Properties

Label 91.8.a.e.1.6
Level $91$
Weight $8$
Character 91.1
Self dual yes
Analytic conductor $28.427$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.4270373191\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 6 x^{11} - 1243 x^{10} + 5598 x^{9} + 567554 x^{8} - 1739560 x^{7} - 117081910 x^{6} + 186018392 x^{5} + 10752389517 x^{4} + 491049966 x^{3} + \cdots + 59402280000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.0691404\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.930860 q^{2} +23.4478 q^{3} -127.134 q^{4} -494.625 q^{5} +21.8266 q^{6} +343.000 q^{7} -237.493 q^{8} -1637.20 q^{9} +O(q^{10})\) \(q+0.930860 q^{2} +23.4478 q^{3} -127.134 q^{4} -494.625 q^{5} +21.8266 q^{6} +343.000 q^{7} -237.493 q^{8} -1637.20 q^{9} -460.427 q^{10} -2424.00 q^{11} -2981.00 q^{12} +2197.00 q^{13} +319.285 q^{14} -11597.9 q^{15} +16052.0 q^{16} +3893.57 q^{17} -1524.00 q^{18} -4518.07 q^{19} +62883.5 q^{20} +8042.60 q^{21} -2256.40 q^{22} +66046.1 q^{23} -5568.70 q^{24} +166529. q^{25} +2045.10 q^{26} -89669.1 q^{27} -43606.8 q^{28} +6156.48 q^{29} -10796.0 q^{30} +17664.1 q^{31} +45341.3 q^{32} -56837.5 q^{33} +3624.37 q^{34} -169657. q^{35} +208143. q^{36} -516218. q^{37} -4205.69 q^{38} +51514.8 q^{39} +117470. q^{40} +470707. q^{41} +7486.53 q^{42} -10789.1 q^{43} +308172. q^{44} +809801. q^{45} +61479.6 q^{46} +326569. q^{47} +376385. q^{48} +117649. q^{49} +155015. q^{50} +91295.7 q^{51} -279312. q^{52} +1.91089e6 q^{53} -83469.3 q^{54} +1.19897e6 q^{55} -81460.3 q^{56} -105939. q^{57} +5730.82 q^{58} +558449. q^{59} +1.47448e6 q^{60} -1.32987e6 q^{61} +16442.8 q^{62} -561560. q^{63} -2.01245e6 q^{64} -1.08669e6 q^{65} -52907.7 q^{66} +2.71304e6 q^{67} -495004. q^{68} +1.54864e6 q^{69} -157926. q^{70} -2.18693e6 q^{71} +388824. q^{72} -2.12208e6 q^{73} -480527. q^{74} +3.90475e6 q^{75} +574398. q^{76} -831432. q^{77} +47953.1 q^{78} -3.62059e6 q^{79} -7.93974e6 q^{80} +1.47801e6 q^{81} +438162. q^{82} +8.69991e6 q^{83} -1.02248e6 q^{84} -1.92586e6 q^{85} -10043.1 q^{86} +144356. q^{87} +575684. q^{88} -2.94631e6 q^{89} +753811. q^{90} +753571. q^{91} -8.39667e6 q^{92} +414184. q^{93} +303990. q^{94} +2.23475e6 q^{95} +1.06315e6 q^{96} -1.35540e7 q^{97} +109515. q^{98} +3.96857e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} + 82 q^{3} + 986 q^{4} + 1026 q^{5} + 309 q^{6} + 4116 q^{7} + 228 q^{8} + 10902 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{2} + 82 q^{3} + 986 q^{4} + 1026 q^{5} + 309 q^{6} + 4116 q^{7} + 228 q^{8} + 10902 q^{9} + 6668 q^{10} + 12168 q^{11} - 183 q^{12} + 26364 q^{13} + 2058 q^{14} - 28790 q^{15} + 85914 q^{16} + 82710 q^{17} - 44965 q^{18} - 10302 q^{19} + 141318 q^{20} + 28126 q^{21} - 97457 q^{22} + 98376 q^{23} - 519981 q^{24} + 272736 q^{25} + 13182 q^{26} + 306652 q^{27} + 338198 q^{28} + 350592 q^{29} + 231528 q^{30} + 55092 q^{31} + 114420 q^{32} + 609912 q^{33} + 812002 q^{34} + 351918 q^{35} + 1472143 q^{36} + 376310 q^{37} + 2825424 q^{38} + 180154 q^{39} + 2169290 q^{40} + 1387272 q^{41} + 105987 q^{42} + 568708 q^{43} + 3392031 q^{44} + 3556226 q^{45} - 1736829 q^{46} + 1359444 q^{47} + 4151249 q^{48} + 1411788 q^{49} + 3983712 q^{50} + 2709260 q^{51} + 2166242 q^{52} + 2061780 q^{53} + 2196651 q^{54} - 2112846 q^{55} + 78204 q^{56} + 2359902 q^{57} + 670268 q^{58} + 395964 q^{59} - 1052376 q^{60} + 444006 q^{61} + 2854353 q^{62} + 3739386 q^{63} + 12026858 q^{64} + 2254122 q^{65} - 4605681 q^{66} - 3094010 q^{67} + 4668954 q^{68} + 3839892 q^{69} + 2287124 q^{70} + 5694366 q^{71} - 9780585 q^{72} + 7052346 q^{73} - 4436259 q^{74} - 16288696 q^{75} - 3051830 q^{76} + 4173624 q^{77} + 678873 q^{78} + 4304160 q^{79} + 3807018 q^{80} - 6689556 q^{81} - 4733665 q^{82} + 2704554 q^{83} - 62769 q^{84} + 9301878 q^{85} + 1510998 q^{86} + 16231802 q^{87} - 70453923 q^{88} - 10986042 q^{89} - 12851300 q^{90} + 9042852 q^{91} - 16505451 q^{92} - 47230934 q^{93} - 24306151 q^{94} - 21839424 q^{95} - 86512741 q^{96} - 24462382 q^{97} + 705894 q^{98} + 11555078 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.930860 0.0822771 0.0411386 0.999153i \(-0.486901\pi\)
0.0411386 + 0.999153i \(0.486901\pi\)
\(3\) 23.4478 0.501392 0.250696 0.968066i \(-0.419340\pi\)
0.250696 + 0.968066i \(0.419340\pi\)
\(4\) −127.134 −0.993230
\(5\) −494.625 −1.76963 −0.884813 0.465946i \(-0.845714\pi\)
−0.884813 + 0.465946i \(0.845714\pi\)
\(6\) 21.8266 0.0412531
\(7\) 343.000 0.377964
\(8\) −237.493 −0.163997
\(9\) −1637.20 −0.748606
\(10\) −460.427 −0.145600
\(11\) −2424.00 −0.549109 −0.274554 0.961572i \(-0.588530\pi\)
−0.274554 + 0.961572i \(0.588530\pi\)
\(12\) −2981.00 −0.497998
\(13\) 2197.00 0.277350
\(14\) 319.285 0.0310978
\(15\) −11597.9 −0.887277
\(16\) 16052.0 0.979737
\(17\) 3893.57 0.192210 0.0961052 0.995371i \(-0.469361\pi\)
0.0961052 + 0.995371i \(0.469361\pi\)
\(18\) −1524.00 −0.0615931
\(19\) −4518.07 −0.151118 −0.0755588 0.997141i \(-0.524074\pi\)
−0.0755588 + 0.997141i \(0.524074\pi\)
\(20\) 62883.5 1.75765
\(21\) 8042.60 0.189509
\(22\) −2256.40 −0.0451791
\(23\) 66046.1 1.13188 0.565939 0.824447i \(-0.308514\pi\)
0.565939 + 0.824447i \(0.308514\pi\)
\(24\) −5568.70 −0.0822270
\(25\) 166529. 2.13158
\(26\) 2045.10 0.0228196
\(27\) −89669.1 −0.876738
\(28\) −43606.8 −0.375406
\(29\) 6156.48 0.0468748 0.0234374 0.999725i \(-0.492539\pi\)
0.0234374 + 0.999725i \(0.492539\pi\)
\(30\) −10796.0 −0.0730026
\(31\) 17664.1 0.106494 0.0532470 0.998581i \(-0.483043\pi\)
0.0532470 + 0.998581i \(0.483043\pi\)
\(32\) 45341.3 0.244607
\(33\) −56837.5 −0.275319
\(34\) 3624.37 0.0158145
\(35\) −169657. −0.668856
