Properties

Label 160.6.n.a
Level 160
Weight 6
Character orbit 160.n
Analytic conductor 25.661
Analytic rank 0
Dimension 14
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 160.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.6614111701\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 4 x^{13} + 8 x^{12} - 4626 x^{11} + 149441 x^{10} - 2113414 x^{9} + 17958066 x^{8} - 97717112 x^{7} + 355171384 x^{6} - 910571904 x^{5} + 2428303248 x^{4} - 9166992192 x^{3} + 32237484304 x^{2} - 66916821408 x + 69451154208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{31}\cdot 5^{3} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} + \beta_{2} ) q^{3} + ( 3 - 4 \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} + ( -5 + 5 \beta_{1} - \beta_{6} - \beta_{8} ) q^{7} + ( 58 \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} + \beta_{11} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{1} + \beta_{2} ) q^{3} + ( 3 - 4 \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} + ( -5 + 5 \beta_{1} - \beta_{6} - \beta_{8} ) q^{7} + ( 58 \beta_{1} - \beta_{2} - \beta_{6} + \beta_{7} + \beta_{11} ) q^{9} + ( -36 \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{12} ) q^{11} + ( -30 + 31 \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{10} - \beta_{12} ) q^{13} + ( -19 + 163 \beta_{1} - 6 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - 5 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{10} + 3 \beta_{11} + \beta_{12} + \beta_{13} ) q^{15} + ( 90 + 90 \beta_{1} - 20 \beta_{2} - \beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} - 2 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} ) q^{17} + ( -402 - 4 \beta_{1} - 6 \beta_{2} + \beta_{3} + 4 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + \beta_{10} + 6 \beta_{13} ) q^{19} + ( 426 + \beta_{1} - 7 \beta_{2} + 6 \beta_{3} - \beta_{4} + 8 \beta_{5} + 12 \beta_{6} - 11 \beta_{7} + 6 \beta_{8} + 6 \beta_{9} - 2 \beta_{10} - 4 \beta_{13} ) q^{21} + ( -217 - 221 \beta_{1} + 39 \beta_{2} - 3 \beta_{3} + 7 \beta_{4} - 10 \beta_{5} - 4 \beta_{6} + 8 \beta_{7} - 13 \beta_{9} + 3 \beta_{10} + 3 \beta_{12} ) q^{23} + ( -305 + 77 \beta_{1} - 13 \beta_{2} + 7 \beta_{3} - \beta_{4} - 6 \beta_{5} + 21 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 16 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + \beta_{12} + 6 \beta_{13} ) q^{25} + ( 150 - 158 \beta_{1} + 6 \beta_{2} + 7 \beta_{3} + 15 \beta_{4} + 4 \beta_{5} + 18 \beta_{6} - 38 \beta_{7} + 14 \beta_{8} - \beta_{10} - 2 \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{27} + ( 17 - 1235 \beta_{1} + 25 \beta_{2} + 17 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} + 11 \beta_{6} + 23 \beta_{7} + 8 \beta_{8} - 8 \beta_{9} - 18 \beta_{11} - 4 \beta_{12} ) q^{29} + ( -358 \beta_{1} + 95 \beta_{2} + 8 \beta_{4} - 26 \beta_{5} + 77 \beta_{6} - 14 \beta_{7} - 9 \beta_{8} + 9 \beta_{9} - 6 \beta_{11} - 10 \beta_{12} ) q^{31} + ( -185 + 211 \beta_{1} + 20 \beta_{2} + 25 \beta_{3} - \beta_{4} - 53 \beta_{5} - 100 \beta_{6} - 11 \beta_{7} - 18 \beta_{8} - 5 \beta_{10} + 7 \beta_{11} + 5 \beta_{12} + 7 \beta_{13} ) q^{33} + ( 249 - 191 \beta_{1} - 124 \beta_{2} + 16 \beta_{3} + 27 \beta_{4} + 7 \beta_{5} + 79 \beta_{6} + 3 \beta_{7} - 26 \beta_{8} + 8 \beta_{9} + 4 \beta_{10} - 6 \beta_{11} - 7 \beta_{12} - 12 \beta_{13} ) q^{35} + ( -52 - 83 \beta_{1} - 225 \beta_{2} + 47 \beta_{3} - 16 \beta_{4} + 88 \beta_{5} + 25 \beta_{7} - 32 \beta_{9} - 16 \beta_{10} - 6 \beta_{11} - 16 \beta_{12} + 6 \beta_{13} ) q^{37} + ( 846 - 28 \beta_{1} - 129 \beta_{2} - 4 \beta_{3} + 28 \beta_{4} + 64 \beta_{5} + 93 \beta_{6} + 40 \beta_{7} + 25 \beta_{8} + 25 \beta_{9} - 4 \beta_{10} - 12 \beta_{13} ) q^{39} + ( 805 + 10 \beta_{1} + 49 \beta_{2} + 46 \beta_{3} - 10 \beta_{4} + 23 \beta_{5} + 5 \beta_{6} - 100 \beta_{7} - 18 \beta_{8} - 18 \beta_{9} - 2 \beta_{10} + 3 \beta_{13} ) q^{41} + ( 259 + 163 \beta_{1} + 219 \beta_{2} + 40 \beta_{3} + 56 \beta_{4} + 32 \beta_{5} - 66 \beta_{6} + 180 \beta_{7} - 26 \beta_{9} - 10 \beta_{10} + 18 \beta_{11} - 10 \beta_{12} - 18 \beta_{13} ) q^{43} + ( 1235 - 2651 \beta_{1} - 36 \beta_{2} + 64 \beta_{3} - 7 \beta_{4} + 15 \beta_{5} + 516 \beta_{6} - 19 \beta_{7} + 40 \beta_{8} - 10 \beta_{9} - 15 \beta_{10} - 10 \beta_{11} + 5 \beta_{12} ) q^{45} + ( -51 - \beta_{1} - 14 \beta_{2} - 19 \beta_{3} + 33 \beta_{4} + 84 \beta_{5} - 331 \beta_{6} - 70 \beta_{7} + 31 \beta_{8} + 5 \beta_{10} - 18 \beta_{11} - 5 \beta_{12} - 18 \beta_{13} ) q^{47} + ( 58 - 624 \beta_{1} + 287 \beta_{2} + 58 \beta_{3} - 8 \beta_{4} - 32 \beta_{5} + 247 \beta_{6} + 139 \beta_{7} + 46 \beta_{8} - 46 \beta_{9} + 15 \beta_{11} + 10 \beta_{12} ) q^{49} + ( 60 - 5994 \beta_{1} + 609 \beta_{2} + 60 \beta_{3} + 106 \beta_{4} - 260 \beta_{5} + 455 \beta_{6} - 4 \beta_{8} + 4 \beta_{9} - 14 \beta_{11} + 12 \beta_{12} ) q^{51} + ( 381 - 282 \beta_{1} + 78 \beta_{2} + 71 \beta_{3} - 28 \beta_{4} - 159 \beta_{5} - 283 \beta_{6} - 40 \beta_{7} - 86 \beta_{8} + 7 \beta_{10} - 18 \beta_{11} - 7 \beta_{12} - 18 \beta_{13} ) q^{53} + ( -4522 + 1802 \beta_{1} - 556 \beta_{2} - 64 \beta_{3} + 47 \beta_{4} - 45 \beta_{5} + 96 \beta_{6} + 13 \beta_{7} - 9 \beta_{8} - 23 \beta_{9} + 21 \beta_{10} + 21 \beta_{11} + 12 \beta_{12} + 27 \beta_{13} ) q^{55} + ( -606 - 676 \beta_{1} - 1504 \beta_{2} + 75 \beta_{3} - 5 \beta_{4} + 156 \beta_{5} + 18 \beta_{6} + 40 \beta_{7} + 58 \beta_{9} + 13 \beta_{10} - 12 \beta_{11} + 13 \beta_{12} + 12 \beta_{13} ) q^{57} + ( 886 - 124 \beta_{1} - 638 \beta_{2} + 97 \beta_{3} + 124 \beta_{4} + 339 \beta_{5} + 620 \beta_{6} - 79 \beta_{7} - 16 \beta_{8} - 16 \beta_{9} + 9 \beta_{10} + 6 \beta_{13} ) q^{59} + ( 1113 + 18 \beta_{1} + 618 \beta_{2} + 43 \beta_{3} - 18 \beta_{4} - 23 \beta_{5} - 511 \beta_{6} - 150 \beta_{7} + 90 \beta_{8} + 90 \beta_{9} + 46 \beta_{10} + 30 \beta_{13} ) q^{61} + ( -1693 - 1641 \beta_{1} + 1213 \beta_{2} - 59 \beta_{3} + 7 \beta_{4} - 218 \beta_{5} - 26 \beta_{6} - 100 \beta_{7} + 39 \beta_{9} - 19 \beta_{10} - 74 \beta_{11} - 19 \beta_{12} + 74 \beta_{13} ) q^{63} + ( 3010 - 4063 \beta_{1} + 357 \beta_{2} + 62 \beta_{3} - 66 \beta_{4} + 57 \beta_{5} + 647 \beta_{6} + 38 \beta_{7} - 86 \beta_{8} + 78 \beta_{9} + 4 \beta_{10} - 6 \beta_{11} - 42 \beta_{12} - 57 \beta_{13} ) q^{65} + ( -8043 + 8019 \beta_{1} + 124 \beta_{2} + 130 \beta_{3} + 154 \beta_{4} - 184 \beta_{5} - 1043 \beta_{6} - 348 \beta_{7} - 70 \beta_{8} - 6 \beta_{10} + 84 \beta_{11} + 6 \beta_{12} + 84 \beta_{13} ) q^{67} + ( 49 + 11890 \beta_{1} - 424 \beta_{2} + 49 \beta_{3} + 26 \beta_{4} - 57 \beta_{5} - 455 \beta_{6} + 218 \beta_{7} - 152 \beta_{8} + 152 \beta_{9} + 146 \beta_{11} + 44 \beta_{12} ) q^{69} + ( -148 + 774 \beta_{1} + 615 \beta_{2} - 148 \beta_{3} + 12 \beta_{4} + 202 \beta_{5} + 829 \beta_{6} - 170 \beta_{7} + 27 \beta_{8} - 27 \beta_{9} + 138 \beta_{11} + 78 \beta_{12} ) q^{71} + ( -1479 + 1525 \beta_{1} + 58 \beta_{2} + 49 \beta_{3} + 3 \beta_{4} - 68 \beta_{5} - 1060 \beta_{6} - 100 \beta_{7} + 330 \beta_{8} + 9 \beta_{10} - 36 \beta_{11} - 9 \beta_{12} - 36 \beta_{13} ) q^{73} + ( -4003 - 5427 \beta_{1} - 1501 \beta_{2} + 209 \beta_{3} + 163 \beta_{4} + 24 \beta_{5} + 792 \beta_{6} - 122 \beta_{7} + 148 \beta_{8} - 4 \beta_{9} - 47 \beta_{10} - 92 \beta_{11} + 31 \beta_{12} + 26 \beta_{13} ) q^{75} + ( -657 - 593 \beta_{1} - 1299 \beta_{2} - 74 \beta_{3} + 10 \beta_{4} - 41 \beta_{5} + 59 \beta_{6} - 75 \beta_{7} + 14 \beta_{9} + 69 \beta_{10} + 48 \beta_{11} + 69 \beta_{12} - 48 \beta_{13} ) q^{77} + ( 10544 + 136 \beta_{1} + 18 \beta_{2} - 208 \beta_{3} - 136 \beta_{4} - 414 \beta_{5} - 108 \beta_{6} + 298 \beta_{7} - 150 \beta_{8} - 150 \beta_{9} - 18 \beta_{10} - 66 \beta_{13} ) q^{79} + ( 12022 + 4 \beta_{1} + 711 \beta_{2} - 28 \beta_{3} - 4 \beta_{4} - 15 \beta_{5} - 773 \beta_{6} + 90 \beta_{7} - 230 \beta_{8} - 230 \beta_{9} - 38 \beta_{10} - 21 \beta_{13} ) q^{81} + ( 2429 + 2133 \beta_{1} + 669 \beta_{2} + 163 \beta_{3} + 133 \beta_{4} + 318 \beta_{5} - 44 \beta_{6} + 376 \beta_{7} + 170 \beta_{9} + 89 \beta_{10} + 36 \beta_{11} + 89 \beta_{12} - 36 \beta_{13} ) q^{83} + ( -3872 - 13966 \beta_{1} + 10 \beta_{2} - 122 \beta_{3} + 56 \beta_{4} - 96 \beta_{5} + 3272 \beta_{6} + 44 \beta_{7} - 18 \beta_{8} - 356 \beta_{9} + 87 \beta_{10} + 72 \beta_{11} + 29 \beta_{12} - 6 \beta_{13} ) q^{85} + ( -9948 + 9916 \beta_{1} - 290 \beta_{2} - 274 \beta_{3} - 242 \beta_{4} + 300 \beta_{5} - 4110 \beta_{6} + 796 \beta_{7} - 246 \beta_{8} - 16 \beta_{10} + 22 \beta_{11} + 16 \beta_{12} + 22 \beta_{13} ) q^{87} + ( -54 + 3182 \beta_{1} + 2670 \beta_{2} - 54 \beta_{3} - 80 \beta_{4} + 74 \beta_{5} + 2664 \beta_{6} - 190 \beta_{7} - 4 \beta_{8} + 4 \beta_{9} - 162 \beta_{11} - 140 \beta_{12} ) q^{89} + ( 252 - 33838 \beta_{1} + 1685 \beta_{2} + 252 \beta_{3} - 64 \beta_{4} - 262 \beta_{5} + 1359 \beta_{6} + 502 \beta_{7} + 56 \beta_{8} - 56 \beta_{9} - 66 \beta_{11} - 138 \beta_{12} ) q^{91} + ( -22289 + 22023 \beta_{1} - 227 \beta_{2} - 175 \beta_{3} + 91 \beta_{4} + 353 \beta_{5} - 1161 \beta_{6} + 185 \beta_{7} + 24 \beta_{8} - 52 \beta_{10} + 140 \beta_{11} + 52 \beta_{12} + 140 \beta_{13} ) q^{93} + ( -11762 + 10444 \beta_{1} - 2417 \beta_{2} - 242 \beta_{3} - 469 \beta_{4} + 159 \beta_{5} + 217 \beta_{6} + 85 \beta_{7} + 128 \beta_{8} + 116 \beta_{9} - 67 \beta_{10} - 27 \beta_{11} - 124 \beta_{12} - 99 \beta_{13} ) q^{95} + ( -3079 - 2919 \beta_{1} - 3976 \beta_{2} - 133 \beta_{3} - 27 \beta_{4} - 444 \beta_{5} - 112 \beta_{6} - 114 \beta_{7} + 106 \beta_{9} - 139 \beta_{10} - 66 \beta_{11} - 139 \beta_{12} + 66 \beta_{13} ) q^{97} + ( 28504 + 12 \beta_{1} - 1313 \beta_{2} + 287 \beta_{3} - 12 \beta_{4} + 253 \beta_{5} + 1555 \beta_{6} - 529 \beta_{7} + 46 \beta_{8} + 46 \beta_{9} - 57 \beta_{10} + 10 \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 10q^{3} + 42q^{5} - 66q^{7} + O(q^{10}) \) \( 14q - 10q^{3} + 42q^{5} - 66q^{7} - 414q^{13} - 278q^{15} + 1222q^{17} - 5672q^{19} + 5924q^{21} - 2902q^{23} - 4466q^{25} + 2168q^{27} - 2444q^{33} + 2618q^{35} - 1790q^{37} + 11076q^{39} + 11644q^{41} + 3982q^{43} + 14704q^{45} + 1278q^{47} + 5882q^{53} - 65608q^{55} - 14552q^{57} + 8504q^{59} + 20564q^{61} - 19422q^{63} + 40798q^{65} - 107926q^{67} - 16418q^{73} - 66586q^{75} - 13348q^{77} + 146544q^{79} + 173806q^{81} + 36398q^{83} - 66262q^{85} - 124384q^{87} - 306620q^{93} - 173768q^{95} - 60314q^{97} + 388628q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 4 x^{13} + 8 x^{12} - 4626 x^{11} + 149441 x^{10} - 2113414 x^{9} + 17958066 x^{8} - 97717112 x^{7} + 355171384 x^{6} - 910571904 x^{5} + 2428303248 x^{4} - 9166992192 x^{3} + 32237484304 x^{2} - 66916821408 x + 69451154208\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-17698243422788399786322792428 \nu^{13} + 33339598605867540664447340809 \nu^{12} - 111566481532271037945221541610 \nu^{11} + 81574043330479363757248078257118 \nu^{10} - 2472391595710102013111236607934330 \nu^{9} + 32355518440523178255983518692998557 \nu^{8} - 254388696751243387855192637041212756 \nu^{7} + 1246643719825304584828348337878941040 \nu^{6} - 3992608702077023462626800882531182472 \nu^{5} + 8586609210797225701138502778301522360 \nu^{4} - 23885609292141888818043668122534492304 \nu^{3} + 93524178076331793639201391781753566912 \nu^{2} - 355219515211437437156695767539235567040 \nu + 518482390237866563439088670755099656144\)\()/ \)\(22\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(53\!\cdots\!57\)\( \nu^{13} + \)\(13\!\cdots\!59\)\( \nu^{12} + \)\(55\!\cdots\!00\)\( \nu^{11} - \)\(25\!\cdots\!32\)\( \nu^{10} + \)\(64\!\cdots\!75\)\( \nu^{9} - \)\(65\!\cdots\!23\)\( \nu^{8} + \)\(37\!\cdots\!44\)\( \nu^{7} - \)\(10\!\cdots\!80\)\( \nu^{6} + \)\(14\!\cdots\!08\)\( \nu^{5} - \)\(26\!\cdots\!00\)\( \nu^{4} + \)\(34\!\cdots\!36\)\( \nu^{3} - \)\(97\!\cdots\!68\)\( \nu^{2} + \)\(74\!\cdots\!40\)\( \nu + \)\(29\!\cdots\!84\)\(\)\()/ \)\(68\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(11\!\cdots\!89\)\( \nu^{13} + \)\(18\!\cdots\!32\)\( \nu^{12} - \)\(29\!\cdots\!50\)\( \nu^{11} + \)\(54\!\cdots\!14\)\( \nu^{10} - \)\(16\!\cdots\!25\)\( \nu^{9} + \)\(22\!\cdots\!46\)\( \nu^{8} - \)\(19\!\cdots\!88\)\( \nu^{7} + \)\(10\!\cdots\!60\)\( \nu^{6} - \)\(37\!\cdots\!16\)\( \nu^{5} + \)\(87\!\cdots\!00\)\( \nu^{4} - \)\(21\!\cdots\!72\)\( \nu^{3} + \)\(83\!\cdots\!36\)\( \nu^{2} - \)\(30\!\cdots\!80\)\( \nu + \)\(60\!\cdots\!32\)\(\)\()/ \)\(13\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(12\!\cdots\!53\)\( \nu^{13} - \)\(46\!\cdots\!14\)\( \nu^{12} + \)\(52\!\cdots\!50\)\( \nu^{11} - \)\(54\!\cdots\!78\)\( \nu^{10} + \)\(18\!\cdots\!25\)\( \nu^{9} - \)\(25\!\cdots\!92\)\( \nu^{8} + \)\(20\!\cdots\!76\)\( \nu^{7} - \)\(10\!\cdots\!20\)\( \nu^{6} + \)\(31\!\cdots\!32\)\( \nu^{5} - \)\(63\!\cdots\!00\)\( \nu^{4} + \)\(22\!\cdots\!44\)\( \nu^{3} - \)\(97\!\cdots\!72\)\( \nu^{2} + \)\(31\!\cdots\!60\)\( \nu - \)\(24\!\cdots\!64\)\(\)\()/ \)\(13\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(13\!\cdots\!41\)\( \nu^{13} + \)\(13\!\cdots\!42\)\( \nu^{12} + \)\(37\!\cdots\!50\)\( \nu^{11} - \)\(63\!\cdots\!66\)\( \nu^{10} + \)\(17\!\cdots\!25\)\( \nu^{9} - \)\(19\!\cdots\!24\)\( \nu^{8} + \)\(13\!\cdots\!72\)\( \nu^{7} - \)\(50\!\cdots\!40\)\( \nu^{6} + \)\(11\!\cdots\!04\)\( \nu^{5} - \)\(19\!\cdots\!00\)\( \nu^{4} + \)\(10\!\cdots\!68\)\( \nu^{3} - \)\(43\!\cdots\!84\)\( \nu^{2} + \)\(89\!\cdots\!20\)\( \nu + \)\(32\!\cdots\!92\)\(\)\()/ \)\(13\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(69\!\cdots\!36\)\( \nu^{13} + \)\(25\!\cdots\!43\)\( \nu^{12} + \)\(59\!\cdots\!50\)\( \nu^{11} + \)\(32\!\cdots\!86\)\( \nu^{10} - \)\(10\!\cdots\!50\)\( \nu^{9} + \)\(13\!\cdots\!79\)\( \nu^{8} - \)\(10\!\cdots\!12\)\( \nu^{7} + \)\(50\!\cdots\!40\)\( \nu^{6} - \)\(14\!\cdots\!84\)\( \nu^{5} + \)\(29\!\cdots\!00\)\( \nu^{4} - \)\(10\!\cdots\!28\)\( \nu^{3} + \)\(47\!\cdots\!64\)\( \nu^{2} - \)\(12\!\cdots\!20\)\( \nu + \)\(15\!\cdots\!68\)\(\)\()/ \)\(68\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(15\!\cdots\!33\)\( \nu^{13} + \)\(55\!\cdots\!04\)\( \nu^{12} + \)\(75\!\cdots\!50\)\( \nu^{11} + \)\(72\!\cdots\!58\)\( \nu^{10} - \)\(22\!\cdots\!25\)\( \nu^{9} + \)\(31\!\cdots\!62\)\( \nu^{8} - \)\(24\!\cdots\!36\)\( \nu^{7} + \)\(11\!\cdots\!20\)\( \nu^{6} - \)\(36\!\cdots\!52\)\( \nu^{5} + \)\(75\!\cdots\!00\)\( \nu^{4} - \)\(22\!\cdots\!84\)\( \nu^{3} + \)\(10\!\cdots\!92\)\( \nu^{2} - \)\(34\!\cdots\!60\)\( \nu + \)\(46\!\cdots\!04\)\(\)\()/ \)\(13\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(95\!\cdots\!18\)\( \nu^{13} + \)\(36\!\cdots\!09\)\( \nu^{12} + \)\(86\!