Properties

Label 63.4.e.c
Level 63
Weight 4
Character orbit 63.e
Analytic conductor 3.717
Analytic rank 0
Dimension 6
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 63.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.71712033036\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.9924270768.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -8 + \beta_{1} + \beta_{2} + 8 \beta_{4} + \beta_{5} ) q^{4} + ( -\beta_{1} - \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{5} + ( -4 - 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{7} + ( -10 + 9 \beta_{2} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -8 + \beta_{1} + \beta_{2} + 8 \beta_{4} + \beta_{5} ) q^{4} + ( -\beta_{1} - \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{5} + ( -4 - 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{7} + ( -10 + 9 \beta_{2} + \beta_{3} ) q^{8} + ( 22 + 11 \beta_{1} + 11 \beta_{2} - 22 \beta_{4} - \beta_{5} ) q^{10} + ( 12 + \beta_{1} + \beta_{2} - 12 \beta_{4} + 3 \beta_{5} ) q^{11} + ( 19 - 5 \beta_{2} + \beta_{3} ) q^{13} + ( 64 + 6 \beta_{1} - \beta_{2} - 3 \beta_{3} - 22 \beta_{4} - \beta_{5} ) q^{14} + ( -19 \beta_{1} + \beta_{3} - 74 \beta_{4} - \beta_{5} ) q^{16} + ( 16 - 16 \beta_{4} - 4 \beta_{5} ) q^{17} + ( 7 \beta_{1} - \beta_{3} + 65 \beta_{4} + \beta_{5} ) q^{19} + ( -150 - 11 \beta_{2} + 3 \beta_{3} ) q^{20} + ( 2 + 13 \beta_{2} + \beta_{3} ) q^{22} + ( -24 \beta_{1} + 4 \beta_{3} + 80 \beta_{4} - 4 \beta_{5} ) q^{23} + ( -53 - 29 \beta_{1} - 29 \beta_{2} + 53 \beta_{4} + \beta_{5} ) q^{25} + ( 16 \beta_{1} - 5 \beta_{3} + 86 \beta_{4} + 5 \beta_{5} ) q^{26} + ( -110 + 49 \beta_{1} - 16 \beta_{2} - \beta_{3} + 126 \beta_{4} - \beta_{5} ) q^{28} + ( -26 - 25 \beta_{2} + 5 \beta_{3} ) q^{29} + ( 39 + 22 \beta_{1} + 22 \beta_{2} - 39 \beta_{4} - 2 \beta_{5} ) q^{31} + ( 218 - 29 \beta_{1} - 29 \beta_{2} - 218 \beta_{4} - 11 \beta_{5} ) q^{32} + ( -24 - 48 \beta_{2} ) q^{34} + ( -100 - 24 \beta_{1} - 47 \beta_{2} + 11 \beta_{3} + 158 \beta_{4} - 4 \beta_{5} ) q^{35} + ( -19 \beta_{1} + \beta_{3} - 81 \beta_{4} - \beta_{5} ) q^{37} + ( -106 + 80 \beta_{1} + 80 \beta_{2} + 106 \beta_{4} + 7 \beta_{5} ) q^{38} + ( -75 \beta_{1} - 3 \beta_{3} + 18 \beta_{4} + 3 \beta_{5} ) q^{40} + ( -82 - 2 \beta_{2} - 14 \beta_{3} ) q^{41} + ( 143 + 69 \beta_{2} + 3 \beta_{3} ) q^{43} + ( -11 \beta_{1} - 11 \beta_{3} - 298 \beta_{4} + 11 \beta_{5} ) q^{44} + ( 360 + 24 \beta_{1} + 24 \beta_{2} - 360 \beta_{4} - 24 \beta_{5} ) q^{46} + ( -72 \beta_{1} + 28 \beta_{3} - 46 \beta_{4} - 28 \beta_{5} ) q^{47} + ( -7 - 46 \beta_{1} - 35 \beta_{2} + 25 \beta_{3} - 95 \beta_{4} - 2 \beta_{5} ) q^{49} + ( 470 + 32 \beta_{2} - 29 \beta_{3} ) q^{50} + ( -74 + 102 \beta_{1} + 102 \beta_{2} + 74 \beta_{4} + 24 \beta_{5} ) q^{52} + ( 154 + 69 \beta_{1} + 69 \beta_{2} - 154 \beta_{4} + 11 \beta_{5} ) q^{53} + ( -350 - 19 \beta_{2} - 25 \beta_{3} ) q^{55} + ( -454 + 81 \beta_{1} + 111 \beta_{2} - 8 \beta_{3} + 522 \beta_{4} + 33 \beta_{5} ) q^{56} + ( -41 \beta_{1} - 25 \beta_{3} + 430 \beta_{4} + 25 \beta_{5} ) q^{58} + ( 358 - 69 \beta_{1} - 69 \beta_{2} - 358 \beta_{4} + 29 \beta_{5} ) q^{59} + ( 100 \beta_{1} + 20 \beta_{3} - 10 \beta_{4} - 20 \beta_{5} ) q^{61} + ( -364 - 33 \beta_{2} + 22 \beta_{3} ) q^{62} + ( -194 - 183 \beta_{2} - 21 \beta_{3} ) q^{64} + ( 50 \beta_{1} - 18 \beta_{3} - 174 \beta_{4} + 18 \beta_{5} ) q^{65} + ( 215 + 17 \beta_{1} + 17 \beta_{2} - 215 \beta_{4} + 47 \beta_{5} ) q^{67} + ( 24 \beta_{1} - 16 \beta_{3} + 640 \beta_{4} + 16 \beta_{5} ) q^{68} + ( 360 - 7 \beta_{1} + 102 \beta_{2} - 47 \beta_{3} + 434 \beta_{4} + 23 \beta_{5} ) q^{70} + ( -66 + 120 \beta_{2} + 12 \beta_{3} ) q^{71} + ( -363 - 101 \beta_{1} - 101 \beta_{2} + 363 \beta_{4} - 23 \beta_{5} ) q^{73} + ( 298 - 108 \beta_{1} - 108 \beta_{2} - 298 \beta_{4} - 19 \beta_{5} ) q^{74} + ( -718 + 186 \beta_{2} + 72 \beta_{3} ) q^{76} + ( -410 + 25 \beta_{1} - 24 \beta_{2} + 5 \beta_{3} + 438 \beta_{4} - 45 \beta_{5} ) q^{77} + ( 36 \beta_{1} + 48 \beta_{3} - 299 \beta_{4} - 48 \beta_{5} ) q^{79} + ( 18 - 121 \beta_{1} - 121 \beta_{2} - 18 \beta_{4} - 51 \beta_{5} ) q^{80} + ( 32 \beta_{1} - 2 \beta_{3} - 52 \beta_{4} + 2 \beta_{5} ) q^{82} + ( -156 + 51 \beta_{2} - 27 \beta_{3} ) q^{83} + ( 624 + 72 \beta_{2} ) q^{85} + ( 50 \beta_{1} + 69 \beta_{3} - 1086 \beta_{4} - 69 \beta_{5} ) q^{86} + ( 258 - 117 \beta_{1} - 117 \beta_{2} - 258 \beta_{4} - 3 \beta_{5} ) q^{88} + ( 170 \beta_{1} - 22 \beta_{3} + 532 \beta_{4} + 22 \beta_{5} ) q^{89} + ( 18 - 49 \beta_{1} - 74 \beta_{2} - 23 \beta_{3} - 287 \beta_{4} - 23 \beta_{5} ) q^{91} + ( 112 - 336 \beta_{2} + 56 \beta_{3} ) q^{92} + ( 984 - 342 \beta_{1} - 342 \beta_{2} - 984 \beta_{4} - 72 \beta_{5} ) q^{94} + ( -246 - 2 \beta_{1} - 2 \beta_{2} + 246 \beta_{4} + 54 \beta_{5} ) q^{95} + ( 24 + 53 \beta_{2} - 49 \beta_{3} ) q^{97} + ( 724 - 313 \beta_{1} - 157 \beta_{2} - 35 \beta_{3} - 26 \beta_{4} - 11 \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + q^{2} - 25q^{4} + 11q^{5} - 13q^{7} - 78q^{8} + O(q^{10}) \) \( 6q + q^{2} - 25q^{4} + 11q^{5} - 13q^{7} - 78q^{8} + 55q^{10} + 35q^{11} + 124q^{13} + 326q^{14} - 241q^{16} + 48q^{17} + 202q^{19} - 878q^{20} - 14q^{22} + 216q^{23} - 130q^{25} + 274q^{26} - 201q^{28} - 106q^{29} + 95q^{31} + 683q^{32} - 48q^{34} - 56q^{35} - 262q^{37} - 398q^{38} - 21q^{40} - 488q^{41} + 720q^{43} - 905q^{44} + 1056q^{46} - 210q^{47} - 303q^{49} + 2756q^{50} - 324q^{52} + 393q^{53} - 2062q^{55} - 1299q^{56} + 1249q^{58} + 1143q^{59} + 70q^{61} - 2118q^{62} - 798q^{64} - 472q^{65} + 628q^{67} + 1944q^{68} + 3251q^{70} - 636q^{71} - 988q^{73} + 1002q^{74} - 4680q^{76} - 1073q^{77} - 861q^{79} + 175q^{80} - 124q^{82} - 1038q^{83} + 3600q^{85} - 3208q^{86} + 891q^{88} + 1766q^{89} - 654q^{91} + 1344q^{92} + 3294q^{94} - 736q^{95} + 38q^{97} + 4267q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 25 x^{4} + 12 x^{3} + 582 x^{2} - 144 x + 36\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 25 \nu^{4} + 625 \nu^{3} - 582 \nu^{2} + 144 \nu - 3600 \)\()/14406\)
\(\beta_{3}\)\(=\)\((\)\( -25 \nu^{5} + 625 \nu^{4} - 1219 \nu^{3} + 14550 \nu^{2} - 3600 \nu + 234060 \)\()/14406\)
\(\beta_{4}\)\(=\)\((\)\( 100 \nu^{5} - 99 \nu^{4} + 2475 \nu^{3} + 1825 \nu^{2} + 57618 \nu + 150 \)\()/14406\)
\(\beta_{5}\)\(=\)\((\)\( -1601 \nu^{5} + 1609 \nu^{4} - 40225 \nu^{3} - 14212 \nu^{2} - 936438 \nu + 231696 \)\()/14406\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} + 16 \beta_{4} + \beta_{2} + \beta_{1} - 16\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 25 \beta_{2} - 10\)
\(\nu^{4}\)\(=\)\(-25 \beta_{5} - 394 \beta_{4} + 25 \beta_{3} - 43 \beta_{1}\)
\(\nu^{5}\)\(=\)\(-43 \beta_{5} - 538 \beta_{4} - 637 \beta_{2} - 637 \beta_{1} + 538\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-\beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−2.27818 + 3.94593i
0.124036 0.214837i
2.65415 4.59712i
−2.27818 3.94593i
0.124036 + 0.214837i
2.65415 + 4.59712i
−2.27818 + 3.94593i 0 −6.38024 11.0509i 8.93660 15.4786i 0 2.26047 18.3818i 21.6905 0 40.7184 + 70.5264i
