Properties

Label 2025.4.a.bl
Level $2025$
Weight $4$
Character orbit 2025.a
Self dual yes
Analytic conductor $119.479$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(119.478867762\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 91 x^{14} + 3268 x^{12} - 59128 x^{10} + 571975 x^{8} - 2881141 x^{6} + 6555196 x^{4} - 4069504 x^{2} + 614656\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{12}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 45)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 3 + \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{11} ) q^{7} + ( 3 \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 3 + \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{11} ) q^{7} + ( 3 \beta_{1} + \beta_{3} ) q^{8} + ( 5 + \beta_{2} - \beta_{6} ) q^{11} + ( 2 \beta_{1} + \beta_{3} + \beta_{11} - \beta_{13} - \beta_{15} ) q^{13} + ( 6 + 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{9} ) q^{14} + ( 8 + 3 \beta_{2} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{16} + ( \beta_{1} + \beta_{3} + \beta_{14} - \beta_{15} ) q^{17} + ( -\beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{19} + ( 11 \beta_{1} + 3 \beta_{3} - \beta_{10} + \beta_{11} + 3 \beta_{12} + \beta_{13} ) q^{22} + ( -\beta_{3} + \beta_{10} - 3 \beta_{11} + \beta_{12} - \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{23} + ( 26 + 6 \beta_{2} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{26} + ( 19 \beta_{1} + 2 \beta_{3} + \beta_{10} - \beta_{11} - 6 \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{28} + ( 29 + 7 \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{8} + 2 \beta_{9} ) q^{29} + ( 5 - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} - 4 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{31} + ( 2 \beta_{1} + \beta_{3} - 6 \beta_{10} + \beta_{12} - 3 \beta_{13} + \beta_{14} - \beta_{15} ) q^{32} + ( 9 + 5 \beta_{2} + 3 \beta_{5} + \beta_{6} + 9 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{34} + ( 23 \beta_{1} + 6 \beta_{3} + 6 \beta_{10} + \beta_{11} + 6 \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{37} + ( 4 \beta_{1} - 2 \beta_{3} + 3 \beta_{10} - 9 \beta_{11} - 2 \beta_{12} - \beta_{14} + \beta_{15} ) q^{38} + ( 40 - 2 \beta_{2} - 4 \beta_{5} + 3 \beta_{6} - \beta_{7} - 2 \beta_{8} - 5 \beta_{9} ) q^{41} + ( 35 \beta_{1} - \beta_{3} - 2 \beta_{10} - 2 \beta_{11} - 9 \beta_{12} - 2 \beta_{13} - \beta_{14} + \beta_{15} ) q^{43} + ( 92 + 29 \beta_{2} + 3 \beta_{4} + 4 \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{44} + ( -11 - 6 \beta_{2} + 2 \beta_{4} - 3 \beta_{5} - 18 \beta_{7} + 5 \beta_{8} - 2 \beta_{9} ) q^{46} + ( 2 \beta_{10} - 6 \beta_{11} - 2 \beta_{12} - 8 \beta_{13} + 3 \beta_{14} ) q^{47} + ( -3 - 3 \beta_{2} - 2 \beta_{5} - 3 \beta_{6} + 11 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{49} + ( 46 \beta_{1} + 9 \beta_{3} - 11 \beta_{10} + 8 \beta_{11} + 18 \beta_{12} - 18 \beta_{13} - \beta_{14} ) q^{52} + ( \beta_{1} - 3 \beta_{3} + 7 \beta_{10} - 12 \beta_{11} - \beta_{12} + 5 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{53} + ( 143 + 20 \beta_{2} - 3 \beta_{4} + \beta_{5} + 7 \beta_{7} - 4 \beta_{8} + 5 \beta_{9} ) q^{56} + ( 92 \beta_{1} - 4 \beta_{10} - 5 \beta_{11} - 21 \beta_{12} + 11 \beta_{13} - \beta_{14} + 6 \beta_{15} ) q^{58} + ( 130 + 27 \beta_{2} - 2 \beta_{4} - 3 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} ) q^{59} + ( -1 + 3 \beta_{2} + 5 \beta_{4} + 6 \beta_{5} - 6 \beta_{6} - 9 \beta_{7} + 2 \beta_{8} + 10 \beta_{9} ) q^{61} + ( 5 \beta_{1} - 6 \beta_{3} + 17 \beta_{10} - 6 \beta_{11} - \beta_{12} + 31 \beta_{13} - 4 \beta_{14} + 7 \beta_{15} ) q^{62} + ( -25 - 9 \beta_{2} + 4 \beta_{4} + \beta_{5} + 2 \beta_{6} + 20 \beta_{7} + \beta_{8} + 5 \beta_{9} ) q^{64} + ( 22 \beta_{1} - 5 \beta_{3} - 6 \beta_{10} - 7 \beta_{11} + 12 \beta_{12} + 18 \beta_{13} - \beta_{14} - \beta_{15} ) q^{67} + ( 23 \beta_{1} + 7 \beta_{3} - 20 \beta_{10} - 3 \beta_{11} - 3 \beta_{12} - \beta_{13} + 9 \beta_{14} + 6 \beta_{15} ) q^{68} + ( 180 - 2 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} + 8 \beta_{8} + 5 \beta_{9} ) q^{71} + ( 89 \beta_{1} - 6 \beta_{3} + 12 \beta_{10} + 7 \beta_{11} - 3 \beta_{12} - 7 \beta_{13} - 5 \beta_{14} - 6 \beta_{15} ) q^{73} + ( 235 + 58 \beta_{2} + 4 \beta_{4} + \beta_{5} - 5 \beta_{6} - \beta_{7} - 4 \beta_{8} - 6 \beta_{9} ) q^{74} + ( -12 + 3 \beta_{2} - 3 \beta_{4} + \beta_{5} + 14 \beta_{6} - 19 \beta_{7} - 6 \beta_{8} - 3 \beta_{9} ) q^{76} + ( \beta_{1} - 8 \beta_{3} - 28 \beta_{10} - 15 \beta_{11} - 5 \beta_{12} - 2 \beta_{13} - 3 \beta_{14} + 6 \beta_{15} ) q^{77} + ( -20 - 8 \beta_{2} + 6 \beta_{4} - 3 \beta_{5} - 12 \beta_{6} + 3 \beta_{7} + 9 \beta_{8} + 9 \beta_{9} ) q^{79} + ( 46 \beta_{1} - 16 \beta_{3} + 42 \beta_{10} - 10 \beta_{11} + 21 \beta_{12} + 17 \beta_{13} - 7 \beta_{14} + \beta_{15} ) q^{82} + ( 22 \beta_{1} + 13 \beta_{3} - 5 \beta_{10} + 15 \beta_{11} + 11 \beta_{12} - 19 \beta_{13} + 2 \beta_{14} - 5 \beta_{15} ) q^{83} + ( 372 + 35 \beta_{2} - 9 \beta_{4} - \beta_{5} + 8 \beta_{6} - 9 \beta_{7} + 4 \beta_{8} + \beta_{9} ) q^{86} + ( 208 \beta_{1} + 16 \beta_{3} - 11 \beta_{10} + 11 \beta_{11} - 18 \beta_{12} - 28 \beta_{13} + 5 \beta_{14} - 7 \beta_{15} ) q^{88} + ( 242 + 23 \beta_{2} - 3 \beta_{4} - 11 \beta_{5} - 3 \beta_{6} - \beta_{7} + 11 \beta_{8} - 4 \beta_{9} ) q^{89} + ( -50 - 12 \beta_{2} + 5 \beta_{4} + 5 \beta_{5} + 8 \beta_{6} + 4 \beta_{7} - 9 \beta_{8} ) q^{91} + ( -20 \beta_{1} - \beta_{3} + 26 \beta_{10} + 21 \beta_{11} + 15 \beta_{12} - 29 \beta_{13} - 15 \beta_{14} ) q^{92} + ( -65 - 12 \beta_{2} - 3 \beta_{4} + \beta_{5} + 16 \beta_{6} + 17 \beta_{7} + 2 \beta_{8} - 11 \beta_{9} ) q^{94} + ( -4 \beta_{1} - 25 \beta_{3} - 29 \beta_{10} - 23 \beta_{11} - 15 \beta_{12} - 34 \beta_{13} + 