Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2100,2,Mod(1349,2100)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2100, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 3, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2100.1349");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2100.bo (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: | \(16.7685844245\) |
| 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: |
\( x^{20} - 3 x^{18} - 16 x^{16} + 87 x^{14} + 91 x^{12} - 1104 x^{10} + 819 x^{8} + 7047 x^{6} + \cdots + 59049 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | no (minimal twist has level 420) |
| 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
\( x^{20} - 3 x^{18} - 16 x^{16} + 87 x^{14} + 91 x^{12} - 1104 x^{10} + 819 x^{8} + 7047 x^{6} + \cdots + 59049 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 29 \nu^{19} - 966 \nu^{17} + 3218 \nu^{15} + 6063 \nu^{13} - 4664 \nu^{11} + 13224 \nu^{9} + \cdots - 2119203 \nu ) / 5038848 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 47 \nu^{18} + 654 \nu^{16} - 1354 \nu^{14} + 1797 \nu^{12} + 18808 \nu^{10} - 57192 \nu^{8} + \cdots + 1673055 ) / 1679616 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 83 \nu^{19} + 930 \nu^{17} - 86 \nu^{15} - 29625 \nu^{13} + 64280 \nu^{11} + 292920 \nu^{9} + \cdots - 1541835 \nu ) / 5038848 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 19 \nu^{19} + 222 \nu^{17} - 466 \nu^{15} - 1677 \nu^{13} + 13744 \nu^{11} - 528 \nu^{9} + \cdots - 448335 \nu ) / 839808 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 31 \nu^{19} + 48 \nu^{17} - 908 \nu^{15} - 3921 \nu^{13} + 26636 \nu^{11} + 1212 \nu^{9} + \cdots - 2512863 \nu ) / 1259712 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 131 \nu^{18} - 138 \nu^{16} + 4094 \nu^{14} - 6303 \nu^{12} - 42728 \nu^{10} + 100920 \nu^{8} + \cdots + 2224179 ) / 1679616 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 53 \nu^{18} - 366 \nu^{16} + 3098 \nu^{14} + 63 \nu^{12} - 42344 \nu^{10} + 74808 \nu^{8} + \cdots + 3759453 ) / 559872 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 22 \nu^{18} + 111 \nu^{16} + 865 \nu^{14} - 2391 \nu^{12} - 10642 \nu^{10} + 43206 \nu^{8} + \cdots + 1581201 ) / 209952 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 49 \nu^{19} + 672 \nu^{17} - 2188 \nu^{15} - 6897 \nu^{13} + 32620 \nu^{11} + 30396 \nu^{9} + \cdots + 426465 \nu ) / 1259712 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 37 \nu^{18} + 138 \nu^{16} + 322 \nu^{14} - 2841 \nu^{12} + 3464 \nu^{10} + 18600 \nu^{8} + \cdots - 37179 ) / 279936 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 221 \nu^{18} - 1194 \nu^{16} - 4130 \nu^{14} + 32097 \nu^{12} - 39208 \nu^{10} - 303240 \nu^{8} + \cdots + 4323699 ) / 1679616 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - \nu^{19} + 3 \nu^{17} + 16 \nu^{15} - 87 \nu^{13} - 91 \nu^{11} + 1104 \nu^{9} - 819 \nu^{7} + \cdots + 19683 \nu ) / 19683 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 38 \nu^{18} - 33 \nu^{16} - 527 \nu^{14} + 309 \nu^{12} + 3134 \nu^{10} - 1290 \nu^{8} + \cdots + 137781 ) / 209952 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 59 \nu^{18} - 246 \nu^{16} - 62 \nu^{14} + 4455 \nu^{12} - 9976 \nu^{10} - 20952 \nu^{8} + \cdots + 1502469 ) / 279936 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 383 \nu^{18} - 150 \nu^{16} - 3374 \nu^{14} + 