\(36\) 208143. 0.743538
\(37\) −516218. −1.67543 −0.837717 0.546104i \(-0.816110\pi\)
−0.837717 + 0.546104i \(0.816110\pi\)
\(38\) −4205.69 −0.0124335
\(39\) 51514.8 0.139061
\(40\) 117470. 0.290214
\(41\) 470707. 1.06661 0.533307 0.845922i \(-0.320949\pi\)
0.533307 + 0.845922i \(0.320949\pi\)
\(42\) 7486.53 0.0155922
\(43\) −10789.1 −0.0206940 −0.0103470 0.999946i \(-0.503294\pi\)
−0.0103470 + 0.999946i \(0.503294\pi\)
\(44\) 308172. 0.545391
\(45\) 809801. 1.32475
\(46\) 61479.6 0.0931277
\(47\) 326569. 0.458809 0.229405 0.973331i \(-0.426322\pi\)
0.229405 + 0.973331i \(0.426322\pi\)
\(48\) 376385. 0.491233
\(49\) 117649. 0.142857
\(50\) 155015. 0.175380
\(51\) 91295.7 0.0963729
\(52\) −279312. −0.275473
\(53\) 1.91089e6 1.76307 0.881536 0.472117i \(-0.156510\pi\)
0.881536 + 0.472117i \(0.156510\pi\)
\(54\) −83469.3 −0.0721355
\(55\) 1.19897e6 0.971717
\(56\) −81460.3 −0.0619852
\(57\) −105939. −0.0757692
\(58\) 5730.82 0.00385673
\(59\) 558449. 0.353999 0.176999 0.984211i \(-0.443361\pi\)
0.176999 + 0.984211i \(0.443361\pi\)
\(60\) 1.47448e6 0.881271
\(61\) −1.32987e6 −0.750162 −0.375081 0.926992i \(-0.622385\pi\)
−0.375081 + 0.926992i \(0.622385\pi\)
\(62\) 16442.8 0.00876203
\(63\) −561560. −0.282946
\(64\) −2.01245e6 −0.959612
\(65\) −1.08669e6 −0.490806
\(66\) −52907.7 −0.0226524
\(67\) 2.71304e6 1.10203 0.551017 0.834494i \(-0.314240\pi\)
0.551017 + 0.834494i \(0.314240\pi\)
\(68\) −495004. −0.190909
\(69\) 1.54864e6 0.567515
\(70\) −157926. −0.0550315
\(71\) −2.18693e6 −0.725153 −0.362577 0.931954i \(-0.618103\pi\)
−0.362577 + 0.931954i \(0.618103\pi\)
\(72\) 388824. 0.122769
\(73\) −2.12208e6 −0.638458 −0.319229 0.947678i \(-0.603424\pi\)
−0.319229 + 0.947678i \(0.603424\pi\)
\(74\) −480527. −0.137850
\(75\) 3.90475e6 1.06876
\(76\) 574398. 0.150095
\(77\) −831432. −0.207544
\(78\) 47953.1 0.0114416
\(79\) −3.62059e6 −0.826200 −0.413100 0.910686i \(-0.635554\pi\)
−0.413100 + 0.910686i \(0.635554\pi\)
\(80\) −7.93974e6 −1.73377
\(81\) 1.47801e6 0.309016
\(82\) 438162. 0.0877580
\(83\) 8.69991e6 1.67010 0.835048 0.550177i \(-0.185440\pi\)
0.835048 + 0.550177i \(0.185440\pi\)
\(84\) −1.02248e6 −0.188226
\(85\) −1.92586e6 −0.340141
\(86\) −10043.1 −0.00170265
\(87\) 144356. 0.0235027
\(88\) 575684. 0.0900523
\(89\) −2.94631e6 −0.443010 −0.221505 0.975159i \(-0.571097\pi\)
−0.221505 + 0.975159i \(0.571097\pi\)
\(90\) 753811. 0.108997
\(91\) 753571. 0.104828
\(92\) −8.39667e6 −1.12422
\(93\) 414184. 0.0533953
\(94\) 303990. 0.0377495
\(95\) 2.23475e6 0.267422
\(96\) 1.06315e6 0.122644
\(97\) −1.35540e7 −1.50788 −0.753940 0.656944i \(-0.771849\pi\)
−0.753940 + 0.656944i \(0.771849\pi\)
\(98\) 109515. 0.0117539
\(99\) 3.96857e6 0.411066
\(100\) −2.11715e7 −2.11715
\(101\) −877418. −0.0847387 −0.0423694 0.999102i \(-0.513491\pi\)
−0.0423694 + 0.999102i \(0.513491\pi\)
\(102\) 84983.5 0.00792929
\(103\) −4.70617e6 −0.424363 −0.212182 0.977230i \(-0.568057\pi\)
−0.212182 + 0.977230i \(0.568057\pi\)
\(104\) −521773. −0.0454847
\(105\) −3.97807e6 −0.335359
\(106\) 1.77877e6 0.145061
\(107\) 1.75215e7 1.38270 0.691348 0.722522i \(-0.257017\pi\)
0.691348 + 0.722522i \(0.257017\pi\)
\(108\) 1.13999e7 0.870803
\(109\) 2.32662e7 1.72081 0.860405 0.509610i \(-0.170210\pi\)
0.860405 + 0.509610i \(0.170210\pi\)
\(110\) 1.11607e6 0.0799501
\(111\) −1.21042e7 −0.840050
\(112\) 5.50584e6 0.370306
\(113\) −1.35712e7 −0.884798 −0.442399 0.896818i \(-0.645872\pi\)
−0.442399 + 0.896818i \(0.645872\pi\)
\(114\) −98614.1 −0.00623407
\(115\) −3.26681e7 −2.00300
\(116\) −782695. −0.0465575
\(117\) −3.59693e6 −0.207626
\(118\) 519838. 0.0291260
\(119\) 1.33550e6 0.0726487
\(120\) 2.75442e6 0.145511
\(121\) −1.36114e7 −0.698480
\(122\) −1.23792e6 −0.0617212
\(123\) 1.10371e7 0.534792
\(124\) −2.24570e6 −0.105773
\(125\) −4.37271e7 −2.00247
\(126\) −522733. −0.0232800
\(127\) 3.70642e7 1.60562 0.802808 0.596238i \(-0.203338\pi\)
0.802808 + 0.596238i \(0.203338\pi\)
\(128\) −7.67700e6 −0.323561
\(129\) −252981. −0.0103758
\(130\) −1.01156e6 −0.0403821
\(131\) 2.90039e7 1.12722 0.563608 0.826042i \(-0.309413\pi\)
0.563608 + 0.826042i \(0.309413\pi\)
\(132\) 7.22595e6 0.273455
\(133\) −1.54970e6 −0.0571171
\(134\) 2.52546e6 0.0906722
\(135\) 4.43526e7 1.55150
\(136\) −924698. −0.0315220
\(137\) −5.47775e7 −1.82004 −0.910018 0.414568i \(-0.863933\pi\)
−0.910018 + 0.414568i \(0.863933\pi\)
\(138\) 1.44156e6 0.0466935
\(139\) 5.17133e6 0.163324 0.0816620 0.996660i \(-0.473977\pi\)
0.0816620 + 0.996660i \(0.473977\pi\)
\(140\) 2.15690e7 0.664328
\(141\) 7.65732e6 0.230044
\(142\) −2.03572e6 −0.0596635
\(143\) −5.32553e6 −0.152295
\(144\) −2.62804e7 −0.733437
\(145\) −3.04515e6 −0.0829509
\(146\) −1.97536e6 −0.0525305
\(147\) 2.75861e6 0.0716275
\(148\) 6.56287e7 1.66409
\(149\) −4.51810e7 −1.11893 −0.559467 0.828853i \(-0.688994\pi\)
−0.559467 + 0.828853i \(0.688994\pi\)
\(150\) 3.63477e6 0.0879342
\(151\) −4.62761e7 −1.09380 −0.546899 0.837199i \(-0.684192\pi\)
−0.546899 + 0.837199i \(0.684192\pi\)
\(152\) 1.07301e6 0.0247829
\(153\) −6.37456e6 −0.143890
\(154\) −773946. −0.0170761
\(155\) −8.73711e6 −0.188455
\(156\) −6.54926e6 −0.138120
\(157\) 6.09778e7 1.25754 0.628771 0.777590i \(-0.283558\pi\)
0.628771 + 0.777590i \(0.283558\pi\)
\(158\) −3.37027e6 −0.0679774
\(159\) 4.48062e7 0.883991
\(160\) −2.24270e7 −0.432863
\(161\) 2.26538e7 0.427810
\(162\) 1.37582e6 0.0254249
\(163\) 9.57829e7 1.73233 0.866166 0.499756i \(-0.166577\pi\)
0.866166 + 0.499756i \(0.166577\pi\)
\(164\) −5.98427e7 −1.05939
\(165\) 2.81133e7 0.487211
\(166\) 8.09840e6 0.137411
\(167\) 9.10292e7 1.51242 0.756211 0.654328i \(-0.227048\pi\)