7 \beta_{14} + \beta_{15} ) q^{97} + ( -39 \beta_{1} - 11 \beta_{3} - 2 \beta_{10} + 15 \beta_{11} + 7 \beta_{12} + 29 \beta_{13} + 9 \beta_{14} + 6 \beta_{15} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 54 q^{4} + O(q^{10}) \) \( 16 q + 54 q^{4} + 90 q^{11} + 102 q^{14} + 146 q^{16} + 4 q^{19} + 468 q^{26} + 516 q^{29} + 38 q^{31} + 212 q^{34} + 576 q^{41} + 1644 q^{44} - 290 q^{46} - 4 q^{49} + 2430 q^{56} + 2202 q^{59} + 20 q^{61} - 322 q^{64} + 2952 q^{71} + 4080 q^{74} - 396 q^{76} - 218 q^{79} + 6108 q^{86} + 4074 q^{89} - 942 q^{91} - 1078 q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 91 x^{14} + 3268 x^{12} - 59128 x^{10} + 571975 x^{8} - 2881141 x^{6} + 6555196 x^{4} - 4069504 x^{2} + 614656\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 11 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 19 \nu \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{14} - 86 \nu^{12} + 2754 \nu^{10} - 40102 \nu^{8} + 252737 \nu^{6} - 416664 \nu^{4} - 951920 \nu^{2} + 182272 \)\()/10368\)
\(\beta_{5}\)\(=\)\((\)\( 13 \nu^{14} - 1022 \nu^{12} + 29466 \nu^{10} - 370126 \nu^{8} + 1713581 \nu^{6} + 1226376 \nu^{4} - 22574480 \nu^{2} + 15782656 \)\()/176256\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{14} + 94 \nu^{12} - 11574 \nu^{10} + 387218 \nu^{8} - 5621959 \nu^{6} + 36266220 \nu^{4} - 85224032 \nu^{2} + 17199616 \)\()/352512\)
\(\beta_{7}\)\(=\)\((\)\( 25 \nu^{14} - 2138 \nu^{12} + 70506 \nu^{10} - 1127470 \nu^{8} + 9049121 \nu^{6} - 34165980 \nu^{4} + 49592896 \nu^{2} - 17007872 \)\()/352512\)
\(\beta_{8}\)\(=\)\((\)\( 11 \nu^{14} - 938 \nu^{12} + 30582 \nu^{10} - 479522 \nu^{8} + 3772331 \nu^{6} - 14336208 \nu^{4} + 22336880 \nu^{2} - 6685568 \)\()/58752\)
\(\beta_{9}\)\(=\)\((\)\( -73 \nu^{14} + 6194 \nu^{12} - 200394 \nu^{10} + 3084622 \nu^{8} - 22981937 \nu^{6} + 73691748 \nu^{4} - 64228384 \nu^{2} + 8294528 \)\()/176256\)
\(\beta_{10}\)\(=\)\((\)\( 101 \nu^{15} - 8407 \nu^{13} + 263820 \nu^{11} - 3850424 \nu^{9} + 25802659 \nu^{7} - 63425913 \nu^{5} + 13608404 \nu^{3} - 100002400 \nu \)\()/2878848\)
\(\beta_{11}\)\(=\)\((\)\( -895 \nu^{15} + 75908 \nu^{13} - 2465730 \nu^{11} + 38436238 \nu^{9} - 295408283 \nu^{7} + 1022927202 \nu^{5} - 1116316120 \nu^{3} - 79369792 \nu \)\()/17273088\)
\(\beta_{12}\)\(=\)\((\)\( 283 \nu^{15} - 25214 \nu^{13} + 875550 \nu^{11} - 15014794 \nu^{9} + 133274387 \nu^{7} - 583184292 \nu^{5} + 1031384800 \nu^{3} - 288310016 \nu \)\()/5757696\)
\(\beta_{13}\)\(=\)\((\)\( -365 \nu^{15} + 31990 \nu^{13} - 1088058 \nu^{11} + 18206894 \nu^{9} - 157923085 \nu^{7} + 689377056 \nu^{5} - 1314355088 \nu^{3} + 640762624 \nu \)\()/5757696\)
\(\beta_{14}\)\(=\)\((\)\( 1831 \nu^{15} - 155204 \nu^{13} + 5037714 \nu^{11} - 78348574 \nu^{9} + 598222883 \nu^{7} - 2054688498 \nu^{5} + 2511702328 \nu^{3} - 1433400128 \nu \)\()/17273088\)
