Properties

Label 25.9.c.c
Level $25$
Weight $9$
Character orbit 25.c
Analytic conductor $10.184$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 25.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.1844652515\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} + 8 x^{10} + 124 x^{9} + 1665 x^{8} - 2456 x^{7} + 4192 x^{6} + 50576 x^{5} + 221184 x^{4} + 133760 x^{3} + 3200 x^{2} - 80000 x + 1000000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{6}\cdot 5^{12} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( -\beta_{5} - \beta_{6} - \beta_{7} ) q^{3} + ( 281 \beta_{1} + \beta_{11} ) q^{4} + ( -533 + 7 \beta_{8} - \beta_{9} ) q^{6} + ( -42 \beta_{2} + 18 \beta_{3} + 4 \beta_{4} ) q^{7} + ( -316 \beta_{5} + 27 \beta_{6} + 60 \beta_{7} ) q^{8} + ( -1586 \beta_{1} - 22 \beta_{10} - 4 \beta_{11} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( -\beta_{5} - \beta_{6} - \beta_{7} ) q^{3} + ( 281 \beta_{1} + \beta_{11} ) q^{4} + ( -533 + 7 \beta_{8} - \beta_{9} ) q^{6} + ( -42 \beta_{2} + 18 \beta_{3} + 4 \beta_{4} ) q^{7} + ( -316 \beta_{5} + 27 \beta_{6} + 60 \beta_{7} ) q^{8} + ( -1586 \beta_{1} - 22 \beta_{10} - 4 \beta_{11} ) q^{9} + ( -9843 + 21 \beta_{8} - 22 \beta_{9} ) q^{11} + ( 140 \beta_{2} - 233 \beta_{3} + 812 \beta_{4} ) q^{12} + ( 906 \beta_{5} - 270 \beta_{6} + 258 \beta_{7} ) q^{13} + ( 24132 \beta_{1} + 60 \beta_{10} + 28 \beta_{11} ) q^{14} + ( -100339 - 288 \beta_{8} + 147 \beta_{9} ) q^{16} + ( -134 \beta_{2} + 1134 \beta_{3} + 1149 \beta_{4} ) q^{17} + ( 3234 \beta_{5} + 1344 \beta_{6} + 2928 \beta_{7} ) q^{18} + ( -14875 \beta_{1} + 369 \beta_{10} - 214 \beta_{11} ) q^{19} + ( -113788 + 410 \beta_{8} - 84 \beta_{9} ) q^{21} + ( 3903 \beta_{2} - 1980 \beta_{3} + 4344 \beta_{4} ) q^{22} + ( -7822 \beta_{5} - 3294 \beta_{6} - 2154 \beta_{7} ) q^{23} + ( 105625 \beta_{1} - 1576 \beta_{10} + 649 \beta_{11} ) q^{24} + ( 448002 + 306 \beta_{8} - 918 \beta_{9} ) q^{26} + ( -23127 \beta_{2} - 2769 \beta_{3} - 6345 \beta_{4} ) q^{27} + ( -22848 \beta_{5} + 1404 \beta_{6} - 7984 \beta_{7} ) q^{28} + ( -401550 \beta_{1} + 978 \beta_{10} - 740 \beta_{11} ) q^{29} + ( 268642 + 2358 \beta_{8} + 1244 \beta_{9} ) q^{31} + ( 55884 \beta_{2} + 16065 \beta_{3} - 34932 \beta_{4} ) q^{32} + ( 40308 \beta_{5} + 13188 \beta_{6} + 6939 \beta_{7} ) q^{33} + ( 238647 \beta_{1} + 1089 \beta_{10} + 149 \beta_{11} ) q^{34} + ( 1206466 - 8528 \beta_{8} + 2062 \beta_{9} ) q^{36} + ( -85098 \beta_{2} - 14094 \beta_{3} - 20464 \beta_{4} ) q^{37} + ( 69031 \beta_{5} - 30132 \beta_{6} - 65976 \beta_{7} ) q^{38} + ( -477282 \beta_{1} + 5988 \beta_{10} + 912 \beta_{11} ) q^{39} + ( 911697 - 2712 \beta_{8} - 4016 \beta_{9} ) q^{41} + ( 98364 \beta_{2} - 29328 \beta_{3} + 64080 \beta_{4} ) q^{42} + ( -7998 \beta_{5} + 4698 \beta_{6} + 64228 \beta_{7} ) q^{43} + ( 609117 \beta_{1} - 15576 \beta_{10} - 3211 \beta_{11} ) q^{44} + ( -4272738 + 19638 \beta_{8} + 2374 \beta_{9} ) q^{46} + ( 58680 \beta_{2} + 50112 \beta_{3} - 2382 \beta_{4} ) q^{47} + ( -295652 \beta_{5} + 61891 \beta_{6} + 58012 \beta_{7} ) q^{48} + ( -3012049 \beta_{1} + 4464 \beta_{10} - 2688 \beta_{11} ) q^{49} + ( -3968473 + 1967 \beta_{8} - 8162 \beta_{9} ) q^{51} + ( -476472 \beta_{2} + 24138 \beta_{3} + 165192 \beta_{4} ) q^{52} + ( -139196 \beta_{5} - 45036 \beta_{6} + 125742 \beta_{7} ) q^{53} + ( 11746725 \beta_{1} + 7959 \beta_{10} + 19551 \beta_{11} ) q^{54} + ( -5393100 + 2976 \beta_{8} + 9100 \beta_{9} ) q^{56} + ( 496106 \beta_{2} - 35618 \beta_{3} + 25733 \beta_{4} ) q^{57} + ( 595374 \beta_{5} - 84528 \beta_{6} - 185232 \beta_{7} ) q^{58} + ( -4117950 \beta_{1} - 28980 \beta_{10} + 9312 \beta_{11} ) q^{59} + ( -1966468 - 24750 \beta_{8} + 5500 \beta_{9} ) q^{61} + ( 145238 \beta_{2} - 122040 \beta_{3} + 264912 \beta_{4} ) q^{62} + ( 384300 \beta_{5} + 44412 \beta_{6} + 266808 \beta_{7} ) q^{63} + ( -5802209 \beta_{1} + 95328 \beta_{10} - 69249 \beta_{11} ) q^{64} + ( 22061319 - 73569 \beta_{8} - 20181 \beta_{9} ) q^{66} + ( -286419 \beta_{2} + 78939 \beta_{3} - 614361 \beta_{4} ) q^{67} + ( -271660 \beta_{5} + 222453 \beta_{6} - 442020 \beta_{7} ) q^{68} + ( -26884142 \beta_{1} - 96184 \beta_{10} - 1808 \beta_{11} ) q^{69} + ( -3989238 + 3000 \beta_{8} + 77000 \beta_{9} ) q^{71} + ( 33864 \beta_{2} + 274458 \beta_{3} - 602184 \beta_{4} ) q^{72} + ( -603222 \beta_{5} - 125694 \beta_{6} + 80197 \beta_{7} ) q^{73} + ( 43162152 \beta_{1} + 5016 \beta_{10} + 78728 \beta_{11} ) q^{74} + ( 36070475 + 223992 \beta_{8} - 110355 \beta_{9} ) q^{76} + ( 431466 \beta_{2} - 359154 \beta_{3} - 13788 \beta_{4} ) q^{77} + ( 80154 \beta_{5} - 370584 \beta_{6} - 807552 \beta_{7} ) q^{78} + ( -2874200 \beta_{1} + 94914 \beta_{10} + 62668 \beta_{11} ) q^{79} + ( 250671 + 39810 \beta_{8} - 8532 \beta_{9} ) q^{81} + ( -2140017 \beta_{2} + 70560 \beta_{3} - 149568 \beta_{4} ) q^{82} + ( -1479865 \beta_{5} + 204687 \beta_{6} - 91563 \beta_{7} ) q^{83} + ( -20968028 \beta_{1} - 204592 \beta_{10} - 26460 \beta_{11} ) q^{84} + ( -8778468 - 211476 \beta_{8} + 76924 \beta_{9} ) q^{86} + ( 977124 \beta_{2} + 259428 \beta_{3} - 385686 \beta_{4} ) q^{87} + ( -331212 \beta_{5} + 434439 \beta_{6} + 938220 \beta_{7} ) q^{88} + ( -10559775 \beta_{1} - 12066 \beta_{10} - 162700 \beta_{11} ) q^{89} + ( -5353128 - 64404 \beta_{8} - 99960 \beta_{9} ) q^{91} + ( 3393176 \beta_{2} - 388746 \beta_{3} + 2134008 \beta_{4} ) q^{92} + ( 2628428 \beta_{5} - 71332 \beta_{6} + 1377566 \beta_{7} ) q^{93} + ( -28131858 \beta_{1} + 207594 \beta_{10} - 111174 \beta_{11} ) q^{94} + ( -131681763 - 18144 \beta_{8} + 249411 \beta_{9} ) q^{96} + ( -2576040 \beta_{2} - 21528 \beta_{3} + 1318124 \beta_{4} ) q^{97} + ( 3696049 \beta_{5} - 367200 \beta_{6} - 804096 \beta_{7} ) q^{98} + ( 44350998 \beta_{1} + 298452 \beta_{10} + 132480 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 6396q^{6} + O(q^{10}) \) \( 12q - 6396q^{6} - 118116q^{11} - 1204068q^{16} - 1365456q^{21} + 5376024q^{26} + 3223704q^{31} + 14477592q^{36} + 10940364q^{41} - 51272856q^{46} - 47621676q^{51} - 64717200q^{56} - 23597616q^{61} + 264735828q^{66} - 47870856q^{71} + 432845700q^{76} + 3008052q^{81} - 105341616q^{86} - 64237536q^{91} - 1580181156q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 4 x^{11} + 8 x^{10} + 124 x^{9} + 1665 x^{8} - 2456 x^{7} + 4192 x^{6} + 50576 x^{5} + 221184 x^{4} + 133760 x^{3} + 3200 x^{2} - 80000 x + 1000000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-449334647 \nu^{11} + 12061467443 \nu^{10} - 45928215186 \nu^{9} + 33350532992 \nu^{8} + 485787845405 \nu^{7} + 18307600018527 \nu^{6} - 29743835378214 \nu^{5} + 25122958585108 \nu^{4} + 391795974445792 \nu^{3} + 2619570367020680 \nu^{2} + 742617488427600 \nu - 93665468020000\)\()/ 5581520471420000 \)
\(\beta_{2}\)\(=\)\((\)\(-381080397548 \nu^{11} + 6650619527307 \nu^{10} - 37023789923414 \nu^{9} + 65801651136208 \nu^{8} - 171099951179440 \nu^{7} + 7803983586776523 \nu^{6} - 33522577801375186 \nu^{5} + 64053508653668192 \nu^{4} + 77705746695051688 \nu^{3} + 314446245222974520 \nu^{2} - 586491243793535600 \nu + 893767035735100000\)\()/ 8372280707130000 \)
\(\beta_{3}\)\(=\)\((\)\(-435927172721 \nu^{11} + 7533818317764 \nu^{10} - 43325832090128 \nu^{9} + 78139416210016 \nu^{8} - 211070404447705 \nu^{7} + 8675411389531896 \nu^{6} - 40733398019692072 \nu^{5} + 74064999620016284 \nu^{4} + 82048639403653576 \nu^{3} + 309514999935977040 \nu^{2} - 1196810894658951200 \nu + 922317979047700000\)\()/ 8372280707130000 \)
\(\beta_{4}\)\(=\)\((\)\(-207349351901 \nu^{11} + 3608923885464 \nu^{10} - 20095877149928 \nu^{9} + 35763821961616 \nu^{8} - 94461295798645 \nu^{7} + 4231435047360396 \nu^{6} - 18220487011478272 \nu^{5} + 34974606965024684 \nu^{4} + 41694360232978696 \nu^{3} + 166630693079745840 \nu^{2} - 355782590076255200 \nu + 477584608191700000\)\()/ 1339564913140800 \)
\(\beta_{5}\)\(=\)\((\)\(-2727925523508 \nu^{11} + 14877294336467 \nu^{10} - 35907038981134 \nu^{9} - 337439888728952 \nu^{8} - 3875122781812200 \nu^{7} + 12988417959142963 \nu^{6} - 20344818011353466 \nu^{5} - 157225538636164648 \nu^{4} - 241887606416273592 \nu^{3} + 211819995842916920 \nu^{2} + 173135939040904400 \nu - 1133723541093860000\)\()/ 8372280707130000 \)
