Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [112,2,Mod(27,112)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(112, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 1, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("112.27");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 112 = 2^{4} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 112.j (of order \(4\), degree \(2\), 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: | \(0.894324502638\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{15} + 6\nu^{11} + 12\nu^{9} - 47\nu^{7} + 20\nu^{5} + 48\nu^{3} - 64\nu ) / 256 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -13\nu^{15} + 48\nu^{13} - 14\nu^{11} + 4\nu^{9} - 477\nu^{7} + 428\nu^{5} + 656\nu^{3} + 3136\nu ) / 2560 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 17\nu^{14} - 72\nu^{12} + 6\nu^{10} - 36\nu^{8} + 513\nu^{6} - 1332\nu^{4} + 3696\nu^{2} - 4544 ) / 3840 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -13\nu^{15} + 48\nu^{13} - 14\nu^{11} + 4\nu^{9} - 477\nu^{7} + 428\nu^{5} - 1904\nu^{3} + 5696\nu ) / 2560 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -29\nu^{15} - 16\nu^{13} + 18\nu^{11} + 452\nu^{9} + 19\nu^{7} + 44\nu^{5} - 2032\nu^{3} + 1088\nu ) / 2560 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 43\nu^{14} - 48\nu^{12} - 126\nu^{10} + 36\nu^{8} + 27\nu^{6} + 492\nu^{4} + 144\nu^{2} - 1216 ) / 3840 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 4\nu^{14} - 9\nu^{12} + 12\nu^{10} - 42\nu^{8} + 36\nu^{6} - 69\nu^{4} + 252\nu^{2} - 448 ) / 240 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -53\nu^{15} + 88\nu^{13} - 94\nu^{11} + 404\nu^{9} - 997\nu^{7} + 948\nu^{5} - 4144\nu^{3} + 3776\nu ) / 2560 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -71\nu^{14} + 216\nu^{12} - 138\nu^{10} + 828\nu^{8} - 2199\nu^{6} + 1356\nu^{4} - 5328\nu^{2} + 14912 ) / 3840 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( -37\nu^{14} + 72\nu^{12} - 126\nu^{10} + 276\nu^{8} - 693\nu^{6} + 1092\nu^{4} - 2256\nu^{2} + 3904 ) / 1920 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 67\nu^{15} - 72\nu^{13} - 14\nu^{11} - 396\nu^{9} + 403\nu^{7} - 652\nu^{5} + 2896\nu^{3} - 1344\nu ) / 2560 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 43\nu^{14} - 108\nu^{12} + 114\nu^{10} - 324\nu^{8} + 747\nu^{6} - 528\nu^{4} + 3024\nu^{2} - 6016 ) / 1920 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 15\nu^{15} - 32\nu^{13} + 26\nu^{11} - 140\nu^{9} + 319\nu^{7} - 116\nu^{5} + 1040\nu^{3} - 2240\nu ) / 512 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( -149\nu^{14} + 264\nu^{12} - 222\nu^{10} + 1332\nu^{8} - 2181\nu^{6} + 1764\nu^{4} - 11952\nu^{2} + 18368 ) / 3840 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( -127\nu^{15} + 312\nu^{13} - 346\nu^{11} + 1116\nu^{9} - 2383\nu^{7} + 2252\nu^{5} - 8656\nu^{3} + 17984\nu ) / 2560 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{11} - \beta_{8} + 2\beta_{5} + 2\beta_{4} + 2\beta_{2} - \beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{14} + \beta_{12} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{6} + \beta_{3} + 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{11} - \beta_{8} + 2\beta_{5} - 4\beta_{4} + 8\beta_{2} - \beta_1 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{14} + 5\beta_{12} + 3\beta_{10} + \beta_{9} - 