0.756211 + 0.654328i \(0.227048\pi\)
\(168\) −1.91006e6 −0.0310789
\(169\) 4.82681e6 0.0769231
\(170\) −1.79271e6 −0.0279858
\(171\) 7.39698e6 0.113127
\(172\) 1.37166e6 0.0205540
\(173\) 5.65303e7 0.830080 0.415040 0.909803i \(-0.363768\pi\)
0.415040 + 0.909803i \(0.363768\pi\)
\(174\) 134375. 0.00193373
\(175\) 5.71196e7 0.805660
\(176\) −3.89101e7 −0.537982
\(177\) 1.30944e7 0.177492
\(178\) −2.74260e6 −0.0364496
\(179\) 1.07498e8 1.40092 0.700462 0.713690i \(-0.252977\pi\)
0.700462 + 0.713690i \(0.252977\pi\)
\(180\) −1.02953e8 −1.31578
\(181\) 360328. 0.00451672 0.00225836 0.999997i \(-0.499281\pi\)
0.00225836 + 0.999997i \(0.499281\pi\)
\(182\) 701469. 0.00862499
\(183\) −3.11826e7 −0.376126
\(184\) −1.56855e7 −0.185625
\(185\) 2.55335e8 2.96489
\(186\) 385547. 0.00439321
\(187\) −9.43802e6 −0.105544
\(188\) −4.15178e7 −0.455703
\(189\) −3.07565e7 −0.331376
\(190\) 2.08024e6 0.0220027
\(191\) −9.36138e7 −0.972127 −0.486063 0.873924i \(-0.661568\pi\)
−0.486063 + 0.873924i \(0.661568\pi\)
\(192\) −4.71876e7 −0.481142
\(193\) 2.52440e7 0.252759 0.126380 0.991982i \(-0.459664\pi\)
0.126380 + 0.991982i \(0.459664\pi\)
\(194\) −1.26169e7 −0.124064
\(195\) −2.54805e7 −0.246086
\(196\) −1.49571e7 −0.141890
\(197\) 2.48755e7 0.231814 0.115907 0.993260i \(-0.463023\pi\)
0.115907 + 0.993260i \(0.463023\pi\)
\(198\) 3.69418e6 0.0338213
\(199\) 9.49592e7 0.854184 0.427092 0.904208i \(-0.359538\pi\)
0.427092 + 0.904208i \(0.359538\pi\)
\(200\) −3.95496e7 −0.349573
\(201\) 6.36149e7 0.552552
\(202\) −816753. −0.00697206
\(203\) 2.11167e6 0.0177170
\(204\) −1.16067e7 −0.0957205
\(205\) −2.32824e8 −1.88751
\(206\) −4.38079e6 −0.0349154
\(207\) −1.08131e8 −0.847330
\(208\) 3.52663e7 0.271730
\(209\) 1.09518e7 0.0829799
\(210\) −3.70303e6 −0.0275924
\(211\) −1.66589e8 −1.22084 −0.610419 0.792079i \(-0.708999\pi\)
−0.610419 + 0.792079i \(0.708999\pi\)
\(212\) −2.42938e8 −1.75114
\(213\) −5.12786e7 −0.363586
\(214\) 1.63100e7 0.113764
\(215\) 5.33656e6 0.0366207
\(216\) 2.12958e7 0.143783
\(217\) 6.05878e6 0.0402510
\(218\) 2.16576e7 0.141583
\(219\) −4.97582e7 −0.320118
\(220\) −1.52430e8 −0.965139
\(221\) 8.55418e6 0.0533096
\(222\) −1.12673e7 −0.0691169
\(223\) 2.41515e8 1.45840 0.729202 0.684299i \(-0.239892\pi\)
0.729202 + 0.684299i \(0.239892\pi\)
\(224\) 1.55521e7 0.0924529
\(225\) −2.72642e8 −1.59571
\(226\) −1.26329e7 −0.0727986
\(227\) 2.97079e8 1.68571 0.842853 0.538143i \(-0.180874\pi\)
0.842853 + 0.538143i \(0.180874\pi\)
\(228\) 1.34684e7 0.0752563
\(229\) 4.88538e7 0.268828 0.134414 0.990925i \(-0.457085\pi\)
0.134414 + 0.990925i \(0.457085\pi\)
\(230\) −3.04094e7 −0.164801
\(231\) −1.94953e7 −0.104061
\(232\) −1.46212e6 −0.00768735
\(233\) 1.21240e8 0.627915 0.313957 0.949437i \(-0.398345\pi\)
0.313957 + 0.949437i \(0.398345\pi\)
\(234\) −3.34824e6 −0.0170829
\(235\) −1.61529e8 −0.811921
\(236\) −7.09976e7 −0.351602
\(237\) −8.48950e7 −0.414250
\(238\) 1.24316e6 0.00597733
\(239\) −1.38404e8 −0.655779 −0.327889 0.944716i \(-0.606337\pi\)
−0.327889 + 0.944716i \(0.606337\pi\)
\(240\) −1.86169e8 −0.869298
\(241\) 1.27947e8 0.588805 0.294402 0.955682i \(-0.404879\pi\)
0.294402 + 0.955682i \(0.404879\pi\)
\(242\) −1.26703e7 −0.0574689
\(243\) 2.30763e8 1.03168
\(244\) 1.69071e8 0.745084
\(245\) −5.81922e7 −0.252804
\(246\) 1.02739e7 0.0440012
\(247\) −9.92619e6 −0.0419125
\(248\) −4.19511e6 −0.0174647
\(249\) 2.03994e8 0.837374
\(250\) −4.07038e7 −0.164757
\(251\) −1.07601e8 −0.429495 −0.214748 0.976670i \(-0.568893\pi\)
−0.214748 + 0.976670i \(0.568893\pi\)
\(252\) 7.13931e7 0.281031
\(253\) −1.60096e8 −0.621524
\(254\) 3.45016e7 0.132105
\(255\) −4.51572e7 −0.170544
\(256\) 2.50448e8 0.932990
\(257\) −4.81773e7 −0.177042 −0.0885210 0.996074i \(-0.528214\pi\)
−0.0885210 + 0.996074i \(0.528214\pi\)
\(258\) −235489. −0.000853694 0
\(259\) −1.77063e8 −0.633255
\(260\) 1.38155e8 0.487483
\(261\) −1.00794e7 −0.0350908
\(262\) 2.69986e7 0.0927441
\(263\) 4.31418e8 1.46236 0.731178 0.682187i \(-0.238971\pi\)
0.731178 + 0.682187i \(0.238971\pi\)
\(264\) 1.34985e7 0.0451516
\(265\) −9.45175e8 −3.11998
\(266\) −1.44255e6 −0.00469943
\(267\) −6.90845e7 −0.222122
\(268\) −3.44919e8 −1.09457
\(269\) 4.66698e8 1.46185 0.730925 0.682458i \(-0.239089\pi\)
0.730925 + 0.682458i \(0.239089\pi\)
\(270\) 4.12861e7 0.127653
\(271\) −2.20036e8 −0.671587 −0.335793 0.941936i \(-0.609004\pi\)
−0.335793 + 0.941936i \(0.609004\pi\)
\(272\) 6.24997e7 0.188316
\(273\) 1.76696e7 0.0525602
\(274\) −5.09901e7 −0.149747
\(275\) −4.03667e8 −1.17047
\(276\) −1.96883e8 −0.563673
\(277\) −1.79201e8 −0.506596 −0.253298 0.967388i \(-0.581515\pi\)
−0.253298 + 0.967388i \(0.581515\pi\)
\(278\) 4.81378e6 0.0134378
\(279\) −2.89197e7 −0.0797221
\(280\) 4.02923e7 0.109691
\(281\) 2.85279e8 0.767003 0.383501 0.923540i \(-0.374718\pi\)
0.383501 + 0.923540i \(0.374718\pi\)
\(282\) 7.12789e6 0.0189273
\(283\) 2.09994e8 0.550750 0.275375 0.961337i \(-0.411198\pi\)
0.275375 + 0.961337i \(0.411198\pi\)
\(284\) 2.78031e8 0.720244
\(285\) 5.24000e7 0.134083
\(286\) −4.95732e6 −0.0125304
\(287\) 1.61453e8 0.403142
\(288\) −7.42329e7 −0.183114
\(289\) −3.95179e8 −0.963055
\(290\) −2.83461e6 −0.00682497
\(291\) −3.17812e8 −0.756039
\(292\) 2.69788e8 0.634136
\(293\) 1.67817e8 0.389762 0.194881 0.980827i \(-0.437568\pi\)
0.194881 + 0.980827i \(0.437568\pi\)
\(294\) 2.56788e6 0.00589331
\(295\) −2.76223e8 −0.626445
\(296\) 1.22599e8 0.274767
\(297\) 2.17358e8 0.481424
\(298\) −4.20572e7 −0.0920626
\(299\) 1.45103e8 0.313926
\(300\) −4.96424e8 −1.06152
\(301\) −3.70066e6 −0.00782161
\(302\) −4.30765e7 −0.0899945
