Properties

Label 177.8.a.c.1.1
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + 12724944 x^{10} + 71331230512 x^{9} + 50741131904 x^{8} - 7480805165760 x^{7} - 10751966150272 x^{6} + 413177144536320 x^{5} + 886760582981376 x^{4} - 10454479722123264 x^{3} - 29180140031461376 x^{2} + 79787300207378432 x + 248723246810300416\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-22.1687\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-22.1687 q^{2} -27.0000 q^{3} +363.451 q^{4} +258.765 q^{5} +598.555 q^{6} +411.003 q^{7} -5219.64 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-22.1687 q^{2} -27.0000 q^{3} +363.451 q^{4} +258.765 q^{5} +598.555 q^{6} +411.003 q^{7} -5219.64 q^{8} +729.000 q^{9} -5736.49 q^{10} +437.879 q^{11} -9813.17 q^{12} +13224.5 q^{13} -9111.39 q^{14} -6986.67 q^{15} +69190.8 q^{16} +24930.4 q^{17} -16161.0 q^{18} +38005.6 q^{19} +94048.5 q^{20} -11097.1 q^{21} -9707.20 q^{22} +79710.8 q^{23} +140930. q^{24} -11165.5 q^{25} -293169. q^{26} -19683.0 q^{27} +149379. q^{28} -64355.4 q^{29} +154885. q^{30} +241791. q^{31} -865756. q^{32} -11822.7 q^{33} -552675. q^{34} +106353. q^{35} +264956. q^{36} +353659. q^{37} -842535. q^{38} -357060. q^{39} -1.35066e6 q^{40} +246146. q^{41} +246008. q^{42} -151362. q^{43} +159147. q^{44} +188640. q^{45} -1.76708e6 q^{46} -512816. q^{47} -1.86815e6 q^{48} -654620. q^{49} +247524. q^{50} -673122. q^{51} +4.80644e6 q^{52} -141087. q^{53} +436346. q^{54} +113308. q^{55} -2.14528e6 q^{56} -1.02615e6 q^{57} +1.42668e6 q^{58} -205379. q^{59} -2.53931e6 q^{60} +846302. q^{61} -5.36020e6 q^{62} +299621. q^{63} +1.03362e7 q^{64} +3.42203e6 q^{65} +262095. q^{66} +3.36339e6 q^{67} +9.06099e6 q^{68} -2.15219e6 q^{69} -2.35771e6 q^{70} -4.07342e6 q^{71} -3.80511e6 q^{72} -4.42831e6 q^{73} -7.84015e6 q^{74} +301467. q^{75} +1.38132e7 q^{76} +179969. q^{77} +7.91556e6 q^{78} -4.19495e6 q^{79} +1.79042e7 q^{80} +531441. q^{81} -5.45672e6 q^{82} -6.96295e6 q^{83} -4.03324e6 q^{84} +6.45114e6 q^{85} +3.35550e6 q^{86} +1.73760e6 q^{87} -2.28557e6 q^{88} -2.48793e6 q^{89} -4.18190e6 q^{90} +5.43529e6 q^{91} +2.89710e7 q^{92} -6.52837e6 q^{93} +1.13685e7 q^{94} +9.83454e6 q^{95} +2.33754e7 q^{96} +8.97300e6 q^{97} +1.45121e7 q^{98} +319214. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + 6521q^{10} - 1764q^{11} - 31482q^{12} + 18192q^{13} - 7827q^{14} + 8586q^{15} + 139226q^{16} - 15507q^{17} + 1458q^{18} + 52083q^{19} + 721q^{20} - 84915q^{21} - 234434q^{22} + 63823q^{23} - 63585q^{24} + 202153q^{25} - 367956q^{26} - 334611q^{27} + 182306q^{28} - 502955q^{29} - 176067q^{30} + 347531q^{31} - 243908q^{32} + 47628q^{33} - 330872q^{34} + 92641q^{35} + 850014q^{36} + 447615q^{37} + 775669q^{38} - 491184q^{39} + 2203270q^{40} + 940335q^{41} + 211329q^{42} + 478562q^{43} - 596924q^{44} - 231822q^{45} - 3078663q^{46} + 703121q^{47} - 3759102q^{48} + 1895082q^{49} - 876967q^{50} + 418689q^{51} + 6278296q^{52} - 1005974q^{53} - 39366q^{54} + 5212846q^{55} + 3425294q^{56} - 1406241q^{57} + 6710166q^{58} - 3491443q^{59} - 19467q^{60} + 11510749q^{61} + 5996234q^{62} + 2292705q^{63} + 29496941q^{64} + 11094180q^{65} + 6329718q^{66} + 14007144q^{67} + 19688159q^{68} - 1723221q^{69} + 30909708q^{70} + 5229074q^{71} + 1716795q^{72} + 5452211q^{73} + 12819662q^{74} - 5458131q^{75} + 41929340q^{76} + 9930777q^{77} + 9934812q^{78} + 15275654q^{79} + 36576105q^{80} + 9034497q^{81} + 32025935q^{82} + 7826609q^{83} - 4922262q^{84} + 11836945q^{85} + 51649136q^{86} + 13579785q^{87} + 30223741q^{88} - 6436185q^{89} + 4753809q^{90} + 11633535q^{91} + 43357972q^{92} - 9383337q^{93} - 4494252q^{94} + 23741055q^{95} + 6585516q^{96} + 26377540q^{97} + 26517816q^{98} - 1285956q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −22.1687 −1.95945 −0.979727 0.200338i \(-0.935796\pi\)
−0.979727 + 0.200338i \(0.935796\pi\)
\(3\) −27.0000 −0.577350
\(4\) 363.451 2.83946
\(5\) 258.765 0.925787 0.462894 0.886414i \(-0.346811\pi\)
0.462894 + 0.886414i \(0.346811\pi\)
\(6\) 598.555 1.13129
\(7\) 411.003 0.452899 0.226450 0.974023i \(-0.427288\pi\)
0.226450 + 0.974023i \(0.427288\pi\)
\(8\) −5219.64 −3.60434
\(9\) 729.000 0.333333
\(10\) −5736.49 −1.81404
\(11\) 437.879 0.0991927 0.0495964 0.998769i \(-0.484207\pi\)
0.0495964 + 0.998769i \(0.484207\pi\)
\(12\) −9813.17 −1.63936
\(13\) 13224.5 1.66946 0.834730 0.550659i \(-0.185624\pi\)
0.834730 + 0.550659i \(0.185624\pi\)
\(14\) −9111.39 −0.887435
\(15\) −6986.67 −0.534504
\(16\) 69190.8 4.22307
\(17\) 24930.4 1.23072 0.615359 0.788247i \(-0.289011\pi\)
0.615359 + 0.788247i \(0.289011\pi\)
\(18\) −16161.0 −0.653151
\(19\) 38005.6 1.27119 0.635595 0.772023i \(-0.280755\pi\)
0.635595 + 0.772023i \(0.280755\pi\)
\(20\) 94048.5 2.62874
\(21\) −11097.1 −0.261481
\(22\) −9707.20 −0.194364
\(23\) 79710.8 1.36606 0.683030 0.730390i \(-0.260662\pi\)
0.683030 + 0.730390i \(0.260662\pi\)
\(24\) 140930. 2.08096
\(25\) −11165.5 −0.142918
\(26\) −293169. −3.27123
\(27\) −19683.0 −0.192450
\(28\) 149379. 1.28599
\(29\) −64355.4 −0.489996 −0.244998 0.969524i \(-0.578787\pi\)
−0.244998 + 0.969524i \(0.578787\pi\)
\(30\) 154885. 1.04734
\(31\) 241791. 1.45772 0.728862 0.684661i \(-0.240050\pi\)
0.728862 + 0.684661i \(0.240050\pi\)
\(32\) −865756. −4.67058
\(33\) −11822.7 −0.0572689
\(34\) −552675. −2.41154
\(35\) 106353. 0.419288
\(36\) 264956. 0.946486
\(37\) 353659. 1.14783 0.573916 0.818914i \(-0.305424\pi\)
0.573916 + 0.818914i \(0.305424\pi\)
\(38\) −842535. −2.49084
\(39\) −357060. −0.963863
\(40\) −1.35066e6 −3.33685
