Properties

Label 6003.2.a.o
Level $6003$
Weight $2$
Character orbit 6003.a
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 4 x^{12} - 11 x^{11} + 58 x^{10} + 24 x^{9} - 298 x^{8} + 97 x^{7} + 641 x^{6} - 402 x^{5} - 547 x^{4} + 352 x^{3} + 219 x^{2} - 88 x - 40\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 667)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{8} ) q^{5} + \beta_{9} q^{7} + ( -\beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{8} ) q^{5} + \beta_{9} q^{7} + ( -\beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} ) q^{8} + ( \beta_{1} - \beta_{4} - \beta_{8} ) q^{10} + ( -1 - \beta_{11} ) q^{11} + ( 1 + \beta_{4} + \beta_{8} ) q^{13} + ( 1 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{9} + 2 \beta_{10} - \beta_{12} ) q^{14} + ( \beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{16} + ( -2 + \beta_{1} + \beta_{4} - \beta_{6} - \beta_{10} + \beta_{11} ) q^{17} + ( -\beta_{1} + 2 \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} ) q^{19} + ( -2 + \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} + \beta_{11} ) q^{20} + ( -2 + 3 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{7} + \beta_{11} ) q^{22} + q^{23} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{12} ) q^{25} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} - \beta_{11} ) q^{26} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{28} - q^{29} + ( -1 + 2 \beta_{1} + \beta_{3} - 2 \beta_{6} + 2 \beta_{7} + \beta_{11} ) q^{31} + ( -2 + \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} ) q^{32} + ( -2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{34} + ( -\beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{12} ) q^{35} + ( 1 - \beta_{3} + \beta_{5} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{12} ) q^{37} + ( 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} - \beta_{9} ) q^{38} + ( 1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{40} + ( -1 + \beta_{2} - \beta_{4} + \beta_{8} - \beta_{9} - 2 \beta_{12} ) q^{41} + ( -\beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} + \beta_{12} ) q^{43} + ( -3 + \beta_{1} - \beta_{4} + 2 \beta_{5} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{44} -\beta_{1} q^{46} + ( -2 + 3 \beta_{1} - \beta_{2} + \beta_{9} + \beta_{11} ) q^{47} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{49} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 3 \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{12} ) q^{50} + ( \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{52} + ( -4 + 3 \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{11} ) q^{53} + ( -\beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{11} ) q^{55} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{12} ) q^{56} + \beta_{1} q^{58} + ( \beta_{1} - \beta_{2} + \beta_{3} - 4 \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{12} ) q^{59} + ( 2 - \beta_{1} + 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{10} - \beta_{11} - 2 \beta_{12} ) q^{61} + ( -2 + \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{11} ) q^{62} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} + \beta_{9} - \beta_{11} + \beta_{12} ) q^{64} + ( 1 