Properties

Label 7623.2.a.dc
Level 7623
Weight 2
Character orbit 7623.a
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 693)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{7} q^{5} + q^{7} + ( \beta_{1} + \beta_{6} - \beta_{7} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + \beta_{7} q^{5} + q^{7} + ( \beta_{1} + \beta_{6} - \beta_{7} ) q^{8} + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{10} + \beta_{14} q^{13} + \beta_{1} q^{14} + ( 2 + \beta_{2} + \beta_{3} ) q^{16} + ( \beta_{6} - \beta_{12} + \beta_{15} ) q^{17} + ( -\beta_{2} + \beta_{8} - \beta_{13} ) q^{19} + ( -\beta_{6} + 2 \beta_{7} + \beta_{9} - 2 \beta_{10} ) q^{20} + ( \beta_{1} + \beta_{10} + \beta_{12} - \beta_{15} ) q^{23} + ( 4 + \beta_{2} + 2 \beta_{5} - \beta_{8} - \beta_{14} ) q^{25} + ( \beta_{7} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{15} ) q^{26} + ( 1 + \beta_{2} ) q^{28} + ( -\beta_{1} + \beta_{6} + 2 \beta_{10} + \beta_{11} + \beta_{15} ) q^{29} + ( 1 - \beta_{3} - \beta_{4} - \beta_{5} ) q^{31} + ( 2 \beta_{1} - 2 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} ) q^{32} + ( -1 + \beta_{2} - 2 \beta_{5} + \beta_{8} + \beta_{14} ) q^{34} + \beta_{7} q^{35} + ( 3 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{8} ) q^{37} + ( \beta_{7} - \beta_{9} + 2 \beta_{11} + \beta_{12} + \beta_{15} ) q^{38} + ( -6 - 2 \beta_{2} - \beta_{3} + \beta_{4} - 5 \beta_{5} + \beta_{13} + \beta_{14} ) q^{40} + ( 2 \beta_{1} + \beta_{6} + \beta_{9} - 2 \beta_{10} ) q^{41} + ( 2 + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{13} ) q^{43} + ( 7 + 2 \beta_{2} + 4 \beta_{5} - \beta_{8} - \beta_{14} ) q^{46} + ( -2 \beta_{1} + \beta_{6} + 2 \beta_{10} - \beta_{11} - \beta_{12} - \beta_{15} ) q^{47} + q^{49} + ( 3 \beta_{1} - \beta_{6} - \beta_{7} - \beta_{9} + 2 \beta_{10} - 3 \beta_{11} - 2 \beta_{15} ) q^{50} + ( -1 - 2 \beta_{3} - 2 \beta_{5} + \beta_{13} ) q^{52} + ( \beta_{1} - \beta_{6} + \beta_{7} - \beta_{9} - 2 \beta_{10} - \beta_{12} ) q^{53} + ( \beta_{1} + \beta_{6} - \beta_{7} ) q^{56} + ( 2 - \beta_{4} + 4 \beta_{5} + 2 \beta_{8} + \beta_{14} ) q^{58} + ( -\beta_{7} - \beta_{9} - \beta_{11} + \beta_{12} - \beta_{15} ) q^{59} + ( -2 - 2 \beta_{4} + \beta_{14} ) q^{61} + ( 2 \beta_{1} - \beta_{6} + 2 \beta_{7} + \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} ) q^{62} + ( 1 + \beta_{3} - \beta_{5} + \beta_{8} - \beta_{13} - \beta_{14} ) q^{64} + ( 4 \beta_{1} + \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - \beta_{11} - 3 \beta_{12} - \beta_{15} ) q^{65} + ( 4 + 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{13} - \beta_{14} ) q^{67} + ( 