
// Make newform 1100.6.b.i in Magma, downloaded from the LMFDB on 13 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1100_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1100_b_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1100_6_b_i();".
// 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_1100_6_b_i();".
// 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
        poly := [36582879760245225, 0, 11029470554922471, 0, 761025943061793, 0, 19011199757266, 0, 184347539486, 0, 841157794, 0, 1950697, 0, 2231, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-10013627964516016502178508354011, 0, -42283378596194345044367536320, 0, -18846392353777716916215892, 0, 893484911202881828572026, 0, 6490191635647275020068, 0, 18003410340768016076, 0, 23185297939504341, 0, 11678598862180, 0], [174532736697702275654540135577, 0, -31077077621037106461797904480, 0, -94231961768888584581079460, 0, 4467424556014409142860130, 0, 32450958178236375100340, 0, 90017051703840080380, 0, 115926489697521705, 0, 58392994310900, 0], [-414924080215088292601665340989145907832609, 0, -23102364599667285803552545796587177203327, 0, -87024048663089552047281381526095667738, 0, 1488454787503013562602188435049290012, 0, 12790860097546787943225038269366532, 0, 38813075628541417699720809706022, 0, 52078720401536568278310355933, 0, 26071504441571960777760995, 0], [120474060830772211408074337915024899407463, 0, 19724925651042727820082486305428559042679, 0, 769306236955546621757325992195412235438, 0, 8485413303562096945291574453528770340, 0, 40329475985805523301984390323844320, 0, 94341139054338407049661164477166, 0, 107319573221886120188684645405, 0, 47512443124349155950149593, 0], [800122425685228380049753508922594761708085, 0, 94444292205062782832599848129348314571330, 0, 2923692817866372688153015981313918966768, 0, 31320354178353629633813058046554399478, 0, 151637773568204999545657698612292916, 0, 366203365713566883966390100441520, 0, 431675917701365584832796580801, 0, 198159958929481659169640402, 0], [-80773204057332880973288476004393416981812, 0, -23605633073383295244691546752084967441785, 0, -970133679589493716818341557268570848724, 0, -11008576416660071677699842272910714588, 0, -53768836797132162523426763840208154, 0, -128907264169837621446878065008788, 0, -149877613374580763979038027688, 0, -67675308395995488835347265, 0], [697607309270727178756263866446272660388986, 0, 165708572639196196367818063809126084820161, 0, 6253110688373938530722975201860637331988, 0, 68497832221951815106767676584506952656, 0, 329081923457797306868506483090793050, 0, 783883262219442777228369323088628, 0, 910917657591225453299416475510, 0, 412582842935335794809450713, 0], [0, -17707763238147346882310396920979073, 0, -6057501184410419972510242848488259, 0, -143415176549794132459559808604208, 0, -1403265207349325271773091091558, 0, -6470241364548016337980221962, 0, -15112496402162926872424016, 0, -17370069951394645782133, 0, -7816117698280095143], [0, 122231908691596332209845031560734033, 0, 6057501184410419972510242848488259, 0, 143415176549794132459559808604208, 0, 1403265207349325271773091091558, 0, 6470241364548016337980221962, 0, 15112496402162926872424016, 0, 17370069951394645782133, 0, 7816117698280095143], [0, -6481449618751364356608744699529448346158264307, 0, -1180093434902656846409577930464937425098377141, 0, -45260163791715978333989165407375232439972712, 0, -499746084764690066090210281796184341224922, 0, -2405875658499984557307778346484686483998, 0, -5726885529203628647802824384101329640, 0, -6641208364935264603065755153420295, 0, -2999588625420019785230301920329], [0, -41679562781918312556261296366711629729509741441, 0, -4148110711526653491514262157424890302444884158, 0, -136130073999113839689928322361491320703205266, 0, -1456197269877215230480190536177140437536996, 0, -6959388212121426077210388652099311292814, 0, -16562684437953184871586057742106185202, 0, -19250029711209628903103633635455811, 0, -8721566958001256909003260706576], [0, -2587040205156777672864170136806487853087052559, 0, -891547044751089829219056710410325180854926090, 0, -40967538130336594611306818918721588192540629, 0, -475923382687727588627227379463102656413705, 0, -2350788465835085643979078689446917779089, 