Properties

Label 138.8.a.h.1.3
Level $138$
Weight $8$
Character 138.1
Self dual yes
Analytic conductor $43.109$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 138.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(43.1091335168\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 2x^{3} - 8367x^{2} - 89140x + 11077220 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-65.5856\) of defining polynomial
Character \(\chi\) \(=\) 138.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +340.987 q^{5} +216.000 q^{6} +1267.12 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +340.987 q^{5} +216.000 q^{6} +1267.12 q^{7} +512.000 q^{8} +729.000 q^{9} +2727.89 q^{10} +5101.25 q^{11} +1728.00 q^{12} -5814.29 q^{13} +10137.0 q^{14} +9206.64 q^{15} +4096.00 q^{16} +6404.73 q^{17} +5832.00 q^{18} -38566.6 q^{19} +21823.2 q^{20} +34212.2 q^{21} +40810.0 q^{22} -12167.0 q^{23} +13824.0 q^{24} +38147.0 q^{25} -46514.3 q^{26} +19683.0 q^{27} +81095.7 q^{28} -208061. q^{29} +73653.1 q^{30} +3653.78 q^{31} +32768.0 q^{32} +137734. q^{33} +51237.9 q^{34} +432071. q^{35} +46656.0 q^{36} +23930.9 q^{37} -308533. q^{38} -156986. q^{39} +174585. q^{40} -1151.50 q^{41} +273698. q^{42} -489302. q^{43} +326480. q^{44} +248579. q^{45} -97336.0 q^{46} +988907. q^{47} +110592. q^{48} +782050. q^{49} +305176. q^{50} +172928. q^{51} -372115. q^{52} -777027. q^{53} +157464. q^{54} +1.73946e6 q^{55} +648766. q^{56} -1.04130e6 q^{57} -1.66449e6 q^{58} +1.61187e6 q^{59} +589225. q^{60} -1.04092e6 q^{61} +29230.2 q^{62} +923731. q^{63} +262144. q^{64} -1.98260e6 q^{65} +1.10187e6 q^{66} -2.06066e6 q^{67} +409903. q^{68} -328509. q^{69} +3.45657e6 q^{70} +3.58392e6 q^{71} +373248. q^{72} -975850. q^{73} +191447. q^{74} +1.02997e6 q^{75} -2.46826e6 q^{76} +6.46390e6 q^{77} -1.25589e6 q^{78} +6.00359e6 q^{79} +1.39668e6 q^{80} +531441. q^{81} -9212.01 q^{82} -5.40252e6 q^{83} +2.18958e6 q^{84} +2.18393e6 q^{85} -3.91441e6 q^{86} -5.61764e6 q^{87} +2.61184e6 q^{88} -4.13911e6 q^{89} +1.98863e6 q^{90} -7.36740e6 q^{91} -778688. q^{92} +98652.1 q^{93} +7.91125e6 q^{94} -1.31507e7 q^{95} +884736. q^{96} +1.18844e7 q^{97} +6.25640e6 q^{98} +3.71881e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{2} + 108 q^{3} + 256 q^{4} + 270 q^{5} + 864 q^{6} + 2022 q^{7} + 2048 q^{8} + 2916 q^{9} + 2160 q^{10} + 4120 q^{11} + 6912 q^{12} + 8036 q^{13} + 16176 q^{14} + 7290 q^{15} + 16384 q^{16} + 37182 q^{17} + 23328 q^{18} + 5702 q^{19} + 17280 q^{20} + 54594 q^{21} + 32960 q^{22} - 48668 q^{23} + 55296 q^{24} + 121480 q^{25} + 64288 q^{26} + 78732 q^{27} + 129408 q^{28} + 217716 q^{29} + 58320 q^{30} + 222852 q^{31} + 131072 q^{32} + 111240 q^{33} + 297456 q^{34} + 68440 q^{35} + 186624 q^{36} + 486428 q^{37} + 45616 q^{38} + 216972 q^{39} + 138240 q^{40} + 338336 q^{41} + 436752 q^{42} + 730974 q^{43} + 263680 q^{44} + 196830 q^{45} - 389344 q^{46} + 338248 q^{47} + 442368 q^{48} - 310552 q^{49} + 971840 q^{50} + 1003914 q^{51} + 514304 q^{52} - 375502 q^{53} + 629856 q^{54} + 424840 q^{55} + 1035264 q^{56} + 153954 q^{57} + 1741728 q^{58} + 71392 q^{59} + 466560 q^{60} + 2101164 q^{61} + 1782816 q^{62} + 1474038 q^{63} + 1048576 q^{64} + 1578780 q^{65} + 889920 q^{66} + 4337162 q^{67} + 2379648 q^{68} - 1314036 q^{69} + 547520 q^{70} + 2288016 q^{71} + 1492992 q^{72} - 1107328 q^{73} + 3891424 q^{74} + 3279960 q^{75} + 364928 q^{76} + 5826200 q^{77} + 1735776 q^{78} + 60610 q^{79} + 1105920 q^{80} + 2125764 q^{81} + 2706688 q^{82} + 1485464 q^{83} + 3494016 q^{84} - 8843820 q^{85} + 5847792 q^{86} + 5878332 q^{87} + 2109440 q^{88} + 1485090 q^{89} + 1574640 q^{90} - 2898412 q^{91} - 3114752 q^{92} + 6017004 q^{93} + 2705984 q^{94} + 8545200 q^{95} + 3538944 q^{96} + 1935444 q^{97} - 2484416 q^{98} + 3003480 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) 27.0000 0.577350
\(4\) 64.0000 0.500000
\(5\) 340.987 1.21995 0.609976 0.792420i \(-0.291179\pi\)
0.609976 + 0.792420i \(0.291179\pi\)
\(6\) 216.000 0.408248
\(7\) 1267.12 1.39629 0.698143 0.715958i \(-0.254010\pi\)
0.698143 + 0.715958i \(0.254010\pi\)
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) 2727.89 0.862636
\(11\) 5101.25 1.15559 0.577793 0.816183i \(-0.303914\pi\)
0.577793 + 0.816183i \(0.303914\pi\)
\(12\) 1728.00 0.288675
\(13\) −5814.29 −0.733998 −0.366999 0.930221i \(-0.619615\pi\)
−0.366999 + 0.930221i \(0.619615\pi\)
\(14\) 10137.0 0.987324
\(15\) 9206.64 0.704339
\(16\) 4096.00 0.250000
\(17\) 6404.73 0.316177 0.158088 0.987425i \(-0.449467\pi\)
0.158088 + 0.987425i \(0.449467\pi\)
\(18\) 5832.00 0.235702
\(19\) −38566.6 −1.28995 −0.644977 0.764202i \(-0.723133\pi\)
−0.644977 + 0.764202i \(0.723133\pi\)
\(20\) 21823.2 0.609976
\(21\) 34212.2 0.806147
\(22\) 40810.0 0.817122
\(23\) −12167.0 −0.208514
\(24\) 13824.0 0.204124
\(25\) 38147.0 0.488281
\(26\) −46514.3 −0.519015
\(27\) 19683.0 0.192450
\(28\) 81095.7 0.698143
\(29\) −208061. −1.58415 −0.792077 0.610422i \(-0.791000\pi\)
−0.792077 + 0.610422i \(0.791000\pi\)
\(30\) 73653.1 0.498043
\(31\) 3653.78 0.0220281 0.0110140 0.999939i \(-0.496494\pi\)
0.0110140 + 0.999939i \(0.496494\pi\)
\(32\) 32768.0 0.176777
\(33\) 137734. 0.667178
\(34\) 51237.9 0.223571
\(35\) 432071. 1.70340
\(36\) 46656.0 0.166667
\(37\) 23930.9 0.0776699 0.0388350 0.999246i \(-0.487635\pi\)
0.0388350 + 0.999246i \(0.487635\pi\)
\(38\) −308533. −0.912135
\(39\) −156986. −0.423774
\(40\) 174585. 0.431318
\(41\) −1151.50 −0.00260928 −0.00130464 0.999999i \(-0.500415\pi\)
−0.00130464 + 0.999999i \(0.500415\pi\)
\(42\) 273698. 0.570032