37.2 0.124036 0.214837i 0 3.96923 + 6.87491i −6.21730 + 10.7687i 0 −18.4385 1.73873i 3.95388 0 1.54234 + 2.67141i
37.3 2.65415 4.59712i 0 −10.0890 17.4746i 2.78070 4.81631i 0 9.67799 + 15.7904i −64.6443 0 −14.7608 25.5664i
46.1 −2.27818 3.94593i 0 −6.38024 + 11.0509i 8.93660 + 15.4786i 0 2.26047 + 18.3818i 21.6905 0 40.7184 70.5264i
46.2 0.124036 + 0.214837i 0 3.96923 6.87491i −6.21730 10.7687i 0 −18.4385 + 1.73873i 3.95388 0 1.54234 2.67141i
46.3 2.65415 + 4.59712i 0 −10.0890 + 17.4746i 2.78070 + 4.81631i 0 9.67799 15.7904i −64.6443 0 −14.7608 + 25.5664i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.4.e.c 6
3.b odd 2 1 21.4.e.b 6
7.b odd 2 1 441.4.e.w 6
7.c even 3 1 inner 63.4.e.c 6
7.c even 3 1 441.4.a.s 3
7.d odd 6 1 441.4.a.t 3
7.d odd 6 1 441.4.e.w 6
12.b even 2 1 336.4.q.k 6
21.c even 2 1 147.4.e.n 6
21.g even 6 1 147.4.a.m 3
21.g even 6 1 147.4.e.n 6
21.h odd 6 1 21.4.e.b 6
21.h odd 6 1 147.4.a.l 3
84.j odd 6 1 2352.4.a.cg 3
84.n even 6 1 336.4.q.k 6
84.n even 6 1 2352.4.a.ci 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.b 6 3.b odd 2 1
21.4.e.b 6 21.h odd 6 1
63.4.e.c 6 1.a even 1 1 trivial
63.4.e.c 6 7.c even 3 1 inner
147.4.a.l 3 21.h odd 6 1
147.4.a.m 3 21.g even 6 1
147.4.e.n 6 21.c even 2 1
147.4.e.n 6 21.g even 6 1
336.4.q.k 6 12.b even 2 1
336.4.q.k 6 84.n even 6 1
441.4.a.s 3 7.c even 3 1
441.4.a.t 3 7.d odd 6 1
441.4.e.w 6 7.b odd 2 1
441.4.e.w 6 7.d odd 6 1
2352.4.a.cg 3 84.j odd 6 1
2352.4.a.ci 3 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} + 25 T_{2}^{4} + 12 T_{2}^{3} + 582 T_{2}^{2} - 144 T_{2} + 36 \) acting on \(S_{4}^{\mathrm{new}}(63, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} + 20 T^{3} - 10 T^{4} - 64 T^{5} + 1060 T^{6} - 512 T^{7} - 640 T^{8} + 10240 T^{9} + 4096 T^{10} - 32768 T^{11} + 262144 T^{12} \)
$3$ \( \)
$5$ \( 1 - 11 T - 62 T^{2} + 1015 T^{3} - 6040 T^{4} + 54313 T^{5} + 121696 T^{6} + 6789125 T^{7} - 94375000 T^{8} + 1982421875 T^{9} - 15136718750 T^{10} - 335693359375 T^{11} + 3814697265625 T^{12} \)
$7$ \( 1 + 13 T + 236 T^{2} + 12145 T^{3} + 80948 T^{4} + 1529437 T^{5} + 40353607 T^{6} \)
$11$ \( 1 - 35 T - 1400 T^{2} + 113593 T^{3} - 198940 T^{4} - 87110135 T^{5} + 3928586038 T^{6} - 115943589685 T^{7} - 352434345340 T^{8} + 267846352063763 T^{9} - 4393799727409400 T^{10} - 146203685929547785 T^{11} + 5559917313492231481 T^{12} \)
$13$ \( ( 1 - 62 T + 7016 T^{2} - 253976 T^{3} + 15414152 T^{4} - 299262158 T^{5} + 10604499373 T^{6} )^{2} \)
