Newspace parameters
| Level: | \( N \) | \(=\) | \( 2401 = 7^{4} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2401.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(19.1720815253\) |
| Analytic rank: | \(1\) |
| Dimension: | \(18\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} - \cdots)\) |
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| Defining polynomial: |
\( x^{18} - 26x^{16} + 249x^{14} - 1126x^{12} + 2746x^{10} - 3811x^{8} + 2997x^{6} - 1246x^{4} + 224x^{2} - 7 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{18} - 26x^{16} + 249x^{14} - 1126x^{12} + 2746x^{10} - 3811x^{8} + 2997x^{6} - 1246x^{4} + 224x^{2} - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 127 \nu^{16} - 4338 \nu^{14} + 57288 \nu^{12} - 368009 \nu^{10} + 1204612 \nu^{8} - 1998993 \nu^{6} + \cdots - 367 ) / 4243 \)
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| \(\beta_{3}\) | \(=\) |
\( ( - 411 \nu^{17} + 10063 \nu^{15} - 87106 \nu^{13} + 330832 \nu^{11} - 621324 \nu^{9} + \cdots + 46825 \nu ) / 4243 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 794 \nu^{16} - 19203 \nu^{14} + 163085 \nu^{12} - 604015 \nu^{10} + 1140236 \nu^{8} - 1198265 \nu^{6} + \cdots + 7862 ) / 4243 \)
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| \(\beta_{5}\) | \(=\) |
\( ( - 667 \nu^{17} + 19108 \nu^{15} - 207629 \nu^{13} + 1084606 \nu^{11} - 2914210 \nu^{9} + \cdots + 12986 \nu ) / 4243 \)
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| \(\beta_{6}\) | \(=\) |
\( ( 1660 \nu^{17} - 38961 \nu^{15} + 311140 \nu^{13} - 994403 \nu^{11} + 1308498 \nu^{9} + \cdots - 12548 \nu ) / 4243 \)
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| \(\beta_{7}\) | \(=\) |
\( ( 1660 \nu^{16} - 38961 \nu^{14} + 311140 \nu^{12} - 994403 \nu^{10} + 1308498 \nu^{8} - 406368 \nu^{6} + \cdots - 16791 ) / 4243 \)
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| \(\beta_{8}\) | \(=\) |
\( ( - 1916 \nu^{16} + 48006 \nu^{14} - 431663 \nu^{12} + 1748177 \nu^{10} - 3601384 \nu^{8} + \cdots - 21291 ) / 4243 \)
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| \(\beta_{9}\) | \(=\) |
\( ( 3682 \nu^{16} - 89552 \nu^{14} + 765227 \nu^{12} - 2830805 \nu^{10} + 5104225 \nu^{8} - 4574943 \nu^{6} + \cdots + 12379 ) / 4243 \)
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| \(\beta_{10}\) | \(=\) |
\( ( - 3661 \nu^{16} + 92042 \nu^{14} - 832195 \nu^{12} + 3399621 \nu^{10} - 7072542 \nu^{8} + \cdots - 16382 ) / 4243 \)
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| \(\beta_{11}\) | \(=\) |
\( \nu^{17} - 25\nu^{15} + 224\nu^{13} - 902\nu^{11} + 1844\nu^{9} - 1967\nu^{7} + 1030\nu^{5} - 216\nu^{3} + 8\nu \)
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| \(\beta_{12}\) | \(=\) |
\( ( 5037 \nu^{16} - 125278 \nu^{14} + 1113517 \nu^{12} - 4431201 \nu^{10} + 8964328 \nu^{8} + \cdots + 41806 ) / 4243 \)
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| \(\beta_{13}\) | \(=\) |
\( ( 5321 \nu^{17} - 131003 \nu^{15} + 1143335 \nu^{13} - 4394024 \nu^{11} + 8381040 \nu^{9} + \cdots - 409 \nu ) / 4243 \)
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| \(\beta_{14}\) | \(=\) |
\( ( - 5753 \nu^{17} + 138576 \nu^{15} - 1163473 \nu^{13} + 4156040 \nu^{11} - 7034047 \nu^{9} + \cdots + 55480 \nu ) / 4243 \)
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| \(\beta_{15}\) | \(=\) |
\( ( - 6697 \nu^{17} + 164239 \nu^{15} - 1424657 \nu^{13} + 5425604 \nu^{11} - 10272826 \nu^{9} + \cdots - 29258 \nu ) / 4243 \)
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| \(\beta_{16}\) | \(=\) |
\( ( - 9960 \nu^{17} + 250738 \nu^{15} - 2274168 \nu^{13} + 9365061 \nu^{11} - 19858678 \nu^{9} + \cdots - 153834 \nu ) / 4243 \)
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| \(\beta_{17}\) | \(=\) |
\( ( 12031 \nu^{16} - 299762 \nu^{14} + 2672414 \nu^{12} - 10690296 \nu^{10} + 21788500 \nu^{8} + \cdots + 90218 ) / 4243 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{12} + 2\beta_{9} - \beta_{7} - \beta_{4} + \beta_{2} + 2 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{15} - 2\beta_{14} - \beta_{13} + \beta_{11} - 2\beta_{6} + \beta_{3} + 8\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( -8\beta_{12} + \beta_{10} + 20\beta_{9} + 2\beta_{8} - 11\beta_{7} - 11\beta_{4} + 9\beta_{2} + 12 \)