\(43\) −489302. −0.938506 −0.469253 0.883064i \(-0.655477\pi\)
−0.469253 + 0.883064i \(0.655477\pi\)
\(44\) 326480. 0.577793
\(45\) 248579. 0.406650
\(46\) −97336.0 −0.147442
\(47\) 988907. 1.38935 0.694677 0.719322i \(-0.255547\pi\)
0.694677 + 0.719322i \(0.255547\pi\)
\(48\) 110592. 0.144338
\(49\) 782050. 0.949617
\(50\) 305176. 0.345267
\(51\) 172928. 0.182545
\(52\) −372115. −0.366999
\(53\) −777027. −0.716920 −0.358460 0.933545i \(-0.616698\pi\)
−0.358460 + 0.933545i \(0.616698\pi\)
\(54\) 157464. 0.136083
\(55\) 1.73946e6 1.40976
\(56\) 648766. 0.493662
\(57\) −1.04130e6 −0.744755
\(58\) −1.66449e6 −1.12017
\(59\) 1.61187e6 1.02176 0.510879 0.859652i \(-0.329320\pi\)
0.510879 + 0.859652i \(0.329320\pi\)
\(60\) 589225. 0.352170
\(61\) −1.04092e6 −0.587171 −0.293586 0.955933i \(-0.594849\pi\)
−0.293586 + 0.955933i \(0.594849\pi\)
\(62\) 29230.2 0.0155762
\(63\) 923731. 0.465429
\(64\) 262144. 0.125000
\(65\) −1.98260e6 −0.895442
\(66\) 1.10187e6 0.471766
\(67\) −2.06066e6 −0.837036 −0.418518 0.908209i \(-0.637450\pi\)
−0.418518 + 0.908209i \(0.637450\pi\)
\(68\) 409903. 0.158088
\(69\) −328509. −0.120386
\(70\) 3.45657e6 1.20449
\(71\) 3.58392e6 1.18838 0.594189 0.804325i \(-0.297473\pi\)
0.594189 + 0.804325i \(0.297473\pi\)
\(72\) 373248. 0.117851
\(73\) −975850. −0.293598 −0.146799 0.989166i \(-0.546897\pi\)
−0.146799 + 0.989166i \(0.546897\pi\)
\(74\) 191447. 0.0549209
\(75\) 1.02997e6 0.281909
\(76\) −2.46826e6 −0.644977
\(77\) 6.46390e6 1.61353
\(78\) −1.25589e6 −0.299653
\(79\) 6.00359e6 1.36999 0.684993 0.728550i \(-0.259805\pi\)
0.684993 + 0.728550i \(0.259805\pi\)
\(80\) 1.39668e6 0.304988
\(81\) 531441. 0.111111
\(82\) −9212.01 −0.00184504
\(83\) −5.40252e6 −1.03711 −0.518553 0.855046i \(-0.673529\pi\)
−0.518553 + 0.855046i \(0.673529\pi\)
\(84\) 2.18958e6 0.403073
\(85\) 2.18393e6 0.385720
\(86\) −3.91441e6 −0.663624
\(87\) −5.61764e6 −0.914611
\(88\) 2.61184e6 0.408561
\(89\) −4.13911e6 −0.622361 −0.311180 0.950351i \(-0.600724\pi\)
−0.311180 + 0.950351i \(0.600724\pi\)
\(90\) 1.98863e6 0.287545
\(91\) −7.36740e6 −1.02487
\(92\) −778688. −0.104257
\(93\) 98652.1 0.0127179
\(94\) 7.91125e6 0.982422
\(95\) −1.31507e7 −1.57368
\(96\) 884736. 0.102062
\(97\) 1.18844e7 1.32214 0.661070 0.750324i \(-0.270103\pi\)
0.661070 + 0.750324i \(0.270103\pi\)
\(98\) 6.25640e6 0.671481
\(99\) 3.71881e6 0.385195
\(100\) 2.44141e6 0.244141
\(101\) 9.54327e6 0.921663 0.460832 0.887488i \(-0.347551\pi\)
0.460832 + 0.887488i \(0.347551\pi\)
\(102\) 1.38342e6 0.129079
\(103\) −3.12347e6 −0.281648 −0.140824 0.990035i \(-0.544975\pi\)
−0.140824 + 0.990035i \(0.544975\pi\)
\(104\) −2.97692e6 −0.259508
\(105\) 1.16659e7 0.983460
\(106\) −6.21621e6 −0.506939
\(107\) 1.02468e7 0.808620 0.404310 0.914622i \(-0.367512\pi\)
0.404310 + 0.914622i \(0.367512\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) −1.26733e7 −0.937337 −0.468668 0.883374i \(-0.655266\pi\)
−0.468668 + 0.883374i \(0.655266\pi\)
\(110\) 1.39157e7 0.996850
\(111\) 646134. 0.0448427
\(112\) 5.19012e6 0.349072
\(113\) 543321. 0.0354227 0.0177114 0.999843i \(-0.494362\pi\)
0.0177114 + 0.999843i \(0.494362\pi\)
\(114\) −8.33039e6 −0.526621
\(115\) −4.14879e6 −0.254377
\(116\) −1.33159e7 −0.792077
\(117\) −4.23862e6 −0.244666
\(118\) 1.28950e7 0.722493
\(119\) 8.11557e6 0.441473
\(120\) 4.71380e6 0.249022
\(121\) 6.53557e6 0.335378
\(122\) −8.32740e6 −0.415193
\(123\) −31090.5 −0.00150647
\(124\) 233842. 0.0110140
\(125\) −1.36320e7 −0.624272
\(126\) 7.38984e6 0.329108
\(127\) −6.08628e6 −0.263657 −0.131828 0.991273i \(-0.542085\pi\)
−0.131828 + 0.991273i \(0.542085\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −1.32111e7 −0.541847
\(130\) −1.58608e7 −0.633173
\(131\) −2.60740e7 −1.01335 −0.506674 0.862138i \(-0.669125\pi\)
−0.506674 + 0.862138i \(0.669125\pi\)
\(132\) 8.81496e6 0.333589
\(133\) −4.88686e7 −1.80115
\(134\) −1.64853e7 −0.591874
\(135\) 6.71164e6 0.234780
\(136\) 3.27922e6 0.111785
\(137\) 4.55380e7 1.51304 0.756522 0.653968i \(-0.226897\pi\)
0.756522 + 0.653968i \(0.226897\pi\)
\(138\) −2.62807e6 −0.0851257
\(139\) −4.36079e7 −1.37725 −0.688626 0.725116i \(-0.741786\pi\)
−0.688626 + 0.725116i \(0.741786\pi\)
\(140\) 2.76526e7 0.851701
\(141\) 2.67005e7 0.802144
\(142\) 2.86714e7 0.840310
\(143\) −2.96601e7 −0.848198
\(144\) 2.98598e6 0.0833333
\(145\) −7.09460e7 −1.93259
\(146\) −7.80680e6 −0.207605
\(147\) 2.11154e7 0.548262
\(148\) 1.53158e6 0.0388350
\(149\) −7.44370e7 −1.84347 −0.921737 0.387816i \(-0.873230\pi\)
−0.921737 + 0.387816i \(0.873230\pi\)
\(150\) 8.23975e6 0.199340
\(151\) 2.77945e7 0.656962 0.328481 0.944511i \(-0.393463\pi\)
0.328481 + 0.944511i \(0.393463\pi\)
\(152\) −1.97461e7 −0.456067
\(153\) 4.66905e6 0.105392
\(154\) 5.17112e7 1.14094
\(155\) 1.24589e6 0.0268732
\(156\) −1.00471e7 −0.211887
\(157\) 3.33077e7 0.686904 0.343452 0.939170i \(-0.388404\pi\)
0.343452 + 0.939170i \(0.388404\pi\)
\(158\) 4.80287e7 0.968726
\(159\) −2.09797e7 −0.413914
\(160\) 1.11735e7 0.215659
\(161\) −1.54171e7 −0.291146
\(162\) 4.25153e6 0.0785674
\(163\) −1.45963e7 −0.263989 −0.131994 0.991250i \(-0.542138\pi\)
−0.131994 + 0.991250i \(0.542138\pi\)
\(164\) −73696.1 −0.00130464
\(165\) 4.69654e7 0.813924
\(166\) −4.32201e7 −0.733344
\(167\) −7.61604e7 −1.26538 −0.632691 0.774404i \(-0.718050\pi\)
−0.632691 + 0.774404i \(0.718050\pi\)
\(168\) 1.75167e7 0.285016
\(169\) −2.89426e7 −0.461247
\(170\) 1.74714e7 0.272745
\(171\) −2.81151e7 −0.429985
\(172\) −3.13153e7 −0.469253
\(173\) −1.17815e8 −1.72997 −0.864983 0.501801i \(-0.832671\pi\)
