Properties

Label 8037.2.a.o
Level $8037$
Weight $2$
Character orbit 8037.a
Self dual yes
Analytic conductor $64.176$
Analytic rank $0$
Dimension $18$
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] = [8037,2,Mod(1,8037)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8037, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8037.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8037.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: \(64.1757681045\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{17} - 28 x^{16} + 25 x^{15} + 322 x^{14} - 247 x^{13} - 1971 x^{12} + 1231 x^{11} + 6953 x^{10} - 3283 x^{9} - 14235 x^{8} + 4562 x^{7} + 15962 x^{6} - 2882 x^{5} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 893)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{10} q^{5} + (\beta_{17} + \beta_{14}) q^{7} + (\beta_{17} + \beta_{15} + \beta_{14} - \beta_{13} - \beta_{10} - \beta_{3} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{10} q^{5} + (\beta_{17} + \beta_{14}) q^{7} + (\beta_{17} + \beta_{15} + \beta_{14} - \beta_{13} - \beta_{10} - \beta_{3} - 1) q^{8} + ( - \beta_{16} + \beta_{12} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4}) q^{10} - \beta_{11} q^{11} + ( - \beta_{7} + 1) q^{13} + ( - \beta_{17} - \beta_{15} - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{8} + \beta_{6} - \beta_{4} + \beta_{2} + \cdots + 1) q^{14}+ \cdots + (2 \beta_{17} + 3 \beta_{15} - \beta_{13} + 6 \beta_{12} - \beta_{11} - \beta_{10} + 4 \beta_{9} + \beta_{8} + \cdots - 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - q^{2} + 21 q^{4} - 5 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} - 6 q^{11} + 21 q^{13} + 9 q^{14} + 23 q^{16} + 4 q^{17} - 18 q^{19} - 9 q^{20} + 28 q^{22} - 9 q^{23} + 11 q^{25} + q^{26} + 7 q^{28} - 12 q^{29} + 26 q^{31} + 7 q^{32} + 26 q^{34} - 9 q^{35} + 8 q^{37} + q^{38} + 16 q^{40} - 12 q^{41} + 28 q^{43} - 2 q^{44} - 33 q^{46} - 18 q^{47} + 17 q^{49} + 29 q^{50} + 30 q^{52} - 5 q^{53} + 28 q^{55} + 77 q^{56} - 6 q^{58} - 30 q^{59} - 16 q^{61} - 16 q^{62} + 28 q^{64} + 22 q^{65} + 45 q^{67} + 96 q^{68} - 2 q^{70} + q^{71} - 24 q^{73} + 19 q^{74} - 21 q^{76} - 2 q^{77} + 33 q^{79} + 25 q^{80} + 18 q^{82} + 13 q^{83} - 7 q^{85} - 3 q^{86} + 27 q^{88} + 6 q^{89} + 42 q^{91} + 11 q^{92} + q^{94} + 5 q^{95} + 44 q^{97} - 58 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - x^{17} - 28 x^{16} + 25 x^{15} + 322 x^{14} - 247 x^{13} - 1971 x^{12} + 1231 x^{11} + 6953 x^{10} - 3283 x^{9} - 14235 x^{8} + 4562 x^{7} + 15962 x^{6} - 2882 x^{5} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1375 \nu^{17} + 5084 \nu^{16} + 36270 \nu^{15} - 122133 \nu^{14} - 385293 \nu^{13} + 1142502 \nu^{12} + 2121571 \nu^{11} - 5271936 \nu^{10} - 6496241 \nu^{9} + \cdots + 15159 ) / 25060 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1245 \nu^{17} - 14641 \nu^{16} - 40655 \nu^{15} + 363742 \nu^{14} + 548917 \nu^{13} - 3562623 \nu^{12} - 3965914 \nu^{11} + 17502894 \nu^{10} + 16545409 \nu^{9} + \cdots - 125826 ) / 25060 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3027 \nu^{17} + 6697 \nu^{16} - 74329 \nu^{15} - 168948 \nu^{14} + 714237 \nu^{13} + 1691399 \nu^{12} - 3386660 \nu^{11} - 8576416 \nu^{10} + 8111547 \nu^{9} + \cdots + 23292 ) / 25060 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2588 \nu^{17} - 5615 \nu^{16} - 79161 \nu^{15} + 139029 \nu^{14} + 1002284 \nu^{13} - 1353473 \nu^{12} - 6792347 \nu^{11} + 6572488 \nu^{10} + 26570780 \nu^{9} + \cdots - 358605 ) / 25060 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1240 \nu^{17} - 3931 \nu^{16} + 38655 \nu^{15} + 100262 \nu^{14} - 492753 \nu^{13} - 1019383 \nu^{12} + 3303026 \nu^{11} + 5294509 \nu^{10} - 12477886 \nu^{9} + \cdots + 66324 ) / 12530 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1023 \nu^{17} + 7752 \nu^{16} + 35806 \nu^{15} - 193633 \nu^{14} - 510113 \nu^{13} + 1911944 \nu^{12} + 3828075 \nu^{11} - 9513046 \nu^{10} - 16327263 \nu^{9} + \cdots + 361577 ) / 12530 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1455 \nu^{17} - 6528 \nu^{16} - 49475 \nu^{15} + 157256 \nu^{14} + 679971 \nu^{13} - 1476119 \nu^{12} - 4872722 \nu^{11} + 6830792 \nu^{10} + 19578362 \nu^{9} + \cdots - 270523 ) / 12530 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2073 \nu^{17} + 1488 \nu^{16} - 54846 \nu^{15} - 36877 \nu^{14} + 589003 \nu^{13} + 363546 \nu^{12} - 3325055 \nu^{11} - 1834374 \nu^{10} + 10698493 \nu^{9} + 5102844 \nu^{8} + \cdots - 13747 ) / 12530 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4595 \nu^{17} + 9469 \nu^{16} - 115125 \nu^{15} - 236848 \nu^{14} + 1140117 \nu^{13} + 2345227 \nu^{12} - 5679104 \nu^{11} - 11727536 \nu^{10} + 14932039 \nu^{9} + \cdots - 33996 ) / 25060 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 6239 \nu^{17} + 4103 \nu^{16} - 153743 \nu^{15} - 100284 \nu^{14} + 1492971 \nu^{13} + 964945 \nu^{12} - 7287684 \nu^{11} - 4649798 \nu^{10} + 18950143 \nu^{9} + \cdots - 142672 ) / 25060 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3067 \nu^{17} + 290 \nu^{16} + 84064 \nu^{15} - 8371 \nu^{14} - 943021 \nu^{13} + 94912 \nu^{12} + 5611143 \nu^{11} - 535462 \nu^{10} - 19185335 \nu^{9} + 1570704 \nu^{8} + \cdots + 261015 ) / 12530 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3252 \nu^{17} + 2949 \nu^{16} - 82884 \nu^{15} - 73532 \nu^{14} + 844628 \nu^{13} + 726240 \nu^{12} - 4427692 \nu^{11} - 3626404 \nu^{10} + 12846929 \nu^{9} + 9675021 \nu^{8} + \cdots - 74371 ) / 12530 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 7089 \nu^{17} + 664 \nu^{16} - 186758 \nu^{15} - 19521 \nu^{14} + 1989179 \nu^{13} + 238418 \nu^{12} - 11064125 \nu^{11} - 1547932 \nu^{10} + 34713719 \nu^{9} + 5656942 \nu^{8} + \cdots + 299 ) / 25060 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 1383 \nu^{17} + 1899 \nu^{16} + 41081 \nu^{15} - 46402 \nu^{14} - 503867 \nu^{13} + 442805 \nu^{12} + 3299938 \nu^{11} - 2083314 \nu^{10} - 12447731 \nu^{9} + \cdots + 121854 ) / 3580 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 8478 \nu^{17} + 1039 \nu^{16} + 223616 \nu^{15} - 23022 \nu^{14} - 2385882 \nu^{13} + 184260 \nu^{12} + 13306628 \nu^{11} - 605434 \nu^{10} - 41938751 \nu^{9} + \cdots + 341599 ) / 12530 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{17} - \beta_{15} - \beta_{14} + \beta_{13} + \beta_{10} + \beta_{3} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{17} - 2\beta_{12} - \beta_{11} - \beta_{9} + \beta_{5} + 8\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 10 \beta_{17} - 10 \beta_{15} - 9 \beta_{14} + 9 \beta_{13} - 2 \beta_{12} + 10 \beta_{10} - \beta_{9} + 8 \beta_{3} + \beta_{2} + 20 \beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{17} - 2 \beta_{15} - \beta_{14} - \beta_{13} - 23 \beta_{12} - 10 \beta_{11} - \beta_{10} - 12 \beta_{9} + 2 \beta_{8} + 3 \beta_{6} + 11 \beta_{5} + \beta_{4} - \beta_{3} + 60 \beta_{2} - \beta _1 + 98 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 82 \beta_{17} - 81 \beta_{15} - 69 \beta_{14} + 67 \beta_{13} - 27 \beta_{12} - \beta_{11} + 79 \beta_{10} - 15 \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{6} + 53 \beta_{3} + 14 \beta_{2} + 114 \beta _1 + 54 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 112 \beta_{17} - 30 \beta_{15} - 16 \beta_{14} - 13 \beta_{13} - 208 \beta_{12} - 80 \beta_{11} - 12 \beta_{10} - 111 \beta_{9} + 29 \beta_{8} + \beta_{7} + 45 \beta_{6} + 94 \beta_{5} + 13 \beta_{4} - 15 \beta_{3} + 441 \beta_{2} - 16 \beta _1 + 639 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 637 \beta_{17} + 2 \beta_{16} - 620 \beta_{15} - 507 \beta_{14} + 476 \beta_{13} - 272 \beta_{12} - 20 \beta_{11} + 587 \beta_{10} - 158 \beta_{9} + 18 \beta_{8} - 16 \beta_{7} + 36 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 338 \beta_{3} + \cdots + 358 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 955 \beta_{17} - 3 \beta_{16} - 321 \beta_{15} - 180 \beta_{14} - 116 \beta_{13} - 1722 \beta_{12} - 601 \beta_{11} - 99 \beta_{10} - 932 \beta_{9} + 298 \beta_{8} + 19 \beta_{7} + 469 \beta_{6} + 735 \beta_{5} + 118 \beta_{4} + \cdots + 4302 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4845 \beta_{17} + 41 \beta_{16} - 4645 \beta_{15} - 3680 \beta_{14} + 3342 \beta_{13} - 2440 \beta_{12} - 254 \beta_{11} + 4280 \beta_{10} - 1444 \beta_{9} + 212 \beta_{8} - 175 \beta_{7} + 431 \beta_{6} + 84 \beta_{5} + \cdots + 2427 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 7786 \beta_{17} - 67 \beta_{16} - 3009 \beta_{15} - 1748 \beta_{14} - 880 \beta_{13} - 13644 \beta_{12} - 4417 \beta_{11} - 679 \beta_{10} - 7457 \beta_{9} + 2677 \beta_{8} + 237 \beta_{7} + 4219 \beta_{6} + \cdots + 29520 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 36459 \beta_{17} + 538 \beta_{16} - 34447 \beta_{15} - 26632 \beta_{14} + 23438 \beta_{13} - 20592 \beta_{12} - 2657 \beta_{11} + 31024 \beta_{10} - 12270 \beta_{9} + 2088 \beta_{8} - 1636 \beta_{7} + \cdots + 17018 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 61837 \beta_{17} - 934 \beta_{16} - 26339 \beta_{15} - 15697 \beta_{14} - 6050 \beta_{13} - 105420 \beta_{12} - 32186 \beta_{11} - 4009 \beta_{10} - 58034 \beta_{9} + 22463 \beta_{8} + 2459 \beta_{7} + \cdots + 205229 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 272599 \beta_{17} + 5784 \beta_{16} - 254027 \beta_{15} - 192725 \beta_{14} + 164755 \beta_{13} - 167527 \beta_{12} - 25047 \beta_{11} + 224540 \beta_{10} - 99951 \beta_{9} + 18714 \beta_{8} + \cdots + 123317 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 482938 \beta_{17} - 10494 \beta_{16} - 221286 \beta_{15} - 134481 \beta_{14} - 38421 \beta_{13} - 801916 \beta_{12} - 233707 \beta_{11} - 19374 \beta_{10} - 443929 \beta_{9} + 181158 \beta_{8} + \cdots + 1440869 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 2029601 \beta_{17} + 55621 \beta_{16} - 1867082 \beta_{15} - 1395823 \beta_{14} + 1161951 \beta_{13} - 1330972 \beta_{12} - 221695 \beta_{11} + 1625134 \beta_{10} - 793015 \beta_{9} + \cdots + 917721 \) Copy content Toggle raw display

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
2.72555
2.37327
2.33471
2.33085
1.52472
1.43818
1.35283
0.230018
0.146947
0.0898969
−0.494084
−1.15272
−1.40846
−1.48566
−1.61470
−2.13101
−2.60234
−2.65800
−2.72555 0 5.42860 0.150400 0 −4.90968 −9.34479 0 −0.409922
1.2 −2.37327 0 3.63241 2.08614 0 4.17378 −3.87415 0 −4.95098
1.3 −2.33471 0 3.45089 −4.12904 0 −0.743390 −3.38741 0 9.64013
1.4 −2.33085 0 3.43286 −1.49796 0 0.334615 −3.33977 0 3.49151
1.5 −1.52472 0 0.324761 1.59729 0 −1.34206 2.55426 0 −2.43541
1.6 −1.43818 0 0.0683575 −1.27662 0 4.10362 2.77805 0 1.83600
1.7 −1.35283 0 −0.169850 −0.632563 0 −0.940374 2.93544 0 0.855750
1.8 −0.230018 0 −1.94709 2.49148 0 −0.810750 0.907901 0 −0.573085
