
// Make newform 3888.2.a.p in Magma, downloaded from the LMFDB on 21 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3888_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_3888_2_a_p();".
// 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_3888_2_a_p();".
// 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_3888_a();"
function MakeCharacter_3888_a()
    N := 3888;
    order := 1;
    char_gens := [2431, 2917, 1217];
    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_3888_a_Hecke(Kf)
    return MakeCharacter_3888_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], [0], [1], [0], [2], [-2], [-4], [1], [-1], [-9], [-2], [0], [0], [-4], [9], [-5], [-10], [4], [-11], [13], [1], [-4], [16], [0], [-13], [-9], [-16], [0], [10], [18], [8], [10], [-18], [0], [-10], [-14], [-20], [9], [-9], [9], [18], [18], [9], [2], [18], [-10], [-16], [14], [-18], [-20], [-14], [1], [-18], [-14], [-10], [-16], [14], [10], [22], [20], [-11], [-9], [-27], [27], [-27], [18], [-5], [9], [18], [18], [20], [-27], [-22], [-16], [25], [-27], [25], [8], [-18], [27], [0], [8], [-5], [1], [36], [-38], [4], [-22], [-2], [16], [0], [-20], [-28], [18], [-9], [-9], [-9], [32], [-19], [36], [13], [-14], [-28], [-2], [45], [-37], [2], [4], [27], [-5], [22], [8], [-18], [-4], [-16], [-2], [9], [13], [18], [36], [20], [-23], [25], [32], [31], [18], [50], [32], [36], [20], [-44], [-37], [18], [-10], [-28], [29], [27], [11], [1], [-40], [53], [-45], [34], [-14], [10], [-9], [32], [-54], [9], [41], [0], [14], [-44], [-8], [23], [47], [-52], [-34], [-18], [-27], [18], [-36], [-38], [36], [-54], [36], [44], [10], [-37], [46], [18], [0], [1], [1], [52], [62], [-5], [-63], [8], [14], [28], [52], [-18], [18], [0], [-18], [-8], [9], [-2], [-45], [1], [-54], [-43], [-47], [18], [44], [7], [40], [2], [63], [-19], [-14], [2], [-14], [-18], [-9], [46], [-18], [38], [41], [-27], [-63], [14], [52], [-44], [-31], [-2], [-2], [0], [45], [20], [0], [-54], [28], [18], [-58], [54], [36], [-18], [-16], [18], [-19], [52], [-44], [-9], [-29], [-25], [13], [-26], [-40], [-72], [55], [-70], [10], [72], [-29], [-10], [36], [-25], [-68], [-54], [-36], [73], [22], [18], [18], [8], [-67], [63], [10], [-58], [0], [-38], [-54], [10], [49], [41], [2], [52], [17], [22], [-47], [-59], [-70], [-38], [-80], [36], [-76], [55], [-70], [-64], [-52], [-16], [-4], [-20], [32], [35], [9], [4], [68], [61], [-72], [-8], [72], [-54], [-62], [-50], [54], [-72], [76], [-43], [5], [10], [-82], [-36], [-37], [18], [-70], [1], [-46], [-81], [-54], [-34], [-43], [-31], [26], [-54], [27], [-31], [18], [-13], [-38], [59], [62], [-2], [63], [-36], [36], [-16], [9], [-68], [-27], [-36], [36], [63], [-18], [-26], [18], [2], [54], [-68], [18], [27], [-54], [79], [50], [0], [5], [81], [-49], [45], [-82], [19], [18], [-34], [-4], [71], [4], [-62], [9], [-94], [-54], [72], [52], [-91], [63], [-81], [-82], [73], [-2], [36], [-80], [-65], [92], [-28], [36], [27], [19], [-90], [23], [-63], [26], [-90], [36], [-88], [27], [54], [-4], [71], [-9], [49], [-74], [36], [9], [-45], [-74], [18], [-72], [43], [-98], [2], [-90], [74], [-82], [58], [26], [-32], [-90], [7], [0], [86], [-83], [40], [68], [9], [-5], [45], [-14], [72], [-80], [65], [-82], [43], [-26], [82], [26], [92], [90], [63], [65]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3888_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3888_2_a_p();".
// 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_3888_2_a_p(:prec:=1)
    chi := MakeCharacter_3888_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_3888_2_a_p();".
// 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_3888_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3888_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-1, 1]>,<7,R![0, 1]>,<11,R![-2, 1]>,<13,R![2, 1]>],Snew);
    return Vf;
end function;