Properties

Label 8004.2.a.f
Level 8004
Weight 2
Character orbit 8004.a
Self dual yes
Analytic conductor 63.912
Analytic rank 1
Dimension 9
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + q^{3} -\beta_{1} q^{5} + ( -1 - \beta_{6} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} -\beta_{1} q^{5} + ( -1 - \beta_{6} ) q^{7} + q^{9} + ( -1 - \beta_{3} ) q^{11} + ( \beta_{1} + \beta_{3} + \beta_{6} - \beta_{7} ) q^{13} -\beta_{1} q^{15} + ( \beta_{1} + \beta_{7} ) q^{17} + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{19} + ( -1 - \beta_{6} ) q^{21} - q^{23} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{25} + q^{27} - q^{29} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{31} + ( -1 - \beta_{3} ) q^{33} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{8} ) q^{35} + ( 1 - 2 \beta_{2} - \beta_{4} + \beta_{6} ) q^{37} + ( \beta_{1} + \beta_{3} + \beta_{6} - \beta_{7} ) q^{39} + ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} ) q^{41} + ( -1 - \beta_{1} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{43} -\beta_{1} q^{45} + ( -2 + 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} + 2 \beta_{8} ) q^{47} + ( -1 - \beta_{3} + 2 \beta_{5} + \beta_{8} ) q^{49} + ( \beta_{1} + \beta_{7} ) q^{51} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{53} + ( -3 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{8} ) q^{55} + ( -1 - \beta_{1} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{57} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{7} + \beta_{8} ) q^{59} + ( -\beta_{2} + 2 \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{61} + ( -1 - \beta_{6} ) q^{63} + ( -2 - 4 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{65} + ( -3 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{67} - q^{69} + ( 3 + 2 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} - 3 \beta_{8} ) q^{71} + ( -5 - \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - \beta_{5} - 4 \beta_{6} ) q^{73} + ( -1 + 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{75} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{8} ) q^{77} + ( 1 - \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{8} ) q^{79} + q^{81} + ( 3 - 2 \beta_{2} + 4 \beta_{3} + \beta_{4} - \beta_{5} + 4 \beta_{6} - 2 \beta_{8} ) q^{83} + ( -5 - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{85} - q^{87} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{89} + ( -2 - 2 \beta_{1} + \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{91} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{93} + ( 2 + 5 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{95} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{8} ) q^{97} + ( -1 - \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 9q^{3} - q^{5} - 5q^{7} + 9q^{9} + O(q^{10}) \) \( 9q + 9q^{3} - q^{5} - 5q^{7} + 9q^{9} - 8q^{11} - q^{13} - q^{15} - 2q^{17} - 11q^{19} - 5q^{21} - 9q^{23} - 10q^{25} + 9q^{27} - 9q^{29} - 8q^{33} + q^{35} - 2q^{37} - q^{39} - 3q^{41} - 19q^{43} - q^{45} - 3q^{47} - 6q^{49} - 2q^{51} - 9q^{53} - 7q^{55} - 11q^{57} - 2q^{59} - 25q^{61} - 5q^{63} - 12q^{65} - 20q^{67} - 9q^{69} + 9q^{71} - 11q^{73} - 10q^{75} - 19q^{77} + 4q^{79} + 9q^{81} - 9q^{83} - 50q^{85} - 9q^{87} - 29q^{89} - 38q^{91} + 23q^{95} - 43q^{97} - 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - x^{8} - 17 x^{7} + 4 x^{6} + 75 x^{5} + x^{4} - 118 x^{3} - 26 x^{2} + 60 x + 24\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -11 \nu^{8} - 2 \nu^{7} + 180 \nu^{6} + 161 \nu^{5} - 559 \nu^{4} - 480 \nu^{3} + 553 \nu^{2} + 258 \nu - 182 \)\()/34\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{8} - 4 \nu^{7} + 88 \nu^{6} + 118 \nu^{5} - 302 \nu^{4} - 382 \nu^{3} + 307 \nu^{2} + 278 \nu - 41 \)\()/17\)
