Properties

Label 6042.2.a.bg
Level 6042
Weight 2
Character orbit 6042.a
Self dual yes
Analytic conductor 48.246
Analytic rank 0
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2456129013\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 5 x^{11} - 34 x^{10} + 169 x^{9} + 453 x^{8} - 2095 x^{7} - 3056 x^{6} + 11545 x^{5} + 11014 x^{4} - 26766 x^{3} - 19370 x^{2} + 18296 x + 7848\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 5 x^{11} - 34 x^{10} + 169 x^{9} + 453 x^{8} - 2095 x^{7} - 3056 x^{6} + 11545 x^{5} + 11014 x^{4} - 26766 x^{3} - 19370 x^{2} + 18296 x + 7848\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-204446 \nu^{11} + 2608739 \nu^{10} + 7340808 \nu^{9} - 137165906 \nu^{8} + 23174311 \nu^{7} + 2214031276 \nu^{6} - 1959913603 \nu^{5} - 13476233233 \nu^{4} + 13238233488 \nu^{3} + 26161780884 \nu^{2} - 16849170666 \nu - 5802453752\)\()/ 669887500 \)
\(\beta_{3}\)\(=\)\((\)\(-1345873 \nu^{11} + 12257282 \nu^{10} + 10822604 \nu^{9} - 340995353 \nu^{8} + 325470868 \nu^{7} + 3363307163 \nu^{6} - 4306515439 \nu^{5} - 14675891754 \nu^{4} + 15577520444 \nu^{3} + 28087836042 \nu^{2} - 16773810608 \nu - 13115755176\)\()/ 1339775000 \)
\(\beta_{4}\)\(=\)\((\)\(-4094329 \nu^{11} + 15259486 \nu^{10} + 135031392 \nu^{9} - 375232369 \nu^{8} - 1847667036 \nu^{7} + 2590798999 \nu^{6} + 12914192153 \nu^{5} - 1970474342 \nu^{4} - 39438600788 \nu^{3} - 18686256534 \nu^{2} + 25762553016 \nu + 12411594952\)\()/ 1339775000 \)
\(\beta_{5}\)\(=\)\((\)\(7020169 \nu^{11} - 62872796 \nu^{10} - 89204962 \nu^{9} + 1961538359 \nu^{8} - 1283733404 \nu^{7} - 22214565039 \nu^{6} + 24690558217 \nu^{5} + 109575504662 \nu^{4} - 112755982232 \nu^{3} - 208346946326 \nu^{2} + 120758335624 \nu + 71971042128\)\()/ 1339775000 \)
\(\beta_{6}\)\(=\)\((\)\(-8194199 \nu^{11} + 58777566 \nu^{10} + 170893902 \nu^{9} - 1833840689 \nu^{8} - 448056866 \nu^{7} + 20509140469 \nu^{6} - 9521117757 \nu^{5} - 97762562352 \nu^{4} + 58800907072 \nu^{3} + 173526880946 \nu^{2} - 64977367004 \nu - 49602823488\)\()/ 1339775000 \)
\(\beta_{7}\)\(=\)\((\)\(4234532 \nu^{11} - 27548113 \nu^{10} - 105119461 \nu^{9} + 886007227 \nu^{8} + 648769138 \nu^{7} - 10137638892 \nu^{6} + 1783597851 \nu^{5} + 48288656186 \nu^{4} - 24245652146 \nu^{3} - 81138473578 \nu^{2} + 38591078772 \nu + 20499466684\)\()/ 669887500 \)
\(\beta_{8}\)\(=\)\((\)\(-5101356 \nu^{11} + 28749929 \nu^{10} + 131490613 \nu^{9} - 829014291 \nu^{8} - 1119132754 \nu^{7} + 8194525736 \nu^{6} + 3349514017 \nu^{5} - 31912312038 \nu^{4} + 564837718 \nu^{3} + 40111343774 \nu^{2} - 12568245276 \nu - 6270409172\)\()/ 669887500 \)
\(\beta_{9}\)\(=\)\((\)\(-1085494 \nu^{11} + 5435946 \nu^{10} + 32741387 \nu^{9} - 164915409 \nu^{8} - 359968771 \nu^{7} + 1727123514 \nu^{6} + 1716947858 \nu^{5} - 7130254537 \nu^{4} - 2801209518 \nu^{3} + 9465077226 \nu^{2} - 328369874 \nu - 2493542228\)\()/ 133977500 \)