\(41\) 246146. 0.557761 0.278881 0.960326i \(-0.410037\pi\)
0.278881 + 0.960326i \(0.410037\pi\)
\(42\) 246008. 0.512361
\(43\) −151362. −0.290321 −0.145160 0.989408i \(-0.546370\pi\)
−0.145160 + 0.989408i \(0.546370\pi\)
\(44\) 159147. 0.281654
\(45\) 188640. 0.308596
\(46\) −1.76708e6 −2.67673
\(47\) −512816. −0.720475 −0.360238 0.932861i \(-0.617304\pi\)
−0.360238 + 0.932861i \(0.617304\pi\)
\(48\) −1.86815e6 −2.43819
\(49\) −654620. −0.794882
\(50\) 247524. 0.280041
\(51\) −673122. −0.710556
\(52\) 4.80644e6 4.74036
\(53\) −141087. −0.130173 −0.0650867 0.997880i \(-0.520732\pi\)
−0.0650867 + 0.997880i \(0.520732\pi\)
\(54\) 436346. 0.377097
\(55\) 113308. 0.0918314
\(56\) −2.14528e6 −1.63240
\(57\) −1.02615e6 −0.733922
\(58\) 1.42668e6 0.960124
\(59\) −205379. −0.130189
\(60\) −2.53931e6 −1.51770
\(61\) 846302. 0.477388 0.238694 0.971095i \(-0.423281\pi\)
0.238694 + 0.971095i \(0.423281\pi\)
\(62\) −5.36020e6 −2.85634
\(63\) 299621. 0.150966
\(64\) 1.03362e7 4.92871
\(65\) 3.42203e6 1.54556
\(66\) 262095. 0.112216
\(67\) 3.36339e6 1.36620 0.683101 0.730324i \(-0.260631\pi\)
0.683101 + 0.730324i \(0.260631\pi\)
\(68\) 9.06099e6 3.49458
\(69\) −2.15219e6 −0.788695
\(70\) −2.35771e6 −0.821576
\(71\) −4.07342e6 −1.35069 −0.675344 0.737503i \(-0.736005\pi\)
−0.675344 + 0.737503i \(0.736005\pi\)
\(72\) −3.80511e6 −1.20145
\(73\) −4.42831e6 −1.33232 −0.666159 0.745810i \(-0.732063\pi\)
−0.666159 + 0.745810i \(0.732063\pi\)
\(74\) −7.84015e6 −2.24912
\(75\) 301467. 0.0825137
\(76\) 1.38132e7 3.60949
\(77\) 179969. 0.0449243
\(78\) 7.91556e6 1.88865
\(79\) −4.19495e6 −0.957265 −0.478633 0.878015i \(-0.658867\pi\)
−0.478633 + 0.878015i \(0.658867\pi\)
\(80\) 1.79042e7 3.90967
\(81\) 531441. 0.111111
\(82\) −5.45672e6 −1.09291
\(83\) −6.96295e6 −1.33666 −0.668329 0.743866i \(-0.732990\pi\)
−0.668329 + 0.743866i \(0.732990\pi\)
\(84\) −4.03324e6 −0.742466
\(85\) 6.45114e6 1.13938
\(86\) 3.35550e6 0.568870
\(87\) 1.73760e6 0.282899
\(88\) −2.28557e6 −0.357524
\(89\) −2.48793e6 −0.374088 −0.187044 0.982352i \(-0.559891\pi\)
−0.187044 + 0.982352i \(0.559891\pi\)
\(90\) −4.18190e6 −0.604679
\(91\) 5.43529e6 0.756097
\(92\) 2.89710e7 3.87887
\(93\) −6.52837e6 −0.841617
\(94\) 1.13685e7 1.41174
\(95\) 9.83454e6 1.17685
\(96\) 2.33754e7 2.69656
\(97\) 8.97300e6 0.998244 0.499122 0.866532i \(-0.333656\pi\)
0.499122 + 0.866532i \(0.333656\pi\)
\(98\) 1.45121e7 1.55754
\(99\) 319214. 0.0330642
\(100\) −4.05810e6 −0.405810
\(101\) −1.29861e6 −0.125417 −0.0627084 0.998032i \(-0.519974\pi\)
−0.0627084 + 0.998032i \(0.519974\pi\)
\(102\) 1.49222e7 1.39230
\(103\) 1.51079e7 1.36230 0.681150 0.732144i \(-0.261480\pi\)
0.681150 + 0.732144i \(0.261480\pi\)
\(104\) −6.90268e7 −6.01729
\(105\) −2.87154e6 −0.242076
\(106\) 3.12772e6 0.255069
\(107\) 6.86606e6 0.541832 0.270916 0.962603i \(-0.412673\pi\)
0.270916 + 0.962603i \(0.412673\pi\)
\(108\) −7.15380e6 −0.546454
\(109\) −1.26929e7 −0.938790 −0.469395 0.882988i \(-0.655528\pi\)
−0.469395 + 0.882988i \(0.655528\pi\)
\(110\) −2.51189e6 −0.179939
\(111\) −9.54879e6 −0.662701
\(112\) 2.84376e7 1.91262
\(113\) −1.90547e7 −1.24230 −0.621151 0.783691i \(-0.713335\pi\)
−0.621151 + 0.783691i \(0.713335\pi\)
\(114\) 2.27485e7 1.43809
\(115\) 2.06264e7 1.26468
\(116\) −2.33900e7 −1.39132
\(117\) 9.64063e6 0.556487
\(118\) 4.55298e6 0.255099
\(119\) 1.02465e7 0.557391
\(120\) 3.64679e7 1.92653
\(121\) −1.92954e7 −0.990161
\(122\) −1.87614e7 −0.935419
\(123\) −6.64593e6 −0.322024
\(124\) 8.78793e7 4.13915
\(125\) −2.31053e7 −1.05810
\(126\) −6.64220e6 −0.295812
\(127\) 3.81929e7 1.65451 0.827254 0.561828i \(-0.189902\pi\)
0.827254 + 0.561828i \(0.189902\pi\)
\(128\) −1.18324e8 −4.98700
\(129\) 4.08678e6 0.167617
\(130\) −7.58620e7 −3.02846
\(131\) 4.40577e7 1.71227 0.856135 0.516752i \(-0.172859\pi\)
0.856135 + 0.516752i \(0.172859\pi\)
\(132\) −4.29698e6 −0.162613
\(133\) 1.56204e7 0.575721
\(134\) −7.45619e7 −2.67701
\(135\) −5.09328e6 −0.178168
\(136\) −1.30128e8 −4.43592
\(137\) −2.64599e7 −0.879156 −0.439578 0.898204i \(-0.644872\pi\)
−0.439578 + 0.898204i \(0.644872\pi\)
\(138\) 4.77113e7 1.54541
\(139\) −2.17850e7 −0.688029 −0.344014 0.938964i \(-0.611787\pi\)
−0.344014 + 0.938964i \(0.611787\pi\)
\(140\) 3.86542e7 1.19055
\(141\) 1.38460e7 0.415967
\(142\) 9.03024e7 2.64661
\(143\) 5.79071e6 0.165598
\(144\) 5.04401e7 1.40769
\(145\) −1.66530e7 −0.453632
\(146\) 9.81698e7 2.61061
\(147\) 1.76747e7 0.458926
\(148\) 1.28538e8 3.25922
\(149\) 5.57470e6 0.138061 0.0690303 0.997615i \(-0.478009\pi\)
0.0690303 + 0.997615i \(0.478009\pi\)
\(150\) −6.68314e6 −0.161682
\(151\) −3.98105e7 −0.940975 −0.470487 0.882407i \(-0.655922\pi\)
−0.470487 + 0.882407i \(0.655922\pi\)
\(152\) −1.98376e8 −4.58180
\(153\) 1.81743e7 0.410240
\(154\) −3.98969e6 −0.0880271
\(155\) 6.25673e7 1.34954
\(156\) −1.29774e8 −2.73685
\(157\) −2.43241e7 −0.501635 −0.250818 0.968034i \(-0.580699\pi\)
−0.250818 + 0.968034i \(0.580699\pi\)
\(158\) 9.29966e7 1.87572
\(159\) 3.80936e6 0.0751556
\(160\) −2.24028e8 −4.32396
\(161\) 3.27614e7 0.618687
\(162\) −1.17814e7 −0.217717
\(163\) 1.62611e7 0.294098 0.147049 0.989129i \(-0.453022\pi\)
0.147049 + 0.989129i \(0.453022\pi\)
\(164\) 8.94618e7 1.58374
\(165\) −3.05931e6 −0.0530189
\(166\) 1.54360e8 2.61912
\(167\) 2.00507e7 0.333136 0.166568 0.986030i \(-0.446732\pi\)
0.166568 + 0.986030i \(0.446732\pi\)
\(168\) 5.79227e7 0.942467
\(169\) 1.12138e8 1.78710
\(170\) −1.43013e8 −2.23257
\(171\) 2.77061e7 0.423730
\(172\) −5.50128e7 −0.824354
