Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1600,2,Mod(207,1600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1600, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1600.207");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1600 = 2^{6} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1600.s (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: | \(12.7760643234\) |
| 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: | \( 2^{15} \) |
| 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
\( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + \cdots + 512 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 75 \nu^{17} + 89 \nu^{16} + 248 \nu^{15} + 6 \nu^{14} - 375 \nu^{13} - 1487 \nu^{12} - 2550 \nu^{11} + \cdots - 28416 ) / 640 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 191 \nu^{17} - 380 \nu^{16} - 1110 \nu^{15} - 1252 \nu^{14} - 997 \nu^{13} + 1614 \nu^{12} + \cdots + 38912 ) / 1280 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 225 \nu^{17} + 299 \nu^{16} + 798 \nu^{15} + 106 \nu^{14} - 1145 \nu^{13} - 4757 \nu^{12} + \cdots - 90496 ) / 640 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 613 \nu^{17} - 951 \nu^{16} - 2732 \nu^{15} - 1810 \nu^{14} + 249 \nu^{13} + 9185 \nu^{12} + \cdots + 196480 ) / 640 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 655 \nu^{17} - 817 \nu^{16} - 2314 \nu^{15} - 198 \nu^{14} + 3215 \nu^{13} + 13511 \nu^{12} + \cdots + 267008 ) / 640 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 129 \nu^{17} - 186 \nu^{16} - 524 \nu^{15} - 232 \nu^{14} + 321 \nu^{13} + 2280 \nu^{12} + \cdots + 46208 ) / 128 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1371 \nu^{17} - 1882 \nu^{16} - 5234 \nu^{15} - 1600 \nu^{14} + 4903 \nu^{13} + 25980 \nu^{12} + \cdots + 513280 ) / 1280 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1413 \nu^{17} + 1966 \nu^{16} + 5582 \nu^{15} + 2000 \nu^{14} - 4409 \nu^{13} - 26220 \nu^{12} + \cdots - 528640 ) / 1280 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 813 \nu^{17} + 1211 \nu^{16} + 3432 \nu^{15} + 1850 \nu^{14} - 1369 \nu^{13} - 13525 \nu^{12} + \cdots - 277120 ) / 640 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 829 \nu^{17} + 1131 \nu^{16} + 3182 \nu^{15} + 1042 \nu^{14} - 2757 \nu^{13} - 15309 \nu^{12} + \cdots - 303232 ) / 640 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 1671 \nu^{17} + 2410 \nu^{16} + 6790 \nu^{15} + 3072 \nu^{14} - 3963 \nu^{13} - 29124 \nu^{12} + \cdots - 588032 ) / 1280 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 856 \nu^{17} + 1201 \nu^{16} + 3432 \nu^{15} + 1466 \nu^{14} - 2108 \nu^{13} - 14917 \nu^{12} + \cdots - 306816 ) / 640 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 1861 \nu^{17} + 2632 \nu^{16} + 7454 \nu^{15} + 3140 \nu^{14} - 4833 \nu^{13} - 32910 \nu^{12} + \cdots - 670720 ) / 1280 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 1039 \nu^{17} + 1484 \nu^{16} + 4198 \nu^{15} + 1844 \nu^{14} - 2547 \nu^{13} - 18238 \nu^{12} + \cdots - 372864 ) / 640 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 2339 \nu^{17} + 3438 \nu^{16} + 9706 \nu^{15} + 4920 \nu^{14} - 4527 \nu^{13} - 39440 \nu^{12} + \cdots - 807680 ) / 1280 