Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [468,2,Mod(19,468)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(468, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("468.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 468.cb (of order \(12\), 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: | \(3.73699881460\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | 16.0.102930383934669717504.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 4 x^{15} + 5 x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} - 12 x^{10} + 32 x^{9} - 36 x^{8} + \cdots + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 52) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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
\( x^{16} - 4 x^{15} + 5 x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} - 12 x^{10} + 32 x^{9} - 36 x^{8} + \cdots + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 19 \nu^{15} + 26 \nu^{14} + 49 \nu^{13} + 12 \nu^{12} - 163 \nu^{11} - 146 \nu^{10} + 172 \nu^{9} + \cdots - 1152 ) / 1408 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{15} - 4 \nu^{14} + 5 \nu^{13} - 2 \nu^{12} + 5 \nu^{11} - 8 \nu^{10} - 12 \nu^{9} + 32 \nu^{8} + \cdots - 512 ) / 128 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 25 \nu^{15} + 6 \nu^{14} + 31 \nu^{13} + 76 \nu^{12} - 61 \nu^{11} - 182 \nu^{10} + 40 \nu^{8} + \cdots - 384 ) / 1408 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3 \nu^{15} - 62 \nu^{14} + 27 \nu^{13} + 56 \nu^{12} + 167 \nu^{11} - 146 \nu^{10} - 400 \nu^{9} + \cdots - 2560 ) / 1408 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 10 \nu^{15} - 37 \nu^{14} - 12 \nu^{13} + 7 \nu^{12} + 106 \nu^{11} + 87 \nu^{10} - 266 \nu^{9} + \cdots + 1216 ) / 704 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 25 \nu^{15} + 72 \nu^{14} - 13 \nu^{13} - 34 \nu^{12} - 149 \nu^{11} - 28 \nu^{10} + 396 \nu^{9} + \cdots + 2432 ) / 1408 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 7 \nu^{15} + 17 \nu^{14} + \nu^{13} + 5 \nu^{12} - 57 \nu^{11} - 9 \nu^{10} + 100 \nu^{9} + \cdots + 544 ) / 352 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 7 \nu^{15} + 28 \nu^{14} - 21 \nu^{13} - 6 \nu^{12} - 57 \nu^{11} + 24 \nu^{10} + 166 \nu^{9} + \cdots + 1600 ) / 352 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 47 \nu^{15} + 78 \nu^{14} + 13 \nu^{13} + 32 \nu^{12} - 191 \nu^{11} - 150 \nu^{10} + 420 \nu^{9} + \cdots + 3200 ) / 704 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 53 \nu^{15} - 139 \nu^{14} + 61 \nu^{13} + 3 \nu^{12} + 271 \nu^{11} - 59 \nu^{10} - 748 \nu^{9} + \cdots - 8704 ) / 704 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 34 \nu^{15} - 83 \nu^{14} + 31 \nu^{13} - 7 \nu^{12} + 173 \nu^{11} - \nu^{10} - 467 \nu^{9} + \cdots - 4608 ) / 352 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 