Properties

Label 91.8.a.c.1.9
Level $91$
Weight $8$
Character 91.1
Self dual yes
Analytic conductor $28.427$
Analytic rank $1$
Dimension $10$
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: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 2 x^{9} - 957 x^{8} + 1224 x^{7} + 310102 x^{6} - 241884 x^{5} - 40367312 x^{4} + 11067840 x^{3} + 1840757376 x^{2} + 541859072 x - 4516262912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(16.1785\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q+14.1785 q^{2} +6.78261 q^{3} +73.0290 q^{4} -4.23520 q^{5} +96.1671 q^{6} +343.000 q^{7} -779.405 q^{8} -2141.00 q^{9} +O(q^{10})\) \(q+14.1785 q^{2} +6.78261 q^{3} +73.0290 q^{4} -4.23520 q^{5} +96.1671 q^{6} +343.000 q^{7} -779.405 q^{8} -2141.00 q^{9} -60.0487 q^{10} -7896.54 q^{11} +495.328 q^{12} -2197.00 q^{13} +4863.22 q^{14} -28.7257 q^{15} -20398.5 q^{16} +31982.0 q^{17} -30356.0 q^{18} -9142.53 q^{19} -309.293 q^{20} +2326.44 q^{21} -111961. q^{22} +28761.2 q^{23} -5286.40 q^{24} -78107.1 q^{25} -31150.1 q^{26} -29355.1 q^{27} +25048.9 q^{28} -13425.2 q^{29} -407.287 q^{30} -172793. q^{31} -189455. q^{32} -53559.2 q^{33} +453455. q^{34} -1452.67 q^{35} -156355. q^{36} -245025. q^{37} -129627. q^{38} -14901.4 q^{39} +3300.94 q^{40} -778150. q^{41} +32985.3 q^{42} +713663. q^{43} -576676. q^{44} +9067.55 q^{45} +407790. q^{46} +132602. q^{47} -138355. q^{48} +117649. q^{49} -1.10744e6 q^{50} +216921. q^{51} -160445. q^{52} +743372. q^{53} -416211. q^{54} +33443.4 q^{55} -267336. q^{56} -62010.2 q^{57} -190349. q^{58} +1.05200e6 q^{59} -2097.81 q^{60} +1.76584e6 q^{61} -2.44994e6 q^{62} -734362. q^{63} -75182.7 q^{64} +9304.74 q^{65} -759387. q^{66} +744278. q^{67} +2.33561e6 q^{68} +195076. q^{69} -20596.7 q^{70} +383299. q^{71} +1.66870e6 q^{72} -3.29444e6 q^{73} -3.47408e6 q^{74} -529770. q^{75} -667670. q^{76} -2.70851e6 q^{77} -211279. q^{78} -2.30144e6 q^{79} +86391.7 q^{80} +4.48325e6 q^{81} -1.10330e7 q^{82} -3.15382e6 q^{83} +169897. q^{84} -135450. q^{85} +1.01186e7 q^{86} -91058.3 q^{87} +6.15460e6 q^{88} -1.36538e6 q^{89} +128564. q^{90} -753571. q^{91} +2.10040e6 q^{92} -1.17199e6 q^{93} +1.88009e6 q^{94} +38720.5 q^{95} -1.28500e6 q^{96} -1.40862e6 q^{97} +1.66808e6 q^{98} +1.69065e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 18 q^{2} - 80 q^{3} + 670 q^{4} - 927 q^{5} - 1419 q^{6} + 3430 q^{7} - 4878 q^{8} + 3612 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 18 q^{2} - 80 q^{3} + 670 q^{4} - 927 q^{5} - 1419 q^{6} + 3430 q^{7} - 4878 q^{8} + 3612 q^{9} + 9420 q^{10} + 876 q^{11} - 8765 q^{12} - 21970 q^{13} - 6174 q^{14} - 5320 q^{15} + 41370 q^{16} + 6294 q^{17} - 16027 q^{18} - 97401 q^{19} - 166650 q^{20} - 27440 q^{21} + 74171 q^{22} - 15255 q^{23} + 196187 q^{24} + 162145 q^{25} + 39546 q^{26} - 181820 q^{27} + 229810 q^{28} - 340533 q^{29} - 325020 q^{30} - 148675 q^{31} - 642762 q^{32} - 624400 q^{33} - 1161518 q^{34} - 317961 q^{35} - 773917 q^{36} - 621782 q^{37} - 805092 q^{38} + 175760 q^{39} - 350478 q^{40} - 2043336 q^{41} - 486717 q^{42} - 1801391 q^{43} - 3953667 q^{44} - 1908807 q^{45} - 2707731 q^{46} - 1624701 q^{47} - 6068625 q^{48} + 1176490 q^{49} - 6891516 q^{50} + 1811700 q^{51} - 1471990 q^{52} - 199965 q^{53} - 2895913 q^{54} + 739086 q^{55} - 1673154 q^{56} + 2159088 q^{57} + 2071092 q^{58} - 8098908 q^{59} + 8096436 q^{60} + 2271618 q^{61} - 8910225 q^{62} + 1238916 q^{63} + 8099930 q^{64} + 2036619 q^{65} - 5999191 q^{66} + 1970272 q^{67} - 1766238 q^{68} - 4622962 q^{69} + 3231060 q^{70} - 7145820 q^{71} + 984975 q^{72} + 1409431 q^{73} - 5498643 q^{74} - 8857892 q^{75} - 2749534 q^{76} + 300468 q^{77} + 3117543 q^{78} - 9011055 q^{79} - 23850522 q^{80} + 11613490 q^{81} + 27962597 q^{82} - 15006567 q^{83} - 3006395 q^{84} - 9416628 q^{85} + 38357850 q^{86} - 15828996 q^{87} + 42205269 q^{88} - 11472777 q^{89} + 53425712 q^{90} - 7535710 q^{91} + 16755837 q^{92} + 36339848 q^{93} + 5133371 q^{94} + 29637939 q^{95} + 65329611 q^{96} + 3228571 q^{97} - 2117682 q^{98} + 19367194 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.1785 1.25321 0.626606 0.779336i \(-0.284444\pi\)
0.626606 + 0.779336i \(0.284444\pi\)
\(3\) 6.78261 0.145035 0.0725175 0.997367i \(-0.476897\pi\)
0.0725175 + 0.997367i \(0.476897\pi\)
\(4\) 73.0290 0.570539
\(5\) −4.23520 −0.0151523 −0.00757616 0.999971i \(-0.502412\pi\)
−0.00757616 + 0.999971i \(0.502412\pi\)
\(6\) 96.1671 0.181759
\(7\) 343.000 0.377964
\(8\) −779.405 −0.538205
\(9\) −2141.00 −0.978965
\(10\) −60.0487 −0.0189891
\(11\) −7896.54 −1.78880 −0.894401 0.447266i \(-0.852398\pi\)
−0.894401 + 0.447266i \(0.852398\pi\)
\(12\) 495.328 0.0827481
\(13\) −2197.00 −0.277350
\(14\) 4863.22 0.473669
\(15\) −28.7257 −0.00219762
\(16\) −20398.5 −1.24502
\(17\) 31982.0 1.57883 0.789413 0.613863i \(-0.210385\pi\)
0.789413 + 0.613863i \(0.210385\pi\)
\(18\) −30356.0 −1.22685
\(19\) −9142.53 −0.305794 −0.152897 0.988242i \(-0.548860\pi\)
−0.152897 + 0.988242i \(0.548860\pi\)
\(20\) −309.293 −0.00864499
\(21\) 2326.44 0.0548181
\(22\) −111961. −2.24175
\(23\) 28761.2 0.492901 0.246451 0.969155i \(-0.420736\pi\)
0.246451 + 0.969155i \(0.420736\pi\)
\(24\) −5286.40 −0.0780586
\(25\) −78107.1 −0.999770
\(26\) −31150.1 −0.347578
\(27\) −29355.1 −0.287019
\(28\) 25048.9 0.215644
\(29\) −13425.2 −0.102218 −0.0511092 0.998693i \(-0.516276\pi\)
−0.0511092 + 0.998693i \(0.516276\pi\)
\(30\) −407.287 −0.00275408
\(31\) −172793. −1.04174 −0.520871 0.853636i \(-0.674393\pi\)
−0.520871 + 0.853636i \(0.674393\pi\)
\(32\) −189455. −1.02207
\(33\) −53559.2 −0.259439
\(34\) 453455. 1.97860
