Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [833,2,Mod(67,833)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(833, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("833.67");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 833 = 7^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 833.j (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: | \(6.65153848837\) |
| 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: |
\( x^{20} - 15 x^{18} + 158 x^{16} - 789 x^{14} + 2811 x^{12} - 5497 x^{10} + 7763 x^{8} - 6130 x^{6} + \cdots + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 119) |
| 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
\( x^{20} - 15 x^{18} + 158 x^{16} - 789 x^{14} + 2811 x^{12} - 5497 x^{10} + 7763 x^{8} - 6130 x^{6} + \cdots + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 105167985248 \nu^{18} + 1578475893214 \nu^{16} - 16365824001665 \nu^{14} + \cdots + 78380272675006 ) / 33764073776255 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 105167985248 \nu^{19} - 1578475893214 \nu^{17} + 16365824001665 \nu^{15} + \cdots - 112144346451261 \nu ) / 33764073776255 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 236129147271 \nu^{19} - 3281139197798 \nu^{17} + 34019237713405 \nu^{15} + \cdots + 11\!\cdots\!48 \nu ) / 33764073776255 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 303278752213 \nu^{19} - 4348221369074 \nu^{17} + 45082871365015 \nu^{15} + \cdots + 638549285555839 \nu ) / 33764073776255 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 460801810894 \nu^{18} - 6375056707571 \nu^{16} + 65591526282169 \nu^{14} + \cdots - 644575884732 ) / 33764073776255 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 472619581157 \nu^{18} + 6984961679586 \nu^{16} - 72420905506335 \nu^{14} + \cdots + 30029587318919 ) / 33764073776255 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 475596342403 \nu^{18} + 6993779433564 \nu^{16} - 72512329018290 \nu^{14} + \cdots - 57671699234524 ) / 33764073776255 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 493117151933 \nu^{19} - 7245067417544 \nu^{17} + 75117712437340 \nu^{15} + \cdots + 225879576443454 \nu ) / 33764073776255 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 598285137181 \nu^{18} - 8823543310758 \nu^{16} + 91483536439005 \nu^{14} + \cdots + 79971156215938 ) / 33764073776255 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 122498334469 \nu^{18} - 1833970855129 \nu^{16} + 19304479249306 \nu^{14} + \cdots - 227538607333 ) / 6752814755251 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - 122498334469 \nu^{19} + 1833970855129 \nu^{17} - 19304479249306 \nu^{15} + \cdots + 227538607333 \nu ) / 6752814755251 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 689548905141 \nu^{18} + 10891933046952 \nu^{16} - 117027224125074 \nu^{14} + \cdots + 60014234912796 ) / 33764073776255 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 978391868587 \nu^{18} - 15396851960764 \nu^{16} + 165239222424698 \nu^{14} + \cdots - 83148106178662 ) / 33764073776255 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( 1425943258811 \nu^{18} - 21174880035929 \nu^{16} + 222383983019251 \nu^{14} + \cdots - 2578664825543 ) / 33764073776255 \)
|
| \(\beta_{16}\) | \(=\) |
\( ( 2143602916404 \nu^{19} - 31923210204788 \nu^{17} + 335272203267446 \nu^{15} + \cdots - 115860704313469 \nu ) / 33764073776255 \)
|
| \(\beta_{17}\) | \(=\) |
\( ( 3734486457336 \nu^{19} - 56489916441197 \nu^{17} + 597033821938674 \nu^{15} + \cdots - 233910577305051 \nu ) / 33764073776255 \)
|
| \(\beta_{18}\) | \(=\) |
