Properties

Label 4025.2.a.v
Level 4025
Weight 2
Character orbit 4025.a
Self dual yes
Analytic conductor 32.140
Analytic rank 0
Dimension 8
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1397868136\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{4} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{3} + \beta_{6} - \beta_{7} ) q^{6} - q^{7} + \beta_{3} q^{8} + ( \beta_{4} - \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{4} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( \beta_{1} + \beta_{3} + \beta_{6} - \beta_{7} ) q^{6} - q^{7} + \beta_{3} q^{8} + ( \beta_{4} - \beta_{6} ) q^{9} + ( \beta_{1} + \beta_{3} - \beta_{7} ) q^{11} + ( 2 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{12} + ( \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{13} -\beta_{1} q^{14} + ( -\beta_{2} + \beta_{3} + \beta_{4} ) q^{16} + ( \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{17} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{18} + \beta_{5} q^{19} + ( -1 - \beta_{4} ) q^{21} + ( 4 + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{22} + q^{23} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} ) q^{24} + ( 3 + \beta_{1} + \beta_{2} + \beta_{6} ) q^{26} + ( -\beta_{1} - \beta_{4} - \beta_{6} ) q^{27} + ( -1 - \beta_{2} ) q^{28} + ( -2 + \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{29} + ( \beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{31} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{32} + ( 1 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{33} + ( -2 - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{6} + 2 \beta_{7} ) q^{34} + ( 3 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{36} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} - 3 \beta_{7} ) q^{37} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{38} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} ) q^{39} + ( -1 - 3 \beta_{2} + 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{41} + ( -\beta_{1} - \beta_{3} - \beta_{6} + \beta_{7} ) q^{42} + ( -2 \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{43} + ( 2 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{6} ) q^{44} + \beta_{1} q^{46} + ( 3 + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{47} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{5} ) q^{48} + q^{49} + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{51} + ( 3 + \beta_{1} + \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{52} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} ) q^{53} + ( -3 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{54} -\beta_{3} q^{56} + ( 2 - \beta_{1} + \beta_{2} + \beta_{5} + \beta_{7} ) q^{57} + ( -\beta_{1} + \beta_{2} - 3 \beta_{3} - 4 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{58} + ( 1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{59} + ( 1 - 2 \beta_{3} + \beta_{4} - \beta_{6} ) q^{61} + ( 3 + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{62} + ( -\beta_{4} + \beta_{6} ) q^{63} + ( -6 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{64} + ( 9 + \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + \beta_{5} - \beta_{6} ) q^{66} + ( 4 + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{67} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} - 4 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{68} + ( 1 + \beta_{4} ) q^{69} + ( 3 - \beta_{1} - \beta_{3} + 3 \beta_{6} ) q^{71} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{72} + ( 4 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{73} + ( -4 