4 \beta_{1} + \beta_{6} - \beta_{7} + \beta_{9} - 2 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} ) q^{68} + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{70} + ( -\beta_{1} + \beta_{6} + 2 \beta_{10} + \beta_{11} - \beta_{15} ) q^{71} + ( -1 - 2 \beta_{2} - 2 \beta_{4} - \beta_{13} - \beta_{14} ) q^{73} + ( 4 \beta_{1} - 2 \beta_{7} - \beta_{9} - 3 \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{15} ) q^{74} + ( -2 + \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{8} + \beta_{13} ) q^{76} + ( 3 + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{14} ) q^{79} + ( -6 \beta_{1} + 2 \beta_{7} + \beta_{9} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{15} ) q^{80} + ( 4 \beta_{2} - \beta_{3} + 3 \beta_{4} - 7 \beta_{5} + \beta_{13} + \beta_{14} ) q^{82} + ( 2 \beta_{7} + 2 \beta_{12} ) q^{83} + ( 4 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + 2 \beta_{13} + \beta_{14} ) q^{85} + ( \beta_{1} - \beta_{7} - 2 \beta_{9} - \beta_{10} + 2 \beta_{12} ) q^{86} + ( \beta_{6} + \beta_{7} - \beta_{9} + 2 \beta_{10} + \beta_{11} + 2 \beta_{15} ) q^{89} + \beta_{14} q^{91} + ( 4 \beta_{1} - 2 \beta_{7} - \beta_{9} + 2 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} ) q^{92} + ( -2 \beta_{4} + 6 \beta_{5} - 2 \beta_{8} - \beta_{14} ) q^{94} + ( -\beta_{6} - \beta_{9} + 4 \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - 2 \beta_{15} ) q^{95} + ( 6 - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} + \beta_{14} ) q^{97} + \beta_{1} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 20q^{4} + 16q^{7} + O(q^{10}) \) \( 16q + 20q^{4} + 16q^{7} - 6q^{10} + 32q^{16} + 10q^{19} + 44q^{25} + 20q^{28} + 30q^{31} + 12q^{34} + 38q^{37} - 68q^{40} + 16q^{43} + 80q^{46} + 16q^{49} + 2q^{52} + 18q^{58} - 28q^{61} + 34q^{64} + 52q^{67} - 6q^{70} - 14q^{73} - 14q^{76} + 54q^{79} + 64q^{82} + 30q^{85} - 60q^{94} + 108q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 26 x^{14} + 268 x^{12} - 1395 x^{10} + 3876 x^{8} - 5635 x^{6} + 4042 x^{4} - 1272 x^{2} + 121\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 7 \nu^{2} + 5 \)
\(\beta_{4}\)\(=\)\((\)\( 8 \nu^{14} - 155 \nu^{12} + 1002 \nu^{10} - 1989 \nu^{8} - 2930 \nu^{6} + 13717 \nu^{4} - 12905 \nu^{2} + 3144 \)\()/307\)
\(\beta_{5}\)\(=\)\((\)\( -15 \nu^{14} + 329 \nu^{12} - 2723 \nu^{10} + 10752 \nu^{8} - 21906 \nu^{6} + 24667 \nu^{4} - 14063 \nu^{2} + 2087 \)\()/614\)
\(\beta_{6}\)\(=\)\((\)\( -47 \nu^{15} + 1563 \nu^{13} - 19625 \nu^{11} + 117562 \nu^{9} - 344202 \nu^{7} + 441351 \nu^{5} - 159537 \nu^{3} - 55925 \nu \)\()/6754\)
