
// Make newform 1600.4.a.r in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1600_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_1600_4_a_r();".
// 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_1600_4_a_r();".
// 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_1600_a();"
function MakeCharacter_1600_a()
    N := 1600;
    order := 1;
    char_gens := [1151, 901, 577];
    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_1600_a_Hecke(Kf)
    return MakeCharacter_1600_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], [-2], [0], [-6], [-60], [50], [30], [-40], [-178], [-166], [20], [10], [-250], [142], [-214], [490], [800], [-250], [-774], [100], [230], [-1320], [982], [874], [310], [1498], [-1402], [1194], [-650], [1510], [-1246], [-2660], [-2770], [560], [2350], [580], [-1310], [1862], [-726], [3250], [1120], [2842], [3180], [4670], [-2990], [-4240], [4060], [5622], [1554], [-1134], [1710], [4440], [-850], [660], [7590], [-762], [150], [-6580], [4530], [6950], [-3882], [1370], [4106], [2220], [9430], [-6470], [-900], [-530], [-414], [-8614], [2270], [8080], [-2374], [1810], [-8120], [11782], [4350], [-7470], [11698], [-3650], [1120], [-4850], [-12580], [-13130], [8560], [-4258], [2550], [6710], [14482], [-162], [-15974], [-10760], [9266], [-2860], [-7160], [1398], [-7446], [-16438], [-7322], [-10878], [16114], [3690], [-2562], [-6050], [-8260], [16870], [-966], [-26290], [-11640], [-25450], [-16694], [15890], [1230], [10840], [-14060], [-17650], [27358], [6786], [-9030], [15600], [-16850], [7990], [18690], [19182], [23380], [-11850], [-25646], [-30280], [-17446], [-16750], [36560], [30142], [11860], [37010], [-11718], [4706], [-28670], [20434], [3930], [-4854], [13140], [-22050], [-14578], [-37054], [6150], [-8200], [-42990], [-32130], [-15440], [-46938], [-31230], [25550], [4318], [-1766], [41906], [25140], [32920], [10150], [-28530], [-9678], [36986], [3350], [-43774], [-8740], [48310], [-2282], [-31580], [-2790], [23954], [-20830], [-1720], [24338], [58140], [37590], [33280], [9786], [21980], [21018], [30862], [-32650], [-27734], [-14580], [24010], [8390], [50502], [49746], [-9830], [-29122], [-42050], [-31180], [5990], [-39138], [58980], [-22650], [16034], [32190], [22350], [-7590], [67230], [35478], [-44134], [-27540], [-4790], [32914], [-34560], [64290], [-44720], [49518], [36874], [18700], [-6130], [33950], [-49002], [50794], [34280], [-7850], [-20446], [1218], [-24326], [27730], [62750], [9880], [32946], [11278], [-33246], [16750], [-14890], [-19760], [-76414], [-63780], [71250], [47480], [70300], [26550], [87718], [46834], [52950], [-32790], [16760], [58380], [-6778], [107700], [49742], [8314], [-33170], [-106080], [45874], [82020], [71520], [-71682], [-64350], [402], [-11694], [18750], [-87710], [67560], [-34438], [-2254], [-24030], [46550], [-23738], [-58526], [-12494], [-75990], [20190], [126920], [38346], [93962], [-8578], [85530], [93378], [68986], [-22130], [78720], [-17850], [-56018], [-63134], [29226], [-106650], [-74100], [104822], [114460], [-135214], [-104118], [56874], [42580], [46270], [99890], [118840], [92950], [-42650], [68994], [87270], [107140], [-84830], [14950], [49300], [-156470], [-13320], [-44934], [72110], [34530], [58280], [126798], [4660], [-134650], [-149854], [-74966], [0], [92010], [18662], [-894], [-48142], [52718], [143034], [-17926], [13880], [40740], [-174330], [-116650], [11060], [69470], [-40250], [191358], [-89170], [-141250], [-143480], [-54898], [-95094], [4490], [-125850], [66450], [-44440], [-148458], [13700], [72474], [-45050], [-56650], [-27050], [146034], [-28390], [-29690], [59550], [105780], [-93830], [-9560], [194750], [7586], [-96780], [200690], [-47460], [200310], [153750], [-153682], [-78174], [-179850], [198120], [125100], [79350], [-26178], [-123350], [142122], [-106214], [-70080], [-9726], [-94290], [241410], [128598], [-82250], [-77220], [29880], [239358], [47150], [212420], [122050], [-197560], [155860], [11150], [-209450], [-48850], [-14862], [-176610], [-14], [-105810], [135480], [-91138], [146380], [-86470], [154782], [217866], [-189450], [192610], [-114960], [156106], [-41820], [86990], [-233320], [225866], [-128060], [-128050], [-203450], [226350], [141626], [118350], [11550], [-110620], [235690], [101602], [47702], [142000], [189470], [-177350], [-114442], [61020], [67350], [-116718], [-225520], [-260966], [4950], [-204498], [13350], [-186350], [94346], [129520], [32230], [88690], [-12562], [29550], [-101380], [-125920]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1600_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1600_4_a_r();".
// 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_1600_4_a_r(:prec:=1)
    chi := MakeCharacter_1600_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_1600_4_a_r();".
// 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_1600_4_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1600_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![2, 1]>,<7,R![6, 1]>,<11,R![60, 1]>,<13,R![-50, 1]>],Snew);
    return Vf;
end function;