Properties

Label 4025.2.a.t
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: \( 2 \)
Twist minimal: no (minimal twist has level 805)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -1 - \beta_{6} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{5} ) q^{6} - q^{7} + ( -\beta_{1} - \beta_{3} ) q^{8} + ( 1 - \beta_{2} + \beta_{3} + 2 \beta_{6} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -1 - \beta_{6} ) q^{3} + ( 1 + \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{5} ) q^{6} - q^{7} + ( -\beta_{1} - \beta_{3} ) q^{8} + ( 1 - \beta_{2} + \beta_{3} + 2 \beta_{6} ) q^{9} + ( 1 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{11} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} ) q^{12} + ( -1 - \beta_{5} ) q^{13} + \beta_{1} q^{14} + ( \beta_{1} + \beta_{4} ) q^{16} + ( -1 + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{17} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{18} + ( \beta_{1} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{19} + ( 1 + \beta_{6} ) q^{21} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{6} - \beta_{7} ) q^{22} - q^{23} + ( -1 + \beta_{2} + \beta_{6} - \beta_{7} ) q^{24} + ( 2 \beta_{1} + \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{26} + ( -3 + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{27} + ( -1 - \beta_{2} ) q^{28} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{29} + ( -2 - \beta_{3} - \beta_{4} - \beta_{7} ) q^{31} + ( -2 + 2 \beta_{1} + \beta_{3} - \beta_{5} ) q^{32} + ( -1 + 3 \beta_{1} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{33} + ( -1 - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{34} + ( -2 - 2 \beta_{1} - 3 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{36} + ( 1 + \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{37} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{38} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{39} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{41} + ( -\beta_{1} + \beta_{5} ) q^{42} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{43} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{44} + \beta_{1} q^{46} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{47} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{48} + q^{49} + ( 1 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{51} + ( -2 - \beta_{1} - \beta_{3} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{52} + ( 2 \beta_{1} - 2 \beta_{2} - \beta_{5} + 3 \beta_{6} ) q^{53} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{54} + ( \beta_{1} + \beta_{3} ) q^{56} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{57} + ( 2 + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{58} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{59} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{7} ) q^{61} + ( -3 + 4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{62} + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{6} ) q^{63} + ( -5 - 3 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{64} + ( -5 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{66} + ( -1 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{67} + ( 1 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{5} + 4 \beta_{6} ) q^{68} + ( 1 + \beta_{6} ) q^{69} + ( 5 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{71} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} ) q^{72} + ( -4 \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{73} + ( 3 - 5 