−0.864983 + 0.501801i \(0.832671\pi\)
\(174\) −4.49411e7 −0.646728
\(175\) 4.83368e7 0.681781
\(176\) 2.08947e7 0.288896
\(177\) 4.35205e7 0.589913
\(178\) −3.31129e7 −0.440075
\(179\) 2.34279e7 0.305314 0.152657 0.988279i \(-0.451217\pi\)
0.152657 + 0.988279i \(0.451217\pi\)
\(180\) 1.59091e7 0.203325
\(181\) −5.66147e7 −0.709667 −0.354833 0.934930i \(-0.615462\pi\)
−0.354833 + 0.934930i \(0.615462\pi\)
\(182\) −5.89392e7 −0.724694
\(183\) −2.81050e7 −0.339004
\(184\) −6.22950e6 −0.0737210
\(185\) 8.16012e6 0.0947535
\(186\) 789217. 0.00899292
\(187\) 3.26721e7 0.365369
\(188\) 6.32900e7 0.694677
\(189\) 2.49407e7 0.268716
\(190\) −1.05206e8 −1.11276
\(191\) −1.26851e8 −1.31727 −0.658636 0.752461i \(-0.728866\pi\)
−0.658636 + 0.752461i \(0.728866\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) 1.36553e8 1.36726 0.683630 0.729828i \(-0.260400\pi\)
0.683630 + 0.729828i \(0.260400\pi\)
\(194\) 9.50755e7 0.934894
\(195\) −5.35301e7 −0.516984
\(196\) 5.00512e7 0.474808
\(197\) −1.16049e8 −1.08146 −0.540729 0.841197i \(-0.681852\pi\)
−0.540729 + 0.841197i \(0.681852\pi\)
\(198\) 2.97505e7 0.272374
\(199\) 1.41367e7 0.127163 0.0635816 0.997977i \(-0.479748\pi\)
0.0635816 + 0.997977i \(0.479748\pi\)
\(200\) 1.95312e7 0.172633
\(201\) −5.56377e7 −0.483263
\(202\) 7.63462e7 0.651714
\(203\) −2.63638e8 −2.21193
\(204\) 1.10674e7 0.0912723
\(205\) −392647. −0.00318320
\(206\) −2.49877e7 −0.199155
\(207\) −8.86974e6 −0.0695048
\(208\) −2.38153e7 −0.183500
\(209\) −1.96738e8 −1.49065
\(210\) 9.33274e7 0.695411
\(211\) 1.88579e8 1.38199 0.690993 0.722861i \(-0.257173\pi\)
0.690993 + 0.722861i \(0.257173\pi\)
\(212\) −4.97297e7 −0.358460
\(213\) 9.67660e7 0.686110
\(214\) 8.19743e7 0.571781
\(215\) −1.66845e8 −1.14493
\(216\) 1.00777e7 0.0680414
\(217\) 4.62978e6 0.0307575
\(218\) −1.01386e8 −0.662797
\(219\) −2.63479e7 −0.169509
\(220\) 1.11325e8 0.704879
\(221\) −3.72390e7 −0.232073
\(222\) 5.16907e6 0.0317086
\(223\) 3.17153e8 1.91515 0.957573 0.288192i \(-0.0930542\pi\)
0.957573 + 0.288192i \(0.0930542\pi\)
\(224\) 4.15210e7 0.246831
\(225\) 2.78091e7 0.162760
\(226\) 4.34657e6 0.0250476
\(227\) 9.71697e6 0.0551366 0.0275683 0.999620i \(-0.491224\pi\)
0.0275683 + 0.999620i \(0.491224\pi\)
\(228\) −6.66431e7 −0.372378
\(229\) 2.54644e8 1.40123 0.700614 0.713540i \(-0.252909\pi\)
0.700614 + 0.713540i \(0.252909\pi\)
\(230\) −3.31903e7 −0.179872
\(231\) 1.74525e8 0.931571
\(232\) −1.06527e8 −0.560083
\(233\) 2.64155e7 0.136808 0.0684042 0.997658i \(-0.478209\pi\)
0.0684042 + 0.997658i \(0.478209\pi\)
\(234\) −3.39089e7 −0.173005
\(235\) 3.37204e8 1.69494
\(236\) 1.03160e8 0.510879
\(237\) 1.62097e8 0.790962
\(238\) 6.49245e7 0.312169
\(239\) −1.27674e7 −0.0604936 −0.0302468 0.999542i \(-0.509629\pi\)
−0.0302468 + 0.999542i \(0.509629\pi\)
\(240\) 3.77104e7 0.176085
\(241\) 1.47494e8 0.678757 0.339379 0.940650i \(-0.389783\pi\)
0.339379 + 0.940650i \(0.389783\pi\)
\(242\) 5.22845e7 0.237148
\(243\) 1.43489e7 0.0641500
\(244\) −6.66192e7 −0.293586
\(245\) 2.66669e8 1.15849
\(246\) −248724. −0.00106524
\(247\) 2.24238e8 0.946823
\(248\) 1.87074e6 0.00778810
\(249\) −1.45868e8 −0.598773
\(250\) −1.09056e8 −0.441427
\(251\) 2.67374e8 1.06724 0.533618 0.845726i \(-0.320832\pi\)
0.533618 + 0.845726i \(0.320832\pi\)
\(252\) 5.91188e7 0.232714
\(253\) −6.20669e7 −0.240956
\(254\) −4.86902e7 −0.186433
\(255\) 5.89661e7 0.222696
\(256\) 1.67772e7 0.0625000
\(257\) −1.37033e8 −0.503568 −0.251784 0.967784i \(-0.581017\pi\)
−0.251784 + 0.967784i \(0.581017\pi\)
\(258\) −1.05689e8 −0.383143
\(259\) 3.03233e7 0.108449
\(260\) −1.26886e8 −0.447721
\(261\) −1.51676e8 −0.528051
\(262\) −2.08592e8 −0.716545
\(263\) 1.60586e8 0.544329 0.272165 0.962251i \(-0.412260\pi\)
0.272165 + 0.962251i \(0.412260\pi\)
\(264\) 7.05197e7 0.235883
\(265\) −2.64956e8 −0.874607
\(266\) −3.90948e8 −1.27360
\(267\) −1.11756e8 −0.359320
\(268\) −1.31882e8 −0.418518
\(269\) 4.67109e8 1.46314 0.731568 0.681768i \(-0.238789\pi\)
0.731568 + 0.681768i \(0.238789\pi\)
\(270\) 5.36931e7 0.166014
\(271\) 5.51685e8 1.68383 0.841916 0.539609i \(-0.181428\pi\)
0.841916 + 0.539609i \(0.181428\pi\)
\(272\) 2.62338e7 0.0790442
\(273\) −1.98920e8 −0.591710
\(274\) 3.64304e8 1.06988
\(275\) 1.94597e8 0.564251
\(276\) −2.10246e7 −0.0601929
\(277\) 4.78153e8 1.35172 0.675862 0.737028i \(-0.263772\pi\)
0.675862 + 0.737028i \(0.263772\pi\)
\(278\) −3.48863e8 −0.973865
\(279\) 2.66361e6 0.00734269
\(280\) 2.21220e8 0.602244
\(281\) 7.37447e8 1.98271 0.991354 0.131218i \(-0.0418887\pi\)
0.991354 + 0.131218i \(0.0418887\pi\)
\(282\) 2.13604e8 0.567201
\(283\) −6.82950e8 −1.79117 −0.895584 0.444893i \(-0.853242\pi\)
−0.895584 + 0.444893i \(0.853242\pi\)
\(284\) 2.29371e8 0.594189
\(285\) −3.55069e8 −0.908565
\(286\) −2.37281e8 −0.599766
\(287\) −1.45909e6 −0.00364331
\(288\) 2.38879e7 0.0589256
\(289\) −3.69318e8 −0.900032
\(290\) −5.67568e8 −1.36655
\(291\) 3.20880e8 0.763338
\(292\) −6.24544e7 −0.146799
\(293\) −4.73242e7 −0.109912 −0.0549562 0.998489i \(-0.517502\pi\)
−0.0549562 + 0.998489i \(0.517502\pi\)
\(294\) 1.68923e8 0.387680
\(295\) 5.49627e8 1.24650
\(296\) 1.22526e7 0.0274605
\(297\) 1.00408e8 0.222393
\(298\) −5.95496e8 −1.30353
\(299\) 7.07425e7 0.153049
\(300\) 6.59180e7 0.140955
\(301\) −6.20004e8 −1.31042
\(302\) 2.22356e8 0.464542
\(303\) 2.57668e8 0.532123
\(304\) −1.57969e8 −0.322488
\(305\) −3.54942e8 −0.716321
\(306\) 3.73524e7 0.0745236
\(307\) 3.47596e8 0.685630 0.342815 0.939403i \(-0.388620\pi\)
0.342815 + 0.939403i \(0.388620\pi\)
\(308\) 4.13689e8 0.806764
