Properties

Label 177.8.a.c.1.14
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 1.14
Root \(15.1891\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+15.1891 q^{2} -27.0000 q^{3} +102.710 q^{4} -418.807 q^{5} -410.107 q^{6} -814.653 q^{7} -384.131 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+15.1891 q^{2} -27.0000 q^{3} +102.710 q^{4} -418.807 q^{5} -410.107 q^{6} -814.653 q^{7} -384.131 q^{8} +729.000 q^{9} -6361.32 q^{10} -2422.96 q^{11} -2773.17 q^{12} +10673.2 q^{13} -12373.9 q^{14} +11307.8 q^{15} -18981.5 q^{16} -26705.4 q^{17} +11072.9 q^{18} +33854.6 q^{19} -43015.7 q^{20} +21995.6 q^{21} -36802.7 q^{22} +15069.2 q^{23} +10371.5 q^{24} +97274.0 q^{25} +162116. q^{26} -19683.0 q^{27} -83673.1 q^{28} +32501.7 q^{29} +171756. q^{30} +26459.3 q^{31} -239144. q^{32} +65420.0 q^{33} -405632. q^{34} +341182. q^{35} +74875.7 q^{36} +497796. q^{37} +514222. q^{38} -288176. q^{39} +160877. q^{40} +366715. q^{41} +334095. q^{42} +493861. q^{43} -248863. q^{44} -305310. q^{45} +228889. q^{46} -825019. q^{47} +512501. q^{48} -159884. q^{49} +1.47751e6 q^{50} +721046. q^{51} +1.09624e6 q^{52} -1.78313e6 q^{53} -298968. q^{54} +1.01475e6 q^{55} +312933. q^{56} -914074. q^{57} +493673. q^{58} -205379. q^{59} +1.16142e6 q^{60} +628605. q^{61} +401894. q^{62} -593882. q^{63} -1.20276e6 q^{64} -4.46999e6 q^{65} +993674. q^{66} +1.85849e6 q^{67} -2.74292e6 q^{68} -406870. q^{69} +5.18226e6 q^{70} +3.64832e6 q^{71} -280032. q^{72} +5.50965e6 q^{73} +7.56110e6 q^{74} -2.62640e6 q^{75} +3.47721e6 q^{76} +1.97387e6 q^{77} -4.37714e6 q^{78} -6.04365e6 q^{79} +7.94959e6 q^{80} +531441. q^{81} +5.57009e6 q^{82} -4.72119e6 q^{83} +2.25917e6 q^{84} +1.11844e7 q^{85} +7.50133e6 q^{86} -877545. q^{87} +930735. q^{88} -875348. q^{89} -4.63740e6 q^{90} -8.69493e6 q^{91} +1.54776e6 q^{92} -714401. q^{93} -1.25313e7 q^{94} -1.41785e7 q^{95} +6.45690e6 q^{96} -1.26140e7 q^{97} -2.42850e6 q^{98} -1.76634e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + 6521q^{10} - 1764q^{11} - 31482q^{12} + 18192q^{13} - 7827q^{14} + 8586q^{15} + 139226q^{16} - 15507q^{17} + 1458q^{18} + 52083q^{19} + 721q^{20} - 84915q^{21} - 234434q^{22} + 63823q^{23} - 63585q^{24} + 202153q^{25} - 367956q^{26} - 334611q^{27} + 182306q^{28} - 502955q^{29} - 176067q^{30} + 347531q^{31} - 243908q^{32} + 47628q^{33} - 330872q^{34} + 92641q^{35} + 850014q^{36} + 447615q^{37} + 775669q^{38} - 491184q^{39} + 2203270q^{40} + 940335q^{41} + 211329q^{42} + 478562q^{43} - 596924q^{44} - 231822q^{45} - 3078663q^{46} + 703121q^{47} - 3759102q^{48} + 1895082q^{49} - 876967q^{50} + 418689q^{51} + 6278296q^{52} - 1005974q^{53} - 39366q^{54} + 5212846q^{55} + 3425294q^{56} - 1406241q^{57} + 6710166q^{58} - 3491443q^{59} - 19467q^{60} + 11510749q^{61} + 