Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [462,2,Mod(89,462)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(462, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 5, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("462.89");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 462.k (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: | \(3.68908857338\) |
| 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: |
\( x^{20} - 6 x^{19} + 19 x^{18} - 42 x^{17} + 62 x^{16} - 42 x^{15} - 25 x^{14} + 6 x^{13} + 445 x^{12} + \cdots + 59049 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{20} - 6 x^{19} + 19 x^{18} - 42 x^{17} + 62 x^{16} - 42 x^{15} - 25 x^{14} + 6 x^{13} + 445 x^{12} + \cdots + 59049 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{19} - 6 \nu^{18} + 19 \nu^{17} - 42 \nu^{16} + 62 \nu^{15} - 42 \nu^{14} - 25 \nu^{13} + \cdots - 118098 ) / 19683 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1501 \nu^{19} - 11100 \nu^{18} + 79886 \nu^{17} - 186672 \nu^{16} + 236878 \nu^{15} + \cdots + 31768362 ) / 25351704 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 538 \nu^{19} + 1727 \nu^{18} - 21322 \nu^{17} + 102482 \nu^{16} - 220028 \nu^{15} + \cdots + 308714733 ) / 8450568 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3941 \nu^{19} + 57282 \nu^{18} - 277064 \nu^{17} + 662019 \nu^{16} - 914977 \nu^{15} + \cdots + 510911631 ) / 33802272 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1232 \nu^{19} + 5597 \nu^{18} - 13781 \nu^{17} + 29807 \nu^{16} - 38128 \nu^{15} + \cdots + 71849511 ) / 4828896 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 25957 \nu^{19} + 143919 \nu^{18} - 321337 \nu^{17} + 259002 \nu^{16} + 376723 \nu^{15} + \cdots - 958365270 ) / 101406816 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 27967 \nu^{19} + 12315 \nu^{18} + 252995 \nu^{17} - 657600 \nu^{16} + 920893 \nu^{15} + \cdots + 638910180 ) / 101406816 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 1795 \nu^{19} + 9627 \nu^{18} - 21937 \nu^{17} + 38256 \nu^{16} - 30839 \nu^{15} + \cdots + 72748368 ) / 4828896 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 16230 \nu^{19} + 123337 \nu^{18} - 452289 \nu^{17} + 1002997 \nu^{16} - 1265262 \nu^{15} + \cdots + 1053650673 ) / 33802272 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 77773 \nu^{19} + 134496 \nu^{18} + 62150 \nu^{17} - 578403 \nu^{16} + 1140403 \nu^{15} + \cdots + 977831757 ) / 101406816 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 2234 \nu^{19} + 12045 \nu^{18} - 27746 \nu^{17} + 36660 \nu^{16} - 6520 \nu^{15} + \cdots + 9848061 ) / 2816856 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 85589 \nu^{19} + 304506 \nu^{18} - 485684 \nu^{17} - 119673 \nu^{16} + 1927079 \nu^{15} + \cdots - 2300135697 ) / 101406816 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 93578 \nu^{19} - 620469 \nu^{18} + 2151005 \nu^{17} - 4614381 \nu^{16} + 5525050 \nu^{15} + \cdots - 4479437457 ) / 101406816 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 125911 \nu^{19} - 566055 \nu^{18} + 1259389 \nu^{17} - 1667448 \nu^{16} + 437723 \nu^{15} + \cdots - 1121931000 ) / 101406816 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 152435 \nu^{19} - 609981 \nu^{18} + 1552223 \nu^{17} - 2696802 \nu^{16} + 2474215 \nu^{15} + \cdots - 3856962582 ) / 101406816 