Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [104,2,Mod(83,104)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(104, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("104.83");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 104 = 2^{3} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 104.m (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: | \(0.830444181021\) |
| Analytic rank: | \(0\) |
| Dimension: | \(20\) |
| Relative dimension: | \(10\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{20} - 2 x^{19} + 2 x^{18} - 7 x^{16} + 14 x^{15} - 14 x^{14} + 8 x^{13} + 16 x^{12} - 40 x^{11} + \cdots + 1024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 2 x^{19} + 2 x^{18} - 7 x^{16} + 14 x^{15} - 14 x^{14} + 8 x^{13} + 16 x^{12} - 40 x^{11} + \cdots + 1024 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9 \nu^{19} + 10 \nu^{18} - 250 \nu^{17} + 272 \nu^{16} + 121 \nu^{15} - 390 \nu^{14} + 950 \nu^{13} + \cdots + 44032 ) / 31232 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 3 \nu^{19} + 20 \nu^{18} + 58 \nu^{17} - 316 \nu^{16} + 125 \nu^{15} - 124 \nu^{14} + \cdots + 38912 ) / 31232 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3 \nu^{19} - 42 \nu^{18} - 42 \nu^{17} - 24 \nu^{16} - 147 \nu^{15} + 774 \nu^{14} - 186 \nu^{13} + \cdots - 24064 ) / 31232 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 62 \nu^{19} - 61 \nu^{18} + 288 \nu^{17} - 192 \nu^{16} + 282 \nu^{15} - 53 \nu^{14} + \cdots + 82432 ) / 62464 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 71 \nu^{19} - 12 \nu^{18} - 176 \nu^{17} + 300 \nu^{16} - 289 \nu^{15} + 692 \nu^{14} + 368 \nu^{13} + \cdots - 18432 ) / 31232 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 8 \nu^{19} + 47 \nu^{18} - 44 \nu^{17} + 12 \nu^{16} - 208 \nu^{15} - 729 \nu^{14} + 804 \nu^{13} + \cdots + 51712 ) / 31232 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 41 \nu^{19} - 64 \nu^{18} - 152 \nu^{17} + 208 \nu^{16} - 239 \nu^{15} + 48 \nu^{14} + 1160 \nu^{13} + \cdots + 85504 ) / 31232 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 66 \nu^{19} + 81 \nu^{18} + 72 \nu^{17} - 118 \nu^{15} + 361 \nu^{14} - 232 \nu^{13} - 912 \nu^{12} + \cdots - 27136 ) / 62464 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 85 \nu^{19} + \nu^{18} + 20 \nu^{17} - 76 \nu^{16} - 91 \nu^{15} + 313 \nu^{14} + 260 \nu^{13} + \cdots + 52224 ) / 31232 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 161 \nu^{19} + 111 \nu^{18} - 116 \nu^{17} - 128 \nu^{16} + 991 \nu^{15} - 729 \nu^{14} + \cdots + 54784 ) / 62464 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( 206 \nu^{19} - 193 \nu^{18} - 304 \nu^{17} + 504 \nu^{16} - 666 \nu^{15} + 1415 \nu^{14} + \cdots - 152064 ) / 62464 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 53 \nu^{19} - 54 \nu^{18} - 46 \nu^{17} + 80 \nu^{16} - 171 \nu^{15} + 122 \nu^{14} + 162 \nu^{13} + \cdots + 7168 ) / 15616 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 7 \nu^{19} + 67 \nu^{18} - 68 \nu^{17} - 46 \nu^{16} + 129 \nu^{15} - 269 \nu^{14} + 220 \nu^{13} + \cdots + 17920 ) / 15616 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( - 222 \nu^{19} + 293 \nu^{18} - 216 \nu^{17} - 80 \nu^{16} + 2122 \nu^{15} - 3491 \nu^{14} + \cdots + 210432 ) / 62464 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 131 \nu^{19} - 234 \nu^{18} - 6 \nu^{17} + 272 \nu^{16} - 733 \nu^{15} + 1318 \nu^{14} + \cdots - 80896 ) / 31232 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 177 \nu^{19} - 453 \nu^{18} + 348 \nu^{17} + 96 \nu^{16} - 927 \nu^{15} + 3571 \nu^{14} + \cdots - 237056 ) / 62464 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 13 \nu^{19} - 38 \nu^{18} + 