Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [714,2,Mod(545,714)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(714, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("714.545");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 714.f (of order \(2\), degree \(1\), 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: | \(5.70131870432\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{20} + 26 x^{18} + 287 x^{16} + 1758 x^{14} + 6551 x^{12} + 15294 x^{10} + 22146 x^{8} + 18872 x^{6} + \cdots + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{20} + 26 x^{18} + 287 x^{16} + 1758 x^{14} + 6551 x^{12} + 15294 x^{10} + 22146 x^{8} + 18872 x^{6} + \cdots + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - \nu^{19} - 26 \nu^{17} - 283 \nu^{15} - 1666 \nu^{13} - 5695 \nu^{11} - 11174 \nu^{9} + \cdots + 444 \nu ) / 48 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - \nu^{19} - 24 \nu^{17} - 237 \nu^{15} - 1238 \nu^{13} - 3635 \nu^{11} - 5748 \nu^{9} + \cdots + 692 \nu ) / 24 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 3 \nu^{19} - 70 \nu^{17} - 671 \nu^{15} - 3406 \nu^{13} - 9805 \nu^{11} - 15772 \nu^{9} + \cdots + 164 \nu ) / 24 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 9 \nu^{19} + 10 \nu^{18} - 222 \nu^{17} + 232 \nu^{16} - 2283 \nu^{15} + 2210 \nu^{14} - 12690 \nu^{13} + \cdots - 88 ) / 96 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 11 \nu^{19} + 2 \nu^{18} - 274 \nu^{17} + 44 \nu^{16} - 2861 \nu^{15} + 382 \nu^{14} - 16286 \nu^{13} + \cdots + 88 ) / 96 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 9 \nu^{19} - 10 \nu^{18} + 222 \nu^{17} - 240 \nu^{16} + 2283 \nu^{15} - 2394 \nu^{14} + 12690 \nu^{13} + \cdots + 56 ) / 96 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 11 \nu^{19} + 2 \nu^{18} + 274 \nu^{17} + 44 \nu^{16} + 2861 \nu^{15} + 382 \nu^{14} + 16286 \nu^{13} + \cdots + 88 ) / 96 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 15 \nu^{19} - 2 \nu^{18} - 354 \nu^{17} - 44 \nu^{16} - 3465 \nu^{15} - 382 \nu^{14} - 18270 \nu^{13} + \cdots - 88 ) / 96 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 9 \nu^{19} + 10 \nu^{18} + 222 \nu^{17} + 232 \nu^{16} + 2283 \nu^{15} + 2210 \nu^{14} + 12690 \nu^{13} + \cdots - 88 ) / 96 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 9 \nu^{19} + 10 \nu^{18} + 222 \nu^{17} + 240 \nu^{16} + 2283 \nu^{15} + 2394 \nu^{14} + 12690 \nu^{13} + \cdots - 56 ) / 96 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( \nu^{19} - 5 \nu^{18} + 28 \nu^{17} - 118 \nu^{16} + 329 \nu^{15} - 1151 \nu^{14} + 2106 \nu^{13} + \cdots - 36 ) / 24 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 2 \nu^{18} + 49 \nu^{16} + 500 \nu^{14} + 2756 \nu^{12} + 8888 \nu^{10} + 16924 \nu^{8} + \cdots + 1589 \nu^{2} ) / 6 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 17 \nu^{19} + 10 \nu^{18} + 414 \nu^{17} + 232 \nu^{16} + 4203 \nu^{15} + 2210 \nu^{14} + 23098 \nu^{13} + \cdots + 200 ) / 96 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 6 \nu^{18} - 146 \nu^{16} - 1480 \nu^{14} - 8111 \nu^{12} - 26057 \nu^{10} - 49619 \nu^{8} + \cdots - 56 ) / 12 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 3 \nu^{19} - 9 \nu^{18} + 74 \nu^{17} - 220 \nu^{16} + 763 \nu^{15} - 2243 \nu^{14} + 4274 \nu^{13} + \cdots - 172 ) / 24 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 3 \nu^{19} - 9 \nu^{18} - 74 \nu^{17} - 220 \nu^{16} - 763 \nu^{15} - 2243 \nu^{14} - 4274 \nu^{13} + \cdots - 172 ) / 24 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 5 \nu^{18} - 120 \nu^{16} - 1197 \nu^{14} - 6440 \nu^{12} - 