
// Make newform 4356.2.a.d in Magma, downloaded from the LMFDB on 28 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_4356_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_4356_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_4356_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_4356_a();"
function MakeCharacter_4356_a()
    N := 4356;
    order := 1;
    char_gens := [2179, 1937, 1333];
    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_4356_a_Hecke( : Kf := Kf)
    return MakeCharacter_4356_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], [-2], [2], [0], [2], [4], [6], [0], [-8], [-8], [10], [8], [2], [8], [2], [-12], [-10], [12], [-8], [-6], [2], [16], [14], [-2], [-16], [4], [0], [10], [-6], [10], [12], [2], [6], [20], [10], [6], [16], [12], [0], [12], [-10], [-16], [22], [24], [-16], [-18], [-4], [-24], [-14], [24], [4], [18], [-4], [30], [-4], [2], [-10], [-14], [-8], [26], [20], [-2], [16], [-18], [14], [8], [-14], [0], [14], [6], [-8], [-4], [-34], [16], [0], [14], [18], [18], [10], [12], [6], [-4], [-18], [-34], [-12], [-42], [-22], [12], [4], [20], [28], [-8], [0], [4], [-20], [2], [-6], [34], [18], [2], [20], [-36], [20], [10], [34], [-44], [-16], [-24], [30], [-14], [-2], [10], [-32], [0], [6], [-16], [16], [-6], [-20], [30], [22], [8], [-44], [-8], [-36], [34], [-40], [-8], [26], [34], [-40], [-32], [54], [-20], [-34], [2], [-14], [38], [-20], [-18], [16], [-12], [-48], [18], [0], [6], [-28], [-8], [48], [38], [-46], [44], [-48], [0], [-8], [-50], [-6], [38], [-24], [-52], [36], [-14], [-20], [-38], [-8], [-56], [-2], [-62], [-46], [36], [-62], [-24], [42], [-28], [18], [26], [-30], [-14], [-42], [-32], [-12], [-42], [-20], [16], [-6], [50], [32], [58], [60], [58], [56], [4], [30], [4], [14], [-34], [-34], [12], [-8], [20], [-54], [-42], [26], [28], [-2], [20], [52], [36], [-4], [22], [-30], [8], [-20], [0], [14], [-50], [0], [-56], [-54], [-14], [-30], [34], [-36], [-28], [-34], [-14], [0], [6], [-24], [50], [38], [-22], [-12], [-4], [8], [-22], [-52], [60], [0], [60], [-38], [56], [22], [8], [-16], [-8], [-36], [-38], [-72], [-14], [32], [8], [-10], [36], [-72], [10], [70], [30], [-34], [38], [-28], [42], [46], [-66], [36], [-30], [18], [-58], [56], [18], [56], [66], [-26], [-6], [72], [12], [-2], [-74], [20], [68], [-72], [72], [22], [66], [-48], [-18], [-56], [72], [24], [-18], [4], [16], [20], [-18], [-36], [-8], [-54], [-16], [46], [54], [36], [-54], [-36], [-60], [-50], [-36], [14], [-48], [6], [-4], [46], [-32], [44], [-12], [26], [28], [0], [74], [36], [78], [26], [-76], [48], [-24], [2], [-60], [4], [-52], [-72], [18], [-18], [-70], [-48], [2], [-36], [66], [24], [10], [58], [-22], [-74], [-4], [-4], [-30], [8], [70], [40], [-64], [2], [14], [42], [-30], [-30], [-58], [-4], [96], [8], [12], [64], [6], [64], [-96], [24], [-60], [-6], [-44], [34], [54], [84], [-32], [72], [64], [-26], [14], [36], [-52], [14], [36], [2], [-42], [-26], [2], [-4], [-6], [48], [-24], [-102], [-14], [88], [-82], [42], [-12], [44], [24], [26], [-14], [-58], [-88], [0], [-82], [-54], [34], [78], [12], [98], [-30], [16], [-68], [-68], [22], [76], [-44], [-54], [-54], [14], [36], [-44], [70], [20], [-86], [14], [-24], [48], [-54], [-70], [20], [24], [-40], [4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4356_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4356_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_4356_2_a_d(:prec:=1)
    chi := MakeCharacter_4356_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_4356_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_4356_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4356_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![-2, 1]>],Snew);
    return Vf;
end function;