Properties

Label 38.10.a.d.1.4
Level $38$
Weight $10$
Character 38.1
Self dual yes
Analytic conductor $19.571$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.5713617742\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 34433 x^{2} - 2723303 x - 48270488\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-67.1081\) of defining polynomial
Character \(\chi\) \(=\) 38.1

$q$-expansion

\(f(q)\) \(=\) \(q-16.0000 q^{2} +265.578 q^{3} +256.000 q^{4} -2367.11 q^{5} -4249.24 q^{6} +5859.40 q^{7} -4096.00 q^{8} +50848.5 q^{9} +O(q^{10})\) \(q-16.0000 q^{2} +265.578 q^{3} +256.000 q^{4} -2367.11 q^{5} -4249.24 q^{6} +5859.40 q^{7} -4096.00 q^{8} +50848.5 q^{9} +37873.7 q^{10} -38743.9 q^{11} +67987.9 q^{12} +179439. q^{13} -93750.4 q^{14} -628650. q^{15} +65536.0 q^{16} +184492. q^{17} -813576. q^{18} +130321. q^{19} -605979. q^{20} +1.55613e6 q^{21} +619902. q^{22} -115333. q^{23} -1.08781e6 q^{24} +3.65006e6 q^{25} -2.87102e6 q^{26} +8.27687e6 q^{27} +1.50001e6 q^{28} +569755. q^{29} +1.00584e7 q^{30} +5.80788e6 q^{31} -1.04858e6 q^{32} -1.02895e7 q^{33} -2.95187e6 q^{34} -1.38698e7 q^{35} +1.30172e7 q^{36} +3.15312e6 q^{37} -2.08514e6 q^{38} +4.76550e7 q^{39} +9.69566e6 q^{40} +54772.2 q^{41} -2.48980e7 q^{42} -1.63249e7 q^{43} -9.91844e6 q^{44} -1.20364e8 q^{45} +1.84533e6 q^{46} +2.84428e7 q^{47} +1.74049e7 q^{48} -6.02101e6 q^{49} -5.84010e7 q^{50} +4.89969e7 q^{51} +4.59363e7 q^{52} +7.33201e7 q^{53} -1.32430e8 q^{54} +9.17109e7 q^{55} -2.40001e7 q^{56} +3.46104e7 q^{57} -9.11608e6 q^{58} -1.45518e7 q^{59} -1.60935e8 q^{60} +9.96035e7 q^{61} -9.29260e7 q^{62} +2.97942e8 q^{63} +1.67772e7 q^{64} -4.24751e8 q^{65} +1.64632e8 q^{66} -2.49050e8 q^{67} +4.72299e7 q^{68} -3.06299e7 q^{69} +2.21917e8 q^{70} -1.33230e8 q^{71} -2.08276e8 q^{72} -2.28114e8 q^{73} -5.04499e7 q^{74} +9.69375e8 q^{75} +3.33622e7 q^{76} -2.27016e8 q^{77} -7.62479e8 q^{78} -6.65788e8 q^{79} -1.55131e8 q^{80} +1.19730e9 q^{81} -876356. q^{82} -3.48852e8 q^{83} +3.98368e8 q^{84} -4.36712e8 q^{85} +2.61199e8 q^{86} +1.51314e8 q^{87} +1.58695e8 q^{88} +2.61939e8 q^{89} +1.92582e9 q^{90} +1.05140e9 q^{91} -2.95252e7 q^{92} +1.54244e9 q^{93} -4.55084e8 q^{94} -3.08484e8 q^{95} -2.78478e8 q^{96} -1.10091e9 q^{97} +9.63361e7 q^{98} -1.97007e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 64q^{2} + 84q^{3} + 1024q^{4} - 1395q^{5} - 1344q^{6} + 12307q^{7} - 16384q^{8} + 16538q^{9} + O(q^{10}) \) \( 4q - 64q^{2} + 84q^{3} + 1024q^{4} - 1395q^{5} - 1344q^{6} + 12307q^{7} - 16384q^{8} + 16538q^{9} + 22320q^{10} - 104249q^{11} + 21504q^{12} + 120486q^{13} - 196912q^{14} - 591090q^{15} + 262144q^{16} - 412139q^{17} - 264608q^{18} + 521284q^{19} - 357120q^{20} + 2437006q^{21} + 1667984q^{22} + 3010300q^{23} - 344064q^{24} + 9760585q^{25} - 1927776q^{26} + 12387978q^{27} + 3150592q^{28} + 6153240q^{29} + 9457440q^{30} + 12774024q^{31} - 4194304q^{32} - 3258022q^{33} + 6594224q^{34} + 9823425q^{35} + 4233728q^{36} + 20506048q^{37} - 8340544q^{38} + 69881444q^{39} + 5713920q^{40} + 11620300q^{41} - 38992096q^{42} + 7698327q^{43} - 26687744q^{44} - 124015815q^{45} - 48164800q^{46} - 31581083q^{47} + 5505024q^{48} + 18970383q^{49} - 156169360q^{50} - 8594812q^{51} + 30844416q^{52} + 72549422q^{53} - 198207648q^{54} + 21332505q^{55} - 50409472q^{56} + 10946964q^{57} - 98451840q^{58} - 149234120q^{59} - 151319040q^{60} + 129004373q^{61} - 204384384q^{62} + 102967551q^{63} + 67108864q^{64} + 124691700q^{65} + 52128352q^{66} + 132595266q^{67} - 105507584q^{68} - 45529972q^{69} - 157174800q^{70} - 47138482q^{71} - 67739648q^{72} - 39332795q^{73} - 328096768q^{74} + 824627010q^{75} + 133448704q^{76} - 165933719q^{77} - 1118103104q^{78} - 307010840q^{79} - 91422720q^{80} + 1305551744q^{81} - 185924800q^{82} - 746568232q^{83} + 623873536q^{84} - 105005985q^{85} - 123173232q^{86} - 82148208q^{87} + 427003904q^{88} + 286943482q^{89} + 1984253040q^{90} + 3155781114q^{91} + 770636800q^{92} + 1151901596q^{93} + 505297328q^{94} - 181797795q^{95} - 88080384q^{96} + 793519958q^{97} - 303526128q^{98} - 1681833809q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.0000 −0.707107
\(3\) 265.578 1.89298 0.946490 0.322733i \(-0.104602\pi\)
0.946490 + 0.322733i \(0.104602\pi\)
\(4\) 256.000 0.500000
\(5\) −2367.11 −1.69376 −0.846881 0.531782i \(-0.821523\pi\)
−0.846881 + 0.531782i \(0.821523\pi\)
\(6\) −4249.24 −1.33854
\(7\) 5859.40 0.922385 0.461192 0.887300i \(-0.347422\pi\)
0.461192 + 0.887300i \(0.347422\pi\)
\(8\) −4096.00 −0.353553
\(9\) 50848.5 2.58337
\(10\) 37873.7 1.19767
\(11\) −38743.9 −0.797878 −0.398939 0.916978i \(-0.630621\pi\)
−0.398939 + 0.916978i \(0.630621\pi\)
\(12\) 67987.9 0.946490
\(13\) 179439. 1.74249 0.871247 0.490845i \(-0.163312\pi\)
0.871247 + 0.490845i \(0.163312\pi\)
\(14\) −93750.4 −0.652225
\(15\) −628650. −3.20626
\(16\) 65536.0 0.250000
\(17\) 184492. 0.535744 0.267872 0.963455i \(-0.413680\pi\)
0.267872 + 0.963455i \(0.413680\pi\)
\(18\) −813576. −1.82672
\(19\) 130321. 0.229416
\(20\) −605979. −0.846881
\(21\) 1.55613e6 1.74606
\(22\) 619902. 0.564185
\(23\) −115333. −0.0859366 −0.0429683 0.999076i \(-0.513681\pi\)
−0.0429683 + 0.999076i \(0.513681\pi\)
\(24\) −1.08781e6 −0.669269
\(25\) 3.65006e6 1.86883
\(26\) −2.87102e6 −1.23213
\(27\) 8.27687e6 2.99729
\(28\) 1.50001e6 0.461192
\(29\) 569755. 0.149588 0.0747941 0.997199i \(-0.476170\pi\)
0.0747941 + 0.997199i \(0.476170\pi\)
\(30\) 1.00584e7 2.26717
\(31\) 5.80788e6 1.12951 0.564755 0.825259i \(-0.308971\pi\)
0.564755 + 0.825259i \(0.308971\pi\)
\(32\) −1.04858e6 −0.176777
\(33\) −1.02895e7 −1.51037
\(34\) −2.95187e6 −0.378828
\(35\) −1.38698e7 −1.56230
\(36\) 1.30172e7 1.29169