\(35\) −1452.67 −0.00572704
\(36\) −156355. −0.558538
\(37\) −245025. −0.795251 −0.397626 0.917548i \(-0.630166\pi\)
−0.397626 + 0.917548i \(0.630166\pi\)
\(38\) −129627. −0.383224
\(39\) −14901.4 −0.0402255
\(40\) 3300.94 0.00815506
\(41\) −778150. −1.76327 −0.881637 0.471928i \(-0.843558\pi\)
−0.881637 + 0.471928i \(0.843558\pi\)
\(42\) 32985.3 0.0686986
\(43\) 713663. 1.36884 0.684421 0.729087i \(-0.260055\pi\)
0.684421 + 0.729087i \(0.260055\pi\)
\(44\) −576676. −1.02058
\(45\) 9067.55 0.0148336
\(46\) 407790. 0.617709
\(47\) 132602. 0.186297 0.0931486 0.995652i \(-0.470307\pi\)
0.0931486 + 0.995652i \(0.470307\pi\)
\(48\) −138355. −0.180572
\(49\) 117649. 0.142857
\(50\) −1.10744e6 −1.25292
\(51\) 216921. 0.228985
\(52\) −160445. −0.158239
\(53\) 743372. 0.685868 0.342934 0.939359i \(-0.388579\pi\)
0.342934 + 0.939359i \(0.388579\pi\)
\(54\) −416211. −0.359696
\(55\) 33443.4 0.0271045
\(56\) −267336. −0.203422
\(57\) −62010.2 −0.0443508
\(58\) −190349. −0.128101
\(59\) 1.05200e6 0.666860 0.333430 0.942775i \(-0.391794\pi\)
0.333430 + 0.942775i \(0.391794\pi\)
\(60\) −2097.81 −0.00125383
\(61\) 1.76584e6 0.996083 0.498042 0.867153i \(-0.334053\pi\)
0.498042 + 0.867153i \(0.334053\pi\)
\(62\) −2.44994e6 −1.30552
\(63\) −734362. −0.370014
\(64\) −75182.7 −0.0358499
\(65\) 9304.74 0.00420250
\(66\) −759387. −0.325132
\(67\) 744278. 0.302325 0.151162 0.988509i \(-0.451698\pi\)
0.151162 + 0.988509i \(0.451698\pi\)
\(68\) 2.33561e6 0.900782
\(69\) 195076. 0.0714879
\(70\) −20596.7 −0.00717719
\(71\) 383299. 0.127097 0.0635483 0.997979i \(-0.479758\pi\)
0.0635483 + 0.997979i \(0.479758\pi\)
\(72\) 1.66870e6 0.526884
\(73\) −3.29444e6 −0.991177 −0.495588 0.868557i \(-0.665048\pi\)
−0.495588 + 0.868557i \(0.665048\pi\)
\(74\) −3.47408e6 −0.996618
\(75\) −529770. −0.145002
\(76\) −667670. −0.174467
\(77\) −2.70851e6 −0.676104
\(78\) −211279. −0.0504110
\(79\) −2.30144e6 −0.525175 −0.262588 0.964908i \(-0.584576\pi\)
−0.262588 + 0.964908i \(0.584576\pi\)
\(80\) 86391.7 0.0188650
\(81\) 4.48325e6 0.937337
\(82\) −1.10330e7 −2.20976
\(83\) −3.15382e6 −0.605429 −0.302715 0.953081i \(-0.597893\pi\)
−0.302715 + 0.953081i \(0.597893\pi\)
\(84\) 169897. 0.0312758
\(85\) −135450. −0.0239229
\(86\) 1.01186e7 1.71545
\(87\) −91058.3 −0.0148252
\(88\) 6.15460e6 0.962743
\(89\) −1.36538e6 −0.205300 −0.102650 0.994718i \(-0.532732\pi\)
−0.102650 + 0.994718i \(0.532732\pi\)
\(90\) 128564. 0.0185896
\(91\) −753571. −0.104828
\(92\) 2.10040e6 0.281219
\(93\) −1.17199e6 −0.151089
\(94\) 1.88009e6 0.233470
\(95\) 38720.5 0.00463349
\(96\) −1.28500e6 −0.148236
\(97\) −1.40862e6 −0.156709 −0.0783543 0.996926i \(-0.524967\pi\)
−0.0783543 + 0.996926i \(0.524967\pi\)
\(98\) 1.66808e6 0.179030
\(99\) 1.69065e7 1.75117
\(100\) −5.70408e6 −0.570408
\(101\) 1.57617e7 1.52222 0.761112 0.648620i \(-0.224654\pi\)
0.761112 + 0.648620i \(0.224654\pi\)
\(102\) 3.07561e6 0.286966
\(103\) 8.44811e6 0.761780 0.380890 0.924620i \(-0.375618\pi\)
0.380890 + 0.924620i \(0.375618\pi\)
\(104\) 1.71235e6 0.149271
\(105\) −9852.93 −0.000830621 0
\(106\) 1.05399e7 0.859538
\(107\) −2.23508e7 −1.76380 −0.881899 0.471437i \(-0.843735\pi\)
−0.881899 + 0.471437i \(0.843735\pi\)
\(108\) −2.14378e6 −0.163756
\(109\) 2.67222e6 0.197642 0.0988210 0.995105i \(-0.468493\pi\)
0.0988210 + 0.995105i \(0.468493\pi\)
\(110\) 474177. 0.0339677
\(111\) −1.66191e6 −0.115339
\(112\) −6.99668e6 −0.470575
\(113\) 1.75901e7 1.14682 0.573409 0.819269i \(-0.305620\pi\)
0.573409 + 0.819269i \(0.305620\pi\)
\(114\) −879210. −0.0555809
\(115\) −121810. −0.00746860
\(116\) −980432. −0.0583196
\(117\) 4.70377e6 0.271516
\(118\) 1.49158e7 0.835716
\(119\) 1.09698e7 0.596740
\(120\) 22389.0 0.00118277
\(121\) 4.28681e7 2.19981
\(122\) 2.50368e7 1.24830
\(123\) −5.27789e6 −0.255736
\(124\) −1.26189e7 −0.594354
\(125\) 661675. 0.0303012
\(126\) −1.04121e7 −0.463706
\(127\) −1.88365e7 −0.815994 −0.407997 0.912983i \(-0.633773\pi\)
−0.407997 + 0.912983i \(0.633773\pi\)
\(128\) 2.31843e7 0.977146
\(129\) 4.84050e6 0.198530
\(130\) 131927. 0.00526662
\(131\) −2.92085e7 −1.13517 −0.567584 0.823316i \(-0.692122\pi\)
−0.567584 + 0.823316i \(0.692122\pi\)
\(132\) −3.91137e6 −0.148020
\(133\) −3.13589e6 −0.115579
\(134\) 1.05527e7 0.378877
\(135\) 124325. 0.00434901
\(136\) −2.49269e7 −0.849732
\(137\) −3.09166e7 −1.02724 −0.513618 0.858019i \(-0.671695\pi\)
−0.513618 + 0.858019i \(0.671695\pi\)
\(138\) 2.76588e6 0.0895894
\(139\) 4.72180e6 0.149127 0.0745635 0.997216i \(-0.476244\pi\)
0.0745635 + 0.997216i \(0.476244\pi\)
\(140\) −106087. −0.00326750
\(141\) 899386. 0.0270196
\(142\) 5.43460e6 0.159279
\(143\) 1.73487e7 0.496124
\(144\) 4.36731e7 1.21883
\(145\) 56858.6 0.00154885
\(146\) −4.67101e7 −1.24215
\(147\) 797968. 0.0207193
\(148\) −1.78939e7 −0.453722
\(149\) −6.35120e7 −1.57291 −0.786455 0.617648i \(-0.788086\pi\)
−0.786455 + 0.617648i \(0.788086\pi\)
\(150\) −7.51133e6 −0.181718
\(151\) 1.03388e7 0.244371 0.122185 0.992507i \(-0.461010\pi\)
0.122185 + 0.992507i \(0.461010\pi\)
\(152\) 7.12573e6 0.164580
\(153\) −6.84733e7 −1.54561
\(154\) −3.84026e7 −0.847301
\(155\) 731813. 0.0157848
\(156\) −1.08823e6 −0.0229502
\(157\) 1.07253e7 0.221188 0.110594 0.993866i \(-0.464725\pi\)
0.110594 + 0.993866i \(0.464725\pi\)
\(158\) −3.26309e7 −0.658156
\(159\) 5.04201e6 0.0994749
\(160\) 802382. 0.0154868
\(161\) 9.86509e6 0.186299
\(162\) 6.35657e7 1.17468
\(163\) −8.36920e7 −1.51366 −0.756828 0.653614i \(-0.773252\pi\)
−0.756828 + 0.653614i \(0.773252\pi\)
\(164\) −5.68275e7 −1.00602
\(165\) 226834. 0.00393110
\(166\) −4.47163e7 −0.758731