5996234q^{62} + 2292705q^{63} + 29496941q^{64} + 11094180q^{65} + 6329718q^{66} + 14007144q^{67} + 19688159q^{68} - 1723221q^{69} + 30909708q^{70} + 5229074q^{71} + 1716795q^{72} + 5452211q^{73} + 12819662q^{74} - 5458131q^{75} + 41929340q^{76} + 9930777q^{77} + 9934812q^{78} + 15275654q^{79} + 36576105q^{80} + 9034497q^{81} + 32025935q^{82} + 7826609q^{83} - 4922262q^{84} + 11836945q^{85} + 51649136q^{86} + 13579785q^{87} + 30223741q^{88} - 6436185q^{89} + 4753809q^{90} + 11633535q^{91} + 43357972q^{92} - 9383337q^{93} - 4494252q^{94} + 23741055q^{95} + 6585516q^{96} + 26377540q^{97} + 26517816q^{98} - 1285956q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 15.1891 1.34254 0.671272 0.741211i \(-0.265748\pi\)
0.671272 + 0.741211i \(0.265748\pi\)
\(3\) −27.0000 −0.577350
\(4\) 102.710 0.802423
\(5\) −418.807 −1.49837 −0.749184 0.662362i \(-0.769554\pi\)
−0.749184 + 0.662362i \(0.769554\pi\)
\(6\) −410.107 −0.775118
\(7\) −814.653 −0.897696 −0.448848 0.893608i \(-0.648166\pi\)
−0.448848 + 0.893608i \(0.648166\pi\)
\(8\) −384.131 −0.265256
\(9\) 729.000 0.333333
\(10\) −6361.32 −2.01162
\(11\) −2422.96 −0.548874 −0.274437 0.961605i \(-0.588491\pi\)
−0.274437 + 0.961605i \(0.588491\pi\)
\(12\) −2773.17 −0.463279
\(13\) 10673.2 1.34738 0.673692 0.739012i \(-0.264707\pi\)
0.673692 + 0.739012i \(0.264707\pi\)
\(14\) −12373.9 −1.20520
\(15\) 11307.8 0.865083
\(16\) −18981.5 −1.15854
\(17\) −26705.4 −1.31834 −0.659171 0.751993i \(-0.729093\pi\)
−0.659171 + 0.751993i \(0.729093\pi\)
\(18\) 11072.9 0.447515
\(19\) 33854.6 1.13235 0.566174 0.824286i \(-0.308423\pi\)
0.566174 + 0.824286i \(0.308423\pi\)
\(20\) −43015.7 −1.20233
\(21\) 21995.6 0.518285
\(22\) −36802.7 −0.736887
\(23\) 15069.2 0.258252 0.129126 0.991628i \(-0.458783\pi\)
0.129126 + 0.991628i \(0.458783\pi\)
\(24\) 10371.5 0.153145
\(25\) 97274.0 1.24511
\(26\) 162116. 1.80892
\(27\) −19683.0 −0.192450
\(28\) −83673.1 −0.720332
\(29\) 32501.7 0.247464 0.123732 0.992316i \(-0.460514\pi\)
0.123732 + 0.992316i \(0.460514\pi\)
\(30\) 171756. 1.16141
\(31\) 26459.3 0.159519 0.0797594 0.996814i \(-0.474585\pi\)
0.0797594 + 0.996814i \(0.474585\pi\)
\(32\) −239144. −1.29014
\(33\) 65420.0 0.316892
\(34\) −405632. −1.76993
\(35\) 341182. 1.34508
\(36\) 74875.7 0.267474
\(37\) 497796. 1.61564 0.807822 0.589427i \(-0.200646\pi\)
0.807822 + 0.589427i \(0.200646\pi\)
\(38\) 514222. 1.52023
\(39\) −288176. −0.777913
\(40\) 160877. 0.397451
\(41\) 366715. 0.830970 0.415485 0.909600i \(-0.363612\pi\)
0.415485 + 0.909600i \(0.363612\pi\)
\(42\) 334095. 0.695820
\(43\) 493861. 0.947251 0.473626 0.880726i \(-0.342945\pi\)
0.473626 + 0.880726i \(0.342945\pi\)
\(44\) −248863. −0.440429
\(45\) −305310. −0.499456
\(46\) 228889. 0.346715