20 \nu^{17} + 50 \nu^{16} - 155 \nu^{15} + 226 \nu^{14} - 188 \nu^{13} + \cdots - 13568 ) / 3904 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{14} - \beta_{13} - \beta_{10} - \beta_{9} + \beta_{7} \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{18} + \beta_{17} + \beta_{13} + \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} + 2\beta_{6} + \beta_{5} - \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{19} - 3 \beta_{18} + \beta_{17} - 2 \beta_{15} + \beta_{12} + 2 \beta_{11} - 2 \beta_{9} + \cdots - 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( - \beta_{18} - 2 \beta_{16} + 2 \beta_{15} + \beta_{13} + \beta_{12} + 2 \beta_{11} - \beta_{10} + \cdots - 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( - \beta_{19} + 3 \beta_{17} + 2 \beta_{16} + 7 \beta_{14} - 9 \beta_{13} - 3 \beta_{10} + \cdots - \beta_{2} \)
|
| \(\nu^{8}\) | \(=\) |
\( - 3 \beta_{18} + 3 \beta_{17} - 2 \beta_{16} + 2 \beta_{15} + 4 \beta_{14} - 3 \beta_{13} + 3 \beta_{12} + \cdots - 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( - \beta_{19} - \beta_{18} - \beta_{17} + 6 \beta_{15} - 13 \beta_{12} - 6 \beta_{11} + 6 \beta_{9} + \cdots + 6 \)
|
| \(\nu^{10}\) | \(=\) |
\( 5 \beta_{18} - 6 \beta_{16} + 6 \beta_{15} - 8 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} + 6 \beta_{11} + \cdots - 6 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 11 \beta_{19} + \beta_{17} - 2 \beta_{16} - 32 \beta_{15} + 17 \beta_{14} - 23 \beta_{13} - 13 \beta_{10} + \cdots + 32 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 8 \beta_{19} + 3 \beta_{18} + 13 \beta_{17} - 6 \beta_{16} + 6 \beta_{15} - 12 \beta_{14} - 5 \beta_{13} + \cdots + 10 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 23 \beta_{19} + 9 \beta_{18} - 15 \beta_{17} - 22 \beta_{15} - 43 \beta_{12} - 26 \beta_{11} + \cdots - 22 \)
|
| \(\nu^{14}\) | \(=\) |
\( 11 \beta_{18} - 10 \beta_{16} + 10 \beta_{15} - 48 \beta_{14} + 13 \beta_{13} - 3 \beta_{12} + 10 \beta_{11} + \cdots - 10 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 45 \beta_{19} + 15 \beta_{17} + 18 \beta_{16} - 64 \beta_{15} - 9 \beta_{14} + 47 \beta_{13} + \cdots + 64 \)
|
| \(\nu^{16}\) | \(=\) |
\( 56 \beta_{19} - 99 \beta_{18} + 75 \beta_{17} + 70 \beta_{16} - 70 \beta_{15} - 140 \beta_{14} + \cdots - 42 \)
|
| \(\nu^{17}\) | \(=\) |
\( - \beta_{19} - 17 \beta_{18} - 97 \beta_{17} - 90 \beta_{15} - 93 \beta_{12} + 42 \beta_{11} + \cdots - 90 \)
|
| \(\nu^{18}\) | \(=\) |
\( 85 \beta_{18} + 74 \beta_{16} + 342 \beta_{15} - 56 \beta_{14} - 77 \beta_{13} - 173 \beta_{12} + \cdots + 74 \)
|
| \(\nu^{19}\) | \(=\) |
\( - 59 \beta_{19} + 97 \beta_{17} + 94 \beta_{16} - 32 \beta_{15} + 321 \beta_{14} - 103 \beta_{13} + \cdots + 32 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/104\mathbb{Z}\right)^\times\).
| \(n\) | \(41\) | \(53\) | \(79\) |
| \(\chi(n)\) | \(-\beta_{15}\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 83.1 |
|
−1.41123 | + | 0.0918109i | 2.19371 | 1.98314 | − | 0.259133i | −0.274863 | − | 0.274863i | −3.09583 | + | 0.201407i | 0.352378 | − | 0.352378i | −2.77488 | + | 0.547770i | 1.81237 | 0.413130 | + | 0.362660i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.2 | −1.33837 | − | 0.456912i | −2.95212 | 1.58246 | + | 1.22303i | 1.04334 | + | 1.04334i | 3.95103 | + | 1.34886i | 2.37322 | − | 2.37322i | −1.55910 | − | 2.35992i | 5.71504 | −0.919655 | − | 1.87308i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.3 | −0.813947 | − | 1.15650i | 1.38809 | −0.674982 | + | 1.88266i | 2.40872 | + | 2.40872i | −1.12983 | − | 1.60533i | 0.127019 | − | 0.127019i | 2.72669 | − | 0.751766i | −1.07320 | 0.825113 | − | 