\(\beta_{6}\)\(=\)\((\)\(4175127571641 \nu^{11} - 21153383834384 \nu^{10} + 49390928549668 \nu^{9} + 514100031045104 \nu^{8} + 6217465685948625 \nu^{7} - 17351460615923476 \nu^{6} + 27648898660009532 \nu^{5} + 230071439819114596 \nu^{4} + 526070134489984584 \nu^{3} - 123042970649069840 \nu^{2} - 198729082677228800 \nu + 1247625114598220000\)\()/ 8372280707130000 \)
\(\beta_{7}\)\(=\)\((\)\(-1550015596341 \nu^{11} + 8356009540724 \nu^{10} - 20060042759548 \nu^{9} - 191658965219744 \nu^{8} - 2220383154537645 \nu^{7} + 7230531059293336 \nu^{6} - 11343535848559652 \nu^{5} - 88917144546491956 \nu^{4} - 149243933399361864 \nu^{3} + 106049680478950640 \nu^{2} + 93910105577124800 \nu - 610883057460620000\)\()/ 1339564913140800 \)
\(\beta_{8}\)\(=\)\((\)\(572380746383 \nu^{11} - 2036265285124 \nu^{10} + 181744291018 \nu^{9} + 95492643662604 \nu^{8} + 909244071727231 \nu^{7} - 1288692597374696 \nu^{6} - 2746934977689118 \nu^{5} + 51157179993258096 \nu^{4} + 83754911423788736 \nu^{3} + 26376040044011120 \nu^{2} - 196135089198878000 \nu + 383974712446124000\)\()/ 111630409428400 \)
\(\beta_{9}\)\(=\)\((\)\(39828320474 \nu^{11} - 141775272712 \nu^{10} + 13469724484 \nu^{9} + 6653307104727 \nu^{8} + 63244284483538 \nu^{7} - 89744414904848 \nu^{6} - 188808282377284 \nu^{5} + 3580518768001248 \nu^{4} + 5851971198380288 \nu^{3} + 1840584763196960 \nu^{2} - 13691088598724000 \nu + 29397487192378875\)\()/ 6976900589275 \)
\(\beta_{10}\)\(=\)\((\)\(3672375956607 \nu^{11} - 24100275366453 \nu^{10} + 80120961677856 \nu^{9} + 333658706564268 \nu^{8} + 4949193516916155 \nu^{7} - 22488461054564617 \nu^{6} + 59286542830870044 \nu^{5} + 116931082369255632 \nu^{4} + 270676000135919688 \nu^{3} - 409047550686121080 \nu^{2} + 635765474055926400 \nu + 271880958159720000\)\()/ 558152047142000 \)
\(\beta_{11}\)\(=\)\((\)\(8224243606507 \nu^{11} - 54768226676503 \nu^{10} + 182306064041706 \nu^{9} + 741572009869568 \nu^{8} + 10989507276185255 \nu^{7} - 51744355689454067 \nu^{6} + 134211765950123694 \nu^{5} + 258903920872190332 \nu^{4} + 575827073848470688 \nu^{3} - 1105783357863599080 \nu^{2} + 1364799133894076400 \nu + 612803755717220000\)\()/ 1116304094284000 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(2 \beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{8} + 12 \beta_{4} + 100 \beta_{1} + 100\)\()/300\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{11} - 6 \beta_{10} + 15 \beta_{6} - 45 \beta_{5} - 15 \beta_{3} - 45 \beta_{2} + 2850 \beta_{1}\)\()/150\)
\(\nu^{3}\)\(=\)\((\)\(17 \beta_{11} - 11 \beta_{10} + 17 \beta_{9} - 11 \beta_{8} + 144 \beta_{7} - 225 \beta_{5} + 2525 \beta_{1} - 2525\)\()/75\)
\(\nu^{4}\)\(=\)\((\)\(73 \beta_{9} - 144 \beta_{8} + 264 \beta_{7} + 390 \beta_{6} - 1470 \beta_{5} - 264 \beta_{4} + 390 \beta_{3} + 1470 \beta_{2} - 54825\)\()/75\)