\(\beta_{7}\)\(=\)\((\)\( -47 \nu^{15} + 1563 \nu^{13} - 19625 \nu^{11} + 117562 \nu^{9} - 344202 \nu^{7} + 441351 \nu^{5} - 166291 \nu^{3} - 22155 \nu \)\()/6754\)
\(\beta_{8}\)\(=\)\((\)\( -63 \nu^{14} + 1259 \nu^{12} - 8735 \nu^{10} + 23300 \nu^{8} - 12308 \nu^{6} - 26935 \nu^{4} + 27141 \nu^{2} - 5111 \)\()/614\)
\(\beta_{9}\)\(=\)\((\)\( 76 \nu^{15} - 2240 \nu^{13} + 25483 \nu^{11} - 139086 \nu^{9} + 363590 \nu^{7} - 378848 \nu^{5} + 57151 \nu^{3} + 48902 \nu \)\()/3377\)
\(\beta_{10}\)\(=\)\((\)\( -15 \nu^{15} + 329 \nu^{13} - 2723 \nu^{11} + 10752 \nu^{9} - 21906 \nu^{7} + 24667 \nu^{5} - 14063 \nu^{3} + 2701 \nu \)\()/614\)
\(\beta_{11}\)\(=\)\((\)\( 353 \nu^{15} - 9871 \nu^{13} + 108453 \nu^{11} - 588520 \nu^{9} + 1624528 \nu^{7} - 2124543 \nu^{5} + 1130541 \nu^{3} - 157219 \nu \)\()/6754\)
\(\beta_{12}\)\(=\)\((\)\( -230 \nu^{15} + 6068 \nu^{13} - 63345 \nu^{11} + 331872 \nu^{9} - 913359 \nu^{7} + 1263820 \nu^{5} - 778773 \nu^{3} + 153982 \nu \)\()/3377\)
\(\beta_{13}\)\(=\)\((\)\( 39 \nu^{14} - 1101 \nu^{12} + 12176 \nu^{10} - 66146 \nu^{8} + 180738 \nu^{6} - 227581 \nu^{4} + 108279 \nu^{2} - 11689 \)\()/307\)
\(\beta_{14}\)\(=\)\((\)\( -63 \nu^{14} + 1566 \nu^{12} - 15182 \nu^{10} + 72420 \nu^{8} - 176246 \nu^{6} + 205157 \nu^{4} - 97194 \nu^{2} + 12388 \)\()/307\)
\(\beta_{15}\)\(=\)\((\)\( -799 \nu^{15} + 19817 \nu^{13} - 191791 \nu^{11} + 914537 \nu^{9} - 2231290 \nu^{7} + 2616448 \nu^{5} - 1256642 \nu^{3} + 173816 \nu \)\()/3377\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + \beta_{6} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{3} + 7 \beta_{2} + 16\)
\(\nu^{5}\)\(=\)\(\beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - 10 \beta_{7} + 8 \beta_{6} + 30 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-\beta_{14} - \beta_{13} + \beta_{8} - \beta_{5} + 11 \beta_{3} + 46 \beta_{2} + 97\)
\(\nu^{7}\)\(=\)\(13 \beta_{12} + 14 \beta_{11} - 12 \beta_{10} - 13 \beta_{9} - 80 \beta_{7} + 58 \beta_{6} + 192 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-13 \beta_{14} - 13 \beta_{13} + 14 \beta_{8} - 14 \beta_{5} + 3 \beta_{4} + 93 \beta_{3} + 307 \beta_{2} + 619\)
\(\nu^{9}\)\(=\)\(\beta_{15} + 123 \beta_{12} + 139 \beta_{11} - 113 \beta_{10} - 119 \beta_{9} - 597 \beta_{7} + 415 \beta_{6} + 1274 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-118 \beta_{14} - 119 \beta_{13} + 140 \beta_{8} - 158 \beta_{5} + 54 \beta_{4} + 716 \beta_{3} + 2088 \beta_{2} + 4064\)
\(\nu^{11}\)\(=\)\(22 \beta_{15} + 1028 \beta_{12} + 1210 \beta_{11} - 982 \beta_{10} - 953 \beta_{9} - 4321 \beta_{7} + 2965 \beta_{6} + 8657 \beta_{1}\)