0, -5690609445474277531277487989535403757, 0, -6683004752304122114247568328948146, 0, -3049569047035759046539842626831], [0, -65868773559030981645065284542727783346610450966, 0, 1374479851290636393967467578959907703209317959, 0, 148582632521245977884940344490947992341608944, 0, 1884369819885989662533071865602493281907188, 0, 9553533028880452837714968774214600062298, 0, 23372150087540570102737855202095032016, 0, 27588538774152650451461930990065898, 0, 12625137100570072594357832624371], [0, -624421354207299070811922582912074831635117612521, 0, -78824598793344547849986751684621514344588064388, 0, -2422508601859954192739554805302238224160553356, 0, -25274701492225341881610076864935026987519506, 0, -119309859297699206554196261578348757570204, 0, -281780842804353029962608889888788818732, 0, -325708996353595625975514285321721801, 0, -146933990752964379483675922584056], [0, -1270314883803361565874543269961805268267650060854, 0, -125963787513079970892466591175282687593765831047, 0, -3638395882525187858198699631243908468992081144, 0, -37080433492191188995198401456633339830648564, 0, -173052070610917613492381053347503849213986, 0, -406299101535534033197735751865301913688, 0, -468249349021097647907990359579944254, 0, -210970999371661450583542448050939]];
    Rf_basisdens := [1, 36067963071986923752007955424, 36067963071986923752007955424, 142674860068246611493058959215261169968, 142674860068246611493058959215261169968, 285349720136493222986117918430522339936, 95116573378831074328705972810174113312, 285349720136493222986117918430522339936, 104524145453448985327534634639754960, 41809658181379594131013853855901984, 110258275811580571882167005327798540584250592, 91881896509650476568472504439832117153542160, 27564568952895142970541751331949635146062648, 110258275811580571882167005327798540584250592, 551291379057902859410835026638992702921252960, 551291379057902859410835026638992702921252960];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1100_b();"
function MakeCharacter_1100_b()
    N := 1100;
    order := 2;
    char_gens := [551, 177, 101];
    v := [2, 1, 2];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1100_b_Hecke();"
function MakeCharacter_1100_b_Hecke( : Kf := false)
    N := 1100;
    order := 2;
    char_gens := [551, 177, 101];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := 6;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0], [121, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, 5, -1, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -13, -1, 1, -1, 0, 0, 0], [44, -2, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 28, 86, 1, -1, -1, 2, 0, -1], [-341, -7, 1, -1, 1, 1, 4, 3, 0, 0, 0, 0, 0, 0, 0, 0], [473, -10, -9, 1, -2, 2, -3, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -116, -278, -30, -5, -4, 1, -1, 2], [1103, -13, -1, 2, 2, 8, -1, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -80, 341, -40, 3, 0, 5, -1, 6], [0, 0, 0, 0, 0, 0, 0, 0, -173, -292, -51, 19, 5, 1, -7, -7], [0, 0, 0, 0, 0, 0, 0, 0, -306, 653, -12, 8, -4, 10, 7, -6], [-2581, 6, -123, -16, -36, -22, -20, -9, 0, 0, 0, 0, 0, 0, 0, 0], [1335, -11, 61, 3, 14, -3, -14, 20, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -917, -310, -52, 10, 26, -11, 8, 5], [3995, 17, 31, 26, 47, -2, -15, -25, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1228, -1326, 44, -13, 9, -5, 12, -29], [8904, 133, -257, 37, 27, 32, 35, 46, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1895, -2176, -27, 27, 3, 6, -7, -38], [1745, 141, -680, 22, -17, 5, -15, -9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1028, -4961, 24, -67, 9, 34, -71, 72], [6742, 6, 343, 23, 38, 18, 58, -34, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3903, -1827, -32, 6, -45, -36, 48, 60], [0, 0, 0, 0, 0, 0, 0, 0, -3079, -9177, -14, -41, -28, 37, -26, -42], [-6332, -4, -600, -46, 72, 21, 72, 69, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1382, 5691, 498, 111, -217, 34, -20, 24], [0, 0, 0, 0, 0, 0, 0, 0, -4644, -15934, -93, -4, -4, -117, 13, 137], [50149, 294, 499, -73, -13, -90, 28, 167, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2759, -14824, 792, 128, 26, 75, 63, -93], [35797, -159, -825, -46, 197, -123, -172, 70, 0, 0, 0, 0, 