
// Make newform 1520.2.a.t in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1520_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_1520_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_1520_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 := [3, 4, -4, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-1, 2, 0, 0], [2, -3, -2, 1], [-5, -2, 2, 0]];
    Rf_basisdens := [1, 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_1520_a();"
function MakeCharacter_1520_a()
    N := 1520;
    order := 1;
    char_gens := [191, 1141, 1217, 401];
    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_1520_a_Hecke(Kf)
    return MakeCharacter_1520_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], [0, 0, 1, 0], [-1, 0, 0, 0], [-1, 1, 0, 0], [-1, -1, 0, 0], [1, 1, 1, 0], [2, 0, 2, 0], [-1, 0, 0, 0], [3, 1, 2, 0], [2, 0, 2, 0], [-2, 1, -2, -1], [-1, 0, 1, 1], [4, -1, 0, 1], [-1, 1, 0, 0], [3, -1, 0, 2], [-3, 0, -1, -1], [0, -1, 0, -1], [5, 2, 0, -1], [6, 1, 3, -1], [4, 1, -2, 1], [6, 0, -2, 0], [4, -1, 0, -1], [-1, 1, -2, 0], [0, 0, -2, -2], [7, 2, -1, -1], [1, -2, -4, -1], [-2, -1, -1, 1], [-4, -1, -3, 3], [2, 3, 2, -1], [-3, 1, -1, 0], [4, 0, -3, 0], [-4, 0, -4, 0], [0, 0, 2, 2], [1, 3, 4, -2], [1, 0, -4, 1], [0, 2, 2, 2], [8, 0, 2, 2], [-7, -3, 0, 2], [6, -2, -1, 0], [-5, -1, -1, 0], [-4, 3, 4, -1], [-2, -2, -4, -2], [2, -2, 4, 0], [-9, 0, -5, -3], [-10, 0, -4, 0], [-4, -4, -4, 0], [2, -2, -6, 0], [4, 0, 3, 0], [-4, 1, 1, -3], [-1, -2, 0, 1], [-2, 0, 4, 0], [-6, 2, 0, -4], [4, -3, -8, -1], [6, 2, 0, -4], [1, 2, -3, 1], [5, 1, 2, -2], [2, -1, 6, 3], [-5, 1, 4, 2], [8, 4, 6, -2], [-2, -3, 4, -3], [-3, -3, -6, -2], [-3, 4, -1, 3], [12, -1, -3, 3], [-1, 3, 8, 0], [14, -2, 0, -2], [-19, 3, -3, -2], [6, 0, -6, -2], [9, 1, -5, 4], [-13, 3, -6, -2], [-14, -2, -4, -2], [-10, -2, -6, 2], [5, 1, -4, 4], [-19, -1, 0, 4], [-7, 6, -5, 1], [4, -2, 0, -2], [20, 1, 5, 1], [2, -6, 0, -2], [-8, -6, 2, 0], [16, 4, 4, -2], [-12, -1, 2, -3], [-22, 0, 0, -2], [0, -1, 0, 1], [-2, -8, 2, 2], [13, -5, 3, 2], [14, -3, 4, 3], [11, 5, 2, -4], [0, -3, -10, -1], [6, 2, 0, -2], [-2, 0, -8, 0], [-13, 1, -6, 0], [-5, -1, -2, 2], [13, -1, 4, 2], [16, -1, -1, -5], [-2, -4, -4, 6], [7, -3, 0, -6], [-1, -1, -6, -6], [6, -2, -2, 2], [6, -4, -8, 0], [-2, -2, -9, 4], [-13, 2, -4, 1], [6, 4, -3, 2], [-20, -2, -2, 4], [2, 6, -1, 0], [6, -9, 6, 3], [3, 5, 8, 2], [18, -4, -4, 0], [1, -5, -2, 4], [-10, 6, 4, 2], [-18, 1, 4, -1], [28, 5, 8, -1], [4, 1, 3, 1], [2, 0, 2, -4], [6, 2, -6, 2], [-7, -1, 4, 2], [19, -1, 4, 0], [4, -5, -4, 5], [-11, 1, -6, -2], [11, -1, 10, -2], [2, -10, 2, 2], [2, -1, 8, 5], [-16, 9, 0, 3], [-9, -5, -11, 4], [25, -2, -5, 1], [10, -4, -9, -2], [3, 3, -4, 0], [23, 0, 8, 3], [6, -6, 4, 6], [7, 7, 0, 0], [29, -5, 0, 0], [12, -2, 12, 0], [8, 4, 0, 0], [2, -4, 1, 