
// Make newform 882.4.a.a in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_882_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_882_4_a_a();".
// 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_882_4_a_a();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_882_a();"
function MakeCharacter_882_a()
    N := 882;
    order := 1;
    char_gens := [785, 199];
    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_882_a_Hecke(Kf)
    return MakeCharacter_882_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 := [[-2], [0], [-22], [0], [26], [54], [-74], [-116], [-58], [208], [252], [50], [-126], [164], [444], [12], [-124], [162], [-860], [-238], [146], [-984], [-656], [954], [-526], [-1306], [508], [498], [-614], [-1232], [-2808], [-520], [-2516], [2672], [-1164], [1672], [-446], [428], [-4], [590], [3534], [-1098], [-4854], [-1498], [620], [-32], [4268], [-3464], [3252], [-418], [2084], [1662], [-6182], [996], [5994], [6414], [-2686], [-5100], [-4426], [7508], [-3412], [4734], [-5836], [-5620], [-6082], [-7308], [-8020], [4590], [-6546], [7994], [-4650], [346], [6784], [-6098], [-2660], [760], [-104], [-4398], [13236], [9490], [-4236], [918], [-11814], [8374], [-3840], [10166], [8200], [-6074], [2006], [-3728], [6380], [-17180], [-2728], [2574], [-7484], [-7920], [-7254], [17862], [592], [-6402], [-8988], [3244], [-9812], [-12156], [6876], [-20002], [-18404], [9846], [9234], [1510], [-17544], [9246], [-29212], [-7096], [488], [8756], [3364], [-21804], [13488], [28946], [20642], [-17602], [-4266], [-26874], [17128], [11968], [-5278], [-6720], [-16804], [-27522], [21132], [-30], [-15480], [28770], [12418], [-12346], [-38098], [-13824], [-22170], [16288], [8720], [-31372], [-17648], [-2382], [-26650], [-24092], [8194], [16962], [48556], [-34274], [7126], [9222], [37652], [21996], [14844], [-19446], [-39200], [15954], [-2546], [-430], [38266], [-28216], [-27712], [-32976], [-8940], [3288], [33944], [54562], [22202], [27462], [-17630], [32114], [-10878], [48042], [-21240], [-54586], [-38652], [-45496], [-37268], [66058], [-27744], [24568], [33742], [-17050], [48910], [68814], [-13446], [-23816], [-8550], [-21336], [7922], [16122], [-38420], [-8450], [11082], [-35186], [-62146], [-4682], [-57030], [-49452], [-83940], [-61392], [-28226], [34170], [51624], [-26658], [52288], [29154], [63912], [-50132], [-50206], [56426], [32472], [-27376], [-14532], [-40106], [-71288], [-33626], [-54826], [-76072], [50002], [65396], [-51024], [82840], [600], [-12478], [-67310], [40438], [-53556], [63510], [57930], [39404], [35608], [-39840], [-70580], [-93024], [22274], [-38532], [64062], [-79312], [-57706], [-21728], [35632], [56274], [23770], [94620], [17028], [-17036], [104924], [-94794], [77050], [77358], [65058], [73298], [-28898], [-116182], [91022], [73596], [26846], [742], [-36320], [-42554], [-93562], [-76414], [13134], [-32056], [-57972], [-116602], [-32636], [88872], [102882], [-82924], [-30942], [80216], [-108622], [48564], [-104590], [-73854], [-4946], [18420], [-119304], [-95400], [-109512], [-6830], [139392], [15722], [27582], [-124], [85732], [3906], [91164], [-142632], [126228], [428], [102326], [160566], [-15824], [50178], [-133260], [-44616], [-163066], [65472], [124336], [96738], [124892], [121214], [57646], [2302], [-64390], [-17878], [103224], [174160], [33244], [-35140], [67126], [-6974], [33240], [-28226], [146266], [158748], [32284], [99958], [111198], [-63608], [-50528], [175210], [80204], [76052], [-45670], [150992], [77162], [-162256], [-41460], [-190028], [-64052], [182372], [210726], [-97246], [-57854], [86636], [-18898], [-113748], [-104094], [-103992], [100872], [34170], [19990], [-171196], [83112], [177534], [-8876], [-79006], [-121692], [-139776], [124174], [-182714], [87584], [-29664], [209240], [-148942], [-226082], [-164534], [-179462], [-102070], [142328], [71990], [36138], [-217704], [48766], [176246], [23588], [240690], [127252], [-74740], [-12242], [160680], [-168874], [137246], [67918], [-50726], [-27762], [-239840], [-65624], [-118344], [174800], [-180424], [245576], [107270], [37732], [93028], [40226], [205746], [107834], [-131060], [186994], [153894], [-102664], [-82178], [254772], [-216304], [-100886], [101928], [220384], [-84286], [65490], [40576], [65134], [-34032], [-8840], [-221420], [-166906], [-107944], [-252370], [73236], [3434], [30130], [64574], [-43204], [177246], [-135268], [177680], [-204854], [-176926], [99864], [57670], [115574], [19702], [161628], [54264], [36220]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_882_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_882_4_a_a();".
// 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_882_4_a_a(:prec:=1)
    chi := MakeCharacter_882_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_882_4_a_a();".
// 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_882_4_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_882_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![22, 1]>,<11,R![-26, 1]>,<13,R![-54, 1]>],Snew);
    return Vf;
end function;