Properties

Label 336.7.bh.b
Level $336$
Weight $7$
Character orbit 336.bh
Analytic conductor $77.298$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 336.bh (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(77.2981720963\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} + 212 x^{6} - 787 x^{5} + 38792 x^{4} - 92833 x^{3} + 1563109 x^{2} + 3107772 x + 38787984\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -9 + 9 \beta_{2} ) q^{3} + ( -7 - 3 \beta_{2} + \beta_{3} - \beta_{6} ) q^{5} + ( -75 - 2 \beta_{1} + 37 \beta_{2} - 3 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{7} -243 \beta_{2} q^{9} +O(q^{10})\) \( q + ( -9 + 9 \beta_{2} ) q^{3} + ( -7 - 3 \beta_{2} + \beta_{3} - \beta_{6} ) q^{5} + ( -75 - 2 \beta_{1} + 37 \beta_{2} - 3 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{7} -243 \beta_{2} q^{9} + ( 267 + 2 \beta_{1} + 267 \beta_{2} - 5 \beta_{3} + 2 \beta_{4} - \beta_{5} + 5 \beta_{6} + 4 \beta_{7} ) q^{11} + ( 862 + 4 \beta_{1} + 1729 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} + 6 \beta_{5} + 5 \beta_{6} + 5 \beta_{7} ) q^{13} + ( 99 - 9 \beta_{2} - 9 \beta_{3} + 9 \beta_{5} + 9 \beta_{6} - 9 \beta_{7} ) q^{15} + ( 652 - 37 \beta_{1} - 574 \beta_{2} - 39 \beta_{3} + 37 \beta_{4} + 24 \beta_{5} - 39 \beta_{6} + 15 \beta_{7} ) q^{17} + ( 4087 - 97 \beta_{1} + 2037 \beta_{2} + \beta_{3} - 194 \beta_{4} - 14 \beta_{5} + 13 \beta_{6} - 14 \beta_{7} ) q^{19} + ( 369 + 9 \beta_{1} - 1332 \beta_{2} + 45 \beta_{4} + 36 \beta_{5} + 9 \beta_{6} ) q^{21} + ( -29 - 311 \beta_{1} - 3808 \beta_{2} + 4 \beta_{3} - 25 \beta_{5} + 29 \beta_{6} + 25 \beta_{7} ) q^{23} + ( 5568 - 330 \beta_{1} + 5568 \beta_{2} - 41 \beta_{3} - 330 \beta_{4} - \beta_{5} + 41 \beta_{6} + 40 \beta_{7} ) q^{25} + ( 2187 + 4374 \beta_{2} ) q^{27} + ( 4218 + 61 \beta_{2} - 122 \beta_{3} - 25 \beta_{4} + 122 \beta_{5} - 61 \beta_{6} + 61 \beta_{7} ) q^{29} + ( -3090 - 90 \beta_{1} + 3328 \beta_{2} - 119 \beta_{3} + 90 \beta_{4} + 282 \beta_{5} - 119 \beta_{6} - 163 \beta_{7} ) q^{31} + ( -4842 - 18 \beta_{1} - 2394 \beta_{2} + 81 \beta_{3} - 36 \beta_{4} - 27 \beta_{5} - 54 \beta_{6} - 27 \beta_{7} ) q^{33} + ( 35213 + 375 \beta_{1} + 9439 \beta_{2} - 189 \beta_{3} - 435 \beta_{4} + 170 \beta_{5} - 66 \beta_{6} + 28 \beta_{7} ) q^{35} + ( -202 + 790 \beta_{1} + 11453 \beta_{2} - 71 \beta_{3} - 273 \beta_{5} + 202 \beta_{6} + 273 \beta_{7} ) q^{37} + ( -23418 - 54 \beta_{1} - 23418 \beta_{2} - 9 \beta_{3} - 54 \beta_{4} - 153 \beta_{5} + 9 \beta_{6} - 144 \beta_{7} ) q^{39} + ( 6232 + 260 \beta_{1} + 12900 \beta_{2} - 346 \beta_{3} + 130 \beta_{4} - 346 \beta_{5} + 436 \beta_{6} + 436 \beta_{7} ) q^{41} + ( 55439 + 257 \beta_{2} + 397 \beta_{3} + 1445 \beta_{4} - 397 \beta_{5} - 257 \beta_{6} + 257 \beta_{7} ) q^{43} + ( -972 + 972 \beta_{2} - 243 \beta_{5} + 243 \beta_{7} ) q^{45} + ( -14171 + 247 \beta_{1} - 7292 \beta_{2} - 56 \beta_{3} + 494 \beta_{4} - 357 \beta_{5} + 413 \beta_{6} - 357 \beta_{7} ) q^{47} + ( -20424 + 2476 \beta_{1} - 47598 \beta_{2} - 644 \beta_{3} + 1082 \beta_{4} + 370 \beta_{5} + 173 \beta_{6} + 413 \beta_{7} ) q^{49} + ( -567 + 999 \beta_{1} + 15984 \beta_{2} + 486 \beta_{3} - 81 \beta_{5} + 567 \beta_{6} + 81 \beta_{7} ) q^{51} + ( -3184 - 1028 \beta_{1} - 3184 \beta_{2} + 1306 \beta_{3} - 1028 \beta_{4} + 149 \beta_{5} - 1306 \beta_{6} - 1157 \beta_{7} ) q^{53} + ( -44850 + 750 \beta_{1} - 90045 \beta_{2} + 358 \beta_{3} + 375 \beta_{4} + 358 \beta_{5} - 345 \beta_{6} - 345 \beta_{7} ) q^{55} + ( -55107 + 243 \beta_{2} - 135 \beta_{3} + 2619 \beta_{4} + 135 \beta_{5} - 243 \beta_{6} + 243 \beta_{7} ) q^{57} + ( -150135 + 95 \beta_{1} + 151215 \beta_{2} - 540 \beta_{3} - 95 \beta_{4} + 377 \beta_{5} - 540 \beta_{6} + 163 \beta_{7} ) q^{59} + ( -65932 - 764 \beta_{1} - 33220 \beta_{2} - 52 \beta_{3} - 1528 \beta_{4} - 456 \beta_{5} + 508 \beta_{6} - 456 \beta_{7} ) q^{61} + ( 8262 + 243 \beta_{1} + 26973 \beta_{2} - 486 \beta_{4} - 729 \beta_{5} + 243 \beta_{6} ) q^{63} + ( -1 - 4395 \beta_{1} + 96458 \beta_{2} + 1586 \beta_{3} + 1585 \beta_{5} + \beta_{6} - 1585 \beta_{7} ) q^{65} + ( -95940 - 4167 \beta_{1} - 95940 \beta_{2} - 842 \beta_{3} - 4167 \beta_{4} + 167 \beta_{5} + 842 \beta_{6} + 1009 \beta_{7} ) q^{67} + ( 34497 + 5598 \beta_{1} + 68508 \beta_{2} + 189 \beta_{3} + 2799 \beta_{4} + 189 \beta_{5} - 486 \beta_{6} - 486 \beta_{7} ) q^{69} + ( -27171 + 1706 \beta_{2} - 187 \beta_{3} - 827 \beta_{4} + 187 \beta_{5} - 1706 \beta_{6} + 1706 \beta_{7} ) q^{71} + ( 74272 - 2284 \beta_{1} - 75354 \beta_{2} + 541 \beta_{3} + 2284 \beta_{4} - 887 \beta_{5} + 541 \beta_{6} + 346 \beta_{7} ) q^{73} + ( -100584 + 2970 \beta_{1} - 50103 \beta_{2} + 729 \beta_{3} + 5940 \beta_{4} - 351 \beta_{5} - 378 \beta_{6} - 351 \beta_{7} ) q^{75} + ( -110762 - 1987 \beta_{1} - 138853 \beta_{2} + 1169 \beta_{3} + 1440 \beta_{4} - 297 \beta_{5} - 860 \beta_{6} - 567 \beta_{7} ) q^{77} + ( -653 + 13248 \beta_{1} - 140140 \beta_{2} + 484 \beta_{3} - 169 \beta_{5} + 653 \beta_{6} + 169 \beta_{7} ) q^{79} + ( -59049 - 59049 \beta_{2} ) q^{81} + ( 78670 - 9514 \beta_{1} + 162183 \beta_{2} - 1609 \beta_{3} - 4757 \beta_{4} - 1609 \beta_{5} + 4843 \beta_{6} + 4843 \beta_{7} ) q^{83} + ( 246796 + 1618 \beta_{2} - 1790 \beta_{3} - 17550 \beta_{4} + 1790 \beta_{5} - 1618 \beta_{6} + 1618 \beta_{7} ) q^{85} + ( -39060 - 225 \beta_{1} + 35766 \beta_{2} + 1647 \beta_{3} + 225 \beta_{4} - 1647 \beta_{5} + 1647 \beta_{6} ) q^{87} + ( -295108 - 1046 \beta_{1} - 147542 \beta_{2} + 126 \beta_{3} - 2092 \beta_{4} - 102 \beta_{5} - 24 \beta_{6} - 102 \beta_{7} ) q^{89} + ( 425729 - 11931 \beta_{1} + 107635 \beta_{2} + 2807 \beta_{3} - 4362 \beta_{4} + 3698 \beta_{5} - 4581 \beta_{6} - 924 \beta_{7} ) q^{91} + ( -3609 + 2430 \beta_{1} - 90252 \beta_{2} - 396 \beta_{3} - 4005 \beta_{5} + 3609 \beta_{6} + 4005 \beta_{7} ) q^{93} + ( -287060 + 20115 \beta_{1} - 287060 \beta_{2} + 7867 \beta_{3} + 20115 \beta_{4} + 5936 \beta_{5} - 7867 \beta_{6} - 1931 \beta_{7} ) q^{95} + ( 134387 + 2492 \beta_{1} + 270401 \beta_{2} + 2652 \beta_{3} + 1246 \beta_{4} + 2652 \beta_{5} + 1627 \beta_{6} + 1627 \beta_{7} ) q^{97} + ( 65853 - 243 \beta_{2} - 972 \beta_{3} + 486 \beta_{4} + 972 \beta_{5} + 243 \beta_{6} - 243 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 108q^{3} - 42q^{5} - 748q^{7} + 972q^{9} + O(q^{10}) \) \( 8q - 108q^{3} - 42q^{5} - 748q^{7} + 972q^{9} + 1070q^{11} + 756q^{15} + 7212q^{17} + 24606q^{19} + 8154q^{21} + 15224q^{23} + 22274q^{25} + 32524q^{29} - 40200q^{31} - 28890q^{33} + 242436q^{35} - 45670q^{37} - 93366q^{39} + 445660q^{43} - 10206q^{45} - 82884q^{47} + 24116q^{49} - 64908q^{51} - 13034q^{53} - 442908q^{57} - 1810362q^{59} - 392856q^{61} - 38394q^{63} - 389004q^{65} - 384094q^{67} - 225688q^{71} + 903078q^{73} - 601398q^{75} - 327674q^{77} + 559592q^{79} - 236196q^{81} + 1953576q^{85} - 439074q^{87} - 1770036q^{89} + 2960718q^{91} + 361800q^{93} - 1160112q^{95} + 520020q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 212 x^{6} - 787 x^{5} + 38792 x^{4} - 92833 x^{3} + 1563109 x^{2} + 3107772 x + 38787984\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(1585013359 \nu^{7} - 18232571539 \nu^{6} + 303603349712 \nu^{5} - 4245938445433 \nu^{4} + 65749585575908 \nu^{3} - 780366646056751 \nu^{2} + 97319550794346895 \nu - 22182278913510768\)\()/ 23802631772961636 \)
\(\beta_{2}\)\(=\)\((\)\(1585013359 \nu^{7} - 18232571539 \nu^{6} + 303603349712 \nu^{5} - 4245938445433 \nu^{4} + 65749585575908 \nu^{3} - 780366646056751 \nu^{2} + 2109023702500351 \nu - 22182278913510768\)\()/ 23802631772961636 \)
\(\beta_{3}\)\(=\)\((\)\(13858370275 \nu^{7} - 1233059081317 \nu^{6} - 15146056834333 \nu^{5} - 230507120454535 \nu^{4} - 1133698910951701 \nu^{3} - 18697714244203045 \nu^{2} - 316744918301136878 \nu + 312743079434231442\)\()/ 11901315886480818 \)
\(\beta_{4}\)\(=\)\((\)\(32076220 \nu^{7} + 62465284 \nu^{6} + 5777520100 \nu^{5} - 8215312820 \nu^{4} + 1220086899616 \nu^{3} + 710067329620 \nu^{2} + 6368992827120 \nu + 129923180639535\)\()/ 11465622241311 \)
