Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [525,2,Mod(26,525)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(525, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("525.26");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 525.t (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: | \(4.19214610612\) |
| 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: |
\( x^{20} - 3 x^{19} + 8 x^{18} - 15 x^{17} + 18 x^{16} - 45 x^{15} + 59 x^{14} - 147 x^{13} + 271 x^{12} + \cdots + 59049 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| 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
\( x^{20} - 3 x^{19} + 8 x^{18} - 15 x^{17} + 18 x^{16} - 45 x^{15} + 59 x^{14} - 147 x^{13} + 271 x^{12} + \cdots + 59049 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1471510 \nu^{19} - 69953127 \nu^{18} + 93678038 \nu^{17} - 218139213 \nu^{16} + \cdots - 1156143537891 ) / 74996560260 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2890481 \nu^{19} - 61404921 \nu^{18} + 102401257 \nu^{17} - 162636525 \nu^{16} + \cdots - 1076681692251 ) / 74996560260 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 797782 \nu^{19} - 4425285 \nu^{18} - 10337801 \nu^{17} - 9682413 \nu^{16} + \cdots + 69903131301 ) / 18749140065 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1369114 \nu^{19} + 43032333 \nu^{18} - 52765180 \nu^{17} + 119927013 \nu^{16} + \cdots + 728009491662 ) / 18749140065 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2231098 \nu^{19} - 38618346 \nu^{18} + 35313338 \nu^{17} - 116786112 \nu^{16} + \cdots - 415240815177 ) / 18749140065 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3771427 \nu^{19} - 21206079 \nu^{18} + 5337977 \nu^{17} - 61097823 \nu^{16} + \cdots - 3633009408 ) / 18749140065 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 16385029 \nu^{19} - 140514804 \nu^{18} + 102757399 \nu^{17} - 407033892 \nu^{16} + \cdots - 2201360498100 ) / 74996560260 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 23044883 \nu^{19} - 34101699 \nu^{18} - 31187623 \nu^{17} - 73217703 \nu^{16} + \cdots - 543176221113 ) / 74996560260 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 30237802 \nu^{19} - 54259353 \nu^{18} + 214884218 \nu^{17} - 236951109 \nu^{16} + \cdots - 1626318995895 ) / 74996560260 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 8368807 \nu^{19} - 7140033 \nu^{18} + 20556833 \nu^{17} - 38614374 \nu^{16} + \cdots + 40868304975 ) / 18749140065 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 37886113 \nu^{19} - 154722717 \nu^{18} + 251556047 \nu^{17} - 527337261 \nu^{16} + \cdots - 2018497456485 ) / 74996560260 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 38985829 \nu^{19} + 89896524 \nu^{18} - 218228033 \nu^{17} + 285292200 \nu^{16} + \cdots + 1324570004424 ) / 74996560260 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 13359008 \nu^{19} + 19113527 \nu^{18} - 18111052 \nu^{17} + 89227951 \nu^{16} + \cdots - 229996215855 ) / 24998853420 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 12455246 \nu^{19} + 44968920 \nu^{18} - 80649223 \nu^{17} + 145859691 \nu^{16} + \cdots + 553787129583 ) / 18749140065 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 512306 \nu^{19} + 1702863 \nu^{18} - 2952415 \nu^{17} + 5590548 \nu^{16} - 6383691 \nu^{15} + \cdots + 19282234437 ) / 635564070 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 31370234 \nu^{19} + 71949081 \nu^{18} - 107067631 \nu^{17} + 265648533 \nu^{16} + \cdots + 376958974500 ) / 37498280130 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 5988796 \nu^{19} - 15464541 \nu^{18} + 28972577 \nu^{17} - 54823470 \nu^{16} + \cdots - 164723228181 ) / 6249713355 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 18932536 \nu^{19} + 53426298 \nu^{18} - 66027095 \nu^{17} + 192757263 \nu^{16} + \cdots + 266926260897 ) / 18749140065 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{19} + 2\beta_{18} + \beta_{15} + \beta_{13} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{5} + 2\beta _1 - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{19} + 3 \beta_{18} + 2 \beta_{15} - \beta_{14} + \beta_{13} - 2 \beta_{11} - \beta_{9} + \beta_{4} + \cdots + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 2 \beta_{18} + \beta_{17} - \beta_{14} - 3 \beta_{13} + \beta_{12} + 3 \beta_{11} - \beta_{10} + \cdots - 1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3 \beta_{18} + \beta_{17} - \beta_{16} + 4 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + 5 \beta_{11} + \cdots + 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{19} - \beta_{18} + 5 \beta_{16} - 4 \beta_{14} - 4 \beta_{13} - 5 \beta_{12} + 8 \beta_{10} + \cdots + 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( 18 \beta_{19} + 53 \beta_{18} + 3 \beta_{17} + 6 \beta_{16} + 9 \beta_{15} - 2 \beta_{14} + 22 \beta_{13} + \cdots - 1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 19 \beta_{19} + 7 \beta_{18} - 9 \beta_{17} - 9 \beta_{16} + 19 \beta_{15} + 6 \beta_{13} + 22 \beta_{11} + \cdots - 13 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2 \beta_{19} - 15 \beta_{18} + 38 \beta_{17} + 19 \beta_{16} + 4 \beta_{15} - 14 \beta_{14} - 52 \beta_{13} + \cdots - 65 \)
|
| \(\nu^{10}\) | \(=\) |
\( 84 \beta_{18} + 49 \beta_{17} + 18 \beta_{15} - 7 \beta_{14} + 11 \beta_{13} - 83 \beta_{12} + 39 \beta_{11} + \cdots - 49 \)
|
| \(\nu^{11}\) | \(=\) |
\( 37 \beta_{19} + 78 \beta_{18} - 72 \beta_{17} + 72 \beta_{16} - 37 \beta_{15} + 30 \beta_{14} + \cdots + 133 \)
|
| \(\nu^{12}\) | \(=\) |
\( 235 \beta_{19} + 466 \beta_{18} + 104 \beta_{16} - 28 \beta_{14} + 232 \beta_{13} + 226 \beta_{12} + \cdots - 7 \)
|
| \(\nu^{13}\) | \(=\) |
\( 100 \beta_{19} - 166 \beta_{18} - 4 \beta_{17} - 8 \beta_{16} + 50 \beta_{15} - 48 \beta_{14} + \cdots - 412 \)
|
| \(\nu^{14}\) | \(=\) |
\( 166 \beta_{19} + 816 \beta_{18} + 564 \beta_{17} + 564 \beta_{16} + 166 \beta_{15} + 74 \beta_{13} + \cdots - 1597 \)
|
| \(\nu^{15}\) | \(=\) |
\( 270 \beta_{19} + 255 \beta_{18} + 472 \beta_{17} + 236 \beta_{16} + 540 \beta_{15} - 480 \beta_{14} + \cdots - 1136 \)
|
| \(\nu^{16}\) | \(=\) |
\( 1274 \beta_{18} - 992 \beta_{17} - 1575 \beta_{15} + 1148 \beta_{14} - 119 \beta_{13} - 50 \beta_{12} + \cdots + 992 \)
|
| \(\nu^{17}\) | \(=\) |
\( 1787 \beta_{19} + 1176 \beta_{18} - 1144 \beta_{17} + 1144 \beta_{16} - 1787 \beta_{15} - 1078 \beta_{14} + \cdots + 1897 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 350 \beta_{19} + 1482 \beta_{18} + 4865 \beta_{16} - 943 \beta_{14} - 2085 \beta_{13} - 1127 \beta_{12} + \cdots + 497 \)
|
| \(\nu^{19}\) | \(=\) |
\( 2684 \beta_{19} + 8072 \beta_{18} + 3601 \beta_{17} + 7202 \beta_{16} + 1342 \beta_{15} + 4310 \beta_{14} + \cdots - 41443 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(176\) | \(451\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1 - \beta_{11}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 26.1 |
|
