Properties

Label 22.5.d.a
Level $22$
Weight $5$
Character orbit 22.d
Analytic conductor $2.274$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 22.d (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.27413918784\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 4 x^{15} + 138 x^{14} - 428 x^{13} + 7783 x^{12} - 18620 x^{11} + 235604 x^{10} - 425164 x^{9} + 4199998 x^{8} - 5446172 x^{7} + 45313276 x^{6} + \cdots + 1499670491 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 11^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{13} - \beta_{10} + 3 \beta_{3} + \beta_{2} + 1) q^{3} + ( - 8 \beta_{5} + 8 \beta_{3} + 8 \beta_{2} + 8) q^{4} + ( - \beta_{15} + \beta_{14} - \beta_{13} - 2 \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \cdots + 4) q^{5}+ \cdots + ( - \beta_{12} - \beta_{11} + 18 \beta_{10} - 8 \beta_{8} + 4 \beta_{6} + 11 \beta_{5} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + ( - \beta_{13} - \beta_{10} + 3 \beta_{3} + \beta_{2} + 1) q^{3} + ( - 8 \beta_{5} + 8 \beta_{3} + 8 \beta_{2} + 8) q^{4} + ( - \beta_{15} + \beta_{14} - \beta_{13} - 2 \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + \cdots + 4) q^{5}+ \cdots + (37 \beta_{15} + 12 \beta_{14} + 156 \beta_{13} + 95 \beta_{12} - 185 \beta_{11} + \cdots + 2569) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{3} + 32 q^{4} + 30 q^{5} - 80 q^{6} + 150 q^{7} + 110 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{3} + 32 q^{4} + 30 q^{5} - 80 q^{6} + 150 q^{7} + 110 q^{9} - 30 q^{11} - 384 q^{12} - 510 q^{13} - 96 q^{14} - 1398 q^{15} - 256 q^{16} + 1770 q^{17} + 1120 q^{18} + 1020 q^{19} - 240 q^{20} + 240 q^{22} - 2424 q^{23} - 640 q^{24} - 858 q^{25} + 480 q^{26} + 2224 q^{27} + 1600 q^{28} + 4890 q^{29} + 3360 q^{30} + 602 q^{31} - 2648 q^{33} - 3904 q^{34} - 8670 q^{35} + 720 q^{36} - 4518 q^{37} - 4800 q^{38} - 1130 q^{39} - 1280 q^{40} + 1290 q^{41} - 3808 q^{42} + 720 q^{44} + 12152 q^{45} + 4480 q^{46} + 642 q^{47} - 128 q^{48} + 9534 q^{49} + 6720 q^{50} - 1500 q^{51} + 4000 q^{52} + 2598 q^{53} + 2582 q^{55} - 3072 q^{56} + 9140 q^{57} - 6496 q^{58} + 6660 q^{59} - 5776 q^{60} - 27410 q^{61} - 19680 q^{62} - 27260 q^{63} + 2048 q^{64} + 2528 q^{66} + 21524 q^{67} + 14160 q^{68} + 11416 q^{69} + 34400 q^{70} - 5562 q^{71} + 10240 q^{72} - 7790 q^{73} + 5760 q^{74} + 3576 q^{75} - 1110 q^{77} - 39424 q^{78} - 2770 q^{79} - 3840 q^{80} - 25464 q^{81} - 17472 q^{82} - 36900 q^{83} - 24480 q^{84} - 24750 q^{85} + 624 q^{86} - 5760 q^{88} + 46596 q^{89} + 55360 q^{90} + 32370 q^{91} + 14112 q^{92} + 20722 q^{93} + 58880 q^{94} + 74250 q^{95} - 3732 q^{97} + 45802 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 138 x^{14} - 428 x^{13} + 7783 x^{12} - 18620 x^{11} + 235604 x^{10} - 425164 x^{9} + 4199998 x^{8} - 5446172 x^{7} + 45313276 x^{6} + \cdots + 1499670491 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 24\!\cdots\!52 \nu^{15} + \cdots + 64\!\cdots\!13 ) / 58\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 20\!\cdots\!00 \nu^{15} + \cdots + 54\!\cdots\!14 ) / 31\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 63\!\cdots\!72 \nu^{15} + \cdots + 61\!\cdots\!46 ) / 31\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 22\!\cdots\!65 \nu^{15} + \cdots + 27\!\cdots\!63 ) / 96\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11\!\cdots\!56 \nu^{15} + \cdots + 14\!\cdots\!38 ) / 31\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 36\!\cdots\!14 \nu^{15} + \cdots + 27\!\cdots\!44 ) / 96\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 76\!