Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1560,2,Mod(781,1560)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1560, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1560.781");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1560.w (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: | \(12.4566627153\) |
| 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: |
\( x^{20} - 2 x^{19} + 2 x^{18} - 5 x^{16} + 10 x^{15} - 12 x^{14} + 16 x^{13} - 2 x^{12} - 40 x^{11} + \cdots + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{9} \) |
| Twist minimal: | yes |
| 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
\( x^{20} - 2 x^{19} + 2 x^{18} - 5 x^{16} + 10 x^{15} - 12 x^{14} + 16 x^{13} - 2 x^{12} - 40 x^{11} + \cdots + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{19} + 12 \nu^{18} + 10 \nu^{17} - 52 \nu^{16} + 3 \nu^{15} - 12 \nu^{14} - 20 \nu^{13} + \cdots - 1536 ) / 2048 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{19} - 12 \nu^{18} - 10 \nu^{17} + 52 \nu^{16} - 3 \nu^{15} + 12 \nu^{14} + 20 \nu^{13} + \cdots + 1536 ) / 2048 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7 \nu^{19} - 4 \nu^{18} + 6 \nu^{17} + 36 \nu^{16} + 21 \nu^{15} - 28 \nu^{14} - 76 \nu^{13} + \cdots + 7680 ) / 4096 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{19} - 2 \nu^{18} + 2 \nu^{17} - 5 \nu^{15} + 10 \nu^{14} - 12 \nu^{13} + 16 \nu^{12} + \cdots - 1024 ) / 512 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 5 \nu^{19} + 32 \nu^{18} + 6 \nu^{17} + 12 \nu^{16} + 81 \nu^{15} - 112 \nu^{14} + 76 \nu^{13} + \cdots + 16896 ) / 2048 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - \nu^{19} + \nu^{15} + 2 \nu^{13} - 12 \nu^{12} - 6 \nu^{11} + 12 \nu^{10} + 12 \nu^{9} + \cdots + 128 \nu ) / 256 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{19} - \nu^{18} + 2 \nu^{16} - 5 \nu^{15} + 5 \nu^{14} - 2 \nu^{13} + 4 \nu^{12} + 14 \nu^{11} + \cdots - 512 ) / 256 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 15 \nu^{19} + 44 \nu^{18} - 38 \nu^{17} + 12 \nu^{16} + 147 \nu^{15} - 108 \nu^{14} + 124 \nu^{13} + \cdots + 27136 ) / 4096 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 5 \nu^{19} - 4 \nu^{18} + 6 \nu^{17} - 28 \nu^{16} + \nu^{15} + 4 \nu^{14} + 12 \nu^{13} + \cdots - 3584 ) / 1024 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 5 \nu^{19} + 2 \nu^{18} + 6 \nu^{17} + 8 \nu^{16} - 17 \nu^{15} - 10 \nu^{14} - 40 \nu^{13} + \cdots + 2816 \nu ) / 1024 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 5 \nu^{19} - 4 \nu^{18} + 18 \nu^{17} + 28 \nu^{16} - 49 \nu^{15} + 4 \nu^{14} - 68 \nu^{13} + \cdots - 3584 ) / 1024 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 11 \nu^{19} + 4 \nu^{18} - 2 \nu^{17} + 4 \nu^{16} - 63 \nu^{15} - 4 \nu^{14} - 44 \nu^{13} + \cdots - 4608 ) / 2048 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 7 \nu^{19} + 4 \nu^{18} - 26 \nu^{17} + 4 \nu^{16} - 11 \nu^{15} - 4 \nu^{14} + 20 \nu^{13} + \cdots + 512 ) / 1024 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 31 \nu^{19} - 60 \nu^{18} - 26 \nu^{17} + 52 \nu^{16} - 163 \nu^{15} + 188 \nu^{14} - 220 \nu^{13} + \cdots - 18944 ) / 4096 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 4 \nu^{19} - 3 \nu^{18} + 6 \nu^{17} + 6 \nu^{16} + 4 \nu^{15} + 7 \nu^{14} + 10 \nu^{13} + \cdots - 768 ) / 512 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 