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 1368 \nu^{17} + 1895 \nu^{16} + 5350 \nu^{15} + 1966 \nu^{14} - 4144 \nu^{13} - 24827 \nu^{12} + \cdots - 496256 ) / 640 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 3419 \nu^{17} - 4780 \nu^{16} - 13550 \nu^{15} - 5268 \nu^{14} + 9767 \nu^{13} + 61766 \nu^{12} + \cdots + 1253888 ) / 1280 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{17} + \beta_{13} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{2} - 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{16} + \beta_{15} - \beta_{13} + \beta_{10} + 2\beta_{9} - \beta_{5} + \beta_{4} - \beta_{3} + 3\beta _1 - 2 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{17} + \beta_{16} + \beta_{15} + 2 \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} - 2 \beta_{10} + \cdots + 3 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 2 \beta_{16} + \beta_{15} - 4 \beta_{14} + \beta_{13} + 4 \beta_{12} + 2 \beta_{11} - \beta_{10} + \cdots + 4 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4 \beta_{17} - 3 \beta_{16} + \beta_{15} + 2 \beta_{14} + 2 \beta_{13} + \beta_{12} - \beta_{11} + \cdots + 6 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - \beta_{17} + 3 \beta_{16} - \beta_{15} + 4 \beta_{14} + 5 \beta_{13} - 7 \beta_{12} - 3 \beta_{11} + \cdots + 7 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 2 \beta_{17} - \beta_{16} + 5 \beta_{15} - 2 \beta_{14} - 10 \beta_{13} + 5 \beta_{12} - 5 \beta_{11} + \cdots - 4 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 7 \beta_{17} - 7 \beta_{16} + 16 \beta_{15} - 8 \beta_{13} + 9 \beta_{12} + 9 \beta_{11} - 7 \beta_{10} + \cdots - 7 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 5 \beta_{17} - 3 \beta_{16} - 11 \beta_{15} + 18 \beta_{14} + 11 \beta_{13} + 17 \beta_{12} - \beta_{11} + \cdots + 49 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 7 \beta_{17} - 3 \beta_{16} + 2 \beta_{15} - 28 \beta_{14} + 34 \beta_{13} - 11 \beta_{12} - 3 \beta_{11} + \cdots + 53 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 7 \beta_{17} + 16 \beta_{16} + 14 \beta_{15} + 12 \beta_{14} - 27 \beta_{13} - 22 \beta_{12} - 18 \beta_{11} + \cdots - 53 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 5 \beta_{17} + \beta_{16} + 57 \beta_{15} + 40 \beta_{14} - 43 \beta_{13} + 39 \beta_{12} - 35 \beta_{11} + \cdots + 7 ) / 4 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 37 \beta_{17} - 8 \beta_{16} - 16 \beta_{15} + 64 \beta_{14} - 11 \beta_{13} + 24 \beta_{12} + \cdots + 219 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 42 \beta_{17} - 28 \beta_{16} + 5 \beta_{15} - 76 \beta_{14} + 115 \beta_{13} + 42 \beta_{12} + \cdots - 80 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 37 \beta_{17} - 9 \beta_{16} + 51 \beta_{15} - 90 \beta_{14} - 11 \beta_{13} - 17 \beta_{12} + \cdots + 181 ) / 4 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 158 \beta_{17} + 292 \beta_{16} + 65 \beta_{15} + 312 \beta_{14} - 63 \beta_{13} - 86 \beta_{12} + \cdots + 218 ) / 4 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 50 \beta_{17} - 123 \beta_{16} + 169 \beta_{15} + 14 \beta_{14} - 164 \beta_{13} + 133 \beta_{12} + \cdots + 68 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).