\cdots\!50 \nu^{15} + \cdots - 23\!\cdots\!38 ) / 19\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 14\!\cdots\!82 \nu^{15} + \cdots - 31\!\cdots\!49 ) / 17\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!18 \nu^{15} + \cdots + 17\!\cdots\!85 ) / 17\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14\!\cdots\!60 \nu^{15} + \cdots + 16\!\cdots\!66 ) / 96\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\!\cdots\!40 \nu^{15} + \cdots - 16\!\cdots\!43 ) / 47\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 63\!\cdots\!06 \nu^{15} + \cdots + 20\!\cdots\!26 ) / 19\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10\!\cdots\!20 \nu^{15} + \cdots + 22\!\cdots\!82 ) / 19\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13\!\cdots\!84 \nu^{15} + \cdots + 11\!\cdots\!23 ) / 19\!\cdots\!42 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 20\!\cdots\!92 \nu^{15} + \cdots - 84\!\cdots\!16 ) / 19\!\cdots\!42 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} - 2 \beta_{13} + \beta_{12} - 4 \beta_{11} + 3 \beta_{10} - 8 \beta_{9} + 2 \beta_{7} + 3 \beta_{6} - 5 \beta_{5} + 5 \beta_{4} - 2 \beta_{3} + 11 \beta_{2} - 10 \beta _1 + 8 ) / 22 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{15} - 5 \beta_{14} + 3 \beta_{13} - 2 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} + \beta_{8} + 8 \beta_{7} + 19 \beta_{6} + 41 \beta_{5} + 11 \beta_{4} - 39 \beta_{2} - 15 \beta _1 - 204 ) / 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 13 \beta_{15} - 21 \beta_{14} + 87 \beta_{13} - 2 \beta_{12} + 59 \beta_{11} - 242 \beta_{10} + 147 \beta_{9} + 35 \beta_{8} - 54 \beta_{7} - 89 \beta_{6} + 565 \beta_{5} - 362 \beta_{4} + 147 \beta_{3} - 463 \beta_{2} + 430 \beta _1 - 644 ) / 22 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 70 \beta_{15} + 202 \beta_{14} - 68 \beta_{13} + 2 \beta_{12} + 97 \beta_{11} - 333 \beta_{10} + 266 \beta_{9} - 47 \beta_{8} - 262 \beta_{7} - 546 \beta_{6} - 1244 \beta_{5} - 463 \beta_{4} - 202 \beta_{3} + 1272 \beta_{2} + \cdots + 4357 ) / 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1426 \beta_{15} + 1465 \beta_{14} - 2664 \beta_{13} - 356 \beta_{12} - 663 \beta_{11} + 7326 \beta_{10} - 2566 \beta_{9} - 920 \beta_{8} + 849 \beta_{7} + 1538 \beta_{6} - 26348 \beta_{5} + 12135 \beta_{4} + \cdots + 30337 ) / 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1186 \beta_{15} - 5716 \beta_{14} - 259 \beta_{13} - 788 \beta_{12} - 3255 \beta_{11} + 18109 \beta_{10} - 9342 \beta_{9} + 1427 \beta_{8} + 7233 \beta_{7} + 14013 \beta_{6} + 23034 \beta_{5} + 19094 \beta_{4} + \cdots - 92504 ) / 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 66328 \beta_{15} - 68005 \beta_{14} + 69680 \beta_{13} + 17335 \beta_{12} - 4648 \beta_{11} - 152756 \beta_{10} + 35376 \beta_{9} + 24821 \beta_{8} - 7695 \beta_{7} + 8177 \beta_{6} + 1018876 \beta_{5} + \cdots - 1240266 ) / 22 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 18107 \beta_{15} + 127200 \beta_{14} + 87194 \beta_{13} + 58295 \beta_{12} + 93976 \beta_{11} - 744827 \beta_{10} + 305261 \beta_{9} - 25825 \beta_{8} - 193850 \beta_{7} - 347107 \beta_{6} + \cdots + 1600827 ) / 11 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2438043 \beta_{15} + 2661543 \beta_{14} - 1490270 \beta_{13} - 563322 \beta_{12} + 752714 \beta_{11} + 1278139 \beta_{10} + 36160 \beta_{9} - 763269 \beta_{8} - 145705 \beta_{7} + \cdots + 45597572 ) / 22 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2717076 \beta_{15} - 1786399 \beta_{14} - 4663669 \beta_{13} - 2957353 \beta_{12} - 2373319 \beta_{11} + 25863483 \beta_{10} - 