\(\beta_{4}\)\(=\)\((\)\( -21 \nu^{8} + 24 \nu^{7} + 322 \nu^{6} - 113 \nu^{5} - 1061 \nu^{4} + 252 \nu^{3} + 895 \nu^{2} - 104 \nu - 162 \)\()/34\)
\(\beta_{5}\)\(=\)\((\)\( 22 \nu^{8} - 13 \nu^{7} - 343 \nu^{6} - 67 \nu^{5} + 1050 \nu^{4} + 229 \nu^{3} - 749 \nu^{2} - 74 \nu + 24 \)\()/34\)
\(\beta_{6}\)\(=\)\((\)\( 4 \nu^{8} + 10 \nu^{7} - 67 \nu^{6} - 210 \nu^{5} + 109 \nu^{4} + 683 \nu^{3} + 193 \nu^{2} - 559 \nu - 297 \)\()/17\)
\(\beta_{7}\)\(=\)\((\)\( -23 \nu^{8} + 36 \nu^{7} + 364 \nu^{6} - 297 \nu^{5} - 1447 \nu^{4} + 854 \nu^{3} + 1861 \nu^{2} - 598 \nu - 702 \)\()/34\)
\(\beta_{8}\)\(=\)\((\)\( -16 \nu^{8} + 28 \nu^{7} + 251 \nu^{6} - 248 \nu^{5} - 1014 \nu^{4} + 685 \nu^{3} + 1319 \nu^{2} - 484 \nu - 495 \)\()/17\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_{1} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{8} - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} - \beta_{2} + 12 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(\beta_{8} - 16 \beta_{7} + 15 \beta_{6} - 4 \beta_{5} + 12 \beta_{4} + 16 \beta_{3} - 4 \beta_{2} + 37 \beta_{1} + 38\)
\(\nu^{5}\)\(=\)\(17 \beta_{8} - 58 \beta_{7} + 39 \beta_{6} - 28 \beta_{5} + 12 \beta_{4} + 41 \beta_{3} - 16 \beta_{2} + 173 \beta_{1} + 69\)
\(\nu^{6}\)\(=\)\(30 \beta_{8} - 246 \beta_{7} + 215 \beta_{6} - 82 \beta_{5} + 147 \beta_{4} + 233 \beta_{3} - 73 \beta_{2} + 612 \beta_{1} + 480\)
\(\nu^{7}\)\(=\)\(238 \beta_{8} - 944 \beta_{7} + 675 \beta_{6} - 402 \beta_{5} + 300 \beta_{4} + 719 \beta_{3} - 258 \beta_{2} + 2611 \beta_{1} + 1298\)
\(\nu^{8}\)\(=\)\(602 \beta_{8} - 3809 \beta_{7} + 3167 \beta_{6} - 1388 \beta_{5} + 1967 \beta_{4} + 3432 \beta_{3} - 1138 \beta_{2} + 9792 \beta_{1} + 6751\)

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
3.95129
1.58421
1.51482
1.05365
−0.509166
−0.808455
−1.09926
−1.91513
−2.77196
0 1.00000 0 −3.95129 0 −1.61455 0 1.00000 0
1.2 0 1.00000 0 −1.58421 0 1.44549 0 1.00000 0
1.3 0 1.00000 0 −1.51482 0 −4.32415 0 1.00000 0
1.4 0 1.00000 0 −1.05365 0 3.84396 0 1.00000 0
1.5 0 1.00000 0 0.509166 0 1.30726 0 1.00000 0
1.6 0 1.00000 0 0.808455 0 −2.11918 0 1.00000 0
1.7 0 1.00000 0 1.09926 0 −1.62400 0 1.00000 0
1.8 0 1.00000 0 1.91513 0 −2.97719 0 1.00000 0
1.9 0 1.00000 0 2.77196 0 1.06236 0 1.00000 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 8004.2.a.f 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8004.2.a.f 9 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(1\)
\(29\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\(T_{5}^{9} + \cdots\)
\(T_{7}^{9} + \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 - T )^{9} \)
$5$ \( 1 + T + 28 T^{2} + 36 T^{3} + 380 T^{4} + 579 T^{5} + 3332 T^{6} + 5506 T^{7} + 21415 T^{8} + 33836 T^{9} + 107075 T^{10} + 137650 T^{11} + 416500 T^{12} + 361875 T^{13} + 1187500 T^{14} + 562500 T^{15} + 2187500 T^{16} + 390625 T^{17} + 1953125 T^{18} \)
$7$ \( 1 + 5 T + 47 T^{2} + 179 T^{3} + 987 T^{4} + 3081 T^{5} + 12739 T^{6} + 33935 T^{7} + 116730 T^{8} + 272488 T^{9} + 817110 T^{10} + 1662815 T^{11} + 4369477 T^{12} + 7397481 T^{13} + 16588509 T^{14} + 21059171 T^{15} + 38706521 T^{16} + 28824005 T^{17} + 40353607 T^{18} \)