73 \nu^{15} - 204 \nu^{14} + 109 \nu^{13} + 6 \nu^{12} + 365 \nu^{11} - 112 \nu^{10} - 1068 \nu^{9} + \cdots - 13568 ) / 704 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( - 82 \nu^{15} + 201 \nu^{14} - 52 \nu^{13} + 13 \nu^{12} - 434 \nu^{11} - 51 \nu^{10} + 1138 \nu^{9} + \cdots + 9728 ) / 704 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 46 \nu^{15} + 129 \nu^{14} - 72 \nu^{13} + 3 \nu^{12} - 230 \nu^{11} + 87 \nu^{10} + 654 \nu^{9} + \cdots + 9760 ) / 352 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} + \beta_{5} - \beta_{4} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{14} - \beta_{13} + 2 \beta_{12} - \beta_{11} - 2 \beta_{10} + \beta_{8} + 2 \beta_{5} + \beta_{4} + \cdots + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{15} + 2\beta_{12} + \beta_{9} + 2\beta_{8} + \beta_{7} + 2\beta_{4} + 3\beta_{3} - 2\beta_{2} + 2\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{15} + 2 \beta_{14} - 2 \beta_{13} + 4 \beta_{12} + 2 \beta_{11} + 2 \beta_{9} + 2 \beta_{8} + \cdots - 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3 \beta_{15} + 4 \beta_{14} + 4 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} + \beta_{9} + 3 \beta_{8} + \cdots - 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( 3 \beta_{15} + 3 \beta_{14} + 5 \beta_{13} + 6 \beta_{12} - 3 \beta_{11} + 4 \beta_{9} + \beta_{8} + \cdots - 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 2 \beta_{15} + 4 \beta_{14} - 4 \beta_{13} + 2 \beta_{12} + 8 \beta_{11} + 4 \beta_{10} + \cdots - 9 \beta_1 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2 \beta_{15} + 7 \beta_{14} + 5 \beta_{13} - 2 \beta_{12} + 5 \beta_{11} + 4 \beta_{9} - 5 \beta_{8} + \cdots + 5 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 3 \beta_{15} + 2 \beta_{14} + 10 \beta_{13} - 10 \beta_{12} - 6 \beta_{11} - 11 \beta_{9} + 5 \beta_{7} + \cdots - 6 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 13 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} - 22 \beta_{11} - 16 \beta_{9} - 2 \beta_{8} - 17 \beta_{7} + \cdots + 17 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 7 \beta_{15} - 20 \beta_{13} - 4 \beta_{12} - 20 \beta_{10} - 27 \beta_{9} + 7 \beta_{8} - 23 \beta_{7} + \cdots + 20 \)
|
| \(\nu^{13}\) | \(=\) |
\( 13 \beta_{15} - 13 \beta_{14} + 13 \beta_{13} + 14 \beta_{12} - 35 \beta_{11} + 16 \beta_{10} - 30 \beta_{9} + \cdots + 2 \)
|
| \(\nu^{14}\) | \(=\) |
\( 30 \beta_{14} - 78 \beta_{13} + 54 \beta_{12} + 30 \beta_{11} - 60 \beta_{10} - 78 \beta_{9} + 45 \beta_{8} + \cdots - 60 \)
|
| \(\nu^{15}\) | \(=\) |
\( 78 \beta_{15} + 15 \beta_{14} + 45 \beta_{13} + 90 \beta_{12} - 27 \beta_{11} - 72 \beta_{10} + \cdots + 61 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).
| \(n\) | \(145\) | \(209\) | \(235\) |
| \(\chi(n)\) | \(-\beta_{4}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
