Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [225,3,Mod(101,225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(225, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([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("225.101");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 225.j (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: | \(6.13080594811\) |
| 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: |
\( x^{20} + 3 x^{18} - 19 x^{16} - 66 x^{14} + 109 x^{12} + 813 x^{10} + 981 x^{8} - 5346 x^{6} + \cdots + 59049 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{10} \) |
| Twist minimal: | no (minimal twist has level 45) |
| 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
\( x^{20} + 3 x^{18} - 19 x^{16} - 66 x^{14} + 109 x^{12} + 813 x^{10} + 981 x^{8} - 5346 x^{6} + \cdots + 59049 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 29 \nu^{19} - 609 \nu^{17} - 286 \nu^{15} + 8916 \nu^{13} + 19627 \nu^{11} - 25071 \nu^{9} + \cdots + 2407887 \nu ) / 393660 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 23 \nu^{18} - 327 \nu^{16} - 1058 \nu^{14} + 3333 \nu^{12} + 13496 \nu^{10} + 8097 \nu^{8} + \cdots + 1154736 ) / 65610 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 17 \nu^{18} - 123 \nu^{16} + 512 \nu^{14} + 1518 \nu^{12} - 2609 \nu^{10} - 13407 \nu^{8} + \cdots - 728271 ) / 26244 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 6 \nu^{18} - 4 \nu^{16} - 126 \nu^{14} - 59 \nu^{12} + 837 \nu^{10} + 2804 \nu^{8} - 1497 \nu^{6} + \cdots + 96957 ) / 7290 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 107 \nu^{19} - 357 \nu^{17} + 2492 \nu^{15} + 16008 \nu^{13} + 8371 \nu^{11} + \cdots + 3287061 \nu ) / 393660 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 191 \nu^{19} + 366 \nu^{17} - 6356 \nu^{15} + 1776 \nu^{13} + 65747 \nu^{11} + 90924 \nu^{9} + \cdots + 12938292 \nu ) / 393660 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 47 \nu^{18} - 87 \nu^{16} + 488 \nu^{14} + 1104 \nu^{12} - 101 \nu^{10} - 15315 \nu^{8} + \cdots - 111537 ) / 26244 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 113 \nu^{19} + 147 \nu^{17} + 1823 \nu^{15} - 2748 \nu^{13} - 17096 \nu^{11} + \cdots - 2598156 \nu ) / 196830 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 14 \nu^{18} - 42 \nu^{16} + 185 \nu^{14} + 681 \nu^{12} + 13 \nu^{10} - 6036 \nu^{8} + \cdots - 45927 ) / 6561 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 101 \nu^{18} + 324 \nu^{16} + 3476 \nu^{14} - 3546 \nu^{12} - 37487 \nu^{10} - 49734 \nu^{8} + \cdots - 5196312 ) / 43740 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 143 \nu^{19} + 57 \nu^{17} + 2393 \nu^{15} - 768 \nu^{13} - 20366 \nu^{11} - 48462 \nu^{9} + \cdots - 3188646 \nu ) / 196830 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 103 \nu^{19} - 132 \nu^{17} - 2578 \nu^{15} + 528 \nu^{13} + 22621 \nu^{11} + 47172 \nu^{9} + \cdots + 2948076 \nu ) / 131220 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 388 \nu^{18} + 1077 \nu^{16} + 10288 \nu^{14} - 9438 \nu^{12} - 102556 \nu^{10} + \cdots - 12997341 ) / 131220 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 89 \nu^{19} - 168 \nu^{17} + 530 \nu^{15} + 3264 \nu^{13} + 7093 \nu^{11} - 28032 \nu^{9} + \cdots + 1285956 \nu ) / 78732 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 539 \nu^{18} - 1734 \nu^{16} + 8594 \nu^{14} + 28806 \nu^{12} - 13283 \nu^{10} - 261906 \nu^{8} + \cdots - 3332988 ) / 131220 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 56 \nu^{19} - 181 \nu^{17} + 836 \nu^{15} + 3214 \nu^{13} - 1412 \nu^{11} - 27019 \nu^{9} + \cdots - 468747 \nu ) / 43740 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 