Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [444,2,Mod(371,444)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(444, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 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("444.371");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 444 = 2^{2} \cdot 3 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 444.c (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: | \(3.54535784974\) |
| 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: |
\( x^{20} - x^{18} + 3x^{16} - 7x^{14} + 12x^{12} - 40x^{10} + 48x^{8} - 112x^{6} + 192x^{4} - 256x^{2} + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| 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
\( x^{20} - x^{18} + 3x^{16} - 7x^{14} + 12x^{12} - 40x^{10} + 48x^{8} - 112x^{6} + 192x^{4} - 256x^{2} + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - \nu^{2} ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{19} + 4 \nu^{18} + 3 \nu^{17} + 12 \nu^{16} + 15 \nu^{15} - 132 \nu^{14} + 53 \nu^{13} + \cdots + 5120 ) / 12288 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{19} - 14 \nu^{18} + 51 \nu^{17} + 54 \nu^{16} - 33 \nu^{15} + 78 \nu^{14} + 197 \nu^{13} + \cdots - 2560 ) / 24576 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{17} - \nu^{15} + 3\nu^{13} - 7\nu^{11} + 12\nu^{9} - 40\nu^{7} + 48\nu^{5} + 16\nu^{3} + 64\nu ) / 256 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{18} - 9\nu^{16} + 19\nu^{14} + 33\nu^{12} + 32\nu^{10} + 56\nu^{8} + 80\nu^{6} - 112\nu^{4} - 1280 ) / 2048 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{19} - 2 \nu^{18} - 21 \nu^{17} + 42 \nu^{16} + 39 \nu^{15} - 78 \nu^{14} - 19 \nu^{13} + \cdots + 3584 ) / 12288 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -3\nu^{18} + 7\nu^{16} - 29\nu^{14} + 49\nu^{12} + 16\nu^{10} + 152\nu^{8} - 368\nu^{6} + 784\nu^{4} + 768 ) / 2048 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( -\nu^{18} + \nu^{16} - 3\nu^{14} + 7\nu^{12} - 12\nu^{10} - 24\nu^{8} - 48\nu^{6} - 80\nu^{4} - 192\nu^{2} ) / 512 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 5 \nu^{18} + \nu^{16} + 5 \nu^{14} + 7 \nu^{12} - 112 \nu^{10} + 168 \nu^{8} - 16 \nu^{6} + \cdots + 1280 ) / 2048 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 2 \nu^{19} + \nu^{18} + 6 \nu^{17} - 21 \nu^{16} + 54 \nu^{15} + 39 \nu^{14} + 26 \nu^{13} + \cdots - 1792 ) / 6144 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - \nu^{19} - 4 \nu^{18} + \nu^{17} - 4 \nu^{16} + 13 \nu^{15} - 4 \nu^{14} - 9 \nu^{13} + 4 \nu^{12} + \cdots - 512 ) / 2048 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - \nu^{19} + 4 \nu^{18} + \nu^{17} + 4 \nu^{16} + 13 \nu^{15} + 4 \nu^{14} - 9 \nu^{13} - 4 \nu^{12} + \cdots + 512 ) / 2048 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - \nu^{19} + 4 \nu^{18} + \nu^{17} + 4 \nu^{16} + 13 \nu^{15} + 4 \nu^{14} - 9 \nu^{13} - 4 \nu^{12} + \cdots + 512 ) / 2048 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 3 \nu^{19} - 7 \nu^{17} - 3 \nu^{15} + 47 \nu^{13} - 48 \nu^{11} + 136 \nu^{9} + 176 \nu^{7} + \cdots + 256 \nu ) / 4096 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 13 \nu^{19} + 62 \nu^{18} + 57 \nu^{17} - 6 \nu^{16} + 93 \nu^{15} - 30 \nu^{14} - 17 \nu^{13} + \cdots + 8704 ) / 24576 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 17 \nu^{19} - 46 \nu^{18} - 45 \nu^{17} + 54 \nu^{16} - 33 \nu^{15} - 114 \nu^{14} + 229 \nu^{13} + \cdots - 512 ) / 24576 