Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1080,2,Mod(541,1080)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 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("1080.541");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1080.k (of order \(2\), degree \(1\), 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: | \(8.62384341830\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - x^{18} + 5x^{16} + 28x^{12} - 28x^{10} + 112x^{8} + 320x^{4} - 256x^{2} + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{20} - x^{18} + 5x^{16} + 28x^{12} - 28x^{10} + 112x^{8} + 320x^{4} - 256x^{2} + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{19} - 3\nu^{17} - \nu^{15} - 20\nu^{13} - 28\nu^{11} - 84\nu^{9} - 448\nu^{5} - 320\nu^{3} - 1024\nu ) / 512 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{19} - \nu^{17} + 5\nu^{15} + 28\nu^{11} - 28\nu^{9} + 112\nu^{7} + 320\nu^{3} - 256\nu ) / 512 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{18} + \nu^{16} - 5\nu^{14} - 28\nu^{10} + 28\nu^{8} - 112\nu^{6} - 320\nu^{2} + 256 ) / 256 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5 \nu^{18} + 61 \nu^{16} + 63 \nu^{14} + 98 \nu^{12} + 84 \nu^{10} + 1260 \nu^{8} + 280 \nu^{6} + \cdots + 16256 ) / 896 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 17 \nu^{19} + 53 \nu^{17} + 7 \nu^{15} + 84 \nu^{13} - 252 \nu^{11} + 1260 \nu^{9} + \cdots + 15872 \nu ) / 3584 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 19 \nu^{19} - 79 \nu^{17} - 21 \nu^{15} - 252 \nu^{13} + 420 \nu^{11} - 2212 \nu^{9} + \cdots - 31232 \nu ) / 3584 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 17 \nu^{18} + 53 \nu^{16} + 7 \nu^{14} + 84 \nu^{12} - 252 \nu^{10} + 1260 \nu^{8} - 224 \nu^{6} + \cdots + 15872 ) / 1792 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 11 \nu^{18} - 17 \nu^{16} + 21 \nu^{14} - 70 \nu^{12} + 252 \nu^{10} - 364 \nu^{8} + 280 \nu^{6} + \cdots - 4736 ) / 896 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 29 \nu^{18} + 71 \nu^{16} + 77 \nu^{14} + 84 \nu^{12} + 588 \nu^{10} + 1092 \nu^{8} + 1344 \nu^{6} + \cdots + 12032 ) / 1792 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 9 \nu^{19} - 23 \nu^{17} - 21 \nu^{15} - 56 \nu^{13} - 196 \nu^{11} - 420 \nu^{9} - 336 \nu^{7} + \cdots - 5248 \nu ) / 896 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( \nu^{18} - \nu^{16} + \nu^{14} - 4\nu^{12} + 16\nu^{10} - 36\nu^{8} + 32\nu^{6} - 16\nu^{4} + 160\nu^{2} - 384 ) / 64 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 9 \nu^{18} - 23 \nu^{16} - 21 \nu^{14} - 56 \nu^{12} - 196 \nu^{10} - 420 \nu^{8} - 336 \nu^{6} + \cdots - 4352 ) / 448 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 5 \nu^{19} - 23 \nu^{17} - 7 \nu^{15} - 56 \nu^{13} + 98 \nu^{11} - 532 \nu^{9} + 112 \nu^{7} + \cdots - 6592 \nu ) / 448 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 16 \nu^{19} - 37 \nu^{17} - 63 \nu^{15} - 77 \nu^{13} - 364 \nu^{11} - 756 \nu^{9} + \cdots - 9280 \nu ) / 896 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 31 \nu^{19} - 3 \nu^{17} - 49 \nu^{15} + 14 \nu^{13} - 700 \nu^{11} + 364 \nu^{9} + \cdots + 6016 \nu ) / 1792 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 9 \nu^{19} + 5 \nu^{17} - 9 \nu^{15} + 12 \nu^{13} - 156 \nu^{11} + 