\(167\) 2.58014e7 0.428683 0.214342 0.976759i \(-0.431239\pi\)
0.214342 + 0.976759i \(0.431239\pi\)
\(168\) −1.81324e6 −0.0295034
\(169\) 4.82681e6 0.0769231
\(170\) −1.92048e6 −0.0299804
\(171\) 1.95741e7 0.299361
\(172\) 5.21181e7 0.780978
\(173\) −1.19610e8 −1.75632 −0.878162 0.478364i \(-0.841230\pi\)
−0.878162 + 0.478364i \(0.841230\pi\)
\(174\) −1.29107e6 −0.0185792
\(175\) −2.67907e7 −0.377878
\(176\) 1.61077e8 2.22710
\(177\) 7.13532e6 0.0967180
\(178\) −1.93590e7 −0.257284
\(179\) 3.97906e6 0.0518555 0.0259278 0.999664i \(-0.491746\pi\)
0.0259278 + 0.999664i \(0.491746\pi\)
\(180\) 662194. 0.00846315
\(181\) −1.44074e8 −1.80596 −0.902982 0.429678i \(-0.858627\pi\)
−0.902982 + 0.429678i \(0.858627\pi\)
\(182\) −1.06845e7 −0.131372
\(183\) 1.19770e7 0.144467
\(184\) −2.24166e7 −0.265282
\(185\) 1.03773e6 0.0120499
\(186\) −1.66170e7 −0.189346
\(187\) −2.52547e8 −2.82421
\(188\) 9.68377e6 0.106290
\(189\) −1.00688e7 −0.108483
\(190\) 548997. 0.00580674
\(191\) −6.58290e7 −0.683597 −0.341799 0.939773i \(-0.611036\pi\)
−0.341799 + 0.939773i \(0.611036\pi\)
\(192\) −509935. −0.00519949
\(193\) −1.22231e8 −1.22386 −0.611931 0.790911i \(-0.709607\pi\)
−0.611931 + 0.790911i \(0.709607\pi\)
\(194\) −1.99721e7 −0.196389
\(195\) 63110.5 0.000609509 0
\(196\) 8.59179e6 0.0815056
\(197\) 2.12664e8 1.98181 0.990903 0.134576i \(-0.0429673\pi\)
0.990903 + 0.134576i \(0.0429673\pi\)
\(198\) 2.39708e8 2.19459
\(199\) 1.33715e8 1.20280 0.601400 0.798948i \(-0.294610\pi\)
0.601400 + 0.798948i \(0.294610\pi\)
\(200\) 6.08770e7 0.538082
\(201\) 5.04815e6 0.0438476
\(202\) 2.23477e8 1.90767
\(203\) −4.60486e6 −0.0386349
\(204\) 1.58416e7 0.130645
\(205\) 3.29562e6 0.0267177
\(206\) 1.19781e8 0.954671
\(207\) −6.15776e7 −0.482533
\(208\) 4.48155e7 0.345308
\(209\) 7.21943e7 0.547005
\(210\) −139700. −0.00104094
\(211\) 9.13240e7 0.669262 0.334631 0.942349i \(-0.391388\pi\)
0.334631 + 0.942349i \(0.391388\pi\)
\(212\) 5.42877e7 0.391315
\(213\) 2.59977e6 0.0184334
\(214\) −3.16900e8 −2.21041
\(215\) −3.02251e6 −0.0207411
\(216\) 2.28795e7 0.154475
\(217\) −5.92680e7 −0.393741
\(218\) 3.78880e7 0.247687
\(219\) −2.23449e7 −0.143755
\(220\) 2.44234e6 0.0154642
\(221\) −7.02644e7 −0.437887
\(222\) −2.35633e7 −0.144544
\(223\) 1.77088e8 1.06935 0.534677 0.845057i \(-0.320433\pi\)
0.534677 + 0.845057i \(0.320433\pi\)
\(224\) −6.49832e7 −0.386307
\(225\) 1.67227e8 0.978740
\(226\) 2.49401e8 1.43721
\(227\) −1.07588e8 −0.610484 −0.305242 0.952275i \(-0.598737\pi\)
−0.305242 + 0.952275i \(0.598737\pi\)
\(228\) −4.52855e6 −0.0253039
\(229\) 1.45669e7 0.0801572 0.0400786 0.999197i \(-0.487239\pi\)
0.0400786 + 0.999197i \(0.487239\pi\)
\(230\) −1.72707e6 −0.00935973
\(231\) −1.83708e7 −0.0980587
\(232\) 1.04637e7 0.0550145
\(233\) 2.07312e8 1.07369 0.536845 0.843681i \(-0.319616\pi\)
0.536845 + 0.843681i \(0.319616\pi\)
\(234\) 6.66922e7 0.340267
\(235\) −561595. −0.00282284
\(236\) 7.68267e7 0.380470
\(237\) −1.56098e7 −0.0761688
\(238\) 1.55535e8 0.747841
\(239\) −9.60165e7 −0.454939 −0.227469 0.973785i \(-0.573045\pi\)
−0.227469 + 0.973785i \(0.573045\pi\)
\(240\) 585962. 0.00273609
\(241\) −3.30651e8 −1.52163 −0.760816 0.648967i \(-0.775201\pi\)
−0.760816 + 0.648967i \(0.775201\pi\)
\(242\) 6.07805e8 2.75683
\(243\) 9.46078e7 0.422966
\(244\) 1.28957e8 0.568305
\(245\) −498267. −0.00216462
\(246\) −7.48324e7 −0.320492
\(247\) 2.00861e7 0.0848120
\(248\) 1.34676e8 0.560671
\(249\) −2.13911e7 −0.0878084
\(250\) 9.38153e6 0.0379738
\(251\) −2.91957e8 −1.16536 −0.582681 0.812701i \(-0.697996\pi\)
−0.582681 + 0.812701i \(0.697996\pi\)
\(252\) −5.36297e7 −0.211107
\(253\) −2.27114e8 −0.881702
\(254\) −2.67073e8 −1.02261
\(255\) −918706. −0.00346965
\(256\) 3.38342e8 1.26042
\(257\) −4.45549e8 −1.63731 −0.818653 0.574289i \(-0.805279\pi\)
−0.818653 + 0.574289i \(0.805279\pi\)
\(258\) 6.86309e7 0.248800
\(259\) −8.40436e7 −0.300577
\(260\) 679516. 0.00239769
\(261\) 2.87434e7 0.100068
\(262\) −4.14132e8 −1.42261
\(263\) 7.86991e7 0.266762 0.133381 0.991065i \(-0.457417\pi\)
0.133381 + 0.991065i \(0.457417\pi\)
\(264\) 4.17443e7 0.139631
\(265\) −3.14833e6 −0.0103925
\(266\) −4.44621e7 −0.144845
\(267\) −9.26083e6 −0.0297756
\(268\) 5.43539e7 0.172488
\(269\) −5.41542e8 −1.69629 −0.848144 0.529766i \(-0.822280\pi\)
−0.848144 + 0.529766i \(0.822280\pi\)
\(270\) 1.76274e6 0.00545022
\(271\) 4.01239e8 1.22465 0.612323 0.790608i \(-0.290235\pi\)
0.612323 + 0.790608i \(0.290235\pi\)
\(272\) −6.52384e8 −1.96568
\(273\) −5.11118e6 −0.0152038
\(274\) −4.38351e8 −1.28734
\(275\) 6.16775e8 1.78839
\(276\) 1.42462e7 0.0407866
\(277\) −2.09709e8 −0.592841 −0.296420 0.955058i \(-0.595793\pi\)
−0.296420 + 0.955058i \(0.595793\pi\)
\(278\) 6.69480e7 0.186888
\(279\) 3.69949e8 1.01983
\(280\) 1.13222e6 0.00308232
\(281\) 4.84561e7 0.130279 0.0651397 0.997876i \(-0.479251\pi\)
0.0651397 + 0.997876i \(0.479251\pi\)
\(282\) 1.27519e7 0.0338613
\(283\) −5.91245e8 −1.55065 −0.775327 0.631560i \(-0.782415\pi\)
−0.775327 + 0.631560i \(0.782415\pi\)
\(284\) 2.79920e7 0.0725136
\(285\) 262626. 0.000672018 0
\(286\) 2.45978e8 0.621749
\(287\) −2.66906e8 −0.666455
\(288\) 4.05623e8 1.00057
\(289\) 6.12508e8 1.49269
\(290\) 806169. 0.00194103
\(291\) −9.55413e6 −0.0227282
\(292\) −2.40589e8 −0.565505
\(293\) 4.23336e8 0.983215 0.491607 0.870817i \(-0.336410\pi\)
0.491607 + 0.870817i \(0.336410\pi\)
\(294\) 1.13140e7 0.0259656
\(295\) −4.45544e6 −0.0101045
\(296\) 1.90974e8 0.428008
\(297\) 2.31804e8 0.513420
\(298\) −9.00503e8 −1.97119
\(299\) −6.31884e7 −0.136706
\(300\) −3.86886e7 −0.0827291
\(301\) 2.44786e8 0.517374
\(302\) 1.46588e8 0.306249