\(309\) −8.43336e7 −0.162610
\(310\) 9.96713e6 0.0190022
\(311\) −8.46459e8 −1.59568 −0.797838 0.602873i \(-0.794023\pi\)
−0.797838 + 0.602873i \(0.794023\pi\)
\(312\) −8.03767e7 −0.149827
\(313\) 5.31447e8 0.979613 0.489806 0.871831i \(-0.337067\pi\)
0.489806 + 0.871831i \(0.337067\pi\)
\(314\) 2.66462e8 0.485715
\(315\) 3.14980e8 0.567801
\(316\) 3.84230e8 0.684993
\(317\) −7.52971e8 −1.32761 −0.663806 0.747905i \(-0.731060\pi\)
−0.663806 + 0.747905i \(0.731060\pi\)
\(318\) −1.67838e8 −0.292681
\(319\) −1.06137e9 −1.83062
\(320\) 8.93876e7 0.152494
\(321\) 2.76663e8 0.466857
\(322\) −1.23336e8 −0.205871
\(323\) −2.47009e8 −0.407853
\(324\) 3.40122e7 0.0555556
\(325\) −2.21798e8 −0.358398
\(326\) −1.16770e8 −0.186668
\(327\) −3.42178e8 −0.541172
\(328\) −589569. −0.000922521 0
\(329\) 1.25306e9 1.93994
\(330\) 3.75723e8 0.575531
\(331\) −1.86489e8 −0.282654 −0.141327 0.989963i \(-0.545137\pi\)
−0.141327 + 0.989963i \(0.545137\pi\)
\(332\) −3.45761e8 −0.518553
\(333\) 1.74456e7 0.0258900
\(334\) −6.09284e8 −0.894760
\(335\) −7.02657e8 −1.02114
\(336\) 1.40133e8 0.201537
\(337\) −7.20597e8 −1.02562 −0.512811 0.858501i \(-0.671396\pi\)
−0.512811 + 0.858501i \(0.671396\pi\)
\(338\) −2.31540e8 −0.326151
\(339\) 1.46697e7 0.0204513
\(340\) 1.39771e8 0.192860
\(341\) 1.86388e7 0.0254553
\(342\) −2.24921e8 −0.304045
\(343\) −5.25761e7 −0.0703491
\(344\) −2.50522e8 −0.331812
\(345\) −1.12017e8 −0.146865
\(346\) −9.42517e8 −1.22327
\(347\) −8.14326e8 −1.04627 −0.523137 0.852249i \(-0.675238\pi\)
−0.523137 + 0.852249i \(0.675238\pi\)
\(348\) −3.59529e8 −0.457306
\(349\) −4.25035e8 −0.535224 −0.267612 0.963527i \(-0.586234\pi\)
−0.267612 + 0.963527i \(0.586234\pi\)
\(350\) 3.86694e8 0.482092
\(351\) −1.14443e8 −0.141258
\(352\) 1.67158e8 0.204281
\(353\) −2.10866e8 −0.255149 −0.127575 0.991829i \(-0.540719\pi\)
−0.127575 + 0.991829i \(0.540719\pi\)
\(354\) 3.48164e8 0.417131
\(355\) 1.22207e9 1.44976
\(356\) −2.64903e8 −0.311180
\(357\) 2.19120e8 0.254885
\(358\) 1.87423e8 0.215890
\(359\) −9.62505e8 −1.09792 −0.548962 0.835847i \(-0.684977\pi\)
−0.548962 + 0.835847i \(0.684977\pi\)
\(360\) 1.27273e8 0.143773
\(361\) 5.93513e8 0.663980
\(362\) −4.52918e8 −0.501810
\(363\) 1.76460e8 0.193631
\(364\) −4.71514e8 −0.512436
\(365\) −3.32752e8 −0.358175
\(366\) −2.24840e8 −0.239712
\(367\) −5.91557e7 −0.0624691 −0.0312345 0.999512i \(-0.509944\pi\)
−0.0312345 + 0.999512i \(0.509944\pi\)
\(368\) −4.98360e7 −0.0521286
\(369\) −839445. −0.000869761 0
\(370\) 6.52809e7 0.0670009
\(371\) −9.84586e8 −1.00103
\(372\) 6.31373e6 0.00635896
\(373\) −7.57694e8 −0.755984 −0.377992 0.925809i \(-0.623385\pi\)
−0.377992 + 0.925809i \(0.623385\pi\)
\(374\) 2.61377e8 0.258355
\(375\) −3.68063e8 −0.360424
\(376\) 5.06320e8 0.491211
\(377\) 1.20973e9 1.16277
\(378\) 1.99526e8 0.190011
\(379\) −2.47156e8 −0.233203 −0.116601 0.993179i \(-0.537200\pi\)
−0.116601 + 0.993179i \(0.537200\pi\)
\(380\) −8.41645e8 −0.786840
\(381\) −1.64330e8 −0.152222
\(382\) −1.01480e9 −0.931452
\(383\) −1.95064e9 −1.77411 −0.887056 0.461662i \(-0.847253\pi\)
−0.887056 + 0.461662i \(0.847253\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) 2.20410e9 1.96843
\(386\) 1.09243e9 0.966799
\(387\) −3.56701e8 −0.312835
\(388\) 7.60604e8 0.661070
\(389\) 1.70660e9 1.46997 0.734987 0.678082i \(-0.237188\pi\)
0.734987 + 0.678082i \(0.237188\pi\)
\(390\) −4.28241e8 −0.365563
\(391\) −7.79264e7 −0.0659274
\(392\) 4.00410e8 0.335740
\(393\) −7.03998e8 −0.585056
\(394\) −9.28392e8 −0.764706
\(395\) 2.04714e9 1.67132
\(396\) 2.38004e8 0.192598
\(397\) 9.51729e8 0.763390 0.381695 0.924288i \(-0.375340\pi\)
0.381695 + 0.924288i \(0.375340\pi\)
\(398\) 1.13093e8 0.0899179
\(399\) −1.31945e9 −1.03989
\(400\) 1.56250e8 0.122070
\(401\) 1.03883e9 0.804522 0.402261 0.915525i \(-0.368225\pi\)
0.402261 + 0.915525i \(0.368225\pi\)
\(402\) −4.45102e8 −0.341718
\(403\) −2.12441e7 −0.0161686
\(404\) 6.10769e8 0.460832
\(405\) 1.81214e8 0.135550
\(406\) −2.10910e9 −1.56407
\(407\) 1.22077e8 0.0897542
\(408\) 8.85390e7 0.0645393
\(409\) 2.67559e9 1.93370 0.966849 0.255348i \(-0.0821899\pi\)
0.966849 + 0.255348i \(0.0821899\pi\)
\(410\) −3.14117e6 −0.00225086
\(411\) 1.22953e9 0.873557
\(412\) −1.99902e8 −0.140824
\(413\) 2.04243e9 1.42667
\(414\) −7.09579e7 −0.0491473
\(415\) −1.84219e9 −1.26522
\(416\) −1.90523e8 −0.129754
\(417\) −1.17741e9 −0.795157
\(418\) −1.57390e9 −1.05405
\(419\) 3.41769e8 0.226978 0.113489 0.993539i \(-0.463797\pi\)
0.113489 + 0.993539i \(0.463797\pi\)
\(420\) 7.46619e8 0.491730
\(421\) 1.83184e9 1.19646 0.598232 0.801323i \(-0.295870\pi\)
0.598232 + 0.801323i \(0.295870\pi\)
\(422\) 1.50863e9 0.977212
\(423\) 7.20913e8 0.463118
\(424\) −3.97838e8 −0.253469
\(425\) 2.44321e8 0.154383
\(426\) 7.74128e8 0.485153
\(427\) −1.31898e9 −0.819860
\(428\) 6.55794e8 0.404310
\(429\) −8.00824e8 −0.489707
\(430\) −1.33476e9 −0.809589
\(431\) −1.20261e9 −0.723529 −0.361764 0.932270i \(-0.617826\pi\)
−0.361764 + 0.932270i \(0.617826\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) 1.83198e9 1.08446 0.542228 0.840231i \(-0.317581\pi\)
0.542228 + 0.840231i \(0.317581\pi\)
\(434\) 3.70382e7 0.0217488
\(435\) −1.91554e9 −1.11578
\(436\) −8.11089e8 −0.468668
\(437\) 4.69240e8 0.268974
\(438\) −2.10784e8 −0.119861
\(439\) −1.50903e9 −0.851277 −0.425639 0.904893i \(-0.639951\pi\)
−0.425639 + 0.904893i \(0.639951\pi\)
\(440\) 8.90603e8 0.498425
\(441\) 5.70115e8 0.316539
\(442\) −2.97912e8 −0.164100
\(443\) 2.98792e9 1.63289 0.816444 0.577425i \(-0.195942\pi\)
0.816444 + 0.577425i \(0.195942\pi\)
