Properties

Label 8034.2.a.v
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $11$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - 5 x^{10} - 31 x^{9} + 168 x^{8} + 300 x^{7} - 1928 x^{6} - 736 x^{5} + 8532 x^{4} - 2065 x^{3} - 10494 x^{2} + 4024 x + 576\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + \beta_{1} q^{5} + q^{6} -\beta_{9} q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} + q^{3} + q^{4} + \beta_{1} q^{5} + q^{6} -\beta_{9} q^{7} + q^{8} + q^{9} + \beta_{1} q^{10} + ( -1 + \beta_{4} ) q^{11} + q^{12} + q^{13} -\beta_{9} q^{14} + \beta_{1} q^{15} + q^{16} + ( 1 - \beta_{2} ) q^{17} + q^{18} + ( -1 - \beta_{9} ) q^{19} + \beta_{1} q^{20} -\beta_{9} q^{21} + ( -1 + \beta_{4} ) q^{22} + ( 2 + \beta_{5} ) q^{23} + q^{24} + ( 2 + \beta_{1} - \beta_{5} + \beta_{6} ) q^{25} + q^{26} + q^{27} -\beta_{9} q^{28} + ( 3 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{29} + \beta_{1} q^{30} + ( 1 + \beta_{9} ) q^{31} + q^{32} + ( -1 + \beta_{4} ) q^{33} + ( 1 - \beta_{2} ) q^{34} + ( 2 + \beta_{3} - \beta_{5} - \beta_{9} ) q^{35} + q^{36} + ( 2 - \beta_{3} + \beta_{4} + \beta_{9} ) q^{37} + ( -1 - \beta_{9} ) q^{38} + q^{39} + \beta_{1} q^{40} + ( -1 - \beta_{5} - \beta_{7} ) q^{41} -\beta_{9} q^{42} + ( -1 + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} ) q^{43} + ( -1 + \beta_{4} ) q^{44} + \beta_{1} q^{45} + ( 2 + \beta_{5} ) q^{46} + ( 1 + \beta_{3} ) q^{47} + q^{48} + ( 3 - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{49} + ( 2 + \beta_{1} - \beta_{5} + \beta_{6} ) q^{50} + ( 1 - \beta_{2} ) q^{51} + q^{52} + ( 2 - 2 \beta_{1} - \beta_{4} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{53} + q^{54} + ( \beta_{2} + \beta_{3} + \beta_{6} + \beta_{8} + \beta_{10} ) q^{55} -\beta_{9} q^{56} + ( -1 - \beta_{9} ) q^{57} + ( 3 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{58} + ( -3 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{59} + \beta_{1} q^{60} + ( -\beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} - \beta_{8} + 2 \beta_{10} ) q^{61} + ( 1 + \beta_{9} ) q^{62} -\beta_{9} q^{63} + q^{64} + \beta_{1} q^{65} + ( -1 + \beta_{4} ) q^{66} + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{67} + ( 1 - \beta_{2} ) q^{68} + ( 2 + \beta_{5} ) q^{69} + ( 2 + \beta_{3} - \beta_{5} - \beta_{9} ) q^{70} + ( 3 + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{9} ) q^{71} + q^{72} + ( -1 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{73} + ( 2 - \beta_{3} + \beta_{4} + \beta_{9} ) q^{74} + ( 2 + \beta_{1} - \beta_{5} + \beta_{6} ) q^{75} + ( -1 - \beta_{9} ) q^{76} + ( 4 + \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + 4 \beta_{9} ) q^{77} + q^{78} + ( 5 - \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{79} + \beta_{1} q^{80} + q^{81} + ( -1 - \beta_{5} - \beta_{7} ) q^{82} + ( 3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{83} -\beta_{9} q^{84} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{7} - \beta_{8} ) q^{85} + ( -1 + \beta_{5} - \beta_{6} - \beta_{8} - \beta_{10} ) q^{86} + ( 3 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{87} + ( -1 + \beta_{4} ) q^{88} + ( 2 - \beta_{3} - 2 \beta_{5} - \beta_{7} - \beta_{8} ) q^{89} + \beta_{1} q^{90} -\beta_{9} q^{91} + ( 2 + \beta_{5} ) q^{92} + ( 1 + \beta_{9} ) q^{93} + ( 1 + \beta_{3} ) q^{94} + ( 2 - \beta_{1} + \beta_{3} - \beta_{5} - \beta_{9} ) q^{95} + q^{96} + ( -2 - \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} ) q^{97} + ( 3 - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{98} + ( -1 + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q + 11q^{2} + 11q^{3} + 11q^{4} + 5q^{5} + 11q^{6} + 4q^{7} + 11q^{8} + 11q^{9} + O(q^{10}) \) \( 11q + 11q^{2} + 11q^{3} + 11q^{4} + 5q^{5} + 11q^{6} + 4q^{7} + 11q^{8} + 11q^{9} + 5q^{10} - 7q^{11} + 11q^{12} + 11q^{13} + 4q^{14} + 5q^{15} + 11q^{16} + 10q^{17} + 11q^{18} - 7q^{19} + 5q^{20} + 4q^{21} - 7q^{22} + 18q^{23} + 11q^{24} + 32q^{25} + 11q^{26} + 11q^{27} + 4q^{28} + 29q^{29} + 5q^{30} + 7q^{31} + 11q^{32} - 7q^{33} + 10q^{34} + 31q^{35} + 11q^{36} + 21q^{37} - 7q^{38} + 11q^{39} + 5q^{40} - 3q^{41} + 4q^{42} - 17q^{43} - 7q^{44} + 5q^{45} + 18q^{46} + 12q^{47} + 11q^{48} + 21q^{49} + 32q^{50} + 10q^{51} + 11q^{52} + 11q^{53} + 11q^{54} + 4q^{55} + 4q^{56} - 7q^{57} + 29q^{58} - 48q^{59} + 5q^{60} - q^{61} + 7q^{62} + 4q^{63} + 11q^{64} + 5q^{65} - 7q^{66} - 9q^{67} + 10q^{68} + 18q^{69} + 31q^{70} + 17q^{71} + 11q^{72} - 23q^{73} + 21q^{74} + 32q^{75} - 7q^{76} + 26q^{77} + 11q^{78} + 41q^{79} + 5q^{80} + 11q^{81} - 3q^{82} + 19q^{83} + 4q^{84} + 17q^{85} - 17q^{86} + 29q^{87} - 7q^{88} + 32q^{89} + 5q^{90} + 4q^{91} + 18q^{92} + 7q^{93} + 12q^{94} + 26q^{95} + 11q^{96} - 16q^{97} + 21q^{98} - 7q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 5 x^{10} - 31 x^{9} + 168 x^{8} + 300 x^{7} - 1928 x^{6} - 736 x^{5} + 8532 x^{4} - 2065 x^{3} - 10494 x^{2} + 4024 x + 576\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-7214441 \nu^{10} - 17965281 \nu^{9} + 354406289 \nu^{8} + 1471530490 \nu^{7} - 7557802336 \nu^{6} - 29633121208 \nu^{5} + 75666608448 \nu^{4} + 192625837968 \nu^{3} - 292454456923 \nu^{2} - 278709400064 \nu + 159253845372\)\()/ 24169157748 \)
\(\beta_{3}\)\(=\)\((\)\(-6954475 \nu^{10} + 45806123 \nu^{9} + 154221401 \nu^{8} - 1300710052 \nu^{7} - 1247220084 \nu^{6} + 11904407208 \nu^{5} + 10889091824 \nu^{4} - 42269912108 \nu^{3} - 58601234445 \nu^{2} + 68038889518 \nu + 3291397224\)\()/ 16112771832 \)