\(47\) −825019. −1.15910 −0.579550 0.814936i \(-0.696772\pi\)
−0.579550 + 0.814936i \(0.696772\pi\)
\(48\) 512501. 0.668884
\(49\) −159884. −0.194142
\(50\) 1.47751e6 1.67161
\(51\) 721046. 0.761145
\(52\) 1.09624e6 1.08117
\(53\) −1.78313e6 −1.64520 −0.822598 0.568624i \(-0.807476\pi\)
−0.822598 + 0.568624i \(0.807476\pi\)
\(54\) −298968. −0.258373
\(55\) 1.01475e6 0.822415
\(56\) 312933. 0.238119
\(57\) −914074. −0.653761
\(58\) 493673. 0.332232
\(59\) −205379. −0.130189
\(60\) 1.16142e6 0.694163
\(61\) 628605. 0.354587 0.177294 0.984158i \(-0.443266\pi\)
0.177294 + 0.984158i \(0.443266\pi\)
\(62\) 401894. 0.214161
\(63\) −593882. −0.299232
\(64\) −1.20276e6 −0.573522
\(65\) −4.46999e6 −2.01888
\(66\) 993674. 0.425442
\(67\) 1.85849e6 0.754915 0.377458 0.926027i \(-0.376798\pi\)
0.377458 + 0.926027i \(0.376798\pi\)
\(68\) −2.74292e6 −1.05787
\(69\) −406870. −0.149102
\(70\) 5.18226e6 1.80583
\(71\) 3.64832e6 1.20973 0.604865 0.796328i \(-0.293227\pi\)
0.604865 + 0.796328i \(0.293227\pi\)
\(72\) −280032. −0.0884185
\(73\) 5.50965e6 1.65765 0.828827 0.559505i \(-0.189009\pi\)
0.828827 + 0.559505i \(0.189009\pi\)
\(74\) 7.56110e6 2.16907
\(75\) −2.62640e6 −0.718863
\(76\) 3.47721e6 0.908622
\(77\) 1.97387e6 0.492722
\(78\) −4.37714e6 −1.04438
\(79\) −6.04365e6 −1.37913 −0.689564 0.724224i \(-0.742198\pi\)
−0.689564 + 0.724224i \(0.742198\pi\)
\(80\) 7.94959e6 1.73592
\(81\) 531441. 0.111111
\(82\) 5.57009e6 1.11561
\(83\) −4.72119e6 −0.906313 −0.453156 0.891431i \(-0.649702\pi\)
−0.453156 + 0.891431i \(0.649702\pi\)
\(84\) 2.25917e6 0.415884
\(85\) 1.11844e7 1.97536
\(86\) 7.50133e6 1.27173
\(87\) −877545. −0.142874
\(88\) 930735. 0.145592
\(89\) −875348. −0.131618 −0.0658091 0.997832i \(-0.520963\pi\)
−0.0658091 + 0.997832i \(0.520963\pi\)
\(90\) −4.63740e6 −0.670541
\(91\) −8.69493e6 −1.20954
\(92\) 1.54776e6 0.207228
\(93\) −714401. −0.0920983
\(94\) −1.25313e7 −1.55614
\(95\) −1.41785e7 −1.69667
\(96\) 6.45690e6 0.744860
\(97\) −1.26140e7 −1.40331 −0.701653 0.712519i \(-0.747554\pi\)
−0.701653 + 0.712519i \(0.747554\pi\)
\(98\) −2.42850e6 −0.260644
\(99\) −1.76634e6 −0.182958
\(100\) 9.99103e6 0.999103
\(101\) 1.43083e6 0.138185 0.0690926 0.997610i \(-0.477990\pi\)
0.0690926 + 0.997610i \(0.477990\pi\)
\(102\) 1.09521e7 1.02187
\(103\) 5.60899e6 0.505772 0.252886 0.967496i \(-0.418620\pi\)
0.252886 + 0.967496i \(0.418620\pi\)
\(104\) −4.09990e6 −0.357401
\(105\) −9.21191e6 −0.776582
\(106\) −2.70842e7 −2.20875
\(107\) 2.17512e7 1.71648 0.858241 0.513247i \(-0.171558\pi\)
0.858241 + 0.513247i \(0.171558\pi\)
\(108\) −2.02164e6 −0.154426
\(109\) −9.32545e6 −0.689727 −0.344863 0.938653i \(-0.612075\pi\)
−0.344863 + 0.938653i \(0.612075\pi\)
\(110\) 1.54132e7 1.10413
\(111\) −1.34405e7 −0.932792
\(112\) 1.54634e7 1.04002