\(444\) 4.13526e7 0.0224214
\(445\) −1.41138e9 −0.759250
\(446\) 2.53722e9 1.35421
\(447\) −2.00980e9 −1.06433
\(448\) 3.32168e8 0.174536
\(449\) 1.91998e9 1.00100 0.500500 0.865737i \(-0.333150\pi\)
0.500500 + 0.865737i \(0.333150\pi\)
\(450\) 2.22473e8 0.115089
\(451\) −5.87410e6 −0.00301525
\(452\) 3.47725e7 0.0177114
\(453\) 7.50452e8 0.379297
\(454\) 7.77358e7 0.0389875
\(455\) −2.51219e9 −1.25029
\(456\) −5.33145e8 −0.263311
\(457\) −1.94014e9 −0.950883 −0.475441 0.879747i \(-0.657712\pi\)
−0.475441 + 0.879747i \(0.657712\pi\)
\(458\) 2.03715e9 0.990818
\(459\) 1.26064e8 0.0608482
\(460\) −2.65522e8 −0.127189
\(461\) −8.14499e8 −0.387202 −0.193601 0.981080i \(-0.562017\pi\)
−0.193601 + 0.981080i \(0.562017\pi\)
\(462\) 1.39620e9 0.658720
\(463\) −6.61468e7 −0.0309724 −0.0154862 0.999880i \(-0.504930\pi\)
−0.0154862 + 0.999880i \(0.504930\pi\)
\(464\) −8.52217e8 −0.396038
\(465\) 3.36390e7 0.0155152
\(466\) 2.11324e8 0.0967382
\(467\) −2.88027e9 −1.30865 −0.654327 0.756212i \(-0.727048\pi\)
−0.654327 + 0.756212i \(0.727048\pi\)
\(468\) −2.71272e8 −0.122333
\(469\) −2.61110e9 −1.16874
\(470\) 2.69763e9 1.19851
\(471\) 8.99308e8 0.396584
\(472\) 8.25278e8 0.361246
\(473\) −2.49605e9 −1.08452
\(474\) 1.29677e9 0.559294
\(475\) −1.47120e9 −0.629860
\(476\) 5.19396e8 0.220737
\(477\) −5.66453e8 −0.238973
\(478\) −1.02139e8 −0.0427754
\(479\) −3.40043e9 −1.41371 −0.706854 0.707360i \(-0.749886\pi\)
−0.706854 + 0.707360i \(0.749886\pi\)
\(480\) 3.01683e8 0.124511
\(481\) −1.39141e8 −0.0570096
\(482\) 1.17995e9 0.479954
\(483\) −4.16260e8 −0.168093
\(484\) 4.18276e8 0.167689
\(485\) 4.05243e9 1.61295
\(486\) 1.14791e8 0.0453609
\(487\) 3.01213e9 1.18174 0.590871 0.806766i \(-0.298784\pi\)
0.590871 + 0.806766i \(0.298784\pi\)
\(488\) −5.32954e8 −0.207596
\(489\) −3.94100e8 −0.152414
\(490\) 2.13335e9 0.819174
\(491\) −835191. −0.000318420 0 −0.000159210 1.00000i \(-0.500051\pi\)
−0.000159210 1.00000i \(0.500051\pi\)
\(492\) −1.98979e6 −0.000753235 0
\(493\) −1.33257e9 −0.500872
\(494\) 1.79390e9 0.669505
\(495\) 1.26807e9 0.469919
\(496\) 1.49659e7 0.00550702
\(497\) 4.54126e9 1.65932
\(498\) −1.16694e9 −0.423396
\(499\) −4.35806e9 −1.57015 −0.785075 0.619401i \(-0.787375\pi\)
−0.785075 + 0.619401i \(0.787375\pi\)
\(500\) −8.72447e8 −0.312136
\(501\) −2.05633e9 −0.730569
\(502\) 2.13899e9 0.754650
\(503\) −2.40286e8 −0.0841862 −0.0420931 0.999114i \(-0.513403\pi\)
−0.0420931 + 0.999114i \(0.513403\pi\)
\(504\) 4.72950e8 0.164554
\(505\) 3.25413e9 1.12438
\(506\) −4.96535e8 −0.170382
\(507\) −7.81449e8 −0.266301
\(508\) −3.89522e8 −0.131828
\(509\) −3.36572e9 −1.13127 −0.565634 0.824656i \(-0.691369\pi\)
−0.565634 + 0.824656i \(0.691369\pi\)
\(510\) 4.71729e8 0.157470
\(511\) −1.23652e9 −0.409947
\(512\) 1.34218e8 0.0441942
\(513\) −7.59107e8 −0.248252
\(514\) −1.09626e9 −0.356076
\(515\) −1.06506e9 −0.343597
\(516\) −8.45513e8 −0.270923
\(517\) 5.04466e9 1.60552
\(518\) 2.42587e8 0.0766854
\(519\) −3.18099e9 −0.998797
\(520\) −1.01509e9 −0.316587
\(521\) 6.69390e8 0.207371 0.103685 0.994610i \(-0.466937\pi\)
0.103685 + 0.994610i \(0.466937\pi\)
\(522\) −1.21341e9 −0.373388
\(523\) 6.04759e9 1.84853 0.924265 0.381752i \(-0.124679\pi\)
0.924265 + 0.381752i \(0.124679\pi\)
\(524\) −1.66874e9 −0.506674
\(525\) 1.30509e9 0.393626
\(526\) 1.28469e9 0.384899
\(527\) 2.34015e7 0.00696476
\(528\) 5.64157e8 0.166794
\(529\) 1.48036e8 0.0434783
\(530\) −2.11965e9 −0.618441
\(531\) 1.17505e9 0.340586
\(532\) −3.12759e9 −0.900573
\(533\) 6.69516e6 0.00191521
\(534\) −8.94048e8 −0.254078
\(535\) 3.49402e9 0.986477
\(536\) −1.05506e9 −0.295937
\(537\) 6.32553e8 0.176273
\(538\) 3.73687e9 1.03459
\(539\) 3.98943e9 1.09736
\(540\) 4.29545e8 0.117390
\(541\) 9.56170e8 0.259624 0.129812 0.991539i \(-0.458563\pi\)
0.129812 + 0.991539i \(0.458563\pi\)
\(542\) 4.41348e9 1.19065
\(543\) −1.52860e9 −0.409726
\(544\) 2.09870e8 0.0558927
\(545\) −4.32141e9 −1.14351
\(546\) −1.59136e9 −0.418402
\(547\) −7.50590e9 −1.96086 −0.980431 0.196864i \(-0.936924\pi\)
−0.980431 + 0.196864i \(0.936924\pi\)
\(548\) 2.91443e9 0.756522
\(549\) −7.58834e8 −0.195724
\(550\) 1.55678e9 0.398986
\(551\) 8.02420e9 2.04348
\(552\) −1.68197e8 −0.0425628
\(553\) 7.60727e9 1.91289
\(554\) 3.82523e9 0.955813
\(555\) 2.20323e8 0.0547060
\(556\) −2.79091e9 −0.688626
\(557\) 2.30925e9 0.566209 0.283105 0.959089i \(-0.408636\pi\)
0.283105 + 0.959089i \(0.408636\pi\)
\(558\) 2.13088e7 0.00519207
\(559\) 2.84494e9 0.688862
\(560\) 1.76976e9 0.425851
\(561\) 8.82148e8 0.210946
\(562\) 5.89957e9 1.40199
\(563\) 2.20676e9 0.521166 0.260583 0.965451i \(-0.416085\pi\)
0.260583 + 0.965451i \(0.416085\pi\)
\(564\) 1.70883e9 0.401072
\(565\) 1.85265e8 0.0432140
\(566\) −5.46360e9 −1.26655
\(567\) 6.73400e8 0.155143
\(568\) 1.83497e9 0.420155
\(569\) −2.56775e9 −0.584332 −0.292166 0.956368i \(-0.594376\pi\)
−0.292166 + 0.956368i \(0.594376\pi\)
\(570\) −2.84055e9 −0.642452
\(571\) −3.30249e9 −0.742360 −0.371180 0.928561i \(-0.621047\pi\)
−0.371180 + 0.928561i \(0.621047\pi\)
\(572\) −1.89825e9 −0.424099
\(573\) −3.42497e9 −0.760528
\(574\) −1.16727e7 −0.00257621
\(575\) −4.64134e8 −0.101814
\(576\) 1.91103e8 0.0416667
\(577\) −4.83666e9 −1.04817 −0.524083 0.851667i \(-0.675592\pi\)
−0.524083 + 0.851667i \(0.675592\pi\)
\(578\) −2.95454e9 −0.636419
\(579\) 3.68694e9 0.789388
\(580\) −4.54054e9 −0.966295
\(581\) −6.84564e9 −1.44810
\(582\) 2.56704e9 0.539761
\(583\) −3.96381e9 −0.828462
\(584\) −4.99635e8 −0.103803
\(585\) −1.44531e9 −0.298481
\(586\) −3.78593e8 −0.0777198