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 160799 \nu^{19} + 572295 \nu^{18} - 946949 \nu^{17} + 498360 \nu^{16} + 1645421 \nu^{15} + \cdots - 523252872 ) / 101406816 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 17987 \nu^{19} - 75384 \nu^{18} + 186410 \nu^{17} - 310323 \nu^{16} + 249247 \nu^{15} + \cdots - 375510087 ) / 11267424 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 83249 \nu^{19} + 337060 \nu^{18} - 747026 \nu^{17} + 1038097 \nu^{16} - 458641 \nu^{15} + \cdots + 1122121269 ) / 33802272 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{19} - \beta_{13} - \beta_{11} - \beta_{10} - \beta_{9} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( - \beta_{17} - 2 \beta_{15} - 2 \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} + \cdots - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 2 \beta_{19} - \beta_{18} - 2 \beta_{15} - \beta_{14} + \beta_{13} - 2 \beta_{12} - \beta_{10} + \cdots + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 4 \beta_{18} - \beta_{17} - 2 \beta_{16} + \beta_{15} + 6 \beta_{14} - 5 \beta_{13} + 7 \beta_{12} + \cdots + 3 \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 2 \beta_{18} + \beta_{17} - 2 \beta_{16} + 5 \beta_{14} - 2 \beta_{13} + 4 \beta_{12} - 6 \beta_{11} + \cdots - 12 \)
|
| \(\nu^{7}\) | \(=\) |
\( 4 \beta_{19} + 2 \beta_{18} - 2 \beta_{17} + 4 \beta_{16} - \beta_{15} + 5 \beta_{14} + 9 \beta_{13} + \cdots + 14 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 10 \beta_{19} - 8 \beta_{17} + 13 \beta_{16} - 8 \beta_{15} + 37 \beta_{13} - 20 \beta_{12} + \cdots + 16 \beta_1 \)
|
| \(\nu^{9}\) | \(=\) |
\( 24 \beta_{19} - 24 \beta_{18} + 13 \beta_{17} + 24 \beta_{16} + 26 \beta_{15} + 22 \beta_{14} + \cdots + 11 \)
|
| \(\nu^{10}\) | \(=\) |
\( 7 \beta_{19} - 24 \beta_{18} - 45 \beta_{15} + 61 \beta_{14} - 61 \beta_{13} + 67 \beta_{12} + \cdots - 129 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 56 \beta_{19} + 64 \beta_{18} + 108 \beta_{17} + 32 \beta_{16} - 108 \beta_{15} - 68 \beta_{14} + \cdots + 172 \)
|
| \(\nu^{12}\) | \(=\) |
\( 20 \beta_{18} + 80 \beta_{17} + 20 \beta_{16} + 38 \beta_{14} + 20 \beta_{13} + 72 \beta_{12} + \cdots + 69 \)
|
| \(\nu^{13}\) | \(=\) |
\( 72 \beta_{19} + 56 \beta_{18} + 216 \beta_{17} + 112 \beta_{16} + 108 \beta_{15} - 82 \beta_{14} + \cdots - 510 \)
|
| \(\nu^{14}\) | \(=\) |
\( 159 \beta_{19} + 4 \beta_{17} + 64 \beta_{16} + 4 \beta_{15} - 189 \beta_{13} + 616 \beta_{12} + \cdots - 979 \beta_1 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 1224 \beta_{19} + 432 \beta_{18} - 651 \beta_{17} - 432 \beta_{16} - 1302 \beta_{15} - 950 \beta_{14} + \cdots + 523 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 514 \beta_{19} + 29 \beta_{18} + 1150 \beta_{15} - 1557 \beta_{14} + 1557 \beta_{13} - 490 \beta_{12} + \cdots + 1596 \)
|
| \(\nu^{17}\) | \(=\) |
\( 820 \beta_{19} - 372 \beta_{18} - 1879 \beta_{17} - 186 \beta_{16} + 1879 \beta_{15} - 74 \beta_{14} + \cdots - 9716 \)
|
| \(\nu^{18}\) | \(=\) |
\( 3390 \beta_{18} + 771 \beta_{17} + 3390 \beta_{16} - 4307 \beta_{14} + 3390 \beta_{13} - 5768 \beta_{12} + \cdots - 4578 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 5768 \beta_{19} - 950 \beta_{18} - 3830 \beta_{17} - 1900 \beta_{16} - 1915 \beta_{15} + 757 \beta_{14} + \cdots + 2384 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/462\mathbb{Z}\right)^\times\).