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{74} + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} + 2 \beta_{7} ) q^{76} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{77} + ( 6 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{78} + ( -1 - 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - \beta_{6} - \beta_{7} ) q^{79} + ( 4 - \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{81} + ( -5 - \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{82} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{83} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{7} ) q^{84} + ( -2 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 5 \beta_{6} - 3 \beta_{7} ) q^{86} + ( -2 - \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{87} + ( 4 - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{88} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{89} + ( 1 + \beta_{5} ) q^{91} + ( -1 - \beta_{2} ) q^{92} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{93} + ( -5 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 6 \beta_{6} - 4 \beta_{7} ) q^{94} + ( 4 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{96} + ( -2 + 3 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{97} -\beta_{1} q^{98} + ( -2 - 6 \beta_{1} + 2 \beta_{2} - 5 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - q^{2} - 7q^{3} + 7q^{4} + q^{6} - 8q^{7} - 3q^{8} + 9q^{9} + O(q^{10}) \) \( 8q - q^{2} - 7q^{3} + 7q^{4} + q^{6} - 8q^{7} - 3q^{8} + 9q^{9} + 6q^{11} - 8q^{12} - 8q^{13} + q^{14} + q^{16} - 4q^{17} + 11q^{18} + 7q^{21} + 8q^{22} - 8q^{23} - 8q^{24} + 2q^{26} - 28q^{27} - 7q^{28} - 3q^{29} - 16q^{31} - 12q^{32} - 5q^{33} - 4q^{34} - 14q^{36} + 7q^{37} - 11q^{38} + 16q^{39} + 7q^{41} - q^{42} + 10q^{43} + 7q^{44} + q^{46} - 19q^{47} + 6q^{48} + 8q^{49} + 7q^{51} - 18q^{52} + q^{53} - 25q^{54} + 3q^{56} + 25q^{57} + 9q^{58} + 21q^{59} - 7q^{61} - 18q^{62} - 9q^{63} - 37q^{64} - 43q^{66} - 11q^{67} + 17q^{68} + 7q^{69} + 8q^{71} - 4q^{72} + 3q^{73} + 12q^{74} + 8q^{76} - 6q^{77} + 50q^{78} - 15q^{79} + 28q^{81} - 41q^{82} - 25q^{83} + 8q^{84} - 12q^{86} - 21q^{87} + 29q^{88} + q^{89} + 8q^{91} - 7q^{92} + 5q^{93} - 36q^{94} + 30q^{96} - 10q^{97} - q^{98} - 18q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 11 x^{6} + 9 x^{5} + 36 x^{4} - 23 x^{3} - 30 x^{2} + 17 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 6 \nu^{2} - \nu + 4 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 7 \nu^{3} + 9 \nu - 2 \)
\(\beta_{6}\)\(=\)\( \nu^{7} - \nu^{6} - 11 \nu^{5} + 8 \nu^{4} + 36 \nu^{3} - 16 \nu^{2} - 30 \nu + 8 \)
\(\beta_{7}\)\(=\)\( 2 \nu^{7} - \nu^{6} - 22 \nu^{5} + 8 \nu^{4} + 72 \nu^{3} - 17 \nu^{2} - 62 \nu + 12 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 6 \beta_{2} + \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(\beta_{5} + 7 \beta_{3} + 26 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(\beta_{7} - 2 \beta_{6} + 8 \beta_{4} + 33 \beta_{2} + 10 \beta_{1} + 71\)
\(\nu^{7}\)\(=\)\(\beta_{7} - \beta_{6} + 11 \beta_{5} + 41 \beta_{3} + \beta_{2} + 138 \beta_{1} + 21\)

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
2.42061
2.28656
1.18794
0.322285
0.181467
−1.17189
−1.94419
−2.28279
−2.42061 0.490228 3.85934 0 −1.18665 −1.00000 −4.50072 −2.75968 0
1.2 −2.28656 −2.13745 3.22836 0 4.88741 −1.00000 −2.80872 1.56870 0
1.3 −1.18794 −1.57054 −0.588791 0 1.86571 −1.00000 3.07534 −0.533418 0
1.4 −0.322285 1.07802 −1.89613 0 −0.347431 −1.00000 1.25566 −1.83786 0
1.5 −0.181467 −3.25071 −1.96707 0 0.589897 −1.00000 0.719892 7.56714 0
