Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [576,3,Mod(319,576)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(576, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("576.319");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 576 = 2^{6} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 576.o (of order \(6\), degree \(2\), 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: | \(15.6948632272\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 10 x^{19} + 50 x^{18} - 130 x^{17} + 203 x^{16} - 296 x^{15} + 1260 x^{14} - 3380 x^{13} + \cdots + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{26}\cdot 3^{4} \) |
| Twist minimal: | no (minimal twist has level 288) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{20} - 10 x^{19} + 50 x^{18} - 130 x^{17} + 203 x^{16} - 296 x^{15} + 1260 x^{14} - 3380 x^{13} + \cdots + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 17\!\cdots\!05 \nu^{19} + \cdots + 13\!\cdots\!08 ) / 40\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 50\!\cdots\!00 \nu^{19} + \cdots - 65\!\cdots\!24 ) / 28\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 36\!\cdots\!32 \nu^{19} + \cdots + 38\!\cdots\!88 ) / 20\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 22981222561 \nu^{19} - 228756996538 \nu^{18} + 1139644587846 \nu^{17} + \cdots - 4347762518976 ) / 11835144571200 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 62\!\cdots\!36 \nu^{19} + \cdots + 74\!\cdots\!28 ) / 28\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 90\!\cdots\!75 \nu^{19} + \cdots + 23\!\cdots\!32 ) / 40\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 32\!\cdots\!73 \nu^{19} + \cdots - 60\!\cdots\!56 ) / 10\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 37\!\cdots\!15 \nu^{19} + \cdots - 19\!\cdots\!08 ) / 93\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 34\!\cdots\!94 \nu^{19} + \cdots - 25\!\cdots\!84 ) / 84\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 39\!\cdots\!04 \nu^{19} + \cdots + 25\!\cdots\!12 ) / 93\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 73\!\cdots\!53 \nu^{19} + \cdots - 38\!\cdots\!20 ) / 14\!\cdots\!00 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 21\!\cdots\!77 \nu^{19} + \cdots + 12\!\cdots\!44 ) / 40\!\cdots\!00 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 11\!\cdots\!21 \nu^{19} + \cdots - 24\!\cdots\!56 ) / 21\!\cdots\!00 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 46\!\cdots\!79 \nu^{19} + \cdots + 45\!\cdots\!76 ) / 80\!\cdots\!80 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 27\!\cdots\!17 \nu^{19} + \cdots + 97\!\cdots\!72 ) / 42\!\cdots\!00 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 56\!\cdots\!35 \nu^{19} + \cdots + 20\!\cdots\!36 ) / 84\!\cdots\!00 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( - 12\!\cdots\!89 \nu^{19} + \cdots - 11\!\cdots\!84 ) / 12\!\cdots\!00 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 51\!\cdots\!23 \nu^{19} + \cdots + 97\!\cdots\!76 ) / 40\!\cdots\!00 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 61\!\cdots\!01 \nu^{19} + \cdots - 87\!