Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3872,2,Mod(1937,3872)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3872, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("3872.1937");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3872 = 2^{5} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3872.c (of order \(2\), degree \(1\), 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: | \(30.9180756626\) |
| 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: |
\( x^{20} - x^{18} - 2 x^{16} - 2 x^{15} - 4 x^{14} - 4 x^{13} + 12 x^{12} + 16 x^{11} + 32 x^{9} + \cdots + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{12} \) |
| Twist minimal: | no (minimal twist has level 88) |
| 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
\( x^{20} - x^{18} - 2 x^{16} - 2 x^{15} - 4 x^{14} - 4 x^{13} + 12 x^{12} + 16 x^{11} + 32 x^{9} + \cdots + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{18} + 3 \nu^{16} + 10 \nu^{14} - 2 \nu^{13} - 12 \nu^{12} - 12 \nu^{11} + 12 \nu^{10} + \cdots - 1024 ) / 768 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 3 \nu^{19} + 10 \nu^{18} + 39 \nu^{17} + 30 \nu^{16} + 18 \nu^{15} - 86 \nu^{14} - 128 \nu^{13} + \cdots - 2560 ) / 12288 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9 \nu^{19} + 34 \nu^{18} - 21 \nu^{17} - 90 \nu^{16} + 42 \nu^{15} - 62 \nu^{14} - 128 \nu^{13} + \cdots + 3584 ) / 12288 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3 \nu^{19} + 2 \nu^{18} + 9 \nu^{17} + 6 \nu^{16} + 30 \nu^{15} + 14 \nu^{14} + 8 \nu^{13} + \cdots + 2560 ) / 3072 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3 \nu^{19} + 2 \nu^{18} + 15 \nu^{17} + 6 \nu^{16} - 6 \nu^{15} + 26 \nu^{14} + 80 \nu^{13} + \cdots + 5632 ) / 3072 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{19} + 2 \nu^{18} + 3 \nu^{17} + 6 \nu^{16} + 10 \nu^{15} + 18 \nu^{14} + 32 \nu^{13} + \cdots + 512 ) / 1024 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 11 \nu^{19} - 54 \nu^{18} - 81 \nu^{17} - 66 \nu^{16} + 34 \nu^{15} + 154 \nu^{14} + 288 \nu^{13} + \cdots + 26112 ) / 12288 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 11 \nu^{19} + 54 \nu^{18} - 15 \nu^{17} + 66 \nu^{16} + 62 \nu^{15} - 154 \nu^{14} - 96 \nu^{13} + \cdots - 26112 ) / 12288 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 5 \nu^{19} + 6 \nu^{18} + \nu^{17} + 18 \nu^{16} + 30 \nu^{15} + 6 \nu^{14} + 52 \nu^{12} + \cdots + 2560 ) / 4096 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 13 \nu^{19} - 42 \nu^{18} + 9 \nu^{17} - 30 \nu^{16} - 82 \nu^{15} - 106 \nu^{14} + 384 \nu^{13} + \cdots + 23040 ) / 12288 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 5 \nu^{19} - 10 \nu^{18} - 31 \nu^{17} + 2 \nu^{16} - 2 \nu^{15} + 6 \nu^{14} + 160 \nu^{13} + \cdots + 6656 ) / 4096 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 7 \nu^{19} + 2 \nu^{18} - 5 \nu^{17} + 6 \nu^{16} - 54 \nu^{15} - 30 \nu^{14} - 68 \nu^{12} + \cdots - 8704 ) / 4096 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 21 \nu^{19} - 38 \nu^{18} + 15 \nu^{17} + 78 \nu^{16} - 30 \nu^{15} - 38 \nu^{14} - 128 \nu^{13} + \cdots + 9728 ) / 12288 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 23 \nu^{19} - 30 \nu^{18} - 21 \nu^{17} + 6 \nu^{16} + 10 \nu^{15} + 34 \nu^{14} + 192 \nu^{13} + \cdots + 19968 ) / 12288 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - \nu^{19} + 2 \nu^{18} + \nu^{17} - 2 \nu^{16} + 2 \nu^{15} - 2 \nu^{14} - 4 \nu^{12} - 20 \nu^{11} + \cdots - 512 ) / 512 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 3 \nu^{19} - 4 \nu^{18} + 3 \nu^{17} + 12 \nu^{16} + 6 \nu^{15} + 38 \nu^{14} + 68 \nu^{13} + \cdots + 1024 ) / 1536 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - \nu^{19} - 2 \nu^{18} + \nu^{17} + 2 \nu^{16} + 2 \nu^{15} + 6 \nu^{14} + 8 \nu^{13} + 12 \nu^{12} + \cdots + 512 ) / 512 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( - \nu^{19} - \nu^{15} + 2 \nu^{14} + 6 \nu^{13} + 6 \nu^{12} + 12 \nu^{11} - 4 \nu^{10} - 12 \nu^{9} + \cdots + 384 ) / 384 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 4 \nu^{19} - 6 \nu^{18} - 3 \nu^{17} - 6 \nu^{16} + 5 \nu^{15} + 20 \nu^{14} + 36 \nu^{13} + \cdots + 2688 ) / 384 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{17} - \beta_{16} + \beta_{15} + \beta_{13} + \beta_{6} - \beta_{4} - \beta_{2} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{17} - \beta_{16} - \beta_{15} - \beta_{13} + \beta_{6} - \beta_{4} + \beta_{2} ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{17} - \beta_{16} + \beta_{15} + \beta_{13} - 2 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + \cdots + 2 \beta_1 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{17} - \beta_{16} + \beta_{15} - 4 \beta_{14} + \beta_{13} + 4 \beta_{12} + 2 \beta_{9} - 2 \beta_{8} + \cdots + 4 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4 \beta_{18} + \beta_{17} - \beta_{16} + \beta_{15} - 4 \beta_{14} + \beta_{13} + 4 \beta_{11} + \cdots + 2 \beta_1 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 4 \beta_{18} + \beta_{17} - \beta_{16} + \beta_{15} + 8 \beta_{14} + \beta_{13} + 4 \beta_{12} + \cdots + 8 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 4 \beta_{18} + \beta_{17} - 9 \beta_{16} + \beta_{15} + 16 \beta_{14} + \beta_{13} + 4 \beta_{12} + \cdots - 2 \beta_1 ) / 4 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 8 \beta_{19} - 4 \beta_{18} + \beta_{17} - 17 \beta_{16} - 7 \beta_{15} - 8 \beta_{14} + 9 \beta_{13} + \cdots + 6 \beta_1 ) / 4 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( - 4 \beta_{18} + \beta_{17} - 9 \beta_{16} + \beta_{15} + 8 \beta_{14} + \beta_{13} + 4 \beta_{12} + \cdots - 24 ) / 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 8 \beta_{19} - 4 \beta_{18} - 15 \beta_{17} - 17 \beta_{16} + \beta_{15} - 16 \beta_{14} + \beta_{13} + \cdots + 32 ) / 4 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( 44 \beta_{18} - 15 \beta_{17} - 9 \beta_{16} + \beta_{15} + 8 \beta_{14} + 33 \beta_{13} + 20 \beta_{12} + \cdots - 56 ) / 4 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 8 \beta_{19} - 4 \beta_{18} - 15 \beta_{17} - 33 \beta_{16} - 15 \beta_{15} + 96 \beta_{14} + \cdots - 112 ) / 4 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 16 \beta_{19} - 20 \beta_{18} - 47 \beta_{17} + 7 \beta_{16} - 15 \beta_{15} + 56 \beta_{14} - 15 \beta_{13} + \cdots - 8 ) / 4 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 40 \beta_{19} + 28 \beta_{18} - 47 \beta_{17} - 49 \beta_{16} - 15 \beta_{15} - 112 \beta_{14} + \cdots + 86 \beta_1 ) / 4 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 64 \beta_{19} - 84 \beta_{18} - 47 \beta_{17} + 23 \beta_{16} - 47 \beta_{15} - 24 \beta_{14} + \cdots - 152 ) / 4 \)
|
| \(\nu^{16}\) | \(=\) |
\( ( - 40 \beta_{19} - 68 \beta_{18} - 111 \beta_{17} + 31 \beta_{16} - 143 \beta_{15} + 480 \beta_{14} + \cdots - 240 ) / 4 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( - 16 \beta_{19} + 76 \beta_{18} - 111 \beta_{17} + 39 \beta_{16} + 81 \beta_{15} + 88 \beta_{14} + \cdots - 328 ) / 4 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 56 \beta_{19} + 412 \beta_{18} - 303 \beta_{17} - 81 \beta_{16} - 79 \beta_{15} - 432 \beta_{14} + \cdots - 32 ) / 4 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 160 \beta_{19} - 532 \beta_{18} - 175 \beta_{17} + 183 \beta_{16} + 17 \beta_{15} - 184 \beta_{14} + \cdots - 440 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3872\mathbb{Z}\right)^\times\).