\(\beta_{15}\)\(=\)\((\)\( 137 \nu^{15} - 12712 \nu^{13} + 468198 \nu^{11} - 8743514 \nu^{9} + 87674005 \nu^{7} - 455445594 \nu^{5} + 1030303640 \nu^{3} - 475010752 \nu \)\()/1016064\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 11\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 19 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{7} - \beta_{6} + \beta_{5} + 27 \beta_{2} + 208\)
\(\nu^{5}\)\(=\)\(-\beta_{15} + \beta_{14} - 3 \beta_{13} + \beta_{12} - 6 \beta_{10} + 33 \beta_{3} + 418 \beta_{1}\)
\(\nu^{6}\)\(=\)\(5 \beta_{9} + \beta_{8} - 20 \beta_{7} - 38 \beta_{6} + 41 \beta_{5} + 4 \beta_{4} + 687 \beta_{2} + 4583\)
\(\nu^{7}\)\(=\)\(-45 \beta_{15} + 57 \beta_{14} - 143 \beta_{13} + 39 \beta_{12} + 30 \beta_{11} - 268 \beta_{10} + 918 \beta_{3} + 9830 \beta_{1}\)
\(\nu^{8}\)\(=\)\(241 \beta_{9} + 29 \beta_{8} - 235 \beta_{7} - 1141 \beta_{6} + 1288 \beta_{5} + 200 \beta_{4} + 17333 \beta_{2} + 108106\)
\(\nu^{9}\)\(=\)\(-1476 \beta_{15} + 2112 \beta_{14} - 4732 \beta_{13} + 1254 \beta_{12} + 1518 \beta_{11} - 8774 \beta_{10} + 24314 \beta_{3} + 238799 \beta_{1}\)
\(\nu^{10}\)\(=\)\(8180 \beta_{9} + 508 \beta_{8} + 212 \beta_{7} - 31636 \beta_{6} + 36676 \beta_{5} + 6868 \beta_{4} + 437433 \beta_{2} + 2633567\)
\(\nu^{11}\)\(=\)\(-42740 \beta_{15} + 66188 \beta_{14} - 136604 \beta_{13} + 36404 \beta_{12} + 52824 \beta_{11} - 255544 \beta_{10} + 631557 \beta_{3} + 5902367 \beta_{1}\)
\(\nu^{12}\)\(=\)\(242180 \beta_{9} + 4228 \beta_{8} + 125815 \beta_{7} - 843953 \beta_{6} + 996029 \beta_{5} + 202384 \beta_{4} + 11058627 \beta_{2} + 65230888\)
\(\nu^{13}\)\(=\)\(-1162845 \beta_{15} + 1911261 \beta_{14} - 3697319 \beta_{13} + 983541 \beta_{12} + 1577928 \beta_{11} - 7033990 \beta_{10} + 16250517 \beta_{3} + 147362278 \beta_{1}\)
\(\nu^{14}\)\(=\)\(6700657 \beta_{9} - 125203 \beta_{8} + 5450348 \beta_{7} - 22023454 \beta_{6} + 26358613 \beta_{5} + 5510372 \beta_{4} + 280010935 \beta_{2} + 1630940951\)
\(\nu^{15}\)\(=\)\(-30553497 \beta_{15} + 52782405 \beta_{14} - 96685123 \beta_{13} + 25248315 \beta_{12} + 43569078 \beta_{11} - 187755644 \beta_{10} + 415965778 \beta_{3} + 3702009370 \beta_{1}\)

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
−5.05435
−5.02371
−4.07626
−3.08740
−2.67506
−2.46385
−0.785333
−0.473990
0.473990
0.785333
2.46385
2.67506
3.08740
4.07626
5.02371
5.05435
−5.05435 0 17.5465 0 0 −21.0117 −48.2513 0 0
1.2 −5.02371 0 17.2377 0 0 5.38197 −46.4074 0 0
1.3 −4.07626 0 8.61587 0 0 −13.3430 −2.51043 0 0
1.4 −3.08740 0 1.53204 0 0 31.3204 19.9692 0 0
1.5 −2.67506 0 −0.844033 0 0 −15.4153 23.6584 0 0
1.6 −2.46385 0 −1.92944 0 0 19.2401 24.4647 0 0
1.7 −0.785333 0 −7.38325 0 0 −20.9136 12.0810 0 0
1.8 −0.473990 0 −7.77533 0 0 −8.20657 7.47735 0 0
1.9 0.473990 0 −7.77533 0 0 8.20657 −7.47735 0 0
1.10 0.785333 0 −7.38325 0 0 20.9136 −12.0810 0 0
1.11 2.46385 0 −1.92944 0 0 −19.2401 −24.4647 0 0
1.12 2.67506 0 −0.844033 0 0 15.4153 −23.6584 0 0
1.13 3.08740 0 1.53204 0 0 −31.3204 −19.9692 0 0
