Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [392,3,Mod(67,392)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(392, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("392.67");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 392 = 2^{3} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 392.k (of order \(6\), 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: | \(10.6812263629\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - x^{15} + 3 x^{14} + 6 x^{13} - 22 x^{12} + 44 x^{11} - 20 x^{10} - 112 x^{9} + 368 x^{8} + \cdots + 65536 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{14} \) |
| Twist minimal: | no (minimal twist has level 56) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{16} - x^{15} + 3 x^{14} + 6 x^{13} - 22 x^{12} + 44 x^{11} - 20 x^{10} - 112 x^{9} + 368 x^{8} + \cdots + 65536 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 3 \nu^{15} - 33 \nu^{14} - 293 \nu^{13} + 642 \nu^{12} - 918 \nu^{11} - 1068 \nu^{10} + \cdots - 2064384 ) / 802816 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 9 \nu^{15} + 71 \nu^{14} - 165 \nu^{13} + 246 \nu^{12} + 234 \nu^{11} - 852 \nu^{10} + \cdots + 753664 ) / 802816 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 10 \nu^{15} + 19 \nu^{14} + 29 \nu^{13} + 159 \nu^{12} - 34 \nu^{11} - 326 \nu^{10} + 1308 \nu^{9} + \cdots + 212992 ) / 200704 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 21 \nu^{15} - 387 \nu^{14} - 151 \nu^{13} + 1066 \nu^{12} - 3730 \nu^{11} + 5588 \nu^{10} + \cdots - 7618560 ) / 802816 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 57 \nu^{15} - 71 \nu^{14} - 251 \nu^{13} + 154 \nu^{12} - 682 \nu^{11} + 948 \nu^{10} + \cdots - 3112960 ) / 802816 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{15} - \nu^{14} + 3 \nu^{13} + 6 \nu^{12} - 22 \nu^{11} + 44 \nu^{10} - 20 \nu^{9} + \cdots - 16384 ) / 16384 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - \nu^{15} - 103 \nu^{14} + 53 \nu^{13} + 178 \nu^{12} - 778 \nu^{11} + 1860 \nu^{10} + \cdots - 1753088 ) / 200704 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 33 \nu^{15} + 3 \nu^{14} + 3 \nu^{13} - 168 \nu^{12} + 58 \nu^{11} - 40 \nu^{10} - 2372 \nu^{9} + \cdots - 770048 ) / 401408 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 43 \nu^{15} - 65 \nu^{14} - 85 \nu^{13} + 106 \nu^{12} - 454 \nu^{11} + 1012 \nu^{10} + \cdots - 2277376 ) / 401408 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 65 \nu^{15} + 313 \nu^{14} + 53 \nu^{13} - 942 \nu^{12} + 3158 \nu^{11} - 6204 \nu^{10} + \cdots + 4882432 ) / 802816 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 25 \nu^{15} - 14 \nu^{14} - 20 \nu^{13} - 29 \nu^{12} - 56 \nu^{11} - 214 \nu^{10} - 8 \nu^{9} + \cdots + 606208 ) / 200704 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 63 \nu^{15} + 69 \nu^{14} - 123 \nu^{13} + 208 \nu^{12} + 102 \nu^{11} - 936 \nu^{10} + \cdots - 1556480 ) / 401408 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 41 \nu^{15} + 271 \nu^{14} - 337 \nu^{13} - 436 \nu^{12} + 1338 \nu^{11} - 3664 \nu^{10} + \cdots + 1671168 ) / 401408 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 37 \nu^{15} + 391 \nu^{14} - 313 \nu^{13} - 368 \nu^{12} + 3042 \nu^{11} - 5272 \nu^{10} + \cdots + 4939776 ) / 401408 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 23 \nu^{15} + 4 \nu^{14} + 92 \nu^{13} - 337 \nu^{12} + 642 \nu^{11} - 46 \nu^{10} + \cdots + 436224 ) / 100352 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{11} + \beta_{9} + \beta_{8} + \beta_{3} + \beta_{2} - \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{13} + 