Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [648,3,Mod(163,648)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(648, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("648.163");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 648 = 2^{3} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 648.b (of order \(2\), degree \(1\), 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: | \(17.6567211305\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 5 x^{19} + 16 x^{18} - 46 x^{17} + 122 x^{16} - 292 x^{15} + 692 x^{14} - 1528 x^{13} + \cdots + 1048576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{6} \) |
| Twist minimal: | no (minimal twist has level 72) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{20} - 5 x^{19} + 16 x^{18} - 46 x^{17} + 122 x^{16} - 292 x^{15} + 692 x^{14} - 1528 x^{13} + \cdots + 1048576 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{19} - 5 \nu^{18} + 16 \nu^{17} - 46 \nu^{16} + 122 \nu^{15} - 292 \nu^{14} + 692 \nu^{13} + \cdots - 1048576 ) / 262144 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{19} + 5 \nu^{18} - 16 \nu^{17} + 46 \nu^{16} - 122 \nu^{15} + 292 \nu^{14} - 692 \nu^{13} + \cdots + 1310720 ) / 262144 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1389 \nu^{19} + 3157 \nu^{18} - 12596 \nu^{17} + 28374 \nu^{16} - 74218 \nu^{15} + \cdots + 329777152 ) / 54788096 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 451 \nu^{19} - 2353 \nu^{18} + 6066 \nu^{17} - 16130 \nu^{16} + 42794 \nu^{15} - 89648 \nu^{14} + \cdots - 456785920 ) / 27394048 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 5 \nu^{19} + 21 \nu^{18} - 60 \nu^{17} + 166 \nu^{16} - 426 \nu^{15} + 972 \nu^{14} + \cdots + 2359296 ) / 262144 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 263 \nu^{19} + 2893 \nu^{18} - 5122 \nu^{17} + 21298 \nu^{16} - 43314 \nu^{15} + \cdots + 635961344 ) / 27394048 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 1947 \nu^{19} - 4547 \nu^{18} + 16876 \nu^{17} - 39562 \nu^{16} + 92230 \nu^{15} + \cdots - 324009984 ) / 54788096 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 9 \nu^{19} + 25 \nu^{18} - 60 \nu^{17} + 174 \nu^{16} - 434 \nu^{15} + 924 \nu^{14} + \cdots + 2621440 ) / 262144 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 211 \nu^{19} + 858 \nu^{18} - 2241 \nu^{17} + 7740 \nu^{16} - 17256 \nu^{15} + 43430 \nu^{14} + \cdots + 182845440 ) / 6848512 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 103 \nu^{19} + 489 \nu^{18} - 1344 \nu^{17} + 4570 \nu^{16} - 10072 \nu^{15} + 24412 \nu^{14} + \cdots + 104955904 ) / 3424256 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 1017 \nu^{19} + 3663 \nu^{18} - 10426 \nu^{17} + 29998 \nu^{16} - 70662 \nu^{15} + \cdots + 508297216 ) / 27394048 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 2321 \nu^{19} + 6861 \nu^{18} - 21256 \nu^{17} + 62030 \nu^{16} - 154826 \nu^{15} + \cdots + 1143996416 ) / 54788096 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 571 \nu^{19} + 1892 \nu^{18} - 5177 \nu^{17} + 14434 \nu^{16} - 35988 \nu^{15} + \cdots + 194772992 ) / 13697024 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 3617 \nu^{19} + 8917 \nu^{18} - 28304 \nu^{17} + 73918 \nu^{16} - 190682 \nu^{15} + \cdots + 1484259328 ) / 54788096 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 983 \nu^{19} - 3109 \nu^{18} + 9298 \nu^{17} - 26122 \nu^{16} + 65394 \nu^{15} - 150080 \nu^{14} + \cdots - 461963264 ) / 13697024 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 1098 \nu^{19} + 3265 \nu^{18} - 10859 \nu^{17} + 29580 \nu^{16} - 74470 \nu^{15} + \cdots + 480575488 ) / 13697024 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 5083 \nu^{19} - 16683 \nu^{18} + 47236 \nu^{17} - 132762 \nu^{16} + 323574 \nu^{15} + \cdots - 2347499520 ) / 54788096 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 1792 \nu^{19} - 5411 \nu^{18} + 16391 \nu^{17} - 46536 \nu^{16} + 114730 \nu^{15} + \cdots - 797900800 ) / 13697024 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{18} + \beta_{15} + \beta_{14} + \beta_{12} - \beta_{9} + \beta_{6} - \beta_{4} - 2\beta_{3} + \beta_{2} - 2\beta _1 + 4 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - \beta_{19} + 2 \beta_{18} - \beta_{15} + 2 \beta_{13} - 2 \beta_{11} + \beta_{10} - 2 \beta_{8} + \cdots - 3 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - \beta_{19} - 6 \beta_{18} - 2 \beta_{17} - 3 \beta_{15} - 2 \beta_{14} - 2 \beta_{11} + 3 \beta_{10} + \cdots + 5 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 5 \beta_{19} + 2 \beta_{18} - 2 \beta_{17} - 4 \beta_{16} + 3 \beta_{15} + 6 \beta_{14} + 8 \beta_{13} + \cdots - 23 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 9 \beta_{19} + 12 \beta_{18} + 2 \beta_{17} - 4 \beta_{16} - \beta_{15} - 8 \beta_{13} - 14 \beta_{12} + \cdots - 37 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - \beta_{19} - 8 \beta_{18} + 22 \beta_{17} - 4 \beta_{16} - 9 \beta_{15} + 24 \beta_{14} - 28 \beta_{13} + \cdots - 41 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 39 \beta_{19} + 36 \beta_{18} + 38 \beta_{17} + 20 \beta_{16} + 19 \beta_{15} - 36 \beta_{14} + \cdots - 165 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 19 \beta_{19} - 80 \beta_{18} - 26 \beta_{17} + 52 \beta_{16} + 115 \beta_{15} - 176 \beta_{14} + \cdots - 745 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 123 \beta_{19} + 8 \beta_{18} + 62 \beta_{17} - 284 \beta_{16} - 53 \beta_{15} + 176 \beta_{14} + \cdots + 335 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 163 \beta_{19} - 488 \beta_{18} - 146 \beta_{17} - 12 \beta_{16} + 19 \beta_{15} - 16 \beta_{14} + \cdots + 831 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 379 \beta_{19} - 1552 \beta_{18} - 354 \beta_{17} + 948 \beta_{16} + 3 \beta_{15} - 88 \beta_{14} + \cdots - 4313 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 549 \beta_{19} + 1872 \beta_{18} + 974 \beta_{17} - 12 \beta_{16} + 307 \beta_{15} + 984 \beta_{14} + \cdots + 11735 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 661 \beta_{19} + 1248 \beta_{18} + 526 \beta_{17} + 916 \beta_{16} + 1555 \beta_{15} - 408 \beta_{14} + \cdots - 745 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 1717 \beta_{19} - 7728 \beta_{18} - 1810 \beta_{17} + 660 \beta_{16} - 10685 \beta_{15} - 4552 \beta_{14} + \cdots - 8553 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 8357 \beta_{19} + 2832 \beta_{18} - 8786 \beta_{17} + 1428 \beta_{16} + 5363 \beta_{15} + \cdots - 28825 ) / 2 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 1397 \beta_{19} + 31824 \beta_{18} - 11314 \beta_{17} + 8852 \beta_{16} + 7907 \beta_{15} + \cdots - 73417 ) / 2 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 22085 \beta_{19} - 15888 \beta_{18} + 4654 \beta_{17} - 24876 \beta_{16} - 92365 \beta_{15} + \cdots + 355239 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).
