Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [44,2,Mod(7,44)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(44, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 7]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("44.7");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 44 = 2^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 44.g (of order \(10\), degree \(4\), 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: | \(0.351341768894\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 5 x^{15} + 13 x^{14} - 25 x^{13} + 35 x^{12} - 30 x^{11} - 2 x^{10} + 60 x^{9} - 116 x^{8} + \cdots + 256 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} - 5 x^{15} + 13 x^{14} - 25 x^{13} + 35 x^{12} - 30 x^{11} - 2 x^{10} + 60 x^{9} - 116 x^{8} + \cdots + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{15} - 5 \nu^{14} + 13 \nu^{13} - 25 \nu^{12} + 35 \nu^{11} - 30 \nu^{10} - 2 \nu^{9} + 60 \nu^{8} + \cdots - 640 ) / 128 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 35 \nu^{15} + 94 \nu^{14} - 210 \nu^{13} + 334 \nu^{12} - 328 \nu^{11} + 111 \nu^{10} + 476 \nu^{9} + \cdots + 2304 ) / 64 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 30 \nu^{15} + 141 \nu^{14} - 309 \nu^{13} + 573 \nu^{12} - 693 \nu^{11} + 429 \nu^{10} + 398 \nu^{9} + \cdots + 9664 ) / 64 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 33 \nu^{15} - 147 \nu^{14} + 323 \nu^{13} - 591 \nu^{12} + 705 \nu^{11} - 424 \nu^{10} - 446 \nu^{9} + \cdots - 9472 ) / 64 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 37 \nu^{15} + 152 \nu^{14} - 334 \nu^{13} + 602 \nu^{12} - 704 \nu^{11} + 405 \nu^{10} + 498 \nu^{9} + \cdots + 9088 ) / 64 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 35 \nu^{15} - 120 \nu^{14} + 265 \nu^{13} - 455 \nu^{12} + 501 \nu^{11} - 248 \nu^{10} - 475 \nu^{9} + \cdots - 5536 ) / 32 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 47 \nu^{15} + 153 \nu^{14} - 338 \nu^{13} + 572 \nu^{12} - 618 \nu^{11} + 291 \nu^{10} + 633 \nu^{9} + \cdots + 6464 ) / 32 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 71 \nu^{15} - 281 \nu^{14} + 619 \nu^{13} - 1107 \nu^{12} + 1281 \nu^{11} - 722 \nu^{10} - 952 \nu^{9} + \cdots - 16128 ) / 64 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 279 \nu^{15} + 987 \nu^{14} - 2179 \nu^{13} + 3775 \nu^{12} - 4213 \nu^{11} + 2162 \nu^{10} + \cdots + 48384 ) / 128 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 307 \nu^{15} - 1097 \nu^{14} + 2421 \nu^{13} - 4209 \nu^{12} + 4711 \nu^{11} - 2444 \nu^{10} + \cdots - 54528 ) / 128 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 293 \nu^{15} + 1079 \nu^{14} - 2383 \nu^{13} + 4179 \nu^{12} - 4733 \nu^{11} + 2528 \nu^{10} + \cdots + 56320 ) / 128 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 162 \nu^{15} - 591 \nu^{14} + 1304 \nu^{13} - 2280 \nu^{12} + 2572 \nu^{11} - 1360 \nu^{10} + \cdots - 30400 ) / 32 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 361 \nu^{15} + 1262 \nu^{14} - 2786 \nu^{13} + 4814 \nu^{12} - 5348 \nu^{11} + 2721 \nu^{10} + \cdots + 60672 ) / 64 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 785 \nu^{15} + 2815 \nu^{14} - 6215 \nu^{13} + 10819 \nu^{12} - 12133 \nu^{11} + 6324 \nu^{10} + \cdots + 141056 ) / 128 \)