\(303\) −2.05735e7 −0.0424874
\(304\) −7.25241e7 −0.148056
\(305\) 6.57788e8 1.32751
\(306\) −5.93382e6 −0.0118388
\(307\) −1.67749e8 −0.330885 −0.165442 0.986219i \(-0.552905\pi\)
−0.165442 + 0.986219i \(0.552905\pi\)
\(308\) 1.05703e8 0.206139
\(309\) −1.10349e8 −0.212772
\(310\) −8.13302e6 −0.0155055
\(311\) 4.39640e8 0.828774 0.414387 0.910101i \(-0.363996\pi\)
0.414387 + 0.910101i \(0.363996\pi\)
\(312\) −1.22344e7 −0.0228057
\(313\) 1.04202e9 1.92075 0.960375 0.278712i \(-0.0899074\pi\)
0.960375 + 0.278712i \(0.0899074\pi\)
\(314\) 5.67618e7 0.103467
\(315\) 2.77762e8 0.500709
\(316\) 4.60299e8 0.820607
\(317\) 3.08564e8 0.544049 0.272024 0.962290i \(-0.412307\pi\)
0.272024 + 0.962290i \(0.412307\pi\)
\(318\) 4.17082e7 0.0727322
\(319\) −1.49233e7 −0.0257394
\(320\) 9.95410e8 1.69815
\(321\) 4.10840e8 0.693274
\(322\) 2.10875e7 0.0351990
\(323\) −1.75914e7 −0.0290464
\(324\) −1.87905e8 −0.306924
\(325\) 3.65865e8 0.591193
\(326\) 8.91604e7 0.142531
\(327\) 5.45542e8 0.862801
\(328\) −1.11790e8 −0.174922
\(329\) 1.12013e8 0.173414
\(330\) 2.61695e7 0.0400864
\(331\) 3.23910e8 0.490938 0.245469 0.969404i \(-0.421058\pi\)
0.245469 + 0.969404i \(0.421058\pi\)
\(332\) −1.10605e9 −1.65879
\(333\) 8.45153e8 1.25424
\(334\) 8.47354e7 0.124438
\(335\) −1.34194e9 −1.95019
\(336\) 1.29100e8 0.185669
\(337\) −1.01210e8 −0.144051 −0.0720256 0.997403i \(-0.522946\pi\)
−0.0720256 + 0.997403i \(0.522946\pi\)
\(338\) 4.49308e6 0.00632901
\(339\) −3.18215e8 −0.443631
\(340\) 2.44841e8 0.337838
\(341\) −4.28178e7 −0.0584768
\(342\) 6.88555e6 0.00930780
\(343\) 4.03536e7 0.0539949
\(344\) 2.56234e6 0.00339377
\(345\) −7.65995e8 −1.00429
\(346\) 5.26217e7 0.0682966
\(347\) −3.55081e8 −0.456220 −0.228110 0.973635i \(-0.573255\pi\)
−0.228110 + 0.973635i \(0.573255\pi\)
\(348\) −1.83525e7 −0.0233436
\(349\) −1.43782e9 −1.81057 −0.905284 0.424806i \(-0.860342\pi\)
−0.905284 + 0.424806i \(0.860342\pi\)
\(350\) 5.31703e7 0.0662874
\(351\) −1.97003e8 −0.243163
\(352\) −1.09907e8 −0.134316
\(353\) 3.80937e8 0.460937 0.230469 0.973080i \(-0.425974\pi\)
0.230469 + 0.973080i \(0.425974\pi\)
\(354\) 1.21891e7 0.0146036
\(355\) 1.08171e9 1.28325
\(356\) 3.74574e8 0.440011
\(357\) 3.13144e7 0.0364255
\(358\) 1.00066e8 0.115264
\(359\) −2.87741e8 −0.328225 −0.164112 0.986442i \(-0.552476\pi\)
−0.164112 + 0.986442i \(0.552476\pi\)
\(360\) −1.92322e8 −0.217256
\(361\) −8.73459e8 −0.977163
\(362\) 335415. 0.000371623 0
\(363\) −3.19157e8 −0.350213
\(364\) −9.58041e7 −0.104119
\(365\) 1.04964e9 1.12983
\(366\) −2.90266e7 −0.0309465
\(367\) −2.01071e8 −0.212334 −0.106167 0.994348i \(-0.533858\pi\)
−0.106167 + 0.994348i \(0.533858\pi\)
\(368\) 1.06017e9 1.10894
\(369\) −7.70642e8 −0.798473
\(370\) 2.37681e8 0.243943
\(371\) 6.55435e8 0.666378
\(372\) −5.26567e7 −0.0530339
\(373\) −3.59090e8 −0.358280 −0.179140 0.983824i \(-0.557331\pi\)
−0.179140 + 0.983824i \(0.557331\pi\)
\(374\) −8.78547e6 −0.00868389
\(375\) −1.02530e9 −1.00402
\(376\) −7.75580e7 −0.0752435
\(377\) 1.35258e7 0.0130007
\(378\) −2.86300e7 −0.0272646
\(379\) −1.48278e9 −1.39907 −0.699537 0.714597i \(-0.746610\pi\)
−0.699537 + 0.714597i \(0.746610\pi\)
\(380\) −2.84112e8 −0.265611
\(381\) 8.69074e8 0.805044
\(382\) −8.71413e7 −0.0799838
\(383\) −1.60399e9 −1.45883 −0.729417 0.684069i \(-0.760209\pi\)
−0.729417 + 0.684069i \(0.760209\pi\)
\(384\) −1.80009e8 −0.162231
\(385\) 4.11247e8 0.367274
\(386\) 2.34986e7 0.0207963
\(387\) 1.76639e7 0.0154917
\(388\) 1.72317e9 1.49767
\(389\) 7.26145e8 0.625460 0.312730 0.949842i \(-0.398756\pi\)
0.312730 + 0.949842i \(0.398756\pi\)
\(390\) −2.37188e7 −0.0202473
\(391\) 2.57155e8 0.217559
\(392\) −2.79409e7 −0.0234282
\(393\) 6.80078e8 0.565178
\(394\) 2.31556e7 0.0190730
\(395\) 1.79084e9 1.46206
\(396\) −5.04539e8 −0.408283
\(397\) −6.04307e7 −0.0484720 −0.0242360 0.999706i \(-0.507715\pi\)
−0.0242360 + 0.999706i \(0.507715\pi\)
\(398\) 8.83937e7 0.0702798
\(399\) −3.63370e7 −0.0286381
\(400\) 2.67313e9 2.08838
\(401\) −1.70581e9 −1.32107 −0.660534 0.750796i \(-0.729670\pi\)
−0.660534 + 0.750796i \(0.729670\pi\)
\(402\) 5.92166e7 0.0454624
\(403\) 3.88080e7 0.0295361
\(404\) 1.11549e8 0.0841651
\(405\) −7.31063e8 −0.546843
\(406\) 1.96567e6 0.00145771
\(407\) 1.25131e9 0.919995
\(408\) −2.16821e7 −0.0158049
\(409\) −9.81645e8 −0.709452 −0.354726 0.934970i \(-0.615426\pi\)
−0.354726 + 0.934970i \(0.615426\pi\)
\(410\) −2.16726e8 −0.155299
\(411\) −1.28441e9 −0.912553
\(412\) 5.98312e8 0.421490
\(413\) 1.91548e8 0.133799
\(414\) −1.00654e8 −0.0697159
\(415\) −4.30320e9 −2.95545
\(416\) 9.96149e7 0.0678419
\(417\) 1.21256e8 0.0818895
\(418\) 1.01946e7 0.00682735
\(419\) 7.44824e8 0.494657 0.247329 0.968932i \(-0.420447\pi\)
0.247329 + 0.968932i \(0.420447\pi\)
\(420\) 5.05746e8 0.333089
\(421\) −2.00261e9 −1.30800 −0.654001 0.756494i \(-0.726911\pi\)
−0.654001 + 0.756494i \(0.726911\pi\)
\(422\) −1.55071e8 −0.100447
\(423\) −5.34659e8 −0.343467
\(424\) −4.53824e8 −0.289139
\(425\) 6.48394e8 0.409711
\(426\) −4.77332e7 −0.0299148
\(427\) −4.56146e8 −0.283535
\(428\) −2.22756e9 −1.37334
\(429\) −1.24872e8 −0.0763597
\(430\) 4.96759e6 0.00301305
\(431\) −1.33664e9 −0.804162 −0.402081 0.915604i \(-0.631713\pi\)
−0.402081 + 0.915604i \(0.631713\pi\)
\(432\) −1.43937e9 −0.858973
\(433\) 1.70086e9 1.00684 0.503421 0.864042i \(-0.332075\pi\)
0.503421 + 0.864042i \(0.332075\pi\)
\(434\) 5.63988e6 0.00331174
\(435\) −7.14022e7 −0.0415910
\(436\) −2.95792e9 −1.70916
\(437\) −2.98401e8 −0.171047
\(438\) −4.63179e7 −0.0263384
\(439\) 9.42702e8 0.531800 0.265900 0.964001i \(-0.414331\pi\)
