Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [186,2,Mod(119,186)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(186, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("186.119");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 186 = 2 \cdot 3 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 186.h (of order \(6\), degree \(2\), 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: | \(1.48521747760\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 2 x^{18} - x^{16} - 24 x^{15} - 28 x^{14} + 24 x^{13} + 37 x^{12} + 72 x^{11} + 426 x^{10} + \cdots + 59049 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{20} - 2 x^{18} - x^{16} - 24 x^{15} - 28 x^{14} + 24 x^{13} + 37 x^{12} + 72 x^{11} + 426 x^{10} + \cdots + 59049 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 7 \nu^{19} + 36 \nu^{18} - 121 \nu^{17} - 270 \nu^{16} + 385 \nu^{15} - 606 \nu^{14} + \cdots - 708588 ) / 524880 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{19} - 2 \nu^{17} - \nu^{15} - 24 \nu^{14} - 28 \nu^{13} + 24 \nu^{12} + 37 \nu^{11} + \cdots - 13122 \nu ) / 19683 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 151 \nu^{19} - 186 \nu^{18} - 1049 \nu^{17} + 1587 \nu^{16} - 601 \nu^{15} - 7569 \nu^{14} + \cdots + 7774785 ) / 3149280 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 349 \nu^{19} + 435 \nu^{18} + 580 \nu^{17} + 237 \nu^{16} - 2176 \nu^{15} - 8595 \nu^{14} + \cdots - 14683518 ) / 3149280 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 20 \nu^{19} - 65 \nu^{18} + 31 \nu^{17} + 4 \nu^{16} - 205 \nu^{15} + 716 \nu^{14} + \cdots + 583929 ) / 209952 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 382 \nu^{19} - 1005 \nu^{18} - 325 \nu^{17} + 714 \nu^{16} - 3677 \nu^{15} + 6690 \nu^{14} + \cdots + 33323319 ) / 3149280 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 89 \nu^{19} - 180 \nu^{18} - 407 \nu^{17} + 279 \nu^{16} + 125 \nu^{15} + 291 \nu^{14} + \cdots + 6790635 ) / 629856 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 145 \nu^{19} + 426 \nu^{18} + 79 \nu^{17} - 609 \nu^{16} - 73 \nu^{15} - 5121 \nu^{14} + \cdots - 6869367 ) / 1049760 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 691 \nu^{19} + 714 \nu^{18} + 841 \nu^{17} + 2757 \nu^{16} - 1411 \nu^{15} - 15219 \nu^{14} + \cdots - 12892365 ) / 3149280 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 746 \nu^{19} - 1047 \nu^{18} + 187 \nu^{17} - 1740 \nu^{16} + 35 \nu^{15} + 24432 \nu^{14} + \cdots + 19230291 ) / 3149280 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 134 \nu^{19} + 176 \nu^{18} + 239 \nu^{17} + 368 \nu^{16} + 121 \nu^{15} - 1916 \nu^{14} + \cdots - 4920750 ) / 524880 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 329 \nu^{19} - 2592 \nu^{18} - 2483 \nu^{17} + 3051 \nu^{16} - 10723 \nu^{15} + 2847 \nu^{14} + \cdots + 37574847 ) / 3149280 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 65 \nu^{19} + 9 \nu^{18} - 4 \nu^{17} + 225 \nu^{16} - 236 \nu^{15} - 1083 \nu^{14} + \cdots - 1180980 ) / 209952 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 974 \nu^{19} + 645 \nu^{18} + 2335 \nu^{17} - 6042 \nu^{16} + 5951 \nu^{15} + 28590 \nu^{14} + \cdots - 21041127 ) / 3149280 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 176 \nu^{19} + 507 \nu^{18} + 368 \nu^{17} + 255 \nu^{16} + 1300 \nu^{15} - 3867 \nu^{14} + \cdots - 7912566 ) / 524880 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 