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} + \beta_{5} ) q^{74} + ( 4 - \beta_{2} + \beta_{3} + 4 \beta_{4} - \beta_{6} + \beta_{7} ) q^{76} + ( -\beta_{1} - \beta_{3} + \beta_{7} ) q^{77} + ( 3 + \beta_{1} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{78} + ( -3 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{79} + ( -1 - 2 \beta_{1} - \beta_{3} - \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{81} + ( -4 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{82} + ( 1 + \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{83} + ( -2 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{84} + ( 1 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} - 6 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{86} + ( -4 - 3 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{87} + ( 5 + \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{88} + ( 3 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{89} + ( -\beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{91} + ( 1 + \beta_{2} ) q^{92} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{4} ) q^{93} + ( 1 + 4 \beta_{1} + 3 \beta_{2} + \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{94} + ( 5 - \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - 4 \beta_{6} + \beta_{7} ) q^{96} + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{97} + \beta_{1} q^{98} + ( 2 + 3 \beta_{1} + \beta_{3} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + q^{2} + 4q^{3} + 5q^{4} - q^{6} - 8q^{7} + O(q^{10}) \) \( 8q + q^{2} + 4q^{3} + 5q^{4} - q^{6} - 8q^{7} + 3q^{11} + 9q^{12} + 5q^{13} - q^{14} - q^{16} + 5q^{17} + 2q^{18} - 2q^{19} - 4q^{21} + 21q^{22} + 8q^{23} - 6q^{24} + 18q^{26} + 7q^{27} - 5q^{28} - 9q^{29} - 3q^{31} + 6q^{32} + 4q^{33} - 10q^{34} + 16q^{36} + 6q^{37} + 4q^{38} - 2q^{39} - 7q^{41} + q^{42} + 8q^{43} + 4q^{44} + q^{46} + 22q^{47} + 9q^{48} + 8q^{49} - 12q^{51} + 11q^{52} + 21q^{53} - 15q^{54} + 8q^{57} + 16q^{58} + 14q^{59} + 8q^{61} + 12q^{62} - 40q^{64} + 55q^{66} + 21q^{67} + 3q^{68} + 4q^{69} + 11q^{71} - q^{72} + 26q^{73} - 41q^{74} + 21q^{76} - 3q^{77} + 17q^{78} - 16q^{79} - 20q^{81} - q^{82} + 20q^{83} - 9q^{84} + 14q^{86} - 29q^{87} + 32q^{88} + 15q^{89} - 5q^{91} + 5q^{92} + 19q^{93} + 21q^{94} + 52q^{96} + q^{97} + q^{98} + 15q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 10 x^{6} + 9 x^{5} + 28 x^{4} - 22 x^{3} - 16 x^{2} + 7 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 4 \nu + 1 \)
\(\beta_{5}\)\(=\)\( \nu^{6} - \nu^{5} - 8 \nu^{4} + 6 \nu^{3} + 17 \nu^{2} - 7 \nu - 7 \)
\(\beta_{6}\)\(=\)\( \nu^{7} - \nu^{6} - 9 \nu^{5} + 8 \nu^{4} + 21 \nu^{3} - 17 \nu^{2} - 5 \nu + 3 \)
\(\beta_{7}\)\(=\)\( \nu^{7} - \nu^{6} - 10 \nu^{5} + 9 \nu^{4} + 27 \nu^{3} - 21 \nu^{2} - 10 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + \beta_{3} + 5 \beta_{2} + 14\)
\(\nu^{5}\)\(=\)\(-\beta_{7} + \beta_{6} + \beta_{4} + 7 \beta_{3} + \beta_{2} + 19 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(-\beta_{7} + \beta_{6} + \beta_{5} + 9 \beta_{4} + 9 \beta_{3} + 24 \beta_{2} + 2 \beta_{1} + 70\)
\(\nu^{7}\)\(=\)\(-10 \beta_{7} + 11 \beta_{6} + \beta_{5} + 10 \beta_{4} + 43 \beta_{3} + 10 \beta_{2} + 94 \beta_{1} + 24\)

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
−2.18675
−2.07739
−0.571357
−0.333224
0.651939
1.09801
2.02014
2.39863
−2.18675 2.66680 2.78189 0 −5.83163 −1.00000 −1.70979 4.11181 0
1.2 −2.07739 −0.298210 2.31556 0 0.619500 −1.00000 −0.655538 −2.91107 0
1.3 −0.571357 −1.62458 −1.67355 0 0.928216 −1.00000 2.09891 −0.360734 0
1.4 −0.333224 0.161242 −1.88896 0 −0.0537298 −1.00000 1.29590 −2.97400 0
