Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [416,2,Mod(47,416)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(416, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 2, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("416.47");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 416 = 2^{5} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 416.u (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: | \(3.32177672409\) |
| 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: |
\( x^{20} - 2 x^{19} + 2 x^{18} - 7 x^{16} + 14 x^{15} - 14 x^{14} + 8 x^{13} + 16 x^{12} - 40 x^{11} + \cdots + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{16} \) |
| Twist minimal: | no (minimal twist has level 104) |
| 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
\( x^{20} - 2 x^{19} + 2 x^{18} - 7 x^{16} + 14 x^{15} - 14 x^{14} + 8 x^{13} + 16 x^{12} - 40 x^{11} + \cdots + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 3 \nu^{19} + 20 \nu^{18} + 58 \nu^{17} - 316 \nu^{16} + 125 \nu^{15} - 124 \nu^{14} + \cdots + 38912 ) / 31232 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 3 \nu^{19} - 42 \nu^{18} - 42 \nu^{17} - 24 \nu^{16} - 147 \nu^{15} + 774 \nu^{14} - 186 \nu^{13} + \cdots - 24064 ) / 31232 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 29 \nu^{19} + 112 \nu^{18} - 212 \nu^{17} - 696 \nu^{16} + 1371 \nu^{15} - 1280 \nu^{14} + \cdots + 167936 ) / 62464 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 66 \nu^{19} + 81 \nu^{18} + 72 \nu^{17} - 118 \nu^{15} + 361 \nu^{14} - 232 \nu^{13} - 912 \nu^{12} + \cdots - 27136 ) / 62464 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 34 \nu^{19} + 105 \nu^{18} - 512 \nu^{17} + 24 \nu^{16} - 182 \nu^{15} - 927 \nu^{14} + \cdots + 186880 ) / 62464 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 65 \nu^{19} + 183 \nu^{18} - 68 \nu^{17} + 208 \nu^{16} - 641 \nu^{15} - 273 \nu^{14} + \cdots + 48640 ) / 62464 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 33 \nu^{19} + 111 \nu^{18} + 108 \nu^{17} - 196 \nu^{16} + 31 \nu^{15} - 777 \nu^{14} + \cdots - 33792 ) / 31232 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 97 \nu^{19} + 118 \nu^{18} - 158 \nu^{17} + 112 \nu^{16} + 463 \nu^{15} - 634 \nu^{14} + \cdots + 29696 ) / 31232 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 72 \nu^{19} + 127 \nu^{18} + 8 \nu^{17} - 156 \nu^{16} + 192 \nu^{15} - 681 \nu^{14} + \cdots + 34816 ) / 31232 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 87 \nu^{19} - 246 \nu^{18} + 364 \nu^{17} - 280 \nu^{16} - 129 \nu^{15} + 2442 \nu^{14} + \cdots - 82944 ) / 62464 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 7 \nu^{19} - 67 \nu^{18} + 68 \nu^{17} + 46 \nu^{16} - 129 \nu^{15} + 269 \nu^{14} - 220 \nu^{13} + \cdots - 17920 ) / 15616 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 57 \nu^{19} + 79 \nu^{18} - 84 \nu^{17} + 26 \nu^{16} + 535 \nu^{15} - 641 \nu^{14} + 124 \nu^{13} + \cdots + 7424 ) / 15616 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 73 \nu^{19} + 80 \nu^{18} - 134 \nu^{17} - 44 \nu^{16} + 439 \nu^{15} - 496 \nu^{14} + \cdots - 31744 ) / 15616 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 81 \nu^{19} - 13 \nu^{18} + 68 \nu^{17} - 166 \nu^{16} + 335 \nu^{15} - 61 \nu^{14} + \cdots - 28928 ) / 15616 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 193 \nu^{19} + 359 \nu^{18} - 260 \nu^{17} - 120 \nu^{16} + 1343 \nu^{15} - 2113 \nu^{14} + \cdots + 133632 ) / 62464 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 47 \nu^{19} - 48 \nu^{18} + 74 \nu^{17} - 132 \nu^{16} + 137 \nu^{15} + 48 \nu^{14} - 230 \nu^{13} + \cdots + 8192 ) / 15616 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 14 \nu^{19} - 