Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [240,2,Mod(61,240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(240, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3, 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("240.61");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 240.s (of order \(4\), 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: | \(1.91640964851\) |
| 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: |
\( x^{20} - 2 x^{18} - 4 x^{17} + 7 x^{16} + 16 x^{15} + 6 x^{14} - 36 x^{13} - 42 x^{12} + 40 x^{11} + \cdots + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| 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
\( x^{20} - 2 x^{18} - 4 x^{17} + 7 x^{16} + 16 x^{15} + 6 x^{14} - 36 x^{13} - 42 x^{12} + 40 x^{11} + \cdots + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - \nu^{19} + 2 \nu^{17} + 4 \nu^{16} - 7 \nu^{15} - 16 \nu^{14} - 6 \nu^{13} + 36 \nu^{12} + \cdots + 512 \nu ) / 256 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 18 \nu^{19} - 33 \nu^{18} - 8 \nu^{17} + 104 \nu^{16} + 94 \nu^{15} - 199 \nu^{14} - 616 \nu^{13} + \cdots + 5632 ) / 512 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 31 \nu^{19} + 39 \nu^{18} - 30 \nu^{17} - 204 \nu^{16} - 67 \nu^{15} + 497 \nu^{14} + 946 \nu^{13} + \cdots - 17920 ) / 512 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 33 \nu^{19} - 44 \nu^{18} + 32 \nu^{17} + 220 \nu^{16} + 89 \nu^{15} - 508 \nu^{14} - 1012 \nu^{13} + \cdots + 18432 ) / 512 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 23 \nu^{19} - 30 \nu^{18} + 13 \nu^{17} + 136 \nu^{16} + 59 \nu^{15} - 318 \nu^{14} - 673 \nu^{13} + \cdots + 9472 ) / 256 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 11 \nu^{19} - 36 \nu^{18} - 44 \nu^{17} + 28 \nu^{16} + 131 \nu^{15} + 12 \nu^{14} - 464 \nu^{13} + \cdots - 5632 ) / 256 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 21 \nu^{19} + 10 \nu^{18} + 104 \nu^{17} + 192 \nu^{16} - 123 \nu^{15} - 594 \nu^{14} + \cdots + 28672 ) / 512 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 39 \nu^{19} - 32 \nu^{18} + 80 \nu^{17} + 284 \nu^{16} - \nu^{15} - 760 \nu^{14} - 1052 \nu^{13} + \cdots + 31744 ) / 512 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 5 \nu^{19} - 31 \nu^{18} - 54 \nu^{17} - 12 \nu^{16} + 129 \nu^{15} + 119 \nu^{14} - 310 \nu^{13} + \cdots - 10752 ) / 256 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 33 \nu^{19} - 16 \nu^{18} + 98 \nu^{17} + 264 \nu^{16} - 47 \nu^{15} - 720 \nu^{14} - 790 \nu^{13} + \cdots + 32256 ) / 512 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 22 \nu^{19} + 17 \nu^{18} - 44 \nu^{17} - 160 \nu^{16} - 10 \nu^{15} + 407 \nu^{14} + 588 \nu^{13} + \cdots - 16896 ) / 256 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 16 \nu^{19} - \nu^{18} - 64 \nu^{17} - 136 \nu^{16} + 68 \nu^{15} + 409 \nu^{14} + 312 \nu^{13} + \cdots - 19968 ) / 256 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 31 \nu^{19} + 10 \nu^{18} + 132 \nu^{17} + 260 \nu^{16} - 161 \nu^{15} - 770 \nu^{14} + \cdots + 39936 ) / 512 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 35 \nu^{19} + 62 \nu^{18} + 8 \nu^{17} - 200 \nu^{16} - 163 \nu^{15} + 426 \nu^{14} + 1204 \nu^{13} + \cdots - 10240 ) / 256 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 31 \nu^{19} + 64 \nu^{18} + 32 \nu^{17} - 164 \nu^{16} - 199 \nu^{15} + 280 \nu^{14} + 1132 \nu^{13} + \cdots - 5120 ) / 256 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 21 \nu^{19} + 4 \nu^{18} - 80 \nu^{17} - 184 \nu^{16} + 59 \nu^{15} + 532 \nu^{14} + 484 \nu^{13} + \cdots - 24064 ) / 256 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 180 \nu^{19} - 217 \nu^{18} + 