Properties

Label 8001.2.a.l
Level 8001
Weight 2
Character orbit 8001.a
Self dual yes
Analytic conductor 63.888
Analytic rank 1
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: 7.7.118870813.1
Defining polynomial: \(x^{7} - 9 x^{5} - 3 x^{4} + 20 x^{3} + 7 x^{2} - 13 x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{4} q^{2} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{4} + ( 1 + \beta_{3} - \beta_{5} ) q^{5} + q^{7} + ( 1 + \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})\) \( q -\beta_{4} q^{2} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{4} + ( 1 + \beta_{3} - \beta_{5} ) q^{5} + q^{7} + ( 1 + \beta_{5} + \beta_{6} ) q^{8} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{10} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{11} + ( -3 - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{13} -\beta_{4} q^{14} + ( -\beta_{1} + 2 \beta_{5} ) q^{16} + ( \beta_{1} - \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{17} + ( -1 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{19} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{20} + ( -2 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{6} ) q^{22} + ( -1 - 3 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{23} + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{25} + ( -1 - \beta_{1} - 2 \beta_{3} + 3 \beta_{4} - \beta_{6} ) q^{26} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{28} + ( 1 - \beta_{2} + \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{29} + ( -5 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{31} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{5} ) q^{32} + ( -\beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{34} + ( 1 + \beta_{3} - \beta_{5} ) q^{35} + ( -5 - 3 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{37} + ( 1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{38} + ( -2 - 2 \beta_{1} + \beta_{3} + \beta_{5} + 2 \beta_{6} ) q^{40} + ( 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{41} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{5} - 3 \beta_{6} ) q^{43} + ( 1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} ) q^{44} + ( -3 - \beta_{3} + 4 \beta_{4} - \beta_{6} ) q^{46} + ( -3 - \beta_{1} + 5 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{47} + q^{49} + ( -3 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 4 \beta_{6} ) q^{50} + ( -5 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} ) q^{52} + ( \beta_{1} - 3 \beta_{2} + \beta_{3} - 3 \beta_{5} ) q^{53} + ( -3 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} ) q^{55} + ( 1 + \beta_{5} + \beta_{6} ) q^{56} + ( -3 + 3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{58} + ( 6 - \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - \beta_{5} - 3 \beta_{6} ) q^{59} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{61} + ( 1 - 2 \beta_{2} + 5 \beta_{4} + \beta_{5} - \beta_{6} ) q^{62} + ( -2 + \beta_{1} - 4 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{64} + ( -1 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} + 2 \beta_{6} ) q^{65} + ( -3 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{67} + ( -3 - 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{68} + ( -\beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{70} + ( -2 - \beta_{1} + 5 \beta_{2} + \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} ) q^{71} + ( -3 + 5 \beta_{1} - \beta_{5} + 2 \beta_{6} ) q^{73} + ( 2 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} ) q^{74} + ( -5 - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{76} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{77} + ( 1 + \beta_{2} - 4 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} ) q^{79} + ( -2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{80} + ( -\beta_{1} + 2 \beta_{2} - 3 \beta_{4} + 4 \beta_{6} ) q^{82} + ( 5 - 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} ) q^{83} + ( -5 + 2 \beta_{2} + \beta_{3} - 6 \beta_{4} - 2 \beta_{5} - 5 \beta_{6} ) q^{85} + ( -2 - \beta_{1} - 4 \beta_{2} - \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{86} + ( 3 - 4 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{88} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + 7 \beta_{6} ) q^{89} + ( -3 - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{91} + ( -6 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} - \beta_{5} - 3 \beta_{6} ) q^{92} + ( -1 - 3 \beta_{2} - \beta_{3} + \beta_{4} + 6 \beta_{5} + 2 \beta_{6} ) q^{94} + ( -1 - 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - \beta_{4} + 4 \beta_{5} - 3 \beta_{6} ) q^{95} + ( 4 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 5 \beta_{6} ) q^{97} -\beta_{4} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 2q^{2} + 4q^{4} + 8q^{5} + 7q^{7} + 9q^{8} + O(q^{10}) \) \( 7q + 2q^{2} + 4q^{4} + 8q^{5} + 7q^{7} + 9q^{8} + 3q^{11} - 23q^{13} + 2q^{14} + 2q^{16} - 3q^{17} - 9q^{19} + 9q^{20} - 19q^{22} - 12q^{23} + 3q^{25} - 18q^{26} + 4q^{28} + 9q^{29} - 33q^{31} - 10q^{32} - 2q^{34} + 8q^{35} - 33q^{37} + 3q^{38} - 9q^{40} + 3q^{41} - 9q^{43} - 2q^{44} - 32q^{46} - 11q^{47} + 7q^{49} - 29q^{50} - 21q^{52} - q^{53} - 16q^{55} + 9q^{56} - 5q^{58} + 30q^{59} - 19q^{61} - 3q^{62} - 21q^{64} - 14q^{65} - 30q^{67} - 24q^{68} - 8q^{71} - 20q^{73} + 9q^{74} - 42q^{76} + 3q^{77} + 8q^{79} - 12q^{80} + 10q^{82} + 34q^{83} - 28q^{85} - 24q^{86} - q^{88} + 12q^{89} - 23q^{91} - 60q^{92} - 3q^{94} - 12q^{95} + 7q^{97} + 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 9 x^{5} - 3 x^{4} + 20 x^{3} + 7 x^{2} - 13 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{5} - 7 \nu^{3} - 4 \nu^{2} + 7 \nu + 5 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - 8 \nu^{3} - 3 \nu^{2} + 11 \nu + 4 \)
\(\beta_{4}\)\(=\)\( -\nu^{6} + 8 \nu^{4} + 3 \nu^{3} - 12 \nu^{2} - 4 \nu + 2 \)
\(\beta_{5}\)\(=\)\( \nu^{6} + 2 \nu^{5} - 8 \nu^{4} - 18 \nu^{3} + 6 \nu^{2} + 22 \nu + 4 \)
\(\beta_{6}\)\(=\)\( 2 \nu^{6} + \nu^{5} - 15 \nu^{4} - 14 \nu^{3} + 15 \nu^{2} + 16 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} - 2 \beta_{3} + 4 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(\beta_{6} + 6 \beta_{5} + 8 \beta_{4} - 7 \beta_{3} - 6 \beta_{2} + 3 \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(11 \beta_{5} + 11 \beta_{4} - 18 \beta_{3} - 3 \beta_{2} + 21 \beta_{1} + 21\)
\(\nu^{6}\)\(=\)\(8 \beta_{6} + 39 \beta_{5} + 54 \beta_{4} - 50 \beta_{3} - 36 \beta_{2} + 32 \beta_{1} + 92\)

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
−0.301070
2.69855
−2.06168
−1.52532
1.20244
−1.14753