509 \nu^{18} - 69 \nu^{16} + 7889 \nu^{14} + 546 \nu^{12} - 54428 \nu^{10} - 156966 \nu^{8} + \cdots - 8975448 ) / 65610 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 6 \nu^{19} - 4 \nu^{17} - 126 \nu^{15} - 59 \nu^{13} + 837 \nu^{11} + 2804 \nu^{9} - 1497 \nu^{7} + \cdots + 89667 \nu ) / 2430 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{14} - \beta_{11} + 2\beta_{5} + \beta_{4} - \beta_{3} - 1 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{19} - \beta_{17} + \beta_{15} - \beta_{13} + 2\beta_{12} - \beta_{9} - \beta_{7} + 2\beta_{6} - \beta_{2} ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2\beta_{16} - \beta_{14} + \beta_{11} + 4\beta_{10} - \beta_{8} - 3\beta_{5} + \beta_{3} + 15 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -3\beta_{19} + 3\beta_{17} + 12\beta_{13} - 3\beta_{12} - 3\beta_{7} - 15\beta_{2} + 5\beta_1 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 2\beta_{18} - 6\beta_{11} - 6\beta_{10} - 3\beta_{8} - 10\beta_{5} + 9\beta_{4} - 7\beta_{3} + 12 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( \beta_{19} - 11 \beta_{17} - \beta_{15} + 16 \beta_{13} + 25 \beta_{12} + 4 \beta_{9} - 8 \beta_{7} + \cdots + \beta_1 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 3 \beta_{18} - \beta_{16} - 2 \beta_{14} + 5 \beta_{11} + 38 \beta_{10} - 26 \beta_{8} + 72 \beta_{5} + \cdots + 33 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 10 \beta_{19} + 18 \beta_{17} + 9 \beta_{15} + 24 \beta_{13} - 48 \beta_{12} + 57 \beta_{9} + \cdots + 42 \beta_1 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - \beta_{18} - 45 \beta_{16} - 58 \beta_{14} + 19 \beta_{11} + 69 \beta_{10} + 21 \beta_{8} + \cdots - 368 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 69 \beta_{19} - 55 \beta_{17} + 25 \beta_{15} + 296 \beta_{13} - 4 \beta_{12} + 50 \beta_{9} + \cdots - 182 \beta_1 ) / 3 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 57 \beta_{18} + 19 \beta_{16} - 127 \beta_{14} + 58 \beta_{11} + 61 \beta_{10} - 322 \beta_{8} + \cdots + 270 ) / 3 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 131 \beta_{19} + 267 \beta_{17} - 243 \beta_{15} + 561 \beta_{13} - 171 \beta_{12} + 417 \beta_{9} + \cdots + 208 \beta_1 ) / 3 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 48 \beta_{18} + 162 \beta_{16} - 535 \beta_{14} + 367 \beta_{11} + 147 \beta_{10} + 303 \beta_{8} + \cdots - 3485 ) / 3 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 119 \beta_{19} - 917 \beta_{17} + 197 \beta_{15} - 620 \beta_{13} - 848 \beta_{12} + 1333 \beta_{9} + \cdots - 1353 \beta_1 ) / 3 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 63 \beta_{18} - 295 \beta_{16} - 278 \beta_{14} + 719 \beta_{11} + 2084 \beta_{10} - 1574 \beta_{8} + \cdots - 10770 ) / 3 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 1014 \beta_{19} + 2886 \beta_{17} - 567 \beta_{15} + 1500 \beta_{13} - 4614 \beta_{12} + \cdots - 2633 \beta_1 ) / 3 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 1253 \beta_{18} + 243 \beta_{16} - 7821 \beta_{14} + 6450 \beta_{11} - 4017 \beta_{10} + 5673 \beta_{8} + \cdots - 13845 ) / 3 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 7120 \beta_{19} - 4120 \beta_{17} - 5804 \beta_{15} + 3575 \beta_{13} - 6640 \beta_{12} + \cdots - 12322 \beta_1 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1 - \beta_{5}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 101.1 |
|
