
// Make newform 3381.2.a.h in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3381_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_3381_2_a_h();".
// 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_3381_2_a_h();".
// 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_3381_a();"
function MakeCharacter_3381_a()
    N := 3381;
    order := 1;
    char_gens := [2255, 346, 442];
    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_3381_a_Hecke(Kf)
    return MakeCharacter_3381_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 := [[0], [1], [0], [0], [-2], [-3], [2], [3], [1], [6], [-3], [-3], [0], [3], [-10], [-4], [-6], [-2], [1], [-6], [1], [-7], [4], [-14], [-10], [16], [-1], [-16], [-9], [-18], [1], [0], [12], [21], [10], [-12], [-2], [-20], [16], [6], [-20], [-5], [-6], [-5], [4], [-8], [12], [16], [2], [15], [2], [-24], [-14], [-26], [-20], [0], [-24], [20], [-7], [-22], [15], [-6], [27], [-22], [1], [2], [-13], [27], [-28], [10], [10], [-10], [37], [-11], [29], [18], [36], [7], [32], [37], [-28], [15], [12], [-7], [8], [-14], [-6], [27], [-10], [-29], [-26], [18], [19], [30], [-35], [-36], [20], [6], [21], [17], [28], [18], [-36], [-40], [-17], [3], [18], [-34], [-40], [-13], [39], [-38], [6], [-7], [16], [48], [-41], [-30], [-26], [0], [-11], [7], [22], [-4], [43], [6], [-10], [26], [1], [35], [29], [-8], [11], [-46], [30], [29], [10], [8], [20], [32], [-12], [-18], [40], [-24], [29], [54], [-17], [0], [-20], [10], [42], [-54], [-9], [-6], [25], [20], [-39], [-36], [25], [-36], [-10], [58], [19], [30], [34], [36], [9], [-1], [35], [-20], [-6], [10], [-18], [31], [5], [-62], [-53], [0], [-40], [25], [-23], [-6], [-47], [22], [-24], [50], [2], [17], [-31], [-36], [-55], [-38], [-56], [-30], [30], [54], [17], [2], [54], [-60], [-62], [19], [14], [34], [-4], [-58], [-4], [14], [10], [-28], [-13], [0], [-40], [-4], [22], [-3], [48], [62], [62], [-16], [-47], [-33], [36], [32], [0], [26], [-32], [-42], [-35], [-30], [21], [68], [17], [-6], [-37], [-34], [47], [-76], [-46], [-36], [-58], [-27], [-23], [74], [-34], [22], [63], [46], [-52], [-2], [-5], [-12], [24], [71], [-52], [-24], [-13], [-67], [-26], [-22], [41], [-66], [-19], [-7], [12], [-53], [68], [-18], [-76], [-2], [-53], [28], [35], [-23], [-22], [56], [22], [-53], [41], [-6], [66], [-11], [-38], [65], [-20], [0], [47], [-50], [-23], [24], [54], [-54], [-18], [36], [-73], [76], [63], [6], [6], [83], [-46], [12], [-16], [-56], [23], [-59], [-46], [-5], [-36], [-1], [60], [-54], [-38], [-4], [-16], [-53], [-2], [20], [-29], [0], [27], [89], [70], [-89], [-12], [-65], [-83], [-35], [24], [74], [37], [34], [-75], [-66], [-63], [-72], [6], [24], [-43], [-13], [-9], [68], [-12], [-17], [12], [-86], [69], [52], [84], [70], [-68], [58], [24], [-16], [53], [-66], [-26], [-18], [82], [88], [58], [-14], [68], [-76], [-63], [-18], [-22], [59], [58], [-30], [41], [60], [36], [-35], [25], [-44], [-40], [-6], [-36], [73], [20], [-30], [-8], [-52], [84], [20], [15], [-50], [37], [44], [27], [78], [60], [-41], [-40], [-22], [-80], [-10], [-71], [60], [30], [-56], [-4], [0], [18], [-19], [98], [74], [31], [46], [38], [-32], [46], [-20], [95], [54], [12], [-67], [-36], [4], [6], [-35], [-44], [44], [30], [108], [18], [-76], [12], [-94]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3381_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3381_2_a_h();".
// 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_3381_2_a_h(:prec:=1)
    chi := MakeCharacter_3381_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_3381_2_a_h();".
// 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_3381_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3381_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![0, 1]>,<5,R![0, 1]>,<11,R![2, 1]>,<13,R![3, 1]>],Snew);
    return Vf;
end function;