
// Make newform 6160.2.a.v in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6160_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_6160_2_a_v();".
// 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_6160_2_a_v();".
// 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 := [-3, 0, 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_6160_a();"
function MakeCharacter_6160_a()
    N := 6160;
    order := 1;
    char_gens := [5391, 1541, 3697, 5281, 5601];
    v := [1, 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_6160_a_Hecke(Kf)
    return MakeCharacter_6160_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], [-1, 1], [1, 0], [-1, 0], [-1, 0], [2, 2], [0, -2], [1, -1], [-3, -1], [-3, 1], [-2, 4], [5, 1], [-3, 1], [-2, -4], [0, 0], [-3, 1], [0, -8], [2, 0], [4, -4], [-6, 2], [8, -2], [-5, 1], [-6, -6], [-12, 2], [11, 3], [-6, 0], [-14, -2], [6, 4], [-7, -3], [6, -4], [4, 8], [9, -5], [-6, -8], [1, 3], [-9, -1], [13, 3], [-4, -6], [-8, 0], [0, -8], [-6, -4], [6, -8], [2, 8], [0, 4], [-22, 2], [-12, 6], [4, -12], [-8, 0], [-2, -6], [0, 8], [-10, 2], [-6, -8], [9, 3], [-13, 3], [0, 0], [-3, -15], [-12, 4], [-6, -6], [-14, 6], [14, -8], [0, 6], [4, -4], [-6, -2], [-14, 10], [6, 8], [29, 1], [-3, 17], [-2, -4], [-10, 12], [6, 8], [20, -6], [-3, 17], [3, 1], [-26, 2], [20, -6], [4, 16], [-6, 6], [0, 14], [-28, 6], [6, 18], [17, -3], [-12, 12], [-10, -12], [-27, -5], [-1, -1], [-26, 6], [0, 12], [-18, 10], [-34, 0], [12, -10], [13, 15], [-9, 5], [0, 8], [-17, -7], [-36, 4], [-2, 0], [12, -4], [-18, -12], [0, -6], [-2, 10], [5, -7], [16, 12], [12, 14], [0, -12], [-30, 0], [10, -6], [17, 5], [3, -15], [36, 2], [-6, -22], [-1, -17], [-14, 12], [-4, -18], [12, 2], [4, -4], [-8, -12], [6, -8], [13, -1], [-6, 18], [-15, -3], [18, 14], [-10, -8], [-34, -8], [6, -8], [-18, 14], [-14, 10], [9, -3], [8, 6], [-24, 0], [4, -20], [2, 0], [-32, -8], [24, 16], [22, 14], [-13, -21], [21, 1], [5, 17], [30, 0], [28, 0], [24, 10], [-12, -10], [-11, -9], [33, -11], [19, -3], [-12, 24], [2, -22], [18, -12], [14, -4], [12, 18], [-20, 12], [-9, 9], [8, 18], [18, -4], [10, -18], [-24, 0], [-2, -10], [-18, -2], [1, 15], [-12, -22], [26, 0], [6, 24], [-6, -18], [6, 10], [40, 8], [-36, 8], [6, 4], [18, 6], [16, -4], [-10, 8], [32, 10], [0, 18], [6, 10], [-22, 12], [-9, -23], [14, 8], [34, 4], [18, 4], [40, 0], [-6, 4], [34, 0], [38, 8], [-11, 3], [57, 3], [5, 25], [30, 16], [-39, -9], [-18, -22], [20, 18], [43, -3], [-46, -8], [-12, 20], [-7, -27], [30, 4], [-44, 0], [-6, 0], [-30, 12], [-33, -1], [-4, 18], [47, -9], [30, -4], [-42, 12], [-21, -5], [28, 8], [-34, -12], [11, 3], [42, -6], [18, 8], [-44, 12], [18, -8], [-42, 8], [22, 6], [14, -18], [-54, -6], [19, -3], [21, 11], [0, -8], [-40, -6], [-2, -18], [-21, -5], [-15, 19], [15, -1], [-1, 