Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [52,3,Mod(23,52)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(52, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 5]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("52.23");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 52 = 2^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 52.i (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: | \(1.41689737467\) |
| 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: |
\( x^{20} - 9 x^{18} + 60 x^{16} - 240 x^{14} + 720 x^{12} - 2432 x^{10} + 11520 x^{8} - 61440 x^{6} + \cdots + 1048576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{12} \) |
| 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
\( x^{20} - 9 x^{18} + 60 x^{16} - 240 x^{14} + 720 x^{12} - 2432 x^{10} + 11520 x^{8} - 61440 x^{6} + \cdots + 1048576 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{19} - 9 \nu^{17} + 60 \nu^{15} - 240 \nu^{13} + 720 \nu^{11} - 2432 \nu^{9} + \cdots - 589824 \nu ) / 262144 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - \nu^{19} + 25 \nu^{17} - 76 \nu^{15} + 304 \nu^{13} - 208 \nu^{11} + 3712 \nu^{9} + \cdots + 196608 \nu ) / 262144 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3 \nu^{19} + 3 \nu^{18} + 17 \nu^{17} + 5 \nu^{16} - 136 \nu^{15} - 44 \nu^{14} + 816 \nu^{13} + \cdots + 1703936 ) / 131072 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{18} - 9 \nu^{16} + 44 \nu^{14} - 160 \nu^{12} + 336 \nu^{10} - 1408 \nu^{8} + 10240 \nu^{6} + \cdots - 229376 ) / 16384 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{18} - 9 \nu^{16} + 44 \nu^{14} - 160 \nu^{12} + 336 \nu^{10} - 1408 \nu^{8} + 10240 \nu^{6} + \cdots - 229376 ) / 16384 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11 \nu^{19} + 163 \nu^{17} - 852 \nu^{15} + 3280 \nu^{13} - 7664 \nu^{11} + 21632 \nu^{9} + \cdots - 262144 ) / 524288 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - \nu^{19} - 44 \nu^{18} + 9 \nu^{17} + 652 \nu^{16} - 60 \nu^{15} - 3408 \nu^{14} + \cdots + 26476544 ) / 524288 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{19} - 112 \nu^{18} - 25 \nu^{17} + 752 \nu^{16} + 76 \nu^{15} - 3392 \nu^{14} - 304 \nu^{13} + \cdots + 6029312 ) / 524288 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 7 \nu^{19} - 45 \nu^{18} - 19 \nu^{17} + 133 \nu^{16} + 40 \nu^{15} - 380 \nu^{14} + 176 \nu^{13} + \cdots - 8257536 ) / 131072 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 11 \nu^{19} + 3 \nu^{18} - 55 \nu^{17} + 5 \nu^{16} + 216 \nu^{15} - 44 \nu^{14} - 464 \nu^{13} + \cdots + 1703936 ) / 131072 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 11 \nu^{19} - 3 \nu^{18} - 55 \nu^{17} - 5 \nu^{16} + 216 \nu^{15} + 44 \nu^{14} - 464 \nu^{13} + \cdots - 1703936 ) / 131072 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 37 \nu^{19} - 48 \nu^{18} + 93 \nu^{17} + 688 \nu^{16} - 188 \nu^{15} - 3648 \nu^{14} + \cdots + 28049408 ) / 524288 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 81 \nu^{19} - 36 \nu^{18} - 313 \nu^{17} + 708 \nu^{16} + 1052 \nu^{15} - 3824 \nu^{14} + \cdots + 34865152 ) / 524288 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 19 \nu^{19} - 24 \nu^{18} + 267 \nu^{17} + 24 \nu^{16} - 1396 \nu^{15} + 32 \nu^{14} + \cdots - 7602176 ) / 262144 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 19 \nu^{19} + 24 \nu^{18} + 267 \nu^{17} - 24 \nu^{16} - 1396 \nu^{15} - 32 \nu^{14} + \cdots + 7602176 ) / 262144 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 45 \nu^{19} - 2 \nu^{18} - 285 \nu^{17} + 18 \nu^{16} + 1300 \nu^{15} - 120 \nu^{14} + \cdots + 1048576 ) / 262144 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 45 \nu^{19} - 2 \nu^{18} + 285 \nu^{17} + 18 \nu^{16} - 1300 \nu^{15} - 120 \nu^{14} + \cdots + 1048576 ) / 262144 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 25 \nu^{19} + 232 \nu^{18} + 289 \nu^{17} - 2024 \nu^{16} - 1500 \nu^{15} + 9760 \nu^{14} + \cdots - 43778048 ) / 524288 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 61 \nu^{19} - 741 \nu^{17} + 3852 \nu^{15} - 13616 \nu^{13} + 30864 \nu^{11} - 102272 \nu^{9} + \cdots - 262144 ) / 524288 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{17} - \beta_{16} + \beta_{13} + \beta_{12} - \beta_{10} - 2\beta_{7} - \beta _1 + 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2 \beta_{19} + \beta_{17} - \beta_{16} + 2 \beta_{15} + 2 \beta_{14} + \beta_{13} - \beta_{12} + \cdots - 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 3 \beta_{17} - 3 \beta_{16} - 4 \beta_{15} + 4 \beta_{14} + \beta_{13} + \beta_{12} - 4 \beta_{11} + \cdots - 10 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 6 \beta_{19} + 3 \beta_{17} - 3 \beta_{16} + 2 \beta_{15} + 2 \beta_{14} - \beta_{13} + \beta_{12} + \cdots + 10 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 8 \beta_{19} + 16 \beta_{18} + 9 \beta_{17} + 9 \beta_{16} + 12 \beta_{15} - 12 \beta_{14} + 5 \beta_{13} + \cdots - 2 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 2 \beta_{19} - \beta_{17} + \beta_{16} - 22 \beta_{15} - 22 \beta_{14} + 3 \beta_{13} - 3 \beta_{12} + \cdots + 82 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 24 \beta_{19} - 48 \beta_{18} + 77 \beta_{17} + 77 \beta_{16} + 60 \beta_{15} - 60 \beta_{14} + \cdots - 10 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 6 \beta_{19} + 19 \beta_{17} - 19 \beta_{16} + 2 \beta_{15} + 2 \beta_{14} + 39 \beta_{13} - 39 \beta_{12} + \cdots - 22 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 56 \beta_{19} - 112 \beta_{18} - 39 \beta_{17} - 39 \beta_{16} + 12 \beta_{15} - 12 \beta_{14} + \cdots + 1182 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 338 \beta_{19} + 103 \beta_{17} - 103 \beta_{16} - 390 \beta_{15} - 390 \beta_{14} + 107 \beta_{13} + \cdots + 2 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 296 \beta_{19} + 592 \beta_{18} - 907 \beta_{17} - 907 \beta_{16} + 220 \beta_{15} - 220 \beta_{14} + \cdots + 550 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 1270 \beta_{19} + 843 \beta_{17} - 843 \beta_{16} + 146 \beta_{15} + 146 \beta_{14} + 959 \beta_{13} + \cdots + 762 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 648 \beta_{19} + 1296 \beta_{18} - 4959 \beta_{17} - 4959 \beta_{16} - 660 \beta_{15} + 660 \beta_{14} + \cdots - 11890 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 9758 \beta_{19} + 415 \beta_{17} - 415 \beta_{16} + 2378 \beta_{15} + 2378 \beta_{14} - 2493 \beta_{13} + \cdots + 18450 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 3176 \beta_{19} + 6352 \beta_{18} - 10851 \beta_{17} - 10851 \beta_{16} + 23996 \beta_{15} + \cdots - 25514 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 7846 \beta_{19} - 11357 \beta_{17} + 11357 \beta_{16} - 3294 \beta_{15} - 3294 \beta_{14} - 2121 \beta_{13} + \cdots + 65226 ) / 2 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 49464 \beta_{19} - 98928 \beta_{18} + 24233 \beta_{17} + 24233 \beta_{16} + 61644 \beta_{15} + \cdots - 293890 ) / 2 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 51954 \beta_{19} - 2665 \beta_{17} + 2665 \beta_{16} + 33178 \beta_{15} + 33178 \beta_{14} + \cdots - 230750 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/52\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(41\) |
| \(\chi(n)\) | \(-1\) | \(1 + \beta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 23.1 |
|