1.6 1.17189 1.97939 −0.626674 0 2.31962 −1.00000 −3.07817 0.917966 0
1.7 1.94419 −3.14297 1.77986 0 −6.11052 −1.00000 −0.427999 6.87827 0
1.8 2.28279 −0.445964 3.21111 0 −1.01804 −1.00000 2.76472 −2.80112 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.t 8
5.b even 2 1 805.2.a.m 8
15.d odd 2 1 7245.2.a.bp 8
35.c odd 2 1 5635.2.a.bb 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.a.m 8 5.b even 2 1
4025.2.a.t 8 1.a even 1 1 trivial
5635.2.a.bb 8 35.c odd 2 1
7245.2.a.bp 8 15.d odd 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

$p$ $F_p(T)$
$2$ \( 1 + T + 5 T^{2} + 5 T^{3} + 16 T^{4} + 17 T^{5} + 46 T^{6} + 41 T^{7} + 102 T^{8} + 82 T^{9} + 184 T^{10} + 136 T^{11} + 256 T^{12} + 160 T^{13} + 320 T^{14} + 128 T^{15} + 256 T^{16} \)
$3$ \( 1 + 7 T + 32 T^{2} + 112 T^{3} + 328 T^{4} + 830 T^{5} + 1863 T^{6} + 3745 T^{7} + 6824 T^{8} + 11235 T^{9} + 16767 T^{10} + 22410 T^{11} + 26568 T^{12} + 27216 T^{13} + 23328 T^{14} + 15309 T^{15} + 6561 T^{16} \)
$5$ 1
$7$ \( ( 1 + T )^{8} \)
$11$ \( 1 - 6 T + 53 T^{2} - 273 T^{3} + 1395 T^{4} - 6203 T^{5} + 24895 T^{6} - 93532 T^{7} + 320928 T^{8} - 1028852 T^{9} + 3012295 T^{10} - 8256193 T^{11} + 20424195 T^{12} - 43966923 T^{13} + 93892733 T^{14} - 116923026 T^{15} + 214358881 T^{16} \)
$13$ \( 1 + 8 T + 106 T^{2} + 647 T^{3} + 4834 T^{4} + 23288 T^{5} + 125401 T^{6} + 484541 T^{7} + 2035168 T^{8} + 6299033 T^{9} + 21192769 T^{10} + 51163736 T^{11} + 138063874 T^{12} + 240226571 T^{13} + 511641754 T^{14} + 501988136 T^{15} + 815730721 T^{16} \)
$17$ \( 1 + 4 T + 51 T^{2} + 209 T^{3} + 1445 T^{4} + 3607 T^{5} + 25137 T^{6} + 43060 T^{7} + 346508 T^{8} + 732020 T^{9} + 7264593 T^{10} + 17721191 T^{11} + 120687845 T^{12} + 296750113 T^{13} + 1231016019 T^{14} + 1641354692 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 + 89 T^{2} - 25 T^{3} + 3758 T^{4} - 2787 T^{5} + 103839 T^{6} - 112070 T^{7} + 2193378 T^{8} - 2129330 T^{9} + 37485879 T^{10} - 19116033 T^{11} + 489746318 T^{12} - 61902475 T^{13} + 4187083409 T^{14} + 16983563041 T^{16} \)
$23$ \( ( 1 + T )^{8} \)
$29$ \( 1 + 3 T + 131 T^{2} + 494 T^{3} + 8858 T^{4} + 36316 T^{5} + 400317 T^{6} + 1617223 T^{7} + 13313754 T^{8} + 46899467 T^{9} + 336666597 T^{10} + 885710924 T^{11} + 6265095098 T^{12} + 10132507606 T^{13} + 77921855051 T^{14} + 51749628927 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 + 16 T + 282 T^{2} + 2915 T^{3} + 31017 T^{4} + 243008 T^{5} + 1914656 T^{6} + 11903421 T^{7} + 73922784 T^{8} + 369006051 T^{9} + 1839984416 T^{10} + 7239451328 T^{11} + 28644850857 T^{12} + 83453975165 T^{13} + 250276038042 T^{14} + 440201825776 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 - 7 T + 108 T^{2} - 228 T^{3} + 3794 T^{4} + 11242 T^{5} + 99020 T^{6} + 722845 T^{7} + 4372490 T^{8} + 26745265 T^{9} + 135558380 T^{10} + 569441026 T^{11} + 7110566834 T^{12} - 15810422196 T^{13} + 277098452172 T^{14} - 664523139931 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 - 7 T + 158 T^{2} - 1315 T^{3} + 13133 T^{4} - 106952 T^{5} + 819746 T^{6} - 5500878 T^{7} + 39191124 T^{8} - 225535998 T^{9} + 1377993026 T^{10} - 7371238792 T^{11} + 37110719213 T^{12} - 152350904315 T^{13} + 750516470078 T^{14} - 1363279917167 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 - 10 T + 187 T^{2} - 2015 T^{3} + 20782 T^{4} - 189837 T^{5} + 1590085 T^{6} - 11446780 T^{7} + 82824466 T^{8} - 492211540 T^{9} + 2940067165 T^{10} - 15093370359 T^{11} + 71049522382 T^{12} - 296222012645 T^{13} + 1182094890163 T^{14} - 2718186111070 