| \(n\) | \(325\) | \(487\) | \(569\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 163.1 |
|
−1.91586 | − | 0.574017i | 0 | 3.34101 | + | 2.19947i | − | 9.32956i | 0 | 5.67213i | −5.13836 | − | 6.13166i | 0 | −5.35532 | + | 17.8741i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.2 | −1.91586 | + | 0.574017i | 0 | 3.34101 | − | 2.19947i | 9.32956i | 0 | − | 5.67213i | −5.13836 | + | 6.13166i | 0 | −5.35532 | − | 17.8741i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.3 | −1.84882 | − | 0.762809i | 0 | 2.83624 | + | 2.82059i | 5.08972i | 0 | 12.6566i | −3.09212 | − | 7.37826i | 0 | 3.88249 | − | 9.40996i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.4 | −1.84882 | + | 0.762809i | 0 | 2.83624 | − | 2.82059i | − | 5.08972i | 0 | − | 12.6566i | −3.09212 | + | 7.37826i | 0 | 3.88249 | + | 9.40996i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.5 | −1.71211 | − | 1.03377i | 0 | 1.86265 | + | 3.53985i | − | 1.74300i | 0 | − | 9.16606i | 0.470322 | − | 7.98616i | 0 | −1.80186 | + | 2.98421i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.6 | −1.71211 | + | 1.03377i | 0 | 1.86265 | − | 3.53985i | 1.74300i | 0 | 9.16606i | 0.470322 | + | 7.98616i | 0 | −1.80186 | − | 2.98421i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.7 | −1.16351 | − | 1.62673i | 0 | −1.29249 | + | 3.78543i | 4.44064i | 0 | 0.813280i | 7.66169 | − | 2.30184i | 0 | 7.22371 | − | 5.16672i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.8 | −1.16351 | + | 1.62673i | 0 | −1.29249 | − | 3.78543i | − | 4.44064i | 0 | − | 0.813280i | 7.66169 | + | 2.30184i | 0 | 7.22371 | + | 5.16672i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.9 | −0.599583 | − | 1.90801i | 0 | −3.28100 | + | 2.28802i | 0.0191684i | 0 | − | 4.70204i | 6.33280 | + | 4.88832i | 0 | 0.0365734 | − | 0.0114930i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.10 | −0.599583 | + | 1.90801i | 0 | −3.28100 | − | 2.28802i | − | 0.0191684i | 0 | 4.70204i | 6.33280 | − | 4.88832i | 0 | 0.0365734 | + | 0.0114930i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.11 | 0.0461485 | − | 1.99947i | 0 | −3.99574 | − | 0.184545i | − | 7.01299i | 0 | 9.32128i | −0.553388 | + | 7.98084i | 0 | −14.0222 | − | 0.323638i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.12 | 0.0461485 | + | 1.99947i | 0 | −3.99574 | + | 0.184545i | 7.01299i | 0 | − | 9.32128i | −0.553388 | − | 7.98084i | 0 | −14.0222 | + | 0.323638i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.13 | 0.403958 | − | 1.95878i | 0 | −3.67364 | − | 1.58253i | 6.75257i | 0 | − | 4.05078i | −4.58382 | + | 6.55657i | 0 | 13.2268 | + | 2.72775i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.14 | 0.403958 | + | 1.95878i | 0 | −3.67364 | + | 1.58253i | − | 6.75257i | 0 | 4.05078i | −4.58382 | − | 6.55657i | 0 | 13.2268 | − | 2.72775i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.15 | 1.12827 | − | 1.65136i | 0 | −1.45401 | − | 3.72637i | − | 1.96774i | 0 | − | 10.0414i | −7.79412 | − | 1.80325i | 0 | −3.24945 | − | 2.22014i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.16 | 1.12827 | + | 1.65136i | 0 | −1.45401 | + | 3.72637i | 1.96774i | 0 | 10.0414i | −7.79412 | + | 1.80325i | 0 | −3.24945 | + | 2.22014i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.17 | 1.37956 | − | 1.44803i | 0 | −0.193602 | − | 3.99531i | − | 5.95598i | 0 | 4.72316i | −6.05243 | − | 5.23145i | 0 | −8.62446 | − | 8.21666i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.18 | 1.37956 | + | 1.44803i | 0 | −0.193602 | + | 3.99531i | 5.95598i | 0 | − | 4.72316i | −6.05243 | + | 5.23145i | 0 | −8.62446 | + | 8.21666i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.19 | 1.78193 | − | 0.908135i | 0 | 2.35058 | − | 3.23647i | 6.25871i | 0 | 6.88599i | 1.24943 | − | 7.90183i | 0 | 5.68375 | + | 11.1526i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.20 | 1.78193 | + | 0.908135i | 0 | 2.35058 | + | 3.23647i | − | 6.25871i | 0 | − | 6.88599i | 1.24943 | + | 7.90183i | 0 | 5.68375 | − | 11.1526i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 648.3.b.e | 20 | |