\cdots\!60 ) / 40\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{19} + 5 \beta_{17} + \beta_{16} - 2 \beta_{15} - \beta_{14} + \beta_{13} - 3 \beta_{11} + \cdots + 10 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{19} + 11 \beta_{17} - 3 \beta_{16} - \beta_{14} - \beta_{13} - \beta_{11} + 13 \beta_{10} + \cdots - 2 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - \beta_{19} + 8 \beta_{17} - 8 \beta_{16} + 7 \beta_{15} - 2 \beta_{14} - 5 \beta_{13} - 3 \beta_{12} + \cdots - 70 ) / 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 5 \beta_{19} - 6 \beta_{17} - 3 \beta_{16} + 6 \beta_{15} + 5 \beta_{14} - 3 \beta_{13} + 9 \beta_{11} + \cdots - 93 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 50 \beta_{19} - 219 \beta_{17} + 35 \beta_{16} + 65 \beta_{15} + 95 \beta_{14} - 32 \beta_{13} + \cdots - 712 ) / 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 16 \beta_{19} - 364 \beta_{17} + 183 \beta_{16} + 121 \beta_{14} + 23 \beta_{13} + 87 \beta_{12} + \cdots - 22 ) / 6 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 436 \beta_{19} - 813 \beta_{17} + 1171 \beta_{16} - 683 \beta_{15} + 197 \beta_{14} + 692 \beta_{13} + \cdots + 7268 ) / 12 \)
|
| \(\nu^{8}\) | \(=\) |
\( 197 \beta_{19} - 2 \beta_{18} + 288 \beta_{17} + 130 \beta_{16} - 260 \beta_{15} - 168 \beta_{14} + \cdots + 2989 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 5408 \beta_{19} - 108 \beta_{18} + 24871 \beta_{17} - 5827 \beta_{16} - 7579 \beta_{15} - 11783 \beta_{14} + \cdots + 88244 ) / 12 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( - 910 \beta_{19} + 38110 \beta_{17} - 23133 \beta_{16} - 17375 \beta_{14} - 329 \beta_{13} + \cdots + 20102 ) / 6 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 71620 \beta_{19} + 3300 \beta_{18} + 78359 \beta_{17} - 152225 \beta_{16} + 86221 \beta_{15} + \cdots - 757060 ) / 12 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( - 77783 \beta_{19} + 5424 \beta_{18} - 115260 \beta_{17} - 51483 \beta_{16} + 102966 \beta_{15} + \cdots - 1010541 ) / 3 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( - 609812 \beta_{19} + 65520 \beta_{18} - 2995593 \beta_{17} + 731669 \beta_{16} + 991517 \beta_{15} + \cdots - 10925740 ) / 12 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( 383714 \beta_{19} - 4412686 \beta_{17} + 2909385 \beta_{16} + 2321047 \beta_{14} - 156835 \beta_{13} + \cdots - 3993490 ) / 6 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 10733416 \beta_{19} - 1075080 \beta_{18} - 8007369 \beta_{17} + 19484455 \beta_{16} - 11447843 \beta_{15} + \cdots + 81306236 ) / 12 \)
|
| \(\nu^{16}\) | \(=\) |
\( 3462089 \beta_{19} - 492030 \beta_{18} + 5004560 \beta_{17} + 2292244 \beta_{16} - 4584488 \beta_{15} + \cdots + 39546037 \)
|
| \(\nu^{17}\) | \(=\) |
\( ( 70569776 \beta_{19} - 15862836 \beta_{18} + 369053695 \beta_{17} - 87970555 \beta_{16} + \cdots + 1365471140 ) / 12 \)
|
| \(\nu^{18}\) | \(=\) |
\( ( - 75508774 \beta_{19} + 529550398 \beta_{17} - 364712457 \beta_{16} - 304819691 \beta_{14} + \cdots + 640539110 ) / 6 \)
|
| \(\nu^{19}\) | \(=\) |
\( ( - 1537320796 \beta_{19} + 219015660 \beta_{18} + 856804631 \beta_{17} - 2489751185 \beta_{16} + \cdots - 8955907876 ) / 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).
| \(n\) | \(65\) | \(127\) | \(325\) |
| \(\chi(n)\) | \(\beta_{4}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 319.1 |
|