\(173\) 1.22817e8 1.80342 0.901712 0.432338i \(-0.142311\pi\)
0.901712 + 0.432338i \(0.142311\pi\)
\(174\) −3.85202e7 −0.554328
\(175\) −4.58903e6 −0.0647274
\(176\) 3.02972e7 0.418898
\(177\) 5.54523e6 0.0751646
\(178\) 5.51542e7 0.733008
\(179\) −4.55080e7 −0.593065 −0.296532 0.955023i \(-0.595830\pi\)
−0.296532 + 0.955023i \(0.595830\pi\)
\(180\) 6.85614e7 0.876245
\(181\) −1.43158e8 −1.79448 −0.897241 0.441541i \(-0.854432\pi\)
−0.897241 + 0.441541i \(0.854432\pi\)
\(182\) −1.20493e8 −1.48154
\(183\) −2.28502e7 −0.275620
\(184\) −4.16061e8 −4.92374
\(185\) 9.15147e7 1.06265
\(186\) 1.44725e8 1.64911
\(187\) 1.09165e7 0.122078
\(188\) −1.86383e8 −2.04576
\(189\) −8.08976e6 −0.0871605
\(190\) −2.18019e8 −2.30599
\(191\) 5.12128e7 0.531817 0.265908 0.963998i \(-0.414328\pi\)
0.265908 + 0.963998i \(0.414328\pi\)
\(192\) −2.79079e8 −2.84559
\(193\) 9.92991e7 0.994248 0.497124 0.867680i \(-0.334389\pi\)
0.497124 + 0.867680i \(0.334389\pi\)
\(194\) −1.98920e8 −1.95601
\(195\) −9.23948e7 −0.892332
\(196\) −2.37922e8 −2.25704
\(197\) −1.34248e8 −1.25105 −0.625527 0.780202i \(-0.715116\pi\)
−0.625527 + 0.780202i \(0.715116\pi\)
\(198\) −7.07655e6 −0.0647878
\(199\) −6.47369e7 −0.582326 −0.291163 0.956674i \(-0.594042\pi\)
−0.291163 + 0.956674i \(0.594042\pi\)
\(200\) 5.82796e7 0.515124
\(201\) −9.08115e7 −0.788777
\(202\) 2.87886e7 0.245748
\(203\) −2.64502e7 −0.221919
\(204\) −2.44647e8 −2.01759
\(205\) 6.36940e7 0.516368
\(206\) −3.34922e8 −2.66936
\(207\) 5.81092e7 0.455353
\(208\) 9.15010e8 7.05025
\(209\) 1.66419e7 0.126093
\(210\) 6.36582e7 0.474337
\(211\) −3.50321e7 −0.256731 −0.128365 0.991727i \(-0.540973\pi\)
−0.128365 + 0.991727i \(0.540973\pi\)
\(212\) −5.12783e7 −0.369622
\(213\) 1.09982e8 0.779820
\(214\) −1.52212e8 −1.06169
\(215\) −3.91673e7 −0.268775
\(216\) 1.02738e8 0.693655
\(217\) 9.93769e7 0.660201
\(218\) 2.81385e8 1.83952
\(219\) 1.19564e8 0.769214
\(220\) 4.11819e7 0.260751
\(221\) 3.29692e8 2.05464
\(222\) 2.11684e8 1.29853
\(223\) 2.00970e8 1.21357 0.606783 0.794867i \(-0.292460\pi\)
0.606783 + 0.794867i \(0.292460\pi\)
\(224\) −3.55828e8 −2.11530
\(225\) −8.13962e6 −0.0476393
\(226\) 4.22417e8 2.43423
\(227\) 2.01923e8 1.14577 0.572883 0.819637i \(-0.305825\pi\)
0.572883 + 0.819637i \(0.305825\pi\)
\(228\) −3.72956e8 −2.08394
\(229\) −8.98434e7 −0.494381 −0.247191 0.968967i \(-0.579507\pi\)
−0.247191 + 0.968967i \(0.579507\pi\)
\(230\) −4.57260e8 −2.47808
\(231\) −4.85917e6 −0.0259370
\(232\) 3.35912e8 1.76611
\(233\) 2.22230e8 1.15095 0.575476 0.817819i \(-0.304817\pi\)
0.575476 + 0.817819i \(0.304817\pi\)
\(234\) −2.13720e8 −1.09041
\(235\) −1.32699e8 −0.667007
\(236\) −7.46452e7 −0.369666
\(237\) 1.13264e8 0.552677
\(238\) −2.27151e8 −1.09218
\(239\) 1.33230e8 0.631262 0.315631 0.948882i \(-0.397784\pi\)
0.315631 + 0.948882i \(0.397784\pi\)
\(240\) −4.83413e8 −2.25725
\(241\) 5.22856e7 0.240615 0.120307 0.992737i \(-0.461612\pi\)
0.120307 + 0.992737i \(0.461612\pi\)
\(242\) 4.27754e8 1.94017
\(243\) −1.43489e7 −0.0641500
\(244\) 3.07589e8 1.35552
\(245\) −1.69393e8 −0.735892
\(246\) 1.47332e8 0.630991
\(247\) 5.02604e8 2.12220
\(248\) −1.26206e9 −5.25412
\(249\) 1.88000e8 0.771720
\(250\) 5.12214e8 2.07330
\(251\) −2.01583e8 −0.804629 −0.402314 0.915502i \(-0.631794\pi\)
−0.402314 + 0.915502i \(0.631794\pi\)
\(252\) 1.08897e8 0.428663
\(253\) 3.49037e7 0.135503
\(254\) −8.46686e8 −3.24193
\(255\) −1.74181e8 −0.657823
\(256\) 1.30006e9 4.84309
\(257\) 2.23672e7 0.0821949 0.0410975 0.999155i \(-0.486915\pi\)
0.0410975 + 0.999155i \(0.486915\pi\)
\(258\) −9.05986e7 −0.328437
\(259\) 1.45355e8 0.519852
\(260\) 1.24374e9 4.38857
\(261\) −4.69151e7 −0.163332
\(262\) −9.76702e8 −3.35512
\(263\) 2.02431e8 0.686170 0.343085 0.939304i \(-0.388528\pi\)
0.343085 + 0.939304i \(0.388528\pi\)
\(264\) 6.17104e7 0.206416
\(265\) −3.65085e7 −0.120513
\(266\) −3.46284e8 −1.12810
\(267\) 6.71742e7 0.215980
\(268\) 1.22243e9 3.87928
\(269\) 6.24827e8 1.95716 0.978581 0.205862i \(-0.0659997\pi\)
0.978581 + 0.205862i \(0.0659997\pi\)
\(270\) 1.12911e8 0.349112
\(271\) −3.54191e8 −1.08105 −0.540525 0.841328i \(-0.681774\pi\)
−0.540525 + 0.841328i \(0.681774\pi\)
\(272\) 1.72496e9 5.19741
\(273\) −1.46753e8 −0.436533
\(274\) 5.86581e8 1.72267
\(275\) −4.88912e6 −0.0141764
\(276\) −7.82216e8 −2.23947
\(277\) −6.00748e8 −1.69829 −0.849147 0.528156i \(-0.822884\pi\)
−0.849147 + 0.528156i \(0.822884\pi\)
\(278\) 4.82946e8 1.34816
\(279\) 1.76266e8 0.485908
\(280\) −5.55125e8 −1.51126
\(281\) −5.54600e8 −1.49110 −0.745551 0.666448i \(-0.767814\pi\)
−0.745551 + 0.666448i \(0.767814\pi\)
\(282\) −3.06948e8 −0.815067
\(283\) 1.85734e8 0.487124 0.243562 0.969885i \(-0.421684\pi\)
0.243562 + 0.969885i \(0.421684\pi\)
\(284\) −1.48049e9 −3.83522
\(285\) −2.65533e8 −0.679456
\(286\) −1.28372e8 −0.324482
\(287\) 1.01166e8 0.252610
\(288\) −6.31136e8 −1.55686
\(289\) 2.11188e8 0.514668
\(290\) 3.69174e8 0.888870
\(291\) −2.42271e8 −0.576336
\(292\) −1.60947e9 −3.78306
\(293\) 8.06147e8 1.87231 0.936154 0.351590i \(-0.114359\pi\)
0.936154 + 0.351590i \(0.114359\pi\)
\(294\) −3.91826e8 −0.899244
\(295\) −5.31450e7 −0.120527
\(296\) −1.84597e9 −4.13717
\(297\) −8.61877e6 −0.0190896
\(298\) −1.23584e8 −0.270523
\(299\) 1.05413e9 2.28058
\(300\) 1.09569e8 0.234294
\(301\) −6.22103e7 −0.131486
\(302\) 8.82546e8 1.84380
\(303\) 3.50626e7 0.0724094
\(304\) 2.62964e9 5.36832
\(305\) 2.18994e8 0.441959
\(306\) −4.02900e8 −0.803845
\(307\) −5.41975e8 −1.06904 −0.534522 0.845155i \(-0.679508\pi\)
−0.534522 + 0.845155i \(0.679508\pi\)
