
// Make newform 1050.4.a.g in Magma, downloaded from the LMFDB on 19 October 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1050_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_1050_4_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_1050_4_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1050_a();"
function MakeCharacter_1050_a()
    N := 1050;
    order := 1;
    char_gens := [701, 127, 451];
    v := [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_1050_a_Hecke( : Kf := Kf)
    return MakeCharacter_1050_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]];
    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], [3], [0], [-7], [-72], [34], [-6], [92], [180], [-114], [56], [34], [6], [-164], [-168], [-654], [-492], [-250], [124], [36], [-1010], [56], [-228], [390], [70], [-1350], [-2000], [-696], [-1114], [462], [-1064], [180], [2718], [-1348], [558], [1928], [2410], [-740], [-3984], [1038], [-2568], [-2698], [-4116], [3310], [-1278], [2936], [-3508], [1888], [-3564], [1334], [-2658], [-588], [5690], [180], [-5310], [-828], [-4134], [-2968], [4786], [-4398], [-4772], [-6522], [6244], [-528], [5830], [-5046], [-5020], [7486], [10032], [5942], [90], [10596], [-4016], [-3278], [4628], [2880], [7974], [12346], [9738], [-430], [-1812], [-10690], [-4116], [-9938], [1784], [11712], [7650], [-3674], [-3102], [-8984], [-3612], [-9288], [5848], [-5952], [10748], [16488], [14058], [-14466], [-18524], [4358], [2140], [-2022], [-7356], [11202], [-10564], [18574], [-13188], [22506], [10596], [14618], [-5168], [-5726], [7806], [-18052], [-6208], [-21510], [11140], [-9312], [-4878], [-9744], [2990], [-33266], [-5370], [-384], [-14524], [24750], [-1042], [-36960], [16288], [7810], [-36700], [-29508], [-15136], [-3422], [31446], [-18718], [1686], [-5492], [17310], [35754], [33644], [28734], [28672], [15912], [17534], [40656], [-23870], [29610], [-45484], [-46164], [2986], [6534], [-29756], [-29952], [36268], [-23604], [34184], [-53922], [-40538], [-3606], [-14064], [-33066], [26368], [55884], [51126], [14184], [51680], [-52094], [-14398], [18918], [28728], [56342], [-17100], [-38522], [7472], [49038], [-38788], [17862], [3040], [54326], [-21224], [-60252], [-17102], [-8814], [50172], [53514], [-35750], [-45380], [-934], [44880], [57958], [7800], [20828], [36114], [-26304], [-22782], [-25630], [12490], [37266], [-30288], [25734], [5888], [81034], [-16918], [34956], [50862], [-76336], [61344], [-28950], [-34972], [-5042], [17970], [-27128], [44724], [-22416], [-5974], [-28208], [-54090], [-16716], [-70062], [-51154], [77744], [60210], [79936], [-36564], [104366], [-18438], [-52716], [-40016], [10032], [-19622], [48044], [-11968], [-50406], [49372], [-52272], [48746], [83778], [110256], [20232], [-33480], [56252], [28168], [11990], [-69054], [19536], [-41096], [-50388], [-60244], [78156], [51802], [-20922], [-1668], [7634], [-4482], [-59712], [25598], [38284], [2070], [-13754], [53224], [-74856], [51014], [9466], [44250], [12404], [82854], [60798], [-30044], [-96582], [107606], [-4844], [104542], [5024], [-10010], [-272], [48600], [-141370], [-116278], [11580], [-65856], [-149200], [-25464], [-127906], [86020], [-118932], [-103826], [41682], [129584], [97470], [110790], [90876], [-78066], [26196], [-139286], [-122622], [-124144], [-8706], [12036], [105244], [36646], [113874], [-45160], [-87648], [-68452], [-85682], [-2016], [-94210], [23508], [-167870], [63408], [-70122], [31722], [-160220], [135396], [-147598], [-87036], [13332], [67174], [43530], [-34756], [37222], [-31806], [-4160], [9270], [-119158], [-181708], [-78644], [9444], [36234], [121142], [102210], [166376], [76788], [28892], [-114948], [-101338], [-93438], [60770], [-25112], [41074], [-151698], [-61806], [85160], [72642], [-4464], [-188458], [119932], [-174336], [-75594], [171596], [8566], [-122106], [172672], [162542], [167634], [-102960], [-26580], [-135666], [140748], [-115166], [192390], [32916], [-142248], [-164468], [179062], [115638], [-126128], [-31174], [-50904], [181460], [-97452], [-173922], [92576], [84634], [-72060], [250164], [40750], [-165498], [-200882], [201330], [139638], [32776], [-151794], [-245644], [-21336], [165440], [-132806], [138148], [-40056], [96770], [-196818], [13560], [-181964], [33540], [55366], [62624], [273198], [-56212], [122502], [140438], [-8130], [-178184], [32490], [124098], [-182104], [103786], [254370], [173356], [62700], [142558], [109290], [-142704], [-15748], [195910], [21738], [96132], [209824], [8874], [-161856], [-738], [59794], [42324], [74508], [-113402], [-116130], [-210984], [-110142], [-26692], [222336]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1050_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1050_4_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_1050_4_a_g(:prec:=1)
    chi := MakeCharacter_1050_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_1050_4_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_1050_4_a_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1050_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<11,R![72, 1]>,<13,R![-34, 1]>],Snew);
    return Vf;
end function;