\(\beta_{10}\)\(=\)\((\)\(1085494 \nu^{11} - 5435946 \nu^{10} - 32741387 \nu^{9} + 164915409 \nu^{8} + 359968771 \nu^{7} - 1727123514 \nu^{6} - 1716947858 \nu^{5} + 7130254537 \nu^{4} + 2801209518 \nu^{3} - 9331099726 \nu^{2} + 194392374 \nu + 1555699728\)\()/ 133977500 \)
\(\beta_{11}\)\(=\)\((\)\(11405033 \nu^{11} - 65451972 \nu^{10} - 329920034 \nu^{9} + 2197952363 \nu^{8} + 2858059572 \nu^{7} - 26085348123 \nu^{6} - 3108483931 \nu^{5} + 127093683434 \nu^{4} - 48828499424 \nu^{3} - 214479866182 \nu^{2} + 98517025968 \nu + 61388109096\)\()/ 1339775000 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} + \beta_{9} + \beta_{1} + 7\)
\(\nu^{3}\)\(=\)\(2 \beta_{10} + \beta_{9} - 2 \beta_{7} + \beta_{5} + \beta_{3} - \beta_{2} + 12 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(\beta_{11} + 17 \beta_{10} + 15 \beta_{9} + \beta_{8} - 6 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 23 \beta_{1} + 76\)
\(\nu^{5}\)\(=\)\(2 \beta_{11} + 56 \beta_{10} + 32 \beta_{9} + \beta_{8} - 43 \beta_{7} + 7 \beta_{6} + 23 \beta_{5} + 2 \beta_{4} + 10 \beta_{3} - 21 \beta_{2} + 171 \beta_{1} + 100\)
\(\nu^{6}\)\(=\)\(23 \beta_{11} + 300 \beta_{10} + 237 \beta_{9} + 9 \beta_{8} - 162 \beta_{7} + 11 \beta_{6} + 52 \beta_{5} - 22 \beta_{4} - 80 \beta_{3} - 110 \beta_{2} + 457 \beta_{1} + 997\)
\(\nu^{7}\)\(=\)\(66 \beta_{11} + 1223 \beta_{10} + 732 \beta_{9} + 8 \beta_{8} - 831 \beta_{7} + 190 \beta_{6} + 412 \beta_{5} + 119 \beta_{4} - 19 \beta_{3} - 399 \beta_{2} + 2707 \beta_{1} + 2061\)
\(\nu^{8}\)\(=\)\(490 \beta_{11} + 5631 \beta_{10} + 4104 \beta_{9} - 37 \beta_{8} - 3455 \beta_{7} + 418 \beta_{6} + 1055 \beta_{5} + 144 \beta_{4} - 1805 \beta_{3} - 2197 \beta_{2} + 8755 \beta_{1} + 14695\)
\(\nu^{9}\)\(=\)\(1785 \beta_{11} + 24783 \beta_{10} + 15275 \beta_{9} - 450 \beta_{8} - 15984 \beta_{7} + 4002 \beta_{6} + 6897 \beta_{5} + 4007 \beta_{4} - 3691 \beta_{3} - 7398 \beta_{2} + 45697 \beta_{1} + 39988\)
\(\nu^{10}\)\(=\)\(10821 \beta_{11} + 108976 \beta_{10} + 75835 \beta_{9} - 4232 \beta_{8} - 69216 \beta_{7} + 11089 \beta_{6} + 19663 \beta_{5} + 13629 \beta_{4} - 38924 \beta_{3} - 39380 \beta_{2} + 165691 \beta_{1} + 234569\)
\(\nu^{11}\)\(=\)\(44889 \beta_{11} + 489981 \beta_{10} + 309441 \beta_{9} - 22465 \beta_{8} - 309458 \beta_{7} + 78304 \beta_{6} + 112990 \beta_{5} + 109070 \beta_{4} - 120071 \beta_{3} - 133634 \beta_{2} + 804261 \beta_{1} + 758488\)

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
−3.46060
−2.85168
−2.80221
−1.58560
−1.52770
−0.365306
0.927745
2.31473
2.39115
3.49639
4.02498
4.43811
1.00000 1.00000 1.00000 −3.46060 1.00000 0.599762 1.00000 1.00000 −3.46060
1.2 1.00000 1.00000 1.00000 −2.85168 1.00000 3.66552 1.00000 1.00000 −2.85168
1.3 1.00000 1.00000 1.00000 −2.80221 1.00000 −4.03420 1.00000 1.00000 −2.80221
1.4 1.00000 1.00000 1.00000 −1.58560 1.00000 4.52427 1.00000 1.00000 −1.58560
1.5 1.00000 1.00000 1.00000 −1.52770 1.00000 −0.0681258 1.00000 1.00000 −1.52770
1.6 1.00000 1.00000 1.00000 −0.365306 1.00000 −3.59210 1.00000 1.00000 −0.365306
