Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [351,2,Mod(64,351)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(351, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("351.64");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 351 = 3^{3} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 351.t (of order \(6\), degree \(2\), not 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: | \(2.80274911095\) |
| 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: |
\( x^{20} - 6x^{16} + 9x^{14} + 54x^{12} + 81x^{10} + 486x^{8} + 729x^{6} - 4374x^{4} + 59049 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{9} \) |
| Twist minimal: | no (minimal twist has level 117) |
| 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
\( x^{20} - 6x^{16} + 9x^{14} + 54x^{12} + 81x^{10} + 486x^{8} + 729x^{6} - 4374x^{4} + 59049 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{18} - 3\nu^{16} - 3\nu^{14} + 9\nu^{12} - 27\nu^{10} - 324\nu^{8} - 486\nu^{6} - 729\nu^{4} + 6561 ) / 6561 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{18} + 6\nu^{16} + 3\nu^{14} + 54\nu^{10} + 81\nu^{8} + 972\nu^{6} + 2916\nu^{4} + 2187\nu^{2} - 13122 ) / 6561 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 4 \nu^{18} - 21 \nu^{16} - 66 \nu^{14} - 99 \nu^{12} - 189 \nu^{10} - 648 \nu^{8} - 4617 \nu^{6} + \cdots - 39366 ) / 19683 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5 \nu^{19} + 15 \nu^{17} - 12 \nu^{15} - 153 \nu^{13} - 27 \nu^{11} + 324 \nu^{9} + 2673 \nu^{7} + \cdots - 98415 \nu ) / 59049 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 4 \nu^{18} - 3 \nu^{16} + 42 \nu^{14} + 117 \nu^{12} + 54 \nu^{10} + 81 \nu^{8} - 972 \nu^{6} + \cdots + 118098 ) / 19683 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 5 \nu^{19} - 6 \nu^{17} + 66 \nu^{15} + 99 \nu^{13} + 27 \nu^{11} - 81 \nu^{9} + \cdots + 157464 \nu ) / 59049 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4 \nu^{19} - 6 \nu^{17} - 96 \nu^{15} - 225 \nu^{13} - 297 \nu^{11} - 810 \nu^{9} + \cdots - 196830 \nu ) / 59049 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 8 \nu^{18} - 24 \nu^{16} + 3 \nu^{14} + 18 \nu^{12} - 54 \nu^{10} - 324 \nu^{8} - 3402 \nu^{6} + \cdots + 98415 ) / 19683 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 8 \nu^{19} - 24 \nu^{17} + 57 \nu^{15} + 180 \nu^{13} + 108 \nu^{11} - 81 \nu^{9} + \cdots + 216513 \nu ) / 59049 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 7 \nu^{18} + 6 \nu^{16} + 87 \nu^{14} + 198 \nu^{12} + 297 \nu^{10} + 324 \nu^{8} - 1701 \nu^{6} + \cdots + 177147 ) / 19683 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 10 \nu^{19} - 21 \nu^{17} + 51 \nu^{15} + 90 \nu^{13} + 216 \nu^{11} - 405 \nu^{9} + \cdots + 137781 \nu ) / 59049 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 10 \nu^{19} - 3 \nu^{17} + 78 \nu^{15} + 225 \nu^{13} - 27 \nu^{11} + 81 \nu^{9} + \cdots + 196830 \nu ) / 59049 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 4 \nu^{18} + 21 \nu^{16} + 30 \nu^{14} + 18 \nu^{12} + 81 \nu^{10} + 324 \nu^{8} + 3159 \nu^{6} + \cdots - 26244 ) / 6561 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 11 \nu^{19} + 15 \nu^{17} - 102 \nu^{15} - 180 \nu^{13} - 108 \nu^{11} - 648 \nu^{9} + \cdots - 216513 \nu ) / 59049 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 13 \nu^{19} + 30 \nu^{17} - 15 \nu^{15} - 117 \nu^{13} - 54 \nu^{11} + 891 \nu^{9} + \cdots - 196830 \nu ) / 59049 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 8 \nu^{18} - 6 \nu^{16} + 111 \nu^{14} + 315 \nu^{12} + 