\(308\) 6.54100e7 0.127561
\(309\) −4.07912e8 −0.786525
\(310\) −1.38703e9 −2.64436
\(311\) −2.23407e8 −0.421149 −0.210575 0.977578i \(-0.567534\pi\)
−0.210575 + 0.977578i \(0.567534\pi\)
\(312\) 1.86372e9 3.47409
\(313\) −6.53332e8 −1.20428 −0.602142 0.798389i \(-0.705686\pi\)
−0.602142 + 0.798389i \(0.705686\pi\)
\(314\) 5.39234e8 0.982932
\(315\) 7.75315e7 0.139763
\(316\) −1.52466e9 −2.71812
\(317\) 3.82141e8 0.673777 0.336889 0.941544i \(-0.390625\pi\)
0.336889 + 0.941544i \(0.390625\pi\)
\(318\) −8.44484e7 −0.147264
\(319\) −2.81799e7 −0.0486040
\(320\) 2.67466e9 4.56293
\(321\) −1.85384e8 −0.312827
\(322\) −7.26276e8 −1.21229
\(323\) 9.47497e8 1.56448
\(324\) 1.93153e8 0.315495
\(325\) −1.47657e8 −0.238596
\(326\) −3.60486e8 −0.576272
\(327\) 3.42709e8 0.542011
\(328\) −1.28479e9 −2.01036
\(329\) −2.10769e8 −0.326303
\(330\) 6.78210e7 0.103888
\(331\) −1.19439e8 −0.181030 −0.0905148 0.995895i \(-0.528851\pi\)
−0.0905148 + 0.995895i \(0.528851\pi\)
\(332\) −2.53069e9 −3.79539
\(333\) 2.57817e8 0.382611
\(334\) −4.44497e8 −0.652764
\(335\) 8.70328e8 1.26481
\(336\) −7.67815e8 −1.10425
\(337\) 6.48790e8 0.923421 0.461710 0.887031i \(-0.347236\pi\)
0.461710 + 0.887031i \(0.347236\pi\)
\(338\) −2.48595e9 −3.50173
\(339\) 5.14477e8 0.717244
\(340\) 2.34467e9 3.23523
\(341\) 1.05875e8 0.144596
\(342\) −6.14208e8 −0.830279
\(343\) −6.07529e8 −0.812901
\(344\) 7.90056e8 1.04641
\(345\) −5.56913e8 −0.730164
\(346\) −2.72270e9 −3.53372
\(347\) −1.32744e9 −1.70553 −0.852766 0.522292i \(-0.825077\pi\)
−0.852766 + 0.522292i \(0.825077\pi\)
\(348\) 6.31531e8 0.803280
\(349\) −2.41076e7 −0.0303575 −0.0151787 0.999885i \(-0.504832\pi\)
−0.0151787 + 0.999885i \(0.504832\pi\)
\(350\) 1.01733e8 0.126830
\(351\) −2.60297e8 −0.321288
\(352\) −3.79096e8 −0.463287
\(353\) −1.15393e9 −1.39626 −0.698132 0.715969i \(-0.745985\pi\)
−0.698132 + 0.715969i \(0.745985\pi\)
\(354\) −1.22931e8 −0.147282
\(355\) −1.05406e9 −1.25045
\(356\) −9.04241e8 −1.06221
\(357\) −2.76655e8 −0.321810
\(358\) 1.00885e9 1.16208
\(359\) −1.33636e8 −0.152438 −0.0762190 0.997091i \(-0.524285\pi\)
−0.0762190 + 0.997091i \(0.524285\pi\)
\(360\) −9.84632e8 −1.11228
\(361\) 5.50557e8 0.615924
\(362\) 3.17361e9 3.51621
\(363\) 5.20977e8 0.571670
\(364\) 1.97546e9 2.14691
\(365\) −1.14589e9 −1.23344
\(366\) 5.06558e8 0.540064
\(367\) 6.96923e8 0.735959 0.367980 0.929834i \(-0.380050\pi\)
0.367980 + 0.929834i \(0.380050\pi\)
\(368\) 5.51525e9 5.76897
\(369\) 1.79440e8 0.185920
\(370\) −2.02876e9 −2.08221
\(371\) −5.79872e7 −0.0589554
\(372\) −2.37274e9 −2.38974
\(373\) −2.37598e7 −0.0237062 −0.0118531 0.999930i \(-0.503773\pi\)
−0.0118531 + 0.999930i \(0.503773\pi\)
\(374\) −2.42005e8 −0.239207
\(375\) 6.23843e8 0.610894
\(376\) 2.67671e9 2.59684
\(377\) −8.51065e8 −0.818028
\(378\) 1.79339e8 0.170787
\(379\) 6.51339e8 0.614568 0.307284 0.951618i \(-0.400580\pi\)
0.307284 + 0.951618i \(0.400580\pi\)
\(380\) 3.57437e9 3.34162
\(381\) −1.03121e9 −0.955231
\(382\) −1.13532e9 −1.04207
\(383\) 7.66231e8 0.696889 0.348445 0.937329i \(-0.386710\pi\)
0.348445 + 0.937329i \(0.386710\pi\)
\(384\) 3.19476e9 2.87925
\(385\) 4.65699e7 0.0415903
\(386\) −2.20133e9 −1.94818
\(387\) −1.10343e8 −0.0967736
\(388\) 3.26124e9 2.83447
\(389\) −9.02040e8 −0.776966 −0.388483 0.921456i \(-0.627001\pi\)
−0.388483 + 0.921456i \(0.627001\pi\)
\(390\) 2.04827e9 1.74848
\(391\) 1.98723e9 1.68124
\(392\) 3.41688e9 2.86502
\(393\) −1.18956e9 −0.988580
\(394\) 2.97610e9 2.45138
\(395\) −1.08551e9 −0.886224
\(396\) 1.16019e8 0.0938846
\(397\) −4.92346e8 −0.394915 −0.197458 0.980311i \(-0.563268\pi\)
−0.197458 + 0.980311i \(0.563268\pi\)
\(398\) 1.43513e9 1.14104
\(399\) −4.21751e8 −0.332393
\(400\) −7.72547e8 −0.603552
\(401\) −8.45019e8 −0.654427 −0.327213 0.944950i \(-0.606110\pi\)
−0.327213 + 0.944950i \(0.606110\pi\)
\(402\) 2.01317e9 1.54557
\(403\) 3.19756e9 2.43361
\(404\) −4.71983e8 −0.356116
\(405\) 1.37519e8 0.102865
\(406\) 5.86367e8 0.434839
\(407\) 1.54860e8 0.113857
\(408\) 3.51345e9 2.56108
\(409\) −1.44863e9 −1.04695 −0.523474 0.852042i \(-0.675364\pi\)
−0.523474 + 0.852042i \(0.675364\pi\)
\(410\) −1.41201e9 −1.01180
\(411\) 7.14417e8 0.507581
\(412\) 5.49097e9 3.86820
\(413\) −8.44113e7 −0.0589624
\(414\) −1.28820e9 −0.892244
\(415\) −1.80177e9 −1.23746
\(416\) −1.14491e10 −7.79734
\(417\) 5.88196e8 0.397233
\(418\) −3.68929e8 −0.247073
\(419\) −1.06290e9 −0.705902 −0.352951 0.935642i \(-0.614822\pi\)
−0.352951 + 0.935642i \(0.614822\pi\)
\(420\) −1.04366e9 −0.687365
\(421\) −1.46411e9 −0.956282 −0.478141 0.878283i \(-0.658689\pi\)
−0.478141 + 0.878283i \(0.658689\pi\)
\(422\) 7.76617e8 0.503052
\(423\) −3.73843e8 −0.240158
\(424\) 7.36424e8 0.469189
\(425\) −2.78360e8 −0.175892
\(426\) −2.43817e9 −1.52802
\(427\) 3.47832e8 0.216208
\(428\) 2.49548e9 1.53851
\(429\) −1.56349e8 −0.0956082
\(430\) 8.68289e8 0.526653
\(431\) 9.71394e8 0.584420 0.292210 0.956354i \(-0.405609\pi\)
0.292210 + 0.956354i \(0.405609\pi\)
\(432\) −1.36188e9 −0.812730
\(433\) −1.99519e9 −1.18107 −0.590535 0.807012i \(-0.701083\pi\)
−0.590535 + 0.807012i \(0.701083\pi\)
\(434\) −2.20306e9 −1.29363
\(435\) 4.49630e8 0.261904
\(436\) −4.61325e9 −2.66566
\(437\) 3.02946e9 1.73652
\(438\) −2.65058e9 −1.50724
\(439\) −3.32793e9 −1.87737 −0.938683 0.344782i \(-0.887953\pi\)
−0.938683 + 0.344782i \(0.887953\pi\)
\(440\) −5.91426e8 −0.330991
\(441\) −4.77218e8 −0.264961
\(442\) −7.30883e9 −4.02596
\(443\) 6.61628e8 0.361577 0.180789 0.983522i \(-0.442135\pi\)
