Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [960,2,Mod(367,960)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(960, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("960.367");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 960 = 2^{6} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 960.bc (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: | \(7.66563859404\) |
| 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: |
\( x^{20} - 2 x^{19} + 3 x^{18} - 6 x^{17} + 2 x^{16} + 4 x^{14} + 20 x^{13} - 24 x^{12} + 40 x^{11} + \cdots + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{17} \) |
| Twist minimal: | no (minimal twist has level 240) |
| 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
\( x^{20} - 2 x^{19} + 3 x^{18} - 6 x^{17} + 2 x^{16} + 4 x^{14} + 20 x^{13} - 24 x^{12} + 40 x^{11} + \cdots + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3 \nu^{19} + 6 \nu^{18} - 35 \nu^{17} + 34 \nu^{16} - 30 \nu^{15} + 88 \nu^{14} + 36 \nu^{13} + \cdots + 11776 ) / 2560 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 15 \nu^{19} + 2 \nu^{18} - 63 \nu^{17} - 2 \nu^{16} - 102 \nu^{15} + 152 \nu^{14} + 260 \nu^{13} + \cdots + 18944 ) / 2560 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 9 \nu^{19} - 6 \nu^{18} + 57 \nu^{17} - 34 \nu^{16} - 22 \nu^{15} - 112 \nu^{14} - 188 \nu^{13} + \cdots - 34304 ) / 2560 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 17 \nu^{19} + 8 \nu^{18} + 17 \nu^{17} + 32 \nu^{16} + 58 \nu^{15} - 60 \nu^{14} - 84 \nu^{13} + \cdots - 5120 ) / 2560 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 27 \nu^{19} - 10 \nu^{18} - 99 \nu^{17} - 70 \nu^{16} - 86 \nu^{15} + 208 \nu^{14} + 644 \nu^{13} + \cdots + 37376 ) / 2560 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 23 \nu^{19} - 8 \nu^{18} + 21 \nu^{17} - 72 \nu^{16} - 106 \nu^{15} - 36 \nu^{14} - 4 \nu^{13} + \cdots - 10752 ) / 2560 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{19} - 2 \nu^{18} + 3 \nu^{17} - 6 \nu^{16} + 2 \nu^{15} + 4 \nu^{13} + 20 \nu^{12} - 24 \nu^{11} + \cdots - 1024 ) / 256 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 29 \nu^{19} + 30 \nu^{18} - 7 \nu^{17} + 50 \nu^{16} + 62 \nu^{15} - 96 \nu^{14} - 108 \nu^{13} + \cdots - 5632 ) / 2560 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 8 \nu^{19} - 7 \nu^{18} + 12 \nu^{17} - 33 \nu^{16} - 12 \nu^{15} - 50 \nu^{14} + 36 \nu^{13} + \cdots - 1280 ) / 640 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 41 \nu^{19} + 38 \nu^{18} - 169 \nu^{17} - 78 \nu^{16} - 186 \nu^{15} + 192 \nu^{14} + 892 \nu^{13} + \cdots + 59904 ) / 2560 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 17 \nu^{19} - 46 \nu^{18} + 75 \nu^{17} - 74 \nu^{16} + 50 \nu^{15} - 168 \nu^{14} - 36 \nu^{13} + \cdots - 11776 ) / 2560 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 2 \nu^{19} - 5 \nu^{18} - 2 \nu^{17} - 3 \nu^{16} + 26 \nu^{14} + 16 \nu^{13} + 28 \nu^{12} + \cdots + 512 ) / 256 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 3 \nu^{18} - 6 \nu^{17} + 3 \nu^{16} - 8 \nu^{15} + 4 \nu^{14} + 14 \nu^{13} + 16 \nu^{12} + \cdots + 1792 ) / 128 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 3 \nu^{19} + 35 \nu^{18} - 29 \nu^{17} + 25 \nu^{16} - 86 \nu^{15} - 62 \nu^{14} + 4 \nu^{13} + \cdots + 6656 ) / 1280 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 35 \nu^{19} + 88 \nu^{18} - 57 \nu^{17} + 72 \nu^{16} - 78 \nu^{15} - 252 \nu^{14} + 20 \nu^{13} + \cdots + 22016 ) / 2560 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 6 \nu^{19} + 11 \nu^{18} - 7 \nu^{17} + 14 \nu^{16} + 2 \nu^{15} - 23 \nu^{14} - 7 \nu^{13} + \cdots + 1024 ) / 320 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 