1441 \nu^{19} - 1533 \nu^{18} - 772 \nu^{17} - 1821 \nu^{16} + 568 \nu^{15} + 27423 \nu^{14} + \cdots + 52671708 ) / 3149280 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 830 \nu^{19} - 1206 \nu^{18} + 76 \nu^{17} - 2151 \nu^{16} - 2482 \nu^{15} + 18831 \nu^{14} + \cdots + 15136227 ) / 1574640 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 863 \nu^{19} + 2103 \nu^{18} + 1802 \nu^{17} + 627 \nu^{16} + 3664 \nu^{15} - 16443 \nu^{14} + \cdots - 39444732 ) / 1574640 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{19} + \beta_{18} - \beta_{15} - \beta_{10} - \beta_{5} - \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{19} - \beta_{18} - 2 \beta_{16} + \beta_{15} - 2 \beta_{12} + 2 \beta_{10} - 2 \beta_{9} + \cdots + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2 \beta_{17} - 2 \beta_{16} + 4 \beta_{14} - 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + \cdots + \beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( - 2 \beta_{19} - 2 \beta_{18} + 4 \beta_{16} - 6 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} + 4 \beta_{11} + \cdots + 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( 6 \beta_{18} + 2 \beta_{16} - 6 \beta_{15} - 4 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - 12 \beta_{6} + \cdots + 7 \)
|
| \(\nu^{7}\) | \(=\) |
\( 4 \beta_{19} + 6 \beta_{18} - 4 \beta_{17} - 6 \beta_{16} + 8 \beta_{15} + 14 \beta_{14} - 4 \beta_{13} + \cdots + 16 \)
|
| \(\nu^{8}\) | \(=\) |
\( 17 \beta_{19} + 19 \beta_{18} + 14 \beta_{17} - 23 \beta_{15} + 20 \beta_{14} - 14 \beta_{13} + \cdots + 10 \)
|
| \(\nu^{9}\) | \(=\) |
\( 23 \beta_{19} - 21 \beta_{18} + 40 \beta_{17} - 36 \beta_{16} + 17 \beta_{15} + 12 \beta_{14} + 16 \beta_{13} + \cdots + 1 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 56 \beta_{18} + 24 \beta_{17} - 24 \beta_{16} - 16 \beta_{14} + 12 \beta_{13} - 48 \beta_{12} + \cdots - 4 \beta_1 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 28 \beta_{19} + 44 \beta_{18} + 28 \beta_{17} - 60 \beta_{16} - 4 \beta_{15} - 16 \beta_{14} + \cdots + 180 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 112 \beta_{19} + 84 \beta_{18} + 220 \beta_{16} + 40 \beta_{15} + 212 \beta_{14} + 84 \beta_{11} + \cdots + 17 \)
|
| \(\nu^{13}\) | \(=\) |
\( 216 \beta_{19} + 588 \beta_{18} + 68 \beta_{17} + 152 \beta_{16} - 520 \beta_{15} + 84 \beta_{14} + \cdots - 152 \)
|
| \(\nu^{14}\) | \(=\) |
\( 465 \beta_{19} - 523 \beta_{18} + 296 \beta_{17} - 516 \beta_{16} + 639 \beta_{15} + 620 \beta_{14} + \cdots + 1216 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 123 \beta_{19} - 313 \beta_{18} + 1240 \beta_{17} - 1382 \beta_{16} - 51 \beta_{15} + 1284 \beta_{14} + \cdots - 67 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 584 \beta_{18} + 1538 \beta_{17} + 1534 \beta_{16} - 1452 \beta_{14} + 2314 \beta_{13} + \cdots - 2236 \beta_1 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 702 \beta_{19} + 2154 \beta_{18} + 676 \beta_{17} + 448 \beta_{16} - 2236 \beta_{15} - 766 \beta_{14} + \cdots - 800 \)
|
| \(\nu^{18}\) | \(=\) |
\( 1280 \beta_{19} + 2106 \beta_{18} - 882 \beta_{16} + 3322 \beta_{15} + 7196 \beta_{14} + 2028 \beta_{11} + \cdots - 33 \)
|
| \(\nu^{19}\) | \(=\) |
\( 796 \beta_{19} + 1986 \beta_{18} + 1768 \beta_{17} + 2042 \beta_{16} - 4880 \beta_{15} + 9266 \beta_{14} + \cdots + 13544 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/186\mathbb{Z}\right)^\times\).
