Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [80,2,Mod(21,80)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(80, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("80.21");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 80 = 2^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 80.l (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: | \(0.638803216170\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} + \cdots + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - \nu^{15} - 70 \nu^{14} + 120 \nu^{13} + 144 \nu^{12} + 89 \nu^{11} - 606 \nu^{10} - 968 \nu^{9} + \cdots - 9472 ) / 384 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 51 \nu^{15} - 68 \nu^{14} + 320 \nu^{13} + 176 \nu^{12} - 277 \nu^{11} - 1368 \nu^{10} + \cdots - 25216 ) / 2688 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 2 \nu^{15} + \nu^{14} + 6 \nu^{13} + 3 \nu^{12} - 14 \nu^{11} - 33 \nu^{10} + 14 \nu^{9} + \cdots - 320 ) / 48 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 163 \nu^{15} + 58 \nu^{14} + 376 \nu^{13} + 568 \nu^{12} - 501 \nu^{11} - 2502 \nu^{10} + \cdots - 7296 ) / 2688 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 121 \nu^{15} - 65 \nu^{14} - 264 \nu^{13} - 372 \nu^{12} + 487 \nu^{11} + 1725 \nu^{10} + 44 \nu^{9} + \cdots + 5056 ) / 1344 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 397 \nu^{15} + 502 \nu^{14} + 772 \nu^{13} + 664 \nu^{12} - 2523 \nu^{11} - 5226 \nu^{10} + \cdots - 8832 ) / 2688 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 61 \nu^{15} + 82 \nu^{14} + 124 \nu^{13} + 112 \nu^{12} - 411 \nu^{11} - 870 \nu^{10} + 532 \nu^{9} + \cdots - 1536 ) / 384 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 177 \nu^{15} - 604 \nu^{14} + 240 \nu^{13} + 328 \nu^{12} + 1607 \nu^{11} - 368 \nu^{10} + \cdots - 48256 ) / 896 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 81 \nu^{15} + 184 \nu^{14} + 56 \nu^{13} + 8 \nu^{12} - 679 \nu^{11} - 588 \nu^{10} + 1680 \nu^{9} + \cdots + 7808 ) / 384 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 396 \nu^{15} - 1201 \nu^{14} + 256 \nu^{13} + 460 \nu^{12} + 3508 \nu^{11} + 489 \nu^{10} + \cdots - 83072 ) / 1344 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 827 \nu^{15} - 2780 \nu^{14} + 904 \nu^{13} + 1480 \nu^{12} + 7821 \nu^{11} - 744 \nu^{10} + \cdots - 210048 ) / 2688 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 526 \nu^{15} + 1205 \nu^{14} + 300 \nu^{13} - 24 \nu^{12} - 4258 \nu^{11} - 3477 \nu^{10} + \cdots + 53696 ) / 1344 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 206 \nu^{15} + 633 \nu^{14} - 178 \nu^{13} - 268 \nu^{12} - 1770 \nu^{11} - 57 \nu^{10} + \cdots + 46400 ) / 448 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 1431 \nu^{15} - 3748 \nu^{14} + 136 \nu^{13} + 856 \nu^{12} + 11857 \nu^{11} + 5328 \nu^{10} + \cdots - 219776 ) / 2688 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 2295 \nu^{15} + 6418 \nu^{14} - 580 \nu^{13} - 1936 \nu^{12} - 19969 \nu^{11} - 7086 \nu^{10} + \cdots + 392960 ) / 2688 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{12} - \beta_{11} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{15} - \beta_{13} - \beta_{12} - \beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - \beta_{15} - 2 \beta_{14} - \beta_{13} - \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + \beta_{8} + \cdots + 2 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 3 \beta_{15} + 3 \beta_{14} - \beta_{13} - 