
// Make newform 2280.2.a.t in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2280_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_2280_2_a_t();".
// 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_2280_2_a_t();".
// 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 := [2, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-1, 2, 0], [-4, -1, 1]];
    Rf_basisdens := [1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_2280_a();"
function MakeCharacter_2280_a()
    N := 2280;
    order := 1;
    char_gens := [1711, 1141, 761, 457, 1921];
    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_2280_a_Hecke(Kf)
    return MakeCharacter_2280_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], [1, 0, 0], [-1, 0, 0], [0, 0, 1], [1, -1, -1], [2, 0, 1], [1, 1, 0], [-1, 0, 0], [1, -1, 0], [1, 1, -1], [3, 1, 0], [2, 0, 1], [-1, -1, -1], [4, 0, 1], [-1, 1, 0], [1, 1, 0], [4, 0, 2], [5, 1, -2], [2, 2, 0], [2, -2, -2], [10, 0, 0], [-2, -2, 0], [5, 3, -2], [-1, -1, -3], [10, 0, 1], [2, 0, 2], [-2, -2, 0], [-2, -2, 0], [10, 0, -2], [5, 1, -2], [-10, -2, 0], [-1, 1, -1], [2, 0, 4], [10, 2, 2], [-4, -2, 4], [-5, 1, -4], [0, -2, -2], [-2, -2, 1], [-9, 1, -2], [9, 1, 6], [2, 2, 6], [0, 2, 2], [-11, 3, 1], [16, 2, -1], [-7, 1, -4], [-10, 2, -2], [0, -4, 0], [-6, 2, 0], [-5, 1, 2], [-5, 3, 2], [6, -4, -4], [-1, 1, 1], [6, -4, -4], [1, -1, -3], [-19, 1, -2], [-15, -1, -6], [3, -1, -3], [-4, 4, 6], [2, -4, -4], [-7, -3, 3], [-4, 0, 5], [3, -1, -2], [-18, -2, -2], [-11, -5, 3], [8, 2, 2], [-5, -1, -8], [9, 3, 4], [0, 2, -5], [-1, 1, -2], [20, 2, -4], [16, 2, -4], [1, -1, 1], [-6, 6, 1], [4, -2, 1], [1, -5, 0], [-14, 6, 4], [-4, -2, -8], [-16, -2, 0], [3, 3, 9], [-8, 2, 6], [5, 3, 3], [16, -6, -4], [-4, 4, 2], [0, 2, 7], [14, -2, -4], [1, -1, -4], [-15, -3, -1], [-10, 4, 2], [-2, 4, 4], [2, -2, -7], [-9, 1, 4], [11, -3, -5], [-28, 0, -2], [-23, -1, 1], [14, -2, -6], [-11, -5, 2], [-13, -1, 1], [-1, -1, -1], [-20, 0, 2], [2, 4, 0], [-18, -2, -2], [-10, -4, -4], [-9, -3, -6], [17, 5, -1], [4, -8, -6], [-8, -6, 6], [7, 1, -4], [-12, 6, 4], [4, -4, -8], [6, 4, 2], [-18, -2, -6], [10, -4, -2], [-15, 1, -8], [4, 0, -2], [-8, 0, -12], [19, 3, 5], [0, 4, -9], [-11, -5, 2], [11, -1, 4], [0, -4, -4], [0, 2, -2], [2, 0, -9], [-31, 1, 4], [2, 2, 4], [24, -4, -2], [-20, -2, -12], [5, -7, 2], [-11, 3, 5], [-2, 2, 7], [0, -2, -4], [16, 4, 0], [-10, 2, 4], [5, -1, 4], [-2, 8, 2], [-24, 2, -4], [-34, 4, 4], [11, 7, 8], [-8, 4, 2], [-3, -3, -2], [20, -2, 4], [8, -4, 4], [-24, -6, 2], [0, -8, -7], [13, -1, 6], [-12, 6, 2], [6, 2, 2], [6, 8, -2], [19, -5, 4], [14, 6, 4], [26, 6, -4], [18, 8, -5], [8, 2, 6], [-14, 2, 9], [16, 0, -4], [-2, -2, 6], [-10, 2, -4], [2, 6, 8], [8, 2, 10], [0, -6, 2], [-5, -9, -3], [-1, 1, 2], [13, 9, 2], [-16, 0, -9], [20, 0, 8], [29, -7, -4], [25, 7, 2], [-34, -2, 0], [-24, -2, 6], [-12, -2, -16], [-10, 4, -8], [16, -4, 6], [36, -2, -4], [-5, 5, 1], [42, 0, 4], [2, 2, 12], [-16, -6, 10], [44, 0, -2], [4, 6, 2], [-2, -2, -10], [12, 2, -4], [22, -6, 3], [36, 0, -4], [-24, -6, 7], [39, -1, -6], [11, 5, 16], [14, -4, 0], [-4, -2, -3], [16, -4, -2], [-26, 4, -6], [-29, -3, -1], [-48, 2, 1], [-1, 9, 4], [0, -4, -8], [21, -3, -3], [35, 5, 0], [1, -3, 10], [11, -1, 2], [22, -8, 0], [11, 7, 4], [21, 3, 16], [37, -3, 1], [10, -6, 4], [24, 2, -5], [-18, -4, -16], [3, -3, -13], [-19, 5, 4], [4, 4, -8], [21, -1, 10], [6, -4, -6], [0, 12, 4], [-6, 8, 2], [-34, -4, 6], [-42, 2, 1], [3, 1, 2], [26, -2, 6], [-18, 4, 10], [14, -6, 5], [19, 