
// Make newform 11.6.a.a in Magma, downloaded from the LMFDB on 21 August 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_11_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_11_6_a_a();".
// 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_11_6_a_a();".
// 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_11_a();"
function MakeCharacter_11_a()
    N := 11;
    order := 1;
    char_gens := [2];
    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_11_a_Hecke( : Kf := Kf)
    return MakeCharacter_11_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 := 6;
    raw_aps := [[-4], [-15], [-19], [10], [-121], [-1148], [686], [-384], [3709], [-5424], [-6443], [12063], [-1528], [-4026], [7168], [-29862], [-6461], [-16980], [29999], [31023], [1924], [65138], [-102714], [17415], [66905], [96730], [-95704], [-32658], [-185438], [72849], [-78184], [-462], [296233], [-399818], [72670], [-303082], [-532987], [282076], [-573588], [-386286], [545079], [-279485], [-444437], [-18476], [270182], [43320], [1029684], [461281], [-855570], [-665805], [1209336], [-571482], [-267080], [1387367], [-885922], [1446874], [-353878], [525260], [-595610], [732318], [2233804], [-1531080], [-1142680], [586956], [-233857], [-935503], [-1058231], [506186], [467636], [304470], [2518683], [-3018412], [994429], [1728958], [454765], [2275573], [389781], [-1619330], [-5543678], [-2704926], [3373372], [-4525514], [-684534], [-4225907], [-2091848], [1562843], [-3004493], [-2445516], [7791036], [-1051961], [3970035], [-8539076], [-1864867], [5157272], [4533404], [1711634], [9738219], [4302789], [2622800], [-2496344], [11432312], [-9815290], [-8191924], [-7542864], [-8694004], [2033793], [3517796], [-8345348], [6150224], [-6862318], [-9455362], [-4636576], [6057042], [-5639935], [1126165], [-14201961], [1607939], [3102355], [6888515], [-12413434], [-8109935], [17806334], [15517902], [-2181056], [22989167], [2340922], [-19269445], [-21466515], [-16770543], [-17537268], [14738654], [-4809460], [8293171], [-352294], [16898460], [-36652], [-31746246], [20198512], [15565993], [-29166672], [-16521482], [5566142], [11880069], [-13285628], [-653987], [24774713], [27129062], [-28423208], [26592183], [-22250016], [28342788], [36644533], [16877164], [2739406], [-31328580], [-24976212], [-11061266], [-20173854], [9107336], [-36770862], [-49574325], [35378738], [27805888], [15683521], [20113965], [-20926899], [-31866272], [1389134]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_11_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_11_6_a_a();".
// 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_11_6_a_a(:prec:=1)
    chi := MakeCharacter_11_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_11_6_a_a();".
// 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_11_6_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_11_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![4, 1]>],Snew);
    return Vf;
end function;