1.7 1.00000 1.00000 1.00000 0.927745 1.00000 1.16139 1.00000 1.00000 0.927745
1.8 1.00000 1.00000 1.00000 2.31473 1.00000 4.85590 1.00000 1.00000 2.31473
1.9 1.00000 1.00000 1.00000 2.39115 1.00000 −1.97837 1.00000 1.00000 2.39115
1.10 1.00000 1.00000 1.00000 3.49639 1.00000 1.05576 1.00000 1.00000 3.49639
1.11 1.00000 1.00000 1.00000 4.02498 1.00000 −3.74027 1.00000 1.00000 4.02498
1.12 1.00000 1.00000 1.00000 4.43811 1.00000 3.55047 1.00000 1.00000 4.43811
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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 6042.2.a.bg 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.bg 12 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(19\) \(1\)
\(53\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\(T_{5}^{12} - \cdots\)
\(T_{7}^{12} - \cdots\)
\(T_{11}^{12} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{12} \)
$3$ \( ( 1 - T )^{12} \)
$5$ \( 1 - 5 T + 26 T^{2} - 106 T^{3} + 403 T^{4} - 1365 T^{5} + 4314 T^{6} - 12805 T^{7} + 35809 T^{8} - 94766 T^{9} + 238410 T^{10} - 572569 T^{11} + 1307498 T^{12} - 2862845 T^{13} + 5960250 T^{14} - 11845750 T^{15} + 22380625 T^{16} - 40015625 T^{17} + 67406250 T^{18} - 106640625 T^{19} + 157421875 T^{20} - 207031250 T^{21} + 253906250 T^{22} - 244140625 T^{23} + 244140625 T^{24} \)
$7$ \( 1 - 6 T + 42 T^{2} - 182 T^{3} + 842 T^{4} - 3126 T^{5} + 12234 T^{6} - 40841 T^{7} + 139737 T^{8} - 421349 T^{9} + 1289820 T^{10} - 3541930 T^{11} + 9887448 T^{12} - 24793510 T^{13} + 63201180 T^{14} - 144522707 T^{15} + 335508537 T^{16} - 686414687 T^{17} + 1439317866 T^{18} - 2574395418 T^{19} + 4853962442 T^{20} - 7344356474 T^{21} + 11863960458 T^{22} - 11863960458 T^{23} + 13841287201 T^{24} \)
$11$ \( 1 - 7 T + 48 T^{2} - 194 T^{3} + 872 T^{4} - 3090 T^{5} + 12958 T^{6} - 43921 T^{7} + 157899 T^{8} - 471790 T^{9} + 1589506 T^{10} - 4941064 T^{11} + 17160504 T^{12} - 54351704 T^{13} + 192330226 T^{14} - 627952490 T^{15} + 2311799259 T^{16} - 7073520971 T^{17} + 22955887438 T^{18} - 60215358390 T^{19} + 186920944232 T^{20} - 457441852054 T^{21} + 1244996380848 T^{22} - 1997181694277 T^{23} + 3138428376721 T^{24} \)
$13$ \( 1 - 9 T + 105 T^{2} - 654 T^{3} + 4381 T^{4} - 20957 T^{5} + 102874 T^{6} - 396562 T^{7} + 1578777 T^{8} - 5043901 T^{9} + 18041717 T^{10} - 52566721 T^{11} + 205552546 T^{12} - 683367373 T^{13} + 3049050173 T^{14} - 11081450497 T^{15} + 45091449897 T^{16} - 147240694666 T^{17} + 496553149066 T^{18} - 1315020670769 T^{19} + 3573716288701 T^{20} - 6935342589942 T^{21} + 14475141644145 T^{22} - 16129443546333 T^{23} + 23298085122481 T^{24} \)
$17$ \( 1 - 25 T + 401 T^{2} - 4764 T^{3} + 46653 T^{4} - 390107 T^{5} + 2874360 T^{6} - 18940060 T^{7} + 113180029 T^{8} - 617776027 T^{9} + 3101562359 T^{10} - 14365828137 T^{11} + 61574351498 T^{12} - 244219078329 T^{13} + 896351521751 T^{14} - 3035133620651 T^{15} + 9452909202109 T^{16} - 26892176771420 T^{17} + 69380062830840 T^{18} - 160075988708011 T^{19} + 325440011894973 T^{20} - 564952643631708 T^{21} + 808413554080049 T^{22} - 856797407690825 T^{23} + 582622237229761 T^{24} \)