10], [-18, -4, -2, -2], [-2, 0, -16, 0], [-15, 0, 12, -3], [-7, -4, -4, 1], [-3, 3, 1, -2], [10, -4, -7, -2], [-11, -2, 3, -3], [-14, 6, 0, 2], [22, -1, 2, 1], [34, 4, 0, 0], [3, -1, -4, 2], [16, 12, 3, -4], [14, 3, -6, 3], [24, 3, 8, -1], [-4, 2, 4, 0], [13, 0, 1, 3], [-2, -4, 8, -2], [-10, -4, -3, 6], [-15, -8, 3, 3], [-17, 2, -8, -7], [9, -1, 4, -8], [-30, 7, 1, 5], [16, 1, 9, 1], [2, -8, -2, 6], [-26, 0, 4, -2], [18, 4, 0, -8], [2, 6, 2, 2], [2, -2, 0, 2], [5, -3, 2, 6], [9, -4, 1, 3], [-19, -3, -14, -6], [-4, 5, -2, 1], [-13, -5, 1, 0], [4, -3, -11, -3], [-4, -8, -2, 4], [-14, 8, -6, 0], [-8, -8, 14, -2], [12, 2, -6, -4], [26, -4, 4, -2], [18, -8, 4, 4], [-3, -3, -4, -4], [-8, 12, -12, -2], [0, -2, 0, -10], [-15, 0, -4, 5], [-20, 0, -12, 0], [-27, 2, -12, -1], [-18, 0, -11, 2], [-1, 2, 0, 5], [-5, -3, -10, -4], [-16, -7, -14, 1], [-11, 5, -15, 0], [-9, 9, -3, -6], [-3, -7, 2, -2], [34, 4, 12, 0], [-17, 5, 3, -6], [2, 0, 7, -6], [-18, 1, -6, -3], [2, 4, 0, -10], [41, 2, 5, 1], [19, 3, -10, -2], [20, 5, 6, 1], [30, 3, -4, -1], [-25, 5, 4, 4], [17, -5, 11, -2], [18, -2, 8, 2], [26, 2, 4, 6], [-32, -10, -8, 8], [15, -5, 8, 6], [30, 8, 6, -4], [-20, 0, -10, -4], [-23, -9, -7, 6], [2, -8, -14, -4], [3, -1, -8, 0], [-20, 8, -2, 2], [-3, 3, 16, -10], [-22, -7, -15, 3], [-14, -2, 12, -6], [0, 11, -2, -5], [2, -2, 12, -2], [-7, 2, 0, 3], [3, -5, -6, 2], [-20, -10, 7, 2], [-22, -2, 8, 4], [-10, -9, -8, 7], [-3, -1, 0, 0], [-16, 8, 0, -6], [16, -10, 1, 2], [-20, -8, -4, 2], [2, -1, 12, -1], [8, -4, -4, 12], [8, -2, 2, 4], [21, -3, 2, 6], [-36, -9, -3, 3], [22, -4, -8, -8], [3, -7, -19, -2], [32, -4, -4, -4], [18, 5, 21, 3], [-6, 4, -8, 2], [-2, 4, 8, -8], [-2, 6, -8, 4], [14, 6, 2, -8], [-30, -1, 4, 3], [-31, -7, 6, -2], [-1, 11, 4, -2], [13, -2, 12, -1], [-30, 6, 8, 2], [3, -3, -8, -2], [2, 0, -6, -2], [36, 5, -1, 1], [0, 6, -8, 2], [-1, 3, -8, 6], [-16, 4, 10, -10], [3, 10, 11, 3], [8, 2, 0, -6], [-3, -7, 4, -2], [-58, 0, -6, -2], [4, 4, 12, 4], [33, -5, -4, 0], [0, 6, 12, -4], [-22, 12, 0, 4], [-19, -1, -6, 0], [32, 5, 6, -1], [26, 10, 12, 2], [0, 4, 8, 0], [-10, 2, 0, -10], [14, 11, -9, 1], [13, 4, -5, -5], [34, -2, -6, 2], [14, 7, -1, -7], [8, 4, 7, -12], [-23, -2, 12, -1], [-7, 6, 3, 9], [0, 4, 8, -6], [-8, -3, 0, -3], [40, 6, 10, 0], [-25, 4, 4, 3], [14, 3, 5, 1], [-8, 0, -20, -6], [-2, -4, -8, 4], [-4, 6, -1, 10], [8, 8, 6, -10], [15, -9, 4, 8], [55, 0, -1, -3], [-3, -3, 4, -10], [-23, -3, -8, -6], [4, 2, 18, 0], [4, -5, -4, -11], [17, -5, -12, 2], [-44, 7, -15, -9], [-39, -1, 12, -6], [9, -3, -6, -6], [-36, -4, -4, 6], [13, -3, 18, 6], [1, 3, -4, -2], [26, -10, -8, -6], [-15, -11, -15, 4], [6, -10, -4, -4], [-26, -3, 2, 1], [-2, -8, -16, 8], [7, -1, -12, 10], [7, 8, 3, -11], [18, -6, 10, 4], [11, -9, -1, 8], [-1, -14, -8, -3], [-22, -15, -2, 3], [14, -6, -4, 