| \(n\) | \(125\) | \(127\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{12}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 119.1 |
|
− | 1.00000i | −1.41564 | + | 0.997981i | −1.00000 | 0.213592 | − | 0.123317i | 0.997981 | + | 1.41564i | 2.27249 | − | 3.93606i | 1.00000i | 1.00807 | − | 2.82556i | −0.123317 | − | 0.213592i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 119.2 | − | 1.00000i | −0.669086 | − | 1.59760i | −1.00000 | −3.69011 | + | 2.13048i | −1.59760 | + | 0.669086i | 0.247791 | − | 0.429186i | 1.00000i | −2.10465 | + | 2.13786i | 2.13048 | + | 3.69011i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 119.3 | − | 1.00000i | 0.0908641 | + | 1.72967i | −1.00000 | −2.08105 | + | 1.20150i | 1.72967 | − | 0.0908641i | −1.68759 | + | 2.92298i | 1.00000i | −2.98349 | + | 0.314329i | 1.20150 | + | 2.08105i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 119.4 | − | 1.00000i | 0.356524 | − | 1.69496i | −1.00000 | 2.22053 | − | 1.28202i | −1.69496 | − | 0.356524i | 0.0797352 | − | 0.138105i | 1.00000i | −2.74578 | − | 1.20859i | −1.28202 | − | 2.22053i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 119.5 | − | 1.00000i | 1.63734 | + | 0.564913i | −1.00000 | 1.60499 | − | 0.926640i | 0.564913 | − | 1.63734i | −0.412428 | + | 0.714346i | 1.00000i | 2.36175 | + | 1.84991i | −0.926640 | − | 1.60499i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 119.6 | 1.00000i | −1.71810 | + | 0.219354i | −1.00000 | 3.69011 | − | 2.13048i | −0.219354 | − | 1.71810i | 0.247791 | − | 0.429186i | − | 1.00000i | 2.90377 | − | 0.753747i | 2.13048 | + | 3.69011i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 119.7 | 1.00000i | −1.28962 | + | 1.15624i | −1.00000 | −2.22053 | + | 1.28202i | −1.15624 | − | 1.28962i | 0.0797352 | − | 0.138105i | − | 1.00000i | 0.326223 | − | 2.98221i | −1.28202 | − | 2.22053i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 119.8 | 1.00000i | 0.156457 | − | 1.72497i | −1.00000 | −0.213592 | + | 0.123317i | 1.72497 | + | 0.156457i | 2.27249 | − | 3.93606i | − | 1.00000i | −2.95104 | − | 0.539768i | −0.123317 | − | 0.213592i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 119.9 | 1.00000i | 1.30790 | + | 1.13552i | −1.00000 | −1.60499 | + | 0.926640i | −1.13552 | + | 1.30790i | −0.412428 | + | 0.714346i | − | 1.00000i | 0.421193 | + | 2.97029i | −0.926640 | − | 1.60499i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 119.10 | 1.00000i | 1.54337 | − | 0.786142i | −1.00000 | 2.08105 | − | 1.20150i | 0.786142 | + | 1.54337i | −1.68759 | + | 2.92298i | − | 1.00000i | 1.76396 | − | 2.42661i | 1.20150 | + | 2.08105i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.1 | − | 1.00000i | −1.71810 | − | 0.219354i | −1.00000 | 3.69011 | + | 2.13048i | −0.219354 | + | 1.71810i | 0.247791 | + | 0.429186i | 1.00000i | 2.90377 | + | 0.753747i | 2.13048 | − | 3.69011i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.2 | − | 1.00000i | −1.28962 | − | 1.15624i | −1.00000 | −2.22053 | − | 1.28202i | −1.15624 | + | 1.28962i | 0.0797352 | + | 0.138105i | 1.00000i | 0.326223 | + | 2.98221i | −1.28202 | + | 2.22053i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.3 | − | 1.00000i | 0.156457 | + | 1.72497i | −1.00000 | −0.213592 | − | 0.123317i | 1.72497 | − | 0.156457i | 2.27249 | + | 3.93606i | 1.00000i | −2.95104 | + | 0.539768i | −0.123317 | + | 0.213592i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.4 | − | 1.00000i | 1.30790 | − | 1.13552i | −1.00000 | −1.60499 | − | 0.926640i | −1.13552 | − | 1.30790i | −0.412428 | − | 0.714346i | 1.00000i | 0.421193 | − | 