Newspace parameters
Level: | \( N \) | \(=\) | \( 325 = 5^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 325.o (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.59513806569\) |
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
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Defining polynomial: | \( x^{20} - 16 x^{18} + 168 x^{16} - 1022 x^{14} + 4518 x^{12} - 12577 x^{10} + 24961 x^{8} - 25050 x^{6} + 17307 x^{4} - 1242 x^{2} + 81 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 3^{2}\cdot 5^{4} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
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 \( x^{20} - 16 x^{18} + 168 x^{16} - 1022 x^{14} + 4518 x^{12} - 12577 x^{10} + 24961 x^{8} - 25050 x^{6} + 17307 x^{4} - 1242 x^{2} + 81 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( - 26863568 \nu^{18} + 405681404 \nu^{16} - 4156515402 \nu^{14} + 23801296135 \nu^{12} - 100725879444 \nu^{10} + 249332478494 \nu^{8} + \cdots - 784835807949 ) / 268382163375 \) |
\(\beta_{3}\) | \(=\) | \( ( 184732631 \nu^{18} - 2787465974 \nu^{16} + 28559714937 \nu^{14} - 163780437865 \nu^{12} + 692094729714 \nu^{10} - 1713181312139 \nu^{8} + \cdots + 555085489302 ) / 483087894075 \) |
\(\beta_{4}\) | \(=\) | \( ( 1541220706 \nu^{18} - 24010628908 \nu^{16} + 246007206354 \nu^{14} - 1431129051095 \nu^{12} + 5961554285988 \nu^{10} + \cdots + 2517866659143 ) / 2415439470375 \) |
\(\beta_{5}\) | \(=\) | \( ( 616989689 \nu^{18} - 9590339552 \nu^{16} + 98260343376 \nu^{14} - 570567788680 \nu^{12} + 2381167527072 \nu^{10} + \cdots + 1970521442442 ) / 805146490125 \) |
\(\beta_{6}\) | \(=\) | \( ( 2247835573 \nu^{18} - 36643617454 \nu^{16} + 387026214582 \nu^{14} - 2393494010915 \nu^{12} + 10686512655534 \nu^{10} + \cdots - 3135575353986 ) / 2415439470375 \) |
\(\beta_{7}\) | \(=\) | \( ( 752247488 \nu^{18} - 11955369104 \nu^{16} + 125160533772 \nu^{14} - 756327386530 \nu^{12} + 3327250262379 \nu^{10} + \cdots - 84588695091 ) / 805146490125 \) |
\(\beta_{8}\) | \(=\) | \( ( 3034965602 \nu^{18} - 49691600891 \nu^{16} + 525170629968 \nu^{14} - 3246813163900 \nu^{12} + 14406712674306 \nu^{10} + \cdots - 370294114329 ) / 2415439470375 \) |
\(\beta_{9}\) | \(=\) | \( ( 507278227 \nu^{18} - 8303208961 \nu^{16} + 88236686808 \nu^{14} - 549318509930 \nu^{12} + 2473855622691 \nu^{10} + \cdots - 740380583214 ) / 268382163375 \) |
\(\beta_{10}\) | \(=\) | \( ( - 3940067489 \nu^{19} + 60296811032 \nu^{17} - 617786818116 \nu^{15} + 3563719409905 \nu^{13} - 14970982793352 \nu^{11} + \cdots - 16110502021512 \nu ) / 7246318411125 \) |
\(\beta_{11}\) | \(=\) | \( ( - 752247488 \nu^{18} + 11955369104 \nu^{16} - 125160533772 \nu^{14} + 756327386530 \nu^{12} - 3327250262379 \nu^{10} + \cdots + 84588695091 ) / 268382163375 \) |
\(\beta_{12}\) | \(=\) | \( ( - 933076765 \nu^{19} + 14250041788 \nu^{17} - 146002546794 \nu^{15} + 841270881110 \nu^{13} - 3538116307668 \nu^{11} + \cdots - 10358741131677 \nu ) / 1449263682225 \) |
