Properties

Label 60.5.k.a
Level $60$
Weight $5$
Character orbit 60.k
Analytic conductor $6.202$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 60.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.20219778503\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} + 8 x^{6} + 28 x^{5} + 97 x^{4} - 168 x^{3} + 288 x^{2} + 864 x + 1296\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2}\cdot 5^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{5} + ( -17 - 18 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 5 \beta_{5} - 2 \beta_{7} ) q^{7} -27 \beta_{1} q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{5} + ( -17 - 18 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 5 \beta_{5} - 2 \beta_{7} ) q^{7} -27 \beta_{1} q^{9} + ( 42 + 8 \beta_{3} - 3 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{11} + ( 45 - 41 \beta_{1} + 13 \beta_{2} + 4 \beta_{3} + 6 \beta_{4} + 5 \beta_{5} + 11 \beta_{6} - \beta_{7} ) q^{13} + ( -15 + 33 \beta_{1} + 6 \beta_{3} - 6 \beta_{4} + 8 \beta_{5} - 3 \beta_{7} ) q^{15} + ( -126 - 131 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} - \beta_{4} + 15 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} ) q^{17} + ( -1 + 12 \beta_{1} - 6 \beta_{2} - \beta_{3} + 11 \beta_{4} + 4 \beta_{5} - \beta_{6} - 15 \beta_{7} ) q^{19} + ( 99 - 16 \beta_{2} + 9 \beta_{3} - 16 \beta_{5} - 9 \beta_{6} ) q^{21} + ( 153 - 158 \beta_{1} + 12 \beta_{2} - 5 \beta_{3} - 11 \beta_{4} - 8 \beta_{5} - 19 \beta_{6} + 3 \beta_{7} ) q^{23} + ( -254 + 188 \beta_{1} - 9 \beta_{2} - 13 \beta_{3} + 17 \beta_{4} + 67 \beta_{5} + 14 \beta_{6} + 6 \beta_{7} ) q^{25} -27 \beta_{5} q^{27} + ( 6 + 502 \beta_{1} + 108 \beta_{2} + 6 \beta_{3} - 13 \beta_{4} - 96 \beta_{5} + 6 \beta_{6} + 37 \beta_{7} ) q^{29} + ( 156 - 80 \beta_{2} - 84 \beta_{3} + 14 \beta_{4} - 108 \beta_{5} + 28 \beta_{6} + 14 \beta_{7} ) q^{31} + ( -27 + 48 \beta_{1} + 41 \beta_{2} + 21 \beta_{3} - 27 \beta_{4} - 3 \beta_{5} - 30 \beta_{6} + 24 \beta_{7} ) q^{33} + ( 148 + 88 \beta_{1} - 146 \beta_{2} - 26 \beta_{3} - 25 \beta_{4} + 60 \beta_{5} - 32 \beta_{6} + 13 \beta_{7} ) q^{35} + ( -60 - 9 \beta_{1} + 45 \beta_{2} - 51 \beta_{3} + 39 \beta_{4} + 99 \beta_{5} + 6 \beta_{6} + 96 \beta_{7} ) q^{37} + ( -18 - 198 \beta_{1} + 3 \beta_{2} - 18 \beta_{3} - 63 \beta_{4} - 39 \beta_{5} - 18 \beta_{6} - 9 \beta_{7} ) q^{39} + ( -457 - 244 \beta_{2} + 35 \beta_{3} + 11 \beta_{4} - 266 \beta_{5} - 79 \beta_{6} + 11 \beta_{7} ) q^{41} + ( -834 + 716 \beta_{1} + 194 \beta_{2} - 118 \beta_{3} + 66 \beta_{4} - 26 \beta_{5} + 40 \beta_{6} - 92 \beta_{7} ) q^{43} + ( 81 - 27 \beta_{1} + 27 \beta_{3} + 27 \beta_{5} - 27 \beta_{6} + 27 \beta_{7} ) q^{45} + ( 615 + 606 \beta_{1} - 30 \beta_{2} + 9 \beta_{3} - 