1.5 0.651939 2.38619 −1.57498 0 1.55565 −1.00000 −2.33067 2.69390 0
1.6 1.09801 0.493654 −0.794374 0 0.542037 −1.00000 −3.06825 −2.75631 0
1.7 2.02014 −1.91409 2.08098 0 −3.86674 −1.00000 0.163592 0.663753 0
1.8 2.39863 2.12900 3.75344 0 5.10670 −1.00000 4.20585 1.53265 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 4025.2.a.v yes 8
5.b even 2 1 4025.2.a.u 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4025.2.a.u 8 5.b even 2 1
4025.2.a.v yes 8 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(1\)
\(23\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4025))\):

\(T_{2}^{8} - \cdots\)
\(T_{3}^{8} - \cdots\)
\(T_{11}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 6 T^{2} - 5 T^{3} + 20 T^{4} - 16 T^{5} + 56 T^{6} - 45 T^{7} + 131 T^{8} - 90 T^{9} + 224 T^{10} - 128 T^{11} + 320 T^{12} - 160 T^{13} + 384 T^{14} - 128 T^{15} + 256 T^{16} \)
$3$ \( 1 - 4 T + 20 T^{2} - 57 T^{3} + 177 T^{4} - 398 T^{5} + 951 T^{6} - 1768 T^{7} + 3437 T^{8} - 5304 T^{9} + 8559 T^{10} - 10746 T^{11} + 14337 T^{12} - 13851 T^{13} + 14580 T^{14} - 8748 T^{15} + 6561 T^{16} \)
$5$ 1
$7$ \( ( 1 + T )^{8} \)
$11$ \( 1 - 3 T + 58 T^{2} - 188 T^{3} + 1705 T^{4} - 5286 T^{5} + 32172 T^{6} - 89557 T^{7} + 420105 T^{8} - 985127 T^{9} + 3892812 T^{10} - 7035666 T^{11} + 24962905 T^{12} - 30277588 T^{13} + 102750538 T^{14} - 58461513 T^{15} + 214358881 T^{16} \)
$13$ \( 1 - 5 T + 74 T^{2} - 262 T^{3} + 2314 T^{4} - 5941 T^{5} + 43418 T^{6} - 86890 T^{7} + 614789 T^{8} - 1129570 T^{9} + 7337642 T^{10} - 13052377 T^{11} + 66090154 T^{12} - 97278766 T^{13} + 357183866 T^{14} - 313742585 T^{15} + 815730721 T^{16} \)
$17$ \( 1 - 5 T + 93 T^{2} - 477 T^{3} + 4414 T^{4} - 20794 T^{5} + 133012 T^{6} - 544109 T^{7} + 2721975 T^{8} - 9249853 T^{9} + 38440468 T^{10} - 102160922 T^{11} + 368661694 T^{12} - 677271789 T^{13} + 2244793917 T^{14} - 2051693365 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 + 2 T + 123 T^{2} + 222 T^{3} + 7083 T^{4} + 11277 T^{5} + 247444 T^{6} + 337556 T^{7} + 5716293 T^{8} + 6413564 T^{9} + 89327284 T^{10} + 77348943 T^{11} + 923063643 T^{12} + 549693978 T^{13} + 5786643363 T^{14} + 1787743478 T^{15} + 16983563041 T^{16} \)
$23$ \( ( 1 - T )^{8} \)
$29$ \( 1 + 9 T + 177 T^{2} + 1073 T^{3} + 12317 T^{4} + 54195 T^{5} + 500732 T^{6} + 1763562 T^{7} + 15551717 T^{8} + 51143298 T^{9} + 421115612 T^{10} + 1321761855 T^{11} + 8711580077 T^{12} + 22008462877 T^{13} + 105283727817 T^{14} + 155248886781 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 + 3 T + 202 T^{2} + 494 T^{3} + 18723 T^{4} + 37846 T^{5} + 1052980 T^{6} + 1772343 T^{7} + 39487773 T^{8} + 54942633 T^{9} + 1011913780 T^{10} + 1127470186 T^{11} + 17291083683 T^{12} + 14142800594 T^{13} + 179275743562 T^{14} + 82537842333 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 - 6 T + 128 T^{2} - 483 T^{3} + 6815 T^{4} - 18492 T^{5} + 279952 T^{6} - 724717 T^{7} + 11064919 T^{8} - 26814529 T^{9} + 383254288 T^{10} - 936675276 T^{11} + 12772407215 T^{12} - 33493131231 T^{13} + 328412980352 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 + 7 T + 146 T^{2} + 290 T^{3} + 6657 T^{4} - 14616 T^{5} + 316450 T^{6} - 398833 T^{7} + 18250885 T^{8} - 16352153 T^{9} + 531952450 T^{10} - 1007349336 T^{11} + 18811090977 T^{12} + 33598298290 T^{13} + 693515219186 T^{14} + 1363279917167 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 - 8 T + 186 T^{2} - 811 T^{3} + 12532 T^{4} - 3752 T^{5} + 314858 T^{6} + 2838902 T^{7} + 3830323 T^{8} + 122072786 T^{9} + 582172442 T^{10} - 298310264 T^{11} + 42844414132 T^{12} - 119223847273 T^{13} + 1175773527114 