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Properties

Label 1045.4.a.b
Level $1045$
Weight $4$
Character orbit 1045.a
Self dual yes
Analytic conductor $61.657$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1045,4,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(61.6569959560\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 8 x^{19} - 82 x^{18} + 700 x^{17} + 2826 x^{16} - 25467 x^{15} - 53768 x^{14} + 498499 x^{13} + 623613 x^{12} - 5673747 x^{11} - 4539454 x^{10} + 37893109 x^{9} + \cdots + 17756160 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + 5 q^{5} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{6} + ( - \beta_{7} - 7) q^{7} + ( - \beta_{5} + \beta_{4} - \beta_{2} + 4 \beta_1 - 7) q^{8} + (\beta_{9} - \beta_{5} + \beta_1 + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{5} - 1) q^{3} + (\beta_{2} - \beta_1 + 4) q^{4} + 5 q^{5} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{6} + ( - \beta_{7} - 7) q^{7} + ( - \beta_{5} + \beta_{4} - \beta_{2} + 4 \beta_1 - 7) q^{8} + (\beta_{9} - \beta_{5} + \beta_1 + 8) q^{9} + (5 \beta_1 - 5) q^{10} + 11 q^{11} + ( - \beta_{19} - \beta_{17} - \beta_{14} - \beta_{13} - \beta_{10} - \beta_{8} - 2 \beta_{7} - \beta_{6} + \cdots - 10) q^{12}+ \cdots + (11 \beta_{9} - 11 \beta_{5} + 11 \beta_1 + 88) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 8 x^{19} - 82 x^{18} + 700 x^{17} + 2826 x^{16} - 25467 x^{15} - 53768 x^{14} + 498499 x^{13} + 623613 x^{12} - 5673747 x^{11} - 4539454 x^{10} + 37893109 x^{9} + \cdots + 17756160 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 39\!\cdots\!97 \nu^{19} + \cdots - 10\!\cdots\!64 ) / 34\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16\!\cdots\!09 \nu^{19} + \cdots + 46\!\cdots\!92 ) / 13\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\!\cdots\!09 \nu^{19} + \cdots + 31\!\cdots\!20 ) / 13\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 26\!\cdots\!93 \nu^{19} + \cdots - 13\!\cdots\!08 ) / 98\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!65 \nu^{19} + \cdots + 15\!\cdots\!32 ) / 19\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26\!\cdots\!21 \nu^{19} + \cdots + 37\!\cdots\!12 ) / 34\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17\!\cdots\!43 \nu^{19} + \cdots + 91\!\cdots\!52 ) / 13\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27\!\cdots\!79 \nu^{19} + \cdots - 40\!\cdots\!20 ) / 19\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 60\!\cdots\!03 \nu^{19} + \cdots - 45\!\cdots\!08 ) / 34\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 25\!\cdots\!31 \nu^{19} + \cdots + 30\!\cdots\!16 ) / 13\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 32\!\cdots\!39 \nu^{19} + \cdots + 16\!\cdots\!72 ) / 17\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 27\!\cdots\!13 \nu^{19} + \cdots + 32\!\cdots\!48 ) / 13\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 19\!\cdots\!71 \nu^{19} + \cdots + 41\!\cdots\!76 ) / 69\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 58\!\cdots\!27 \nu^{19} + \cdots + 25\!\cdots\!88 ) / 13\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 62\!\cdots\!67 \nu^{19} + \cdots + 46\!\cdots\!20 ) / 13\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 35\!\cdots\!61 \nu^{19} + \cdots - 35\!\cdots\!28 ) / 69\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 85\!\cdots\!05 \nu^{19} + \cdots - 48\!\cdots\!48 ) / 13\!\cdots\!52 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + 2\beta_{2} + 20\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - \beta_{18} - \beta_{17} + \beta_{16} + \beta_{12} - \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 6 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 31 \beta_{2} + 48 \beta _1 + 224 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6 \beta_{19} - 6 \beta_{18} - 4 \beta_{17} + 3 \beta_{16} - \beta_{15} + 2 \beta_{14} + 4 \beta_{13} + 2 \beta_{12} + 3 \beta_{11} + \beta_{10} - 6 \beta_{9} + 10 \beta_{8} + 17 \beta_{7} + 12 \beta_{6} - 63 \beta_{5} + 40 \beta_{4} - 16 \beta_{3} + 106 \beta_{2} + 