| \(n\) | \(577\) | \(901\) | \(1151\) |
| \(\chi(n)\) | \(\beta_{6}\) | \(\beta_{6}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 207.1 |
|
0 | −2.96561 | 0 | 0 | 0 | −0.115101 | − | 0.115101i | 0 | 5.79486 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 207.2 | 0 | −2.55161 | 0 | 0 | 0 | 2.40368 | + | 2.40368i | 0 | 3.51070 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 207.3 | 0 | −1.39319 | 0 | 0 | 0 | −2.13436 | − | 2.13436i | 0 | −1.05903 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 207.4 | 0 | −0.496487 | 0 | 0 | 0 | 1.55426 | + | 1.55426i | 0 | −2.75350 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 207.5 | 0 | 0.614566 | 0 | 0 | 0 | 2.83610 | + | 2.83610i | 0 | −2.62231 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 207.6 | 0 | 0.692712 | 0 | 0 | 0 | −0.343872 | − | 0.343872i | 0 | −2.52015 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 207.7 | 0 | 1.28110 | 0 | 0 | 0 | −1.13975 | − | 1.13975i | 0 | −1.35879 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 207.8 | 0 | 1.96251 | 0 | 0 | 0 | −1.60205 | − | 1.60205i | 0 | 0.851447 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 207.9 | 0 | 2.85601 | 0 | 0 | 0 | −0.458895 | − | 0.458895i | 0 | 5.15678 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 943.1 | 0 | −2.96561 | 0 | 0 | 0 | −0.115101 | + | 0.115101i | 0 | 5.79486 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 943.2 | 0 | −2.55161 | 0 | 0 | 0 | 2.40368 | − | 2.40368i | 0 | 3.51070 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 943.3 | 0 | −1.39319 | 0 | 0 | 0 | −2.13436 | + | 2.13436i | 0 | −1.05903 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 943.4 | 0 | −0.496487 | 0 | 0 | 0 | 1.55426 | − | 1.55426i | 0 | −2.75350 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 943.5 | 0 | 0.614566 | 0 | 0 | 0 | 2.83610 | − | 2.83610i | 0 | −2.62231 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 943.6 | 0 | 0.692712 | 0 | 0 | 0 | −0.343872 | + | 0.343872i | 0 | −2.52015 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 943.7 | 0 | 1.28110 | 0 | 0 | 0 | −1.13975 | + | 1.13975i | 0 | −1.35879 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 943.8 | 0 | 1.96251 | 0 | 0 | 0 | −1.60205 | + | 1.60205i | 0 | 0.851447 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 943.9 | 0 | 2.85601 | 0 | 0 | 0 | −0.458895 | + | 0.458895i | 0 | 5.15678 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 80.s | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1600.2.s.d | 18 | |
| 4.b | odd | 2 | 1 | 400.2.s.d | 18 | ||
| 5.b | even | 2 | 1 | 320.2.s.b | 18 | ||
| 5.c | odd | 4 | 1 | 320.2.j.b | 18 | ||
| 5.c | odd | 4 | 1 | 1600.2.j.d | 18 | ||
| 16.e | even | 4 | 1 | 400.2.j.d | 18 | ||
| 16.f | odd | 4 | 1 | 1600.2.j.d | 18 | ||
| 20.d | odd | 2 | 1 | 80.2.s.b | yes | 18 | |
| 20.e | even | 4 | 1 | 80.2.j.b | ✓ | 18 | |
| 20.e | even | 4 | 1 | 400.2.j.d | 18 | ||
| 40.e | odd | 2 | 1 | 640.2.s.d | 18 | ||
| 40.f | even | 2 | 1 | 640.2.s.c | 18 | ||
| 40.i | odd | 4 | 1 | 640.2.j.c | 18 | ||
| 40.k | even | 4 | 1 | 640.2.j.d | 18 | ||
| 60.h | even | 2 | 1 | 720.2.z.g | 18 | ||
| 60.l | odd | 4 | 1 | 720.2.bd.g | 18 | ||
| 80.i | odd | 4 | 1 | 400.2.s.d | 18 | ||
