Properties

Label 8034.2.a.t
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: \(x^{11} - 4 x^{10} - 24 x^{9} + 88 x^{8} + 220 x^{7} - 637 x^{6} - 977 x^{5} + 1739 x^{4} + 1872 x^{3} - 1494 x^{2} - 1161 x - 162\)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 4 x^{10} - 24 x^{9} + 88 x^{8} + 220 x^{7} - 637 x^{6} - 977 x^{5} + 1739 x^{4} + 1872 x^{3} - 1494 x^{2} - 1161 x - 162\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-5953 \nu^{10} + 71878 \nu^{9} - 597285 \nu^{8} + 825299 \nu^{7} + 11387240 \nu^{6} - 30641537 \nu^{5} - 54431851 \nu^{4} + 176099749 \nu^{3} + 73406730 \nu^{2} - 265050639 \nu - 16896519\)\()/16688889\)
\(\beta_{3}\)\(=\)\((\)\(-32168 \nu^{10} - 265732 \nu^{9} + 2755953 \nu^{8} + 3489646 \nu^{7} - 42680480 \nu^{6} - 10039690 \nu^{5} + 212224462 \nu^{4} + 23110703 \nu^{3} - 312848580 \nu^{2} - 42135030 \nu + 42233265\)\()/16688889\)
\(\beta_{4}\)\(=\)\((\)\(53471 \nu^{10} - 338801 \nu^{9} - 262824 \nu^{8} + 4406315 \nu^{7} - 4924159 \nu^{6} - 2522459 \nu^{5} + 23092565 \nu^{4} - 84254744 \nu^{3} - 26952456 \nu^{2} + 115529085 \nu + 33384069\)\()/16688889\)
\(\beta_{5}\)\(=\)\((\)\(35999 \nu^{10} - 355853 \nu^{9} + 249954 \nu^{8} + 6151373 \nu^{7} - 12119914 \nu^{6} - 30684362 \nu^{5} + 69103229 \nu^{4} + 50477308 \nu^{3} - 98979894 \nu^{2} - 8715126 \nu + 10853631\)\()/5562963\)
\(\beta_{6}\)\(=\)\((\)\(-57370 \nu^{10} + 225044 \nu^{9} + 1255382 \nu^{8} - 4321489 \nu^{7} - 10803735 \nu^{6} + 25564583 \nu^{5} + 48171502 \nu^{4} - 55233325 \nu^{3} - 83365948 \nu^{2} + 46532766 \nu + 18345483\)\()/5562963\)
\(\beta_{7}\)\(=\)\((\)\(-176675 \nu^{10} + 631196 \nu^{9} + 4961220 \nu^{8} - 15848045 \nu^{7} - 51857339 \nu^{6} + 134095505 \nu^{5} + 242900272 \nu^{4} - 436355941 \nu^{3} - 452083029 \nu^{2} + 474682383 \nu + 184701816\)\()/16688889\)
\(\beta_{8}\)\(=\)\((\)\(-75437 \nu^{10} + 263873 \nu^{9} + 1986210 \nu^{8} - 6019115 \nu^{7} - 19487438 \nu^{6} + 44806799 \nu^{5} + 83571139 \nu^{4} - 121626778 \nu^{3} - 120826686 \nu^{2} + 108695202 \nu + 22884885\)\()/5562963\)
\(\beta_{9}\)\(=\)\((\)\(-168806 \nu^{10} + 844770 \nu^{9} + 2991638 \nu^{8} - 16491977 \nu^{7} - 18171259 \nu^{6} + 101055744 \nu^{5} + 62639412 \nu^{4} - 227337411 \nu^{3} - 110775703 \nu^{2} + 169506057 \nu + 58191552\)\()/5562963\)
\(\beta_{10}\)\(=\)\((\)\(77942 \nu^{10} - 390059 \nu^{9} - 1408296 \nu^{8} + 7828925 \nu^{7} + 8470799 \nu^{6} - 49927739 \nu^{5} - 26369470 \nu^{4} + 117061726 \nu^{3} + 44577900 \nu^{2} - 84578166 \nu - 31061394\)\()/2384127\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{10} + \beta_{8} + \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 9 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(-\beta_{10} - 13 \beta_{9} + 9 \beta_{8} + \beta_{7} + 14 \beta_{6} - 13 \beta_{5} + \beta_{4} + 2 \beta_{3} + 17 \beta_{1} + 47\)
\(\nu^{5}\)\(=\)\(12 \beta_{10} - 6 \beta_{9} + 13 \beta_{8} - 2 \beta_{7} + 26 \beta_{6} - 33 \beta_{5} + 30 \beta_{4} + 16 \beta_{3} + 8 \beta_{2} + 102 \beta_{1} + 74\)
