Properties

Label 1336.2.a.e
Level $1336$
Weight $2$
Character orbit 1336.a
Self dual yes
Analytic conductor $10.668$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1336 = 2^{3} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1336.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.6680137100\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 5 x^{11} - 15 x^{10} + 100 x^{9} + 36 x^{8} - 641 x^{7} + 129 x^{6} + 1804 x^{5} - 433 x^{4} - 2399 x^{3} + 148 x^{2} + 1224 x + 224\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + \beta_{9} q^{5} + ( 1 + \beta_{11} ) q^{7} + ( 2 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} - \beta_{9} + \beta_{11} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + \beta_{9} q^{5} + ( 1 + \beta_{11} ) q^{7} + ( 2 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} - \beta_{9} + \beta_{11} ) q^{9} + ( 1 + \beta_{2} - \beta_{6} + \beta_{10} ) q^{11} + ( -1 + \beta_{1} + \beta_{7} ) q^{13} + ( 2 - \beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} ) q^{15} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{17} + ( -\beta_{3} - \beta_{8} - \beta_{11} ) q^{19} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{21} + ( 2 - \beta_{2} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{23} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{25} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{27} + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{29} + ( -\beta_{1} + \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} ) q^{31} + ( 1 + 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{7} + 2 \beta_{8} + \beta_{10} ) q^{33} + ( -2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{35} + ( -1 - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{10} ) q^{37} + ( 3 - 2 \beta_{1} + \beta_{2} + \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{39} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - \beta_{10} ) q^{41} + ( 3 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} + 2 \beta_{8} ) q^{43} + ( -2 + \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} + 3 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{45} + ( 3 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{47} + ( 1 - \beta_{1} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{9} + \beta_{11} ) q^{49} + ( 2 + 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - 3 \beta_{9} + \beta_{10} ) q^{51} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} - 4 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + 2 \beta_{10} ) q^{53} + ( 3 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + 5 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{55} + ( 2 - 3 \beta_{3} - \beta_{4} + 4 \beta_{5} + 3 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} ) q^{57} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - \beta_{9} + \beta_{10} ) q^{59} + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} ) q^{61} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{63} + ( 5 - \beta_{1} + 2 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{65} + ( 5 - 2 \beta_{3} + \beta_{5} + 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{10} ) q^{67} + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} - 2 \beta_{8} ) q^{69} + ( 2 - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{71} + ( -1 + \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{73} + ( 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 5 \beta_{6} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{75} + ( 2 + \beta_{1} - \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{77} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} ) q^{79} + ( 6 + 4 \beta_{2} - 5 \beta_{3} + 2 \beta_{5} + 3 \beta_{6} + \beta_{8} - 4 \beta_{9} + 3 \beta_{11} ) q^{81} + ( 4 - \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 3 \beta_{10} + 2 \beta_{11} ) q^{83} + ( -3 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{11} ) q^{85} + ( -\beta_{2} - \beta_{5} - 2 \beta_{6} - 3 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{87} + ( 1 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{89} + ( -\beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{91} + ( -5 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{8} + 3 \beta_{9} - 3 \beta_{11} ) q^{93} + ( 5 - \beta_{1} - 3 \beta_{3} - \beta_{4} - \beta_{5} + 4 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{11} ) q^{95} + ( -1 - 3 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - 3 \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} + \beta_{11} ) q^{97} + ( 5 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{8} - 3 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 5q^{3} - 2q^{5} + 10q^{7} + 19q^{9} + O(q^{10}) \) \( 12q + 5q^{3} - 2q^{5} + 10q^{7} + 19q^{9} + 14q^{11} - 9q^{13} + 8q^{15} + 4q^{17} + 9q^{19} + 7q^{21} + 15q^{23} + 18q^{25} + 20q^{27} + 17q^{29} + 11q^{31} + 8q^{33} + 19q^{35} - 29q^{37} + 26q^{39} + 14q^{41} + 15q^{43} - 26q^{45} + 17q^{47} + 20q^{49} + 36q^{51} - 7q^{53} + 13q^{55} + 3q^{57} + 32q^{59} - 8q^{61} + 50q^{63} + 37q^{65} + 39q^{67} + q^{69} + 35q^{71} - 12q^{73} + 29q^{75} + 13q^{77} + 36q^{79} + 44q^{81} + 25q^{83} - 38q^{85} + 14q^{87} + 23q^{89} - 2q^{91} - 25q^{93} + 38q^{95} - 2q^{97} + 43q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 5 x^{11} - 15 x^{10} + 100 x^{9} + 36 x^{8} - 641 x^{7} + 129 x^{6} + 1804 x^{5} - 433 x^{4} - 2399 x^{3} + 148 x^{2} + 1224 x + 224\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-4134 \nu^{11} + 110987 \nu^{10} - 343875 \nu^{9} - 1750092 \nu^{8} + 7226582 \nu^{7} + 6214148 \nu^{6} - 38734849 \nu^{5} - 4221695 \nu^{4} + 72747321 \nu^{3} + 3377831 \nu^{2} - 43596932 \nu - 7879652\)\()/243358\)
\(\beta_{3}\)\(=\)\((\)\(-26659 \nu^{11} + 333527 \nu^{10} - 500007 \nu^{9} - 5651176 \nu^{8} + 15580284 \nu^{7} + 25075571 \nu^{6} - 91240819 \nu^{5} - 37971272 \nu^{4} + 181435679 \nu^{3} + 34242297 \nu^{2} - 112484744 \nu - 23226840\)\()/973432\)
\(\beta_{4}\)\(=\)\((\)\(28693 \nu^{11} - 134357 \nu^{10} - 387059 \nu^{9} + 2394592 \nu^{8} + 500524 \nu^{7} - 11991941 \nu^{6} + 3606689 \nu^{5} + 22278164 \nu^{4} - 3899557 \nu^{3} - 17935139 \nu^{2} - 2303608 \nu + 3831112\)\()/973432\)
\(\beta_{5}\)\(=\)\((\)\(41307 \nu^{11} - 307651 \nu^{10} - 114673 \nu^{9} + 5338476 \nu^{8} - 7428812 \nu^{7} - 25143667 \nu^{6} + 48730959 \nu^{5} + 42238968 \nu^{4} - 91641495 \nu^{3} - 33729889 \nu^{2} + 53347564 \nu + 13360032\)\()/973432\)
\(\beta_{6}\)\(=\)\((\)\(12411 \nu^{11} - 65032 \nu^{10} - 134173 \nu^{9} + 1111702 \nu^{8} - 358916 \nu^{7} - 4950043 \nu^{6} + 4244680 \nu^{5} + 6497698 \nu^{4} - 5950306 \nu^{3} - 1861465 \nu^{2} + 1275146 \nu - 173194\)\()/243358\)
