Properties

Label 230.6.a.i
Level $230$
Weight $6$
Character orbit 230.a
Self dual yes
Analytic conductor $36.888$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.8882785570\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 3 x^{5} - 1156 x^{4} + 593 x^{3} + 338133 x^{2} + 408388 x - 13033476\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + ( 2 + \beta_{1} ) q^{3} + 16 q^{4} -25 q^{5} + ( 8 + 4 \beta_{1} ) q^{6} + ( 17 + 2 \beta_{1} - \beta_{2} ) q^{7} + 64 q^{8} + ( 147 + 6 \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})\) \( q + 4 q^{2} + ( 2 + \beta_{1} ) q^{3} + 16 q^{4} -25 q^{5} + ( 8 + 4 \beta_{1} ) q^{6} + ( 17 + 2 \beta_{1} - \beta_{2} ) q^{7} + 64 q^{8} + ( 147 + 6 \beta_{1} + 3 \beta_{3} + \beta_{4} ) q^{9} -100 q^{10} + ( 50 + 7 \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{11} + ( 32 + 16 \beta_{1} ) q^{12} + ( 82 + 16 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} + 5 \beta_{5} ) q^{13} + ( 68 + 8 \beta_{1} - 4 \beta_{2} ) q^{14} + ( -50 - 25 \beta_{1} ) q^{15} + 256 q^{16} + ( -125 + 7 \beta_{1} + 5 \beta_{2} - 8 \beta_{3} - 5 \beta_{4} - 27 \beta_{5} ) q^{17} + ( 588 + 24 \beta_{1} + 12 \beta_{3} + 4 \beta_{4} ) q^{18} + ( 446 + 14 \beta_{1} + 8 \beta_{2} - 21 \beta_{3} - 9 \beta_{4} - 3 \beta_{5} ) q^{19} -400 q^{20} + ( 1014 - 5 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} + 9 \beta_{4} + 25 \beta_{5} ) q^{21} + ( 200 + 28 \beta_{1} - 8 \beta_{4} + 4 \beta_{5} ) q^{22} -529 q^{23} + ( 128 + 64 \beta_{1} ) q^{24} + 625 q^{25} + ( 328 + 64 \beta_{1} - 8 \beta_{3} + 12 \beta_{4} + 20 \beta_{5} ) q^{26} + ( 2512 + 109 \beta_{1} - 2 \beta_{2} + 20 \beta_{3} + 31 \beta_{4} - 4 \beta_{5} ) q^{27} + ( 272 + 32 \beta_{1} - 16 \beta_{2} ) q^{28} + ( 569 - 22 \beta_{1} + \beta_{2} + 10 \beta_{3} - 33 \beta_{4} + 23 \beta_{5} ) q^{29} + ( -200 - 100 \beta_{1} ) q^{30} + ( 5167 + 24 \beta_{1} + 23 \beta_{2} + 35 \beta_{3} + 2 \beta_{4} + 12 \beta_{5} ) q^{31} + 1024 q^{32} + ( 2472 + 54 \beta_{1} + 34 \beta_{2} - 35 \beta_{3} - 31 \beta_{4} + 3 \beta_{5} ) q^{33} + ( -500 + 28 \beta_{1} + 20 \beta_{2} - 32 \beta_{3} - 20 \beta_{4} - 108 \beta_{5} ) q^{34} + ( -425 - 50 \beta_{1} + 25 \beta_{2} ) q^{35} + ( 2352 + 96 \beta_{1} + 48 \beta_{3} + 16 \beta_{4} ) q^{36} + ( 4675 - 272 \beta_{1} - 45 \beta_{2} - 4 \beta_{3} + 11 \beta_{4} + 48 \beta_{5} ) q^{37} + ( 1784 + 56 \beta_{1} + 32 \beta_{2} - 84 \beta_{3} - 36 \beta_{4} - 12 \beta_{5} ) q^{38} + ( 6454 + \beta_{1} - 102 \beta_{2} + 163 \beta_{3} + 69 \beta_{4} - 37 \beta_{5} ) q^{39} -1600 q^{40} + ( 5585 - 75 \beta_{1} - 19 \beta_{2} + 35 \beta_{3} + 5 \beta_{4} - 183 \beta_{5} ) q^{41} + ( 4056 - 20 \beta_{1} - 16 \beta_{2} + 16 \beta_{3} + 36 \beta_{4} + 100 \beta_{5} ) q^{42} + ( 2782 + 146 \beta_{1} + 4 \beta_{2} - 111 \beta_{3} + 28 \beta_{4} + 43 \beta_{5} ) q^{43} + ( 