\(113\) 1.50397e7 0.980537 0.490268 0.871572i \(-0.336899\pi\)
0.490268 + 0.871572i \(0.336899\pi\)
\(114\) −1.38840e7 −0.877703
\(115\) −6.31110e6 −0.386957
\(116\) 3.33825e6 0.198571
\(117\) 7.78074e6 0.449128
\(118\) −3.11953e6 −0.174784
\(119\) 2.17556e7 1.18347
\(120\) −4.34367e6 −0.229468
\(121\) −1.36164e7 −0.698738
\(122\) 9.54797e6 0.476049
\(123\) −9.90131e6 −0.479761
\(124\) 2.71764e6 0.128002
\(125\) −8.01973e6 −0.367261
\(126\) −9.02056e6 −0.401732
\(127\) 1.78413e7 0.772883 0.386441 0.922314i \(-0.373704\pi\)
0.386441 + 0.922314i \(0.373704\pi\)
\(128\) 1.23415e7 0.520156
\(129\) −1.33342e7 −0.546896
\(130\) −6.78954e7 −2.71043
\(131\) 3.78349e6 0.147043 0.0735214 0.997294i \(-0.476576\pi\)
0.0735214 + 0.997294i \(0.476576\pi\)
\(132\) 6.71930e6 0.254282
\(133\) −2.75797e7 −1.01650
\(134\) 2.82289e7 1.01351
\(135\) 8.24337e6 0.288361
\(136\) 1.02584e7 0.349697
\(137\) −2.85640e7 −0.949069 −0.474534 0.880237i \(-0.657384\pi\)
−0.474534 + 0.880237i \(0.657384\pi\)
\(138\) −6.18000e6 −0.200176
\(139\) 3.44323e7 1.08746 0.543731 0.839260i \(-0.317011\pi\)
0.543731 + 0.839260i \(0.317011\pi\)
\(140\) 3.50429e7 1.07932
\(141\) 2.22755e7 0.669207
\(142\) 5.54148e7 1.62412
\(143\) −2.58607e7 −0.739544
\(144\) −1.38375e7 −0.386180
\(145\) −1.36119e7 −0.370793
\(146\) 8.36868e7 2.22547
\(147\) 4.31687e6 0.112088
\(148\) 5.11287e7 1.29643
\(149\) 4.80665e6 0.119039 0.0595196 0.998227i \(-0.481043\pi\)
0.0595196 + 0.998227i \(0.481043\pi\)
\(150\) −3.98927e7 −0.965105
\(151\) −5.09354e6 −0.120393 −0.0601964 0.998187i \(-0.519173\pi\)
−0.0601964 + 0.998187i \(0.519173\pi\)
\(152\) −1.30046e7 −0.300362
\(153\) −1.94682e7 −0.439447
\(154\) 2.99815e7 0.661500
\(155\) −1.10813e7 −0.239018
\(156\) −2.95986e7 −0.624215
\(157\) 2.01246e7 0.415028 0.207514 0.978232i \(-0.433463\pi\)
0.207514 + 0.978232i \(0.433463\pi\)
\(158\) −9.17980e7 −1.85154
\(159\) 4.81445e7 0.949854
\(160\) 1.00155e8 1.93310
\(161\) −1.22762e7 −0.231832
\(162\) 8.07213e6 0.149172
\(163\) −6.76023e7 −1.22266 −0.611328 0.791377i \(-0.709365\pi\)
−0.611328 + 0.791377i \(0.709365\pi\)
\(164\) 3.76654e7 0.666790
\(165\) −2.73983e7 −0.474821
\(166\) −7.17108e7 −1.21676
\(167\) 1.10791e8 1.84076 0.920382 0.391020i \(-0.127878\pi\)
0.920382 + 0.391020i \(0.127878\pi\)
\(168\) −8.44920e6 −0.137478
\(169\) 5.11680e7 0.815446
\(170\) 1.69881e8 2.65201
\(171\) 2.46800e7 0.377449
\(172\) 5.07245e7 0.760096
\(173\) 1.14352e8 1.67912 0.839559 0.543269i \(-0.182814\pi\)
0.839559 + 0.543269i \(0.182814\pi\)
\(174\) −1.33292e7 −0.191814
\(175\) −7.92445e7 −1.11773
\(176\) 4.59915e7 0.635892
\(177\) 5.54523e6 0.0751646
\(178\) −1.32958e7 −0.176703
\(179\) −1.89246e7 −0.246627 −0.123313 0.992368i \(-0.539352\pi\)
−0.123313 + 0.992368i \(0.539352\pi\)
