
// Make newform 1638.2.a.d in Magma, downloaded from the LMFDB on 03 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1638_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_1638_2_a_d();".
// 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_1638_2_a_d();".
// 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_1638_a();"
function MakeCharacter_1638_a()
    N := 1638;
    order := 1;
    char_gens := [911, 703, 379];
    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_1638_a_Hecke( : Kf := Kf)
    return MakeCharacter_1638_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]];
    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], [0], [-2], [1], [4], [-1], [-6], [-4], [0], [6], [-8], [10], [6], [4], [-4], [-10], [-4], [-6], [-8], [0], [-10], [-8], [-4], [6], [-2], [-10], [-12], [-16], [-6], [-2], [-16], [20], [-10], [-12], [-14], [16], [18], [16], [12], [-18], [-24], [-22], [0], [-6], [-6], [20], [12], [-8], [12], [-6], [-18], [24], [30], [4], [-6], [-24], [-18], [8], [-10], [-10], [28], [-2], [-12], [16], [10], [18], [-8], [34], [-8], [2], [6], [16], [-28], [14], [16], [-4], [-34], [18], [30], [22], [-28], [-30], [32], [34], [-20], [-24], [-2], [-14], [14], [-24], [-12], [4], [8], [32], [32], [24], [14], [26], [4], [-46], [12], [-22], [12], [22], [-20], [22], [28], [-18], [-8], [2], [-28], [-30], [6], [28], [16], [30], [-28], [-8], [14], [0], [-38], [18], [38], [-36], [20], [-50], [-6], [-16], [28], [2], [40], [8], [16], [-26], [-26], [-50], [-18], [-12], [30], [30], [20], [2], [32], [-20], [-38], [4], [-22], [18], [28], [-56], [-6], [18], [12], [0], [12], [40], [-16], [-34], [-22], [54], [-12], [14], [16], [-4], [30], [-4], [40], [2], [-6], [-42], [36], [-30], [-16], [-46], [-12], [-62], [32], [26], [-12], [-22], [-56], [36], [6], [-42], [56], [22], [-26], [-44], [-38], [44], [-46], [20], [52], [-66], [16], [-38], [58], [46], [30], [24], [-30], [-4], [42], [-30], [-28], [-34], [16], [-16], [6], [-60], [-46], [70], [24], [12], [12], [38], [-8], [58], [56], [-30], [38], [-56], [-18], [48], [-52], [-26], [42], [-56], [-44], [60], [-34], [-52], [72], [-10], [4], [-44], [62], [-46], [8], [-8], [-60], [36], [4], [-22], [14], [32], [16], [-60], [40], [-32], [-38], [70], [-24], [50], [30], [60], [-50], [4], [6], [-2], [40], [48], [-70], [-38], [-26], [-36], [-46], [-34], [16], [70], [18], [-80], [70], [-56], [-30], [-56], [-60], [74], [-22], [-84], [-24], [-40], [-32], [42], [28], [-32], [-46], [-78], [56], [-86], [-42], [84], [54], [36], [-26], [30], [-84], [22], [-20], [4], [-54], [-54], [16], [-8], [44], [2], [8], [-6], [8], [86], [0], [50], [78], [44], [0], [26], [28], [-16], [-74], [6], [-52], [26], [30], [-80], [-74], [-30], [-40], [20], [-16], [-42], [-38], [-66], [-12], [60], [-64], [12], [-78], [6], [30], [84], [-6], [6], [70], [-32], [-6], [16], [-6], [88], [-12], [22], [20], [34], [26], [-28], [-34], [-6], [-36], [12], [-90], [-40], [58], [58], [-24], [36], [68], [-22], [30], [-40], [-22], [-32], [-60], [-16], [-74], [92], [70], [20], [96], [-10], [-6], [74], [-66], [-58], [-16], [-50], [-60], [60], [-32], [-62], [88], [40], [-54], [-66], [-60], [92], [72], [90], [24], [34], [92], [-30], [34], [86], [16], [30], [22], [52], [-78], [14], [84], [-20], [-38], [94], [-32], [-52], [90], [-50], [-80], [28], [-10], [-16], [-18], [-22], [8], [-84], [-98], [38], [-24], [22], [76], [-32]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1638_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1638_2_a_d();".
// 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_1638_2_a_d(:prec:=1)
    chi := MakeCharacter_1638_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_1638_2_a_d();".
// 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_1638_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1638_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]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;