43 \nu^{19} + 76 \nu^{18} + 2 \nu^{17} - 68 \nu^{16} + 127 \nu^{15} - 204 \nu^{14} + 236 \nu^{13} + \cdots + 12800 ) / 4096 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 53 \nu^{19} + 76 \nu^{18} - 18 \nu^{17} - 76 \nu^{16} + 289 \nu^{15} - 236 \nu^{14} + 420 \nu^{13} + \cdots + 38400 ) / 4096 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 75 \nu^{19} - 44 \nu^{18} - 2 \nu^{17} + 36 \nu^{16} - 223 \nu^{15} + 172 \nu^{14} - 428 \nu^{13} + \cdots - 16896 ) / 4096 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{11} - 2\beta_{10} - 2\beta_{9} + 2\beta_{7} - 2\beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2\beta_{18} - 2\beta_{11} - 2\beta_{8} + 2\beta_{6} - 2\beta_{4} - \beta_{3} - \beta_{2} - \beta _1 + 2 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 2 \beta_{18} + 2 \beta_{17} + 2 \beta_{15} - 2 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \cdots + 3 \beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 2 \beta_{19} - 2 \beta_{18} + 2 \beta_{17} - 6 \beta_{16} - 2 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} + \cdots + 2 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 2 \beta_{18} - 2 \beta_{17} + 2 \beta_{15} - 2 \beta_{14} + 6 \beta_{13} - 2 \beta_{12} - 2 \beta_{10} + \cdots - 4 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 2 \beta_{19} + 2 \beta_{18} - 2 \beta_{17} - 2 \beta_{16} + 2 \beta_{15} - 2 \beta_{14} - 6 \beta_{13} + \cdots - 6 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 4 \beta_{19} - 10 \beta_{18} + 6 \beta_{17} + 12 \beta_{16} + 2 \beta_{15} + 2 \beta_{14} + 6 \beta_{13} + \cdots + 16 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 10 \beta_{19} + 2 \beta_{18} + 14 \beta_{17} - 2 \beta_{16} - 6 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} + \cdots + 22 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 8 \beta_{19} - 10 \beta_{18} + 2 \beta_{17} - 10 \beta_{15} - 14 \beta_{14} + 10 \beta_{13} - 14 \beta_{12} + \cdots - 4 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 10 \beta_{19} + 2 \beta_{18} + 2 \beta_{17} - 22 \beta_{16} - 18 \beta_{15} - 6 \beta_{14} + \cdots + 18 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 12 \beta_{19} + 6 \beta_{18} - 18 \beta_{17} + 36 \beta_{16} - 14 \beta_{15} + 18 \beta_{14} + 14 \beta_{13} + \cdots + 40 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 42 \beta_{19} + 10 \beta_{18} + 30 \beta_{17} + 46 \beta_{16} + 26 \beta_{15} - 6 \beta_{14} - 6 \beta_{13} + \cdots + 70 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 114 \beta_{18} + 66 \beta_{17} - 24 \beta_{16} - 74 \beta_{15} - 22 \beta_{14} - 38 \beta_{13} + \cdots - 108 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 10 \beta_{19} - 30 \beta_{18} + 66 \beta_{17} - 38 \beta_{16} - 34 \beta_{15} - 30 \beta_{14} + \cdots - 6 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 4 \beta_{19} + 22 \beta_{18} - 58 \beta_{17} - 132 \beta_{16} - 38 \beta_{15} - 174 \beta_{14} + \cdots - 192 ) / 2 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 110 \beta_{19} - 246 \beta_{18} + 86 \beta_{17} - 10 \beta_{16} + 18 \beta_{15} - 38 \beta_{14} + \cdots + 30 ) / 2 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 200 \beta_{19} + 302 \beta_{18} + 154 \beta_{17} + 112 \beta_{16} + 14 \beta_{15} + 138 \beta_{14} + \cdots - 244 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times\).