3\beta_{7} - \beta_{6} - 3\beta_{3} + 1 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 6\beta_{15} + 18\beta_{13} + \beta_{11} + 5\beta_{8} + 8\beta_{5} + 8\beta_{4} + 8\beta_{2} + 5\beta_1 ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -\beta_{14} + \beta_{12} - 3\beta_{10} - 3\beta_{9} - 9\beta_{7} - \beta_{6} - 3\beta_{3} + 5 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 12\beta_{15} + 12\beta_{13} - 11\beta_{11} - 25\beta_{8} + 2\beta_{5} + 8\beta_{4} - 16\beta_{2} - \beta_1 ) / 6 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 3\beta_{14} + 17\beta_{12} - 5\beta_{10} + 9\beta_{9} - 15\beta_{7} + 3\beta_{6} + 5\beta_{3} - 7 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 18\beta_{15} + 6\beta_{13} + \beta_{11} - 31\beta_{8} + 20\beta_{5} - 16\beta_{4} - 16\beta_{2} + 29\beta_1 ) / 6 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 3\beta_{14} + 37\beta_{12} - 11\beta_{10} + 21\beta_{9} - 13\beta_{7} - 21\beta_{6} - 19\beta_{3} - 11 ) / 2 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( -95\beta_{11} - 97\beta_{8} - 22\beta_{5} + 92\beta_{4} - 64\beta_{2} + 95\beta_1 ) / 6 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( -49\beta_{14} - 43\beta_{12} - 53\beta_{10} + 25\beta_{9} - 67\beta_{7} - 17\beta_{6} - 43\beta_{3} - 71 ) / 2 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 126 \beta_{15} - 6 \beta_{13} - 215 \beta_{11} - 331 \beta_{8} - 256 \beta_{5} + 8 \beta_{4} + \cdots + 173 \beta_1 ) / 6 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( -33\beta_{14} - 15\beta_{12} - 123\beta_{10} + 69\beta_{9} - 57\beta_{7} + 111\beta_{6} - 75\beta_{3} - 67 ) / 2 \)
|
| \(\nu^{15}\) | \(=\) |
\( ( 228 \beta_{15} + 132 \beta_{13} + 37 \beta_{11} - 337 \beta_{8} - 142 \beta_{5} + 176 \beta_{4} + \cdots + 455 \beta_1 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) | \(85\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(\beta_{12}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 27.1 |
|
−1.41298 | + | 0.0591148i | −2.23450 | + | 2.23450i | 1.99301 | − | 0.167056i | −0.584038 | + | 0.584038i | 3.02521 | − | 3.28940i | −1.82596 | − | 1.91465i | −2.80620 | + | 0.353863i | − | 6.98602i | 0.790708 | − | 0.859758i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 27.2 | −1.41298 | + | 0.0591148i | 2.23450 | − | 2.23450i | 1.99301 | − | 0.167056i | 0.584038 | − | 0.584038i | −3.02521 | + | 3.28940i | −1.82596 | + | 1.91465i | −2.80620 | + | 0.353863i | − | 6.98602i | −0.790708 | + | 0.859758i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 27.3 | −0.576222 | − | 1.29150i | −1.28999 | + | 1.28999i | −1.33594 | + | 1.48838i | −0.599312 | + | 0.599312i | 2.40933 | + | 0.922696i | −0.152445 | + | 2.64136i | 2.69204 | + | 0.867721i | − | 0.328129i | 1.11935 | + | 0.428674i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 27.4 | −0.576222 | − | 1.29150i | 1.28999 | − | 1.28999i | −1.33594 | + | 1.48838i | 0.599312 | − | 0.599312i | −2.40933 | − | 0.922696i | −0.152445 | − | 2.64136i | 2.69204 | + | 0.867721i | − | 0.328129i | −1.11935 | − | 0.428674i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 27.5 | 0.192769 | + | 1.40101i | −1.03649 | + | 1.03649i | −1.92568 | + | 0.540143i | −1.68683 | + | 1.68683i | −1.65195 | − | 1.25234i | 1.38554 | − | 2.25395i | −1.12796 | − | 2.59378i | 0.851361i | −2.68844 | − | 