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{12} ) q^{65} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} ) q^{67} + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{12} ) q^{68} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{7} + \beta_{8} + 3 \beta_{9} - 4 \beta_{10} + \beta_{11} + 4 \beta_{12} ) q^{70} + ( -2 + 3 \beta_{1} - \beta_{3} + \beta_{4} + 3 \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{12} ) q^{71} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{73} + ( -1 + 2 \beta_{4} + \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{12} ) q^{74} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{12} ) q^{76} + ( -2 + 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} - \beta_{12} ) q^{77} + ( 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} - \beta_{6} + 2 \beta_{8} + \beta_{10} + \beta_{12} ) q^{79} + ( -1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{4} + 3 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{12} ) q^{80} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} + 2 \beta_{9} - 2 \beta_{10} + \beta_{11} + 3 \beta_{12} ) q^{82} + ( -1 - 2 \beta_{1} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{8} - \beta_{9} - \beta_{11} - 2 \beta_{12} ) q^{83} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 3 \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{85} + ( 2 - \beta_{2} + 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + 3 \beta_{9} ) q^{86} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{10} - \beta_{12} ) q^{88} + ( -4 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - 3 \beta_{8} + 3 \beta_{10} - 2 \beta_{12} ) q^{89} + ( -2 + \beta_{2} - 2 \beta_{4} + \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} - 3 \beta_{12} ) q^{91} + ( 1 + \beta_{2} ) q^{92} + ( -6 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} - \beta_{9} + 2 \beta_{10} - \beta_{11} - \beta_{12} ) q^{94} + ( -1 + 4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 5 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{10} + 2 \beta_{11} - 2 \beta_{12} ) q^{95} + ( 4 + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{7} + \beta_{8} + \beta_{11} ) q^{97} + ( 3 \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} - \beta_{6} + 2 \beta_{8} + 4 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 4q^{2} + 12q^{4} - 16q^{5} + q^{7} - 6q^{8} + O(q^{10}) \) \( 13q - 4q^{2} + 12q^{4} - 16q^{5} + q^{7} - 6q^{8} + 10q^{10} - 10q^{11} + 7q^{13} + 12q^{14} + 2q^{16} - 26q^{17} - 25q^{20} - 15q^{22} + 13q^{23} + 19q^{25} + 15q^{26} + 5q^{28} - 13q^{29} - 6q^{31} - 16q^{32} + 11q^{34} - q^{35} + 15q^{37} - 8q^{38} + 14q^{40} - 9q^{41} + q^{43} - 29q^{44} - 4q^{46} - 15q^{47} + 4q^{49} - 31q^{50} - 8q^{52} - 43q^{53} - 3q^{55} + 5q^{56} + 4q^{58} + 9q^{59} + 20q^{61} - 11q^{62} - 16q^{64} + 25q^{65} + q^{67} - 21q^{68} - 2q^{70} - 17q^{71} + 26q^{73} - 11q^{74} + 8q^{76} - 17q^{77} + 5q^{79} - 10q^{80} - 25q^{82} - 4q^{83} + 20q^{85} + 13q^{86} - 32q^{88} - 48q^{89} - 9q^{91} + 12q^{92} - 65q^{94} - 8q^{95} + 30q^{97} - 2q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13} - 4 x^{12} - 11 x^{11} + 58 x^{10} + 24 x^{9} - 298 x^{8} + 97 x^{7} + 641 x^{6} - 402 x^{5} - 547 x^{4} + 352 x^{3} + 219 x^{2} - 88 x - 40\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - \nu^{3} - 6 \nu^{2} + 4 \nu + 4 \)