214 \nu^{17} + 1228 \nu^{16} + 376 \nu^{15} - 2983 \nu^{14} + \cdots + 110080 ) / 512 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{18} + \beta_{15} + \beta_{14} - \beta_{12} - \beta_{9} - \beta_{5} - \beta_{4} + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{19} - \beta_{16} - \beta_{13} + 2 \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \cdots - 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( - \beta_{19} - \beta_{18} - \beta_{17} - \beta_{12} + \beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{7} + \cdots - 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{19} + \beta_{17} + 2 \beta_{16} - \beta_{13} - 2 \beta_{12} - \beta_{11} + \beta_{10} - 5 \beta_{9} + \cdots - 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 2 \beta_{19} - \beta_{17} - 4 \beta_{16} + \beta_{15} - \beta_{14} - 2 \beta_{13} + 4 \beta_{12} + \cdots - 5 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 2 \beta_{19} - \beta_{18} - 4 \beta_{17} + 2 \beta_{16} - 3 \beta_{15} - 4 \beta_{14} + \beta_{12} + \cdots - 1 \)
|
| \(\nu^{9}\) | \(=\) |
\( 7 \beta_{19} + 5 \beta_{18} + 3 \beta_{17} + 7 \beta_{16} - 2 \beta_{15} + 2 \beta_{13} - 5 \beta_{12} + \cdots + 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( \beta_{19} + 2 \beta_{18} - 5 \beta_{17} - 8 \beta_{16} - 10 \beta_{14} + 5 \beta_{13} + 8 \beta_{12} + \cdots + 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( 10 \beta_{19} + 2 \beta_{18} + 3 \beta_{17} + 8 \beta_{16} + 7 \beta_{15} + 9 \beta_{14} + 10 \beta_{13} + \cdots + 13 \)
|
| \(\nu^{12}\) | \(=\) |
\( - \beta_{18} - 8 \beta_{17} - 4 \beta_{16} - 3 \beta_{15} + 4 \beta_{14} + 10 \beta_{13} + 13 \beta_{12} + \cdots + 11 \)
|
| \(\nu^{13}\) | \(=\) |
\( 21 \beta_{19} + 7 \beta_{18} + 21 \beta_{17} + 7 \beta_{16} + 6 \beta_{15} + 26 \beta_{13} - 19 \beta_{12} + \cdots + 30 \)
|
| \(\nu^{14}\) | \(=\) |
\( - \beta_{19} + 6 \beta_{18} + 5 \beta_{17} - 16 \beta_{16} + 28 \beta_{15} + 38 \beta_{14} + 15 \beta_{13} + \cdots - 54 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 18 \beta_{19} - 22 \beta_{18} + 9 \beta_{17} - 4 \beta_{16} - 15 \beta_{15} + 15 \beta_{14} + \cdots + 35 \)
|
| \(\nu^{16}\) | \(=\) |
\( - 28 \beta_{19} - 35 \beta_{18} + 24 \beta_{17} + 7 \beta_{15} + 28 \beta_{14} - 6 \beta_{13} + \cdots - 23 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 37 \beta_{19} + 121 \beta_{18} + 59 \beta_{17} - 19 \beta_{16} - 86 \beta_{15} - 32 \beta_{14} + \cdots - 134 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 55 \beta_{19} - 118 \beta_{18} - 85 \beta_{17} + 32 \beta_{16} - 44 \beta_{15} - 30 \beta_{14} + \cdots + 86 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 22 \beta_{19} + 30 \beta_{18} + 111 \beta_{17} + 20 \beta_{16} - 105 \beta_{15} - 39 \beta_{14} + \cdots + 101 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(97\) | \(161\) | \(181\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-\beta_{9}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 61.1 |
|
−1.38431 | + | 0.289262i | −0.707107 | − | 0.707107i | 1.83266 | − | 0.800859i | −0.707107 | + | 0.707107i | 1.18340 | + | 0.774320i | − | 2.60796i | −2.30531 | + | 1.63876i | 1.00000i | 0.774320 | − | 1.18340i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.2 | −1.13207 | + | 0.847599i | 0.707107 | + | 0.707107i | 0.563151 | − | 1.91908i | 0.707107 | − | 0.707107i | −1.39984 | − | 0.201149i | − | 4.27253i | 0.989085 | + | 2.64985i | 1.00000i | −0.201149 | + | 1.39984i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.3 | −1.04932 | − | 0.948122i | −0.707107 | − | 0.707107i | 0.202128 | + | 1.98976i | −0.707107 | + | 0.707107i | 0.0715547 | + | 1.41240i | 0.740019i | 1.67444 | − | 2.27953i | 1.00000i | 1.41240 | − | 0.0715547i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.4 | −0.720859 | − | 1.21670i | 0.707107 | + | 0.707107i | −0.960724 | + | 1.75414i | 0.707107 | − | 0.707107i | 0.350613 | − | 1.37006i | 0.0588949i | 2.82681 | − | 0.0955746i | 1.00000i | −1.37006 | − | 0.350613i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.5 | −0.491956 | + | 1.32589i | 0.707107 | + | 0.707107i | −1.51596 | − | 1.30456i | 0.707107 | − | 0.707107i | −1.28541 | + | 0.589679i | 3.46600i | 2.47548 | − | 1.36821i | 1.00000i | 0.589679 | + | 1.28541i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.6 | −0.0861743 | + | 1.41159i | −0.707107 | − | 0.707107i | −1.98515 | − | 0.243285i | −0.707107 | + | 0.707107i | 1.05908 | − | 0.937207i | 2.76462i | 0.514486 | − | 2.78124i | 1.00000i | −0.937207 | − | 1.05908i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.7 | 1.15787 | + | 0.811989i | 0.707107 | + | 0.707107i | 0.681349 | + | 1.88036i | 0.707107 | − | 0.707107i | 0.244579 | + | 1.39290i | − | 2.18060i | −0.737916 | + | 2.73047i | 1.00000i | 1.39290 | − | 0.244579i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.8 | 1.18701 | − | 0.768775i | 0.707107 | + | 0.707107i | 0.817970 | − | 1.82508i | 0.707107 | − | 0.707107i | 1.38295 | + | 0.295735i | 4.92824i | −0.432142 | − | 2.79522i | 1.00000i | 0.295735 | − | 1.38295i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.9 | 1.19834 | + | 0.750988i | −0.707107 | − | 0.707107i | 0.872033 | + | 1.79988i | −0.707107 | + | 0.707107i | −0.316325 | − | 1.37838i | 3.79862i | −0.306697 | + | 2.81175i | 1.00000i | −1.37838 | + | 0.316325i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.10 | 1.32147 | − | 0.503713i | −0.707107 | − | 0.707107i | 1.49255 | − | 1.33128i | −0.707107 | + | 0.707107i | −1.29060 | − | 0.578239i | − | 2.69529i | 1.30176 | − | 2.51106i | 1.00000i | −0.578239 | + | 1.29060i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.1 | −1.38431 | − | 0.289262i | −0.707107 | + | 0.707107i | 1.83266 | + | 0.800859i | −0.707107 | − | 0.707107i | 1.18340 | − | 0.774320i | 2.60796i | −2.30531 | − | 1.63876i | − | 1.00000i | 0.774320 | + | 1.18340i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.2 | −1.13207 | − | 0.847599i | 0.707107 | − | 0.707107i | 0.563151 | + | 1.91908i | 0.707107 | + | 0.707107i | −1.39984 | + | 0.201149i | 4.27253i | 0.989085 | − | 2.64985i | − | 1.00000i | −0.201149 | − | 1.39984i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.3 | −1.04932 | + | 0.948122i | −0.707107 | + | 0.707107i | 0.202128 | − | 1.98976i | −0.707107 | − | 0.707107i | 0.0715547 | − | 1.41240i | − | 0.740019i | 1.67444 | + | 2.27953i | − | 1.00000i | 1.41240 | + | 0.0715547i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.4 | −0.720859 | + | 1.21670i | 0.707107 | − | 0.707107i | −0.960724 | − | 1.75414i | 0.707107 | + | 0.707107i | 0.350613 | + | 1.37006i | − | 0.0588949i | 2.82681 | + | 0.0955746i | − | 1.00000i | −1.37006 | + | 0.350613i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.5 | −0.491956 | − | 1.32589i | 0.707107 | − | 0.707107i | −1.51596 | + | 1.30456i | 0.707107 | + | 0.707107i | −1.28541 | − | 0.589679i | − | 3.46600i | 2.47548 | + | 1.36821i | − | 1.00000i | 0.589679 | − | 1.28541i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.6 | −0.0861743 | − | 1.41159i | −0.707107 | + | 0.707107i | −1.98515 | + | 0.243285i | −0.707107 | − | 0.707107i | 1.05908 | + | 0.937207i | − | 2.76462i | 0.514486 | + | 2.78124i | − | 1.00000i | −0.937207 | + | 1.05908i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.7 | 1.15787 | − | 0.811989i | 0.707107 | − | 0.707107i | 0.681349 | − | 1.88036i | 