Properties

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

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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: \(1\)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -\beta_{3} q^{3} + ( \beta_{1} + \beta_{2} ) q^{4} + ( \beta_{6} + \beta_{7} ) q^{6} + q^{7} + ( -1 - \beta_{2} - \beta_{5} - \beta_{6} ) q^{8} + \beta_{6} q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} -\beta_{3} q^{3} + ( \beta_{1} + \beta_{2} ) q^{4} + ( \beta_{6} + \beta_{7} ) q^{6} + q^{7} + ( -1 - \beta_{2} - \beta_{5} - \beta_{6} ) q^{8} + \beta_{6} q^{9} + ( -1 + \beta_{3} - \beta_{7} ) q^{11} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{12} + ( -1 + \beta_{2} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{13} -\beta_{1} q^{14} + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{16} + ( \beta_{4} + \beta_{5} ) q^{17} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{18} + ( -2 \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{19} -\beta_{3} q^{21} + ( \beta_{1} + 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{22} + q^{23} + ( 1 + 2 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{24} + ( 1 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - \beta_{6} ) q^{26} + ( -\beta_{1} + 2 \beta_{3} - \beta_{6} ) q^{27} + ( \beta_{1} + \beta_{2} ) q^{28} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{29} + ( -1 + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{31} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{32} + ( -3 + 3 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} ) q^{33} + ( -\beta_{2} - \beta_{3} + \beta_{5} ) q^{34} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} ) q^{36} + ( -2 - \beta_{1} + \beta_{2} - \beta_{6} + \beta_{7} ) q^{37} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{38} + ( 2 - 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{6} ) q^{39} + ( -2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{41} + ( \beta_{6} + \beta_{7} ) q^{42} + ( -3 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{7} ) q^{43} + ( \beta_{2} - \beta_{3} - 2 \beta_{6} ) q^{44} -\beta_{1} q^{46} + ( 1 - 2 \beta_{1} - 4 \beta_{2} - \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{47} + ( -4 - \beta_{1} + \beta_{2} - \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{48} + q^{49} + ( -\beta_{1} - \beta_{4} - \beta_{5} ) q^{51} + ( 1 - 5 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{52} + ( -1 + \beta_{1} - \beta_{2} + 4 \beta_{4} + \beta_{6} ) q^{53} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{54} + ( -1 - \beta_{2} - \beta_{5} - \beta_{6} ) q^{56} + ( -1 - \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{57} + ( -3 + \beta_{1} - \beta_{2} + 3 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{58} + ( 3 - \beta_{1} - \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{59} + ( -1 - 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{61} + ( 1 - \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + \beta_{4} - 4 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{62} + \beta_{6} q^{63} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{64} + ( -5 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{66} + ( -3 - 2 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} - \beta_{4} + 4 \beta_{5} + 2 \beta_{6} ) q^{67} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{68} -\beta_{3} q^{69} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{71} + ( -3 - 2 \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{72} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 6 \beta_{4} + \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{73} + ( 