\(\beta_{4}\)\(=\)\((\)\( 6 \nu^{12} - 22 \nu^{11} - 67 \nu^{10} + 313 \nu^{9} + 147 \nu^{8} - 1530 \nu^{7} + 604 \nu^{6} + 2882 \nu^{5} - 2433 \nu^{4} - 1566 \nu^{3} + 1894 \nu^{2} + 153 \nu - 363 \)\()/19\)
\(\beta_{5}\)\(=\)\((\)\( -11 \nu^{12} + 34 \nu^{11} + 145 \nu^{10} - 482 \nu^{9} - 621 \nu^{8} + 2387 \nu^{7} + 818 \nu^{6} - 4758 \nu^{5} + 233 \nu^{4} + 3232 \nu^{3} - 236 \nu^{2} - 613 \nu - 28 \)\()/19\)
\(\beta_{6}\)\(=\)\((\)\( -11 \nu^{12} + 34 \nu^{11} + 145 \nu^{10} - 482 \nu^{9} - 621 \nu^{8} + 2387 \nu^{7} + 818 \nu^{6} - 4777 \nu^{5} + 233 \nu^{4} + 3365 \nu^{3} - 198 \nu^{2} - 765 \nu - 104 \)\()/19\)
\(\beta_{7}\)\(=\)\((\)\( -11 \nu^{12} + 34 \nu^{11} + 145 \nu^{10} - 482 \nu^{9} - 621 \nu^{8} + 2387 \nu^{7} + 818 \nu^{6} - 4777 \nu^{5} + 233 \nu^{4} + 3384 \nu^{3} - 217 \nu^{2} - 860 \nu - 47 \)\()/19\)
\(\beta_{8}\)\(=\)\((\)\( -9 \nu^{12} + 33 \nu^{11} + 110 \nu^{10} - 479 \nu^{9} - 382 \nu^{8} + 2447 \nu^{7} + 44 \nu^{6} - 5121 \nu^{5} + 1379 \nu^{4} + 3869 \nu^{3} - 865 \nu^{2} - 942 \nu + 3 \)\()/19\)
\(\beta_{9}\)\(=\)\((\)\( 10 \nu^{12} - 24 \nu^{11} - 156 \nu^{10} + 357 \nu^{9} + 910 \nu^{8} - 1923 \nu^{7} - 2464 \nu^{6} + 4512 \nu^{5} + 3203 \nu^{4} - 4377 \nu^{3} - 2024 \nu^{2} + 1338 \nu + 516 \)\()/19\)
\(\beta_{10}\)\(=\)\((\)\( -31 \nu^{12} + 82 \nu^{11} + 457 \nu^{10} - 1196 \nu^{9} - 2422 \nu^{8} + 6176 \nu^{7} + 5613 \nu^{6} - 13307 \nu^{5} - 6078 \nu^{4} + 10979 \nu^{3} + 4344 \nu^{2} - 2985 \nu - 1326 \)\()/38\)
\(\beta_{11}\)\(=\)\((\)\( -21 \nu^{12} + 58 \nu^{11} + 301 \nu^{10} - 839 \nu^{9} - 1512 \nu^{8} + 4272 \nu^{7} + 3092 \nu^{6} - 8947 \nu^{5} - 2400 \nu^{4} + 6887 \nu^{3} + 1313 \nu^{2} - 1628 \nu - 449 \)\()/19\)
\(\beta_{12}\)\(=\)\((\)\( -31 \nu^{12} + 82 \nu^{11} + 457 \nu^{10} - 1196 \nu^{9} - 2422 \nu^{8} + 6195 \nu^{7} + 5575 \nu^{6} - 13478 \nu^{5} - 5774 \nu^{4} + 11397 \nu^{3} + 3736 \nu^{2} - 3175 \nu - 1136 \)\()/19\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{7} - \beta_{6} + \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{7} - \beta_{6} + \beta_{3} + 7 \beta_{2} + \beta_{1} + 14\)
\(\nu^{5}\)\(=\)\(7 \beta_{7} - 8 \beta_{6} + \beta_{5} + 9 \beta_{2} + 27 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(\beta_{12} - \beta_{11} + \beta_{9} + 9 \beta_{7} - 10 \beta_{6} + \beta_{5} + 9 \beta_{3} + 44 \beta_{2} + 12 \beta_{1} + 74\)
\(\nu^{7}\)\(=\)\(3 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 43 \beta_{7} - 54 \beta_{6} + 11 \beta_{5} + 2 \beta_{3} + 67 \beta_{2} + 151 \beta_{1} + 28\)
\(\nu^{8}\)\(=\)\(16 \beta_{12} - 13 \beta_{11} - 4 \beta_{10} + 15 \beta_{9} + 65 \beta_{7} - 80 \beta_{6} + 14 \beta_{5} + 64 \beta_{3} + 274 \beta_{2} + 106 \beta_{1} + 412\)
\(\nu^{9}\)\(=\)\(45 \beta_{12} - 28 \beta_{11} - 30 \beta_{10} + 33 \beta_{9} + 261 \beta_{7} - 352 \beta_{6} + 91 \beta_{5} + 2 \beta_{4} + 33 \beta_{3} + 473 \beta_{2} + 869 \beta_{1} + 264\)
\(\nu^{10}\)\(=\)\(169 \beta_{12} - 117 \beta_{11} - 66 \beta_{10} + 156 \beta_{9} + 2 \beta_{8} + 445 \beta_{7} - 599 \beta_{6} + 137 \beta_{5} + 5 \beta_{4} + 428 \beta_{3} + 1720 \beta_{2} + 837 \beta_{1} + 2367\)