\(\nu^{5}\)\(=\)\((\)\(-747 \beta_{11} + 586 \beta_{10} + 747 \beta_{9} - 586 \beta_{8} - 7212 \beta_{4} + 1125 \beta_{3} + 15000 \beta_{2} - 163975 \beta_{1} - 163975\)\()/75\)
\(\nu^{6}\)\(=\)\((\)\(-1583 \beta_{11} + 2424 \beta_{10} - 7320 \beta_{7} - 6210 \beta_{6} + 28380 \beta_{5} - 7320 \beta_{4} + 6210 \beta_{3} + 28380 \beta_{2} - 865775 \beta_{1}\)\()/25\)
\(\nu^{7}\)\(=\)\((\)\(-36613 \beta_{11} + 33104 \beta_{10} - 36613 \beta_{9} + 33104 \beta_{8} - 367476 \beta_{7} - 99225 \beta_{6} + 875700 \beta_{5} - 9996725 \beta_{1} + 9996725\)\()/75\)
\(\nu^{8}\)\(=\)\((\)\(-290297 \beta_{9} + 386016 \beta_{8} - 1424736 \beta_{7} - 920760 \beta_{6} + 4776480 \beta_{5} + 1424736 \beta_{4} - 920760 \beta_{3} - 4776480 \beta_{2} + 133078425\)\()/75\)
\(\nu^{9}\)\(=\)\((\)\(1898923 \beta_{11} - 1876624 \beta_{10} - 1898923 \beta_{9} + 1876624 \beta_{8} + 19240308 \beta_{4} - 6627825 \beta_{3} - 49553100 \beta_{2} + 588239275 \beta_{1} + 588239275\)\()/75\)
\(\nu^{10}\)\(=\)\((\)\(17056301 \beta_{11} - 20989728 \beta_{10} + 85237080 \beta_{7} + 47378670 \beta_{6} - 265898760 \beta_{5} + 85237080 \beta_{4} - 47378670 \beta_{3} - 265898760 \beta_{2} + 7082438925 \beta_{1}\)\()/75\)
\(\nu^{11}\)\(=\)\((\)\(101596857 \beta_{11} - 105815456 \beta_{10} + 101596857 \beta_{9} - 105815456 \beta_{8} + 1030566804 \beta_{7} + 402640425 \beta_{6} - 2776867500 \beta_{5} + 33828064025 \beta_{1} - 33828064025\)\()/75\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
2.82685 + 2.82685i
0.908990 + 0.908990i
−1.51110 1.51110i
−3.96059 3.96059i
−1.54050 1.54050i
5.27634 + 5.27634i
2.82685 2.82685i
0.908990 0.908990i
−1.51110 + 1.51110i
−3.96059 + 3.96059i
−1.54050 + 1.54050i
5.27634 5.27634i
−21.8383 21.8383i −42.5089 + 42.5089i 697.824i 0 1856.64 −432.570 432.570i 9648.70 9648.70i 2946.99i 0
7.2 −18.1270 18.1270i 95.3921 95.3921i 401.173i 0 −3458.34 −1478.49 1478.49i 2631.55 2631.55i 11638.3i 0
7.3 −0.0371264 0.0371264i −36.2471 + 36.2471i 255.997i 0 2.69145 1325.17 + 1325.17i −19.0086 + 19.0086i 3933.30i 0
7.4 0.0371264 + 0.0371264i 36.2471 36.2471i 255.997i 0 2.69145 −1325.17 1325.17i 19.0086 19.0086i 3933.30i 0
7.5 18.1270 + 18.1270i −95.3921 + 95.3921i 401.173i 0 −3458.34 1478.49 + 1478.49i −2631.55 + 2631.55i 11638.3i 0
7.6 21.8383 + 21.8383i 42.5089 42.5089i 697.824i 0 1856.64 432.570 + 432.570i −9648.70 + 9648.70i 2946.99i 0
18.1 −21.8383 + 21.8383i −42.5089 42.5089i 697.824i 0 1856.64 −432.570 + 432.570i 9648.70 + 9648.70i 2946.99i 0
18.2 −18.1270 + 18.1270i 95.3921 + 95.3921i 401.173i 0 −3458.34 −1478.49 + 1478.49i 2631.55 + 2631.55i 11638.3i 0
18.3 −0.0371264 + 0.0371264i −36.2471 36.2471i 255.997i 0 2.69145 1325.17 1325.17i −19.0086 19.0086i 3933.30i 0
