Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [380,2,Mod(151,380)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(380, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("380.151");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 380 = 2^{2} \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 380.f (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: | \(3.03431527681\) |
| 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: |
\( x^{20} - x^{18} + x^{16} - 9x^{14} + 20x^{12} - 24x^{10} + 80x^{8} - 144x^{6} + 64x^{4} - 256x^{2} + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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
\( x^{20} - x^{18} + x^{16} - 9x^{14} + 20x^{12} - 24x^{10} + 80x^{8} - 144x^{6} + 64x^{4} - 256x^{2} + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - \nu^{19} - 3 \nu^{17} + 11 \nu^{15} - 19 \nu^{13} + 72 \nu^{11} + 144 \nu^{9} + 160 \nu^{7} + \cdots + 1024 \nu ) / 1536 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{18} + 3 \nu^{16} - 19 \nu^{14} + 43 \nu^{12} - 64 \nu^{10} + 104 \nu^{8} - 240 \nu^{6} + \cdots + 1280 ) / 512 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - \nu^{19} - 3 \nu^{17} + 19 \nu^{15} - 11 \nu^{13} + 32 \nu^{11} - 200 \nu^{9} + 336 \nu^{7} + \cdots - 1792 \nu ) / 1024 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - \nu^{19} + 3 \nu^{17} + 5 \nu^{15} - 13 \nu^{13} + 18 \nu^{11} - 120 \nu^{9} + 16 \nu^{7} + \cdots - 896 \nu ) / 768 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{19} - 3 \nu^{17} + 7 \nu^{15} - 23 \nu^{13} + 66 \nu^{11} - 156 \nu^{9} + 8 \nu^{7} + \cdots - 1024 \nu ) / 768 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - \nu^{18} + 5 \nu^{16} - 5 \nu^{14} + 13 \nu^{12} + 8 \nu^{10} + 40 \nu^{8} - 112 \nu^{6} + \cdots - 256 ) / 512 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - \nu^{19} + 3 \nu^{17} + 6 \nu^{16} + 5 \nu^{15} + 6 \nu^{14} - 13 \nu^{13} + 18 \nu^{12} + \cdots - 1664 \nu ) / 768 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 5 \nu^{19} - 15 \nu^{17} + 31 \nu^{15} - 119 \nu^{13} + 96 \nu^{11} - 168 \nu^{9} + \cdots - 4864 \nu ) / 3072 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 5 \nu^{19} + 9 \nu^{17} - 41 \nu^{15} + 49 \nu^{13} - 72 \nu^{11} + 264 \nu^{9} - 400 \nu^{7} + \cdots + 1280 \nu ) / 3072 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( -\nu^{19} - 3\nu^{17} + 3\nu^{15} + 5\nu^{13} + 16\nu^{11} - 56\nu^{9} + 16\nu^{7} - 176\nu^{5} + 512\nu^{3} ) / 512 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( -\nu^{19} + \nu^{17} - \nu^{15} + 9\nu^{13} - 20\nu^{11} + 24\nu^{9} - 80\nu^{7} + 144\nu^{5} - 64\nu^{3} + 256\nu ) / 512 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( \nu^{19} - 3 \nu^{17} - 6 \nu^{16} - 5 \nu^{15} - 6 \nu^{14} + 13 \nu^{13} + 30 \nu^{12} - 18 \nu^{11} + \cdots + 2304 ) / 768 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 5 \nu^{18} + 9 \nu^{16} - 41 \nu^{14} + 49 \nu^{12} - 72 \nu^{10} + 264 \nu^{8} - 400 \nu^{6} + \cdots + 1280 ) / 1536 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( \nu^{19} - 2 \nu^{18} - 3 \nu^{17} - 5 \nu^{15} + 4 \nu^{14} + 13 \nu^{13} + 4 \nu^{12} - 18 \nu^{11} + \cdots - 1024 ) / 768 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( -\nu^{18} + \nu^{16} - \nu^{14} + 9\nu^{12} - 20\nu^{10} + 24\nu^{8} - 80\nu^{6} + 144\nu^{4} - 64\nu^{2} + 256 ) / 256 