Properties

Label 8048.2.a.q
Level 8048
Weight 2
Character orbit 8048.a
Self dual yes
Analytic conductor 64.264
Analytic rank 1
Dimension 12
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8048 = 2^{4} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8048.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.2636035467\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 5 x^{11} - 13 x^{10} + 94 x^{9} + 15 x^{8} - 616 x^{7} + 368 x^{6} + 1643 x^{5} - 1463 x^{4} - 1556 x^{3} + 1284 x^{2} + 576 x - 208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1006)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} -\beta_{9} q^{5} + ( -1 - \beta_{10} ) q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} -\beta_{9} q^{5} + ( -1 - \beta_{10} ) q^{7} + ( 1 + \beta_{2} ) q^{9} + ( -2 + \beta_{8} ) q^{11} + \beta_{11} q^{13} + ( 1 + \beta_{3} + \beta_{4} + \beta_{10} ) q^{15} + ( 1 - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{17} + ( -1 + \beta_{6} ) q^{19} + ( \beta_{1} - \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{21} + ( -1 - \beta_{3} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{23} + ( -\beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{25} + ( -1 - \beta_{4} - \beta_{5} - \beta_{6} ) q^{27} + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{29} + ( -1 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{8} + \beta_{10} ) q^{31} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} - 3 \beta_{8} + \beta_{10} + \beta_{11} ) q^{33} + ( 2 \beta_{1} + \beta_{5} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{35} + ( 2 + \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{37} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{39} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{10} + \beta_{11} ) q^{41} + ( \beta_{1} + \beta_{2} - 2 \beta_{6} + 2 \beta_{10} + \beta_{11} ) q^{43} + ( -\beta_{2} + \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{45} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{47} + ( 1 + \beta_{1} + 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{49} + ( \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{51} + ( 1 - \beta_{3} - \beta_{5} - \beta_{8} + \beta_{10} ) q^{53} + ( -1 - \beta_{2} - \beta_{5} + 2 \beta_{8} + 2 \beta_{9} ) q^{55} + ( -\beta_{8} + 2 \beta_{10} + \beta_{11} ) q^{57} + ( -1 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - \beta_{6} - 4 \beta_{8} + \beta_{10} + \beta_{11} ) q^{59} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{61} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{7} - 2 \beta_{10} ) q^{63} + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} ) q^{65} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} - \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{67} + ( \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{69} + ( -2 - 3 \beta_{2} + \beta_{4} + 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} ) q^{71} + ( -2 - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{6} + 2 \beta_{8} - 2 \beta_{10} - \beta_{11} ) q^{73} + ( 4 + \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{75} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{77} + ( -2 + \beta_{3} - \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{11} ) q^{79} + ( -3 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{8} - \beta_{10} ) q^{81} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{8} - \beta_{10} - \beta_{11} ) q^{83} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{11} ) q^{85} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{87} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{10} ) q^{89} + ( 3 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} - 3 \beta_{11} ) q^{91} + ( -3 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{93} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{5} + \beta_{8} ) q^{95} + ( -7 + \beta_{1} - \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{97} + ( -\beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + 4 \beta_{8} + \beta_{9} - \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 5q^{3} + 5q^{5} - 8q^{7} + 15q^{9} + O(q^{10}) \) \( 12q - 5q^{3} + 5q^{5} - 8q^{7} + 15q^{9} - 18q^{11} + 4q^{13} - 2q^{15} - 2q^{17} - 6q^{19} + q^{21} - 13q^{23} + q^{25} - 8q^{27} + 20q^{29} - 7q^{31} - 8q^{33} - q^{35} + 10q^{37} - 7q^{39} + 2q^{41} - 8q^{43} - 7q^{45} - 12q^{47} + 4q^{49} - 2q^{51} + 12q^{53} - 8q^{55} - 10q^{57} - 6q^{59} - 10q^{61} - 7q^{63} + 4q^{65} - 7q^{67} - 12q^{69} - 22q^{71} - 23q^{73} + 34q^{75} - 19q^{77} - 13q^{79} - 28q^{81} - q^{83} - 28q^{85} + 22q^{87} - 3q^{89} + 21q^{91} - 33q^{93} + 2q^{95} - 70q^{97} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 5 x^{11} - 13 x^{10} + 94 x^{9} + 15 x^{8} - 616 x^{7} + 368 x^{6} + 1643 x^{5} - 1463 x^{4} - 1556 x^{3} + 1284 x^{2} + 576 x - 208\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{11} + 543 \nu^{10} - 917 \nu^{9} - 9770 \nu^{8} + 14883 \nu^{7} + 57076 \nu^{6} - 82700 \nu^{5} - 116237 \nu^{4} + 172813 \nu^{3} + 42460 \nu^{2} - 79464 \nu + 1864 \)\()/7192\)
\(\beta_{4}\)\(=\)\((\)\( 115 \nu^{11} - 485 \nu^{10} - 2969 \nu^{9} + 10988 \nu^{8} + 28617 \nu^{7} - 87062 \nu^{6} - 124940 \nu^{5} + 283161 \nu^{4} + 244729 \nu^{3} - 325906 \nu^{2} - 184320 \nu + 56136 \)\()/7192\)
\(\beta_{5}\)\(=\)\((\)\( 577 \nu^{11} - 1339 \nu^{10} - 11285 \nu^{9} + 24612 \nu^{8} + 77557 \nu^{7} - 157540 \nu^{6} - 221932 \nu^{5} + 406595 \nu^{4} + 245943 \nu^{3} - 348940 \nu^{2} - 127588 \nu + 65052 \)\()/3596\)