\(\beta_{5}\)\(=\)\((\)\(-85151344841 \nu^{7} - 799568126245 \nu^{6} - 32979536564113 \nu^{5} - 205148902375939 \nu^{4} - 4947341926250533 \nu^{3} - 20889480107476681 \nu^{2} - 336404097761466014 \nu - 1772851718373789258\)\()/ 11901315886480818 \)
\(\beta_{6}\)\(=\)\((\)\(41883584669 \nu^{7} + 456963608136 \nu^{6} + 4283641236335 \nu^{5} + 19286324891880 \nu^{4} + 212218491969083 \nu^{3} + 8199002660156988 \nu^{2} - 88533841034081394 \nu - 44383880102164944\)\()/ 3967105295493606 \)
\(\beta_{7}\)\(=\)\((\)\(-661077962011 \nu^{7} - 1518878968019 \nu^{6} - 138652672166732 \nu^{5} + 353235182738359 \nu^{4} - 21344721106790528 \nu^{3} + 29812237950379993 \nu^{2} - 714157673331781651 \nu + 177777443566348128\)\()/ 23802631772961636 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{7} - 2 \beta_{6} - 3 \beta_{4} + 2 \beta_{3} - 423 \beta_{2} - 3 \beta_{1} - 423\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(8 \beta_{7} - 8 \beta_{6} + 16 \beta_{5} + 157 \beta_{4} - 16 \beta_{3} + 8 \beta_{2} + 883\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(414 \beta_{7} + 406 \beta_{6} - 414 \beta_{5} - 8 \beta_{3} + 62983 \beta_{2} + 975 \beta_{1} - 406\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-3968 \beta_{7} - 2272 \beta_{6} - 1696 \beta_{5} - 27421 \beta_{4} + 2272 \beta_{3} - 337267 \beta_{2} - 27421 \beta_{1} - 337267\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(608 \beta_{7} - 608 \beta_{6} + 77226 \beta_{5} + 237963 \beta_{4} - 77226 \beta_{3} + 608 \beta_{2} + 10825873\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(559536 \beta_{7} + 862968 \beta_{6} - 559536 \beta_{5} + 303432 \beta_{3} + 86136107 \beta_{2} + 5056597 \beta_{1} - 862968\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
5.73828 9.93899i
−7.08935 + 12.2791i
−2.30325 + 3.98935i
4.15432 7.19549i
5.73828 + 9.93899i
−7.08935 12.2791i
−2.30325 3.98935i
4.15432 + 7.19549i
0 −13.5000 + 7.79423i 0 −175.367 101.248i 0 −284.280 + 191.921i 0 121.500 210.444i 0
145.2 0 −13.5000 + 7.79423i 0 −68.9069 39.7834i 0 −284.244 191.975i 0 121.500 210.444i 0
145.3 0 −13.5000 + 7.79423i 0 71.9311 + 41.5295i 0 −77.0894 + 334.225i 0 121.500 210.444i 0
145.4 0 −13.5000 + 7.79423i 0 151.343 + 87.3778i 0 271.614 209.463i 0 121.500 210.444i 0
241.1 0 −13.5000 7.79423i 0 −175.367 + 101.248i 0 −284.280 191.921i 0 121.500 + 210.444i 0
241.2 0 −13.5000 7.79423i 0 −68.9069 + 39.7834i 0 −284.244 + 191.975i 0 121.500 + 210.444i 0
241.3 0 −13.5000 7.79423i 0 71.9311 41.5295i 0 −77.0894 334.225i 0 121.500 + 210.444i 0