\(303\) 1.06906e8 0.220776
\(304\) 1.86494e8 0.380721
\(305\) −7.47867e6 −0.0150930
\(306\) −9.70846e8 −1.93698
\(307\) 2.05699e8 0.405741 0.202870 0.979206i \(-0.434973\pi\)
0.202870 + 0.979206i \(0.434973\pi\)
\(308\) −1.97800e8 −0.385744
\(309\) 5.73003e7 0.110485
\(310\) 1.03760e7 0.0197817
\(311\) −9.59049e6 −0.0180792 −0.00903961 0.999959i \(-0.502877\pi\)
−0.00903961 + 0.999959i \(0.502877\pi\)
\(312\) 1.16142e7 0.0216496
\(313\) −1.75652e8 −0.323778 −0.161889 0.986809i \(-0.551759\pi\)
−0.161889 + 0.986809i \(0.551759\pi\)
\(314\) 1.52068e8 0.277195
\(315\) 3.11017e6 0.00560657
\(316\) −1.68072e8 −0.299633
\(317\) 2.37691e8 0.419087 0.209544 0.977799i \(-0.432802\pi\)
0.209544 + 0.977799i \(0.432802\pi\)
\(318\) 7.14879e7 0.124663
\(319\) 1.06013e8 0.182849
\(320\) 318414. 0.000543210 0
\(321\) −1.51597e8 −0.255813
\(322\) 1.39872e8 0.233472
\(323\) −2.92396e8 −0.482795
\(324\) 3.27408e8 0.534787
\(325\) 1.71601e8 0.277286
\(326\) −1.18662e9 −1.89693
\(327\) 1.81246e7 0.0286650
\(328\) 6.06494e8 0.949004
\(329\) 4.54824e7 0.0704137
\(330\) 3.21616e6 0.00492650
\(331\) −2.39476e8 −0.362965 −0.181482 0.983394i \(-0.558090\pi\)
−0.181482 + 0.983394i \(0.558090\pi\)
\(332\) −2.30320e8 −0.345421
\(333\) 5.24598e8 0.778523
\(334\) 3.65825e8 0.537231
\(335\) −3.15217e6 −0.00458092
\(336\) −4.74558e7 −0.0682498
\(337\) −1.11824e9 −1.59158 −0.795791 0.605571i \(-0.792945\pi\)
−0.795791 + 0.605571i \(0.792945\pi\)
\(338\) 6.84368e7 0.0964009
\(339\) 1.19307e8 0.166329
\(340\) −9.89179e6 −0.0136489
\(341\) 1.36447e9 1.86347
\(342\) 2.77531e8 0.375163
\(343\) 4.03536e7 0.0539949
\(344\) −5.56232e8 −0.736718
\(345\) −826187. −0.00108321
\(346\) −1.69588e9 −2.20104
\(347\) −2.13142e8 −0.273852 −0.136926 0.990581i \(-0.543722\pi\)
−0.136926 + 0.990581i \(0.543722\pi\)
\(348\) −6.64989e6 −0.00845838
\(349\) 7.18720e8 0.905046 0.452523 0.891753i \(-0.350524\pi\)
0.452523 + 0.891753i \(0.350524\pi\)
\(350\) −3.79851e8 −0.473561
\(351\) 6.44932e7 0.0796048
\(352\) 1.49604e9 1.82829
\(353\) −4.62333e8 −0.559428 −0.279714 0.960083i \(-0.590240\pi\)
−0.279714 + 0.960083i \(0.590240\pi\)
\(354\) 1.01168e8 0.121208
\(355\) −1.62335e6 −0.00192581
\(356\) −9.97122e7 −0.117131
\(357\) 7.44040e7 0.0865481
\(358\) 5.64170e7 0.0649859
\(359\) −5.22700e7 −0.0596241 −0.0298120 0.999556i \(-0.509491\pi\)
−0.0298120 + 0.999556i \(0.509491\pi\)
\(360\) −7.06729e6 −0.00798352
\(361\) −8.10286e8 −0.906490
\(362\) −2.04274e9 −2.26326
\(363\) 2.90758e8 0.319050
\(364\) −5.50325e7 −0.0598088
\(365\) 1.39526e7 0.0150186
\(366\) 1.69815e8 0.181048
\(367\) 1.14754e9 1.21182 0.605910 0.795533i \(-0.292809\pi\)
0.605910 + 0.795533i \(0.292809\pi\)
\(368\) −5.86685e8 −0.613674
\(369\) 1.66602e9 1.72618
\(370\) 1.47134e7 0.0151011
\(371\) 2.54977e8 0.259234
\(372\) −8.55891e7 −0.0862021
\(373\) 1.45073e9 1.44746 0.723731 0.690082i \(-0.242426\pi\)
0.723731 + 0.690082i \(0.242426\pi\)
\(374\) −3.58073e9 −3.53933
\(375\) 4.48788e6 0.00439473
\(376\) −1.03350e8 −0.100266
\(377\) 2.94953e7 0.0283503
\(378\) −1.42760e8 −0.135952
\(379\) 4.04404e8 0.381574 0.190787 0.981631i \(-0.438896\pi\)
0.190787 + 0.981631i \(0.438896\pi\)
\(380\) 2.82772e6 0.00264359
\(381\) −1.27761e8 −0.118348
\(382\) −9.33354e8 −0.856692
\(383\) −1.75951e9 −1.60028 −0.800138 0.599816i \(-0.795240\pi\)
−0.800138 + 0.599816i \(0.795240\pi\)
\(384\) 1.57250e8 0.141720
\(385\) 1.14711e7 0.0102445
\(386\) −1.73305e9 −1.53376
\(387\) −1.52795e9 −1.34005
\(388\) −1.02870e8 −0.0894084
\(389\) −1.35193e9 −1.16448 −0.582238 0.813019i \(-0.697823\pi\)
−0.582238 + 0.813019i \(0.697823\pi\)
\(390\) 894810. 0.000763844 0
\(391\) 9.19840e8 0.778205
\(392\) −9.16962e7 −0.0768865
\(393\) −1.98110e8 −0.164639
\(394\) 3.01524e9 2.48362
\(395\) 9.74705e6 0.00795763
\(396\) 1.23466e9 0.999114
\(397\) −1.06139e8 −0.0851350 −0.0425675 0.999094i \(-0.513554\pi\)
−0.0425675 + 0.999094i \(0.513554\pi\)
\(398\) 1.89587e9 1.50736
\(399\) −2.12695e7 −0.0167630
\(400\) 1.59327e9 1.24474
\(401\) 1.62632e9 1.25951 0.629755 0.776794i \(-0.283155\pi\)
0.629755 + 0.776794i \(0.283155\pi\)
\(402\) 7.15751e7 0.0549504
\(403\) 3.79626e8 0.288927
\(404\) 1.15106e9 0.868488
\(405\) −1.89875e7 −0.0142028
\(406\) −6.52899e7 −0.0484177
\(407\) 1.93485e9 1.42255
\(408\) −1.69070e8 −0.123241
\(409\) −3.64660e8 −0.263546 −0.131773 0.991280i \(-0.542067\pi\)
−0.131773 + 0.991280i \(0.542067\pi\)
\(410\) 4.67269e7 0.0334829
\(411\) −2.09696e8 −0.148985
\(412\) 6.16957e8 0.434625
\(413\) 3.60837e8 0.252049
\(414\) −8.73077e8 −0.604716
\(415\) 1.33571e7 0.00917366
\(416\) 4.16234e8 0.283472
\(417\) 3.20262e7 0.0216286
\(418\) 1.02361e9 0.685513
\(419\) −2.17097e7 −0.0144180 −0.00720899 0.999974i \(-0.502295\pi\)
−0.00720899 + 0.999974i \(0.502295\pi\)
\(420\) −719550. −0.000473902 0
\(421\) −8.42655e8 −0.550380 −0.275190 0.961390i \(-0.588741\pi\)
−0.275190 + 0.961390i \(0.588741\pi\)
\(422\) 1.29483e9 0.838727
\(423\) −2.83900e8 −0.182378
\(424\) −5.79388e8 −0.369138
\(425\) −2.49802e9 −1.57846
\(426\) 3.68608e7 0.0231010
\(427\) 6.05681e8 0.376484
\(428\) −1.63225e9 −1.00632
\(429\) 1.17669e8 0.0719554
\(430\) −4.28545e7 −0.0259930
\(431\) 7.78311e8 0.468255 0.234128 0.972206i \(-0.424777\pi\)
0.234128 + 0.972206i \(0.424777\pi\)
\(432\) 5.98800e8 0.357346
\(433\) 1.08344e8 0.0641353 0.0320677 0.999486i \(-0.489791\pi\)
0.0320677 + 0.999486i \(0.489791\pi\)
\(434\) −8.40329e8 −0.493441
\(435\) 385650. 0.000224637 0
\(436\) 1.95149e8 0.112762
\(437\) −2.62950e8 −0.150726
\(438\) −3.16816e8 −0.180156
\(439\) 2.99962e9 1.69216 0.846079 0.533058i \(-0.178957\pi\)