\(\beta_{6}\)\(=\)\((\)\( -1269 \nu^{11} + 3163 \nu^{10} + 25539 \nu^{9} - 60212 \nu^{8} - 183731 \nu^{7} + 402142 \nu^{6} + 568804 \nu^{5} - 1096351 \nu^{4} - 729423 \nu^{3} + 1023786 \nu^{2} + 396344 \nu - 193432 \)\()/7192\)
\(\beta_{7}\)\(=\)\((\)\( -1533 \nu^{11} + 3651 \nu^{10} + 30291 \nu^{9} - 66456 \nu^{8} - 214779 \nu^{7} + 419298 \nu^{6} + 663784 \nu^{5} - 1062087 \nu^{4} - 869847 \nu^{3} + 882834 \nu^{2} + 471292 \nu - 131744 \)\()/7192\)
\(\beta_{8}\)\(=\)\((\)\( 969 \nu^{11} - 2445 \nu^{10} - 18341 \nu^{9} + 44290 \nu^{8} + 120335 \nu^{7} - 276728 \nu^{6} - 319484 \nu^{5} + 683499 \nu^{4} + 302929 \nu^{3} - 523112 \nu^{2} - 136116 \nu + 72944 \)\()/3596\)
\(\beta_{9}\)\(=\)\((\)\( 1163 \nu^{11} - 3185 \nu^{10} - 21833 \nu^{9} + 58386 \nu^{8} + 141625 \nu^{7} - 369564 \nu^{6} - 370074 \nu^{5} + 924969 \nu^{4} + 351689 \nu^{3} - 713098 \nu^{2} - 199010 \nu + 85748 \)\()/3596\)
\(\beta_{10}\)\(=\)\((\)\( 2572 \nu^{11} - 6743 \nu^{10} - 48993 \nu^{9} + 123571 \nu^{8} + 325094 \nu^{7} - 783065 \nu^{6} - 882020 \nu^{5} + 1969696 \nu^{4} + 887759 \nu^{3} - 1550579 \nu^{2} - 452248 \nu + 209308 \)\()/3596\)
\(\beta_{11}\)\(=\)\((\)\( -646 \nu^{11} + 1630 \nu^{10} + 12527 \nu^{9} - 30126 \nu^{8} - 85018 \nu^{7} + 192876 \nu^{6} + 237562 \nu^{5} - 490727 \nu^{4} - 249300 \nu^{3} + 391294 \nu^{2} + 125805 \nu - 53424 \)\()/899\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{4} + 6 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{10} + 2 \beta_{8} + \beta_{5} + \beta_{4} - \beta_{3} + 8 \beta_{2} + 24\)
\(\nu^{5}\)\(=\)\(-3 \beta_{9} + 6 \beta_{8} + \beta_{7} + 12 \beta_{6} + 9 \beta_{5} + 15 \beta_{4} + 38 \beta_{1} + 11\)
\(\nu^{6}\)\(=\)\(-2 \beta_{11} - 18 \beta_{10} + 39 \beta_{8} + 6 \beta_{6} + 10 \beta_{5} + 24 \beta_{4} - 12 \beta_{3} + 57 \beta_{2} - \beta_{1} + 162\)
\(\nu^{7}\)\(=\)\(-\beta_{11} - 9 \beta_{10} - 42 \beta_{9} + 111 \beta_{8} + 14 \beta_{7} + 124 \beta_{6} + 71 \beta_{5} + 179 \beta_{4} + \beta_{3} - 2 \beta_{2} + 252 \beta_{1} + 116\)
\(\nu^{8}\)\(=\)\(-32 \beta_{11} - 220 \beta_{10} - 7 \beta_{9} + 517 \beta_{8} + 7 \beta_{7} + 116 \beta_{6} + 85 \beta_{5} + 353 \beta_{4} - 110 \beta_{3} + 407 \beta_{2} - 12 \beta_{1} + 1195\)
\(\nu^{9}\)\(=\)\(-21 \beta_{11} - 197 \beta_{10} - 440 \beta_{9} + 1468 \beta_{8} + 156 \beta_{7} + 1222 \beta_{6} + 563 \beta_{5} + 1943 \beta_{4} + 15 \beta_{3} - 20 \beta_{2} + 1751 \beta_{1} + 1222\)
\(\nu^{10}\)\(=\)\(-373 \beta_{11} - 2361 \beta_{10} - 167 \beta_{9} + 5950 \beta_{8} + 155 \beta_{7} + 1609 \beta_{6} + 741 \beta_{5} + 4347 \beta_{4} - 921 \beta_{3} + 2987 \beta_{2} - 60 \beta_{1} + 9512\)
\(\nu^{11}\)\(=\)\(-323 \beta_{11} - 2948 \beta_{10} - 4203 \beta_{9} + 17093 \beta_{8} + 1615 \beta_{7} + 11846 \beta_{6} + 4629 \beta_{5} + 20163 \beta_{4} + 142 \beta_{3} - 21 \beta_{2} + 12753 \beta_{1} + 12839\)