\cdots\!50\)\( \nu^{11} + \)\(44\!\cdots\!18\)\( \nu^{10} - \)\(14\!\cdots\!00\)\( \nu^{9} + \)\(19\!\cdots\!77\)\( \nu^{8} - \)\(14\!\cdots\!56\)\( \nu^{7} + \)\(70\!\cdots\!20\)\( \nu^{6} - \)\(20\!\cdots\!92\)\( \nu^{5} + \)\(41\!\cdots\!00\)\( \nu^{4} - \)\(14\!\cdots\!64\)\( \nu^{3} + \)\(65\!\cdots\!32\)\( \nu^{2} - \)\(16\!\cdots\!60\)\( \nu + \)\(21\!\cdots\!84\)\(\)\()/ \)\(34\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-\)\(96\!\cdots\!91\)\( \nu^{13} - \)\(18\!\cdots\!17\)\( \nu^{12} - \)\(31\!\cdots\!00\)\( \nu^{11} + \)\(44\!\cdots\!16\)\( \nu^{10} - \)\(11\!\cdots\!25\)\( \nu^{9} + \)\(12\!\cdots\!49\)\( \nu^{8} - \)\(78\!\cdots\!72\)\( \nu^{7} + \)\(26\!\cdots\!40\)\( \nu^{6} - \)\(59\!\cdots\!04\)\( \nu^{5} + \)\(14\!\cdots\!00\)\( \nu^{4} - \)\(84\!\cdots\!68\)\( \nu^{3} + \)\(25\!\cdots\!84\)\( \nu^{2} - \)\(44\!\cdots\!20\)\( \nu + \)\(16\!\cdots\!08\)\(\)\()/ \)\(34\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(67\!\cdots\!51\)\( \nu^{13} + \)\(63\!\cdots\!62\)\( \nu^{12} + \)\(95\!\cdots\!50\)\( \nu^{11} - \)\(30\!\cdots\!26\)\( \nu^{10} + \)\(85\!\cdots\!75\)\( \nu^{9} - \)\(10\!\cdots\!64\)\( \nu^{8} + \)\(71\!\cdots\!92\)\( \nu^{7} - \)\(28\!\cdots\!40\)\( \nu^{6} + \)\(63\!\cdots\!44\)\( \nu^{5} - \)\(68\!\cdots\!00\)\( \nu^{4} + \)\(55\!\cdots\!48\)\( \nu^{3} - \)\(27\!\cdots\!24\)\( \nu^{2} + \)\(61\!\cdots\!20\)\( \nu + \)\(24\!\cdots\!12\)\(\)\()/ \)\(13\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(96\!\cdots\!11\)\( \nu^{13} - \)\(69\!\cdots\!68\)\( \nu^{12} - \)\(64\!\cdots\!50\)\( \nu^{11} - \)\(44\!\cdots\!86\)\( \nu^{10} + \)\(15\!\cdots\!75\)\( \nu^{9} - \)\(23\!\cdots\!54\)\( \nu^{8} + \)\(20\!\cdots\!12\)\( \nu^{7} - \)\(10\!\cdots\!40\)\( \nu^{6} + \)\(34\!\cdots\!84\)\( \nu^{5} - \)\(72\!\cdots\!00\)\( \nu^{4} + \)\(19\!\cdots\!28\)\( \nu^{3} - \)\(99\!\cdots\!64\)\( \nu^{2} + \)\(31\!\cdots\!20\)\( \nu - \)\(47\!\cdots\!68\)\(\)\()/ \)\(13\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(12\!\cdots\!43\)\( \nu^{13} + \)\(19\!\cdots\!84\)\( \nu^{12} - \)\(72\!\cdots\!50\)\( \nu^{11} + \)\(56\!\cdots\!18\)\( \nu^{10} - \)\(16\!\cdots\!75\)\( \nu^{9} + \)\(21\!\cdots\!02\)\( \nu^{8} - \)\(17\!\cdots\!56\)\( \nu^{7} + \)\(84\!\cdots\!20\)\( \nu^{6} - \)\(26\!\cdots\!92\)\( \nu^{5} + \)\(58\!\cdots\!00\)\( \nu^{4} - \)\(17\!\cdots\!64\)\( \nu^{3} + \)\(79\!\cdots\!32\)\( \nu^{2} - \)\(24\!\cdots\!60\)\( \nu + \)\(35\!\cdots\!84\)\(\)\()/ \)\(13\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(22\!\cdots\!67\)\( \nu^{13} - \)\(12\!\cdots\!46\)\( \nu^{12} - \)\(64\!\cdots\!50\)\( \nu^{11} - \)\(10\!\cdots\!42\)\( \nu^{10} + \)\(30\!\cdots\!75\)\( \nu^{9} - \)\(37\!\cdots\!88\)\( \nu^{8} + \)\(26\!\cdots\!64\)\( \nu^{7} - \)\(11\!\cdots\!80\)\( \nu^{6} + \)\(28\!\cdots\!48\)\( \nu^{5} - \)\(56\!\cdots\!00\)\( \nu^{4} + \)\(24\!\cdots\!16\)\( \nu^{3} - \)\(10\!\cdots\!08\)\( \nu^{2} + \)\(24\!\cdots\!40\)\( \nu - \)\(18\!\cdots\!96\)\(\)\()/ \)\(13\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{13} + \beta_{12} + 3 \beta_{11} - \beta_{10} + 2 \beta_{8} - 6 \beta_{7} + 47 \beta_{6} - 12 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} + 5 \beta_{2} - 58 \beta_{1} + 62\)\()/160\)
\(\nu^{2}\)\(=\)\((\)\(-17 \beta_{12} - 36 \beta_{11} - 8 \beta_{9} + 8 \beta_{8} + 34 \beta_{7} - 343 \beta_{6} + 78 \beta_{5} - 52 \beta_{4} + 9 \beta_{3} - 369 \beta_{2} - 7771 \beta_{1} + 9\)\()/80\)
\(\nu^{3}\)\(=\)\((\)\(-1347 \beta_{13} + 589 \beta_{12} + 1347 \beta_{11} + 589 \beta_{10} - 98 \beta_{9} + 2788 \beta_{7} - 1141 \beta_{6} - 2654 \beta_{5} + 1730 \beta_{4} - 1430 \beta_{3} + 23031 \beta_{2} + 151146 \beta_{1} + 151446\)\()/160\)
\(\nu^{4}\)\(=\)\((\)\(3975 \beta_{13} - 1802 \beta_{10} - 351 \beta_{9} - 351 \beta_{8} - 8407 \beta_{7} + 38099 \beta_{6} + 1367 \beta_{5} + 95 \beta_{4} + 5152 \beta_{3} - 34844 \beta_{2} - 95 \beta_{1} - 577119\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(-649107 \beta_{13} - 290279 \beta_{12} - 649107 \beta_{11} + 290279 \beta_{10} + 36622 \beta_{8} + 1443134 \beta_{7} - 11517663 \beta_{6} + 1218028 \beta_{5} - 781168 \beta_{4} - 820184 \beta_{3} - 529905 \beta_{2} - 85301178 \beta_{1} + 85262162\)\()/160\)
\(\nu^{6}\)\(=\)\((\)\(4550703 \beta_{12} + 10121334 \beta_{11} + 561642 \beta_{9} - 561642 \beta_{8} - 2468176 \beta_{7} + 87011117 \beta_{6} - 21064612 \beta_{5} + 12764028 \beta_{4} + 87259 \beta_{3} + 95311701 \beta_{2} + 1385752939 \beta_{1} + 87259\)\()/80\)
\(\nu^{7}\)\(=\)\((\)\(322070793 \beta_{13} - 144504171 \beta_{12} - 322070793 \beta_{11} - 144504171 \beta_{10} - 27779418 \beta_{9} - 596632972 \beta_{7} + 263310439 \beta_{6} + 734365386 \beta_{5} - 407814610 \beta_{4} + 396562870 \beta_{3} - 5762721989 \beta_{2} - 43334847614 \beta_{1} - 43346099354\)\()/160\)