$17$ \( 1 - 48 T - 10035 T^{2} + 125232 T^{3} + 74409318 T^{4} + 234420432 T^{5} - 437742983351 T^{6} + 1151707582416 T^{7} + 1796060047467942 T^{8} + 14850996949472304 T^{9} - 5846614150600651635 T^{10} - \)\(13\!\cdots\!64\)\( T^{11} + \)\(14\!\cdots\!09\)\( T^{12} \)
$19$ \( 1 - 202 T + 7946 T^{2} - 627636 T^{3} + 247297462 T^{4} - 17185599794 T^{5} + 349471935958 T^{6} - 117876028987046 T^{7} + 11634326968854022 T^{8} - 202530415883220444 T^{9} + 17587000346899715306 T^{10} - \)\(30\!\cdots\!98\)\( T^{11} + \)\(10\!\cdots\!41\)\( T^{12} \)
$23$ \( 1 - 216 T + 10827 T^{2} - 387864 T^{3} + 53856198 T^{4} + 24653558952 T^{5} - 5413409425505 T^{6} + 299959851768984 T^{7} + 7972650149090022 T^{8} - 698602275885685032 T^{9} + \)\(23\!\cdots\!67\)\( T^{10} - \)\(57\!\cdots\!12\)\( T^{11} + \)\(32\!\cdots\!69\)\( T^{12} \)
$29$ \( ( 1 + 53 T + 52695 T^{2} + 3410210 T^{3} + 1285178355 T^{4} + 31525636013 T^{5} + 14507145975869 T^{6} )^{2} \)
$31$ \( 1 - 95 T - 70347 T^{2} + 3756594 T^{3} + 3398738767 T^{4} - 83374434539 T^{5} - 110906046363338 T^{6} - 2483807779351349 T^{7} + 3016393166469901327 T^{8} + 99322925971043714574 T^{9} - \)\(55\!\cdots\!67\)\( T^{10} - \)\(22\!\cdots\!45\)\( T^{11} + \)\(69\!\cdots\!41\)\( T^{12} \)
$37$ \( 1 + 262 T - 97404 T^{2} - 9678072 T^{3} + 12194182072 T^{4} + 680381910454 T^{5} - 605701122868778 T^{6} + 34463384910226462 T^{7} + 31286934978284739448 T^{8} - \)\(12\!\cdots\!44\)\( T^{9} - \)\(64\!\cdots\!24\)\( T^{10} + \)\(87\!\cdots\!66\)\( T^{11} + \)\(16\!\cdots\!29\)\( T^{12} \)
$41$ \( ( 1 + 244 T + 187983 T^{2} + 33933832 T^{3} + 12955976343 T^{4} + 1159025434804 T^{5} + 327381934393961 T^{6} )^{2} \)
$43$ \( ( 1 - 360 T + 166158 T^{2} - 38975294 T^{3} + 13210724106 T^{4} - 2275690697640 T^{5} + 502592611936843 T^{6} )^{2} \)
$47$ \( 1 + 210 T - 20853 T^{2} - 83809446 T^{3} - 12756928590 T^{4} + 2596137940074 T^{5} + 3698984470026571 T^{6} + 269538829352302902 T^{7} - \)\(13\!\cdots\!10\)\( T^{8} - \)\(93\!\cdots\!82\)\( T^{9} - \)\(24\!\cdots\!73\)\( T^{10} + \)\(25\!\cdots\!30\)\( T^{11} + \)\(12\!\cdots\!89\)\( T^{12} \)
$53$ \( 1 - 393 T - 211446 T^{2} + 23899125 T^{3} + 46453564620 T^{4} + 3425920762143 T^{5} - 9724787230272680 T^{6} + 510040805305563411 T^{7} + \)\(10\!\cdots\!80\)\( T^{8} + \)\(78\!\cdots\!25\)\( T^{9} - \)\(10\!\cdots\!86\)\( T^{10} - \)\(28\!\cdots\!01\)\( T^{11} + \)\(10\!\cdots\!89\)\( T^{12} \)