204 \nu^{9} - 256 \nu^{7} + \cdots + 2048 \nu ) / 512 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - 31 \nu^{18} - 3 \nu^{16} - 49 \nu^{14} + 14 \nu^{12} - 700 \nu^{10} + 364 \nu^{8} - 952 \nu^{6} + \cdots + 6016 ) / 896 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( -41\nu^{18} + \nu^{16} - 77\nu^{14} - 868\nu^{10} + 252\nu^{8} - 1456\nu^{6} - 1120\nu^{4} - 8640\nu^{2} + 4864 ) / 896 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{19} - \beta_{13} + \beta_{9} - \beta_{8} - \beta_{4} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{16} - \beta_{11} + \beta_{7} + \beta_{3} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{13} + 2\beta_{12} - 2\beta_{10} + 2\beta_{8} - 2 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -2\beta_{17} + \beta_{16} + \beta_{11} + \beta_{7} + 4\beta_{6} + \beta_{3} - 2\beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -\beta_{19} + 2\beta_{18} + 2\beta_{12} + 2\beta_{10} - \beta_{9} + 3\beta_{8} - 2\beta_{5} - 3\beta_{4} - 2 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 2 \beta_{16} + 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{11} - 6 \beta_{7} + 2 \beta_{6} + \cdots - 3 \beta_1 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 5 \beta_{19} + 6 \beta_{18} + \beta_{13} - 4 \beta_{12} + 4 \beta_{10} + 3 \beta_{9} + \beta_{8} + \cdots - 16 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 6 \beta_{17} + 5 \beta_{16} - 6 \beta_{15} + 10 \beta_{14} - 3 \beta_{11} - 7 \beta_{7} + \cdots - 14 \beta_1 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 8 \beta_{19} - 8 \beta_{18} + 9 \beta_{13} - 10 \beta_{12} - 2 \beta_{10} - 4 \beta_{9} + 18 \beta_{8} + \cdots + 22 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 14 \beta_{17} - 21 \beta_{16} + 8 \beta_{14} + 11 \beta_{11} - 5 \beta_{7} + 20 \beta_{6} + \cdots + 13 \beta_1 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( 13 \beta_{19} - 6 \beta_{18} - 12 \beta_{13} - 30 \beta_{12} + 2 \beta_{10} - 7 \beta_{9} - 55 \beta_{8} + \cdots + 30 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 24 \beta_{17} - 14 \beta_{16} + 14 \beta_{15} - 26 \beta_{14} - 14 \beta_{11} - 2 \beta_{7} + \cdots + 15 \beta_1 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( \beta_{19} - 22 \beta_{18} + 27 \beta_{13} - 12 \beta_{12} - 36 \beta_{10} - 7 \beta_{9} - 29 \beta_{8} + \cdots - 80 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 38 \beta_{17} + 63 \beta_{16} - 42 \beta_{15} - 58 \beta_{14} + 63 \beta_{11} + 107 \beta_{7} + \cdots + 6 \beta_1 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 48 \beta_{19} - 104 \beta_{18} + 43 \beta_{13} + 82 \beta_{12} - 6 \beta_{10} - 84 \beta_{9} + \cdots - 126 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 102 \beta_{17} - 175 \beta_{16} + 112 \beta_{15} - 152 \beta_{14} + 49 \beta_{11} + 33 \beta_{7} + \cdots + 31 \beta_1 ) / 2 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 81 \beta_{19} + 174 \beta_{18} + 4 \beta_{13} + 86 \beta_{12} + 118 \beta_{10} - 61 \beta_{9} + \cdots - 54 ) / 2 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 136 \beta_{17} + 142 \beta_{16} - 70 \beta_{15} - 30 \beta_{14} - 114 \beta_{11} - 206 \beta_{7} + \cdots - 163 \beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times\).
