
// Make newform 1521.4.a.o in Magma, downloaded from the LMFDB on 25 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_4_a_o();".
// 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_4_a_o();".
// 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_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 := 4;
    raw_aps := [[0, 1], [0, 0], [0, 1], [0, 8], [0, -8], [0, 0], [117, 0], [0, -66], [-78, 0], [141, 0], [0, 90], [0, -83], [0, 157], [-104, 0], [0, 174], [-93, 0], [0, -164], [145, 0], [0, 454], [0, -610], [0, -265], [1276, 0], [0, 456], [0, 564], [0, 116], [429, 0], [-182, 0], [1506, 0], [0, 896], [687, 0], [-286, 0], [1974, 0], [0, -489], [236, 0], [0, 27], [0, 1022], [1211, 0], [0, 580], [0, -528], [2574, 0], [3744, 0], [637, 0], [2598, 0], [0, 645], [0, 1184], [2522, 0], [1042, 0], [0, 1390], [0, -1390], [0, -1448], [-5850, 0], [0, 3108], [0, -2839], [-3978, 0], [2067, 0], [2052, 0], [-3330, 0], [0, 1620], [-377, 0], [0, -21], [7124, 0], [0, -4805], [0, 1282], [4914, 0], [-518, 0], [0, -2261], [0, -4304], [3575, 0], [6966, 0], [0, -3840], [0, 3251], [0, -4116], [2, 0], [3499, 0], [0, -3186], [0, 4252], [-1209, 0], [0, -6752], [0, 1721], [0, 25], [9462, 0], [0, 4081], [0, -5732], [6617, 0], [-13988, 0], [-2004, 0], [0, 5244], [0, -1457], [0, -11309], [0, -4984], [5460, 0], [0, 1474], [0, 6252], [11388, 0], [0, -10206], [-3876, 0], [0, -9853], [-2121, 0], [-11464, 0], [0, 2751], [6554, 0], [0, 10457], [-12168, 0], [-7722, 0], [-11440, 0], [0, 8917], [0, -8124], [0, 15553], [10554, 0], [-14831, 0], [-7954, 0], [0, -14561], [0, 10037], [0, 4740], [0, -7428], [6201, 0], [0, -9712], [-13494, 0], [11334, 0], [-13236, 0], [0, -6843], [-8021, 0], [21630, 0], [0, -15322], [0, 480], [30186, 0], [0, 6859], [18408, 0], [-21112, 0], [0, -13833], [0, 1828], [0, -17380], [-28496, 0], [17422, 0], [0, 23860], [0, 8124], [0, -116], [0, 3986], [-31278, 0], [-8049, 0], [0, 8098], [0, -4640], [40300, 0], [0, 22820], [12311, 0], [0, 12394], [0, -447], [13923, 0], [-22358, 0], [0, 1288], [0, -9673], [17355, 0], [46982, 0], [8916, 0], [30836, 0], [27480, 0], [-28442, 0], [0, -4029], [-38465, 0], [0, -2820], [0, -12566], [6474, 0], [0, 4354], [-34998, 0], [0, -14559], [0, -32586], [59282, 0], [-37711, 0], [0, 840], [21021, 0], [0, -16730], [0, -837], [45786, 0], [0, -17499], [15106, 0], [49905, 0], [0, 4194], [0, -28231], [12220, 0], [-8006, 0], [0, 23954], [-4368, 0], [-47591, 0], [0, 9192], [0, 25108], [-429, 0], [12766, 0], [0, -12650], [0, -2813], [0, 12940], [27809, 0], [0, -16626], [56240, 0], [0, -37656], [-1716, 0], [44382, 0], [0, -39599], [66053, 0], [0, 24835], [47838, 0], [0, -27425], [8426, 0], [0, 49711], [8309, 0], [0, -18910], [-14289, 0], [0, 46864], [31536, 0], [0, -4549], [40196, 0], [55133, 0], [-27510, 0], [60010, 0], [0, -29102], [0, -10854], [0, -32479], [32398, 0], [-15615, 0], [0, 25208], [0, 6804], [-63295, 0], [0, 15478], [0, 42511], [0, 24816], [-33696, 0], [-68861, 0], [77985, 0], [62112, 0], [-17134, 0], [0, -112], [12181, 0], [-78076, 