432 \nu^{10} + 648 \nu^{8} + \cdots + 295245 ) / 19683 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 7 \nu^{18} - 15 \nu^{16} - 195 \nu^{14} - 387 \nu^{12} - 459 \nu^{10} - 324 \nu^{8} - 486 \nu^{6} + \cdots - 334611 ) / 19683 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 8 \nu^{19} + 78 \nu^{17} + 240 \nu^{15} + 306 \nu^{13} + 783 \nu^{11} + 2268 \nu^{9} + \cdots + 137781 \nu ) / 59049 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 14 \nu^{19} + 78 \nu^{17} + 258 \nu^{15} + 441 \nu^{13} + 783 \nu^{11} + 2025 \nu^{9} + \cdots + 196830 \nu ) / 59049 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{15} - \beta_{12} - 2\beta_{11} - \beta_{7} + \beta_{6} + \beta_{4} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{17} + 2\beta_{16} + 2\beta_{13} - \beta_{10} + 2\beta_{8} - \beta_{5} + \beta_{3} - \beta_{2} + \beta _1 - 1 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{19} - \beta_{18} - \beta_{15} - \beta_{14} - \beta_{9} - \beta_{6} + \beta_{4} \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{17} + 2\beta_{10} - 2\beta_{5} - 2\beta_{3} - 2\beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( - \beta_{19} + 2 \beta_{18} - 2 \beta_{15} - 2 \beta_{14} - 3 \beta_{12} - 3 \beta_{11} + \cdots - 4 \beta_{4} \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{17} + \beta_{16} + \beta_{13} - 5\beta_{10} + 7\beta_{8} + 7\beta_{5} - 4\beta_{3} + 7\beta_{2} + 2\beta _1 - 5 \)
|
| \(\nu^{7}\) | \(=\) |
\( -3\beta_{18} + 3\beta_{15} - 3\beta_{12} + 3\beta_{11} - 12\beta_{9} - 9\beta_{7} + 9\beta_{6} - 6\beta_{4} \)
|
| \(\nu^{8}\) | \(=\) |
\( 12 \beta_{17} + 3 \beta_{16} + 3 \beta_{13} + 6 \beta_{10} + 3 \beta_{8} + 18 \beta_{5} - 6 \beta_{3} + \cdots - 21 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 24 \beta_{19} + 12 \beta_{18} + 24 \beta_{15} - 21 \beta_{14} - 9 \beta_{12} - 27 \beta_{11} + \cdots + 3 \beta_{4} \)
|
| \(\nu^{10}\) | \(=\) |
\( - 15 \beta_{17} + 24 \beta_{16} - 30 \beta_{13} + 24 \beta_{10} + 6 \beta_{8} - 165 \beta_{5} + \cdots + 51 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 18 \beta_{19} + 45 \beta_{18} - 72 \beta_{15} + 45 \beta_{14} - 99 \beta_{12} + 72 \beta_{11} + \cdots - 99 \beta_{6} \)
|
| \(\nu^{12}\) | \(=\) |
\( 54 \beta_{17} + 36 \beta_{16} - 45 \beta_{13} - 81 \beta_{10} - 180 \beta_{8} + 423 \beta_{5} + \cdots - 567 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 9 \beta_{19} - 9 \beta_{18} + 306 \beta_{15} + 144 \beta_{14} + 216 \beta_{12} + 135 \beta_{11} + \cdots - 441 \beta_{4} \)
|
| \(\nu^{14}\) | \(=\) |
\( - 549 \beta_{17} - 504 \beta_{16} - 342 \beta_{13} + 117 \beta_{10} - 207 \beta_{8} - 477 \beta_{5} + \cdots - 450 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 54 \beta_{19} + 162 \beta_{18} + 1296 \beta_{15} + 135 \beta_{14} + 378 \beta_{12} + \cdots - 108 \beta_{4} \)
|
| \(\nu^{16}\) | \(=\) |
\( - 351 \beta_{17} - 729 \beta_{16} + 108 \beta_{10} - 891 \beta_{8} + 2241 \beta_{5} + 1593 \beta_{3} + \cdots + 1323 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 1026 \beta_{19} + 1080 \beta_{18} - 270 \beta_{15} + 3294 \beta_{14} + 2916 \beta_{12} + \cdots + 2538 \beta_{4} \)
|
| \(\nu^{18}\) | \(=\) |
\( - 1998 \beta_{17} + 1917 \beta_{16} - 27 \beta_{13} - 3024 \beta_{10} - 2862 \beta_{8} - 4806 \beta_{5} + \cdots + 5238 \)
|
| \(\nu^{19}\) | \(=\) |
\( 7776 \beta_{19} - 6723 \beta_{18} - 3969 \beta_{15} - 4131 \beta_{14} - 3078 \beta_{12} + \cdots + 6723 \beta_{4} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/351\mathbb{Z}\right)^\times\).
