
// Make newform 4368.2.a.d in Magma, downloaded from the LMFDB on 24 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_4368_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_4368_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_4368_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_4368_a();"
function MakeCharacter_4368_a()
    N := 4368;
    order := 1;
    char_gens := [3823, 1093, 1457, 1249, 2017];
    v := [1, 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_4368_a_Hecke( : Kf := Kf)
    return MakeCharacter_4368_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], [-1], [-2], [1], [4], [1], [-2], [4], [4], [-2], [0], [-2], [2], [-4], [12], [6], [0], [-10], [-4], [8], [-6], [-8], [-8], [-6], [2], [-6], [8], [0], [14], [10], [8], [-12], [22], [-4], [18], [16], [22], [-12], [-12], [2], [24], [-26], [12], [2], [-6], [8], [-4], [24], [8], [-26], [26], [24], [2], [4], [-18], [20], [-6], [24], [22], [22], [28], [14], [12], [-24], [-6], [2], [12], [18], [-16], [-34], [-14], [-24], [24], [-26], [-28], [12], [14], [14], [-10], [26], [-20], [-26], [40], [-30], [32], [24], [-18], [18], [-2], [-24], [-4], [-12], [-32], [8], [20], [16], [6], [-18], [-36], [-2], [-20], [-38], [4], [26], [28], [18], [24], [-22], [12], [-22], [-8], [14], [22], [28], [-40], [-22], [-20], [48], [30], [0], [22], [18], [34], [44], [-28], [-42], [6], [-8], [0], [30], [-4], [-40], [8], [22], [-14], [34], [-34], [-44], [42], [-30], [4], [34], [8], [-28], [30], [12], [-26], [6], [-4], [0], [-10], [22], [-20], [16], [36], [12], [40], [-6], [42], [-34], [20], [-46], [24], [-12], [-42], [4], [-24], [46], [-38], [26], [-20], [-18], [-36], [-30], [-24], [6], [-60], [42], [-56], [-10], [-56], [44], [-58], [2], [24], [18], [30], [-52], [50], [52], [-38], [-44], [36], [6], [-32], [-18], [2], [30], [2], [-48], [18], [40], [-42], [10], [-56], [-22], [-40], [-24], [6], [-20], [34], [10], [56], [32], [28], [-6], [32], [30], [48], [-14], [-26], [16], [-66], [-56], [36], [-10], [-18], [-4], [-32], [-28], [-50], [-4], [64], [10], [68], [-12], [2], [34], [64], [40], [52], [20], [40], [38], [-46], [-72], [8], [-72], [-20], [-20], [-34], [18], [16], [50], [-14], [-52], [-42], [44], [2], [-6], [8], [72], [70], [-26], [-70], [-52], [-30], [-54], [36], [-10], [-58], [52], [10], [-24], [26], [32], [60], [-58], [42], [-52], [-16], [32], [72], [70], [44], [-52], [2], [18], [56], [30], [-2], [-36], [78], [-16], [-34], [74], [-24], [42], [44], [20], [58], [42], [0], [-48], [4], [42], [-32], [14], [0], [-26], [24], [18], [-14], [-52], [-48], [66], [16], [32], [-46], [-6], [-20], [74], [42], [-40], [-74], [10], [-92], [-28], [-84], [54], [-18], [30], [-40], [32], [-52], [-8], [22], [-86], [-70], [0], [46], [-22], [38], [64], [-54], [0], [54], [-12], [-28], [-62], [-20], [-14], [-54], [16], [6], [-50], [-60], [72], [-38], [-48], [-2], [-66], [52], [12], [-44], [90], [-10], [-8], [10], [-48], [-76], [0], [-58], [80], [-66], [8], [-60], [-14], [-66], [2], [-42], [30], [-88], [-18], [-4], [4], [-16], [-10], [-4], [32], [10], [38], [52], [-12], [24], [-38], [-32], [22], [92], [-30], [-50], [66], [-8], [-54], [-18], [-72], [-66], [-34], [20], [-32], [34], [-50], [-48], [-28], [58], [18], [8], [16], [-54], [-56], [78], [-58], [48], [12], [74], [6], [96], [30], [68], [-80]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4368_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4368_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_4368_2_a_d(:prec:=1)
    chi := MakeCharacter_4368_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_4368_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_4368_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4368_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<11,R![-4, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;