0.846079 + 0.533058i \(0.178957\pi\)
\(440\) −2.60660e7 −0.0145878
\(441\) −2.51886e8 −0.139852
\(442\) −9.96242e8 −0.548765
\(443\) −1.32632e9 −0.724827 −0.362413 0.932017i \(-0.618047\pi\)
−0.362413 + 0.932017i \(0.618047\pi\)
\(444\) −1.21368e8 −0.0658055
\(445\) 5.78266e6 0.00311077
\(446\) 2.51083e9 1.34013
\(447\) −4.30777e8 −0.228127
\(448\) −2.57877e7 −0.0135500
\(449\) −2.05878e9 −1.07337 −0.536683 0.843784i \(-0.680323\pi\)
−0.536683 + 0.843784i \(0.680323\pi\)
\(450\) 2.37102e9 1.22657
\(451\) 6.14469e9 3.15415
\(452\) 1.28459e9 0.654305
\(453\) 7.01239e7 0.0354423
\(454\) −1.52544e9 −0.765065
\(455\) 3.19153e6 0.00158840
\(456\) 4.83311e7 0.0238698
\(457\) −1.73861e9 −0.852111 −0.426055 0.904697i \(-0.640097\pi\)
−0.426055 + 0.904697i \(0.640097\pi\)
\(458\) 2.06536e8 0.100454
\(459\) −9.38835e8 −0.453153
\(460\) −8.89563e6 −0.00426113
\(461\) 3.26227e9 1.55084 0.775419 0.631448i \(-0.217539\pi\)
0.775419 + 0.631448i \(0.217539\pi\)
\(462\) −2.60470e8 −0.122888
\(463\) 4.02274e9 1.88360 0.941799 0.336177i \(-0.109134\pi\)
0.941799 + 0.336177i \(0.109134\pi\)
\(464\) 2.73855e8 0.127264
\(465\) 4.96361e6 0.00228935
\(466\) 2.93937e9 1.34556
\(467\) −3.53786e9 −1.60743 −0.803715 0.595015i \(-0.797146\pi\)
−0.803715 + 0.595015i \(0.797146\pi\)
\(468\) 3.43512e8 0.154910
\(469\) 2.55287e8 0.114268
\(470\) −7.96256e6 −0.00353761
\(471\) 7.27456e7 0.0320799
\(472\) −8.19935e8 −0.358907
\(473\) −5.63547e9 −2.44859
\(474\) −2.21322e8 −0.0954556
\(475\) 7.14096e8 0.305724
\(476\) 8.01115e8 0.340463
\(477\) −1.59156e9 −0.671441
\(478\) −1.36137e9 −0.570135
\(479\) −3.37375e9 −1.40261 −0.701307 0.712860i \(-0.747400\pi\)
−0.701307 + 0.712860i \(0.747400\pi\)
\(480\) 5.44225e6 0.00224613
\(481\) 5.38320e8 0.220563
\(482\) −4.68812e9 −1.90693
\(483\) 6.69111e7 0.0270199
\(484\) 3.13062e9 1.25508
\(485\) 5.96580e6 0.00237450
\(486\) 1.34139e9 0.530066
\(487\) 1.44875e9 0.568384 0.284192 0.958767i \(-0.408275\pi\)
0.284192 + 0.958767i \(0.408275\pi\)
\(488\) −1.37630e9 −0.536097
\(489\) −5.67650e8 −0.219533
\(490\) −7.06467e6 −0.00271272
\(491\) 3.69191e9 1.40756 0.703779 0.710419i \(-0.251495\pi\)
0.703779 + 0.710419i \(0.251495\pi\)
\(492\) −3.85439e8 −0.145908
\(493\) −4.29366e8 −0.161385
\(494\) 2.84791e8 0.106287
\(495\) −7.16023e7 −0.0265344
\(496\) 3.52471e9 1.29699
\(497\) 1.31472e8 0.0480380
\(498\) −3.03294e8 −0.110043
\(499\) 3.77246e9 1.35917 0.679583 0.733599i \(-0.262161\pi\)
0.679583 + 0.733599i \(0.262161\pi\)
\(500\) 4.83214e7 0.0172880
\(501\) 1.75001e8 0.0621740
\(502\) −4.13951e9 −1.46045
\(503\) 2.38391e9 0.835223 0.417611 0.908626i \(-0.362867\pi\)
0.417611 + 0.908626i \(0.362867\pi\)
\(504\) 5.72365e8 0.199143
\(505\) −6.67541e7 −0.0230652
\(506\) −3.22013e9 −1.10496
\(507\) 3.27384e7 0.0111565
\(508\) −1.37561e9 −0.465556
\(509\) −1.98641e9 −0.667663 −0.333831 0.942633i \(-0.608342\pi\)
−0.333831 + 0.942633i \(0.608342\pi\)
\(510\) −1.30258e7 −0.00434821
\(511\) −1.12999e9 −0.374630
\(512\) 1.82957e9 0.602427
\(513\) 2.68380e8 0.0877687
\(514\) −6.31721e9 −2.05189
\(515\) −3.57795e7 −0.0115427
\(516\) 3.53497e8 0.113269
\(517\) −1.04709e9 −0.333249
\(518\) −1.19161e9 −0.376686
\(519\) −8.11265e8 −0.254728
\(520\) −7.25216e6 −0.00226181
\(521\) 1.17398e9 0.363688 0.181844 0.983327i \(-0.441793\pi\)
0.181844 + 0.983327i \(0.441793\pi\)
\(522\) 4.07537e8 0.125407
\(523\) −4.23345e9 −1.29401 −0.647006 0.762485i \(-0.723979\pi\)
−0.647006 + 0.762485i \(0.723979\pi\)
\(524\) −2.13307e9 −0.647658
\(525\) −1.81711e8 −0.0548055
\(526\) 1.11583e9 0.334310
\(527\) −5.52626e9 −1.64473
\(528\) 1.09253e9 0.323008
\(529\) −2.57762e9 −0.757049
\(530\) −4.46385e7 −0.0130240
\(531\) −2.25233e9 −0.652832
\(532\) −2.29011e8 −0.0659425
\(533\) 1.70960e9 0.489044
\(534\) −1.31304e8 −0.0373151
\(535\) 9.46600e7 0.0267257
\(536\) −5.80094e8 −0.162713
\(537\) 2.69884e7 0.00752086
\(538\) −7.67824e9 −2.12581
\(539\) −9.29020e8 −0.255543
\(540\) 9.07933e6 0.00248128
\(541\) −4.48035e9 −1.21653 −0.608263 0.793736i \(-0.708133\pi\)
−0.608263 + 0.793736i \(0.708133\pi\)
\(542\) 5.68895e9 1.53474
\(543\) −9.77195e8 −0.261928
\(544\) −6.05916e9 −1.61368
\(545\) −1.13174e7 −0.00299473
\(546\) −7.24687e7 −0.0190536
\(547\) 6.39802e9 1.67144 0.835718 0.549159i \(-0.185052\pi\)
0.835718 + 0.549159i \(0.185052\pi\)
\(548\) −2.25781e9 −0.586078
\(549\) −3.78065e9 −0.975131
\(550\) 8.74493e9 2.24123
\(551\) 1.22741e8 0.0312578
\(552\) −1.52043e8 −0.0384752
\(553\) −7.89393e8 −0.198498
\(554\) −2.97335e9 −0.742955
\(555\) 7.03853e6 0.00174766
\(556\) 3.44829e8 0.0850828
\(557\) −2.53397e9 −0.621311 −0.310655 0.950523i \(-0.600548\pi\)
−0.310655 + 0.950523i \(0.600548\pi\)
\(558\) 5.24531e9 1.27806
\(559\) −1.56792e9 −0.379649
\(560\) 2.96324e7 0.00713030
\(561\) −1.71293e9 −0.409609
\(562\) 6.87033e8 0.163268
\(563\) 4.29257e9 1.01377 0.506883 0.862015i \(-0.330798\pi\)
0.506883 + 0.862015i \(0.330798\pi\)
\(564\) 6.56813e7 0.0154157
\(565\) −7.44978e7 −0.0173770
\(566\) −8.38295e9 −1.94330
\(567\) 1.53776e9 0.354280
\(568\) −2.98745e8 −0.0684041
\(569\) 3.31946e9 0.755396 0.377698 0.925929i \(-0.376716\pi\)
0.377698 + 0.925929i \(0.376716\pi\)
\(570\) 3.72364e6 0.000842180 0
\(571\) 1.51995e8 0.0341668 0.0170834 0.999854i \(-0.494562\pi\)
0.0170834 + 0.999854i \(0.494562\pi\)
\(572\) 1.26696e9 0.283058
\(573\) −4.46492e8 −0.0991455
\(574\) −3.78431e9 −0.835209
\(575\) −2.24645e9 −0.492788
\(576\) 1.60966e8 0.0350958
\(577\) −5.73397e8 −0.124263 −0.0621313 0.998068i \(-0.519790\pi\)
−0.0621313 + 0.998068i \(0.519790\pi\)
\(578\) 8.68443e9 1.87066
\(579\) −8.29049e8 −0.177503
\(580\) 4.15233e6 0.000883678 0