\(\beta_{13}\) | \(=\) | \( ( 5109145513 \nu^{19} - 78781632454 \nu^{17} + 807177912177 \nu^{15} - 4670732251835 \nu^{13} + 19560544640994 \nu^{11} + \cdots + 31141040114619 \nu ) / 7246318411125 \) |
\(\beta_{14}\) | \(=\) | \( ( 752247488 \nu^{19} - 11955369104 \nu^{17} + 125160533772 \nu^{15} - 756327386530 \nu^{13} + 3327250262379 \nu^{11} + \cdots - 889735185216 \nu ) / 805146490125 \) |
\(\beta_{15}\) | \(=\) | \( ( 2666577601 \nu^{19} - 43219439509 \nu^{17} + 456347434890 \nu^{15} - 2810921453033 \nu^{13} + 12538938914913 \nu^{11} + \cdots - 3618038185722 \nu ) / 1449263682225 \) |
\(\beta_{16}\) | \(=\) | \( ( - 6140567453 \nu^{19} + 93599061194 \nu^{17} - 958993771047 \nu^{15} + 5520042575035 \nu^{13} - 23239536397134 \nu^{11} + \cdots - 67441792514994 \nu ) / 2415439470375 \) |
\(\beta_{17}\) | \(=\) | \( ( - 31801642196 \nu^{19} + 508643497118 \nu^{17} - 5336018421279 \nu^{15} + 32406411228190 \nu^{13} - 142835648184933 \nu^{11} + \cdots + 3640232347587 \nu ) / 7246318411125 \) |
\(\beta_{18}\) | \(=\) | \( ( - 51124092454 \nu^{19} + 814385618257 \nu^{17} - 8531748012306 \nu^{15} + 51653059068920 \nu^{13} - 227391939174972 \nu^{11} + \cdots + 5786180186193 \nu ) / 7246318411125 \) |
\(\beta_{19}\) | \(=\) | \( ( - 66704068546 \nu^{19} + 1062239293243 \nu^{17} - 11127239788269 \nu^{15} + 67350140699030 \nu^{13} - 296467419138753 \nu^{11} + \cdots + 7542959242032 \nu ) / 2415439470375 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{11} + 3\beta_{7} \) |
\(\nu^{3}\) | \(=\) | \( \beta_{18} + \beta_{15} + 5\beta_{14} - \beta_{10} + 5\beta_1 \) |
\(\nu^{4}\) | \(=\) | \( 6\beta_{11} + \beta_{9} - \beta_{8} + 15\beta_{7} + \beta_{6} + \beta_{4} - 6\beta_{2} - 15 \) |
\(\nu^{5}\) | \(=\) | \( -\beta_{19} + 10\beta_{18} + \beta_{16} + 7\beta_{15} + 28\beta_{14} - 10\beta_{12} \) |
\(\nu^{6}\) | \(=\) | \( -10\beta_{5} + 11\beta_{4} - 8\beta_{3} - 37\beta_{2} - 86 \) |
\(\nu^{7}\) | \(=\) | \( 11\beta_{16} - \beta_{13} - 80\beta_{12} + 42\beta_{10} - 167\beta_1 \) |
\(\nu^{8}\) | \(=\) | \( -236\beta_{11} - 81\beta_{9} + 91\beta_{8} - 527\beta_{7} - 52\beta_{6} - 81\beta_{5} - 52\beta_{3} \) |
\(\nu^{9}\) | \(=\) | \( 91\beta_{19} - 590\beta_{18} + 19\beta_{17} - 249\beta_{15} - 1041\beta_{14} + 249\beta_{10} - 1041\beta_1 \) |
\(\nu^{10}\) | \(=\) | \( -1540\beta_{11} - 609\beta_{9} + 681\beta_{8} - 3354\beta_{7} - 321\beta_{6} - 681\beta_{4} + 1540\beta_{2} + 3354 \) |
\(\nu^{11}\) | \(=\) | \( 681 \beta_{19} - 4192 \beta_{18} + 216 \beta_{17} - 681 \beta_{16} - 1501 \beta_{15} - 6683 \beta_{14} + 216 \beta_{13} + 4192 \beta_{12} \) |
\(\nu^{12}\) | \(=\) | \( 4408\beta_{5} - 4873\beta_{4} + 1966\beta_{3} + 10194\beta_{2} + 21843 \) |
\(\nu^{13}\) | \(=\) | \( -4873\beta_{16} + 1977\beta_{13} + 29221\beta_{12} - 9253\beta_{10} + 43732\beta_1 \) |
\(\nu^{14}\) | \(=\) | \( 68080 \beta_{11} + 31198 \beta_{9} - 34094 \beta_{8} + 144314 \beta_{7} + 12149 \beta_{6} + 31198 \beta_{5} + 12149 \beta_{3} \) |