3\beta_{12} + \beta_{11} - 3\beta_{9} + \beta_{6} - 3\beta_{3} + \beta_{2} - 1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 4 \beta_{15} - \beta_{13} - \beta_{12} + 4 \beta_{10} + 3 \beta_{8} + 4 \beta_{7} - 11 \beta_{6} + \cdots - 9 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 8 \beta_{14} - \beta_{11} + 4 \beta_{10} + 3 \beta_{9} + 3 \beta_{8} + 4 \beta_{5} - 8 \beta_{4} + \cdots - 3 \beta_1 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4 \beta_{15} - 3 \beta_{13} - 11 \beta_{12} + 3 \beta_{11} + 39 \beta_{9} - 28 \beta_{7} + 31 \beta_{6} + \cdots - 3 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 4 \beta_{15} + 8 \beta_{14} - 15 \beta_{13} + 9 \beta_{12} - 44 \beta_{10} + 29 \beta_{8} - 44 \beta_{7} + \cdots + 1 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 8 \beta_{14} - 3 \beta_{11} - 12 \beta_{10} + 25 \beta_{9} + 25 \beta_{8} - 92 \beta_{5} + \cdots + 31 \beta_1 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 52 \beta_{15} - 41 \beta_{13} + 79 \beta_{12} + 41 \beta_{11} + 21 \beta_{9} + 92 \beta_{7} + \cdots + 103 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 124 \beta_{15} - 104 \beta_{14} + 75 \beta_{13} + 259 \beta_{12} - 308 \beta_{10} - 497 \beta_{8} + \cdots + 155 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 248 \beta_{14} - 177 \beta_{11} - 548 \beta_{10} - 61 \beta_{9} - 61 \beta_{8} + 140 \beta_{5} + \cdots - 1115 \beta_1 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 804 \beta_{15} - 227 \beta_{13} - 411 \beta_{12} + 227 \beta_{11} - 57 \beta_{9} + 788 \beta_{7} + \cdots - 6259 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 300 \beta_{15} + 1608 \beta_{14} - 1959 \beta_{13} + 257 \beta_{12} + 2436 \beta_{10} + 69 \beta_{8} + \cdots + 1673 ) / 4 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 600 \beta_{14} - 219 \beta_{11} + 948 \beta_{10} + 1185 \beta_{9} + 1185 \beta_{8} + \cdots - 2921 \beta_1 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 1164 \beta_{15} + 4479 \beta_{13} + 3351 \beta_{12} - 4479 \beta_{11} - 1203 \beta_{9} - 4260 \beta_{7} + \cdots - 3377 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 2148 \beta_{15} + 2328 \beta_{14} + 979 \beta_{13} - 16757 \beta_{12} - 3540 \beta_{10} + \cdots - 115997 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(295\) | \(297\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
−1.94657 | − | 0.459219i | −2.61182 | + | 4.52380i | 3.57824 | + | 1.78780i | 5.42814 | − | 3.13394i | 7.16148 | − | 7.60647i | 0 | −6.14428 | − | 5.12327i | −9.14316 | − | 15.8364i | −12.0054 | + | 3.60771i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.2 | −1.56937 | − | 1.23978i | 0.0487183 | − | 0.0843825i | 0.925871 | + | 3.89137i | −3.00119 | + | 1.73274i | −0.181073 | + | 0.0720276i | 0 | 3.37142 | − | 7.25490i | 4.49525 | + | 7.78601i | 6.85820 | + | 1.00151i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.3 | −0.944308 | + | 1.76303i | −1.72064 | + | 2.98023i | −2.21656 | − | 3.32969i | −4.22869 | + | 2.44143i | −3.62943 | − | 5.84780i | 0 | 7.96347 | − | 0.763618i | −1.42120 | − | 2.46158i | −0.311142 | − | 9.76078i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.4 | −0.288997 | − | 1.97901i | 0.0487183 | − | 0.0843825i | −3.83296 | + | 1.14386i | 3.00119 | − | 1.73274i | −0.181073 | − | 0.0720276i | 0 | 3.37142 | + | 7.25490i | 4.49525 | + | 7.78601i | −4.29644 | − | 5.43862i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.5 | −0.109554 | + | 1.99700i | 2.28374 | − | 3.95555i | −3.97600 | − | 0.437557i | −4.96451 | + | 2.86626i | 7.64902 | + | 4.99396i | 0 | 1.30939 | − | 7.89212i | −5.93090 | − | 10.2726i | −5.18003 | − | 10.2281i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.6 | 0.575587 | − | 1.91538i | −2.61182 | + | 4.52380i | −3.33740 | − | 2.20494i | −5.42814 | + | 3.13394i | 7.16148 | + | 7.60647i | 0 | −6.14428 | + | 5.12327i | −9.14316 | − | 15.8364i | 2.87833 | + | 12.2008i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.7 | 1.78423 | + | 0.903622i | 2.28374 | − | 3.95555i | 2.36693 | + | 3.22453i | 4.96451 | − | 2.86626i | 7.64902 | − | 4.99396i | 0 | 1.30939 | + | 7.89212i | −5.93090 | − | 10.2726i | 11.4478 | − | 0.628019i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.8 | 1.99898 | + | 0.0637211i | −1.72064 | + | 2.98023i | 3.99188 | + | 0.254755i | 4.22869 | − | 2.44143i | −3.62943 | + | 5.84780i | 0 | 7.96347 | + | 0.763618i | −1.42120 | − | 2.46158i | 8.60865 | − | 4.61093i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 275.1 | −1.94657 | + | 0.459219i | −2.61182 | − | 4.52380i | 3.57824 | − | 1.78780i | 5.42814 | + | 3.13394i | 7.16148 | + | 7.60647i | 0 | −6.14428 | + | 5.12327i | −9.14316 | + | 15.8364i | −12.0054 | − | 3.60771i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 275.2 | −1.56937 | + | 1.23978i | 0.0487183 | + | 0.0843825i | 0.925871 | − | 3.89137i | −3.00119 | − | 1.73274i | −0.181073 | − | 0.0720276i | 0 | 3.37142 | + | 7.25490i | 4.49525 | − | 7.78601i | 6.85820 | − | 1.00151i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 275.3 | −0.944308 | − | 1.76303i | −1.72064 | − | 2.98023i | −2.21656 | + | 3.32969i | −4.22869 | − | 2.44143i | −3.62943 | + | 5.84780i | 0 | 7.96347 | + | 0.763618i | −1.42120 | + | 2.46158i | −0.311142 | + | 9.76078i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 275.4 | −0.288997 | + | 1.97901i | 0.0487183 | + | 0.0843825i | −3.83296 | − | 1.14386i | 3.00119 | + | 1.73274i | −0.181073 | + | 0.0720276i | 0 | 3.37142 | − | 7.25490i | 4.49525 | − | 7.78601i | −4.29644 | + | 5.43862i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 275.5 | −0.109554 | − | 1.99700i | 2.28374 | + | 3.95555i | −3.97600 | + | 0.437557i | −4.96451 | − | 2.86626i | 7.64902 | − | 4.99396i | 0 | 1.30939 | + | 7.89212i | −5.93090 | + | 10.2726i | −5.18003 | + | 10.2281i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 275.6 | 0.575587 | + | 1.91538i | −2.61182 | − | 4.52380i | −3.33740 | + | 2.20494i | −5.42814 | − | 3.13394i | 7.16148 | − | 7.60647i | 0 | −6.14428 | − | 5.12327i | −9.14316 | + | 15.8364i | 2.87833 | − | 12.2008i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 275.7 | 1.78423 | − | 0.903622i | 2.28374 | + | 3.95555i | 2.36693 | − | 3.22453i | 4.96451 | + | 2.86626i | 7.64902 | + | 4.99396i | 0 | 1.30939 | − | 7.89212i | −5.93090 | + | 10.2726i | 11.4478 | + | 0.628019i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 275.8 | 1.99898 | − | 0.0637211i | −1.72064 | − | 2.98023i | 3.99188 | − | 0.254755i | 4.22869 | + | 2.44143i | −3.62943 | − | 5.84780i | 0 | 7.96347 | − | 0.763618i | −1.42120 | + | 2.46158i | 8.60865 | + | 4.61093i | |||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
| 56.k | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 392.3.k.n | 16 | |
| 7.b | odd | 2 | 1 | 392.3.k.o | 16 | ||
| 7.c | even | 3 | 1 | 392.3.g.m | 8 | ||
| 7.c | even | 3 | 1 | inner | 392.3.k.n | 16 | |
| 7.d | odd | 6 | 1 | 56.3.g.b | ✓ | 8 | |