\(180\) −3.13584e7 −0.400775
\(181\) 9.30564e7 1.16646 0.583232 0.812306i \(-0.301788\pi\)
0.583232 + 0.812306i \(0.301788\pi\)
\(182\) −1.32069e8 −1.62386
\(183\) −1.69723e7 −0.204721
\(184\) −5.78857e6 −0.0685029
\(185\) −2.08480e8 −2.42083
\(186\) −1.08511e7 −0.123646
\(187\) 6.47062e7 0.723603
\(188\) −8.47378e7 −0.930089
\(189\) 1.60348e7 0.172762
\(190\) −2.15360e8 −2.27786
\(191\) 1.28269e8 1.33200 0.666001 0.745951i \(-0.268005\pi\)
0.666001 + 0.745951i \(0.268005\pi\)
\(192\) 3.24746e7 0.331123
\(193\) −7.49379e7 −0.750327 −0.375164 0.926959i \(-0.622413\pi\)
−0.375164 + 0.926959i \(0.622413\pi\)
\(194\) −1.91596e8 −1.88400
\(195\) 1.20690e8 1.16560
\(196\) −1.64217e7 −0.155784
\(197\) −1.42886e8 −1.33155 −0.665775 0.746152i \(-0.731899\pi\)
−0.665775 + 0.746152i \(0.731899\pi\)
\(198\) −2.68292e7 −0.245629
\(199\) −7.59209e7 −0.682929 −0.341465 0.939895i \(-0.610923\pi\)
−0.341465 + 0.939895i \(0.610923\pi\)
\(200\) −3.73660e7 −0.330272
\(201\) −5.01792e7 −0.435851
\(202\) 2.17330e7 0.185520
\(203\) −2.64776e7 −0.222148
\(204\) 7.40587e7 0.610760
\(205\) −1.53583e8 −1.24510
\(206\) 8.51958e7 0.679021
\(207\) 1.09855e7 0.0860841
\(208\) −2.02593e8 −1.56100
\(209\) −8.20284e7 −0.621516
\(210\) −1.39921e8 −1.04260
\(211\) −6.90924e7 −0.506339 −0.253170 0.967422i \(-0.581473\pi\)
−0.253170 + 0.967422i \(0.581473\pi\)
\(212\) −1.83146e8 −1.32014
\(213\) −9.85046e7 −0.698438
\(214\) 3.30381e8 2.30445
\(215\) −2.06832e8 −1.41933
\(216\) 7.56085e6 0.0510485
\(217\) −2.15551e7 −0.143199
\(218\) −1.41646e8 −0.925988
\(219\) −1.48760e8 −0.957047
\(220\) 1.04225e8 0.659925
\(221\) −2.85031e8 −1.77631
\(222\) −2.04150e8 −1.25231
\(223\) −2.23657e8 −1.35057 −0.675284 0.737558i \(-0.735979\pi\)
−0.675284 + 0.737558i \(0.735979\pi\)
\(224\) 1.94820e8 1.15815
\(225\) 7.09127e7 0.415036
\(226\) 2.28440e8 1.31641
\(227\) −2.62570e7 −0.148989 −0.0744945 0.997221i \(-0.523734\pi\)
−0.0744945 + 0.997221i \(0.523734\pi\)
\(228\) −9.38847e7 −0.524593
\(229\) 2.78378e8 1.53183 0.765916 0.642940i \(-0.222286\pi\)
0.765916 + 0.642940i \(0.222286\pi\)
\(230\) −9.58602e7 −0.519507
\(231\) −5.32946e7 −0.284473
\(232\) −1.24849e7 −0.0656413
\(233\) 4.33659e7 0.224596 0.112298 0.993675i \(-0.464179\pi\)
0.112298 + 0.993675i \(0.464179\pi\)
\(234\) 1.18183e8 0.602974
\(235\) 3.45523e8 1.73676
\(236\) −2.10945e7 −0.104467
\(237\) 1.63179e8 0.796240
\(238\) 3.30449e8 1.58886
\(239\) −3.28872e8 −1.55824 −0.779120 0.626875i \(-0.784334\pi\)
−0.779120 + 0.626875i \(0.784334\pi\)
\(240\) −2.14639e8 −1.00223
\(241\) 3.31564e7 0.152583 0.0762917 0.997086i \(-0.475692\pi\)
0.0762917 + 0.997086i \(0.475692\pi\)
\(242\) −2.06822e8 −0.938086
\(243\) −1.43489e7 −0.0641500
\(244\) 6.45641e7 0.284529
\(245\) 6.69604e7 0.290896
