
// Make newform 400.6.a.b in Magma, downloaded from the LMFDB on 06 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_400_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_400_6_a_b();".
// 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_400_6_a_b();".
// 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_400_a();"
function MakeCharacter_400_a()
    N := 400;
    order := 1;
    char_gens := [351, 101, 177];
    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_400_a_Hecke(Kf)
    return MakeCharacter_400_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 := 6;
    raw_aps := [[0], [-18], [0], [242], [-656], [206], [-1690], [1364], [2198], [-2218], [1700], [846], [-1818], [10534], [12074], [-32586], [-8668], [-34670], [-47566], [-948], [63102], [-46536], [-88778], [-104934], [36254], [42486], [147934], [-18390], [-145006], [-82746], [-274446], [-202608], [48142], [111156], [243178], [-368852], [-319546], [69862], [-343422], [1142], [86684], [-651418], [-29140], [646406], [-431138], [131608], [-1210784], [34886], [-124182], [-456386], [-252666], [-65064], [-1406002], [548400], [493830], [-1071810], [999394], [-1007604], [-1022858], [-1181722], [-917506], [512302], [1409458], [2786036], [-1550858], [-377322], [1630632], [3367166], [837202], [-1519098], [3518510], [3570888], [-3582310], [-635530], [-67060], [4451286], [5798250], [-333874], [-2556894], [-3055106], [3543468], [1972938], [-1373956], [-5188130], [2940824], [-1283466], [-4952634], [-7913146], [-6185298], [491934], [447442], [-8184872], [6215242], [-827856], [10400404], [2038206], [3661334], [3247130], [-4973570], [2425438], [-731254], [-7719922], [3105758], [-482498], [-13850200], [-7097642], [1569258], [13047726], [5246880], [-5573162], [1982490], [-969810], [11294598], [18072796], [-4626340], [19905774], [12107838], [-2531238], [-13704346], [9833196], [6686874], [727566], [-18695138], [-17985018], [-1200056], [-11519438], [-610738], [-3972776], [-12321982], [-8145786], [-7166532], [-5650410], [-9095004], [11288038], [15293322], [1774018], [14644110], [-5970742], [34049958], [-26354022], [9496576], [-15988678], [31834678], [12757458], [6186130], [5667536], [17601022], [-162162], [7105204], [4089558], [-809194], [3904110], [35828990], [27757058], [20191362], [-27517868], [-13178576], [40068758], [-30425770], [-32634946], [30153410], [-303066], [-7398630], [6184136], [-1639282], [-11301954], [-3126428], [35582742], [21813554], [-2363554], [-47397940], [-48957410], [18825524], [26939710], [-43101104], [27049674], [45188648], [51341542], [27123334], [-12545246], [-70176358], [-46262688], [-890042], [850846], [-57213914], [-49933490], [79549534], [-36164674], [-44100322], [-33791660], [-42801962], [84863526], [85501816], [42474994], [-43129558], [-79578298], [-12402930], [35634294], [14756894], [-102075410], [-23384682], [-83273476], [-48017402], [81161298], [15062196], [-55072906], [-24963808], [-12502450], [-89953830], [-94402568], [118967942], [-22791398], [-34980514], [-6790038], [-57194480], [24063318], [23998498], [-111422510], [-5401086], [27559870], [-135791734], [71510720], [45766610], [22636014], [55294458], [34106650], [-60484834], [66259904], [118065434], [86534600], [113639486], [31793220], [-52110500], [134935382], [41579886], [88087122], [24992070], [-75249210], [16972780], [-111498492], [-20400538], [33426416], [43920646], [115761094], [-126305122], [-138938104], [16681762], [-12236008], [156455540], [151900894], [-73708962], [24411410], [44325930], [107365358], [-187805002], [-23921340], [112917110], [-27419070], [151980966], [148084326], [-171783434], [-21038438], [-118255938], [12507838], [-225252298], [-206416732], [154660310], [-20959350], [-93358026], [200786206], [14307086], [72279538], [-16019802], [40198096], [23179798], [62192726], [6697538], [-124737850], [127396166], [-11842752], [72877350], [181561860], [-254818982], [-192092426], [-81495110], [75845188], [-214912442], [-290140146], [46602960], [86125590], [89927714], [-36934182], [275413614], [30905856], [-62408938], [314702562], [250957588], [43978398], [-314037524], [245578818], [-260204946], [195348006], [-236775464], [-144442850], [-255775728], [215989982], [67555698], [-283180154], [-352095096], [188856142], [191802374], [-168998514], [-103800606], [277000222], [174603346], [-280283606], [160090996], [337226388], [-281053210], [52959446], [1220296], [-316432418], [37303026], [-209295122], [95135998], [-207810402], [274293380], [191706470], [216443770], [1983846], [344108354], [166604998], [248426216], [233148806], [445426496], [-344600406], [-292801150], [-424794906], [-289296650], [-444071310], [-457529138], [154291830], [-130057222], [238204780], [146631070], [29156548], [171487018], [43521346], [-342171708], [147190302], [-261198976], [238063062], [354077570], [179744078], [328800974], [-185387762], [44577336], [320024824], [492385774], [-363518906], [-74736026], [225544682], [482195194], [-483381228], [-33002646], [503556086], [-187465554], [346407086], [-372225130], [-306129888], [-205858580], [-516505634], [82295754], [433052172], [314705374], [-524925756], [-449550124], [-663408338], [45148422], [241067422], [-479405570], [91714990], [-616033158], [-228580242], [8494932], [128233238], [-393684868], [30290910], [-139624554], [398879066], [180017590], [188708646], [-559820844], [67512994], [-183061540], [-47627762], [31999672], [-698630502], [544102096], [-111590790], [633900546], [563036550], [362155162], [540821454], [-316124214], [-307277652], [328235222], [392212434], [-54706346], [206774604], [-575457490], [19410942], [-494641826], [39737840], [-367680466], [-294373170], [158285536], [16303090], [348718974], [438177558], [36762354], [-522384082], [-322956502], [540275572], [370358166], [-751152426], [427624734], [896945118], [-670673528], [603007352]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_400_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_400_6_a_b();".
// 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_400_6_a_b(:prec:=1)
    chi := MakeCharacter_400_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_400_6_a_b();".
// 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_400_6_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_400_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![18, 1]>],Snew);
    return Vf;
end function;