\(\beta_{4}\)\(=\)\((\)\(12175783 \nu^{10} - 58843755 \nu^{9} - 96960025 \nu^{8} + 1153915696 \nu^{7} - 4060162144 \nu^{6} - 2903894740 \nu^{5} + 53455821804 \nu^{4} - 26795724084 \nu^{3} - 160415090299 \nu^{2} + 34242147922 \nu + 68016964824\)\()/ 24169157748 \)
\(\beta_{5}\)\(=\)\((\)\(6501563 \nu^{10} + 42249738 \nu^{9} - 439905770 \nu^{8} - 1187809591 \nu^{7} + 8087991802 \nu^{6} + 11048729236 \nu^{5} - 51187504290 \nu^{4} - 39173547342 \nu^{3} + 83266816921 \nu^{2} + 42230737499 \nu + 16835685894\)\()/ 12084578874 \)
\(\beta_{6}\)\(=\)\((\)\(6501563 \nu^{10} + 42249738 \nu^{9} - 439905770 \nu^{8} - 1187809591 \nu^{7} + 8087991802 \nu^{6} + 11048729236 \nu^{5} - 51187504290 \nu^{4} - 39173547342 \nu^{3} + 95351395795 \nu^{2} + 30146158625 \nu - 67756366224\)\()/ 12084578874 \)
\(\beta_{7}\)\(=\)\((\)\(-64472003 \nu^{10} + 67818039 \nu^{9} + 2616704621 \nu^{8} - 1691813840 \nu^{7} - 36952460260 \nu^{6} + 10235837864 \nu^{5} + 196673190432 \nu^{4} + 3251983908 \nu^{3} - 197764359613 \nu^{2} - 33353267558 \nu - 332744495184\)\()/ 48338315496 \)
\(\beta_{8}\)\(=\)\((\)\(-13289341 \nu^{10} + 82204613 \nu^{9} + 372751607 \nu^{8} - 2717567812 \nu^{7} - 2917748212 \nu^{6} + 30309554056 \nu^{5} + 1318194840 \nu^{4} - 127738310148 \nu^{3} + 40432304461 \nu^{2} + 142993848946 \nu - 20159257776\)\()/ 5370923944 \)
\(\beta_{9}\)\(=\)\((\)\(125904587 \nu^{10} - 456748671 \nu^{9} - 4328210285 \nu^{8} + 14601473048 \nu^{7} + 51523740940 \nu^{6} - 155126675336 \nu^{5} - 239310756048 \nu^{4} + 597489887604 \nu^{3} + 307612462405 \nu^{2} - 504759302554 \nu - 33312963024\)\()/ 48338315496 \)
\(\beta_{10}\)\(=\)\((\)\(11080037 \nu^{10} - 37445815 \nu^{9} - 379154155 \nu^{8} + 1228551980 \nu^{7} + 4492255116 \nu^{6} - 14049457206 \nu^{5} - 21039436504 \nu^{4} + 64779403690 \nu^{3} + 28287160275 \nu^{2} - 88710185906 \nu + 9170866704\)\()/ 4028192958 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{5} + \beta_{1} + 7\)
\(\nu^{3}\)\(=\)\(-2 \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{3} + 12 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{10} - \beta_{9} - 5 \beta_{7} + 17 \beta_{6} - 19 \beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 21 \beta_{1} + 79\)
\(\nu^{5}\)\(=\)\(-10 \beta_{10} - 28 \beta_{9} - 17 \beta_{8} - 20 \beta_{7} + 44 \beta_{6} - 22 \beta_{5} + \beta_{4} - 49 \beta_{3} + 6 \beta_{2} + 168 \beta_{1} + 47\)
\(\nu^{6}\)\(=\)\(-29 \beta_{10} - 15 \beta_{9} + 7 \beta_{8} - 117 \beta_{7} + 282 \beta_{6} - 326 \beta_{5} - 27 \beta_{4} - 79 \beta_{3} + 56 \beta_{2} + 391 \beta_{1} + 1035\)
\(\nu^{7}\)\(=\)\(-274 \beta_{10} - 341 \beta_{9} - 240 \beta_{8} - 350 \beta_{7} + 842 \beta_{6} - 448 \beta_{5} + 18 \beta_{4} - 979 \beta_{3} + 209 \beta_{2} + 2537 \beta_{1} + 1196\)
\(\nu^{8}\)\(=\)\(-677 \beta_{10} - 121 \beta_{9} + 244 \beta_{8} - 2165 \beta_{7} + 4726 \beta_{6} - 5433 \beta_{5} - 451 \beta_{4} - 2044 \beta_{3} + 1277 \beta_{2} + 7030 \beta_{1} + 14810\)
