Properties

Label 4004.2.a.i
Level 4004
Weight 2
Character orbit 4004.a
Self dual yes
Analytic conductor 31.972
Analytic rank 0
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} -\beta_{7} q^{5} + q^{7} + ( 1 + \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} -\beta_{7} q^{5} + q^{7} + ( 1 + \beta_{3} + \beta_{4} ) q^{9} - q^{11} + q^{13} + ( -1 + 2 \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{8} ) q^{15} + ( -1 + \beta_{1} - \beta_{3} - \beta_{7} ) q^{17} + ( 1 + \beta_{3} + \beta_{4} - \beta_{5} ) q^{19} + \beta_{1} q^{21} + ( -1 - \beta_{4} - \beta_{6} ) q^{23} + ( 5 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{8} ) q^{25} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{8} ) q^{27} + ( -\beta_{4} + \beta_{5} - \beta_{6} ) q^{29} + ( 1 - \beta_{1} + \beta_{2} - \beta_{6} ) q^{31} -\beta_{1} q^{33} -\beta_{7} q^{35} + ( 1 + \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{8} ) q^{37} + \beta_{1} q^{39} + ( 2 + \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{41} + ( 1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} ) q^{43} + ( 4 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{7} ) q^{45} + ( 2 - \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{47} + q^{49} + ( 3 - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{51} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{53} + \beta_{7} q^{55} + ( -2 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{57} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{59} + ( 5 - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{8} ) q^{61} + ( 1 + \beta_{3} + \beta_{4} ) q^{63} -\beta_{7} q^{65} + ( 2 - \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{67} + ( 3 - 4 \beta_{1} + \beta_{3} - \beta_{4} - 3 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{69} + ( 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{71} + ( 2 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{73} + ( -4 + 8 \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{75} - q^{77} + ( 1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{79} + ( -1 + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{81} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{83} + ( 6 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{85} + ( 4 - \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{87} + ( -\beta_{4} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{89} + q^{91} + ( -1 + \beta_{1} - 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{93} + ( -1 + \beta_{2} + 2 \beta_{3} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{95} + ( 3 - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{97} + ( -1 - \beta_{3} - \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 4q^{3} + 4q^{5} + 9q^{7} + 13q^{9} + O(q^{10}) \) \( 9q + 4q^{3} + 4q^{5} + 9q^{7} + 13q^{9} - 9q^{11} + 9q^{13} - 5q^{17} + 17q^{19} + 4q^{21} - 8q^{23} + 35q^{25} + 4q^{27} - 3q^{29} + 9q^{31} - 4q^{33} + 4q^{35} + 7q^{37} + 4q^{39} + 14q^{41} + 21q^{43} + 43q^{45} + q^{47} + 9q^{49} + 25q^{51} - 8q^{53} - 4q^{55} + 8q^{57} - 4q^{59} + 30q^{61} + 13q^{63} + 4q^{65} + 15q^{67} + 9q^{69} - q^{71} + 18q^{73} - 32q^{75} - 9q^{77} + 13q^{79} + 13q^{81} + 2q^{83} + 45q^{85} + 21q^{87} + 9q^{91} - 2q^{93} + 11q^{95} + 29q^{97} - 13q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 4 x^{8} - 12 x^{7} + 60 x^{6} + 15 x^{5} - 233 x^{4} + 74 x^{3} + 271 x^{2} - 67 x - 87\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -10 \nu^{8} - 189 \nu^{7} + 627 \nu^{6} + 2658 \nu^{5} - 6837 \nu^{4} - 8299 \nu^{3} + 17667 \nu^{2} + 5318 \nu - 5712 \)\()/681\)