\(\nu^{8}\)\(=\)\((\)\(-1015819605 \beta_{13} + 456147646 \beta_{10} + 49378493 \beta_{9} + 49378493 \beta_{8} + 2105820101 \beta_{7} - 9530822297 \beta_{6} - 230167781 \beta_{5} + 13998595 \beta_{4} - 1273984576 \beta_{3} + 8698986772 \beta_{2} - 13998595 \beta_{1} + 137706097181\)\()/16\)
\(\nu^{9}\)\(=\)\((\)\(160815790533 \beta_{13} + 72189394841 \beta_{12} + 160815790533 \beta_{11} - 72189394841 \beta_{10} - 14855380258 \beta_{8} - 368450766226 \beta_{7} + 2882226273177 \beta_{6} - 297336337332 \beta_{5} + 198840392312 \beta_{4} + 203775166976 \beta_{3} + 131585772135 \beta_{2} + 21734308994502 \beta_{1} - 21729374219838\)\()/160\)
\(\nu^{10}\)\(=\)\((\)\(-1141073051377 \beta_{12} - 2541616650546 \beta_{11} - 120090369758 \beta_{9} + 120090369758 \beta_{8} + 567954791864 \beta_{7} - 21748890401923 \beta_{6} + 5264577690148 \beta_{5} - 3184174585092 \beta_{4} - 37301571341 \beta_{3} - 23829293506979 \beta_{2} - 343928025881101 \beta_{1} - 37301571341\)\()/80\)
\(\nu^{11}\)\(=\)\((\)\(-80396898736167 \beta_{13} + 36092621151669 \beta_{12} + 80396898736167 \beta_{11} + 36092621151669 \beta_{10} + 7522750781062 \beta_{9} + 148599757458868 \beta_{7} - 65796954147081 \beta_{6} - 184367396857174 \beta_{5} + 101889575298750 \beta_{4} - 99483670723130 \beta_{3} + 1441378962706091 \beta_{2} + 10871613161477026 \beta_{1} + 10874019066052646\)\()/160\)
\(\nu^{12}\)\(=\)\((\)\(254232041165115 \beta_{13} - 114135193791106 \beta_{10} - 11945785122003 \beta_{9} - 11945785122003 \beta_{8} - 526520672238731 \beta_{7} + 2383259696234647 \beta_{6} + 56664374886891 \beta_{5} - 3772606785165 \beta_{4} + 318441629622336 \beta_{3} - 2175180653618252 \beta_{2} + 3772606785165 \beta_{1} - 34390883573292819\)\()/16\)
\(\nu^{13}\)\(=\)\((\)\(-40202344576531227 \beta_{13} - 18048302994489639 \beta_{12} - 40202344576531227 \beta_{11} + 18048302994489639 \beta_{10} + 3770934495970462 \beta_{8} + 92208397241437614 \beta_{7} - 720803456067542983 \beta_{6} + 74302779885282508 \beta_{5} - 49753870264335208 \beta_{4} - 50951304283788064 \beta_{3} - 32903001289298425 \beta_{2} - 5438315933376697018 \beta_{1} + 5437118499357244162\)\()/160\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(-1\) \(\beta_{1}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
2.75256 + 2.75256i
−2.20370 2.20370i
4.57273 + 4.57273i
6.86993 + 6.86993i
2.04998 + 2.04998i
3.77108 + 3.77108i
−15.8126 15.8126i
2.75256 2.75256i
−2.20370 + 2.20370i
4.57273 4.57273i
6.86993 6.86993i
2.04998 2.04998i
3.77108 3.77108i
−15.8126 + 15.8126i
0 −18.9245 + 18.9245i 0 −36.4318 + 42.3996i 0 −112.521 112.521i 0 473.272i 0
63.2 0 −10.9598 + 10.9598i 0 −14.9228 53.8731i 0 75.2427 + 75.2427i 0 2.76340i 0
63.3 0 −9.28112 + 9.28112i 0 42.6946 + 36.0856i 0 105.819 + 105.819i 0 70.7216i 0
63.4 0 −1.47740 + 1.47740i 0 55.2474 8.52775i 0 −156.265 156.265i 0 238.635i 0
63.5 0 5.65790 5.65790i 0 −54.5514 + 12.2124i 0 −23.8754 23.8754i 0 178.976i 0
63.6 0 13.4331 13.4331i 0 15.0352 + 53.8418i 0 76.4413 + 76.4413i 0 117.896i 0
63.7 0 16.5519 16.5519i 0 13.9288 54.1386i 0 2.15894 + 2.15894i 0 304.928i 0
127.1 0 −18.9245 18.9245i 0 −36.4318 42.3996i 0 −112.521 + 112.521i 0 473.272i 0
127.2 0 −10.9598 10.9598i 0 −14.9228 + 53.8731i 0 75.2427 75.2427i 0 2.76340i 0
127.3 0 −9.28112 9.28112i 0 42.6946 36.0856i 0 105.819 105.819i 0 70.7216i 0
127.4 0 −1.47740 1.47740i 0 55.2474 + 8.52775i 0 −156.265 + 156.265i 0 238.635i 0
127.5 0 5.65790 + 5.65790i 0 −54.5514 12.2124i 0 −23.8754 + 23.8754i 0 178.976i 0
127.6 0 13.4331 + 13.4331i 0 15.0352 53.8418i 0 76.4413 76.4413i 0 117.896i 0
127.7 0 16.5519 + 16.5519i 0 13.9288 + 54.1386i 0 2.15894 2.15894i 0 304.928i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.6.n.a 14
4.b odd 2 1 160.6.n.b yes 14
5.c odd 4 1 160.6.n.b yes 14
20.e even 4 1 inner 160.6.n.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.n.a 14 1.a even 1 1 trivial
160.6.n.a 14 20.e even 4 1 inner
160.6.n.b yes 14 4.b odd 2 1
160.6.n.b yes 14 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{14} + \cdots\) acting on \(S_{6}^{\mathrm{new}}(160, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 10 T + 50 T^{2} + 254 T^{3} - 91369 T^{4} - 1354100 T^{5} - 8940292 T^{6} - 156709276 T^{7} + 3134936461 T^{8} + 38721912918 T^{9} + 290596015278 T^{10} + 2245195781154 T^{11} + 9414782746155 T^{12} + 473122978927848 T^{13} + 19255041373064328 T^{14} + 114968883879467064 T^{15} + 555933506377706595 T^{16} + 32216105460571098678 T^{17} + \)\(10\!\cdots\!78\)\( T^{18} + \)\(32\!\cdots\!74\)\( T^{19} + \)\(64\!\cdots\!89\)\( T^{20} - \)\(78\!\cdots\!32\)\( T^{21} - \)\(10\!\cdots\!92\)\( T^{22} - \)\(40\!\cdots\!00\)\( T^{23} - \)\(65\!\cdots\!81\)\( T^{24} + \)\(44\!\cdots\!78\)\( T^{25} + \)\(21\!\cdots\!50\)\( T^{26} + \)\(10\!\cdots\!30\)\( T^{27} + \)\(25\!\cdots\!49\)\( T^{28} \)