3 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{74} + ( -3 - \beta_{2} + 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{76} + ( -1 + \beta_{3} - \beta_{7} ) q^{77} + ( -3 - 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{78} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} ) q^{79} + ( -6 + \beta_{1} + \beta_{3} - 3 \beta_{6} + \beta_{7} ) q^{81} + ( -2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{82} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 5 \beta_{6} ) q^{83} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{84} + ( -4 + 4 \beta_{1} - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{86} + ( -4 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{87} + ( 1 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{88} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{89} + ( -1 + \beta_{2} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{91} + ( \beta_{1} + \beta_{2} ) q^{92} + ( -2 + 5 \beta_{1} + 3 \beta_{2} - \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{93} + ( 2 \beta_{1} + \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{94} + ( 1 + 3 \beta_{1} + 4 \beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{96} + ( -5 - \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + \beta_{5} + 2 \beta_{7} ) q^{97} -\beta_{1} q^{98} + ( 1 - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 3q^{2} - 2q^{3} + 5q^{4} - q^{6} + 8q^{7} - 12q^{8} + O(q^{10}) \) \( 8q - 3q^{2} - 2q^{3} + 5q^{4} - q^{6} + 8q^{7} - 12q^{8} - 5q^{11} - 3q^{12} - 9q^{13} - 3q^{14} - q^{16} + q^{17} - 6q^{18} - 4q^{19} - 2q^{21} + 5q^{22} + 8q^{23} + 16q^{24} + 22q^{26} + q^{27} + 5q^{28} - 5q^{29} - q^{31} - 2q^{32} - 12q^{33} - 2q^{34} + 16q^{36} - 18q^{37} - 14q^{38} + 14q^{39} - q^{41} - q^{42} - 20q^{43} - 3q^{46} - 10q^{47} - 31q^{48} + 8q^{49} - 4q^{51} - 11q^{52} - 11q^{53} + 29q^{54} - 12q^{56} - 8q^{57} - 24q^{58} + 20q^{59} - 6q^{61} - 2q^{62} + 8q^{64} - 37q^{66} - 23q^{67} - 9q^{68} - 2q^{69} + 3q^{71} - 29q^{72} + 8q^{73} + 35q^{74} - 29q^{76} - 5q^{77} - 31q^{78} + 4q^{79} - 44q^{81} - 27q^{82} - 4q^{83} - 3q^{84} - 18q^{86} - 27q^{87} + 4q^{88} - 17q^{89} - 9q^{91} + 5q^{92} + 7q^{93} + 13q^{94} + 22q^{96} - 41q^{97} - 3q^{98} + 7q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} - 6 x^{6} + 19 x^{5} + 12 x^{4} - 34 x^{3} - 12 x^{2} + 17 x + 7\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - \nu^{3} - 5 \nu^{2} + 2 \nu + 4 \)
\(\beta_{4}\)\(=\)\( \nu^{6} - 2 \nu^{5} - 7 \nu^{4} + 11 \nu^{3} + 15 \nu^{2} - 13 \nu - 9 \)
\(\beta_{5}\)\(=\)\( -\nu^{7} + 3 \nu^{6} + 5 \nu^{5} - 17 \nu^{4} - 5 \nu^{3} + 23 \nu^{2} - 2 \nu - 5 \)
\(\beta_{6}\)\(=\)\( \nu^{7} - 3 \nu^{6} - 5 \nu^{5} + 17 \nu^{4} + 6 \nu^{3} - 24 \nu^{2} - \nu + 6 \)
\(\beta_{7}\)\(=\)\( -\nu^{7} + 3 \nu^{6} + 6 \nu^{5} - 18 \nu^{4} - 11 \nu^{3} + 26 \nu^{2} + 5 \nu - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{2} + 4 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{3} + 6 \beta_{2} + 7 \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\(\beta_{7} + 7 \beta_{6} + 6 \beta_{5} + \beta_{3} + 9 \beta_{2} + 21 \beta_{1} + 8\)
\(\nu^{6}\)\(=\)\(2 \beta_{7} + 10 \beta_{6} + 8 \beta_{5} + \beta_{4} + 9 \beta_{3} + 34 \beta_{2} + 45 \beta_{1} + 33\)
\(\nu^{7}\)\(=\)\(11 \beta_{7} + 43 \beta_{6} + 31 \beta_{5} + 3 \beta_{4} + 15 \beta_{3} + 63 \beta_{2} + 122 \beta_{1} + 56\)

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
2.55976
2.33027
1.38128
1.07669
−0.468649
−0.673262
−1.34053
−1.86556
−2.55976 −2.51872 4.55237 0 6.44731 1.00000 −6.53345 3.34393 0
1.2 −2.33027 1.65734 3.43017 0 −3.86204 1.00000 −3.33268 −0.253237 0