2 \beta_{10} - 3 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + \cdots + \beta_1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( \beta_{15} - 5 \beta_{13} - \beta_{12} - 4 \beta_{11} - 4 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} + \cdots - 5 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( \beta_{14} - 2 \beta_{13} + \beta_{12} + 2 \beta_{11} - 6 \beta_{10} - 3 \beta_{9} - 4 \beta_{8} + \cdots - 5 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 4 \beta_{15} + 2 \beta_{14} - 4 \beta_{13} + 7 \beta_{12} - \beta_{11} + 2 \beta_{10} - 10 \beta_{9} + \cdots - 7 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( \beta_{15} + 10 \beta_{14} - 5 \beta_{13} + \beta_{12} + 4 \beta_{11} - 18 \beta_{10} - 2 \beta_{9} + \cdots - 9 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 5 \beta_{15} - 2 \beta_{14} - 7 \beta_{13} + 2 \beta_{12} - \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + \cdots - 16 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 9 \beta_{15} - 3 \beta_{14} + 11 \beta_{13} + 22 \beta_{12} + 30 \beta_{11} + 2 \beta_{10} + 17 \beta_{9} + \cdots + 6 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 19 \beta_{15} + 4 \beta_{14} + 5 \beta_{13} + 19 \beta_{12} + 12 \beta_{11} - 20 \beta_{10} + \cdots + 27 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 22 \beta_{15} - \beta_{14} + 8 \beta_{13} - 19 \beta_{12} + 36 \beta_{11} - 48 \beta_{10} + 57 \beta_{9} + \cdots - 5 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 58 \beta_{15} - 34 \beta_{14} + 26 \beta_{13} + 75 \beta_{12} + 23 \beta_{11} + 38 \beta_{10} + \cdots + 53 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 101 \beta_{15} - 32 \beta_{14} + 97 \beta_{13} + 29 \beta_{12} + 60 \beta_{11} - 112 \beta_{10} + \cdots + 67 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 69 \beta_{15} + 74 \beta_{14} - 51 \beta_{13} - 16 \beta_{12} - 11 \beta_{11} - 62 \beta_{10} + \cdots - 26 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(21\) | \(31\) |
| \(\chi(n)\) | \(1\) | \(\beta_{10}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 21.1 |
|
−1.37735 | + | 0.320793i | −0.720673 | + | 0.720673i | 1.79418 | − | 0.883688i | 0.707107 | + | 0.707107i | 0.761432 | − | 1.22381i | 4.02840i | −2.18774 | + | 1.79271i | 1.96126i | −1.20077 | − | 0.747098i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 21.2 | −1.17275 | + | 0.790349i | 1.37027 | − | 1.37027i | 0.750696 | − | 1.85377i | −0.707107 | − | 0.707107i | −0.523995 | + | 2.68998i | − | 2.73482i | 0.584744 | + | 2.76732i | − | 0.755274i | 1.38812 | + | 0.270400i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 21.3 | −0.376912 | − | 1.36306i | 1.82762 | − | 1.82762i | −1.71587 | + | 1.02751i | −0.707107 | − | 0.707107i | −3.18001 | − | 1.80230i | 4.50961i | 2.04729 | + | 1.95156i | − | 3.68037i | −0.697313 | + | 1.23035i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 21.4 | −0.257150 | + | 1.39064i | −1.66366 | + | 1.66366i | −1.86775 | − | 0.715205i | −0.707107 | − | 0.707107i | −1.88574 | − | 2.74137i | 2.89402i | 1.47488 | − | 2.41345i | − | 2.53555i | 1.16516 | − | 0.801497i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 21.5 | 0.114638 | + | 1.40956i | 1.42313 | − | 1.42313i | −1.97372 | + | 0.323179i | 0.707107 | + | 0.707107i | 2.16913 | + | 1.84284i | 0.690576i | −0.681804 | − | 2.74502i | − | 1.05061i | −0.915648 | + | 1.07777i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 