], [-11], [0], [-1], [12], [10], [14], [1], [4], [-2], [-1], [-8], [-2], [-12], [-2], [-6], [-3], [-16], [11], [-4], [4], [13], [-5], [-8], [0], [17], [-11], [-18], [3], [-4], [2], [-10], [15], [-18], [-14], [-17], [-3], [-6], [0], [4], [-20], [6], [-22], [-15], [-2], [-12], [28], [-12], [12], [-14], [-28], [-2], [20], [2], [-26], [-30], [20], [-1], [-18], [8], [6], [-4], [-16], [-18], [-14], [-1], [-6], [34], [-18], [21], [-17], [6], [-30], [-28], [25], [18], [-5], [-2], [37], [13], [-35], [21], [-8], [38], [-22], [-36], [21], [7], [34], [-17], [20], [-8], [1], [6], [-44], [-46], [18], [16], [-11], [8], [25], [-29], [-3], [-45], [25], [-12], [29], [-24], [39], [14], [-42], [-29], [-18], [-29], [20], [34], [-18], [-4], [-43], [-14], [30], [30], [50], [-20], [-4], [5], [21], [35], [-18], [12], [-29], [-40], [26], [21], [-47], [24], [-8], [10], [24], [38], [-2], [37], [19], [-16], [-20], [-3], [42], [-32], [-40], [18], [17], [-36], [47], [-36], [9], [-23], [-2], [58], [36], [-58], [21], [48], [-16], [14], [43], [38], [4], [54], [36], [-41], [-25], [-12], [-26], [-54], [-30], [-42], [36], [-29], [-38], [15], [-52], [-19], [60], [-41], [-58], [-35], [-14], [24], [60], [-10], [64], [17], [48], [-4], [-4], [18], [-50], [36], [5], [22], [-43], [-52], [22], [66], [-1], [25], [-50], [5], [-14], [-18], [-12], [70], [35], [-8], [51], [-32], [69], [72], [-27], [39], [-32], [44], [-21], [-72], [18], [-57], [41], [12], [34], [-54], [36], [52], [41], [54], [-48], [-14], [12], [-10], [14], [-7], [25], [-70], [-78], [64], [56], [-21], [28], [36], [0], [27], [34], [6], [50], [-63], [38], [-52], [-48], [-12], [-14], [2], [-42], [10], [39], [-32], [-64], [-64], [3], [73], [-20], [36], [44], [39], [-30], [80], [35], [-24], [-81], [-76], [-4], [3], [-37], [-21], [-69], [64], [-58], [54], [-8], [58], [19], [53], [30], [-12], [-6], [-66], [-27], [-74], [48], [-13], [48], [50], [-18], [36], [54], [-17], [-58], [-40], [48], [66], [25], [-54], [0], [18], [-62], [5], [3], [82], [4], [-9], [-9], [51], [-47], [71], [24], [-84], [-52], [-42], [-11], [0], [54], [-7], [-14], [6], [-9], [-50], [-48], [68], [35], [12], [-12], [6], [-20], [-26], [84], [32], [-40], [-51], [77], [-68], [-48], [46], [83], [81], [3], [-84], [2], [78], [81], [11], [-55], [-68], [-70], [62], [-19], [47], [-82], [-36], [-75], [13], [-55], [48], [14], [1], [11], [-22], [-24], [-25], [51], [5], [-84], [-12], [66], [70], [28], [34], [-6], [-46], [-88], [25], [31], [-4], [-4], [24], [77], [36], [46], [-83], [10], [-33], [67], [20], [-56], [-63], [84], [-59], [-7], [-48], [0], [51], [-26], [-12], [67], [-98], [28], [-68], [71], [-50], [36], [46], [-17], [0]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : 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![-1, 1]>,<17,R![-5, 1]>,<29,R![3, 1]>],Snew);
    return Vf;
end function;