3, 3], [47, 1, 2], [23, 3, 4], [30, 4, 8], [-15, 3, -4], [-1, 7, -11], [-4, 4, -9], [19, 1, -6], [0, -10, 4], [15, -9, -6], [28, 4, 2], [-28, 0, -16], [15, 1, 3], [-12, -6, -10], [-29, -7, -8], [22, -2, -8], [47, -1, 5], [-18, 6, 3], [11, -3, 4], [-31, -7, 2], [-12, -10, 0], [11, -3, 11], [-18, 2, 10], [-14, 2, -16], [-32, -4, -8], [6, -6, 9], [-12, -2, -10], [-9, -9, 4], [-23, -1, -1], [-16, 0, -13], [2, 2, 4], [7, 13, 8], [-31, -9, 6], [-2, 0, 6], [-2, -12, -4], [3, 5, -14], [-34, 4, 6], [15, 3, -8], [-53, -3, 5], [27, 3, 10], [-26, -2, -12], [7, -5, -4], [-20, -2, -8], [-10, 6, -4], [28, 0, 4], [-8, 6, 12], [18, 0, -1], [-21, -1, -4], [15, 5, -8], [5, -3, 9], [28, -2, 6], [-36, 0, 4], [7, -5, -8], [16, -14, -8], [-6, 2, 0], [4, 6, 18], [-22, -2, -10], [-32, 2, -1], [-8, 0, 11], [25, 7, -10], [34, 0, 2], [-4, -2, 10], [-13, -3, 1], [14, -6, -8], [-34, 2, -4], [21, 11, -8], [28, -2, 0], [-12, -8, -17], [-23, 7, -1], [-22, -8, -8], [-39, -7, 8], [12, 4, 16], [-7, 5, 7], [28, 6, -6], [21, 3, 6], [-3, 9, 0], [28, 0, 18], [24, 10, -11], [-10, 12, 6], [38, 6, -4], [-71, 1, -4], [-44, -8, -4], [-38, 2, 7], [-14, 8, 0], [25, 1, 16], [38, 2, -4], [-35, -1, 2], [14, 6, -14], [-28, -10, 3], [12, 0, 20], [-6, 0, -12], [-31, -1, 11], [-34, 0, -6], [-35, 3, 4], [-50, 4, -4], [11, -5, -3], [38, -2, 2], [-19, -5, 6], [22, 12, 0], [-9, -7, -5], [-20, 4, -4], [-10, -4, -14], [-32, -6, 8], [7, 5, -8], [-10, -12, -2], [25, -7, -7], [-12, 8, 6], [13, -3, 0], [-10, -4, -20], [-44, -8, 0], [24, -4, 14], [25, -9, 2], [-11, 13, 16], [-48, -10, 4], [41, 1, 0], [36, -4, -2], [-57, 1, 2], [10, -6, -6], [-7, 7, -6], [-42, 4, 0], [-55, -3, 4], [31, 3, 18], [2, 6, 13], [-10, -4, 11], [-4, 6, 16], [-15, -7, 13], [-33, -3, -4], [9, -15, -8], [-58, 6, 8], [-3, -7, -10], [20, -8, -16], [52, 4, 0], [-36, -10, 0], [29, -1, -12], [54, -4, -5], [-6, 0, 18], [18, 2, 14], [16, 2, 22], [53, 9, -4], [9, 7, -9], [7, 9, 7], [-29, -1, 0], [-38, 6, 12], [72, -2, -4], [-2, 4, -4], [-14, -2, 8], [-10, 6, 6], [-22, -6, -7], [22, -4, 13], [-14, -8, 8], [-12, -8, 2], [10, -8, -14], [-43, 3, -5], [16, 12, 4], [-15, 7, -14], [53, 5, -7], [18, 6, 8], [-34, 8, 2], [0, 12, 12], [-65, 1, -1], [-30, -16, 12], [-48, -6, 6], [-12, -18, -1], [-19, 5, 13], [11, 7, -4], [-22, -2, 7], [34, -8, -4], [19, 1, 4], [-5, -11, -2], [22, -14, -14], [-34, 0, 4], [18, -6, 7], [47, 1, -6], [-34, -4, 10], [17, -15, -16], [-31, -9, -13], [-14, -6, 3], [-56, -8, 6], [16, -14, -7], [7, -3, 12], [13, -7, -17], [21, -1, -16], [-4, 14, 8], [-18, -12, 16], [19, -9, -4], [18, -6, 6], [25, 13, 10], [3, -9, -9], [-10, 10, -4], [8, 14, -10], [41, 5, 19], [-18, -10, 10], [-27, -5, -21], [20, 6, 25], [-11, -3, -16], [-8, -4, -12], [24, 4, -2], [2, -8, -12], [-40, -14, 8], [26, -2, 4], [16, 4, -8], [-48, 10, 0], [-22, -2, -20], [9, -7, -1], [34, 0, -9], [-3, 3, 0], [20, 0, 0], [-60, 6, 7], [-31, 9, 10], [18, 2, 16], [10, -16, -16], [-44, -8, -6], [-53, -3, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2280_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2280_2_a_t();".
// 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_2280_2_a_t(:prec:=3)
    chi := MakeCharacter_2280_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_2280_2_a_t();".
// 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_2280_2_a_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2280_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![16, -14, 0, 1]>,<11,R![96, -26, -4, 1]>,<13,R![36, -2, -6, 1]>],Snew);
    return Vf;
end function;