# Properties

 Label 8046.2.a.e Level 8046 Weight 2 Character orbit 8046.a Self dual yes Analytic conductor 64.248 Analytic rank 1 Dimension 8 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$8046 = 2 \cdot 3^{3} \cdot 149$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 8046.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.2476334663$$ Analytic rank: $$1$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + \beta_{1} q^{5} + ( -1 - \beta_{5} ) q^{7} - q^{8} +O(q^{10})$$ $$q - q^{2} + q^{4} + \beta_{1} q^{5} + ( -1 - \beta_{5} ) q^{7} - q^{8} -\beta_{1} q^{10} + ( 1 - \beta_{6} ) q^{11} + ( -1 + \beta_{1} - \beta_{4} + \beta_{5} ) q^{13} + ( 1 + \beta_{5} ) q^{14} + q^{16} + ( -1 - \beta_{1} + \beta_{3} ) q^{17} + ( -1 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{19} + \beta_{1} q^{20} + ( -1 + \beta_{6} ) q^{22} + ( 3 + \beta_{2} - \beta_{3} - \beta_{7} ) q^{23} + ( \beta_{4} + \beta_{5} ) q^{25} + ( 1 - \beta_{1} + \beta_{4} - \beta_{5} ) q^{26} + ( -1 - \beta_{5} ) q^{28} + ( -\beta_{1} - \beta_{2} + 2 \beta_{6} ) q^{29} + ( -1 - \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{31} - q^{32} + ( 1 + \beta_{1} - \beta_{3} ) q^{34} + ( -1 - 3 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{35} + ( -\beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{37} + ( 1 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{38} -\beta_{1} q^{40} + ( -2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{41} + ( -2 - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{43} + ( 1 - \beta_{6} ) q^{44} + ( -3 - \beta_{2} + \beta_{3} + \beta_{7} ) q^{46} + ( 3 + 2 \beta_{2} - \beta_{7} ) q^{47} + ( \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} ) q^{49} + ( -\beta_{4} - \beta_{5} ) q^{50} + ( -1 + \beta_{1} - \beta_{4} + \beta_{5} ) q^{52} + ( -2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{53} + ( -1 - \beta_{2} ) q^{55} + ( 1 + \beta_{5} ) q^{56} + ( \beta_{1} + \beta_{2} - 2 \beta_{6} ) q^{58} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{59} + ( -3 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} ) q^{61} + ( 1 + \beta_{3} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{62} + q^{64} + ( 5 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{65} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{67} + ( -1 - \beta_{1} + \beta_{3} ) q^{68} + ( 1 + 3 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{70} + ( 1 + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{71} + ( -\beta_{1} + 3 \beta_{4} - \beta_{7} ) q^{73} + ( \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{74} + ( -1 + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{76} + ( 2 + \beta_{1} + \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{77} + ( -4 - 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{79} + \beta_{1} q^{80} + ( 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{82} + ( 3 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{83} + ( -5 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{85} + ( 2 + \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{86} + ( -1 + \beta_{6} ) q^{88} + ( -4 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{89} + ( -3 - 3 \beta_{1} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{91} + ( 3 + \beta_{2} - \beta_{3} - \beta_{7} ) q^{92} + ( -3 - 2 \beta_{2} + \beta_{7} ) q^{94} + ( -1 + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{95} + ( -1 - 3 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{97} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{2} + 8q^{4} - 5q^{7} - 8q^{8} + O(q^{10})$$ $$8q - 8q^{2} + 8q^{4} - 5q^{7} - 8q^{8} + 6q^{11} - 8q^{13} + 5q^{14} + 8q^{16} - 5q^{17} - 14q^{19} - 6q^{22} + 21q^{23} - 6q^{25} + 8q^{26} - 5q^{28} + 3q^{29} - 4q^{31} - 8q^{32} + 5q^{34} - 2q^{35} - 3q^{37} + 14q^{38} + 7q^{41} - 12q^{43} + 6q^{44} - 21q^{46} + 25q^{47} - 7q^{49} + 6q^{50} - 8q^{52} + 3q^{53} - 9q^{55} + 5q^{56} - 3q^{58} + 2q^{59} - 17q^{61} + 4q^{62} + 8q^{64} + 32q^{65} - 14q^{67} - 5q^{68} + 2q^{70} + 7q^{71} - 10q^{73} + 3q^{74} - 14q^{76} + 12q^{77} - 33q^{79} - 7q^{82} + 13q^{83} - 33q^{85} + 12q^{86} - 6q^{88} - 22q^{89} - 22q^{91} + 21q^{92} - 25q^{94} - 14q^{95} - 11q^{97} + 7q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 17 x^{6} - 2 x^{5} + 71 x^{4} - 18 x^{3} - 81 x^{2} + 36 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 17 \nu^{5} + 5 \nu^{4} - 71 \nu^{3} - 18 \nu^{2} + 63 \nu + 12$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{6} - 14 \nu^{5} - 33 \nu^{4} + 25 \nu^{3} + 64 \nu^{2} - 21 \nu - 9$$$$)/6$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{7} + 2 \nu^{6} - 31 \nu^{5} - 35 \nu^{4} + 93 \nu^{3} + 55 \nu^{2} - 75 \nu - 18$$$$)/6$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{7} - 2 \nu^{6} + 31 \nu^{5} + 35 \nu^{4} - 93 \nu^{3} - 49 \nu^{2} + 75 \nu - 12$$$$)/6$$ $$\beta_{6}$$ $$=$$ $$($$$$-2 \nu^{7} - \nu^{6} + 34 \nu^{5} + 21 \nu^{4} - 137 \nu^{3} - 35 \nu^{2} + 144 \nu - 3$$$$)/6$$ $$\beta_{7}$$ $$=$$ $$($$$$5 \nu^{7} + 6 \nu^{6} - 79 \nu^{5} - 103 \nu^{4} + 247 \nu^{3} + 192 \nu^{2} - 213 \nu - 48$$$$)/6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{4} + 5$$ $$\nu^{3}$$ $$=$$ $$-\beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 9 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$-\beta_{7} - 2 \beta_{6} + 11 \beta_{5} + 13 \beta_{4} + 3 \beta_{2} + 6 \beta_{1} + 46$$ $$\nu^{5}$$ $$=$$ $$-14 \beta_{7} - 24 \beta_{6} + 30 \beta_{5} + 34 \beta_{4} + 26 \beta_{3} + 12 \beta_{2} + 94 \beta_{1} + 53$$ $$\nu^{6}$$ $$=$$ $$-16 \beta_{7} - 38 \beta_{6} + 132 \beta_{5} + 154 \beta_{4} + 10 \beta_{3} + 50 \beta_{2} + 129 \beta_{1} + 494$$ $$\nu^{7}$$ $$=$$ $$-172 \beta_{7} - 276 \beta_{6} + 405 \beta_{5} + 483 \beta_{4} + 300 \beta_{3} + 142 \beta_{2} + 1052 \beta_{1} + 911$$

## 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
 −2.92810 −2.60917 −1.29304 −0.181211 0.854224 1.17278 1.40004 3.58447
−1.00000 0 1.00000 −2.92810 0 −3.28444 −1.00000 0 2.92810
1.2 −1.00000 0 1.00000 −2.60917 0 −0.922875 −1.00000 0 2.60917
1.3 −1.00000 0 1.00000 −1.29304 0 −0.779884 −1.00000 0 1.29304
1.4 −1.00000 0 1.00000 −0.181211 0 3.43581 −1.00000 0 0.181211
1.5 −1.00000 0 1.00000 0.854224 0 0.727081 −1.00000 0 −0.854224
1.6 −1.00000 0 1.00000 1.17278 0 1.96150 −1.00000 0 −1.17278
1.7 −1.00000 0 1.00000 1.40004 0 −2.13611 −1.00000 0 −1.40004
1.8 −1.00000 0 1.00000 3.58447 0 −4.00108 −1.00000 0 −3.58447
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 8046.2.a.e 8
3.b odd 2 1 8046.2.a.f yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8046.2.a.e 8 1.a even 1 1 trivial
8046.2.a.f yes 8 3.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$149$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8046))$$:

 $$T_{5}^{8} - 17 T_{5}^{6} - 2 T_{5}^{5} + 71 T_{5}^{4} - 18 T_{5}^{3} - 81 T_{5}^{2} + 36 T_{5} + 9$$ $$T_{11}^{8} - \cdots$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{8}$$
$3$ 
$5$ $$1 + 23 T^{2} - 2 T^{3} + 261 T^{4} - 68 T^{5} + 1964 T^{6} - 734 T^{7} + 11099 T^{8} - 3670 T^{9} + 49100 T^{10} - 8500 T^{11} + 163125 T^{12} - 6250 T^{13} + 359375 T^{14} + 390625 T^{16}$$
$7$ $$1 + 5 T + 44 T^{2} + 164 T^{3} + 854 T^{4} + 2581 T^{5} + 10192 T^{6} + 25906 T^{7} + 84279 T^{8} + 181342 T^{9} + 499408 T^{10} + 885283 T^{11} + 2050454 T^{12} + 2756348 T^{13} + 5176556 T^{14} + 4117715 T^{15} + 5764801 T^{16}$$
$11$ $$1 - 6 T + 69 T^{2} - 320 T^{3} + 2012 T^{4} - 7675 T^{5} + 34896 T^{6} - 115513 T^{7} + 435155 T^{8} - 1270643 T^{9} + 4222416 T^{10} - 10215425 T^{11} + 29457692 T^{12} - 51536320 T^{13} + 122237709 T^{14} - 116923026 T^{15} + 214358881 T^{16}$$
$13$ $$1 + 8 T + 84 T^{2} + 448 T^{3} + 2738 T^{4} + 10836 T^{5} + 50511 T^{6} + 166032 T^{7} + 700113 T^{8} + 2158416 T^{9} + 8536359 T^{10} + 23806692 T^{11} + 78200018 T^{12} + 166339264 T^{13} + 405451956 T^{14} + 501988136 T^{15} + 815730721 T^{16}$$
$17$ $$1 + 5 T + 98 T^{2} + 386 T^{3} + 4323 T^{4} + 14035 T^{5} + 119347 T^{6} + 330216 T^{7} + 2354845 T^{8} + 5613672 T^{9} + 34491283 T^{10} + 68953955 T^{11} + 361061283 T^{12} + 548064802 T^{13} + 2365481762 T^{14} + 2051693365 T^{15} + 6975757441 T^{16}$$
$19$ $$1 + 14 T + 189 T^{2} + 1675 T^{3} + 13551 T^{4} + 88651 T^{5} + 528103 T^{6} + 2701748 T^{7} + 12614699 T^{8} + 51333212 T^{9} + 190645183 T^{10} + 608057209 T^{11} + 1765979871 T^{12} + 4147465825 T^{13} + 8891671509 T^{14} + 12514204346 T^{15} + 16983563041 T^{16}$$
$23$ $$1 - 21 T + 315 T^{2} - 3351 T^{3} + 29840 T^{4} - 220880 T^{5} + 1441926 T^{6} - 8201727 T^{7} + 41898423 T^{8} - 188639721 T^{9} + 762778854 T^{10} - 2687446960 T^{11} + 8350455440 T^{12} - 21568185393 T^{13} + 46631305035 T^{14} - 71501334387 T^{15} + 78310985281 T^{16}$$
$29$ $$1 - 3 T + 116 T^{2} - 257 T^{3} + 5913 T^{4} - 5002 T^{5} + 188655 T^{6} + 144119 T^{7} + 5241187 T^{8} + 4179451 T^{9} + 158658855 T^{10} - 121993778 T^{11} + 4182152553 T^{12} - 5271365293 T^{13} + 68999505236 T^{14} - 51749628927 T^{15} + 500246412961 T^{16}$$
$31$ $$1 + 4 T + 143 T^{2} + 277 T^{3} + 9440 T^{4} + 5823 T^{5} + 413774 T^{6} - 77032 T^{7} + 14007581 T^{8} - 2387992 T^{9} + 397636814 T^{10} + 173472993 T^{11} + 8718038240 T^{12} + 7930274827 T^{13} + 126913026383 T^{14} + 110050456444 T^{15} + 852891037441 T^{16}$$
$37$ $$1 + 3 T + 157 T^{2} + 169 T^{3} + 10925 T^{4} - 4578 T^{5} + 520980 T^{6} - 602386 T^{7} + 20795125 T^{8} - 22288282 T^{9} + 713221620 T^{10} - 231889434 T^{11} + 20475208925 T^{12} + 11719128733 T^{13} + 402819046213 T^{14} + 284795631399 T^{15} + 3512479453921 T^{16}$$
$41$ $$1 - 7 T + 221 T^{2} - 1651 T^{3} + 22651 T^{4} - 175892 T^{5} + 1468078 T^{6} - 11131140 T^{7} + 69043319 T^{8} - 456376740 T^{9} + 2467839118 T^{10} - 12122652532 T^{11} + 64006312411 T^{12} - 191278587851 T^{13} + 1049773037261 T^{14} - 1363279917167 T^{15} + 7984925229121 T^{16}$$
$43$ $$1 + 12 T + 268 T^{2} + 2841 T^{3} + 34765 T^{4} + 312940 T^{5} + 2760526 T^{6} + 20722959 T^{7} + 144745419 T^{8} + 891087237 T^{9} + 5104212574 T^{10} + 24880920580 T^{11} + 118854616765 T^{12} + 417650986563 T^{13} + 1694125297132 T^{14} + 3261823333284 T^{15} + 11688200277601 T^{16}$$