| \(n\) | \(391\) | \(521\) | \(781\) | \(937\) | \(1081\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 781.1 |
|
−1.39528 | − | 0.230662i | 1.00000i | 1.89359 | + | 0.643673i | − | 1.00000i | 0.230662 | − | 1.39528i | −4.28425 | −2.49361 | − | 1.33488i | −1.00000 | −0.230662 | + | 1.39528i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.2 | −1.39528 | + | 0.230662i | − | 1.00000i | 1.89359 | − | 0.643673i | 1.00000i | 0.230662 | + | 1.39528i | −4.28425 | −2.49361 | + | 1.33488i | −1.00000 | −0.230662 | − | 1.39528i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.3 | −1.31330 | − | 0.524636i | − | 1.00000i | 1.44951 | + | 1.37801i | 1.00000i | −0.524636 | + | 1.31330i | 0.433192 | −1.18069 | − | 2.57021i | −1.00000 | 0.524636 | − | 1.31330i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.4 | −1.31330 | + | 0.524636i | 1.00000i | 1.44951 | − | 1.37801i | − | 1.00000i | −0.524636 | − | 1.31330i | 0.433192 | −1.18069 | + | 2.57021i | −1.00000 | 0.524636 | + | 1.31330i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.5 | −1.26346 | − | 0.635342i | 1.00000i | 1.19268 | + | 1.60546i | − | 1.00000i | 0.635342 | − | 1.26346i | 4.21598 | −0.486889 | − | 2.78621i | −1.00000 | −0.635342 | + | 1.26346i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.6 | −1.26346 | + | 0.635342i | − | 1.00000i | 1.19268 | − | 1.60546i | 1.00000i | 0.635342 | + | 1.26346i | 4.21598 | −0.486889 | + | 2.78621i | −1.00000 | −0.635342 | − | 1.26346i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.7 | −0.741368 | − | 1.20431i | 1.00000i | −0.900747 | + | 1.78568i | − | 1.00000i | 1.20431 | − | 0.741368i | 1.16629 | 2.81831 | − | 0.239062i | −1.00000 | −1.20431 | + | 0.741368i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.8 | −0.741368 | + | 1.20431i | − | 1.00000i | −0.900747 | − | 1.78568i | 1.00000i | 1.20431 | + | 0.741368i | 1.16629 | 2.81831 | + | 0.239062i | −1.00000 | −1.20431 | − | 0.741368i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.9 | −0.182570 | − | 1.40238i | 1.00000i | −1.93334 | + | 0.512066i | − | 1.00000i | 1.40238 | − | 0.182570i | −0.184370 | 1.07108 | + | 2.61778i | −1.00000 | −1.40238 | + | 0.182570i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.10 | −0.182570 | + | 1.40238i | − | 1.00000i | −1.93334 | − | 0.512066i | 1.00000i | 1.40238 | + | 0.182570i | −0.184370 | 1.07108 | − | 2.61778i | −1.00000 | −1.40238 | − | 0.182570i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.11 | 0.0820714 | − | 1.41183i | − | 1.00000i | −1.98653 | − | 0.231742i | 1.00000i | −1.41183 | − | 0.0820714i | −4.19666 | −0.490217 | + | 2.78562i | −1.00000 | 1.41183 | + | 0.0820714i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.12 | 0.0820714 | + | 1.41183i | 1.00000i | −1.98653 | + | 0.231742i | − | 1.00000i | −1.41183 | + | 0.0820714i | −4.19666 | −0.490217 | − | 2.78562i | −1.00000 | 1.41183 | − | 0.0820714i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.13 | 0.514103 | − | 1.31746i | − | 1.00000i | −1.47140 | − | 1.35462i | 1.00000i | −1.31746 | − | 0.514103i | 3.05712 | −2.54110 | + | 1.24209i | −1.00000 | 1.31746 | + | 0.514103i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.14 | 0.514103 | + | 1.31746i | 1.00000i | −1.47140 | + | 1.35462i | − | 1.00000i | −1.31746 | + | 0.514103i | 3.05712 | −2.54110 | − | 1.24209i | −1.00000 | 1.31746 | − | 0.514103i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.15 | 0.705594 | − | 1.22562i | 1.00000i | −1.00427 | − | 