$19$ \( ( 1 + T )^{12} \)
$23$ \( 1 - 22 T + 359 T^{2} - 4313 T^{3} + 43687 T^{4} - 379849 T^{5} + 2940135 T^{6} - 20552284 T^{7} + 131883227 T^{8} - 783898094 T^{9} + 4359914946 T^{10} - 22776567696 T^{11} + 112444041402 T^{12} - 523861057008 T^{13} + 2306395006434 T^{14} - 9537688109698 T^{15} + 36906334126907 T^{16} - 132281549257412 T^{17} + 435245498505015 T^{18} - 1293319541217503 T^{19} + 3421172013971047 T^{20} - 7768371428889919 T^{21} + 14872117525699991 T^{22} - 20961814674106394 T^{23} + 21914624432020321 T^{24} \)
$29$ \( 1 - T + 94 T^{2} - 211 T^{3} + 7101 T^{4} - 10302 T^{5} + 378037 T^{6} - 607867 T^{7} + 16144533 T^{8} - 19970146 T^{9} + 601798241 T^{10} - 699227997 T^{11} + 18215670146 T^{12} - 20277611913 T^{13} + 506112320681 T^{14} - 487051890794 T^{15} + 11418721444773 T^{16} - 12468050609183 T^{17} + 224865223800877 T^{18} - 177708225735318 T^{19} + 3552249778436061 T^{20} - 3061007800908359 T^{21} + 39546479930218894 T^{22} - 12200509765705829 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 - 23 T + 445 T^{2} - 5917 T^{3} + 70703 T^{4} - 698985 T^{5} + 6397901 T^{6} - 51549683 T^{7} + 391319819 T^{8} - 2693480316 T^{9} + 17667357270 T^{10} - 106680145964 T^{11} + 617566877258 T^{12} - 3307084524884 T^{13} + 16978330336470 T^{14} - 80241472093956 T^{15} + 361392070562699 T^{16} - 1475823658609133 T^{17} + 5678160688173581 T^{18} - 19230904574377335 T^{19} + 60301955020191023 T^{20} - 156443244324690307 T^{21} + 364734587706456445 T^{22} - 584394968617311113 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 + T + 153 T^{2} + 172 T^{3} + 14557 T^{4} + 17741 T^{5} + 1027620 T^{6} + 1357490 T^{7} + 57856685 T^{8} + 79884153 T^{9} + 2726402803 T^{10} + 3714046391 T^{11} + 109011214090 T^{12} + 137419716467 T^{13} + 3732445437307 T^{14} + 4046372001909 T^{15} + 108432742616285 T^{16} + 94133728187930 T^{17} + 2636591772416580 T^{18} + 1684186432216553 T^{19} + 51131163410727997 T^{20} + 22353419244753244 T^{21} + 735713408979930897 T^{22} + 177917621779460413 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 + 15 T + 388 T^{2} + 4794 T^{3} + 72092 T^{4} + 745880 T^{5} + 8424369 T^{6} + 74587795 T^{7} + 691255809 T^{8} + 5325124642 T^{9} + 42167491899 T^{10} + 285108637746 T^{11} + 1967933018180 T^{12} + 11689454147586 T^{13} + 70883553882219 T^{14} + 367012915451282 T^{15} + 1953323706095649 T^{16} + 8641458569666795 T^{17} + 40016630914648929 T^{18} + 145263317802360280 T^{19} + 575649229617791132 T^{20} + 1569468993484649034 T^{21} + 5207991812339131588 T^{22} + 8254935475743726615 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 - 2 T + 233 T^{2} - 287 T^{3} + 25751 T^{4} - 2581 T^{5} + 1822165 T^{6} + 2603366 T^{7} + 94853259 T^{8} + 308963208 T^{9} + 4122552186 T^{10} + 19951457638 T^{11} + 173311159930 T^{12} + 857912678434 T^{13} + 7622598991914 T^{14} + 24564737778456 T^{15} + 324284416722459 