\(37\) 3.15312e6 0.276587 0.138294 0.990391i \(-0.455838\pi\)
0.138294 + 0.990391i \(0.455838\pi\)
\(38\) −2.08514e6 −0.162221
\(39\) 4.76550e7 3.29851
\(40\) 9.69566e6 0.598836
\(41\) 54772.2 0.00302714 0.00151357 0.999999i \(-0.499518\pi\)
0.00151357 + 0.999999i \(0.499518\pi\)
\(42\) −2.48980e7 −1.23465
\(43\) −1.63249e7 −0.728187 −0.364094 0.931362i \(-0.618621\pi\)
−0.364094 + 0.931362i \(0.618621\pi\)
\(44\) −9.91844e6 −0.398939
\(45\) −1.20364e8 −4.37562
\(46\) 1.84533e6 0.0607663
\(47\) 2.84428e7 0.850220 0.425110 0.905142i \(-0.360235\pi\)
0.425110 + 0.905142i \(0.360235\pi\)
\(48\) 1.74049e7 0.473245
\(49\) −6.02101e6 −0.149206
\(50\) −5.84010e7 −1.32146
\(51\) 4.89969e7 1.01415
\(52\) 4.59363e7 0.871247
\(53\) 7.33201e7 1.27639 0.638193 0.769877i \(-0.279682\pi\)
0.638193 + 0.769877i \(0.279682\pi\)
\(54\) −1.32430e8 −2.11941
\(55\) 9.17109e7 1.35142
\(56\) −2.40001e7 −0.326112
\(57\) 3.46104e7 0.434279
\(58\) −9.11608e6 −0.105775
\(59\) −1.45518e7 −0.156345 −0.0781725 0.996940i \(-0.524909\pi\)
−0.0781725 + 0.996940i \(0.524909\pi\)
\(60\) −1.60935e8 −1.60313
\(61\) 9.96035e7 0.921065 0.460533 0.887643i \(-0.347658\pi\)
0.460533 + 0.887643i \(0.347658\pi\)
\(62\) −9.29260e7 −0.798684
\(63\) 2.97942e8 2.38286
\(64\) 1.67772e7 0.125000
\(65\) −4.24751e8 −2.95137
\(66\) 1.64632e8 1.06799
\(67\) −2.49050e8 −1.50991 −0.754954 0.655778i \(-0.772341\pi\)
−0.754954 + 0.655778i \(0.772341\pi\)
\(68\) 4.72299e7 0.267872
\(69\) −3.06299e7 −0.162676
\(70\) 2.21917e8 1.10471
\(71\) −1.33230e8 −0.622215 −0.311108 0.950375i \(-0.600700\pi\)
−0.311108 + 0.950375i \(0.600700\pi\)
\(72\) −2.08276e8 −0.913360
\(73\) −2.28114e8 −0.940153 −0.470077 0.882626i \(-0.655774\pi\)
−0.470077 + 0.882626i \(0.655774\pi\)
\(74\) −5.04499e7 −0.195577
\(75\) 9.69375e8 3.53766
\(76\) 3.33622e7 0.114708
\(77\) −2.27016e8 −0.735950
\(78\) −7.62479e8 −2.33240
\(79\) −6.65788e8 −1.92315 −0.961577 0.274534i \(-0.911476\pi\)
−0.961577 + 0.274534i \(0.911476\pi\)
\(80\) −1.55131e8 −0.423441
\(81\) 1.19730e9 3.09044
\(82\) −876356. −0.00214051
\(83\) −3.48852e8 −0.806844 −0.403422 0.915014i \(-0.632179\pi\)
−0.403422 + 0.915014i \(0.632179\pi\)
\(84\) 3.98368e8 0.873028
\(85\) −4.36712e8 −0.907422
\(86\) 2.61199e8 0.514906
\(87\) 1.51314e8 0.283167
\(88\) 1.58695e8 0.282092
\(89\) 2.61939e8 0.442533 0.221267 0.975213i \(-0.428981\pi\)
0.221267 + 0.975213i \(0.428981\pi\)
\(90\) 1.92582e9 3.09403
\(91\) 1.05140e9 1.60725
\(92\) −2.95252e7 −0.0429683
\(93\) 1.54244e9 2.13814
\(94\) −4.55084e8 −0.601197
\(95\) −3.08484e8 −0.388576
\(96\) −2.78478e8 −0.334635
\(97\) −1.10091e9 −1.26264 −0.631321 0.775521i \(-0.717487\pi\)
−0.631321 + 0.775521i \(0.717487\pi\)
\(98\) 9.63361e7 0.105505
\(99\) −1.97007e9 −2.06122
\(100\) 9.34416e8 0.934416
\(101\) 1.22467e9 1.17104 0.585521 0.810658i \(-0.300890\pi\)
0.585521 + 0.810658i \(0.300890\pi\)
\(102\) −7.83951e8 −0.717114
\(103\) 1.37095e9 1.20020 0.600100 0.799925i \(-0.295128\pi\)
0.600100 + 0.799925i \(0.295128\pi\)
\(104\) −7.34981e8 −0.616065
\(105\) −3.68352e9 −2.95740
\(106\) −1.17312e9 −0.902541
\(107\) 1.42672e9 1.05223 0.526115 0.850413i \(-0.323648\pi\)
0.526115 + 0.850413i \(0.323648\pi\)
\(108\) 2.11888e9 1.49865
\(109\) −2.50340e9 −1.69868 −0.849340 0.527846i \(-0.823000\pi\)
−0.849340 + 0.527846i \(0.823000\pi\)
\(110\) −1.46737e9 −0.955595
\(111\) 8.37398e8 0.523575
\(112\) 3.84002e8 0.230596
\(113\) −3.27341e9 −1.88863 −0.944317 0.329036i \(-0.893276\pi\)
−0.944317 + 0.329036i \(0.893276\pi\)
\(114\) −5.53766e8 −0.307082
\(115\) 2.73005e8 0.145556
\(116\) 1.45857e8 0.0747941
\(117\) 9.12420e9 4.50151
\(118\) 2.32830e8 0.110553
\(119\) 1.08101e9 0.494162
\(120\) 2.57495e9 1.13358
\(121\) −8.56858e8 −0.363391
\(122\) −1.59366e9 −0.651291
\(123\) 1.45463e7 0.00573032
\(124\) 1.48682e9 0.564755
\(125\) −4.01683e9 −1.47160
\(126\) −4.76707e9 −1.68494
\(127\) −4.27729e9 −1.45899 −0.729494 0.683987i \(-0.760244\pi\)
−0.729494 + 0.683987i \(0.760244\pi\)
\(128\) −2.68435e8 −0.0883883
\(129\) −4.33554e9 −1.37844
\(130\) 6.79601e9 2.08693
\(131\) 3.80463e9 1.12873 0.564367 0.825524i \(-0.309120\pi\)
0.564367 + 0.825524i \(0.309120\pi\)
\(132\) −2.63412e9 −0.755183
\(133\) 7.63603e8 0.211610
\(134\) 3.98481e9 1.06767
\(135\) −1.95922e10 −5.07670
\(136\) −7.55678e8 −0.189414
\(137\) 1.42380e8 0.0345308 0.0172654 0.999851i \(-0.494504\pi\)
0.0172654 + 0.999851i \(0.494504\pi\)
\(138\) 4.90078e8 0.115029
\(139\) 2.12706e9 0.483295 0.241648 0.970364i \(-0.422312\pi\)
0.241648 + 0.970364i \(0.422312\pi\)
\(140\) −3.55067e9 −0.781151
\(141\) 7.55377e9 1.60945
\(142\) 2.13169e9 0.439972
\(143\) −6.95216e9 −1.39030
\(144\) 3.33241e9 0.645843
\(145\) −1.34867e9 −0.253367
\(146\) 3.64982e9 0.664789
\(147\) −1.59905e9 −0.282444
\(148\) 8.07198e8 0.138294
\(149\) 5.74897e9 0.955547 0.477773 0.878483i \(-0.341444\pi\)
0.477773 + 0.878483i \(0.341444\pi\)
\(150\) −1.55100e10 −2.50150
\(151\) 3.66097e9 0.573059 0.286530 0.958071i \(-0.407498\pi\)
0.286530 + 0.958071i \(0.407498\pi\)
\(152\) −5.33795e8 −0.0811107
\(153\) 9.38114e9 1.38403
\(154\) 3.63226e9 0.520395
\(155\) −1.37479e10 −1.91312
\(156\) 1.21997e10 1.64925
\(157\) −1.22861e9 −0.161386 −0.0806930 0.996739i \(-0.525713\pi\)
−0.0806930 + 0.996739i \(0.525713\pi\)
\(158\) 1.06526e10 1.35988
\(159\) 1.94722e10 2.41617
\(160\) 2.48209e9 0.299418
\(161\) −6.75782e8 −0.0792666
\(162\) −1.91568e10 −2.18527
\(163\) −6.60342e8 −0.0732697 −0.0366349 0.999329i \(-0.511664\pi\)
−0.0366349 + 0.999329i \(0.511664\pi\)
\(164\) 1.40217e7 0.00151357
\(165\) 2.43564e10 2.55820
\(166\) 5.58163e9 0.570525
\(167\) −1.05713e10 −1.05173 −0.525865 0.850568i \(-0.676258\pi\)
−0.525865 + 0.850568i \(0.676258\pi\)