\(\nu^{6}\)\(=\)\(-24 \beta_{10} - 168 \beta_{9} + 85 \beta_{8} + 6 \beta_{7} + 194 \beta_{6} - 168 \beta_{5} + 23 \beta_{4} + 46 \beta_{3} - 16 \beta_{2} + 253 \beta_{1} + 538\)
\(\nu^{7}\)\(=\)\(111 \beta_{10} - 166 \beta_{9} + 137 \beta_{8} - 48 \beta_{7} + 470 \beta_{6} - 489 \beta_{5} + 373 \beta_{4} + 257 \beta_{3} + 13 \beta_{2} + 1260 \beta_{1} + 1171\)
\(\nu^{8}\)\(=\)\(-379 \beta_{10} - 2174 \beta_{9} + 823 \beta_{8} - 26 \beta_{7} + 2689 \beta_{6} - 2268 \beta_{5} + 386 \beta_{4} + 876 \beta_{3} - 465 \beta_{2} + 3565 \beta_{1} + 6803\)
\(\nu^{9}\)\(=\)\(852 \beta_{10} - 3280 \beta_{9} + 1337 \beta_{8} - 824 \beta_{7} + 7508 \beta_{6} - 7242 \beta_{5} + 4382 \beta_{4} + 4211 \beta_{3} - 1103 \beta_{2} + 16113 \beta_{1} + 17861\)
\(\nu^{10}\)\(=\)\(-5248 \beta_{10} - 28433 \beta_{9} + 7861 \beta_{8} - 1367 \beta_{7} + 37184 \beta_{6} - 31807 \beta_{5} + 5490 \beta_{4} + 15512 \beta_{3} - 9866 \beta_{2} + 49080 \beta_{1} + 91048\)

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
−3.18653
−2.18067
−2.06747
−1.52965
−0.387444
−0.203778
1.30889
1.64076
3.03614
3.71131
3.85844
−1.00000 1.00000 1.00000 −3.18653 −1.00000 1.47944 −1.00000 1.00000 3.18653
1.2 −1.00000 1.00000 1.00000 −2.18067 −1.00000 −1.20699 −1.00000 1.00000 2.18067
1.3 −1.00000 1.00000 1.00000 −2.06747 −1.00000 −2.54251 −1.00000 1.00000 2.06747
1.4 −1.00000 1.00000 1.00000 −1.52965 −1.00000 −0.354974 −1.00000 1.00000 1.52965
1.5 −1.00000 1.00000 1.00000 −0.387444 −1.00000 5.18739 −1.00000 1.00000 0.387444
1.6 −1.00000 1.00000 1.00000 −0.203778 −1.00000 0.561930 −1.00000 1.00000 0.203778
1.7 −1.00000 1.00000 1.00000 1.30889 −1.00000 2.83911 −1.00000 1.00000 −1.30889
1.8 −1.00000 1.00000 1.00000 1.64076 −1.00000 −3.72013 −1.00000 1.00000 −1.64076
1.9 −1.00000 1.00000 1.00000 3.03614 −1.00000 −4.00525 −1.00000 1.00000 −3.03614
1.10 −1.00000 1.00000 1.00000 3.71131 −1.00000 1.88167 −1.00000 1.00000 −3.71131
1.11 −1.00000 1.00000 1.00000 3.85844 −1.00000 3.88029 −1.00000 1.00000 −3.85844
\(n\): 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.t 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.t 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

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{11} \)
$3$ \( ( -1 + T )^{11} \)
$5$ \( -162 - 1161 T - 1494 T^{2} + 1872 T^{3} + 1739 T^{4} - 977 T^{5} - 637 T^{6} + 220 T^{7} + 88 T^{8} - 24 T^{9} - 4 T^{10} + T^{11} \)
$7$ \( 1451 + 1114 T - 9037 T^{2} + 997 T^{3} + 7580 T^{4} - 1985 T^{5} - 1738 T^{6} + 487 T^{7} + 145 T^{8} - 39 T^{9} - 4 T^{10} + T^{11} \)
$11$ \( 3960 + 25576 T + 55702 T^{2} + 47837 T^{3} + 6881 T^{4} - 10528 T^{5} - 2925 T^{6} + 1010 T^{7} + 234 T^{8} - 53 T^{9} - 5 T^{10} + T^{11} \)
$13$ \( ( 1 + T )^{11} \)
$17$ \( -15928 + 36516 T + 142186 T^{2} - 268453 T^{3} + 94121 T^{4} + 25299 T^{5} - 16339 T^{6} + 438 T^{7} + 727 T^{8} - 72 T^{9} - 8 T^{10} + T^{11} \)
$19$ \( 7472 - 87896 T - 10716 T^{2} + 93132 T^{3} - 4333 T^{4} - 20974 T^{5} + 1334 T^{6} + 1861 T^{7} - 96 T^{8} - 72 T^{9} + 2 T^{10} + T^{11} \)