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - \nu^{19} + 2 \nu^{18} + 3 \nu^{17} - 12 \nu^{16} + 5 \nu^{15} + 8 \nu^{14} - 13 \nu^{13} + \cdots + 256 ) / 768 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 2 \nu^{19} - 9 \nu^{18} - 6 \nu^{17} + 9 \nu^{16} - 10 \nu^{15} - 33 \nu^{14} + 26 \nu^{13} + \cdots - 768 ) / 1536 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 17 \nu^{19} + 45 \nu^{17} - 29 \nu^{15} + 37 \nu^{13} - 192 \nu^{11} - 72 \nu^{9} + \cdots - 7936 \nu ) / 3072 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{18} - \beta_{15} - \beta_{14} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{19} - \beta_{10} + \beta_{6} - 2\beta_{5} + \beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 2 \beta_{19} + \beta_{17} + \beta_{16} - \beta_{15} + \beta_{14} - \beta_{13} + 2 \beta_{10} + \cdots - 1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2\beta_{12} - 2\beta_{11} + 2\beta_{10} + 2\beta_{6} - 2\beta_{5} + 2\beta_{4} + \beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( \beta_{18} - 4\beta_{16} + 3\beta_{15} + 3\beta_{14} + 4\beta_{8} - 4\beta_{7} + 4 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( \beta_{19} - 8\beta_{12} + 3\beta_{10} - 3\beta_{6} - 2\beta_{5} + 4\beta_{4} + 3\beta_1 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 2 \beta_{19} + 2 \beta_{18} - 3 \beta_{17} - 7 \beta_{16} - 3 \beta_{15} + 3 \beta_{14} + 7 \beta_{13} + \cdots - 9 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 2\beta_{19} + 2\beta_{12} - 2\beta_{11} - 4\beta_{10} - 6\beta_{5} - 2\beta_{4} + 8\beta_{2} - 5\beta_1 ) / 2 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 8 \beta_{19} - 3 \beta_{18} + 10 \beta_{17} - 6 \beta_{16} + \beta_{15} + 9 \beta_{14} + 2 \beta_{13} + \cdots + 6 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 5 \beta_{19} - 12 \beta_{12} + 4 \beta_{11} + 15 \beta_{10} - 16 \beta_{9} + 17 \beta_{6} + \cdots - 3 \beta_1 ) / 2 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 2 \beta_{19} + 12 \beta_{18} + 17 \beta_{17} + \beta_{16} + 19 \beta_{15} - 19 \beta_{14} + 7 \beta_{13} + \cdots - 17 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 20 \beta_{19} - 14 \beta_{12} + 30 \beta_{11} - 18 \beta_{10} - 48 \beta_{9} + 6 \beta_{6} + \cdots - 15 \beta_1 ) / 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( - 24 \beta_{19} - 35 \beta_{18} + 4 \beta_{17} + 48 \beta_{16} + 19 \beta_{15} - 21 \beta_{14} + \cdots - 16 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( - 3 \beta_{19} + 160 \beta_{12} - 40 \beta_{11} - 129 \beta_{10} - 32 \beta_{9} + 17 \beta_{6} + \cdots - 29 \beta_1 ) / 2 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( 26 \beta_{19} - 74 \beta_{18} - 71 \beta_{17} - 43 \beta_{16} + 77 \beta_{15} - 5 \beta_{14} + \cdots + 283 ) / 2 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 150 \beta_{19} - 70 \beta_{12} - 90 \beta_{11} - 128 \beta_{10} - 128 \beta_{9} + 76 \beta_{6} + \cdots + 131 \beta_1 ) / 2 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 48 \beta_{19} + 33 \beta_{18} - 50 \beta_{17} - 178 \beta_{16} - 183 \beta_{15} - 295 \beta_{14} + \cdots - 142 ) / 2 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( 137 \beta_{19} - 164 \beta_{12} - 196 \beta_{11} - 445 \beta_{10} - 208 \beta_{9} + 237 \beta_{6} + \cdots + 61 \beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).