241.4 0 −13.5000 7.79423i 0 151.343 87.3778i 0 271.614 + 209.463i 0 121.500 + 210.444i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 241.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.7.bh.b 8
4.b odd 2 1 21.7.f.b 8
7.d odd 6 1 inner 336.7.bh.b 8
12.b even 2 1 63.7.m.c 8
28.d even 2 1 147.7.f.a 8
28.f even 6 1 21.7.f.b 8
28.f even 6 1 147.7.d.a 8
28.g odd 6 1 147.7.d.a 8
28.g odd 6 1 147.7.f.a 8
84.j odd 6 1 63.7.m.c 8
84.j odd 6 1 441.7.d.d 8
84.n even 6 1 441.7.d.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.b 8 4.b odd 2 1
21.7.f.b 8 28.f even 6 1
63.7.m.c 8 12.b even 2 1
63.7.m.c 8 84.j odd 6 1
147.7.d.a 8 28.f even 6 1
147.7.d.a 8 28.g odd 6 1
147.7.f.a 8 28.d even 2 1
147.7.f.a 8 28.g odd 6 1
336.7.bh.b 8 1.a even 1 1 trivial
336.7.bh.b 8 7.d odd 6 1 inner
441.7.d.d 8 84.j odd 6 1
441.7.d.d 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(98\!\cdots\!00\)\( T_{5}^{2} - \)\(24\!\cdots\!00\)\( T_{5} + \)\(54\!\cdots\!00\)\( \)">\(T_{5}^{8} + \cdots\) acting on \(S_{7}^{\mathrm{new}}(336, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 243 + 27 T + T^{2} )^{4} \)
$5$ \( 54693259182810000 - 24205120650000 T - 9840546578700 T^{2} + 4356625500 T^{3} + 1536505749 T^{4} - 1767906 T^{5} - 41505 T^{6} + 42 T^{7} + T^{8} \)
$7$ \( \)\(19\!\cdots\!01\)\( + 1218053371237015852 T + 3705229535984494 T^{2} + 5061633633224 T^{3} + 8234236703 T^{4} + 43023176 T^{5} + 267694 T^{6} + 748 T^{7} + T^{8} \)
$11$ \( \)\(34\!\cdots\!00\)\( + 18488408967989050800 T + 113509429921169980 T^{2} + 43785002211340 T^{3} + 345504620141 T^{4} - 343064750 T^{5} + 1409111 T^{6} - 1070 T^{7} + T^{8} \)
$13$ \( \)\(34\!\cdots\!00\)\( + 53257929335845404768 T^{2} + 105787465152777 T^{4} + 22315686 T^{6} + T^{8} \)
$17$ \( \)\(22\!\cdots\!00\)\( + \)\(46\!\cdots\!60\)\( T - \)\(83\!\cdots\!72\)\( T^{2} - 23263548118111082880 T^{3} + 3514480347514032 T^{4} + 547654326480 T^{5} - 58598892 T^{6} - 7212 T^{7} + T^{8} \)
$19$ \( \)\(12\!\cdots\!96\)\( - \)\(11\!\cdots\!92\)\( T + \)\(36\!\cdots\!68\)\( T^{2} - 27399467456974423980 T^{3} - 6359170662393867 T^{4} + 670975723710 T^{5} + 174549627 T^{6} - 24606 T^{7} + T^{8} \)
$23$ \( \)\(63\!\cdots\!00\)\( + \)\(71\!\cdots\!00\)\( T + \)\(15\!\cdots\!00\)\( T^{2} - 51980122560355404800 T^{3} + 101423010754761920 T^{4} - 1256314418240 T^{5} + 520738856 T^{6} - 15224 T^{7} + T^{8} \)
$29$ \( ( -39242852020022400 + 9430137809600 T - 442580539 T^{2} - 16262 T^{3} + T^{4} )^{2} \)
$31$ \( \)\(42\!\cdots\!21\)\( - \)\(16\!\cdots\!80\)\( T - \)\(92\!\cdots\!06\)\( T^{2} + \)\(43\!\cdots\!80\)\( T^{3} + 2063880899710463727 T^{4} - 70137742150800 T^{5} - 1206039954 T^{6} + 40200 T^{7} + T^{8} \)