\( ( 11767895547054 \nu^{19} - 176194784496911 \nu^{17} + \cdots - 21863907259012 \nu ) / 33764073776255 \)
|
| \(\beta_{19}\) | \(=\) |
\( ( - 20944702587753 \nu^{19} + 313328976477377 \nu^{17} + \cdots + 38806614014209 \nu ) / 33764073776255 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{15} - \beta_{14} - \beta_{13} + 3\beta_{11} - \beta_{10} - \beta_{8} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{19} + 2\beta_{18} - \beta_{17} - \beta_{16} - 6\beta_{12} - \beta_{9} + \beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( -8\beta_{15} - 7\beta_{14} - 9\beta_{13} + 18\beta_{11} - 2\beta_{7} + 2\beta_{6} + 8\beta_{2} - 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{19} + 20\beta_{18} - 11\beta_{17} - 14\beta_{16} - 46\beta_{12} - 20\beta_{5} + 9\beta_{4} - 46\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 52\beta_{10} + 75\beta_{8} - 25\beta_{7} + 69\beta_{2} - 133 \)
|
| \(\nu^{7}\) | \(=\) |
\( 102\beta_{9} - 175\beta_{5} + 75\beta_{4} - 142\beta_{3} - 379\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 596\beta_{15} + 412\beta_{14} + 629\beta_{13} - 1068\beta_{11} + 412\beta_{10} + 629\beta_{8} - 244\beta_{6} \)
|
| \(\nu^{9}\) | \(=\) |
\( -629\beta_{19} - 1502\beta_{18} + 900\beta_{17} + 1301\beta_{16} + 3193\beta_{12} + 900\beta_{9} - 1301\beta_{3} \)
|
| \(\nu^{10}\) | \(=\) |
\( 5123 \beta_{15} + 3394 \beta_{14} + 5324 \beta_{13} - 8880 \beta_{11} + 2201 \beta_{7} - 2201 \beta_{6} + \cdots + 8880 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 5324 \beta_{19} - 12849 \beta_{18} + 7796 \beta_{17} + 11455 \beta_{16} + 27123 \beta_{12} + \cdots + 27123 \beta_1 \)
|
| \(\nu^{12}\) | \(=\) |
\( -28517\beta_{10} - 45296\beta_{8} + 19251\beta_{7} - 43902\beta_{2} + 74967 \)
|
| \(\nu^{13}\) | \(=\) |
\( -67019\beta_{9} + 109843\beta_{5} - 45296\beta_{4} + 99183\beta_{3} + 231184\beta_1 \)
|
| \(\nu^{14}\) | \(=\) |
\( - 375663 \beta_{15} - 241844 \beta_{14} - 386323 \beta_{13} + 637193 \beta_{11} + \cdots + 166202 \beta_{6} \)
|
| \(\nu^{15}\) | \(=\) |
\( 386323 \beta_{19} + 938848 \beta_{18} - 574248 \beta_{17} - 852546 \beta_{16} - 1973427 \beta_{12} + \cdots + 852546 \beta_{3} \)
|
| \(\nu^{16}\) | \(=\) |
\( - 3212296 \beta_{15} - 2059729 \beta_{14} - 3298598 \beta_{13} + 5432335 \beta_{11} - 1426794 \beta_{7} + \cdots - 5432335 \)
|
| \(\nu^{17}\) | \(=\) |
\( 3298598 \beta_{19} + 8023990 \beta_{18} - 4913317 \beta_{17} - 7304753 \beta_{16} + \cdots - 16856546 \beta_1 \)
|
| \(\nu^{18}\) | \(=\) |
\( 17575783\beta_{10} + 28179134\beta_{8} - 12218070\beta_{7} + 27459897\beta_{2} - 46375558 \)
|
| \(\nu^{19}\) | \(=\) |
\( 42011923\beta_{9} - 68576338\beta_{5} + 28179134\beta_{4} - 62499388\beta_{3} - 144026512\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/833\mathbb{Z}\right)^\times\).
| \(n\) | \(52\) | \(785\) |
| \(\chi(n)\) | \(-\beta_{11}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
−1.22628 | − | 2.12399i | −1.01061 | − | 0.583476i | −2.00755 | + | 3.47717i | −2.29376 | + | 1.32430i | 2.86203i | 0 | 4.94216 | −0.819111 | − | 1.41874i | 5.62560 | + | 3.24794i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.2 | −1.22628 | − | 2.12399i | 1.01061 | + | 0.583476i | −2.00755 | + | 3.47717i | 2.29376 | − | 1.32430i | − | 2.86203i | 0 | 4.94216 | −0.819111 | − | 1.41874i | −5.62560 | − | 3.24794i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.3 | −0.717211 | − | 1.24225i | −2.53174 | − | 1.46170i | −0.0287840 | + | 0.0498554i | 1.51432 | − | 0.874295i | 4.19339i | 0 | −2.78627 | 2.77312 | + | 4.80319i | −2.17218 | − | 