−3.19328 | − | 1.84364i | −2.49550 | + | 1.66507i | 4.79800 | + | 8.31039i | 0 | 11.0386 | − | 0.716233i | 1.28370 | − | 2.22343i | − | 20.6340i | 3.45506 | − | 8.31039i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.2 | −2.46092 | − | 1.42081i | 0.600288 | − | 2.93933i | 2.03740 | + | 3.52889i | 0 | −5.65349 | + | 6.38055i | 4.86464 | − | 8.42581i | − | 0.212574i | −8.27931 | − | 3.52889i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.3 | −1.98614 | − | 1.14670i | 2.99446 | − | 0.182154i | 0.629835 | + | 1.09091i | 0 | −6.15630 | − | 3.07197i | −3.49042 | + | 6.04559i | 6.28466i | 8.93364 | − | 1.09091i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.4 | −1.15863 | − | 0.668935i | −2.98279 | − | 0.320841i | −1.10505 | − | 1.91401i | 0 | 3.24133 | + | 2.36703i | −4.10376 | + | 7.10792i | 8.30831i | 8.79412 | + | 1.91401i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.5 | −0.457947 | − | 0.264396i | 0.546115 | + | 2.94987i | −1.86019 | − | 3.22194i | 0 | 0.529842 | − | 1.49528i | −1.38329 | + | 2.39593i | 4.08247i | −8.40352 | + | 3.22194i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.6 | 0.457947 | + | 0.264396i | −0.546115 | − | 2.94987i | −1.86019 | − | 3.22194i | 0 | 0.529842 | − | 1.49528i | 1.38329 | − | 2.39593i | − | 4.08247i | −8.40352 | + | 3.22194i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.7 | 1.15863 | + | 0.668935i | 2.98279 | + | 0.320841i | −1.10505 | − | 1.91401i | 0 | 3.24133 | + | 2.36703i | 4.10376 | − | 7.10792i | − | 8.30831i | 8.79412 | + | 1.91401i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.8 | 1.98614 | + | 1.14670i | −2.99446 | + | 0.182154i | 0.629835 | + | 1.09091i | 0 | −6.15630 | − | 3.07197i | 3.49042 | − | 6.04559i | − | 6.28466i | 8.93364 | − | 1.09091i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.9 | 2.46092 | + | 1.42081i | −0.600288 | + | 2.93933i | 2.03740 | + | 3.52889i | 0 | −5.65349 | + | 6.38055i | −4.86464 | + | 8.42581i | 0.212574i | −8.27931 | − | 3.52889i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 101.10 | 3.19328 | + | 1.84364i | 2.49550 | − | 1.66507i | 4.79800 | + | 8.31039i | 0 | 11.0386 | − | 0.716233i | −1.28370 | + | 2.22343i | 20.6340i | 3.45506 | − | 8.31039i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 176.1 | −3.19328 | + | 1.84364i | −2.49550 | − | 1.66507i | 4.79800 | − | 8.31039i | 0 | 11.0386 | + | 0.716233i | 1.28370 | + | 2.22343i | 20.6340i | 3.45506 | + | 8.31039i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 176.2 | −2.46092 | + | 1.42081i | 0.600288 | + | 2.93933i | 2.03740 | − | 3.52889i | 0 | −5.65349 | − | 6.38055i | 4.86464 | + | 8.42581i | 0.212574i | −8.27931 | + | 3.52889i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 176.3 | −1.98614 | + | 1.14670i | 2.99446 | + | 0.182154i | 0.629835 | − | 1.09091i | 0 | −6.15630 | + | 3.07197i | −3.49042 | − | 6.04559i | − | 6.28466i | 8.93364 | + | 1.09091i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 176.4 | −1.15863 | + | 0.668935i | −2.98279 | + | 0.320841i | −1.10505 | + | 1.91401i | 0 | 3.24133 | − | 2.36703i | −4.10376 | − | 7.10792i | − | 8.30831i | 8.79412 | − | 1.91401i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 176.5 | −0.457947 | + | 0.264396i | 0.546115 | − | 2.94987i | −1.86019 | + | 3.22194i | 0 | 0.529842 | + | 1.49528i | −1.38329 | − | 2.39593i | − | 4.08247i | −8.40352 | − | 3.22194i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 176.6 | 0.457947 | − | 0.264396i | −0.546115 | + | 2.94987i | −1.86019 | + | 3.22194i | 0 | 0.529842 | + | 1.49528i | 1.38329 | + | 2.39593i | 4.08247i | −8.40352 | − | 3.22194i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 176.7 | 1.15863 | − | 0.668935i | 2.98279 | − | 0.320841i | −1.10505 | + | 1.91401i | 0 | 3.24133 | − | 2.36703i | 4.10376 | + | 7.10792i | 8.30831i | 8.79412 | − | 1.91401i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 176.8 | 1.98614 | − | 1.14670i | −2.99446 | − | 