| \(n\) | \(28\) | \(326\) |
| \(\chi(n)\) | \(-1\) | \(-\beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 64.1 |
|
−2.14539 | + | 1.23864i | 0 | 2.06847 | − | 3.58269i | −0.771397 | − | 0.445366i | 0 | −0.850723 | + | 0.491165i | 5.29379i | 0 | 2.20660 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.2 | −1.97712 | + | 1.14149i | 0 | 1.60600 | − | 2.78168i | 2.78501 | + | 1.60793i | 0 | −2.09815 | + | 1.21137i | 2.76698i | 0 | −7.34174 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.3 | −1.41717 | + | 0.818205i | 0 | 0.338918 | − | 0.587023i | −0.950358 | − | 0.548689i | 0 | −2.77942 | + | 1.60470i | − | 2.16360i | 0 | 1.79576 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.4 | −0.929969 | + | 0.536918i | 0 | −0.423439 | + | 0.733417i | −1.10543 | − | 0.638222i | 0 | 0.890926 | − | 0.514376i | − | 3.05708i | 0 | 1.37069 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.5 | −0.784270 | + | 0.452798i | 0 | −0.589947 | + | 1.02182i | 1.94254 | + | 1.12153i | 0 | 2.97576 | − | 1.71806i | − | 2.87970i | 0 | −2.03130 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.6 | 0.784270 | − | 0.452798i | 0 | −0.589947 | + | 1.02182i | −1.94254 | − | 1.12153i | 0 | −2.97576 | + | 1.71806i | 2.87970i | 0 | −2.03130 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.7 | 0.929969 | − | 0.536918i | 0 | −0.423439 | + | 0.733417i | 1.10543 | + | 0.638222i | 0 | −0.890926 | + | 0.514376i | 3.05708i | 0 | 1.37069 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.8 | 1.41717 | − | 0.818205i | 0 | 0.338918 | − | 0.587023i | 0.950358 | + | 0.548689i | 0 | 2.77942 | − | 1.60470i | 2.16360i | 0 | 1.79576 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.9 | 1.97712 | − | 1.14149i | 0 | 1.60600 | − | 2.78168i | −2.78501 | − | 1.60793i | 0 | 2.09815 | − | 1.21137i | − | 2.76698i | 0 | −7.34174 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 64.10 | 2.14539 | − | 1.23864i | 0 | 2.06847 | − | 3.58269i | 0.771397 | + | 0.445366i | 0 | 0.850723 | − | 0.491165i | − | 5.29379i | 0 | 2.20660 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.1 | −2.14539 | − | 1.23864i | 0 | 2.06847 | + | 3.58269i | −0.771397 | + | 0.445366i | 0 | −0.850723 | − | 0.491165i | − | 5.29379i | 0 | 2.20660 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.2 | −1.97712 | − | 1.14149i | 0 | 1.60600 | + | 2.78168i | 2.78501 | − | 1.60793i | 0 | −2.09815 | − | 1.21137i | − | 2.76698i | 0 | −7.34174 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.3 | −1.41717 | − | 0.818205i | 0 | 0.338918 | + | 0.587023i | −0.950358 | + | 0.548689i | 0 | −2.77942 | − | 1.60470i | 2.16360i | 0 | 1.79576 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.4 | −0.929969 | − | 0.536918i | 0 | −0.423439 | − | 0.733417i | −1.10543 | + | 0.638222i | 0 | 0.890926 | + | 0.514376i | 3.05708i | 0 | 1.37069 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.5 | −0.784270 | − | 0.452798i | 0 | −0.589947 | − | 1.02182i | 1.94254 | − | 1.12153i | 0 | 2.97576 | + | 1.71806i | 2.87970i | 0 | −2.03130 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.6 | 0.784270 | + | 0.452798i | 0 | −0.589947 | − | 1.02182i | −1.94254 | + | 1.12153i | 0 | −2.97576 | − | 1.71806i | − | 2.87970i | 0 | −2.03130 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.7 | 0.929969 | + | 0.536918i | 0 | −0.423439 | − | 0.733417i | 1.10543 | − | 0.638222i | 0 | −0.890926 | − | 0.514376i | − | 3.05708i | 0 | 1.37069 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.8 | 1.41717 | + | 0.818205i | 0 | 0.338918 | + | 0.587023i | 0.950358 | − | 0.548689i | 0 | 2.77942 | + | 1.60470i | − | 2.16360i | 0 | 1.79576 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.9 | 1.97712 | + | 1.14149i | 0 | 1.60600 | + | 2.78168i | −2.78501 | + | 1.60793i | 0 | 2.09815 | + | 1.21137i | 2.76698i | 0 | −7.34174 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.10 | 2.14539 | + | 1.23864i | 0 | 2.06847 | + | 3.58269i | 0.771397 | − | 0.445366i | 0 | 0.850723 | + | 0.491165i | 5.29379i | 0 | 2.20660 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