18.4 0.0371264 0.0371264i 36.2471 + 36.2471i 255.997i 0 2.69145 −1325.17 + 1325.17i 19.0086 + 19.0086i 3933.30i 0
18.5 18.1270 18.1270i −95.3921 95.3921i 401.173i 0 −3458.34 1478.49 1478.49i −2631.55 2631.55i 11638.3i 0
18.6 21.8383 21.8383i 42.5089 + 42.5089i 697.824i 0 1856.64 432.570 432.570i −9648.70 9648.70i 2946.99i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.9.c.c 12
5.b even 2 1 inner 25.9.c.c 12
5.c odd 4 2 inner 25.9.c.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.9.c.c 12 1.a even 1 1 trivial
25.9.c.c 12 5.b even 2 1 inner
25.9.c.c 12 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 1341657 T_{2}^{8} + 392912788608 T_{2}^{4} + 2985984 \) acting on \(S_{9}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 85209 T^{4} - 180773760 T^{8} - 192729711800320 T^{12} - 776417387174952960 T^{16} + \)\(15\!\cdots\!44\)\( T^{20} + \)\(79\!\cdots\!36\)\( T^{24} \)
$3$ \( 1 - 31969251 T^{4} + 3952987763669190 T^{8} - \)\(12\!\cdots\!95\)\( T^{12} + \)\(73\!\cdots\!90\)\( T^{16} - \)\(10\!\cdots\!31\)\( T^{20} + \)\(63\!\cdots\!21\)\( T^{24} \)
$5$ 1
$7$ \( 1 + 40557324189894 T^{4} + \)\(23\!\cdots\!15\)\( T^{8} + \)\(91\!\cdots\!80\)\( T^{12} + \)\(25\!\cdots\!15\)\( T^{16} + \)\(49\!\cdots\!94\)\( T^{20} + \)\(13\!\cdots\!01\)\( T^{24} \)
$11$ \( ( 1 + 29529 T + 761875590 T^{2} + 11079570629205 T^{3} + 163314798933614790 T^{4} + \)\(13\!\cdots\!69\)\( T^{5} + \)\(98\!\cdots\!41\)\( T^{6} )^{4} \)
$13$ \( 1 - 1511876723838358746 T^{4} + \)\(12\!\cdots\!15\)\( T^{8} - \)\(84\!\cdots\!20\)\( T^{12} + \)\(55\!\cdots\!15\)\( T^{16} - \)\(29\!\cdots\!06\)\( T^{20} + \)\(86\!\cdots\!41\)\( T^{24} \)
$17$ \( 1 + \)\(19\!\cdots\!89\)\( T^{4} + \)\(17\!\cdots\!90\)\( T^{8} + \)\(10\!\cdots\!05\)\( T^{12} + \)\(41\!\cdots\!90\)\( T^{16} + \)\(10\!\cdots\!69\)\( T^{20} + \)\(13\!\cdots\!81\)\( T^{24} \)
$19$ \( ( 1 - 42327890571 T^{2} + \)\(10\!\cdots\!90\)\( T^{4} - \)\(18\!\cdots\!95\)\( T^{6} + \)\(29\!\cdots\!90\)\( T^{8} - \)\(35\!\cdots\!31\)\( T^{10} + \)\(23\!\cdots\!41\)\( T^{12} )^{2} \)
$23$ \( 1 + \)\(10\!\cdots\!34\)\( T^{4} + \)\(92\!\cdots\!15\)\( T^{8} + \)\(79\!\cdots\!80\)\( T^{12} + \)\(34\!\cdots\!15\)\( T^{16} + \)\(14\!\cdots\!94\)\( T^{20} + \)\(53\!\cdots\!61\)\( T^{24} \)
$29$ \( ( 1 - 2002059089466 T^{2} + \)\(17\!\cdots\!15\)\( T^{4} - \)\(10\!\cdots\!20\)\( T^{6} + \)\(44\!\cdots\!15\)\( T^{8} - \)\(12\!\cdots\!06\)\( T^{10} + \)\(15\!\cdots\!61\)\( T^{12} )^{2} \)
$31$ \( ( 1 - 805926 T + 1543939264815 T^{2} - 886676176458176020 T^{3} + \)\(13\!\cdots\!15\)\( T^{4} - \)\(58\!\cdots\!06\)\( T^{5} + \)\(62\!\cdots\!21\)\( T^{6} )^{4} \)