$5$ \( 1 - 42 T + 3115 T^{2} - 77660 T^{3} - 1600075 T^{4} + 15516250 T^{5} - 16330865625 T^{6} + 925536875000 T^{7} - 51033955078125 T^{8} + 151525878906250 T^{9} - 48830413818359375 T^{10} - 7406234741210937500 T^{11} + \)\(92\!\cdots\!75\)\( T^{12} - \)\(39\!\cdots\!50\)\( T^{13} + \)\(29\!\cdots\!25\)\( T^{14} \)
$7$ \( 1 + 66 T + 2178 T^{2} - 3104786 T^{3} - 390660977 T^{4} + 28598040732 T^{5} + 7558178349116 T^{6} + 730117525814980 T^{7} - 100452887411613603 T^{8} - 13091475719468548578 T^{9} + \)\(44\!\cdots\!50\)\( T^{10} + \)\(38\!\cdots\!06\)\( T^{11} + \)\(37\!\cdots\!31\)\( T^{12} - \)\(47\!\cdots\!84\)\( T^{13} - \)\(77\!\cdots\!76\)\( T^{14} - \)\(79\!\cdots\!88\)\( T^{15} + \)\(10\!\cdots\!19\)\( T^{16} + \)\(18\!\cdots\!58\)\( T^{17} + \)\(35\!\cdots\!50\)\( T^{18} - \)\(17\!\cdots\!46\)\( T^{19} - \)\(22\!\cdots\!47\)\( T^{20} + \)\(27\!\cdots\!40\)\( T^{21} + \)\(48\!\cdots\!16\)\( T^{22} + \)\(30\!\cdots\!24\)\( T^{23} - \)\(70\!\cdots\!73\)\( T^{24} - \)\(93\!\cdots\!98\)\( T^{25} + \)\(11\!\cdots\!78\)\( T^{26} + \)\(56\!\cdots\!62\)\( T^{27} + \)\(14\!\cdots\!49\)\( T^{28} \)
$11$ \( 1 - 1517670 T^{2} + 1118279059571 T^{4} - 531239593921187868 T^{6} + \)\(18\!\cdots\!61\)\( T^{8} - \)\(48\!\cdots\!18\)\( T^{10} + \)\(10\!\cdots\!67\)\( T^{12} - \)\(18\!\cdots\!88\)\( T^{14} + \)\(27\!\cdots\!67\)\( T^{16} - \)\(32\!\cdots\!18\)\( T^{18} + \)\(31\!\cdots\!61\)\( T^{20} - \)\(24\!\cdots\!68\)\( T^{22} + \)\(13\!\cdots\!71\)\( T^{24} - \)\(46\!\cdots\!70\)\( T^{26} + \)\(78\!\cdots\!01\)\( T^{28} \)
$13$ \( 1 + 414 T + 85698 T^{2} + 485371686 T^{3} + 407510115667 T^{4} - 25330305055892 T^{5} + 72383288599671444 T^{6} + \)\(13\!\cdots\!64\)\( T^{7} + \)\(19\!\cdots\!85\)\( T^{8} - \)\(26\!\cdots\!74\)\( T^{9} + \)\(24\!\cdots\!22\)\( T^{10} + \)\(12\!\cdots\!02\)\( T^{11} - \)\(99\!\cdots\!17\)\( T^{12} - \)\(27\!\cdots\!44\)\( T^{13} + \)\(44\!\cdots\!64\)\( T^{14} - \)\(10\!\cdots\!92\)\( T^{15} - \)\(13\!\cdots\!33\)\( T^{16} + \)\(63\!\cdots\!14\)\( T^{17} + \)\(45\!\cdots\!22\)\( T^{18} - \)\(18\!\cdots\!82\)\( T^{19} + \)\(49\!\cdots\!65\)\( T^{20} + \)\(13\!\cdots\!48\)\( T^{21} + \)\(26\!\cdots\!44\)\( T^{22} - \)\(33\!\cdots\!56\)\( T^{23} + \)\(20\!\cdots\!83\)\( T^{24} + \)\(89\!\cdots\!02\)\( T^{25} + \)\(58\!\cdots\!98\)\( T^{26} + \)\(10\!\cdots\!02\)\( T^{27} + \)\(94\!\cdots\!49\)\( T^{28} \)
$17$ \( 1 - 1222 T + 746642 T^{2} - 1023656774 T^{3} - 7517818418661 T^{4} + 9805314295428420 T^{5} - 5845038493789599340 T^{6} + \)\(85\!\cdots\!80\)\( T^{7} + \)\(21\!\cdots\!05\)\( T^{8} - \)\(32\!\cdots\!70\)\( T^{9} + \)\(19\!\cdots\!70\)\( T^{10} - \)\(33\!\cdots\!50\)\( T^{11} - \)\(20\!\cdots\!37\)\( T^{12} + \)\(66\!\cdots\!24\)\( T^{13} - \)\(44\!\cdots\!04\)\( T^{14} + \)\(93\!\cdots\!68\)\( T^{15} - \)\(41\!\cdots\!13\)\( T^{16} - \)\(96\!\cdots\!50\)\( T^{17} + \)\(80\!\cdots\!70\)\( T^{18} - \)\(19\!\cdots\!90\)\( T^{19} + \)\(17\!\cdots\!45\)\( T^{20} + \)\(99\!\cdots\!40\)\( T^{21} - \)\(96\!\cdots\!40\)\( T^{22} + \)\(22\!\cdots\!40\)\( T^{23} - \)\(25\!\cdots\!89\)\( T^{24} - \)\(48\!\cdots\!82\)\( T^{25} + \)\(50\!\cdots\!42\)\( T^{26} - \)\(11\!\cdots\!54\)\( T^{27} + \)\(13\!\cdots\!49\)\( T^{28} \)
$19$ \( ( 1 + 2836 T + 12192485 T^{2} + 21773970856 T^{3} + 58437600122669 T^{4} + 77435325554759820 T^{5} + \)\(17\!\cdots\!85\)\( T^{6} + \)\(20\!\cdots\!00\)\( T^{7} + \)\(43\!\cdots\!15\)\( T^{8} + \)\(47\!\cdots\!20\)\( T^{9} + \)\(88\!\cdots\!31\)\( T^{10} + \)\(81\!\cdots\!56\)\( T^{11} + \)\(11\!\cdots\!15\)\( T^{12} + \)\(65\!\cdots\!36\)\( T^{13} + \)\(57\!\cdots\!99\)\( T^{14} )^{2} \)
$23$ \( 1 + 2902 T + 4210802 T^{2} - 8585545830 T^{3} - 77137172452561 T^{4} - 5298422918089420 T^{5} + \)\(34\!\cdots\!32\)\( T^{6} + \)\(17\!\cdots\!48\)\( T^{7} + \)\(43\!\cdots\!33\)\( T^{8} - \)\(14\!\cdots\!50\)\( T^{9} - \)\(10\!\cdots\!54\)\( T^{10} - \)\(57\!\cdots\!98\)\( T^{11} - \)\(19\!\cdots\!21\)\( T^{12} + \)\(51\!\cdots\!28\)\( T^{13} + \)\(13\!\cdots\!40\)\( T^{14} + \)\(33\!\cdots\!04\)\( T^{15} - \)\(79\!\cdots\!29\)\( T^{16} - \)\(15\!\cdots\!86\)\( T^{17} - \)\(18\!\cdots\!54\)\( T^{18} - \)\(15\!\cdots\!50\)\( T^{19} + \)\(30\!\cdots\!17\)\( T^{20} + \)\(81\!\cdots\!36\)\( T^{21} + \)\(10\!\cdots\!32\)\( T^{22} - \)\(10\!\cdots\!60\)\( T^{23} - \)\(94\!\cdots\!89\)\( T^{24} - \)\(67\!\cdots\!10\)\( T^{25} + \)\(21\!\cdots\!02\)\( T^{26} + \)\(94\!\cdots\!86\)\( T^{27} + \)\(20\!\cdots\!49\)\( T^{28} \)
$29$ \( 1 - 127109206 T^{2} + 8400089254893139 T^{4} - \)\(38\!\cdots\!76\)\( T^{6} + \)\(13\!\cdots\!57\)\( T^{8} - \)\(39\!\cdots\!02\)\( T^{10} + \)\(98\!\cdots\!23\)\( T^{12} - \)\(21\!\cdots\!92\)\( T^{14} + \)\(41\!\cdots\!23\)\( T^{16} - \)\(69\!\cdots\!02\)\( T^{18} + \)\(10\!\cdots\!57\)\( T^{20} - \)\(12\!\cdots\!76\)\( T^{22} + \)\(11\!\cdots\!39\)\( T^{24} - \)\(70\!\cdots\!06\)\( T^{26} + \)\(23\!\cdots\!01\)\( T^{28} \)