1.3 −1.38128 1.77230 −0.0920619 0 −2.44804 1.00000 2.88973 0.141045 0
1.4 −1.07669 −0.452781 −0.840734 0 0.487505 1.00000 3.05860 −2.79499 0
1.5 0.468649 −2.11571 −1.78037 0 −0.991526 1.00000 −1.77166 1.47624 0
1.6 0.673262 −0.897707 −1.54672 0 −0.604392 1.00000 −2.38787 −2.19412 0
1.7 1.34053 2.02792 −0.202970 0 2.71850 1.00000 −2.95316 1.11246 0
1.8 1.86556 −1.47264 1.48032 0 −2.74730 1.00000 −0.969501 −0.831331 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.s 8
5.b even 2 1 4025.2.a.w yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4025.2.a.s 8 1.a even 1 1 trivial
4025.2.a.w yes 8 5.b even 2 1

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

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 10 T^{2} + 23 T^{3} + 52 T^{4} + 96 T^{5} + 172 T^{6} + 267 T^{7} + 407 T^{8} + 534 T^{9} + 688 T^{10} + 768 T^{11} + 832 T^{12} + 736 T^{13} + 640 T^{14} + 384 T^{15} + 256 T^{16} \)
$3$ \( 1 + 2 T + 14 T^{2} + 23 T^{3} + 103 T^{4} + 152 T^{5} + 511 T^{6} + 650 T^{7} + 1787 T^{8} + 1950 T^{9} + 4599 T^{10} + 4104 T^{11} + 8343 T^{12} + 5589 T^{13} + 10206 T^{14} + 4374 T^{15} + 6561 T^{16} \)
$5$ \( \)
$7$ \( ( 1 - T )^{8} \)
$11$ \( 1 + 5 T + 70 T^{2} + 288 T^{3} + 2261 T^{4} + 7814 T^{5} + 44648 T^{6} + 129823 T^{7} + 592081 T^{8} + 1428053 T^{9} + 5402408 T^{10} + 10400434 T^{11} + 33103301 T^{12} + 46382688 T^{13} + 124009270 T^{14} + 97435855 T^{15} + 214358881 T^{16} \)
$13$ \( 1 + 9 T + 90 T^{2} + 552 T^{3} + 3242 T^{4} + 15387 T^{5} + 68244 T^{6} + 272378 T^{7} + 1016775 T^{8} + 3540914 T^{9} + 11533236 T^{10} + 33805239 T^{11} + 92594762 T^{12} + 204953736 T^{13} + 434412810 T^{14} + 564736653 T^{15} + 815730721 T^{16} \)
$17$ \( 1 - T + 103 T^{2} - 159 T^{3} + 4808 T^{4} - 9344 T^{5} + 137670 T^{6} - 281203 T^{7} + 2746785 T^{8} - 4780451 T^{9} + 39786630 T^{10} - 45907072 T^{11} + 401568968 T^{12} - 225757263 T^{13} + 2486169607 T^{14} - 410338673 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 + 4 T + 91 T^{2} + 340 T^{3} + 3601 T^{4} + 13255 T^{5} + 87650 T^{6} + 332944 T^{7} + 1718231 T^{8} + 6325936 T^{9} + 31641650 T^{10} + 90916045 T^{11} + 469285921 T^{12} + 841873660 T^{13} + 4281175171 T^{14} + 3575486956 T^{15} + 16983563041 T^{16} \)
$23$ \( ( 1 - T )^{8} \)
$29$ \( 1 + 5 T + 93 T^{2} + 333 T^{3} + 5333 T^{4} + 16991 T^{5} + 225916 T^{6} + 634890 T^{7} + 7414465 T^{8} + 18411810 T^{9} + 189995356 T^{10} + 414393499 T^{11} + 3771929573 T^{12} + 6830212617 T^{13} + 55318568853 T^{14} + 86249381545 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 + T + 100 T^{2} - 148 T^{3} + 4385 T^{4} - 19004 T^{5} + 119190 T^{6} - 1048099 T^{7} + 2877455 T^{8} - 32491069 T^{9} + 114541590 T^{10} - 566148164 T^{11} + 4049639585 T^{12} - 4237114348 T^{13} + 88750368100 T^{14} + 27512614111 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 + 18 T + 376 T^{2} + 4505 T^{3} + 54939 T^{4} + 489840 T^{5} + 4312804 T^{6} + 29910143 T^{7} + 203070623 T^{8} + 1106675291 T^{9} + 5904228676 T^{10} + 24811865520 T^{11} + 102964531179 T^{12} + 312394526285 T^{13} + 964713129784 T^{14} + 1708773788394 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 + T + 218 T^{2} + 322 T^{3} + 23375 T^{4} + 37522 T^{5} + 1614950 T^{6} + 2440663 T^{7} + 78223615 T^{8} + 100067183 T^{9} + 2714730950 T^{10} + 2586053762 T^{11} + 66052163375 T^{12} + 37305696722 T^{13} + 1035522724538 T^{14} + 194754273881 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 + 20 T + 370 T^{2} + 4633 T^{3} + 53740 T^{4} + 509876 T^{5} + 4469206 T^{6} + 33824174 T^{7} + 236968703 T^{8} + 1454439482 T^{9} + 8263561894 T^{10} + 40538711132 T^{11} + 