\(\beta_{7}\)\(=\)\((\)\(15928 \nu^{11} - 102912 \nu^{10} - 108909 \nu^{9} + 1828134 \nu^{8} - 1744024 \nu^{7} - 9086400 \nu^{6} + 13627972 \nu^{5} + 16669301 \nu^{4} - 27880538 \nu^{3} - 13839681 \nu^{2} + 17581492 \nu + 5796342\)\()/243358\)
\(\beta_{8}\)\(=\)\((\)\(-70349 \nu^{11} + 489671 \nu^{10} + 295481 \nu^{9} - 8486560 \nu^{8} + 11005132 \nu^{7} + 39588421 \nu^{6} - 76436047 \nu^{5} - 62809378 \nu^{4} + 149898571 \nu^{3} + 42811521 \nu^{2} - 90263824 \nu - 18920800\)\()/486716\)
\(\beta_{9}\)\(=\)\((\)\(194845 \nu^{11} - 951441 \nu^{10} - 2586835 \nu^{9} + 17268420 \nu^{8} + 1949556 \nu^{7} - 89411357 \nu^{6} + 40411013 \nu^{5} + 168228596 \nu^{4} - 86413565 \nu^{3} - 112352907 \nu^{2} + 42986204 \nu + 13689440\)\()/973432\)
\(\beta_{10}\)\(=\)\((\)\(49825 \nu^{11} - 360853 \nu^{10} - 146621 \nu^{9} + 6201508 \nu^{8} - 8855778 \nu^{7} - 28321401 \nu^{6} + 58720147 \nu^{5} + 42956708 \nu^{4} - 111353595 \nu^{3} - 28079769 \nu^{2} + 63733948 \nu + 12190868\)\()/243358\)
\(\beta_{11}\)\(=\)\((\)\(207127 \nu^{11} - 1165859 \nu^{10} - 2032305 \nu^{9} + 20591672 \nu^{8} - 10923772 \nu^{7} - 100394311 \nu^{6} + 109890519 \nu^{5} + 171582340 \nu^{4} - 215847827 \nu^{3} - 112277337 \nu^{2} + 121867592 \nu + 26939384\)\()/973432\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{11} - \beta_{9} - \beta_{7} + \beta_{5} - \beta_{3} + \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 7 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(12 \beta_{11} - 13 \beta_{9} + \beta_{8} - 9 \beta_{7} + 3 \beta_{6} + 11 \beta_{5} - 14 \beta_{3} + 13 \beta_{2} + 42\)
\(\nu^{5}\)\(=\)\(4 \beta_{11} + 10 \beta_{9} + 28 \beta_{8} + 15 \beta_{7} + 13 \beta_{6} - 12 \beta_{5} - 17 \beta_{4} - 13 \beta_{3} - 12 \beta_{2} + 64 \beta_{1} + 22\)
\(\nu^{6}\)\(=\)\(131 \beta_{11} + 5 \beta_{10} - 149 \beta_{9} + 14 \beta_{8} - 86 \beta_{7} + 41 \beta_{6} + 113 \beta_{5} - 2 \beta_{4} - 157 \beta_{3} + 150 \beta_{2} - 3 \beta_{1} + 409\)
\(\nu^{7}\)\(=\)\(62 \beta_{11} + 11 \beta_{10} + 98 \beta_{9} + 333 \beta_{8} + 191 \beta_{7} + 137 \beta_{6} - 142 \beta_{5} - 212 \beta_{4} - 132 \beta_{3} - 132 \beta_{2} + 649 \beta_{1} + 208\)
\(\nu^{8}\)\(=\)\(1418 \beta_{11} + 114 \beta_{10} - 1661 \beta_{9} + 168 \beta_{8} - 863 \beta_{7} + 433 \beta_{6} + 1163 \beta_{5} - 23 \beta_{4} - 1675 \beta_{3} + 1675 \beta_{2} - 70 \beta_{1} + 4196\)
\(\nu^{9}\)\(=\)\(731 \beta_{11} + 268 \beta_{10} + 1008 \beta_{9} + 3765 \beta_{8} + 2278 \beta_{7} + 1331 \beta_{6} - 1718 \beta_{5} - 2388 \beta_{4} - 1229 \beta_{3} - 1452 \beta_{2} + 6834 \beta_{1} + 1859\)
\(\nu^{10}\)\(=\)\(15372 \beta_{11} + 1796 \beta_{10} - 18337 \beta_{9} + 1900 \beta_{8} - 8924 \beta_{7} + 4192 \beta_{6} + 12015 \beta_{5} - 84 \beta_{4} - 17624 \beta_{3} + 18443 \beta_{2} - 1159 \beta_{1} + 43958\)
\(\nu^{11}\)\(=\)\(7975 \beta_{11} + 4409 \beta_{10} + 10718 \beta_{9} + 41683 \beta_{8} + 26303 \beta_{7} + 12366 \beta_{6} - 20812 \beta_{5} - 25790 \beta_{4} - 10887 \beta_{3} - 16117 \beta_{2} + 73015 \beta_{1} + 15931\)

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
−3.31023
−2.02356
−1.26807
−1.22013
−0.945593
−0.203398
1.18487
1.85311
1.94180
2.51502
3.16421
3.31199
0 −3.31023 0 −2.56191 0 1.56163 0 7.95764 0
1.2 0 −2.02356 0 −0.0986065 0 4.09952 0 1.09480 0
1.3 0 −1.26807 0 −0.339536 0 −4.08348 0 −1.39199 0