$37$ \( \)\(25\!\cdots\!00\)\( - \)\(12\!\cdots\!00\)\( T + \)\(21\!\cdots\!00\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + 10133981768848670525 T^{4} - 93188930684650 T^{5} + 5175060655 T^{6} + 45670 T^{7} + T^{8} \)
$41$ \( \)\(53\!\cdots\!16\)\( + \)\(87\!\cdots\!08\)\( T^{2} + 50479416076831845984 T^{4} + 11994986352 T^{6} + T^{8} \)
$43$ \( ( -\)\(14\!\cdots\!84\)\( + 2845100102192540 T - 1577056827 T^{2} - 222830 T^{3} + T^{4} )^{2} \)
$47$ \( \)\(30\!\cdots\!00\)\( - \)\(12\!\cdots\!00\)\( T + \)\(18\!\cdots\!00\)\( T^{2} - \)\(87\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!20\)\( T^{4} - 1025758292062320 T^{5} - 10085910828 T^{6} + 82884 T^{7} + T^{8} \)
$53$ \( \)\(23\!\cdots\!00\)\( - \)\(38\!\cdots\!00\)\( T + \)\(25\!\cdots\!64\)\( T^{2} + \)\(27\!\cdots\!44\)\( T^{3} + \)\(20\!\cdots\!21\)\( T^{4} + 946587441999554 T^{5} + 50746432799 T^{6} + 13034 T^{7} + T^{8} \)
$59$ \( \)\(13\!\cdots\!36\)\( + \)\(41\!\cdots\!92\)\( T + \)\(55\!\cdots\!32\)\( T^{2} + \)\(43\!\cdots\!28\)\( T^{3} + \)\(21\!\cdots\!13\)\( T^{4} + 715051889313496398 T^{5} + 1487447487327 T^{6} + 1810362 T^{7} + T^{8} \)
$61$ \( \)\(78\!\cdots\!00\)\( + \)\(11\!\cdots\!40\)\( T + \)\(61\!\cdots\!72\)\( T^{2} + \)\(15\!\cdots\!00\)\( T^{3} - \)\(51\!\cdots\!48\)\( T^{4} - 1477092202992000 T^{5} + 47685396912 T^{6} + 392856 T^{7} + T^{8} \)
$67$ \( \)\(93\!\cdots\!00\)\( - \)\(14\!\cdots\!40\)\( T + \)\(25\!\cdots\!04\)\( T^{2} - \)\(80\!\cdots\!20\)\( T^{3} + \)\(67\!\cdots\!57\)\( T^{4} + 12184102069585486 T^{5} + 192425428651 T^{6} + 384094 T^{7} + T^{8} \)
$71$ \( ( -\)\(25\!\cdots\!12\)\( - 44439430271275520 T - 187952640388 T^{2} + 112844 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(12\!\cdots\!24\)\( + \)\(24\!\cdots\!32\)\( T + \)\(16\!\cdots\!20\)\( T^{2} + \)\(44\!\cdots\!04\)\( T^{3} - \)\(63\!\cdots\!51\)\( T^{4} - 18164126696429538 T^{5} + 291963532599 T^{6} - 903078 T^{7} + T^{8} \)
$79$ \( \)\(58\!\cdots\!25\)\( - \)\(18\!\cdots\!80\)\( T + \)\(57\!\cdots\!34\)\( T^{2} - \)\(12\!\cdots\!12\)\( T^{3} + \)\(36\!\cdots\!07\)\( T^{4} - 233175179655084368 T^{5} + 779488395718 T^{6} - 559592 T^{7} + T^{8} \)
$83$ \( \)\(36\!\cdots\!24\)\( + \)\(24\!\cdots\!96\)\( T^{2} + \)\(12\!\cdots\!25\)\( T^{4} + 2038066317246 T^{6} + T^{8} \)
$89$ \( \)\(13\!\cdots\!44\)\( + \)\(38\!\cdots\!40\)\( T + \)\(51\!\cdots\!48\)\( T^{2} + \)\(40\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!44\)\( T^{4} + 673833907778767344 T^{5} + 1425031860636 T^{6} + 1770036 T^{7} + T^{8} \)
$97$ \( \)\(22\!\cdots\!00\)\( + \)\(33\!\cdots\!12\)\( T^{2} + \)\(14\!\cdots\!01\)\( T^{4} + 2348711138742 T^{6} + T^{8} \)
show more
show less