1.72958i | − | 1.00000i | 1.22562 | + | 0.705594i | −2.07003 | −2.82841 | + | 0.0104776i | −1.00000 | −1.22562 | − | 0.705594i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.16 | 0.705594 | + | 1.22562i | − | 1.00000i | −1.00427 | + | 1.72958i | 1.00000i | 1.22562 | − | 0.705594i | −2.07003 | −2.82841 | − | 0.0104776i | −1.00000 | −1.22562 | + | 0.705594i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.17 | 1.20967 | − | 0.732592i | − | 1.00000i | 0.926619 | − | 1.77239i | 1.00000i | −0.732592 | − | 1.20967i | −2.74234 | −0.177535 | − | 2.82285i | −1.00000 | 0.732592 | + | 1.20967i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.18 | 1.20967 | + | 0.732592i | 1.00000i | 0.926619 | + | 1.77239i | − | 1.00000i | −0.732592 | + | 1.20967i | −2.74234 | −0.177535 | + | 2.82285i | −1.00000 | 0.732592 | − | 1.20967i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.19 | 1.38454 | − | 0.288202i | 1.00000i | 1.83388 | − | 0.798052i | − | 1.00000i | 0.288202 | + | 1.38454i | −3.39492 | 2.30907 | − | 1.63346i | −1.00000 | −0.288202 | − | 1.38454i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 781.20 | 1.38454 | + | 0.288202i | − | 1.00000i | 1.83388 | + | 0.798052i | 1.00000i | 0.288202 | − | 1.38454i | −3.39492 | 2.30907 | + | 1.63346i | −1.00000 | −0.288202 | + | 1.38454i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1560.2.w.f | ✓ | 20 |
| 4.b | odd | 2 | 1 | 6240.2.w.f | 20 | ||
| 8.b | even | 2 | 1 | inner | 1560.2.w.f | ✓ | 20 |
| 8.d | odd | 2 | 1 | 6240.2.w.f | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1560.2.w.f | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 1560.2.w.f | ✓ | 20 | 8.b | even | 2 | 1 | inner |
| 6240.2.w.f | 20 | 4.b | odd | 2 | 1 | ||
| 6240.2.w.f | 20 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1560, [\chi])\):
|
\( T_{7}^{10} + 8 T_{7}^{9} - 12 T_{7}^{8} - 228 T_{7}^{7} - 275 T_{7}^{6} + 1636 T_{7}^{5} + 3358 T_{7}^{4} + \cdots + 416 \)
|
|
\( T_{11}^{20} + 124 T_{11}^{18} + 6590 T_{11}^{16} + 195972 T_{11}^{14} + 3566297 T_{11}^{12} + \cdots + 198246400 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} + 2 T^{19} + \cdots + 1024 \)
|
| $3$ |
\( (T^{2} + 1)^{10} \)
|
| $5$ |
\( (T^{2} + 1)^{10} \)
|
| $7$ |
\( (T^{10} + 8 T^{9} + \cdots + 416)^{2} \)
|
| $11$ |
\( T^{20} + \cdots + 198246400 \)
|
| $13$ |
\( (T^{2} + 1)^{10} \)
|
| $17$ |
\( (T^{10} - 8 T^{9} + \cdots - 3488)^{2} \)
|
| $19$ |
\( T^{20} + \cdots + 7487094784 \)
|
| $23$ |
\( (T^{10} - 24 T^{9} + \cdots - 3400352)^{2} \)
|
| $29$ |
\( T^{20} + \cdots + 1701250662400 \)
|
| $31$ |
\( (T^{10} + 4 T^{9} + \cdots - 16384)^{2} \)
|
| $37$ |
\( T^{20} + \cdots + 251442073600 \)
|
| $41$ |
\( (T^{10} - 8 T^{9} + \cdots - 50990720)^{2} \)
|
| $43$ |
\( T^{20} + 296 T^{18} + \cdots + 67108864 \)
|
| $47$ |
\( (T^{10} + 4 T^{9} + \cdots - 497096704)^{2} \)
|
| $53$ |
\( T^{20} + \cdots + 91447449849856 \)
|
| $59$ |
\( T^{20} + \cdots + 261369043615744 \)
|
| $61$ |
\( T^{20} + \cdots + 318928165273600 \)
|
| $67$ |
\( T^{20} + \cdots + 16\!\cdots\!00 \)
|
| $71$ |
\( (T^{10} + 4 T^{9} + \cdots + 104160256)^{2} \)
|
| $73$ |
\( (T^{10} - 20 T^{9} + \cdots + 4096)^{2} \)
|
| $79$ |
\( (T^{10} + 4 T^{9} + \cdots - 1678716928)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 35\!\cdots\!24 \)
|
| $89$ |
\( (T^{10} - 28 T^{9} + \cdots - 2344371904)^{2} \)
|
| $97$ |
\( (T^{10} + 24 T^{9} + \cdots + 176269088)^{2} \)
|
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