$11$ \( 1 + 8 T + 95 T^{2} + 557 T^{3} + 3888 T^{4} + 18097 T^{5} + 93740 T^{6} + 359179 T^{7} + 1493566 T^{8} + 4770750 T^{9} + 16429226 T^{10} + 43460659 T^{11} + 124767940 T^{12} + 264958177 T^{13} + 626166288 T^{14} + 986759477 T^{15} + 1851281245 T^{16} + 1714871048 T^{17} + 2357947691 T^{18} \)
$13$ \( 1 + T + 64 T^{2} + 22 T^{3} + 2056 T^{4} - 466 T^{5} + 44369 T^{6} - 24869 T^{7} + 727482 T^{8} - 452444 T^{9} + 9457266 T^{10} - 4202861 T^{11} + 97478693 T^{12} - 13309426 T^{13} + 763378408 T^{14} + 106189798 T^{15} + 4015905088 T^{16} + 815730721 T^{17} + 10604499373 T^{18} \)
$17$ \( 1 + 2 T + 89 T^{2} + 229 T^{3} + 4280 T^{4} + 11246 T^{5} + 136975 T^{6} + 341214 T^{7} + 3152871 T^{8} + 6956984 T^{9} + 53598807 T^{10} + 98610846 T^{11} + 672958175 T^{12} + 939277166 T^{13} + 6076987960 T^{14} + 5527503301 T^{15} + 36520141897 T^{16} + 13951514882 T^{17} + 118587876497 T^{18} \)
$19$ \( 1 + 11 T + 152 T^{2} + 1069 T^{3} + 8804 T^{4} + 47609 T^{5} + 305389 T^{6} + 1397998 T^{7} + 7647564 T^{8} + 30527568 T^{9} + 145303716 T^{10} + 504677278 T^{11} + 2094663151 T^{12} + 6204452489 T^{13} + 21799575596 T^{14} + 50292046789 T^{15} + 135868504328 T^{16} + 186819193451 T^{17} + 322687697779 T^{18} \)
$23$ \( ( 1 + T )^{9} \)
$29$ \( ( 1 + T )^{9} \)
$31$ \( 1 + 199 T^{2} + 101 T^{3} + 18680 T^{4} + 15967 T^{5} + 1110738 T^{6} + 1111403 T^{7} + 46725240 T^{8} + 44236770 T^{9} + 1448482440 T^{10} + 1068058283 T^{11} + 33089995758 T^{12} + 14745859807 T^{13} + 534792540680 T^{14} + 89637871781 T^{15} + 5475010208089 T^{16} + 26439622160671 T^{18} \)
$37$ \( 1 + 2 T + 156 T^{2} + 477 T^{3} + 13177 T^{4} + 43010 T^{5} + 802457 T^{6} + 2429458 T^{7} + 37726953 T^{8} + 101904822 T^{9} + 1395897261 T^{10} + 3325928002 T^{11} + 40646854421 T^{12} + 80607664610 T^{13} + 913745321389 T^{14} + 1223851497093 T^{15} + 14809372832748 T^{16} + 7024958907842 T^{17} + 129961739795077 T^{18} \)
$41$ \( 1 + 3 T + 248 T^{2} + 652 T^{3} + 30406 T^{4} + 70619 T^{5} + 2398670 T^{6} + 4904464 T^{7} + 134000435 T^{8} + 237983212 T^{9} + 5494017835 T^{10} + 8244403984 T^{11} + 165318735070 T^{12} + 199552416059 T^{13} + 3522723647606 T^{14} + 3097067965132 T^{15} + 48299059922488 T^{16} + 23954775687363 T^{17} + 327381934393961 T^{18} \)
$43$ \( 1 + 19 T + 425 T^{2} + 5251 T^{3} + 67970 T^{4} + 626243 T^{5} + 5981877 T^{6} + 44535666 T^{7} + 349092239 T^{8} + 2219303996 T^{9} + 15010966277 T^{10} + 82346446434 T^{11} + 475601094639 T^{12} + 2141000194643 T^{13} + 9992163870710 T^{14} + 33193477370299 T^{15} + 115522909720475 T^{16} + 222075805274419 T^{17} + 502592611936843 T^{18} \)
$47$ \( 1 + 3 T + 236 T^{2} + 887 T^{3} + 29987 T^{4} + 119173 T^{5} + 2564809 T^{6} + 9903133 T^{7} + 160339521 T^{8} + 557577112 T^{9} + 7535957487 T^{10} + 21876020797 T^{11} + 266286164807 T^{12} + 581526223813 T^{13} + 6877368724909 T^{14} + 9561163996823 T^{15} + 119563056429268 T^{16} + 71433859985283 T^{17} + 1119130473102767 T^{18} \)
$53$ \( 1 + 9 T + 338 T^{2} + 2726 T^{3} + 56502 T^{4} + 400121 T^{5} + 5994720 T^{6} + 36962368 T^{7} + 441826927 T^{8} + 2340190448 T^{9} + 23416827131 T^{10} + 103827291712 T^{11} + 892475929440 T^{12} + 3157147148201 T^{13} + 23628881745486 T^{14} + 60420048437654 T^{15} + 397052365264906 T^{16} + 560337213702249 T^{17} + 3299763591802133 T^{18} \)
