Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [135,3,Mod(44,135)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(135, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([5, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("135.44");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 135.h (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: | \(3.67848356886\) |
| 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: | \( 3^{12} \) |
| 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
\( x^{20} - 3 x^{18} - 19 x^{16} + 66 x^{14} + 109 x^{12} - 813 x^{10} + 981 x^{8} + 5346 x^{6} + \cdots + 59049 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 29 \nu^{19} + 609 \nu^{17} - 286 \nu^{15} - 8916 \nu^{13} + 19627 \nu^{11} + 25071 \nu^{9} + \cdots - 2407887 \nu ) / 393660 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 23 \nu^{18} - 327 \nu^{16} + 1058 \nu^{14} + 3333 \nu^{12} - 13496 \nu^{10} + 8097 \nu^{8} + \cdots + 1154736 ) / 65610 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - \nu^{19} + 3 \nu^{17} + 19 \nu^{15} - 66 \nu^{13} - 109 \nu^{11} + 813 \nu^{9} - 981 \nu^{7} + \cdots + 19683 \nu ) / 6561 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 6 \nu^{18} - 4 \nu^{16} + 126 \nu^{14} - 59 \nu^{12} - 837 \nu^{10} + 2804 \nu^{8} + 1497 \nu^{6} + \cdots + 89667 ) / 7290 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 11 \nu^{18} - 33 \nu^{16} - 128 \nu^{14} + 483 \nu^{12} - 340 \nu^{10} - 3597 \nu^{8} + \cdots - 20412 \nu^{2} ) / 13122 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 338 \nu^{19} - 213 \nu^{17} - 7448 \nu^{15} + 7332 \nu^{13} + 48236 \nu^{11} - 166587 \nu^{9} + \cdots - 6436341 \nu ) / 393660 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 163 \nu^{18} + 93 \nu^{16} + 2908 \nu^{14} - 3477 \nu^{12} - 13366 \nu^{10} + 68457 \nu^{8} + \cdots + 1614006 ) / 65610 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 344 \nu^{18} + 1509 \nu^{16} + 7454 \nu^{14} - 22776 \nu^{12} - 26588 \nu^{10} + 208221 \nu^{8} + \cdots + 6055803 ) / 131220 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 270 \nu^{19} - 281 \nu^{18} - 105 \nu^{17} - 219 \nu^{16} - 5760 \nu^{15} + 4556 \nu^{14} + \cdots + 2998377 ) / 262440 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 270 \nu^{19} + 281 \nu^{18} - 105 \nu^{17} + 219 \nu^{16} - 5760 \nu^{15} - 4556 \nu^{14} + \cdots - 2735937 ) / 262440 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 235 \nu^{19} - 219 \nu^{18} - 435 \nu^{17} - 234 \nu^{16} - 2764 \nu^{15} + 6348 \nu^{14} + \cdots + 8936082 ) / 157464 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 235 \nu^{19} - 219 \nu^{18} + 435 \nu^{17} - 234 \nu^{16} + 2764 \nu^{15} + 6348 \nu^{14} + \cdots + 8936082 ) / 157464 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 223 \nu^{18} + 63 \nu^{16} + 4678 \nu^{14} - 4662 \nu^{12} - 30121 \nu^{10} + 106497 \nu^{8} + \cdots + 3855681 ) / 43740 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 133 \nu^{19} - 87 \nu^{17} + 2203 \nu^{15} + 1428 \nu^{13} - 19276 \nu^{11} + 40332 \nu^{9} + \cdots + 2991816 \nu ) / 65610 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 1700 \nu^{19} - 843 \nu^{18} - 1995 \nu^{17} - 657 \nu^{16} - 22580 \nu^{15} + 13668 \nu^{14} + \cdots + 8995131 ) / 787320 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 6 \nu^{19} - 4 \nu^{17} + 126 \nu^{15} - 59 \nu^{13} - 837 \nu^{11} + 2804 \nu^{9} + \cdots + 96957 \nu ) / 2430 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 509 \nu^{18} - 69 \nu^{16} - 7889 \nu^{14} + 546 \nu^{12} + 54428 \nu^{10} - 156966 \nu^{8} + \cdots - 8975448 ) / 65610 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 2056 \nu^{19} - 843 \nu^{18} + 2271 \nu^{17} - 657 \nu^{16} + 34636 \nu^{15} + 13668 \nu^{14} + \cdots + 8207811 ) / 787320 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{8} + \beta_{6} - 2\beta_{5} ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{19} - \beta_{17} - \beta_{16} - 2\beta_{11} - \beta_{7} + \beta_{4} - 2\beta_{2} + \beta _1 + 1 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{14} - \beta_{13} - \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + \beta_{9} - 3 \beta_{8} - 3 \beta_{5} + \cdots + 11 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -3\beta_{19} - 3\beta_{17} + 4\beta_{15} + 3\beta_{10} - 12\beta_{7} - 2\beta_{4} + 3\beta_{2} + 8\beta _1 - 3 ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 2 \beta_{18} - 3 \beta_{14} - 3 \beta_{13} - 3 \beta_{12} + 6 \beta_{11} - 6 \beta_{10} + 3 \beta_{9} + \cdots + 4 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 11 \beta_{19} - \beta_{17} + \beta_{16} + 8 \beta_{15} - 19 \beta_{11} - 9 \beta_{10} + 16 \beta_{7} + \cdots + 8 ) / 3 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 3 \beta_{18} - 3 \beta_{14} - 23 \beta_{11} + 23 \beta_{10} + 4 \beta_{9} - 29 \beta_{8} + 5 \beta_{6} + \cdots + 113 ) / 3 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 15 \beta_{19} + 10 \beta_{17} - 24 \beta_{16} + 17 \beta_{15} - 33 \beta_{13} + 33 \beta_{12} + \cdots - 21 ) / 3 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( \beta_{18} - 48 \beta_{14} + 9 \beta_{13} + 9 \beta_{12} + 27 \beta_{11} - 27 \beta_{10} + 3 \beta_{9} + \cdots + 505 ) / 3 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 26 \beta_{19} + 69 \beta_{17} - 106 \beta_{16} - 49 \beta_{15} - 81 \beta_{13} + 81 \beta_{12} + \cdots + 124 ) / 3 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 57 \beta_{18} - 13 \beta_{14} + 82 \beta_{13} + 82 \beta_{12} - 309 \beta_{11} + 309 \beta_{10} + \cdots + 48 ) / 3 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 144 \beta_{19} - 131 \beta_{17} - 654 \beta_{16} - 59 \beta_{15} - 411 \beta_{13} + 411 \beta_{12} + \cdots + 99 ) / 3 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 48 \beta_{18} - 99 \beta_{14} - 69 \beta_{13} - 69 \beta_{12} - 204 \beta_{11} + 204 \beta_{10} + \cdots + 2202 ) / 3 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 323 \beta_{19} - 119 \beta_{17} - 791 \beta_{16} + 150 \beta_{15} - 594 \beta_{13} + 594 \beta_{12} + \cdots + 476 ) / 3 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 63 \beta_{18} - 469 \beta_{14} + 28 \beta_{13} + 28 \beta_{12} - 1105 \beta_{11} + 1105 \beta_{10} + \cdots - 10676 ) / 3 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 51 \beta_{19} - 1014 \beta_{17} - 3402 \beta_{16} + 1103 \beta_{15} - 2835 \beta_{13} + 2835 \beta_{12} + \cdots + 1515 ) / 3 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( 1253 \beta_{18} - 2040 \beta_{14} + 669 \beta_{13} + 669 \beta_{12} - 3633 \beta_{11} + 3633 \beta_{10} + \cdots + 26663 ) / 3 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 6545 \beta_{19} + 7120 \beta_{17} - 4861 \beta_{16} - 17 \beta_{15} - 10665 \beta_{13} + 10665 \beta_{12} + \cdots - 1493 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).