183726365740 T^{12} + 681090116419 T^{13} + 2338904328130 T^{14} + 5436372222140 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 + 10 T + 203 T^{2} + 2104 T^{3} + 23744 T^{4} + 210815 T^{5} + 1848114 T^{6} + 13961441 T^{7} + 101566631 T^{8} + 656187727 T^{9} + 4082483826 T^{10} + 21887445745 T^{11} + 115863145664 T^{12} + 482541894728 T^{13} + 2188180711787 T^{14} + 5066231204630 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 + 11 T + 185 T^{2} + 1808 T^{3} + 21781 T^{4} + 178945 T^{5} + 1785701 T^{6} + 12903950 T^{7} + 107340417 T^{8} + 683909350 T^{9} + 5016034109 T^{10} + 26640794765 T^{11} + 171862566661 T^{12} + 756097451344 T^{13} + 4100406808865 T^{14} + 12921822538207 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 - 20 T + 491 T^{2} - 6579 T^{3} + 96053 T^{4} - 991671 T^{5} + 10783675 T^{6} - 90024096 T^{7} + 781129041 T^{8} - 5311421664 T^{9} + 37537972675 T^{10} - 203668398309 T^{11} + 1163908876133 T^{12} - 4703486963121 T^{13} + 20710642017731 T^{14} - 49773029696380 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 + 6 T + 156 T^{2} + 1025 T^{3} + 19201 T^{4} + 111096 T^{5} + 1701891 T^{6} + 8894564 T^{7} + 113263297 T^{8} + 542568404 T^{9} + 6332736411 T^{10} + 25216681176 T^{11} + 265853993041 T^{12} + 865711208525 T^{13} + 8037178400316 T^{14} + 18856457016126 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 23 T + 411 T^{2} + 5199 T^{3} + 61873 T^{4} + 653083 T^{5} + 6639648 T^{6} + 60135256 T^{7} + 519181851 T^{8} + 4029062152 T^{9} + 29805379872 T^{10} + 196423202329 T^{11} + 1246810309633 T^{12} + 7019300431293 T^{13} + 37178395071459 T^{14} + 139396366922429 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 - 3 T + 313 T^{2} - 317 T^{3} + 49061 T^{4} - 8243 T^{5} + 5430652 T^{6} - 422566 T^{7} + 449286567 T^{8} - 30002186 T^{9} + 27375916732 T^{10} - 2950260373 T^{11} + 1246722481541 T^{12} - 571940704267 T^{13} + 40095388867273 T^{14} - 27285360475173 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 - 8 T + 242 T^{2} - 1479 T^{3} + 32419 T^{4} - 179732 T^{5} + 3196385 T^{6} - 16886574 T^{7} + 259809365 T^{8} - 1232719902 T^{9} + 17033535665 T^{10} - 69918803444 T^{11} + 920642574979 T^{12} - 3066072886047 T^{13} + 36622882761938 T^{14} - 88379188152776 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 - 4 T + 560 T^{2} - 2142 T^{3} + 141251 T^{4} - 497005 T^{5} + 21068076 T^{6} - 64753492 T^{7} + 2039964787 T^{8} - 5115525868 T^{9} + 131485862316 T^{10} - 245042848195 T^{11} + 5501737891331 T^{12} - 6591054806658 T^{13} + 136128975091760 T^{14} - 76815635944636 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 + 4 T + 271 T^{2} + 1905 T^{3} + 42756 T^{4} + 390631 T^{5} + 4863288 T^{6} + 48279102 T^{7} + 452154785 T^{8} + 4007165466 T^{9} + 33503191032 T^{10} + 223357727597 T^{11} + 2029127972676 T^{12} + 7503872424915 T^{13} + 88600841182999 T^{14} + 108544203958508 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 + 17 T + 720 T^{2} + 9700 T^{3} + 221878 T^{4} + 2426901 T^{5} + 38946962 T^{6} + 347718674 T^{7} + 4307242151 T^{8} + 30946961986 T^{9} + 308498886002 T^{10} + 1710889971069 T^{11} + 13921122948598 T^{12} + 54165376655300 T^{13} + 357826529491920 T^{14} + 751932693223993 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 + 41 T + 1234 T^{2} + 26542 T^{3} + 486246 T^{4} + 7410777 T^{5} + 100057472 T^{6} + 1171754020 T^{7} + 12324003755 T^{8} + 113660139940 T^{9} + 941440754048 T^{10} + 6763616076921 T^{11} + 43047008769126 T^{12} + 227925185101294 T^{13} + 1027887454082386 T^{14} + 3312729663602633 T^{15} + 7837433594376961 T^{16} \)
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