800 + 112 \beta_{1} - 32 \beta_{4} + 16 \beta_{5} ) q^{44} + ( -3675 - 150 \beta_{1} - 75 \beta_{3} - 25 \beta_{4} ) q^{45} -2116 q^{46} + ( 3016 - 352 \beta_{1} - 114 \beta_{2} + 65 \beta_{3} + 60 \beta_{4} - 56 \beta_{5} ) q^{47} + ( 512 + 256 \beta_{1} ) q^{48} + ( 1114 - 27 \beta_{1} - 11 \beta_{2} - 66 \beta_{3} - 19 \beta_{4} + 299 \beta_{5} ) q^{49} + 2500 q^{50} + ( 950 - 176 \beta_{1} + 234 \beta_{2} - 96 \beta_{3} - 139 \beta_{4} + 30 \beta_{5} ) q^{51} + ( 1312 + 256 \beta_{1} - 32 \beta_{3} + 48 \beta_{4} + 80 \beta_{5} ) q^{52} + ( -1179 - 384 \beta_{1} - 153 \beta_{2} - 51 \beta_{3} - 51 \beta_{4} - 3 \beta_{5} ) q^{53} + ( 10048 + 436 \beta_{1} - 8 \beta_{2} + 80 \beta_{3} + 124 \beta_{4} - 16 \beta_{5} ) q^{54} + ( -1250 - 175 \beta_{1} + 50 \beta_{4} - 25 \beta_{5} ) q^{55} + ( 1088 + 128 \beta_{1} - 64 \beta_{2} ) q^{56} + ( 1722 - 509 \beta_{1} + 104 \beta_{2} - 20 \beta_{3} - 255 \beta_{4} - 149 \beta_{5} ) q^{57} + ( 2276 - 88 \beta_{1} + 4 \beta_{2} + 40 \beta_{3} - 132 \beta_{4} + 92 \beta_{5} ) q^{58} + ( 1375 - 314 \beta_{1} + 203 \beta_{2} - 203 \beta_{3} + \beta_{4} - 93 \beta_{5} ) q^{59} + ( -800 - 400 \beta_{1} ) q^{60} + ( 4388 - 191 \beta_{1} - 92 \beta_{2} + 104 \beta_{3} + 220 \beta_{4} - 43 \beta_{5} ) q^{61} + ( 20668 + 96 \beta_{1} + 92 \beta_{2} + 140 \beta_{3} + 8 \beta_{4} + 48 \beta_{5} ) q^{62} + ( -2399 + 433 \beta_{1} - 79 \beta_{2} + 252 \beta_{3} + 202 \beta_{4} - 61 \beta_{5} ) q^{63} + 4096 q^{64} + ( -2050 - 400 \beta_{1} + 50 \beta_{3} - 75 \beta_{4} - 125 \beta_{5} ) q^{65} + ( 9888 + 216 \beta_{1} + 136 \beta_{2} - 140 \beta_{3} - 124 \beta_{4} + 12 \beta_{5} ) q^{66} + ( -3397 - 1682 \beta_{1} - 167 \beta_{2} - 50 \beta_{3} + 71 \beta_{4} - 158 \beta_{5} ) q^{67} + ( -2000 + 112 \beta_{1} + 80 \beta_{2} - 128 \beta_{3} - 80 \beta_{4} - 432 \beta_{5} ) q^{68} + ( -1058 - 529 \beta_{1} ) q^{69} + ( -1700 - 200 \beta_{1} + 100 \beta_{2} ) q^{70} + ( 3609 - 1611 \beta_{1} + 309 \beta_{2} + 202 \beta_{3} - 145 \beta_{4} + 190 \beta_{5} ) q^{71} + ( 9408 + 384 \beta_{1} + 192 \beta_{3} + 64 \beta_{4} ) q^{72} + ( 120 - 178 \beta_{1} + 174 \beta_{2} - 64 \beta_{3} + 60 \beta_{4} - 143 \beta_{5} ) q^{73} + ( 18700 - 1088 \beta_{1} - 180 \beta_{2} - 16 \beta_{3} + 44 \beta_{4} + 192 \beta_{5} ) q^{74} + ( 1250 + 625 \beta_{1} ) q^{75} + ( 7136 + 224 \beta_{1} + 128 \beta_{2} - 336 \beta_{3} - 144 \beta_{4} - 48 \beta_{5} ) q^{76} + ( 17638 - 613 \beta_{1} - 150 \beta_{2} - 123 \beta_{3} - 61 \beta_{4} + 710 \beta_{5} ) q^{77} + ( 25816 + 4 \beta_{1} - 408 \beta_{2} + 652 \beta_{3} + 276 \beta_{4} - 148 \beta_{5} ) q^{78} + ( 12518 - 2410 \beta_{1} - 244 \beta_{2} + 49 \beta_{3} - 16 \beta_{4} + 217 \beta_{5} ) q^{79} -6400 q^{80} + ( 18117 + 2847 \beta_{1} - 484 \beta_{2} + 305 \beta_{3} + 509 \beta_{4} - 54 \beta_{5} ) q^{81} + ( 