| \(n\) | \(485\) | \(1695\) | \(2785\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1937.1 |
|
0 | − | 2.85679i | 0 | − | 2.60791i | 0 | 0.196362 | 0 | −5.16127 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.2 | 0 | − | 2.75783i | 0 | 2.90574i | 0 | 2.36352 | 0 | −4.60564 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.3 | 0 | − | 2.55485i | 0 | 2.69541i | 0 | 3.94908 | 0 | −3.52726 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.4 | 0 | − | 2.44793i | 0 | − | 0.200474i | 0 | −2.34615 | 0 | −2.99235 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.5 | 0 | − | 1.87996i | 0 | − | 0.493106i | 0 | 0.174770 | 0 | −0.534260 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.6 | 0 | − | 1.28633i | 0 | 1.58193i | 0 | −1.86825 | 0 | 1.34535 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.7 | 0 | − | 0.868641i | 0 | − | 3.67418i | 0 | −1.19528 | 0 | 2.24546 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.8 | 0 | − | 0.612207i | 0 | − | 1.46030i | 0 | 4.42813 | 0 | 2.62520 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.9 | 0 | − | 0.540427i | 0 | − | 2.90090i | 0 | 2.60451 | 0 | 2.70794 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.10 | 0 | − | 0.321222i | 0 | − | 1.28733i | 0 | −3.30669 | 0 | 2.89682 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.11 | 0 | 0.321222i | 0 | 1.28733i | 0 | −3.30669 | 0 | 2.89682 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.12 | 0 | 0.540427i | 0 | 2.90090i | 0 | 2.60451 | 0 | 2.70794 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.13 | 0 | 0.612207i | 0 | 1.46030i | 0 | 4.42813 | 0 | 2.62520 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.14 | 0 | 0.868641i | 0 | 3.67418i | 0 | −1.19528 | 0 | 2.24546 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.15 | 0 | 1.28633i | 0 | − | 1.58193i | 0 | −1.86825 | 0 | 1.34535 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.16 | 0 | 1.87996i | 0 | 0.493106i | 0 | 0.174770 | 0 | −0.534260 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.17 | 0 | 2.44793i | 0 | 0.200474i | 0 | −2.34615 | 0 | −2.99235 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.18 | 0 | 2.55485i | 0 | − | 2.69541i | 0 | 3.94908 | 0 | −3.52726 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.19 | 0 | 2.75783i | 0 | − | 2.90574i | 0 | 2.36352 | 0 | −4.60564 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1937.20 | 0 | 2.85679i | 0 | 2.60791i | 0 | 0.196362 | 0 | −5.16127 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3872.2.c.i | 20 | |
| 4.b | odd | 2 | 1 | 968.2.c.i | 20 | ||
| 8.b | even | 2 | 1 | inner | 3872.2.c.i | 20 | |
| 8.d | odd | 2 | 1 | 968.2.c.i | 20 | ||
| 11.b | odd | 2 | 1 | 3872.2.c.h | 20 | ||
| 11.d | odd | 10 | 2 | 352.2.w.a | 40 | ||
| 44.c | even | 2 | 1 | 968.2.c.h | 20 | ||
| 44.g | even | 10 | 2 | 88.2.o.a | ✓ | 40 | |
| 44.g | even | 10 | 2 | 968.2.o.j | 40 | ||
| 44.h | odd | 10 | 2 | 968.2.o.d | 40 | ||
| 44.h | odd | 10 | 2 | 968.2.o.i | 40 | ||
| 88.b | odd | 2 | 1 | 3872.2.c.h | 20 | ||
| 88.g | even | 2 | 1 | 968.2.c.h | 20 | ||