$37$ \( 1 - \)\(62\!\cdots\!46\)\( T^{4} + \)\(17\!\cdots\!15\)\( T^{8} - \)\(27\!\cdots\!20\)\( T^{12} + \)\(26\!\cdots\!15\)\( T^{16} - \)\(14\!\cdots\!06\)\( T^{20} + \)\(35\!\cdots\!41\)\( T^{24} \)
$41$ \( ( 1 - 2735091 T + 21344589626790 T^{2} - 35893800635342135895 T^{3} + \)\(17\!\cdots\!90\)\( T^{4} - \)\(17\!\cdots\!31\)\( T^{5} + \)\(50\!\cdots\!61\)\( T^{6} )^{4} \)
$43$ \( 1 - \)\(10\!\cdots\!06\)\( T^{4} + \)\(53\!\cdots\!15\)\( T^{8} - \)\(38\!\cdots\!20\)\( T^{12} + \)\(10\!\cdots\!15\)\( T^{16} - \)\(37\!\cdots\!06\)\( T^{20} + \)\(65\!\cdots\!01\)\( T^{24} \)
$47$ \( 1 + \)\(59\!\cdots\!74\)\( T^{4} + \)\(50\!\cdots\!15\)\( T^{8} + \)\(14\!\cdots\!80\)\( T^{12} + \)\(16\!\cdots\!15\)\( T^{16} + \)\(61\!\cdots\!94\)\( T^{20} + \)\(33\!\cdots\!21\)\( T^{24} \)
$53$ \( 1 - \)\(26\!\cdots\!26\)\( T^{4} - \)\(78\!\cdots\!85\)\( T^{8} + \)\(87\!\cdots\!80\)\( T^{12} - \)\(11\!\cdots\!85\)\( T^{16} - \)\(59\!\cdots\!06\)\( T^{20} + \)\(33\!\cdots\!21\)\( T^{24} \)
$59$ \( ( 1 - 552614752247226 T^{2} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(26\!\cdots\!20\)\( T^{6} + \)\(32\!\cdots\!15\)\( T^{8} - \)\(25\!\cdots\!06\)\( T^{10} + \)\(10\!\cdots\!21\)\( T^{12} )^{2} \)
$61$ \( ( 1 + 5899404 T + 492347465676915 T^{2} + \)\(20\!\cdots\!80\)\( T^{3} + \)\(94\!\cdots\!15\)\( T^{4} + \)\(21\!\cdots\!44\)\( T^{5} + \)\(70\!\cdots\!41\)\( T^{6} )^{4} \)
$67$ \( 1 - \)\(14\!\cdots\!11\)\( T^{4} + \)\(44\!\cdots\!90\)\( T^{8} - \)\(29\!\cdots\!95\)\( T^{12} + \)\(12\!\cdots\!90\)\( T^{16} - \)\(11\!\cdots\!31\)\( T^{20} + \)\(20\!\cdots\!81\)\( T^{24} \)
$71$ \( ( 1 + 11967714 T + 567005178199215 T^{2} - \)\(39\!\cdots\!20\)\( T^{3} + \)\(36\!\cdots\!15\)\( T^{4} + \)\(49\!\cdots\!94\)\( T^{5} + \)\(26\!\cdots\!81\)\( T^{6} )^{4} \)
$73$ \( 1 + \)\(12\!\cdots\!09\)\( T^{4} + \)\(17\!\cdots\!90\)\( T^{8} + \)\(11\!\cdots\!05\)\( T^{12} + \)\(73\!\cdots\!90\)\( T^{16} + \)\(22\!\cdots\!69\)\( T^{20} + \)\(75\!\cdots\!61\)\( T^{24} \)
$79$ \( ( 1 - 4370568718882566 T^{2} + \)\(11\!\cdots\!15\)\( T^{4} - \)\(21\!\cdots\!20\)\( T^{6} + \)\(26\!\cdots\!15\)\( T^{8} - \)\(23\!\cdots\!06\)\( T^{10} + \)\(12\!\cdots\!61\)\( T^{12} )^{2} \)
$83$ \( 1 + \)\(23\!\cdots\!89\)\( T^{4} + \)\(35\!\cdots\!90\)\( T^{8} - \)\(11\!\cdots\!95\)\( T^{12} + \)\(92\!\cdots\!90\)\( T^{16} + \)\(15\!\cdots\!69\)\( T^{20} + \)\(17\!\cdots\!81\)\( T^{24} \)
$89$ \( ( 1 - 10560432796497411 T^{2} + \)\(76\!\cdots\!90\)\( T^{4} - \)\(35\!\cdots\!95\)\( T^{6} + \)\(11\!\cdots\!90\)\( T^{8} - \)\(25\!\cdots\!31\)\( T^{10} + \)\(37\!\cdots\!81\)\( T^{12} )^{2} \)
$97$ \( 1 + \)\(46\!\cdots\!74\)\( T^{4} - \)\(19\!\cdots\!85\)\( T^{8} - \)\(15\!\cdots\!20\)\( T^{12} - \)\(72\!\cdots\!85\)\( T^{16} + \)\(66\!\cdots\!94\)\( T^{20} + \)\(53\!\cdots\!21\)\( T^{24} \)
show more
show less