−1.41419 | − | 0.00757716i | 0 | 1.99989 | + | 0.0214311i | 1.52798 | − | 1.52798i | 0 | −1.97429 | − | 0.529008i | −2.82806 | − | 0.0454612i | 0 | −2.17244 | + | 2.14928i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.2 | −0.0921725 | + | 1.41121i | 0 | −1.98301 | − | 0.260149i | −0.894007 | + | 0.894007i | 0 | 4.37156 | + | 1.17136i | 0.549903 | − | 2.77446i | 0 | −1.17923 | − | 1.34403i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.3 | 0.785427 | + | 1.17605i | 0 | −0.766209 | + | 1.84741i | −0.894007 | + | 0.894007i | 0 | −4.37156 | − | 1.17136i | −2.77446 | + | 0.549903i | 0 | −1.75358 | − | 0.349224i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 19.4 | 1.22094 | − | 0.713659i | 0 | 0.981383 | − | 1.74267i | 1.52798 | − | 1.52798i | 0 | 1.97429 | + | 0.529008i | −0.0454612 | − | 2.82806i | 0 | 0.775114 | − | 2.95603i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.1 | −0.902074 | − | 1.08916i | 0 | −0.372527 | + | 1.96500i | 0.166404 | + | 0.166404i | 0 | 0.684384 | − | 2.55416i | 2.47624 | − | 1.36683i | 0 | 0.0311314 | − | 0.331348i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.2 | −0.236640 | + | 1.39427i | 0 | −1.88800 | − | 0.659882i | 0.166404 | + | 0.166404i | 0 | −0.684384 | + | 2.55416i | 1.36683 | − | 2.47624i | 0 | −0.271390 | + | 0.192635i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.3 | 0.526485 | − | 1.31256i | 0 | −1.44563 | − | 1.38209i | 2.19962 | + | 2.19962i | 0 | −0.152604 | + | 0.569525i | −2.57517 | + | 1.16983i | 0 | 4.04520 | − | 1.72907i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 163.4 | 1.11223 | + | 0.873468i | 0 | 0.474107 | + | 1.94299i | 2.19962 | + | 2.19962i | 0 | 0.152604 | − | 0.569525i | −1.16983 | + | 2.57517i | 0 | 0.525184 | + | 4.36778i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 271.1 | −1.41419 | + | 0.00757716i | 0 | 1.99989 | − | 0.0214311i | 1.52798 | + | 1.52798i | 0 | −1.97429 | + | 0.529008i | −2.82806 | + | 0.0454612i | 0 | −2.17244 | − | 2.14928i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 271.2 | −0.0921725 | − | 1.41121i | 0 | −1.98301 | + | 0.260149i | −0.894007 | − | 0.894007i | 0 | 4.37156 | − | 1.17136i | 0.549903 | + | 2.77446i | 0 | −1.17923 | + | 1.34403i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 271.3 | 0.785427 | − | 1.17605i | 0 | −0.766209 | − | 1.84741i | −0.894007 | − | 0.894007i | 0 | −4.37156 | + | 1.17136i | −2.77446 | − | 0.549903i | 0 | −1.75358 | + | 0.349224i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 271.4 | 1.22094 | + | 0.713659i | 0 | 0.981383 | + | 1.74267i | 1.52798 | + | 1.52798i | 0 | 1.97429 | − | 0.529008i | −0.0454612 | + | 2.82806i | 0 | 0.775114 | + | 2.95603i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 379.1 | −0.902074 | + | 1.08916i | 0 | −0.372527 | − | 1.96500i | 0.166404 | − | 0.166404i | 0 | 0.684384 | + | 2.55416i | 2.47624 | + | 1.36683i | 0 | 0.0311314 | + | 0.331348i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 379.2 | −0.236640 | − | 1.39427i | 0 | −1.88800 | + | 0.659882i | 0.166404 | − | 0.166404i | 0 | −0.684384 | − | 2.55416i | 1.36683 | + | 2.47624i | 0 | −0.271390 | − | 0.192635i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 379.3 | 0.526485 | + | 1.31256i | 0 | −1.44563 | + | 1.38209i | 2.19962 | − | 2.19962i | 0 | −0.152604 | − | 0.569525i | −2.57517 | − | 1.16983i | 0 | 4.04520 | + | 1.72907i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 379.4 | 1.11223 | − | 0.873468i | 0 | 0.474107 | − | 1.94299i | 2.19962 | − | 2.19962i | 0 | 0.152604 | + | 0.569525i | −1.16983 | − | 2.57517i | 0 | 0.525184 | − | 4.36778i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 13.f | odd | 12 | 1 | inner |
| 52.l | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 468.2.cb.f | 16 | |
| 3.b | odd | 2 | 1 | 52.2.l.b | ✓ | 16 | |
| 4.b | odd | 2 | 1 | inner | 468.2.cb.f | 16 | |
| 12.b | even | 2 | 1 | 52.2.l.b | ✓ | 16 | |
| 13.f | odd | 12 | 1 | inner | 468.2.cb.f | 16 | |
| 24.f | even | 2 | 1 | 832.2.bu.n | 16 | ||
| 24.h | odd | 2 | 1 | 832.2.bu.n | 16 | ||
| 39.d | odd | 2 | 1 | 676.2.l.k | 16 | ||
| 39.f | even | 4 | 1 | 676.2.l.i | 16 | ||
| 39.f | even | 4 | 1 | 676.2.l.m | 16 | ||
| 39.h | odd | 6 | 1 | 676.2.f.i | 16 | ||
| 39.h | odd | 6 | 1 | 676.2.l.i | 16 | ||
| 39.i | odd | 6 | 1 | 676.2.f.h | 16 | ||
| 39.i | odd | 6 | 1 | 676.2.l.m | 16 | ||
| 39.k | even | 12 | 1 | 52.2.l.b | ✓ | 16 | |
| 39.k | even | 12 | 1 | 676.2.f.h | 16 | ||
| 39.k | even | 12 | 1 | 676.2.f.i | 16 | ||
| 39.k | even | 12 | 1 | 676.2.l.k | 16 | ||
| 52.l | even | 12 | 1 | inner | 468.2.cb.f | 16 | |
| 156.h | even | 2 | 1 | 676.2.l.k | 16 | ||
| 156.l | odd | 4 | 1 | 676.2.l.i | 16 | ||
| 156.l | odd | 4 | 1 | 676.2.l.m | 16 | ||
| 156.p | even | 6 | 1 | 676.2.f.h | 16 | ||
| 156.p | even | 6 | 1 | 676.2.l.m | 16 | ||
| 156.r | even | 6 | 1 | 676.2.f.i | 16 | ||
| 156.r | even | 6 | 1 | 676.2.l.i | 16 | ||
| 156.v | odd | 12 | 1 | 52.2.l.b | ✓ | 16 | |
| 156.v | odd | 12 | 1 | 676.2.f.h | 16 | ||
| 156.v | odd | 12 | 1 | 676.2.f.i | 16 | ||
| 156.v | odd | 12 | 1 | 676.2.l.k | 16 | ||
| 312.bo | even | 12 | 1 | 832.2.bu.n | 16 | ||
| 312.bq | odd | 12 | 1 | 832.2.bu.n | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 52.2.l.b | ✓ | 16 | 3.b | odd | 2 | 1 | |
| 52.2.l.b | ✓ | 16 | 12.b | even | 2 | 1 | |
| 52.2.l.b | ✓ | 16 | 39.k | even | 12 | 1 | |
| 52.2.l.b | ✓ | 16 | 156.v | odd | 12 | 1 | |
| 468.2.cb.f | 16 | 1.a | even | 1 | 1 | trivial | |
| 468.2.cb.f | 16 | 4.b | odd | 2 | 1 | inner | |
| 468.2.cb.f | 16 | 13.f | odd | 12 | 1 | inner | |
| 468.2.cb.f | 16 | 52.l | even | 12 | 1 | inner | |
| 676.2.f.h | 16 | 39.i | odd | 6 | 1 | ||