| 80.i | odd | 4 | 1 | 640.2.s.d | 18 | ||
| 80.j | even | 4 | 1 | 320.2.s.b | 18 | ||
| 80.k | odd | 4 | 1 | 320.2.j.b | 18 | ||
| 80.k | odd | 4 | 1 | 640.2.j.c | 18 | ||
| 80.q | even | 4 | 1 | 80.2.j.b | ✓ | 18 | |
| 80.q | even | 4 | 1 | 640.2.j.d | 18 | ||
| 80.s | even | 4 | 1 | 640.2.s.c | 18 | ||
| 80.s | even | 4 | 1 | inner | 1600.2.s.d | 18 | |
| 80.t | odd | 4 | 1 | 80.2.s.b | yes | 18 | |
| 240.bf | even | 4 | 1 | 720.2.z.g | 18 | ||
| 240.bm | 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 | 20.e | even | 4 | 1 | |
| 80.2.j.b | ✓ | 18 | 80.q | even | 4 | 1 | |
| 80.2.s.b | yes | 18 | 20.d | odd | 2 | 1 | |
| 80.2.s.b | yes | 18 | 80.t | odd | 4 | 1 | |
| 320.2.j.b | 18 | 5.c | odd | 4 | 1 | ||
| 320.2.j.b | 18 | 80.k | odd | 4 | 1 | ||
| 320.2.s.b | 18 | 5.b | even | 2 | 1 | ||
| 320.2.s.b | 18 | 80.j | even | 4 | 1 | ||
| 400.2.j.d | 18 | 16.e | even | 4 | 1 | ||
| 400.2.j.d | 18 | 20.e | even | 4 | 1 | ||
| 400.2.s.d | 18 | 4.b | odd | 2 | 1 | ||
| 400.2.s.d | 18 | 80.i | odd | 4 | 1 | ||
| 640.2.j.c | 18 | 40.i | odd | 4 | 1 | ||
| 640.2.j.c | 18 | 80.k | odd | 4 | 1 | ||
| 640.2.j.d | 18 | 40.k | even | 4 | 1 | ||
| 640.2.j.d | 18 | 80.q | even | 4 | 1 | ||
| 640.2.s.c | 18 | 40.f | even | 2 | 1 | ||
| 640.2.s.c | 18 | 80.s | even | 4 | 1 | ||
| 640.2.s.d | 18 | 40.e | odd | 2 | 1 | ||
| 640.2.s.d | 18 | 80.i | odd | 4 | 1 | ||
| 720.2.z.g | 18 | 60.h | even | 2 | 1 | ||
| 720.2.z.g | 18 | 240.bf | even | 4 | 1 | ||
| 720.2.bd.g | 18 | 60.l | odd | 4 | 1 | ||
| 720.2.bd.g | 18 | 240.bm | odd | 4 | 1 | ||
| 1600.2.j.d | 18 | 5.c | odd | 4 | 1 | ||
| 1600.2.j.d | 18 | 16.f | odd | 4 | 1 | ||
| 1600.2.s.d | 18 | 1.a | even | 1 | 1 | trivial | |
| 1600.2.s.d | 18 | 80.s | even | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{9} - 16T_{3}^{7} + 4T_{3}^{6} + 76T_{3}^{5} - 40T_{3}^{4} - 104T_{3}^{3} + 72T_{3}^{2} + 20T_{3} - 16 \)
acting on \(S_{2}^{\mathrm{new}}(1600, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{18} \)
|
| $3$ |
\( (T^{9} - 16 T^{7} + \cdots - 16)^{2} \)
|
| $5$ |
\( T^{18} \)
|
| $7$ |
\( T^{18} - 2 T^{17} + \cdots + 288 \)
|
| $11$ |
\( T^{18} - 2 T^{17} + \cdots + 5431808 \)
|
| $13$ |
\( T^{18} + 112 T^{16} + \cdots + 67108864 \)
|
| $17$ |
\( T^{18} - 6 T^{17} + \cdots + 512 \)
|
| $19$ |
\( T^{18} - 2 T^{17} + \cdots + 4608 \)
|
| $23$ |
\( T^{18} + \cdots + 17700587552 \)
|
| $29$ |
\( T^{18} - 14 T^{17} + \cdots + 82330112 \)
|
| $31$ |
\( T^{18} + 196 T^{16} + \cdots + 16384 \)
|
| $37$ |
\( T^{18} + \cdots + 574297214976 \)
|
| $41$ |
\( T^{18} + \cdots + 242788765696 \)
|
| $43$ |
\( T^{18} + \cdots + 337207844416 \)
|
| $47$ |
\( T^{18} + \cdots + 16870640672 \)
|
| $53$ |
\( (T^{9} + 6 T^{8} + \cdots - 220832)^{2} \)
|
| $59$ |
\( T^{18} + \cdots + 144166720393728 \)
|
| $61$ |
\( T^{18} + \cdots + 121236758528 \)
|
| $67$ |
\( T^{18} + \cdots + 555525752896 \)
|
| $71$ |
\( (T^{9} + 12 T^{8} + \cdots + 27648)^{2} \)
|
| $73$ |
\( T^{18} + \cdots + 35535647232 \)
|
| $79$ |
\( (T^{9} - 8 T^{8} + \cdots - 45002752)^{2} \)
|
| $83$ |
\( (T^{9} - 20 T^{8} + \cdots - 8413744)^{2} \)
|
| $89$ |
\( (T^{9} - 6 T^{8} + \cdots - 251904)^{2} \)
|
| $97$ |
\( T^{18} + \cdots + 380349381734912 \)
|
| show more | |
| show less |