| \(n\) | \(217\) | \(271\) | \(541\) | \(1001\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 541.1 |
|
−1.34522 | − | 0.436333i | 0 | 1.61923 | + | 1.17393i | − | 1.00000i | 0 | −3.61492 | −1.66599 | − | 2.28571i | 0 | −0.436333 | + | 1.34522i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.2 | −1.34522 | + | 0.436333i | 0 | 1.61923 | − | 1.17393i | 1.00000i | 0 | −3.61492 | −1.66599 | + | 2.28571i | 0 | −0.436333 | − | 1.34522i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.3 | −1.19357 | − | 0.758543i | 0 | 0.849224 | + | 1.81075i | − | 1.00000i | 0 | 4.63236 | 0.359923 | − | 2.80543i | 0 | −0.758543 | + | 1.19357i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.4 | −1.19357 | + | 0.758543i | 0 | 0.849224 | − | 1.81075i | 1.00000i | 0 | 4.63236 | 0.359923 | + | 2.80543i | 0 | −0.758543 | − | 1.19357i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.5 | −1.17425 | − | 0.788128i | 0 | 0.757708 | + | 1.85091i | 1.00000i | 0 | 2.12726 | 0.569020 | − | 2.77060i | 0 | 0.788128 | − | 1.17425i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.6 | −1.17425 | + | 0.788128i | 0 | 0.757708 | − | 1.85091i | − | 1.00000i | 0 | 2.12726 | 0.569020 | + | 2.77060i | 0 | 0.788128 | + | 1.17425i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.7 | −0.662801 | − | 1.24928i | 0 | −1.12139 | + | 1.65605i | 1.00000i | 0 | −1.52861 | 2.81212 | + | 0.303298i | 0 | 1.24928 | − | 0.662801i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.8 | −0.662801 | + | 1.24928i | 0 | −1.12139 | − | 1.65605i | − | 1.00000i | 0 | −1.52861 | 2.81212 | − | 0.303298i | 0 | 1.24928 | + | 0.662801i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.9 | −0.444539 | − | 1.34253i | 0 | −1.60477 | + | 1.19361i | − | 1.00000i | 0 | −1.61609 | 2.31584 | + | 1.62384i | 0 | −1.34253 | + | 0.444539i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.10 | −0.444539 | + | 1.34253i | 0 | −1.60477 | − | 1.19361i | 1.00000i | 0 | −1.61609 | 2.31584 | − | 1.62384i | 0 | −1.34253 | − | 0.444539i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.11 | 0.444539 | − | 1.34253i | 0 | −1.60477 | − | 1.19361i | − | 1.00000i | 0 | −1.61609 | −2.31584 | + | 1.62384i | 0 | −1.34253 | − | 0.444539i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.12 | 0.444539 | + | 1.34253i | 0 | −1.60477 | + | 1.19361i | 1.00000i | 0 | −1.61609 | −2.31584 | − | 1.62384i | 0 | −1.34253 | + | 0.444539i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.13 | 0.662801 | − | 1.24928i | 0 | −1.12139 | − | 1.65605i | 1.00000i | 0 | −1.52861 | −2.81212 | + | 0.303298i | 0 | 1.24928 | + | 0.662801i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.14 | 0.662801 | + | 1.24928i | 0 | −1.12139 | + | 1.65605i | − | 1.00000i | 0 | −1.52861 | −2.81212 | − | 0.303298i | 0 | 1.24928 | − | 0.662801i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.15 | 1.17425 | − | 0.788128i | 0 | 0.757708 | − | 1.85091i | 1.00000i | 0 | 2.12726 | −0.569020 | − | 2.77060i | 0 | 0.788128 | + | 1.17425i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.16 | 1.17425 | + | 0.788128i | 0 | 0.757708 | + | 1.85091i | − | 1.00000i | 0 | 2.12726 | −0.569020 | + | 2.77060i | 0 | 0.788128 | − | 1.17425i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.17 | 1.19357 | − | 0.758543i | 0 | 0.849224 | − | 1.81075i | − | 1.00000i | 0 | 4.63236 | −0.359923 | − | 2.80543i | 0 | −0.758543 | − | 1.19357i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.18 | 1.19357 | + | 0.758543i | 0 | 0.849224 | + | 1.81075i | 1.00000i | 0 | 4.63236 | −0.359923 | + | 2.80543i | 0 | −0.758543 | + | 1.19357i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.19 | 1.34522 | − | 0.436333i | 0 | 1.61923 | − | 1.17393i | − | 1.00000i | 0 | −3.61492 | 1.66599 | − | 2.28571i | 0 | −0.436333 | − | 1.34522i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 541.20 | 1.34522 | + | 0.436333i | 0 | 1.61923 | + | 1.17393i | 1.00000i | 0 | −3.61492 | 1.66599 | + | 2.28571i | 0 | −0.436333 | + | 1.34522i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 24.h | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1080.2.k.d | ✓ | 20 |
| 3.b | odd | 2 | 1 | inner | 1080.2.k.d | ✓ | 20 |
| 4.b | odd | 2 | 1 | 4320.2.k.d | 20 | ||
| 8.b | even | 2 | 1 | inner | 1080.2.k.d | ✓ | 20 |
| 8.d | odd | 2 | 1 | 4320.2.k.d | 20 | ||
| 12.b | even | 2 | 1 | 4320.2.k.d | 20 | ||
| 24.f | even | 2 | 1 | 4320.2.k.d | 20 | ||
| 24.h | odd | 2 | 1 | inner | 1080.2.k.d | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1080.2.k.d | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 1080.2.k.d | ✓ | 20 | 3.b | odd | 2 | 1 | inner |
| 1080.2.k.d | ✓ | 20 | 8.b | even | 2 | 1 | inner |
| 1080.2.k.d | ✓ | 20 | 24.h | odd | 2 | 1 | inner |
| 4320.2.k.d | 20 | 4.b | odd | 2 | 1 | ||
| 4320.2.k.d | 20 | 8.d | odd | 2 | 1 | ||
| 4320.2.k.d | 20 | 12.b | even | 2 | 1 | ||
| 4320.2.k.d | 20 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1080, [\chi])\):
|
\( T_{7}^{5} - 22T_{7}^{3} - 18T_{7}^{2} + 76T_{7} + 88 \)
|
|
\( T_{17}^{10} - 119T_{17}^{8} + 5122T_{17}^{6} - 94474T_{17}^{4} + 633189T_{17}^{2} - 34839 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - T^{18} + \cdots + 1024 \)
|
| $3$ |
\( T^{20} \)
|
| $5$ |
\( (T^{2} + 1)^{10} \)
|
| $7$ |
\( (T^{5} - 22 T^{3} + \cdots + 88)^{4} \)
|
| $11$ |
\( (T^{10} + 72 T^{8} + \cdots + 7744)^{2} \)
|
| $13$ |
\( (T^{10} + 80 T^{8} + \cdots + 45504)^{2} \)
|
| $17$ |
\( (T^{10} - 119 T^{8} + \cdots - 34839)^{2} \)
|
| $19$ |
\( (T^{10} + 147 T^{8} + \cdots + 5464351)^{2} \)
|
| $23$ |
\( (T^{10} - 171 T^{8} + \cdots - 21608791)^{2} \)
|
| $29$ |
\( (T^{10} + 216 T^{8} + \cdots + 14400)^{2} \)
|
| $31$ |
\( (T^{5} - 5 T^{4} + \cdots - 4309)^{4} \)
|
| $37$ |
\( (T^{10} + 264 T^{8} + \cdots + 5177344)^{2} \)
|
| $41$ |
\( (T^{10} - 312 T^{8} + \cdots - 288803776)^{2} \)
|
| $43$ |
\( (T^{10} + 284 T^{8} + \cdots + 154840000)^{2} \)
|
| $47$ |
\( (T^{10} - 316 T^{8} + \cdots - 12640000)^{2} \)
|
| $53$ |
\( (T^{10} + 269 T^{8} + \cdots + 38626225)^{2} \)
|
| $59$ |
\( (T^{10} + 280 T^{8} + \cdots + 732736)^{2} \)
|
| $61$ |
\( (T^{10} + 243 T^{8} + \cdots + 1234375)^{2} \)
|
| $67$ |
\( (T^{10} + 312 T^{8} + \cdots + 72806400)^{2} \)
|
| $71$ |
\( (T^{10} - 176 T^{8} + \cdots - 126400)^{2} \)
|
| $73$ |
\( (T^{5} + 10 T^{4} + \cdots + 6696)^{4} \)
|
| $79$ |
\( (T^{5} + T^{4} - 174 T^{3} + \cdots - 745)^{4} \)
|
| $83$ |
\( (T^{10} + 269 T^{8} + \cdots + 268992801)^{2} \)
|
| $89$ |
\( (T^{10} - 284 T^{8} + \cdots - 154840000)^{2} \)
|
| $97$ |
\( (T^{5} - 10 T^{4} + \cdots + 12928)^{4} \)
|
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