| \(n\) | \(21\) | \(77\) | \(191\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 151.1 |
|
−1.40065 | − | 0.195405i | 1.89649 | 1.92363 | + | 0.547388i | −1.00000 | −2.65631 | − | 0.370584i | 2.89109i | −2.58737 | − | 1.14259i | 0.596662 | 1.40065 | + | 0.195405i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.2 | −1.40065 | + | 0.195405i | 1.89649 | 1.92363 | − | 0.547388i | −1.00000 | −2.65631 | + | 0.370584i | − | 2.89109i | −2.58737 | + | 1.14259i | 0.596662 | 1.40065 | − | 0.195405i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.3 | −1.31446 | − | 0.521712i | −2.15859 | 1.45563 | + | 1.37154i | −1.00000 | 2.83739 | + | 1.12616i | − | 1.66163i | −1.19783 | − | 2.56227i | 1.65950 | 1.31446 | + | 0.521712i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.4 | −1.31446 | + | 0.521712i | −2.15859 | 1.45563 | − | 1.37154i | −1.00000 | 2.83739 | − | 1.12616i | 1.66163i | −1.19783 | + | 2.56227i | 1.65950 | 1.31446 | − | 0.521712i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.5 | −0.976481 | − | 1.02298i | −0.502080 | −0.0929701 | + | 1.99784i | −1.00000 | 0.490271 | + | 0.513617i | 3.57756i | 2.13453 | − | 1.85574i | −2.74792 | 0.976481 | + | 1.02298i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.6 | −0.976481 | + | 1.02298i | −0.502080 | −0.0929701 | − | 1.99784i | −1.00000 | 0.490271 | − | 0.513617i | − | 3.57756i | 2.13453 | + | 1.85574i | −2.74792 | 0.976481 | − | 1.02298i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.7 | −0.611950 | − | 1.27496i | 3.21061 | −1.25103 | + | 1.56042i | −1.00000 | −1.96474 | − | 4.09340i | − | 3.02772i | 2.75504 | + | 0.640115i | 7.30804 | 0.611950 | + | 1.27496i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.8 | −0.611950 | + | 1.27496i | 3.21061 | −1.25103 | − | 1.56042i | −1.00000 | −1.96474 | + | 4.09340i | 3.02772i | 2.75504 | − | 0.640115i | 7.30804 | 0.611950 | − | 1.27496i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.9 | −0.482046 | − | 1.32952i | 0.428612 | −1.53526 | + | 1.28178i | −1.00000 | −0.206611 | − | 0.569850i | 1.38367i | 2.44423 | + | 1.42329i | −2.81629 | 0.482046 | + | 1.32952i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.10 | −0.482046 | + | 1.32952i | 0.428612 | −1.53526 | − | 1.28178i | −1.00000 | −0.206611 | + | 0.569850i | − | 1.38367i | 2.44423 | − | 1.42329i | −2.81629 | 0.482046 | − | 1.32952i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.11 | 0.482046 | − | 1.32952i | −0.428612 | −1.53526 | − | 1.28178i | −1.00000 | −0.206611 | + | 0.569850i | − | 1.38367i | −2.44423 | + | 1.42329i | −2.81629 | −0.482046 | + | 1.32952i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.12 | 0.482046 | + | 1.32952i | −0.428612 | −1.53526 | + | 1.28178i | −1.00000 | −0.206611 | − | 0.569850i | 1.38367i | −2.44423 | − | 1.42329i | −2.81629 | −0.482046 | − | 1.32952i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.13 | 0.611950 | − | 1.27496i | −3.21061 | −1.25103 | − | 1.56042i | −1.00000 | −1.96474 | + | 4.09340i | 3.02772i | −2.75504 | + | 0.640115i | 7.30804 | −0.611950 | + | 1.27496i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.14 | 0.611950 | + | 1.27496i | −3.21061 | −1.25103 | + | 1.56042i | −1.00000 | −1.96474 | − | 4.09340i | − | 3.02772i | −2.75504 | − | 0.640115i | 7.30804 | −0.611950 | − | 1.27496i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.15 | 0.976481 | − | 1.02298i | 0.502080 | −0.0929701 | − | 1.99784i | −1.00000 | 0.490271 | − | 