0], [0, 16106], [-68367, 0], [938, 0], [0, -25570], [0, 3440], [0, 28293], [9750, 0], [61332, 0], [0, 57058], [46982, 0], [-107144, 0], [0, -25845], [0, 14171], [79716, 0], [0, -41878], [0, -16180], [0, -1756], [38142, 0], [0, 49148], [0, -49620], [0, -2660], [-115453, 0], [28182, 0], [0, -39164], [60010, 0], [0, 35060], [-60606, 0], [0, 48983], [-15808, 0], [-23064, 0], [0, -62063], [85997, 0], [0, 34143], [116170, 0], [0, -51024], [0, 30577], [0, 38254], [18213, 0], [-26845, 0], [0, -11844], [0, 26827], [117286, 0], [26626, 0], [0, 19954], [0, 62908], [0, 34055], [0, 80387], [-86970, 0], [-29304, 0], [0, 79178], [6888, 0], [0, -73503], [0, -88444], [72774, 0], [-89, 0], [0, -10633], [0, -57712], [61815, 0], [6966, 0], [-139284, 0], [0, 6007], [0, -39658], [-123082, 0], [-21021, 0], [-122458, 0], [-67821, 0], [19836, 0], [0, 56038], [-98579, 0], [0, -13125], [-43784, 0], [-912, 0], [-24802, 0], [0, -18747], [36192, 0], [-99799, 0], [0, -7288], [130793, 0], [159924, 0], [0, 17564], [-83085, 0], [7048, 0], [0, -25514], [44917, 0], [0, 78690], [0, -30926], [0, -54436], [19305, 0], [-89726, 0], [0, -50428], [-91509, 0], [0, -29112], [0, 47572], [171355, 0], [0, -76874], [0, 92208], [-91260, 0], [-173487, 0], [0, -56321], [-114207, 0], [-113930, 0], [0, -28738], [0, -31396], [0, 42862], [0, -44101], [0, 84648], [0, -14353], [132496, 0], [0, -21672], [-111789, 0], [0, -5559], [-44876, 0], [0, -81600], [61308, 0], [25963, 0], [0, -38368], [0, 79006], [-116454, 0], [0, -129312], [0, -71700], [0, -14923], [-46618, 0], [-102661, 0], [-187761, 0], [0, -76240], [0, 97072], [99255, 0], [0, 9842], [0, 96615], [17901, 0], [-177582, 0], [0, -59604], [-50024, 0], [-80494, 0], [0, 23496], [0, -22352], [142987, 0], [200250, 0], [-44798, 0], [0, -106218], [-186825, 0], [-53870, 0], [104873, 0], [0, -2274], [70356, 0], [0, 22684], [148185, 0], [-212693, 0], [0, 13096], [0, -40711], [0, 19040], [0, -66052], [0, 117736], [0, 100482], [0, -50532], [-64246, 0], [0, 34930], [45810, 0], [0, 108389], [0, 131015], [0, -52528], [-152696, 0], [0, 94060], [128806, 0], [0, -99620], [-167622, 0], [139960, 0], [0, -40321], [0, -76688], [85098, 0], [0, -27656], [0, -87095], [0, -63409], [71398, 0], [0, 153152], [0, -28001], [0, -56412], [0, 89016], [69433, 0], [-127077, 0], [-95136, 0], [114010, 0], [-143975, 0], [-247821, 0], [0, -127646], [126154, 0], [0, 53708], [-157404, 0], [-153231, 0], [0, 139047], [0, -18108], [189798, 0], [0, 116], [0, -104311], [-217152, 0], [0, 163177], [0, -119774], [306798, 0]];
    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_4_a_o();".
// 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_4_a_o(:prec:=2)
    chi := MakeCharacter_1521_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_4_a_o();".
// 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_4_a_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1521_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![-3, 0, 1]>,<5,R![-3, 0, 1]>,<7,R![-192, 0, 1]>],Snew);
    return Vf;
end function;