
// Make newform 2738.2.a.l in Magma, downloaded from the LMFDB on 08 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2738_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_2738_2_a_l();".
// 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_2738_2_a_l();".
// 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
        poly := [-3, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_2738_a();"
function MakeCharacter_2738_a()
    N := 2738;
    order := 1;
    char_gens := [1371];
    v := [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_2738_a_Hecke( : Kf := Kf)
    return MakeCharacter_2738_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], [1, 1], [0, 1], [2, -2], [0, -1], [1, -1], [6, 0], [-2, 0], [3, -3], [-3, 3], [-2, 1], [0, 0], [3, 3], [4, -2], [0, 2], [-6, 0], [-6, -2], [4, -5], [8, -5], [6, 0], [-10, -1], [7, -7], [12, -4], [0, 4], [-2, 8], [6, -8], [-2, 1], [3, -1], [-2, 0], [-6, 4], [-10, -4], [-6, -2], [-6, 3], [5, -3], [6, 6], [-16, -2], [2, -4], [10, -8], [-6, 5], [-18, 4], [0, 6], [20, 0], [-6, 5], [4, 0], [-6, 0], [-8, -8], [8, 1], [8, -6], [0, -6], [20, 2], [-3, 5], [6, 5], [-8, 0], [12, 4], [6, -4], [0, 6], [-12, -4], [-4, -8], [16, -7], [12, 0], [-20, -2], [-12, 10], [11, 3], [-9, -3], [-14, 10], [0, 4], [4, 4], [-10, 7], [-24, 6], [-4, 14], [-24, 6], [-6, 4], [-10, 6], [-4, -6], [-1, 11], [0, 16], [21, 7], [20, -6], [-6, 6], [-8, -10], [-3, 17], [-17, 9], [0, -4], [5, 13], [-5, -11], [12, 9], [6, 4], [-2, 2], [6, -8], [-29, 1], [0, 0], [-3, 7], [28, 4], [-3, 9], [10, 14], [18, -5], [-18, 6], [6, 12], [-14, 14], [4, -13], [16, 2], [0, -5], [-18, 20], [0, -8], [-19, 1], [10, -14], [18, 12], [30, -5], [-30, 4], [-22, -1], [25, -5], [-16, -6], [15, -25], [-4, -1], [-17, 1], [-9, -5], [22, -6], [9, -13], [-3, 3], [-33, 7], [19, 5], [23, -1], [-18, 14], [-12, 0], [8, 0], [-24, -7], [16, 11], [-24, -12], [-26, 3], [14, -12], [-28, 11], [12, -18], [14, 0], [-8, 1], [33, -7], [22, 0], [-24, 20], [20, -4], [3, -11], [-6, 18], [32, -17], [6, -4], [8, 16], [-30, -2], [7, 9], [-36, 10], [-8, -15], [-6, -16], [-2, -20], [12, 0], [2, -4], [6, -15], [-8, -8], [18, -8], [-26, 0], [36, -8], [28, -8], [-30, 5], [-25, 3], [0, 6], [-24, 12], [6, 19], [-14, 9], [12, -15], [-18, 0], [-12, 0], [-53, 1], [34, 8], [38, 3], [12, -3], [-18, 2], [7, -11], [42, 3], [17, -3], [2, 2], [24, 0], [4, 10], [-24, 12], [-16, 18], [14, -10], [28, 12], [-18, 0], [-8, 23], [6, -8], [-24, 6], [30, -18], [26, 8], [10, -6], [-32, 24], [-12, 0], [-8, -20], [36, 4], [16, -18], [-6, -26], [0, -11], [-6, 28], [22, 6], [-14, 16], [-15, 1], [6, -27], [12, 21], [62, 0], [2, 2], [5, -19], [15, -13], [-12, -11], [-34, -6], [-12, -13], [12, -28], [44, 4], [10, 16], [6, 16], [19, -3], [-12, -15], [42, -3], [26, -15], [-2, 19], [-30, 24], [-18, 