0.180789 + 0.983522i \(0.442135\pi\)
\(444\) −3.47052e9 −1.88171
\(445\) −6.43791e8 −0.346326
\(446\) −4.45523e9 −2.37793
\(447\) −1.50517e8 −0.0797094
\(448\) 4.24823e9 2.23221
\(449\) 2.39627e9 1.24932 0.624661 0.780896i \(-0.285237\pi\)
0.624661 + 0.780896i \(0.285237\pi\)
\(450\) 1.80445e8 0.0933470
\(451\) 1.07782e8 0.0553259
\(452\) −6.92544e9 −3.52747
\(453\) 1.07488e9 0.543272
\(454\) −4.47637e9 −2.24507
\(455\) 1.40646e9 0.699985
\(456\) 5.35614e9 2.64530
\(457\) 5.13759e8 0.251798 0.125899 0.992043i \(-0.459818\pi\)
0.125899 + 0.992043i \(0.459818\pi\)
\(458\) 1.99171e9 0.968717
\(459\) −4.90706e8 −0.236852
\(460\) 7.49668e9 3.59101
\(461\) 1.60015e9 0.760687 0.380344 0.924845i \(-0.375806\pi\)
0.380344 + 0.924845i \(0.375806\pi\)
\(462\) 1.07722e8 0.0508225
\(463\) 7.65148e8 0.358271 0.179136 0.983824i \(-0.442670\pi\)
0.179136 + 0.983824i \(0.442670\pi\)
\(464\) −4.45280e9 −2.06929
\(465\) −1.68932e9 −0.779158
\(466\) −4.92655e9 −2.25524
\(467\) 2.99058e9 1.35877 0.679386 0.733782i \(-0.262246\pi\)
0.679386 + 0.733782i \(0.262246\pi\)
\(468\) 3.50389e9 1.58012
\(469\) 1.38236e9 0.618752
\(470\) 2.94176e9 1.30697
\(471\) 6.56751e8 0.289619
\(472\) 1.07200e9 0.469245
\(473\) −6.62784e7 −0.0287977
\(474\) −2.51091e9 −1.08295
\(475\) −4.24350e8 −0.181676
\(476\) 3.72409e9 1.58269
\(477\) −1.02853e8 −0.0433911
\(478\) −2.95354e9 −1.23693
\(479\) −1.96100e9 −0.815275 −0.407637 0.913144i \(-0.633647\pi\)
−0.407637 + 0.913144i \(0.633647\pi\)
\(480\) 6.04875e9 2.49644
\(481\) 4.67695e9 1.91626
\(482\) −1.15910e9 −0.471473
\(483\) −8.84557e8 −0.357199
\(484\) −7.01294e9 −2.81152
\(485\) 2.32190e9 0.924161
\(486\) 3.18096e8 0.125699
\(487\) −4.21150e9 −1.65229 −0.826143 0.563460i \(-0.809470\pi\)
−0.826143 + 0.563460i \(0.809470\pi\)
\(488\) −4.41739e9 −1.72066
\(489\) −4.39049e8 −0.169798
\(490\) 3.75522e9 1.44195
\(491\) 1.16149e9 0.442824 0.221412 0.975180i \(-0.428934\pi\)
0.221412 + 0.975180i \(0.428934\pi\)
\(492\) −2.41547e9 −0.914373
\(493\) −1.60441e9 −0.603047
\(494\) −1.11421e10 −4.15835
\(495\) 8.26015e7 0.0306105
\(496\) 1.67297e10 6.15607
\(497\) −1.67419e9 −0.611725
\(498\) −4.16771e9 −1.51215
\(499\) 2.79801e9 1.00808 0.504042 0.863679i \(-0.331846\pi\)
0.504042 + 0.863679i \(0.331846\pi\)
\(500\) −8.39763e9 −3.00443
\(501\) −5.41368e8 −0.192336
\(502\) 4.46883e9 1.57663
\(503\) −2.19333e9 −0.768452 −0.384226 0.923239i \(-0.625532\pi\)
−0.384226 + 0.923239i \(0.625532\pi\)
\(504\) −1.56391e9 −0.544133
\(505\) −3.36037e8 −0.116109
\(506\) −7.73769e8 −0.265512
\(507\) −3.02772e9 −1.03178
\(508\) 1.38812e10 4.69791
\(509\) −1.63668e9 −0.550112 −0.275056 0.961428i \(-0.588696\pi\)
−0.275056 + 0.961428i \(0.588696\pi\)
\(510\) 3.86136e9 1.28897
\(511\) −1.82005e9 −0.603405
\(512\) −1.36750e10 −4.50281
\(513\) −7.48065e8 −0.244641
\(514\) −4.95851e8 −0.161057
\(515\) 3.90939e9 1.26120
\(516\) 1.48534e9 0.475941
\(517\) −2.24551e8 −0.0714659
\(518\) −3.22232e9 −1.01863
\(519\) −3.31606e9 −1.04121
\(520\) −1.78618e10 −5.57074
\(521\) −8.21908e8 −0.254619 −0.127310 0.991863i \(-0.540634\pi\)
−0.127310 + 0.991863i \(0.540634\pi\)
\(522\) 1.04005e9 0.320041
\(523\) 8.10572e8 0.247763 0.123881 0.992297i \(-0.460466\pi\)
0.123881 + 0.992297i \(0.460466\pi\)
\(524\) 1.60128e10 4.86192
\(525\) 1.23904e8 0.0373704
\(526\) −4.48763e9 −1.34452
\(527\) 6.02797e9 1.79405
\(528\) −8.18024e8 −0.241851
\(529\) 2.94899e9 0.866120
\(530\) 8.09346e8 0.236139
\(531\) −1.49721e8 −0.0433963
\(532\) 5.67725e9 1.63474
\(533\) 3.25514e9 0.931160
\(534\) −1.48916e9 −0.423202
\(535\) 1.77670e9 0.501621
\(536\) −1.75557e10 −4.92425
\(537\) 1.22872e9 0.342406
\(538\) −1.38516e10 −3.83497
\(539\) −2.86644e8 −0.0788465
\(540\) −1.85116e9 −0.505900
\(541\) 5.84572e9 1.58726 0.793629 0.608402i \(-0.208189\pi\)
0.793629 + 0.608402i \(0.208189\pi\)
\(542\) 7.85196e9 2.11827
\(543\) 3.86525e9 1.03604
\(544\) −2.15837e10 −5.74816
\(545\) −3.28449e9 −0.869120
\(546\) 3.25332e9 0.855366
\(547\) −7.90478e8 −0.206507 −0.103253 0.994655i \(-0.532925\pi\)
−0.103253 + 0.994655i \(0.532925\pi\)
\(548\) −9.61686e9 −2.49633
\(549\) 6.16954e8 0.159129
\(550\) 1.08385e8 0.0277780
\(551\) −2.44587e9 −0.622877
\(552\) 1.12337e10 2.84272
\(553\) −1.72414e9 −0.433544
\(554\) 1.33178e10 3.32773
\(555\) −2.47090e9 −0.613521
\(556\) −7.91779e9 −1.95363
\(557\) 4.59101e9 1.12568 0.562840 0.826566i \(-0.309709\pi\)
0.562840 + 0.826566i \(0.309709\pi\)
\(558\) −3.90759e9 −0.952114
\(559\) −2.00168e9 −0.484679
\(560\) 7.35867e9 1.77068
\(561\) −2.94746e8 −0.0704820
\(562\) 1.22947e10 2.92175
\(563\) −1.87013e9 −0.441663 −0.220832 0.975312i \(-0.570877\pi\)
−0.220832 + 0.975312i \(0.570877\pi\)
\(564\) 5.03235e9 1.18112
\(565\) −4.93069e9 −1.15011
\(566\) −4.11749e9 −0.954497
\(567\) 2.18424e8 0.0503221
\(568\) 2.12618e10 4.86833
\(569\) −3.29999e8 −0.0750965 −0.0375482 0.999295i \(-0.511955\pi\)
−0.0375482 + 0.999295i \(0.511955\pi\)
\(570\) 5.88651e9 1.33136
\(571\) 8.31103e9 1.86822 0.934111 0.356982i \(-0.116194\pi\)
0.934111 + 0.356982i \(0.116194\pi\)
\(572\) 2.10464e9 0.470210
\(573\) −1.38275e9 −0.307045
\(574\) −2.24273e9 −0.494977
\(575\) −8.90008e8 −0.195234
\(576\) 7.53513e9 1.64290
\(577\) −3.65097e9 −0.791212 −0.395606 0.918420i \(-0.629465\pi\)
−0.395606 + 0.918420i \(0.629465\pi\)
\(578\) −4.68177e9 −1.00847
\(579\) −2.68108e9 −0.574029
\(580\) −6.05253e9 −1.28807
\(581\) −2.86179e9 −0.605371
\(582\) 5.37083e9 1.12930
\(583\) −6.17792e7 −0.0129123
\(584\) 2.31142e10 4.80212
\(585\) 2.49466e9 0.515188
\(586\) −1.78712e10 −3.66870