| 7.d | odd | 6 | 1 | 392.3.k.o | 16 | ||
| 8.d | odd | 2 | 1 | inner | 392.3.k.n | 16 | |
| 21.g | even | 6 | 1 | 504.3.g.b | 8 | ||
| 28.f | even | 6 | 1 | 224.3.g.b | 8 | ||
| 28.g | odd | 6 | 1 | 1568.3.g.m | 8 | ||
| 56.e | even | 2 | 1 | 392.3.k.o | 16 | ||
| 56.j | odd | 6 | 1 | 224.3.g.b | 8 | ||
| 56.k | odd | 6 | 1 | 392.3.g.m | 8 | ||
| 56.k | odd | 6 | 1 | inner | 392.3.k.n | 16 | |
| 56.m | even | 6 | 1 | 56.3.g.b | ✓ | 8 | |
| 56.m | even | 6 | 1 | 392.3.k.o | 16 | ||
| 56.p | even | 6 | 1 | 1568.3.g.m | 8 | ||
| 84.j | odd | 6 | 1 | 2016.3.g.b | 8 | ||
| 112.v | even | 12 | 2 | 1792.3.d.j | 16 | ||
| 112.x | odd | 12 | 2 | 1792.3.d.j | 16 | ||
| 168.ba | even | 6 | 1 | 2016.3.g.b | 8 | ||
| 168.be | odd | 6 | 1 | 504.3.g.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.3.g.b | ✓ | 8 | 7.d | odd | 6 | 1 | |
| 56.3.g.b | ✓ | 8 | 56.m | even | 6 | 1 | |
| 224.3.g.b | 8 | 28.f | even | 6 | 1 | ||
| 224.3.g.b | 8 | 56.j | odd | 6 | 1 | ||
| 392.3.g.m | 8 | 7.c | even | 3 | 1 | ||
| 392.3.g.m | 8 | 56.k | odd | 6 | 1 | ||
| 392.3.k.n | 16 | 1.a | even | 1 | 1 | trivial | |
| 392.3.k.n | 16 | 7.c | even | 3 | 1 | inner | |
| 392.3.k.n | 16 | 8.d | odd | 2 | 1 | inner | |
| 392.3.k.n | 16 | 56.k | odd | 6 | 1 | inner | |
| 392.3.k.o | 16 | 7.b | odd | 2 | 1 | ||
| 392.3.k.o | 16 | 7.d | odd | 6 | 1 | ||
| 392.3.k.o | 16 | 56.e | even | 2 | 1 | ||
| 392.3.k.o | 16 | 56.m | even | 6 | 1 | ||
| 504.3.g.b | 8 | 21.g | even | 6 | 1 | ||
| 504.3.g.b | 8 | 168.be | odd | 6 | 1 | ||
| 1568.3.g.m | 8 | 28.g | odd | 6 | 1 | ||
| 1568.3.g.m | 8 | 56.p | even | 6 | 1 | ||
| 1792.3.d.j | 16 | 112.v | even | 12 | 2 | ||
| 1792.3.d.j | 16 | 112.x | odd | 12 | 2 | ||
| 2016.3.g.b | 8 | 84.j | odd | 6 | 1 | ||
| 2016.3.g.b | 8 | 168.ba | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\):
|
\( T_{3}^{8} + 4T_{3}^{7} + 38T_{3}^{6} + 72T_{3}^{5} + 796T_{3}^{4} + 1696T_{3}^{3} + 6576T_{3}^{2} - 640T_{3} + 64 \)
|
|
\( T_{5}^{16} - 108 T_{5}^{14} + 7500 T_{5}^{12} - 315824 T_{5}^{10} + 9739280 T_{5}^{8} + \cdots + 136651472896 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + T^{15} + \cdots + 65536 \)
|
| $3$ |
\( (T^{8} + 4 T^{7} + 38 T^{6} + \cdots + 64)^{2} \)
|
| $5$ |
\( T^{16} + \cdots + 136651472896 \)
|
| $7$ |
\( T^{16} \)
|
| $11$ |
\( (T^{8} - 16 T^{7} + \cdots + 746496)^{2} \)
|
| $13$ |
\( (T^{8} + 908 T^{6} + \cdots + 133448704)^{2} \)
|
| $17$ |
\( (T^{8} + 40 T^{7} + \cdots + 565504)^{2} \)
|
| $19$ |
\( (T^{8} - 28 T^{7} + \cdots + 1658944)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 24\!\cdots\!56 \)
|
| $29$ |
\( (T^{8} + 3344 T^{6} + \cdots + 13389266944)^{2} \)
|
| $31$ |
\( T^{16} + \cdots + 38\!\cdots\!76 \)
|
| $37$ |
\( T^{16} + \cdots + 30\!\cdots\!16 \)
|
| $41$ |
\( (T^{4} + 64 T^{3} + \cdots + 837776)^{4} \)
|
| $43$ |
\( (T^{4} - 2716 T^{2} + \cdots - 217952)^{4} \)
|
| $47$ |
\( T^{16} + \cdots + 56\!\cdots\!76 \)
|
| $53$ |
\( T^{16} + \cdots + 281474976710656 \)
|
| $59$ |
\( (T^{8} + \cdots + 126626768156736)^{2} \)
|
| $61$ |
\( T^{16} + \cdots + 10\!\cdots\!36 \)
|
| $67$ |
\( (T^{8} + 152 T^{7} + \cdots + 626959908864)^{2} \)
|
| $71$ |
\( (T^{8} + \cdots + 221437256269824)^{2} \)
|
| $73$ |
\( (T^{8} + \cdots + 2981506703616)^{2} \)
|
| $79$ |
\( T^{16} + \cdots + 79\!\cdots\!76 \)
|
| $83$ |
\( (T^{4} + 36 T^{3} + \cdots + 3180872)^{4} \)
|
| $89$ |
\( (T^{8} + 256 T^{7} + \cdots + 382834237696)^{2} \)
|
| $97$ |
\( (T^{4} + 32 T^{3} + \cdots + 9539216)^{4} \)
|
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