22340 - 300 \beta_{1} - 76 \beta_{2} + 140 \beta_{3} + 20 \beta_{4} - 732 \beta_{5} ) q^{82} + ( 9277 + 1478 \beta_{1} + 237 \beta_{2} + 103 \beta_{3} - 11 \beta_{4} + 1029 \beta_{5} ) q^{83} + ( 16224 - 80 \beta_{1} - 64 \beta_{2} + 64 \beta_{3} + 144 \beta_{4} + 400 \beta_{5} ) q^{84} + ( 3125 - 175 \beta_{1} - 125 \beta_{2} + 200 \beta_{3} + 125 \beta_{4} + 675 \beta_{5} ) q^{85} + ( 11128 + 584 \beta_{1} + 16 \beta_{2} - 444 \beta_{3} + 112 \beta_{4} + 172 \beta_{5} ) q^{86} + ( -12428 + 630 \beta_{1} + 586 \beta_{2} - 1065 \beta_{3} - 636 \beta_{4} - 8 \beta_{5} ) q^{87} + ( 3200 + 448 \beta_{1} - 128 \beta_{4} + 64 \beta_{5} ) q^{88} + ( 5806 + 128 \beta_{1} + 294 \beta_{2} - 453 \beta_{3} - 136 \beta_{4} - 1231 \beta_{5} ) q^{89} + ( -14700 - 600 \beta_{1} - 300 \beta_{3} - 100 \beta_{4} ) q^{90} + ( 2942 + 2477 \beta_{1} - 324 \beta_{2} + 182 \beta_{3} + 54 \beta_{4} - 365 \beta_{5} ) q^{91} -8464 q^{92} + ( 17502 + 7877 \beta_{1} + 190 \beta_{2} - 115 \beta_{3} - 29 \beta_{4} - 643 \beta_{5} ) q^{93} + ( 12064 - 1408 \beta_{1} - 456 \beta_{2} + 260 \beta_{3} + 240 \beta_{4} - 224 \beta_{5} ) q^{94} + ( -11150 - 350 \beta_{1} - 200 \beta_{2} + 525 \beta_{3} + 225 \beta_{4} + 75 \beta_{5} ) q^{95} + ( 2048 + 1024 \beta_{1} ) q^{96} + ( 11718 + 3583 \beta_{1} - 142 \beta_{2} - 546 \beta_{3} + 2 \beta_{4} - 1045 \beta_{5} ) q^{97} + ( 4456 - 108 \beta_{1} - 44 \beta_{2} - 264 \beta_{3} - 76 \beta_{4} + 1196 \beta_{5} ) q^{98} + ( -924 - 270 \beta_{1} + 528 \beta_{2} - 348 \beta_{3} - 357 \beta_{4} - 984 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 24 q^{2} + 15 q^{3} + 96 q^{4} - 150 q^{5} + 60 q^{6} + 106 q^{7} + 384 q^{8} + 899 q^{9} + O(q^{10}) \) \( 6 q + 24 q^{2} + 15 q^{3} + 96 q^{4} - 150 q^{5} + 60 q^{6} + 106 q^{7} + 384 q^{8} + 899 q^{9} - 600 q^{10} + 321 q^{11} + 240 q^{12} + 527 q^{13} + 424 q^{14} - 375 q^{15} + 1536 q^{16} - 660 q^{17} + 3596 q^{18} + 2749 q^{19} - 2400 q^{20} + 6002 q^{21} + 1284 q^{22} - 3174 q^{23} + 960 q^{24} + 3750 q^{25} + 2108 q^{26} + 15372 q^{27} + 1696 q^{28} + 3337 q^{29} - 1500 q^{30} + 31094 q^{31} + 6144 q^{32} + 15087 q^{33} - 2640 q^{34} - 2650 q^{35} + 14384 q^{36} + 27037 q^{37} + 10996 q^{38} + 38528 q^{39} - 9600 q^{40} + 33608 q^{41} + 24008 q^{42} + 17024 q^{43} + 5136 q^{44} - 22475 q^{45} - 12696 q^{46} + 16864 q^{47} + 3840 q^{48} + 6002 q^{49} + 15000 q^{50} + 5719 q^{51} + 8432 q^{52} - 8475 q^{53} + 61488 q^{54} - 8025 q^{55} + 6784 q^{56} + 9566 q^{57} + 13348 q^{58} + 7899 q^{59} - 6000 q^{60} + 25437 q^{61} + 124376 q^{62} - 13333 q^{63} + 24576 q^{64} - 13175 q^{65} + 60348 q^{66} - 25517 q^{67} - 10560 q^{68} - 7935 q^{69} - 10600 q^{70} + 17204 q^{71} + 57536 q^{72} + 760 q^{73} + 108148 q^{74} + 9375 q^{75} + 43984 q^{76} + 102330 q^{77} + 154112 q^{78} + 66972 q^{79} - 38400 q^{80} + 115874 