\(581\) −1.08176e9 −0.228831
\(582\) −1.35463e8 −0.0284833
\(583\) −5.87007e9 −1.22688
\(584\) 2.56770e9 0.533457
\(585\) −1.99214e7 −0.00411410
\(586\) 6.00225e9 1.23218
\(587\) −4.92920e9 −1.00587 −0.502936 0.864324i \(-0.667747\pi\)
−0.502936 + 0.864324i \(0.667747\pi\)
\(588\) 5.82748e7 0.0118212
\(589\) 1.57976e9 0.318558
\(590\) −6.31714e7 −0.0126630
\(591\) 1.44241e9 0.287431
\(592\) 4.99814e9 0.990107
\(593\) −1.67170e7 −0.00329205 −0.00164603 0.999999i \(-0.500524\pi\)
−0.00164603 + 0.999999i \(0.500524\pi\)
\(594\) 3.28662e9 0.643424
\(595\) −4.64594e7 −0.00904200
\(596\) −4.63822e9 −0.897406
\(597\) 9.06935e8 0.174448
\(598\) −8.95914e8 −0.171322
\(599\) 7.62725e8 0.145002 0.0725010 0.997368i \(-0.476902\pi\)
0.0725010 + 0.997368i \(0.476902\pi\)
\(600\) 4.12905e8 0.0780407
\(601\) 9.05252e9 1.70102 0.850509 0.525960i \(-0.176294\pi\)
0.850509 + 0.525960i \(0.176294\pi\)
\(602\) 3.47070e9 0.648379
\(603\) −1.59350e9 −0.295965
\(604\) 7.55030e8 0.139423
\(605\) −1.81555e8 −0.0333323
\(606\) 1.51576e9 0.276679
\(607\) 6.84926e9 1.24303 0.621517 0.783401i \(-0.286517\pi\)
0.621517 + 0.783401i \(0.286517\pi\)
\(608\) 1.73210e9 0.312544
\(609\) −3.12330e7 −0.00560342
\(610\) −1.06036e8 −0.0189147
\(611\) −2.91326e8 −0.0516696
\(612\) −5.00054e9 −0.881834
\(613\) −1.74317e9 −0.305653 −0.152826 0.988253i \(-0.548838\pi\)
−0.152826 + 0.988253i \(0.548838\pi\)
\(614\) 2.91650e9 0.508479
\(615\) 2.23529e7 0.00387500
\(616\) 2.11103e9 0.363883
\(617\) 7.09852e9 1.21666 0.608331 0.793684i \(-0.291839\pi\)
0.608331 + 0.793684i \(0.291839\pi\)
\(618\) 8.12430e8 0.138461
\(619\) 9.12322e9 1.54608 0.773038 0.634360i \(-0.218736\pi\)
0.773038 + 0.634360i \(0.218736\pi\)
\(620\) 5.34436e7 0.00900585
\(621\) −8.44289e8 −0.141472
\(622\) −1.35979e8 −0.0226571
\(623\) −4.68325e8 −0.0775959
\(624\) 3.03966e8 0.0500817
\(625\) 6.09931e9 0.999311
\(626\) −2.49048e9 −0.405763
\(627\) 4.89666e8 0.0793348
\(628\) 7.83258e8 0.126196
\(629\) −7.83638e9 −1.25556
\(630\) 4.40975e7 0.00702622
\(631\) 3.81738e9 0.604871 0.302435 0.953170i \(-0.402200\pi\)
0.302435 + 0.953170i \(0.402200\pi\)
\(632\) 1.79375e9 0.282652
\(633\) 6.19416e8 0.0970665
\(634\) 3.37009e9 0.525205
\(635\) 7.97764e7 0.0123642
\(636\) 3.68213e8 0.0567543
\(637\) −2.58475e8 −0.0396214
\(638\) 1.50310e9 0.229148
\(639\) −8.20643e8 −0.124423
\(640\) −9.81903e7 −0.0148060
\(641\) 1.10940e10 1.66373 0.831866 0.554976i \(-0.187273\pi\)
0.831866 + 0.554976i \(0.187273\pi\)
\(642\) −2.14941e9 −0.320587
\(643\) 2.55934e9 0.379655 0.189827 0.981817i \(-0.439207\pi\)
0.189827 + 0.981817i \(0.439207\pi\)
\(644\) 7.20438e8 0.106291
\(645\) −2.05005e7 −0.00300819
\(646\) −4.14573e9 −0.605044
\(647\) 2.73789e9 0.397421 0.198710 0.980058i \(-0.436325\pi\)
0.198710 + 0.980058i \(0.436325\pi\)
\(648\) −3.49427e9 −0.504480
\(649\) −8.30717e9 −1.19288
\(650\) 2.43304e9 0.347499
\(651\) −4.01992e8 −0.0571062
\(652\) −6.11194e9 −0.863600
\(653\) −1.05682e10 −1.48526 −0.742632 0.669700i \(-0.766423\pi\)
−0.742632 + 0.669700i \(0.766423\pi\)
\(654\) 2.56979e8 0.0359233
\(655\) 1.23704e8 0.0172004
\(656\) 1.58731e10 2.19532
\(657\) 7.05338e9 0.970327
\(658\) 6.44870e8 0.0882433
\(659\) 1.20741e10 1.64344 0.821722 0.569888i \(-0.193013\pi\)
0.821722 + 0.569888i \(0.193013\pi\)
\(660\) 1.65655e7 0.00224285
\(661\) 5.35648e9 0.721397 0.360699 0.932682i \(-0.382538\pi\)
0.360699 + 0.932682i \(0.382538\pi\)
\(662\) −3.39541e9 −0.454872
\(663\) −4.76576e8 −0.0635090
\(664\) 2.45810e9 0.325845
\(665\) 1.32811e7 0.00175129
\(666\) 7.43799e9 0.975654
\(667\) −3.86126e8 −0.0503836
\(668\) 1.88425e9 0.244580
\(669\) 1.20112e9 0.155094
\(670\) −4.46929e7 −0.00574086
\(671\) −1.39440e10 −1.78180
\(672\) −4.40756e8 −0.0560281
\(673\) 1.55353e9 0.196457 0.0982287 0.995164i \(-0.468682\pi\)
0.0982287 + 0.995164i \(0.468682\pi\)
\(674\) −1.58549e10 −1.99459
\(675\) 2.29284e9 0.286953
\(676\) 3.52497e8 0.0438876
\(677\) −9.46059e9 −1.17181 −0.585906 0.810379i \(-0.699261\pi\)
−0.585906 + 0.810379i \(0.699261\pi\)
\(678\) 1.69159e9 0.208445
\(679\) −4.83157e8 −0.0592303
\(680\) 1.05570e8 0.0128754
\(681\) −7.29729e8 −0.0885415
\(682\) 1.93460e10 2.33532
\(683\) −2.99091e9 −0.359196 −0.179598 0.983740i \(-0.557480\pi\)
−0.179598 + 0.983740i \(0.557480\pi\)
\(684\) 1.42948e9 0.170797
\(685\) 1.30938e8 0.0155650
\(686\) 5.72152e8 0.0676671
\(687\) 9.88016e7 0.0116256
\(688\) −1.45576e10 −1.70424
\(689\) −1.63319e9 −0.190226
\(690\) −1.17141e7 −0.00135749
\(691\) −4.13933e9 −0.477262 −0.238631 0.971110i \(-0.576699\pi\)
−0.238631 + 0.971110i \(0.576699\pi\)
\(692\) −8.73496e9 −1.00205
\(693\) 5.79891e9 0.661882
\(694\) −3.02203e9 −0.343194
\(695\) −1.99978e7 −0.00225962
\(696\) 7.09712e7 0.00797903
\(697\) −2.48868e10 −2.78390
\(698\) 1.01904e10 1.13421
\(699\) 1.40612e9 0.155723
\(700\) −1.95650e9 −0.215594
\(701\) 4.65534e9 0.510432 0.255216 0.966884i \(-0.417853\pi\)
0.255216 + 0.966884i \(0.417853\pi\)
\(702\) 9.14415e8 0.0997616
\(703\) 2.24015e9 0.243183
\(704\) 5.93683e8 0.0641284
\(705\) −3.80908e6 −0.000409410 0
\(706\) −6.55518e9 −0.701081
\(707\) 5.40627e9 0.575347
\(708\) 5.21086e8 0.0551814
\(709\) −1.78640e10 −1.88243 −0.941213 0.337813i \(-0.890313\pi\)
−0.941213 + 0.337813i \(0.890313\pi\)
\(710\) −2.30166e7 −0.00241345
\(711\) 4.92737e9 0.514128
\(712\) 1.06418e9 0.110493
\(713\) −4.96973e9 −0.513475
\(714\) 1.05494e9 0.108463
\(715\) −7.34752e7 −0.00751744
\(716\) 2.90587e8 0.0295856
\(717\) −6.51243e8 −0.0659821
\(718\) −7.41108e8 −0.0747216
\(719\) −1.18738e10 −1.19134 −0.595672 0.803228i \(-0.703114\pi\)
−0.595672 + 0.803228i \(0.703114\pi\)