|
| \(\nu\) | \(=\) |
\( -\beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + 2\beta_{9} + \beta_{6} + \beta_{4} + \beta_{2} + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( - \beta_{15} + \beta_{13} + 3 \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{8} - \beta_{7} + \beta_{6} + \cdots + 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{13} - 2\beta_{11} + 3\beta_{6} - 2\beta_{4} - \beta_{3} - 2\beta_{2} - 2\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{14} - 2 \beta_{12} + 4 \beta_{11} + \beta_{10} - 5 \beta_{9} + \beta_{8} - \beta_{7} + \cdots + 2 \beta_1 \)
|
| \(\nu^{7}\) | \(=\) |
\( 6 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} - \beta_{11} + \beta_{10} + 5 \beta_{9} + 3 \beta_{8} + \cdots + 10 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 5 \beta_{15} - 2 \beta_{14} - \beta_{13} + 3 \beta_{12} + 2 \beta_{11} - \beta_{10} - 8 \beta_{9} + \cdots + 6 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 7 \beta_{15} + 6 \beta_{14} + 3 \beta_{13} + \beta_{12} + 3 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} + \cdots + 4 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 2 \beta_{15} + 14 \beta_{14} + 9 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{9} + \cdots + 2 \beta_1 \)
|
| \(\nu^{11}\) | \(=\) |
\( 26 \beta_{15} + 21 \beta_{14} + 40 \beta_{13} - 24 \beta_{12} - 20 \beta_{11} + 19 \beta_{10} + \cdots + 28 \)
|
| \(\nu^{12}\) | \(=\) |
\( 20 \beta_{15} - \beta_{14} - \beta_{13} - 30 \beta_{12} + \beta_{11} - 23 \beta_{10} - 21 \beta_{9} + \cdots + 10 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 21 \beta_{15} + 8 \beta_{14} + 11 \beta_{13} + 31 \beta_{12} + 24 \beta_{11} - 11 \beta_{10} + \cdots + 50 \)
|
| \(\nu^{14}\) | \(=\) |
\( 7 \beta_{15} + 45 \beta_{13} + 19 \beta_{12} - 33 \beta_{11} - 9 \beta_{10} + 52 \beta_{9} + 62 \beta_{8} + \cdots + 32 \)
|
| \(\nu^{15}\) | \(=\) |
\( 20 \beta_{15} + 20 \beta_{14} + 61 \beta_{13} - 56 \beta_{12} + 22 \beta_{11} + 40 \beta_{10} + \cdots + 68 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/44\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) | \(23\) |
| \(\chi(n)\) | \(-\beta_{12}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
