Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [160,3,Mod(17,160)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(160, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 2, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("160.17");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 160 = 2^{5} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 160.m (of order \(4\), 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: | \(4.35968422976\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - x^{18} - 3x^{16} + 11x^{14} + x^{12} - 40x^{10} + 4x^{8} + 176x^{6} - 192x^{4} - 256x^{2} + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{34} \) |
| Twist minimal: | no (minimal twist has level 40) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{20} - x^{18} - 3x^{16} + 11x^{14} + x^{12} - 40x^{10} + 4x^{8} + 176x^{6} - 192x^{4} - 256x^{2} + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 3 \nu^{18} - 25 \nu^{16} + 85 \nu^{14} - 61 \nu^{12} + 185 \nu^{10} - 20 \nu^{8} - 252 \nu^{6} + \cdots + 2816 ) / 1600 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2 \nu^{19} - 19 \nu^{17} + 7 \nu^{15} + 93 \nu^{13} + 35 \nu^{11} + 35 \nu^{9} - 132 \nu^{7} + \cdots - 1408 \nu ) / 9600 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 13 \nu^{18} + 47 \nu^{16} + 29 \nu^{14} + 59 \nu^{12} + 65 \nu^{10} - 620 \nu^{8} - 908 \nu^{6} + \cdots - 704 ) / 1600 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 5 \nu^{19} + 234 \nu^{18} - 215 \nu^{17} + 246 \nu^{16} - 805 \nu^{15} - 1278 \nu^{14} + \cdots - 118272 ) / 38400 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 5 \nu^{19} - 234 \nu^{18} - 215 \nu^{17} - 246 \nu^{16} - 805 \nu^{15} + 1278 \nu^{14} + \cdots + 118272 ) / 38400 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 25 \nu^{19} + 82 \nu^{18} - 45 \nu^{17} - 50 \nu^{16} + 185 \nu^{15} + 10 \nu^{14} + 95 \nu^{13} + \cdots + 16896 ) / 12800 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 7 \nu^{18} - 7 \nu^{16} - 69 \nu^{14} + 61 \nu^{12} + 215 \nu^{10} - 360 \nu^{8} - 212 \nu^{6} + \cdots - 4736 ) / 640 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 25 \nu^{19} + 82 \nu^{18} + 45 \nu^{17} - 50 \nu^{16} - 185 \nu^{15} + 10 \nu^{14} - 95 \nu^{13} + \cdots + 16896 ) / 12800 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 105 \nu^{19} + 122 \nu^{18} - 489 \nu^{17} - 250 \nu^{16} + 357 \nu^{15} - 590 \nu^{14} + \cdots - 54784 ) / 38400 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 165 \nu^{19} - 166 \nu^{18} - 183 \nu^{17} + 950 \nu^{16} + 1179 \nu^{15} - 830 \nu^{14} + \cdots - 130048 ) / 38400 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 105 \nu^{19} - 122 \nu^{18} - 489 \nu^{17} + 250 \nu^{16} + 357 \nu^{15} + 590 \nu^{14} + \cdots + 54784 ) / 38400 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 165 \nu^{19} - 166 \nu^{18} + 183 \nu^{17} + 950 \nu^{16} - 1179 \nu^{15} - 830 \nu^{14} + \cdots - 130048 ) / 38400 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 5 \nu^{19} + 5 \nu^{17} - 17 \nu^{15} - 15 \nu^{13} + 59 \nu^{11} - 70 \nu^{9} - 660 \nu^{7} + \cdots - 1792 \nu ) / 768 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 73 \nu^{19} - 84 \nu^{18} + 56 \nu^{17} + 204 \nu^{16} + 232 \nu^{15} + 228 \nu^{14} + \cdots + 12672 ) / 9600 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 6 \nu^{19} + 13 \nu^{17} - 9 \nu^{15} + 29 \nu^{13} + 115 \nu^{11} - 25 \nu^{9} + 44 \nu^{7} + \cdots + 896 \nu ) / 640 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 73 \nu^{19} - 84 \nu^{18} - 56 \nu^{17} + 204 \nu^{16} - 232 \nu^{15} + 228 \nu^{14} + 732 \nu^{13} + \cdots + 12672 ) / 9600 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 375 \nu^{19} - 38 \nu^{18} + 435 \nu^{17} + 550 \nu^{16} + 345 \nu^{15} - 1390 \nu^{14} + \cdots + 27136 ) / 38400 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 375 \nu^{19} + 38 \nu^{18} + 435 \nu^{17} - 550 \nu^{16} + 345 \nu^{15} + 1390 \nu^{14} + \cdots - 27136 ) / 38400 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 10 \nu^{19} - 23 \nu^{17} - \nu^{15} + 93 \nu^{13} - 125 \nu^{11} - 165 \nu^{9} + 440 \nu^{7} + \cdots + 192 \nu ) / 800 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{18} + \beta_{17} + \beta_{16} - \beta_{14} - \beta_{8} + \beta_{6} + 2\beta_{2} ) / 16 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{18} - \beta_{17} + \beta_{16} + \beta_{14} - 2 \beta_{11} + 2 \beta_{9} + \beta_{8} - 2 \beta_{7} + \cdots + 2 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2 \beta_{19} + \beta_{18} + \beta_{17} - \beta_{16} + 2 \beta_{15} + \beta_{14} + 2 \beta_{13} + \cdots - 2 \beta_{2} ) / 16 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{18} - \beta_{17} - \beta_{16} - \beta_{14} + 4 \beta_{12} - 4 \beta_{11} + 4 \beta_{10} + \cdots + 10 ) / 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 2 \beta_{19} - \beta_{18} - \beta_{17} + \beta_{16} - \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \cdots + 12 \beta_{2} ) / 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( \beta_{18} - \beta_{17} + 5 \beta_{16} + 5 \beta_{14} + 4 \beta_{11} - 4 \beta_{9} + 11 \beta_{8} + \cdots - 42 ) / 16 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 6 \beta_{19} - \beta_{18} - \beta_{17} - 5 \beta_{16} + 10 \beta_{15} + 5 \beta_{14} + \cdots - 70 \beta_{2} ) / 16 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - \beta_{18} + \beta_{17} - 11 \beta_{16} - 11 \beta_{14} + 8 \beta_{12} + 22 \beta_{11} + 8 \beta_{10} + \cdots - 26 ) / 16 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 12 \beta_{19} + 7 \beta_{18} + 7 \beta_{17} - 3 \beta_{16} - 4 \beta_{15} + 3 \beta_{14} + \cdots + 194 \beta_{2} ) / 16 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 3 \beta_{18} - 3 \beta_{17} + 7 \beta_{16} + 7 \beta_{14} - 2 \beta_{12} + 16 \beta_{11} - 2 \beta_{10} + \cdots + 28 ) / 8 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 8 \beta_{19} + 13 \beta_{18} + 13 \beta_{17} - 27 \beta_{16} + 40 \beta_{15} + 27 \beta_{14} + \cdots + 146 \beta_{2} ) / 16 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 55 \beta_{18} - 55 \beta_{17} - 25 \beta_{16} - 25 \beta_{14} + 10 \beta_{11} - 10 \beta_{9} + \cdots + 414 ) / 16 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 14 \beta_{19} + 11 \beta_{18} + 11 \beta_{17} + 29 \beta_{16} + 30 \beta_{15} - 29 \beta_{14} + \cdots + 858 \beta_{2} ) / 16 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 71 \beta_{18} - 71 \beta_{17} + 41 \beta_{16} + 41 \beta_{14} - 4 \beta_{12} - 68 \beta_{11} + \cdots - 586 ) / 16 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 62 \beta_{19} - 47 \beta_{18} - 47 \beta_{17} - 53 \beta_{16} + 56 \beta_{15} + 53 \beta_{14} + \cdots + 244 \beta_{2} ) / 8 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 31 \beta_{18} - 31 \beta_{17} - 21 \beta_{16} - 21 \beta_{14} + 272 \beta_{12} - 12 \beta_{11} + \cdots + 554 ) / 16 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 358 \beta_{19} - 227 \beta_{18} - 227 \beta_{17} + 185 \beta_{16} + 70 \beta_{15} + \cdots - 1410 \beta_{2} ) / 16 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 139 \beta_{18} + 139 \beta_{17} + 15 \beta_{16} + 15 \beta_{14} - 120 \beta_{12} + 314 \beta_{11} + \cdots - 4478 ) / 16 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 668 \beta_{19} - 71 \beta_{18} - 71 \beta_{17} - 301 \beta_{16} + 220 \beta_{15} + \cdots - 4562 \beta_{2} ) / 16 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(97\) | \(101\) |