0, 0, 0, 0], [-5307, -435, -1139, -29, -247, -334, -232, -124, 0, 0, 0, 0, 0, 0, 0, 0], [17449, -641, -513, -74, 209, 75, -65, -221, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -202, -12955, 956, 104, -109, 173, 41, -72], [0, 0, 0, 0, 0, 0, 0, 0, -622, 17640, 367, 219, 488, 300, 90, -321], [0, 0, 0, 0, 0, 0, 0, 0, -2746, -31916, -954, -356, -236, 15, -57, 21], [0, 0, 0, 0, 0, 0, 0, 0, 3652, 27652, 287, -314, -176, 80, 169, 157], [111462, -222, 834, 162, 322, 406, 242, 269, 0, 0, 0, 0, 0, 0, 0, 0], [71398, -633, 128, -358, -237, -185, -349, 41, 0, 0, 0, 0, 0, 0, 0, 0], [115282, 47, -962, 180, 51, 524, 139, 325, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 751, -1537, 666, -50, 427, -164, 31, -267], [0, 0, 0, 0, 0, 0, 0, 0, 13840, -23010, -860, 461, 348, 113, -50, -71], [146274, 343, 1575, -338, -636, -179, -324, -438, 0, 0, 0, 0, 0, 0, 0, 0], [181411, 785, -2318, -368, -516, -144, -474, -296, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7587, 16465, -391, -303, -862, -384, -130, 565], [0, 0, 0, 0, 0, 0, 0, 0, 3431, -56899, -807, 361, -660, 313, 326, -34], [290982, -96, 1910, 187, 269, 292, 919, -93, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 7716, 37338, 784, 289, -1043, -163, 140, -387], [225808, -1201, 6277, -356, -305, -175, -262, 441, 0, 0, 0, 0, 0, 0, 0, 0], [454655, -41, 544, 160, 110, 276, -906, -451, 0, 0, 0, 0, 0, 0, 0, 0], [136488, -1145, -5461, 535, -74, -225, 375, 67, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 22483, -99217, -2932, -119, -154, 266, -205, 249], [0, 0, 0, 0, 0, 0, 0, 0, -13054, 37991, -5434, 188, -363, -1076, 96, -125], [367266, -1618, 1695, -651, 262, -327, -1100, 56, 0, 0, 0, 0, 0, 0, 0, 0], [396406, 910, -8274, 167, -578, -263, -353, 239, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 36157, -107774, 4947, 256, -259, 379, -253, -10], [491108, 1502, 2536, -279, -489, 28, 1501, -128, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 32804, 122168, 2793, 236, 712, 743, -509, -113], [0, 0, 0, 0, 0, 0, 0, 0, 28527, 90405, 490, 216, 519, 80, -57, -235], [0, 0, 0, 0, 0, 0, 0, 0, -7528, -180753, -571, -1742, 1317, 1020, 284, 37], [-228690, -3195, -11236, -1084, -153, 12, 1115, -208, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 27282, 80870, 9216, -1212, 2322, 350, 462, 800], [0, 0, 0, 0, 0, 0, 0, 0, 1064, -202846, -1496, -125, 313, -1526, 849, 496], [414005, 1526, -8447, 270, 568, 160, 2184, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 39399, -189324, 1050, 594, 1122, -457, -1140, 23], [0, 0, 0, 0, 0, 0, 0, 0, 11859, -265066, 5319, -166, 1372, 309, -930, 771], [504987, -4665, 772, 1259, 1695, -3, 1443, -608, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 11766, 62705, -1388, 288, 2098, -800, -566, 774], [98622, -6598, 6545, -196, 687, 755, 847, -973, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -30621, -169389, -201, 1163, -616, 1376, 961, -1274], [0, 0, 0, 0, 0, 0, 0, 0, -37765, -64809, -11007, 274, -1886, 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-1871703, 6774021, 1776, 51774, 30730, 38478, -54143, 5387], [65704784, -60896, 195959, 3887, 2980, -53673, -42502, 589, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 633626, -7250077, -445686, -43294, -782, 5101, -33881, -7710], [11159135, 294182, 85581, -22850, 49167, 83097, 86113, -45147, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 878504, -5897124, -441272, 14219, 139426, 10733, -23894, -7163], [-15014756, 209636, -1097029, 32144, 17285, 51483, 13832, -16554, 0, 0, 0, 0, 0, 0, 0, 0], [32919133, -197320, 396103, 14778, -85305, 2503, -43682, -13010, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4017209, -3255255, 197528, 16940, 7727, 85709, 35328, -21178], [0, 0, 0, 0, 0, 0, 0, 0, -953426, 1652374, -85730, -29266, -88747, -107667, -28602, 9894], [0, 0, 0, 0, 0, 0, 0, 0, 1687858, 5722072, 9716, 19298, -56870, 16824, 61178, -2246], [0, 0, 0, 0, 0, 0, 0, 0, -2394352, 5926643, -157334, 29588, 51545, -28051, 7733, 24144], [0, 0, 0, 0, 0, 0, 0, 0, -1437729, -9921641, -250716, -13165, 127124, 49982, 22081, -56255], [-20950812, -230194, 42173, -63884, -9576, 4165, 29459, 19400, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2118005, 3957906, 236645, -43329, -112267, 