33 \nu^{19} - 96 \nu^{18} + 83 \nu^{17} - 184 \nu^{16} + 82 \nu^{15} + 156 \nu^{14} + 236 \nu^{13} + \cdots - 27648 ) / 2560 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 7 \nu^{19} - 26 \nu^{18} + 30 \nu^{17} - 34 \nu^{16} + 25 \nu^{15} + 12 \nu^{14} - 26 \nu^{13} + \cdots - 11136 ) / 640 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 35 \nu^{19} + 49 \nu^{18} - 11 \nu^{17} + 71 \nu^{16} - 4 \nu^{15} - 86 \nu^{14} - 220 \nu^{13} + \cdots - 10752 ) / 1280 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{16} - \beta_{14} - \beta_{8} - \beta_{7} + \beta_{6} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{19} - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - 2 \beta_{8} - \beta_{7} + \cdots - 1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - \beta_{19} - \beta_{16} + \beta_{15} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} + \cdots + 1 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - \beta_{19} + 2 \beta_{16} + \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - 2 \beta_{9} + \cdots + 5 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - \beta_{19} - 4 \beta_{17} + 3 \beta_{16} - \beta_{15} - 4 \beta_{14} + 4 \beta_{13} + 4 \beta_{12} + \cdots + 1 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( \beta_{19} - 4 \beta_{17} + 4 \beta_{16} + 2 \beta_{15} - 7 \beta_{14} - \beta_{13} + 5 \beta_{12} + \cdots - 1 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - \beta_{19} - 4 \beta_{18} - 3 \beta_{16} + 3 \beta_{15} + 2 \beta_{14} - 4 \beta_{13} + 4 \beta_{12} + \cdots - 19 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - \beta_{19} - 8 \beta_{18} - 4 \beta_{17} - 4 \beta_{16} - 6 \beta_{15} + 3 \beta_{14} - 3 \beta_{13} + \cdots + 17 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 7 \beta_{19} + 4 \beta_{18} - 16 \beta_{17} + 7 \beta_{16} + \beta_{15} - 6 \beta_{14} - 12 \beta_{13} + \cdots - 5 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 17 \beta_{19} - 12 \beta_{17} + 8 \beta_{16} - 6 \beta_{15} - 7 \beta_{14} - 17 \beta_{13} + 9 \beta_{12} + \cdots - 1 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 3 \beta_{19} - 36 \beta_{18} - 23 \beta_{16} - 13 \beta_{15} + 2 \beta_{14} - 28 \beta_{13} + 12 \beta_{12} + \cdots - 7 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 7 \beta_{19} - 32 \beta_{18} + 12 \beta_{17} - 24 \beta_{16} - 18 \beta_{15} + 23 \beta_{14} - 47 \beta_{13} + \cdots + 25 ) / 4 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 3 \beta_{19} + 20 \beta_{18} - 17 \beta_{16} - 35 \beta_{15} + 38 \beta_{14} - 28 \beta_{13} + \cdots - 41 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 33 \beta_{19} + 32 \beta_{18} + 4 \beta_{17} + 8 \beta_{16} - 14 \beta_{15} + \beta_{14} - 73 \beta_{13} + \cdots + 239 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 13 \beta_{19} - 68 \beta_{18} + 16 \beta_{17} + 33 \beta_{16} - 53 \beta_{15} - 38 \beta_{14} + \cdots - 55 ) / 4 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 7 \beta_{19} - 160 \beta_{18} + 92 \beta_{17} - 64 \beta_{16} - 202 \beta_{15} + 111 \beta_{14} + \cdots - 55 ) / 4 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 171 \beta_{19} + 100 \beta_{18} + 112 \beta_{17} - 161 \beta_{16} + 101 \beta_{15} + 142 \beta_{14} + \cdots + 207 ) / 4 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 15 \beta_{19} + 224 \beta_{18} + 100 \beta_{17} + 296 \beta_{16} + 34 \beta_{15} - 95 \beta_{14} + \cdots + 351 ) / 4 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 269 \beta_{19} - 228 \beta_{18} - 144 \beta_{17} + 241 \beta_{16} - 405 \beta_{15} - 86 \beta_{14} + \cdots - 1111 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).