0.182154i | 0.629835 | − | 1.09091i | 0 | −6.15630 | + | 3.07197i | 3.49042 | + | 6.04559i | 6.28466i | 8.93364 | + | 1.09091i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 176.9 | 2.46092 | − | 1.42081i | −0.600288 | − | 2.93933i | 2.03740 | − | 3.52889i | 0 | −5.65349 | − | 6.38055i | −4.86464 | − | 8.42581i | − | 0.212574i | −8.27931 | + | 3.52889i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 176.10 | 3.19328 | − | 1.84364i | 2.49550 | + | 1.66507i | 4.79800 | − | 8.31039i | 0 | 11.0386 | + | 0.716233i | −1.28370 | − | 2.22343i | − | 20.6340i | 3.45506 | + | 8.31039i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
| 45.h | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 225.3.j.e | 20 | |
| 3.b | odd | 2 | 1 | 675.3.j.e | 20 | ||
| 5.b | even | 2 | 1 | inner | 225.3.j.e | 20 | |
| 5.c | odd | 4 | 2 | 45.3.h.a | ✓ | 20 | |
| 9.c | even | 3 | 1 | 675.3.j.e | 20 | ||
| 9.d | odd | 6 | 1 | inner | 225.3.j.e | 20 | |
| 15.d | odd | 2 | 1 | 675.3.j.e | 20 | ||
| 15.e | even | 4 | 2 | 135.3.h.a | 20 | ||
| 45.h | odd | 6 | 1 | inner | 225.3.j.e | 20 | |
| 45.j | even | 6 | 1 | 675.3.j.e | 20 | ||
| 45.k | odd | 12 | 2 | 135.3.h.a | 20 | ||
| 45.k | odd | 12 | 2 | 405.3.d.a | 20 | ||
| 45.l | even | 12 | 2 | 45.3.h.a | ✓ | 20 | |
| 45.l | even | 12 | 2 | 405.3.d.a | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.3.h.a | ✓ | 20 | 5.c | odd | 4 | 2 | |
| 45.3.h.a | ✓ | 20 | 45.l | even | 12 | 2 | |
| 135.3.h.a | 20 | 15.e | even | 4 | 2 | ||
| 135.3.h.a | 20 | 45.k | odd | 12 | 2 | ||
| 225.3.j.e | 20 | 1.a | even | 1 | 1 | trivial | |
| 225.3.j.e | 20 | 5.b | even | 2 | 1 | inner | |
| 225.3.j.e | 20 | 9.d | odd | 6 | 1 | inner | |
| 225.3.j.e | 20 | 45.h | odd | 6 | 1 | inner | |
| 405.3.d.a | 20 | 45.k | odd | 12 | 2 | ||
| 405.3.d.a | 20 | 45.l | even | 12 | 2 | ||
| 675.3.j.e | 20 | 3.b | odd | 2 | 1 | ||
| 675.3.j.e | 20 | 9.c | even | 3 | 1 | ||
| 675.3.j.e | 20 | 15.d | odd | 2 | 1 | ||
| 675.3.j.e | 20 | 45.j | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} - 29 T_{2}^{18} + 561 T_{2}^{16} - 6012 T_{2}^{14} + 46527 T_{2}^{12} - 219603 T_{2}^{10} + \cdots + 83521 \)
acting on \(S_{3}^{\mathrm{new}}(225, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - 29 T^{18} + \cdots + 83521 \)
|
| $3$ |
\( T^{20} + \cdots + 3486784401 \)
|
| $5$ |
\( T^{20} \)
|
| $7$ |
\( T^{20} + \cdots + 245780494502544 \)
|
| $11$ |
\( (T^{10} + 12 T^{9} + \cdots + 21579372)^{2} \)
|
| $13$ |
\( T^{20} + \cdots + 14\!\cdots\!84 \)
|
| $17$ |
\( (T^{10} + 704 T^{8} + \cdots + 1532096164)^{2} \)
|
| $19$ |
\( (T^{5} - 2 T^{4} + \cdots + 7946)^{4} \)
|
| $23$ |
\( T^{20} + \cdots + 15352201216 \)
|
| $29$ |
\( (T^{10} + \cdots + 19167316554672)^{2} \)
|
| $31$ |
\( (T^{10} + \cdots + 116686747082896)^{2} \)
|
| $37$ |
\( (T^{10} + \cdots - 2777948395200)^{2} \)
|
| $41$ |
\( (T^{10} + \cdots + 957338655721083)^{2} \)
|
| $43$ |
\( T^{20} + \cdots + 26\!\cdots\!24 \)
|
| $47$ |
\( T^{20} + \cdots + 12\!\cdots\!96 \)
|
| $53$ |
\( (T^{10} + \cdots + 37\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{10} + \cdots + 70779040697088)^{2} \)
|
| $61$ |
\( (T^{10} + \cdots + 457796433007684)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 42\!\cdots\!09 \)
|
| $71$ |
\( (T^{10} + \cdots + 38\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{10} + \cdots - 10\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{10} + \cdots + 18\!\cdots\!36)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 42\!\cdots\!96 \)
|
| $89$ |
\( (T^{10} + \cdots + 27\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{20} + \cdots + 76\!\cdots\!00 \)
|
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