5187 \nu^{12} + 42872 \nu^{10} - 57576 \nu^{8} + \cdots - 4139991 ) / 1679616 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 131 \nu^{19} + 138 \nu^{17} - 4094 \nu^{15} + 6303 \nu^{13} + 42728 \nu^{11} - 100920 \nu^{9} + \cdots - 2224179 \nu ) / 1679616 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 127 \nu^{19} - 48 \nu^{17} - 4084 \nu^{15} + 3777 \nu^{13} + 48628 \nu^{11} - 119868 \nu^{9} + \cdots - 5570289 \nu ) / 1259712 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( 65 \nu^{19} - 118 \nu^{17} - 902 \nu^{15} + 3073 \nu^{13} + 6224 \nu^{11} - 33136 \nu^{9} + \cdots - 277749 \nu ) / 279936 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{16} - \beta_{15} + \beta_{12} + \beta_{11} + \beta_{8} - \beta_{7} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{17} + \beta_{13} - 2\beta_{10} + \beta_{5} + \beta_{4} - 2\beta_{2} + \beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{14} - \beta_{11} + \beta_{9} - \beta_{8} - 2\beta_{7} - 3\beta_{3} + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( -3\beta_{19} + 2\beta_{18} + 2\beta_{17} - 5\beta_{13} - 4\beta_{6} - 3\beta_{5} + 4\beta_{4} + 2\beta_{2} + 6\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{16} + \beta_{15} - 5\beta_{14} + 4\beta_{12} + 9\beta_{11} + 3\beta_{9} - 12\beta_{7} - 10\beta_{3} - 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{19} - 12 \beta_{18} + 12 \beta_{17} - 10 \beta_{13} - 20 \beta_{10} + 12 \beta_{6} + 9 \beta_{5} + \cdots + \beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 19 \beta_{16} + 8 \beta_{15} - 9 \beta_{14} - 21 \beta_{12} - 30 \beta_{11} + 7 \beta_{9} - 5 \beta_{8} + \cdots + 12 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 19 \beta_{19} - 2 \beta_{18} + 37 \beta_{17} - 25 \beta_{13} + 16 \beta_{10} - 26 \beta_{6} + \cdots - 9 \beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 32 \beta_{16} + 65 \beta_{15} - 29 \beta_{14} - 48 \beta_{12} + 26 \beta_{11} - 6 \beta_{9} + \cdots + 12 \)
|
| \(\nu^{11}\) | \(=\) |
\( 34 \beta_{19} - 144 \beta_{18} + 68 \beta_{17} - 101 \beta_{13} - 84 \beta_{10} + 168 \beta_{6} + \cdots - 36 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( - 28 \beta_{16} - 58 \beta_{15} - 10 \beta_{14} - 120 \beta_{12} - 242 \beta_{11} - 42 \beta_{9} + \cdots - 183 \)
|
| \(\nu^{13}\) | \(=\) |
\( - 70 \beta_{19} + 20 \beta_{18} + 296 \beta_{17} + 268 \beta_{13} + 184 \beta_{10} - 268 \beta_{6} + \cdots - 303 \beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( 137 \beta_{16} + 493 \beta_{15} - 330 \beta_{14} - 403 \beta_{12} - 37 \beta_{11} - 162 \beta_{9} + \cdots + 312 \)
|
| \(\nu^{15}\) | \(=\) |
\( 42 \beta_{19} - 228 \beta_{18} - 703 \beta_{17} - 907 \beta_{13} + 302 \beta_{10} + 96 \beta_{6} + \cdots - 91 \beta_1 \)
|
| \(\nu^{16}\) | \(=\) |
\( 1824 \beta_{16} - 1326 \beta_{15} - 232 \beta_{14} + 1056 \beta_{12} + 193 \beta_{11} + 491 \beta_{9} + \cdots - 3606 \)
|
| \(\nu^{17}\) | \(=\) |
\( - 849 \beta_{19} + 1834 \beta_{18} + 94 \beta_{17} + 3011 \beta_{13} + 1716 \beta_{10} + \cdots - 2550 \beta_1 \)
|
| \(\nu^{18}\) | \(=\) |
\( 2062 \beta_{16} + 5951 \beta_{15} - 775 \beta_{14} - 3268 \beta_{12} - 3105 \beta_{11} + 1305 \beta_{9} + \cdots + 2109 \)
|
| \(\nu^{19}\) | \(=\) |
\( 467 \beta_{19} + 6744 \beta_{18} - 14316 \beta_{17} - 11798 \beta_{13} + 12332 \beta_{10} + \cdots - 1159 \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).