\(\beta_{3}\)\(=\)\((\)\( -46 \nu^{8} + 84 \nu^{7} + 705 \nu^{6} - 1257 \nu^{5} - 2712 \nu^{4} + 5000 \nu^{3} + 2817 \nu^{2} - 5365 \nu - 1623 \)\()/681\)
\(\beta_{4}\)\(=\)\((\)\( 46 \nu^{8} - 84 \nu^{7} - 705 \nu^{6} + 1257 \nu^{5} + 2712 \nu^{4} - 5000 \nu^{3} - 2136 \nu^{2} + 5365 \nu - 1101 \)\()/681\)
\(\beta_{5}\)\(=\)\((\)\( -77 \nu^{8} + 111 \nu^{7} + 1491 \nu^{6} - 1734 \nu^{5} - 9129 \nu^{4} + 7126 \nu^{3} + 20334 \nu^{2} - 6449 \nu - 11703 \)\()/681\)
\(\beta_{6}\)\(=\)\((\)\( 176 \nu^{8} - 351 \nu^{7} - 2727 \nu^{6} + 4839 \nu^{5} + 11235 \nu^{4} - 14926 \nu^{3} - 15249 \nu^{2} + 6374 \nu + 6417 \)\()/681\)
\(\beta_{7}\)\(=\)\((\)\( 298 \nu^{8} - 633 \nu^{7} - 4656 \nu^{6} + 8913 \nu^{5} + 19464 \nu^{4} - 29312 \nu^{3} - 26214 \nu^{2} + 18175 \nu + 9774 \)\()/681\)
\(\beta_{8}\)\(=\)\((\)\( -143 \nu^{8} + 271 \nu^{7} + 2315 \nu^{6} - 3804 \nu^{5} - 10533 \nu^{4} + 12553 \nu^{3} + 16717 \nu^{2} - 8215 \nu - 7044 \)\()/227\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{8} + 2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 8 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-\beta_{8} - 3 \beta_{7} + 2 \beta_{6} - \beta_{5} + 11 \beta_{4} + 10 \beta_{3} + \beta_{2} + 26\)
\(\nu^{5}\)\(=\)\(13 \beta_{8} - 2 \beta_{7} + 27 \beta_{6} - 15 \beta_{5} + 17 \beta_{4} + 11 \beta_{3} + \beta_{2} + 74 \beta_{1} - 18\)
\(\nu^{6}\)\(=\)\(-13 \beta_{8} - 46 \beta_{7} + 35 \beta_{6} - 15 \beta_{5} + 116 \beta_{4} + 95 \beta_{3} + 15 \beta_{2} + 5 \beta_{1} + 209\)
\(\nu^{7}\)\(=\)\(139 \beta_{8} - 43 \beta_{7} + 311 \beta_{6} - 171 \beta_{5} + 211 \beta_{4} + 109 \beta_{3} + 15 \beta_{2} + 727 \beta_{1} - 212\)
\(\nu^{8}\)\(=\)\(-133 \beta_{8} - 552 \beta_{7} + 466 \beta_{6} - 182 \beta_{5} + 1220 \beta_{4} + 920 \beta_{3} + 171 \beta_{2} + 135 \beta_{1} + 1876\)

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
−3.11788
−1.90247
−0.980714
−0.554873
0.848270
1.89826
2.10249
2.42810
3.27882
0 −3.11788 0 4.40203 0 1.00000 0 6.72118 0
1.2 0 −1.90247 0 −3.63968 0 1.00000 0 0.619386 0
1.3 0 −0.980714 0 −1.42373 0 1.00000 0 −2.03820 0
1.4 0 −0.554873 0 3.11431 0 1.00000 0 −2.69212 0
1.5 0 0.848270 0 −0.844151 0 1.00000 0 −2.28044 0
1.6 0 1.89826 0 3.07273 0 1.00000 0 0.603374 0
1.7 0 2.10249 0 −4.18458 0 1.00000 0 1.42045 0
1.8 0 2.42810 0 0.790216 0 1.00000 0 2.89567 0
1.9 0 3.27882 0 2.71285 0 1.00000 0 7.75069 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

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 4004.2.a.i 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4004.2.a.i 9 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)
\(13\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{9} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4004))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 - 4 T + 15 T^{2} - 36 T^{3} + 87 T^{4} - 161 T^{5} + 299 T^{6} - 473 T^{7} + 815 T^{8} - 1323 T^{9} + 2445 T^{10} - 4257 T^{11} + 8073 T^{12} - 13041 T^{13} + 21141 T^{14} - 26244 T^{15} + 32805 T^{16} - 26244 T^{17} + 19683 T^{18} \)
$5$ \( 1 - 4 T + 13 T^{2} - 24 T^{3} + 49 T^{4} - 63 T^{5} + 131 T^{6} - 541 T^{7} + 1767 T^{8} - 4913 T^{9} + 8835 T^{10} - 13525 T^{11} + 16375 T^{12} - 39375 T^{13} + 153125 T^{14} - 375000 T^{15} + 1015625 T^{16} - 1562500 T^{17} + 1953125 T^{18} \)