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
3.16835
2.49390
2.44300
2.31191
1.74625
1.29494
0.264006
−0.665271
−0.867509
−2.17256
−2.25121
−2.76581
0 −3.16835 0 0.310718 0 −0.631809 0 7.03843 0
1.2 0 −2.49390 0 1.94778 0 1.87622 0 3.21954 0
1.3 0 −2.44300 0 −1.37594 0 −4.82526 0 2.96827 0
1.4 0 −2.31191 0 0.313584 0 2.05708 0 2.34493 0
1.5 0 −1.74625 0 −2.58578 0 −0.635213 0 0.0494057 0
1.6 0 −1.29494 0 3.51641 0 −1.29880 0 −1.32313 0
1.7 0 −0.264006 0 1.70058 0 −2.77338 0 −2.93030 0
1.8 0 0.665271 0 2.52521 0 3.86663 0 −2.55741 0
1.9 0 0.867509 0 −3.26212 0 −1.33041 0 −2.24743 0
1.10 0 2.17256 0 2.95208 0 −2.27467 0 1.72004 0
1.11 0 2.25121 0 1.33593 0 −4.39363 0 2.06796 0
1.12 0 2.76581 0 −2.37847 0 2.36325 0 4.64970 0
\(n\): 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 8048.2.a.q 12
4.b odd 2 1 1006.2.a.j 12
12.b even 2 1 9054.2.a.bi 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1006.2.a.j 12 4.b odd 2 1
8048.2.a.q 12 1.a even 1 1 trivial
9054.2.a.bi 12 12.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(503\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8048))\):

\(T_{3}^{12} + \cdots\)
\(T_{5}^{12} - \cdots\)
\(T_{7}^{12} + \cdots\)
\(T_{13}^{12} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 5 T + 23 T^{2} + 71 T^{3} + 219 T^{4} + 553 T^{5} + 1403 T^{6} + 3112 T^{7} + 6916 T^{8} + 13793 T^{9} + 27414 T^{10} + 49644 T^{11} + 89792 T^{12} + 148932 T^{13} + 246726 T^{14} + 372411 T^{15} + 560196 T^{16} + 756216 T^{17} + 1022787 T^{18} + 1209411 T^{19} + 1436859 T^{20} + 1397493 T^{21} + 1358127 T^{22} + 885735 T^{23} + 531441 T^{24} \)
$5$ \( 1 - 5 T + 42 T^{2} - 154 T^{3} + 791 T^{4} - 2412 T^{5} + 9636 T^{6} - 25656 T^{7} + 86256 T^{8} - 204402 T^{9} + 600308 T^{10} - 1277475 T^{11} + 3342092 T^{12} - 6387375 T^{13} + 15007700 T^{14} - 25550250 T^{15} + 53910000 T^{16} - 80175000 T^{17} + 150562500 T^{18} - 188437500 T^{19} + 308984375 T^{20} - 300781250 T^{21} + 410156250 T^{22} - 244140625 T^{23} + 244140625 T^{24} \)
$7$ \( 1 + 8 T + 72 T^{2} + 399 T^{3} + 2199 T^{4} + 9669 T^{5} + 41150 T^{6} + 152258 T^{7} + 542867 T^{8} + 1744965 T^{9} + 5421535 T^{10} + 15427129 T^{11} + 42558007 T^{12} + 107989903 T^{13} + 265655215 T^{14} + 598522995 T^{15} + 1303423667 T^{16} + 2559000206 T^{17} + 4841256350 T^{18} + 7962837267 T^{19} + 12676797399 T^{20} + 16101089193 T^{21} + 20338217928 T^{22} + 15818613944 T^{23} + 13841287201 T^{24} \)
$11$ \( 1 + 18 T + 236 T^{2} + 2219 T^{3} + 17491 T^{4} + 115780 T^{5} + 677529 T^{6} + 3514943 T^{7} + 16587530 T^{8} + 71330085 T^{9} + 283931035 T^{10} + 1045560197 T^{11} + 3596181460 T^{12} + 11501162167 T^{13} + 34355655235 T^{14} + 94940343135 T^{15} + 242858026730 T^{16} + 566085085093 T^{17} + 1200283952769 T^{18} + 2256224658380 T^{19} + 3749351187571 T^{20} + 5232285926329 T^{21} + 6121232205836 T^{22} + 5135610070998 T^{23} + 3138428376721 T^{24} \)