315 \nu^{18} + 104 \nu^{17} + 720 \nu^{16} - 358 \nu^{15} + 125 \nu^{14} + \cdots - 6656 ) / 62464 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 179 \nu^{19} + 541 \nu^{18} - 732 \nu^{17} - 312 \nu^{16} + 2125 \nu^{15} - 3035 \nu^{14} + \cdots + 259584 ) / 62464 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 237 \nu^{19} + 342 \nu^{18} + 98 \nu^{17} - 432 \nu^{16} + 1075 \nu^{15} - 1562 \nu^{14} + \cdots + 66560 ) / 31232 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{19} + \beta_{18} - 2\beta_{15} - \beta_{10} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{3} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{18} + \beta_{17} + \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + 2 \beta_{10} + \cdots - \beta_1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{19} + \beta_{17} - 2\beta_{12} - 2\beta_{11} - \beta_{10} - 3\beta_{9} + \beta_{8} - \beta_{5} + \beta_{3} - 2 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - \beta_{19} + \beta_{18} - \beta_{17} + 3 \beta_{16} + \beta_{15} - \beta_{14} + \beta_{12} - 3 \beta_{9} + \cdots + 5 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( \beta_{19} - \beta_{18} + 4 \beta_{15} - 2 \beta_{14} + 2 \beta_{11} - 3 \beta_{10} - \beta_{8} + \cdots - 2 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 3 \beta_{18} - 3 \beta_{17} + 9 \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} + \cdots - 11 \beta_1 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - \beta_{19} + 3 \beta_{17} + 4 \beta_{16} - 4 \beta_{13} - 2 \beta_{12} - 10 \beta_{11} - 7 \beta_{10} + \cdots - 10 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 7 \beta_{19} + 3 \beta_{18} - 3 \beta_{17} + 5 \beta_{16} + 11 \beta_{15} + 5 \beta_{14} - 5 \beta_{12} + \cdots - 5 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 9 \beta_{19} + 9 \beta_{18} - 16 \beta_{16} + 12 \beta_{15} + 18 \beta_{14} - 16 \beta_{13} + \cdots - 14 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 5 \beta_{18} - 5 \beta_{17} - 9 \beta_{15} + \beta_{14} - 7 \beta_{13} + \beta_{12} + 9 \beta_{11} + \cdots - 13 \beta_1 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( \beta_{19} - 11 \beta_{17} - 12 \beta_{16} + 12 \beta_{13} - 14 \beta_{12} + 66 \beta_{11} - 33 \beta_{10} + \cdots + 66 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - \beta_{19} + 45 \beta_{18} - 45 \beta_{17} + 3 \beta_{16} - 67 \beta_{15} + 3 \beta_{14} - 3 \beta_{12} + \cdots + 29 ) / 4 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 9 \beta_{19} + 79 \beta_{18} - 8 \beta_{16} - 28 \beta_{15} - 18 \beta_{14} - 8 \beta_{13} + 170 \beta_{11} + \cdots - 170 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 27 \beta_{18} - 27 \beta_{17} + 33 \beta_{15} - 41 \beta_{14} + 15 \beta_{13} - 41 \beta_{12} + \cdots + 37 \beta_1 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 31 \beta_{19} - 93 \beta_{17} + 36 \beta_{16} - 36 \beta_{13} + 110 \beta_{12} + 150 \beta_{11} + \cdots + 150 ) / 4 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 9 \beta_{19} + 19 \beta_{18} - 19 \beta_{17} + 85 \beta_{16} - 53 \beta_{15} - 187 \beta_{14} + \cdots + 235 ) / 4 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 247 \beta_{19} - 135 \beta_{18} - 96 \beta_{16} + 268 \beta_{15} - 78 \beta_{14} - 96 \beta_{13} + \cdots - 382 ) / 4 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 5 \beta_{18} - 5 \beta_{17} + 7 \beta_{15} - 47 \beta_{14} - 151 \beta_{13} - 47 \beta_{12} + \cdots - 445 \beta_1 ) / 4 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 127 \beta_{19} + 197 \beta_{17} + 68 \beta_{16} - 68 \beta_{13} - 14 \beta_{12} - 430 \beta_{11} + \cdots - 430 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).