| 13.b | even | 2 | 1 | inner |
| 117.t | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 351.2.t.c | 20 | |
| 3.b | odd | 2 | 1 | 117.2.t.c | ✓ | 20 | |
| 9.c | even | 3 | 1 | inner | 351.2.t.c | 20 | |
| 9.c | even | 3 | 1 | 1053.2.b.i | 10 | ||
| 9.d | odd | 6 | 1 | 117.2.t.c | ✓ | 20 | |
| 9.d | odd | 6 | 1 | 1053.2.b.j | 10 | ||
| 13.b | even | 2 | 1 | inner | 351.2.t.c | 20 | |
| 39.d | odd | 2 | 1 | 117.2.t.c | ✓ | 20 | |
| 117.n | odd | 6 | 1 | 117.2.t.c | ✓ | 20 | |
| 117.n | odd | 6 | 1 | 1053.2.b.j | 10 | ||
| 117.t | even | 6 | 1 | inner | 351.2.t.c | 20 | |
| 117.t | even | 6 | 1 | 1053.2.b.i | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 117.2.t.c | ✓ | 20 | 3.b | odd | 2 | 1 | |
| 117.2.t.c | ✓ | 20 | 9.d | odd | 6 | 1 | |
| 117.2.t.c | ✓ | 20 | 39.d | odd | 2 | 1 | |
| 117.2.t.c | ✓ | 20 | 117.n | odd | 6 | 1 | |
| 351.2.t.c | 20 | 1.a | even | 1 | 1 | trivial | |
| 351.2.t.c | 20 | 9.c | even | 3 | 1 | inner | |
| 351.2.t.c | 20 | 13.b | even | 2 | 1 | inner | |
| 351.2.t.c | 20 | 117.t | even | 6 | 1 | inner | |
| 1053.2.b.i | 10 | 9.c | even | 3 | 1 | ||
| 1053.2.b.i | 10 | 117.t | even | 6 | 1 | ||
| 1053.2.b.j | 10 | 9.d | odd | 6 | 1 | ||
| 1053.2.b.j | 10 | 117.n | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(351, [\chi])\):
|
\( T_{2}^{20} - 16 T_{2}^{18} + 165 T_{2}^{16} - 1012 T_{2}^{14} + 4501 T_{2}^{12} - 12987 T_{2}^{10} + \cdots + 6561 \)
|
|
\( T_{5}^{20} - 19 T_{5}^{18} + 249 T_{5}^{16} - 1618 T_{5}^{14} + 7456 T_{5}^{12} - 19407 T_{5}^{10} + \cdots + 6561 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - 16 T^{18} + \cdots + 6561 \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} - 19 T^{18} + \cdots + 6561 \)
|
| $7$ |
\( T^{20} - 30 T^{18} + \cdots + 531441 \)
|
| $11$ |
\( T^{20} + \cdots + 276922881 \)
|
| $13$ |
\( T^{20} + \cdots + 137858491849 \)
|
| $17$ |
\( (T^{5} - 3 T^{4} - 33 T^{3} + \cdots - 81)^{4} \)
|
| $19$ |
\( (T^{10} + 129 T^{8} + \cdots + 700569)^{2} \)
|
| $23$ |
\( (T^{10} + 12 T^{9} + \cdots + 531441)^{2} \)
|
| $29$ |
\( (T^{10} + 6 T^{9} + \cdots + 6561)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 138251528157681 \)
|
| $37$ |
\( (T^{10} + 231 T^{8} + \cdots + 41641209)^{2} \)
|
| $41$ |
\( T^{20} + \cdots + 273245607441 \)
|
| $43$ |
\( (T^{10} - 2 T^{9} + \cdots + 32041)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 49241109874401 \)
|
| $53$ |
\( (T^{5} + 27 T^{4} + \cdots + 243)^{4} \)
|
| $59$ |
\( T^{20} + \cdots + 75017234240001 \)
|
| $61$ |
\( (T^{10} + T^{9} + \cdots + 118091689)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 282429536481 \)
|
| $71$ |
\( (T^{10} + 97 T^{8} + \cdots + 178929)^{2} \)
|
| $73$ |
\( (T^{10} + 369 T^{8} + \cdots + 56746089)^{2} \)
|
| $79$ |
\( (T^{10} + 7 T^{9} + \cdots + 1481089)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 20\!\cdots\!81 \)
|
| $89$ |
\( (T^{10} + 637 T^{8} + \cdots + 5851791009)^{2} \)
|
| $97$ |
\( T^{20} + \cdots + 47048089623921 \)
|
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