1.4 0 −1.22013 0 3.37141 0 3.01986 0 −1.51128 0
1.5 0 −0.945593 0 −3.79127 0 −1.93894 0 −2.10585 0
1.6 0 −0.203398 0 1.32800 0 0.559903 0 −2.95863 0
1.7 0 1.18487 0 −0.696822 0 2.51795 0 −1.59609 0
1.8 0 1.85311 0 3.00198 0 0.497697 0 0.434028 0
1.9 0 1.94180 0 −3.68233 0 −2.51484 0 0.770577 0
1.10 0 2.51502 0 3.24915 0 3.64134 0 3.32531 0
1.11 0 3.16421 0 1.18478 0 −2.23231 0 7.01222 0
1.12 0 3.31199 0 −2.96484 0 4.87167 0 7.96927 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(167\) \(-1\)

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 1336.2.a.e 12
4.b odd 2 1 2672.2.a.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1336.2.a.e 12 1.a even 1 1 trivial
2672.2.a.n 12 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{12} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1336))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 5 T + 21 T^{2} - 65 T^{3} + 180 T^{4} - 416 T^{5} + 858 T^{6} - 1532 T^{7} + 2456 T^{8} - 3338 T^{9} + 4105 T^{10} - 4482 T^{11} + 6566 T^{12} - 13446 T^{13} + 36945 T^{14} - 90126 T^{15} + 198936 T^{16} - 372276 T^{17} + 625482 T^{18} - 909792 T^{19} + 1180980 T^{20} - 1279395 T^{21} + 1240029 T^{22} - 885735 T^{23} + 531441 T^{24} \)
$5$ \( 1 + 2 T + 23 T^{2} + 48 T^{3} + 303 T^{4} + 618 T^{5} + 3004 T^{6} + 5842 T^{7} + 23765 T^{8} + 43696 T^{9} + 154685 T^{10} + 262994 T^{11} + 843222 T^{12} + 1314970 T^{13} + 3867125 T^{14} + 5462000 T^{15} + 14853125 T^{16} + 18256250 T^{17} + 46937500 T^{18} + 48281250 T^{19} + 118359375 T^{20} + 93750000 T^{21} + 224609375 T^{22} + 97656250 T^{23} + 244140625 T^{24} \)
$7$ \( 1 - 10 T + 82 T^{2} - 476 T^{3} + 2479 T^{4} - 10943 T^{5} + 44794 T^{6} - 164522 T^{7} + 569513 T^{8} - 1817643 T^{9} + 5531118 T^{10} - 15745437 T^{11} + 42979650 T^{12} - 110218059 T^{13} + 271024782 T^{14} - 623451549 T^{15} + 1367400713 T^{16} - 2765121254 T^{17} + 5269969306 T^{18} - 9012031049 T^{19} + 14290941679 T^{20} - 19208316932 T^{21} + 23162970418 T^{22} - 19773267430 T^{23} + 13841287201 T^{24} \)
$11$ \( 1 - 14 T + 160 T^{2} - 1234 T^{3} + 8382 T^{4} - 46483 T^{5} + 238446 T^{6} - 1078238 T^{7} + 4676555 T^{8} - 18581954 T^{9} + 71789378 T^{10} - 256135639 T^{11} + 884942924 T^{12} - 2817492029 T^{13} + 8686514738 T^{14} - 24732580774 T^{15} + 68469441755 T^{16} - 173651308138 T^{17} + 422421634206 T^{18} - 905822169593 T^{19} + 1796756140542 T^{20} - 2909707450694 T^{21} + 4149987936160 T^{22} - 3994363388554 T^{23} + 3138428376721 T^{24} \)
$13$ \( 1 + 9 T + 124 T^{2} + 784 T^{3} + 6135 T^{4} + 29412 T^{5} + 166800 T^{6} + 624997 T^{7} + 2854483 T^{8} + 8529550 T^{9} + 35235252 T^{10} + 92130964 T^{11} + 416218410 T^{12} + 1197702532 T^{13} + 5954757588 T^{14} + 18739421350 T^{15} + 81526888963 T^{16} + 232057011121 T^{17} + 805111741200 T^{18} + 1845559382004 T^{19} + 5004507973335 T^{20} + 8313927508432 T^{21} + 17094452989276 T^{22} + 16129443546333 T^{23} + 23298085122481 T^{24} \)
$17$ \( 1 - 4 T + 97 T^{2} - 264 T^{3} + 4341 T^{4} - 7296 T^{5} + 129101 T^{6} - 117284 T^{7} + 3130755 T^{8} - 1704552 T^{9} + 67580850 T^{10} - 32335528 T^{11} + 1258614414 T^{12} - 549703976 T^{13} + 19530865650 T^{14} - 8374463976 T^{15} + 261483788355 T^{16} - 166526508388 T^{17} + 3116184295469 T^{18} - 2993830958208 T^{19} + 30281763051381 T^{20} - 31307199395208 T^{21} + 195551408343553 T^{22} - 137087585230532 T^{23} + 582622237229761 T^{24} \)