21.6 | 0.562546 | − | 1.29751i | 0.209571 | − | 0.209571i | −1.36708 | − | 1.45982i | 0.707107 | + | 0.707107i | −0.154028 | − | 0.389815i | − | 1.73696i | −2.66319 | + | 0.952595i | 2.91216i | 1.31526 | − | 0.519701i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 21.7 | 1.09971 | + | 0.889181i | −0.120009 | + | 0.120009i | 0.418713 | + | 1.95568i | −0.707107 | − | 0.707107i | −0.238684 | + | 0.0252650i | − | 2.66881i | −1.27849 | + | 2.52299i | 2.97120i | −0.148864 | − | 1.40636i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 21.8 | 1.40727 | − | 0.139945i | −2.32624 | + | 2.32624i | 1.96083 | − | 0.393883i | 0.707107 | + | 0.707107i | −2.94811 | + | 3.59920i | − | 0.982011i | 2.70430 | − | 0.828709i | − | 7.82281i | 1.09405 | + | 0.896135i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.1 | −1.37735 | − | 0.320793i | −0.720673 | − | 0.720673i | 1.79418 | + | 0.883688i | 0.707107 | − | 0.707107i | 0.761432 | + | 1.22381i | − | 4.02840i | −2.18774 | − | 1.79271i | − | 1.96126i | −1.20077 | + | 0.747098i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.2 | −1.17275 | − | 0.790349i | 1.37027 | + | 1.37027i | 0.750696 | + | 1.85377i | −0.707107 | + | 0.707107i | −0.523995 | − | 2.68998i | 2.73482i | 0.584744 | − | 2.76732i | 0.755274i | 1.38812 | − | 0.270400i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.3 | −0.376912 | + | 1.36306i | 1.82762 | + | 1.82762i | −1.71587 | − | 1.02751i | −0.707107 | + | 0.707107i | −3.18001 | + | 1.80230i | − | 4.50961i | 2.04729 | − | 1.95156i | 3.68037i | −0.697313 | − | 1.23035i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.4 | −0.257150 | − | 1.39064i | −1.66366 | − | 1.66366i | −1.86775 | + | 0.715205i | −0.707107 | + | 0.707107i | −1.88574 | + | 2.74137i | − | 2.89402i | 1.47488 | + | 2.41345i | 2.53555i | 1.16516 | + | 0.801497i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.5 | 0.114638 | − | 1.40956i | 1.42313 | + | 1.42313i | −1.97372 | − | 0.323179i | 0.707107 | − | 0.707107i | 2.16913 | − | 1.84284i | − | 0.690576i | −0.681804 | + | 2.74502i | 1.05061i | −0.915648 | − | 1.07777i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.6 | 0.562546 | + | 1.29751i | 0.209571 | + | 0.209571i | −1.36708 | + | 1.45982i | 0.707107 | − | 0.707107i | −0.154028 | + | 0.389815i | 1.73696i | −2.66319 | − | 0.952595i | − | 2.91216i | 1.31526 | + | 0.519701i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.7 | 1.09971 | − | 0.889181i | −0.120009 | − | 0.120009i | 0.418713 | − | 1.95568i | −0.707107 | + | 0.707107i | −0.238684 | − | 0.0252650i | 2.66881i | −1.27849 | − | 2.52299i | − | 2.97120i | −0.148864 | + | 1.40636i | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 61.8 | 1.40727 | + | 0.139945i | −2.32624 | − | 2.32624i | 1.96083 | + | 0.393883i | 0.707107 | − | 0.707107i | −2.94811 | − | 3.59920i | 0.982011i | 2.70430 | + | 0.828709i | 7.82281i | 1.09405 | − | 0.896135i | |||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 80.2.l.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | 720.2.t.c | 16 | ||
| 4.b | odd | 2 | 1 | 320.2.l.a | 16 | ||
| 5.b | even | 2 | 1 | 400.2.l.h | 16 | ||
| 5.c | odd | 4 | 1 | 400.2.q.g | 16 | ||
| 5.c | odd | 4 | 1 | 400.2.q.h | 16 | ||
| 8.b | even | 2 | 1 | 640.2.l.b | 16 | ||
| 8.d | odd | 2 | 1 | 640.2.l.a | 16 | ||
| 12.b | even | 2 | 1 | 2880.2.t.c | 16 | ||
| 16.e | even | 4 | 1 | inner | 80.2.l.a | ✓ | 16 |
| 16.e | even | 4 | 1 | 640.2.l.b | 16 | ||