\(\nu^{11}\)\(=\)\(462 \beta_{12} - 266 \beta_{11} - 312 \beta_{10} + 364 \beta_{9} + 7 \beta_{8} + 1603 \beta_{7} - 2290 \beta_{6} + 680 \beta_{5} + 41 \beta_{4} + 361 \beta_{3} + 3268 \beta_{2} + 5135 \beta_{1} + 2139\)
\(\nu^{12}\)\(=\)\(1506 \beta_{12} - 912 \beta_{11} - 728 \beta_{10} + 1397 \beta_{9} + 48 \beta_{8} + 3002 \beta_{7} - 4350 \beta_{6} + 1157 \beta_{5} + 105 \beta_{4} + 2823 \beta_{3} + 10918 \beta_{2} + 6258 \beta_{1} + 13929\)

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
2.60496
2.43510
2.24788
1.46878
1.30511
1.18857
0.775068
−0.397523
−0.552582
−0.806558
−1.78056
−2.22202
−2.26622
−2.60496 0 4.78583 −1.68070 0 2.80821 −7.25699 0 4.37815
1.2 −2.43510 0 3.92972 −3.49556 0 −4.87125 −4.69905 0 8.51205
1.3 −2.24788 0 3.05297 0.269352 0 −0.523800 −2.36696 0 −0.605470
1.4 −1.46878 0 0.157302 −4.31834 0 1.97692 2.70651 0 6.34267
1.5 −1.30511 0 −0.296685 −0.572849 0 1.21746 2.99743 0 0.747631
1.6 −1.18857 0 −0.587311 2.45282 0 −2.24002 3.07519 0 −2.91534
1.7 −0.775068 0 −1.39927 −3.48646 0 0.0624539 2.63467 0 2.70224
1.8 0.397523 0 −1.84198 3.16986 0 −1.67469 −1.52727 0 1.26009
1.9 0.552582 0 −1.69465 −3.05371 0 −3.72348 −2.04160 0 −1.68743
1.10 0.806558 0 −1.34946 −2.04515 0 4.05430 −2.70154 0 −1.64953
1.11 1.78056 0 1.17038 −0.267809 0 2.76268 −1.47718 0 −0.476849
1.12 2.22202 0 2.93739 −2.84468 0 −1.80107 2.08291 0 −6.32095
1.13 2.26622 0 3.13576 −0.126764 0 2.95229 2.57389 0 −0.287275
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(-1\)
\(29\) \(1\)

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 6003.2.a.o 13
3.b odd 2 1 667.2.a.c 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
667.2.a.c 13 3.b odd 2 1
6003.2.a.o 13 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\(T_{2}^{13} + \cdots\)
\(T_{5}^{13} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 40 - 88 T - 219 T^{2} + 352 T^{3} + 547 T^{4} - 402 T^{5} - 641 T^{6} + 97 T^{7} + 298 T^{8} + 24 T^{9} - 58 T^{10} - 11 T^{11} + 4 T^{12} + T^{13} \)
$3$ \( T^{13} \)
$5$ \( -64 - 736 T - 1214 T^{2} + 7871 T^{3} + 27980 T^{4} + 31627 T^{5} + 12522 T^{6} - 2653 T^{7} - 3861 T^{8} - 1022 T^{9} + 72 T^{10} + 86 T^{11} + 16 T^{12} + T^{13} \)
$7$ \( -896 + 13136 T + 21248 T^{2} - 28577 T^{3} - 21628 T^{4} + 18795 T^{5} + 7651 T^{6} - 5513 T^{7} - 1117 T^{8} + 769 T^{9} + 62 T^{10} - 47 T^{11} - T^{12} + T^{13} \)
$11$ \( 28240 - 84324 T - 104116 T^{2} + 89963 T^{3} + 87193 T^{4} - 34927 T^{5} - 30043 T^{6} + 5638 T^{7} + 4849 T^{8} - 258 T^{9} - 363 T^{10} - 14 T^{11} + 10 T^{12} + T^{13} \)
$13$ \( 1694 + 10769 T + 18807 T^{2} - 1139 T^{3} - 25616 T^{4} - 6800 T^{5} + 14358 T^{6} + 2573 T^{7} - 4014 T^{8} + 174 T^{9} + 316 T^{10} - 36 T^{11} - 7 T^{12} + T^{13} \)
$17$ \( 551222 - 2085187 T + 2028189 T^{2} + 392345 T^{3} - 1064836 T^{4} - 51852 T^{5} + 232399 T^{6} + 30260 T^{7} - 20566 T^{8} - 5041 T^{9} + 320 T^{10} + 222 T^{11} + 26 T^{12} + T^{13} \)
$19$ \( 11924 - 31165 T - 249084 T^{2} + 645162 T^{3} + 450800 T^{4} - 329303 T^{5} - 301024 T^{6} - 41990 T^{7} + 17175 T^{8} + 4413 T^{9} - 239 T^{10} - 116 T^{11} + T^{13} \)
$23$ \( ( -1 + T )^{13} \)
$29$ \( ( 1 + T )^{13} \)