q^{81} + 134432 q^{82} + 58523 q^{83} + 96032 q^{84} + 16500 q^{85} + 68096 q^{86} - 70854 q^{87} + 20544 q^{88} + 38406 q^{89} - 89900 q^{90} + 25111 q^{91} - 50784 q^{92} + 130338 q^{93} + 67456 q^{94} - 68725 q^{95} + 15360 q^{96} + 82861 q^{97} + 24008 q^{98} - 2973 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 1156 x^{4} + 593 x^{3} + 338133 x^{2} + 408388 x - 13033476\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 473 \nu^{5} - 432595 \nu^{4} + 3301386 \nu^{3} + 342922345 \nu^{2} - 1629413963 \nu - 39620612346 \)\()/ 192701508 \)
\(\beta_{3}\)\(=\)\((\)\( -457 \nu^{5} + 10559 \nu^{4} - 948996 \nu^{3} + 25766089 \nu^{2} + 628918249 \nu - 10563544722 \)\()/96350754\)
\(\beta_{4}\)\(=\)\((\)\( 457 \nu^{5} - 10559 \nu^{4} + 948996 \nu^{3} + 6350829 \nu^{2} - 693152085 \nu - 1833585626 \)\()/32116918\)
\(\beta_{5}\)\(=\)\((\)\( 8222 \nu^{5} - 84553 \nu^{4} - 7593858 \nu^{3} + 46230502 \nu^{2} + 1479281155 \nu - 1814631102 \)\()/96350754\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + 3 \beta_{3} + 2 \beta_{1} + 386\)
\(\nu^{3}\)\(=\)\(-4 \beta_{5} + 25 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 571 \beta_{1} + 1160\)
\(\nu^{4}\)\(=\)\(-22 \beta_{5} + 1014 \beta_{4} + 2404 \beta_{3} - 468 \beta_{2} + 2573 \beta_{1} + 224818\)
\(\nu^{5}\)\(=\)\(7798 \beta_{5} + 27895 \beta_{4} + 9701 \beta_{3} - 6660 \beta_{2} + 362674 \beta_{1} + 1433662\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−24.9052
−19.3306
−7.76257
5.93160
20.8018
28.2649
4.00000 −22.9052 16.0000 −25.0000 −91.6207 10.2657 64.0000 281.647 −100.000
1.2 4.00000 −17.3306 16.0000 −25.0000 −69.3223 −200.644 64.0000 57.3489 −100.000
1.3 4.00000 −5.76257 16.0000 −25.0000 −23.0503 50.4462 64.0000 −209.793 −100.000
1.4 4.00000 7.93160 16.0000 −25.0000 31.7264 221.199 64.0000 −180.090 −100.000
1.5 4.00000 22.8018 16.0000 −25.0000 91.2074 −73.3684 64.0000 276.924 −100.000
1.6 4.00000 30.2649 16.0000 −25.0000 121.060 98.1017 64.0000 672.963 −100.000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(23\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.6.a.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.6.a.i 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 15 T_{3}^{5} - 1066 T_{3}^{4} + 9561 T_{3}^{3} + 307311 T_{3}^{2} - 900468 T_{3} - 12520800 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(230))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -4 + T )^{6} \)
$3$ \( -12520800 - 900468 T + 307311 T^{2} + 9561 T^{3} - 1066 T^{4} - 15 T^{5} + T^{6} \)
$5$ \( ( 25 + T )^{6} \)
$7$ \( 165428983104 - 18748948820 T + 217369064 T^{2} + 4322851 T^{3} - 47804 T^{4} - 106 T^{5} + T^{6} \)
$11$ \( -221236397079552 + 1314020686896 T + 28636746048 T^{2} + 40641750 T^{3} - 292689 T^{4} - 321 T^{5} + T^{6} \)
$13$ \( 6606225862820316 - 75466026870480 T + 121409606835 T^{2} + 794899305 T^{3} - 1347474 T^{4} - 527 T^{5} + T^{6} \)