2.97029i | −0.926640 | + | 1.60499i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.5 | − | 1.00000i | 1.54337 | + | 0.786142i | −1.00000 | 2.08105 | + | 1.20150i | 0.786142 | − | 1.54337i | −1.68759 | − | 2.92298i | 1.00000i | 1.76396 | + | 2.42661i | 1.20150 | − | 2.08105i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.6 | 1.00000i | −1.41564 | − | 0.997981i | −1.00000 | 0.213592 | + | 0.123317i | 0.997981 | − | 1.41564i | 2.27249 | + | 3.93606i | − | 1.00000i | 1.00807 | + | 2.82556i | −0.123317 | + | 0.213592i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.7 | 1.00000i | −0.669086 | + | 1.59760i | −1.00000 | −3.69011 | − | 2.13048i | −1.59760 | − | 0.669086i | 0.247791 | + | 0.429186i | − | 1.00000i | −2.10465 | − | 2.13786i | 2.13048 | − | 3.69011i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.8 | 1.00000i | 0.0908641 | − | 1.72967i | −1.00000 | −2.08105 | − | 1.20150i | 1.72967 | + | 0.0908641i | −1.68759 | − | 2.92298i | − | 1.00000i | −2.98349 | − | 0.314329i | 1.20150 | − | 2.08105i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.9 | 1.00000i | 0.356524 | + | 1.69496i | −1.00000 | 2.22053 | + | 1.28202i | −1.69496 | + | 0.356524i | 0.0797352 | + | 0.138105i | − | 1.00000i | −2.74578 | + | 1.20859i | −1.28202 | + | 2.22053i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 161.10 | 1.00000i | 1.63734 | − | 0.564913i | −1.00000 | 1.60499 | + | 0.926640i | 0.564913 | + | 1.63734i | −0.412428 | − | 0.714346i | − | 1.00000i | 2.36175 | − | 1.84991i | −0.926640 | + | 1.60499i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 31.e | odd | 6 | 1 | inner |
| 93.g | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 186.2.h.a | ✓ | 20 |
| 3.b | odd | 2 | 1 | inner | 186.2.h.a | ✓ | 20 |
| 31.e | odd | 6 | 1 | inner | 186.2.h.a | ✓ | 20 |
| 93.g | even | 6 | 1 | inner | 186.2.h.a | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 186.2.h.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 186.2.h.a | ✓ | 20 | 3.b | odd | 2 | 1 | inner |
| 186.2.h.a | ✓ | 20 | 31.e | odd | 6 | 1 | inner |
| 186.2.h.a | ✓ | 20 | 93.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(186, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{10} \)
|
| $3$ |
\( T^{20} + 2 T^{18} + \cdots + 59049 \)
|
| $5$ |
\( T^{20} - 34 T^{18} + \cdots + 20736 \)
|
| $7$ |
\( (T^{10} - T^{9} + 17 T^{8} + \cdots + 1)^{2} \)
|
| $11$ |
\( T^{20} + 43 T^{18} + \cdots + 4782969 \)
|
| $13$ |
\( (T^{10} - 17 T^{8} + \cdots + 1728)^{2} \)
|
| $17$ |
\( T^{20} + 58 T^{18} + \cdots + 11664 \)
|
| $19$ |
\( (T^{10} - 2 T^{9} + \cdots + 272484)^{2} \)
|
| $23$ |
\( (T^{10} - 92 T^{8} + \cdots - 836352)^{2} \)
|
| $29$ |
\( (T^{10} - 233 T^{8} + \cdots - 9720000)^{2} \)
|
| $31$ |
\( (T^{10} + 7 T^{9} + \cdots + 28629151)^{2} \)
|
| $37$ |
\( (T^{10} + 18 T^{9} + \cdots + 8748)^{2} \)
|
| $41$ |
\( T^{20} + \cdots + 6990080303376 \)
|
| $43$ |
\( (T^{10} - 13 T^{8} + \cdots + 768)^{2} \)
|
| $47$ |
\( (T^{10} + 240 T^{8} + \cdots + 627264)^{2} \)
|
| $53$ |
\( T^{20} + \cdots + 673246809 \)
|
| $59$ |
\( T^{20} + \cdots + 88\!\cdots\!61 \)
|
| $61$ |
\( (T^{10} + 212 T^{8} + \cdots + 14520000)^{2} \)
|
| $67$ |
\( (T^{10} + 20 T^{9} + \cdots + 4)^{2} \)
|
| $71$ |
\( T^{20} + \cdots + 64\!\cdots\!16 \)
|
| $73$ |
\( (T^{10} + 18 T^{9} + \cdots + 135636528)^{2} \)
|
| $79$ |
\( (T^{10} + 6 T^{9} + \cdots + 855613632)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 39\!\cdots\!09 \)
|
| $89$ |
\( (T^{10} - 628 T^{8} + \cdots - 4423680000)^{2} \)
|
| $97$ |
\( (T^{5} - 31 T^{4} + \cdots + 712)^{4} \)
|
| show more | |
| show less |