$13$ \( 1 - 4 T + 95 T^{2} - 278 T^{3} + 4083 T^{4} - 8607 T^{5} + 110337 T^{6} - 159728 T^{7} + 2185482 T^{8} - 2023466 T^{9} + 34878800 T^{10} - 21192537 T^{11} + 480640372 T^{12} - 275502981 T^{13} + 5894517200 T^{14} - 4445554802 T^{15} + 62419551402 T^{16} - 59305888304 T^{17} + 532575624633 T^{18} - 540076485819 T^{19} + 3330628533843 T^{20} - 2948050825694 T^{21} + 13096556725655 T^{22} - 7168641576148 T^{23} + 23298085122481 T^{24} \)
$17$ \( 1 + 2 T + 86 T^{2} + 211 T^{3} + 3520 T^{4} + 11022 T^{5} + 97722 T^{6} + 347875 T^{7} + 2230039 T^{8} + 7425255 T^{9} + 46049968 T^{10} + 127773955 T^{11} + 844910160 T^{12} + 2172157235 T^{13} + 13308440752 T^{14} + 36480277815 T^{15} + 186255087319 T^{16} + 493932753875 T^{17} + 2358771517818 T^{18} + 4522752853806 T^{19} + 24554666192320 T^{20} + 25022041940867 T^{21} + 173375475438614 T^{22} + 68543792615266 T^{23} + 582622237229761 T^{24} \)
$19$ \( 1 + 6 T + 179 T^{2} + 1019 T^{3} + 15567 T^{4} + 81493 T^{5} + 860771 T^{6} + 4072782 T^{7} + 33490409 T^{8} + 141658777 T^{9} + 962981822 T^{10} + 3604155843 T^{11} + 20956227758 T^{12} + 68478961017 T^{13} + 347636437742 T^{14} + 971637551443 T^{15} + 4364503591289 T^{16} + 10084611437418 T^{17} + 40495730034251 T^{18} + 72844289626327 T^{19} + 264383125859247 T^{20} + 328818764036801 T^{21} + 1097460860146379 T^{22} + 698941553389314 T^{23} + 2213314919066161 T^{24} \)
$23$ \( 1 + 13 T + 237 T^{2} + 2002 T^{3} + 21372 T^{4} + 133225 T^{5} + 1071017 T^{6} + 5181303 T^{7} + 35330634 T^{8} + 137041097 T^{9} + 885747404 T^{10} + 2981518951 T^{11} + 20222903567 T^{12} + 68574935873 T^{13} + 468560376716 T^{14} + 1667379027199 T^{15} + 9886959949194 T^{16} + 33348643294929 T^{17} + 158548953729113 T^{18} + 453607870176575 T^{19} + 1673662377425532 T^{20} + 3605907628248926 T^{21} + 9818083157634813 T^{22} + 12386526852881051 T^{23} + 21914624432020321 T^{24} \)
$29$ \( 1 - 20 T + 399 T^{2} - 4927 T^{3} + 59113 T^{4} - 549200 T^{5} + 4994143 T^{6} - 38035759 T^{7} + 285978915 T^{8} - 1866713053 T^{9} + 12107702506 T^{10} - 69364622861 T^{11} + 396392131942 T^{12} - 2011574062969 T^{13} + 10182577807546 T^{14} - 45527264649617 T^{15} + 202267452980115 T^{16} - 780157120177091 T^{17} + 2970632724808903 T^{18} - 9473632068902800 T^{19} + 29571066209363593 T^{20} - 71476708223106563 T^{21} + 167862186086780199 T^{22} - 244010195314116580 T^{23} + 353814783205469041 T^{24} \)