$19$ \( 1 - 9 T + 127 T^{2} - 999 T^{3} + 8513 T^{4} - 55475 T^{5} + 374218 T^{6} - 2092084 T^{7} + 11981297 T^{8} - 59771665 T^{9} + 302921079 T^{10} - 1367448582 T^{11} + 6310019386 T^{12} - 25981523058 T^{13} + 109354509519 T^{14} - 409973850235 T^{15} + 1561414606337 T^{16} - 5180207100316 T^{17} + 17605415496058 T^{18} - 49587534721025 T^{19} + 144581072168033 T^{20} - 322365010081221 T^{21} + 778645414740727 T^{22} - 1048412330083971 T^{23} + 2213314919066161 T^{24} \)
$23$ \( 1 - 15 T + 268 T^{2} - 2898 T^{3} + 30599 T^{4} - 259246 T^{5} + 2053716 T^{6} - 14360493 T^{7} + 93286443 T^{8} - 558477374 T^{9} + 3118163168 T^{10} - 16394322468 T^{11} + 80741669146 T^{12} - 377069416764 T^{13} + 1649508315872 T^{14} - 6794994209458 T^{15} + 26105371495563 T^{16} - 92429058597099 T^{17} + 304023673813524 T^{18} - 882687377832962 T^{19} + 2396237838613319 T^{20} - 5219740412919774 T^{21} + 11102305005257932 T^{22} - 14292146368708905 T^{23} + 21914624432020321 T^{24} \)
$29$ \( 1 - 17 T + 308 T^{2} - 3084 T^{3} + 32784 T^{4} - 239556 T^{5} + 1971086 T^{6} - 12005398 T^{7} + 88342328 T^{8} - 493352408 T^{9} + 3403529628 T^{10} - 17607784369 T^{11} + 110114806362 T^{12} - 510625746701 T^{13} + 2862368417148 T^{14} - 12032371878712 T^{15} + 62482850090168 T^{16} - 246244507182302 T^{17} + 1172447920496606 T^{18} - 4132311369078804 T^{19} + 16400078402513424 T^{20} - 44740038189579996 T^{21} + 129577827856461908 T^{22} - 207408666016999093 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 - 11 T + 268 T^{2} - 2246 T^{3} + 31422 T^{4} - 208276 T^{5} + 2177752 T^{6} - 11625436 T^{7} + 102606128 T^{8} - 450844454 T^{9} + 3710634278 T^{10} - 14293752563 T^{11} + 118137620686 T^{12} - 443106329453 T^{13} + 3565919541158 T^{14} - 13431107129114 T^{15} + 94758913936688 T^{16} - 332826362684836 T^{17} + 1932762916305112 T^{18} - 5730217216582636 T^{19} + 26799542178471102 T^{20} - 59383391372867066 T^{21} + 219660380910854668 T^{22} - 279493245860453141 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 + 29 T + 631 T^{2} + 9945 T^{3} + 134321 T^{4} + 1544182 T^{5} + 15965302 T^{6} + 148019388 T^{7} + 1262977605 T^{8} + 9884449483 T^{9} + 72006725027 T^{10} + 485946774853 T^{11} + 3066780197650 T^{12} + 17980030669561 T^{13} + 98577206561963 T^{14} + 500677019662399 T^{15} + 2367023371164405 T^{16} + 10264250076638316 T^{17} + 40962596969060518 T^{18} + 146592095894990206 T^{19} + 471799752730122641 T^{20} + 1292469502262040765 T^{21} + 3034216738995662719 T^{22} + 5159611031604351977 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 - 14 T + 330 T^{2} - 3554 T^{3} + 49735 T^{4} - 449836 T^{5} + 4743290 T^{6} - 37513844 T^{7} + 327083851 T^{8} - 2326331990 T^{9} + 17697320220 T^{10} - 115153987482 T^{11} + 790833981402 T^{12} - 4721313486762 T^{13} + 29749195289820 T^{14} - 160333127082790 T^{15} + 924260789885611 T^{16} - 4346211450746644 T^{17} + 22531121945292890 T^{18} - 87607483545533516 T^{19} + 397130256270332935 T^{20} - 1163515394836137394 T^{21} + 4429477572350292330 T^{22} - 7704606444027478174 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 - 15 T + 258 T^{2} - 1904 T^{3} + 16807 T^{4} - 43956 T^{5} + 334214 T^{6} + 930811 T^{7} + 18567651 T^{8} - 73256050 T^{9} + 1947741960 