0.513617i | − | 3.57756i | −2.13453 | − | 1.85574i | −2.74792 | −0.976481 | + | 1.02298i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.16 | 0.976481 | + | 1.02298i | 0.502080 | −0.0929701 | + | 1.99784i | −1.00000 | 0.490271 | + | 0.513617i | 3.57756i | −2.13453 | + | 1.85574i | −2.74792 | −0.976481 | − | 1.02298i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.17 | 1.31446 | − | 0.521712i | 2.15859 | 1.45563 | − | 1.37154i | −1.00000 | 2.83739 | − | 1.12616i | 1.66163i | 1.19783 | − | 2.56227i | 1.65950 | −1.31446 | + | 0.521712i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.18 | 1.31446 | + | 0.521712i | 2.15859 | 1.45563 | + | 1.37154i | −1.00000 | 2.83739 | + | 1.12616i | − | 1.66163i | 1.19783 | + | 2.56227i | 1.65950 | −1.31446 | − | 0.521712i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.19 | 1.40065 | − | 0.195405i | −1.89649 | 1.92363 | − | 0.547388i | −1.00000 | −2.65631 | + | 0.370584i | − | 2.89109i | 2.58737 | − | 1.14259i | 0.596662 | −1.40065 | + | 0.195405i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 151.20 | 1.40065 | + | 0.195405i | −1.89649 | 1.92363 | + | 0.547388i | −1.00000 | −2.65631 | − | 0.370584i | 2.89109i | 2.58737 | + | 1.14259i | 0.596662 | −1.40065 | − | 0.195405i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 19.b | odd | 2 | 1 | inner |
| 76.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 380.2.f.a | ✓ | 20 |
| 4.b | odd | 2 | 1 | inner | 380.2.f.a | ✓ | 20 |
| 19.b | odd | 2 | 1 | inner | 380.2.f.a | ✓ | 20 |
| 76.d | even | 2 | 1 | inner | 380.2.f.a | ✓ | 20 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 380.2.f.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 380.2.f.a | ✓ | 20 | 4.b | odd | 2 | 1 | inner |
| 380.2.f.a | ✓ | 20 | 19.b | odd | 2 | 1 | inner |
| 380.2.f.a | ✓ | 20 | 76.d | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 19T_{3}^{8} + 110T_{3}^{6} - 218T_{3}^{4} + 80T_{3}^{2} - 8 \)
acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - T^{18} + \cdots + 1024 \)
|
| $3$ |
\( (T^{10} - 19 T^{8} + \cdots - 8)^{2} \)
|
| $5$ |
\( (T + 1)^{20} \)
|
| $7$ |
\( (T^{10} + 35 T^{8} + \cdots + 5184)^{2} \)
|
| $11$ |
\( (T^{10} + 60 T^{8} + \cdots + 4096)^{2} \)
|
| $13$ |
\( (T^{10} + 65 T^{8} + \cdots + 648)^{2} \)
|
| $17$ |
\( (T^{5} - 5 T^{4} + \cdots - 672)^{4} \)
|
| $19$ |
\( T^{20} + \cdots + 6131066257801 \)
|
| $23$ |
\( (T^{10} + 163 T^{8} + \cdots + 9684544)^{2} \)
|
| $29$ |
\( (T^{10} + 197 T^{8} + \cdots + 23011328)^{2} \)
|
| $31$ |
\( (T^{10} - 152 T^{8} + \cdots - 401408)^{2} \)
|
| $37$ |
\( (T^{10} + 104 T^{8} + \cdots + 288)^{2} \)
|
| $41$ |
\( (T^{10} + 128 T^{8} + \cdots + 32768)^{2} \)
|
| $43$ |
\( (T^{10} + 160 T^{8} + \cdots + 2304)^{2} \)
|
| $47$ |
\( (T^{10} + 168 T^{8} + \cdots + 215296)^{2} \)
|
| $53$ |
\( (T^{10} + 265 T^{8} + \cdots + 17310728)^{2} \)
|
| $59$ |
\( (T^{10} - 195 T^{8} + \cdots - 49840128)^{2} \)
|
| $61$ |
\( (T^{5} + 6 T^{4} + \cdots + 2656)^{4} \)
|
| $67$ |
\( (T^{10} - 283 T^{8} + \cdots - 71472968)^{2} \)
|
| $71$ |
\( (T^{10} - 600 T^{8} + \cdots - 3612672)^{2} \)
|
| $73$ |
\( (T^{5} + 9 T^{4} + \cdots + 29936)^{4} \)
|
| $79$ |
\( (T^{10} - 644 T^{8} + \cdots - 9152503808)^{2} \)
|
| $83$ |
\( (T^{10} + 604 T^{8} + \cdots + 1885643776)^{2} \)
|
| $89$ |
\( (T^{10} + 580 T^{8} + \cdots + 1763704832)^{2} \)
|
| $97$ |
\( (T^{10} + 344 T^{8} + \cdots + 691488)^{2} \)
|
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