6], [46, -4, 4, 2], [-3, -5, -16, -8], [-14, 2, 2, 2], [-25, -1, -9, -12], [23, -7, 12, 10], [-24, -2, -15, -2], [-8, 6, 12, -6], [37, 1, -1, -8], [0, 4, -1, -12], [10, -12, -6, 4], [-28, 6, -16, -6], [-22, -2, 0, 10], [43, -3, -4, 4], [25, -2, -20, 7], [-12, 6, 0, -12], [38, -8, 15, 6], [25, 3, 2, 4], [-22, 0, -22, -4], [27, 3, -4, -4], [-2, 8, 22, 2], [16, 0, -16, 6], [2, 14, -4, 2], [20, 9, 10, -3], [18, -14, -10, -10], [10, -3, 24, 5], [-8, -5, 9, 7], [12, -10, 14, 4], [26, -1, 0, -13], [4, 13, 24, 1], [6, 0, -1, 6], [-16, -4, -15, 0], [20, 0, -14, 2], [-47, 2, -8, -5], [-17, 9, 11, -2], [-39, -11, -16, 4], [-45, -7, -8, 0], [-33, -7, -24, -6], [13, 1, 24, 6], [-30, -7, 2, -3], [21, 2, 1, -3], [14, 4, 16, -4], [-5, -1, -6, -10], [-9, -15, 11, -2], [4, 6, -10, 4], [-8, -7, -2, -1], [-20, -3, 6, -3], [21, 4, -9, -5], [-4, 9, -4, -3], [18, -4, -24, 4], [32, 8, 3, -8], [16, 21, 6, -7], [-10, -2, -6, -10], [10, -5, 6, 1], [11, 7, 13, -4], [13, 2, -8, -1], [56, 3, 1, -9], [-14, 8, 6, 0], [51, 5, 9, 2], [6, 18, 0, 0], [28, -14, -4, 2], [-34, -6, 6, -2], [30, -2, -7, 0], [10, 0, -14, 8], [13, 0, 4, -15], [-44, -9, 3, -1], [10, -6, 20, 4], [25, 1, -6, 10], [5, -13, -9, 2], [-8, 8, 2, -18], [-12, 11, 1, -5], [6, 5, 0, 13], [-41, 7, -12, 0], [22, 15, 8, -7], [-39, 11, -4, 4], [64, -1, 2, -3], [6, -8, 4, 6], [10, 4, 26, 4], [-22, -18, -8, 8], [10, 8, 12, -14], [-30, -20, -4, 8], [30, -10, 0, 2], [-1, -6, 3, 15], [42, 0, 0, -12], [-8, 6, -8, 0], [-25, 3, -8, 6], [-18, -6, -4, 2], [-24, -3, -4, -3], [-38, -5, 3, 9], [-59, 11, 0, 2], [-32, 2, -22, -4], [-19, -9, -12, -4], [12, 9, -5, 1], [48, 3, 6, 5], [35, -7, 11, 2], [-14, 0, -28, -2], [51, 5, 22, 4], [20, 0, -10, 12], [3, -3, 17, 10], [-8, 7, 8, 3], [12, 4, 2, 6], [-54, 3, -6, 1], [6, -8, 0, 4], [0, -15, -18, -1], [8, -16, 0, -6], [10, 10, -13, 0], [-53, -9, -19, 4], [24, 3, -10, -7], [8, -4, -16, 12], [-28, -2, 6, -8], [2, 16, -20, 0], [-14, 0, -5, 10], [32, 10, -4, -2], [49, -6, 17, 9], [30, 4, 0, 4], [10, 1, 1, -5], [39, -13, 24, 12], [8, 14, 6, -16], [2, 6, 8, 6], [16, 7, 8, -13], [4, -2, -3, -2], [66, -2, 10, 2], [12, 13, 5, -3], [-14, -5, -14, 7], [-7, 0, 3, 3], [13, -3, -8, 6], [-48, -6, -4, 14], [45, 17, 15, -8], [-1, -7, 3, 6], [-6, 10, 7, -4], [3, 0, 8, -13], [-40, -8, 20, -4], [5, -15, -8, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1520_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1520_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_1520_2_a_t(:prec:=4)
    chi := MakeCharacter_1520_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_1520_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_1520_2_a_t( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1520_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, -16, -8, 2, 1]>,<7,R![32, -48, -16, 4, 1]>],Snew);
    return Vf;
end function;