T^{10} - 8094843132 T^{11} + 100281619882 T^{12} - 348078254676 T^{13} + 3601374884040 T^{14} - 5824368767350 T^{15} + 63479103806451 T^{16} + 136837075837273 T^{17} + 2112688030058486 T^{18} - 11948058869819292 T^{19} + 196443582065640007 T^{20} - 956936333127749072 T^{21} + 5575762436827336242 T^{22} - 13939406092068340605 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 - 17 T + 416 T^{2} - 5222 T^{3} + 75152 T^{4} - 755072 T^{5} + 8230488 T^{6} - 69868476 T^{7} + 639405486 T^{8} - 4787353750 T^{9} + 38964096514 T^{10} - 265590831261 T^{11} + 1982556678638 T^{12} - 12482769069267 T^{13} + 86071689199426 T^{14} - 497037428386250 T^{15} + 3120094801329966 T^{16} - 16023986117299332 T^{17} + 88718202414750552 T^{18} - 382536932814238336 T^{19} + 1789465815204662672 T^{20} - 5844099330542649274 T^{21} + 21881239010105300384 T^{22} - 42026706656428209151 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 + 7 T + 90 T^{2} + 488 T^{3} + 12865 T^{4} + 79310 T^{5} + 853506 T^{6} + 3921193 T^{7} + 65117699 T^{8} + 321866540 T^{9} + 3600066580 T^{10} + 15238294818 T^{11} + 207282814582 T^{12} + 807629625354 T^{13} + 10112587023220 T^{14} + 47918524875580 T^{15} + 513809966723219 T^{16} + 1639825239783149 T^{17} + 18917415209768274 T^{18} + 93166340500472470 T^{19} + 800970917142159265 T^{20} + 1610284632799440904 T^{21} + 15739872332896174410 T^{22} + 64883251505605341179 T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( 1 - 32 T + 710 T^{2} - 11554 T^{3} + 163099 T^{4} - 1999788 T^{5} + 22569414 T^{6} - 232762054 T^{7} + 2268422075 T^{8} - 20712796662 T^{9} + 180762854212 T^{10} - 1489434396946 T^{11} + 11760399511698 T^{12} - 87876629419814 T^{13} + 629235495511972 T^{14} - 4253973465644898 T^{15} + 27487289183144075 T^{16} - 166407248289750146 T^{17} + 951989926484656374 T^{18} - 4976775375523218372 T^{19} + 23947897542827150779 T^{20} - \)\(10\!\cdots\!06\)\( T^{21} + \)\(36\!\cdots\!10\)\( T^{22} - \)\(96\!\cdots\!88\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 + 8 T + 593 T^{2} + 4174 T^{3} + 164810 T^{4} + 1024441 T^{5} + 28619615 T^{6} + 157483913 T^{7} + 3485187899 T^{8} + 16996065526 T^{9} + 315625078990 T^{10} + 1362184283202 T^{11} + 21896493361504 T^{12} + 83093241275322 T^{13} + 1174440918921790 T^{14} + 3857783949157006 T^{15} + 48255357504678059 T^{16} + 133010330386805813 T^{17} + 1474493278867691015 T^{18} + 3219554613676189261 T^{19} + 31595282255081881610 T^{20} + 48811365791489704534 T^{21} + \)\(42\!\cdots\!93\)\( T^{22} + \)\(34\!\cdots\!88\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 - 39 T + 1027 T^{2} - 19953 T^{3} + 327827 T^{4} - 4668260 T^{5} + 60334624 T^{6} - 711808264 T^{7} + 7810781633 T^{8} - 79587200761 T^{9} + 760914720237 T^{10} - 6811012689499 T^{11} + 57514505185654 T^{12} - 456337850196433 T^{13} + 3415746179143893 T^{14} - 23936885262480643 T^{15} + 157396005791160593 T^{16} - 961030208596484248 T^{17} + 5457772475814919456 T^{18} - 28292977558665147980 T^{19} + \)\(13\!\cdots\!07\)\( T^{20} - \)\(54\!\cdots\!91\)\( T^{21} + \)\(18\!\cdots\!23\)\( T^{22} - \)\(47\!\cdots\!37\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 - 35 T + 965 T^{2} - 18827 T^{3} + 320068 T^{4} - 4564809 T^{5} + 59233681 T^{6} - 683322497 T^{7} + 7374100007 T^{8} - 73103273734 T^{9} + 693289648154 T^{10} - 6177140226134 T^{11} + 53453933493352 T^{12} - 438576956055514 T^{13} + 3494873116344314 T^{14} - 26164465805409674 T^{15} + 187388277039981767 T^{16} - 1232870505286009447 T^{17} + 7587851353785943201 T^{18} - 41517486355104662319 T^{19} + \)\(20\!