1.13462
−2.09968 0 2.40865 3.29054 0 1.00000 −0.858029 0 −6.90907
1.2 −0.840819 0 −1.29302 −2.74724 0 1.00000 2.76884 0 2.30993
1.3 −0.692358 0 −1.52064 2.78145 0 1.00000 2.43754 0 −1.92576
1.4 −0.246202 0 −1.93938 0.318209 0 1.00000 0.969884 0 −0.0783436
1.5 1.24280 0 −0.455452 3.33400 0 1.00000 −3.05163 0 4.14349
1.6 2.15625 0 2.64943 0.239094 0 1.00000 1.40032 0 0.515547
1.7 2.48001 0 4.15043 0.783950 0 1.00000 5.33307 0 1.94420
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 8001.2.a.l 7
3.b odd 2 1 2667.2.a.j 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2667.2.a.j 7 3.b odd 2 1
8001.2.a.l 7 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(127\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\( T_{2}^{7} - 2 T_{2}^{6} - 7 T_{2}^{5} + 11 T_{2}^{4} + 14 T_{2}^{3} - 9 T_{2}^{2} - 11 T_{2} - 2 \)
\( T_{5}^{7} - 8 T_{5}^{6} + 13 T_{5}^{5} + 42 T_{5}^{4} - 149 T_{5}^{3} + 138 T_{5}^{2} - 46 T_{5} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 7 T^{2} - 13 T^{3} + 28 T^{4} - 41 T^{5} + 73 T^{6} - 94 T^{7} + 146 T^{8} - 164 T^{9} + 224 T^{10} - 208 T^{11} + 224 T^{12} - 128 T^{13} + 128 T^{14} \)
$3$ 1
$5$ \( 1 - 8 T + 48 T^{2} - 198 T^{3} + 701 T^{4} - 2022 T^{5} + 5344 T^{6} - 12315 T^{7} + 26720 T^{8} - 50550 T^{9} + 87625 T^{10} - 123750 T^{11} + 150000 T^{12} - 125000 T^{13} + 78125 T^{14} \)
$7$ \( ( 1 - T )^{7} \)
$11$ \( 1 - 3 T + 31 T^{2} - 93 T^{3} + 656 T^{4} - 1715 T^{5} + 9199 T^{6} - 22080 T^{7} + 101189 T^{8} - 207515 T^{9} + 873136 T^{10} - 1361613 T^{11} + 4992581 T^{12} - 5314683 T^{13} + 19487171 T^{14} \)
$13$ \( 1 + 23 T + 289 T^{2} + 2532 T^{3} + 17108 T^{4} + 93828 T^{5} + 430561 T^{6} + 1679255 T^{7} + 5597293 T^{8} + 15856932 T^{9} + 37586276 T^{10} + 72316452 T^{11} + 107303677 T^{12} + 111016607 T^{13} + 62748517 T^{14} \)
$17$ \( 1 + 3 T + 50 T^{2} + 146 T^{3} + 1373 T^{4} + 4645 T^{5} + 30241 T^{6} + 99122 T^{7} + 514097 T^{8} + 1342405 T^{9} + 6745549 T^{10} + 12194066 T^{11} + 70992850 T^{12} + 72412707 T^{13} + 410338673 T^{14} \)
$19$ \( 1 + 9 T + 127 T^{2} + 828 T^{3} + 6701 T^{4} + 34195 T^{5} + 201852 T^{6} + 825304 T^{7} + 3835188 T^{8} + 12344395 T^{9} + 45962159 T^{10} + 107905788 T^{11} + 314464573 T^{12} + 423412929 T^{13} + 893871739 T^{14} \)
$23$ \( 1 + 12 T + 150 T^{2} + 1067 T^{3} + 8328 T^{4} + 46995 T^{5} + 292164 T^{6} + 1356265 T^{7} + 6719772 T^{8} + 24860355 T^{9} + 101326776 T^{10} + 298590347 T^{11} + 965451450 T^{12} + 1776430668 T^{13} + 3404825447 T^{14} \)
$29$ \( 1 - 9 T + 107 T^{2} - 652 T^{3} + 4764 T^{4} - 22810 T^{5} + 134487 T^{6} - 612145 T^{7} + 3900123 T^{8} - 19183210 T^{9} + 116189196 T^{10} - 461147212 T^{11} + 2194692943 T^{12} - 5353409889 T^{13} + 17249876309 T^{14} \)
$31$ \( 1 + 33 T + 645 T^{2} + 8883 T^{3} + 95284 T^{4} + 826887 T^{5} + 5965621 T^{6} + 36144101 T^{7} + 184934251 T^{8} + 794638407 T^{9} + 2838605644 T^{10} + 8203637043 T^{11} + 18465802395 T^{12} + 29287621473 T^{13} + 27512614111 T^{14} \)
$37$ \( 1 + 33 T + 653 T^{2} + 9315 T^{3} + 105122 T^{4} + 971473 T^{5} + 7548399 T^{6} + 49726081 T^{7} + 279290763 T^{8} + 1329946537 T^{9} + 5324744666 T^{10} + 17457809715 T^{11} + 45281603921 T^{12} + 84668971497 T^{13} + 94931877133 T^{14} \)
