Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [640,2,Mod(543,640)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(640, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("640.543");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 640 = 2^{7} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 640.j (of order \(4\), degree \(2\), not minimal) |
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: | no |
| Analytic conductor: | \(5.11042572936\) |
| Analytic rank: | \(0\) |
| Dimension: | \(18\) |
| Relative dimension: | \(9\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{18} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + \cdots + 512 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{13} \) |
| Twist minimal: | no (minimal twist has level 80) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + \cdots + 512 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 75 \nu^{17} + 89 \nu^{16} + 248 \nu^{15} + 6 \nu^{14} - 375 \nu^{13} - 1487 \nu^{12} - 2550 \nu^{11} + \cdots - 28416 ) / 640 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 100 \nu^{17} - 111 \nu^{16} - 342 \nu^{15} - 14 \nu^{14} + 460 \nu^{13} + 1963 \nu^{12} + \cdots + 41344 ) / 640 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 191 \nu^{17} - 380 \nu^{16} - 1110 \nu^{15} - 1252 \nu^{14} - 997 \nu^{13} + 1614 \nu^{12} + \cdots + 38912 ) / 1280 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 182 \nu^{17} - 271 \nu^{16} - 762 \nu^{15} - 418 \nu^{14} + 286 \nu^{13} + 2911 \nu^{12} + \cdots + 58368 ) / 640 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 330 \nu^{17} - 407 \nu^{16} - 1174 \nu^{15} - 78 \nu^{14} + 1610 \nu^{13} + 6791 \nu^{12} + \cdots + 135168 ) / 640 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 357 \nu^{17} - 493 \nu^{16} - 1406 \nu^{15} - 506 \nu^{14} + 1101 \nu^{13} + 6607 \nu^{12} + \cdots + 134656 ) / 640 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 228 \nu^{17} + 359 \nu^{16} + 1013 \nu^{15} + 672 \nu^{14} - 134 \nu^{13} - 3459 \nu^{12} + \cdots - 71232 ) / 320 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 129 \nu^{17} + 186 \nu^{16} + 524 \nu^{15} + 232 \nu^{14} - 321 \nu^{13} - 2280 \nu^{12} + \cdots - 46208 ) / 128 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 1413 \nu^{17} + 1966 \nu^{16} + 5582 \nu^{15} + 2000 \nu^{14} - 4409 \nu^{13} - 26220 \nu^{12} + \cdots - 528640 ) / 1280 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 1671 \nu^{17} - 2410 \nu^{16} - 6790 \nu^{15} - 3072 \nu^{14} + 3963 \nu^{13} + 29124 \nu^{12} + \cdots + 588032 ) / 1280 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 1861 \nu^{17} - 2632 \nu^{16} - 7454 \nu^{15} - 3140 \nu^{14} + 4833 \nu^{13} + 32910 \nu^{12} + \cdots + 670720 ) / 1280 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 1039 \nu^{17} + 1484 \nu^{16} + 4198 \nu^{15} + 1844 \nu^{14} - 2547 \nu^{13} - 18238 \nu^{12} + \cdots - 372864 ) / 640 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 593 \nu^{17} - 812 \nu^{16} - 2294 \nu^{15} - 774 \nu^{14} + 1929 \nu^{13} + 10958 \nu^{12} + \cdots + 218944 ) / 320 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 1177 \nu^{17} - 1672 \nu^{16} - 4754 \nu^{15} - 2092 \nu^{14} + 2901 \nu^{13} + 20614 \nu^{12} + \cdots + 418432 ) / 640 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 2339 \nu^{17} - 3438 \nu^{16} - 9706 \nu^{15} - 4920 \nu^{14} + 4527 \nu^{13} + 39440 \nu^{12} + \cdots + 807680 ) / 1280 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 3419 \nu^{17} - 4780 \nu^{16} - 13550 \nu^{15} - 5268 \nu^{14} + 9767 \nu^{13} + 61766 \nu^{12} + \cdots + 1253888 ) / 1280 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 1899 \nu^{17} - 2719 \nu^{16} - 7688 \nu^{15} - 3394 \nu^{14} + 4727 \nu^{13} + 33453 \nu^{12} + \cdots + 677504 ) / 640 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{16} - \beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{3} - 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - \beta_{16} - \beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} + \beta_{10} + 2 \beta_{7} - \beta_{5} + \cdots - 2 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{12} + \beta_{10} - \beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2 \beta_{17} - \beta_{16} + \beta_{15} - 3 \beta_{14} + 2 \beta_{13} - \beta_{12} - 3 \beta_{11} + \cdots + 6 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2 \beta_{17} + \beta_{16} - 2 \beta_{15} + 2 \beta_{14} - 4 \beta_{13} - \beta_{11} + 4 \beta_{10} + \cdots + 5 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 2 \beta_{17} - \beta_{16} - \beta_{15} + 4 \beta_{14} + \beta_{13} - 2 \beta_{12} - \beta_{11} + \cdots + 2 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 2 \beta_{17} + 3 \beta_{16} - 2 \beta_{15} - 6 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} + 7 \beta_{11} + \cdots - 1 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 2 \beta_{17} - \beta_{16} - 5 \beta_{15} - \beta_{14} + \beta_{12} - 3 \beta_{11} - \beta_{10} + \cdots + 4 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 13 \beta_{17} - 11 \beta_{16} + \beta_{15} + 5 \beta_{14} - 8 \beta_{13} + 4 \beta_{12} - \beta_{11} + \cdots + 20 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 12 \beta_{17} + 15 \beta_{16} - 25 \beta_{15} + 3 \beta_{14} - 2 \beta_{13} - 31 \beta_{12} - 11 \beta_{11} + \cdots + 30 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 50 \beta_{17} + 19 \beta_{16} + 14 \beta_{15} + 10 \beta_{14} + 36 \beta_{13} + 2 \beta_{12} + \cdots - 25 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 10 \beta_{17} - 30 \beta_{16} - 19 \beta_{15} + 13 \beta_{14} - 9 \beta_{13} + 7 \beta_{12} + 12 \beta_{11} + \cdots + 13 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 80 \beta_{17} + 13 \beta_{16} - 40 \beta_{15} - 28 \beta_{13} + 64 \beta_{12} + 67 \beta_{11} + \cdots + 163 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 76 \beta_{17} - 3 \beta_{16} - 39 \beta_{15} + 3 \beta_{14} - 84 \beta_{13} - 79 \beta_{12} - 81 \beta_{11} + \cdots - 114 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 4 \beta_{17} - 66 \beta_{16} - 30 \beta_{15} + 56 \beta_{14} + 32 \beta_{13} - 101 \beta_{12} + \cdots + 86 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 126 \beta_{17} + 145 \beta_{16} - 25 \beta_{15} + 99 \beta_{14} + 118 \beta_{13} + 213 \beta_{12} + \cdots + 258 ) / 4 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 162 \beta_{17} - 113 \beta_{16} - 198 \beta_{15} - 70 \beta_{14} - 108 \beta_{13} + 84 \beta_{12} + \cdots + 39 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).
| \(n\) | \(257\) | \(261\) | \(511\) |
| \(\chi(n)\) | \(-\beta_{8}\) | \(\beta_{8}\) | \(-1\) |
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 543.1 |
|