\(720\) −1.84964e8 −0.0184682
\(721\) 2.89770e9 0.287926
\(722\) −1.14886e10 −1.13602
\(723\) −2.24268e9 −0.220690
\(724\) −1.05215e10 −1.03037
\(725\) 1.04861e9 0.102195
\(726\) 4.12250e9 0.399837
\(727\) −7.11104e8 −0.0686377 −0.0343189 0.999411i \(-0.510926\pi\)
−0.0343189 + 0.999411i \(0.510926\pi\)
\(728\) 5.87337e8 0.0564192
\(729\) −9.16319e9 −0.875992
\(730\) 1.97827e8 0.0188215
\(731\) 2.28243e10 2.16116
\(732\) 8.74667e8 0.0824240
\(733\) −1.05216e10 −0.986772 −0.493386 0.869810i \(-0.664241\pi\)
−0.493386 + 0.869810i \(0.664241\pi\)
\(734\) 1.62704e10 1.51867
\(735\) −3.37956e6 −0.000313945 0
\(736\) −5.44897e9 −0.503781
\(737\) −5.87722e9 −0.540799
\(738\) 2.36216e10 2.16327
\(739\) −2.00163e9 −0.182444 −0.0912218 0.995831i \(-0.529077\pi\)
−0.0912218 + 0.995831i \(0.529077\pi\)
\(740\) 7.57844e7 0.00687494
\(741\) 1.36237e8 0.0123007
\(742\) 3.61518e9 0.324875
\(743\) 1.94561e9 0.174018 0.0870092 0.996208i \(-0.472269\pi\)
0.0870092 + 0.996208i \(0.472269\pi\)
\(744\) 9.13452e8 0.0813169
\(745\) 2.68986e8 0.0238332
\(746\) 2.05692e10 1.81398
\(747\) 6.75231e9 0.592694
\(748\) −1.84432e10 −1.61132
\(749\) −7.66631e9 −0.666653
\(750\) 6.36313e7 0.00550753
\(751\) 1.36710e10 1.17777 0.588884 0.808218i \(-0.299568\pi\)
0.588884 + 0.808218i \(0.299568\pi\)
\(752\) −2.70487e9 −0.231945
\(753\) −1.98023e9 −0.169018
\(754\) 4.18198e8 0.0355289
\(755\) −4.37868e7 −0.00370279
\(756\) −7.35315e8 −0.0618938
\(757\) −1.83919e10 −1.54096 −0.770481 0.637463i \(-0.779984\pi\)
−0.770481 + 0.637463i \(0.779984\pi\)
\(758\) 5.73383e9 0.478193
\(759\) −1.54043e9 −0.127878
\(760\) −3.01789e7 −0.00249377
\(761\) 5.42592e9 0.446300 0.223150 0.974784i \(-0.428366\pi\)
0.223150 + 0.974784i \(0.428366\pi\)
\(762\) −1.81145e9 −0.148315
\(763\) 9.16571e8 0.0747016
\(764\) −4.80742e9 −0.390019
\(765\) 2.89998e8 0.0234197
\(766\) −2.49471e10 −2.00548
\(767\) −2.31125e9 −0.184954
\(768\) 2.29484e9 0.182805
\(769\) −5.18738e9 −0.411345 −0.205672 0.978621i \(-0.565938\pi\)
−0.205672 + 0.978621i \(0.565938\pi\)
\(770\) 1.62643e8 0.0128386
\(771\) −3.02199e9 −0.237467
\(772\) −8.92644e9 −0.698261
\(773\) −1.79029e10 −1.39411 −0.697053 0.717020i \(-0.745506\pi\)
−0.697053 + 0.717020i \(0.745506\pi\)
\(774\) −2.16640e10 −1.67936
\(775\) 1.34963e10 1.04150
\(776\) 1.09789e9 0.0843414
\(777\) −5.70035e8 −0.0435941
\(778\) −1.91683e10 −1.45933
\(779\) 7.11426e9 0.539199
\(780\) 4.60890e6 0.000347749 0
\(781\) −3.02674e9 −0.227351
\(782\) 1.30419e10 0.975255
\(783\) 3.94100e8 0.0293386
\(784\) −2.39986e9 −0.177861
\(785\) −4.54239e7 −0.00335151
\(786\) −2.80890e9 −0.206328
\(787\) −1.13894e10 −0.832890 −0.416445 0.909161i \(-0.636724\pi\)
−0.416445 + 0.909161i \(0.636724\pi\)
\(788\) 1.55306e10 1.13070
\(789\) 5.33786e8 0.0386899
\(790\) 1.38198e8 0.00997259
\(791\) 6.03342e9 0.433457
\(792\) −1.31770e10 −0.942491
\(793\) −3.87954e9 −0.276264
\(794\) −1.50489e9 −0.106692
\(795\) −2.13539e7 −0.00150728
\(796\) 9.76505e9 0.686244
\(797\) 1.82526e10 1.27709 0.638544 0.769585i \(-0.279537\pi\)
0.638544 + 0.769585i \(0.279537\pi\)
\(798\) −3.01569e8 −0.0210076
\(799\) 4.24086e9 0.294131
\(800\) 1.47978e10 1.02184
\(801\) 2.92327e9 0.200981
\(802\) 2.30588e10 1.57843
\(803\) 2.60146e10 1.77302
\(804\) 3.68661e8 0.0250168
\(805\) −4.17807e7 −0.00282286
\(806\) 5.38252e9 0.362087
\(807\) −3.67307e9 −0.246021
\(808\) −1.22847e10 −0.819269
\(809\) −3.19068e9 −0.211867 −0.105934 0.994373i \(-0.533783\pi\)
−0.105934 + 0.994373i \(0.533783\pi\)
\(810\) −2.69214e8 −0.0177992
\(811\) 8.78229e9 0.578142 0.289071 0.957308i \(-0.406654\pi\)
0.289071 + 0.957308i \(0.406654\pi\)
\(812\) −3.36288e8 −0.0220427
\(813\) 2.72145e9 0.177616
\(814\) 2.74332e10 1.78275
\(815\) 3.54453e8 0.0229354
\(816\) −4.42487e9 −0.285092
\(817\) −6.52468e9 −0.418584
\(818\) −5.17032e9 −0.330279
\(819\) 1.61339e9 0.102623
\(820\) 2.40676e8 0.0152435
\(821\) −1.84405e10 −1.16298 −0.581489 0.813554i \(-0.697530\pi\)
−0.581489 + 0.813554i \(0.697530\pi\)
\(822\) −2.97316e9 −0.186710
\(823\) −7.73606e9 −0.483749 −0.241875 0.970308i \(-0.577762\pi\)
−0.241875 + 0.970308i \(0.577762\pi\)
\(824\) −6.58450e9 −0.409994
\(825\) 4.18335e9 0.259379
\(826\) 5.11611e9 0.315871
\(827\) −2.71320e10 −1.66806 −0.834030 0.551719i \(-0.813972\pi\)
−0.834030 + 0.551719i \(0.813972\pi\)
\(828\) −4.49695e9 −0.275304
\(829\) 1.50861e10 0.919680 0.459840 0.888002i \(-0.347907\pi\)
0.459840 + 0.888002i \(0.347907\pi\)
\(830\) 1.89383e8 0.0114965
\(831\) −1.42238e9 −0.0859826
\(832\) 1.65176e8 0.00994298
\(833\) 3.76265e9 0.225546
\(834\) 4.54082e8 0.0271052
\(835\) −1.09274e8 −0.00649554
\(836\) 5.27228e9 0.312088
\(837\) 5.07236e9 0.299000
\(838\) −3.07810e8 −0.0180688
\(839\) 1.73769e10 1.01580 0.507898 0.861417i \(-0.330423\pi\)
0.507898 + 0.861417i \(0.330423\pi\)
\(840\) 7.67942e6 0.000447045 0
\(841\) −1.70696e10 −0.989551
\(842\) −1.19476e10 −0.689742
\(843\) 3.28659e8 0.0188951
\(844\) 6.66930e9 0.381840
\(845\) −2.04425e7 −0.00116556
\(846\) −4.02526e9 −0.228559
\(847\) 1.47038e10 0.831451
\(848\) −1.51637e10 −0.853923
\(849\) −4.01019e9 −0.224899
\(850\) −3.54181e10 −1.97815
\(851\) −7.04721e9 −0.391980
\(852\) 1.89859e8 0.0105170
\(853\) −4.19213e9 −0.231267 −0.115633 0.993292i \(-0.536890\pi\)
−0.115633 + 0.993292i \(0.536890\pi\)
\(854\) 8.58764e9 0.471814
\(855\) −8.29004e7 −0.00453602
\(856\) 1.74203e10 0.949286
\(857\) −2.55881e8 −0.0138869 −0.00694343 0.999976i \(-0.502210\pi\)
−0.00694343 + 0.999976i \(0.502210\pi\)
\(858\) 1.66837e9 0.0901753
\(859\) 3.29449e9 0.177342 0.0886711 0.996061i \(-0.471738\pi\)
0.0886711 + 0.996061i \(0.471738\pi\)
