Properties

 Label 6012.2.a.h Level $6012$ Weight $2$ Character orbit 6012.a Self dual yes Analytic conductor $48.006$ Analytic rank $1$ Dimension $9$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$6012 = 2^{2} \cdot 3^{2} \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6012.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$48.0060616952$$ Analytic rank: $$1$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Defining polynomial: $$x^{9} - 29 x^{7} - 7 x^{6} + 266 x^{5} + 69 x^{4} - 901 x^{3} - 199 x^{2} + 875 x + 391$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2004) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{5} -\beta_{6} q^{7} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{5} -\beta_{6} q^{7} + ( -1 + \beta_{5} ) q^{11} + ( 1 - \beta_{4} + \beta_{6} + \beta_{7} ) q^{13} + ( -1 + \beta_{2} - \beta_{3} - \beta_{8} ) q^{17} + ( -\beta_{2} - \beta_{7} ) q^{19} + ( -2 - \beta_{1} - \beta_{2} - \beta_{5} + \beta_{8} ) q^{23} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{25} + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} ) q^{29} + ( 1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{31} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{35} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{7} + \beta_{8} ) q^{37} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{41} + ( -2 - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{43} + ( -4 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{47} + ( 2 - \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{49} + ( -2 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{53} + ( 1 - 2 \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} ) q^{55} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{7} + \beta_{8} ) q^{59} + ( 3 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{61} + ( -2 + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{65} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{67} + ( -2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} ) q^{71} + ( -3 + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{73} + ( -2 + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{77} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{79} + ( -1 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{83} + ( 1 - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{85} + ( -4 - \beta_{2} - 2 \beta_{4} + \beta_{7} ) q^{89} + ( -4 + \beta_{1} - 2 \beta_{4} - 2 \beta_{6} + \beta_{8} ) q^{91} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{95} + ( 4 - \beta_{1} + 5 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9q - 9q^{5} + 2q^{7} + O(q^{10})$$ $$9q - 9q^{5} + 2q^{7} - 7q^{11} + 6q^{13} - 7q^{17} + 2q^{19} - 19q^{23} + 22q^{25} - 13q^{29} + 12q^{31} - 4q^{35} + 15q^{37} - 18q^{41} - 6q^{43} - 