| 16.f | odd | 4 | 1 | 320.2.l.a | 16 | ||
| 16.f | odd | 4 | 1 | 640.2.l.a | 16 | ||
| 20.d | odd | 2 | 1 | 1600.2.l.i | 16 | ||
| 20.e | even | 4 | 1 | 1600.2.q.g | 16 | ||
| 20.e | even | 4 | 1 | 1600.2.q.h | 16 | ||
| 32.g | even | 8 | 1 | 5120.2.a.s | 8 | ||
| 32.g | even | 8 | 1 | 5120.2.a.v | 8 | ||
| 32.h | odd | 8 | 1 | 5120.2.a.t | 8 | ||
| 32.h | odd | 8 | 1 | 5120.2.a.u | 8 | ||
| 48.i | odd | 4 | 1 | 720.2.t.c | 16 | ||
| 48.k | even | 4 | 1 | 2880.2.t.c | 16 | ||
| 80.i | odd | 4 | 1 | 400.2.q.g | 16 | ||
| 80.j | even | 4 | 1 | 1600.2.q.g | 16 | ||
| 80.k | odd | 4 | 1 | 1600.2.l.i | 16 | ||
| 80.q | even | 4 | 1 | 400.2.l.h | 16 | ||
| 80.s | even | 4 | 1 | 1600.2.q.h | 16 | ||
| 80.t | odd | 4 | 1 | 400.2.q.h | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 80.2.l.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 80.2.l.a | ✓ | 16 | 16.e | even | 4 | 1 | inner |
| 320.2.l.a | 16 | 4.b | odd | 2 | 1 | ||
| 320.2.l.a | 16 | 16.f | odd | 4 | 1 | ||
| 400.2.l.h | 16 | 5.b | even | 2 | 1 | ||
| 400.2.l.h | 16 | 80.q | even | 4 | 1 | ||
| 400.2.q.g | 16 | 5.c | odd | 4 | 1 | ||
| 400.2.q.g | 16 | 80.i | odd | 4 | 1 | ||
| 400.2.q.h | 16 | 5.c | odd | 4 | 1 | ||
| 400.2.q.h | 16 | 80.t | odd | 4 | 1 | ||
| 640.2.l.a | 16 | 8.d | odd | 2 | 1 | ||
| 640.2.l.a | 16 | 16.f | odd | 4 | 1 | ||
| 640.2.l.b | 16 | 8.b | even | 2 | 1 | ||
| 640.2.l.b | 16 | 16.e | even | 4 | 1 | ||
| 720.2.t.c | 16 | 3.b | odd | 2 | 1 | ||
| 720.2.t.c | 16 | 48.i | odd | 4 | 1 | ||
| 1600.2.l.i | 16 | 20.d | odd | 2 | 1 | ||
| 1600.2.l.i | 16 | 80.k | odd | 4 | 1 | ||
| 1600.2.q.g | 16 | 20.e | even | 4 | 1 | ||
| 1600.2.q.g | 16 | 80.j | even | 4 | 1 | ||
| 1600.2.q.h | 16 | 20.e | even | 4 | 1 | ||
| 1600.2.q.h | 16 | 80.s | even | 4 | 1 | ||
| 2880.2.t.c | 16 | 12.b | even | 2 | 1 | ||
| 2880.2.t.c | 16 | 48.k | even | 4 | 1 | ||
| 5120.2.a.s | 8 | 32.g | even | 8 | 1 | ||
| 5120.2.a.t | 8 | 32.h | odd | 8 | 1 | ||
| 5120.2.a.u | 8 | 32.h | odd | 8 | 1 | ||
| 5120.2.a.v | 8 | 32.g | even | 8 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(80, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + 2 T^{14} + \cdots + 256 \)
|
| $3$ |
\( T^{16} - 8 T^{13} + \cdots + 16 \)
|
| $5$ |
\( (T^{4} + 1)^{4} \)
|
| $7$ |
\( T^{16} + 64 T^{14} + \cdots + 204304 \)
|
| $11$ |
\( T^{16} + 8 T^{15} + \cdots + 1290496 \)
|
| $13$ |
\( T^{16} + 128 T^{13} + \cdots + 20647936 \)
|
| $17$ |
\( (T^{8} - 72 T^{6} + \cdots + 13888)^{2} \)
|
| $19$ |
\( T^{16} + 8 T^{15} + \cdots + 614656 \)
|
| $23$ |
\( T^{16} + 128 T^{14} + \cdots + 1731856 \)
|
| $29$ |
\( T^{16} + \cdots + 3017085184 \)
|
| $31$ |
\( (T^{8} - 96 T^{6} + \cdots - 20224)^{2} \)
|
| $37$ |
\( T^{16} + 16 T^{15} + \cdots + 18939904 \)
|
| $41$ |
\( T^{16} + \cdots + 110660014336 \)
|
| $43$ |
\( T^{16} - 8 T^{15} + \cdots + 53640976 \)
|
| $47$ |
\( (T^{8} + 20 T^{7} + \cdots + 575044)^{2} \)
|
| $53$ |
\( T^{16} + \cdots + 383725735936 \)
|
| $59$ |
\( T^{16} + \cdots + 12227051776 \)
|
| $61$ |
\( T^{16} + \cdots + 1393986371584 \)
|
| $67$ |
\( T^{16} + \cdots + 46120451769616 \)
|
| $71$ |
\( T^{16} + \cdots + 3333516427264 \)
|
| $73$ |
\( T^{16} + \cdots + 15847788544 \)
|
| $79$ |
\( (T^{8} - 8 T^{7} + \cdots + 4352)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 2050640656 \)
|
| $89$ |
\( T^{16} + \cdots + 684153962496 \)
|
| $97$ |
\( (T^{8} - 440 T^{6} + \cdots - 8549312)^{2} \)
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