\(168\) −6.37390e9 −0.617324
\(169\) 2.15938e10 2.03629
\(170\) 6.98738e9 0.641645
\(171\) 6.62663e9 0.592666
\(172\) −4.17918e9 −0.364094
\(173\) −1.23439e10 −1.04772 −0.523861 0.851804i \(-0.675509\pi\)
−0.523861 + 0.851804i \(0.675509\pi\)
\(174\) −2.42103e9 −0.200230
\(175\) 2.13872e10 1.72378
\(176\) −2.53912e9 −0.199469
\(177\) −3.86465e9 −0.295958
\(178\) −4.19103e9 −0.312918
\(179\) 3.15457e9 0.229669 0.114834 0.993385i \(-0.463366\pi\)
0.114834 + 0.993385i \(0.463366\pi\)
\(180\) −3.08131e10 −2.18781
\(181\) 1.16218e10 0.804857 0.402429 0.915451i \(-0.368166\pi\)
0.402429 + 0.915451i \(0.368166\pi\)
\(182\) −1.68225e10 −1.13650
\(183\) 2.64525e10 1.74356
\(184\) 4.72404e8 0.0303832
\(185\) −7.46376e9 −0.468474
\(186\) −2.46791e10 −1.51189
\(187\) −7.14793e9 −0.427458
\(188\) 7.28135e9 0.425110
\(189\) 4.84975e10 2.76466
\(190\) 4.93574e9 0.274765
\(191\) −3.25043e9 −0.176722 −0.0883610 0.996089i \(-0.528163\pi\)
−0.0883610 + 0.996089i \(0.528163\pi\)
\(192\) 4.45565e9 0.236622
\(193\) 2.45765e9 0.127501 0.0637503 0.997966i \(-0.479694\pi\)
0.0637503 + 0.997966i \(0.479694\pi\)
\(194\) 1.76146e10 0.892823
\(195\) −1.12804e11 −5.58689
\(196\) −1.54138e9 −0.0746031
\(197\) −1.10866e10 −0.524445 −0.262223 0.965007i \(-0.584455\pi\)
−0.262223 + 0.965007i \(0.584455\pi\)
\(198\) 3.15211e10 1.45750
\(199\) −1.66765e10 −0.753817 −0.376908 0.926251i \(-0.623013\pi\)
−0.376908 + 0.926251i \(0.623013\pi\)
\(200\) −1.49507e10 −0.660732
\(201\) −6.61422e10 −2.85823
\(202\) −1.95947e10 −0.828051
\(203\) 3.33842e9 0.137978
\(204\) 1.25432e10 0.507076
\(205\) −1.29652e8 −0.00512726
\(206\) −2.19352e10 −0.848669
\(207\) −5.86451e9 −0.222006
\(208\) 1.17597e10 0.435624
\(209\) −5.04914e9 −0.183046
\(210\) 5.89363e10 2.09120
\(211\) 2.51893e10 0.874872 0.437436 0.899249i \(-0.355887\pi\)
0.437436 + 0.899249i \(0.355887\pi\)
\(212\) 1.87699e10 0.638193
\(213\) −3.53830e10 −1.17784
\(214\) −2.28275e10 −0.744039
\(215\) 3.86428e10 1.23338
\(216\) −3.39021e10 −1.05970
\(217\) 3.40307e10 1.04184
\(218\) 4.00545e10 1.20115
\(219\) −6.05819e10 −1.77969
\(220\) 2.34780e10 0.675708
\(221\) 3.31050e10 0.933530
\(222\) −1.33984e10 −0.370223
\(223\) 4.72569e10 1.27966 0.639829 0.768518i \(-0.279005\pi\)
0.639829 + 0.768518i \(0.279005\pi\)
\(224\) −6.14403e9 −0.163056
\(225\) 1.85600e11 4.82789
\(226\) 5.23746e10 1.33547
\(227\) −3.66710e10 −0.916657 −0.458328 0.888783i \(-0.651552\pi\)
−0.458328 + 0.888783i \(0.651552\pi\)
\(228\) 8.86025e9 0.217140
\(229\) 1.94037e10 0.466258 0.233129 0.972446i \(-0.425104\pi\)
0.233129 + 0.972446i \(0.425104\pi\)
\(230\) −4.36808e9 −0.102924
\(231\) −6.02904e10 −1.39314
\(232\) −2.33372e9 −0.0528874
\(233\) −3.57852e10 −0.795430 −0.397715 0.917509i \(-0.630197\pi\)
−0.397715 + 0.917509i \(0.630197\pi\)
\(234\) −1.45987e11 −3.18305
\(235\) −6.73270e10 −1.44007
\(236\) −3.72527e9 −0.0781725
\(237\) −1.76819e11 −3.64049
\(238\) −1.72962e10 −0.349425
\(239\) 5.18718e9 0.102835 0.0514174 0.998677i \(-0.483626\pi\)
0.0514174 + 0.998677i \(0.483626\pi\)
\(240\) −4.11992e10 −0.801565
\(241\) 1.46244e10 0.279254 0.139627 0.990204i \(-0.455410\pi\)
0.139627 + 0.990204i \(0.455410\pi\)
\(242\) 1.37097e10 0.256957
\(243\) 1.55063e11 2.85285
\(244\) 2.54985e10 0.460533
\(245\) 1.42524e10 0.252720
\(246\) −2.32741e8 −0.00405195
\(247\) 2.33846e10 0.399756
\(248\) −2.37891e10 −0.399342
\(249\) −9.26473e10 −1.52734
\(250\) 6.42693e10 1.04057
\(251\) 1.36902e8 0.00217710 0.00108855 0.999999i \(-0.499654\pi\)
0.00108855 + 0.999999i \(0.499654\pi\)
\(252\) 7.62731e10 1.19143
\(253\) 4.46845e9 0.0685669
\(254\) 6.84366e10 1.03166
\(255\) −1.15981e11 −1.71773
\(256\) 4.29497e9 0.0625000
\(257\) 5.04580e10 0.721491 0.360745 0.932664i \(-0.382522\pi\)
0.360745 + 0.932664i \(0.382522\pi\)
\(258\) 6.93686e10 0.974707
\(259\) 1.84754e10 0.255120
\(260\) −1.08736e11 −1.47569
\(261\) 2.89712e10 0.386442
\(262\) −6.08741e10 −0.798135
\(263\) 2.49089e10 0.321036 0.160518 0.987033i \(-0.448684\pi\)
0.160518 + 0.987033i \(0.448684\pi\)
\(264\) 4.21459e10 0.533995
\(265\) −1.73556e11 −2.16189
\(266\) −1.22177e10 −0.149631
\(267\) 6.95653e10 0.837706
\(268\) −6.37569e10 −0.754954
\(269\) 1.21965e11 1.42020 0.710101 0.704100i \(-0.248649\pi\)
0.710101 + 0.704100i \(0.248649\pi\)
\(270\) 3.13476e11 3.58977
\(271\) −1.00248e10 −0.112905 −0.0564524 0.998405i \(-0.517979\pi\)
−0.0564524 + 0.998405i \(0.517979\pi\)
\(272\) 1.20909e10 0.133936
\(273\) 2.79230e11 3.04249
\(274\) −2.27808e9 −0.0244170
\(275\) −1.41418e11 −1.49110
\(276\) −7.84124e9 −0.0813381
\(277\) 4.21205e10 0.429868 0.214934 0.976629i \(-0.431046\pi\)
0.214934 + 0.976629i \(0.431046\pi\)
\(278\) −3.40329e10 −0.341741
\(279\) 2.95322e11 2.91794
\(280\) 5.68108e10 0.552357
\(281\) −1.07825e11 −1.03167 −0.515836 0.856687i \(-0.672519\pi\)
−0.515836 + 0.856687i \(0.672519\pi\)
\(282\) −1.20860e11 −1.13805
\(283\) 4.94720e10 0.458480 0.229240 0.973370i \(-0.426376\pi\)
0.229240 + 0.973370i \(0.426376\pi\)
\(284\) −3.41070e10 −0.311108
\(285\) −8.19264e10 −0.735566
\(286\) 1.11235e11 0.983088
\(287\) 3.20933e8 0.00279219
\(288\) −5.33185e10 −0.456680
\(289\) −8.45506e10 −0.712979
\(290\) 2.15787e10 0.179157
\(291\) −2.92378e11 −2.39016
\(292\) −5.83971e10 −0.470077
\(293\) −3.72945e9 −0.0295624 −0.0147812 0.999891i \(-0.504705\pi\)
−0.0147812 + 0.999891i \(0.504705\pi\)
\(294\) 2.55847e10 0.199718
\(295\) 3.44458e10 0.264811
\(296\) −1.29152e10 −0.0977884
\(297\) −3.20678e11 −2.39147
\(298\) −9.19835e10 −0.675674
\(299\) −2.06952e10 −0.149744
\(300\) 2.48160e11 1.76883
\(301\) −9.56543e10 −0.671669
\(302\) −5.85755e10 −0.405214
\(303\) 3.25244e11 2.21676
\(304\) 8.54072e9 0.0573539
