
// Make newform 2352.4.a.bi in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2352_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_2352_4_a_bi();".
// 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_2352_4_a_bi();".
// 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_2352_a();"
function MakeCharacter_2352_a()
    N := 2352;
    order := 1;
    char_gens := [1471, 1765, 785, 2257];
    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_2352_a_Hecke(Kf)
    return MakeCharacter_2352_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], [3], [12], [0], [-20], [-84], [-96], [-12], [176], [58], [264], [258], [0], [-156], [408], [-722], [-492], [-492], [-412], [-296], [240], [-776], [-924], [-744], [-168], [-1524], [408], [820], [-918], [-110], [-16], [-1692], [1126], [-1092], [1070], [120], [1836], [-916], [-504], [-1836], [-2372], [-1092], [-2512], [-2430], [-1762], [-3096], [-156], [-5040], [-2172], [2700], [-3802], [4408], [3096], [924], [-2760], [2360], [4020], [-4800], [6446], [-2602], [6900], [-4452], [2436], [7488], [-1752], [-1562], [7092], [366], [6364], [-10500], [408], [11936], [2448], [11374], [5892], [10488], [4514], [-6036], [-6770], [12504], [-9492], [5182], [5720], [13608], [-12864], [13252], [226], [-11334], [-1596], [-12728], [3012], [4296], [8184], [12164], [-972], [-7728], [11604], [-10848], [18132], [6950], [-17012], [3926], [18828], [11990], [15716], [-13872], [-8820], [-16872], [6056], [10752], [-20256], [-28190], [29318], [-24348], [25184], [32318], [3948], [-13848], [-3158], [24596], [-15468], [13470], [-9564], [-13852], [324], [24922], [-17886], [6792], [-1512], [-11244], [1996], [656], [-1056], [-18702], [17904], [-7560], [-14292], [-26364], [17220], [16442], [31332], [-25810], [-12368], [-6316], [-23868], [48216], [-27300], [8640], [-24372], [-2176], [-27574], [16968], [1860], [-2280], [-36084], [-24152], [-36336], [432], [22176], [-43524], [-1868], [-9238], [30616], [27540], [-16402], [-55176], [-27096], [-16812], [-37294], [-23004], [1940], [12348], [43240], [-29338], [-9144], [49560], [-2900], [17954], [5712], [-21660], [62336], [-38052], [-7166], [1272], [14120], [-23196], [63174], [-7212], [25990], [-51144], [39072], [56068], [1044], [33780], [49036], [-29832], [-17970], [6586], [-29736], [-1440], [71798], [5880], [76116], [-22608], [-31500], [-87876], [28632], [-61004], [-61990], [82668], [-27826], [23772], [48312], [42996], [76704], [16728], [47496], [52320], [-26984], [-46982], [-19902], [-59136], [-72094], [-31296], [76692], [-36846], [-14184], [67208], [-2448], [-13012], [-35190], [11772], [-47448], [-53894], [-40572], [17568], [-35352], [-34546], [88660], [-86352], [15044], [-107532], [-118512], [78294], [67704], [119016], [-57792], [-33420], [-30652], [1328], [-13242], [8664], [36184], [-1680], [-6900], [120380], [60626], [-78612], [-128436], [-124728], [59312], [-99388], [-65532], [-101052], [110448], [-14292], [-40682], [1512], [-57220], [37442], [58812], [-110676], [122208], [13056], [-58128], [-75696], [-124228], [-112218], [26970], [-75948], [-17040], [-46640], [10584], [10500], [-9756], [25432], [127602], [-61010], [-3432], [-105336], [61046], [-92556], [-25642], [-28140], [57206], [-21852], [-139200], [-81900], [110028], [-105252], [-17880], [139306], [13448], [87652], [152692], [5406], [-31268], [104916], [51584], [70398], [-1776], [26578], [150142], [133260], [-162584], [29808], [-117012], [58192], [-19152], [-45554], [48060], [-126666], [-60228], [136008], [111866], [-37032], [7036], [-24204], [-9512], [106178], [-121290], [165178], [206808], [47052], [-106868], [141204], [48890], [-200664], [147168], [-109128], [-74194], [-28586], [-31500], [-59712], [-72262], [-62204], [-87540], [154260], [-24528], [-137748], [-5556], [160922], [24502], [182712], [-176114], [-231336], [-16728], [-107004], [52786], [-62512], [-154082], [75192], [-11864], [-85756], [-65004], [130374], [71652], [-15784], [-14166], [-162308], [-151716], [155408], [-76466], [-100704], [-5146], [153396], [-139400], [210912], [82416], [28728], [-5388], [194714], [-85920], [-201566], [160860], [-130272], [-182328], [-81996], [-21188], [169176], [219998], [31932], [-155980], [156732], [114640], [9078], [-48120], [47040], [-190276], [-257006], [75780], [176546], [-61144], [137856], [243588], [141528], [89494], [38866], [196236], [93900], [144576], [59954], [48508], [291348], [19910], [-147324], [145552], [194136], [57456], [-79728], [-223238], [289812], [-216736], [204540], [-59640], [156060], [-38380], [-214954], [198588], [-301464]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2352_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2352_4_a_bi();".
// 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_2352_4_a_bi(:prec:=1)
    chi := MakeCharacter_2352_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_2352_4_a_bi();".
// 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_2352_4_a_bi( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2352_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![-12, 1]>,<11,R![20, 1]>],Snew);
    return Vf;
end function;