9468051 \beta_{9} - 236795 \beta_{8} + \cdots - 7547092 ) / 11 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 77811612 \beta_{15} - 93939692 \beta_{14} + 19463149 \beta_{13} + 13065524 \beta_{12} - 38772543 \beta_{11} + 84648077 \beta_{10} - 32253477 \beta_{9} + 23740684 \beta_{8} + \cdots - 1526859144 ) / 22 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 14322956 \beta_{15} - 2170156 \beta_{14} + 16325990 \beta_{13} + 11323158 \beta_{12} + 4305643 \beta_{11} - 70687957 \beta_{10} + 25248414 \beta_{9} + 4522317 \beta_{8} + \cdots - 104590147 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2179600762 \beta_{15} + 3051330737 \beta_{14} + 326168257 \beta_{13} - 113166782 \beta_{12} + 1585596008 \beta_{11} - 6534893955 \beta_{10} + 1864271297 \beta_{9} + \cdots + 46663603783 ) / 22 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 7130399021 \beta_{15} + 3431668430 \beta_{14} - 5699114548 \beta_{13} - 4644317402 \beta_{12} - 361974838 \beta_{11} + 19683189118 \beta_{10} + \cdots + 78938670464 ) / 11 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 51112645034 \beta_{15} - 91072961036 \beta_{14} - 39024604999 \beta_{13} - 9812286729 \beta_{12} - 57679374942 \beta_{11} + 311972097559 \beta_{10} + \cdots - 1283826177161 ) / 22 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/22\mathbb{Z}\right)^\times\).

\(n\) \(13\)
\(\chi(n)\) \(1 + \beta_{2} + \beta_{3} - \beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−0.309017 2.89265i
−0.309017 + 2.01862i
−0.309017 + 4.73267i
−0.309017 3.85864i
0.809017 + 5.77971i
0.809017 3.49146i
0.809017 + 3.04259i
0.809017 5.33084i
0.809017 5.77971i
0.809017 + 3.49146i
0.809017 3.04259i
0.809017 + 5.33084i
−0.309017 + 2.89265i
−0.309017 2.01862i
−0.309017 4.73267i
−0.309017 + 3.85864i
−1.66251 + 2.28825i −1.86966 + 5.75422i −2.47214 7.60845i −23.9466 + 17.3982i −10.0587 13.8447i −38.7210 + 12.5812i 21.5200 + 6.99226i 35.9150 + 26.0937i 83.7204i
7.2 −1.66251 + 2.28825i 3.38726 10.4249i −2.47214 7.60845i 17.3145 12.5798i 18.2234 + 25.0824i 27.3556 8.88838i 21.5200 + 6.99226i −31.6748 23.0131i 60.5339i
7.3 1.66251 2.28825i 0.309288 0.951890i −2.47214 7.60845i 35.4031 25.7219i −1.66396 2.29025i −65.6073 + 21.3171i −21.5200 6.99226i 64.7199 + 47.0218i 123.774i
7.4 1.66251 2.28825i 3.26328 10.0434i −2.47214 7.60845i −24.6252 + 17.8912i −17.5564 24.1643i 86.5218 28.1126i −21.5200 6.99226i −24.6896 17.9380i 86.0928i
13.1 −2.68999 0.874032i −3.20187 + 2.32629i 6.47214 + 4.70228i 13.3838 + 41.1910i 10.6463 3.45918i −7.80030 + 10.7362i −13.3001 18.3060i −20.1901 + 62.1386i 122.501i
13.2 −2.68999 0.874032i 4.50928 3.27619i 6.47214 + 4.70228i −10.0117 30.8128i −14.9934 + 4.87166i 50.5008 69.5083i −13.3001 18.3060i −15.4301 + 47.4891i 91.6368i
13.3 2.68999 + 0.874032i −13.1019 + 9.51912i 6.47214 + 4.70228i 8.38512 + 25.8068i −43.5642 + 14.1549i 26.6976 36.7461i 13.3001 + 18.3060i 56.0169 172.402i 76.7489i
13.4 2.68999 + 0.874032i 5.70436 4.14446i 6.47214 + 4.70228i −0.903101 2.77946i 18.9671 6.16277i −3.94724 + 5.43292i 13.3001 + 18.3060i −9.66723 + 29.7527i 8.26607i
17.1 −2.68999 + 0.874032i −3.20187 2.32629i 6.47214 4.70228i 13.3838 41.1910i 10.6463 + 3.45918i −7.80030 10.7362i −13.3001 + 18.3060i −20.1901 62.1386i 122.501i
17.2 −2.68999 + 0.874032i 4.50928 + 3.27619i 6.47214 4.70228i −10.0117 + 30.8128i −14.9934 4.87166i 50.5008 + 69.5083i −13.3001 + 18.3060i −15.4301 47.4891i 91.6368i
17.3 2.68999 0.874032i −13.1019 9.51912i 6.47214 4.70228i 8.38512 25.8068i −43.5642 14.1549i 26.6976 + 36.7461i 13.3001 18.3060i 56.0169 + 172.402i 76.7489i