−1.40958 | − | 0.114404i | 0.704424 | + | 0.228881i | 1.97382 | + | 0.322523i | 1.09089 | − | 0.792578i | −0.966756 | − | 0.403215i | 0.503194 | + | 1.54867i | −2.74536 | − | 0.680436i | −1.98322 | − | 1.44090i | −1.62837 | + | 0.992398i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 7.2 | −0.0737040 | − | 1.41229i | 1.70537 | + | 0.554109i | −1.98914 | + | 0.208183i | −2.39991 | + | 1.74363i | 0.656871 | − | 2.44932i | −0.815620 | − | 2.51022i | 0.440622 | + | 2.79390i | 0.174207 | + | 0.126569i | 2.63940 | + | 3.26086i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 7.3 | 0.544389 | + | 1.30524i | −0.704424 | − | 0.228881i | −1.40728 | + | 1.42111i | 1.09089 | − | 0.792578i | −0.0847364 | − | 1.04404i | −0.503194 | − | 1.54867i | −2.62099 | − | 1.06320i | −1.98322 | − | 1.44090i | 1.62837 | + | 0.992398i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 7.4 | 1.36594 | − | 0.366325i | −1.70537 | − | 0.554109i | 1.73161 | − | 1.00076i | −2.39991 | + | 1.74363i | −2.53243 | − | 0.132161i | 0.815620 | + | 2.51022i | 1.99868 | − | 2.00132i | 0.174207 | + | 0.126569i | −2.63940 | + | 3.26086i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 19.1 | −1.40958 | + | 0.114404i | 0.704424 | − | 0.228881i | 1.97382 | − | 0.322523i | 1.09089 | + | 0.792578i | −0.966756 | + | 0.403215i | 0.503194 | − | 1.54867i | −2.74536 | + | 0.680436i | −1.98322 | + | 1.44090i | −1.62837 | − | 0.992398i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 19.2 | −0.0737040 | + | 1.41229i | 1.70537 | − | 0.554109i | −1.98914 | − | 0.208183i | −2.39991 | − | 1.74363i | 0.656871 | + | 2.44932i | −0.815620 | + | 2.51022i | 0.440622 | − | 2.79390i | 0.174207 | − | 0.126569i | 2.63940 | − | 3.26086i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 19.3 | 0.544389 | − | 1.30524i | −0.704424 | + | 0.228881i | −1.40728 | − | 1.42111i | 1.09089 | + | 0.792578i | −0.0847364 | + | 1.04404i | −0.503194 | + | 1.54867i | −2.62099 | + | 1.06320i | −1.98322 | + | 1.44090i | 1.62837 | − | 0.992398i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 19.4 | 1.36594 | + | 0.366325i | −1.70537 | + | 0.554109i | 1.73161 | + | 1.00076i | −2.39991 | − | 1.74363i | −2.53243 | + | 0.132161i | 0.815620 | − | 2.51022i | 1.99868 | + | 2.00132i | 0.174207 | − | 0.126569i | −2.63940 | − | 3.26086i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 35.1 | −1.40874 | + | 0.124276i | −1.59814 | − | 2.19965i | 1.96911 | − | 0.350146i | −0.720859 | − | 2.21858i | 2.52473 | + | 2.90013i | 1.04462 | + | 0.758960i | −2.73046 | + | 0.737979i | −1.35736 | + | 4.17752i | 1.29122 | + | 3.03582i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 35.2 | −1.06665 | − | 0.928579i | 1.59814 | + | 2.19965i | 0.275480 | + | 1.98094i | −0.720859 | − | 2.21858i | 0.337896 | − | 3.83025i | −1.04462 | − | 0.758960i | 1.54562 | − | 2.36877i | −1.35736 | + | 4.17752i | −1.29122 | + | 3.03582i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 35.3 | −0.656642 | + | 1.25253i | 0.539857 | + | 0.743049i | −1.13764 | − | 1.64492i | 0.529876 | + | 1.63079i | −1.28518 | + | 0.188268i | −1.93399 | − | 1.40513i | 2.80733 | − | 0.344804i | 0.666375 | − | 2.05089i | −2.39055 | − | 0.407162i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 35.4 | 0.204982 | − | 1.39928i | −0.539857 | − | 0.743049i | −1.91596 | − | 0.573655i | 0.529876 | + | 1.63079i | −1.15039 | + | 0.603098i | 1.93399 | + | 1.40513i | −1.19544 | + | 2.56338i | 0.666375 | − | 2.05089i | 2.39055 | − | 0.407162i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 39.1 | −1.40874 | − | 0.124276i | −1.59814 | + | 2.19965i | 1.96911 | + | 0.350146i | −0.720859 | + | 2.21858i | 2.52473 | − | 2.90013i | 1.04462 | − | 0.758960i | −2.73046 | − | 0.737979i | −1.35736 | − | 4.17752i | 1.29122 | − | 3.03582i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 39.2 | −1.06665 | + | 0.928579i | 1.59814 | − | 2.19965i | 0.275480 | − | 1.98094i | −0.720859 | + | 2.21858i | 0.337896 | + | 3.83025i | −1.04462 | + | 0.758960i | 1.54562 | + | 2.36877i | −1.35736 | − | 4.17752i | −1.29122 | − | 3.03582i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 39.3 | −0.656642 | − | 1.25253i | 0.539857 | − | 0.743049i | −1.13764 | + | 1.64492i | 0.529876 | − | 1.63079i | −1.28518 | − | 0.188268i | −1.93399 | + | 1.40513i | 2.80733 | + | 0.344804i | 0.666375 | + | 2.05089i | −2.39055 | + | 0.407162i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 39.4 | 0.204982 | + | 1.39928i | −0.539857 | + | 0.743049i | −1.91596 | + | 0.573655i | 0.529876 | − | 1.63079i | −1.15039 | − | 0.603098i | 1.93399 | − | 1.40513i | −1.19544 | − | 2.56338i | 0.666375 | + | 2.05089i | 2.39055 | + | 0.407162i | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 11.d | odd | 10 | 1 | inner |
| 44.g | even | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 44.2.g.a | ✓ | 16 |
| 3.b | odd | 2 | 1 | 396.2.r.a | 16 | ||
| 4.b | odd | 2 | 1 | inner | 44.2.g.a | ✓ | 16 |
| 8.b | even | 2 | 1 | 704.2.u.c | 16 | ||
| 8.d | odd | 2 | 1 | 704.2.u.c | 16 | ||
| 11.b | odd | 2 | 1 | 484.2.g.i | 16 | ||
| 11.c | even | 5 | 1 | 484.2.c.d | 16 | ||
| 11.c | even | 5 | 1 | 484.2.g.f | 16 | ||
| 11.c | even | 5 | 1 | 484.2.g.i | 16 | ||
| 11.c | even | 5 | 1 | 484.2.g.j | 16 | ||
| 11.d | odd | 10 | 1 | inner | 44.2.g.a | ✓ | 16 |
| 11.d | odd | 10 | 1 | 484.2.c.d | 16 | ||
| 11.d | odd | 10 | 1 | 484.2.g.f | 16 | ||
| 11.d | odd | 10 | 1 | 484.2.g.j | 16 | ||
| 12.b | even | 2 | 1 | 396.2.r.a | 16 | ||
| 33.f | even | 10 | 1 | 396.2.r.a | 16 | ||
| 44.c | even | 2 | 1 | 484.2.g.i | 16 | ||
| 44.g | even | 10 | 1 | inner | 44.2.g.a | ✓ | 16 |
| 44.g | even | 10 | 1 | 484.2.c.d | 16 | ||
| 44.g | even | 10 | 1 | 484.2.g.f | 16 | ||
| 44.g | even | 10 | 1 | 484.2.g.j | 16 | ||
| 44.h | odd | 10 | 1 | 484.2.c.d | 16 | ||
| 44.h | odd | 10 | 1 | 484.2.g.f | 16 | ||
| 44.h | odd | 10 | 1 | 484.2.g.i | 16 | ||
| 44.h | odd | 10 | 1 | 484.2.g.j | 16 | ||
| 88.k | even | 10 | 1 | 704.2.u.c | 16 | ||