−1.96169 | + | 0.389575i | −3.81294 | + | 2.20140i | 3.69646 | − | 1.52845i | − | 7.10309i | 6.62221 | − | 5.80390i | 2.77485 | − | 4.80618i | −6.65587 | + | 4.43840i | 5.19237 | − | 8.99344i | 2.76718 | + | 13.9341i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.2 | −1.77801 | − | 0.915786i | 2.93761 | − | 1.69603i | 2.32267 | + | 3.25656i | − | 0.495498i | −6.77631 | + | 0.325344i | 3.56184 | − | 6.16928i | −1.14743 | − | 7.91728i | 1.25303 | − | 2.17031i | −0.453770 | + | 0.881002i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.3 | −1.31823 | + | 1.50409i | 3.81294 | − | 2.20140i | −0.524554 | − | 3.96546i | − | 7.10309i | −1.71522 | + | 8.63695i | −2.77485 | + | 4.80618i | 6.65587 | + | 4.43840i | 5.19237 | − | 8.99344i | 10.6837 | + | 9.36348i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.4 | −1.24404 | − | 1.56600i | −2.56937 | + | 1.48343i | −0.904738 | + | 3.89634i | 2.94969i | 5.51945 | + | 2.17821i | −3.55296 | + | 6.15391i | 7.22721 | − | 3.43037i | −0.0988878 | + | 0.171279i | 4.61922 | − | 3.66953i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.5 | −0.0959134 | + | 1.99770i | −2.93761 | + | 1.69603i | −3.98160 | − | 0.383212i | − | 0.495498i | −3.10640 | − | 6.03113i | −3.56184 | + | 6.16928i | 1.14743 | − | 7.91728i | 1.25303 | − | 2.17031i | 0.989855 | + | 0.0475249i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.6 | 0.127690 | − | 1.99592i | −0.590249 | + | 0.340781i | −3.96739 | − | 0.509720i | − | 7.13975i | 0.604802 | + | 1.22160i | 4.53442 | − | 7.85384i | −1.52396 | + | 7.85351i | −4.26774 | + | 7.39194i | −14.2504 | − | 0.911679i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.7 | 0.359919 | − | 1.96735i | 4.31646 | − | 2.49211i | −3.74092 | − | 1.41617i | 4.86044i | −3.34927 | − | 9.38894i | −5.11655 | + | 8.86212i | −4.13253 | + | 6.84998i | 7.92123 | − | 13.7200i | 9.56219 | + | 1.74936i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.8 | 0.734180 | + | 1.86037i | 2.56937 | − | 1.48343i | −2.92196 | + | 2.73169i | 2.94969i | 4.64611 | + | 3.69088i | 3.55296 | − | 6.15391i | −7.22721 | − | 3.43037i | −0.0988878 | + | 0.171279i | −5.48751 | + | 2.16560i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.9 | 1.79236 | + | 0.887377i | 0.590249 | − | 0.340781i | 2.42513 | + | 3.18100i | − | 7.13975i | 1.36034 | − | 0.0870289i | −4.53442 | + | 7.85384i | 1.52396 | + | 7.85351i | −4.26774 | + | 7.39194i | 6.33565 | − | 12.7970i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 23.10 | 1.88373 | + | 0.671975i | −4.31646 | + | 2.49211i | 3.09690 | + | 2.53164i | 4.86044i | −9.80570 | + | 1.79391i | 5.11655 | − | 8.86212i | 4.13253 | + | 6.84998i | 7.92123 | − | 13.7200i | −3.26610 | + | 9.15578i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.1 | −1.96169 | − | 0.389575i | −3.81294 | − | 2.20140i | 3.69646 | + | 1.52845i | 7.10309i | 6.62221 | + | 5.80390i | 2.77485 | + | 4.80618i | −6.65587 | − | 4.43840i | 5.19237 | + | 8.99344i | 2.76718 | − | 13.9341i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.2 | −1.77801 | + | 0.915786i | 2.93761 | + | 1.69603i | 2.32267 | − | 3.25656i | 0.495498i | −6.77631 | − | 0.325344i | 3.56184 | + | 6.16928i | −1.14743 | + | 7.91728i | 1.25303 | + | 2.17031i | −0.453770 | − | 0.881002i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.3 | −1.31823 | − | 1.50409i | 3.81294 | + | 2.20140i | −0.524554 | + | 3.96546i | 7.10309i | −1.71522 | − | 8.63695i | −2.77485 | − | 4.80618i | 6.65587 | − | 4.43840i | 5.19237 | + | 8.99344i | 10.6837 | − | 9.36348i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.4 | −1.24404 | + | 1.56600i | −2.56937 | − | 1.48343i | −0.904738 | − | 3.89634i | − | 2.94969i | 5.51945 | − | 2.17821i | −3.55296 | − | 6.15391i | 7.22721 | + | 3.43037i | −0.0988878 | − | 0.171279i | 4.61922 | + | 3.66953i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.5 | −0.0959134 | − | 1.99770i | −2.93761 | − | 1.69603i | −3.98160 | + | 0.383212i | 0.495498i | −3.10640 | + | 6.03113i | −3.56184 | − | 6.16928i | 1.14743 | + | 7.91728i | 1.25303 | + | 2.17031i | 0.989855 | − | 0.0475249i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.6 | 0.127690 | + | 1.99592i | −0.590249 | − | 0.340781i | −3.96739 | + | 0.509720i | 7.13975i | 0.604802 | − | 1.22160i | 4.53442 | + | 7.85384i | −1.52396 | − | 7.85351i | −4.26774 | − | 7.39194i | −14.2504 | + | 0.911679i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.7 | 0.359919 | + | 1.96735i | 4.31646 | + | 2.49211i | −3.74092 | + | 1.41617i | − | 4.86044i | −3.34927 | + | 9.38894i | −5.11655 | − | 8.86212i | −4.13253 | − | 6.84998i | 7.92123 | + | 13.7200i | 9.56219 | − | 1.74936i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.8 | 0.734180 | − | 1.86037i | 2.56937 | + | 1.48343i | −2.92196 | − | 2.73169i | − | 2.94969i | 4.64611 | − | 3.69088i | 3.55296 | + | 6.15391i | −7.22721 | + | 3.43037i | −0.0988878 | − | 0.171279i | −5.48751 | − | 2.16560i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.9 | 1.79236 | − | 0.887377i | 0.590249 | + | 0.340781i | 2.42513 | − | 3.18100i | 7.13975i | 1.36034 | + | 0.0870289i | −4.53442 | − | 7.85384i | 1.52396 | − | 7.85351i | −4.26774 | − | 7.39194i | 6.33565 | + | 12.7970i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 43.10 | 1.88373 | − | 0.671975i | −4.31646 | − | 2.49211i | 3.09690 | − | 2.53164i | − | 4.86044i | −9.80570 | − | 1.79391i | 5.11655 | + | 8.86212i | 4.13253 | − | 6.84998i | 7.92123 | + | 13.7200i | −3.26610 | − | 9.15578i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 13.e | even | 6 | 1 | inner |
| 52.i | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 52.3.i.b | ✓ | 20 |
| 4.b | odd | 2 | 1 | inner | 52.3.i.b | ✓ | 20 |
| 13.e | even | 6 | 1 | inner | 52.3.i.b | ✓ | 20 |
| 52.i | odd | 6 | 1 | inner | 52.3.i.b | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 52.3.i.b | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 52.3.i.b | ✓ | 20 | 4.b | odd | 2 | 1 | inner |
| 52.3.i.b | ✓ | 20 | 13.e | even | 6 | 1 | inner |
| 52.3.i.b | ✓ | 20 | 52.i | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{20} - 65 T_{3}^{18} + 2714 T_{3}^{16} - 68321 T_{3}^{14} + 1256170 T_{3}^{12} - 15406093 T_{3}^{10} + \cdots + 513294336 \)
acting on \(S_{3}^{\mathrm{new}}(52, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + 3 T^{19} + \cdots + 1048576 \)
|
| $3$ |
\( T^{20} + \cdots + 513294336 \)
|
| $5$ |
\( (T^{10} + 134 T^{8} + \cdots + 129792)^{2} \)
|
| $7$ |
\( T^{20} + \cdots + 46\!\cdots\!00 \)
|
| $11$ |
\( T^{20} + \cdots + 20\!\cdots\!64 \)
|
| $13$ |
\( (T^{10} + \cdots + 137858491849)^{2} \)
|
| $17$ |
\( (T^{10} + 11 T^{9} + \cdots + 102515625)^{2} \)
|
| $19$ |
\( T^{20} + \cdots + 16\!\cdots\!00 \)
|
| $23$ |
\( T^{20} + \cdots + 51\!\cdots\!00 \)
|
| $29$ |
\( (T^{10} + \cdots + 56306849356521)^{2} \)
|
| $31$ |
\( (T^{10} + \cdots - 168817287168)^{2} \)
|
| $37$ |
\( (T^{10} + \cdots + 329933380923)^{2} \)
|
| $41$ |
\( (T^{10} + \cdots + 1558973646387)^{2} \)
|
| $43$ |
\( T^{20} + \cdots + 50\!\cdots\!36 \)
|
| $47$ |
\( (T^{10} + \cdots - 22\!\cdots\!72)^{2} \)
|
| $53$ |
\( (T^{5} - 54 T^{4} + \cdots - 31792392)^{4} \)
|
| $59$ |
\( T^{20} + \cdots + 20\!\cdots\!84 \)
|
| $61$ |
\( (T^{10} + \cdots + 599485402640881)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 67\!\cdots\!04 \)
|
| $71$ |
\( T^{20} + \cdots + 14\!\cdots\!04 \)
|
| $73$ |
\( (T^{10} + \cdots + 29\!\cdots\!88)^{2} \)
|
| $79$ |
\( (T^{10} + \cdots + 98\!\cdots\!16)^{2} \)
|
| $83$ |
\( (T^{10} + \cdots - 67\!\cdots\!88)^{2} \)
|
| $89$ |
\( (T^{10} + \cdots + 66\!\cdots\!92)^{2} \)
|
| $97$ |
\( (T^{10} + \cdots + 35\!\cdots\!12)^{2} \)
|
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