\(\nu^{9}\)\(=\)\(-5611 \beta_{10} - 3868 \beta_{9} - 2996 \beta_{8} - 6017 \beta_{7} + 15514 \beta_{6} - 8891 \beta_{5} + 395 \beta_{4} - 18203 \beta_{3} + 5188 \beta_{2} + 40156 \beta_{1} + 25238\)
\(\nu^{10}\)\(=\)\(-14206 \beta_{10} + 969 \beta_{9} + 6288 \beta_{8} - 37184 \beta_{7} + 80285 \beta_{6} - 89691 \beta_{5} - 5778 \beta_{4} - 45119 \beta_{3} + 26759 \beta_{2} + 124641 \beta_{1} + 224907\)

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
−3.73039
−3.11723
−2.84496
−1.31499
−0.111668
0.542814
1.54900
2.64390
3.14620
4.01421
4.22313
1.00000 1.00000 1.00000 −3.73039 1.00000 −3.81724 1.00000 1.00000 −3.73039
1.2 1.00000 1.00000 1.00000 −3.11723 1.00000 −0.395748 1.00000 1.00000 −3.11723
1.3 1.00000 1.00000 1.00000 −2.84496 1.00000 −0.764995 1.00000 1.00000 −2.84496
1.4 1.00000 1.00000 1.00000 −1.31499 1.00000 3.43679 1.00000 1.00000 −1.31499
1.5 1.00000 1.00000 1.00000 −0.111668 1.00000 −0.538332 1.00000 1.00000 −0.111668
1.6 1.00000 1.00000 1.00000 0.542814 1.00000 3.05565 1.00000 1.00000 0.542814
1.7 1.00000 1.00000 1.00000 1.54900 1.00000 −5.16602 1.00000 1.00000 1.54900
1.8 1.00000 1.00000 1.00000 2.64390 1.00000 4.97730 1.00000 1.00000 2.64390
1.9 1.00000 1.00000 1.00000 3.14620 1.00000 1.95604 1.00000 1.00000 3.14620
1.10 1.00000 1.00000 1.00000 4.01421 1.00000 2.23635 1.00000 1.00000 4.01421
1.11 1.00000 1.00000 1.00000 4.22313 1.00000 −0.979791 1.00000 1.00000 4.22313
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(13\) \(-1\)
\(103\) \(1\)

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 8034.2.a.v 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.v 11 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{11} - \cdots\)
\(T_{7}^{11} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{11} \)
$3$ \( ( -1 + T )^{11} \)
$5$ \( 576 + 4024 T - 10494 T^{2} - 2065 T^{3} + 8532 T^{4} - 736 T^{5} - 1928 T^{6} + 300 T^{7} + 168 T^{8} - 31 T^{9} - 5 T^{10} + T^{11} \)
$7$ \( -720 - 3881 T - 5641 T^{2} + 1043 T^{3} + 5695 T^{4} - 127 T^{5} - 2142 T^{6} + 389 T^{7} + 180 T^{8} - 41 T^{9} - 4 T^{10} + T^{11} \)
$11$ \( 21310 - 91739 T + 98465 T^{2} + 17493 T^{3} - 46188 T^{4} - 2224 T^{5} + 7671 T^{6} + 739 T^{7} - 425 T^{8} - 54 T^{9} + 7 T^{10} + T^{11} \)
$13$ \( ( -1 + T )^{11} \)
$17$ \( 2976848 - 1124900 T - 2509776 T^{2} + 260105 T^{3} + 509453 T^{4} - 33333 T^{5} - 36967 T^{6} + 2830 T^{7} + 1047 T^{8} - 94 T^{9} - 10 T^{10} + T^{11} \)
$19$ \( -5248 + 1024 T + 9866 T^{2} - 351 T^{3} - 6531 T^{4} - 442 T^{5} + 1799 T^{6} + 203 T^{7} - 204 T^{8} - 26 T^{9} + 7 T^{10} + T^{11} \)
$23$ \( -1574784 + 496936 T + 1495881 T^{2} - 663576 T^{3} - 224539 T^{4} + 152171 T^{5} - 7777 T^{6} - 7654 T^{7} + 1313 T^{8} + 22 T^{9} - 18 T^{10} + T^{11} \)