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 + 19 T + 204 T^{2} + 1818 T^{3} + 15942 T^{4} + 130342 T^{5} + 1055049 T^{6} + 8462151 T^{7} + 62857704 T^{8} + 397721097 T^{9} + 2330603241 T^{10} + 13532497466 T^{11} + 77791874502 T^{12} + 416949222726 T^{13} + 2198959927116 T^{14} + 9625839288797 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 - T + 184 T^{2} - 630 T^{3} + 18404 T^{4} - 93240 T^{5} + 1334752 T^{6} - 7959493 T^{7} + 76724166 T^{8} - 421853129 T^{9} + 3749318368 T^{10} - 13881291480 T^{11} + 145216412324 T^{12} - 263463160590 T^{13} + 4078242447736 T^{14} - 1174711139837 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 - 21 T + 444 T^{2} - 4604 T^{3} + 46566 T^{4} - 171266 T^{5} + 163748 T^{6} + 19348241 T^{7} - 146115662 T^{8} + 1141546219 T^{9} + 570006788 T^{10} - 35174439814 T^{11} + 564257032326 T^{12} - 3291511472596 T^{13} + 18728156936604 T^{14} - 52261681181199 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 + 7 T + 224 T^{2} + 1286 T^{3} + 20480 T^{4} + 103824 T^{5} + 1113080 T^{6} + 6058615 T^{7} + 57316462 T^{8} + 369575515 T^{9} + 4141770680 T^{10} + 23566075344 T^{11} + 283562823680 T^{12} + 1086150843086 T^{13} + 11540563856864 T^{14} + 21999199852147 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 11 T + 274 T^{2} + 2206 T^{3} + 32906 T^{4} + 190468 T^{5} + 2318198 T^{6} + 10429181 T^{7} + 142198330 T^{8} + 698755127 T^{9} + 10406390822 T^{10} + 57285727084 T^{11} + 663092787626 T^{12} + 2978375986042 T^{13} + 24785596714306 T^{14} + 66667827658553 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 - 8 T + 180 T^{2} - 1131 T^{3} + 24643 T^{4} - 164934 T^{5} + 2608896 T^{6} - 14487455 T^{7} + 197532400 T^{8} - 1028609305 T^{9} + 13151444736 T^{10} - 59031692874 T^{11} + 626220054883 T^{12} - 2040583395981 T^{13} + 23058051105780 T^{14} - 72760961267128 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 - 3 T + 269 T^{2} - 1309 T^{3} + 40920 T^{4} - 215618 T^{5} + 4537865 T^{6} - 22059098 T^{7} + 378794378 T^{8} - 1610314154 T^{9} + 24182282585 T^{10} - 83879067506 T^{11} + 1162056021720 T^{12} - 2713650715237 T^{13} + 40708906871741 T^{14} - 33142195557291 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 + 15 T + 229 T^{2} + 1061 T^{3} + 8607 T^{4} + 10105 T^{5} + 1259311 T^{6} + 12848707 T^{7} + 191850408 T^{8} + 1015047853 T^{9} + 7859359951 T^{10} + 4982159095 T^{11} + 335243347167 T^{12} + 3264756839339 T^{13} + 55667027314309 T^{14} + 288058634792385 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 + 25 T + 632 T^{2} + 8426 T^{3} + 113366 T^{4} + 877076 T^{5} + 7714544 T^{6} + 29380439 T^{7} + 346252610 T^{8} + 2438576437 T^{9} + 53145493616 T^{10} + 501500654812 T^{11} + 5380160018486 T^{12} + 33190356457918 T^{13} + 206626315969208 T^{14} + 678401274740675 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 - T + 330 T^{2} - 1456 T^{3} + 57344 T^{4} - 365870 T^{5} + 7551134 T^{6} - 47530569 T^{7} + 773518510 T^{8} - 4230220641 T^{9} + 59812532414 T^{10} - 257927008030 T^{11} + 3597891067904 T^{12} - 8130390557744 T^{13} + 164003826017130 T^{14} - 44231334895529 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 + 10 T + 574 T^{2} + 4842 T^{3} + 155521 T^{4} + 1122152 T^{5} + 26343286 T^{6} + 162485876 T^{7} + 3054647092 T^{8} + 15761129972 T^{9} + 247863977974 T^{10} + 1024157832296 T^{11} + 13768162310401 T^{12} + 41579901524394 T^{13} + 478125930829246 T^{14} + 807982844781130 T^{15} + 7837433594376961 T^{16} \)
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