4.74625i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.4 | −0.209223 | − | 1.39865i | −1.86790 | −1.91245 | + | 0.585261i | −1.55756 | − | 1.55756i | 0.390808 | + | 2.61254i | −1.24165 | + | 1.24165i | 1.21871 | + | 2.55240i | 0.489042 | −1.85261 | + | 2.50437i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.5 | −0.0918109 | + | 1.41123i | 2.19371 | −1.98314 | − | 0.259133i | 0.274863 | + | 0.274863i | −0.201407 | + | 3.09583i | −0.352378 | + | 0.352378i | 0.547770 | − | 2.77488i | 1.81237 | −0.413130 | + | 0.362660i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.6 | 0.456912 | + | 1.33837i | −2.95212 | −1.58246 | + | 1.22303i | −1.04334 | − | 1.04334i | −1.34886 | − | 3.95103i | −2.37322 | + | 2.37322i | −2.35992 | − | 1.55910i | 5.71504 | 0.919655 | − | 1.87308i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.7 | 0.549370 | − | 1.30315i | 0.238218 | −1.39639 | − | 1.43182i | 0.328612 | + | 0.328612i | 0.130870 | − | 0.310433i | 2.68064 | − | 2.68064i | −2.63300 | + | 1.03310i | −2.94325 | 0.608760 | − | 0.247700i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.8 | 1.15650 | + | 0.813947i | 1.38809 | 0.674982 | + | 1.88266i | −2.40872 | − | 2.40872i | 1.60533 | + | 1.12983i | −0.127019 | + | 0.127019i | −0.751766 | + | 2.72669i | −1.07320 | −0.825113 | − | 4.74625i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.9 | 1.30315 | − | 0.549370i | 0.238218 | 1.39639 | − | 1.43182i | −0.328612 | − | 0.328612i | 0.310433 | − | 0.130870i | −2.68064 | + | 2.68064i | 1.03310 | − | 2.63300i | −2.94325 | −0.608760 | − | 0.247700i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 83.10 | 1.39865 | + | 0.209223i | −1.86790 | 1.91245 | + | 0.585261i | 1.55756 | + | 1.55756i | −2.61254 | − | 0.390808i | 1.24165 | − | 1.24165i | 2.55240 | + | 1.21871i | 0.489042 | 1.85261 | + | 2.50437i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.1 | −1.41123 | − | 0.0918109i | 2.19371 | 1.98314 | + | 0.259133i | −0.274863 | + | 0.274863i | −3.09583 | − | 0.201407i | 0.352378 | + | 0.352378i | −2.77488 | − | 0.547770i | 1.81237 | 0.413130 | − | 0.362660i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.2 | −1.33837 | + | 0.456912i | −2.95212 | 1.58246 | − | 1.22303i | 1.04334 | − | 1.04334i | 3.95103 | − | 1.34886i | 2.37322 | + | 2.37322i | −1.55910 | + | 2.35992i | 5.71504 | −0.919655 | + | 1.87308i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.3 | −0.813947 | + | 1.15650i | 1.38809 | −0.674982 | − | 1.88266i | 2.40872 | − | 2.40872i | −1.12983 | + | 1.60533i | 0.127019 | + | 0.127019i | 2.72669 | + | 0.751766i | −1.07320 | 0.825113 | + | 4.74625i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.4 | −0.209223 | + | 1.39865i | −1.86790 | −1.91245 | − | 0.585261i | −1.55756 | + | 1.55756i | 0.390808 | − | 2.61254i | −1.24165 | − | 1.24165i | 1.21871 | − | 2.55240i | 0.489042 | −1.85261 | − | 2.50437i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.5 | −0.0918109 | − | 1.41123i | 2.19371 | −1.98314 | + | 0.259133i | 0.274863 | − | 0.274863i | −0.201407 | − | 3.09583i | −0.352378 | − | 0.352378i | 0.547770 | + | 2.77488i | 1.81237 | −0.413130 | − | 0.362660i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.6 | 0.456912 | − | 1.33837i | −2.95212 | −1.58246 | − | 1.22303i | −1.04334 | + | 1.04334i | −1.34886 | + | 3.95103i | −2.37322 | − | 2.37322i | −2.35992 | + | 1.55910i | 5.71504 | 0.919655 | + | 1.87308i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.7 | 0.549370 | + | 1.30315i | 0.238218 | −1.39639 | + | 1.43182i | 0.328612 | − | 0.328612i | 0.130870 | + | 0.310433i | 2.68064 | + | 2.68064i | −2.63300 | − | 1.03310i | −2.94325 | 0.608760 | + | 0.247700i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.8 | 1.15650 | − | 0.813947i | 1.38809 | 0.674982 | − | 1.88266i | −2.40872 | + | 2.40872i | 1.60533 | − | 1.12983i | −0.127019 | − | 0.127019i | −0.751766 | − | 2.72669i | −1.07320 | −0.825113 | + | 4.74625i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.9 | 1.30315 | + | 0.549370i | 0.238218 | 1.39639 | + | 1.43182i | −0.328612 | + | 0.328612i | 0.310433 | + | 0.130870i | −2.68064 | − | 2.68064i | 1.03310 | + | 2.63300i | −2.94325 | −0.608760 | + | 0.247700i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 99.10 | 1.39865 | − | 0.209223i | −1.86790 | 1.91245 | − | 0.585261i | 1.55756 | − | 1.55756i | −2.61254 | + | 0.390808i | 1.24165 | + | 1.24165i | 2.55240 | − | 1.21871i | 0.489042 | 1.85261 | − | 2.50437i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
| 13.d | odd | 4 | 1 | inner |
| 104.m | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 104.2.m.b | ✓ | 20 |
| 3.b | odd | 2 | 1 | 936.2.w.h | 20 | ||
| 4.b | odd | 2 | 1 | 416.2.u.b | 20 | ||
| 8.b | even | 2 | 1 | 416.2.u.b | 20 | ||
| 8.d | odd | 2 | 1 | inner | 104.2.m.b | ✓ | 20 |
| 13.d | odd | 4 | 1 | inner | 104.2.m.b | ✓ | 20 |
| 24.f | even | 2 | 1 | 936.2.w.h | 20 | ||
| 39.f | even | 4 | 1 | 936.2.w.h | 20 | ||
| 52.f | even | 4 | 1 | 416.2.u.b | 20 | ||
| 104.j | odd | 4 | 1 | 416.2.u.b | 20 | ||
| 104.m | even | 4 | 1 | inner | 104.2.m.b | ✓ | 20 |
| 312.w | odd | 4 | 1 | 936.2.w.h | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.2.m.b | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
| 104.2.m.b | ✓ | 20 | 8.d | odd | 2 | 1 | inner |
| 104.2.m.b | ✓ | 20 | 13.d | odd | 4 | 1 | inner |
| 104.2.m.b | ✓ | 20 | 104.m | even | 4 | 1 | inner |
| 416.2.u.b | 20 | 4.b | odd | 2 | 1 | ||
| 416.2.u.b | 20 | 8.b | even | 2 | 1 | ||
| 416.2.u.b | 20 | 52.f | even | 4 | 1 | ||
| 416.2.u.b | 20 | 104.j | odd | 4 | 1 | ||
| 936.2.w.h | 20 | 3.b | odd | 2 | 1 | ||
| 936.2.w.h | 20 | 24.f | even | 2 | 1 | ||
| 936.2.w.h | 20 | 39.f | even | 4 | 1 | ||
| 936.2.w.h | 20 | 312.w | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{5} + T_{3}^{4} - 9T_{3}^{3} - 3T_{3}^{2} + 18T_{3} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(104, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} - 2 T^{19} + \cdots + 1024 \)
|
| $3$ |
\( (T^{5} + T^{4} - 9 T^{3} + \cdots - 4)^{4} \)
|
| $5$ |
\( T^{20} + 163 T^{16} + \cdots + 16 \)
|
| $7$ |
\( T^{20} + 343 T^{16} + \cdots + 16 \)
|
| $11$ |
\( (T^{10} + 4 T^{9} + \cdots + 5408)^{2} \)
|
| $13$ |
\( T^{20} + \cdots + 137858491849 \)
|
| $17$ |
\( (T^{10} + 71 T^{8} + \cdots + 355216)^{2} \)
|
| $19$ |
\( (T^{10} + 6 T^{9} + \cdots + 8192)^{2} \)
|
| $23$ |
\( (T^{10} - 94 T^{8} + \cdots - 2048)^{2} \)
|
| $29$ |
\( (T^{10} + 186 T^{8} + \cdots + 21632)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 92829679353856 \)
|
| $37$ |
\( T^{20} + \cdots + 911076029249296 \)
|
| $41$ |
\( (T^{10} + 12 T^{9} + \cdots + 512)^{2} \)
|
| $43$ |
\( (T^{10} + 111 T^{8} + \cdots + 795664)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 1416468496 \)
|
| $53$ |
\( (T^{10} + 230 T^{8} + \cdots + 3442688)^{2} \)
|
| $59$ |
\( (T^{10} - 10 T^{9} + \cdots + 1384448)^{2} \)
|
| $61$ |
\( (T^{10} + 392 T^{8} + \cdots + 8388608)^{2} \)
|
| $67$ |
\( (T^{10} + 8 T^{9} + \cdots + 78575648)^{2} \)
|
| $71$ |
\( T^{20} + \cdots + 97459622042896 \)
|
| $73$ |
\( (T^{10} - 12 T^{9} + \cdots + 512)^{2} \)
|
| $79$ |
\( (T^{10} + 442 T^{8} + \cdots + 336338048)^{2} \)
|
| $83$ |
\( (T^{10} - 8 T^{9} + \cdots + 414950432)^{2} \)
|
| $89$ |
\( (T^{10} + \cdots + 117546549248)^{2} \)
|
| $97$ |
\( (T^{10} - 6 T^{9} + \cdots + 7270250528)^{2} \)
|
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