\(305\) −2.35772e11 −1.56007
\(306\) −1.50098e11 −0.978654
\(307\) 1.97134e11 1.26660 0.633300 0.773907i \(-0.281700\pi\)
0.633300 + 0.773907i \(0.281700\pi\)
\(308\) −5.81161e10 −0.367975
\(309\) 3.64093e11 2.27195
\(310\) 2.19966e11 1.35278
\(311\) −2.55635e11 −1.54953 −0.774763 0.632252i \(-0.782131\pi\)
−0.774763 + 0.632252i \(0.782131\pi\)
\(312\) −1.95195e11 −1.16620
\(313\) 5.67565e10 0.334246 0.167123 0.985936i \(-0.446552\pi\)
0.167123 + 0.985936i \(0.446552\pi\)
\(314\) 1.96578e10 0.114117
\(315\) −7.05260e11 −4.03601
\(316\) −1.70442e11 −0.961577
\(317\) 1.17647e11 0.654354 0.327177 0.944963i \(-0.393903\pi\)
0.327177 + 0.944963i \(0.393903\pi\)
\(318\) −3.11555e11 −1.70849
\(319\) −2.20745e10 −0.119353
\(320\) −3.97134e10 −0.211720
\(321\) 3.78904e11 1.99185
\(322\) 1.08125e10 0.0560500
\(323\) 2.40432e10 0.122908
\(324\) 3.06509e11 1.54522
\(325\) 6.54963e11 3.25643
\(326\) 1.05655e10 0.0518095
\(327\) −6.64848e11 −3.21557
\(328\) −2.24347e8 −0.00107026
\(329\) 1.66658e11 0.784230
\(330\) −3.89702e11 −1.80892
\(331\) −2.89438e11 −1.32535 −0.662673 0.748909i \(-0.730578\pi\)
−0.662673 + 0.748909i \(0.730578\pi\)
\(332\) −8.93061e10 −0.403422
\(333\) 1.60331e11 0.714529
\(334\) 1.69141e11 0.743685
\(335\) 5.89528e11 2.55743
\(336\) 1.01982e11 0.436514
\(337\) 9.82619e10 0.415002 0.207501 0.978235i \(-0.433467\pi\)
0.207501 + 0.978235i \(0.433467\pi\)
\(338\) −3.45501e11 −1.43987
\(339\) −8.69346e11 −3.57515
\(340\) −1.11798e11 −0.453711
\(341\) −2.25020e11 −0.901210
\(342\) −1.06026e11 −0.419078
\(343\) −2.71728e11 −1.06001
\(344\) 6.68669e10 0.257453
\(345\) 7.25041e10 0.275535
\(346\) 1.97503e11 0.740852
\(347\) −2.54903e11 −0.943827 −0.471914 0.881645i \(-0.656437\pi\)
−0.471914 + 0.881645i \(0.656437\pi\)
\(348\) 3.87364e10 0.141584
\(349\) 4.14692e11 1.49628 0.748138 0.663543i \(-0.230948\pi\)
0.748138 + 0.663543i \(0.230948\pi\)
\(350\) −3.42195e11 −1.21890
\(351\) 1.48519e12 5.22277
\(352\) 4.06259e10 0.141046
\(353\) −1.12437e11 −0.385410 −0.192705 0.981257i \(-0.561726\pi\)
−0.192705 + 0.981257i \(0.561726\pi\)
\(354\) 6.18344e10 0.209274
\(355\) 3.15370e11 1.05388
\(356\) 6.70565e10 0.221267
\(357\) 2.87093e11 0.935438
\(358\) −5.04732e10 −0.162400
\(359\) 1.53457e11 0.487596 0.243798 0.969826i \(-0.421607\pi\)
0.243798 + 0.969826i \(0.421607\pi\)
\(360\) 4.93010e11 1.54702
\(361\) 1.69836e10 0.0526316
\(362\) −1.85948e11 −0.569120
\(363\) −2.27562e11 −0.687893
\(364\) 2.69160e11 0.803625
\(365\) 5.39969e11 1.59240
\(366\) −4.23240e11 −1.23288
\(367\) 3.10410e11 0.893180 0.446590 0.894739i \(-0.352638\pi\)
0.446590 + 0.894739i \(0.352638\pi\)
\(368\) −7.55846e9 −0.0214841
\(369\) 2.78509e9 0.00782024
\(370\) 1.19420e11 0.331261
\(371\) 4.29612e11 1.17732
\(372\) 3.94865e11 1.06907
\(373\) −4.01840e11 −1.07489 −0.537444 0.843300i \(-0.680610\pi\)
−0.537444 + 0.843300i \(0.680610\pi\)
\(374\) 1.14367e11 0.302258
\(375\) −1.06678e12 −2.78570
\(376\) −1.16502e11 −0.300598
\(377\) 1.02236e11 0.260656
\(378\) −7.75960e11 −1.95491
\(379\) −4.14457e11 −1.03182 −0.515909 0.856644i \(-0.672546\pi\)
−0.515909 + 0.856644i \(0.672546\pi\)
\(380\) −7.89718e10 −0.194288
\(381\) −1.13595e12 −2.76184
\(382\) 5.20069e10 0.124961
\(383\) 2.40472e11 0.571044 0.285522 0.958372i \(-0.407833\pi\)
0.285522 + 0.958372i \(0.407833\pi\)
\(384\) −7.12905e10 −0.167317
\(385\) 5.37371e11 1.24652
\(386\) −3.93224e10 −0.0901565
\(387\) −8.30098e11 −1.88118
\(388\) −2.81834e11 −0.631321
\(389\) −2.15388e11 −0.476923 −0.238461 0.971152i \(-0.576643\pi\)
−0.238461 + 0.971152i \(0.576643\pi\)
\(390\) 1.80487e12 3.95053
\(391\) −2.12780e10 −0.0460400
\(392\) 2.46621e10 0.0527524
\(393\) 1.01042e12 2.13667
\(394\) 1.77385e11 0.370839
\(395\) 1.57599e12 3.25737
\(396\) −5.04338e11 −1.03061
\(397\) −9.46770e11 −1.91288 −0.956439 0.291932i \(-0.905702\pi\)
−0.956439 + 0.291932i \(0.905702\pi\)
\(398\) 2.66824e11 0.533029
\(399\) 2.02796e11 0.400573
\(400\) 2.39210e11 0.467208
\(401\) 2.94360e10 0.0568498 0.0284249 0.999596i \(-0.490951\pi\)
0.0284249 + 0.999596i \(0.490951\pi\)
\(402\) 1.05828e12 2.02107
\(403\) 1.04216e12 1.96816
\(404\) 3.13515e11 0.585521
\(405\) −2.83414e12 −5.23448
\(406\) −5.34148e10 −0.0975650
\(407\) −1.22164e11 −0.220683
\(408\) −2.00691e11 −0.358557
\(409\) −9.56914e11 −1.69090 −0.845450 0.534054i \(-0.820668\pi\)
−0.845450 + 0.534054i \(0.820668\pi\)
\(410\) 2.07443e9 0.00362552
\(411\) 3.78130e10 0.0653662
\(412\) 3.50963e11 0.600100
\(413\) −8.52651e10 −0.144210
\(414\) 9.38322e10 0.156982
\(415\) 8.25769e11 1.36660
\(416\) −1.88155e11 −0.308032
\(417\) 5.64899e11 0.914868
\(418\) 8.07863e10 0.129433
\(419\) −8.06056e11 −1.27762 −0.638811 0.769364i \(-0.720573\pi\)
−0.638811 + 0.769364i \(0.720573\pi\)
\(420\) −9.42980e11 −1.47870
\(421\) −1.29752e11 −0.201300 −0.100650 0.994922i \(-0.532092\pi\)
−0.100650 + 0.994922i \(0.532092\pi\)
\(422\) −4.03028e11 −0.618628
\(423\) 1.44627e12 2.19644
\(424\) −3.00319e11 −0.451270
\(425\) 6.73407e11 1.00121
\(426\) 5.66128e11 0.832859
\(427\) 5.83617e11 0.849577
\(428\) 3.65239e11 0.526115
\(429\) −1.84634e12 −2.63180
\(430\) −6.18285e11 −0.872129
\(431\) −1.30045e12 −1.81530 −0.907648 0.419732i \(-0.862124\pi\)
−0.907648 + 0.419732i \(0.862124\pi\)
\(432\) 5.42433e11 0.749323
\(433\) −7.31189e11 −0.999618 −0.499809 0.866136i \(-0.666596\pi\)
−0.499809 + 0.866136i \(0.666596\pi\)
\(434\) −5.44491e11 −0.736694
\(435\) −3.58177e11 −0.479618
\(436\) −6.40871e11 −0.849340
\(437\) −1.50303e10 −0.0197152
\(438\) 9.69311e11 1.25843
\(439\) 1.03899e12 1.33513 0.667563 0.744553i \(-0.267337\pi\)
0.667563 + 0.744553i \(0.267337\pi\)
\(440\) −3.75648e11 −0.477797