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| \(\nu^{5}\) | \(=\) |
\( 10\beta_{15} - 20\beta_{14} - 10\beta_{13} + 11\beta_{11} - 22\beta_{6} + 2\beta_{5} + 9\beta_{3} + 71\beta_1 \)
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| \(\nu^{6}\) | \(=\) |
\( - \beta_{17} - 69 \beta_{12} + 9 \beta_{10} + 184 \beta_{9} + 23 \beta_{8} - 103 \beta_{7} - 101 \beta_{4} + \cdots + 101 \)
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| \(\nu^{7}\) | \(=\) |
\( \beta_{16} + 89 \beta_{15} - 184 \beta_{14} - 90 \beta_{13} + 101 \beta_{11} - 213 \beta_{6} + \cdots + 645 \beta_1 \)
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| \(\nu^{8}\) | \(=\) |
\( - 22 \beta_{17} - 606 \beta_{12} + 69 \beta_{10} + 1675 \beta_{9} + 215 \beta_{8} - 941 \beta_{7} + \cdots + 910 \)
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| \(\nu^{9}\) | \(=\) |
\( 22 \beta_{16} + 774 \beta_{15} - 1675 \beta_{14} - 804 \beta_{13} + 903 \beta_{11} - 2010 \beta_{6} + \cdots + 5889 \beta_1 \)
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| \(\nu^{10}\) | \(=\) |
\( - 330 \beta_{17} - 5333 \beta_{12} + 496 \beta_{10} + 15224 \beta_{9} + 1917 \beta_{8} - 8554 \beta_{7} + \cdots + 8306 \)
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| \(\nu^{11}\) | \(=\) |
\( 330 \beta_{16} + 6680 \beta_{15} - 15224 \beta_{14} - 7189 \beta_{13} + 8040 \beta_{11} + \cdots + 53835 \beta_1 \)
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| \(\nu^{12}\) | \(=\) |
\( - 4219 \beta_{17} - 46901 \beta_{12} + 3300 \beta_{10} + 138350 \beta_{9} + 16877 \beta_{8} + \cdots + 76018 \)
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| \(\nu^{13}\) | \(=\) |
\( 4219 \beta_{16} + 57402 \beta_{15} - 138350 \beta_{14} - 64324 \beta_{13} + 71525 \beta_{11} + \cdots + 492340 \beta_1 \)
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| \(\nu^{14}\) | \(=\) |
\( - 49558 \beta_{17} - 412035 \beta_{12} + 18985 \beta_{10} + 1257414 \beta_{9} + 147961 \beta_{8} + \cdots + 696207 \)
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| \(\nu^{15}\) | \(=\) |
\( 49558 \beta_{16} + 491404 \beta_{15} - 1257414 \beta_{14} - 575649 \beta_{13} + 636033 \beta_{11} + \cdots + 4503594 \beta_1 \)
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| \(\nu^{16}\) | \(=\) |
\( - 552953 \beta_{17} - 3615651 \beta_{12} + 72254 \beta_{10} + 11430058 \beta_{9} + 1294432 \beta_{8} + \cdots + 6377846 \)
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| \(\nu^{17}\) | \(=\) |
\( 552953 \beta_{16} + 4190135 \beta_{15} - 11430058 \beta_{14} - 5151497 \beta_{13} + \cdots + 41202849 \beta_1 \)
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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 |
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−2.44614 | −0.707302 | 3.98362 | −4.16494 | 1.73016 | 0 | −4.85223 | −2.49972 | 10.1880 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.44614 | 0.707302 | 3.98362 | 4.16494 | −1.73016 | 0 | −4.85223 | −2.49972 | −10.1880 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.67667 | −2.99719 | 0.811236 | 2.10413 | 5.02531 | 0 | 1.99317 | 5.98316 | −3.52794 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.67667 | 2.99719 | 0.811236 | −2.10413 | −5.02531 | 0 | 1.99317 | 5.98316 | 3.52794 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.57500 | −0.803517 | 0.480628 | −2.40115 | 1.26554 | 0 | 2.39301 | −2.35436 | 3.78181 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.57500 | 0.803517 | 0.480628 | 2.40115 | −1.26554 | 0 | 2.39301 | −2.35436 | −3.78181 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.364381 | −1.30216 | −1.86723 | 0.254380 | 0.474483 | 0 | 1.40914 | −1.30438 | −0.0926912 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −0.364381 | 1.30216 | −1.86723 | −0.254380 | −0.474483 | 0 | 1.40914 | −1.30438 | 0.0926912 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0.206475 | −0.197302 | −1.95737 | −3.82458 | −0.0407379 | 0 | −0.817096 | −2.96107 | −0.789679 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0.206475 | 0.197302 | −1.95737 | 3.82458 | 0.0407379 | 0 | −0.817096 | −2.96107 | 0.789679 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0.351716 | −1.50169 | −1.87630 | 1.06999 | −0.528170 | 0 | −1.36336 | −0.744913 | 0.376334 