0 | −2.98096 | − | 0.337489i | 0 | 1.92225 | − | 3.32943i | 0 | 11.0596 | − | 6.38529i | 0 | 8.77220 | + | 2.01208i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.2 | 0 | −2.90180 | + | 0.761299i | 0 | −0.704345 | + | 1.21996i | 0 | 0.768192 | − | 0.443516i | 0 | 7.84085 | − | 4.41827i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.3 | 0 | −2.73375 | − | 1.23556i | 0 | −4.49547 | + | 7.78638i | 0 | −8.46628 | + | 4.88801i | 0 | 5.94678 | + | 6.75543i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.4 | 0 | −1.64999 | + | 2.50550i | 0 | 0.966952 | − | 1.67481i | 0 | −8.96285 | + | 5.17470i | 0 | −3.55504 | − | 8.26811i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.5 | 0 | −0.998802 | − | 2.82885i | 0 | −1.18939 | + | 2.06008i | 0 | −6.27165 | + | 3.62094i | 0 | −7.00479 | + | 5.65092i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.6 | 0 | 0.998802 | + | 2.82885i | 0 | −1.18939 | + | 2.06008i | 0 | 6.27165 | − | 3.62094i | 0 | −7.00479 | + | 5.65092i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.7 | 0 | 1.64999 | − | 2.50550i | 0 | 0.966952 | − | 1.67481i | 0 | 8.96285 | − | 5.17470i | 0 | −3.55504 | − | 8.26811i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.8 | 0 | 2.73375 | + | 1.23556i | 0 | −4.49547 | + | 7.78638i | 0 | 8.46628 | − | 4.88801i | 0 | 5.94678 | + | 6.75543i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.9 | 0 | 2.90180 | − | 0.761299i | 0 | −0.704345 | + | 1.21996i | 0 | −0.768192 | + | 0.443516i | 0 | 7.84085 | − | 4.41827i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 319.10 | 0 | 2.98096 | + | 0.337489i | 0 | 1.92225 | − | 3.32943i | 0 | −11.0596 | + | 6.38529i | 0 | 8.77220 | + | 2.01208i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 511.1 | 0 | −2.98096 | + | 0.337489i | 0 | 1.92225 | + | 3.32943i | 0 | 11.0596 | + | 6.38529i | 0 | 8.77220 | − | 2.01208i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 511.2 | 0 | −2.90180 | − | 0.761299i | 0 | −0.704345 | − | 1.21996i | 0 | 0.768192 | + | 0.443516i | 0 | 7.84085 | + | 4.41827i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 511.3 | 0 | −2.73375 | + | 1.23556i | 0 | −4.49547 | − | 7.78638i | 0 | −8.46628 | − | 4.88801i | 0 | 5.94678 | − | 6.75543i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 511.4 | 0 | −1.64999 | − | 2.50550i | 0 | 0.966952 | + | 1.67481i | 0 | −8.96285 | − | 5.17470i | 0 | −3.55504 | + | 8.26811i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 511.5 | 0 | −0.998802 | + | 2.82885i | 0 | −1.18939 | − | 2.06008i | 0 | −6.27165 | − | 3.62094i | 0 | −7.00479 | − | 5.65092i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 511.6 | 0 | 0.998802 | − | 2.82885i | 0 | −1.18939 | − | 2.06008i | 0 | 6.27165 | + | 3.62094i | 0 | −7.00479 | − | 5.65092i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 511.7 | 0 | 1.64999 | + | 2.50550i | 0 | 0.966952 | + | 1.67481i | 0 | 8.96285 | + | 5.17470i | 0 | −3.55504 | + | 8.26811i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 511.8 | 0 | 2.73375 | − | 1.23556i | 0 | −4.49547 | − | 7.78638i | 0 | 8.46628 | + | 4.88801i | 0 | 5.94678 | − | 6.75543i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 511.9 | 0 | 2.90180 | + | 0.761299i | 0 | −0.704345 | − | 1.21996i | 0 | −0.768192 | − | 0.443516i | 0 | 7.84085 | + | 4.41827i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 511.10 | 0 | 2.98096 | − | 0.337489i | 0 | 1.92225 | + | 3.32943i | 0 | −11.0596 | − | 6.38529i | 0 | 8.77220 | − | 2.01208i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 9.c | even | 3 | 1 | inner |
| 36.f | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 576.3.o.h | 20 | |
| 3.b | odd | 2 | 1 | 1728.3.o.h | 20 | ||
| 4.b | odd | 2 | 1 | inner | 576.3.o.h | 20 | |