\(\nu^{15}\) | \(=\) | \( - 34094 \beta_{19} + 201560 \beta_{18} - 16153 \beta_{17} + 58284 \beta_{15} + 289727 \beta_{14} - 58284 \beta_{10} + 289727 \beta_1 \) |
\(\nu^{16}\) | \(=\) | \( 457193 \beta_{11} + 217713 \beta_{9} - 235654 \beta_{8} + 962210 \beta_{7} + 76225 \beta_{6} + 235654 \beta_{4} - 457193 \beta_{2} - 962210 \) |
\(\nu^{17}\) | \(=\) | \( - 235654 \beta_{19} + 1381868 \beta_{18} - 123547 \beta_{17} + 235654 \beta_{16} + 373989 \beta_{15} + 1934880 \beta_{14} - 123547 \beta_{13} - 1381868 \beta_{12} \) |
\(\nu^{18}\) | \(=\) | \( -1505415\beta_{5} + 1617522\beta_{4} - 486096\beta_{3} - 3081094\beta_{2} - 6453318 \) |
\(\nu^{19}\) | \(=\) | \( 1617522\beta_{16} - 907212\beta_{13} - 9439075\beta_{12} + 2435764\beta_{10} - 12989495\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).
\(n\) | \(27\) | \(301\) |
\(\chi(n)\) | \(-1\) | \(-\beta_{7}\) |
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.
Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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74.1 |
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−2.25583 | − | 1.30241i | 0.117191 | + | 0.0676602i | 2.39253 | + | 4.14398i | 0 | −0.176242 | − | 0.305261i | −2.81660 | + | 1.62616i | − | 7.25455i | −1.49084 | − | 2.58222i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
74.2 | −1.87176 | − | 1.08066i | −1.69816 | − | 0.980433i | 1.33565 | + | 2.31341i | 0 | 2.11903 | + | 3.67026i | 2.66453 | − | 1.53837i | − | 1.45089i | 0.422497 | + | 0.731787i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
74.3 | −1.56608 | − | 0.904178i | 1.60910 | + | 0.929015i | 0.635076 | + | 1.09998i | 0 | −1.67999 | − | 2.90983i | −3.60988 | + | 2.08417i | 1.31983i | 0.226138 | + | 0.391682i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
74.4 | −0.949161 | − | 0.547998i | 2.91389 | + | 1.68234i | −0.399395 | − | 0.691773i | 0 | −1.84383 | − | 3.19362i | 1.37843 | − | 0.795836i | 3.06747i | 4.16051 | + | 7.20621i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
74.5 | −0.232843 | − | 0.134432i | −0.522064 | − | 0.301414i | −0.963856 | − | 1.66945i | 0 | 0.0810394 | + | 0.140364i | −1.23923 | + | 0.715471i | 1.05602i | −1.31830 | − | 2.28336i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
74.6 | 0.232843 | + | 0.134432i | 0.522064 | + | 0.301414i | −0.963856 | − | 1.66945i | 0 | 0.0810394 | + | 0.140364i | 1.23923 | − | 0.715471i | − | 1.05602i | −1.31830 | − | 2.28336i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
74.7 | 0.949161 | + | 0.547998i | −2.91389 | − | 1.68234i | −0.399395 | − | 0.691773i | 0 | −1.84383 | − | 3.19362i | −1.37843 | + | 0.795836i | − | 3.06747i | 4.16051 | + | 7.20621i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
74.8 | 1.56608 | + | 0.904178i | −1.60910 | − | 0.929015i | 0.635076 | + | 1.09998i | 0 | −1.67999 | − | 2.90983i | 3.60988 | − | 2.08417i | − | 1.31983i | 0.226138 | + | 0.391682i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