| 88.k | even | 10 | 2 | 88.2.o.a | ✓ | 40 | |
| 88.k | even | 10 | 2 | 968.2.o.j | 40 | ||
| 88.l | odd | 10 | 2 | 968.2.o.d | 40 | ||
| 88.l | odd | 10 | 2 | 968.2.o.i | 40 | ||
| 88.p | odd | 10 | 2 | 352.2.w.a | 40 | ||
| 132.n | odd | 10 | 2 | 792.2.br.b | 40 | ||
| 264.r | odd | 10 | 2 | 792.2.br.b | 40 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 88.2.o.a | ✓ | 40 | 44.g | even | 10 | 2 | |
| 88.2.o.a | ✓ | 40 | 88.k | even | 10 | 2 | |
| 352.2.w.a | 40 | 11.d | odd | 10 | 2 | ||
| 352.2.w.a | 40 | 88.p | odd | 10 | 2 | ||
| 792.2.br.b | 40 | 132.n | odd | 10 | 2 | ||
| 792.2.br.b | 40 | 264.r | odd | 10 | 2 | ||
| 968.2.c.h | 20 | 44.c | even | 2 | 1 | ||
| 968.2.c.h | 20 | 88.g | even | 2 | 1 | ||
| 968.2.c.i | 20 | 4.b | odd | 2 | 1 | ||
| 968.2.c.i | 20 | 8.d | odd | 2 | 1 | ||
| 968.2.o.d | 40 | 44.h | odd | 10 | 2 | ||
| 968.2.o.d | 40 | 88.l | odd | 10 | 2 | ||
| 968.2.o.i | 40 | 44.h | odd | 10 | 2 | ||
| 968.2.o.i | 40 | 88.l | odd | 10 | 2 | ||
| 968.2.o.j | 40 | 44.g | even | 10 | 2 | ||
| 968.2.o.j | 40 | 88.k | even | 10 | 2 | ||
| 3872.2.c.h | 20 | 11.b | odd | 2 | 1 | ||
| 3872.2.c.h | 20 | 88.b | odd | 2 | 1 | ||
| 3872.2.c.i | 20 | 1.a | even | 1 | 1 | trivial | |
| 3872.2.c.i | 20 | 8.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 \(S_{2}^{\mathrm{new}}(3872, [\chi])\):
|
\( T_{3}^{20} + 35 T_{3}^{18} + 503 T_{3}^{16} + 3822 T_{3}^{14} + 16493 T_{3}^{12} + 40565 T_{3}^{10} + \cdots + 121 \)
|
|
\( T_{5}^{20} + 51 T_{5}^{18} + 1082 T_{5}^{16} + 12427 T_{5}^{14} + 84082 T_{5}^{12} + 341655 T_{5}^{10} + \cdots + 4096 \)
|
|
\( T_{7}^{10} - 5 T_{7}^{9} - 22 T_{7}^{8} + 111 T_{7}^{7} + 176 T_{7}^{6} - 775 T_{7}^{5} - 701 T_{7}^{4} + \cdots + 64 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} + 35 T^{18} + \cdots + 121 \)
|
| $5$ |
\( T^{20} + 51 T^{18} + \cdots + 4096 \)
|
| $7$ |
\( (T^{10} - 5 T^{9} + \cdots + 64)^{2} \)
|
| $11$ |
\( T^{20} \)
|
| $13$ |
\( T^{20} + 119 T^{18} + \cdots + 5308416 \)
|
| $17$ |
\( (T^{10} + T^{9} + \cdots - 34029)^{2} \)
|
| $19$ |
\( T^{20} + 147 T^{18} + \cdots + 31036041 \)
|
| $23$ |
\( (T^{10} - 2 T^{9} + \cdots - 451584)^{2} \)
|
| $29$ |
\( T^{20} + \cdots + 527896576 \)
|
| $31$ |
\( (T^{10} + T^{9} + \cdots - 590144)^{2} \)
|
| $37$ |
\( T^{20} + \cdots + 478554034176 \)
|
| $41$ |
\( (T^{10} - T^{9} + \cdots + 685431)^{2} \)
|
| $43$ |
\( T^{20} + \cdots + 258242846976 \)
|
| $47$ |
\( (T^{10} + T^{9} + \cdots + 251136)^{2} \)
|
| $53$ |
\( T^{20} + \cdots + 4813029376 \)
|
| $59$ |
\( T^{20} + \cdots + 91511695081 \)
|
| $61$ |
\( T^{20} + \cdots + 47\!\cdots\!96 \)
|
| $67$ |
\( T^{20} + \cdots + 22416209899776 \)
|
| $71$ |
\( (T^{10} - 17 T^{9} + \cdots - 2283264)^{2} \)
|
| $73$ |
\( (T^{10} + T^{9} + \cdots + 4889699)^{2} \)
|
| $79$ |
\( (T^{10} + 29 T^{9} + \cdots + 36844544)^{2} \)
|
| $83$ |
\( T^{20} + \cdots + 17926799881 \)
|
| $89$ |
\( (T^{10} + 4 T^{9} + \cdots - 6007233744)^{2} \)
|
| $97$ |
\( (T^{10} - 5 T^{9} + \cdots - 140501)^{2} \)
|
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