
// Make newform 1521.2.a.g in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1521_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_1521_2_a_g();".
// 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_1521_2_a_g();".
// 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 := [-4, -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_1521_a();"
function MakeCharacter_1521_a()
    N := 1521;
    order := 1;
    char_gens := [677, 847];
    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_1521_a_Hecke(Kf)
    return MakeCharacter_1521_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, -1], [-1, -1], [2, 0], [0, 0], [0, -1], [4, -2], [-2, 0], [-2, 3], [1, -1], [6, -1], [0, -1], [3, -1], [-2, 4], [-4, -3], [6, 2], [7, 2], [3, -1], [-14, 0], [-7, 2], [7, 1], [4, 2], [-8, -2], [-7, 1], [6, -1], [-5, -1], [2, -4], [5, 5], [2, 5], [7, 1], [2, 6], [4, -1], [-7, 7], [4, 1], [0, 6], [11, -6], [3, 5], [-4, 4], [-14, 4], [8, 2], [2, 3], [6, -2], [-1, 8], [-4, 6], [5, -11], [15, -3], [8, 0], [-4, 2], [-10, 4], [-26, 0], [16, -6], [-10, 5], [0, 12], [-12, 11], [0, 6], [12, -6], [-19, 7], [10, 3], [-10, 5], [-9, 3], [-22, -1], [23, -5], [16, -2], [-9, 3], [0, 9], [-11, -5], [-3, 2], [-12, 10], [-1, -5], [-10, -3], [0, -6], [-3, 9], [19, -6], [19, -3], [4, -12], [-20, 9], [11, -9], [-16, -1], [-3, -6], [28, -4], [25, -4], [8, -2], [15, 4], [-9, 3], [16, -12], [2, -4], [-3, -10], [24, 5], [-19, -3], [26, 0], [-4, 4], [4, -2], [30, -4], [-8, -8], [22, -4], [6, -5], [20, 7], [12, -14], [1, -12], [-7, -7], [32, -7], [14, -6], [-10, -12], [4, 6], [-22, -3], [-30, 6], [-6, -15], [26, -10], [-18, -7], [-16, 0], [-13, 14], [-2, 5], [9, -13], [-17, 1], [2, -11], [-5, 17], [22, 10], [-20, 2], [-4, -4], [15, 6], [-17, 4], [0, -10], [10, -18], [-31, 13], [-4, -6], [5, 8], [14, -6], [-9, 19], [33, 8], [4, -4], [22, 6], [16, -18], [30, 0], [0, 6], [-18, 0], [18, -4], [27, -11], [16, 10], [-4, -13], [1, -1], [6, 8], [8, 0], [-4, -12], [1, -8], [-6, 10], [-1, -2], [-34, 1], [23, -1], [-18, 8], [22, -1], [20, 7], [1, -5], [-34, -6], [28, 0], [-20, 12], [24, 8], [0, -5], [-6, 1], [-18, 8], [20, -12], [-6, 16], [-4, -2], [-12, -16], [-44, 11], [2, 10], [-16, 22], [-5, 10], [25, -15], [6, -17], [-28, 6], [-26, 7], [-2, 22], [25, 10], [40, 8], [46, -7], [-5, 15], [30, -13], [17, 3], [31, -5], [-24, 22], [42, -2], [29, 8], [38, -12], [-30, 4], [10, -25], [17, -7], [-4, 24], [53, -2], [-20, 2], [-27, -10], [0, -18], [-21, 11], [42, 8], [-34, 0], [0, -10], [-21, -8], [-43, 8], [22, 1], [34, 2], [-44, 5], [25, 3], [-23, 12], [-15, -6], [-22, 16], [-20, -11], [-57, 3], [-36, 0], [16, -29], [4, 20], [-17, 4], [-20, -14], [-41, -9], [22, 0], [24, -8], [46, -3], [41, -13], [22, 9], [42, 10], [8, 6], [29, -8], [-1, -15], [-8, 23], [-33, 1], [20, -6], [-61, 6], [2, 9], [-20, 24], [59, 5], [0, 20], [29, 8], [7, -15], [-24, 4], [6, 9], [-25, 5], [2, -14], [13, -19], [18, 11], [-52, 12], [26, -20], [40, -6], [0, 10], [0, -2], [-5, -26], [-46, 17], [12, -8], [45, 3], [-18, 22], [3, 3], [-32, -6], [-23, 1], [-26, 12], [14, 2], [-19, 14], [10, -8], [18, 20], [-2, 8], [8, 26], [-28, 10], [-2, -11], [28, 16], [16, 4], [33, 8], [19, -10], [-36, -9], [19, 7], [30, -12], [46, 13], [20, -2], [-36, -3], [-37, -10], [-3, 19], [-18, -11], [44, -24], [-27, 29], [0, 14], [54, -2], [38, 11], [-29, -4], [12, -22], [54, -4], [24, 0], [10, 18], [-5, 26], [-28, -10], [10, -24], [9, -22], [-2, 19], [-15, 17], [54, -11], [-40, -2], [-54, 12], [8, 11], [10, -18], [43, -9], [12, 5], [-24, 0], [18, -9], [2, -30], [27, 11], [-21, 28], [-38, 15], [0, 10], [14, -2], [-21, -1], [9, -18], [-58, 4], [18, 11], [30, -10], [-18, -1], [64, -6], [10, -12], [28, -33], [-7, 29], [6, -24], [-3, -26], [-20, 30], [-2, -20], [9, 1], [30, 9], [-7, 9], [-45, -5], [8, 1], [-35, 9], [-54, 4], [-23, 18], [-15, 29], [-36, -10], [-42, -8], [-54, 3], [-34, -9], [-18, 15], [-7, -23], [76, -10], [-3, -23], [22, 16], [39, -6], [10, 12], [38, -3], [33, -9], [-38, 12], [-32, 19], [42, -21], [-5, -17], [-22, -12], [50, -16], [25, -24], [32, 16], [-30, 4], [-36, 10], [47, 3], [22, -16], [-4, -7], [43, 7], [62, -1], [-4, -29], [-40, -2], [-4, -24], [-8, -25], [-28, 6], [-62, 3], [12, -5], [6, 2], [-38, -6], [25, -37], [-17, 27], [0, 6], [-3, -17], [50, -3], [-32, -20], [-19, 5], [-14, 12], [36, 5], [40, 2], [34, -29], [40, -22], [52, -14], [-45, 35], [10, 17], [-39, 14], [32, -10], [54, 9], [-8, -6], [46, 0], [-63, 23], [-36, -18], [-8, 6], [-7, 9], [-53, 9], [66, 10], [-74, 9], [2, 15], [-64, 2], [0, -6], [-32, -16], [41, 1], [64, 6], [-62, 24], [28, 0], [-24, 13], [43, 11], [-22, -4], [-59, -1], [-30, 1], [-28, 29], [41, 1], [-31, 13], [-28, 31], [-73, 7], [44, -28], [-33, -4], [16, 17], [34, -12], [64, -20], [33, -38], [-16, 5], [56, -12], [-11, 13], [-4, 30], [24, -4], [42, -25], [27, -14], [-48, 18], [-84, -6], [63, 11], [22, 15], [-38, 34], [-20, 31], [-43, 1], [46, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1521_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1521_2_a_g();".
// 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_1521_2_a_g(:prec:=2)
    chi := MakeCharacter_1521_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_1521_2_a_g();".
// 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_1521_2_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1521_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-4, 1, 1]>,<5,R![-2, -3, 1]>,<7,R![-2, 3, 1]>],Snew);
    return Vf;
end function;