\(587\) 3.05418e9 0.623248 0.311624 0.950206i \(-0.399127\pi\)
0.311624 + 0.950206i \(0.399127\pi\)
\(588\) 6.42390e9 1.30310
\(589\) 9.18944e9 1.85304
\(590\) 1.17815e9 0.236168
\(591\) 3.62470e9 0.722297
\(592\) 2.44699e10 4.84738
\(593\) −4.80493e9 −0.946227 −0.473113 0.881002i \(-0.656870\pi\)
−0.473113 + 0.881002i \(0.656870\pi\)
\(594\) 1.91067e8 0.0374053
\(595\) 2.65143e9 0.516026
\(596\) 2.02613e9 0.392018
\(597\) 1.74790e9 0.336206
\(598\) −2.33687e10 −4.46870
\(599\) 3.86139e9 0.734090 0.367045 0.930203i \(-0.380369\pi\)
0.367045 + 0.930203i \(0.380369\pi\)
\(600\) −1.57355e9 −0.297407
\(601\) 1.96007e9 0.368308 0.184154 0.982897i \(-0.441045\pi\)
0.184154 + 0.982897i \(0.441045\pi\)
\(602\) 1.37912e9 0.257641
\(603\) 2.45191e9 0.455401
\(604\) −1.44691e10 −2.67186
\(605\) −4.99299e9 −0.916678
\(606\) −7.77292e8 −0.141883
\(607\) −4.97911e9 −0.903632 −0.451816 0.892111i \(-0.649224\pi\)
−0.451816 + 0.892111i \(0.649224\pi\)
\(608\) −3.29036e10 −5.93719
\(609\) 7.14157e8 0.128125
\(610\) −4.85480e9 −0.865999
\(611\) −6.78171e9 −1.20280
\(612\) 6.60546e9 1.16486
\(613\) 2.72401e9 0.477636 0.238818 0.971064i \(-0.423240\pi\)
0.238818 + 0.971064i \(0.423240\pi\)
\(614\) 1.20149e10 2.09474
\(615\) −1.71974e9 −0.298125
\(616\) −9.39375e8 −0.161922
\(617\) 7.00273e9 1.20024 0.600122 0.799909i \(-0.295119\pi\)
0.600122 + 0.799909i \(0.295119\pi\)
\(618\) 9.04288e9 1.54116
\(619\) 8.54432e9 1.44797 0.723986 0.689815i \(-0.242308\pi\)
0.723986 + 0.689815i \(0.242308\pi\)
\(620\) 2.27401e10 3.83197
\(621\) −1.56895e9 −0.262898
\(622\) 4.95265e9 0.825223
\(623\) −1.02255e9 −0.169424
\(624\) −2.47053e10 −4.07046
\(625\) −5.10655e9 −0.836657
\(626\) 1.44835e10 2.35974
\(627\) −4.49331e8 −0.0727997
\(628\) −8.84062e9 −1.42437
\(629\) 8.81687e9 1.41266
\(630\) −1.71877e9 −0.273859
\(631\) 9.15178e9 1.45012 0.725058 0.688688i \(-0.241813\pi\)
0.725058 + 0.688688i \(0.241813\pi\)
\(632\) 2.18961e10 3.45030
\(633\) 9.45868e8 0.148224
\(634\) −8.47157e9 −1.32024
\(635\) 9.88299e9 1.53172
\(636\) 1.38451e9 0.213401
\(637\) −8.65699e9 −1.32702
\(638\) 6.24711e8 0.0952373
\(639\) −2.96952e9 −0.450229
\(640\) −3.06183e10 −4.61690
\(641\) 7.04336e9 1.05627 0.528137 0.849159i \(-0.322891\pi\)
0.528137 + 0.849159i \(0.322891\pi\)
\(642\) 4.10971e9 0.612969
\(643\) −6.27887e9 −0.931415 −0.465708 0.884939i \(-0.654200\pi\)
−0.465708 + 0.884939i \(0.654200\pi\)
\(644\) 1.19071e10 1.75674
\(645\) 1.05752e9 0.155178
\(646\) −2.10048e10 −3.06552
\(647\) −4.23121e9 −0.614185 −0.307093 0.951680i \(-0.599356\pi\)
−0.307093 + 0.951680i \(0.599356\pi\)
\(648\) −2.77393e9 −0.400482
\(649\) −8.99312e7 −0.0129138
\(650\) 3.27337e9 0.467517
\(651\) −2.68318e9 −0.381167
\(652\) 5.91009e9 0.835080
\(653\) 3.22584e9 0.453364 0.226682 0.973969i \(-0.427212\pi\)
0.226682 + 0.973969i \(0.427212\pi\)
\(654\) −7.59740e9 −1.06205
\(655\) 1.14006e10 1.58520
\(656\) 1.70310e10 2.35547
\(657\) −3.22824e9 −0.444106
\(658\) 4.67247e9 0.639375
\(659\) 1.14067e10 1.55261 0.776303 0.630360i \(-0.217093\pi\)
0.776303 + 0.630360i \(0.217093\pi\)
\(660\) −1.11191e9 −0.150545
\(661\) −5.33531e9 −0.718546 −0.359273 0.933233i \(-0.616975\pi\)
−0.359273 + 0.933233i \(0.616975\pi\)
\(662\) 2.64782e9 0.354719
\(663\) −8.90167e9 −1.18624
\(664\) 3.63441e10 4.81776
\(665\) 4.04202e9 0.532995
\(666\) −5.71547e9 −0.749708
\(667\) −5.12982e9 −0.669363
\(668\) 7.28743e9 0.945926
\(669\) −5.42618e9 −0.700653
\(670\) −1.92940e10 −2.47834
\(671\) 3.70578e8 0.0473534
\(672\) 9.60735e9 1.22127
\(673\) −4.70152e9 −0.594547 −0.297273 0.954792i \(-0.596077\pi\)
−0.297273 + 0.954792i \(0.596077\pi\)
\(674\) −1.43828e10 −1.80940
\(675\) 2.19770e8 0.0275046
\(676\) 4.07565e10 5.07439
\(677\) −1.06147e10 −1.31477 −0.657384 0.753555i \(-0.728337\pi\)
−0.657384 + 0.753555i \(0.728337\pi\)
\(678\) −1.14053e10 −1.40541
\(679\) 3.68793e9 0.452104
\(680\) −3.36726e10 −4.10672
\(681\) −5.45192e9 −0.661508
\(682\) −2.34712e9 −0.283328
\(683\) 1.13835e10 1.36711 0.683556 0.729898i \(-0.260432\pi\)
0.683556 + 0.729898i \(0.260432\pi\)
\(684\) 1.00698e10 1.20316
\(685\) −6.84690e9 −0.813911
\(686\) 1.34681e10 1.59284
\(687\) 2.42577e9 0.285431
\(688\) −1.04729e10 −1.22605
\(689\) −1.86580e9 −0.217319
\(690\) 1.23460e10 1.43072
\(691\) −1.20923e10 −1.39423 −0.697117 0.716958i \(-0.745534\pi\)
−0.697117 + 0.716958i \(0.745534\pi\)
\(692\) 4.46380e10 5.12075
\(693\) 1.31198e8 0.0149748
\(694\) 2.94275e10 3.34191
\(695\) −5.63721e9 −0.636968
\(696\) −9.06962e9 −1.01966
\(697\) 6.13652e9 0.686447
\(698\) 5.34435e8 0.0594841
\(699\) −6.00021e9 −0.664502
\(700\) −1.66789e9 −0.183791
\(701\) −7.42546e9 −0.814161 −0.407080 0.913392i \(-0.633453\pi\)
−0.407080 + 0.913392i \(0.633453\pi\)
\(702\) 5.77044e9 0.629549
\(703\) 1.34410e10 1.45911
\(704\) 4.52603e9 0.488892
\(705\) 3.58287e9 0.385097
\(706\) 2.55811e10 2.73592
\(707\) −5.33734e8 −0.0568011
\(708\) 2.01542e9 0.213427
\(709\) −1.74012e10 −1.83366 −0.916828 0.399283i \(-0.869259\pi\)
−0.916828 + 0.399283i \(0.869259\pi\)
\(710\) 2.33671e10 2.45020
\(711\) −3.05812e9 −0.319088
\(712\) 1.29861e10 1.34834
\(713\) 1.92734e10 1.99134
\(714\) 6.13308e9 0.630572
\(715\) 1.49844e9 0.153309
\(716\) −1.65399e10 −1.68398
\(717\) −3.59721e9 −0.364459
\(718\) 2.96254e9 0.298695
\(719\) −1.32945e10 −1.33389 −0.666947 0.745105i \(-0.732399\pi\)
−0.666947 + 0.745105i \(0.732399\pi\)
\(720\) 1.30521e10 1.30322
\(721\) 6.20937e9 0.616985
\(722\) −1.22051e10 −1.20687
\(723\) −1.41171e9 −0.138919
\(724\) −5.20307e10 −5.09536