| \(n\) | \(56\) | \(82\) |
| \(\chi(n)\) | \(1 + \beta_{5}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 44.1 |
|
−1.84364 | − | 3.19328i | 0 | −4.79800 | + | 8.31039i | 3.30943 | + | 3.74802i | 0 | 2.22343 | − | 1.28370i | 20.6340 | 0 | 5.86709 | − | 17.4779i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 44.2 | −1.42081 | − | 2.46092i | 0 | −2.03740 | + | 3.52889i | 1.22790 | − | 4.84688i | 0 | 8.42581 | − | 4.86464i | 0.212574 | 0 | −13.6724 | + | 3.86473i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 44.3 | −1.14670 | − | 1.98614i | 0 | −0.629835 | + | 1.09091i | −0.662169 | + | 4.95596i | 0 | −6.04559 | + | 3.49042i | −6.28466 | 0 | 10.6025 | − | 4.36783i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 44.4 | −0.668935 | − | 1.15863i | 0 | 1.10505 | − | 1.91401i | −4.41869 | − | 2.33992i | 0 | −7.10792 | + | 4.10376i | −8.30831 | 0 | 0.244718 | + | 6.68487i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 44.5 | −0.264396 | − | 0.457947i | 0 | 1.86019 | − | 3.22194i | 4.68146 | − | 1.75610i | 0 | −2.39593 | + | 1.38329i | −4.08247 | 0 | −2.04196 | − | 1.67955i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 44.6 | 0.264396 | + | 0.457947i | 0 | 1.86019 | − | 3.22194i | 0.819902 | + | 4.93232i | 0 | 2.39593 | − | 1.38329i | 4.08247 | 0 | −2.04196 | + | 1.67955i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 44.7 | 0.668935 | + | 1.15863i | 0 | 1.10505 | − | 1.91401i | −4.23577 | − | 2.65674i | 0 | 7.10792 | − | 4.10376i | 8.30831 | 0 | 0.244718 | − | 6.68487i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 44.8 | 1.14670 | + | 1.98614i | 0 | −0.629835 | + | 1.09091i | 3.96090 | − | 3.05143i | 0 | 6.04559 | − | 3.49042i | 6.28466 | 0 | 10.6025 | + | 4.36783i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 44.9 | 1.42081 | + | 2.46092i | 0 | −2.03740 | + | 3.52889i | −3.58357 | + | 3.48684i | 0 | −8.42581 | + | 4.86464i | −0.212574 | 0 | −13.6724 | − | 3.86473i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 44.10 | 1.84364 | + | 3.19328i | 0 | −4.79800 | + | 8.31039i | 4.90060 | + | 0.992036i | 0 | −2.22343 | + | 1.28370i | −20.6340 | 0 | 5.86709 | + | 17.4779i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.1 | −1.84364 | + | 3.19328i | 0 | −4.79800 | − | 8.31039i | 3.30943 | − | 3.74802i | 0 | 2.22343 | + | 1.28370i | 20.6340 | 0 | 5.86709 | + | 17.4779i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.2 | −1.42081 | + | 2.46092i | 0 | −2.03740 | − | 3.52889i | 1.22790 | + | 4.84688i | 0 | 8.42581 | + | 4.86464i | 0.212574 | 0 | −13.6724 | − | 3.86473i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.3 | −1.14670 | + | 1.98614i | 0 | −0.629835 | − | 1.09091i | −0.662169 | − | 4.95596i | 0 | −6.04559 | − | 3.49042i | −6.28466 | 0 | 10.6025 | + | 4.36783i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.4 | −0.668935 | + | 1.15863i | 0 | 1.10505 | + | 1.91401i | −4.41869 | + | 2.33992i | 0 | −7.10792 | − | 4.10376i | −8.30831 | 0 | 0.244718 | − | 6.68487i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.5 | −0.264396 | + | 0.457947i | 0 | 1.86019 | + | 3.22194i | 4.68146 | + | 1.75610i | 0 | −2.39593 | − | 1.38329i | −4.08247 | 0 | −2.04196 | + | 1.67955i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.6 | 0.264396 | − | 0.457947i | 0 | 1.86019 | + | 3.22194i | 0.819902 | − | 4.93232i | 0 | 2.39593 | + | 1.38329i | 4.08247 | 0 | −2.04196 | − | 1.67955i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.7 | 0.668935 | − | 1.15863i | 0 | 1.10505 | + | 1.91401i | −4.23577 | + | 2.65674i | 0 | 7.10792 | + | 4.10376i | 8.30831 | 0 | 0.244718 | + | 6.68487i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.8 | 1.14670 | − | 1.98614i | 0 | −0.629835 | − | 1.09091i | 3.96090 | + | 3.05143i | 0 | 6.04559 | + | 3.49042i | 6.28466 | 0 | 10.6025 | − | 4.36783i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.9 | 1.42081 | − | 2.46092i | 0 | −2.03740 | − | 3.52889i | −3.58357 | − | 3.48684i | 0 | −8.42581 | − | 4.86464i | −0.212574 | 0 | −13.6724 | + | 3.86473i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 89.10 | 1.84364 | − | 3.19328i | 0 | −4.79800 | − | 8.31039i | 4.90060 | − | 0.992036i | 0 | −2.22343 | − | 1.28370i | −20.6340 | 0 | 5.86709 | − | 17.4779i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 135.3.h.a | 20 | |
| 3.b | odd | 2 | 1 | 45.3.h.a | ✓ | 20 | |
| 5.b | even | 2 | 1 | inner | 135.3.h.a | 20 | |
| 5.c | odd | 4 | 2 | 675.3.j.e | 20 | ||
| 9.c | even | 3 | 1 | 45.3.h.a | ✓ | 20 | |
| 9.c | even | 3 | 1 | 405.3.d.a | 20 | ||
| 9.d | odd | 6 | 1 | inner | 135.3.h.a | 20 | |
| 9.d | odd | 6 | 1 | 405.3.d.a | 20 | ||
| 15.d | odd | 2 | 1 | 45.3.h.a | ✓ | 20 | |
| 15.e | even | 4 | 2 | 225.3.j.e | 20 | ||
| 45.h | odd | 6 | 1 | inner | 135.3.h.a | 20 | |
| 45.h | odd | 6 | 1 | 405.3.d.a | 20 | ||
| 45.j | even | 6 | 1 | 45.3.h.a | ✓ | 20 | |
| 45.j | even | 6 | 1 | 405.3.d.a | 20 | ||
| 45.k | odd | 12 | 2 | 225.3.j.e | 20 | ||
| 45.l | even | 12 | 2 | 675.3.j.e | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.3.h.a | ✓ | 20 | 3.b | odd | 2 | 1 | |
| 45.3.h.a | ✓ | 20 | 9.c | even | 3 | 1 | |
| 45.3.h.a | ✓ | 20 | 15.d | odd | 2 | 1 | |
| 45.3.h.a | ✓ | 20 | 45.j | even | 6 | 1 | |
| 135.3.h.a | 20 | 1.a | even | 1 | 1 | trivial | |
| 135.3.h.a | 20 | 5.b | even | 2 | 1 | inner | |
| 135.3.h.a | 20 | 9.d | odd | 6 | 1 | inner | |
| 135.3.h.a | 20 | 45.h | odd | 6 | 1 | inner | |
| 225.3.j.e | 20 | 15.e | even | 4 | 2 | ||
| 225.3.j.e | 20 | 45.k | odd | 12 | 2 | ||
| 405.3.d.a | 20 | 9.c | even | 3 | 1 | ||
| 405.3.d.a | 20 | 9.d | odd | 6 | 1 | ||
| 405.3.d.a | 20 | 45.h | odd | 6 | 1 | ||
| 405.3.d.a | 20 | 45.j | even | 6 | 1 | ||
| 675.3.j.e | 20 | 5.c | odd | 4 | 2 | ||
| 675.3.j.e | 20 | 45.l | even | 12 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(135, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + 29 T^{18} + \cdots + 83521 \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( T^{20} + \cdots + 95367431640625 \)
|
| $7$ |
\( T^{20} + \cdots + 245780494502544 \)
|
| $11$ |
\( (T^{10} - 12 T^{9} + \cdots + 21579372)^{2} \)
|
| $13$ |
\( T^{20} + \cdots + 14\!\cdots\!84 \)
|
| $17$ |
\( (T^{10} - 704 T^{8} + \cdots - 1532096164)^{2} \)
|
| $19$ |
\( (T^{5} + 2 T^{4} + \cdots - 7946)^{4} \)
|
| $23$ |
\( T^{20} + \cdots + 15352201216 \)
|
| $29$ |
\( (T^{10} + \cdots + 19167316554672)^{2} \)
|
| $31$ |
\( (T^{10} + \cdots + 116686747082896)^{2} \)
|
| $37$ |
\( (T^{10} + \cdots + 2777948395200)^{2} \)
|
| $41$ |
\( (T^{10} + \cdots + 957338655721083)^{2} \)
|
| $43$ |
\( T^{20} + \cdots + 26\!\cdots\!24 \)
|
| $47$ |
\( T^{20} + \cdots + 12\!\cdots\!96 \)
|
| $53$ |
\( (T^{10} + \cdots - 37\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{10} + \cdots + 70779040697088)^{2} \)
|
| $61$ |
\( (T^{10} + \cdots + 457796433007684)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 42\!\cdots\!09 \)
|
| $71$ |
\( (T^{10} + \cdots + 38\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{10} + \cdots + 10\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{10} + \cdots + 18\!\cdots\!36)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 42\!\cdots\!96 \)
|
| $89$ |
\( (T^{10} + \cdots + 27\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{20} + \cdots + 76\!\cdots\!00 \)
|
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