1.25411i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.4 | −0.717211 | − | 1.24225i | 2.53174 | + | 1.46170i | −0.0287840 | + | 0.0498554i | −1.51432 | + | 0.874295i | − | 4.19339i | 0 | −2.78627 | 2.77312 | + | 4.80319i | 2.17218 | + | 1.25411i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.5 | −0.274428 | − | 0.475324i | −0.974258 | − | 0.562488i | 0.849378 | − | 1.47117i | −0.987893 | + | 0.570360i | 0.617451i | 0 | −2.03009 | −0.867214 | − | 1.50206i | 0.542212 | + | 0.313046i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.6 | −0.274428 | − | 0.475324i | 0.974258 | + | 0.562488i | 0.849378 | − | 1.47117i | 0.987893 | − | 0.570360i | − | 0.617451i | 0 | −2.03009 | −0.867214 | − | 1.50206i | −0.542212 | − | 0.313046i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.7 | 0.558220 | + | 0.966865i | −1.69010 | − | 0.975779i | 0.376781 | − | 0.652604i | 1.29801 | − | 0.749406i | − | 2.17880i | 0 | 3.07419 | 0.404290 | + | 0.700250i | 1.44915 | + | 0.836667i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.8 | 0.558220 | + | 0.966865i | 1.69010 | + | 0.975779i | 0.376781 | − | 0.652604i | −1.29801 | + | 0.749406i | 2.17880i | 0 | 3.07419 | 0.404290 | + | 0.700250i | −1.44915 | − | 0.836667i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.9 | 1.15970 | + | 2.00867i | −0.115629 | − | 0.0667582i | −1.68983 | + | 2.92687i | 2.18741 | − | 1.26290i | − | 0.309679i | 0 | −3.19999 | −1.49109 | − | 2.58264i | 5.07349 | + | 2.92918i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 67.10 | 1.15970 | + | 2.00867i | 0.115629 | + | 0.0667582i | −1.68983 | + | 2.92687i | −2.18741 | + | 1.26290i | 0.309679i | 0 | −3.19999 | −1.49109 | − | 2.58264i | −5.07349 | − | 2.92918i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.1 | −1.22628 | + | 2.12399i | −1.01061 | + | 0.583476i | −2.00755 | − | 3.47717i | −2.29376 | − | 1.32430i | − | 2.86203i | 0 | 4.94216 | −0.819111 | + | 1.41874i | 5.62560 | − | 3.24794i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.2 | −1.22628 | + | 2.12399i | 1.01061 | − | 0.583476i | −2.00755 | − | 3.47717i | 2.29376 | + | 1.32430i | 2.86203i | 0 | 4.94216 | −0.819111 | + | 1.41874i | −5.62560 | + | 3.24794i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.3 | −0.717211 | + | 1.24225i | −2.53174 | + | 1.46170i | −0.0287840 | − | 0.0498554i | 1.51432 | + | 0.874295i | − | 4.19339i | 0 | −2.78627 | 2.77312 | − | 4.80319i | −2.17218 | + | 1.25411i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.4 | −0.717211 | + | 1.24225i | 2.53174 | − | 1.46170i | −0.0287840 | − | 0.0498554i | −1.51432 | − | 0.874295i | 4.19339i | 0 | −2.78627 | 2.77312 | − | 4.80319i | 2.17218 | − | 1.25411i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.5 | −0.274428 | + | 0.475324i | −0.974258 | + | 0.562488i | 0.849378 | + | 1.47117i | −0.987893 | − | 0.570360i | − | 0.617451i | 0 | −2.03009 | −0.867214 | + | 1.50206i | 0.542212 | − | 0.313046i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.6 | −0.274428 | + | 0.475324i | 0.974258 | − | 0.562488i | 0.849378 | + | 1.47117i | 0.987893 | + | 0.570360i | 0.617451i | 0 | −2.03009 | −0.867214 | + | 1.50206i | −0.542212 | + | 0.313046i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.7 | 0.558220 | − | 0.966865i | −1.69010 | + | 0.975779i | 0.376781 | + | 0.652604i | 1.29801 | + | 0.749406i | 2.17880i | 0 | 3.07419 | 0.404290 | − | 0.700250i | 1.44915 | − | 0.836667i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.8 | 0.558220 | − | 0.966865i | 1.69010 | − | 0.975779i | 0.376781 | + | 0.652604i | −1.29801 | − | 