0.707107 | + | 0.707107i | 0.244579 | − | 1.39290i | 2.18060i | −0.737916 | − | 2.73047i | − | 1.00000i | 1.39290 | + | 0.244579i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.8 | 1.18701 | + | 0.768775i | 0.707107 | − | 0.707107i | 0.817970 | + | 1.82508i | 0.707107 | + | 0.707107i | 1.38295 | − | 0.295735i | − | 4.92824i | −0.432142 | + | 2.79522i | − | 1.00000i | 0.295735 | + | 1.38295i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.9 | 1.19834 | − | 0.750988i | −0.707107 | + | 0.707107i | 0.872033 | − | 1.79988i | −0.707107 | − | 0.707107i | −0.316325 | + | 1.37838i | − | 3.79862i | −0.306697 | − | 2.81175i | − | 1.00000i | −1.37838 | − | 0.316325i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 181.10 | 1.32147 | + | 0.503713i | −0.707107 | + | 0.707107i | 1.49255 | + | 1.33128i | −0.707107 | − | 0.707107i | −1.29060 | + | 0.578239i | 2.69529i | 1.30176 | + | 2.51106i | − | 1.00000i | −0.578239 | − | 1.29060i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 16.e | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 240.2.s.c | ✓ | 20 |
| 3.b | odd | 2 | 1 | 720.2.t.d | 20 | ||
| 4.b | odd | 2 | 1 | 960.2.s.c | 20 | ||
| 8.b | even | 2 | 1 | 1920.2.s.e | 20 | ||
| 8.d | odd | 2 | 1 | 1920.2.s.f | 20 | ||
| 12.b | even | 2 | 1 | 2880.2.t.d | 20 | ||
| 16.e | even | 4 | 1 | inner | 240.2.s.c | ✓ | 20 |
| 16.e | even | 4 | 1 | 1920.2.s.e | 20 | ||
| 16.f | odd | 4 | 1 | 960.2.s.c | 20 | ||
| 16.f | odd | 4 | 1 | 1920.2.s.f | 20 | ||
| 48.i | odd | 4 | 1 | 720.2.t.d | 20 | ||
| 48.k | even | 4 | 1 | 2880.2.t.d | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 240.2.s.c | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 240.2.s.c | ✓ | 20 | 16.e | even | 4 | 1 | inner |
| 720.2.t.d | 20 | 3.b | odd | 2 | 1 | ||
| 720.2.t.d | 20 | 48.i | odd | 4 | 1 | ||
| 960.2.s.c | 20 | 4.b | odd | 2 | 1 | ||
| 960.2.s.c | 20 | 16.f | odd | 4 | 1 | ||
| 1920.2.s.e | 20 | 8.b | even | 2 | 1 | ||
| 1920.2.s.e | 20 | 16.e | even | 4 | 1 | ||
| 1920.2.s.f | 20 | 8.d | odd | 2 | 1 | ||
| 1920.2.s.f | 20 | 16.f | odd | 4 | 1 | ||
| 2880.2.t.d | 20 | 12.b | even | 2 | 1 | ||
| 2880.2.t.d | 20 | 48.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{20} + 96 T_{7}^{18} + 3880 T_{7}^{16} + 86368 T_{7}^{14} + 1161104 T_{7}^{12} + 9697664 T_{7}^{10} + \cdots + 262144 \)
acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - 2 T^{18} + \cdots + 1024 \)
|
| $3$ |
\( (T^{4} + 1)^{5} \)
|
| $5$ |
\( (T^{4} + 1)^{5} \)
|
| $7$ |
\( T^{20} + 96 T^{18} + \cdots + 262144 \)
|
| $11$ |
\( T^{20} - 8 T^{19} + \cdots + 1048576 \)
|
| $13$ |
\( T^{20} + \cdots + 167981940736 \)
|
| $17$ |
\( (T^{10} + 12 T^{9} + \cdots - 20032)^{2} \)
|
| $19$ |
\( T^{20} + \cdots + 19716653056 \)
|
| $23$ |
\( T^{20} + \cdots + 217558810624 \)
|
| $29$ |
\( T^{20} + \cdots + 19723262623744 \)
|
| $31$ |
\( (T^{10} - 204 T^{8} + \cdots - 28698368)^{2} \)
|
| $37$ |
\( T^{20} - 16 T^{19} + \cdots + 18939904 \)
|
| $41$ |
\( T^{20} + \cdots + 3311118843904 \)
|
| $43$ |
\( T^{20} + \cdots + 3288334336 \)
|
| $47$ |
\( (T^{10} - 166 T^{8} + \cdots - 12544)^{2} \)
|
| $53$ |
\( T^{20} + \cdots + 4398046511104 \)
|
| $59$ |
\( T^{20} + \cdots + 16\!\cdots\!04 \)
|
| $61$ |
\( T^{20} + \cdots + 74350019584 \)
|
| $67$ |
\( T^{20} + \cdots + 210453397504 \)
|
| $71$ |
\( T^{20} + \cdots + 84783728164864 \)
|
| $73$ |
\( T^{20} + \cdots + 12\!\cdots\!76 \)
|
| $79$ |
\( (T^{10} - 28 T^{9} + \cdots + 46268416)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 76\!\cdots\!36 \)
|
| $89$ |
\( T^{20} + \cdots + 15\!\cdots\!36 \)
|
| $97$ |
\( (T^{10} - 28 T^{9} + \cdots - 37943296)^{2} \)
|
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