0.265900 + 0.964001i \(0.414331\pi\)
\(440\) −2.84748e8 −0.159359
\(441\) −1.92615e8 −0.106944
\(442\) 7.96274e6 0.00438616
\(443\) −6.17406e8 −0.337410 −0.168705 0.985667i \(-0.553958\pi\)
−0.168705 + 0.985667i \(0.553958\pi\)
\(444\) 1.53885e9 0.834364
\(445\) 1.45732e9 0.783961
\(446\) 2.24817e8 0.119993
\(447\) −1.05940e9 −0.561025
\(448\) −6.90271e8 −0.362699
\(449\) −2.97982e9 −1.55356 −0.776780 0.629772i \(-0.783148\pi\)
−0.776780 + 0.629772i \(0.783148\pi\)
\(450\) −2.53791e8 −0.131290
\(451\) −1.14099e9 −0.585687
\(452\) 1.72536e9 0.878808
\(453\) −1.08507e9 −0.548422
\(454\) 2.76539e8 0.138695
\(455\) −3.72735e8 −0.185507
\(456\) 2.51598e7 0.0124259
\(457\) −1.82029e9 −0.892141 −0.446070 0.894998i \(-0.647177\pi\)
−0.446070 + 0.894998i \(0.647177\pi\)
\(458\) 4.54760e7 0.0221184
\(459\) −3.49133e8 −0.168518
\(460\) 4.15321e9 1.98944
\(461\) 3.33514e9 1.58548 0.792740 0.609560i \(-0.208654\pi\)
0.792740 + 0.609560i \(0.208654\pi\)
\(462\) −1.81473e7 −0.00856182
\(463\) 4.70190e8 0.220161 0.110080 0.993923i \(-0.464889\pi\)
0.110080 + 0.993923i \(0.464889\pi\)
\(464\) 9.88240e7 0.0459250
\(465\) −2.04866e8 −0.0944897
\(466\) 1.12858e8 0.0516630
\(467\) −1.86051e9 −0.845325 −0.422663 0.906287i \(-0.638904\pi\)
−0.422663 + 0.906287i \(0.638904\pi\)
\(468\) 4.57290e8 0.206220
\(469\) 9.30574e8 0.416530
\(470\) −1.50361e8 −0.0668025
\(471\) 1.42980e9 0.630523
\(472\) −1.32628e8 −0.0580548
\(473\) 2.61528e7 0.0113633
\(474\) −7.90253e7 −0.0340833
\(475\) −7.52391e8 −0.322119
\(476\) −1.69786e8 −0.0721569
\(477\) −3.12851e9 −1.31985
\(478\) −1.28835e8 −0.0539556
\(479\) −1.06883e8 −0.0444358 −0.0222179 0.999753i \(-0.507073\pi\)
−0.0222179 + 0.999753i \(0.507073\pi\)
\(480\) −5.25863e8 −0.217034
\(481\) −1.13413e9 −0.464682
\(482\) 1.19101e8 0.0484452
\(483\) 5.31182e8 0.214501
\(484\) 1.73046e9 0.693751
\(485\) 6.70416e9 2.66838
\(486\) 2.14807e8 0.0848833
\(487\) 2.49058e9 0.977121 0.488560 0.872530i \(-0.337522\pi\)
0.488560 + 0.872530i \(0.337522\pi\)
\(488\) 3.15836e8 0.123025
\(489\) 2.24590e9 0.868578
\(490\) −5.41688e7 −0.0208000
\(491\) 8.23614e8 0.314006 0.157003 0.987598i \(-0.449817\pi\)
0.157003 + 0.987598i \(0.449817\pi\)
\(492\) −1.40318e9 −0.531172
\(493\) 2.39707e7 0.00900984
\(494\) −9.23989e6 −0.00344844
\(495\) −1.96296e9 −0.727433
\(496\) 2.83544e8 0.104336
\(497\) −7.50115e8 −0.274082
\(498\) 1.89890e8 0.0688967
\(499\) −2.73013e8 −0.0983629 −0.0491814 0.998790i \(-0.515661\pi\)
−0.0491814 + 0.998790i \(0.515661\pi\)
\(500\) 5.55917e9 1.98891
\(501\) 2.13443e9 0.758317
\(502\) −1.00161e8 −0.0353376
\(503\) 3.81756e9 1.33751 0.668756 0.743482i \(-0.266827\pi\)
0.668756 + 0.743482i \(0.266827\pi\)
\(504\) 1.33367e8 0.0464024
\(505\) 4.33993e8 0.149956
\(506\) −1.49027e8 −0.0511372
\(507\) 1.13178e8 0.0385687
\(508\) −4.71210e9 −1.59475
\(509\) 2.03749e9 0.684830 0.342415 0.939549i \(-0.388755\pi\)
0.342415 + 0.939549i \(0.388755\pi\)
\(510\) −4.20350e7 −0.0140319
\(511\) −7.27875e8 −0.241315
\(512\) 1.21579e9 0.400325
\(513\) 4.05131e8 0.132490
\(514\) −4.48463e7 −0.0145665
\(515\) 2.32779e9 0.750964
\(516\) 3.21623e7 0.0103056
\(517\) −7.91603e8 −0.251936
\(518\) −1.64821e8 −0.0521024
\(519\) 1.32551e9 0.416196
\(520\) 2.58082e8 0.0804908
\(521\) 2.57357e9 0.797269 0.398634 0.917110i \(-0.369484\pi\)
0.398634 + 0.917110i \(0.369484\pi\)
\(522\) −9.38250e6 −0.00288717
\(523\) 3.57873e9 1.09389 0.546944 0.837169i \(-0.315791\pi\)
0.546944 + 0.837169i \(0.315791\pi\)
\(524\) −3.68737e9 −1.11959
\(525\) 1.33933e9 0.403952
\(526\) 4.01590e8 0.120319
\(527\) 6.87764e7 0.0204693
\(528\) −9.12356e8 −0.269740
\(529\) 9.57259e8 0.281148
\(530\) −8.79825e8 −0.256703
\(531\) −9.14293e8 −0.265005
\(532\) 1.97018e8 0.0567304
\(533\) 1.03414e9 0.295826
\(534\) −6.43079e7 −0.0182755
\(535\) −8.66656e9 −2.44686
\(536\) −6.44330e8 −0.180731
\(537\) 2.52059e9 0.702413
\(538\) 4.34430e8 0.120277
\(539\) −2.85181e8 −0.0784441
\(540\) −5.63870e9 −1.54099
\(541\) −2.56998e8 −0.0697812 −0.0348906 0.999391i \(-0.511108\pi\)
−0.0348906 + 0.999391i \(0.511108\pi\)
\(542\) −2.04823e8 −0.0552562
\(543\) 8.44890e6 0.00226465
\(544\) 1.76540e8 0.0470161
\(545\) −1.15081e10 −3.04519
\(546\) 1.64479e7 0.00432450
\(547\) −3.40405e9 −0.889283 −0.444641 0.895709i \(-0.646669\pi\)
−0.444641 + 0.895709i \(0.646669\pi\)
\(548\) 6.96405e9 1.80772
\(549\) 2.17727e9 0.561576
\(550\) −3.75757e8 −0.0963026
\(551\) −2.78154e7 −0.00708361
\(552\) −3.67791e8 −0.0930709
\(553\) −1.24186e9 −0.312274
\(554\) −1.66811e8 −0.0416813
\(555\) 5.98704e9 1.48657
\(556\) −6.57449e8 −0.162218
\(557\) 4.01934e9 0.985511 0.492756 0.870168i \(-0.335990\pi\)
0.492756 + 0.870168i \(0.335990\pi\)
\(558\) −2.69201e7 −0.00655930
\(559\) −2.37036e7 −0.00573949
\(560\) −2.72333e9 −0.655303
\(561\) −2.21301e8 −0.0529192
\(562\) 2.65554e8 0.0631068
\(563\) 2.08931e9 0.493428 0.246714 0.969088i \(-0.420649\pi\)
0.246714 + 0.969088i \(0.420649\pi\)
\(564\) −9.73502e8 −0.228486
\(565\) 6.71267e9 1.56576
\(566\) 1.95475e8 0.0453141
\(567\) 5.06959e8 0.116797
\(568\) 5.19381e8 0.118923
\(569\) −3.02694e9 −0.688829 −0.344415 0.938818i \(-0.611923\pi\)
−0.344415 + 0.938818i \(0.611923\pi\)
\(570\) 4.87770e7 0.0110320
\(571\) −4.05575e9 −0.911684 −0.455842 0.890061i \(-0.650662\pi\)
−0.455842 + 0.890061i \(0.650662\pi\)
\(572\) 6.77053e8 0.151264
\(573\) −2.19504e9 −0.487417
\(574\) 1.50290e8 0.0331694
\(575\) 1.09986e10 2.41268
\(576\) 3.29479e9 0.718371
\(577\) −7.14490e9 −1.54839 −0.774196 0.632946i \(-0.781846\pi\)
−0.774196 + 0.632946i \(0.781846\pi\)
\(578\) −3.67856e8 −0.0792374
\(579\) 5.91915e8 0.126731