| \(n\) | \(511\) | \(577\) | \(641\) | \(901\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{4}\) | \(1\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 367.1 |
|
0 | − | 1.00000i | 0 | −2.09611 | − | 0.778677i | 0 | 2.44055 | + | 2.44055i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 367.2 | 0 | − | 1.00000i | 0 | −1.37678 | + | 1.76195i | 0 | −0.159531 | − | 0.159531i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 367.3 | 0 | − | 1.00000i | 0 | −1.09619 | − | 1.94894i | 0 | 2.09269 | + | 2.09269i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 367.4 | 0 | − | 1.00000i | 0 | −0.453294 | + | 2.18964i | 0 | −1.25143 | − | 1.25143i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 367.5 | 0 | − | 1.00000i | 0 | 0.454390 | − | 2.18941i | 0 | −0.328507 | − | 0.328507i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 367.6 | 0 | − | 1.00000i | 0 | 0.927338 | − | 2.03471i | 0 | −1.72177 | − | 1.72177i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 367.7 | 0 | − | 1.00000i | 0 | 1.24985 | + | 1.85415i | 0 | −1.96536 | − | 1.96536i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 367.8 | 0 | − | 1.00000i | 0 | 1.92841 | + | 1.13192i | 0 | −1.21767 | − | 1.21767i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 367.9 | 0 | − | 1.00000i | 0 | 2.23072 | + | 0.154491i | 0 | 1.27936 | + | 1.27936i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 367.10 | 0 | − | 1.00000i | 0 | 2.23165 | − | 0.140415i | 0 | 2.83167 | + | 2.83167i | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 463.1 | 0 | 1.00000i | 0 | −2.09611 | + | 0.778677i | 0 | 2.44055 | − | 2.44055i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 463.2 | 0 | 1.00000i | 0 | −1.37678 | − | 1.76195i | 0 | −0.159531 | + | 0.159531i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 463.3 | 0 | 1.00000i | 0 | −1.09619 | + | 1.94894i | 0 | 2.09269 | − | 2.09269i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 463.4 | 0 | 1.00000i | 0 | −0.453294 | − | 2.18964i | 0 | −1.25143 | + | 1.25143i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 463.5 | 0 | 1.00000i | 0 | 0.454390 | + | 2.18941i | 0 | −0.328507 | + | 0.328507i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 463.6 | 0 | 1.00000i | 0 | 0.927338 | + | 2.03471i | 0 | −1.72177 | + | 1.72177i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 463.7 | 0 | 1.00000i | 0 | 1.24985 | − | 1.85415i | 0 | −1.96536 | + | 1.96536i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 463.8 | 0 | 1.00000i | 0 | 1.92841 | − | 1.13192i | 0 | −1.21767 | + | 1.21767i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 463.9 | 0 | 1.00000i | 0 | 2.23072 | − | 0.154491i | 0 | 1.27936 | − | 1.27936i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 463.10 | 0 | 1.00000i | 0 | 2.23165 | + | 0.140415i | 0 | 2.83167 | − | 2.83167i | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 80.j | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 960.2.bc.f | 20 | |
| 4.b | odd | 2 | 1 | 240.2.bc.f | yes | 20 | |
| 5.c | odd | 4 | 1 | 960.2.y.f | 20 | ||
| 8.b | even | 2 | 1 | 1920.2.bc.l | 20 | ||
| 8.d | odd | 2 | 1 | 1920.2.bc.k | 20 | ||
| 12.b | even | 2 | 1 | 720.2.bd.h | 20 | ||
| 16.e | even | 4 | 1 | 240.2.y.f | ✓ | 20 | |