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - \nu^{19} + \nu^{17} - 3 \nu^{15} + 7 \nu^{13} - 12 \nu^{11} + 40 \nu^{9} - 48 \nu^{7} + \cdots + 256 \nu ) / 512 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{17} + \beta_{14} + \beta_{6} - \beta_{5} + \beta_{2} + \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{3} + \beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{18} + \beta_{17} + 2 \beta_{15} - \beta_{14} + 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{9} + \cdots - \beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{18} + 2\beta_{17} - 2\beta_{15} + 2\beta_{14} + 2\beta_{13} - 2\beta_{12} - 2\beta_{9} - 2\beta_{3} - \beta_{2} \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{17} + 4 \beta_{16} + 4 \beta_{15} - \beta_{14} + 4 \beta_{13} - 4 \beta_{12} - 4 \beta_{8} + \cdots + \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 4\beta_{11} - 8\beta_{10} + 4\beta_{9} - 6\beta_{3} - 7\beta_{2} - 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 6 \beta_{18} + 5 \beta_{17} + 8 \beta_{16} - 6 \beta_{15} + 3 \beta_{14} + 2 \beta_{13} + \cdots - \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 2 \beta_{18} - 6 \beta_{17} - 2 \beta_{15} + 2 \beta_{14} + 10 \beta_{13} - 2 \beta_{12} - 16 \beta_{11} + \cdots + 16 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 24 \beta_{18} - 7 \beta_{17} + 4 \beta_{16} - 4 \beta_{15} - \beta_{14} - 12 \beta_{13} + \cdots + 17 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( 8 \beta_{18} + 16 \beta_{17} + 8 \beta_{15} + 8 \beta_{14} - 16 \beta_{13} - 8 \beta_{12} + 4 \beta_{11} + \cdots - 4 \)
|
| \(\nu^{13}\) | \(=\) |
\( 16 \beta_{19} - 14 \beta_{18} - 43 \beta_{17} + 40 \beta_{16} + 18 \beta_{15} + 11 \beta_{14} + \cdots + 31 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( 10 \beta_{18} - 22 \beta_{17} + 38 \beta_{15} + 10 \beta_{14} - 6 \beta_{13} - 10 \beta_{12} - 16 \beta_{10} + \cdots + 32 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 16 \beta_{19} + 32 \beta_{18} + 25 \beta_{17} + 36 \beta_{16} + 20 \beta_{15} + 23 \beta_{14} + \cdots - 15 \beta_1 \)
|
| \(\nu^{16}\) | \(=\) |
\( 80 \beta_{18} - 32 \beta_{17} + 48 \beta_{15} + 80 \beta_{14} + 64 \beta_{13} - 80 \beta_{12} + \cdots + 12 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 64 \beta_{19} - 118 \beta_{18} + 53 \beta_{17} + 8 \beta_{16} + 74 \beta_{15} - 61 \beta_{14} + \cdots + 31 \beta_1 \)
|
| \(\nu^{18}\) | \(=\) |
\( - 14 \beta_{18} + 122 \beta_{17} + 110 \beta_{15} - 14 \beta_{14} - 246 \beta_{13} + 14 \beta_{12} + \cdots - 208 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 416 \beta_{19} - 40 \beta_{18} + 217 \beta_{17} + 260 \beta_{16} - 20 \beta_{15} - 177 \beta_{14} + \cdots - 47 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/444\mathbb{Z}\right)^\times\).
| \(n\) | \(149\) | \(223\) | \(409\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 371.1 |
|
−1.40476 | − | 0.163239i | −1.72087 | − | 0.196528i | 1.94671 | + | 0.458624i | 3.43563i | 2.38532 | + | 0.556987i | 4.57374i | −2.65979 | − | 0.962036i | 2.92275 | + | 0.676395i | 0.560830 | − | 4.82624i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.2 | −1.40476 | + | 0.163239i | −1.72087 | + | 0.196528i | 1.94671 | − | 0.458624i | − | 3.43563i | 2.38532 | − | 0.556987i | − | 4.57374i | −2.65979 | + | 0.962036i | 2.92275 | − | 0.676395i | 0.560830 | + | 4.82624i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.3 | −1.24215 | − | 0.676063i | 0.910797 | − | 1.47324i | 