T^{16} + 382716782219138 T^{17} + 11518566500181085 T^{18} - 701563835267167 T^{19} + 300982845348503351 T^{20} - 144244079625873941 T^{21} + 5035475378995230017 T^{22} - 1858587478942445414 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 - 11 T + 376 T^{2} - 3620 T^{3} + 66852 T^{4} - 558038 T^{5} + 7433127 T^{6} - 54174315 T^{7} + 586439367 T^{8} - 3799206258 T^{9} + 35897346273 T^{10} - 211671258188 T^{11} + 1825396231912 T^{12} - 9948549134836 T^{13} + 79297237917057 T^{14} - 394444991324334 T^{15} + 2861637036801927 T^{16} - 12424608652895205 T^{17} + 80123276500803783 T^{18} - 282714952896931594 T^{19} + 1591832135912046372 T^{20} - 4051252312632016540 T^{21} + 19777273720672098424 T^{22} - 27193751365924135333 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( ( 1 - T )^{12} \)
$59$ \( 1 - 3 T + 248 T^{2} - 369 T^{3} + 39031 T^{4} - 19410 T^{5} + 4366205 T^{6} + 1216027 T^{7} + 397020031 T^{8} + 288715740 T^{9} + 29711899243 T^{10} + 27402516177 T^{11} + 1904841582242 T^{12} + 1616748454443 T^{13} + 103427121264883 T^{14} + 59296149965460 T^{15} + 4810835039858191 T^{16} + 869367250540073 T^{17} + 184168856886002405 T^{18} - 48304725320336790 T^{19} + 5730938810134252951 T^{20} - 3196645457083672491 T^{21} + \)\(12\!\cdots\!48\)\( T^{22} - 90467665334213527977 T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 - 16 T + 524 T^{2} - 6836 T^{3} + 129571 T^{4} - 1439034 T^{5} + 20509529 T^{6} - 199095192 T^{7} + 2345272305 T^{8} - 20188893860 T^{9} + 205080277363 T^{10} - 1573519403630 T^{11} + 14070845097590 T^{12} - 95984683621430 T^{13} + 763103712067723 T^{14} - 4582495317236660 T^{15} + 32472267436733505 T^{16} - 168155062710084792 T^{17} + 1056658612047785969 T^{18} - 4522513794290643714 T^{19} + 24839708252370696451 T^{20} - 79941182690614187876 T^{21} + \)\(37\!\cdots\!24\)\( T^{22} - \)\(69\!\cdots\!76\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 + 2 T + 441 T^{2} + 807 T^{3} + 100313 T^{4} + 185435 T^{5} + 15467773 T^{6} + 28350280 T^{7} + 1794818515 T^{8} + 3152450438 T^{9} + 164806364754 T^{10} + 268293161168 T^{11} + 12238599670182 T^{12} + 17975641798256 T^{13} + 739815771380706 T^{14} + 948140451084194 T^{15} + 36167605068805315 T^{16} + 38276424818479960 T^{17} + 1399189721337339637 T^{18} + 1123868056533070505 T^{19} + 40733866938739328633 T^{20} + 21955673257810022229 T^{21} + \)\(80\!\cdots\!09\)\( T^{22} + \)\(24\!\cdots\!66\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 + 4 T + 295 T^{2} + 1171 T^{3} + 48916 T^{4} + 157817 T^{5} + 5913703 T^{6} + 16676588 T^{7} + 596288815 T^{8} + 1528431904 T^{9} + 52773216194 T^{10} + 126752953158 T^{11} + 4029652229688 T^{12} + 8999459674218 T^{13} + 266029782833954 T^{14} + 547042591192544 T^{15} + 15152701150648015 T^{16} + 30088389544134388 T^{17} + 757547033324469463 T^{18} + 1435364578036792447 T^{19} + 31587679734417645076 T^{20} + 53688594341303815301 T^{21} + \)\(96\!\cdots\!95\)\( T^{22} + \)\(92\!