\(860\) −2.20731e8 −0.0118336
\(861\) −1.81032e9 −0.0966593
\(862\) 1.10353e10 0.586823
\(863\) −3.39290e10 −1.79694 −0.898469 0.439038i \(-0.855319\pi\)
−0.898469 + 0.439038i \(0.855319\pi\)
\(864\) 5.56149e9 0.293355
\(865\) 5.06571e8 0.0266124
\(866\) 1.53615e9 0.0803751
\(867\) 4.15441e9 0.216492
\(868\) −4.32828e9 −0.224645
\(869\) 1.81734e10 0.939435
\(870\) 5.46793e6 0.000281518 0
\(871\) −1.63518e9 −0.0838497
\(872\) −2.08274e9 −0.106372
\(873\) 3.01585e9 0.153412
\(874\) −3.72823e9 −0.188892
\(875\) 2.26954e8 0.0114528
\(876\) −1.63183e9 −0.0820180
\(877\) 2.89806e10 1.45081 0.725403 0.688324i \(-0.241653\pi\)
0.725403 + 0.688324i \(0.241653\pi\)
\(878\) 4.25300e10 2.12063
\(879\) 2.87132e9 0.142601
\(880\) −6.82195e8 −0.0337458
\(881\) −3.24339e10 −1.59803 −0.799013 0.601313i \(-0.794644\pi\)
−0.799013 + 0.601313i \(0.794644\pi\)
\(882\) −3.57136e9 −0.175264
\(883\) −1.73371e9 −0.0847451 −0.0423725 0.999102i \(-0.513492\pi\)
−0.0423725 + 0.999102i \(0.513492\pi\)
\(884\) −5.13134e9 −0.249832
\(885\) −3.02195e7 −0.00146550
\(886\) −1.88051e10 −0.908361
\(887\) 2.31382e10 1.11326 0.556631 0.830760i \(-0.312094\pi\)
0.556631 + 0.830760i \(0.312094\pi\)
\(888\) 1.29530e9 0.0620762
\(889\) −6.46092e9 −0.308417
\(890\) 8.19892e7 0.00389845
\(891\) −3.54022e10 −1.67671
\(892\) 1.29325e10 0.610108
\(893\) −1.21231e9 −0.0569686
\(894\) −6.10776e9 −0.285891
\(895\) −1.68521e7 −0.000785731 0
\(896\) 7.95222e9 0.369326
\(897\) −4.28582e8 −0.0198272
\(898\) −2.91904e10 −1.34516
\(899\) 2.31979e9 0.106485
\(900\) 1.22124e10 0.558410
\(901\) 2.37745e10 1.08287
\(902\) 8.71223e10 3.95282
\(903\) 1.66029e9 0.0750373
\(904\) −1.37098e10 −0.617224
\(905\) 6.10181e8 0.0273646
\(906\) 9.94250e8 0.0444167
\(907\) −3.04752e10 −1.35619 −0.678096 0.734973i \(-0.737195\pi\)
−0.678096 + 0.734973i \(0.737195\pi\)
\(908\) −7.85706e9 −0.348305
\(909\) −3.37458e10 −1.49020
\(910\) 4.52510e7 0.00199060
\(911\) 8.72533e9 0.382356 0.191178 0.981555i \(-0.438769\pi\)
0.191178 + 0.981555i \(0.438769\pi\)
\(912\) 1.26491e9 0.0552178
\(913\) 2.49042e10 1.08299
\(914\) −2.46509e10 −1.06787
\(915\) −5.07249e7 −0.00218901
\(916\) 1.06381e9 0.0457328
\(917\) −1.00185e10 −0.429053
\(918\) −1.33112e10 −0.567897
\(919\) 3.73817e10 1.58875 0.794373 0.607431i \(-0.207800\pi\)
0.794373 + 0.607431i \(0.207800\pi\)
\(920\) 9.49389e7 0.00401964
\(921\) 1.39518e9 0.0588466
\(922\) 4.62539e10 1.94353
\(923\) −8.42109e8 −0.0352503
\(924\) −1.34160e9 −0.0559463
\(925\) 1.91382e10 0.795069
\(926\) 5.70363e10 2.36055
\(927\) −1.80874e10 −0.745756
\(928\) 2.54349e9 0.104475
\(929\) −3.60450e10 −1.47499 −0.737497 0.675350i \(-0.763993\pi\)
−0.737497 + 0.675350i \(0.763993\pi\)
\(930\) 7.03763e7 0.00286904
\(931\) −1.07561e9 −0.0436848
\(932\) 1.51398e10 0.612582
\(933\) −6.50486e7 −0.00262212
\(934\) −5.01615e10 −2.01445
\(935\) 1.06959e9 0.0427933
\(936\) −3.66614e9 −0.146131
\(937\) 3.50259e10 1.39092 0.695458 0.718567i \(-0.255202\pi\)
0.695458 + 0.718567i \(0.255202\pi\)
\(938\) 3.61958e9 0.143202
\(939\) −1.19138e9 −0.0469592
\(940\) −4.10127e7 −0.00161054
\(941\) −3.32485e9 −0.130079 −0.0650396 0.997883i \(-0.520717\pi\)
−0.0650396 + 0.997883i \(0.520717\pi\)
\(942\) 1.03142e9 0.0402030
\(943\) −2.23805e10 −0.869120
\(944\) −2.14592e10 −0.830257
\(945\) 4.26435e7 0.00164377
\(946\) −7.99023e10 −3.06860
\(947\) 2.58279e10 0.988245 0.494123 0.869392i \(-0.335489\pi\)
0.494123 + 0.869392i \(0.335489\pi\)
\(948\) −1.13997e9 −0.0434573
\(949\) 7.23788e9 0.274903
\(950\) 1.01248e10 0.383136
\(951\) 1.61216e9 0.0607823
\(952\) −8.54993e9 −0.321169
\(953\) 3.55136e10 1.32914 0.664569 0.747227i \(-0.268615\pi\)
0.664569 + 0.747227i \(0.268615\pi\)
\(954\) −2.25658e10 −0.841458
\(955\) 2.78799e8 0.0103581
\(956\) −7.01199e9 −0.259560
\(957\) 7.19045e8 0.0265194
\(958\) −4.78346e10 −1.75777
\(959\) −1.06044e10 −0.388259
\(960\) 2.15968e6 7.87844e−5 0
\(961\) 2.34477e9 0.0852253
\(962\) 7.63255e9 0.276412
\(963\) 4.78529e10 1.72670
\(964\) −2.41471e10 −0.868151
\(965\) 5.17675e8 0.0185443
\(966\) 9.48697e8 0.0338616
\(967\) 2.74905e10 0.977664 0.488832 0.872378i \(-0.337423\pi\)
0.488832 + 0.872378i \(0.337423\pi\)
\(968\) −3.34116e10 −1.18395
\(969\) −1.98321e9 −0.0700222
\(970\) 8.45859e7 0.00297575
\(971\) −1.03435e10 −0.362576 −0.181288 0.983430i \(-0.558027\pi\)
−0.181288 + 0.983430i \(0.558027\pi\)
\(972\) 6.90912e9 0.241318
\(973\) 1.61958e9 0.0563647
\(974\) 2.05410e10 0.712305
\(975\) 1.16390e9 0.0402162
\(976\) −3.60203e10 −1.24015
\(977\) 3.21256e10 1.10210 0.551050 0.834472i \(-0.314227\pi\)
0.551050 + 0.834472i \(0.314227\pi\)
\(978\) −8.04841e9 −0.275121
\(979\) 1.07818e10 0.367240
\(980\) −3.63880e7 −0.00123500
\(981\) −5.72121e9 −0.193485
\(982\) 5.23457e10 1.76397
\(983\) 4.63147e10 1.55518 0.777592 0.628769i \(-0.216441\pi\)
0.777592 + 0.628769i \(0.216441\pi\)
\(984\) 4.11361e9 0.137639
\(985\) −9.00673e8 −0.0300290
\(986\) −6.08775e9 −0.202250
\(987\) 3.08489e8 0.0102125
\(988\) 1.46687e9 0.0483885
\(989\) 2.05258e10 0.674704
\(990\) −1.01521e9 −0.0332532
\(991\) 5.46722e10 1.78447 0.892234 0.451573i \(-0.149137\pi\)
0.892234 + 0.451573i \(0.149137\pi\)
\(992\) 3.27365e10 1.06474
\(993\) −1.62427e9 −0.0526426
\(994\) 1.86407e9 0.0602018
\(995\) −5.66309e8 −0.0182252
\(996\) −1.56217e9 −0.0500981
\(997\) −1.10862e10 −0.354282 −0.177141 0.984185i \(-0.556685\pi\)
−0.177141 + 0.984185i \(0.556685\pi\)
\(998\) 5.34877e10 1.70332
\(999\) 7.19274e9 0.228252
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.c.1.9 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.c.1.9 10 1.1 even 1 trivial