\(587\) 8.36248e9 1.70648 0.853241 0.521517i \(-0.174634\pi\)
0.853241 + 0.521517i \(0.174634\pi\)
\(588\) 1.35138e9 0.274131
\(589\) −1.40914e8 −0.0284152
\(590\) 4.39702e9 0.881406
\(591\) −3.13332e9 −0.624380
\(592\) 9.80209e7 0.0194175
\(593\) −7.18982e9 −1.41588 −0.707941 0.706272i \(-0.750376\pi\)
−0.707941 + 0.706272i \(0.750376\pi\)
\(594\) 8.03263e8 0.157255
\(595\) 2.76730e9 0.538576
\(596\) −4.76397e9 −0.921737
\(597\) 3.81690e8 0.0734177
\(598\) 5.65940e8 0.108222
\(599\) 7.64624e9 1.45363 0.726815 0.686833i \(-0.241000\pi\)
0.726815 + 0.686833i \(0.241000\pi\)
\(600\) 5.27344e8 0.0996700
\(601\) 1.72237e9 0.323643 0.161821 0.986820i \(-0.448263\pi\)
0.161821 + 0.986820i \(0.448263\pi\)
\(602\) −4.96003e9 −0.926609
\(603\) −1.50222e9 −0.279012
\(604\) 1.77885e9 0.328481
\(605\) 2.22854e9 0.409145
\(606\) 2.06135e9 0.376268
\(607\) −4.96333e9 −0.900768 −0.450384 0.892835i \(-0.648713\pi\)
−0.450384 + 0.892835i \(0.648713\pi\)
\(608\) −1.26375e9 −0.228034
\(609\) −7.11823e9 −1.27706
\(610\) −2.83953e9 −0.506515
\(611\) −5.74979e9 −1.01978
\(612\) 2.98819e8 0.0526961
\(613\) 5.74407e9 1.00718 0.503591 0.863942i \(-0.332012\pi\)
0.503591 + 0.863942i \(0.332012\pi\)
\(614\) 2.78076e9 0.484814
\(615\) −1.06015e7 −0.00183782
\(616\) 3.30951e9 0.570469
\(617\) −1.05462e10 −1.80758 −0.903788 0.427981i \(-0.859225\pi\)
−0.903788 + 0.427981i \(0.859225\pi\)
\(618\) −6.74669e8 −0.114982
\(619\) 3.60119e9 0.610279 0.305140 0.952308i \(-0.401297\pi\)
0.305140 + 0.952308i \(0.401297\pi\)
\(620\) 7.97370e7 0.0134366
\(621\) −2.39483e8 −0.0401286
\(622\) −6.77167e9 −1.12831
\(623\) −5.24475e9 −0.868994
\(624\) −6.43014e8 −0.105944
\(625\) −7.62856e9 −1.24986
\(626\) 4.25157e9 0.692691
\(627\) −5.31193e9 −0.860628
\(628\) 2.13169e9 0.343452
\(629\) 1.53271e8 0.0245574
\(630\) 2.51984e9 0.401496
\(631\) −2.99729e9 −0.474926 −0.237463 0.971397i \(-0.576316\pi\)
−0.237463 + 0.971397i \(0.576316\pi\)
\(632\) 3.07384e9 0.484363
\(633\) 5.09162e9 0.797890
\(634\) −6.02377e9 −0.938763
\(635\) −2.07534e9 −0.321648
\(636\) −1.34270e9 −0.206957
\(637\) −4.54707e9 −0.697017
\(638\) −8.49096e9 −1.29445
\(639\) 2.61268e9 0.396126
\(640\) 7.15101e8 0.107829
\(641\) 1.30075e9 0.195069 0.0975347 0.995232i \(-0.468904\pi\)
0.0975347 + 0.995232i \(0.468904\pi\)
\(642\) 2.21331e9 0.330118
\(643\) −9.31677e9 −1.38206 −0.691030 0.722826i \(-0.742843\pi\)
−0.691030 + 0.722826i \(0.742843\pi\)
\(644\) −9.86691e8 −0.145573
\(645\) −4.50482e9 −0.661027
\(646\) −1.97607e9 −0.288396
\(647\) −9.62984e9 −1.39783 −0.698914 0.715205i \(-0.746333\pi\)
−0.698914 + 0.715205i \(0.746333\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) 8.22256e9 1.18073
\(650\) −1.77438e9 −0.253425
\(651\) 1.25004e8 0.0177579
\(652\) −9.34162e8 −0.131994
\(653\) 4.51671e9 0.634784 0.317392 0.948294i \(-0.397193\pi\)
0.317392 + 0.948294i \(0.397193\pi\)
\(654\) −2.73742e9 −0.382666
\(655\) −8.89089e9 −1.23623
\(656\) −4.71655e6 −0.000652321 0
\(657\) −7.11395e8 −0.0978660
\(658\) 1.00245e10 1.37174
\(659\) 6.60804e9 0.899443 0.449722 0.893169i \(-0.351523\pi\)
0.449722 + 0.893169i \(0.351523\pi\)
\(660\) 3.00578e9 0.406962
\(661\) 1.31054e10 1.76500 0.882502 0.470308i \(-0.155857\pi\)
0.882502 + 0.470308i \(0.155857\pi\)
\(662\) −1.49191e9 −0.199867
\(663\) −1.00545e9 −0.133987
\(664\) −2.76609e9 −0.366672
\(665\) −1.66635e10 −2.19731
\(666\) 1.39565e8 0.0183070
\(667\) 2.53148e9 0.330319
\(668\) −4.87427e9 −0.632691
\(669\) 8.56313e9 1.10571
\(670\) −5.62125e9 −0.722057
\(671\) −5.31002e9 −0.678527
\(672\) 1.12107e9 0.142508
\(673\) −1.48862e9 −0.188249 −0.0941243 0.995560i \(-0.530005\pi\)
−0.0941243 + 0.995560i \(0.530005\pi\)
\(674\) −5.76477e9 −0.725225
\(675\) 7.50847e8 0.0939698
\(676\) −1.85232e9 −0.230623
\(677\) 1.38673e10 1.71763 0.858817 0.512282i \(-0.171200\pi\)
0.858817 + 0.512282i \(0.171200\pi\)
\(678\) 1.17357e8 0.0144613
\(679\) 1.50590e10 1.84609
\(680\) 1.11817e9 0.136373
\(681\) 2.62358e8 0.0318332
\(682\) 1.49111e8 0.0179996
\(683\) −6.45650e9 −0.775398 −0.387699 0.921786i \(-0.626730\pi\)
−0.387699 + 0.921786i \(0.626730\pi\)
\(684\) −1.79936e9 −0.214992
\(685\) 1.55279e10 1.84584
\(686\) −4.20609e8 −0.0497443
\(687\) 6.87538e9 0.809000
\(688\) −2.00418e9 −0.234626
\(689\) 4.51786e9 0.526218
\(690\) −8.96138e8 −0.103849
\(691\) −8.22490e8 −0.0948325 −0.0474163 0.998875i \(-0.515099\pi\)
−0.0474163 + 0.998875i \(0.515099\pi\)
\(692\) −7.54013e9 −0.864983
\(693\) 4.71218e9 0.537843
\(694\) −6.51461e9 −0.739827
\(695\) −1.48697e10 −1.68018
\(696\) −2.87623e9 −0.323364
\(697\) −7.37506e6 −0.000824994 0
\(698\) −3.40028e9 −0.378460
\(699\) 7.13218e8 0.0789864
\(700\) 3.09355e9 0.340890
\(701\) 1.78860e10 1.96111 0.980553 0.196252i \(-0.0628772\pi\)
0.980553 + 0.196252i \(0.0628772\pi\)
\(702\) −9.15541e8 −0.0998845
\(703\) −9.22934e8 −0.100191
\(704\) 1.33726e9 0.144448
\(705\) 9.10451e9 0.978576
\(706\) −1.68693e9 −0.180418
\(707\) 1.20925e10 1.28691
\(708\) 2.78531e9 0.294956
\(709\) 1.14149e10 1.20285 0.601424 0.798930i \(-0.294600\pi\)
0.601424 + 0.798930i \(0.294600\pi\)
\(710\) 9.77657e9 1.02514
\(711\) 4.37661e9 0.456662
\(712\) −2.11923e9 −0.220038
\(713\) −4.44555e7 −0.00459317
\(714\) 1.75296e9 0.180231
\(715\) −1.01137e10 −1.03476
\(716\) 1.49938e9 0.152657
\(717\) −3.44720e8 −0.0349260
\(718\) −7.70004e9 −0.776349
\(719\) −4.68698e9 −0.470264 −0.235132 0.971963i \(-0.575552\pi\)
−0.235132 + 0.971963i \(0.575552\pi\)
\(720\) 1.01818e9 0.101663
\(721\) −3.95781e9 −0.393262
\(722\) 4.74811e9 0.469505
\(723\) 3.98234e9 0.391881
\(724\) −3.62334e9 −0.354833