| \(n\) | \(261\) | \(287\) | \(353\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-\beta_{11}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 47.1 |
|
0 | −2.19371 | 0 | −0.274863 | + | 0.274863i | 0 | −0.352378 | − | 0.352378i | 0 | 1.81237 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.2 | 0 | −2.19371 | 0 | 0.274863 | − | 0.274863i | 0 | 0.352378 | + | 0.352378i | 0 | 1.81237 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.3 | 0 | −1.38809 | 0 | −2.40872 | + | 2.40872i | 0 | 0.127019 | + | 0.127019i | 0 | −1.07320 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.4 | 0 | −1.38809 | 0 | 2.40872 | − | 2.40872i | 0 | −0.127019 | − | 0.127019i | 0 | −1.07320 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.5 | 0 | −0.238218 | 0 | −0.328612 | + | 0.328612i | 0 | 2.68064 | + | 2.68064i | 0 | −2.94325 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.6 | 0 | −0.238218 | 0 | 0.328612 | − | 0.328612i | 0 | −2.68064 | − | 2.68064i | 0 | −2.94325 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.7 | 0 | 1.86790 | 0 | −1.55756 | + | 1.55756i | 0 | 1.24165 | + | 1.24165i | 0 | 0.489042 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.8 | 0 | 1.86790 | 0 | 1.55756 | − | 1.55756i | 0 | −1.24165 | − | 1.24165i | 0 | 0.489042 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.9 | 0 | 2.95212 | 0 | −1.04334 | + | 1.04334i | 0 | 2.37322 | + | 2.37322i | 0 | 5.71504 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 47.10 | 0 | 2.95212 | 0 | 1.04334 | − | 1.04334i | 0 | −2.37322 | − | 2.37322i | 0 | 5.71504 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.1 | 0 | −2.19371 | 0 | −0.274863 | − | 0.274863i | 0 | −0.352378 | + | 0.352378i | 0 | 1.81237 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.2 | 0 | −2.19371 | 0 | 0.274863 | + | 0.274863i | 0 | 0.352378 | − | 0.352378i | 0 | 1.81237 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.3 | 0 | −1.38809 | 0 | −2.40872 | − | 2.40872i | 0 | 0.127019 | − | 0.127019i | 0 | −1.07320 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.4 | 0 | −1.38809 | 0 | 2.40872 | + | 2.40872i | 0 | −0.127019 | + | 0.127019i | 0 | −1.07320 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.5 | 0 | −0.238218 | 0 | −0.328612 | − | 0.328612i | 0 | 2.68064 | − | 2.68064i | 0 | −2.94325 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.6 | 0 | −0.238218 | 0 | 0.328612 | + | 0.328612i | 0 | −2.68064 | + | 2.68064i | 0 | −2.94325 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.7 | 0 | 1.86790 | 0 | −1.55756 | − | 1.55756i | 0 | 1.24165 | − | 1.24165i | 0 | 0.489042 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.8 | 0 | 1.86790 | 0 | 1.55756 | + | 1.55756i | 0 | −1.24165 | + | 1.24165i | 0 | 0.489042 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.9 | 0 | 2.95212 | 0 | −1.04334 | − | 1.04334i | 0 | 2.37322 | − | 2.37322i | 0 | 5.71504 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 239.10 | 0 | 2.95212 | 0 | 1.04334 | + | 1.04334i | 0 | −2.37322 | + | 2.37322i | 0 | 5.71504 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