514 \beta _1 + 554 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 67 \beta_{19} - 60 \beta_{18} - 46 \beta_{17} + 48 \beta_{16} - 5 \beta_{15} + 11 \beta_{14} + 24 \beta_{13} + 39 \beta_{12} + 10 \beta_{11} + 15 \beta_{10} - 51 \beta_{9} + 119 \beta_{8} + 146 \beta_{7} + 121 \beta_{6} - 384 \beta_{5} + 165 \beta_{4} + \cdots + 5936 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 420 \beta_{19} - 372 \beta_{18} - 204 \beta_{17} + 187 \beta_{16} - 80 \beta_{15} + 161 \beta_{14} + 310 \beta_{13} + 130 \beta_{12} + 175 \beta_{11} + 137 \beta_{10} - 298 \beta_{9} + 709 \beta_{8} + 1098 \beta_{7} + 769 \beta_{6} + \cdots + 23126 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3377 \beta_{19} - 2777 \beta_{18} - 1582 \beta_{17} + 1819 \beta_{16} - 462 \beta_{15} + 962 \beta_{14} + 2014 \beta_{13} + 1367 \beta_{12} + 798 \beta_{11} + 1287 \beta_{10} - 1936 \beta_{9} + 5850 \beta_{8} + \cdots + 187075 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 21643 \beta_{19} - 17550 \beta_{18} - 7475 \beta_{17} + 8277 \beta_{16} - 4543 \beta_{15} + 9024 \beta_{14} + 17486 \beta_{13} + 6093 \beta_{12} + 8123 \beta_{11} + 9641 \beta_{10} - 11268 \beta_{9} + \cdots + 927526 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 155005 \beta_{19} - 119346 \beta_{18} - 48702 \beta_{17} + 64603 \beta_{16} - 28242 \beta_{15} + 56925 \beta_{14} + 116132 \beta_{13} + 49187 \beta_{12} + 44159 \beta_{11} + 76038 \beta_{10} + \cdots + 6575409 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1002188 \beta_{19} - 760767 \beta_{18} - 237489 \beta_{17} + 325670 \beta_{16} - 226505 \beta_{15} + 444272 \beta_{14} + 867742 \beta_{13} + 258394 \beta_{12} + 356225 \beta_{11} + \cdots + 36909652 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6818979 \beta_{19} - 4993113 \beta_{18} - 1399582 \beta_{17} + 2267171 \beta_{16} - 1454239 \beta_{15} + 2871821 \beta_{14} + 5767424 \beta_{13} + 1840975 \beta_{12} + 2121360 \beta_{11} + \cdots + 246131231 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 44172983 \beta_{19} - 31931267 \beta_{18} - 6866475 \beta_{17} + 12257988 \beta_{16} - 10557314 \beta_{15} + 20512025 \beta_{14} + 40228592 \beta_{13} + 10584947 \beta_{12} + \cdots + 1470459851 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 292956562 \beta_{19} - 206552694 \beta_{18} - 37604907 \beta_{17} + 80298409 \beta_{16} - 68567621 \beta_{15} + 133630846 \beta_{14} + 265796356 \beta_{13} + \cdots + 9540322513 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1895755282 \beta_{19} - 1322082746 \beta_{18} - 180712613 \beta_{17} + 455725798 \beta_{16} - 472381015 \beta_{15} + 911624935 \beta_{14} + 1790194732 \beta_{13} + \cdots + 58802396908 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 12398941408 \beta_{19} - 8498991098 \beta_{18} - 916947006 \beta_{17} + 2894980946 \beta_{16} - 3074530997 \beta_{15} + 5940401999 \beta_{14} + 11744810552 \beta_{13} + \cdots + 377068449485 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 80070537363 \beta_{19} - 54399311658 \beta_{18} - 4121592075 \beta_{17} + 16995242528 \beta_{16} - 20576464899 \beta_{15} + 39520278569 \beta_{14} + \cdots + 2361234897287 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 519479969859 \beta_{19} - 348726275330 \beta_{18} - 18319042043 \beta_{17} + 106513938337 \beta_{16} - 133699833268 \beta_{15} + 256854127331 \beta_{14} + \cdots + 15072375790850 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 3347512819449 \beta_{19} - 2231480755745 \beta_{18} - 65788200504 \beta_{17} + 640316557656 \beta_{16} - 879941448401 \beta_{15} + 1684327872066 \beta_{14} + \cdots + 95172808147204 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−4.47888
−4.19424
−3.95689
−3.42835
−2.55697
−2.53684
−2.41218
−0.822685
−0.731640
−0.259637
0.999641
1.12533
1.38435
2.13989
3.57546
3.74913
4.23917
4.30367
5.47116
6.39053
−5.47888 −2.61641 22.0181 5.00000 14.3350 −5.28814 −76.8037 −20.1544 −27.3944
1.2 −5.19424 8.38824 18.9801 5.00000 −43.5706 −4.76095 −57.0335 43.3626 −25.9712
1.3 −4.95689 −6.17893 16.5708 5.00000 30.6283 −36.0727 −42.4843 11.1792 −24.7845