$31$ \( 70389088 + 10958948 T - 79126796 T^{2} - 19326283 T^{3} + 17892174 T^{4} + 4055097 T^{5} - 1548828 T^{6} - 319328 T^{7} + 60842 T^{8} + 11461 T^{9} - 1055 T^{10} - 184 T^{11} + 6 T^{12} + T^{13} \)
$37$ \( 99546086 - 72085251 T - 39532079 T^{2} + 41980940 T^{3} - 1943603 T^{4} - 5855756 T^{5} + 1107781 T^{6} + 271126 T^{7} - 81066 T^{8} - 2411 T^{9} + 2002 T^{10} - 77 T^{11} - 15 T^{12} + T^{13} \)
$41$ \( -171294376 - 314976948 T - 137075870 T^{2} + 53056527 T^{3} + 44382309 T^{4} + 24367 T^{5} - 4106190 T^{6} - 325311 T^{7} + 146794 T^{8} + 14842 T^{9} - 2044 T^{10} - 221 T^{11} + 9 T^{12} + T^{13} \)
$43$ \( -47372716 + 38461693 T + 34159787 T^{2} - 25614530 T^{3} - 8829533 T^{4} + 5442551 T^{5} + 900290 T^{6} - 479235 T^{7} - 33550 T^{8} + 18166 T^{9} + 301 T^{10} - 250 T^{11} - T^{12} + T^{13} \)
$47$ \( 739168 + 4879108 T + 10325732 T^{2} + 6852691 T^{3} - 907492 T^{4} - 2530137 T^{5} - 646509 T^{6} + 125278 T^{7} + 57914 T^{8} - 307 T^{9} - 1642 T^{10} - 73 T^{11} + 15 T^{12} + T^{13} \)
$53$ \( -530954104 - 1215296188 T - 904394778 T^{2} - 135577733 T^{3} + 133482632 T^{4} + 63808656 T^{5} + 6275962 T^{6} - 2101551 T^{7} - 640081 T^{8} - 56826 T^{9} + 1448 T^{10} + 626 T^{11} + 43 T^{12} + T^{13} \)
$59$ \( 91791625948 - 53207825807 T - 77514625261 T^{2} - 17512029164 T^{3} + 1577190619 T^{4} + 765411478 T^{5} + 6560117 T^{6} - 13003195 T^{7} - 388921 T^{8} + 113444 T^{9} + 3325 T^{10} - 519 T^{11} - 9 T^{12} + T^{13} \)
$61$ \( -7874543600 + 24605188824 T - 25329136134 T^{2} + 9358504145 T^{3} - 585939498 T^{4} - 359966270 T^{5} + 59884210 T^{6} + 3192699 T^{7} - 1113148 T^{8} + 14724 T^{9} + 7978 T^{10} - 302 T^{11} - 20 T^{12} + T^{13} \)
$67$ \( -6179333792 + 6672721832 T + 1060851908 T^{2} - 2381971029 T^{3} + 85196160 T^{4} + 223259254 T^{5} - 10995914 T^{6} - 7503540 T^{7} + 134542 T^{8} + 97745 T^{9} - 229 T^{10} - 526 T^{11} - T^{12} + T^{13} \)
$71$ \( 59462318104 - 136367503573 T - 52685499559 T^{2} + 12359709856 T^{3} + 5117805783 T^{4} - 192742640 T^{5} - 153967040 T^{6} - 2756751 T^{7} + 1902698 T^{8} + 79657 T^{9} - 9665 T^{10} - 512 T^{11} + 17 T^{12} + T^{13} \)
$73$ \( 7929287008 + 20760504448 T + 11687557826 T^{2} + 71095319 T^{3} - 1139898096 T^{4} - 78940096 T^{5} + 46674939 T^{6} + 2033475 T^{7} - 926900 T^{8} - 5906 T^{9} + 8319 T^{10} - 178 T^{11} - 26 T^{12} + T^{13} \)
$79$ \( 20724897088 - 1133451457 T - 21029026700 T^{2} - 8453522565 T^{3} + 825410120 T^{4} + 711144419 T^{5} + 15894561 T^{6} - 17654249 T^{7} - 438668 T^{8} + 169325 T^{9} + 2700 T^{10} - 688 T^{11} - 5 T^{12} + T^{13} \)
$83$ \( -27841625168 + 51218652012 T - 4060898852 T^{2} - 8130367619 T^{3} + 421479911 T^{4} + 394464622 T^{5} - 13378524 T^{6} - 8568223 T^{7} + 199243 T^{8} + 92895 T^{9} - 1437 T^{10} - 490 T^{11} + 4 T^{12} + T^{13} \)
$89$ \( -376563331102 - 4651925023 T + 79648412264 T^{2} + 5552628684 T^{3} - 5668714931 T^{4} - 692672169 T^{5} + 138618204 T^{6} + 26046556 T^{7} - 148743 T^{8} - 252778 T^{9} - 12494 T^{10} + 437 T^{11} + 48 T^{12} + T^{13} \)
$97$ \( -32043114320 - 24260058712 T + 5037415018 T^{2} + 5364680079 T^{3} - 324960318 T^{4} - 384464395 T^{5} + 27003706 T^{6} + 10249936 T^{7} - 1084349 T^{8} - 70657 T^{9} + 12050 T^{10} - 148 T^{11} - 30 T^{12} + T^{13} \)
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