
// Make newform 3822.2.a.x 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_x();".
// 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_x();".
// 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], [2], [-4], [4], [0], [4], [4], [0], [8], [-6], [4], [8], [-14], [-14], [16], [-10], [-8], [4], [0], [-2], [-6], [6], [2], [8], [2], [16], [4], [10], [16], [-6], [-4], [-2], [6], [-18], [14], [2], [14], [0], [-18], [-2], [-2], [0], [12], [12], [-14], [22], [24], [14], [-12], [-14], [-24], [-14], [-12], [-18], [22], [20], [-14], [20], [-4], [16], [-18], [-10], [-34], [-18], [26], [12], [8], [26], [18], [-38], [-22], [-24], [-38], [-18], [30], [4], [0], [-24], [-16], [-14], [-18], [-14], [-2], [22], [4], [28], [30], [-8], [-22], [14], [24], [-18], [-2], [-8], [-20], [32], [10], [-44], [-30], [4], [-22], [-48], [16], [16], [-28], [10], [40], [6], [12], [4], [-30], [20], [-48], [-36], [-34], [6], [-2], [-6], [16], [-52], [4], [12], [-40], [26], [-26], [-30], [0], [-40], [-2], [-40], [14], [-6], [44], [-18], [6], [4], [-46], [-40], [48], [10], [-30], [-26], [-14], [36], [32], [-56], [-50], [40], [44], [4], [-36], [24], [-56], [32], [-30], [32], [-18], [-4], [-36], [-18], [46], [-8], [10], [-42], [-46], [-12], [30], [-52], [38], [58], [50], [10], [30], [-42], [34], [-28], [-12], [-50], [48], [48], [42], [42], [-44], [34], [-6], [56], [20], [-12], [-50], [14], [22], [22], [54], [-48], [56], [-2], [62], [-18], [-16], [44], [54], [40], [6], [-34], [-32], [2], [38], [16], [-24], [-34], [-26], [-8], [50], [-8], [6], [6], [-32], [62], [40], [12], [22], [30], [-36], [-18], [48], [46], [-64], [12], [18], [24], [-66], [14], [18], [22], [-24], [0], [-8], [-10], [24], [48], [4], [-72], [-40], [-58], [32], [-8], [-4], [-24], [-44], [2], [60], [34], [36], [78], [42], [64], [6], [-70], [-30], [68], [32], [-18], [-12], [58], [44], [30], [-74], [-42], [40], [-40], [8], [-12], [-80], [58], [12], [-4], [-72], [12], [26], [-52], [-48], [-66], [26], [56], [42], [56], [-60], [-30], [-68], [-22], [-18], [-26], [-74], [-20], [68], [72], [-30], [0], [42], [60], [34], [-42], [-22], [48], [-58], [-24], [-54], [66], [40], [-16], [-40], [12], [-24], [-62], [-90], [0], [18], [70], [-40], [-6], [64], [34], [36], [24], [84], [-32], [-52], [-54], [60], [74], [72], [52], [-36], [-54], [-46], [28], [-6], [6], [-8], [-6], [38], [38], [78], [-66], [78], [28], [-2], [42], [26], [50], [30], [74], [-32], [-66], [-56], [20], [-90], [-60], [-12], [-28], [-54], [-66], [-16], [-58], [46], [8], [8], [-28], [-22], [2], [0], [48], [-38], [98], [0], [38], [-14], [-52], [86], [12], [-10], [24], [-22], [-18], [12], [-90], [34], [52], [-8], [-48], [26], [-28], [58], [76], [2], [86], [78], [-40], [-36], [22], [94], [100], [18], [4], [20], [4], [-48], [42], [88], [98], [78], [-24], [-78], [4], [36], [4], [10], [-32], [28], [-70], [58], [-102], [-62], [100], [-60]];
    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_x();".
// 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_x(: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_x();".
// 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_x( : 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![-2, 1]>,<29,R![0, 1]>],Snew);
    return Vf;
end function;