1.4 −4.42835 −1.78158 11.6103 5.00000 7.88948 24.1111 −15.9878 −23.8260 −22.1418
1.5 −3.55697 3.83974 4.65205 5.00000 −13.6579 −6.66228 11.9086 −12.2564 −17.7849
1.6 −3.53684 1.54358 4.50922 5.00000 −5.45941 −15.5595 12.3463 −24.6173 −17.6842
1.7 −3.41218 −9.99772 3.64299 5.00000 34.1140 −3.40924 14.8669 72.9544 −17.0609
1.8 −1.82269 5.45780 −4.67782 5.00000 −9.94786 19.7082 23.1077 2.78762 −9.11343
1.9 −1.73164 −4.39818 −5.00142 5.00000 7.61606 22.5500 22.5138 −7.65602 −8.65820
1.10 −1.25964 7.52739 −6.41331 5.00000 −9.48178 −31.9473 18.1555 29.6616 −6.29819
1.11 −0.000359199 0 −7.63742 −8.00000 5.00000 0.00274335 −26.4307 0.00574718 31.3302 −0.00179599
1.12 0.125334 −3.53244 −7.98429 5.00000 −0.442735 −25.7104 −2.00338 −14.5219 0.626671
1.13 0.384350 −2.95996 −7.85228 5.00000 −1.13766 3.33433 −6.09282 −18.2387 1.92175
1.14 1.13989 3.45749 −6.70066 5.00000 3.94115 1.89071 −16.7571 −15.0457 5.69943
1.15 2.57546 8.83286 −1.36700 5.00000 22.7487 −30.7821 −24.1243 51.0194 12.8773
1.16 2.74913 −7.66455 −0.442295 5.00000 −21.0708 25.2191 −23.2090 31.7453 13.7456
1.17 3.23917 2.30947 2.49220 5.00000 7.48077 −5.43134 −17.8407 −21.6663 16.1958
1.18 3.30367 −7.25904 2.91425 5.00000 −23.9815 −9.92986 −16.8016 25.6936 16.5184
1.19 4.47116 0.109820 11.9912 5.00000 0.491023 −7.59378 17.8455 −26.9879 22.3558
1.20 5.39053 −8.44019 21.0578 5.00000 −45.4970 −18.2352 70.3882 44.2368 26.9526
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(-1\)
\(19\) \(-1\)

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 1045.4.a.b ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.4.a.b ✓ 20 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 12 T_{2}^{19} - 44 T_{2}^{18} - 1004 T_{2}^{17} - 727 T_{2}^{16} + 32533 T_{2}^{15} + 74163 T_{2}^{14} - 521840 T_{2}^{13} - 1700735 T_{2}^{12} + 4319428 T_{2}^{11} + 18581116 T_{2}^{10} - 16253206 T_{2}^{9} + \cdots - 3840 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1045))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 12 T^{19} - 44 T^{18} + \cdots - 3840 \) Copy content Toggle raw display
$3$ \( T^{20} + 21 T^{19} + \cdots - 751355754688 \) Copy content Toggle raw display
$5$ \( (T - 5)^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + 131 T^{19} + \cdots + 27\!\cdots\!72 \) Copy content Toggle raw display
$11$ \( (T - 11)^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + 223 T^{19} + \cdots - 33\!\cdots\!20 \) Copy content Toggle raw display
$17$ \( T^{20} + 471 T^{19} + \cdots + 92\!\cdots\!60 \) Copy content Toggle raw display
$19$ \( (T - 19)^{20} \) Copy content Toggle raw display
$23$ \( T^{20} + 653 T^{19} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{20} - 51 T^{19} + \cdots + 19\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{20} + 90 T^{19} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{20} + 96 T^{19} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{20} + 1284 T^{19} + \cdots + 62\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{20} + 1592 T^{19} + \cdots + 77\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{20} + 2030 T^{19} + \cdots - 48\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{20} + 943 T^{19} + \cdots - 13\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{20} + 515 T^{19} + \cdots - 18\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{20} - 446 T^{19} + \cdots + 99\!\cdots\!80 \) Copy content Toggle raw display
$67$ \( T^{20} + 1719 T^{19} + \cdots + 55\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{20} + 90 T^{19} + \cdots + 25\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{20} + 3763 T^{19} + \cdots - 68\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{20} - 2 T^{19} + \cdots - 39\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + 3436 T^{19} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{20} - 1700 T^{19} + \cdots + 43\!\cdots\!08 \) Copy content Toggle raw display
$97$ \( T^{20} + 1956 T^{19} + \cdots - 10\!\cdots\!24 \) Copy content Toggle raw display
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