\(441\) −3.06159e11 −0.385455
\(442\) −5.29680e11 −0.660105
\(443\) −4.22964e10 −0.0521779 −0.0260889 0.999660i \(-0.508305\pi\)
−0.0260889 + 0.999660i \(0.508305\pi\)
\(444\) 2.14374e11 0.261787
\(445\) −6.20038e11 −0.749546
\(446\) −7.56111e11 −0.904854
\(447\) 1.52680e12 1.80883
\(448\) 9.83045e10 0.115298
\(449\) −1.99335e11 −0.231459 −0.115730 0.993281i \(-0.536921\pi\)
−0.115730 + 0.993281i \(0.536921\pi\)
\(450\) −2.96960e12 −3.41383
\(451\) −2.12209e9 −0.00241529
\(452\) −8.37994e11 −0.944317
\(453\) 9.72271e11 1.08479
\(454\) 5.86736e11 0.648174
\(455\) −2.48878e12 −2.72230
\(456\) −1.41764e11 −0.153541
\(457\) 6.57751e11 0.705405 0.352702 0.935736i \(-0.385263\pi\)
0.352702 + 0.935736i \(0.385263\pi\)
\(458\) −3.10460e11 −0.329694
\(459\) 1.52701e12 1.60578
\(460\) 6.98893e10 0.0727781
\(461\) 8.30559e11 0.856478 0.428239 0.903665i \(-0.359134\pi\)
0.428239 + 0.903665i \(0.359134\pi\)
\(462\) 9.64647e11 0.985098
\(463\) −1.09774e12 −1.11016 −0.555080 0.831797i \(-0.687312\pi\)
−0.555080 + 0.831797i \(0.687312\pi\)
\(464\) 3.73395e10 0.0373970
\(465\) −3.65112e12 −3.62150
\(466\) 5.72564e11 0.562454
\(467\) 6.31199e11 0.614102 0.307051 0.951693i \(-0.400658\pi\)
0.307051 + 0.951693i \(0.400658\pi\)
\(468\) 2.33580e12 2.25076
\(469\) −1.45929e12 −1.39272
\(470\) 1.07723e12 1.01828
\(471\) −3.26292e11 −0.305500
\(472\) 5.96044e10 0.0552763
\(473\) 6.32491e11 0.581004
\(474\) 2.82910e12 2.57422
\(475\) 4.75680e11 0.428739
\(476\) 2.76739e11 0.247081
\(477\) 3.72822e12 3.29738
\(478\) −8.29948e10 −0.0727152
\(479\) −1.03045e12 −0.894370 −0.447185 0.894442i \(-0.647573\pi\)
−0.447185 + 0.894442i \(0.647573\pi\)
\(480\) 6.59188e11 0.566792
\(481\) 5.65792e11 0.481952
\(482\) −2.33990e11 −0.197463
\(483\) −1.79473e11 −0.150050
\(484\) −2.19356e11 −0.181696
\(485\) 2.60598e12 2.13862
\(486\) −2.48100e12 −2.01727
\(487\) −6.97434e11 −0.561853 −0.280927 0.959729i \(-0.590642\pi\)
−0.280927 + 0.959729i \(0.590642\pi\)
\(488\) −4.07976e11 −0.325646
\(489\) −1.75372e11 −0.138698
\(490\) −2.28038e11 −0.178700
\(491\) 1.25643e12 0.975600 0.487800 0.872955i \(-0.337799\pi\)
0.487800 + 0.872955i \(0.337799\pi\)
\(492\) 3.72385e9 0.00286516
\(493\) 1.05115e11 0.0801409
\(494\) −3.74154e11 −0.282670
\(495\) 4.66336e12 3.49121
\(496\) 3.80625e11 0.282377
\(497\) −7.80650e11 −0.573922
\(498\) 1.48236e12 1.07999
\(499\) −7.26198e11 −0.524327 −0.262164 0.965023i \(-0.584436\pi\)
−0.262164 + 0.965023i \(0.584436\pi\)
\(500\) −1.02831e12 −0.735798
\(501\) −2.80750e12 −1.99090
\(502\) −2.19044e9 −0.00153944
\(503\) 1.01445e12 0.706602 0.353301 0.935510i \(-0.385059\pi\)
0.353301 + 0.935510i \(0.385059\pi\)
\(504\) −1.22037e12 −0.842470
\(505\) −2.89892e12 −1.98347
\(506\) −7.14952e10 −0.0484841
\(507\) 5.73483e12 3.85465
\(508\) −1.09499e12 −0.729494
\(509\) −1.88479e12 −1.24461 −0.622306 0.782774i \(-0.713804\pi\)
−0.622306 + 0.782774i \(0.713804\pi\)
\(510\) 1.85569e12 1.21462
\(511\) −1.33661e12 −0.867183
\(512\) −6.87195e10 −0.0441942
\(513\) 1.07865e12 0.687626
\(514\) −8.07328e11 −0.510171
\(515\) −3.24518e12 −2.03285
\(516\) −1.10990e12 −0.689222
\(517\) −1.10198e12 −0.678372
\(518\) −2.95606e11 −0.180397
\(519\) −3.27827e12 −1.98332
\(520\) 1.73978e12 1.04347
\(521\) 1.02179e12 0.607562 0.303781 0.952742i \(-0.401751\pi\)
0.303781 + 0.952742i \(0.401751\pi\)
\(522\) −4.63539e11 −0.273256
\(523\) −1.23513e12 −0.721862 −0.360931 0.932593i \(-0.617541\pi\)
−0.360931 + 0.932593i \(0.617541\pi\)
\(524\) 9.73985e11 0.564367
\(525\) 5.67996e12 3.26309
\(526\) −3.98542e11 −0.227007
\(527\) 1.07151e12 0.605127
\(528\) −6.74334e11 −0.377592
\(529\) −1.78785e12 −0.992615
\(530\) 2.77690e12 1.52869
\(531\) −7.39940e11 −0.403898
\(532\) 1.95482e11 0.105805
\(533\) 9.82827e9 0.00527478
\(534\) −1.11304e12 −0.592348
\(535\) −3.37719e12 −1.78223
\(536\) 1.02011e12 0.533833
\(537\) 8.37784e11 0.434758
\(538\) −1.95144e12 −1.00423
\(539\) 2.33277e11 0.119048
\(540\) −5.01561e12 −2.53835
\(541\) −8.75167e11 −0.439241 −0.219621 0.975585i \(-0.570482\pi\)
−0.219621 + 0.975585i \(0.570482\pi\)
\(542\) 1.60396e11 0.0798358
\(543\) 3.08648e12 1.52358
\(544\) −1.93454e11 −0.0947070
\(545\) 5.92582e12 2.87716
\(546\) −4.46767e12 −2.15137
\(547\) 1.97687e12 0.944138 0.472069 0.881562i \(-0.343507\pi\)
0.472069 + 0.881562i \(0.343507\pi\)
\(548\) 3.64493e10 0.0172654
\(549\) 5.06469e12 2.37945
\(550\) 2.26268e12 1.05437
\(551\) 7.42510e10 0.0343179
\(552\) 1.25460e11 0.0575147
\(553\) −3.90112e12 −1.77389
\(554\) −6.73929e11 −0.303962
\(555\) −1.98221e12 −0.886811
\(556\) 5.44526e11 0.241648
\(557\) 2.44569e12 1.07660 0.538298 0.842755i \(-0.319068\pi\)
0.538298 + 0.842755i \(0.319068\pi\)
\(558\) −4.72515e12 −2.06330
\(559\) −2.92933e12 −1.26886
\(560\) −9.08973e11 −0.390575
\(561\) −1.89833e12 −0.809169
\(562\) 1.72520e12 0.729503
\(563\) −4.54810e11 −0.190784 −0.0953920 0.995440i \(-0.530410\pi\)
−0.0953920 + 0.995440i \(0.530410\pi\)
\(564\) 1.93376e12 0.804725
\(565\) 7.74852e12 3.19890
\(566\) −7.91552e11 −0.324195
\(567\) 7.01547e12 2.85058
\(568\) 5.45711e11 0.219986
\(569\) 4.19643e12 1.67832 0.839160 0.543885i \(-0.183047\pi\)
0.839160 + 0.543885i \(0.183047\pi\)
\(570\) 1.31082e12 0.520124
\(571\) 3.03648e12 1.19539 0.597694 0.801725i \(-0.296084\pi\)
0.597694 + 0.801725i \(0.296084\pi\)
\(572\) −1.77975e12 −0.695149
\(573\) −8.63241e11 −0.334531
\(574\) −5.13492e9 −0.00197438
\(575\) −4.20972e11 −0.160601
\(576\) 8.53097e11 0.322922
\(577\) −5.21026e12 −1.95690 −0.978449 0.206487i \(-0.933797\pi\)
−0.978449 + 0.206487i \(0.933797\pi\)
\(578\) 1.35281e12 0.504152
\(579\) 6.52697e11 0.241356
\(580\) −3.45260e11 −0.126683
\(581\) −2.04406e12 −0.744221