| 8.b | even | 2 | 1 | 288.3.o.b | ✓ | 20 | |
| 8.d | odd | 2 | 1 | 288.3.o.b | ✓ | 20 | |
| 9.c | even | 3 | 1 | inner | 576.3.o.h | 20 | |
| 9.d | odd | 6 | 1 | 1728.3.o.h | 20 | ||
| 12.b | even | 2 | 1 | 1728.3.o.h | 20 | ||
| 24.f | even | 2 | 1 | 864.3.o.b | 20 | ||
| 24.h | odd | 2 | 1 | 864.3.o.b | 20 | ||
| 36.f | odd | 6 | 1 | inner | 576.3.o.h | 20 | |
| 36.h | even | 6 | 1 | 1728.3.o.h | 20 | ||
| 72.j | odd | 6 | 1 | 864.3.o.b | 20 | ||
| 72.j | odd | 6 | 1 | 2592.3.g.h | 10 | ||
| 72.l | even | 6 | 1 | 864.3.o.b | 20 | ||
| 72.l | even | 6 | 1 | 2592.3.g.h | 10 | ||
| 72.n | even | 6 | 1 | 288.3.o.b | ✓ | 20 | |
| 72.n | even | 6 | 1 | 2592.3.g.g | 10 | ||
| 72.p | odd | 6 | 1 | 288.3.o.b | ✓ | 20 | |
| 72.p | odd | 6 | 1 | 2592.3.g.g | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 288.3.o.b | ✓ | 20 | 8.b | even | 2 | 1 | |
| 288.3.o.b | ✓ | 20 | 8.d | odd | 2 | 1 | |
| 288.3.o.b | ✓ | 20 | 72.n | even | 6 | 1 | |
| 288.3.o.b | ✓ | 20 | 72.p | odd | 6 | 1 | |
| 576.3.o.h | 20 | 1.a | even | 1 | 1 | trivial | |
| 576.3.o.h | 20 | 4.b | odd | 2 | 1 | inner | |
| 576.3.o.h | 20 | 9.c | even | 3 | 1 | inner | |
| 576.3.o.h | 20 | 36.f | odd | 6 | 1 | inner | |
| 864.3.o.b | 20 | 24.f | even | 2 | 1 | ||
| 864.3.o.b | 20 | 24.h | odd | 2 | 1 | ||
| 864.3.o.b | 20 | 72.j | odd | 6 | 1 | ||
| 864.3.o.b | 20 | 72.l | even | 6 | 1 | ||
| 1728.3.o.h | 20 | 3.b | odd | 2 | 1 | ||
| 1728.3.o.h | 20 | 9.d | odd | 6 | 1 | ||
| 1728.3.o.h | 20 | 12.b | even | 2 | 1 | ||
| 1728.3.o.h | 20 | 36.h | even | 6 | 1 | ||
| 2592.3.g.g | 10 | 72.n | even | 6 | 1 | ||
| 2592.3.g.g | 10 | 72.p | odd | 6 | 1 | ||
| 2592.3.g.h | 10 | 72.j | odd | 6 | 1 | ||
| 2592.3.g.h | 10 | 72.l | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(576, [\chi])\):
|
\( T_{5}^{10} + 7 T_{5}^{9} + 78 T_{5}^{8} - 21 T_{5}^{7} + 1374 T_{5}^{6} + 1407 T_{5}^{5} + 9729 T_{5}^{4} + \cdots + 50176 \)
|
|
\( T_{7}^{20} - 419 T_{7}^{18} + 112758 T_{7}^{16} - 18336423 T_{7}^{14} + 2182164342 T_{7}^{12} + \cdots + 47\!\cdots\!00 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} + \cdots + 3486784401 \)
|
| $5$ |
\( (T^{10} + 7 T^{9} + \cdots + 50176)^{2} \)
|
| $7$ |
\( T^{20} + \cdots + 47\!\cdots\!00 \)
|
| $11$ |
\( T^{20} + \cdots + 43\!\cdots\!01 \)
|
| $13$ |
\( (T^{10} + 13 T^{9} + \cdots + 583696)^{2} \)
|
| $17$ |
\( (T^{5} - 18 T^{4} + \cdots - 50832)^{4} \)
|
| $19$ |
\( (T^{10} + \cdots + 1691581169664)^{2} \)
|
| $23$ |
\( T^{20} + \cdots + 80\!\cdots\!56 \)
|
| $29$ |
\( (T^{10} + \cdots + 104284535520400)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 15\!\cdots\!56 \)
|
| $37$ |
\( (T^{5} + 24 T^{4} + \cdots + 14835456)^{4} \)
|
| $41$ |
\( (T^{10} + \cdots + 13190269049281)^{2} \)
|
| $43$ |
\( T^{20} + \cdots + 95\!\cdots\!21 \)
|
| $47$ |
\( T^{20} + \cdots + 24\!\cdots\!36 \)
|
| $53$ |
\( (T^{5} - 48 T^{4} + \cdots - 2275179264)^{4} \)
|
| $59$ |
\( T^{20} + \cdots + 13\!\cdots\!25 \)
|
| $61$ |
\( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 37\!\cdots\!81 \)
|
| $71$ |
\( (T^{10} + \cdots + 10\!\cdots\!24)^{2} \)
|
| $73$ |
\( (T^{5} + 42 T^{4} + \cdots + 2701063440)^{4} \)
|
| $79$ |
\( T^{20} + \cdots + 10\!\cdots\!00 \)
|
| $83$ |
\( T^{20} + \cdots + 22\!\cdots\!76 \)
|
| $89$ |
\( (T^{5} + 60 T^{4} + \cdots - 1531650240)^{4} \)
|
| $97$ |
\( (T^{10} + \cdots + 52\!\cdots\!25)^{2} \)
|
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