$59$ \( 1 - 1143 T + 557208 T^{2} - 118327563 T^{3} - 14314666608 T^{4} + 27063102119841 T^{5} - 16891447327378130 T^{6} + 5558192850270824739 T^{7} - \)\(60\!\cdots\!28\)\( T^{8} - \)\(10\!\cdots\!57\)\( T^{9} + \)\(99\!\cdots\!48\)\( T^{10} - \)\(41\!\cdots\!57\)\( T^{11} + \)\(75\!\cdots\!21\)\( T^{12} \)
$61$ \( 1 - 70 T - 335143 T^{2} - 129510330 T^{3} + 42145697866 T^{4} + 25171752927730 T^{5} - 316289217432887 T^{6} + 5713509651289083130 T^{7} + \)\(21\!\cdots\!26\)\( T^{8} - \)\(15\!\cdots\!30\)\( T^{9} - \)\(88\!\cdots\!03\)\( T^{10} - \)\(42\!\cdots\!70\)\( T^{11} + \)\(13\!\cdots\!81\)\( T^{12} \)
$67$ \( 1 - 628 T - 202942 T^{2} + 436381932 T^{3} - 77667044702 T^{4} - 73528811914784 T^{5} + 76060129771959310 T^{6} - 22114746057926180192 T^{7} - \)\(70\!\cdots\!38\)\( T^{8} + \)\(11\!\cdots\!04\)\( T^{9} - \)\(16\!\cdots\!62\)\( T^{10} - \)\(15\!\cdots\!04\)\( T^{11} + \)\(74\!\cdots\!09\)\( T^{12} \)
$71$ \( ( 1 + 318 T + 742929 T^{2} + 256167372 T^{3} + 265902461319 T^{4} + 40735890286878 T^{5} + 45848500718449031 T^{6} )^{2} \)
$73$ \( 1 + 988 T - 186552 T^{2} - 102237300 T^{3} + 281568890272 T^{4} - 16988127696596 T^{5} - 164639785652996186 T^{6} - 6608670472146686132 T^{7} + \)\(42\!\cdots\!08\)\( T^{8} - \)\(60\!\cdots\!00\)\( T^{9} - \)\(42\!\cdots\!92\)\( T^{10} + \)\(88\!\cdots\!16\)\( T^{11} + \)\(34\!\cdots\!69\)\( T^{12} \)
$79$ \( 1 + 861 T - 479895 T^{2} - 258646666 T^{3} + 325257480351 T^{4} - 27564282842211 T^{5} - 246706047980056146 T^{6} - 13590266448240869229 T^{7} + \)\(79\!\cdots\!71\)\( T^{8} - \)\(30\!\cdots\!54\)\( T^{9} - \)\(28\!\cdots\!95\)\( T^{10} + \)\(25\!\cdots\!39\)\( T^{11} + \)\(14\!\cdots\!61\)\( T^{12} \)
$83$ \( ( 1 + 519 T + 1583745 T^{2} + 545598870 T^{3} + 905564802315 T^{4} + 169682053778511 T^{5} + 186940255267540403 T^{6} )^{2} \)
$89$ \( 1 - 1766 T + 725929 T^{2} + 728159446 T^{3} - 335534377858 T^{4} - 846551335831238 T^{5} + 1249625385561159997 T^{6} - \)\(59\!\cdots\!22\)\( T^{7} - \)\(16\!\cdots\!38\)\( T^{8} + \)\(25\!\cdots\!14\)\( T^{9} + \)\(17\!\cdots\!09\)\( T^{10} - \)\(30\!\cdots\!34\)\( T^{11} + \)\(12\!\cdots\!81\)\( T^{12} \)
$97$ \( ( 1 - 19 T + 2168419 T^{2} + 10094878 T^{3} + 1979057473987 T^{4} - 15826468093651 T^{5} + 760231058654565217 T^{6} )^{2} \)
show more
show less