20267 \nu^{10} - 37752 \nu^{8} + \cdots - 68 ) / 8 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 23 \nu^{19} + 34 \nu^{18} - 562 \nu^{17} + 820 \nu^{16} - 5729 \nu^{15} + 8222 \nu^{14} - 31646 \nu^{13} + \cdots + 536 ) / 96 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 23 \nu^{19} + 34 \nu^{18} + 562 \nu^{17} + 820 \nu^{16} + 5729 \nu^{15} + 8222 \nu^{14} + 31646 \nu^{13} + \cdots + 536 ) / 96 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{10} - \beta_{9} + \beta_{6} + \beta_{4} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{13} - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{4} - 6 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{19} - \beta_{18} - \beta_{16} + \beta_{15} - \beta_{13} - \beta_{11} - 5 \beta_{10} + \cdots - 2 \beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2 \beta_{19} + 2 \beta_{18} + 2 \beta_{14} - 7 \beta_{13} + 7 \beta_{11} + 6 \beta_{10} + 6 \beta_{9} + \cdots + 26 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 7 \beta_{19} + 7 \beta_{18} + 8 \beta_{16} - 8 \beta_{15} + 7 \beta_{13} + 7 \beta_{11} + \cdots + 16 \beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 19 \beta_{19} - 19 \beta_{18} - 2 \beta_{17} - \beta_{16} - \beta_{15} - 16 \beta_{14} + 42 \beta_{13} + \cdots - 128 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 43 \beta_{19} - 43 \beta_{18} - 53 \beta_{16} + 53 \beta_{15} - 39 \beta_{13} - 39 \beta_{11} + \cdots - 110 \beta_1 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 136 \beta_{19} + 136 \beta_{18} + 28 \beta_{17} + 8 \beta_{16} + 8 \beta_{15} + 98 \beta_{14} + \cdots + 668 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 256 \beta_{19} + 256 \beta_{18} + 335 \beta_{16} - 335 \beta_{15} + 204 \beta_{13} + 204 \beta_{11} + \cdots + 718 \beta_1 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 883 \beta_{19} - 883 \beta_{18} - 262 \beta_{17} - 41 \beta_{16} - 41 \beta_{15} - 554 \beta_{14} + \cdots - 3602 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 1508 \beta_{19} - 1508 \beta_{18} - 2085 \beta_{16} + 2085 \beta_{15} - 1048 \beta_{13} + \cdots - 4566 \beta_1 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 5489 \beta_{19} + 5489 \beta_{18} + 2058 \beta_{17} + 149 \beta_{16} + 149 \beta_{15} + 3054 \beta_{14} + \cdots + 19848 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 8847 \beta_{19} + 8847 \beta_{18} + 12880 \beta_{16} - 12880 \beta_{15} + 5379 \beta_{13} + \cdots + 28588 \beta_1 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 33387 \beta_{19} - 33387 \beta_{18} - 14714 \beta_{17} - 207 \beta_{16} - 207 \beta_{15} + \cdots - 111122 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 51838 \beta_{19} - 51838 \beta_{18} - 79134 \beta_{16} + 79134 \beta_{15} - 27798 \beta_{13} + \cdots - 177128 \beta_1 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 200756 \beta_{19} + 200756 \beta_{18} + 99464 \beta_{17} - 2820 \beta_{16} - 2820 \beta_{15} + \cdots + 629846 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 303801 \beta_{19} + 303801 \beta_{18} + 483897 \beta_{16} - 483897 \beta_{15} + 145193 \beta_{13} + \cdots + 1089106 \beta_1 ) / 2 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 1199690 \beta_{19} - 1199690 \beta_{18} - 648776 \beta_{17} + 39088 \beta_{16} + 39088 \beta_{15} + \cdots - 3605058 ) / 2 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 1782215 \beta_{19} - 1782215 \beta_{18} - 2946468 \beta_{16} + 2946468 \beta_{15} - 767867 \beta_{13} + \cdots - 6657168 \beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/714\mathbb{Z}\right)^\times\).