74.9 | 1.87176 | + | 1.08066i | 1.69816 | + | 0.980433i | 1.33565 | + | 2.31341i | 0 | 2.11903 | + | 3.67026i | −2.66453 | + | 1.53837i | 1.45089i | 0.422497 | + | 0.731787i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
74.10 | 2.25583 | + | 1.30241i | −0.117191 | − | 0.0676602i | 2.39253 | + | 4.14398i | 0 | −0.176242 | − | 0.305261i | 2.81660 | − | 1.62616i | 7.25455i | −1.49084 | − | 2.58222i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
224.1 | −2.25583 | + | 1.30241i | 0.117191 | − | 0.0676602i | 2.39253 | − | 4.14398i | 0 | −0.176242 | + | 0.305261i | −2.81660 | − | 1.62616i | 7.25455i | −1.49084 | + | 2.58222i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
224.2 | −1.87176 | + | 1.08066i | −1.69816 | + | 0.980433i | 1.33565 | − | 2.31341i | 0 | 2.11903 | − | 3.67026i | 2.66453 | + | 1.53837i | 1.45089i | 0.422497 | − | 0.731787i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
224.3 | −1.56608 | + | 0.904178i | 1.60910 | − | 0.929015i | 0.635076 | − | 1.09998i | 0 | −1.67999 | + | 2.90983i | −3.60988 | − | 2.08417i | − | 1.31983i | 0.226138 | − | 0.391682i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
224.4 | −0.949161 | + | 0.547998i | 2.91389 | − | 1.68234i | −0.399395 | + | 0.691773i | 0 | −1.84383 | + | 3.19362i | 1.37843 | + | 0.795836i | − | 3.06747i | 4.16051 | − | 7.20621i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
224.5 | −0.232843 | + | 0.134432i | −0.522064 | + | 0.301414i | −0.963856 | + | 1.66945i | 0 | 0.0810394 | − | 0.140364i | −1.23923 | − | 0.715471i | − | 1.05602i | −1.31830 | + | 2.28336i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
224.6 | 0.232843 | − | 0.134432i | 0.522064 | − | 0.301414i | −0.963856 | + | 1.66945i | 0 | 0.0810394 | − | 0.140364i | 1.23923 | + | 0.715471i | 1.05602i | −1.31830 | + | 2.28336i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
224.7 | 0.949161 | − | 0.547998i | −2.91389 | + | 1.68234i | −0.399395 | + | 0.691773i | 0 | −1.84383 | + | 3.19362i | −1.37843 | − | 0.795836i | 3.06747i | 4.16051 | − | 7.20621i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
224.8 | 1.56608 | − | 0.904178i | −1.60910 | + | 0.929015i | 0.635076 | − | 1.09998i | 0 | −1.67999 | + | 2.90983i | 3.60988 | + | 2.08417i | 1.31983i | 0.226138 | − | 0.391682i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
224.9 | 1.87176 | − | 1.08066i | 1.69816 | − | 0.980433i | 1.33565 | − | 2.31341i | 0 | 2.11903 | − | 3.67026i | −2.66453 | − | 1.53837i | − | 1.45089i | 0.422497 | − | 0.731787i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
224.10 | 2.25583 | − | 1.30241i | −0.117191 | + | 0.0676602i | 2.39253 | − | 4.14398i | 0 | −0.176242 | + | 0.305261i | 2.81660 | + | 1.62616i | − | 7.25455i | −1.49084 | + | 2.58222i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
13.c | even | 3 | 1 | inner |
65.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 325.2.o.c | 20 | |