$7$ \( ( 1 - T )^{9} \)
$11$ \( ( 1 + T )^{9} \)
$13$ \( ( 1 - T )^{9} \)
$17$ \( 1 + 5 T + 112 T^{2} + 435 T^{3} + 5849 T^{4} + 18779 T^{5} + 193018 T^{6} + 525275 T^{7} + 4474639 T^{8} + 10441272 T^{9} + 76068863 T^{10} + 151804475 T^{11} + 948297434 T^{12} + 1568440859 T^{13} + 8304743593 T^{14} + 10499842515 T^{15} + 45957931376 T^{16} + 34878787205 T^{17} + 118587876497 T^{18} \)
$19$ \( 1 - 17 T + 232 T^{2} - 2147 T^{3} + 17621 T^{4} - 118587 T^{5} + 739064 T^{6} - 4008751 T^{7} + 20396271 T^{8} - 91757468 T^{9} + 387529149 T^{10} - 1447159111 T^{11} + 5069239976 T^{12} - 15454376427 T^{13} + 43631340479 T^{14} - 101007506507 T^{15} + 207378243448 T^{16} - 288720571697 T^{17} + 322687697779 T^{18} \)
$23$ \( 1 + 8 T + 142 T^{2} + 900 T^{3} + 8348 T^{4} + 43931 T^{5} + 277858 T^{6} + 1305284 T^{7} + 6658435 T^{8} + 30989946 T^{9} + 153144005 T^{10} + 690495236 T^{11} + 3380698286 T^{12} + 12293694971 T^{13} + 53730591364 T^{14} + 133232300100 T^{15} + 483485213474 T^{16} + 626487882248 T^{17} + 1801152661463 T^{18} \)
$29$ \( 1 + 3 T + 133 T^{2} + 432 T^{3} + 8825 T^{4} + 31164 T^{5} + 408405 T^{6} + 1502576 T^{7} + 14859460 T^{8} + 51506786 T^{9} + 430924340 T^{10} + 1263666416 T^{11} + 9960589545 T^{12} + 22041705084 T^{13} + 181010889925 T^{14} + 256963674672 T^{15} + 2294233549097 T^{16} + 1500739238883 T^{17} + 14507145975869 T^{18} \)
$31$ \( 1 - 9 T + 189 T^{2} - 1422 T^{3} + 17963 T^{4} - 115249 T^{5} + 1088445 T^{6} - 6024758 T^{7} + 46368142 T^{8} - 220727252 T^{9} + 1437412402 T^{10} - 5789792438 T^{11} + 32425864995 T^{12} - 106434871729 T^{13} + 514265439413 T^{14} - 1262030234382 T^{15} + 5199884066979 T^{16} - 7676019336969 T^{17} + 26439622160671 T^{18} \)
$37$ \( 1 - 7 T + 147 T^{2} - 663 T^{3} + 8476 T^{4} - 29797 T^{5} + 380684 T^{6} - 1550701 T^{7} + 18320960 T^{8} - 73203984 T^{9} + 677875520 T^{10} - 2122909669 T^{11} + 19282786652 T^{12} - 55844375317 T^{13} + 587759379532 T^{14} - 1701076609167 T^{15} + 13954985938551 T^{16} - 24587356177447 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 - 14 T + 215 T^{2} - 1941 T^{3} + 17113 T^{4} - 117718 T^{5} + 752369 T^{6} - 4616331 T^{7} + 25703150 T^{8} - 172452312 T^{9} + 1053829150 T^{10} - 7760052411 T^{11} + 51854023849 T^{12} - 332642933398 T^{13} + 1982647167713 T^{14} - 9219952331781 T^{15} + 41872168884415 T^{16} - 111788953207694 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 - 21 T + 395 T^{2} - 4694 T^{3} + 49931 T^{4} - 403182 T^{5} + 3019603 T^{6} - 18398443 T^{7} + 117340899 T^{8} - 694152558 T^{9} + 5045658657 T^{10} - 34018721107 T^{11} + 240079575721 T^{12} - 1378399024782 T^{13} + 7340278567433 T^{14} - 29672478152006 T^{15} + 107368351387265 T^{16} - 245452205829621 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 - T + 97 T^{2} + 200 T^{3} + 5791 T^{4} + 21672 T^{5} + 416053 T^{6} + 1080856 T^{7} + 22279878 T^{8} + 55018994 T^{9} + 1047154266 T^{10} + 2387610904 T^{11} + 43195870619 T^{12} + 105752446632 T^{13} + 1328136935537 T^{14} + 2155843065800 T^{15} + 49142442684911 T^{16} - 23811286661761 T^{17} + 1119130473102767 T^{18} \)
$53$ \( 1 + 8 T + 287 T^{2} + 2286 T^{3} + 41111 T^{4} + 313016 T^{5} + 3907455 T^{6} + 27566774 T^{7} + 272219877 T^{8} + 1718117526 T^{9} + 14427653481 T^{10} + 77435068166 T^{11} + 581730178035 T^{12} + 2469846800696 T^{13} + 17192434912723 T^{14} + 50667729540894 T^{15} + 337142097133219 T^{16} + 498077523290888 T^{17} + 3299763591802133 T^{18} \)