−2.36764 | − | 1.36696i | 1.63084 | + | 0.583411i | 2.73715 | + | 4.74088i | 0 | −3.06374 | − | 3.61059i | 2.24882 | − | 1.39384i | − | 9.49844i | 2.31926 | + | 1.90290i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.2 | −1.94891 | − | 1.12521i | −0.742502 | − | 1.56483i | 1.53217 | + | 2.65380i | 0 | −0.313682 | + | 3.88518i | −1.42897 | + | 2.22667i | − | 2.39522i | −1.89738 | + | 2.32378i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.3 | −1.46613 | − | 0.846473i | −0.803015 | + | 1.53466i | 0.433034 | + | 0.750036i | 0 | 2.47637 | − | 1.57028i | 1.71236 | + | 2.01688i | 1.91969i | −1.71033 | − | 2.46470i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.4 | −0.780577 | − | 0.450666i | −1.71408 | − | 0.248842i | −0.593800 | − | 1.02849i | 0 | 1.22583 | + | 0.966719i | 0.105498 | − | 2.64365i | 2.87309i | 2.87616 | + | 0.853070i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.5 | −0.766266 | − | 0.442404i | 1.39625 | + | 1.02493i | −0.608557 | − | 1.05405i | 0 | −0.616465 | − | 1.40308i | −2.63771 | − | 0.206062i | 2.84653i | 0.899028 | + | 2.86212i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.6 | 0.766266 | + | 0.442404i | −0.189492 | − | 1.72165i | −0.608557 | − | 1.05405i | 0 | 0.616465 | − | 1.40308i | −2.63771 | − | 0.206062i | − | 2.84653i | −2.92819 | + | 0.652481i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.7 | 0.780577 | + | 0.450666i | −0.641538 | + | 1.60886i | −0.593800 | − | 1.02849i | 0 | −1.22583 | + | 0.966719i | 0.105498 | − | 2.64365i | − | 2.87309i | −2.17686 | − | 2.06429i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.8 | 1.46613 | + | 0.846473i | −1.73056 | − | 0.0718963i | 0.433034 | + | 0.750036i | 0 | −2.47637 | − | 1.57028i | 1.71236 | + | 2.01688i | − | 1.91969i | 2.98966 | + | 0.248842i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.9 | 1.94891 | + | 1.12521i | 0.983931 | + | 1.42544i | 1.53217 | + | 2.65380i | 0 | 0.313682 | + | 3.88518i | −1.42897 | + | 2.22667i | 2.39522i | −1.06376 | + | 2.80507i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 26.10 | 2.36764 | + | 1.36696i | 0.310170 | − | 1.70405i | 2.73715 | + | 4.74088i | 0 | 3.06374 | − | 3.61059i | 2.24882 | − | 1.39384i | 9.49844i | −2.80759 | − | 1.05709i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.1 | −2.36764 | + | 1.36696i | 1.63084 | − | 0.583411i | 2.73715 | − | 4.74088i | 0 | −3.06374 | + | 3.61059i | 2.24882 | + | 1.39384i | 9.49844i | 2.31926 | − | 1.90290i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.2 | −1.94891 | + | 1.12521i | −0.742502 | + | 1.56483i | 1.53217 | − | 2.65380i | 0 | −0.313682 | − | 3.88518i | −1.42897 | − | 2.22667i | 2.39522i | −1.89738 | − | 2.32378i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.3 | −1.46613 | + | 0.846473i | −0.803015 | − | 1.53466i | 0.433034 | − | 0.750036i | 0 | 2.47637 | + | 1.57028i | 1.71236 | − | 2.01688i | − | 1.91969i | −1.71033 | + | 2.46470i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.4 | −0.780577 | + | 0.450666i | −1.71408 | + | 0.248842i | −0.593800 | + | 1.02849i | 0 | 1.22583 | − | 0.966719i | 0.105498 | + | 2.64365i | − | 2.87309i | 2.87616 | − | 0.853070i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.5 | −0.766266 | + | 0.442404i | 1.39625 | − | 1.02493i | −0.608557 | + | 1.05405i | 0 | −0.616465 | + | 1.40308i | −2.63771 | + | 0.206062i | − | 2.84653i | 0.899028 | − | 2.86212i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.6 | 0.766266 | − | 0.442404i | −0.189492 | + | 1.72165i | −0.608557 | + | 1.05405i | 0 | 0.616465 | + | 1.40308i | −2.63771 | + | 0.206062i | 2.84653i | −2.92819 | − | 0.652481i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.7 | 0.780577 | − | 0.450666i | −0.641538 | − | 1.60886i | −0.593800 | + | 1.02849i | 0 | −1.22583 | − | 0.966719i | 0.105498 | + | 2.64365i | 2.87309i | −2.17686 | + | 2.06429i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.8 | 1.46613 | − | 0.846473i | −1.73056 | + | 0.0718963i | 0.433034 | − | 0.750036i | 0 | −2.47637 | + | 1.57028i | 1.71236 | − | 