| \(n\) | \(701\) | \(1051\) | \(1177\) | \(1501\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) | \(\beta_{7}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1349.1 |
|
0 | −1.70278 | − | 0.317079i | 0 | 0 | 0 | 1.99797 | + | 1.73439i | 0 | 2.79892 | + | 1.07983i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1349.2 | 0 | −1.64368 | − | 0.546177i | 0 | 0 | 0 | 2.41433 | − | 1.08214i | 0 | 2.40338 | + | 1.79548i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1349.3 | 0 | −1.56233 | + | 0.747749i | 0 | 0 | 0 | −2.60608 | − | 0.456468i | 0 | 1.88174 | − | 2.33646i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1349.4 | 0 | −0.368412 | − | 1.69242i | 0 | 0 | 0 | −1.93884 | + | 1.80025i | 0 | −2.72854 | + | 1.24701i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1349.5 | 0 | −0.268793 | + | 1.71107i | 0 | 0 | 0 | 0.615143 | − | 2.57325i | 0 | −2.85550 | − | 0.919845i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1349.6 | 0 | 0.268793 | − | 1.71107i | 0 | 0 | 0 | −0.615143 | + | 2.57325i | 0 | −2.85550 | − | 0.919845i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1349.7 | 0 | 0.368412 | + | 1.69242i | 0 | 0 | 0 | 1.93884 | − | 1.80025i | 0 | −2.72854 | + | 1.24701i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1349.8 | 0 | 1.56233 | − | 0.747749i | 0 | 0 | 0 | 2.60608 | + | 0.456468i | 0 | 1.88174 | − | 2.33646i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1349.9 | 0 | 1.64368 | + | 0.546177i | 0 | 0 | 0 | −2.41433 | + | 1.08214i | 0 | 2.40338 | + | 1.79548i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1349.10 | 0 | 1.70278 | + | 0.317079i | 0 | 0 | 0 | −1.99797 | − | 1.73439i | 0 | 2.79892 | + | 1.07983i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1949.1 | 0 | −1.70278 | + | 0.317079i | 0 | 0 | 0 | 1.99797 | − | 1.73439i | 0 | 2.79892 | − | 1.07983i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1949.2 | 0 | −1.64368 | + | 0.546177i | 0 | 0 | 0 | 2.41433 | + | 1.08214i | 0 | 2.40338 | − | 1.79548i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1949.3 | 0 | −1.56233 | − | 0.747749i | 0 | 0 | 0 | −2.60608 | + | 0.456468i | 0 | 1.88174 | + | 2.33646i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1949.4 | 0 | −0.368412 | + | 1.69242i | 0 | 0 | 0 | −1.93884 | − | 1.80025i | 0 | −2.72854 | − | 1.24701i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1949.5 | 0 | −0.268793 | − | 1.71107i | 0 | 0 | 0 | 0.615143 | + | 2.57325i | 0 | −2.85550 | + | 0.919845i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1949.6 | 0 | 0.268793 | + | 1.71107i | 0 | 0 | 0 | −0.615143 | − | 2.57325i | 0 | −2.85550 | + | 0.919845i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1949.7 | 0 | 0.368412 | − | 1.69242i | 0 | 0 | 0 | 1.93884 | + | 1.80025i | 0 | −2.72854 | − | 1.24701i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1949.8 | 0 | 1.56233 | + | 0.747749i | 0 | 0 | 0 | 2.60608 | − | 0.456468i | 0 | 1.88174 | + | 2.33646i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1949.9 | 0 | 1.64368 | − | 0.546177i | 0 | 0 | 0 | −2.41433 | − | 1.08214i | 0 | 2.40338 | − | 1.79548i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1949.10 | 0 | 1.70278 | − | 0.317079i | 0 | 0 | 0 | −1.99797 | + | 1.73439i | 0 | 2.79892 | − | 1.07983i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 21.g | even | 6 | 1 | inner |
| 105.p | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2100.2.bo.h | 20 | |
| 3.b | odd | 2 | 1 | 2100.2.bo.g | 20 | ||
| 5.b | even | 2 | 1 | inner | 2100.2.bo.h | 20 | |
| 5.c | odd | 4 | 1 | 420.2.bh.a | ✓ | 10 | |