1.14 4.07626 0 8.61587 0 0 13.3430 2.51043 0 0
1.15 5.02371 0 17.2377 0 0 −5.38197 46.4074 0 0
1.16 5.05435 0 17.5465 0 0 21.0117 48.2513 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2025.4.a.bl 16
3.b odd 2 1 2025.4.a.bk 16
5.b even 2 1 inner 2025.4.a.bl 16
5.c odd 4 2 405.4.b.f 16
9.d odd 6 2 225.4.e.g 32
15.d odd 2 1 2025.4.a.bk 16
15.e even 4 2 405.4.b.e 16
45.h odd 6 2 225.4.e.g 32
45.k odd 12 4 135.4.j.a 32
45.l even 12 4 45.4.j.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.j.a 32 45.l even 12 4
135.4.j.a 32 45.k odd 12 4
225.4.e.g 32 9.d odd 6 2
225.4.e.g 32 45.h odd 6 2
405.4.b.e 16 15.e even 4 2
405.4.b.f 16 5.c odd 4 2
2025.4.a.bk 16 3.b odd 2 1
2025.4.a.bk 16 15.d odd 2 1
2025.4.a.bl 16 1.a even 1 1 trivial
2025.4.a.bl 16 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2025))\):

\(T_{2}^{16} - \cdots\)
\(93\!\cdots\!42\)\( T_{7}^{6} + \)\(88\!\cdots\!69\)\( T_{7}^{4} - \)\(39\!\cdots\!00\)\( T_{7}^{2} + \)\(57\!\cdots\!16\)\( \)">\(T_{7}^{16} - \cdots\)
\(T_{11}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 614656 - 4069504 T^{2} + 6555196 T^{4} - 2881141 T^{6} + 571975 T^{8} - 59128 T^{10} + 3268 T^{12} - 91 T^{14} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( T^{16} \)
$7$ \( 5787171908657455716 - 390455956286750700 T^{2} + 8888477253929469 T^{4} - 93497855511042 T^{6} + 525057427827 T^{8} - 1665664668 T^{10} + 2969739 T^{12} - 2742 T^{14} + T^{16} \)
$11$ \( ( 137637673416 + 26757336852 T - 809472312 T^{2} - 158752116 T^{3} + 4235328 T^{4} + 160551 T^{5} - 3975 T^{6} - 45 T^{7} + T^{8} )^{2} \)
$13$ \( \)\(53\!\cdots\!96\)\( - \)\(53\!\cdots\!48\)\( T^{2} + 78699104335949476032 T^{4} - 303501872607601680 T^{6} + 478016713374336 T^{8} - 349151939964 T^{10} + 121119660 T^{12} - 18483 T^{14} + T^{16} \)
$17$ \( \)\(18\!\cdots\!56\)\( - \)\(60\!\cdots\!56\)\( T^{2} + \)\(29\!\cdots\!64\)\( T^{4} - 33539383820058451168 T^{6} + 16572039849099712 T^{8} - 4131909482668 T^{10} + 547814617 T^{12} - 36958 T^{14} + T^{16} \)
$19$ \( ( 11815095359200 - 5704507496360 T - 241429092524 T^{2} + 3427815988 T^{3} + 150069100 T^{4} - 445916 T^{5} - 24341 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$23$ \( \)\(30\!\cdots\!64\)\( - \)\(35\!\cdots\!40\)\( T^{2} + \)\(59\!\cdots\!01\)\( T^{4} - \)\(17\!\cdots\!26\)\( T^{6} + 2343990096556801783 T^{8} - 162205058882176 T^{10} + 6186573343 T^{12} - 123130 T^{14} + T^{16} \)
$29$ \( ( -1696760290826316 - 389207524968612 T - 24156887205807 T^{2} - 280639491210 T^{3} + 2316962691 T^{4} + 17565264 T^{5} - 80745 T^{6} - 258 T^{7} + T^{8} )^{2} \)
$31$ \( ( -127965906581806160 + 5383914360201224 T - 53884731790376 T^{2} - 311543511460 T^{3} + 5330728816 T^{4} + 5857382 T^{5} - 137654 T^{6} - 19 T^{7} + T^{8} )^{2} \)
$37$ \( \)\(12\!\cdots\!24\)\( - \)\(75\!\cdots\!00\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{4} - \)\(17\!\cdots\!92\)\( T^{6} + 95318152008399860976 T^{8} - 2923068725661888 T^{10} + 47940529104 T^{12} - 369900 T^{14} + T^{16} \)