25q^{47} + 19q^{49} - 17q^{53} - 3q^{55} - 3q^{59} + 14q^{61} - 14q^{65} - 4q^{67} - 17q^{71} - 20q^{73} - 14q^{77} - 8q^{79} + q^{83} + 5q^{85} - 36q^{89} - 41q^{91} - 5q^{95} + 31q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - 29 x^{7} - 7 x^{6} + 266 x^{5} + 69 x^{4} - 901 x^{3} - 199 x^{2} + 875 x + 391$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-1006 \nu^{8} + 2871 \nu^{7} + 22263 \nu^{6} - 53775 \nu^{5} - 157887 \nu^{4} + 312589 \nu^{3} + 454514 \nu^{2} - 550699 \nu - 575149$$$$)/51607$$ $$\beta_{3}$$ $$=$$ $$($$$$1976 \nu^{8} - 9538 \nu^{7} - 25159 \nu^{6} + 160208 \nu^{5} - 1570 \nu^{4} - 711563 \nu^{3} + 508013 \nu^{2} + 557721 \nu - 255465$$$$)/51607$$ $$\beta_{4}$$ $$=$$ $$($$$$2012 \nu^{8} - 5742 \nu^{7} - 44526 \nu^{6} + 107550 \nu^{5} + 315774 \nu^{4} - 625178 \nu^{3} - 857421 \nu^{2} + 1101398 \nu + 789049$$$$)/51607$$ $$\beta_{5}$$ $$=$$ $$($$$$2980 \nu^{8} - 18354 \nu^{7} - 43479 \nu^{6} + 348793 \nu^{5} + 150155 \nu^{4} - 1846062 \nu^{3} - 203222 \nu^{2} + 2331119 \nu + 1064431$$$$)/51607$$ $$\beta_{6}$$ $$=$$ $$($$$$3832 \nu^{8} - 14527 \nu^{7} - 71773 \nu^{6} + 255110 \nu^{5} + 401245 \nu^{4} - 1246613 \nu^{3} - 814593 \nu^{2} + 1505084 \nu + 875617$$$$)/51607$$ $$\beta_{7}$$ $$=$$ $$($$$$3879 \nu^{8} - 3837 \nu^{7} - 87023 \nu^{6} + 28674 \nu^{5} + 546052 \nu^{4} + 57429 \nu^{3} - 948686 \nu^{2} - 418506 \nu + 172141$$$$)/51607$$ $$\beta_{8}$$ $$=$$ $$($$$$-3886 \nu^{8} + 8833 \nu^{7} + 75020 \nu^{6} - 124516 \nu^{5} - 361191 \nu^{4} + 420341 \nu^{3} + 179180 \nu^{2} - 75695 \nu + 292783$$$$)/51607$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + 2 \beta_{2} + 7$$ $$\nu^{3}$$ $$=$$ $$\beta_{8} + 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} + 4 \beta_{2} + 10 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{8} + 4 \beta_{7} + \beta_{5} + 14 \beta_{4} + 2 \beta_{3} + 31 \beta_{2} + 5 \beta_{1} + 79$$ $$\nu^{5}$$ $$=$$ $$22 \beta_{8} + 38 \beta_{7} - 23 \beta_{6} + 36 \beta_{5} + 17 \beta_{4} - 12 \beta_{3} + 91 \beta_{2} + 123 \beta_{1} + 91$$ $$\nu^{6}$$ $$=$$ $$115 \beta_{8} + 102 \beta_{7} - 21 \beta_{6} + 38 \beta_{5} + 207 \beta_{4} + 46 \beta_{3} + 486 \beta_{2} + 163 \beta_{1} + 1059$$ $$\nu^{7}$$ $$=$$ $$431 \beta_{8} + 648 \beta_{7} - 397 \beta_{6} + 555 \beta_{5} + 464 \beta_{4} - 97 \beta_{3} + 1703 \beta_{2} + 1714 \beta_{1} + 2087$$ $$\nu^{8}$$ $$=$$ $$2125 \beta_{8} + 2069 \beta_{7} - 679 \beta_{6} + 965 \beta_{5} + 3251 \beta_{4} + 758 \beta_{3} + 7981 \beta_{2} + 3699 \beta_{1} + 15652$$

Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −3.21153 −3.19525 −2.42964 −0.907808 −0.529979 1.61974 1.79204 2.69402 4.16840
0 0 0 −4.21153 0 −5.13720 0 0 0
1.2 0 0 0 −4.19525 0 1.43344 0 0 0
1.3 0 0 0 −3.42964 0 3.44225 0 0 0
1.4 0 0 0 −1.90781 0 2.81337 0 0 0