$59$ \( 1 + 2 T + 358 T^{2} + 678 T^{3} + 64211 T^{4} + 112668 T^{5} + 7411179 T^{6} + 11665706 T^{7} + 601636249 T^{8} + 825027460 T^{9} + 35496538691 T^{10} + 40608322586 T^{11} + 1522100531841 T^{12} + 1365238829148 T^{13} + 45906004163089 T^{14} + 28598401808598 T^{15} + 890937231565202 T^{16} + 293660875208642 T^{17} + 8662995818654939 T^{18} \)
$61$ \( 1 + 25 T + 651 T^{2} + 10695 T^{3} + 168445 T^{4} + 2084217 T^{5} + 24446885 T^{6} + 240989729 T^{7} + 2248681682 T^{8} + 18052608588 T^{9} + 137169582602 T^{10} + 896722781609 T^{11} + 5548978404185 T^{12} + 28857737191497 T^{13} + 142268023921945 T^{14} + 551010403790895 T^{15} + 2045925586249671 T^{16} + 4792682824932025 T^{17} + 11694146092834141 T^{18} \)
$67$ \( 1 + 20 T + 542 T^{2} + 7475 T^{3} + 119860 T^{4} + 1291619 T^{5} + 15632406 T^{6} + 140272099 T^{7} + 1408424239 T^{8} + 10890441078 T^{9} + 94364424013 T^{10} + 629681452411 T^{11} + 4701649325778 T^{12} + 26027570754899 T^{13} + 161825995325020 T^{14} + 676176406713275 T^{15} + 3284905690085066 T^{16} + 8121353551132820 T^{17} + 27206534396294947 T^{18} \)
$71$ \( 1 - 9 T + 162 T^{2} - 624 T^{3} + 13976 T^{4} - 35278 T^{5} + 1370561 T^{6} - 4013065 T^{7} + 111571406 T^{8} - 203499152 T^{9} + 7921569826 T^{10} - 20229860665 T^{11} + 490538858071 T^{12} - 896473282318 T^{13} + 25215909409576 T^{14} - 79934577166704 T^{15} + 1473409465659342 T^{16} - 5811781781211849 T^{17} + 45848500718449031 T^{18} \)
$73$ \( 1 + 11 T + 216 T^{2} + 783 T^{3} + 18999 T^{4} + 100595 T^{5} + 2714667 T^{6} + 13756013 T^{7} + 196354509 T^{8} + 632886124 T^{9} + 14333879157 T^{10} + 73305793277 T^{11} + 1056051612339 T^{12} + 2856721053395 T^{13} + 39386287195407 T^{14} + 118494699184287 T^{15} + 2386238080124952 T^{16} + 8871061010834891 T^{17} + 58871586708267913 T^{18} \)
$79$ \( 1 - 4 T + 142 T^{2} - 1027 T^{3} + 18895 T^{4} - 127472 T^{5} + 2128855 T^{6} - 13201936 T^{7} + 196561307 T^{8} - 1316269664 T^{9} + 15528343253 T^{10} - 82393282576 T^{11} + 1049608540345 T^{12} - 4965044725232 T^{13} + 58140980659105 T^{14} - 249650816820067 T^{15} + 2726955076034578 T^{16} - 6068435239626244 T^{17} + 119851595982618319 T^{18} \)
$83$ \( 1 + 9 T + 464 T^{2} + 3538 T^{3} + 104322 T^{4} + 688973 T^{5} + 15400354 T^{6} + 89528422 T^{7} + 1673708815 T^{8} + 8563628116 T^{9} + 138917831645 T^{10} + 616761299158 T^{11} + 8805722212598 T^{12} + 32697501794333 T^{13} + 410928597959046 T^{14} + 1156715040979522 T^{15} + 12591127659186928 T^{16} + 20270630089251369 T^{17} + 186940255267540403 T^{18} \)
$89$ \( 1 + 29 T + 820 T^{2} + 15028 T^{3} + 260838 T^{4} + 3659454 T^{5} + 48835451 T^{6} + 561496779 T^{7} + 6163562054 T^{8} + 59456950122 T^{9} + 548557022806 T^{10} + 4447615986459 T^{11} + 34427479056019 T^{12} + 229602344796414 T^{13} + 1456534898558262 T^{14} + 7468634840561908 T^{15} + 36269694614333780 T^{16} + 114161075365360349 T^{17} + 350356403707485209 T^{18} \)
$97$ \( 1 + 43 T + 1419 T^{2} + 33653 T^{3} + 675911 T^{4} + 11386603 T^{5} + 168876851 T^{6} + 2198016667 T^{7} + 25611739306 T^{8} + 265951175060 T^{9} + 2484338712682 T^{10} + 20681138819803 T^{11} + 154129342232723 T^{12} + 1008047776622443 T^{13} + 5804277740449127 T^{14} + 28032006881875637 T^{15} + 114652765674442347 T^{16} + 337009644558209323 T^{17} + 760231058654565217 T^{18} \)
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