$23$ \( -11088 - 10499 T + 25197 T^{2} + 23417 T^{3} - 12598 T^{4} - 13852 T^{5} + 301 T^{6} + 1933 T^{7} + 103 T^{8} - 88 T^{9} - 3 T^{10} + T^{11} \)
$29$ \( 609222 - 7711213 T - 1734846 T^{2} + 2084344 T^{3} + 372905 T^{4} - 201192 T^{5} - 27694 T^{6} + 8439 T^{7} + 798 T^{8} - 152 T^{9} - 7 T^{10} + T^{11} \)
$31$ \( -3919184 + 6585208 T + 441300 T^{2} - 2155824 T^{3} + 238863 T^{4} + 194012 T^{5} - 32596 T^{6} - 6205 T^{7} + 1404 T^{8} + 34 T^{9} - 20 T^{10} + T^{11} \)
$37$ \( 576400 - 360160 T - 1937044 T^{2} + 823074 T^{3} + 549161 T^{4} - 198579 T^{5} - 44853 T^{6} + 13877 T^{7} + 492 T^{8} - 219 T^{9} - T^{10} + T^{11} \)
$41$ \( -18250 + 33697 T + 246900 T^{2} - 640812 T^{3} + 584381 T^{4} - 241808 T^{5} + 35530 T^{6} + 6448 T^{7} - 3332 T^{8} + 522 T^{9} - 37 T^{10} + T^{11} \)
$43$ \( 107474032 + 120955064 T + 24822404 T^{2} - 12770316 T^{3} - 4247537 T^{4} + 331347 T^{5} + 191996 T^{6} + 2622 T^{7} - 3108 T^{8} - 146 T^{9} + 16 T^{10} + T^{11} \)
$47$ \( 28302372 - 11091897 T - 33087159 T^{2} + 30731028 T^{3} - 10107176 T^{4} + 1087709 T^{5} + 152054 T^{6} - 50471 T^{7} + 3845 T^{8} + 113 T^{9} - 28 T^{10} + T^{11} \)
$53$ \( -2054968 - 2647788 T + 4016336 T^{2} + 1006447 T^{3} - 1294191 T^{4} - 193579 T^{5} + 117346 T^{6} + 19900 T^{7} - 1491 T^{8} - 274 T^{9} + 5 T^{10} + T^{11} \)
$59$ \( -65166240 - 244200949 T - 72673247 T^{2} + 150040544 T^{3} - 52404333 T^{4} + 6625013 T^{5} + 74116 T^{6} - 97454 T^{7} + 7577 T^{8} + 74 T^{9} - 31 T^{10} + T^{11} \)
$61$ \( 23303527296 - 7651926952 T - 2164372808 T^{2} + 450438430 T^{3} + 62076895 T^{4} - 10105047 T^{5} - 760504 T^{6} + 107460 T^{7} + 4110 T^{8} - 536 T^{9} - 8 T^{10} + T^{11} \)
$67$ \( 12509517 + 80896939 T + 53808441 T^{2} - 16993491 T^{3} - 7871186 T^{4} + 740426 T^{5} + 357340 T^{6} - 3696 T^{7} - 5703 T^{8} - 176 T^{9} + 22 T^{10} + T^{11} \)
$71$ \( -8558660 + 12516063 T + 28216931 T^{2} - 40001122 T^{3} - 5068065 T^{4} + 3175057 T^{5} + 64480 T^{6} - 80436 T^{7} + 4224 T^{8} + 409 T^{9} - 42 T^{10} + T^{11} \)
$73$ \( 2206561775 - 2463104497 T - 203172770 T^{2} + 211653742 T^{3} + 3249910 T^{4} - 6153355 T^{5} + 32957 T^{6} + 79025 T^{7} - 903 T^{8} - 459 T^{9} + 4 T^{10} + T^{11} \)
$79$ \( 95618960 + 965046344 T + 393889612 T^{2} - 325072792 T^{3} + 18413383 T^{4} + 10494689 T^{5} - 1038553 T^{6} - 85347 T^{7} + 12604 T^{8} - 89 T^{9} - 33 T^{10} + T^{11} \)
$83$ \( 22143750 + 375788675 T - 677211831 T^{2} + 8498740 T^{3} + 39456024 T^{4} - 1466164 T^{5} - 803092 T^{6} + 37654 T^{7} + 6592 T^{8} - 344 T^{9} - 18 T^{10} + T^{11} \)
$89$ \( 3961390960 - 2462631864 T - 399165964 T^{2} + 363699960 T^{3} - 31707543 T^{4} - 10105703 T^{5} + 2074685 T^{6} - 69710 T^{7} - 14119 T^{8} + 1629 T^{9} - 67 T^{10} + T^{11} \)
$97$ \( -2398891376 - 3161970720 T + 746474572 T^{2} + 234194030 T^{3} - 42841151 T^{4} - 5866547 T^{5} + 814420 T^{6} + 72899 T^{7} - 6092 T^{8} - 446 T^{9} + 15 T^{10} + T^{11} \)
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