1.5 0 0 0 −1.52998 0 −1.05249 0 0 0
1.6 0 0 0 0.619742 0 1.05844 0 0 0
1.7 0 0 0 0.792043 0 3.80237 0 0 0
1.8 0 0 0 1.69402 0 −4.12928 0 0 0
1.9 0 0 0 3.16840 0 −0.230890 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$167$$ $$1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6012.2.a.h 9
3.b odd 2 1 2004.2.a.d 9
12.b even 2 1 8016.2.a.bb 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.2.a.d 9 3.b odd 2 1
6012.2.a.h 9 1.a even 1 1 trivial
8016.2.a.bb 9 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{9} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6012))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 9 T + 52 T^{2} + 234 T^{3} + 886 T^{4} + 2925 T^{5} + 8664 T^{6} + 23344 T^{7} + 58229 T^{8} + 134906 T^{9} + 291145 T^{10} + 583600 T^{11} + 1083000 T^{12} + 1828125 T^{13} + 2768750 T^{14} + 3656250 T^{15} + 4062500 T^{16} + 3515625 T^{17} + 1953125 T^{18}$$
$7$ $$1 - 2 T + 24 T^{2} - 3 T^{3} + 204 T^{4} + 469 T^{5} + 1446 T^{6} + 5119 T^{7} + 15489 T^{8} + 32962 T^{9} + 108423 T^{10} + 250831 T^{11} + 495978 T^{12} + 1126069 T^{13} + 3428628 T^{14} - 352947 T^{15} + 19765032 T^{16} - 11529602 T^{17} + 40353607 T^{18}$$
$11$ $$1 + 7 T + 76 T^{2} + 448 T^{3} + 2798 T^{4} + 13617 T^{5} + 64556 T^{6} + 259240 T^{7} + 1013497 T^{8} + 3395532 T^{9} + 11148467 T^{10} + 31368040 T^{11} + 85924036 T^{12} + 199366497 T^{13} + 450620698 T^{14} + 793659328 T^{15} + 1481024996 T^{16} + 1500512167 T^{17} + 2357947691 T^{18}$$
$13$ $$1 - 6 T + 72 T^{2} - 290 T^{3} + 2079 T^{4} - 5583 T^{5} + 32513 T^{6} - 50538 T^{7} + 355779 T^{8} - 363982 T^{9} + 4625127 T^{10} - 8540922 T^{11} + 71431061 T^{12} - 159456063 T^{13} + 771918147 T^{14} - 1399774610 T^{15} + 4517893224 T^{16} - 4894384326 T^{17} + 10604499373 T^{18}$$
$17$ $$1 + 7 T + 95 T^{2} + 412 T^{3} + 3239 T^{4} + 7217 T^{5} + 46207 T^{6} - 42146 T^{7} + 182922 T^{8} - 2723810 T^{9} + 3109674 T^{10} - 12180194 T^{11} + 227014991 T^{12} + 602771057 T^{13} + 4598916823 T^{14} + 9944678428 T^{15} + 38982173935 T^{16} + 48830302087 T^{17} + 118587876497 T^{18}$$
$19$ $$1 - 2 T + 102 T^{2} - 170 T^{3} + 4503 T^{4} - 4717 T^{5} + 117347 T^{6} - 26098 T^{7} + 2285365 T^{8} + 603238 T^{9} + 43421935 T^{10} - 9421378 T^{11} + 804883073 T^{12} - 614724157 T^{13} + 11149873797 T^{14} - 7997799770 T^{15} + 91174917378 T^{16} - 33967126082 T^{17} + 322687697779 T^{18}$$
$23$ $$1 + 19 T + 254 T^{2} + 2284 T^{3} + 17484 T^{4} + 110157 T^{5} + 659522 T^{6} + 3574832 T^{7} + 19162755 T^{8} + 93129576 T^{9} + 440743365 T^{10} + 1891086128 T^{11} + 8024404174 T^{12} + 30826445037 T^{13} + 112533021012 T^{14} + 338113970476 T^{15} + 864825663538 T^{16} + 1487908720339 T^{17} + 1801152661463 T^{18}$$