5.b | even | 2 | 1 | inner | 325.2.o.c | 20 | |
5.c | odd | 4 | 1 | 325.2.e.c | ✓ | 10 | |
5.c | odd | 4 | 1 | 325.2.e.d | yes | 10 | |
13.c | even | 3 | 1 | inner | 325.2.o.c | 20 | |
65.n | even | 6 | 1 | inner | 325.2.o.c | 20 | |
65.q | odd | 12 | 1 | 325.2.e.c | ✓ | 10 | |
65.q | odd | 12 | 1 | 325.2.e.d | yes | 10 | |
65.q | odd | 12 | 1 | 4225.2.a.bn | 5 | ||
65.q | odd | 12 | 1 | 4225.2.a.bp | 5 | ||
65.r | odd | 12 | 1 | 4225.2.a.bm | 5 | ||
65.r | odd | 12 | 1 | 4225.2.a.bo | 5 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
325.2.e.c | ✓ | 10 | 5.c | odd | 4 | 1 | |
325.2.e.c | ✓ | 10 | 65.q | odd | 12 | 1 | |
325.2.e.d | yes | 10 | 5.c | odd | 4 | 1 | |
325.2.e.d | yes | 10 | 65.q | odd | 12 | 1 | |
325.2.o.c | 20 | 1.a | even | 1 | 1 | trivial | |
325.2.o.c | 20 | 5.b | even | 2 | 1 | inner | |
325.2.o.c | 20 | 13.c | even | 3 | 1 | inner | |
325.2.o.c | 20 | 65.n | even | 6 | 1 | inner | |
4225.2.a.bm | 5 | 65.r | odd | 12 | 1 | ||
4225.2.a.bn | 5 | 65.q | odd | 12 | 1 | ||
4225.2.a.bo | 5 | 65.r | odd | 12 | 1 | ||
4225.2.a.bp | 5 | 65.q | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{20} - 16 T_{2}^{18} + 168 T_{2}^{16} - 1022 T_{2}^{14} + 4518 T_{2}^{12} - 12577 T_{2}^{10} + 24961 T_{2}^{8} - 25050 T_{2}^{6} + 17307 T_{2}^{4} - 1242 T_{2}^{2} + 81 \)
acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{20} - 16 T^{18} + 168 T^{16} + \cdots + 81 \)
$3$
\( T^{20} - 19 T^{18} + 258 T^{16} - 1583 T^{14} + \cdots + 1 \)
$5$
\( T^{20} \)
$7$
\( T^{20} - 42 T^{18} + 1139 T^{16} + \cdots + 81450625 \)
$11$
\( (T^{10} + 3 T^{9} + 26 T^{8} + 61 T^{7} + \cdots + 9)^{2} \)
$13$
\( T^{20} - 20 T^{18} + \cdots + 137858491849 \)
$17$
\( T^{20} - 88 T^{18} + \cdots + 57178852641 \)
$19$
\( (T^{10} + 4 T^{9} + 24 T^{8} + 34 T^{7} + \cdots + 225)^{2} \)
$23$
\( T^{20} - 169 T^{18} + \cdots + 79502005521 \)
$29$
\( (T^{10} + T^{9} + 66 T^{8} + 233 T^{7} + \cdots + 881721)^{2} \)
$31$
\( (T^{5} - 57 T^{3} - 29 T^{2} + 495 T + 225)^{4} \)
$37$
\( T^{20} - 183 T^{18} + \cdots + 682740290961 \)
$41$
\( (T^{10} + 6 T^{9} + 110 T^{8} + 354 T^{7} + \cdots + 50625)^{2} \)
$43$
\( T^{20} + \cdots + 201364301903521 \)
$47$
\( (T^{10} + 184 T^{8} + 8044 T^{6} + \cdots + 4100625)^{2} \)
$53$
\( (T^{10} + 298 T^{8} + 25443 T^{6} + \cdots + 36905625)^{2} \)
$59$
\( (T^{10} + 12 T^{9} + 231 T^{8} + \cdots + 4782969)^{2} \)
$61$
\( (T^{10} + 5 T^{9} + 72 T^{8} + 145 T^{7} + \cdots + 403225)^{2} \)
$67$
\( T^{20} - 518 T^{18} + \cdots + 65\!\cdots\!81 \)
$71$
\( (T^{10} + 19 T^{9} + 368 T^{8} + \cdots + 101787921)^{2} \)
$73$
\( (T^{10} + 302 T^{8} + 31801 T^{6} + \cdots + 102515625)^{2} \)
$79$
\( (T^{5} - 14 T^{4} - 95 T^{3} + 1350 T^{2} + \cdots - 16875)^{4} \)
$83$
\( (T^{10} + 563 T^{8} + 110723 T^{6} + \cdots + 2891750625)^{2} \)
$89$
\( (T^{10} + 10 T^{9} + 139 T^{8} + \cdots + 2537649)^{2} \)
$97$
\( T^{20} - 633 T^{18} + \cdots + 28\!\cdots\!81 \)
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