
// Make newform 3822.2.a.m 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_m();".
// 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_m();".
// 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], [5], [1], [3], [5], [9], [-1], [-2], [3], [-12], [7], [-4], [-10], [14], [-11], [4], [-10], [-11], [4], [6], [16], [-14], [14], [-7], [16], [-11], [16], [-4], [3], [-1], [12], [20], [7], [-7], [6], [-5], [4], [-6], [-2], [21], [-10], [18], [19], [5], [-2], [4], [-28], [-16], [-18], [-26], [-1], [-14], [20], [-16], [8], [28], [-6], [16], [-18], [-20], [-2], [6], [6], [8], [23], [-6], [-24], [18], [-20], [-32], [-2], [-30], [15], [6], [14], [30], [-33], [-31], [-6], [10], [-16], [21], [6], [-9], [26], [-3], [11], [-5], [-3], [40], [18], [18], [-32], [-11], [25], [38], [25], [-20], [-24], [31], [-6], [12], [-2], [38], [-24], [-9], [20], [21], [5], [-23], [25], [-33], [-32], [-9], [36], [5], [-2], [-10], [11], [6], [7], [4], [-2], [-38], [-36], [17], [22], [-26], [-18], [-54], [-32], [-40], [-25], [-51], [-23], [54], [0], [17], [-40], [46], [-39], [13], [0], [-40], [6], [-24], [-42], [18], [-37], [19], [24], [4], [-41], [18], [-24], [-12], [18], [29], [-24], [1], [44], [29], [-7], [2], [26], [28], [6], [-39], [-12], [-24], [-30], [-17], [58], [20], [42], [-52], [-53], [9], [52], [-30], [30], [-10], [-26], [-28], [-15], [-2], [-17], [56], [-23], [-52], [15], [34], [59], [46], [-20], [8], [-30], [-52], [-19], [8], [-40], [-24], [18], [-50], [-24], [25], [50], [45], [0], [-38], [-66], [-1], [-13], [-26], [-45], [-54], [26], [-20], [-18], [15], [8], [-27], [-4], [-13], [-32], [33], [75], [-72], [-8], [13], [-40], [30], [23], [43], [-56], [-38], [26], [-44], [-68], [53], [50], [-12], [66], [-44], [-6], [10], [67], [39], [-22], [-30], [-44], [24], [23], [-40], [76], [-12], [-7], [50], [22], [50], [21], [18], [32], [-36], [60], [-10], [-26], [30], [-2], [29], [16], [-16], [-40], [-1], [7], [4], [84], [-56], [17], [58], [8], [-23], [-24], [23], [48], [-8], [-75], [-39], [-43], [-57], [-40], [-6], [10], [16], [14], [-51], [-65], [-38], [36], [-86], [-6], [-27], [38], [20], [71], [36], [-50], [78], [-72], [-54], [-49], [-38], [-76], [68], [38], [-53], [34], [-16], [2], [-30], [91], [-1], [-90], [-72], [-27], [-27], [41], [47], [-51], [72], [-68], [20], [70], [-29], [64], [26], [21], [14], [78], [-65], [-42], [-16], [72], [-81], [-68], [-12], [-18], [84], [74], [-76], [8], [56], [11], [45], [-40], [16], [-22], [53], [15], [-87], [4], [58], [6], [33], [29], [-55], [0], [-70], [38], [-73], [-5], [82], [-16], [-41], [-1], [-1], [40], [-66], [21], [-27], [74], [32], [-57], [-75], [-63], [-64], [-12], [26], [22], [-52], [-26], [-26], [-38], [-60], [63], [47], [0], [8], [-80], [-61], [40], [78], [-55], [-98], [99], [41], [60], [-88], [3], [-16], [-59], [57], [24], [12], [-17], [-14], [84], [-7], [82], [48], [-36], [-83], [70], [-48], [-90], [-11], [44]];
    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_m();".
// 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_m(: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_m();".
// 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_m( : 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![-5, 1]>,<17,R![-3, 1]>,<29,R![1, 1]>],Snew);
    return Vf;
end function;