
// Make newform 261.2.a.a in Magma, downloaded from the LMFDB on 30 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_261_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_261_2_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_261_2_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
        poly := [-1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_261_a();"
function MakeCharacter_261_a()
    N := 261;
    order := 1;
    char_gens := [146, 118];
    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_261_a_Hecke(Kf)
    return MakeCharacter_261_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+1)]];
    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], [0, 0], [-2, 0], [-1, 2], [-5, 2], [1, -4], [1, -4], [0, 0], [-6, 4], [-1, 0], [4, -8], [-4, 0], [2, 0], [-6, 4], [-1, 6], [8, 0], [-6, -4], [-6, 8], [5, 2], [4, -8], [-2, 0], [6, 4], [-4, -8], [-5, 12], [8, 0], [-3, -8], [-12, 8], [-2, 4], [-9, -4], [-3, 4], [4, 8], [-17, 2], [6, 0], [-13, 6], [-12, 16], [4, -8], [12, 0], [-2, -4], [14, -4], [2, -16], [-2, -12], [-11, 4], [12, -16], [-4, 16], [16, 0], [-7, 6], [-12, -8], [-11, -2], [-4, 8], [24, 0], [10, 8], [4, 8], [19, -16], [-17, 2], [20, -16], [20, 0], [-3, 0], [14, 4], [-25, -4], [0, 0], [-16, 8], [23, -8], [16, -16], [-1, -2], [-7, -8], [3, 0], [12, -8], [-18, 16], [0, 0], [-2, 16], [-28, 0], [-16, -8], [6, -12], [2, -16], [-2, 12], [18, -12], [-5, 0], [2, 16], [-30, 8], [-8, 0], [-30, 12], [-22, 8], [-10, 28], [10, -16], [-19, 14], [-23, -2], [-3, -12], [-15, 0], [10, 0], [13, -18], [-16, -16], [-8, -8], [4, -24], [8, 0], [-13, 6], [-1, 6], [-2, 0], [-22, 8], [-9, -10], [-2, 8], [-13, 22], [-20, 16], [-19, 30], [3, -4], [0, 8], [22, -16], [26, -20], [2, 24], [3, 22], [8, 16], [-4, -24], [-15, 12], [10, -32], [18, 4], [-11, 14], [-17, 20], [-3, -6], [-6, 4], [15, 8], [9, -26], [-19, 28], [17, -8], [-35, 0], [-16, 0], [-3, 18], [32, -16], [-10, -16], [-10, -4], [-18, 4], [6, 8], [-20, -8], [-5, 22], [12, -8], [10, -24], [-26, -8], [16, -32], [10, 0], [16, -8], [-26, 32], [33, -28], [-15, 10], [-28, 0], [30, -28], [-16, 16], [-2, 8], [23, -10], [6, -24], [20, -16], [22, -36], [24, 16], [-10, 16], [-17, -12], [8, 24], [5, 26], [22, -20], [15, -10], [29, 14], [-2, -8], [7, -8], [-14, 32], [-1, 2], [42, 8], [-4, 24], [-12, 8], [-8, 16], [36, -16], [-41, 2], [-8, 16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_261_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_261_2_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_261_2_a_a(:prec:=2)
    chi := MakeCharacter_261_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(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_261_2_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_261_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_261_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1, 1]>,<5,R![2, 1]>],Snew);
    return Vf;
end function;