
// Make newform 8496.2.a.bb in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8496_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_8496_2_a_bb();".
// 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_8496_2_a_bb();".
// 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_8496_a();"
function MakeCharacter_8496_a()
    N := 8496;
    order := 1;
    char_gens := [5311, 6373, 7553, 3601];
    v := [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_8496_a_Hecke(Kf)
    return MakeCharacter_8496_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, 0], [0, 0], [1, -2], [3, 1], [1, -2], [-5, 2], [0, 3], [0, 5], [-4, 1], [-7, -1], [5, -9], [-2, -3], [-5, 5], [5, -6], [-9, 3], [-5, 2], [-1, 0], [-5, -3], [-7, 6], [-3, 8], [-1, -3], [3, 0], [-1, 1], [7, -11], [3, 0], [0, 6], [2, -2], [4, 5], [-6, -3], [9, 0], [-11, 8], [-12, 14], [-11, 2], [-13, -2], [-16, 5], [3, 9], [9, 0], [13, -9], [-1, -13], [-13, -1], [-3, -4], [13, -15], [9, -12], [8, 0], [12, -14], [11, -9], [4, 3], [12, 4], [-24, -3], [-11, -3], [12, 6], [1, 4], [-16, 12], [9, -10], [12, -12], [1, -1], [9, -4], [9, 2], [-3, 4], [-6, 6], [14, -15], [-20, -1], [16, -16], [12, -25], [-14, 11], [23, -10], [0, -18], [-18, -5], [23, -13], [-27, 0], [8, -13], [-12, 16], [-15, 24], [30, -7], [-15, 12], [-17, 2], [-30, 4], [3, 0], [14, -5], [-18, -12], [-14, 28], [1, 0], [13, 1], [8, 7], [34, -1], [4, 1], [3, 16], [17, -27], [-12, 19], [-6, -3], [0, -24], [-18, -5], [-6, -19], [-1, 17], [7, -12], [-3, -11], [-17, 29], [-13, 17], [-19, -8], [-15, 18], [-18, -9], [18, -12], [-7, 22], [1, -17], [-9, -12], [15, 4], [-3, 20], [-13, 5], [-6, -14], [24, 1], [5, -14], [-30, 14], [-24, -3], [-1, 18], [33, -12], [23, -8], [18, -3], [40, -8], [-37, 5], [0, -3], [19, -19], [-6, 4], [34, 10], [39, 0], [-24, 18], [-25, -11], [-24, 6], [14, -1], [-26, 0], [23, 4], [-24, 39], [-24, 1], [16, 18], [-42, -1], [-27, 6], [-2, -21], [-20, 14], [8, 6], [0, -2], [-12, -7], [-6, 14], [23, -29], [-15, -6], [-9, -10], [-29, 27], [-31, -10], [-12, -13], [9, -32], [-18, -4], [26, 11], [-26, -16], [12, -6], [16, -12], [-25, -4], [15, -8], [-21, 39], [0, -22], [45, -6], [27, -40], [-23, 28], [17, -13], [0, 8], [-6, 15], [40, -11], [11, -16], [-23, -22], [-12, 19], [39, -23], [-12, 4], [6, 0], [19, 16], [-14, 0], [22, -5], [1, -22], [-36, -8], [-9, 6], [51, -2], [-40, 34], [4, 27], [-26, 20], [18, -16], [12, 12], [17, 28], [-28, 2], [-27, 40], [22, -47], [-20, 30], [-18, 22], [-21, 48], [-4, 31], [39, -32], [19, -14], [16, -12], [-35, 37], [-28, 20], [-6, 3], [-22, 9], [-23, 29], [45, -13], [14, 7], [28, -8], [33, -44], [-42, 3], [22, -9], [-24, 6], [-17, 23], [19, -6], [-6, -21], [22, 25], [-2, 30], [42, 9], [13, -32], [-10, 33], [-42, 0], [-5, -2], [34, -13], [-1, 12], [-15, -9], [-12, 