| \(\chi(n)\) | \(1\) | \(\beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | −3.60765 | − | 3.60765i | 0 | −2.34539 | − | 4.41578i | 0 | −1.47907 | − | 1.47907i | 0 | 17.0303i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.2 | 0 | −2.57493 | − | 2.57493i | 0 | 4.90427 | − | 0.973739i | 0 | 4.07624 | + | 4.07624i | 0 | 4.26050i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.3 | 0 | −2.52630 | − | 2.52630i | 0 | −3.09141 | + | 3.92978i | 0 | 5.20520 | + | 5.20520i | 0 | 3.76437i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.4 | 0 | −0.977390 | − | 0.977390i | 0 | −0.801246 | + | 4.93538i | 0 | −8.39950 | − | 8.39950i | 0 | − | 7.08942i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.5 | 0 | −0.130791 | − | 0.130791i | 0 | −4.38731 | − | 2.39823i | 0 | 1.59713 | + | 1.59713i | 0 | − | 8.96579i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.6 | 0 | 0.130791 | + | 0.130791i | 0 | 4.38731 | + | 2.39823i | 0 | 1.59713 | + | 1.59713i | 0 | − | 8.96579i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.7 | 0 | 0.977390 | + | 0.977390i | 0 | 0.801246 | − | 4.93538i | 0 | −8.39950 | − | 8.39950i | 0 | − | 7.08942i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.8 | 0 | 2.52630 | + | 2.52630i | 0 | 3.09141 | − | 3.92978i | 0 | 5.20520 | + | 5.20520i | 0 | 3.76437i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.9 | 0 | 2.57493 | + | 2.57493i | 0 | −4.90427 | + | 0.973739i | 0 | 4.07624 | + | 4.07624i | 0 | 4.26050i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 17.10 | 0 | 3.60765 | + | 3.60765i | 0 | 2.34539 | + | 4.41578i | 0 | −1.47907 | − | 1.47907i | 0 | 17.0303i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.1 | 0 | −3.60765 | + | 3.60765i | 0 | −2.34539 | + | 4.41578i | 0 | −1.47907 | + | 1.47907i | 0 | − | 17.0303i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.2 | 0 | −2.57493 | + | 2.57493i | 0 | 4.90427 | + | 0.973739i | 0 | 4.07624 | − | 4.07624i | 0 | − | 4.26050i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.3 | 0 | −2.52630 | + | 2.52630i | 0 | −3.09141 | − | 3.92978i | 0 | 5.20520 | − | 5.20520i | 0 | − | 3.76437i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.4 | 0 | −0.977390 | + | 0.977390i | 0 | −0.801246 | − | 4.93538i | 0 | −8.39950 | + | 8.39950i | 0 | 7.08942i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.5 | 0 | −0.130791 | + | 0.130791i | 0 | −4.38731 | + | 2.39823i | 0 | 1.59713 | − | 1.59713i | 0 | 8.96579i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.6 | 0 | 0.130791 | − | 0.130791i | 0 | 4.38731 | − | 2.39823i | 0 | 1.59713 | − | 1.59713i | 0 | 8.96579i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.7 | 0 | 0.977390 | − | 0.977390i | 0 | 0.801246 | + | 4.93538i | 0 | −8.39950 | + | 8.39950i | 0 | 7.08942i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.8 | 0 | 2.52630 | − | 2.52630i | 0 | 3.09141 | + | 3.92978i | 0 | 5.20520 | − | 5.20520i | 0 | − | 3.76437i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.9 | 0 | 2.57493 | − | 2.57493i | 0 | −4.90427 | − | 0.973739i | 0 | 4.07624 | − | 4.07624i | 0 | − | 4.26050i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 113.10 | 0 | 3.60765 | − | 3.60765i | 0 | 2.34539 | − | 4.41578i | 0 | −1.47907 | + | 1.47907i | 0 | − | 17.0303i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 40.i | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 160.3.m.a | 20 | |