$31$ \( 1 - 265679454 T^{2} + 34933996110970539 T^{4} - \)\(30\!\cdots\!64\)\( T^{6} + \)\(19\!\cdots\!17\)\( T^{8} - \)\(92\!\cdots\!38\)\( T^{10} + \)\(36\!\cdots\!43\)\( T^{12} - \)\(11\!\cdots\!88\)\( T^{14} + \)\(29\!\cdots\!43\)\( T^{16} - \)\(62\!\cdots\!38\)\( T^{18} + \)\(10\!\cdots\!17\)\( T^{20} - \)\(13\!\cdots\!64\)\( T^{22} + \)\(12\!\cdots\!39\)\( T^{24} - \)\(80\!\cdots\!54\)\( T^{26} + \)\(24\!\cdots\!01\)\( T^{28} \)
$37$ \( 1 + 1790 T + 1602050 T^{2} - 1085215470730 T^{3} - 5589943426343837 T^{4} - 5189725886098796788 T^{5} + \)\(58\!\cdots\!80\)\( T^{6} + \)\(37\!\cdots\!60\)\( T^{7} + \)\(20\!\cdots\!01\)\( T^{8} - \)\(14\!\cdots\!14\)\( T^{9} - \)\(11\!\cdots\!78\)\( T^{10} - \)\(12\!\cdots\!50\)\( T^{11} - \)\(40\!\cdots\!45\)\( T^{12} + \)\(67\!\cdots\!72\)\( T^{13} + \)\(47\!\cdots\!96\)\( T^{14} + \)\(47\!\cdots\!04\)\( T^{15} - \)\(19\!\cdots\!05\)\( T^{16} - \)\(41\!\cdots\!50\)\( T^{17} - \)\(25\!\cdots\!78\)\( T^{18} - \)\(23\!\cdots\!98\)\( T^{19} + \)\(22\!\cdots\!49\)\( T^{20} + \)\(29\!\cdots\!80\)\( T^{21} + \)\(31\!\cdots\!80\)\( T^{22} - \)\(19\!\cdots\!16\)\( T^{23} - \)\(14\!\cdots\!13\)\( T^{24} - \)\(19\!\cdots\!90\)\( T^{25} + \)\(19\!\cdots\!50\)\( T^{26} + \)\(15\!\cdots\!30\)\( T^{27} + \)\(59\!\cdots\!49\)\( T^{28} \)
$41$ \( ( 1 - 5822 T + 527400583 T^{2} - 3511555483012 T^{3} + 138605008918596701 T^{4} - \)\(95\!\cdots\!50\)\( T^{5} + \)\(23\!\cdots\!95\)\( T^{6} - \)\(14\!\cdots\!20\)\( T^{7} + \)\(26\!\cdots\!95\)\( T^{8} - \)\(12\!\cdots\!50\)\( T^{9} + \)\(21\!\cdots\!01\)\( T^{10} - \)\(63\!\cdots\!12\)\( T^{11} + \)\(11\!\cdots\!83\)\( T^{12} - \)\(14\!\cdots\!22\)\( T^{13} + \)\(28\!\cdots\!01\)\( T^{14} )^{2} \)
$43$ \( 1 - 3982 T + 7928162 T^{2} - 5090675647610 T^{3} + 44126507369726119 T^{4} - 95654814508201483332 T^{5} + \)\(12\!\cdots\!96\)\( T^{6} - \)\(15\!\cdots\!12\)\( T^{7} + \)\(46\!\cdots\!93\)\( T^{8} - \)\(33\!\cdots\!78\)\( T^{9} + \)\(35\!\cdots\!38\)\( T^{10} - \)\(10\!\cdots\!70\)\( T^{11} + \)\(78\!\cdots\!59\)\( T^{12} - \)\(84\!\cdots\!64\)\( T^{13} + \)\(18\!\cdots\!08\)\( T^{14} - \)\(12\!\cdots\!52\)\( T^{15} + \)\(16\!\cdots\!91\)\( T^{16} - \)\(34\!\cdots\!90\)\( T^{17} + \)\(16\!\cdots\!38\)\( T^{18} - \)\(22\!\cdots\!54\)\( T^{19} + \)\(47\!\cdots\!57\)\( T^{20} - \)\(23\!\cdots\!84\)\( T^{21} + \)\(28\!\cdots\!96\)\( T^{22} - \)\(30\!\cdots\!76\)\( T^{23} + \)\(20\!\cdots\!31\)\( T^{24} - \)\(35\!\cdots\!70\)\( T^{25} + \)\(80\!\cdots\!62\)\( T^{26} - \)\(59\!\cdots\!26\)\( T^{27} + \)\(22\!\cdots\!49\)\( T^{28} \)
$47$ \( 1 - 1278 T + 816642 T^{2} - 351784473730 T^{3} + 13143771115043551 T^{4} + 55735540634425394844 T^{5} - \)\(20\!\cdots\!24\)\( T^{6} + \)\(35\!\cdots\!24\)\( T^{7} + \)\(21\!\cdots\!21\)\( T^{8} - \)\(27\!\cdots\!06\)\( T^{9} + \)\(23\!\cdots\!58\)\( T^{10} - \)\(14\!\cdots\!22\)\( T^{11} - \)\(33\!\cdots\!25\)\( T^{12} + \)\(57\!\cdots\!44\)\( T^{13} - \)\(35\!\cdots\!52\)\( T^{14} + \)\(13\!\cdots\!08\)\( T^{15} - \)\(17\!\cdots\!25\)\( T^{16} - \)\(16\!\cdots\!46\)\( T^{17} + \)\(64\!\cdots\!58\)\( T^{18} - \)\(17\!\cdots\!42\)\( T^{19} + \)\(31\!\cdots\!29\)\( T^{20} + \)\(11\!\cdots\!32\)\( T^{21} - \)\(15\!\cdots\!24\)\( T^{22} + \)\(97\!\cdots\!08\)\( T^{23} + \)\(52\!\cdots\!99\)\( T^{24} - \)\(32\!\cdots\!90\)\( T^{25} + \)\(17\!\cdots\!42\)\( T^{26} - \)\(62\!\cdots\!46\)\( T^{27} + \)\(11\!\cdots\!49\)\( T^{28} \)
$53$ \( 1 - 5882 T + 17298962 T^{2} + 13844528488830 T^{3} - 174311180941592381 T^{4} - \)\(81\!\cdots\!52\)\( T^{5} + \)\(14\!\cdots\!36\)\( T^{6} - \)\(35\!\cdots\!12\)\( T^{7} - \)\(10\!\cdots\!67\)\( T^{8} + \)\(15\!\cdots\!82\)\( T^{9} + \)\(26\!\cdots\!38\)\( T^{10} - \)\(97\!\cdots\!30\)\( T^{11} + \)\(23\!\cdots\!39\)\( T^{12} + \)\(29\!\cdots\!36\)\( T^{13} - \)\(56\!\cdots\!72\)\( T^{14} + \)\(12\!\cdots\!48\)\( T^{15} + \)\(41\!\cdots\!11\)\( T^{16} - \)\(71\!\cdots\!10\)\( T^{17} + \)\(82\!\cdots\!38\)\( T^{18} + \)\(19\!\cdots\!26\)\( T^{19} - \)\(56\!\cdots\!83\)\( T^{20} - \)\(78\!\cdots\!84\)\( T^{21} + \)\(13\!\cdots\!36\)\( T^{22} - \)\(31\!\cdots\!36\)\( T^{23} - \)\(28\!\cdots\!69\)\( T^{24} + \)\(94\!\cdots\!10\)\( T^{25} + \)\(49\!\cdots\!62\)\( T^{26} - \)\(70\!\cdots\!26\)\( T^{27} + \)\(50\!\cdots\!49\)\( T^{28} \)
$59$ \( ( 1 - 4252 T + 2005910877 T^{2} - 20518241068280 T^{3} + 2395790088022200413 T^{4} - \)\(22\!\cdots\!48\)\( T^{5} + \)\(23\!\cdots\!29\)\( T^{6} - \)\(16\!\cdots\!28\)\( T^{7} + \)\(16\!\cdots\!71\)\( T^{8} - \)\(11\!\cdots\!48\)\( T^{9} + \)\(87\!\cdots\!87\)\( T^{10} - \)\(53\!\cdots\!80\)\( T^{11} + \)\(37\!\cdots\!23\)\( T^{12} - \)\(56\!\cdots\!52\)\( T^{13} + \)\(95\!\cdots\!99\)\( T^{14} )^{2} \)
$61$ \( ( 1 - 10282 T + 2916253203 T^{2} + 3083320715268 T^{3} + 3046127491767085861 T^{4} + \)\(70\!\cdots\!38\)\( T^{5} + \)\(15\!\cdots\!39\)\( T^{6} + \)\(10\!\cdots\!44\)\( T^{7} + \)\(12\!\cdots\!39\)\( T^{8} + \)\(50\!\cdots\!38\)\( T^{9} + \)\(18\!\cdots\!61\)\( T^{10} + \)\(15\!\cdots\!68\)\( T^{11} + \)\(12\!\cdots\!03\)\( T^{12} - \)\(37\!\cdots\!82\)\( T^{13} + \)\(30\!\cdots\!01\)\( T^{14} )^{2} \)