$31$ \( 1 + 7 T + 282 T^{2} + 1830 T^{3} + 38139 T^{4} + 230297 T^{5} + 3280516 T^{6} + 18364168 T^{7} + 200366767 T^{8} + 1028922293 T^{9} + 9171046402 T^{10} + 42467891207 T^{11} + 322887365354 T^{12} + 1316504627417 T^{13} + 8813375592322 T^{14} + 30652624030763 T^{15} + 185042917026607 T^{16} + 525750538661368 T^{17} + 2911470025579396 T^{18} + 6336072491920967 T^{19} + 32528411276962299 T^{20} + 48384508554027930 T^{21} + 231135176928585882 T^{22} + 177859338274833817 T^{23} + 787662783788549761 T^{24} \)
$37$ \( 1 - 10 T + 243 T^{2} - 1891 T^{3} + 28373 T^{4} - 204733 T^{5} + 2343397 T^{6} - 15984122 T^{7} + 146808103 T^{8} - 936951483 T^{9} + 7321030232 T^{10} - 43550538569 T^{11} + 299802706494 T^{12} - 1611369927053 T^{13} + 10022490387608 T^{14} - 47459403468399 T^{15} + 275142021126583 T^{16} - 1108402268650754 T^{17} + 6012515569671373 T^{18} - 19435688001070489 T^{19} + 99659579546100533 T^{20} - 245757649952490607 T^{21} + 1168486002497537307 T^{22} - 1779176217794604130 T^{23} + 6582952005840035281 T^{24} \)
$41$ \( 1 - 2 T + 310 T^{2} - 785 T^{3} + 47098 T^{4} - 138456 T^{5} + 4673492 T^{6} - 14922727 T^{7} + 340248619 T^{8} - 1111165915 T^{9} + 19293798486 T^{10} - 60634376507 T^{11} + 878545992372 T^{12} - 2486009436787 T^{13} + 32432875254966 T^{14} - 76582666027715 T^{15} + 961461277874059 T^{16} - 1728890458780127 T^{17} + 22199574169479572 T^{18} - 26964897744467736 T^{19} + 376074008441140858 T^{20} - 256994818499259385 T^{21} + 4161024386147244310 T^{22} - 1100658063432496882 T^{23} + 22563490300366186081 T^{24} \)
$43$ \( 1 + 8 T + 333 T^{2} + 1866 T^{3} + 49961 T^{4} + 201111 T^{5} + 4798493 T^{6} + 14405750 T^{7} + 347174076 T^{8} + 838886304 T^{9} + 20293592746 T^{10} + 42545902371 T^{11} + 968100841052 T^{12} + 1829473801953 T^{13} + 37522852987354 T^{14} + 66697333372128 T^{15} + 1186919078202876 T^{16} + 2117766877747250 T^{17} + 30333016341085157 T^{18} + 54665712698339877 T^{19} + 583954174069223561 T^{20} + 937837813874149038 T^{21} + 7196623610323654917 T^{22} + 7434349915769781656 T^{23} + 39959630797262576401 T^{24} \)
$47$ \( 1 + 12 T + 302 T^{2} + 3187 T^{3} + 47279 T^{4} + 415791 T^{5} + 4717830 T^{6} + 35652736 T^{7} + 339864289 T^{8} + 2268004051 T^{9} + 19491144185 T^{10} + 119470683497 T^{11} + 965417437421 T^{12} + 5615122124359 T^{13} + 43055937504665 T^{14} + 235470984586973 T^{15} + 1658429313611809 T^{16} + 8176776987489152 T^{17} + 50854505455616070 T^{18} + 210649333880431233 T^{19} + 1125773822081398319 T^{20} + 3566668817778518429 T^{21} + 15884937935220674798 T^{22} + 29665910581008147636 T^{23} + \)\(11\!\cdots\!41\)\( T^{24} \)