$41$ \( 1 - 3 T + 170 T^{2} - 50 T^{3} + 11626 T^{4} + 29278 T^{5} + 496448 T^{6} + 2150172 T^{7} + 20354368 T^{8} + 49216318 T^{9} + 801275546 T^{10} - 141288050 T^{11} + 19695554170 T^{12} - 14250312723 T^{13} + 194754273881 T^{14} \)
$43$ \( 1 + 9 T + 217 T^{2} + 1559 T^{3} + 22058 T^{4} + 131173 T^{5} + 1390533 T^{6} + 6894030 T^{7} + 59792919 T^{8} + 242538877 T^{9} + 1753765406 T^{10} + 5329910759 T^{11} + 31900832131 T^{12} + 56892267441 T^{13} + 271818611107 T^{14} \)
$47$ \( 1 + 11 T + 145 T^{2} + 1339 T^{3} + 14705 T^{4} + 119758 T^{5} + 953008 T^{6} + 6346454 T^{7} + 44791376 T^{8} + 264545422 T^{9} + 1526717215 T^{10} + 6533892859 T^{11} + 33255026015 T^{12} + 118571368619 T^{13} + 506623120463 T^{14} \)
$53$ \( 1 + T + 216 T^{2} - 139 T^{3} + 22869 T^{4} - 30467 T^{5} + 1664224 T^{6} - 2175415 T^{7} + 88203872 T^{8} - 85581803 T^{9} + 3404668113 T^{10} - 1096776859 T^{11} + 90330226488 T^{12} + 22164361129 T^{13} + 1174711139837 T^{14} \)
$59$ \( 1 - 30 T + 570 T^{2} - 7389 T^{3} + 75460 T^{4} - 617301 T^{5} + 4593588 T^{6} - 33613111 T^{7} + 271021692 T^{8} - 2148824781 T^{9} + 15497899340 T^{10} - 89535180429 T^{11} + 407506850430 T^{12} - 1265416009230 T^{13} + 2488651484819 T^{14} \)
$61$ \( 1 + 19 T + 427 T^{2} + 5649 T^{3} + 75542 T^{4} + 765723 T^{5} + 7510227 T^{6} + 59966461 T^{7} + 458123847 T^{8} + 2849255283 T^{9} + 17146598702 T^{10} + 78215155809 T^{11} + 360642620527 T^{12} + 978887112859 T^{13} + 3142742836021 T^{14} \)
$67$ \( 1 + 30 T + 675 T^{2} + 11360 T^{3} + 157485 T^{4} + 1838995 T^{5} + 18608402 T^{6} + 162480644 T^{7} + 1246762934 T^{8} + 8255248555 T^{9} + 47365661055 T^{10} + 228916734560 T^{11} + 911334447225 T^{12} + 2713751465070 T^{13} + 6060711605323 T^{14} \)
$71$ \( 1 + 8 T + 246 T^{2} + 1787 T^{3} + 34468 T^{4} + 225343 T^{5} + 3268094 T^{6} + 18726838 T^{7} + 232034674 T^{8} + 1135954063 T^{9} + 12336476348 T^{10} + 45410673947 T^{11} + 443840420346 T^{12} + 1024802271368 T^{13} + 9095120158391 T^{14} \)
$73$ \( 1 + 20 T + 510 T^{2} + 6961 T^{3} + 102634 T^{4} + 1069123 T^{5} + 11655826 T^{6} + 97527765 T^{7} + 850875298 T^{8} + 5697356467 T^{9} + 39926370778 T^{10} + 197680155601 T^{11} + 1057266512430 T^{12} + 3026684525780 T^{13} + 11047398519097 T^{14} \)
$79$ \( 1 - 8 T + 273 T^{2} - 2858 T^{3} + 45901 T^{4} - 424499 T^{5} + 5436714 T^{6} - 39832374 T^{7} + 429500406 T^{8} - 2649298259 T^{9} + 22630983139 T^{10} - 111319331498 T^{11} + 840036396927 T^{12} - 1944699644168 T^{13} + 19203908986159 T^{14} \)
$83$ \( 1 - 34 T + 962 T^{2} - 18636 T^{3} + 304656 T^{4} - 4073018 T^{5} + 46715214 T^{6} - 458242049 T^{7} + 3877362762 T^{8} - 28059021002 T^{9} + 174198340272 T^{10} - 884433270156 T^{11} + 3789357098566 T^{12} - 11115972694546 T^{13} + 27136050989627 T^{14} \)
$89$ \( 1 - 12 T + 151 T^{2} - 2659 T^{3} + 35240 T^{4} - 317069 T^{5} + 3805221 T^{6} - 40673821 T^{7} + 338664669 T^{8} - 2511503549 T^{9} + 24843107560 T^{10} - 166831618819 T^{11} + 843192976799 T^{12} - 5963775491532 T^{13} + 44231334895529 T^{14} \)
$97$ \( 1 - 7 T + 326 T^{2} - 2842 T^{3} + 55405 T^{4} - 456619 T^{5} + 7188211 T^{6} - 47880078 T^{7} + 697256467 T^{8} - 4296328171 T^{9} + 50566647565 T^{10} - 251600216602 T^{11} + 2799472923782 T^{12} - 5830804034503 T^{13} + 80798284478113 T^{14} \)
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