0 | − | 2.85601i | 0 | −1.43498 | + | 1.71489i | 0 | −0.458895 | + | 0.458895i | 0 | −5.15678 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 543.2 | 0 | − | 1.96251i | 0 | 1.72581 | + | 1.42182i | 0 | −1.60205 | + | 1.60205i | 0 | −0.851447 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 543.3 | 0 | − | 1.28110i | 0 | 0.841703 | − | 2.07160i | 0 | −1.13975 | + | 1.13975i | 0 | 1.35879 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 543.4 | 0 | − | 0.692712i | 0 | −2.22257 | + | 0.245325i | 0 | −0.343872 | + | 0.343872i | 0 | 2.52015 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 543.5 | 0 | − | 0.614566i | 0 | 2.07551 | − | 0.832020i | 0 | 2.83610 | − | 2.83610i | 0 | 2.62231 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 543.6 | 0 | 0.496487i | 0 | 0.987189 | + | 2.00635i | 0 | 1.55426 | − | 1.55426i | 0 | 2.75350 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 543.7 | 0 | 1.39319i | 0 | −0.535339 | − | 2.17104i | 0 | −2.13436 | + | 2.13436i | 0 | 1.05903 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 543.8 | 0 | 2.55161i | 0 | −1.66635 | − | 1.49107i | 0 | 2.40368 | − | 2.40368i | 0 | −3.51070 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 543.9 | 0 | 2.96561i | 0 | 2.22902 | + | 0.177336i | 0 | −0.115101 | + | 0.115101i | 0 | −5.79486 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.1 | 0 | − | 2.96561i | 0 | 2.22902 | − | 0.177336i | 0 | −0.115101 | − | 0.115101i | 0 | −5.79486 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.2 | 0 | − | 2.55161i | 0 | −1.66635 | + | 1.49107i | 0 | 2.40368 | + | 2.40368i | 0 | −3.51070 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.3 | 0 | − | 1.39319i | 0 | −0.535339 | + | 2.17104i | 0 | −2.13436 | − | 2.13436i | 0 | 1.05903 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.4 | 0 | − | 0.496487i | 0 | 0.987189 | − | 2.00635i | 0 | 1.55426 | + | 1.55426i | 0 | 2.75350 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.5 | 0 | 0.614566i | 0 | 2.07551 | + | 0.832020i | 0 | 2.83610 | + | 2.83610i | 0 | 2.62231 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.6 | 0 | 0.692712i | 0 | −2.22257 | − | 0.245325i | 0 | −0.343872 | − | 0.343872i | 0 | 2.52015 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.7 | 0 | 1.28110i | 0 | 0.841703 | + | 2.07160i | 0 | −1.13975 | − | 1.13975i | 0 | 1.35879 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.8 | 0 | 1.96251i | 0 | 1.72581 | − | 1.42182i | 0 | −1.60205 | − | 1.60205i | 0 | −0.851447 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 607.9 | 0 | 2.85601i | 0 | −1.43498 | − | 1.71489i | 0 | −0.458895 | − | 0.458895i | 0 | −5.15678 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 80.j | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 640.2.j.d | 18 | |
| 4.b | odd | 2 | 1 | 640.2.j.c | 18 | ||
| 5.c | odd | 4 | 1 | 640.2.s.d | 18 | ||
| 8.b | even | 2 | 1 | 80.2.j.b | ✓ | 18 | |
| 8.d | odd | 2 | 1 | 320.2.j.b | 18 | ||
| 16.e | even | 4 | 1 | 320.2.s.b | 18 | ||
| 16.e | even | 4 | 1 | 640.2.s.c | 18 | ||
| 16.f | odd | 4 | 1 | 80.2.s.b | yes | 18 | |
| 16.f | odd | 4 | 1 | 640.2.s.d | 18 | ||
| 20.e | even | 4 | 1 | 640.2.s.c | 18 | ||
| 24.h | odd | 2 | 1 | 720.2.bd.g | 18 | ||
| 40.e | odd | 2 | 1 | 1600.2.j.d | 18 | ||
| 40.f | even | 2 | 1 | 400.2.j.d | 18 | ||
| 40.i | odd | 4 | 1 | 80.2.s.b | yes | 18 | |
| 40.i | odd | 4 | 1 | 400.2.s.d | 18 | ||
| 40.k | even | 4 | 1 | 320.2.s.b | 18 | ||
| 40.k | even | 4 | 1 | 1600.2.s.d | 18 | ||
| 48.k | even | 4 | 1 | 720.2.z.g | 18 | ||