1.08588 | + | 1.67954i | 0.380368i | −2.12735 | + | 1.21424i | 5.08906i | −0.213348 | − | 2.82037i | −1.34090 | − | 2.68365i | 0.257153 | − | 0.472475i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.4 | −1.24215 | + | 0.676063i | 0.910797 | + | 1.47324i | 1.08588 | − | 1.67954i | − | 0.380368i | −2.12735 | − | 1.21424i | − | 5.08906i | −0.213348 | + | 2.82037i | −1.34090 | + | 2.68365i | 0.257153 | + | 0.472475i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.5 | −1.03439 | − | 0.964383i | 0.504486 | + | 1.65695i | 0.139930 | + | 1.99510i | − | 4.11408i | 1.07610 | − | 2.20046i | − | 2.35617i | 1.77930 | − | 2.19866i | −2.49099 | + | 1.67182i | −3.96755 | + | 4.25557i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.6 | −1.03439 | + | 0.964383i | 0.504486 | − | 1.65695i | 0.139930 | − | 1.99510i | 4.11408i | 1.07610 | + | 2.20046i | 2.35617i | 1.77930 | + | 2.19866i | −2.49099 | − | 1.67182i | −3.96755 | − | 4.25557i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.7 | −0.697257 | − | 1.23038i | 1.62127 | − | 0.609505i | −1.02767 | + | 1.71578i | 0.766433i | −1.88036 | − | 1.56979i | − | 2.47611i | 2.82761 | + | 0.0680797i | 2.25701 | − | 1.97634i | 0.943004 | − | 0.534401i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.8 | −0.697257 | + | 1.23038i | 1.62127 | + | 0.609505i | −1.02767 | − | 1.71578i | − | 0.766433i | −1.88036 | + | 1.56979i | 2.47611i | 2.82761 | − | 0.0680797i | 2.25701 | + | 1.97634i | 0.943004 | + | 0.534401i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.9 | −0.421398 | − | 1.34997i | −0.275793 | − | 1.70995i | −1.64485 | + | 1.13775i | 2.74566i | −2.19217 | + | 1.09288i | 0.706946i | 2.22907 | + | 1.74105i | −2.84788 | + | 0.943185i | 3.70656 | − | 1.15701i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.10 | −0.421398 | + | 1.34997i | −0.275793 | + | 1.70995i | −1.64485 | − | 1.13775i | − | 2.74566i | −2.19217 | − | 1.09288i | − | 0.706946i | 2.22907 | − | 1.74105i | −2.84788 | − | 0.943185i | 3.70656 | + | 1.15701i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.11 | 0.421398 | − | 1.34997i | 0.275793 | + | 1.70995i | −1.64485 | − | 1.13775i | 2.74566i | 2.42461 | + | 0.348257i | − | 0.706946i | −2.22907 | + | 1.74105i | −2.84788 | + | 0.943185i | 3.70656 | + | 1.15701i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.12 | 0.421398 | + | 1.34997i | 0.275793 | − | 1.70995i | −1.64485 | + | 1.13775i | − | 2.74566i | 2.42461 | − | 0.348257i | 0.706946i | −2.22907 | − | 1.74105i | −2.84788 | − | 0.943185i | 3.70656 | − | 1.15701i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.13 | 0.697257 | − | 1.23038i | −1.62127 | + | 0.609505i | −1.02767 | − | 1.71578i | 0.766433i | −0.380517 | + | 2.41975i | 2.47611i | −2.82761 | + | 0.0680797i | 2.25701 | − | 1.97634i | 0.943004 | + | 0.534401i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.14 | 0.697257 | + | 1.23038i | −1.62127 | − | 0.609505i | −1.02767 | + | 1.71578i | − | 0.766433i | −0.380517 | − | 2.41975i | − | 2.47611i | −2.82761 | − | 0.0680797i | 2.25701 | + | 1.97634i | 0.943004 | − | 0.534401i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.15 | 1.03439 | − | 0.964383i | −0.504486 | − | 1.65695i | 0.139930 | − | 1.99510i | − | 4.11408i | −2.11977 | − | 1.22742i | 2.35617i | −1.77930 | − | 2.19866i | −2.49099 | + | 1.67182i | −3.96755 | − | 4.25557i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.16 | 1.03439 | + | 0.964383i | −0.504486 | + | 1.65695i | 0.139930 | + | 1.99510i | 