27536, -8781, 36761], [0, 0, 0, 0, 0, 0, 0, 0, -2655969, -5049588, 80210, 82490, 51575, -107774, 28195, -34677], [-42403329, 48455, -1422728, -39321, 65136, 25411, 79141, 54829, 0, 0, 0, 0, 0, 0, 0, 0], [39308679, 47214, -677652, 7166, 46420, 73814, 124041, 48833, 0, 0, 0, 0, 0, 0, 0, 0], [42006312, 6723, 246174, -23808, -7015, 1847, 304, -73760, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -4015094, 1018849, 161204, -57565, 93918, -7526, -56810, 49287], [0, 0, 0, 0, 0, 0, 0, 0, 706405, -8070421, 111473, -86270, 33874, 5386, 25936, -45006], [-23261652, 77690, 399687, 29250, 78963, 91357, 58085, -12987, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -3625994, -326698, -87537, 80433, -151933, 9070, 4333, -25968], [0, 0, 0, 0, 0, 0, 0, 0, 1550524, 9536394, 7850, -121931, -14029, -69910, -51418, 127914], [-77074859, -12147, 349974, -24490, -54044, -52822, 34890, 72518, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -914128, -13358833, 34581, 125599, -82761, -2310, -34288, 15496], [0, 0, 0, 0, 0, 0, 0, 0, -4024527, 8073654, 321203, -30947, -44068, -41643, 41691, -46923], [0, 0, 0, 0, 0, 0, 0, 0, -580965, -196833, 229248, -14712, 75541, 31642, -509, 18027], [-49226602, -91296, -306782, 122631, 90646, 11299, 11551, -41160, 0, 0, 0, 0, 0, 0, 0, 0], [-7328695, -262871, -96702, -11917, -143937, -38146, -101755, -50229, 0, 0, 0, 0, 0, 0, 0, 0], [-8856732, 291122, 319707, -18325, -79779, -25992, -97614, 10692, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 737931, -81513, 521069, -11718, -46937, 55259, 52420, -65714], [-43593752, 118678, 357975, 20549, -87749, 7015, 130126, -61660, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2242342, 14090416, -367737, 28542, -117860, -37700, -25582, 123980], [-67241688, 336852, 751399, -31794, -7314, -18759, -54463, -52568, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 823146, 1000981, -94473, -2641, 99034, 11991, -52023, -144], [-6979880, 175920, -309090, 22603, 93074, 18786, 189543, 54691, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1381229, 4331104, -142820, -30374, 17392, 26295, -84333, 25563], [0, 0, 0, 0, 0, 0, 0, 0, 251, 26486133, -179690, 55787, -14059, 98925, -42943, -31202], [-88338433, -180941, 34629, 122544, -14757, 62208, 5035, 27462, 0, 0, 0, 0, 0, 0, 0, 0], [28009899, -40799, 347007, 15322, -12591, 23301, 12021, 54684, 0, 0, 0, 0, 0, 0, 0, 0], [-33123788, -141183, 582273, 50495, -27911, -123422, -60987, 55829, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2094160, -12601989, -653553, 97149, -258778, 19941, -4206, -9000], [0, 0, 0, 0, 0, 0, 0, 0, 1838146, -14342297, -506650, -72120, 71785, -80409, 33981, 10290], [41024250, 163871, -275176, 31047, -148454, 15101, 76055, -62582, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1264805, 6383728, -37615, 30032, 33928, -22183, 785, 73236], [-77378557, 386317, 7499, -101040, -31560, -156977, -90625, -34003, 0, 0, 0, 0, 0, 0, 0, 0], [-22287349, -223686, -471398, 48819, 13353, 31267, 25046, -38165, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1658783, -6544109, -596476, -106935, -39595, 62437, 20440, -65019], [-104930887, -216421, -305028, -100429, 53268, -18743, 43434, 104760, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -2695387, 5033325, -61656, -25863, -95410, -172327, 5454, 13766], [0, 0, 0, 0, 0, 0, 0, 0, 5106751, 1068350, 128605, -86754, 109165, -83729, 29823, 30712], [0, 0, 0, 0, 0, 0, 0, 0, 1091185, 2003489, 351500, -164906, 74876, 21440, 93497, -27141], [-51801495, 63152, 98121, 52177, 135847, -21774, -63099, -61504, 0, 0, 0, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1100_b_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1100_6_b_i();".
// 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_1100_6_b_i(:prec:=16)
    chi := MakeCharacter_1100_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1100_6_b_i();".
// 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_1100_6_b_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1100_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![649668040507055376, 0, 55707429442387896, 0, 1749655004415201, 0, 26670782667673, 0, 214678632158, 0, 924020173, 0, 2087326, 0, 2321, 0, 1]>],Snew);
    return Vf;
end function;