51 \beta_{4} + 186 \beta_{5} + 21 \beta_{6} - 39 \beta_{7} ) q^{47} + ( 35 - 369 \beta_{1} + 174 \beta_{2} + 35 \beta_{3} + 101 \beta_{4} - 104 \beta_{5} + 35 \beta_{6} + 39 \beta_{7} ) q^{49} + ( 291 - 121 \beta_{2} + 45 \beta_{3} - 6 \beta_{4} - 109 \beta_{5} - 21 \beta_{6} - 6 \beta_{7} ) q^{51} + ( 540 - 445 \beta_{1} + 543 \beta_{2} + 95 \beta_{3} - 25 \beta_{4} + 35 \beta_{5} + 10 \beta_{6} + 60 \beta_{7} ) q^{53} + ( 1422 - 498 \beta_{1} - 331 \beta_{2} - 4 \beta_{3} + 36 \beta_{4} + 505 \beta_{5} + 55 \beta_{6} - 71 \beta_{7} ) q^{55} + ( 63 + 18 \beta_{1} - 36 \beta_{2} + 45 \beta_{3} - 27 \beta_{4} + 76 \beta_{5} - 9 \beta_{6} - 81 \beta_{7} ) q^{57} + ( -41 - 1042 \beta_{1} + 178 \beta_{2} - 41 \beta_{3} - 48 \beta_{4} - 260 \beta_{5} - 41 \beta_{6} - 116 \beta_{7} ) q^{59} + ( -1345 - 464 \beta_{2} + 75 \beta_{3} - 49 \beta_{4} - 366 \beta_{5} + 121 \beta_{6} - 49 \beta_{7} ) q^{61} + ( -432 + 486 \beta_{1} + 135 \beta_{2} + 54 \beta_{3} + 27 \beta_{5} + 27 \beta_{6} + 27 \beta_{7} ) q^{63} + ( -1940 - 2515 \beta_{1} - 95 \beta_{2} + 155 \beta_{3} + 25 \beta_{4} + 825 \beta_{5} + 60 \beta_{6} - 40 \beta_{7} ) q^{65} + ( -76 - 196 \beta_{1} - 78 \beta_{2} + 120 \beta_{3} - 36 \beta_{4} + 666 \beta_{5} - 42 \beta_{6} - 198 \beta_{7} ) q^{67} + ( 33 - 552 \beta_{1} + 230 \beta_{2} + 33 \beta_{3} + 105 \beta_{4} - 164 \beta_{5} + 33 \beta_{6} + 27 \beta_{7} ) q^{69} + ( -326 - 548 \beta_{2} + 34 \beta_{3} - 62 \beta_{4} - 424 \beta_{5} + 214 \beta_{6} - 62 \beta_{7} ) q^{71} + ( 499 - 297 \beta_{1} + 670 \beta_{2} + 202 \beta_{3} - 30 \beta_{4} + 86 \beta_{5} + 56 \beta_{6} + 116 \beta_{7} ) q^{73} + ( 1953 + 414 \beta_{1} - 337 \beta_{2} - 99 \beta_{3} - 54 \beta_{4} + 186 \beta_{5} + 27 \beta_{6} - 72 \beta_{7} ) q^{75} + ( 2435 + 2406 \beta_{1} + 80 \beta_{2} + 29 \beta_{3} + 189 \beta_{4} + 120 \beta_{5} - 109 \beta_{6} + 51 \beta_{7} ) q^{77} + ( -40 - 1168 \beta_{1} + 708 \beta_{2} - 40 \beta_{3} - 58 \beta_{4} - 788 \beta_{5} - 40 \beta_{6} - 102 \beta_{7} ) q^{79} -729 q^{81} + ( 1503 - 1658 \beta_{1} + 282 \beta_{2} - 155 \beta_{3} + 39 \beta_{4} - 58 \beta_{5} - 19 \beta_{6} - 97 \beta_{7} ) q^{83} + ( 43 + 1093 \beta_{1} - 341 \beta_{2} - 96 \beta_{3} - 130 \beta_{4} + 345 \beta_{5} - 167 \beta_{6} + 203 \beta_{7} ) q^{85} + ( -2481 - 2370 \beta_{1} + 57 \beta_{2} - 111 \beta_{3} + 3 \beta_{4} + 383 \beta_{5} + 54 \beta_{6} + 168 \beta_{7} ) q^{87} + ( 17 + 5104 \beta_{1} - 10 \beta_{2} + 17 \beta_{3} + 75 \beta_{4} + 44 \beta_{5} + 17 \beta_{6} - 7 \beta_{7} ) q^{89} + ( 3818 - 572 \beta_{2} + 102 \beta_{3} + 116 \beta_{4} - 804 \beta_{5} - 566 \beta_{6} + 116 \beta_{7} ) q^{91} + ( -2034 + 1656 \beta_{1} - 26 \beta_{2} - 378 \beta_{3} + 126 \beta_{4} - 126 \beta_{5} - 