\(582\) 4.67805e12 1.69010
\(583\) −2.84071e12 −1.01840
\(584\) 9.34354e11 0.332394
\(585\) −2.15979e13 −7.62449
\(586\) 5.96712e10 0.0209038
\(587\) −2.61266e12 −0.908264 −0.454132 0.890935i \(-0.650050\pi\)
−0.454132 + 0.890935i \(0.650050\pi\)
\(588\) −4.09356e11 −0.141222
\(589\) 7.56888e11 0.259127
\(590\) −5.51132e11 −0.187250
\(591\) −2.94435e12 −0.992764
\(592\) 2.06643e11 0.0691469
\(593\) 3.90508e12 1.29683 0.648417 0.761286i \(-0.275431\pi\)
0.648417 + 0.761286i \(0.275431\pi\)
\(594\) 5.13085e12 1.69103
\(595\) −2.55887e12 −0.836993
\(596\) 1.47174e12 0.477773
\(597\) −4.42890e12 −1.42696
\(598\) 3.31123e11 0.105885
\(599\) 1.12506e12 0.357070 0.178535 0.983934i \(-0.442864\pi\)
0.178535 + 0.983934i \(0.442864\pi\)
\(600\) −3.97056e12 −1.25075
\(601\) −7.83252e11 −0.244887 −0.122444 0.992475i \(-0.539073\pi\)
−0.122444 + 0.992475i \(0.539073\pi\)
\(602\) 1.53047e12 0.474942
\(603\) −1.26638e13 −3.90066
\(604\) 9.37207e11 0.286530
\(605\) 2.02827e12 0.615499
\(606\) −5.20391e12 −1.56748
\(607\) 2.19285e12 0.655633 0.327816 0.944741i \(-0.393687\pi\)
0.327816 + 0.944741i \(0.393687\pi\)
\(608\) −1.36651e11 −0.0405554
\(609\) 8.86611e11 0.261189
\(610\) 3.77235e12 1.10313
\(611\) 5.10374e12 1.48150
\(612\) 2.40157e12 0.692013
\(613\) −6.85830e11 −0.196175 −0.0980877 0.995178i \(-0.531273\pi\)
−0.0980877 + 0.995178i \(0.531273\pi\)
\(614\) −3.15415e12 −0.895621
\(615\) −3.44326e10 −0.00970581
\(616\) 9.29858e11 0.260198
\(617\) 4.20809e12 1.16897 0.584483 0.811406i \(-0.301297\pi\)
0.584483 + 0.811406i \(0.301297\pi\)
\(618\) −5.82549e12 −1.60651
\(619\) −5.42304e12 −1.48469 −0.742344 0.670019i \(-0.766286\pi\)
−0.742344 + 0.670019i \(0.766286\pi\)
\(620\) −3.51945e12 −0.956560
\(621\) −9.54596e11 −0.257577
\(622\) 4.09016e12 1.09568
\(623\) 1.53481e12 0.408186
\(624\) 3.12312e12 0.824627
\(625\) 2.37923e12 0.623701
\(626\) −9.08105e11 −0.236348
\(627\) −1.34094e12 −0.346502
\(628\) −3.14524e11 −0.0806930
\(629\) 5.81725e11 0.148180
\(630\) 1.12842e13 2.85389
\(631\) −4.74716e12 −1.19207 −0.596035 0.802959i \(-0.703258\pi\)
−0.596035 + 0.802959i \(0.703258\pi\)
\(632\) 2.72707e12 0.679938
\(633\) 6.68971e12 1.65612
\(634\) −1.88234e12 −0.462698
\(635\) 1.01248e13 2.47118
\(636\) 4.98488e12 1.20809
\(637\) −1.08040e12 −0.259991
\(638\) 3.53192e11 0.0843953
\(639\) −6.77457e12 −1.60741
\(640\) 6.35415e11 0.149709
\(641\) −5.71232e12 −1.33645 −0.668223 0.743961i \(-0.732945\pi\)
−0.668223 + 0.743961i \(0.732945\pi\)
\(642\) −6.06247e12 −1.40845
\(643\) 8.29997e12 1.91482 0.957409 0.288736i \(-0.0932350\pi\)
0.957409 + 0.288736i \(0.0932350\pi\)
\(644\) −1.73000e11 −0.0396333
\(645\) 1.02627e13 2.33476
\(646\) −3.84691e11 −0.0869091
\(647\) −1.16040e12 −0.260339 −0.130169 0.991492i \(-0.541552\pi\)
−0.130169 + 0.991492i \(0.541552\pi\)
\(648\) −4.90414e12 −1.09264
\(649\) 5.63795e11 0.124744
\(650\) −1.04794e13 −2.30264
\(651\) 9.03779e12 1.97219
\(652\) −1.69047e11 −0.0366349
\(653\) −1.07293e12 −0.230921 −0.115461 0.993312i \(-0.536834\pi\)
−0.115461 + 0.993312i \(0.536834\pi\)
\(654\) 1.06376e13 2.27375
\(655\) −9.00596e12 −1.91181
\(656\) 3.58955e9 0.000756786 0
\(657\) −1.15992e13 −2.42877
\(658\) −2.66652e12 −0.554535
\(659\) −9.14394e12 −1.88864 −0.944319 0.329031i \(-0.893278\pi\)
−0.944319 + 0.329031i \(0.893278\pi\)
\(660\) 6.23523e12 1.27910
\(661\) 8.78371e12 1.78966 0.894832 0.446403i \(-0.147295\pi\)
0.894832 + 0.446403i \(0.147295\pi\)
\(662\) 4.63101e12 0.937161
\(663\) 8.79195e12 1.76715
\(664\) 1.42890e12 0.285262
\(665\) −1.80753e12 −0.358416
\(666\) −2.56530e12 −0.505248
\(667\) −6.57115e10 −0.0128551
\(668\) −2.70625e12 −0.525865
\(669\) 1.25504e13 2.42237
\(670\) −9.43246e12 −1.80837
\(671\) −3.85903e12 −0.734897
\(672\) −1.63172e12 −0.308662
\(673\) 7.23901e12 1.36023 0.680114 0.733106i \(-0.261930\pi\)
0.680114 + 0.733106i \(0.261930\pi\)
\(674\) −1.57219e12 −0.293451
\(675\) 3.02111e13 5.60144
\(676\) 5.52801e12 1.01814
\(677\) −5.18390e12 −0.948435 −0.474218 0.880408i \(-0.657269\pi\)
−0.474218 + 0.880408i \(0.657269\pi\)
\(678\) 1.39095e13 2.52801
\(679\) −6.45070e12 −1.16464
\(680\) 1.78877e12 0.320822
\(681\) −9.73901e12 −1.73521
\(682\) 3.60032e12 0.637252
\(683\) 2.31773e12 0.407539 0.203769 0.979019i \(-0.434681\pi\)
0.203769 + 0.979019i \(0.434681\pi\)
\(684\) 1.69642e12 0.296333
\(685\) −3.37029e11 −0.0584871
\(686\) 4.34764e12 0.749541
\(687\) 5.15320e12 0.882616
\(688\) −1.06987e12 −0.182047
\(689\) 1.31565e13 2.22409
\(690\) −1.16007e12 −0.194833
\(691\) −2.82266e12 −0.470985 −0.235492 0.971876i \(-0.575670\pi\)
−0.235492 + 0.971876i \(0.575670\pi\)
\(692\) −3.16005e12 −0.523861
\(693\) −1.15434e13 −1.90123
\(694\) 4.07845e12 0.667387
\(695\) −5.03497e12 −0.818587
\(696\) −6.19783e11 −0.100115
\(697\) 1.01050e10 0.00162177
\(698\) −6.63508e12 −1.05803
\(699\) −9.50376e12 −1.50573
\(700\) 5.47512e12 0.861891
\(701\) −9.84845e12 −1.54041 −0.770206 0.637795i \(-0.779847\pi\)
−0.770206 + 0.637795i \(0.779847\pi\)
\(702\) −2.37631e13 −3.69305
\(703\) 4.10918e11 0.0634535
\(704\) −6.50015e11 −0.0997347
\(705\) −1.78806e13 −2.72603
\(706\) 1.79899e12 0.272526
\(707\) 7.17582e12 1.08015
\(708\) −9.89350e11 −0.147979
\(709\) 4.56333e12 0.678225 0.339112 0.940746i \(-0.389873\pi\)
0.339112 + 0.940746i \(0.389873\pi\)
\(710\) −5.04592e12 −0.745209
\(711\) −3.38544e13 −4.96823
\(712\) −1.07290e12 −0.156459
\(713\) −6.69840e11 −0.0970662
\(714\) −4.59348e12 −0.661455
\(715\) 1.64565e13 2.35483
\(716\) 8.07571e11 0.114834
\(717\) 1.37760e12 0.194664
\(718\) −2.45530e12 −0.344783
\(719\) −9.06278e12 −1.26468 −0.632341 0.774690i \(-0.717906\pi\)
−0.632341 + 0.774690i \(0.717906\pi\)
\(720\) −7.88816e12 −1.09391