| 4.b | odd | 2 | 1 | 40.3.i.a | ✓ | 20 | |
| 5.b | even | 2 | 1 | 800.3.m.b | 20 | ||
| 5.c | odd | 4 | 1 | inner | 160.3.m.a | 20 | |
| 5.c | odd | 4 | 1 | 800.3.m.b | 20 | ||
| 8.b | even | 2 | 1 | inner | 160.3.m.a | 20 | |
| 8.d | odd | 2 | 1 | 40.3.i.a | ✓ | 20 | |
| 12.b | even | 2 | 1 | 360.3.u.b | 20 | ||
| 20.d | odd | 2 | 1 | 200.3.i.b | 20 | ||
| 20.e | even | 4 | 1 | 40.3.i.a | ✓ | 20 | |
| 20.e | even | 4 | 1 | 200.3.i.b | 20 | ||
| 24.f | even | 2 | 1 | 360.3.u.b | 20 | ||
| 40.e | odd | 2 | 1 | 200.3.i.b | 20 | ||
| 40.f | even | 2 | 1 | 800.3.m.b | 20 | ||
| 40.i | odd | 4 | 1 | inner | 160.3.m.a | 20 | |
| 40.i | odd | 4 | 1 | 800.3.m.b | 20 | ||
| 40.k | even | 4 | 1 | 40.3.i.a | ✓ | 20 | |
| 40.k | even | 4 | 1 | 200.3.i.b | 20 | ||
| 60.l | odd | 4 | 1 | 360.3.u.b | 20 | ||
| 120.q | odd | 4 | 1 | 360.3.u.b | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.3.i.a | ✓ | 20 | 4.b | odd | 2 | 1 | |
| 40.3.i.a | ✓ | 20 | 8.d | odd | 2 | 1 | |
| 40.3.i.a | ✓ | 20 | 20.e | even | 4 | 1 | |
| 40.3.i.a | ✓ | 20 | 40.k | even | 4 | 1 | |
| 160.3.m.a | 20 | 1.a | even | 1 | 1 | trivial | |
| 160.3.m.a | 20 | 5.c | odd | 4 | 1 | inner | |
| 160.3.m.a | 20 | 8.b | even | 2 | 1 | inner | |
| 160.3.m.a | 20 | 40.i | odd | 4 | 1 | inner | |
| 200.3.i.b | 20 | 20.d | odd | 2 | 1 | ||
| 200.3.i.b | 20 | 20.e | even | 4 | 1 | ||
| 200.3.i.b | 20 | 40.e | odd | 2 | 1 | ||
| 200.3.i.b | 20 | 40.k | even | 4 | 1 | ||
| 360.3.u.b | 20 | 12.b | even | 2 | 1 | ||
| 360.3.u.b | 20 | 24.f | even | 2 | 1 | ||
| 360.3.u.b | 20 | 60.l | odd | 4 | 1 | ||
| 360.3.u.b | 20 | 120.q | odd | 4 | 1 | ||
| 800.3.m.b | 20 | 5.b | even | 2 | 1 | ||
| 800.3.m.b | 20 | 5.c | odd | 4 | 1 | ||
| 800.3.m.b | 20 | 40.f | even | 2 | 1 | ||
| 800.3.m.b | 20 | 40.i | odd | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(160, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} + 1020 T^{16} + \cdots + 82944 \)
|
| $5$ |
\( T^{20} + \cdots + 95367431640625 \)
|
| $7$ |
\( (T^{10} - 2 T^{9} + \cdots + 5671712)^{2} \)
|
| $11$ |
\( (T^{10} + 552 T^{8} + \cdots + 473497600)^{2} \)
|
| $13$ |
\( T^{20} + \cdots + 23\!\cdots\!00 \)
|
| $17$ |
\( (T^{10} + 6 T^{9} + \cdots + 2344207392)^{2} \)
|
| $19$ |
\( (T^{10} + \cdots - 384717505536)^{2} \)
|
| $23$ |
\( (T^{10} - 2 T^{9} + \cdots + 415411488)^{2} \)
|
| $29$ |
\( (T^{10} - 2548 T^{8} + \cdots - 31360000)^{2} \)
|
| $31$ |
\( (T^{5} - 34 T^{4} + \cdots + 1252704)^{4} \)
|
| $37$ |
\( T^{20} + \cdots + 59\!\cdots\!44 \)
|
| $41$ |
\( (T^{5} + 2 T^{4} + \cdots + 74680800)^{4} \)
|
| $43$ |
\( T^{20} + \cdots + 81\!\cdots\!64 \)
|
| $47$ |
\( (T^{10} + \cdots + 418650446283552)^{2} \)
|
| $53$ |
\( T^{20} + \cdots + 30\!\cdots\!84 \)
|
| $59$ |
\( (T^{10} + \cdots - 26\!\cdots\!16)^{2} \)
|
| $61$ |
\( (T^{10} + \cdots + 26\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 23\!\cdots\!44 \)
|
| $71$ |
\( (T^{5} + 62 T^{4} + \cdots - 423110304)^{4} \)
|
| $73$ |
\( (T^{10} + \cdots + 37\!\cdots\!88)^{2} \)
|
| $79$ |
\( (T^{10} + \cdots + 66\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 23\!\cdots\!64 \)
|
| $89$ |
\( (T^{10} + \cdots + 51\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{10} + \cdots + 90\!\cdots\!32)^{2} \)
|
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