\(725\) −7.93689e9 −0.773512
\(726\) 1.41168e9 0.136917
\(727\) −1.85231e10 −1.78790 −0.893951 0.448164i \(-0.852078\pi\)
−0.893951 + 0.448164i \(0.852078\pi\)
\(728\) −3.77211e9 −0.362347
\(729\) 3.87420e8 0.0370370
\(730\) −2.66201e9 −0.253268
\(731\) −3.13385e9 −0.296734
\(732\) −1.79872e9 −0.169502
\(733\) 1.69649e9 0.159107 0.0795534 0.996831i \(-0.474651\pi\)
0.0795534 + 0.996831i \(0.474651\pi\)
\(734\) −4.73245e8 −0.0441723
\(735\) 7.20006e9 0.668853
\(736\) −3.98688e8 −0.0368605
\(737\) −1.05119e10 −0.967266
\(738\) −6.71556e6 −0.000615014 0
\(739\) −8.17586e9 −0.745209 −0.372604 0.927990i \(-0.621535\pi\)
−0.372604 + 0.927990i \(0.621535\pi\)
\(740\) 5.22247e8 0.0473768
\(741\) 6.05441e9 0.546649
\(742\) −7.87669e9 −0.707832
\(743\) 1.95551e9 0.174904 0.0874519 0.996169i \(-0.472128\pi\)
0.0874519 + 0.996169i \(0.472128\pi\)
\(744\) 5.05099e7 0.00449646
\(745\) −2.53820e10 −2.24895
\(746\) −6.06155e9 −0.534562
\(747\) −3.93843e9 −0.345702
\(748\) 2.09102e9 0.182685
\(749\) 1.29839e10 1.12907
\(750\) −2.94451e9 −0.254858
\(751\) 1.41485e10 1.21891 0.609453 0.792823i \(-0.291389\pi\)
0.609453 + 0.792823i \(0.291389\pi\)
\(752\) 4.05056e9 0.347338
\(753\) 7.21909e9 0.616169
\(754\) 9.67780e9 0.822199
\(755\) 9.47757e9 0.801461
\(756\) 1.59621e9 0.134358
\(757\) −3.50371e9 −0.293557 −0.146779 0.989169i \(-0.546890\pi\)
−0.146779 + 0.989169i \(0.546890\pi\)
\(758\) −1.97725e9 −0.164899
\(759\) −1.67581e9 −0.139116
\(760\) −6.73316e9 −0.556380
\(761\) 1.18352e9 0.0973485 0.0486743 0.998815i \(-0.484500\pi\)
0.0486743 + 0.998815i \(0.484500\pi\)
\(762\) −1.31464e9 −0.107637
\(763\) −1.60585e10 −1.30879
\(764\) −8.11844e9 −0.658636
\(765\) 1.59208e9 0.128573
\(766\) −1.56051e10 −1.25449
\(767\) −9.37189e9 −0.749969
\(768\) 4.52985e8 0.0360844
\(769\) 1.26642e10 1.00424 0.502118 0.864799i \(-0.332554\pi\)
0.502118 + 0.864799i \(0.332554\pi\)
\(770\) 1.76328e10 1.39189
\(771\) −3.69988e9 −0.290735
\(772\) 8.73941e9 0.683630
\(773\) −6.91080e9 −0.538146 −0.269073 0.963120i \(-0.586717\pi\)
−0.269073 + 0.963120i \(0.586717\pi\)
\(774\) −2.85361e9 −0.221208
\(775\) 1.39381e8 0.0107559
\(776\) 6.08483e9 0.467447
\(777\) 8.18729e8 0.0626133
\(778\) 1.36528e10 1.03943
\(779\) 4.44095e7 0.00336585
\(780\) −3.42593e9 −0.258492
\(781\) 1.82825e10 1.37327
\(782\) −6.23411e8 −0.0466177
\(783\) −4.09526e9 −0.304870
\(784\) 3.20328e9 0.237404
\(785\) 1.13575e10 0.837990
\(786\) −5.63199e9 −0.413697
\(787\) 1.12082e10 0.819644 0.409822 0.912166i \(-0.365591\pi\)
0.409822 + 0.912166i \(0.365591\pi\)
\(788\) −7.42714e9 −0.540729
\(789\) 4.33581e9 0.314269
\(790\) 1.63771e10 1.18180
\(791\) 6.88453e8 0.0494603
\(792\) 1.90403e9 0.136187
\(793\) 6.05224e9 0.430983
\(794\) 7.61383e9 0.539798
\(795\) −7.15381e9 −0.504955
\(796\) 9.04747e8 0.0635816
\(797\) 2.00641e9 0.140383 0.0701917 0.997534i \(-0.477639\pi\)
0.0701917 + 0.997534i \(0.477639\pi\)
\(798\) −1.05556e10 −0.735314
\(799\) 6.33368e9 0.439281
\(800\) 1.25000e9 0.0863167
\(801\) −3.01741e9 −0.207454
\(802\) 8.31061e9 0.568883
\(803\) −4.97805e9 −0.339278
\(804\) −3.56082e9 −0.241631
\(805\) −5.25701e9 −0.355184
\(806\) −1.69953e8 −0.0114329
\(807\) 1.26119e10 0.844742
\(808\) 4.88615e9 0.325857
\(809\) −2.85624e9 −0.189660 −0.0948300 0.995493i \(-0.530231\pi\)
−0.0948300 + 0.995493i \(0.530231\pi\)
\(810\) 1.44971e9 0.0958484
\(811\) 1.97522e10 1.30030 0.650148 0.759808i \(-0.274707\pi\)
0.650148 + 0.759808i \(0.274707\pi\)
\(812\) −1.68728e10 −1.10597
\(813\) 1.48955e10 0.972161
\(814\) 9.76619e8 0.0634658
\(815\) −4.97714e9 −0.322054
\(816\) 7.08312e8 0.0456362
\(817\) 1.88707e10 1.21063
\(818\) 2.14048e10 1.36733
\(819\) −5.37084e9 −0.341624
\(820\) −2.51294e7 −0.00159160
\(821\) 1.99137e10 1.25589 0.627943 0.778260i \(-0.283897\pi\)
0.627943 + 0.778260i \(0.283897\pi\)
\(822\) 9.83621e9 0.617698
\(823\) 2.60455e10 1.62867 0.814336 0.580393i \(-0.197101\pi\)
0.814336 + 0.580393i \(0.197101\pi\)
\(824\) −1.59922e9 −0.0995776
\(825\) 5.25412e9 0.325770
\(826\) 1.63395e10 1.00881
\(827\) 9.95427e9 0.611984 0.305992 0.952034i \(-0.401012\pi\)
0.305992 + 0.952034i \(0.401012\pi\)
\(828\) −5.67664e8 −0.0347524
\(829\) −3.29779e9 −0.201040 −0.100520 0.994935i \(-0.532051\pi\)
−0.100520 + 0.994935i \(0.532051\pi\)
\(830\) −1.47375e10 −0.894644
\(831\) 1.29101e10 0.780418
\(832\) −1.52418e9 −0.0917498
\(833\) 5.00882e9 0.300247
\(834\) −9.41931e9 −0.562261
\(835\) −2.59697e10 −1.54370
\(836\) −1.25912e10 −0.745326
\(837\) 7.19174e7 0.00423930
\(838\) 2.73415e9 0.160498
\(839\) 1.21672e9 0.0711252 0.0355626 0.999367i \(-0.488678\pi\)
0.0355626 + 0.999367i \(0.488678\pi\)
\(840\) 5.97295e9 0.347705
\(841\) 2.60394e10 1.50954
\(842\) 1.46547e10 0.846028
\(843\) 1.99111e10 1.14472
\(844\) 1.20690e10 0.690993
\(845\) −9.86903e9 −0.562699
\(846\) 5.76730e9 0.327474
\(847\) 8.28135e9 0.468284
\(848\) −3.18270e9 −0.179230
\(849\) −1.84396e10 −1.03413
\(850\) 1.95457e9 0.109165
\(851\) −2.91167e8 −0.0161953
\(852\) 6.19302e9 0.343055
\(853\) −1.25896e10 −0.694527 −0.347264 0.937768i \(-0.612889\pi\)
−0.347264 + 0.937768i \(0.612889\pi\)
\(854\) −1.05518e10 −0.579728
\(855\) −9.58687e9 −0.524560
\(856\) 5.24636e9 0.285890
\(857\) −3.98461e9 −0.216248 −0.108124 0.994137i \(-0.534484\pi\)
−0.108124 + 0.994137i \(0.534484\pi\)
\(858\) −6.40659e9 −0.346275
\(859\) −1.29735e10 −0.698361 −0.349181 0.937055i \(-0.613540\pi\)
−0.349181 + 0.937055i \(0.613540\pi\)
\(860\) −1.06781e10 −0.572466
\(861\) −3.93955e7 −0.00210346
\(862\) −9.62091e9 −0.511612