\(\nu^{12}\)\(=\)\(-931 \beta_{14} - 953 \beta_{13} + 1232 \beta_{8} - 1612 \beta_{5} + 654 \beta_{4} + 5274 \beta_{3} + 14405 \beta_{2} + 27164\)
\(\nu^{13}\)\(=\)\(301 \beta_{15} + 8091 \beta_{12} + 9881 \beta_{11} - 8194 \beta_{10} - 7158 \beta_{9} - 30783 \beta_{7} + 21190 \beta_{6} + 59771 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-6857 \beta_{14} - 7158 \beta_{13} + 10182 \beta_{8} - 15290 \beta_{5} + 6692 \beta_{4} + 37941 \beta_{3} + 100361 \beta_{2} + 183724\)
\(\nu^{15}\)\(=\)\(3325 \beta_{15} + 61672 \beta_{12} + 77902 \beta_{11} - 66615 \beta_{10} - 51956 \beta_{9} - 217373 \beta_{7} + 151508 \beta_{6} + 417076 \beta_{1}\)

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
−2.68084
−2.60390
−2.10399
−2.08213
−1.31288
−0.899827
−0.759363
−0.400970
0.400970
0.759363
0.899827
1.31288
2.08213
2.10399
2.60390
2.68084
−2.68084 0 5.18691 1.17039 0 1.00000 −8.54361 0 −3.13763
1.2 −2.60390 0 4.78032 4.31860 0 1.00000 −7.23969 0 −11.2452
1.3 −2.10399 0 2.42677 −3.75046 0 1.00000 −0.897910 0 7.89093
1.4 −2.08213 0 2.33525 −1.33222 0 1.00000 −0.698024 0 2.77386
1.5 −1.31288 0 −0.276342 −3.45937 0 1.00000 2.98857 0 4.54174
1.6 −0.899827 0 −1.19031 0.803961 0 1.00000 2.87073 0 −0.723426
1.7 −0.759363 0 −1.42337 2.86525 0 1.00000 2.59958 0 −2.17577
1.8 −0.400970 0 −1.83922 2.30562 0 1.00000 1.53941 0 −0.924485
1.9 0.400970 0 −1.83922 −2.30562 0 1.00000 −1.53941 0 −0.924485
1.10 0.759363 0 −1.42337 −2.86525 0 1.00000 −2.59958 0 −2.17577
1.11 0.899827 0 −1.19031 −0.803961 0 1.00000 −2.87073 0 −0.723426
1.12 1.31288 0 −0.276342 3.45937 0 1.00000 −2.98857 0 4.54174
1.13 2.08213 0 2.33525 1.33222 0 1.00000 0.698024 0 2.77386
1.14 2.10399 0 2.42677 3.75046 0 1.00000 0.897910 0 7.89093
1.15 2.60390 0 4.78032 −4.31860 0 1.00000 7.23969 0 −11.2452
1.16 2.68084 0 5.18691 −1.17039 0 1.00000 8.54361 0 −3.13763
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7623.2.a.dc 16
3.b odd 2 1 inner 7623.2.a.dc 16
11.b odd 2 1 7623.2.a.db 16
11.c even 5 2 693.2.m.k 32
33.d even 2 1 7623.2.a.db 16
33.h odd 10 2 693.2.m.k 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
693.2.m.k 32 11.c even 5 2
693.2.m.k 32 33.h odd 10 2
7623.2.a.db 16 11.b odd 2 1
7623.2.a.db 16 33.d even 2 1
7623.2.a.dc 16 1.a even 1 1 trivial
7623.2.a.dc 16 3.b odd 2 1 inner

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7623))\):

\(T_{2}^{16} - \cdots\)
\(T_{5}^{16} - \cdots\)