\(725\) 7.18558e8 0.0700291
\(726\) −1.15494e10 −1.12016
\(727\) −1.92881e9 −0.186174 −0.0930870 0.995658i \(-0.529673\pi\)
−0.0930870 + 0.995658i \(0.529673\pi\)
\(728\) −2.83702e10 −2.72523
\(729\) 3.87420e8 0.0370370
\(730\) 2.54029e10 2.41687
\(731\) −3.77353e9 −0.357303
\(732\) −8.30491e9 −0.782611
\(733\) −1.50113e10 −1.40785 −0.703924 0.710276i \(-0.748570\pi\)
−0.703924 + 0.710276i \(0.748570\pi\)
\(734\) −1.54499e10 −1.44208
\(735\) 4.57361e9 0.424867
\(736\) −6.90101e10 −6.38029
\(737\) 1.47276e9 0.135517
\(738\) −3.97795e9 −0.364303
\(739\) 1.34293e9 0.122404 0.0612022 0.998125i \(-0.480507\pi\)
0.0612022 + 0.998125i \(0.480507\pi\)
\(740\) 3.32611e10 3.01735
\(741\) −1.35703e10 −1.22525
\(742\) 1.28550e9 0.115520
\(743\) −2.17047e10 −1.94130 −0.970650 0.240497i \(-0.922690\pi\)
−0.970650 + 0.240497i \(0.922690\pi\)
\(744\) 3.40757e10 3.03347
\(745\) 1.44254e9 0.127815
\(746\) 5.26723e8 0.0464511
\(747\) −5.07599e9 −0.445553
\(748\) 3.96762e9 0.346636
\(749\) 2.82197e9 0.245395
\(750\) −1.38298e10 −1.19702
\(751\) 2.45348e9 0.211370 0.105685 0.994400i \(-0.466297\pi\)
0.105685 + 0.994400i \(0.466297\pi\)
\(752\) −3.54821e10 −3.04262
\(753\) 5.44274e9 0.464553
\(754\) 1.88670e10 1.60289
\(755\) −1.03016e10 −0.871143
\(756\) −2.94023e9 −0.247489
\(757\) 9.08326e9 0.761038 0.380519 0.924773i \(-0.375745\pi\)
0.380519 + 0.924773i \(0.375745\pi\)
\(758\) −1.44393e10 −1.20422
\(759\) −9.42400e8 −0.0782328
\(760\) −5.13327e10 −4.24177
\(761\) 2.38536e10 1.96204 0.981021 0.193902i \(-0.0621145\pi\)
0.981021 + 0.193902i \(0.0621145\pi\)
\(762\) 2.28605e10 1.87173
\(763\) −5.21682e9 −0.425177
\(764\) 1.86133e10 1.51007
\(765\) 4.70288e9 0.379795
\(766\) −1.69863e10 −1.36552
\(767\) −2.71603e9 −0.217345
\(768\) −3.51015e10 −2.79616
\(769\) 1.44982e10 1.14967 0.574834 0.818270i \(-0.305067\pi\)
0.574834 + 0.818270i \(0.305067\pi\)
\(770\) −1.03239e9 −0.0814943
\(771\) −6.03913e8 −0.0474553
\(772\) 3.60903e10 2.82313
\(773\) −1.48061e10 −1.15296 −0.576479 0.817112i \(-0.695574\pi\)
−0.576479 + 0.817112i \(0.695574\pi\)
\(774\) 2.44616e9 0.189623
\(775\) −2.69971e9 −0.208335
\(776\) −4.68358e10 −3.59801
\(777\) −3.92458e9 −0.300137
\(778\) 1.99970e10 1.52243
\(779\) 9.35492e9 0.709021
\(780\) −3.35810e10 −2.53374
\(781\) −1.78367e9 −0.133978
\(782\) −4.40542e10 −3.29430
\(783\) 1.26671e9 0.0942997
\(784\) −4.52937e10 −3.35684
\(785\) −6.29424e9 −0.464408
\(786\) 2.63709e10 1.93708
\(787\) 1.47489e10 1.07857 0.539286 0.842122i \(-0.318694\pi\)
0.539286 + 0.842122i \(0.318694\pi\)
\(788\) −4.87926e10 −3.55232
\(789\) −5.46564e9 −0.396161
\(790\) 2.40643e10 1.73651
\(791\) −7.83153e9 −0.562638
\(792\) −1.66618e9 −0.119175
\(793\) 1.11919e10 0.796979
\(794\) 1.09147e10 0.773818
\(795\) 9.85730e8 0.0695781
\(796\) −2.35287e10 −1.65349
\(797\) 1.53056e10 1.07090 0.535448 0.844568i \(-0.320143\pi\)
0.535448 + 0.844568i \(0.320143\pi\)
\(798\) 9.34967e9 0.651308
\(799\) −1.27847e10 −0.886702
\(800\) 9.66656e9 0.667509
\(801\) −1.81370e9 −0.124696
\(802\) 1.87330e10 1.28232
\(803\) −1.93906e9 −0.132156
\(804\) −3.30055e10 −2.23970
\(805\) 8.47751e9 0.572773
\(806\) −7.08857e10 −4.76855
\(807\) −1.68703e10 −1.12997
\(808\) 6.77830e9 0.452044
\(809\) 2.54329e8 0.0168879 0.00844396 0.999964i \(-0.497312\pi\)
0.00844396 + 0.999964i \(0.497312\pi\)
\(810\) −3.04861e9 −0.201560
\(811\) 1.23140e10 0.810637 0.405318 0.914176i \(-0.367161\pi\)
0.405318 + 0.914176i \(0.367161\pi\)
\(812\) −9.61336e9 −0.630129
\(813\) 9.56317e9 0.624144
\(814\) −3.43304e9 −0.223097
\(815\) 4.20780e9 0.272272
\(816\) −4.65738e10 −3.00073
\(817\) −5.75262e9 −0.369053
\(818\) 3.21141e10 2.05144
\(819\) 3.96232e9 0.252032
\(820\) 2.31496e10 1.46621
\(821\) 5.18812e9 0.327197 0.163598 0.986527i \(-0.447690\pi\)
0.163598 + 0.986527i \(0.447690\pi\)
\(822\) −1.58377e10 −0.994581
\(823\) −4.12489e9 −0.257937 −0.128968 0.991649i \(-0.541167\pi\)
−0.128968 + 0.991649i \(0.541167\pi\)
\(824\) −7.88576e10 −4.91019
\(825\) 1.32006e8 0.00818476
\(826\) 1.87129e9 0.115534
\(827\) 5.89785e9 0.362597 0.181298 0.983428i \(-0.441970\pi\)
0.181298 + 0.983428i \(0.441970\pi\)
\(828\) 2.11198e10 1.29296
\(829\) −2.77208e10 −1.68991 −0.844957 0.534835i \(-0.820374\pi\)
−0.844957 + 0.534835i \(0.820374\pi\)
\(830\) 3.99429e10 2.42475
\(831\) 1.62202e10 0.980511
\(832\) 1.36691e11 8.22828
\(833\) −1.63200e10 −0.978277
\(834\) −1.30395e10 −0.778361
\(835\) 5.18842e9 0.308413
\(836\) 6.04850e9 0.358035
\(837\) −4.75918e9 −0.280539
\(838\) 2.35632e10 1.38318
\(839\) 1.84332e10 1.07754 0.538771 0.842452i \(-0.318889\pi\)
0.538771 + 0.842452i \(0.318889\pi\)
\(840\) 1.49884e10 0.872524
\(841\) −1.31083e10 −0.759904
\(842\) 3.24574e10 1.87379
\(843\) 1.49742e10 0.860888
\(844\) −1.27325e10 −0.728977
\(845\) 2.90174e10 1.65447
\(846\) 8.28761e9 0.470579
\(847\) −7.93047e9 −0.448443
\(848\) −9.76194e9 −0.549731
\(849\) −5.01483e9 −0.281241
\(850\) 6.17087e9 0.344652
\(851\) 2.81904e10 1.56801
\(852\) 3.99732e10 2.21427
\(853\) −3.15892e10 −1.74268 −0.871339 0.490682i \(-0.836748\pi\)
−0.871339 + 0.490682i \(0.836748\pi\)
\(854\) −7.71099e9 −0.423650
\(855\) 7.16938e9 0.392284
\(856\) −3.58383e10 −1.95294
\(857\) −1.55096e10 −0.841722 −0.420861 0.907125i \(-0.638272\pi\)
−0.420861 + 0.907125i \(0.638272\pi\)
\(858\) 3.46606e9 0.187340
\(859\) −2.86456e10 −1.54199 −0.770997 0.636839i \(-0.780241\pi\)
−0.770997 + 0.636839i \(0.780241\pi\)
\(860\) −1.42354e10 −0.763177
\(861\) −2.73149e9 −0.145844
\(862\) −2.15345e10 −1.14514