$67$ \( 1 + 107926 T + 5824010738 T^{2} + 286091118212770 T^{3} + 7910959270847152663 T^{4} - \)\(10\!\cdots\!32\)\( T^{5} - \)\(16\!\cdots\!76\)\( T^{6} - \)\(10\!\cdots\!52\)\( T^{7} - \)\(49\!\cdots\!51\)\( T^{8} - \)\(12\!\cdots\!38\)\( T^{9} - \)\(20\!\cdots\!90\)\( T^{10} + \)\(24\!\cdots\!26\)\( T^{11} + \)\(60\!\cdots\!11\)\( T^{12} + \)\(31\!\cdots\!32\)\( T^{13} + \)\(11\!\cdots\!96\)\( T^{14} + \)\(43\!\cdots\!24\)\( T^{15} + \)\(10\!\cdots\!39\)\( T^{16} + \)\(61\!\cdots\!18\)\( T^{17} - \)\(68\!\cdots\!90\)\( T^{18} - \)\(57\!\cdots\!66\)\( T^{19} - \)\(30\!\cdots\!99\)\( T^{20} - \)\(86\!\cdots\!36\)\( T^{21} - \)\(18\!\cdots\!76\)\( T^{22} - \)\(15\!\cdots\!24\)\( T^{23} + \)\(15\!\cdots\!87\)\( T^{24} + \)\(77\!\cdots\!10\)\( T^{25} + \)\(21\!\cdots\!38\)\( T^{26} + \)\(53\!\cdots\!82\)\( T^{27} + \)\(66\!\cdots\!49\)\( T^{28} \)
$71$ \( 1 - 12494857902 T^{2} + 77600272039468829243 T^{4} - \)\(32\!\cdots\!32\)\( T^{6} + \)\(99\!\cdots\!41\)\( T^{8} - \)\(25\!\cdots\!58\)\( T^{10} + \)\(55\!\cdots\!91\)\( T^{12} - \)\(10\!\cdots\!64\)\( T^{14} + \)\(17\!\cdots\!91\)\( T^{16} - \)\(26\!\cdots\!58\)\( T^{18} + \)\(34\!\cdots\!41\)\( T^{20} - \)\(35\!\cdots\!32\)\( T^{22} + \)\(28\!\cdots\!43\)\( T^{24} - \)\(14\!\cdots\!02\)\( T^{26} + \)\(38\!\cdots\!01\)\( T^{28} \)
$73$ \( 1 + 16418 T + 134775362 T^{2} - 257655666175822 T^{3} - 4365114300655601877 T^{4} + \)\(39\!\cdots\!72\)\( T^{5} + \)\(40\!\cdots\!12\)\( T^{6} + \)\(72\!\cdots\!32\)\( T^{7} - \)\(10\!\cdots\!35\)\( T^{8} - \)\(45\!\cdots\!66\)\( T^{9} + \)\(74\!\cdots\!94\)\( T^{10} + \)\(14\!\cdots\!50\)\( T^{11} + \)\(46\!\cdots\!07\)\( T^{12} - \)\(13\!\cdots\!16\)\( T^{13} - \)\(14\!\cdots\!04\)\( T^{14} - \)\(27\!\cdots\!88\)\( T^{15} + \)\(20\!\cdots\!43\)\( T^{16} + \)\(12\!\cdots\!50\)\( T^{17} + \)\(13\!\cdots\!94\)\( T^{18} - \)\(17\!\cdots\!38\)\( T^{19} - \)\(80\!\cdots\!15\)\( T^{20} + \)\(11\!\cdots\!24\)\( T^{21} + \)\(13\!\cdots\!12\)\( T^{22} + \)\(28\!\cdots\!96\)\( T^{23} - \)\(63\!\cdots\!73\)\( T^{24} - \)\(78\!\cdots\!54\)\( T^{25} + \)\(84\!\cdots\!62\)\( T^{26} + \)\(21\!\cdots\!74\)\( T^{27} + \)\(27\!\cdots\!49\)\( T^{28} \)
$79$ \( ( 1 - 73272 T + 14910422889 T^{2} - 713614712327088 T^{3} + 91878682859930077941 T^{4} - \)\(30\!\cdots\!92\)\( T^{5} + \)\(35\!\cdots\!29\)\( T^{6} - \)\(97\!\cdots\!08\)\( T^{7} + \)\(10\!\cdots\!71\)\( T^{8} - \)\(29\!\cdots\!92\)\( T^{9} + \)\(26\!\cdots\!59\)\( T^{10} - \)\(63\!\cdots\!88\)\( T^{11} + \)\(41\!\cdots\!11\)\( T^{12} - \)\(62\!\cdots\!72\)\( T^{13} + \)\(26\!\cdots\!99\)\( T^{14} )^{2} \)
$83$ \( 1 - 36398 T + 662407202 T^{2} + 351755925637942 T^{3} - 18959358682913357257 T^{4} - \)\(10\!\cdots\!44\)\( T^{5} + \)\(11\!\cdots\!08\)\( T^{6} - \)\(60\!\cdots\!36\)\( T^{7} + \)\(11\!\cdots\!21\)\( T^{8} + \)\(15\!\cdots\!90\)\( T^{9} - \)\(10\!\cdots\!50\)\( T^{10} + \)\(86\!\cdots\!90\)\( T^{11} - \)\(70\!\cdots\!85\)\( T^{12} - \)\(38\!\cdots\!20\)\( T^{13} + \)\(31\!\cdots\!80\)\( T^{14} - \)\(15\!\cdots\!60\)\( T^{15} - \)\(10\!\cdots\!65\)\( T^{16} + \)\(52\!\cdots\!30\)\( T^{17} - \)\(25\!\cdots\!50\)\( T^{18} + \)\(14\!\cdots\!70\)\( T^{19} + \)\(43\!\cdots\!29\)\( T^{20} - \)\(88\!\cdots\!52\)\( T^{21} + \)\(64\!\cdots\!08\)\( T^{22} - \)\(23\!\cdots\!92\)\( T^{23} - \)\(17\!\cdots\!93\)\( T^{24} + \)\(12\!\cdots\!94\)\( T^{25} + \)\(92\!\cdots\!02\)\( T^{26} - \)\(20\!\cdots\!14\)\( T^{27} + \)\(21\!\cdots\!49\)\( T^{28} \)
$89$ \( 1 - 30334038558 T^{2} + \)\(36\!\cdots\!35\)\( T^{4} - \)\(20\!\cdots\!52\)\( T^{6} + \)\(39\!\cdots\!69\)\( T^{8} - \)\(12\!\cdots\!10\)\( T^{10} + \)\(42\!\cdots\!95\)\( T^{12} - \)\(38\!\cdots\!60\)\( T^{14} + \)\(13\!\cdots\!95\)\( T^{16} - \)\(11\!\cdots\!10\)\( T^{18} + \)\(12\!\cdots\!69\)\( T^{20} - \)\(19\!\cdots\!52\)\( T^{22} + \)\(10\!\cdots\!35\)\( T^{24} - \)\(27\!\cdots\!58\)\( T^{26} + \)\(28\!\cdots\!01\)\( T^{28} \)
$97$ \( 1 + 60314 T + 1818889298 T^{2} - 623535693824102 T^{3} - \)\(16\!\cdots\!77\)\( T^{4} - \)\(33\!\cdots\!64\)\( T^{5} + \)\(29\!\cdots\!52\)\( T^{6} + \)\(51\!\cdots\!52\)\( T^{7} + \)\(12\!\cdots\!01\)\( T^{8} - \)\(44\!\cdots\!50\)\( T^{9} + \)\(18\!\cdots\!50\)\( T^{10} + \)\(91\!\cdots\!50\)\( T^{11} + \)\(44\!\cdots\!75\)\( T^{12} - \)\(82\!\cdots\!00\)\( T^{13} - \)\(73\!\cdots\!00\)\( T^{14} - \)\(70\!\cdots\!00\)\( T^{15} + \)\(32\!\cdots\!75\)\( T^{16} + \)\(57\!\cdots\!50\)\( T^{17} + \)\(99\!\cdots\!50\)\( T^{18} - \)\(20\!\cdots\!50\)\( T^{19} + \)\(51\!\cdots\!49\)\( T^{20} + \)\(17\!\cdots\!36\)\( T^{21} + \)\(88\!\cdots\!52\)\( T^{22} - \)\(84\!\cdots\!48\)\( T^{23} - \)\(36\!\cdots\!73\)\( T^{24} - \)\(11\!\cdots\!86\)\( T^{25} + \)\(29\!\cdots\!98\)\( T^{26} + \)\(83\!\cdots\!98\)\( T^{27} + \)\(11\!\cdots\!49\)\( T^{28} \)
show more
show less