2.03810i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 27.6 | 0.192769 | + | 1.40101i | 1.03649 | − | 1.03649i | −1.92568 | + | 0.540143i | 1.68683 | − | 1.68683i | 1.65195 | + | 1.25234i | 1.38554 | + | 2.25395i | −1.12796 | − | 2.59378i | 0.851361i | 2.68844 | + | 2.03810i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 27.7 | 0.796431 | − | 1.16863i | −1.50619 | + | 1.50619i | −0.731395 | − | 1.86147i | 2.54054 | − | 2.54054i | 0.560603 | + | 2.95975i | 2.59286 | − | 0.526369i | −2.75787 | − | 0.627801i | − | 1.53721i | −0.945586 | − | 4.99231i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 27.8 | 0.796431 | − | 1.16863i | 1.50619 | − | 1.50619i | −0.731395 | − | 1.86147i | −2.54054 | + | 2.54054i | −0.560603 | − | 2.95975i | 2.59286 | + | 0.526369i | −2.75787 | − | 0.627801i | − | 1.53721i | 0.945586 | + | 4.99231i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.1 | −1.41298 | − | 0.0591148i | −2.23450 | − | 2.23450i | 1.99301 | + | 0.167056i | −0.584038 | − | 0.584038i | 3.02521 | + | 3.28940i | −1.82596 | + | 1.91465i | −2.80620 | − | 0.353863i | 6.98602i | 0.790708 | + | 0.859758i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.2 | −1.41298 | − | 0.0591148i | 2.23450 | + | 2.23450i | 1.99301 | + | 0.167056i | 0.584038 | + | 0.584038i | −3.02521 | − | 3.28940i | −1.82596 | − | 1.91465i | −2.80620 | − | 0.353863i | 6.98602i | −0.790708 | − | 0.859758i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.3 | −0.576222 | + | 1.29150i | −1.28999 | − | 1.28999i | −1.33594 | − | 1.48838i | −0.599312 | − | 0.599312i | 2.40933 | − | 0.922696i | −0.152445 | − | 2.64136i | 2.69204 | − | 0.867721i | 0.328129i | 1.11935 | − | 0.428674i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.4 | −0.576222 | + | 1.29150i | 1.28999 | + | 1.28999i | −1.33594 | − | 1.48838i | 0.599312 | + | 0.599312i | −2.40933 | + | 0.922696i | −0.152445 | + | 2.64136i | 2.69204 | − | 0.867721i | 0.328129i | −1.11935 | + | 0.428674i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.5 | 0.192769 | − | 1.40101i | −1.03649 | − | 1.03649i | −1.92568 | − | 0.540143i | −1.68683 | − | 1.68683i | −1.65195 | + | 1.25234i | 1.38554 | + | 2.25395i | −1.12796 | + | 2.59378i | − | 0.851361i | −2.68844 | + | 2.03810i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.6 | 0.192769 | − | 1.40101i | 1.03649 | + | 1.03649i | −1.92568 | − | 0.540143i | 1.68683 | + | 1.68683i | 1.65195 | − | 1.25234i | 1.38554 | − | 2.25395i | −1.12796 | + | 2.59378i | − | 0.851361i | 2.68844 | − | 2.03810i | ||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.7 | 0.796431 | + | 1.16863i | −1.50619 | − | 1.50619i | −0.731395 | + | 1.86147i | 2.54054 | + | 2.54054i | 0.560603 | − | 2.95975i | 2.59286 | + | 0.526369i | −2.75787 | + | 0.627801i | 1.53721i | −0.945586 | + | 4.99231i | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.8 | 0.796431 | + | 1.16863i | 1.50619 | + | 1.50619i | −0.731395 | + | 1.86147i | −2.54054 | − | 2.54054i | −0.560603 | + | 2.95975i | 2.59286 | − | 0.526369i | −2.75787 | + | 0.627801i | 1.53721i | 0.945586 | − | 4.99231i | |||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 16.f | odd | 4 | 1 | inner |
| 112.j | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 112.2.j.d | ✓ | 16 |
| 4.b | odd | 2 | 1 | 448.2.j.d | 16 | ||
| 7.b | odd | 2 | 1 | inner | 112.2.j.d | ✓ | 16 |
| 7.c | even | 3 | 2 | 784.2.w.e | 32 | ||