7], [-24, 0], [-34, -4], [20, 8], [18, 20], [-5, 5], [-36, -15], [-41, -11], [51, -5], [-30, 6], [-46, 18], [30, -28], [-16, 14], [-16, -15], [-22, 12], [6, 21], [-28, 3], [-18, -6], [41, -3], [39, -3], [24, 12], [-18, -15], [-24, 20], [28, -2], [-34, 14], [-32, 17], [-21, 15], [-27, 15], [-29, 9], [-36, 24], [-1, 19], [36, -24], [16, 7], [-42, -11], [54, 0], [10, -24], [-63, 11], [45, -7], [8, -10], [20, -20], [-12, 10], [-20, -6], [-11, 23], [54, -16], [-4, 0], [44, -22], [24, -6], [71, 3], [0, -20], [48, -8], [32, -9], [-21, 25], [28, 5], [28, 16], [22, 16], [-41, -15], [23, -33], [8, 32], [-21, 15], [28, -3], [5, 13], [-6, 6], [18, -20], [-44, 8], [-12, -22], [-10, 24], [22, -24], [6, 22], [-8, -8], [12, 8], [-50, 7], [30, 2], [6, -24], [-42, 8], [42, -24], [24, -8], [-40, 8], [36, 0], [26, 0], [42, -30], [-6, 6], [35, -9], [-8, 8], [36, -6], [14, -38], [-42, 8], [4, 12], [-20, 0], [-12, -18], [-23, 3], [48, 0], [-17, -11], [0, 22], [-48, 18], [54, 7], [32, 9], [-33, 5], [22, 2], [48, -24], [-9, 9], [35, -21], [60, -4], [16, -10], [-70, -7], [24, -47], [-34, 0], [6, 11], [-26, 16], [-43, 33], [28, -42], [18, 27], [-54, -8], [-22, -10], [0, 15], [22, -23], [-48, 0], [28, -32], [12, 19], [20, 24], [-18, 24], [-2, 4], [-52, 18], [-10, 6], [15, -37], [9, 3], [-62, -9], [-33, 45], [6, 8], [32, 16], [68, -4], [39, 9], [-48, 2], [29, -7], [26, 5], [-66, 0], [40, 16], [2, -28], [-33, -21], [-81, 5], [18, 0], [-54, 21], [18, -21], [-71, -9], [-42, 9], [18, -25], [-78, 6], [-1, 11], [-38, 14], [57, -21], [10, -5], [16, 12], [12, 12], [-26, 46], [-24, 8], [-42, -2], [25, -5], [8, 20], [60, -24], [66, -20], [-25, 27], [18, -22], [29, -11], [-45, 25], [18, -10], [22, -31], [-18, -27], [-2, -20], [0, -24], [14, -46], [16, -35], [-32, -10], [-78, 3], [26, -12], [48, -13], [60, -30], [-32, 22], [-42, -6], [-58, 0], [70, 3], [6, -21], [-4, -9], [-6, 30], [-28, 4], [36, -14], [-5, 9], [42, -16], [63, -15], [-52, -12], [-20, 29], [54, -31], [-55, 1], [39, -5], [38, -20], [0, 4], [48, 22], [34, -52], [4, -36], [-18, -4], [-24, 26], [-16, 28], [-18, -12], [21, -5], [24, -15], [4, 17], [12, -20], [24, -16], [-22, 16], [-6, 12], [-60, 19], [18, -7], [8, -43], [-60, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2738_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2738_2_a_l();".
// 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_2738_2_a_l(:prec:=2)
    chi := MakeCharacter_2738_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_2738_2_a_l();".
// 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_2738_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2738_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, -3, 1]>,<5,R![-3, -1, 1]>,<7,R![-12, -2, 1]>,<13,R![-3, -1, 1]>,<17,R![-6, 1]>],Snew);
    return Vf;
end function;