T^{14} - 2174548888856 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 - 22 T + 477 T^{2} - 6490 T^{3} + 84170 T^{4} - 846631 T^{5} + 8082338 T^{6} - 63730055 T^{7} + 476894657 T^{8} - 2995312585 T^{9} + 17853884642 T^{10} - 87899770313 T^{11} + 410722749770 T^{12} - 1488449095430 T^{13} + 5141685711933 T^{14} - 11145708650186 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 - 21 T + 545 T^{2} - 7508 T^{3} + 111457 T^{4} - 1140983 T^{5} + 12195509 T^{6} - 97831066 T^{7} + 813054453 T^{8} - 5185046498 T^{9} + 34257184781 T^{10} - 169866126091 T^{11} + 879449340817 T^{12} - 3139811761444 T^{13} + 12079576815305 T^{14} - 24668933936577 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 - 14 T + 265 T^{2} - 2791 T^{3} + 35713 T^{4} - 324717 T^{5} + 3355461 T^{6} - 26197372 T^{7} + 229201335 T^{8} - 1545644948 T^{9} + 11680359741 T^{10} - 66690052743 T^{11} + 432747313393 T^{12} - 1995353718509 T^{13} + 11177841414865 T^{14} - 34841120787466 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 - 8 T + 404 T^{2} - 2889 T^{3} + 76073 T^{4} - 473002 T^{5} + 8636309 T^{6} - 45493588 T^{7} + 643227315 T^{8} - 2775108868 T^{9} + 32135705789 T^{10} - 107362466962 T^{11} + 1053294662393 T^{12} - 2440038713589 T^{13} + 20814231241844 T^{14} - 25141942688168 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 - 21 T + 559 T^{2} - 7909 T^{3} + 124053 T^{4} - 1337889 T^{5} + 15557612 T^{6} - 135484528 T^{7} + 1266969303 T^{8} - 9077463376 T^{9} + 69838120268 T^{10} - 402387509307 T^{11} + 2499807013413 T^{12} - 10678139471263 T^{13} + 50566235632471 T^{14} - 127274943711783 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 - 11 T + 393 T^{2} - 3817 T^{3} + 76733 T^{4} - 635255 T^{5} + 9436196 T^{6} - 66559266 T^{7} + 796900451 T^{8} - 4725707886 T^{9} + 47567864036 T^{10} - 227364752305 T^{11} + 1949914518173 T^{12} - 6886743432767 T^{13} + 50343411580953 T^{14} - 100046321742301 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 - 26 T + 716 T^{2} - 12063 T^{3} + 196639 T^{4} - 2464044 T^{5} + 29416957 T^{6} - 289161782 T^{7} + 2699414091 T^{8} - 21108810086 T^{9} + 156762963853 T^{10} - 958555004748 T^{11} + 5584201711999 T^{12} - 25007462626359 T^{13} + 108355306022924 T^{14} - 287232361496522 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 + 16 T + 468 T^{2} + 5518 T^{3} + 90791 T^{4} + 852531 T^{5} + 10480272 T^{6} + 84705860 T^{7} + 905161519 T^{8} + 6691762940 T^{9} + 65407377552 T^{10} + 420331031709 T^{11} + 3536316804071 T^{12} + 16979197209682 T^{13} + 113764929183828 T^{14} + 307262543778544 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 - 20 T + 605 T^{2} - 9107 T^{3} + 155840 T^{4} - 1898117 T^{5} + 23665964 T^{6} - 239462516 T^{7} + 2378527291 T^{8} - 19875388828 T^{9} + 163034825996 T^{10} - 1085318625079 T^{11} + 7395904744640 T^{12} - 35872843135801 T^{13} + 197798925888245 T^{14} - 542721019792540 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 - 15 T + 358 T^{2} - 5282 T^{3} + 75100 T^{4} - 930075 T^{5} + 10738476 T^{6} - 109583110 T^{7} + 1120131273 T^{8} - 9752896790 T^{9} + 85059468396 T^{10} - 655674042675 T^{11} + 4711942299100 T^{12} - 29495002009618 T^{13} + 177919302164038 T^{14} - 663470023432935 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 - T + 546 T^{2} + 252 T^{3} + 136930 T^{4} + 233989 T^{5} + 21403884 T^{6} + 49846696 T^{7} + 2395310017 T^{8} + 4835129512 T^{9} + 201389144556 T^{10} + 213555442597 T^{11} + 12122314447330 T^{12} + 2164009744764 T^{13} + 454802714691234 T^{14} - 80798284478113 T^{15} + 7837433594376961 T^{16} \)
show more
show less