\cdots\!84\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 - 35 T + 1022 T^{2} - 19802 T^{3} + 337228 T^{4} - 4649414 T^{5} + 59031846 T^{6} - 663910779 T^{7} + 7186192935 T^{8} - 72635374546 T^{9} + 715614460732 T^{10} - 6614507243496 T^{11} + 58598570501080 T^{12} - 482859028775208 T^{13} + 3813509461240828 T^{14} - 28256395499761282 T^{15} + 204075238840627335 T^{16} - 1376334576231400947 T^{17} + 8933538740821399494 T^{18} - 51363929338268859158 T^{19} + \)\(27\!\cdots\!68\)\( T^{20} - \)\(11\!\cdots\!26\)\( T^{21} + \)\(43\!\cdots\!78\)\( T^{22} - \)\(10\!\cdots\!95\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 - 4 T + 426 T^{2} - 1191 T^{3} + 93776 T^{4} - 169540 T^{5} + 13766285 T^{6} - 9364565 T^{7} + 1510680577 T^{8} + 618883155 T^{9} + 136819141585 T^{10} + 159025960167 T^{11} + 11137472530060 T^{12} + 12563050853193 T^{13} + 853888262631985 T^{14} + 305133531858045 T^{15} + 58841130839276737 T^{16} - 28815294657101435 T^{17} + 3346411192626909485 T^{18} - 3255830729513396860 T^{19} + \)\(14\!\cdots\!36\)\( T^{20} - \)\(14\!\cdots\!29\)\( T^{21} + \)\(40\!\cdots\!26\)\( T^{22} - \)\(29\!\cdots\!16\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 - 39 T + 1409 T^{2} - 34372 T^{3} + 751607 T^{4} - 13605761 T^{5} + 223273646 T^{6} - 3231604216 T^{7} + 42940334829 T^{8} - 516698001793 T^{9} + 5759209830497 T^{10} - 58811067007505 T^{11} + 558833153717942 T^{12} - 4881318561622915 T^{13} + 39675196522293833 T^{14} - 295441200351214091 T^{15} + 2037876194162162109 T^{16} - 12729420348914150888 T^{17} + 72997169186697933374 T^{18} - \)\(36\!\cdots\!47\)\( T^{19} + \)\(16\!\cdots\!87\)\( T^{20} - \)\(64\!\cdots\!16\)\( T^{21} + \)\(21\!\cdots\!41\)\( T^{22} - \)\(50\!\cdots\!13\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 - 11 T + 813 T^{2} - 8213 T^{3} + 318501 T^{4} - 2929739 T^{5} + 79424657 T^{6} - 661176333 T^{7} + 14041054675 T^{8} - 105095625576 T^{9} + 1854779740530 T^{10} - 12370218718928 T^{11} + 187815033681166 T^{12} - 1100949465984592 T^{13} + 14691710324738130 T^{14} - 74089158066687144 T^{15} + 880967236313026675 T^{16} - 3692047949743820517 T^{17} + 39472568569994625377 T^{18} - \)\(12\!\cdots\!31\)\( T^{19} + \)\(12\!\cdots\!81\)\( T^{20} - \)\(28\!\cdots\!17\)\( T^{21} + \)\(25\!\cdots\!13\)\( T^{22} - \)\(30\!\cdots\!79\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 + 21 T + 608 T^{2} + 11944 T^{3} + 214638 T^{4} + 3522096 T^{5} + 51788336 T^{6} + 716763737 T^{7} + 9197836255 T^{8} + 110748578026 T^{9} + 1260503811952 T^{10} + 13463903393776 T^{11} + 136885072090724 T^{12} + 1305998629196272 T^{13} + 11860080366656368 T^{14} + 101077236952723498 T^{15} + 814277830410882655 T^{16} + 6155094093497860409 T^{17} + 43138234069856708144 T^{18} + \)\(28\!\cdots\!48\)\( T^{19} + \)\(16\!\cdots\!18\)\( T^{20} + \)\(90\!\cdots\!48\)\( T^{21} + \)\(44\!\cdots\!92\)\( T^{22} + \)\(15\!\cdots\!13\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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