| 88.p | odd | 10 | 1 | 704.2.u.c | 16 | ||
| 132.n | odd | 10 | 1 | 396.2.r.a | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 44.2.g.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 44.2.g.a | ✓ | 16 | 4.b | odd | 2 | 1 | inner |
| 44.2.g.a | ✓ | 16 | 11.d | odd | 10 | 1 | inner |
| 44.2.g.a | ✓ | 16 | 44.g | even | 10 | 1 | inner |
| 396.2.r.a | 16 | 3.b | odd | 2 | 1 | ||
| 396.2.r.a | 16 | 12.b | even | 2 | 1 | ||
| 396.2.r.a | 16 | 33.f | even | 10 | 1 | ||
| 396.2.r.a | 16 | 132.n | odd | 10 | 1 | ||
| 484.2.c.d | 16 | 11.c | even | 5 | 1 | ||
| 484.2.c.d | 16 | 11.d | odd | 10 | 1 | ||
| 484.2.c.d | 16 | 44.g | even | 10 | 1 | ||
| 484.2.c.d | 16 | 44.h | odd | 10 | 1 | ||
| 484.2.g.f | 16 | 11.c | even | 5 | 1 | ||
| 484.2.g.f | 16 | 11.d | odd | 10 | 1 | ||
| 484.2.g.f | 16 | 44.g | even | 10 | 1 | ||
| 484.2.g.f | 16 | 44.h | odd | 10 | 1 | ||
| 484.2.g.i | 16 | 11.b | odd | 2 | 1 | ||
| 484.2.g.i | 16 | 11.c | even | 5 | 1 | ||
| 484.2.g.i | 16 | 44.c | even | 2 | 1 | ||
| 484.2.g.i | 16 | 44.h | odd | 10 | 1 | ||
| 484.2.g.j | 16 | 11.c | even | 5 | 1 | ||
| 484.2.g.j | 16 | 11.d | odd | 10 | 1 | ||
| 484.2.g.j | 16 | 44.g | even | 10 | 1 | ||
| 484.2.g.j | 16 | 44.h | odd | 10 | 1 | ||
| 704.2.u.c | 16 | 8.b | even | 2 | 1 | ||
| 704.2.u.c | 16 | 8.d | odd | 2 | 1 | ||
| 704.2.u.c | 16 | 88.k | even | 10 | 1 | ||
| 704.2.u.c | 16 | 88.p | odd | 10 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(44, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + 5 T^{15} + \cdots + 256 \)
|
| $3$ |
\( T^{16} - T^{14} + \cdots + 121 \)
|
| $5$ |
\( (T^{8} + 3 T^{7} + \cdots + 256)^{2} \)
|
| $7$ |
\( T^{16} + 11 T^{14} + \cdots + 30976 \)
|
| $11$ |
\( T^{16} + \cdots + 214358881 \)
|
| $13$ |
\( (T^{8} + 5 T^{7} + 16 T^{6} + \cdots + 16)^{2} \)
|
| $17$ |
\( (T^{8} + 5 T^{7} - 24 T^{6} + \cdots + 1)^{2} \)
|
| $19$ |
\( T^{16} + 106 T^{14} + \cdots + 1771561 \)
|
| $23$ |
\( (T^{8} + 88 T^{6} + \cdots + 2816)^{2} \)
|
| $29$ |
\( (T^{8} + 5 T^{7} + \cdots + 400)^{2} \)
|
| $31$ |
\( T^{16} + \cdots + 428888770816 \)
|
| $37$ |
\( (T^{8} - 9 T^{7} + \cdots + 13456)^{2} \)
|
| $41$ |
\( (T^{8} - 5 T^{7} + \cdots + 2588881)^{2} \)
|
| $43$ |
\( (T^{8} - 143 T^{6} + \cdots + 148016)^{2} \)
|
| $47$ |
\( T^{16} + \cdots + 1206517709056 \)
|
| $53$ |
\( (T^{8} - 19 T^{7} + \cdots + 15376)^{2} \)
|
| $59$ |
\( T^{16} + \cdots + 17080137481 \)
|
| $61$ |
\( (T^{8} + 5 T^{7} + \cdots + 633616)^{2} \)
|
| $67$ |
\( (T^{8} + 185 T^{6} + \cdots + 1760000)^{2} \)
|
| $71$ |
\( T^{16} + \cdots + 21908736256 \)
|
| $73$ |
\( (T^{8} + 15 T^{7} + \cdots + 6561)^{2} \)
|
| $79$ |
\( T^{16} + \cdots + 28606986496 \)
|
| $83$ |
\( T^{16} + \cdots + 435963075625 \)
|
| $89$ |
\( (T^{4} + 9 T^{3} + \cdots + 116)^{4} \)
|
| $97$ |
\( (T^{4} + 17 T^{3} + \cdots + 3721)^{4} \)
|
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