
// Make newform 196.4.a.d in Magma, downloaded from the LMFDB on 11 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_196_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_196_4_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_196_4_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_196_a();"
function MakeCharacter_196_a()
    N := 196;
    order := 1;
    char_gens := [99, 101];
    v := [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_196_a_Hecke( : Kf := Kf)
    return MakeCharacter_196_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 := 4;
    raw_aps := [[0], [10], [8], [0], [-40], [12], [58], [-26], [-64], [-62], [-252], [26], [-6], [416], [396], [-450], [-274], [576], [-476], [-448], [158], [-936], [-530], [390], [-214], [-1432], [-764], [324], [-1334], [1798], [-384], [1814], [1666], [-1126], [2694], [-2648], [556], [-328], [4268], [3476], [2268], [276], [-3000], [3278], [-2362], [1036], [3524], [1336], [-1290], [-5524], [6314], [-3960], [7018], [2394], [2766], [7968], [2900], [-2640], [-1522], [-4534], [-4834], [4656], [7238], [-1096], [3818], [1998], [-7936], [2766], [8352], [5924], [-2226], [3880], [2584], [10534], [-4472], [-2468], [-1046], [-2124], [11598], [2770], [-9438], [5550], [-3000], [-12926], [408], [-14452], [10258], [-5498], [16316], [8944], [9422], [-13820], [-13264], [-5940], [-8252], [-4704], [10788], [14586], [-26], [11214], [-5424], [-17618], [3562], [2838], [-360], [-22018], [-1454], [-13818], [6696], [-10010], [-2880], [6522], [6614], [-5266], [3344], [-4882], [15898], [-6132], [-24198], [17456], [656], [-18214], [-30252], [-10836], [-9578], [12442], [-25174], [-34188], [5204], [32880], [-3912], [-16008], [9960], [12378], [-34670], [10898], [25808], [21054], [24276], [21526], [12806], [13214], [32248], [-14316], [-25168], [-9356], [-2372], [-11694], [20506], [-28136], [-37070], [6198], [-31876], [132], [38244], [-7008], [36664], [-45510], [3838], [16832], [-40928], [-24070], [-17152], [32910], [-6822], [48420], [-49216], [-35264]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_196_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_196_4_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_196_4_a_d(:prec:=1)
    chi := MakeCharacter_196_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_196_4_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_196_4_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_196_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![-10, 1]>,<5,R![-8, 1]>],Snew);
    return Vf;
end function;