
// Make newform 3822.2.a.b in Magma, downloaded from the LMFDB on 19 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_b();".
// 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_b();".
// 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], [-4], [-1], [-6], [4], [0], [-4], [-4], [12], [-12], [-8], [-2], [8], [4], [-10], [-14], [8], [2], [16], [0], [-4], [2], [-14], [-14], [10], [-8], [10], [8], [-12], [-10], [0], [14], [4], [10], [-18], [2], [14], [-6], [2], [-12], [14], [-6], [26], [-16], [20], [8], [-2], [14], [24], [26], [4], [2], [-4], [-14], [-20], [-2], [18], [-4], [30], [20], [-28], [0], [18], [14], [14], [30], [6], [24], [16], [-10], [10], [-6], [-18], [-36], [22], [-30], [18], [-20], [0], [16], [-32], [30], [6], [30], [38], [-6], [-4], [-28], [-6], [16], [-6], [-42], [8], [26], [-10], [-8], [-20], [16], [6], [-4], [18], [-12], [38], [4], [36], [12], [28], [-42], [-16], [-14], [20], [-12], [10], [36], [-48], [-16], [-26], [10], [-34], [-6], [40], [-28], [16], [-20], [24], [-26], [-22], [-14], [0], [0], [-2], [28], [10], [-42], [-28], [6], [-42], [-44], [-34], [40], [-24], [-10], [30], [10], [42], [12], [-24], [0], [46], [-56], [36], [-44], [-56], [-24], [-36], [-48], [-42], [-40], [30], [28], [52], [-46], [-14], [32], [46], [-2], [-6], [12], [18], [0], [30], [-10], [2], [-22], [-54], [34], [-10], [-36], [-52], [54], [36], [-16], [-22], [50], [-60], [10], [38], [64], [-20], [20], [42], [-26], [-50], [-26], [-2], [-20], [24], [-46], [-62], [-62], [-8], [48], [30], [-40], [6], [58], [-32], [18], [-42], [-32], [-12], [10], [58], [-56], [18], [-48], [50], [46], [0], [42], [-16], [36], [14], [46], [-56], [66], [24], [54], [8], [28], [50], [-48], [58], [74], [46], [-26], [40], [-8], [-16], [-14], [32], [60], [-36], [56], [36], [-2], [4], [-24], [40], [48], [68], [42], [20], [-22], [-4], [-42], [-50], [8], [70], [-26], [54], [24], [-56], [18], [-48], [74], [-32], [42], [-2], [-14], [-80], [16], [-8], [-4], [-56], [-38], [12], [20], [32], [68], [22], [-28], [-60], [-34], [-66], [-40], [-30], [-12], [28], [6], [40], [42], [-10], [26], [70], [12], [-52], [8], [54], [-80], [-6], [12], [74], [38], [-74], [24], [-26], [-72], [54], [66], [8], [-32], [16], [16], [-40], [-82], [22], [88], [2], [-18], [16], [38], [-72], [-54], [36], [60], [56], [32], [48], [-42], [32], [2], [44], [-20], [-16], [-26], [-42], [-76], [42], [26], [-48], [-34], [30], [42], [46], [-6], [54], [-84], [38], [14], [46], [-22], [-2], [54], [-44], [-50], [-16], [-4], [-42], [24], [-20], [-52], [-54], [-54], [-16], [86], [6], [16], [8], [56], [86], [-14], [28], [76], [70], [-62], [56], [-30], [6], [12], [50], [76], [42], [72], [-42], [38], [-12], [14], [-10], [-100], [8], [64], [58], [12], [-54], [-100], [-90], [-38], [22], [-16], [56], [18], [10], [52], [-66], [-68], [-72], [76], [36], [50], [88], [-78], [30], [-48], [14], [8], [-36], [-88], [-58], [-24], [-84], [-34], [38], [-6], [-42], [20], [-20]];
    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_b();".
// 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_b(: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_b();".
// 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_b( : 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![4, 1]>,<17,R![6, 1]>,<29,R![4, 1]>],Snew);
    return Vf;
end function;