$41$ \( ( -50914609409062035 - 9102671053428924 T - 366728419206576 T^{2} - 3271477051440 T^{3} + 19704394530 T^{4} + 65723148 T^{5} - 270816 T^{6} - 288 T^{7} + T^{8} )^{2} \)
$43$ \( \)\(22\!\cdots\!56\)\( - \)\(63\!\cdots\!08\)\( T^{2} + \)\(19\!\cdots\!24\)\( T^{4} - \)\(15\!\cdots\!44\)\( T^{6} + 36326817147213144288 T^{8} - 1855032885866763 T^{10} + 37172815443 T^{12} - 322017 T^{14} + T^{16} \)
$47$ \( \)\(89\!\cdots\!44\)\( - \)\(10\!\cdots\!48\)\( T^{2} + \)\(11\!\cdots\!01\)\( T^{4} - \)\(23\!\cdots\!58\)\( T^{6} + \)\(19\!\cdots\!19\)\( T^{8} - 7114794582860476 T^{10} + 103054946131 T^{12} - 594514 T^{14} + T^{16} \)
$53$ \( \)\(45\!\cdots\!44\)\( - \)\(42\!\cdots\!08\)\( T^{2} + \)\(58\!\cdots\!68\)\( T^{4} - \)\(33\!\cdots\!04\)\( T^{6} + \)\(94\!\cdots\!72\)\( T^{8} - 14763428906730784 T^{10} + 127074664420 T^{12} - 564124 T^{14} + T^{16} \)
$59$ \( ( -7915694152982015232 - 133548560466028992 T + 4567215170808288 T^{2} - 15464340977472 T^{3} - 76233486096 T^{4} + 336624651 T^{5} - 10707 T^{6} - 1101 T^{7} + T^{8} )^{2} \)
$61$ \( ( -49277468581095097322 - 1204794883975463200 T - 6710949081021689 T^{2} + 20182587065930 T^{3} + 151586239273 T^{4} - 61355824 T^{5} - 750167 T^{6} - 10 T^{7} + T^{8} )^{2} \)
$67$ \( \)\(26\!\cdots\!89\)\( - \)\(97\!\cdots\!40\)\( T^{2} + \)\(57\!\cdots\!72\)\( T^{4} - \)\(13\!\cdots\!04\)\( T^{6} + \)\(13\!\cdots\!82\)\( T^{8} - 682238458110843504 T^{10} + 1756284593724 T^{12} - 2156100 T^{14} + T^{16} \)
$71$ \( ( -\)\(15\!\cdots\!24\)\( + 29482081723807632 T + 9514852456002840 T^{2} - 19250853383952 T^{3} - 121562934084 T^{4} + 336942828 T^{5} + 324162 T^{6} - 1476 T^{7} + T^{8} )^{2} \)
$73$ \( \)\(43\!\cdots\!96\)\( - \)\(16\!\cdots\!60\)\( T^{2} + \)\(24\!\cdots\!52\)\( T^{4} - \)\(18\!\cdots\!28\)\( T^{6} + \)\(80\!\cdots\!20\)\( T^{8} - 2129210402936425920 T^{10} + 3323642992497 T^{12} - 2819826 T^{14} + T^{16} \)
$79$ \( ( \)\(78\!\cdots\!32\)\( - 1762270868813547680 T - 42665226765100928 T^{2} + 476180853712 T^{3} + 405337308784 T^{4} - 44693534 T^{5} - 1142846 T^{6} + 109 T^{7} + T^{8} )^{2} \)
$83$ \( \)\(15\!\cdots\!76\)\( - \)\(14\!\cdots\!84\)\( T^{2} + \)\(25\!\cdots\!77\)\( T^{4} - \)\(12\!\cdots\!34\)\( T^{6} + \)\(26\!\cdots\!39\)\( T^{8} - 282388628108496040 T^{10} + 1370593986895 T^{12} - 2411074 T^{14} + T^{16} \)
$89$ \( ( \)\(15\!\cdots\!00\)\( + 9171892537616518200 T + 101208493454651010 T^{2} - 328435007142033 T^{3} - 908745918615 T^{4} + 2429925534 T^{5} - 266796 T^{6} - 2037 T^{7} + T^{8} )^{2} \)
$97$ \( \)\(17\!\cdots\!96\)\( - \)\(21\!\cdots\!68\)\( T^{2} + \)\(27\!\cdots\!20\)\( T^{4} - \)\(13\!\cdots\!08\)\( T^{6} + \)\(31\!\cdots\!24\)\( T^{8} - 39332894338327456611 T^{10} + 25767039322287 T^{12} - 8277621 T^{14} + T^{16} \)
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