| \(n\) | \(239\) | \(409\) | \(547\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 545.1 |
|
− | 1.00000i | −1.71560 | + | 0.238187i | −1.00000 | 3.83647 | 0.238187 | + | 1.71560i | 2.07795 | − | 1.63772i | 1.00000i | 2.88653 | − | 0.817264i | − | 3.83647i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.2 | − | 1.00000i | −1.69530 | − | 0.354914i | −1.00000 | −1.25674 | −0.354914 | + | 1.69530i | 1.90520 | + | 1.83581i | 1.00000i | 2.74807 | + | 1.20337i | 1.25674i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.3 | − | 1.00000i | −1.32549 | − | 1.11493i | −1.00000 | 0.921622 | −1.11493 | + | 1.32549i | −2.20893 | − | 1.45623i | 1.00000i | 0.513861 | + | 2.95566i | − | 0.921622i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.4 | − | 1.00000i | −0.910071 | + | 1.47369i | −1.00000 | −4.16745 | 1.47369 | + | 0.910071i | 1.67652 | − | 2.04677i | 1.00000i | −1.34354 | − | 2.68233i | 4.16745i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.5 | − | 1.00000i | 0.0831242 | + | 1.73006i | −1.00000 | −0.954297 | 1.73006 | − | 0.0831242i | −1.97889 | + | 1.75613i | 1.00000i | −2.98618 | + | 0.287619i | 0.954297i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.6 | − | 1.00000i | 0.466154 | − | 1.66814i | −1.00000 | −2.78778 | −1.66814 | − | 0.466154i | −0.937546 | + | 2.47407i | 1.00000i | −2.56540 | − | 1.55522i | 2.78778i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.7 | − | 1.00000i | 0.552839 | + | 1.64145i | −1.00000 | 2.14107 | 1.64145 | − | 0.552839i | 2.62191 | − | 0.354348i | 1.00000i | −2.38874 | + | 1.81492i | − | 2.14107i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.8 | − | 1.00000i | 1.16723 | − | 1.27967i | −1.00000 | −0.709466 | −1.27967 | − | 1.16723i | 1.16052 | − | 2.37764i | 1.00000i | −0.275128 | − | 2.98736i | 0.709466i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.9 | − | 1.00000i | 1.65085 | − | 0.524099i | −1.00000 | 3.11389 | −0.524099 | − | 1.65085i | −2.51157 | − | 0.831863i | 1.00000i | 2.45064 | − | 1.73042i | − | 3.11389i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.10 | − | 1.00000i | 1.72625 | − | 0.141628i | −1.00000 | −0.137323 | −0.141628 | − | 1.72625i | 0.194830 | + | 2.63857i | 1.00000i | 2.95988 | − | 0.488972i | 0.137323i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.11 | 1.00000i | −1.71560 | − | 0.238187i | −1.00000 | 3.83647 | 0.238187 | − | 1.71560i | 2.07795 | + | 1.63772i | − | 1.00000i | 2.88653 | + | 0.817264i | 3.83647i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.12 | 1.00000i | −1.69530 | + | 0.354914i | −1.00000 | −1.25674 | −0.354914 | − | 1.69530i | 1.90520 | − | 1.83581i | − | 1.00000i | 2.74807 | − | 1.20337i | − | 1.25674i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.13 | 1.00000i | −1.32549 | + | 1.11493i | −1.00000 | 0.921622 | −1.11493 | − | 1.32549i | −2.20893 | + | 1.45623i | − | 1.00000i | 0.513861 | − | 2.95566i | 0.921622i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.14 | 1.00000i | −0.910071 | − | 1.47369i | −1.00000 | −4.16745 | 1.47369 | − | 0.910071i | 1.67652 | + | 2.04677i | − | 1.00000i | −1.34354 | + | 2.68233i | − | 4.16745i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.15 | 1.00000i | 0.0831242 | − | 1.73006i | −1.00000 | −0.954297 | 1.73006 | + | 0.0831242i | −1.97889 | − | 1.75613i | − | 1.00000i | −2.98618 | − | 0.287619i | − | 0.954297i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.16 | 1.00000i | 0.466154 | + | 1.66814i | −1.00000 | −2.78778 | −1.66814 | + | 0.466154i | −0.937546 | − | 2.47407i | − | 1.00000i | −2.56540 | + | 1.55522i | − | 2.78778i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.17 | 1.00000i | 0.552839 | − | 1.64145i | −1.00000 | 2.14107 | 1.64145 | + | 0.552839i | 2.62191 | + | 0.354348i | − | 1.00000i | −2.38874 | − | 1.81492i | 2.14107i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.18 | 1.00000i | 1.16723 | + | 1.27967i | −1.00000 | −0.709466 | −1.27967 | + | 1.16723i | 1.16052 | + | 2.37764i | − | 1.00000i | −0.275128 | + | 2.98736i | − | 0.709466i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.19 | 1.00000i | 1.65085 | + | 0.524099i | −1.00000 | 3.11389 | −0.524099 | + | 1.65085i | −2.51157 | + | 0.831863i | − | 1.00000i | 2.45064 | + | 1.73042i | 3.11389i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 545.20 | 1.00000i | 1.72625 | + | 0.141628i | −1.00000 | −0.137323 | −0.141628 | + | 1.72625i | 0.194830 | − | 2.63857i | − | 1.00000i | 2.95988 | + | 0.488972i | − | 0.137323i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 714.2.f.b | yes | 20 |
| 3.b | odd | 2 | 1 | 714.2.f.a | ✓ | 20 | |
| 7.b | odd | 2 | 1 | 714.2.f.a | ✓ | 20 | |
| 21.c | even | 2 | 1 | inner | 714.2.f.b | yes | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 714.2.f.a | ✓ | 20 | 3.b | odd | 2 | 1 | |
| 714.2.f.a | ✓ | 20 | 7.b | odd | 2 | 1 | |
| 714.2.f.b | yes | 20 | 1.a | even | 1 | 1 | trivial |
| 714.2.f.b | yes | 20 | 21.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{10} - 29T_{5}^{8} + 246T_{5}^{6} + 52T_{5}^{5} - 656T_{5}^{4} - 384T_{5}^{3} + 360T_{5}^{2} + 288T_{5} + 32 \)
acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{10} \)
|
| $3$ |
\( T^{20} - 2 T^{18} + \cdots + 59049 \)
|
| $5$ |
\( (T^{10} - 29 T^{8} + \cdots + 32)^{2} \)
|
| $7$ |
\( T^{20} + \cdots + 282475249 \)
|
| $11$ |
\( T^{20} + 110 T^{18} + \cdots + 160000 \)
|
| $13$ |
\( T^{20} + 146 T^{18} + \cdots + 13075456 \)
|
| $17$ |
\( (T - 1)^{20} \)
|
| $19$ |
\( T^{20} + 184 T^{18} + \cdots + 84934656 \)
|
| $23$ |
\( T^{20} + \cdots + 2945449984 \)
|
| $29$ |
\( T^{20} + \cdots + 8282456064 \)
|
| $31$ |
\( T^{20} + \cdots + 3397386240000 \)
|
| $37$ |
\( (T^{10} - 10 T^{9} + \cdots + 4384)^{2} \)
|
| $41$ |
\( (T^{10} - 16 T^{9} + \cdots + 4631552)^{2} \)
|
| $43$ |
\( (T^{10} - 10 T^{9} + \cdots - 331776)^{2} \)
|
| $47$ |
\( (T^{10} - 8 T^{9} + \cdots - 40288256)^{2} \)
|
| $53$ |
\( T^{20} + \cdots + 40282095616 \)
|
| $59$ |
\( (T^{10} + 18 T^{9} + \cdots - 54814720)^{2} \)
|
| $61$ |
\( T^{20} + \cdots + 82196707737600 \)
|
| $67$ |
\( (T^{10} + 22 T^{9} + \cdots - 16883712)^{2} \)
|
| $71$ |
\( T^{20} + \cdots + 12032573440000 \)
|
| $73$ |
\( T^{20} + \cdots + 24616839479296 \)
|
| $79$ |
\( (T^{10} - 617 T^{8} + \cdots - 655735680)^{2} \)
|
| $83$ |
\( (T^{10} - 16 T^{9} + \cdots + 62792128)^{2} \)
|
| $89$ |
\( (T^{10} - 18 T^{9} + \cdots + 52761600)^{2} \)
|
| $97$ |
\( T^{20} + \cdots + 12\!\cdots\!56 \)
|
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