
// Make newform 1520.2.a.n in Magma, downloaded from the LMFDB on 24 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_n();".
// 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_n();".
// 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_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, 1], [1, 0], [0, 1], [-4, 0], [-2, 3], [6, -1], [1, 0], [0, -3], [2, -3], [0, 2], [-6, 0], [2, 4], [8, -2], [4, -4], [-2, -1], [0, -1], [6, 2], [0, 1], [0, -4], [6, -3], [0, 2], [-8, 2], [2, 0], [6, 0], [-2, -6], [8, -4], [8, -1], [2, -1], [14, -2], [-8, -2], [-4, 8], [-2, -5], [4, -8], [-2, 6], [0, -2], [-10, -4], [0, 6], [-8, 6], [10, 4], [12, 4], [-18, 0], [-4, 3], [6, -8], [-10, -4], [-4, -5], [-16, 3], [8, 6], [-8, -7], [6, -8], [10, 0], [4, -1], [18, 2], [4, -4], [14, 0], [12, 4], [-26, 0], [4, 7], [6, -2], [2, 0], [16, 2], [6, -11], [-12, -8], [-4, 3], [-18, 9], [-10, 9], [0, 1], [-26, 0], [24, -6], [14, -4], [14, -7], [-4, 1], [4, -4], [-26, 1], [-8, -1], [-24, 4], [-2, -2], [-2, -2], [-14, 4], [-6, 12], [12, -14], [-6, -7], [16, 0], [30, -6], [8, 2], [8, -4], [2, -12], [14, -3], [14, -4], [20, 4], [-8, -4], [-32, 0], [-8, 10], [-4, 2], [4, -18], [-16, 9], [-18, 8], [2, -2], [24, 3], [-26, 10], [4, -8], [14, 10], [-12, 8], [2, -2], [4, -10], [-18, 3], [-8, 6], [18, -4], [-24, 14], [18, 0], [16, 2], [-10, 6], [2, 4], [36, 0], [0, 8], [-14, 14], [8, 10], [-8, 1], [-2, -2], [32, -7], [2, -5], [6, 6], [38, 1], [-4, 0], [12, 2], [-26, 2], [38, 0], [28, -1], [32, -3], [-42, 2], [-20, -2], [-24, 2], [40, -2], [22, 8], [-2, 17], [30, -1], [-2, 5], [-24, 5], [22, -13], [-10, -1], [8, 5], [-2, -6], [32, -1], [8, 1], [2, -9], [0, -8], [6, -12], [30, -6], [12, 16], [8, -4], [14, -7], [2, 16], [32, -10], [-8, -8], [-8, 11], [16, -8], [20, -7], [14, -13], [38, 1], [42, 5], [24, 4], [-2, 18], [36, -8], [36, 8], [-2, 12], [-24, 2], [-8, 2], [14, -12], [-14, 6], [14, 0], [-32, 1], [26, -19], [16, 0], [22, -1], [8, 10], [26, -4], [-20, -6], [-10, 8], [-8, -2], [-2, -20], [-24, -1], [32, 1], [-22, -8], [-34, 2], [-24, 21], [-10, 8], [-34, -3], [4, -8], [2, 22], [-52, 3], [6, -18], [-8, 0], [-20, 16], [-6, -21], [0, 24], [14, -12], [-34, 7], [-10, 22], [14, -9], [48, 3], [-18, -16], [16, -8], [38, 7], [-22, -8], [28, -10], [14, 14], [60, -1], [0, 1], [6, -9], [-16, 19], [-10, -11], [14, -18], [-20, 12], [16, -5], [0, -6], [-6, 14], [-24, 17], [2, -4], [16, 10], [-2, 8], [22, 8], [0, -6], [-30, -2], [-24, 17], [32, -25], [-10, 32], [6, -24], [0, -12], [-8, 16], [-20, -6], [-10, 6], [32, -21], [32, -12], [18, 8], [-16, 18], [-16, 11], [18, 16], [38, -8], [12, 2], [-32, 2], [20, -24], [-28, 24], [32, -21], [18, -19], [-26, 24], [-32, 0], [4, -20], [4, -12], [-4, -12], [0, 21], [-10, 6], [-26, 13], [-8, 17], [2, -18], [-34, 24], [4, 12], [-10, -4], [-12, -16], [-46, 12], [-46, 16], [-8, 10], [-20, 16], [-26, 16], [54, -17], [-30, -12], [-24, 15], [10, -9], [-34, 1], [-32, -17], [-18, 22], [-6, -7], [-24, -5], [6, 25], [-76, 1], [-18, -6], [-48, -1], [24, -28], [18, 21], [10, -22], [52, -16], [-64, 2], [-4, -13], [8, -27], [-30, -15], [16, -36], [-40, -12], [6, 21], [-22, 32], [-12, -7], [-30, -12], [-26, -10], [0, 0], [-10, 18], [-52, 16], [-22, 4], [62, -8], [32, -10], [6, -16], [-32, -3], [16, -32], [-30, 24], [14, -11], [12, 15], [-32, 1], [12, -30], [6, -24], [72, 1], [10, -1], [-32, 0], [30, 14], [-4, 28], [14, -10], [-6, 14], [-60, -8], [52, -16], [2, -32], [4, 22], [16, -12], [-2, 3], [50, 12], [0, 9], [30, -5], [18, -7], [-24, 4], [-50, 3], [-14, 34], [40, -9], [56, -5], [-24, 26], [14, 12], [22, 18], [14, -15], [28, 15], [0, -22], [68, 2], [0, 18], [-62, 3], [-58, 2], [62, -11], [40, 15], [-26, -23], [14, 23], [-30, 29], [-24, -14], [-10, 7], [44, -12], [-10, -20], [-64, 15], [8, 8], [22, 26], [12, -24], [-2, 22], [38, -18], [-8, 4], [-10, 20], [-34, 2], [-60, -5], [4, -20], [-10, -9], [56, 2], [-50, 4], [26, 0], [8, -16], [-8, 39], [56, -4], [-50, -16], [54, -4], [-16, 24], [-62, 4], [-36, 6], [-8, -3], [-48, 17], [34, -7], [-72, 16], [-50, 30], [-8, -19], [-32, 8], [50, -16], [14, -1], [46, -20], [38, 16], [-6, 28], [-40, 11], [50, 0], [-40, 27], [16, 18], [44, -31], [-34, -8], [-8, 6], [88, -4], [2, 4], [38, -3], [-28, -22], [56, -8], [16, -10], [38, -4], [-32, 10], [-54, -2], [-44, 16], [-42, 0], [-10, -4], [34, -4], [48, -18], [38, 20], [30, -32], [-76, 15], [14, 6], [-6, -18], [-80, 7], [-52, 6], [-18, -8], [54, -16], [-8, 37], [-28, 48], [-58, 5], [-18, 12], [40, 0], [0, -18], [-10, -5], [48, -24], [6, 12], [6, -11], [-16, 19], [16, 25], [38, -22], [62, -7], [-8, -15], [-66, -3], [36, 4], [36, -37]];
    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_n();".
// 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_n(:prec:=2)
    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_n();".
// 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_n( : 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, -1, 1]>,<7,R![-4, -1, 1]>],Snew);
    return Vf;
end function;