| 13.d | odd | 4 | 1 | inner |
| 104.m | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 416.2.u.b | 20 | |
| 4.b | odd | 2 | 1 | 104.2.m.b | ✓ | 20 | |
| 8.b | even | 2 | 1 | 104.2.m.b | ✓ | 20 | |
| 8.d | odd | 2 | 1 | inner | 416.2.u.b | 20 | |
| 12.b | even | 2 | 1 | 936.2.w.h | 20 | ||
| 13.d | odd | 4 | 1 | inner | 416.2.u.b | 20 | |
| 24.h | odd | 2 | 1 | 936.2.w.h | 20 | ||
| 52.f | even | 4 | 1 | 104.2.m.b | ✓ | 20 | |
| 104.j | odd | 4 | 1 | 104.2.m.b | ✓ | 20 | |
| 104.m | even | 4 | 1 | inner | 416.2.u.b | 20 | |
| 156.l | odd | 4 | 1 | 936.2.w.h | 20 | ||
| 312.y | even | 4 | 1 | 936.2.w.h | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.2.m.b | ✓ | 20 | 4.b | odd | 2 | 1 | |
| 104.2.m.b | ✓ | 20 | 8.b | even | 2 | 1 | |
| 104.2.m.b | ✓ | 20 | 52.f | even | 4 | 1 | |
| 104.2.m.b | ✓ | 20 | 104.j | odd | 4 | 1 | |
| 416.2.u.b | 20 | 1.a | even | 1 | 1 | trivial | |
| 416.2.u.b | 20 | 8.d | odd | 2 | 1 | inner | |
| 416.2.u.b | 20 | 13.d | odd | 4 | 1 | inner | |
| 416.2.u.b | 20 | 104.m | even | 4 | 1 | inner | |
| 936.2.w.h | 20 | 12.b | even | 2 | 1 | ||
| 936.2.w.h | 20 | 24.h | odd | 2 | 1 | ||
| 936.2.w.h | 20 | 156.l | odd | 4 | 1 | ||
| 936.2.w.h | 20 | 312.y | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} - T_{3}^{4} - 9T_{3}^{3} + 3T_{3}^{2} + 18T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(416, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( (T^{5} - T^{4} - 9 T^{3} + \cdots + 4)^{4} \)
|
| $5$ |
\( T^{20} + 163 T^{16} + \cdots + 16 \)
|
| $7$ |
\( T^{20} + 343 T^{16} + \cdots + 16 \)
|
| $11$ |
\( (T^{10} - 4 T^{9} + \cdots + 5408)^{2} \)
|
| $13$ |
\( T^{20} + \cdots + 137858491849 \)
|
| $17$ |
\( (T^{10} + 71 T^{8} + \cdots + 355216)^{2} \)
|
| $19$ |
\( (T^{10} - 6 T^{9} + \cdots + 8192)^{2} \)
|
| $23$ |
\( (T^{10} - 94 T^{8} + \cdots - 2048)^{2} \)
|
| $29$ |
\( (T^{10} + 186 T^{8} + \cdots + 21632)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 92829679353856 \)
|
| $37$ |
\( T^{20} + \cdots + 911076029249296 \)
|
| $41$ |
\( (T^{10} + 12 T^{9} + \cdots + 512)^{2} \)
|
| $43$ |
\( (T^{10} + 111 T^{8} + \cdots + 795664)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 1416468496 \)
|
| $53$ |
\( (T^{10} + 230 T^{8} + \cdots + 3442688)^{2} \)
|
| $59$ |
\( (T^{10} + 10 T^{9} + \cdots + 1384448)^{2} \)
|
| $61$ |
\( (T^{10} + 392 T^{8} + \cdots + 8388608)^{2} \)
|
| $67$ |
\( (T^{10} - 8 T^{9} + \cdots + 78575648)^{2} \)
|
| $71$ |
\( T^{20} + \cdots + 97459622042896 \)
|
| $73$ |
\( (T^{10} - 12 T^{9} + \cdots + 512)^{2} \)
|
| $79$ |
\( (T^{10} + 442 T^{8} + \cdots + 336338048)^{2} \)
|
| $83$ |
\( (T^{10} + 8 T^{9} + \cdots + 414950432)^{2} \)
|
| $89$ |
\( (T^{10} + \cdots + 117546549248)^{2} \)
|
| $97$ |
\( (T^{10} - 6 T^{9} + \cdots + 7270250528)^{2} \)
|
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