\( T_{13}^{8} - 81 T_{13}^{6} + 60 T_{13}^{5} + 2016 T_{13}^{4} - 2280 T_{13}^{3} - 16016 T_{13}^{2} + 12640 T_{13} + 42176 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T^{2} + 20 T^{4} + 53 T^{6} + 136 T^{8} + 321 T^{10} + 742 T^{12} + 1732 T^{14} + 3753 T^{16} + 6928 T^{18} + 11872 T^{20} + 20544 T^{22} + 34816 T^{24} + 54272 T^{26} + 81920 T^{28} + 98304 T^{30} + 65536 T^{32} \)
$3$ \( \)
$5$ \( 1 + 18 T^{2} + 187 T^{4} + 1440 T^{6} + 9271 T^{8} + 50940 T^{10} + 254033 T^{12} + 1194274 T^{14} + 5790136 T^{16} + 29856850 T^{18} + 158770625 T^{20} + 795937500 T^{22} + 3621484375 T^{24} + 14062500000 T^{26} + 45654296875 T^{28} + 109863281250 T^{30} + 152587890625 T^{32} \)
$7$ \( ( 1 - T )^{16} \)
$11$ \( \)
$13$ \( ( 1 + 23 T^{2} + 60 T^{3} + 430 T^{4} + 1620 T^{5} + 6513 T^{6} + 25120 T^{7} + 110114 T^{8} + 326560 T^{9} + 1100697 T^{10} + 3559140 T^{11} + 12281230 T^{12} + 22277580 T^{13} + 111016607 T^{14} + 815730721 T^{16} )^{2} \)
$17$ \( 1 + 83 T^{2} + 3996 T^{4} + 141086 T^{6} + 4032975 T^{8} + 98649908 T^{10} + 2136156760 T^{12} + 41705258555 T^{14} + 742088310776 T^{16} + 12052819722395 T^{18} + 178413948751960 T^{20} + 2381168961193652 T^{22} + 28133055365616975 T^{24} + 284428515438747614 T^{26} + 2328158459970124956 T^{28} + 13975359604430277107 T^{30} + 48661191875666868481 T^{32} \)
$19$ \( ( 1 - 5 T + 53 T^{2} - 385 T^{3} + 2238 T^{4} - 13185 T^{5} + 71811 T^{6} - 329805 T^{7} + 1585730 T^{8} - 6266295 T^{9} + 25923771 T^{10} - 90435915 T^{11} + 291658398 T^{12} - 953298115 T^{13} + 2493431693 T^{14} - 4469358695 T^{15} + 16983563041 T^{16} )^{2} \)
$23$ \( 1 + 141 T^{2} + 10058 T^{4} + 508504 T^{6} + 20887458 T^{8} + 735324351 T^{10} + 22617452235 T^{12} + 614206741660 T^{14} + 14898756254016 T^{16} + 324915366338140 T^{18} + 6329290450894635 T^{20} + 108854394003633039 T^{22} + 1635717415995505698 T^{24} + 21065546658185371096 T^{26} + \)\(22\!\cdots\!18\)\( T^{28} + \)\(16\!\cdots\!69\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( 1 + 170 T^{2} + 11642 T^{4} + 378370 T^{6} + 3597338 T^{8} - 148553960 T^{10} - 5835168271 T^{12} - 93550702700 T^{14} - 1409800693980 T^{16} - 78676140970700 T^{18} - 4127103649881151 T^{20} - 88363359834901160 T^{22} + 1799555430708297818 T^{24} + \)\(15\!\cdots\!70\)\( T^{26} + \)\(41\!\cdots\!22\)\( T^{28} + \)\(50\!\cdots\!70\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( ( 1 - 15 T + 214 T^{2} - 1830 T^{3} + 15705 T^{4} - 104570 T^{5} + 751866 T^{6} - 4460195 T^{7} + 27640604 T^{8} - 138266045 T^{9} + 722543226 T^{10} - 3115244870 T^{11} + 14503897305 T^{12} - 52391346330 T^{13} + 189925787734 T^{14} - 412689211665 T^{15} + 852891037441 T^{16} )^{2} \)