| 5.c | odd | 4 | 1 | 2100.2.bi.k | 10 | ||
| 7.d | odd | 6 | 1 | 2100.2.bo.g | 20 | ||
| 15.d | odd | 2 | 1 | 2100.2.bo.g | 20 | ||
| 15.e | even | 4 | 1 | 420.2.bh.b | yes | 10 | |
| 15.e | even | 4 | 1 | 2100.2.bi.j | 10 | ||
| 21.g | even | 6 | 1 | inner | 2100.2.bo.h | 20 | |
| 35.i | odd | 6 | 1 | 2100.2.bo.g | 20 | ||
| 35.k | even | 12 | 1 | 420.2.bh.b | yes | 10 | |
| 35.k | even | 12 | 1 | 2100.2.bi.j | 10 | ||
| 35.k | even | 12 | 1 | 2940.2.d.a | 10 | ||
| 35.l | odd | 12 | 1 | 2940.2.d.b | 10 | ||
| 105.p | even | 6 | 1 | inner | 2100.2.bo.h | 20 | |
| 105.w | odd | 12 | 1 | 420.2.bh.a | ✓ | 10 | |
| 105.w | odd | 12 | 1 | 2100.2.bi.k | 10 | ||
| 105.w | odd | 12 | 1 | 2940.2.d.b | 10 | ||
| 105.x | even | 12 | 1 | 2940.2.d.a | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 420.2.bh.a | ✓ | 10 | 5.c | odd | 4 | 1 | |
| 420.2.bh.a | ✓ | 10 | 105.w | odd | 12 | 1 | |
| 420.2.bh.b | yes | 10 | 15.e | even | 4 | 1 | |
| 420.2.bh.b | yes | 10 | 35.k | even | 12 | 1 | |
| 2100.2.bi.j | 10 | 15.e | even | 4 | 1 | ||
| 2100.2.bi.j | 10 | 35.k | even | 12 | 1 | ||
| 2100.2.bi.k | 10 | 5.c | odd | 4 | 1 | ||
| 2100.2.bi.k | 10 | 105.w | odd | 12 | 1 | ||
| 2100.2.bo.g | 20 | 3.b | odd | 2 | 1 | ||
| 2100.2.bo.g | 20 | 7.d | odd | 6 | 1 | ||
| 2100.2.bo.g | 20 | 15.d | odd | 2 | 1 | ||
| 2100.2.bo.g | 20 | 35.i | odd | 6 | 1 | ||
| 2100.2.bo.h | 20 | 1.a | even | 1 | 1 | trivial | |
| 2100.2.bo.h | 20 | 5.b | even | 2 | 1 | inner | |
| 2100.2.bo.h | 20 | 21.g | even | 6 | 1 | inner | |
| 2100.2.bo.h | 20 | 105.p | even | 6 | 1 | inner | |
| 2940.2.d.a | 10 | 35.k | even | 12 | 1 | ||
| 2940.2.d.a | 10 | 105.x | even | 12 | 1 | ||
| 2940.2.d.b | 10 | 35.l | odd | 12 | 1 | ||
| 2940.2.d.b | 10 | 105.w | odd | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2100, [\chi])\):
|
\( T_{11}^{10} + 6 T_{11}^{9} - 4 T_{11}^{8} - 96 T_{11}^{7} + 84 T_{11}^{6} + 2280 T_{11}^{5} + \cdots + 1728 \)
|
|
\( T_{13}^{10} - 81T_{13}^{8} + 2183T_{13}^{6} - 22251T_{13}^{4} + 65704T_{13}^{2} - 3888 \)
|
|
\( T_{19}^{10} + 3 T_{19}^{9} - 12 T_{19}^{8} - 45 T_{19}^{7} + 152 T_{19}^{6} + 519 T_{19}^{5} + \cdots + 192 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{20} \)
|
| $3$ |
\( T^{20} - 3 T^{18} + \cdots + 59049 \)
|
| $5$ |
\( T^{20} \)
|
| $7$ |
\( T^{20} + \cdots + 282475249 \)
|
| $11$ |
\( (T^{10} + 6 T^{9} + \cdots + 1728)^{2} \)
|
| $13$ |
\( (T^{10} - 81 T^{8} + \cdots - 3888)^{2} \)
|
| $17$ |
\( T^{20} + \cdots + 4857532416 \)
|
| $19$ |
\( (T^{10} + 3 T^{9} + \cdots + 192)^{2} \)
|
| $23$ |
\( T^{20} + \cdots + 6063945650064 \)
|
| $29$ |
\( (T^{10} + 142 T^{8} + \cdots + 8748)^{2} \)
|
| $31$ |
\( (T^{10} - 15 T^{9} + \cdots + 1978032)^{2} \)
|
| $37$ |
\( T^{20} + \cdots + 157351936 \)
|
| $41$ |
\( (T^{5} + 4 T^{4} + \cdots + 1338)^{4} \)
|
| $43$ |
\( (T^{10} + 253 T^{8} + \cdots + 2430481)^{2} \)
|
| $47$ |
\( T^{20} + \cdots + 87\!\cdots\!36 \)
|
| $53$ |
\( T^{20} + \cdots + 12\!\cdots\!64 \)
|
| $59$ |
\( (T^{10} + 160 T^{8} + \cdots + 125081856)^{2} \)
|
| $61$ |
\( (T^{10} - 42 T^{9} + \cdots + 1338672)^{2} \)
|
| $67$ |
\( T^{20} + \cdots + 20\!\cdots\!61 \)
|
| $71$ |
\( (T^{10} + 288 T^{8} + \cdots + 88259328)^{2} \)
|
| $73$ |
\( T^{20} + \cdots + 2472844310784 \)
|
| $79$ |
\( (T^{10} + T^{9} + \cdots + 138485824)^{2} \)
|
| $83$ |
\( (T^{10} + 774 T^{8} + \cdots + 829094436)^{2} \)
|
| $89$ |
\( (T^{10} + 28 T^{9} + \cdots + 272484)^{2} \)
|
| $97$ |
\( (T^{10} - 212 T^{8} + \cdots - 1051392)^{2} \)
|
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