\(863\) −3.05356e10 −1.61722 −0.808609 0.588346i \(-0.799779\pi\)
−0.808609 + 0.588346i \(0.799779\pi\)
\(864\) 1.70407e10 0.898853
\(865\) 3.17808e10 1.66959
\(866\) 4.42307e10 2.31425
\(867\) −5.70208e9 −0.297144
\(868\) 3.61186e10 1.87462
\(869\) −1.83688e9 −0.0949537
\(870\) −9.96770e9 −0.513189
\(871\) 4.44790e10 2.28082
\(872\) 6.62524e10 3.38372
\(873\) 6.54132e9 0.332748
\(874\) −6.71592e10 −3.40263
\(875\) −9.49633e9 −0.479212
\(876\) 4.34557e10 2.18415
\(877\) −1.42738e10 −0.714562 −0.357281 0.933997i \(-0.616296\pi\)
−0.357281 + 0.933997i \(0.616296\pi\)
\(878\) 7.37759e10 3.67861
\(879\) −2.17660e10 −1.08098
\(880\) 7.83987e9 0.387810
\(881\) −5.85758e9 −0.288604 −0.144302 0.989534i \(-0.546094\pi\)
−0.144302 + 0.989534i \(0.546094\pi\)
\(882\) 1.05793e10 0.519178
\(883\) 3.15376e10 1.54158 0.770790 0.637089i \(-0.219862\pi\)
0.770790 + 0.637089i \(0.219862\pi\)
\(884\) 1.19827e11 5.83405
\(885\) 1.43491e9 0.0695864
\(886\) −1.46674e10 −0.708494
\(887\) 1.06379e10 0.511829 0.255914 0.966699i \(-0.417624\pi\)
0.255914 + 0.966699i \(0.417624\pi\)
\(888\) 4.98412e10 2.38860
\(889\) 1.56974e10 0.749325
\(890\) 1.42720e10 0.678609
\(891\) 2.32707e8 0.0110214
\(892\) 7.30426e10 3.44587
\(893\) −1.94899e10 −0.915861
\(894\) 3.33677e9 0.156187
\(895\) −1.17759e10 −0.549052
\(896\) −4.86316e10 −2.25861
\(897\) −2.84616e10 −1.31670
\(898\) −5.31222e10 −2.44799
\(899\) −1.55606e10 −0.714278
\(900\) −2.95835e9 −0.135270
\(901\) −3.51737e9 −0.160207
\(902\) −2.38939e9 −0.108408
\(903\) 1.67968e9 0.0759135
\(904\) 9.94585e10 4.47767
\(905\) −3.70442e10 −1.66131
\(906\) −2.38287e10 −1.06452
\(907\) −1.21567e10 −0.540993 −0.270497 0.962721i \(-0.587188\pi\)
−0.270497 + 0.962721i \(0.587188\pi\)
\(908\) 7.33891e10 3.25335
\(909\) −9.46690e8 −0.0418056
\(910\) −3.11795e10 −1.37159
\(911\) −1.79794e10 −0.787881 −0.393941 0.919136i \(-0.628888\pi\)
−0.393941 + 0.919136i \(0.628888\pi\)
\(912\) −7.10003e10 −3.09940
\(913\) −3.04893e9 −0.132587
\(914\) −1.13894e10 −0.493387
\(915\) −5.91283e9 −0.255165
\(916\) −3.26537e10 −1.40378
\(917\) 1.81078e10 0.775486
\(918\) 1.08783e10 0.464100
\(919\) 3.12747e10 1.32919 0.664597 0.747202i \(-0.268603\pi\)
0.664597 + 0.747202i \(0.268603\pi\)
\(920\) −1.07662e11 −4.55834
\(921\) 1.46333e10 0.617212
\(922\) −3.54731e10 −1.49053
\(923\) −5.38688e10 −2.25492
\(924\) −1.76607e9 −0.0736472
\(925\) −3.94876e9 −0.164046
\(926\) −1.69623e10 −0.702016
\(927\) 1.10136e10 0.454100
\(928\) 5.57161e10 2.28856
\(929\) −4.19833e9 −0.171799 −0.0858997 0.996304i \(-0.527376\pi\)
−0.0858997 + 0.996304i \(0.527376\pi\)
\(930\) 3.74499e10 1.52672
\(931\) −2.48792e10 −1.01045
\(932\) 8.07697e10 3.26808
\(933\) 6.03200e9 0.243151
\(934\) −6.62972e10 −2.66245
\(935\) 2.82482e9 0.113019
\(936\) −5.03206e10 −2.00576
\(937\) −4.93627e10 −1.96024 −0.980122 0.198393i \(-0.936428\pi\)
−0.980122 + 0.198393i \(0.936428\pi\)
\(938\) −3.06451e10 −1.21242
\(939\) 1.76400e10 0.695294
\(940\) −4.82296e10 −1.89394
\(941\) 5.81164e9 0.227371 0.113685 0.993517i \(-0.463734\pi\)
0.113685 + 0.993517i \(0.463734\pi\)
\(942\) −1.45593e10 −0.567496
\(943\) 1.96205e10 0.761936
\(944\) −1.42103e10 −0.549797
\(945\) −2.09335e9 −0.0806921
\(946\) 1.46931e9 0.0564278
\(947\) −3.48866e10 −1.33485 −0.667427 0.744676i \(-0.732604\pi\)
−0.667427 + 0.744676i \(0.732604\pi\)
\(948\) 4.11658e10 1.56930
\(949\) −5.85619e10 −2.22425
\(950\) 9.40729e9 0.355985
\(951\) −1.03178e10 −0.389006
\(952\) −5.34829e10 −2.00903
\(953\) 8.20632e9 0.307131 0.153565 0.988138i \(-0.450924\pi\)
0.153565 + 0.988138i \(0.450924\pi\)
\(954\) 2.28011e9 0.0850229
\(955\) 1.32521e10 0.492349
\(956\) 4.84226e10 1.79244
\(957\) 7.60857e8 0.0280615
\(958\) 4.34729e10 1.59749
\(959\) −1.08751e10 −0.398169
\(960\) −7.22159e10 −2.63441
\(961\) 3.09505e10 1.12496
\(962\) −1.03682e11 −3.75482
\(963\) 5.00536e9 0.180611
\(964\) 1.90032e10 0.683216
\(965\) 2.56952e10 0.920462
\(966\) 1.96095e10 0.699916
\(967\) 1.16190e10 0.413213 0.206607 0.978424i \(-0.433758\pi\)
0.206607 + 0.978424i \(0.433758\pi\)
\(968\) 1.00715e11 3.56887
\(969\) −2.55824e10 −0.903251
\(970\) −5.14735e10 −1.81085
\(971\) −4.87190e9 −0.170778 −0.0853889 0.996348i \(-0.527213\pi\)
−0.0853889 + 0.996348i \(0.527213\pi\)
\(972\) −5.21512e9 −0.182151
\(973\) −8.95371e9 −0.311607
\(974\) 9.33634e10 3.23758
\(975\) 3.98674e9 0.137753
\(976\) 5.85563e10 2.01604
\(977\) −2.06779e10 −0.709375 −0.354688 0.934985i \(-0.615413\pi\)
−0.354688 + 0.934985i \(0.615413\pi\)
\(978\) 9.73313e9 0.332711
\(979\) −1.08941e9 −0.0371068
\(980\) −6.15660e10 −2.08954
\(981\) −9.25314e9 −0.312930
\(982\) −2.57487e10 −0.867692
\(983\) −1.54356e8 −0.00518306 −0.00259153 0.999997i \(-0.500825\pi\)
−0.00259153 + 0.999997i \(0.500825\pi\)
\(984\) 3.46893e10 1.16068
\(985\) −3.47388e10 −1.15821
\(986\) 3.55676e10 1.18164
\(987\) 5.69076e9 0.188391
\(988\) 1.82672e11 6.02590
\(989\) −1.20652e10 −0.396596
\(990\) −1.83117e9 −0.0599798
\(991\) −2.23061e10 −0.728057 −0.364029 0.931388i \(-0.618599\pi\)
−0.364029 + 0.931388i \(0.618599\pi\)
\(992\) −2.09332e11 −6.80841
\(993\) 3.22487e9 0.104518
\(994\) 3.71145e10 1.19865
\(995\) −1.67517e10 −0.539110
\(996\) 6.83287e10 2.19127
\(997\) 3.27229e10 1.04573 0.522865 0.852416i \(-0.324863\pi\)
0.522865 + 0.852416i \(0.324863\pi\)
\(998\) −6.20281e10 −1.97529
\(999\) −6.96107e9 −0.220900
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 177.8.a.c.1.1 17
3.2 odd 2 531.8.a.c.1.17 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.1 17 1.1 even 1 trivial
531.8.a.c.1.17 17 3.2 odd 2