$37$ \( ( 1 - 19 T + 294 T^{2} - 3256 T^{3} + 31994 T^{4} - 263281 T^{5} + 1993923 T^{6} - 13472988 T^{7} + 86070388 T^{8} - 498500556 T^{9} + 2729680587 T^{10} - 13335972493 T^{11} + 59961907034 T^{12} - 225783923992 T^{13} + 754323564246 T^{14} - 1803705665527 T^{15} + 3512479453921 T^{16} )^{2} \)
$41$ \( 1 + 284 T^{2} + 40621 T^{4} + 3925468 T^{6} + 292611087 T^{8} + 18270233504 T^{10} + 1002225656887 T^{12} + 49066240684968 T^{14} + 2136999595406616 T^{16} + 82480350591431208 T^{18} + 2832050174430666007 T^{20} + 86785513651410690464 T^{22} + \)\(23\!\cdots\!27\)\( T^{24} + \)\(52\!\cdots\!68\)\( T^{26} + \)\(91\!\cdots\!01\)\( T^{28} + \)\(10\!\cdots\!24\)\( T^{30} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( ( 1 - 8 T + 182 T^{2} - 714 T^{3} + 12126 T^{4} - 18648 T^{5} + 591955 T^{6} - 533592 T^{7} + 28125884 T^{8} - 22944456 T^{9} + 1094524795 T^{10} - 1482646536 T^{11} + 41456380926 T^{12} - 104964028302 T^{13} + 1150488074918 T^{14} - 2174548888856 T^{15} + 11688200277601 T^{16} )^{2} \)
$47$ \( 1 + 189 T^{2} + 16293 T^{4} + 1005791 T^{6} + 62126478 T^{8} + 3858063829 T^{10} + 216619715755 T^{12} + 11267038690895 T^{14} + 550848868000706 T^{16} + 24888888468187055 T^{18} + 1057035111195074155 T^{20} + 41586900765817234741 T^{22} + \)\(14\!\cdots\!58\)\( T^{24} + \)\(52\!\cdots\!59\)\( T^{26} + \)\(18\!\cdots\!13\)\( T^{28} + \)\(48\!\cdots\!41\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 + 351 T^{2} + 64830 T^{4} + 8433028 T^{6} + 868191366 T^{8} + 74745980881 T^{10} + 5523680770247 T^{12} + 355312335854452 T^{14} + 20065793083018168 T^{16} + 998072351415155668 T^{18} + 43584498167699318807 T^{20} + \)\(16\!\cdots\!49\)\( T^{22} + \)\(54\!\cdots\!26\)\( T^{24} + \)\(14\!\cdots\!72\)\( T^{26} + \)\(31\!\cdots\!30\)\( T^{28} + \)\(48\!\cdots\!19\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( 1 + 543 T^{2} + 139357 T^{4} + 22696633 T^{6} + 2682074118 T^{8} + 251509115123 T^{10} + 19990430200355 T^{12} + 1399738718709125 T^{14} + 87411313846785426 T^{16} + 4872490479826464125 T^{18} + \)\(24\!\cdots\!55\)\( T^{20} + \)\(10\!\cdots\!43\)\( T^{22} + \)\(39\!\cdots\!78\)\( T^{24} + \)\(11\!\cdots\!33\)\( T^{26} + \)\(24\!\cdots\!17\)\( T^{28} + \)\(33\!\cdots\!23\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( ( 1 + 14 T + 369 T^{2} + 3982 T^{3} + 63690 T^{4} + 570114 T^{5} + 6849895 T^{6} + 51546370 T^{7} + 501244586 T^{8} + 3144328570 T^{9} + 25488459295 T^{10} + 129405045834 T^{11} + 881841613290 T^{12} + 3363182470582 T^{13} + 19011018139209 T^{14} + 43998399704294 T^{15} + 191707312997281 T^{16} )^{2} \)