| 3.b | odd | 2 | 1 | 648.3.b.f | 20 | ||
| 4.b | odd | 2 | 1 | 2592.3.b.f | 20 | ||
| 8.b | even | 2 | 1 | 2592.3.b.f | 20 | ||
| 8.d | odd | 2 | 1 | inner | 648.3.b.e | 20 | |
| 9.c | even | 3 | 2 | 216.3.p.b | 40 | ||
| 9.d | odd | 6 | 2 | 72.3.p.b | ✓ | 40 | |
| 12.b | even | 2 | 1 | 2592.3.b.e | 20 | ||
| 24.f | even | 2 | 1 | 648.3.b.f | 20 | ||
| 24.h | odd | 2 | 1 | 2592.3.b.e | 20 | ||
| 36.f | odd | 6 | 2 | 864.3.t.b | 40 | ||
| 36.h | even | 6 | 2 | 288.3.t.b | 40 | ||
| 72.j | odd | 6 | 2 | 288.3.t.b | 40 | ||
| 72.l | even | 6 | 2 | 72.3.p.b | ✓ | 40 | |
| 72.n | even | 6 | 2 | 864.3.t.b | 40 | ||
| 72.p | odd | 6 | 2 | 216.3.p.b | 40 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 72.3.p.b | ✓ | 40 | 9.d | odd | 6 | 2 | |
| 72.3.p.b | ✓ | 40 | 72.l | even | 6 | 2 | |
| 216.3.p.b | 40 | 9.c | even | 3 | 2 | ||
| 216.3.p.b | 40 | 72.p | odd | 6 | 2 | ||
| 288.3.t.b | 40 | 36.h | even | 6 | 2 | ||
| 288.3.t.b | 40 | 72.j | odd | 6 | 2 | ||
| 648.3.b.e | 20 | 1.a | even | 1 | 1 | trivial | |
| 648.3.b.e | 20 | 8.d | odd | 2 | 1 | inner | |
| 648.3.b.f | 20 | 3.b | odd | 2 | 1 | ||
| 648.3.b.f | 20 | 24.f | even | 2 | 1 | ||
| 864.3.t.b | 40 | 36.f | odd | 6 | 2 | ||
| 864.3.t.b | 40 | 72.n | even | 6 | 2 | ||
| 2592.3.b.e | 20 | 12.b | even | 2 | 1 | ||
| 2592.3.b.e | 20 | 24.h | odd | 2 | 1 | ||
| 2592.3.b.f | 20 | 4.b | odd | 2 | 1 | ||
| 2592.3.b.f | 20 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(648, [\chi])\):
|
\( T_{5}^{20} + 309 T_{5}^{18} + 39765 T_{5}^{16} + 2787201 T_{5}^{14} + 116198475 T_{5}^{12} + \cdots + 598855680 \)
|
|
\( T_{11}^{10} + 8 T_{11}^{9} - 507 T_{11}^{8} - 5048 T_{11}^{7} + 75578 T_{11}^{6} + 933000 T_{11}^{5} + \cdots + 3045670813 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + 5 T^{19} + \cdots + 1048576 \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} + \cdots + 598855680 \)
|
| $7$ |
\( T^{20} + \cdots + 962854534840320 \)
|
| $11$ |
\( (T^{10} + 8 T^{9} + \cdots + 3045670813)^{2} \)
|
| $13$ |
\( T^{20} + \cdots + 11\!\cdots\!20 \)
|
| $17$ |
\( (T^{10} - T^{9} + \cdots - 50105504)^{2} \)
|
| $19$ |
\( (T^{10} + \cdots + 225954683776)^{2} \)
|
| $23$ |
\( T^{20} + \cdots + 21\!\cdots\!20 \)
|
| $29$ |
\( T^{20} + \cdots + 54\!\cdots\!20 \)
|
| $31$ |
\( T^{20} + \cdots + 39\!\cdots\!20 \)
|
| $37$ |
\( T^{20} + \cdots + 32\!\cdots\!20 \)
|
| $41$ |
\( (T^{10} + \cdots - 64714166076887)^{2} \)
|
| $43$ |
\( (T^{10} + \cdots - 22163791443611)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 79\!\cdots\!20 \)
|
| $53$ |
\( T^{20} + \cdots + 18\!\cdots\!80 \)
|
| $59$ |
\( (T^{10} + \cdots - 32\!\cdots\!91)^{2} \)
|
| $61$ |
\( T^{20} + \cdots + 13\!\cdots\!20 \)
|
| $67$ |
\( (T^{10} + \cdots - 22\!\cdots\!99)^{2} \)
|
| $71$ |
\( T^{20} + \cdots + 56\!\cdots\!00 \)
|
| $73$ |
\( (T^{10} + \cdots - 70\!\cdots\!52)^{2} \)
|
| $79$ |
\( T^{20} + \cdots + 90\!\cdots\!80 \)
|
| $83$ |
\( (T^{10} + \cdots - 9924464138528)^{2} \)
|
| $89$ |
\( (T^{10} + \cdots + 48\!\cdots\!64)^{2} \)
|
| $97$ |
\( (T^{10} + \cdots - 22\!\cdots\!07)^{2} \)
|
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