
// Make newform 1368.2.a.f in Magma, downloaded from the LMFDB on 23 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1368_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_1368_2_a_f();".
// 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_1368_2_a_f();".
// 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_1368_a();"
function MakeCharacter_1368_a()
    N := 1368;
    order := 1;
    char_gens := [343, 685, 1217, 1009];
    v := [1, 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_1368_a_Hecke( : Kf := Kf)
    return MakeCharacter_1368_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 := [[0], [0], [0], [0], [2], [-2], [4], [1], [2], [6], [4], [2], [-2], [4], [10], [6], [4], [2], [4], [4], [-6], [8], [-2], [6], [-10], [4], [0], [12], [-2], [6], [12], [-6], [4], [12], [-20], [-8], [2], [-4], [-8], [10], [-12], [-2], [-10], [-6], [-8], [0], [12], [16], [24], [2], [-16], [6], [-18], [-18], [-22], [-10], [-14], [-16], [18], [-6], [-12], [-14], [-24], [-30], [-2], [18], [8], [-18], [-22], [22], [-4], [14], [-8], [2], [0], [-36], [16], [38], [-10], [-26], [-6], [22], [12], [2], [-16], [18], [34], [-14], [-28], [-8], [6], [30], [-28], [-30], [4], [30], [-14], [-18], [4], [14], [-32], [12], [16], [42], [-4], [22], [22], [8], [-16], [2], [-12], [26], [36], [4], [-32], [30], [20], [18], [-12], [40], [10], [-6], [30], [36], [-36], [20], [-26], [-30], [-32], [-18], [-44], [44], [40], [26], [-52], [-10], [26], [-28], [42], [48], [24], [-4], [16], [12], [30], [4], [6], [46], [20], [-36], [-22], [-52], [-44], [24], [52], [40], [-16], [-8], [-38], [-30], [-38], [-6], [-32], [0], [-18], [-8], [20], [26], [-10], [36], [28], [-38], [-46], [-10], [36], [-16], [52], [-60], [-44], [14], [16], [-48], [-50], [-14], [-14], [-44], [38], [4], [-2], [2], [-34], [-6], [-8], [-42], [50], [6], [2], [-66], [0], [-42], [6], [-4], [-50], [-58], [-54], [12], [40], [-36], [-36], [8], [22], [24], [48], [-4], [-4], [-46], [32], [38], [20], [-40], [-66], [32], [-50], [-16], [8], [-6], [-38], [20], [-52], [-62], [-30], [-8], [36], [14], [20], [-6], [50], [-24], [42], [-36], [-44], [-20], [-40], [-54], [18], [-18], [32], [-24], [56], [18], [-14], [-48], [50], [-26], [64], [66], [-42], [-16], [-18], [38], [64], [72], [-42], [-6], [72], [4], [54], [-40], [64], [0], [-22], [-20], [6], [64], [22], [48], [-10], [10], [10], [-42], [60], [-72], [-2], [10], [-20], [-14], [30], [58], [-56], [42], [20], [-34], [42], [36], [-46], [48], [64], [-48], [0], [-84], [70], [30], [-56], [84], [12], [38], [48], [-74], [74], [-42], [-34], [-16], [-58], [-72], [18], [78], [-14], [0], [26], [60], [-20], [58], [22], [16], [4], [6], [80], [28], [12], [28], [50], [46], [88], [42], [-60], [-2], [-54], [-18], [22], [-80], [-62], [20], [-2], [-20], [14], [48], [42], [36], [-64], [-60], [-68], [-18], [8], [-60], [-70], [-34], [-42], [-42], [36], [36], [62], [-20], [-12], [-84], [-68], [2], [-68], [-52], [-38], [-18], [-16], [-22], [78], [-56], [-2], [-32], [42], [34], [-32], [-26], [-18], [64], [8], [24], [12], [-68], [32], [50], [4], [40], [-50], [-34], [18], [-68], [-12], [94], [44], [-54], [92], [-16], [-22], [24], [40], [-22], [26], [-8], [46], [-14], [-16], [-54], [2], [-72], [-48], [-68], [58], [-60], [48], [-100], [-52], [28], [-82], [-46], [-14], [-64], [50], [42], [-68], [24], [28], [2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1368_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1368_2_a_f();".
// 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_1368_2_a_f(:prec:=1)
    chi := MakeCharacter_1368_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_1368_2_a_f();".
// 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_1368_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1368_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![0, 1]>,<11,R![-2, 1]>],Snew);
    return Vf;
end function;