| 7.d | odd | 6 | 2 | 784.2.w.e | 32 | ||
| 8.b | even | 2 | 1 | 896.2.j.h | 16 | ||
| 8.d | odd | 2 | 1 | 896.2.j.g | 16 | ||
| 16.e | even | 4 | 1 | 448.2.j.d | 16 | ||
| 16.e | even | 4 | 1 | 896.2.j.g | 16 | ||
| 16.f | odd | 4 | 1 | inner | 112.2.j.d | ✓ | 16 |
| 16.f | odd | 4 | 1 | 896.2.j.h | 16 | ||
| 28.d | even | 2 | 1 | 448.2.j.d | 16 | ||
| 56.e | even | 2 | 1 | 896.2.j.g | 16 | ||
| 56.h | odd | 2 | 1 | 896.2.j.h | 16 | ||
| 112.j | even | 4 | 1 | inner | 112.2.j.d | ✓ | 16 |
| 112.j | even | 4 | 1 | 896.2.j.h | 16 | ||
| 112.l | odd | 4 | 1 | 448.2.j.d | 16 | ||
| 112.l | odd | 4 | 1 | 896.2.j.g | 16 | ||
| 112.u | odd | 12 | 2 | 784.2.w.e | 32 | ||
| 112.v | even | 12 | 2 | 784.2.w.e | 32 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 112.2.j.d | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 112.2.j.d | ✓ | 16 | 7.b | odd | 2 | 1 | inner |
| 112.2.j.d | ✓ | 16 | 16.f | odd | 4 | 1 | inner |
| 112.2.j.d | ✓ | 16 | 112.j | even | 4 | 1 | inner |
| 448.2.j.d | 16 | 4.b | odd | 2 | 1 | ||
| 448.2.j.d | 16 | 16.e | even | 4 | 1 | ||
| 448.2.j.d | 16 | 28.d | even | 2 | 1 | ||
| 448.2.j.d | 16 | 112.l | odd | 4 | 1 | ||
| 784.2.w.e | 32 | 7.c | even | 3 | 2 | ||
| 784.2.w.e | 32 | 7.d | odd | 6 | 2 | ||
| 784.2.w.e | 32 | 112.u | odd | 12 | 2 | ||
| 784.2.w.e | 32 | 112.v | even | 12 | 2 | ||
| 896.2.j.g | 16 | 8.d | odd | 2 | 1 | ||
| 896.2.j.g | 16 | 16.e | even | 4 | 1 | ||
| 896.2.j.g | 16 | 56.e | even | 2 | 1 | ||
| 896.2.j.g | 16 | 112.l | odd | 4 | 1 | ||
| 896.2.j.h | 16 | 8.b | even | 2 | 1 | ||
| 896.2.j.h | 16 | 16.f | odd | 4 | 1 | ||
| 896.2.j.h | 16 | 56.h | odd | 2 | 1 | ||
| 896.2.j.h | 16 | 112.j | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(112, [\chi])\):
|
\( T_{3}^{16} + 136T_{3}^{12} + 3992T_{3}^{8} + 38368T_{3}^{4} + 104976 \)
|
|
\( T_{5}^{16} + 200T_{5}^{12} + 5592T_{5}^{8} + 5344T_{5}^{4} + 1296 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{8} + 2 T^{7} + 4 T^{6} + \cdots + 16)^{2} \)
|
| $3$ |
\( T^{16} + 136 T^{12} + \cdots + 104976 \)
|
| $5$ |
\( T^{16} + 200 T^{12} + \cdots + 1296 \)
|
| $7$ |
\( (T^{8} - 4 T^{7} + \cdots + 2401)^{2} \)
|
| $11$ |
\( (T^{2} + 4 T + 8)^{8} \)
|
| $13$ |
\( T^{16} + 2440 T^{12} + \cdots + 18974736 \)
|
| $17$ |
\( (T^{8} + 88 T^{6} + \cdots + 14400)^{2} \)
|
| $19$ |
\( T^{16} + 968 T^{12} + \cdots + 1296 \)
|
| $23$ |
\( (T^{4} - 16 T^{2} - 16 T + 4)^{4} \)
|
| $29$ |
\( (T^{8} + 128 T^{5} + \cdots + 1296)^{2} \)
|
| $31$ |
\( (T^{8} - 120 T^{6} + \cdots + 166464)^{2} \)
|
| $37$ |
\( (T^{8} - 64 T^{5} + \cdots + 400)^{2} \)
|
| $41$ |
\( (T^{8} - 216 T^{6} + \cdots + 419904)^{2} \)
|
| $43$ |
\( (T^{8} + 64 T^{5} + \cdots + 256)^{2} \)
|
| $47$ |
\( (T^{8} - 312 T^{6} + \cdots + 3968064)^{2} \)
|
| $53$ |
\( (T^{8} - 128 T^{5} + \cdots + 1948816)^{2} \)
|
| $59$ |
\( T^{16} + \cdots + 506250000 \)
|
| $61$ |
\( T^{16} + \cdots + 56528804622096 \)
|
| $67$ |
\( (T^{8} + 24 T^{7} + \cdots + 256)^{2} \)
|
| $71$ |
\( (T^{4} - 8 T^{3} + \cdots - 1728)^{4} \)
|
| $73$ |
\( (T^{8} - 96 T^{6} + \cdots + 9216)^{2} \)
|
| $79$ |
\( (T^{8} + 384 T^{6} + \cdots + 36192256)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 57524882954256 \)
|
| $89$ |
\( (T^{8} - 512 T^{6} + \cdots + 5308416)^{2} \)
|
| $97$ |
\( (T^{8} + 312 T^{6} + \cdots + 1742400)^{2} \)
|
| show more | |
| show less |