$53$ \( 1 - 12 T + 545 T^{2} - 5561 T^{3} + 137317 T^{4} - 1216285 T^{5} + 21437635 T^{6} - 166788154 T^{7} + 2329758205 T^{8} - 15995580255 T^{9} + 186586431716 T^{10} - 1128590015573 T^{11} + 11317232997794 T^{12} - 59815270825369 T^{13} + 524121286690244 T^{14} - 2381374001623635 T^{15} + 18382912851146605 T^{16} - 69750054288589922 T^{17} + 475151483891689915 T^{18} - 1428783538716645545 T^{19} + 8549313908216858437 T^{20} - 18349985334011661613 T^{21} + 95313671349204611705 T^{22} - \)\(11\!\cdots\!64\)\( T^{23} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( 1 + 6 T + 277 T^{2} + 1124 T^{3} + 34762 T^{4} + 84929 T^{5} + 3002529 T^{6} + 4076375 T^{7} + 229711951 T^{8} + 229401713 T^{9} + 16358116930 T^{10} + 15943411277 T^{11} + 1033672281260 T^{12} + 940661265343 T^{13} + 56942605033330 T^{14} + 47114294414227 T^{15} + 2783502636281311 T^{16} + 2914299539336125 T^{17} + 126648275492578089 T^{18} + 211358681954192851 T^{19} + 5104119672001406602 T^{20} + 9737207300168151436 T^{21} + \)\(14\!\cdots\!77\)\( T^{22} + \)\(18\!\cdots\!54\)\( T^{23} + \)\(17\!\cdots\!81\)\( T^{24} \)
$61$ \( 1 + 10 T + 365 T^{2} + 3179 T^{3} + 61581 T^{4} + 449878 T^{5} + 6491041 T^{6} + 40615545 T^{7} + 512777084 T^{8} + 2961728657 T^{9} + 35403589926 T^{10} + 197962455527 T^{11} + 2258050315556 T^{12} + 12075709787147 T^{13} + 131736758114646 T^{14} + 672256132294517 T^{15} + 7099829973507644 T^{16} + 34303739070099045 T^{17} + 334420862312599801 T^{18} + 1413850861583455438 T^{19} + 11805528041685561261 T^{20} + 37175690429119734239 T^{21} + \)\(26\!\cdots\!65\)\( T^{22} + \)\(43\!\cdots\!10\)\( T^{23} + \)\(26\!\cdots\!21\)\( T^{24} \)
$67$ \( 1 + 7 T + 554 T^{2} + 3317 T^{3} + 145357 T^{4} + 743955 T^{5} + 24141915 T^{6} + 105743608 T^{7} + 2874222370 T^{8} + 10870454891 T^{9} + 263904792235 T^{10} + 880117558364 T^{11} + 19537385096368 T^{12} + 58967876410388 T^{13} + 1184668612342915 T^{14} + 3269430624381833 T^{15} + 57918802758776770 T^{16} + 142767100065566056 T^{17} + 2183838573361513635 T^{18} + 4508896702338072465 T^{19} + 59024779406600665837 T^{20} + 90244074592510339199 T^{21} + \)\(10\!\cdots\!46\)\( T^{22} + \)\(85\!\cdots\!81\)\( T^{23} + \)\(81\!\cdots\!61\)\( T^{24} \)
$71$ \( 1 + 22 T + 767 T^{2} + 12443 T^{3} + 245151 T^{4} + 3126102 T^{5} + 44600111 T^{6} + 466232143 T^{7} + 5338265595 T^{8} + 47751004997 T^{9} + 474162762402 T^{10} + 3842492624367 T^{11} + 35488548426314 T^{12} + 272816976330057 T^{13} + 2390254485268482 T^{14} + 17090609949481267 T^{15} + 135654302393415195 T^{16} + 841189716780229193 T^{17} + 5713286882008115231 T^{18} + 28432273317386421882 T^{19} + \)\(15\!\cdots\!11\)\( T^{20} + \)\(57\!\cdots\!33\)\( T^{21} + \)\(24\!\cdots\!67\)\( T^{22} + \)\(50\!\cdots\!62\)\( T^{23} + \)\(16\!\cdots\!41\)\( T^{24} \)