15], [-19, -10], [33, 16], [31, -4], [28, -20], [17, 19], [6, -38], [-27, 21], [32, -31], [0, -46], [12, 12], [-24, 3], [-9, 15], [11, 30], [-8, -7], [-11, -8], [-6, 8], [37, 4], [-29, 2], [-39, 25], [-21, -6], [33, -17], [51, 6], [-32, 0], [11, -24], [-46, -14], [40, -13], [3, -25], [-25, 40], [51, -18], [7, -7], [-9, 6], [-48, 1], [56, -29], [-49, -13], [15, -18], [27, 8], [33, 2], [21, 6], [46, -6], [15, -16], [42, 5], [9, 30], [1, 8], [-4, 46], [-12, 30], [-15, 4], [-48, -12], [48, -17], [-70, -2], [1, -36], [-7, 40], [-14, -12], [51, -45], [10, -9], [-31, 16], [31, -30], [-8, 9], [25, -31], [5, 0], [-5, 0], [5, 20], [-75, 12], [48, -14], [62, -38], [-42, 36], [33, 6], [1, 7], [-13, -2], [-44, 55], [-37, -6], [46, -32], [23, 34], [-12, -6], [32, -59], [-17, 40], [-19, 52], [-9, 6], [3, 9], [-51, 44], [-9, 9], [45, -34], [23, 31], [-5, -10], [-51, 26], [-54, -12], [16, 32], [-23, 44], [-41, 28], [57, -35], [-3, 45], [20, 18], [75, -16], [0, -11], [41, 8], [3, -3], [-16, 23], [59, -20], [27, 25], [9, -42], [38, -42], [-27, 14], [42, 0], [-36, -6], [-5, -47], [38, 4], [15, -16], [-18, -24], [-3, -34], [-32, 42], [16, -40], [-57, 31], [0, -33], [54, -38], [-4, -12], [-7, 32], [40, -73], [5, -13], [29, -15], [-18, 12], [17, -15], [-29, 42], [16, 6], [-27, 68], [-20, 62], [-68, -4], [24, 6], [-5, 14], [-15, -24], [-48, 12], [-27, 20], [27, -28], [12, -25], [35, 19], [1, 41], [37, -37], [5, -9], [-4, 32], [18, -36], [27, 39], [-48, -7], [-29, 22], [-39, 54], [-8, 43], [20, -20], [-39, 2], [25, -39], [-15, -45], [76, -8], [54, -54], [14, 4], [-1, -43], [-7, 52], [-51, 6], [0, -9], [26, 22], [55, -9], [6, 45], [54, -39], [-39, 45], [-18, -40], [43, 18], [-42, -22], [-27, 15], [30, 3], [-57, -6], [-35, 50], [-11, -10], [-17, 18], [15, -33], [-22, 21], [-29, 2], [-37, 34], [-22, -41], [58, -5], [-24, 10], [15, 9], [-26, 38], [19, 9], [42, -64], [-22, 67], [55, -74], [-15, 64], [34, -52], [56, -30], [-15, -11], [-6, 23], [-67, 30], [24, -46], [48, -54], [-21, 24], [3, 12], [-16, -45], [-25, 7], [-36, 11], [-41, -18], [-9, 30], [60, -24], [32, -52], [49, 3], [-29, 52], [-7, 49], [-14, 1], [-55, 22], [-6, 19], [10, -26], [48, -36], [-40, 40], [30, 23], [71, -23], [19, -51], [15, -42]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8496_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8496_2_a_bb();".
// 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_8496_2_a_bb(:prec:=2)
    chi := MakeCharacter_8496_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_8496_2_a_bb();".
// 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_8496_2_a_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8496_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-5, 0, 1]>,<7,R![11, -7, 1]>,<11,R![-5, 0, 1]>],Snew);
    return Vf;
end function;