| 80.i | odd | 4 | 1 | 320.2.j.b | 18 | ||
| 80.j | even | 4 | 1 | 400.2.j.d | 18 | ||
| 80.j | even | 4 | 1 | inner | 640.2.j.d | 18 | |
| 80.k | odd | 4 | 1 | 400.2.s.d | 18 | ||
| 80.q | even | 4 | 1 | 1600.2.s.d | 18 | ||
| 80.s | even | 4 | 1 | 80.2.j.b | ✓ | 18 | |
| 80.t | odd | 4 | 1 | 640.2.j.c | 18 | ||
| 80.t | odd | 4 | 1 | 1600.2.j.d | 18 | ||
| 120.w | even | 4 | 1 | 720.2.z.g | 18 | ||
| 240.z | odd | 4 | 1 | 720.2.bd.g | 18 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 80.2.j.b | ✓ | 18 | 8.b | even | 2 | 1 | |
| 80.2.j.b | ✓ | 18 | 80.s | even | 4 | 1 | |
| 80.2.s.b | yes | 18 | 16.f | odd | 4 | 1 | |
| 80.2.s.b | yes | 18 | 40.i | odd | 4 | 1 | |
| 320.2.j.b | 18 | 8.d | odd | 2 | 1 | ||
| 320.2.j.b | 18 | 80.i | odd | 4 | 1 | ||
| 320.2.s.b | 18 | 16.e | even | 4 | 1 | ||
| 320.2.s.b | 18 | 40.k | even | 4 | 1 | ||
| 400.2.j.d | 18 | 40.f | even | 2 | 1 | ||
| 400.2.j.d | 18 | 80.j | even | 4 | 1 | ||
| 400.2.s.d | 18 | 40.i | odd | 4 | 1 | ||
| 400.2.s.d | 18 | 80.k | odd | 4 | 1 | ||
| 640.2.j.c | 18 | 4.b | odd | 2 | 1 | ||
| 640.2.j.c | 18 | 80.t | odd | 4 | 1 | ||
| 640.2.j.d | 18 | 1.a | even | 1 | 1 | trivial | |
| 640.2.j.d | 18 | 80.j | even | 4 | 1 | inner | |
| 640.2.s.c | 18 | 16.e | even | 4 | 1 | ||
| 640.2.s.c | 18 | 20.e | even | 4 | 1 | ||
| 640.2.s.d | 18 | 5.c | odd | 4 | 1 | ||
| 640.2.s.d | 18 | 16.f | odd | 4 | 1 | ||
| 720.2.z.g | 18 | 48.k | even | 4 | 1 | ||
| 720.2.z.g | 18 | 120.w | even | 4 | 1 | ||
| 720.2.bd.g | 18 | 24.h | odd | 2 | 1 | ||
| 720.2.bd.g | 18 | 240.z | odd | 4 | 1 | ||
| 1600.2.j.d | 18 | 40.e | odd | 2 | 1 | ||
| 1600.2.j.d | 18 | 80.t | odd | 4 | 1 | ||
| 1600.2.s.d | 18 | 40.k | even | 4 | 1 | ||
| 1600.2.s.d | 18 | 80.q | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(640, [\chi])\):
|
\( T_{3}^{18} + 32 T_{3}^{16} + 408 T_{3}^{14} + 2656 T_{3}^{12} + 9464 T_{3}^{10} + 18624 T_{3}^{8} + \cdots + 256 \)
|
|
\( T_{7}^{18} - 2 T_{7}^{17} + 2 T_{7}^{16} + 32 T_{7}^{15} + 200 T_{7}^{14} - 120 T_{7}^{13} + 352 T_{7}^{12} + \cdots + 288 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} \)
|
| $3$ |
\( T^{18} + 32 T^{16} + \cdots + 256 \)
|
| $5$ |
\( T^{18} - 4 T^{17} + \cdots + 1953125 \)
|
| $7$ |
\( T^{18} - 2 T^{17} + \cdots + 288 \)
|
| $11$ |
\( T^{18} - 2 T^{17} + \cdots + 5431808 \)
|
| $13$ |
\( (T^{9} - 56 T^{7} + \cdots + 8192)^{2} \)
|
| $17$ |
\( T^{18} + 6 T^{17} + \cdots + 512 \)
|
| $19$ |
\( T^{18} + 2 T^{17} + \cdots + 4608 \)
|
| $23$ |
\( T^{18} + \cdots + 17700587552 \)
|
| $29$ |
\( T^{18} - 14 T^{17} + \cdots + 82330112 \)
|
| $31$ |
\( T^{18} + 196 T^{16} + \cdots + 16384 \)
|
| $37$ |
\( (T^{9} + 4 T^{8} + \cdots - 757824)^{2} \)
|
| $41$ |
\( T^{18} + \cdots + 242788765696 \)
|
| $43$ |
\( (T^{9} - 22 T^{8} + \cdots - 580696)^{2} \)
|
| $47$ |
\( T^{18} + \cdots + 16870640672 \)
|
| $53$ |
\( T^{18} + \cdots + 48766772224 \)
|
| $59$ |
\( T^{18} + \cdots + 144166720393728 \)
|
| $61$ |
\( T^{18} + \cdots + 121236758528 \)
|
| $67$ |
\( (T^{9} + 6 T^{8} + \cdots + 745336)^{2} \)
|
| $71$ |
\( (T^{9} - 12 T^{8} + \cdots - 27648)^{2} \)
|
| $73$ |
\( T^{18} + \cdots + 35535647232 \)
|
| $79$ |
\( (T^{9} - 8 T^{8} + \cdots - 45002752)^{2} \)
|
| $83$ |
\( T^{18} + \cdots + 70791088097536 \)
|
| $89$ |
\( (T^{9} + 6 T^{8} + \cdots + 251904)^{2} \)
|
| $97$ |
\( T^{18} + \cdots + 380349381734912 \)
|
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