\(721\) 8.03293e12 1.10705
\(722\) −2.71737e11 −0.0372161
\(723\) 3.88390e12 0.528623
\(724\) 2.97517e12 0.402429
\(725\) 2.07964e12 0.279555
\(726\) 3.64100e12 0.486414
\(727\) −1.15554e11 −0.0153420 −0.00767100 0.999971i \(-0.502442\pi\)
−0.00767100 + 0.999971i \(0.502442\pi\)
\(728\) −4.30655e12 −0.568249
\(729\) 1.76147e13 2.30995
\(730\) −8.63951e12 −1.12599
\(731\) −3.01182e12 −0.390122
\(732\) 6.77183e12 0.871779
\(733\) 1.08986e13 1.39444 0.697222 0.716855i \(-0.254419\pi\)
0.697222 + 0.716855i \(0.254419\pi\)
\(734\) −4.96657e12 −0.631574
\(735\) 3.78511e12 0.478394
\(736\) 1.20935e11 0.0151916
\(737\) 9.64918e12 1.20472
\(738\) −4.45614e10 −0.00552975
\(739\) −8.91416e12 −1.09946 −0.549731 0.835341i \(-0.685270\pi\)
−0.549731 + 0.835341i \(0.685270\pi\)
\(740\) −1.91072e12 −0.234237
\(741\) 6.21044e12 0.756729
\(742\) −6.87379e12 −0.832490
\(743\) 6.16929e12 0.742652 0.371326 0.928503i \(-0.378903\pi\)
0.371326 + 0.928503i \(0.378903\pi\)
\(744\) −6.31785e12 −0.755946
\(745\) −1.36084e13 −1.61847
\(746\) 6.42943e12 0.760060
\(747\) −1.77386e13 −2.08438
\(748\) −1.82987e12 −0.213729
\(749\) 8.35971e12 0.970561
\(750\) 1.70685e13 1.96979
\(751\) 6.30977e12 0.723826 0.361913 0.932212i \(-0.382124\pi\)
0.361913 + 0.932212i \(0.382124\pi\)
\(752\) 1.86403e12 0.212555
\(753\) 3.63582e10 0.00412121
\(754\) −1.63578e12 −0.184312
\(755\) −8.66589e12 −0.970626
\(756\) 1.24154e13 1.38233
\(757\) −9.13291e12 −1.01083 −0.505414 0.862877i \(-0.668660\pi\)
−0.505414 + 0.862877i \(0.668660\pi\)
\(758\) 6.63131e12 0.729605
\(759\) 1.18672e12 0.129796
\(760\) 1.26355e12 0.137382
\(761\) 5.56499e12 0.601497 0.300748 0.953704i \(-0.402764\pi\)
0.300748 + 0.953704i \(0.402764\pi\)
\(762\) 1.81752e13 1.95291
\(763\) −1.46684e13 −1.56684
\(764\) −8.32110e11 −0.0883610
\(765\) −2.22061e13 −2.34421
\(766\) −3.84755e12 −0.403789
\(767\) −2.61117e12 −0.272430
\(768\) 1.14065e12 0.118311
\(769\) 1.09213e13 1.12618 0.563089 0.826396i \(-0.309613\pi\)
0.563089 + 0.826396i \(0.309613\pi\)
\(770\) −8.59794e12 −0.881426
\(771\) 1.34005e13 1.36577
\(772\) 6.29158e11 0.0637503
\(773\) −1.06326e13 −1.07111 −0.535553 0.844502i \(-0.679897\pi\)
−0.535553 + 0.844502i \(0.679897\pi\)
\(774\) 1.32816e13 1.33019
\(775\) 2.11991e13 2.11086
\(776\) 4.50934e12 0.446412
\(777\) 4.90665e12 0.482937
\(778\) 3.44621e12 0.337235
\(779\) 7.13797e9 0.000694475 0
\(780\) −2.88779e13 −2.79344
\(781\) 5.16186e12 0.496451
\(782\) 3.40448e11 0.0325552
\(783\) 4.71579e12 0.448359
\(784\) −3.94593e11 −0.0373016
\(785\) 2.90825e12 0.273349
\(786\) −1.61668e13 −1.51085
\(787\) −2.50979e12 −0.233212 −0.116606 0.993178i \(-0.537202\pi\)
−0.116606 + 0.993178i \(0.537202\pi\)
\(788\) −2.83817e12 −0.262223
\(789\) 6.61525e12 0.607715
\(790\) −2.52159e13 −2.30331
\(791\) −1.91803e13 −1.74205
\(792\) 8.06941e12 0.728750
\(793\) 1.78727e13 1.60495
\(794\) 1.51483e13 1.35261
\(795\) −4.60927e13 −4.09242
\(796\) −4.26918e12 −0.376908
\(797\) −1.17969e13 −1.03563 −0.517815 0.855493i \(-0.673254\pi\)
−0.517815 + 0.855493i \(0.673254\pi\)
\(798\) −3.24474e12 −0.283248
\(799\) 5.24746e12 0.455500
\(800\) −3.82737e12 −0.330366
\(801\) 1.33192e13 1.14323
\(802\) −4.70976e11 −0.0401989
\(803\) 8.83802e12 0.750127
\(804\) −1.69324e13 −1.42911
\(805\) 1.59965e12 0.134259
\(806\) −1.66745e13 −1.39170
\(807\) 3.23912e13 2.68841
\(808\) −5.01624e12 −0.414026
\(809\) −6.47962e12 −0.531840 −0.265920 0.963995i \(-0.585676\pi\)
−0.265920 + 0.963995i \(0.585676\pi\)
\(810\) 4.53462e13 3.70133
\(811\) −5.08224e12 −0.412536 −0.206268 0.978496i \(-0.566132\pi\)
−0.206268 + 0.978496i \(0.566132\pi\)
\(812\) 8.54636e11 0.0689889
\(813\) −2.66235e12 −0.213727
\(814\) 1.95463e12 0.156046
\(815\) 1.56310e12 0.124102
\(816\) 3.21106e12 0.253538
\(817\) −2.12748e12 −0.167058
\(818\) 1.53106e13 1.19565
\(819\) 5.34624e13 4.15213
\(820\) −3.31908e10 −0.00256363
\(821\) 8.99329e11 0.0690835 0.0345418 0.999403i \(-0.489003\pi\)
0.0345418 + 0.999403i \(0.489003\pi\)
\(822\) −6.05008e11 −0.0462209
\(823\) −7.62233e12 −0.579147 −0.289573 0.957156i \(-0.593513\pi\)
−0.289573 + 0.957156i \(0.593513\pi\)
\(824\) −5.61540e12 −0.424334
\(825\) −3.75574e13 −2.82262
\(826\) 1.36424e12 0.101972
\(827\) 2.54111e12 0.188907 0.0944537 0.995529i \(-0.469890\pi\)
0.0944537 + 0.995529i \(0.469890\pi\)
\(828\) −1.50131e12 −0.111003
\(829\) 2.04803e13 1.50605 0.753027 0.657990i \(-0.228593\pi\)
0.753027 + 0.657990i \(0.228593\pi\)
\(830\) −1.32123e13 −0.966334
\(831\) 1.11863e13 0.813731
\(832\) 3.01048e12 0.217812
\(833\) −1.11083e12 −0.0799363
\(834\) −9.03838e12 −0.646909
\(835\) 2.50234e13 1.78138
\(836\) −1.29258e12 −0.0915228
\(837\) 4.80710e13 3.38547
\(838\) 1.28969e13 0.903415
\(839\) 1.60416e13 1.11769 0.558843 0.829273i \(-0.311245\pi\)
0.558843 + 0.829273i \(0.311245\pi\)
\(840\) 1.50877e13 1.04560
\(841\) −1.41825e13 −0.977623
\(842\) 2.07603e12 0.142340
\(843\) −2.86360e13 −1.95294
\(844\) 6.44846e12 0.437436
\(845\) −5.11148e13 −3.44899
\(846\) −2.31404e13 −1.55312
\(847\) −5.02068e12 −0.335187
\(848\) 4.80511e12 0.319096
\(849\) 1.31387e13 0.867894
\(850\) −1.07745e13 −0.707966
\(851\) −3.63658e11 −0.0237690
\(852\) −9.05805e12 −0.588920
\(853\) 1.69954e13 1.09916 0.549581 0.835440i \(-0.314787\pi\)
0.549581 + 0.835440i \(0.314787\pi\)
\(854\) −9.33787e12 −0.600741
\(855\) −1.56859e13 −1.00384
\(856\) −5.84383e12 −0.372019
\(857\) −4.81872e12 −0.305153 −0.152577 0.988292i \(-0.548757\pi\)
−0.152577 + 0.988292i \(0.548757\pi\)
\(858\) 2.95414e13 1.86097
\(859\) 1.43473e13 0.899084 0.449542 0.893259i \(-0.351587\pi\)
0.449542 + 0.893259i \(0.351587\pi\)
\(860\) 9.89256e12 0.616688
\(861\) 8.52326e10 0.00528556