17.4 2.68999 0.874032i 5.70436 + 4.14446i 6.47214 4.70228i −0.903101 + 2.77946i 18.9671 + 6.16277i −3.94724 5.43292i 13.3001 18.3060i −9.66723 29.7527i 8.26607i
19.1 −1.66251 2.28825i −1.86966 5.75422i −2.47214 + 7.60845i −23.9466 17.3982i −10.0587 + 13.8447i −38.7210 12.5812i 21.5200 6.99226i 35.9150 26.0937i 83.7204i
19.2 −1.66251 2.28825i 3.38726 + 10.4249i −2.47214 + 7.60845i 17.3145 + 12.5798i 18.2234 25.0824i 27.3556 + 8.88838i 21.5200 6.99226i −31.6748 + 23.0131i 60.5339i
19.3 1.66251 + 2.28825i 0.309288 + 0.951890i −2.47214 + 7.60845i 35.4031 + 25.7219i −1.66396 + 2.29025i −65.6073 21.3171i −21.5200 + 6.99226i 64.7199 47.0218i 123.774i
19.4 1.66251 + 2.28825i 3.26328 + 10.0434i −2.47214 + 7.60845i −24.6252 17.8912i −17.5564 + 24.1643i 86.5218 + 28.1126i −21.5200 + 6.99226i −24.6896 + 17.9380i 86.0928i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.5.d.a 16
3.b odd 2 1 198.5.j.a 16
4.b odd 2 1 176.5.n.c 16
11.c even 5 1 242.5.b.e 16
11.d odd 10 1 inner 22.5.d.a 16
11.d odd 10 1 242.5.b.e 16
33.f even 10 1 198.5.j.a 16
44.g even 10 1 176.5.n.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.5.d.a 16 1.a even 1 1 trivial
22.5.d.a 16 11.d odd 10 1 inner
176.5.n.c 16 4.b odd 2 1
176.5.n.c 16 44.g even 10 1
198.5.j.a 16 3.b odd 2 1
198.5.j.a 16 33.f even 10 1
242.5.b.e 16 11.c even 5 1
242.5.b.e 16 11.d odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(22, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 8 T^{6} + 64 T^{4} - 512 T^{2} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + 2 T^{15} + \cdots + 3117763586961 \) Copy content Toggle raw display
$5$ \( T^{16} - 30 T^{15} + \cdots + 88\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( T^{16} - 150 T^{15} + \cdots + 65\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{16} + 30 T^{15} + \cdots + 21\!\cdots\!21 \) Copy content Toggle raw display
$13$ \( T^{16} + 510 T^{15} + \cdots + 48\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{16} - 1770 T^{15} + \cdots + 61\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( T^{16} - 1020 T^{15} + \cdots + 23\!\cdots\!21 \) Copy content Toggle raw display
$23$ \( (T^{8} + 1212 T^{7} + \cdots + 70\!\cdots\!76)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} - 4890 T^{15} + \cdots + 39\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( T^{16} - 602 T^{15} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{16} + 4518 T^{15} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{16} - 1290 T^{15} + \cdots + 98\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( T^{16} + 33713554 T^{14} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{16} - 642 T^{15} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{16} - 2598 T^{15} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{16} - 6660 T^{15} + \cdots + 47\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( T^{16} + 27410 T^{15} + \cdots + 95\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{8} - 10762 T^{7} + \cdots - 18\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 5562 T^{15} + \cdots + 52\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( T^{16} + 7790 T^{15} + \cdots + 87\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{16} + 2770 T^{15} + \cdots + 81\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{16} + 36900 T^{15} + \cdots + 80\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{8} - 23298 T^{7} + \cdots - 37\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 3732 T^{15} + \cdots + 18\!\cdots\!21 \) Copy content Toggle raw display
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