| \(n\) | \(155\) | \(199\) | \(211\) |
| \(\chi(n)\) | \(-1\) | \(1 + \beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 89.1 |
|
−0.866025 | − | 0.500000i | −1.44390 | − | 0.956629i | 0.500000 | + | 0.866025i | 0.0938814 | − | 0.162607i | 0.772144 | + | 1.55042i | −2.64498 | − | 0.0638828i | − | 1.00000i | 1.16972 | + | 2.76256i | −0.162607 | + | 0.0938814i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.2 | −0.866025 | − | 0.500000i | −1.20144 | + | 1.24762i | 0.500000 | + | 0.866025i | −0.798258 | + | 1.38262i | 1.66428 | − | 0.479752i | 0.157053 | − | 2.64109i | − | 1.00000i | −0.113106 | − | 2.99787i | 1.38262 | − | 0.798258i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.3 | −0.866025 | − | 0.500000i | −0.530717 | + | 1.64874i | 0.500000 | + | 0.866025i | −0.417958 | + | 0.723925i | 1.28398 | − | 1.16249i | −1.42670 | + | 2.22812i | − | 1.00000i | −2.43668 | − | 1.75003i | 0.723925 | − | 0.417958i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.4 | −0.866025 | − | 0.500000i | 0.0733649 | − | 1.73050i | 0.500000 | + | 0.866025i | 1.79481 | − | 3.10870i | −0.928784 | + | 1.46197i | 0.833981 | − | 2.51087i | − | 1.00000i | −2.98924 | − | 0.253915i | −3.10870 | + | 1.79481i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.5 | −0.866025 | − | 0.500000i | 1.60269 | + | 0.656793i | 0.500000 | + | 0.866025i | 1.92560 | − | 3.33524i | −1.05958 | − | 1.37015i | 1.58064 | + | 2.12169i | − | 1.00000i | 2.13725 | + | 2.10527i | −3.33524 | + | 1.92560i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.6 | 0.866025 | + | 0.500000i | −1.69321 | − | 0.364755i | 0.500000 | + | 0.866025i | 0.417958 | − | 0.723925i | −1.28398 | − | 1.16249i | −1.42670 | + | 2.22812i | 1.00000i | 2.73391 | + | 1.23521i | 0.723925 | − | 0.417958i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.7 | 0.866025 | + | 0.500000i | −1.68119 | + | 0.416664i | 0.500000 | + | 0.866025i | 0.798258 | − | 1.38262i | −1.66428 | − | 0.479752i | 0.157053 | − | 2.64109i | 1.00000i | 2.65278 | − | 1.40098i | 1.38262 | − | 0.798258i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.8 | 0.866025 | + | 0.500000i | 0.106512 | + | 1.72877i | 0.500000 | + | 0.866025i | −0.0938814 | + | 0.162607i | −0.772144 | + | 1.55042i | −2.64498 | − | 0.0638828i | 1.00000i | −2.97731 | + | 0.368271i | −0.162607 | + | 0.0938814i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.9 | 0.866025 | + | 0.500000i | 0.232547 | − | 1.71637i | 0.500000 | + | 0.866025i | −1.92560 | + | 3.33524i | 1.05958 | − | 1.37015i | 1.58064 | + | 2.12169i | 1.00000i | −2.89184 | − | 0.798273i | −3.33524 | + | 1.92560i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.10 | 0.866025 | + | 0.500000i | 1.53534 | + | 0.801712i | 0.500000 | + | 0.866025i | −1.79481 | + | 3.10870i | 0.928784 | + | 1.46197i | 0.833981 | − | 2.51087i | 1.00000i | 1.71451 | + | 2.46180i | −3.10870 | + | 1.79481i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.1 | −0.866025 | + | 0.500000i | −1.44390 | + | 0.956629i | 0.500000 | − | 0.866025i | 0.0938814 | + | 0.162607i | 0.772144 | − | 1.55042i | −2.64498 | + | 0.0638828i | 1.00000i | 1.16972 | − | 2.76256i | −0.162607 | − | 0.0938814i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.2 | −0.866025 | + | 0.500000i | −1.20144 | − | 1.24762i | 0.500000 | − | 0.866025i | −0.798258 | − | 1.38262i | 1.66428 | + | 0.479752i | 0.157053 | + | 2.64109i | 1.00000i | −0.113106 | + | 2.99787i | 1.38262 | + | 0.798258i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.3 | −0.866025 | + | 0.500000i | −0.530717 | − | 1.64874i | 0.500000 | − | 0.866025i | −0.417958 | − | 0.723925i | 1.28398 | + | 1.16249i | −1.42670 | − | 2.22812i | 1.00000i | −2.43668 | + | 1.75003i | 0.723925 | + | 0.417958i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.4 | −0.866025 | + | 0.500000i | 0.0733649 | + | 1.73050i | 0.500000 | − | 0.866025i | 1.79481 | + | 3.10870i | −0.928784 | − | 1.46197i | 0.833981 | + | 2.51087i | 1.00000i | −2.98924 | + | 0.253915i | −3.10870 | − | 1.79481i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.5 | −0.866025 | + | 0.500000i | 1.60269 | − | 0.656793i | 0.500000 | − | 0.866025i | 1.92560 | + | 3.33524i | −1.05958 | + | 1.37015i | 1.58064 | − | 2.12169i | 1.00000i | 2.13725 | − | 2.10527i | −3.33524 | − | 1.92560i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.6 | 0.866025 | − | 0.500000i | −1.69321 | + | 0.364755i | 0.500000 | − | 0.866025i | 0.417958 | + | 0.723925i | −1.28398 | + | 1.16249i | −1.42670 | − | 2.22812i | − | 1.00000i | 2.73391 | − | 1.23521i | 0.723925 | + | 0.417958i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.7 | 0.866025 | − | 0.500000i | −1.68119 | − | 0.416664i | 0.500000 | − | 0.866025i | 0.798258 | + | 1.38262i | −1.66428 | + | 0.479752i | 0.157053 | + | 2.64109i | − | 1.00000i | 2.65278 | + | 1.40098i | 1.38262 | + | 0.798258i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.8 | 0.866025 | − | 0.500000i | 0.106512 | − | 1.72877i | 0.500000 | − | 0.866025i | −0.0938814 | − | 0.162607i | −0.772144 | − | 1.55042i | −2.64498 | + | 0.0638828i | − | 1.00000i | −2.97731 | − | 0.368271i | −0.162607 | − | 0.0938814i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.9 | 0.866025 | − | 0.500000i | 0.232547 | + | 1.71637i | 0.500000 | − | 0.866025i | −1.92560 | − | 3.33524i | 1.05958 | + | 1.37015i | 1.58064 | − | 2.12169i | − | 1.00000i | −2.89184 | + | 0.798273i | −3.33524 | − | 1.92560i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 353.10 | 0.866025 | − | 0.500000i | 1.53534 | − | 0.801712i | 0.500000 | − | 0.866025i | −1.79481 | − | 3.10870i | 0.928784 | − | 1.46197i | 0.833981 | + | 2.51087i | − | 1.00000i | 1.71451 | − | 2.46180i | −3.10870 | − | 1.79481i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 462.2.k.g | ✓ | 20 |
| 3.b | odd | 2 | 1 | inner | 462.2.k.g | ✓ | 20 |
| 7.d | odd | 6 | 1 | inner | 462.2.k.g | ✓ | 20 |
| 21.g | even | 6 | 1 | inner | 462.2.k.g | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 462.2.k.g | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 462.2.k.g | ✓ | 20 | 3.b | odd | 2 | 1 | inner |
| 462.2.k.g | ✓ | 20 | 7.d | odd | 6 | 1 | inner |
| 462.2.k.g | ✓ | 20 | 21.g | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(462, [\chi])\):
|
\( T_{5}^{20} + 31 T_{5}^{18} + 677 T_{5}^{16} + 7444 T_{5}^{14} + 59212 T_{5}^{12} + 170564 T_{5}^{10} + \cdots + 144 \)
|
|
\( T_{17}^{20} + 59 T_{17}^{18} + 2561 T_{17}^{16} + 42632 T_{17}^{14} + 486784 T_{17}^{12} + \cdots + 241864704 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - T^{2} + 1)^{5} \)
|
| $3$ |
\( T^{20} + 6 T^{19} + \cdots + 59049 \)
|
| $5$ |
\( T^{20} + 31 T^{18} + \cdots + 144 \)
|
| $7$ |
\( (T^{10} + 3 T^{9} + \cdots + 16807)^{2} \)
|
| $11$ |
\( (T^{4} - T^{2} + 1)^{5} \)
|
| $13$ |
\( (T^{10} + 68 T^{8} + \cdots + 9408)^{2} \)
|
| $17$ |
\( T^{20} + \cdots + 241864704 \)
|
| $19$ |
\( (T^{10} - 18 T^{9} + \cdots + 363312)^{2} \)
|
| $23$ |
\( T^{20} + \cdots + 2379503694096 \)
|
| $29$ |
\( (T^{10} + 196 T^{8} + \cdots + 52881984)^{2} \)
|
| $31$ |
\( (T^{10} - 18 T^{9} + \cdots + 845107968)^{2} \)
|
| $37$ |
\( (T^{10} - 8 T^{9} + \cdots + 256)^{2} \)
|
| $41$ |
\( (T^{10} - 31 T^{8} + \cdots - 12)^{2} \)
|
| $43$ |
\( (T^{5} - 8 T^{4} + \cdots - 3712)^{4} \)
|
| $47$ |
\( T^{20} + \cdots + 17\!\cdots\!84 \)
|
| $53$ |
\( T^{20} + \cdots + 11\!\cdots\!96 \)
|
| $59$ |
\( T^{20} + \cdots + 435854640545424 \)
|
| $61$ |
\( (T^{10} - 21 T^{9} + \cdots + 29862075)^{2} \)
|
| $67$ |
\( (T^{10} + 5 T^{9} + \cdots + 6215049)^{2} \)
|
| $71$ |
\( (T^{10} + 300 T^{8} + \cdots + 1679616)^{2} \)
|
| $73$ |
\( (T^{10} - 6 T^{9} + \cdots + 119675568)^{2} \)
|
| $79$ |
\( (T^{10} - 3 T^{9} + \cdots + 94848121)^{2} \)
|
| $83$ |
\( (T^{10} - 667 T^{8} + \cdots - 3378820800)^{2} \)
|
| $89$ |
\( T^{20} + \cdots + 34\!\cdots\!84 \)
|
| $97$ |
\( (T^{10} + 263 T^{8} + \cdots + 240267)^{2} \)
|
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