\(580\) 3.87141e8 0.0823894
\(581\) 2.98407e9 0.631237
\(582\) −2.95838e8 −0.0622047
\(583\) −4.63200e9 −0.968118
\(584\) 5.03981e8 0.104705
\(585\) 1.77913e9 0.367420
\(586\) 1.56214e8 0.0320685
\(587\) 2.15170e9 0.439085 0.219543 0.975603i \(-0.429544\pi\)
0.219543 + 0.975603i \(0.429544\pi\)
\(588\) −3.50712e8 −0.0711426
\(589\) −7.98075e7 −0.0160931
\(590\) −2.57125e8 −0.0515421
\(591\) 5.83275e8 0.116230
\(592\) −8.28635e9 −1.64149
\(593\) −5.66161e9 −1.11493 −0.557466 0.830200i \(-0.688226\pi\)
−0.557466 + 0.830200i \(0.688226\pi\)
\(594\) 2.02330e8 0.0396102
\(595\) −6.60570e8 −0.128561
\(596\) 5.74402e9 1.11136
\(597\) 2.22659e9 0.428281
\(598\) 1.35071e8 0.0258290
\(599\) 5.97313e9 1.13556 0.567778 0.823182i \(-0.307803\pi\)
0.567778 + 0.823182i \(0.307803\pi\)
\(600\) −9.27352e8 −0.175273
\(601\) −1.48842e8 −0.0279681 −0.0139841 0.999902i \(-0.504451\pi\)
−0.0139841 + 0.999902i \(0.504451\pi\)
\(602\) −3.44479e6 −0.000643540 0
\(603\) −4.44180e9 −0.824989
\(604\) 5.88324e9 1.08639
\(605\) 6.73254e9 1.23605
\(606\) −1.91511e7 −0.00349574
\(607\) 8.30448e9 1.50714 0.753568 0.657370i \(-0.228331\pi\)
0.753568 + 0.657370i \(0.228331\pi\)
\(608\) −2.04855e8 −0.0369645
\(609\) 4.95141e7 0.00888318
\(610\) 6.12308e8 0.109223
\(611\) 7.17472e8 0.127251
\(612\) 8.10420e8 0.142916
\(613\) −3.33245e9 −0.584321 −0.292160 0.956369i \(-0.594374\pi\)
−0.292160 + 0.956369i \(0.594374\pi\)
\(614\) −1.56151e8 −0.0272243
\(615\) −5.45921e9 −0.946382
\(616\) 1.97460e8 0.0340366
\(617\) 5.78863e9 0.992152 0.496076 0.868279i \(-0.334774\pi\)
0.496076 + 0.868279i \(0.334774\pi\)
\(618\) −1.02720e8 −0.0175063
\(619\) −8.22318e9 −1.39355 −0.696775 0.717290i \(-0.745382\pi\)
−0.696775 + 0.717290i \(0.745382\pi\)
\(620\) 1.11078e9 0.187179
\(621\) −5.92229e9 −0.992360
\(622\) 4.09243e8 0.0681891
\(623\) −1.01058e9 −0.167442
\(624\) 8.26917e8 0.136243
\(625\) 8.61841e9 1.41204
\(626\) 9.69974e8 0.158034
\(627\) 2.56795e8 0.0416055
\(628\) −7.75232e9 −1.24903
\(629\) −2.00993e9 −0.322036
\(630\) 2.58557e8 0.0411969
\(631\) −5.56524e9 −0.881823 −0.440911 0.897551i \(-0.645345\pi\)
−0.440911 + 0.897551i \(0.645345\pi\)
\(632\) 8.59868e8 0.135495
\(633\) −3.90615e9 −0.612119
\(634\) 2.87230e8 0.0447628
\(635\) −1.83329e10 −2.84134
\(636\) −5.69636e9 −0.878007
\(637\) 2.58475e8 0.0396214
\(638\) −1.38915e7 −0.00211776
\(639\) 3.58044e9 0.542854
\(640\) 3.79724e9 0.572583
\(641\) −2.29706e9 −0.344484 −0.172242 0.985055i \(-0.555101\pi\)
−0.172242 + 0.985055i \(0.555101\pi\)
\(642\) 3.82434e8 0.0570406
\(643\) 5.78591e9 0.858289 0.429145 0.903236i \(-0.358815\pi\)
0.429145 + 0.903236i \(0.358815\pi\)
\(644\) −2.88006e9 −0.424914
\(645\) 1.25131e8 0.0183614
\(646\) −1.63751e7 −0.00238985
\(647\) −1.34228e9 −0.194840 −0.0974201 0.995243i \(-0.531059\pi\)
−0.0974201 + 0.995243i \(0.531059\pi\)
\(648\) −3.51019e8 −0.0506778
\(649\) −1.35368e9 −0.194384
\(650\) 3.40569e8 0.0486417
\(651\) 1.42065e8 0.0201815
\(652\) −1.21772e10 −1.72061
\(653\) 7.74737e8 0.108883 0.0544413 0.998517i \(-0.482662\pi\)
0.0544413 + 0.998517i \(0.482662\pi\)
\(654\) 5.07823e8 0.0709888
\(655\) −1.43461e10 −1.99475
\(656\) 7.55580e9 1.04500
\(657\) 3.47428e9 0.477953
\(658\) 1.04268e8 0.0142680
\(659\) −1.02191e10 −1.39095 −0.695477 0.718549i \(-0.744807\pi\)
−0.695477 + 0.718549i \(0.744807\pi\)
\(660\) −3.57414e9 −0.483913
\(661\) −7.88600e9 −1.06207 −0.531033 0.847351i \(-0.678196\pi\)
−0.531033 + 0.847351i \(0.678196\pi\)
\(662\) 3.01515e8 0.0403930
\(663\) 2.00577e8 0.0267290
\(664\) −2.06617e9 −0.273891
\(665\) 7.66519e8 0.101076
\(666\) 7.86719e8 0.103195
\(667\) 4.06612e8 0.0530566
\(668\) −1.15729e10 −1.50218
\(669\) 5.66301e9 0.731233
\(670\) −1.24916e9 −0.160456
\(671\) 3.22361e9 0.411920
\(672\) 3.64662e8 0.0463552
\(673\) 9.48355e9 1.19927 0.599637 0.800272i \(-0.295312\pi\)
0.599637 + 0.800272i \(0.295312\pi\)
\(674\) −9.42119e7 −0.0118521
\(675\) −1.49325e10 −1.86883
\(676\) −6.13649e8 −0.0764023
\(677\) −1.15389e10 −1.42924 −0.714621 0.699512i \(-0.753401\pi\)
−0.714621 + 0.699512i \(0.753401\pi\)
\(678\) −2.96214e8 −0.0365007
\(679\) −4.64902e9 −0.569925
\(680\) 4.57379e8 0.0557822
\(681\) 6.96586e9 0.845201
\(682\) −3.98573e7 −0.00481130
\(683\) −1.07688e10 −1.29328 −0.646641 0.762795i \(-0.723827\pi\)
−0.646641 + 0.762795i \(0.723827\pi\)
\(684\) −9.40404e8 −0.112362
\(685\) 2.70943e10 3.22078
\(686\) 3.75635e7 0.00444255
\(687\) 1.14551e9 0.134788
\(688\) −1.73187e8 −0.0202747
\(689\) 4.19822e9 0.488988
\(690\) −7.13033e8 −0.0826301
\(691\) 1.58123e10 1.82315 0.911576 0.411132i \(-0.134866\pi\)
0.911576 + 0.411132i \(0.134866\pi\)
\(692\) −7.18689e9 −0.824460
\(693\) 1.36122e9 0.155368
\(694\) −3.30530e8 −0.0375364
\(695\) −2.55787e9 −0.289022
\(696\) −3.42836e7 −0.00385438
\(697\) 1.83273e9 0.205014
\(698\) −1.33841e9 −0.148968
\(699\) 2.84282e9 0.314832
\(700\) −7.26181e9 −0.800206
\(701\) 1.58969e10 1.74301 0.871506 0.490385i \(-0.163144\pi\)
0.871506 + 0.490385i \(0.163144\pi\)
\(702\) −1.83382e8 −0.0200068
\(703\) 2.33231e9 0.253188
\(704\) 4.87818e9 0.526931
\(705\) −3.78751e9 −0.407091
\(706\) 3.54599e8 0.0379246
\(707\) −3.00954e8 −0.0320282
\(708\) −1.66474e9 −0.176291
\(709\) 1.17268e10 1.23571 0.617856 0.786291i \(-0.288001\pi\)
0.617856 + 0.786291i \(0.288001\pi\)
\(710\) 1.00692e9 0.105582
\(711\) 5.92764e9 0.618498
\(712\) 6.99729e8 0.0726524
\(713\) 1.16664e9 0.120538
\(714\) 2.91493e7 0.00299699
\(715\) 2.63414e9 0.269506
\(716\) −1.36666e10 −1.39144
\(717\) −3.24528e9 −0.328802
\(718\) −2.67846e8 −0.0270054
\(719\) 1.50068e10 1.50570 0.752849 0.658194i \(-0.228679\pi\)