4.11408i | −2.11977 | + | 1.22742i | − | 2.35617i | −1.77930 | + | 2.19866i | −2.49099 | − | 1.67182i | −3.96755 | + | 4.25557i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.17 | 1.24215 | − | 0.676063i | −0.910797 | + | 1.47324i | 1.08588 | − | 1.67954i | 0.380368i | −0.135342 | + | 2.44575i | − | 5.08906i | 0.213348 | − | 2.82037i | −1.34090 | − | 2.68365i | 0.257153 | + | 0.472475i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.18 | 1.24215 | + | 0.676063i | −0.910797 | − | 1.47324i | 1.08588 | + | 1.67954i | − | 0.380368i | −0.135342 | − | 2.44575i | 5.08906i | 0.213348 | + | 2.82037i | −1.34090 | + | 2.68365i | 0.257153 | − | 0.472475i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.19 | 1.40476 | − | 0.163239i | 1.72087 | + | 0.196528i | 1.94671 | − | 0.458624i | 3.43563i | 2.44948 | − | 0.00483835i | − | 4.57374i | 2.65979 | − | 0.962036i | 2.92275 | + | 0.676395i | 0.560830 | + | 4.82624i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 371.20 | 1.40476 | + | 0.163239i | 1.72087 | − | 0.196528i | 1.94671 | + | 0.458624i | − | 3.43563i | 2.44948 | + | 0.00483835i | 4.57374i | 2.65979 | + | 0.962036i | 2.92275 | − | 0.676395i | 0.560830 | − | 4.82624i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 4.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 444.2.c.c | ✓ | 20 |
| 3.b | odd | 2 | 1 | inner | 444.2.c.c | ✓ | 20 |
| 4.b | odd | 2 | 1 | inner | 444.2.c.c | ✓ | 20 |
| 12.b | even | 2 | 1 | inner | 444.2.c.c | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 444.2.c.c | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 444.2.c.c | ✓ | 20 | 3.b | odd | 2 | 1 | inner |
| 444.2.c.c | ✓ | 20 | 4.b | odd | 2 | 1 | inner |
| 444.2.c.c | ✓ | 20 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(444, [\chi])\):
|
\( T_{5}^{10} + 37T_{5}^{8} + 443T_{5}^{6} + 1814T_{5}^{4} + 1138T_{5}^{2} + 128 \)
|
|
\( T_{11}^{10} - 70T_{11}^{8} + 1406T_{11}^{6} - 10643T_{11}^{4} + 26240T_{11}^{2} - 72 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - T^{18} + \cdots + 1024 \)
|
| $3$ |
\( T^{20} + 3 T^{18} + \cdots + 59049 \)
|
| $5$ |
\( (T^{10} + 37 T^{8} + \cdots + 128)^{2} \)
|
| $7$ |
\( (T^{10} + 59 T^{8} + \cdots + 9216)^{2} \)
|
| $11$ |
\( (T^{10} - 70 T^{8} + \cdots - 72)^{2} \)
|
| $13$ |
\( (T^{5} - 8 T^{4} - 20 T^{3} + \cdots - 8)^{4} \)
|
| $17$ |
\( (T^{10} + 101 T^{8} + \cdots + 184832)^{2} \)
|
| $19$ |
\( (T^{10} + 103 T^{8} + \cdots + 46656)^{2} \)
|
| $23$ |
\( (T^{10} - 42 T^{8} + \cdots - 18)^{2} \)
|
| $29$ |
\( (T^{10} + 249 T^{8} + \cdots + 12964232)^{2} \)
|
| $31$ |
\( (T^{10} + 157 T^{8} + \cdots + 51984)^{2} \)
|
| $37$ |
\( (T - 1)^{20} \)
|
| $41$ |
\( (T^{10} + 313 T^{8} + \cdots + 25176608)^{2} \)
|
| $43$ |
\( (T^{10} + 240 T^{8} + \cdots + 11614464)^{2} \)
|
| $47$ |
\( (T^{10} - 220 T^{8} + \cdots - 1179648)^{2} \)
|
| $53$ |
\( (T^{10} + 301 T^{8} + \cdots + 525917312)^{2} \)
|
| $59$ |
\( (T^{10} - 200 T^{8} + \cdots - 1179648)^{2} \)
|
| $61$ |
\( (T^{5} + 9 T^{4} + \cdots + 7472)^{4} \)
|
| $67$ |
\( (T^{10} + 297 T^{8} + \cdots + 107246736)^{2} \)
|
| $71$ |
\( (T^{10} - 220 T^{8} + \cdots - 2654208)^{2} \)
|
| $73$ |
\( (T^{5} - 12 T^{4} + \cdots + 326)^{4} \)
|
| $79$ |
\( (T^{10} + 177 T^{8} + \cdots + 576)^{2} \)
|
| $83$ |
\( (T^{10} - 111 T^{8} + \cdots - 294912)^{2} \)
|
| $89$ |
\( (T^{10} + 349 T^{8} + \cdots + 28516352)^{2} \)
|
| $97$ |
\( (T^{5} - 12 T^{4} + \cdots - 8896)^{4} \)
|
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