$59$ \( 1 + 4 T + 267 T^{2} + 620 T^{3} + 35132 T^{4} + 34249 T^{5} + 3125936 T^{6} - 33172 T^{7} + 218144930 T^{8} - 76875498 T^{9} + 12870550870 T^{10} - 115471732 T^{11} + 642001609744 T^{12} + 415007496889 T^{13} + 25116720472468 T^{14} + 26151930857420 T^{15} + 664469946446673 T^{16} + 587321750417284 T^{17} + 8662995818654939 T^{18} \)
$61$ \( 1 - 30 T + 716 T^{2} - 11943 T^{3} + 175473 T^{4} - 2139867 T^{5} + 23818644 T^{6} - 232195520 T^{7} + 2101070175 T^{8} - 16987691396 T^{9} + 128165280675 T^{10} - 863999529920 T^{11} + 5406379633764 T^{12} - 29628258243147 T^{13} + 148203846725373 T^{14} - 615307830993423 T^{15} + 2250203870591036 T^{16} - 5751219389918430 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 - 15 T + 274 T^{2} - 3475 T^{3} + 50073 T^{4} - 537804 T^{5} + 5945176 T^{6} - 55267900 T^{7} + 531241911 T^{8} - 4317339305 T^{9} + 35593208037 T^{10} - 248097603100 T^{11} + 1788088969288 T^{12} - 10837353478284 T^{13} + 67604814482811 T^{14} - 314342878037275 T^{15} + 1660634979858502 T^{16} - 6091015163349615 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 + T + 308 T^{2} + 263 T^{3} + 52152 T^{4} + 15517 T^{5} + 6102418 T^{6} - 606011 T^{7} + 542916157 T^{8} - 122916004 T^{9} + 38547047147 T^{10} - 3054901451 T^{11} + 2184122528798 T^{12} + 394313054077 T^{13} + 94094169113352 T^{14} + 33690374671223 T^{15} + 2801297008784428 T^{16} + 645753531245761 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 - 18 T + 507 T^{2} - 6843 T^{3} + 113603 T^{4} - 1227540 T^{5} + 15477139 T^{6} - 140656093 T^{7} + 1485962414 T^{8} - 11757700036 T^{9} + 108475256222 T^{10} - 749556319597 T^{11} + 6020870182363 T^{12} - 34859976757140 T^{13} + 235507152179579 T^{14} - 1035580110495627 T^{15} + 5601031049182179 T^{16} - 14516281654093458 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 - 13 T + 347 T^{2} - 3078 T^{3} + 56981 T^{4} - 453136 T^{5} + 7041845 T^{6} - 49917247 T^{7} + 659623475 T^{8} - 4209775716 T^{9} + 52110254525 T^{10} - 311533538527 T^{11} + 3471904216955 T^{12} - 17649683904016 T^{13} + 175333750671419 T^{14} - 748223188093638 T^{15} + 6663756418197173 T^{16} - 19722414528785293 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 - 2 T + 364 T^{2} - 1367 T^{3} + 69441 T^{4} - 357405 T^{5} + 9123390 T^{6} - 55363134 T^{7} + 925368647 T^{8} - 5606818522 T^{9} + 76805597701 T^{10} - 381396630126 T^{11} + 5216635797930 T^{12} - 16961841217005 T^{13} + 273530921290563 T^{14} - 446927490395423 T^{15} + 9877522560224228 T^{16} - 4504584464278082 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 + 459 T^{2} - 602 T^{3} + 107035 T^{4} - 232167 T^{5} + 16743355 T^{6} - 43127025 T^{7} + 1933853371 T^{8} - 4847269619 T^{9} + 172112950019 T^{10} - 341609165025 T^{11} + 11803546230995 T^{12} - 14566677866247 T^{13} + 597689803123715 T^{14} - 299182737158522 T^{15} + 20302182717047811 T^{16} + 350356403707485209 T^{18} \)
$97$ \( 1 - 29 T + 869 T^{2} - 15245 T^{3} + 269006 T^{4} - 3475931 T^{5} + 46249846 T^{6} - 489985103 T^{7} + 5557828502 T^{8} - 52338872712 T^{9} + 539109364694 T^{10} - 4610269834127 T^{11} + 42210985698358 T^{12} - 307721672235611 T^{13} + 2310046053174542 T^{14} - 12698658215142605 T^{15} + 70213709211480197 T^{16} - 227285574236931869 T^{17} + 760231058654565217 T^{18} \)
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