$29$ $$1 + 13 T + 154 T^{2} + 1330 T^{3} + 10268 T^{4} + 65265 T^{5} + 395422 T^{6} + 2088442 T^{7} + 11323883 T^{8} + 58112060 T^{9} + 328392607 T^{10} + 1756379722 T^{11} + 9643947158 T^{12} + 46160694465 T^{13} + 210608477932 T^{14} + 791115016930 T^{15} + 2656480951586 T^{16} + 6503203368493 T^{17} + 14507145975869 T^{18}$$
$31$ $$1 - 12 T + 237 T^{2} - 2081 T^{3} + 24736 T^{4} - 175357 T^{5} + 1571692 T^{6} - 9333683 T^{7} + 68244056 T^{8} - 343757830 T^{9} + 2115565736 T^{10} - 8969669363 T^{11} + 46822276372 T^{12} - 161945871997 T^{13} + 708170679136 T^{14} - 1846895160161 T^{15} + 6520489544307 T^{16} - 10234692449292 T^{17} + 26439622160671 T^{18}$$
$37$ $$1 - 15 T + 242 T^{2} - 2138 T^{3} + 21014 T^{4} - 145273 T^{5} + 1219856 T^{6} - 7859170 T^{7} + 60067423 T^{8} - 344194216 T^{9} + 2222494651 T^{10} - 10759203730 T^{11} + 61789365968 T^{12} - 272264990953 T^{13} + 1457193912398 T^{14} - 5485523062442 T^{15} + 22973514266186 T^{16} - 52687191808815 T^{17} + 129961739795077 T^{18}$$
$41$ $$1 + 18 T + 391 T^{2} + 4825 T^{3} + 61116 T^{4} + 575535 T^{5} + 5406478 T^{6} + 41258085 T^{7} + 314515424 T^{8} + 2011563376 T^{9} + 12895132384 T^{10} + 69354840885 T^{11} + 372619870238 T^{12} + 1626324357135 T^{13} + 7080667580316 T^{14} + 22919252962825 T^{15} + 76148921087471 T^{16} + 143728654124178 T^{17} + 327381934393961 T^{18}$$
$43$ $$1 + 6 T + 97 T^{2} + 675 T^{3} + 5484 T^{4} + 22849 T^{5} + 166420 T^{6} + 187403 T^{7} + 1595534 T^{8} - 10660100 T^{9} + 68607962 T^{10} + 346508147 T^{11} + 13231554940 T^{12} + 78116184049 T^{13} + 806194301412 T^{14} + 4266920058075 T^{15} + 26366405277379 T^{16} + 70129201665606 T^{17} + 502592611936843 T^{18}$$
$47$ $$1 + 25 T + 448 T^{2} + 5687 T^{3} + 64249 T^{4} + 630769 T^{5} + 5789701 T^{6} + 47606857 T^{7} + 365856543 T^{8} + 2574158528 T^{9} + 17195257521 T^{10} + 105163547113 T^{11} + 601104126923 T^{12} + 3077951504689 T^{13} + 14735187354743 T^{14} + 61301397576023 T^{15} + 226967157967424 T^{16} + 595282166544025 T^{17} + 1119130473102767 T^{18}$$
$53$ $$1 + 17 T + 318 T^{2} + 3079 T^{3} + 33015 T^{4} + 222369 T^{5} + 1931189 T^{6} + 10783739 T^{7} + 96671639 T^{8} + 529914954 T^{9} + 5123596867 T^{10} + 30291522851 T^{11} + 287509624753 T^{12} + 1754598369489 T^{13} + 13806724201395 T^{14} + 68244067916191 T^{15} + 373558142468166 T^{16} + 1058414736993137 T^{17} + 3299763591802133 T^{18}$$
$59$ $$1 + 3 T + 248 T^{2} + 587 T^{3} + 35733 T^{4} + 67161 T^{5} + 3580941 T^{6} + 5779345 T^{7} + 272819099 T^{8} + 381917456 T^{9} + 16096326841 T^{10} + 20117899945 T^{11} + 735450081639 T^{12} + 813814082121 T^{13} + 25546389976167 T^{14} + 24759973247267 T^{15} + 617185568235112 T^{16} + 440491312812963 T^{17} + 8662995818654939 T^{18}$$