$17$ \( -2240582438923781280 + 17454762350043760 T + 17999095696326 T^{2} - 7133346883 T^{3} - 8384902 T^{4} + 660 T^{5} + T^{6} \)
$19$ \( 29724256493958812928 - 49151645113971152 T + 3758647852400 T^{2} + 23645030182 T^{3} - 6968603 T^{4} - 2749 T^{5} + T^{6} \)
$23$ \( ( 529 + T )^{6} \)
$29$ \( -\)\(13\!\cdots\!32\)\( - 4793561882935386456 T + 2217505556969140 T^{2} + 251950573481 T^{3} - 87036309 T^{4} - 3337 T^{5} + T^{6} \)
$31$ \( \)\(68\!\cdots\!60\)\( + 9072856885157236843 T - 514138920749626 T^{2} - 1317255697385 T^{3} + 338057117 T^{4} - 31094 T^{5} + T^{6} \)
$37$ \( -\)\(36\!\cdots\!08\)\( + 71573167094015114176 T - 25531366298453072 T^{2} + 2331531034152 T^{3} + 121090838 T^{4} - 27037 T^{5} + T^{6} \)
$41$ \( \)\(18\!\cdots\!46\)\( - \)\(26\!\cdots\!19\)\( T - 39294948984849698 T^{2} + 6593621850535 T^{3} + 62704443 T^{4} - 33608 T^{5} + T^{6} \)
$43$ \( \)\(13\!\cdots\!20\)\( - \)\(25\!\cdots\!68\)\( T - 29400205966928384 T^{2} + 8406828306720 T^{3} - 355118044 T^{4} - 17024 T^{5} + T^{6} \)
$47$ \( \)\(29\!\cdots\!00\)\( - \)\(86\!\cdots\!60\)\( T + 4040917609232900 T^{2} + 11306573830198 T^{3} - 553869551 T^{4} - 16864 T^{5} + T^{6} \)
$53$ \( -\)\(80\!\cdots\!36\)\( + \)\(48\!\cdots\!88\)\( T + 625064470555621824 T^{2} - 20094793340616 T^{3} - 1703143386 T^{4} + 8475 T^{5} + T^{6} \)
$59$ \( -\)\(29\!\cdots\!88\)\( + \)\(39\!\cdots\!12\)\( T + 1762685487550225584 T^{2} + 2938893533204 T^{3} - 2529430030 T^{4} - 7899 T^{5} + T^{6} \)
$61$ \( \)\(57\!\cdots\!92\)\( - \)\(21\!\cdots\!32\)\( T + 1142353925192735088 T^{2} + 46393894255178 T^{3} - 2530002649 T^{4} - 25437 T^{5} + T^{6} \)
$67$ \( \)\(13\!\cdots\!20\)\( + \)\(14\!\cdots\!44\)\( T + 3888015748633695168 T^{2} - 39250915635288 T^{3} - 4371029228 T^{4} + 25517 T^{5} + T^{6} \)
$71$ \( -\)\(31\!\cdots\!80\)\( - \)\(11\!\cdots\!49\)\( T + 34803123027317476358 T^{2} + 104701875710777 T^{3} - 10703341295 T^{4} - 17204 T^{5} + T^{6} \)
$73$ \( \)\(72\!\cdots\!48\)\( - \)\(46\!\cdots\!60\)\( T + 330926548507971016 T^{2} + 30838423107594 T^{3} - 2197845799 T^{4} - 760 T^{5} + T^{6} \)
$79$ \( \)\(93\!\cdots\!56\)\( - \)\(80\!\cdots\!60\)\( T + 9439683733291161600 T^{2} + 493768556639040 T^{3} - 7425921544 T^{4} - 66972 T^{5} + T^{6} \)
$83$ \( \)\(62\!\cdots\!04\)\( - \)\(33\!\cdots\!24\)\( T + 27663368410864357664 T^{2} + 1013570434626332 T^{3} - 14376428924 T^{4} - 58523 T^{5} + T^{6} \)
$89$ \( \)\(77\!\cdots\!48\)\( + \)\(23\!\cdots\!48\)\( T + 74495494486893530592 T^{2} - 37873901031480 T^{3} - 17764458804 T^{4} - 38406 T^{5} + T^{6} \)
$97$ \( \)\(27\!\cdots\!76\)\( - \)\(18\!\cdots\!12\)\( T + \)\(17\!\cdots\!08\)\( T^{2} + 2628743862316612 T^{3} - 30340704599 T^{4} - 82861 T^{5} + T^{6} \)
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