| 676.2.f.h | 16 | 39.k | even | 12 | 1 | ||
| 676.2.f.h | 16 | 156.p | even | 6 | 1 | ||
| 676.2.f.h | 16 | 156.v | odd | 12 | 1 | ||
| 676.2.f.i | 16 | 39.h | odd | 6 | 1 | ||
| 676.2.f.i | 16 | 39.k | even | 12 | 1 | ||
| 676.2.f.i | 16 | 156.r | even | 6 | 1 | ||
| 676.2.f.i | 16 | 156.v | odd | 12 | 1 | ||
| 676.2.l.i | 16 | 39.f | even | 4 | 1 | ||
| 676.2.l.i | 16 | 39.h | odd | 6 | 1 | ||
| 676.2.l.i | 16 | 156.l | odd | 4 | 1 | ||
| 676.2.l.i | 16 | 156.r | even | 6 | 1 | ||
| 676.2.l.k | 16 | 39.d | odd | 2 | 1 | ||
| 676.2.l.k | 16 | 39.k | even | 12 | 1 | ||
| 676.2.l.k | 16 | 156.h | even | 2 | 1 | ||
| 676.2.l.k | 16 | 156.v | odd | 12 | 1 | ||
| 676.2.l.m | 16 | 39.f | even | 4 | 1 | ||
| 676.2.l.m | 16 | 39.i | odd | 6 | 1 | ||
| 676.2.l.m | 16 | 156.l | odd | 4 | 1 | ||
| 676.2.l.m | 16 | 156.p | even | 6 | 1 | ||
| 832.2.bu.n | 16 | 24.f | even | 2 | 1 | ||
| 832.2.bu.n | 16 | 24.h | odd | 2 | 1 | ||
| 832.2.bu.n | 16 | 312.bo | even | 12 | 1 | ||
| 832.2.bu.n | 16 | 312.bq | odd | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(468, [\chi])\):
|
\( T_{5}^{8} - 6T_{5}^{7} + 18T_{5}^{6} - 18T_{5}^{5} + 5T_{5}^{4} + 72T_{5}^{2} - 24T_{5} + 4 \)
|
|
\( T_{7}^{16} - 30 T_{7}^{14} + 207 T_{7}^{12} + 2790 T_{7}^{10} - 1399 T_{7}^{8} - 91512 T_{7}^{6} + \cdots + 43264 \)
|
|
\( T_{17}^{8} + 6T_{17}^{7} + 2T_{17}^{6} - 60T_{17}^{5} + 39T_{17}^{4} + 300T_{17}^{3} + 290T_{17}^{2} - 30T_{17} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} - 2 T^{15} + \cdots + 256 \)
|
| $3$ |
\( T^{16} \)
|
| $5$ |
\( (T^{8} - 6 T^{7} + 18 T^{6} + \cdots + 4)^{2} \)
|
| $7$ |
\( T^{16} - 30 T^{14} + \cdots + 43264 \)
|
| $11$ |
\( T^{16} + 18 T^{14} + \cdots + 692224 \)
|
| $13$ |
\( (T^{8} + 6 T^{7} + \cdots + 28561)^{2} \)
|
| $17$ |
\( (T^{8} + 6 T^{7} + 2 T^{6} + \cdots + 1)^{2} \)
|
| $19$ |
\( T^{16} - 54 T^{14} + \cdots + 77228944 \)
|
| $23$ |
\( T^{16} + 106 T^{14} + \cdots + 77228944 \)
|
| $29$ |
\( (T^{8} - 4 T^{7} + \cdots + 51529)^{2} \)
|
| $31$ |
\( T^{16} + \cdots + 1235663104 \)
|
| $37$ |
\( (T^{4} + 4 T^{3} + 5 T^{2} + \cdots + 1)^{4} \)
|
| $41$ |
\( (T^{4} + 12 T^{3} + \cdots + 81)^{4} \)
|
| $43$ |
\( T^{16} + \cdots + 5671027857664 \)
|
| $47$ |
\( T^{16} + \cdots + 5671027857664 \)
|
| $53$ |
\( (T^{4} - 8 T^{3} + \cdots - 128)^{4} \)
|
| $59$ |
\( T^{16} + \cdots + 171720267307264 \)
|
| $61$ |
\( (T^{8} - 2 T^{7} + \cdots + 6889)^{2} \)
|
| $67$ |
\( T^{16} + \cdots + 2205735869584 \)
|
| $71$ |
\( T^{16} + \cdots + 5067731344 \)
|
| $73$ |
\( (T^{8} - 10 T^{7} + \cdots + 676)^{2} \)
|
| $79$ |
\( (T^{8} + 160 T^{6} + \cdots + 53248)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 538409280507904 \)
|
| $89$ |
\( (T^{8} - 26 T^{7} + \cdots + 2896804)^{2} \)
|
| $97$ |
\( (T^{8} + 14 T^{7} + \cdots + 2116)^{2} \)
|
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