252 \beta_{7} ) q^{93} + ( -4655 + 2110 \beta_{1} - 410 \beta_{2} - 85 \beta_{3} + 55 \beta_{4} + 190 \beta_{5} + 75 \beta_{6} + 75 \beta_{7} ) q^{95} + ( -1915 - 1799 \beta_{1} - 70 \beta_{2} - 116 \beta_{3} - 256 \beta_{4} - 658 \beta_{5} + 186 \beta_{6} + 46 \beta_{7} ) q^{97} + ( 81 - 918 \beta_{1} + 162 \beta_{2} + 81 \beta_{3} + 108 \beta_{4} + 81 \beta_{6} + 216 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 12q^{5} - 140q^{7} + O(q^{10}) \) \( 8q + 12q^{5} - 140q^{7} + 288q^{11} + 300q^{13} - 144q^{15} - 1020q^{17} + 792q^{21} + 1320q^{23} - 2036q^{25} + 1472q^{31} - 180q^{33} + 1416q^{35} - 300q^{37} - 3480q^{41} - 6360q^{43} + 648q^{45} + 4800q^{47} + 2232q^{51} + 3900q^{53} + 11172q^{55} + 360q^{57} - 11544q^{61} - 3780q^{63} - 16380q^{65} - 920q^{67} - 3600q^{71} + 2960q^{73} + 15912q^{75} + 19800q^{77} - 5832q^{81} + 12720q^{83} + 1396q^{85} - 19620q^{87} + 32400q^{91} - 14760q^{93} - 37200q^{95} - 15600q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 8 x^{6} + 28 x^{5} + 97 x^{4} - 168 x^{3} + 288 x^{2} + 864 x + 1296\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 343 \nu^{7} - 2023 \nu^{6} + 4880 \nu^{5} + 5404 \nu^{4} + 17563 \nu^{3} - 175455 \nu^{2} + 180684 \nu + 145584 \)\()/486540\)
\(\beta_{2}\)\(=\)\((\)\( -331 \nu^{7} + 2731 \nu^{6} - 12650 \nu^{5} + 15812 \nu^{4} - 15391 \nu^{3} + 43155 \nu^{2} - 579618 \nu + 160812 \)\()/162180\)
\(\beta_{3}\)\(=\)\((\)\( -229 \nu^{7} + 2548 \nu^{6} - 9530 \nu^{5} + 20612 \nu^{4} + 6887 \nu^{3} + 125892 \nu^{2} - 191286 \nu + 1136700 \)\()/97308\)
\(\beta_{4}\)\(=\)\((\)\( 1491 \nu^{7} - 1336 \nu^{6} + 8320 \nu^{5} + 4108 \nu^{4} + 478691 \nu^{3} + 141980 \nu^{2} + 1738968 \nu - 335952 \)\()/324360\)
\(\beta_{5}\)\(=\)\((\)\( 839 \nu^{7} - 3234 \nu^{6} + 5960 \nu^{5} + 27972 \nu^{4} + 55399 \nu^{3} - 80710 \nu^{2} + 211512 \nu + 579672 \)\()/108120\)
\(\beta_{6}\)\(=\)\((\)\( -417 \nu^{7} + 1844 \nu^{6} - 4082 \nu^{5} - 6068 \nu^{4} - 46009 \nu^{3} + 79652 \nu^{2} + 83142 \nu - 68832 \)\()/32436\)
\(\beta_{7}\)\(=\)\((\)\( -12511 \nu^{7} + 71956 \nu^{6} - 161960 \nu^{5} - 314188 \nu^{4} - 346951 \nu^{3} + 4007280 \nu^{2} - 6911928 \nu - 10034928 \)\()/973080\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-3 \beta_{7} + 3 \beta_{6} + 3 \beta_{4} - 3 \beta_{3} + 4 \beta_{2} - 18 \beta_{1} + 15\)\()/30\)
\(\nu^{2}\)\(=\)\((\)\(-15 \beta_{7} - 3 \beta_{6} - 8 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 264 \beta_{1} - 3\)\()/30\)
\(\nu^{3}\)\(=\)\((\)\(-19 \beta_{7} - 11 \beta_{6} - 36 \beta_{5} + 7 \beta_{4} + 15 \beta_{3} - 4 \beta_{2} - 158 \beta_{1} - 143\)\()/10\)
\(\nu^{4}\)\(=\)\((\)\(-63 \beta_{7} - 75 \beta_{6} - 170 \beta_{5} - 63 \beta_{4} + 327 \beta_{3} - 296 \beta_{2} - 3489\)\()/30\)