0.752849 + 0.658194i \(0.228679\pi\)
\(720\) 1.29989e10 1.29791
\(721\) −1.61422e9 −0.160394
\(722\) −8.13068e8 −0.0803982
\(723\) 3.00008e9 0.295222
\(724\) −4.58098e7 −0.00448614
\(725\) 1.02524e9 0.0999173
\(726\) −2.97091e8 −0.0288145
\(727\) 8.31455e9 0.802543 0.401271 0.915959i \(-0.368568\pi\)
0.401271 + 0.915959i \(0.368568\pi\)
\(728\) −1.78968e8 −0.0171916
\(729\) 2.17846e9 0.208259
\(730\) 9.77064e8 0.0929594
\(731\) −4.20081e7 −0.00397761
\(732\) 3.96435e9 0.373579
\(733\) −2.20982e8 −0.0207249 −0.0103625 0.999946i \(-0.503299\pi\)
−0.0103625 + 0.999946i \(0.503299\pi\)
\(734\) −1.87169e8 −0.0174702
\(735\) −1.36448e9 −0.126754
\(736\) 2.99462e9 0.276866
\(737\) −6.57642e9 −0.605136
\(738\) −7.17360e8 −0.0656961
\(739\) −1.17435e10 −1.07039 −0.535196 0.844728i \(-0.679762\pi\)
−0.535196 + 0.844728i \(0.679762\pi\)
\(740\) −3.24616e10 −2.94482
\(741\) −2.32747e8 −0.0210146
\(742\) 6.10118e8 0.0548277
\(743\) −1.70101e10 −1.52141 −0.760703 0.649100i \(-0.775146\pi\)
−0.760703 + 0.649100i \(0.775146\pi\)
\(744\) −9.83660e7 −0.00875669
\(745\) 2.23477e10 1.98009
\(746\) −3.34262e8 −0.0294782
\(747\) −1.42435e10 −1.25024
\(748\) 1.19989e9 0.104830
\(749\) 6.00986e9 0.522610
\(750\) −9.54414e8 −0.0826080
\(751\) 1.69564e10 1.46081 0.730406 0.683013i \(-0.239331\pi\)
0.730406 + 0.683013i \(0.239331\pi\)
\(752\) 5.24209e9 0.449513
\(753\) −2.52301e9 −0.215346
\(754\) 1.25906e7 0.00106966
\(755\) 2.28893e10 1.93561
\(756\) 3.91018e9 0.329132
\(757\) 2.09466e10 1.75500 0.877502 0.479573i \(-0.159208\pi\)
0.877502 + 0.479573i \(0.159208\pi\)
\(758\) −1.38026e9 −0.115112
\(759\) −3.75389e9 −0.311627
\(760\) −5.30739e8 −0.0438564
\(761\) −1.82678e10 −1.50259 −0.751293 0.659969i \(-0.770569\pi\)
−0.751293 + 0.659969i \(0.770569\pi\)
\(762\) 8.08986e8 0.0662367
\(763\) 7.98031e9 0.650405
\(764\) 1.19014e10 0.965546
\(765\) 3.15302e9 0.254631
\(766\) −1.49309e9 −0.120029
\(767\) 1.22691e9 0.0981815
\(768\) 5.87245e9 0.467794
\(769\) 1.33445e10 1.05818 0.529090 0.848566i \(-0.322533\pi\)
0.529090 + 0.848566i \(0.322533\pi\)
\(770\) 3.82814e8 0.0302183
\(771\) −1.12965e9 −0.0887676
\(772\) −3.20935e9 −0.251048
\(773\) 7.46551e9 0.581342 0.290671 0.956823i \(-0.406122\pi\)
0.290671 + 0.956823i \(0.406122\pi\)
\(774\) 1.64426e7 0.00127461
\(775\) 2.94159e9 0.227000
\(776\) 3.21899e9 0.247288
\(777\) −4.15174e9 −0.317509
\(778\) 6.75939e8 0.0514611
\(779\) −2.12669e9 −0.161184
\(780\) 3.23943e9 0.244421
\(781\) 5.30111e9 0.398188
\(782\) 2.39375e8 0.0179001
\(783\) −5.52046e8 −0.0410969
\(784\) 1.88850e9 0.139962
\(785\) −3.01612e10 −2.22538
\(786\) 6.33057e8 0.0465012
\(787\) −1.97846e10 −1.44682 −0.723412 0.690417i \(-0.757427\pi\)
−0.723412 + 0.690417i \(0.757427\pi\)
\(788\) −3.16250e9 −0.230245
\(789\) 1.01158e10 0.733214
\(790\) 1.66702e9 0.120294
\(791\) −4.65493e9 −0.334422
\(792\) −9.42510e8 −0.0674137
\(793\) −2.92173e9 −0.208058
\(794\) −5.62525e7 −0.00398813
\(795\) −2.21623e10 −1.56433
\(796\) −1.20725e10 −0.848401
\(797\) 9.89344e9 0.692218 0.346109 0.938194i \(-0.387503\pi\)
0.346109 + 0.938194i \(0.387503\pi\)
\(798\) −3.38246e7 −0.00235626
\(799\) 1.27152e9 0.0881880
\(800\) 7.55066e9 0.521399
\(801\) 4.82370e9 0.331639
\(802\) −1.58787e9 −0.108694
\(803\) 5.14393e9 0.350583
\(804\) −8.08759e9 −0.548811
\(805\) −1.12051e10 −0.757063
\(806\) 3.61248e7 0.00243015
\(807\) 1.09430e10 0.732960
\(808\) 2.08381e8 0.0138969
\(809\) 2.41288e10 1.60219 0.801097 0.598534i \(-0.204250\pi\)
0.801097 + 0.598534i \(0.204250\pi\)
\(810\) −6.80517e8 −0.0449927
\(811\) 1.84984e10 1.21776 0.608881 0.793262i \(-0.291619\pi\)
0.608881 + 0.793262i \(0.291619\pi\)
\(812\) −2.68465e8 −0.0175971
\(813\) −5.15937e9 −0.336728
\(814\) 1.16480e9 0.0756946
\(815\) −4.73767e10 −3.06558
\(816\) 1.46548e9 0.0944201
\(817\) 4.87458e7 0.00312723
\(818\) −9.13773e8 −0.0583716
\(819\) −1.23375e9 −0.0784752
\(820\) 2.95997e10 1.87473
\(821\) 2.81863e10 1.77761 0.888807 0.458283i \(-0.151535\pi\)
0.888807 + 0.458283i \(0.151535\pi\)
\(822\) −1.19561e9 −0.0750822
\(823\) 2.04648e10 1.27970 0.639851 0.768499i \(-0.278996\pi\)
0.639851 + 0.768499i \(0.278996\pi\)
\(824\) 1.11769e9 0.0695944
\(825\) −9.46511e9 −0.586863
\(826\) 1.78304e8 0.0110086
\(827\) 7.73565e9 0.475584 0.237792 0.971316i \(-0.423576\pi\)
0.237792 + 0.971316i \(0.423576\pi\)
\(828\) 1.37470e10 0.841594
\(829\) 2.90659e10 1.77192 0.885959 0.463763i \(-0.153501\pi\)
0.885959 + 0.463763i \(0.153501\pi\)
\(830\) −4.00567e9 −0.243166
\(831\) −4.20187e9 −0.254003
\(832\) −4.42136e9 −0.266148
\(833\) 4.58075e8 0.0274586
\(834\) 1.12873e8 0.00673763
\(835\) −4.50253e10 −2.67642
\(836\) −1.39234e9 −0.0824182
\(837\) −1.58392e9 −0.0933674
\(838\) 6.93326e8 0.0406990
\(839\) −2.26036e10 −1.32133 −0.660666 0.750680i \(-0.729726\pi\)
−0.660666 + 0.750680i \(0.729726\pi\)
\(840\) 9.44766e8 0.0549980
\(841\) −1.72120e10 −0.997803
\(842\) −1.86415e9 −0.107619
\(843\) 6.68915e9 0.384569
\(844\) 2.11790e10 1.21257
\(845\) −2.38746e9 −0.136125
\(846\) −4.97692e8 −0.0282595
\(847\) −4.66871e9 −0.264001
\(848\) 3.06736e10 1.72735
\(849\) 4.92390e9 0.276142
\(850\) 6.03564e8 0.0337099
\(851\) −3.40942e10 −1.89639
\(852\) 6.51923e9 0.361125
\(853\) −2.79034e9 −0.153934 −0.0769671 0.997034i \(-0.524524\pi\)
−0.0769671 + 0.997034i \(0.524524\pi\)
\(854\) −4.24608e8 −0.0233284
\(855\) −3.65873e9 −0.200193
\(856\) −4.16123e9 −0.226759
\(857\) −1.71802e10 −0.932387 −0.466194 0.884683i \(-0.654375\pi\)
−0.466194 + 0.884683i \(0.654375\pi\)
\(858\) −1.16238e8 −0.00628266
\(859\) 3.72890e9 0.200726 0.100363 0.994951i \(-0.468000\pi\)
0.100363 + 0.994951i \(0.468000\pi\)