\(863\) −2.90007e10 −1.53593 −0.767963 0.640495i \(-0.778729\pi\)
−0.767963 + 0.640495i \(0.778729\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) −4.01732e10 −2.11048
\(866\) 1.46558e10 0.766826
\(867\) −9.97159e9 −0.519634
\(868\) 2.96306e8 0.0153788
\(869\) 3.06258e10 1.58314
\(870\) −1.53243e10 −0.788976
\(871\) 1.19813e10 0.614383
\(872\) −6.48871e9 −0.331399
\(873\) 8.66375e9 0.440713
\(874\) 3.75392e9 0.190193
\(875\) −1.72734e10 −0.871663
\(876\) −1.68627e9 −0.0847544
\(877\) −2.05862e10 −1.03057 −0.515286 0.857018i \(-0.672314\pi\)
−0.515286 + 0.857018i \(0.672314\pi\)
\(878\) −1.20722e10 −0.601944
\(879\) −1.27775e9 −0.0634579
\(880\) 7.12482e9 0.352440
\(881\) 4.47154e9 0.220314 0.110157 0.993914i \(-0.464865\pi\)
0.110157 + 0.993914i \(0.464865\pi\)
\(882\) 4.56092e9 0.223827
\(883\) −6.15100e9 −0.300665 −0.150333 0.988635i \(-0.548034\pi\)
−0.150333 + 0.988635i \(0.548034\pi\)
\(884\) −2.38329e9 −0.116037
\(885\) 1.48399e10 0.719665
\(886\) 2.39034e10 1.15463
\(887\) 2.96262e10 1.42542 0.712712 0.701457i \(-0.247467\pi\)
0.712712 + 0.701457i \(0.247467\pi\)
\(888\) 3.30821e8 0.0158543
\(889\) −7.71205e9 −0.368140
\(890\) −1.12911e10 −0.536871
\(891\) 2.71101e9 0.128398
\(892\) 2.02978e10 0.957573
\(893\) −3.81388e10 −1.79220
\(894\) −1.60784e10 −0.752595
\(895\) 7.98860e9 0.372469
\(896\) 2.65734e9 0.123415
\(897\) 1.91005e9 0.0883630
\(898\) 1.53598e10 0.707813
\(899\) −7.60208e8 −0.0348958
\(900\) 1.77979e9 0.0813802
\(901\) −4.97665e9 −0.226673
\(902\) −4.69928e7 −0.00213210
\(903\) −1.67401e10 −0.756573
\(904\) 2.78180e8 0.0125238
\(905\) −1.93049e10 −0.865759
\(906\) 6.00362e9 0.268204
\(907\) 3.02880e10 1.34786 0.673930 0.738795i \(-0.264605\pi\)
0.673930 + 0.738795i \(0.264605\pi\)
\(908\) 6.21886e8 0.0275683
\(909\) 6.95704e9 0.307221
\(910\) −2.00975e10 −0.884091
\(911\) 9.96481e9 0.436671 0.218336 0.975874i \(-0.429937\pi\)
0.218336 + 0.975874i \(0.429937\pi\)
\(912\) −4.26516e9 −0.186189
\(913\) −2.75596e10 −1.19846
\(914\) −1.55211e10 −0.672376
\(915\) −9.58342e9 −0.413568
\(916\) 1.62972e10 0.700614
\(917\) −3.30389e10 −1.41492
\(918\) 1.00851e9 0.0430262
\(919\) −3.72161e10 −1.58171 −0.790855 0.612003i \(-0.790364\pi\)
−0.790855 + 0.612003i \(0.790364\pi\)
\(920\) −2.12418e9 −0.0899360
\(921\) 9.38508e9 0.395849
\(922\) −6.51599e9 −0.273793
\(923\) −2.08380e10 −0.872267
\(924\) 1.11696e10 0.465786
\(925\) 9.12891e8 0.0379248
\(926\) −5.29174e8 −0.0219008
\(927\) −2.27701e9 −0.0938827
\(928\) −6.81773e9 −0.280041
\(929\) 2.36879e9 0.0969328 0.0484664 0.998825i \(-0.484567\pi\)
0.0484664 + 0.998825i \(0.484567\pi\)
\(930\) 2.69112e8 0.0109709
\(931\) −3.01610e10 −1.22496
\(932\) 1.69059e9 0.0684042
\(933\) −2.28544e10 −0.921263
\(934\) −2.30422e10 −0.925357
\(935\) 1.11408e10 0.445733
\(936\) −2.17017e9 −0.0865025
\(937\) 2.37255e10 0.942166 0.471083 0.882089i \(-0.343863\pi\)
0.471083 + 0.882089i \(0.343863\pi\)
\(938\) −2.08888e10 −0.826425
\(939\) 1.43491e10 0.565580
\(940\) 2.15811e10 0.847472
\(941\) −2.53391e10 −0.991351 −0.495676 0.868508i \(-0.665079\pi\)
−0.495676 + 0.868508i \(0.665079\pi\)
\(942\) 7.19447e9 0.280427
\(943\) 1.40103e7 0.000544073 0
\(944\) 6.60223e9 0.255440
\(945\) 8.50446e9 0.327820
\(946\) −1.99684e10 −0.766874
\(947\) 3.19577e10 1.22278 0.611392 0.791328i \(-0.290610\pi\)
0.611392 + 0.791328i \(0.290610\pi\)
\(948\) 1.03742e10 0.395481
\(949\) 5.67387e9 0.215500
\(950\) −1.17696e10 −0.445378
\(951\) −2.03302e10 −0.766497
\(952\) 4.15517e9 0.156084
\(953\) 1.06573e10 0.398861 0.199431 0.979912i \(-0.436091\pi\)
0.199431 + 0.979912i \(0.436091\pi\)
\(954\) −4.53162e9 −0.168980
\(955\) −4.32544e10 −1.60701
\(956\) −8.17113e8 −0.0302468
\(957\) −2.86570e10 −1.05691
\(958\) −2.72034e10 −0.999642
\(959\) 5.77021e10 2.11264
\(960\) 2.41347e9 0.0880424
\(961\) −2.74993e10 −0.999515
\(962\) −1.11313e9 −0.0403119
\(963\) 7.46991e9 0.269540
\(964\) 9.43961e9 0.339379
\(965\) 4.65628e10 1.66799
\(966\) −3.33008e9 −0.118860
\(967\) 5.33089e10 1.89586 0.947932 0.318472i \(-0.103170\pi\)
0.947932 + 0.318472i \(0.103170\pi\)
\(968\) 3.34621e9 0.118574
\(969\) −6.66924e9 −0.235474
\(970\) 3.24195e10 1.14053
\(971\) −5.29648e10 −1.85661 −0.928304 0.371823i \(-0.878733\pi\)
−0.928304 + 0.371823i \(0.878733\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) −5.52565e10 −1.92304
\(974\) 2.40970e10 0.835617
\(975\) −5.98853e9 −0.206921
\(976\) −4.26363e9 −0.146793
\(977\) −1.79920e10 −0.617233 −0.308616 0.951187i \(-0.599866\pi\)
−0.308616 + 0.951187i \(0.599866\pi\)
\(978\) −3.15280e9 −0.107773
\(979\) −2.11146e10 −0.719191
\(980\) 1.70668e10 0.579243
\(981\) −9.23881e9 −0.312446
\(982\) −6.68153e6 −0.000225157 0
\(983\) 1.56089e10 0.524124 0.262062 0.965051i \(-0.415597\pi\)
0.262062 + 0.965051i \(0.415597\pi\)
\(984\) −1.59184e7 −0.000532618 0
\(985\) −3.95712e10 −1.31933
\(986\) −1.06606e10 −0.354170
\(987\) 3.38327e10 1.12002
\(988\) 1.43512e10 0.473412
\(989\) 5.95333e9 0.195692
\(990\) 1.01445e10 0.332283
\(991\) 4.73053e10 1.54402 0.772009 0.635612i \(-0.219252\pi\)
0.772009 + 0.635612i \(0.219252\pi\)
\(992\) 1.19727e8 0.00389405
\(993\) −5.03521e9 −0.163191
\(994\) 3.63301e10 1.17331
\(995\) 4.82042e9 0.155133
\(996\) −9.33555e9 −0.299386
\(997\) 3.63716e10 1.16233 0.581166 0.813785i \(-0.302597\pi\)
0.581166 + 0.813785i \(0.302597\pi\)
\(998\) −3.48645e10 −1.11026
\(999\) 4.71032e8 0.0149476
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 138.8.a.h.1.3 4
3.2 odd 2 414.8.a.i.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.8.a.h.1.3 4 1.1 even 1 trivial
414.8.a.i.1.2 4 3.2 odd 2