2.01688i | 1.91969i | 2.98966 | − | 0.248842i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.9 | 1.94891 | − | 1.12521i | 0.983931 | − | 1.42544i | 1.53217 | − | 2.65380i | 0 | 0.313682 | − | 3.88518i | −1.42897 | − | 2.22667i | − | 2.39522i | −1.06376 | − | 2.80507i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.10 | 2.36764 | − | 1.36696i | 0.310170 | + | 1.70405i | 2.73715 | − | 4.74088i | 0 | 3.06374 | + | 3.61059i | 2.24882 | + | 1.39384i | − | 9.49844i | −2.80759 | + | 1.05709i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
| 21.g | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 525.2.t.h | ✓ | 20 |
| 3.b | odd | 2 | 1 | inner | 525.2.t.h | ✓ | 20 |
| 5.b | even | 2 | 1 | 525.2.t.i | yes | 20 | |
| 5.c | odd | 4 | 2 | 525.2.q.g | 40 | ||
| 7.d | odd | 6 | 1 | inner | 525.2.t.h | ✓ | 20 |
| 15.d | odd | 2 | 1 | 525.2.t.i | yes | 20 | |
| 15.e | even | 4 | 2 | 525.2.q.g | 40 | ||
| 21.g | even | 6 | 1 | inner | 525.2.t.h | ✓ | 20 |
| 35.i | odd | 6 | 1 | 525.2.t.i | yes | 20 | |
| 35.k | even | 12 | 2 | 525.2.q.g | 40 | ||
| 105.p | even | 6 | 1 | 525.2.t.i | yes | 20 | |
| 105.w | odd | 12 | 2 | 525.2.q.g | 40 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 525.2.q.g | 40 | 5.c | odd | 4 | 2 | ||
| 525.2.q.g | 40 | 15.e | even | 4 | 2 | ||
| 525.2.q.g | 40 | 35.k | even | 12 | 2 | ||
| 525.2.q.g | 40 | 105.w | odd | 12 | 2 | ||
| 525.2.t.h | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 525.2.t.h | ✓ | 20 | 3.b | odd | 2 | 1 | inner |
| 525.2.t.h | ✓ | 20 | 7.d | odd | 6 | 1 | inner |
| 525.2.t.h | ✓ | 20 | 21.g | even | 6 | 1 | inner |
| 525.2.t.i | yes | 20 | 5.b | even | 2 | 1 | |
| 525.2.t.i | yes | 20 | 15.d | odd | 2 | 1 | |
| 525.2.t.i | yes | 20 | 35.i | odd | 6 | 1 | |
| 525.2.t.i | yes | 20 | 105.p | even | 6 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\):
|
\( T_{2}^{20} - 17 T_{2}^{18} + 190 T_{2}^{16} - 1211 T_{2}^{14} + 5569 T_{2}^{12} - 15953 T_{2}^{10} + \cdots + 4761 \)
|
|
\( T_{13}^{10} + 72T_{13}^{8} + 1950T_{13}^{6} + 24651T_{13}^{4} + 142515T_{13}^{2} + 286443 \)
|
|
\( T_{37}^{10} + 12 T_{37}^{9} + 136 T_{37}^{8} + 842 T_{37}^{7} + 5983 T_{37}^{6} + 30371 T_{37}^{5} + \cdots + 1630729 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - 17 T^{18} + \cdots + 4761 \)
|
| $3$ |
\( T^{20} + 3 T^{19} + \cdots + 59049 \)
|
| $5$ |
\( T^{20} \)
|
| $7$ |
\( (T^{10} + T^{8} + \cdots + 16807)^{2} \)
|
| $11$ |
\( T^{20} - 50 T^{18} + \cdots + 76176 \)
|
| $13$ |
\( (T^{10} + 72 T^{8} + \cdots + 286443)^{2} \)
|
| $17$ |
\( T^{20} + \cdots + 1586058134544 \)
|
| $19$ |
\( (T^{10} - 36 T^{8} + \cdots + 11907)^{2} \)
|
| $23$ |
\( T^{20} + \cdots + 1628747698176 \)
|
| $29$ |
\( (T^{10} + 158 T^{8} + \cdots + 1130496)^{2} \)
|
| $31$ |
\( (T^{10} - 21 T^{9} + \cdots + 21168)^{2} \)
|
| $37$ |
\( (T^{10} + 12 T^{9} + \cdots + 1630729)^{2} \)
|
| $41$ |
\( (T^{10} - 261 T^{8} + \cdots - 603612)^{2} \)
|
| $43$ |
\( (T^{5} + 9 T^{4} + \cdots + 16928)^{4} \)
|
| $47$ |
\( T^{20} + \cdots + 133333061904 \)
|
| $53$ |
\( T^{20} + \cdots + 14814784656 \)
|
| $59$ |
\( T^{20} + \cdots + 4638113562384 \)
|
| $61$ |
\( (T^{10} + 45 T^{9} + \cdots + 30509163)^{2} \)
|
| $67$ |
\( (T^{10} + 10 T^{9} + \cdots + 1459264)^{2} \)
|
| $71$ |
\( (T^{10} + 293 T^{8} + \cdots + 2760000)^{2} \)
|
| $73$ |
\( (T^{10} - 24 T^{9} + \cdots + 6858432)^{2} \)
|
| $79$ |
\( (T^{10} - 23 T^{9} + \cdots + 113018161)^{2} \)
|
| $83$ |
\( (T^{10} - 630 T^{8} + \cdots - 8547749532)^{2} \)
|
| $89$ |
\( T^{20} + \cdots + 128470708898064 \)
|
| $97$ |
\( (T^{10} + 297 T^{8} + \cdots + 194069547)^{2} \)
|
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