\(860\) −6.78456e8 −0.0363728
\(861\) 3.78571e9 0.202132
\(862\) −1.24422e9 −0.0661641
\(863\) 1.86053e10 0.985369 0.492685 0.870208i \(-0.336016\pi\)
0.492685 + 0.870208i \(0.336016\pi\)
\(864\) −4.06572e9 −0.214456
\(865\) −2.79613e10 −1.46893
\(866\) 1.58326e9 0.0828400
\(867\) −9.26607e9 −0.482869
\(868\) −7.70274e8 −0.0399785
\(869\) 8.77632e9 0.453673
\(870\) −6.64654e7 −0.00342199
\(871\) 5.96056e9 0.305649
\(872\) −5.52557e9 −0.282208
\(873\) 2.21906e10 1.12881
\(874\) −2.77769e8 −0.0140732
\(875\) −1.49984e10 −0.756861
\(876\) 6.32593e9 0.317951
\(877\) 1.56029e10 0.781098 0.390549 0.920582i \(-0.372285\pi\)
0.390549 + 0.920582i \(0.372285\pi\)
\(878\) 8.77523e8 0.0437550
\(879\) 3.93494e9 0.195424
\(880\) 1.92459e10 0.952027
\(881\) −2.33038e10 −1.14818 −0.574092 0.818791i \(-0.694645\pi\)
−0.574092 + 0.818791i \(0.694645\pi\)
\(882\) −1.79298e8 −0.00879902
\(883\) 1.37344e10 0.671349 0.335675 0.941978i \(-0.391036\pi\)
0.335675 + 0.941978i \(0.391036\pi\)
\(884\) −1.08752e9 −0.0529487
\(885\) −6.47683e9 −0.314095
\(886\) −5.74718e8 −0.0277611
\(887\) −3.05740e10 −1.47102 −0.735511 0.677512i \(-0.763058\pi\)
−0.735511 + 0.677512i \(0.763058\pi\)
\(888\) 2.87467e9 0.137766
\(889\) 1.27130e10 0.606866
\(890\) 1.35656e9 0.0645021
\(891\) −3.58270e9 −0.169683
\(892\) −3.07047e10 −1.44853
\(893\) −1.47546e9 −0.0693342
\(894\) −9.86149e8 −0.0461595
\(895\) −5.31712e10 −2.47911
\(896\) −2.63321e9 −0.122295
\(897\) 3.40235e9 0.157400
\(898\) −2.77380e9 −0.127823
\(899\) 1.08749e8 0.00499189
\(900\) 3.46619e10 1.58491
\(901\) 7.44019e9 0.338881
\(902\) −1.06211e9 −0.0481886
\(903\) −8.67723e7 −0.00392170
\(904\) 3.22307e9 0.145104
\(905\) −1.78227e8 −0.00799290
\(906\) −1.01005e9 −0.0451226
\(907\) −8.11332e9 −0.361055 −0.180527 0.983570i \(-0.557780\pi\)
−0.180527 + 0.983570i \(0.557780\pi\)
\(908\) −3.77687e10 −1.67430
\(909\) 1.43651e9 0.0634359
\(910\) −3.46964e8 −0.0152630
\(911\) 9.37494e9 0.410823 0.205411 0.978676i \(-0.434147\pi\)
0.205411 + 0.978676i \(0.434147\pi\)
\(912\) −1.70053e9 −0.0742339
\(913\) −2.10886e10 −0.917064
\(914\) −1.69443e9 −0.0734028
\(915\) 1.54237e10 0.665602
\(916\) −6.21095e9 −0.267008
\(917\) 9.94835e9 0.426048
\(918\) −3.24994e8 −0.0138652
\(919\) 1.42577e10 0.605962 0.302981 0.952997i \(-0.402018\pi\)
0.302981 + 0.952997i \(0.402018\pi\)
\(920\) 7.75845e9 0.328487
\(921\) −3.93336e9 −0.165903
\(922\) 3.10455e9 0.130449
\(923\) −4.80468e9 −0.201121
\(924\) 2.47850e9 0.103356
\(925\) −8.59655e10 −3.57132
\(926\) 4.37681e8 0.0181142
\(927\) 7.70495e9 0.317681
\(928\) 2.79143e8 0.0114659
\(929\) 7.88274e9 0.322569 0.161284 0.986908i \(-0.448436\pi\)
0.161284 + 0.986908i \(0.448436\pi\)
\(930\) −1.90702e8 −0.00777435
\(931\) −5.31546e8 −0.0215882
\(932\) −1.54137e10 −0.623664
\(933\) 1.03086e10 0.415541
\(934\) −1.73188e9 −0.0695509
\(935\) 4.66829e9 0.186774
\(936\) 8.54247e8 0.0340501
\(937\) −4.64928e10 −1.84628 −0.923139 0.384466i \(-0.874386\pi\)
−0.923139 + 0.384466i \(0.874386\pi\)
\(938\) 8.66234e8 0.0342709
\(939\) 2.44331e10 0.963049
\(940\) 2.05358e10 0.806425
\(941\) 1.45357e10 0.568685 0.284342 0.958723i \(-0.408225\pi\)
0.284342 + 0.958723i \(0.408225\pi\)
\(942\) 1.33094e9 0.0518776
\(943\) 3.10884e10 1.20728
\(944\) 8.96423e9 0.346826
\(945\) 1.52130e10 0.586411
\(946\) 2.43445e7 0.000934938 0
\(947\) 2.94544e9 0.112700 0.0563502 0.998411i \(-0.482054\pi\)
0.0563502 + 0.998411i \(0.482054\pi\)
\(948\) 1.07930e10 0.411446
\(949\) −4.66222e9 −0.177076
\(950\) −7.00370e8 −0.0265030
\(951\) 7.23515e9 0.272782
\(952\) −3.17171e8 −0.0119142
\(953\) 3.44178e9 0.128812 0.0644062 0.997924i \(-0.479485\pi\)
0.0644062 + 0.997924i \(0.479485\pi\)
\(954\) −2.91220e9 −0.108593
\(955\) 4.63037e10 1.72030
\(956\) 1.75958e10 0.651339
\(957\) −3.49919e8 −0.0129055
\(958\) −9.94928e7 −0.00365605
\(959\) −1.87887e10 −0.687909
\(960\) 2.33402e10 0.851441
\(961\) −2.72006e10 −0.988659
\(962\) −1.05572e9 −0.0382327
\(963\) −2.86861e10 −1.03509
\(964\) −1.62664e10 −0.584819
\(965\) −1.24863e10 −0.447289
\(966\) 4.94456e8 0.0176485
\(967\) −3.90155e10 −1.38754 −0.693768 0.720198i \(-0.744051\pi\)
−0.693768 + 0.720198i \(0.744051\pi\)
\(968\) 3.23262e9 0.114549
\(969\) −4.12480e8 −0.0145636
\(970\) 6.24063e9 0.219547
\(971\) 2.36083e10 0.827556 0.413778 0.910378i \(-0.364209\pi\)
0.413778 + 0.910378i \(0.364209\pi\)
\(972\) −2.93376e10 −1.02469
\(973\) 1.77377e9 0.0617307
\(974\) 2.31838e9 0.0803947
\(975\) 8.57873e9 0.296420
\(976\) −2.13471e10 −0.734962
\(977\) 1.53065e10 0.525105 0.262552 0.964918i \(-0.415436\pi\)
0.262552 + 0.964918i \(0.415436\pi\)
\(978\) 2.09062e9 0.0714642
\(979\) 7.14185e9 0.243260
\(980\) 7.39818e9 0.251092
\(981\) −3.80915e10 −1.28821
\(982\) 7.66669e8 0.0258356
\(983\) −1.63433e10 −0.548785 −0.274393 0.961618i \(-0.588477\pi\)
−0.274393 + 0.961618i \(0.588477\pi\)
\(984\) −2.62123e9 −0.0877045
\(985\) −1.23040e10 −0.410224
\(986\) 2.23134e7 0.000741304 0
\(987\) 2.62646e9 0.0869483
\(988\) 1.26195e9 0.0416287
\(989\) −7.12577e8 −0.0234231
\(990\) −1.82724e9 −0.0598511
\(991\) 1.13554e10 0.370633 0.185316 0.982679i \(-0.440669\pi\)
0.185316 + 0.982679i \(0.440669\pi\)
\(992\) 8.00914e8 0.0260492
\(993\) 7.59498e9 0.246153
\(994\) −6.98252e8 −0.0225507
\(995\) −4.69693e10 −1.51159
\(996\) −2.59345e10 −0.831705
\(997\) −6.06817e10 −1.93921 −0.969605 0.244674i \(-0.921319\pi\)
−0.969605 + 0.244674i \(0.921319\pi\)
\(998\) −2.54137e8 −0.00809302
\(999\) 4.62888e10 1.46892
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 91.8.a.e.1.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.e.1.6 12 1.1 even 1 trivial