1.9 −0.146947 0 −1.97841 −2.45635 0 −1.38215 0.584613 0 0.360953
1.10 −0.0898969 0 −1.99192 −2.73841 0 4.94566 0.358861 0 0.246175
1.11 0.494084 0 −1.75588 3.82270 0 −3.40967 −1.85572 0 1.88874
1.12 1.15272 0 −0.671235 1.54521 0 2.91562 −3.07919 0 1.78119
1.13 1.40846 0 −0.0162457 −0.976908 0 −4.66797 −2.83980 0 −1.37593
1.14 1.48566 0 0.207175 −2.74559 0 0.513695 −2.66352 0 −4.07900
1.15 1.61470 0 0.607250 −2.02448 0 −1.29797 −2.24887 0 −3.26892
1.16 2.13101 0 2.54118 1.76024 0 2.12155 1.15327 0 3.75109
1.17 2.60234 0 4.77220 −3.17530 0 1.94962 7.21421 0 −8.26324
1.18 2.65800 0 5.06495 3.19976 0 1.44585 8.14662 0 8.50495
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(19\) \(1\)
\(47\) \(1\)

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 8037.2.a.o 18
3.b odd 2 1 893.2.a.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
893.2.a.c 18 3.b odd 2 1
8037.2.a.o 18 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):

\( T_{2}^{18} + T_{2}^{17} - 28 T_{2}^{16} - 25 T_{2}^{15} + 322 T_{2}^{14} + 247 T_{2}^{13} - 1971 T_{2}^{12} - 1231 T_{2}^{11} + 6953 T_{2}^{10} + 3283 T_{2}^{9} - 14235 T_{2}^{8} - 4562 T_{2}^{7} + 15962 T_{2}^{6} + 2882 T_{2}^{5} + \cdots + 9 \) Copy content Toggle raw display
\( T_{5}^{18} + 5 T_{5}^{17} - 38 T_{5}^{16} - 216 T_{5}^{15} + 507 T_{5}^{14} + 3677 T_{5}^{13} - 2465 T_{5}^{12} - 31997 T_{5}^{11} - 3827 T_{5}^{10} + 154097 T_{5}^{9} + 88150 T_{5}^{8} - 409912 T_{5}^{7} - 362661 T_{5}^{6} + \cdots + 24064 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + T^{17} - 28 T^{16} - 25 T^{15} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + 5 T^{17} - 38 T^{16} + \cdots + 24064 \) Copy content Toggle raw display
$7$ \( T^{18} - 3 T^{17} - 67 T^{16} + \cdots - 27072 \) Copy content Toggle raw display
$11$ \( T^{18} + 6 T^{17} - 76 T^{16} - 482 T^{15} + \cdots - 309 \) Copy content Toggle raw display
$13$ \( T^{18} - 21 T^{17} + 101 T^{16} + \cdots - 45342325 \) Copy content Toggle raw display
$17$ \( T^{18} - 4 T^{17} - 133 T^{16} + \cdots - 1488035 \) Copy content Toggle raw display
$19$ \( (T + 1)^{18} \) Copy content Toggle raw display
$23$ \( T^{18} + 9 T^{17} + \cdots + 16972790873 \) Copy content Toggle raw display
$29$ \( T^{18} + 12 T^{17} - 165 T^{16} + \cdots + 45111135 \) Copy content Toggle raw display
$31$ \( T^{18} - 26 T^{17} + \cdots - 258948800 \) Copy content Toggle raw display
$37$ \( T^{18} - 8 T^{17} + \cdots + 120115873600 \) Copy content Toggle raw display
$41$ \( T^{18} + 12 T^{17} + \cdots + 7449927507 \) Copy content Toggle raw display
$43$ \( T^{18} - 28 T^{17} + \cdots - 1126351309 \) Copy content Toggle raw display
$47$ \( (T + 1)^{18} \) Copy content Toggle raw display
$53$ \( T^{18} + 5 T^{17} + \cdots + 4105395112896 \) Copy content Toggle raw display
$59$ \( T^{18} + 30 T^{17} + \cdots - 4575239705 \) Copy content Toggle raw display
$61$ \( T^{18} + 16 T^{17} + \cdots + 3691665544139 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 419907802851136 \) Copy content Toggle raw display
$71$ \( T^{18} - T^{17} + \cdots - 70936699290979 \) Copy content Toggle raw display
$73$ \( T^{18} + 24 T^{17} + \cdots + 2805216398912 \) Copy content Toggle raw display
$79$ \( T^{18} - 33 T^{17} + \cdots + 56513898187485 \) Copy content Toggle raw display
$83$ \( T^{18} - 13 T^{17} + \cdots - 1326422638784 \) Copy content Toggle raw display
$89$ \( T^{18} - 6 T^{17} + \cdots + 51\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{18} - 44 T^{17} + \cdots - 19859654504448 \) Copy content Toggle raw display
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