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0.351716 | 1.50169 | −1.87630 | −1.06999 | 0.528170 | 0 | −1.36336 | −0.744913 | −0.376334 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 1.36853 | −3.03764 | −0.127136 | 3.15896 | −4.15710 | 0 | −2.91104 | 6.22729 | 4.32313 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.14 | 1.36853 | 3.03764 | −0.127136 | −3.15896 | 4.15710 | 0 | −2.91104 | 6.22729 | −4.32313 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.15 | 2.04106 | −1.12259 | 2.16591 | 2.04996 | −2.29127 | 0 | 0.338625 | −1.73979 | 4.18408 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.16 | 2.04106 | 1.12259 | 2.16591 | −2.04996 | 2.29127 | 0 | 0.338625 | −1.73979 | −4.18408 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.17 | 2.09443 | −1.18059 | 2.38663 | 2.10761 | −2.47265 | 0 | 0.809772 | −1.60622 | 4.41423 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.18 | 2.09443 | 1.18059 | 2.38663 | −2.10761 | 2.47265 | 0 | 0.809772 | −1.60622 | −4.41423 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2401.2.a.g | ✓ | 18 |
| 7.b | odd | 2 | 1 | inner | 2401.2.a.g | ✓ | 18 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2401.2.a.g | ✓ | 18 | 1.a | even | 1 | 1 | trivial |
| 2401.2.a.g | ✓ | 18 | 7.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2401))\):
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\( T_{2}^{9} - 11T_{2}^{7} + T_{2}^{6} + 38T_{2}^{5} - 5T_{2}^{4} - 43T_{2}^{3} + 8T_{2}^{2} + 5T_{2} - 1 \)
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\( T_{3}^{18} - 26 T_{3}^{16} + 249 T_{3}^{14} - 1126 T_{3}^{12} + 2746 T_{3}^{10} - 3811 T_{3}^{8} + \cdots - 7 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{9} - 11 T^{7} + T^{6} + \cdots - 1)^{2} \)
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| $3$ |
\( T^{18} - 26 T^{16} + \cdots - 7 \)
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| $5$ |
\( T^{18} - 62 T^{16} + \cdots - 89383 \)
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| $7$ |
\( T^{18} \)
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| $11$ |
\( (T^{9} + 14 T^{8} + \cdots - 379)^{2} \)
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| $13$ |
\( T^{18} - 112 T^{16} + \cdots - 49268863 \)
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| $17$ |
\( T^{18} + \cdots - 189935767 \)
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| $19$ |
\( T^{18} + \cdots - 2228608543 \)
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| $23$ |
\( (T^{9} + 21 T^{8} + \cdots - 591347)^{2} \)
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| $29$ |
\( (T^{9} + 3 T^{8} + \cdots + 237341)^{2} \)
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| $31$ |
\( T^{18} + \cdots - 377437543 \)
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| $37$ |
\( (T^{9} - 123 T^{7} + \cdots + 1049)^{2} \)
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| $41$ |
\( T^{18} - 497 T^{16} + \cdots - 2362927 \)
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| $43$ |
\( (T^{9} + 24 T^{8} + \cdots - 11519033)^{2} \)
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| $47$ |
\( T^{18} + \cdots - 3684665050087 \)
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| $53$ |
\( (T^{9} + 35 T^{8} + \cdots + 3141473)^{2} \)
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| $59$ |
\( T^{18} + \cdots - 601997676343 \)
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| $61$ |
\( T^{18} + \cdots - 758069346223 \)
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| $67$ |
\( (T^{9} + 7 T^{8} + \cdots + 2367637)^{2} \)
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| $71$ |
\( (T^{9} + 52 T^{8} + \cdots - 4904089)^{2} \)
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| $73$ |
\( T^{18} - 404 T^{16} + \cdots - 19452223 \)
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| $79$ |
\( (T^{9} + 14 T^{8} + \cdots - 17179)^{2} \)
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| $83$ |
\( T^{18} + \cdots - 13\!\cdots\!47 \)
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| $89$ |
\( T^{18} + \cdots - 602817497012983 \)
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| $97$ |
\( T^{18} + \cdots - 159139638348943 \)
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