\(\nu^{5}\)\(=\)\((\)\(861 \beta_{7} - 189 \beta_{6} + 336 \beta_{5} - 525 \beta_{4} + 1197 \beta_{3} - 2072 \beta_{2} + 9978 \beta_{1} - 8781\)\()/30\)
\(\nu^{6}\)\(=\)\((\)\(2201 \beta_{7} + 413 \beta_{6} + 2312 \beta_{5} - 549 \beta_{4} + 413 \beta_{3} - 1486 \beta_{2} + 22880 \beta_{1} + 413\)\()/10\)
\(\nu^{7}\)\(=\)\((\)\(24513 \beta_{7} + 9753 \beta_{6} + 40244 \beta_{5} - 2373 \beta_{4} - 17133 \beta_{3} + 7380 \beta_{2} + 204138 \beta_{1} + 187005\)\()/30\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(1\) \(\beta_{1}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−1.84806 1.84806i
1.62332 + 1.62332i
3.17086 + 3.17086i
−0.946115 0.946115i
−1.84806 + 1.84806i
1.62332 1.62332i
3.17086 3.17086i
−0.946115 + 0.946115i
0 −3.67423 3.67423i 0 4.94005 24.5071i 0 −52.2300 + 52.2300i 0 27.0000i 0
13.2 0 −3.67423 3.67423i 0 11.5321 + 22.1813i 0 −9.71439 + 9.71439i 0 27.0000i 0
13.3 0 3.67423 + 3.67423i 0 −21.4961 + 12.7640i 0 −29.2390 + 29.2390i 0 27.0000i 0
13.4 0 3.67423 + 3.67423i 0 11.0239 22.4382i 0 21.1834 21.1834i 0 27.0000i 0
37.1 0 −3.67423 + 3.67423i 0 4.94005 + 24.5071i 0 −52.2300 52.2300i 0 27.0000i 0
37.2 0 −3.67423 + 3.67423i 0 11.5321 22.1813i 0 −9.71439 9.71439i 0 27.0000i 0
37.3 0 3.67423 3.67423i 0 −21.4961 12.7640i 0 −29.2390 29.2390i 0 27.0000i 0
37.4 0 3.67423 3.67423i 0 11.0239 + 22.4382i 0 21.1834 + 21.1834i 0 27.0000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.5.k.a 8
3.b odd 2 1 180.5.l.b 8
4.b odd 2 1 240.5.bg.d 8
5.b even 2 1 300.5.k.d 8
5.c odd 4 1 inner 60.5.k.a 8
5.c odd 4 1 300.5.k.d 8
15.d odd 2 1 900.5.l.k 8
15.e even 4 1 180.5.l.b 8
15.e even 4 1 900.5.l.k 8
20.e even 4 1 240.5.bg.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.k.a 8 1.a even 1 1 trivial
60.5.k.a 8 5.c odd 4 1 inner
180.5.l.b 8 3.b odd 2 1
180.5.l.b 8 15.e even 4 1
240.5.bg.d 8 4.b odd 2 1
240.5.bg.d 8 20.e even 4 1
300.5.k.d 8 5.b even 2 1
300.5.k.d 8 5.c odd 4 1
900.5.l.k 8 15.d odd 2 1
900.5.l.k 8 15.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 729 + T^{4} )^{2} \)
$5$ \( 152587890625 - 2929687500 T + 425781250 T^{2} + 8437500 T^{3} + 365250 T^{4} + 13500 T^{5} + 1090 T^{6} - 12 T^{7} + T^{8} \)
$7$ \( 1580189787136 + 172367518720 T + 9400947200 T^{2} + 69856960 T^{3} + 3113188 T^{4} + 245480 T^{5} + 9800 T^{6} + 140 T^{7} + T^{8} \)
$11$ \( ( 311125216 + 2512176 T - 30374 T^{2} - 144 T^{3} + T^{4} )^{2} \)
$13$ \( 672162221025000000 - 19553541750000000 T + 284411250000000 T^{2} - 2045439000000 T^{3} + 7332412500 T^{4} - 1215000 T^{5} + 45000 T^{6} - 300 T^{7} + T^{8} \)
$17$ \( 356691957722668096 - 6902530053603840 T + 66787209676800 T^{2} + 2462232259680 T^{3} + 26901465428 T^{4} + 151983240 T^{5} + 520200 T^{6} + 1020 T^{7} + T^{8} \)