$67$ \( ( 1 - 26 T + 589 T^{2} - 8044 T^{3} + 101599 T^{4} - 918934 T^{5} + 8384403 T^{6} - 59948252 T^{7} + 526057568 T^{8} - 4016532884 T^{9} + 37637585067 T^{10} - 276381346642 T^{11} + 2047333742479 T^{12} - 10860406360708 T^{13} + 53279987097541 T^{14} - 157578501738398 T^{15} + 406067677556641 T^{16} )^{2} \)
$71$ \( 1 + 834 T^{2} + 339866 T^{4} + 89812338 T^{6} + 17223424482 T^{8} + 2541454758064 T^{10} + 298486048517537 T^{12} + 28463160728293788 T^{14} + 2227198780428295156 T^{16} + \)\(14\!\cdots\!08\)\( T^{18} + \)\(75\!\cdots\!97\)\( T^{20} + \)\(32\!\cdots\!44\)\( T^{22} + \)\(11\!\cdots\!02\)\( T^{24} + \)\(29\!\cdots\!38\)\( T^{26} + \)\(55\!\cdots\!06\)\( T^{28} + \)\(68\!\cdots\!54\)\( T^{30} + \)\(41\!\cdots\!21\)\( T^{32} \)
$73$ \( ( 1 + 7 T + 301 T^{2} + 2487 T^{3} + 52814 T^{4} + 420273 T^{5} + 6264227 T^{6} + 45136561 T^{7} + 535074018 T^{8} + 3294968953 T^{9} + 33382065683 T^{10} + 163493341641 T^{11} + 1499824700174 T^{12} + 5155729051791 T^{13} + 45551602112989 T^{14} + 77331789633679 T^{15} + 806460091894081 T^{16} )^{2} \)
$79$ \( ( 1 - 27 T + 616 T^{2} - 10464 T^{3} + 155660 T^{4} - 1975347 T^{5} + 22887955 T^{6} - 234448440 T^{7} + 2202874636 T^{8} - 18521426760 T^{9} + 142843727155 T^{10} - 973923109533 T^{11} + 6062969608460 T^{12} - 32198318159136 T^{13} + 149741872600936 T^{14} - 518505542626293 T^{15} + 1517108809906561 T^{16} )^{2} \)
$83$ \( 1 + 692 T^{2} + 253584 T^{4} + 63764956 T^{6} + 12182483532 T^{8} + 1863587574932 T^{10} + 235286312480688 T^{12} + 24958715395566236 T^{14} + 2245860181831703846 T^{16} + \)\(17\!\cdots\!04\)\( T^{18} + \)\(11\!\cdots\!48\)\( T^{20} + \)\(60\!\cdots\!08\)\( T^{22} + \)\(27\!\cdots\!12\)\( T^{24} + \)\(98\!\cdots\!44\)\( T^{26} + \)\(27\!\cdots\!24\)\( T^{28} + \)\(50\!\cdots\!68\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( 1 + 656 T^{2} + 226181 T^{4} + 53166932 T^{6} + 9489558447 T^{8} + 1367605904636 T^{10} + 166086358267207 T^{12} + 17553705254600032 T^{14} + 1650023239040136456 T^{16} + \)\(13\!\cdots\!72\)\( T^{18} + \)\(10\!\cdots\!87\)\( T^{20} + \)\(67\!\cdots\!96\)\( T^{22} + \)\(37\!\cdots\!07\)\( T^{24} + \)\(16\!\cdots\!32\)\( T^{26} + \)\(55\!\cdots\!01\)\( T^{28} + \)\(12\!\cdots\!96\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \)
$97$ \( ( 1 - 54 T + 1829 T^{2} - 44646 T^{3} + 872014 T^{4} - 14158746 T^{5} + 196508043 T^{6} - 2365703818 T^{7} + 24885324098 T^{8} - 229473270346 T^{9} + 1848944176587 T^{10} - 12922305188058 T^{11} + 77198772441934 T^{12} - 383390393114022 T^{13} + 1523505797015141 T^{14} - 4363107361818102 T^{15} + 7837433594376961 T^{16} )^{2} \)
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