$47$ $$1 - 25 T + 534 T^{2} - 7809 T^{3} + 99499 T^{4} - 1049944 T^{5} + 9811105 T^{6} - 80167517 T^{7} + 583909437 T^{8} - 3767873299 T^{9} + 21672730945 T^{10} - 109008335912 T^{11} + 485523379819 T^{12} - 1790955159663 T^{13} + 5756100985686 T^{14} - 12665578011575 T^{15} + 23811286661761 T^{16}$$
$53$ $$1 - 3 T + 259 T^{2} - 671 T^{3} + 33140 T^{4} - 76472 T^{5} + 2800566 T^{6} - 5841713 T^{7} + 172037347 T^{8} - 309610789 T^{9} + 7866789894 T^{10} - 11384921944 T^{11} + 261490540340 T^{12} - 280609175803 T^{13} + 5740569532411 T^{14} - 3524133419511 T^{15} + 62259690411361 T^{16}$$
$59$ $$1 - 2 T + 187 T^{2} - 781 T^{3} + 15293 T^{4} - 93219 T^{5} + 839319 T^{6} - 6407678 T^{7} + 44695837 T^{8} - 378053002 T^{9} + 2921669439 T^{10} - 19145225001 T^{11} + 185310801773 T^{12} - 558355877519 T^{13} + 7887759790867 T^{14} - 4977302969638 T^{15} + 146830437604321 T^{16}$$
$61$ $$1 + 17 T + 287 T^{2} + 3211 T^{3} + 40377 T^{4} + 388503 T^{5} + 3878930 T^{6} + 31804176 T^{7} + 272510093 T^{8} + 1940054736 T^{9} + 14433498530 T^{10} + 88182799443 T^{11} + 559053522057 T^{12} + 2711998722511 T^{13} + 14786347441607 T^{14} + 53426628212357 T^{15} + 191707312997281 T^{16}$$
$67$ $$1 + 14 T + 325 T^{2} + 3419 T^{3} + 55257 T^{4} + 492865 T^{5} + 6126075 T^{6} + 46584916 T^{7} + 484079105 T^{8} + 3121189372 T^{9} + 27499950675 T^{10} + 148235555995 T^{11} + 1113490493097 T^{12} + 4616077740833 T^{13} + 29398974204925 T^{14} + 84849962474522 T^{15} + 406067677556641 T^{16}$$
$71$ $$1 - 7 T + 376 T^{2} - 2079 T^{3} + 67454 T^{4} - 298691 T^{5} + 7720851 T^{6} - 28228478 T^{7} + 635043539 T^{8} - 2004221938 T^{9} + 38920809891 T^{10} - 106904794501 T^{11} + 1714119530174 T^{12} - 3750992820729 T^{13} + 48165706754296 T^{14} - 63665841108737 T^{15} + 645753531245761 T^{16}$$
$73$ $$1 + 10 T + 380 T^{2} + 3079 T^{3} + 71273 T^{4} + 493354 T^{5} + 8654984 T^{6} + 51366911 T^{7} + 741758925 T^{8} + 3749784503 T^{9} + 46122409736 T^{10} + 191923093018 T^{11} + 2024027830793 T^{12} + 6382987434847 T^{13} + 57507005989820 T^{14} + 110473985190970 T^{15} + 806460091894081 T^{16}$$
$79$ $$1 + 33 T + 891 T^{2} + 16593 T^{3} + 267540 T^{4} + 3546960 T^{5} + 42232098 T^{6} + 436784437 T^{7} + 4134028575 T^{8} + 34505970523 T^{9} + 263570523618 T^{10} + 1748789611440 T^{11} + 10420704670740 T^{12} + 51057596828607 T^{13} + 216590922869211 T^{14} + 633728996543247 T^{15} + 1517108809906561 T^{16}$$
$83$ $$1 - 13 T + 566 T^{2} - 6044 T^{3} + 145132 T^{4} - 1282046 T^{5} + 22184174 T^{6} - 163089004 T^{7} + 2235297087 T^{8} - 13536387332 T^{9} + 152826774686 T^{10} - 733057236202 T^{11} + 6887721043372 T^{12} - 23807561646292 T^{13} + 185048251326854 T^{14} - 352768662865151 T^{15} + 2252292232139041 T^{16}$$
$89$ $$1 + 22 T + 483 T^{2} + 6152 T^{3} + 66677 T^{4} + 418457 T^{5} + 1196132 T^{6} - 23297564 T^{7} - 288951823 T^{8} - 2073483196 T^{9} + 9474561572 T^{10} + 294999212833 T^{11} + 4183464403157 T^{12} + 34353133730248 T^{13} + 240041963534163 T^{14} + 973089367701638 T^{15} + 3936588805702081 T^{16}$$
$97$ $$1 + 11 T + 555 T^{2} + 4180 T^{3} + 124875 T^{4} + 596251 T^{5} + 16127779 T^{6} + 47076750 T^{7} + 1614911481 T^{8} + 4566444750 T^{9} + 151746272611 T^{10} + 544182188923 T^{11} + 11055093964875 T^{12} + 35895082274260 T^{13} + 462299462735595 T^{14} + 888781129259243 T^{15} + 7837433594376961 T^{16}$$