$73$ \( 1 + 23 T + 914 T^{2} + 15862 T^{3} + 359607 T^{4} + 5033434 T^{5} + 83581420 T^{6} + 980185074 T^{7} + 13078758902 T^{8} + 131435921598 T^{9} + 1474371678576 T^{10} + 12844486325361 T^{11} + 123956533139464 T^{12} + 937647501751353 T^{13} + 7856926675131504 T^{14} + 51130807912289166 T^{15} + 371413747279891382 T^{16} + 2031993832792002882 T^{17} + 12648729527835950380 T^{18} + 55606351317572489098 T^{19} + \)\(29\!\cdots\!67\)\( T^{20} + \)\(93\!\cdots\!06\)\( T^{21} + \)\(39\!\cdots\!86\)\( T^{22} + \)\(72\!\cdots\!71\)\( T^{23} + \)\(22\!\cdots\!21\)\( T^{24} \)
$79$ \( 1 + 13 T + 740 T^{2} + 9277 T^{3} + 264468 T^{4} + 3122234 T^{5} + 60133963 T^{6} + 655134600 T^{7} + 9654177811 T^{8} + 95352923720 T^{9} + 1149494991777 T^{10} + 10118852614358 T^{11} + 103879296399120 T^{12} + 799389356534282 T^{13} + 7173998243680257 T^{14} + 47012710157985080 T^{15} + 376031007726852691 T^{16} + 2015886113136305400 T^{17} + 14617812056063959723 T^{18} + 59959097569491159206 T^{19} + \)\(40\!\cdots\!48\)\( T^{20} + \)\(11\!\cdots\!63\)\( T^{21} + \)\(70\!\cdots\!40\)\( T^{22} + \)\(97\!\cdots\!27\)\( T^{23} + \)\(59\!\cdots\!41\)\( T^{24} \)
$83$ \( 1 + T + 591 T^{2} - 693 T^{3} + 167307 T^{4} - 517795 T^{5} + 30927255 T^{6} - 144179528 T^{7} + 4250890052 T^{8} - 24232595755 T^{9} + 465398797158 T^{10} - 2800671733828 T^{11} + 42176704430232 T^{12} - 232455753907724 T^{13} + 3206132313621462 T^{14} - 13855883228964185 T^{15} + 201740104623522692 T^{16} - 567929020680556504 T^{17} + 10111368296978272095 T^{18} - 14050911522173912465 T^{19} + \)\(37\!\cdots\!87\)\( T^{20} - \)\(12\!\cdots\!79\)\( T^{21} + \)\(91\!\cdots\!59\)\( T^{22} + \)\(12\!\cdots\!67\)\( T^{23} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( 1 + 3 T + 559 T^{2} + 2334 T^{3} + 165630 T^{4} + 762205 T^{5} + 33875739 T^{6} + 156779452 T^{7} + 5245514383 T^{8} + 23317169957 T^{9} + 641955466630 T^{10} + 2655079688665 T^{11} + 63398187090724 T^{12} + 236302092291185 T^{13} + 5084929251176230 T^{14} + 16437881987416333 T^{15} + 329115327587152303 T^{16} + 875465780349641948 T^{17} + 16835608500477895179 T^{18} + 33713344614046681445 T^{19} + \)\(65\!\cdots\!30\)\( T^{20} + \)\(81\!\cdots\!06\)\( T^{21} + \)\(17\!\cdots\!59\)\( T^{22} + \)\(83\!\cdots\!67\)\( T^{23} + \)\(24\!\cdots\!21\)\( T^{24} \)
$97$ \( 1 + 70 T + 2935 T^{2} + 87469 T^{3} + 2060013 T^{4} + 39911519 T^{5} + 658581287 T^{6} + 9431186506 T^{7} + 119768221401 T^{8} + 1373580747687 T^{9} + 14608461788988 T^{10} + 147941064750717 T^{11} + 1466383547078142 T^{12} + 14350283280819549 T^{13} + 137451016972588092 T^{14} + 1253630061733737351 T^{15} + 10602994527279342681 T^{16} + 80988807554248972042 T^{17} + \)\(54\!\cdots\!23\)\( T^{18} + \)\(32\!\cdots\!47\)\( T^{19} + \)\(16\!\cdots\!93\)\( T^{20} + \)\(66\!\cdots\!73\)\( T^{21} + \)\(21\!\cdots\!15\)\( T^{22} + \)\(50\!\cdots\!10\)\( T^{23} + \)\(69\!\cdots\!41\)\( T^{24} \)
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