\(862\) 2.08073e13 1.28361
\(863\) 1.16081e13 0.712380 0.356190 0.934414i \(-0.384076\pi\)
0.356190 + 0.934414i \(0.384076\pi\)
\(864\) −8.67893e12 −0.529852
\(865\) 2.92194e13 1.77459
\(866\) 1.16990e13 0.706837
\(867\) −2.24548e13 −1.34965
\(868\) 8.71186e12 0.520921
\(869\) 2.57952e13 1.53444
\(870\) 5.73083e12 0.339141
\(871\) −4.46893e13 −2.63101
\(872\) 1.02539e13 0.600574
\(873\) −5.59798e13 −3.26188
\(874\) 2.40485e11 0.0139408
\(875\) −2.35362e13 −1.35738
\(876\) −1.55090e13 −0.889845
\(877\) 6.32859e11 0.0361251 0.0180625 0.999837i \(-0.494250\pi\)
0.0180625 + 0.999837i \(0.494250\pi\)
\(878\) −1.66239e13 −0.944077
\(879\) −9.90458e11 −0.0559611
\(880\) 6.01036e12 0.337854
\(881\) 1.78534e13 0.998457 0.499229 0.866470i \(-0.333617\pi\)
0.499229 + 0.866470i \(0.333617\pi\)
\(882\) 4.89855e12 0.272558
\(883\) −3.78633e12 −0.209602 −0.104801 0.994493i \(-0.533421\pi\)
−0.104801 + 0.994493i \(0.533421\pi\)
\(884\) 8.47488e12 0.466765
\(885\) 9.14803e12 0.501283
\(886\) 6.76742e11 0.0368953
\(887\) −1.59149e13 −0.863274 −0.431637 0.902048i \(-0.642064\pi\)
−0.431637 + 0.902048i \(0.642064\pi\)
\(888\) −3.42998e12 −0.185112
\(889\) −2.50624e13 −1.34575
\(890\) 9.92061e12 0.530009
\(891\) −4.63881e13 −2.46579
\(892\) 1.20978e13 0.639829
\(893\) 3.70669e12 0.195054
\(894\) −2.44288e13 −1.27904
\(895\) −7.46721e12 −0.389004
\(896\) −1.57287e12 −0.0815281
\(897\) −5.49619e12 −0.283462
\(898\) 3.18936e12 0.163666
\(899\) 3.30907e12 0.168961
\(900\) 4.75137e13 2.41394
\(901\) 1.35270e13 0.683815
\(902\) 3.39534e10 0.00170787
\(903\) −2.54037e13 −1.27146
\(904\) 1.34079e13 0.667733
\(905\) −2.75100e13 −1.36324
\(906\) −1.55563e13 −0.767062
\(907\) −8.91543e12 −0.437431 −0.218715 0.975789i \(-0.570187\pi\)
−0.218715 + 0.975789i \(0.570187\pi\)
\(908\) −9.38778e12 −0.458328
\(909\) 6.22726e13 3.02524
\(910\) 3.98206e13 1.92496
\(911\) −2.65789e13 −1.27851 −0.639256 0.768994i \(-0.720757\pi\)
−0.639256 + 0.768994i \(0.720757\pi\)
\(912\) 2.26822e12 0.108570
\(913\) 1.35159e13 0.643763
\(914\) −1.05240e13 −0.498796
\(915\) −6.26158e13 −2.95317
\(916\) 4.96736e12 0.233129
\(917\) 2.22928e13 1.04113
\(918\) −2.44322e13 −1.13546
\(919\) 1.52684e13 0.706110 0.353055 0.935603i \(-0.385143\pi\)
0.353055 + 0.935603i \(0.385143\pi\)
\(920\) −1.11823e12 −0.0514619
\(921\) 5.23545e13 2.39765
\(922\) −1.32889e13 −0.605621
\(923\) −2.39067e13 −1.08421
\(924\) −1.54343e13 −0.696569
\(925\) 1.15091e13 0.516896
\(926\) 1.75639e13 0.785001
\(927\) 6.97107e13 3.10056
\(928\) −5.97431e11 −0.0264437
\(929\) −3.74604e13 −1.65007 −0.825033 0.565084i \(-0.808844\pi\)
−0.825033 + 0.565084i \(0.808844\pi\)
\(930\) 5.84180e13 2.56079
\(931\) −7.84664e11 −0.0342303
\(932\) −9.16102e12 −0.397715
\(933\) −6.78910e13 −2.93322
\(934\) −1.00992e13 −0.434235
\(935\) 1.69199e13 0.724012
\(936\) −3.73727e13 −1.59152
\(937\) −2.62942e13 −1.11438 −0.557188 0.830386i \(-0.688120\pi\)
−0.557188 + 0.830386i \(0.688120\pi\)
\(938\) 2.33486e13 0.984799
\(939\) 1.50733e13 0.632721
\(940\) −1.72357e13 −0.720036
\(941\) 3.58875e13 1.49207 0.746036 0.665906i \(-0.231955\pi\)
0.746036 + 0.665906i \(0.231955\pi\)
\(942\) 5.22066e12 0.216021
\(943\) −6.31704e9 −0.000260143 0
\(944\) −9.53670e11 −0.0390863
\(945\) −1.14799e14 −4.68267
\(946\) −1.01199e13 −0.410832
\(947\) 3.11985e13 1.26054 0.630272 0.776374i \(-0.282943\pi\)
0.630272 + 0.776374i \(0.282943\pi\)
\(948\) −4.52655e13 −1.82025
\(949\) −4.09325e13 −1.63821
\(950\) −7.61088e12 −0.303165
\(951\) 3.12443e13 1.23868
\(952\) −4.42782e12 −0.174713
\(953\) 3.83515e13 1.50614 0.753068 0.657942i \(-0.228573\pi\)
0.753068 + 0.657942i \(0.228573\pi\)
\(954\) −5.96515e13 −2.33160
\(955\) 7.69411e12 0.299325
\(956\) 1.32792e12 0.0514174
\(957\) −5.86250e12 −0.225933
\(958\) 1.64872e13 0.632415
\(959\) 8.34263e11 0.0318507
\(960\) −1.05470e13 −0.400782
\(961\) 7.29182e12 0.275791
\(962\) −9.05267e12 −0.340792
\(963\) 7.25464e13 2.71830
\(964\) 3.74383e12 0.139627
\(965\) −5.81751e12 −0.215956
\(966\) 2.87156e12 0.106101
\(967\) 2.52076e13 0.927071 0.463536 0.886078i \(-0.346581\pi\)
0.463536 + 0.886078i \(0.346581\pi\)
\(968\) 3.50969e12 0.128478
\(969\) 6.38533e12 0.232662
\(970\) −4.16957e13 −1.51223
\(971\) −9.11601e12 −0.329093 −0.164546 0.986369i \(-0.552616\pi\)
−0.164546 + 0.986369i \(0.552616\pi\)
\(972\) 3.96961e13 1.42643
\(973\) 1.24633e13 0.445784
\(974\) 1.11589e13 0.397290
\(975\) 1.73944e14 6.16435
\(976\) 6.52762e12 0.230266
\(977\) 4.10489e13 1.44137 0.720685 0.693262i \(-0.243827\pi\)
0.720685 + 0.693262i \(0.243827\pi\)
\(978\) 2.80595e12 0.0980744
\(979\) −1.01486e13 −0.353087
\(980\) 3.64860e12 0.126360
\(981\) −1.27294e14 −4.38832
\(982\) −2.01029e13 −0.689854
\(983\) 3.76886e12 0.128742 0.0643708 0.997926i \(-0.479496\pi\)
0.0643708 + 0.997926i \(0.479496\pi\)
\(984\) −5.95816e10 −0.00202598
\(985\) 2.62431e13 0.888286
\(986\) −1.68184e12 −0.0566682
\(987\) 4.42606e13 1.48453
\(988\) 5.98647e12 0.199878
\(989\) 1.88280e12 0.0625779
\(990\) −7.46138e13 −2.46866
\(991\) −3.54987e13 −1.16918 −0.584590 0.811329i \(-0.698745\pi\)
−0.584590 + 0.811329i \(0.698745\pi\)
\(992\) −6.09000e12 −0.199671
\(993\) −7.68682e13 −2.50885
\(994\) 1.24904e13 0.405824
\(995\) 3.94750e13 1.27679
\(996\) −2.37177e13 −0.763670
\(997\) −3.48328e13 −1.11650 −0.558252 0.829672i \(-0.688528\pi\)
−0.558252 + 0.829672i \(0.688528\pi\)
\(998\) 1.16192e13 0.370755
\(999\) 2.60980e13 0.829014
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 38.10.a.d.1.4 4
3.2 odd 2 342.10.a.l.1.3 4
4.3 odd 2 304.10.a.e.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.10.a.d.1.4 4 1.1 even 1 trivial
304.10.a.e.1.1 4 4.3 odd 2
342.10.a.l.1.3 4 3.2 odd 2