\cdots\!48\)\( T^{20} - \)\(86\!\cdots\!37\)\( T^{21} + \)\(31\!\cdots\!65\)\( T^{22} - \)\(80\!\cdots\!85\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 + 12 T + 369 T^{2} + 3938 T^{3} + 76243 T^{4} + 768288 T^{5} + 11261961 T^{6} + 104835486 T^{7} + 1302550371 T^{8} + 11200420910 T^{9} + 123899449814 T^{10} + 976028791990 T^{11} + 9844713826450 T^{12} + 71250101815270 T^{13} + 660260168058806 T^{14} + 4357154141145470 T^{15} + 36990139350297411 T^{16} + 217331467964949198 T^{17} + 1704320154431892729 T^{18} + 8487583713439995936 T^{19} + 61486936786280417683 T^{20} + \)\(23\!\cdots\!94\)\( T^{21} + \)\(15\!\cdots\!81\)\( T^{22} + \)\(37\!\cdots\!24\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 - 36 T + 1156 T^{2} - 26026 T^{3} + 521709 T^{4} - 8832620 T^{5} + 135692808 T^{6} - 1863508050 T^{7} + 23540265499 T^{8} - 272195166550 T^{9} + 2918160542324 T^{10} - 28938160763594 T^{11} + 267029484011502 T^{12} - 2286114700323926 T^{13} + 18212239944644084 T^{14} - 134202832720645450 T^{15} + 916895247947555419 T^{16} - 5734119369840511950 T^{17} + 32985219429219592968 T^{18} - \)\(16\!\cdots\!80\)\( T^{19} + \)\(79\!\cdots\!49\)\( T^{20} - \)\(31\!\cdots\!94\)\( T^{21} + \)\(10\!\cdots\!56\)\( T^{22} - \)\(26\!\cdots\!44\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 - 25 T + 783 T^{2} - 14775 T^{3} + 287637 T^{4} - 4408358 T^{5} + 66832046 T^{6} - 866074036 T^{7} + 10989644649 T^{8} - 123351930903 T^{9} + 1350839330659 T^{10} - 13295933048579 T^{11} + 127458076112610 T^{12} - 1103562443032057 T^{13} + 9305932148909851 T^{14} - 70531030515233661 T^{15} + 521550083428174329 T^{16} - 3411500827651045148 T^{17} + 21850094072254182974 T^{18} - \)\(11\!\cdots\!66\)\( T^{19} + \)\(64\!\cdots\!17\)\( T^{20} - \)\(27\!\cdots\!25\)\( T^{21} + \)\(12\!\cdots\!67\)\( T^{22} - \)\(32\!\cdots\!75\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 - 23 T + 825 T^{2} - 15753 T^{3} + 330433 T^{4} - 5305955 T^{5} + 84153798 T^{6} - 1153158478 T^{7} + 15132045743 T^{8} - 179133937149 T^{9} + 2021024761073 T^{10} - 20856731697130 T^{11} + 205426750778462 T^{12} - 1856249121044570 T^{13} + 16008537132459233 T^{14} - 126283872537993381 T^{15} + 949418460830330063 T^{16} - 6439305495270358622 T^{17} + 41822863169311219878 T^{18} - \)\(23\!\cdots\!95\)\( T^{19} + \)\(13\!\cdots\!73\)\( T^{20} - \)\(55\!\cdots\!77\)\( T^{21} + \)\(25\!\cdots\!25\)\( T^{22} - \)\(63\!\cdots\!47\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 + 2 T + 599 T^{2} + 1134 T^{3} + 186500 T^{4} + 298077 T^{5} + 39910011 T^{6} + 53566567 T^{7} + 6513962669 T^{8} + 7655493132 T^{9} + 851815353522 T^{10} + 900196244348 T^{11} + 91127580291916 T^{12} + 87319035701756 T^{13} + 8014730661288498 T^{14} + 6986961883261836 T^{15} + 576676431547410989 T^{16} + 459994337228387719 T^{17} + 33243921879408444219 T^{18} + 24084110242382488701 T^{19} + \)\(14\!\cdots\!00\)\( T^{20} + \)\(86\!\cdots\!78\)\( T^{21} + \)\(44\!\cdots\!51\)\( T^{22} + \)\(14\!\cdots\!06\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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