$61$ $$1 - 14 T + 286 T^{2} - 2662 T^{3} + 34623 T^{4} - 221617 T^{5} + 2297933 T^{6} - 9978522 T^{7} + 111413285 T^{8} - 393019054 T^{9} + 6796210385 T^{10} - 37130080362 T^{11} + 521587130273 T^{12} - 3068473744897 T^{13} + 29242457729523 T^{14} - 137147236548982 T^{15} + 898824451102006 T^{16} - 2683902381961934 T^{17} + 11694146092834141 T^{18}$$
$67$ $$1 + 4 T + 402 T^{2} + 1172 T^{3} + 77435 T^{4} + 162163 T^{5} + 9543569 T^{6} + 14628924 T^{7} + 845476133 T^{8} + 1048306460 T^{9} + 56646900911 T^{10} + 65669239836 T^{11} + 2870352443147 T^{12} + 3267766234723 T^{13} + 104546937660545 T^{14} + 106017223902068 T^{15} + 2436406065339846 T^{16} + 1624270710226564 T^{17} + 27206534396294947 T^{18}$$
$71$ $$1 + 17 T + 319 T^{2} + 2841 T^{3} + 39999 T^{4} + 355099 T^{5} + 4942665 T^{6} + 38075859 T^{7} + 407671928 T^{8} + 2653132208 T^{9} + 28944706888 T^{10} + 191940405219 T^{11} + 1769034172815 T^{12} + 9023662511419 T^{13} + 72167369810649 T^{14} + 363932906619561 T^{15} + 2901343330526729 T^{16} + 10977810031177937 T^{17} + 45848500718449031 T^{18}$$
$73$ $$1 + 20 T + 498 T^{2} + 7086 T^{3} + 112237 T^{4} + 1284803 T^{5} + 15841175 T^{6} + 152923942 T^{7} + 1573749217 T^{8} + 13055799498 T^{9} + 114883692841 T^{10} + 814931686918 T^{11} + 6162486374975 T^{12} + 36486145231523 T^{13} + 232675336383541 T^{14} + 1072354327483854 T^{15} + 5501604462510306 T^{16} + 16129201837881620 T^{17} + 58871586708267913 T^{18}$$
$79$ $$1 + 8 T + 471 T^{2} + 3237 T^{3} + 106842 T^{4} + 664023 T^{5} + 15704132 T^{6} + 88577445 T^{7} + 1658985638 T^{8} + 8277689232 T^{9} + 131059865402 T^{10} + 552811834245 T^{11} + 7742749537148 T^{12} + 25863749635863 T^{13} + 328758859781958 T^{14} + 786874093521477 T^{15} + 9045041132480889 T^{16} + 12136870479252488 T^{17} + 119851595982618319 T^{18}$$
$83$ $$1 - T + 488 T^{2} - 1161 T^{3} + 113615 T^{4} - 399973 T^{5} + 16903319 T^{6} - 70850779 T^{7} + 1823276283 T^{8} - 7479201516 T^{9} + 151331931489 T^{10} - 488091016531 T^{11} + 9665098061053 T^{12} - 18982047025333 T^{13} + 447534102654445 T^{14} - 379577773481409 T^{15} + 13242392882937976 T^{16} - 2252292232139041 T^{17} + 186940255267540403 T^{18}$$
$89$ $$1 + 36 T + 1226 T^{2} + 26142 T^{3} + 519315 T^{4} + 7977763 T^{5} + 115284093 T^{6} + 1381129790 T^{7} + 15698593261 T^{8} + 151798632666 T^{9} + 1397174800229 T^{10} + 10939929066590 T^{11} + 81271711758117 T^{12} + 500542728786883 T^{13} + 2899885832757435 T^{14} + 12992084908302462 T^{15} + 54227616581918554 T^{16} + 141717197005274916 T^{17} + 350356403707485209 T^{18}$$
$97$ $$1 - 31 T + 713 T^{2} - 13605 T^{3} + 225439 T^{4} - 3322243 T^{5} + 44076123 T^{6} - 534128975 T^{7} + 5960297404 T^{8} - 60929995076 T^{9} + 578148848188 T^{10} - 5025619525775 T^{11} + 40227087406779 T^{12} - 294115784097283 T^{13} + 1935921400197823 T^{14} - 11332584127059045 T^{15} + 57609176832894569 T^{16} - 242960441425685791 T^{17} + 760231058654565217 T^{18}$$