$19$ \( 5658689440000000000 + 553343536000000 T^{2} + 18265050000 T^{4} + 235000 T^{6} + T^{8} \)
$23$ \( \)\(69\!\cdots\!56\)\( - 4925226014150922240 T + 17344277950924800 T^{2} - 26459061377280 T^{3} + 54945885968 T^{4} - 246118560 T^{5} + 871200 T^{6} - 1320 T^{7} + T^{8} \)
$29$ \( \)\(33\!\cdots\!56\)\( + 2281077411268531456 T^{2} + 4621594592196 T^{4} + 3665716 T^{6} + T^{8} \)
$31$ \( ( -1071194996864 + 3533902784 T - 2785464 T^{2} - 736 T^{3} + T^{4} )^{2} \)
$37$ \( \)\(28\!\cdots\!96\)\( - \)\(83\!\cdots\!00\)\( T + 12263317109971920000 T^{2} + 29215074875745600 T^{3} + 32639542595028 T^{4} + 3152007000 T^{5} + 45000 T^{6} + 300 T^{7} + T^{8} \)
$41$ \( ( -3221711600000 - 16008792000 T - 9134900 T^{2} + 1740 T^{3} + T^{4} )^{2} \)
$43$ \( \)\(17\!\cdots\!36\)\( + \)\(24\!\cdots\!20\)\( T + \)\(16\!\cdots\!00\)\( T^{2} + 535202802013271040 T^{3} + 105215920711488 T^{4} + 28671180480 T^{5} + 20224800 T^{6} + 6360 T^{7} + T^{8} \)
$47$ \( \)\(24\!\cdots\!76\)\( - \)\(44\!\cdots\!00\)\( T + 40036587662177280000 T^{2} - 16498566355622400 T^{3} + 10695518268048 T^{4} - 12965788800 T^{5} + 11520000 T^{6} - 4800 T^{7} + T^{8} \)
$53$ \( \)\(43\!\cdots\!36\)\( + \)\(96\!\cdots\!00\)\( T + 10551467982183120000 T^{2} + 16727644443098400 T^{3} + 157106201387588 T^{4} + 50409958200 T^{5} + 7605000 T^{6} - 3900 T^{7} + T^{8} \)
$59$ \( \)\(17\!\cdots\!16\)\( + \)\(41\!\cdots\!76\)\( T^{2} + 262928039331876 T^{4} + 34454956 T^{6} + T^{8} \)
$61$ \( ( -162789583002624 - 160960390272 T - 23280756 T^{2} + 5772 T^{3} + T^{4} )^{2} \)
$67$ \( \)\(41\!\cdots\!96\)\( + \)\(84\!\cdots\!60\)\( T + \)\(85\!\cdots\!00\)\( T^{2} + 4482326578732180480 T^{3} + 1272768375531328 T^{4} + 99690050240 T^{5} + 423200 T^{6} + 920 T^{7} + T^{8} \)
$71$ \( ( -301327179200000 - 275496864000 T - 58485200 T^{2} + 1800 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(66\!\cdots\!36\)\( + \)\(37\!\cdots\!20\)\( T + \)\(10\!\cdots\!00\)\( T^{2} + 1510300402711578560 T^{3} + 1128904875279688 T^{4} + 144735141280 T^{5} + 4380800 T^{6} - 2960 T^{7} + T^{8} \)
$79$ \( \)\(71\!\cdots\!36\)\( + \)\(14\!\cdots\!84\)\( T^{2} + 7957202957833536 T^{4} + 156443664 T^{6} + T^{8} \)
$83$ \( \)\(14\!\cdots\!96\)\( + \)\(10\!\cdots\!60\)\( T + 38246079843532800 T^{2} - 51175335116436480 T^{3} + 150847816022928 T^{4} - 152003655360 T^{5} + 80899200 T^{6} - 12720 T^{7} + T^{8} \)
$89$ \( \)\(36\!\cdots\!00\)\( + \)\(71\!\cdots\!00\)\( T^{2} + 4440915658410000 T^{4} + 112438600 T^{6} + T^{8} \)
$97$ \( \)\(19\!\cdots\!56\)\( + \)\(11\!\cdots\!00\)\( T + \)\(36\!\cdots\!00\)\( T^{2} + 50038563220316568000 T^{3} + 3866571291108168 T^{4} + 343976407200 T^{5} + 121680000 T^{6} + 15600 T^{7} + T^{8} \)
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