| 16.e | even | 4 | 1 | 1920.2.y.l | 20 | ||
| 16.f | odd | 4 | 1 | 960.2.y.f | 20 | ||
| 16.f | odd | 4 | 1 | 1920.2.y.k | 20 | ||
| 20.e | even | 4 | 1 | 240.2.y.f | ✓ | 20 | |
| 40.i | odd | 4 | 1 | 1920.2.y.k | 20 | ||
| 40.k | even | 4 | 1 | 1920.2.y.l | 20 | ||
| 48.i | odd | 4 | 1 | 720.2.z.h | 20 | ||
| 60.l | odd | 4 | 1 | 720.2.z.h | 20 | ||
| 80.i | odd | 4 | 1 | 1920.2.bc.k | 20 | ||
| 80.j | even | 4 | 1 | inner | 960.2.bc.f | 20 | |
| 80.s | even | 4 | 1 | 1920.2.bc.l | 20 | ||
| 80.t | odd | 4 | 1 | 240.2.bc.f | yes | 20 | |
| 240.bf | even | 4 | 1 | 720.2.bd.h | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 240.2.y.f | ✓ | 20 | 16.e | even | 4 | 1 | |
| 240.2.y.f | ✓ | 20 | 20.e | even | 4 | 1 | |
| 240.2.bc.f | yes | 20 | 4.b | odd | 2 | 1 | |
| 240.2.bc.f | yes | 20 | 80.t | odd | 4 | 1 | |
| 720.2.z.h | 20 | 48.i | odd | 4 | 1 | ||
| 720.2.z.h | 20 | 60.l | odd | 4 | 1 | ||
| 720.2.bd.h | 20 | 12.b | even | 2 | 1 | ||
| 720.2.bd.h | 20 | 240.bf | even | 4 | 1 | ||
| 960.2.y.f | 20 | 5.c | odd | 4 | 1 | ||
| 960.2.y.f | 20 | 16.f | odd | 4 | 1 | ||
| 960.2.bc.f | 20 | 1.a | even | 1 | 1 | trivial | |
| 960.2.bc.f | 20 | 80.j | even | 4 | 1 | inner | |
| 1920.2.y.k | 20 | 16.f | odd | 4 | 1 | ||
| 1920.2.y.k | 20 | 40.i | odd | 4 | 1 | ||
| 1920.2.y.l | 20 | 16.e | even | 4 | 1 | ||
| 1920.2.y.l | 20 | 40.k | even | 4 | 1 | ||
| 1920.2.bc.k | 20 | 8.d | odd | 2 | 1 | ||
| 1920.2.bc.k | 20 | 80.i | odd | 4 | 1 | ||
| 1920.2.bc.l | 20 | 8.b | even | 2 | 1 | ||
| 1920.2.bc.l | 20 | 80.s | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(960, [\chi])\):
|
\( T_{7}^{20} - 4 T_{7}^{19} + 8 T_{7}^{18} + 32 T_{7}^{17} + 140 T_{7}^{16} - 448 T_{7}^{15} + \cdots + 25600 \)
|
|
\( T_{11}^{20} + 8 T_{11}^{19} + 32 T_{11}^{18} + 24 T_{11}^{17} + 1164 T_{11}^{16} + 8656 T_{11}^{15} + \cdots + 6390784 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( (T^{2} + 1)^{10} \)
|
| $5$ |
\( T^{20} - 8 T^{19} + \cdots + 9765625 \)
|
| $7$ |
\( T^{20} - 4 T^{19} + \cdots + 25600 \)
|
| $11$ |
\( T^{20} + 8 T^{19} + \cdots + 6390784 \)
|
| $13$ |
\( (T^{10} - 4 T^{9} + \cdots - 36608)^{2} \)
|
| $17$ |
\( T^{20} + \cdots + 37171840000 \)
|
| $19$ |
\( T^{20} - 16 T^{19} + \cdots + 6553600 \)
|
| $23$ |
\( T^{20} + \cdots + 8971878400 \)
|
| $29$ |
\( T^{20} + \cdots + 8641718502400 \)
|
| $31$ |
\( T^{20} + 232 T^{18} + \cdots + 98406400 \)
|
| $37$ |
\( (T^{10} - 12 T^{9} + \cdots - 491264)^{2} \)
|
| $41$ |
\( T^{20} + \cdots + 1717986918400 \)
|
| $43$ |
\( (T^{10} - 4 T^{9} + \cdots + 29696)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 163840000 \)
|
| $53$ |
\( T^{20} + \cdots + 20\!\cdots\!00 \)
|
| $59$ |
\( T^{20} + \cdots + 1603768960000 \)
|
| $61$ |
\( T^{20} + \cdots + 55960453571584 \)
|
| $67$ |
\( (T^{10} - 8 T^{9} + \cdots + 56394752)^{2} \)
|
| $71$ |
\( (T^{10} - 312 T^{8} + \cdots + 147865600)^{2} \)
|
| $73$ |
\( T^{20} + \cdots + 631014400 \)
|
| $79$ |
\( (T^{10} - 24 T^{9} + \cdots + 222791680)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 10\!\cdots\!04 \)
|
| $89$ |
\( (T^{10} + 20 T^{9} + \cdots - 7429120)^{2} \)
|
| $97$ |
\( T^{20} + \cdots + 1895578240000 \)
|
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