39], [34, 10], [6, -6], [13, 3], [-24, 12], [-7, -27], [9, 29], [-66, 2], [22, -16], [30, -14], [-13, -13], [-5, 21], [7, 13], [30, 16], [-35, 19], [-42, 0], [20, -10], [-30, -4], [-24, 4], [12, -4], [42, -10], [7, -7], [-20, -16], [14, 0], [-54, 4], [-18, -10], [-38, 2], [36, 20], [16, 36], [24, -24], [-22, 8], [9, 37], [-39, -9], [38, -8], [42, -14], [36, -12], [-52, -2], [34, -6], [-30, 12], [2, -24], [-8, 4], [36, -24], [2, 8], [2, 32], [39, -1], [-20, 4], [-60, 6], [-12, -2], [28, 24], [-6, -12], [-22, -4], [28, -24], [-1, -1], [-5, -35], [26, -24], [52, 12], [24, 28], [5, 1], [2, -36], [-39, 7], [0, 24], [34, 22], [-18, 12], [-22, -16], [40, -12], [0, -28], [56, 6], [6, 28], [4, 4], [-15, 5], [18, -24], [-63, 3], [18, 16], [-9, 29], [-10, 32], [36, 2], [34, -12], [-36, 2], [-3, -5], [-2, 46], [38, -12], [-24, 18], [19, -27], [-42, 10], [-44, 0], [14, -24], [-18, -22], [26, -2], [-18, 18], [8, 26], [6, -16], [18, -32], [-54, -8], [10, 6], [-6, -22], [5, -39], [-6, 10], [57, -13], [41, 13], [-24, 18], [-35, -13], [-34, -16], [54, 20], [-11, -9], [-18, 18], [-4, 26], [-26, -16], [-41, -19], [-24, 20], [24, 26], [23, -17], [-3, -7], [-38, -2], [6, -26], [-2, 2], [-45, 5], [56, -6], [6, -36], [-28, 18], [-14, -8], [-25, -17], [-6, -16], [-24, 10], [40, -24], [3, -5], [-12, 20], [-10, 42], [-20, 36], [66, -10], [18, 0], [-53, -15], [-4, -18], [0, -26], [52, -12], [23, -41], [-48, 2], [-18, 32], [-45, -19], [6, 10], [-51, -5], [-46, 24], [-63, -11], [3, -47], [84, 4], [-23, -37], [32, 18], [-66, 10], [-38, -18], [-4, -18], [12, -32], [-14, 10], [54, 2], [-57, 7], [58, 6], [71, 15], [-42, 26], [-57, 21], [14, 24], [-15, -35], [26, 16], [30, 10], [24, 2], [46, -30], [-30, -32], [13, -45], [-36, -24], [-8, -8], [62, 12], [10, 36], [30, 2], [74, -6], [30, -8], [18, -12], [85, 3], [60, 4], [-10, 6], [-14, -6], [-60, 6], [-2, -12], [-39, 21], [26, -36], [-24, -14], [52, 12], [-51, 9], [0, -42], [-26, 6], [-25, 15], [12, -6], [55, 1], [72, 8], [14, 0], [30, -8], [-36, 20], [-14, -42], [-58, 4], [6, 28], [9, 15], [34, 2], [-39, -11], [42, 28], [-78, 0], [2, -18], [-87, 3], [51, -3], [-1, 39], [-30, -20], [48, -12], [-24, -18], [-68, 0], [-36, 16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6160_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6160_2_a_v();".
// 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_6160_2_a_v(:prec:=2)
    chi := MakeCharacter_6160_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_6160_2_a_v();".
// 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_6160_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6160_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 2, 1]>,<13,R![-8, -4, 1]>,<17,R![-12, 0, 1]>,<19,R![-2, -2, 1]>,<23,R![6, 6, 1]>],Snew);
    return Vf;
end function;