$29$ \( -1253072 - 10038986 T + 6779965 T^{2} + 3594075 T^{3} - 3004262 T^{4} + 475582 T^{5} + 65673 T^{6} - 24442 T^{7} + 1351 T^{8} + 212 T^{9} - 29 T^{10} + T^{11} \)
$31$ \( 5248 + 1024 T - 9866 T^{2} - 351 T^{3} + 6531 T^{4} - 442 T^{5} - 1799 T^{6} + 203 T^{7} + 204 T^{8} - 26 T^{9} - 7 T^{10} + T^{11} \)
$37$ \( 197100 - 344344 T - 1878521 T^{2} - 715707 T^{3} + 500756 T^{4} + 153670 T^{5} - 52493 T^{6} - 5772 T^{7} + 1961 T^{8} + 6 T^{9} - 21 T^{10} + T^{11} \)
$41$ \( 412480 + 13172 T - 1817660 T^{2} + 508049 T^{3} + 551396 T^{4} - 230093 T^{5} - 6619 T^{6} + 10583 T^{7} - 249 T^{8} - 179 T^{9} + 3 T^{10} + T^{11} \)
$43$ \( -15301064 - 35745244 T + 4166194 T^{2} + 19753495 T^{3} - 5335966 T^{4} - 880539 T^{5} + 245858 T^{6} + 17124 T^{7} - 3576 T^{8} - 200 T^{9} + 17 T^{10} + T^{11} \)
$47$ \( -64632 + 144374 T - 15645 T^{2} - 133594 T^{3} + 64623 T^{4} + 20039 T^{5} - 13598 T^{6} - 433 T^{7} + 738 T^{8} - 33 T^{9} - 12 T^{10} + T^{11} \)
$53$ \( 185326464 + 41169212 T - 69886700 T^{2} - 11286115 T^{3} + 8224751 T^{4} + 445269 T^{5} - 392486 T^{6} + 19910 T^{7} + 3923 T^{8} - 298 T^{9} - 11 T^{10} + T^{11} \)
$59$ \( -44802048 + 161687488 T - 145915628 T^{2} - 514253 T^{3} + 23566943 T^{4} + 2236580 T^{5} - 643394 T^{6} - 109959 T^{7} - 1606 T^{8} + 672 T^{9} + 48 T^{10} + T^{11} \)
$61$ \( -8568886528 - 7979004352 T + 478945316 T^{2} + 488671713 T^{3} - 11309458 T^{4} - 10961501 T^{5} + 121828 T^{6} + 113698 T^{7} - 580 T^{8} - 550 T^{9} + T^{10} + T^{11} \)
$67$ \( -701192 - 7764306 T + 17043897 T^{2} + 25716867 T^{3} - 3382729 T^{4} - 1777488 T^{5} + 148172 T^{6} + 40283 T^{7} - 2211 T^{8} - 343 T^{9} + 9 T^{10} + T^{11} \)
$71$ \( -9134800896 + 5985102848 T - 743928320 T^{2} - 240471900 T^{3} + 57029061 T^{4} + 1312114 T^{5} - 1051841 T^{6} + 32045 T^{7} + 7351 T^{8} - 371 T^{9} - 17 T^{10} + T^{11} \)
$73$ \( 3761566272 + 2081708242 T - 144546445 T^{2} - 229461653 T^{3} - 23173040 T^{4} + 5523409 T^{5} + 974724 T^{6} - 11431 T^{7} - 8888 T^{8} - 254 T^{9} + 23 T^{10} + T^{11} \)
$79$ \( -281088 + 1368272 T + 27627852 T^{2} - 1132420 T^{3} - 5102321 T^{4} + 580005 T^{5} + 243405 T^{6} - 49323 T^{7} + 562 T^{8} + 495 T^{9} - 41 T^{10} + T^{11} \)
$83$ \( -10699776 - 93255840 T + 293708214 T^{2} - 122235453 T^{3} + 7947119 T^{4} + 3699200 T^{5} - 592985 T^{6} - 11912 T^{7} + 6876 T^{8} - 241 T^{9} - 19 T^{10} + T^{11} \)
$89$ \( -2868488416 + 2723576080 T + 858648766 T^{2} - 1122989805 T^{3} + 102711058 T^{4} + 19192558 T^{5} - 2474093 T^{6} - 63683 T^{7} + 16370 T^{8} - 242 T^{9} - 32 T^{10} + T^{11} \)
$97$ \( -2335712096 - 3710603236 T - 1994216448 T^{2} - 449197919 T^{3} - 24348264 T^{4} + 6951511 T^{5} + 1107877 T^{6} + 7553 T^{7} - 7912 T^{8} - 353 T^{9} + 16 T^{10} + T^{11} \)
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