0.749406i | − | 2.17880i | 0 | 3.07419 | 0.404290 | − | 0.700250i | −1.44915 | + | 0.836667i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.9 | 1.15970 | − | 2.00867i | −0.115629 | + | 0.0667582i | −1.68983 | − | 2.92687i | 2.18741 | + | 1.26290i | 0.309679i | 0 | −3.19999 | −1.49109 | + | 2.58264i | 5.07349 | − | 2.92918i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 373.10 | 1.15970 | − | 2.00867i | 0.115629 | − | 0.0667582i | −1.68983 | − | 2.92687i | −2.18741 | − | 1.26290i | − | 0.309679i | 0 | −3.19999 | −1.49109 | + | 2.58264i | −5.07349 | + | 2.92918i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
| 17.b | even | 2 | 1 | inner |
| 119.j | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 833.2.j.a | 20 | |
| 7.b | odd | 2 | 1 | 119.2.j.a | ✓ | 20 | |
| 7.c | even | 3 | 1 | 833.2.b.d | 10 | ||
| 7.c | even | 3 | 1 | inner | 833.2.j.a | 20 | |
| 7.d | odd | 6 | 1 | 119.2.j.a | ✓ | 20 | |
| 7.d | odd | 6 | 1 | 833.2.b.c | 10 | ||
| 17.b | even | 2 | 1 | inner | 833.2.j.a | 20 | |
| 119.d | odd | 2 | 1 | 119.2.j.a | ✓ | 20 | |
| 119.h | odd | 6 | 1 | 119.2.j.a | ✓ | 20 | |
| 119.h | odd | 6 | 1 | 833.2.b.c | 10 | ||
| 119.j | even | 6 | 1 | 833.2.b.d | 10 | ||
| 119.j | even | 6 | 1 | inner | 833.2.j.a | 20 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 119.2.j.a | ✓ | 20 | 7.b | odd | 2 | 1 | |
| 119.2.j.a | ✓ | 20 | 7.d | odd | 6 | 1 | |
| 119.2.j.a | ✓ | 20 | 119.d | odd | 2 | 1 | |
| 119.2.j.a | ✓ | 20 | 119.h | odd | 6 | 1 | |
| 833.2.b.c | 10 | 7.d | odd | 6 | 1 | ||
| 833.2.b.c | 10 | 119.h | odd | 6 | 1 | ||
| 833.2.b.d | 10 | 7.c | even | 3 | 1 | ||
| 833.2.b.d | 10 | 119.j | even | 6 | 1 | ||
| 833.2.j.a | 20 | 1.a | even | 1 | 1 | trivial | |
| 833.2.j.a | 20 | 7.c | even | 3 | 1 | inner | |
| 833.2.j.a | 20 | 17.b | even | 2 | 1 | inner | |
| 833.2.j.a | 20 | 119.j | even | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(833, [\chi])\):
|
\( T_{2}^{10} + T_{2}^{9} + 8T_{2}^{8} + 5T_{2}^{7} + 47T_{2}^{6} + 31T_{2}^{5} + 87T_{2}^{4} + 22T_{2}^{3} + 94T_{2}^{2} + 40T_{2} + 25 \)
|
|
\( T_{13}^{5} - 4T_{13}^{4} - 24T_{13}^{3} + 59T_{13}^{2} + 119T_{13} - 245 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{10} + T^{9} + 8 T^{8} + \cdots + 25)^{2} \)
|
| $3$ |
\( T^{20} - 15 T^{18} + \cdots + 1 \)
|
| $5$ |
\( T^{20} - 20 T^{18} + \cdots + 160000 \)
|
| $7$ |
\( T^{20} \)
|
| $11$ |
\( T^{20} - 44 T^{18} + \cdots + 5764801 \)
|
| $13$ |
\( (T^{5} - 4 T^{4} + \cdots - 245)^{4} \)
|
| $17$ |
\( T^{20} + \cdots + 2015993900449 \)
|
| $19$ |
\( (T^{10} - 5 T^{9} + \cdots + 153664)^{2} \)
|
| $23$ |
\( T^{20} + \cdots + 377801998336 \)
|
| $29$ |
\( (T^{10} + 175 T^{8} + \cdots + 24010000)^{2} \)
|
| $31$ |
\( T^{20} + \cdots + 53974440976 \)
|
| $37$ |
\( T^{20} + \cdots + 14757890560000 \)
|
| $41$ |
\( (T^{10} + 182 T^{8} + \cdots + 2972176)^{2} \)
|
| $43$ |
\( (T^{5} + 13 T^{4} + \cdots - 1892)^{4} \)
|
| $47$ |
\( (T^{10} + 15 T^{9} + \cdots + 2458624)^{2} \)
|
| $53$ |
\( (T^{10} - 20 T^{9} + \cdots + 32041)^{2} \)
|
| $59$ |
\( (T^{10} - 7 T^{9} + \cdots + 13868176)^{2} \)
|
| $61$ |
\( T^{20} + \cdots + 655360000 \)
|
| $67$ |
\( (T^{10} - 6 T^{9} + \cdots + 256)^{2} \)
|
| $71$ |
\( (T^{10} + 547 T^{8} + \cdots + 782824441)^{2} \)
|
| $73$ |
\( T^{20} + \cdots + 37583962091776 \)
|
| $79$ |
\( T^{20} + \cdots + 751274631121 \)
|
| $83$ |
\( (T^{5} + 24 T^{4} + \cdots - 7448)^{4} \)
|
| $89$ |
\( (T^{10} + 8 T^{9} + \cdots + 153664)^{2} \)
|
| $97$ |
\( (T^{10} + 465 T^{8} + \cdots + 736579600)^{2} \)
|
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