\(246\) −1.50392e8 −0.644100
\(247\) 3.61336e8 1.52571
\(248\) −1.01638e7 −0.0423133
\(249\) 1.27472e8 0.523260
\(250\) −1.21813e8 −0.493063
\(251\) 2.94819e8 1.17679 0.588394 0.808574i \(-0.299760\pi\)
0.588394 + 0.808574i \(0.299760\pi\)
\(252\) −6.09977e7 −0.240111
\(253\) −3.65122e7 −0.141748
\(254\) 2.70994e8 1.03763
\(255\) −3.01979e8 −1.14047
\(256\) 3.41411e8 1.27186
\(257\) −2.56922e8 −0.944139 −0.472069 0.881561i \(-0.656493\pi\)
−0.472069 + 0.881561i \(0.656493\pi\)
\(258\) −2.02536e8 −0.734231
\(259\) −4.05531e8 −1.45036
\(260\) −4.59114e8 −1.61999
\(261\) 2.36937e7 0.0824881
\(262\) 5.74680e7 0.197411
\(263\) 3.01656e8 1.02251 0.511255 0.859429i \(-0.329181\pi\)
0.511255 + 0.859429i \(0.329181\pi\)
\(264\) −2.51299e7 −0.0840575
\(265\) 7.46787e8 2.46511
\(266\) −4.18913e8 −1.36470
\(267\) 2.36344e7 0.0759898
\(268\) 1.90886e8 0.605762
\(269\) −2.61569e8 −0.819318 −0.409659 0.912239i \(-0.634352\pi\)
−0.409659 + 0.912239i \(0.634352\pi\)
\(270\) 1.25210e8 0.387137
\(271\) −1.44509e8 −0.441064 −0.220532 0.975380i \(-0.570779\pi\)
−0.220532 + 0.975380i \(0.570779\pi\)
\(272\) 5.06909e8 1.52735
\(273\) 2.34763e8 0.698330
\(274\) −4.33863e8 −1.27417
\(275\) −2.35691e8 −0.683406
\(276\) −4.17897e7 −0.119643
\(277\) −4.66473e7 −0.131870 −0.0659352 0.997824i \(-0.521003\pi\)
−0.0659352 + 0.997824i \(0.521003\pi\)
\(278\) 5.22997e8 1.45996
\(279\) 1.92888e7 0.0531730
\(280\) −1.31059e8 −0.356790
\(281\) 6.78292e8 1.82366 0.911832 0.410564i \(-0.134668\pi\)
0.911832 + 0.410564i \(0.134668\pi\)
\(282\) 3.38346e8 0.898440
\(283\) 2.33845e8 0.613305 0.306652 0.951822i \(-0.400791\pi\)
0.306652 + 0.951822i \(0.400791\pi\)
\(284\) 3.74719e8 0.970715
\(285\) 3.82820e8 0.979575
\(286\) −3.92802e8 −0.992870
\(287\) −2.98746e8 −0.745959
\(288\) −1.74336e8 −0.430045
\(289\) 3.02840e8 0.738024
\(290\) −2.06753e8 −0.497805
\(291\) 3.40579e8 0.810199
\(292\) 5.65897e8 1.33014
\(293\) 3.28205e8 0.762270 0.381135 0.924519i \(-0.375533\pi\)
0.381135 + 0.924519i \(0.375533\pi\)
\(294\) 6.55695e7 0.150483
\(295\) 8.60141e7 0.195071
\(296\) −1.91219e8 −0.428559
\(297\) 4.76912e7 0.105631
\(298\) 7.30089e7 0.159815
\(299\) 1.60837e8 0.347965
\(300\) −2.69758e8 −0.576832
\(301\) −4.02325e8 −0.850344
\(302\) −7.73665e7 −0.161632
\(303\) −3.86323e7 −0.0797813
\(304\) −6.42612e8 −1.31187
\(305\) −2.63264e8 −0.531302
\(306\) −2.95706e8 −0.589977
\(307\) 1.82586e8 0.360150 0.180075 0.983653i \(-0.442366\pi\)
0.180075 + 0.983653i \(0.442366\pi\)
\(308\) 2.02737e8 0.395371
\(309\) −1.51443e8 −0.292007
\(310\) −1.68316e8 −0.320892
\(311\) −5.84419e8 −1.10170 −0.550850 0.834605i \(-0.685696\pi\)
−0.550850 + 0.834605i \(0.685696\pi\)
\(312\) 1.10697e8 0.206346
\(313\) 9.32347e8 1.71859 0.859295 0.511480i \(-0.170903\pi\)
0.859295 + 0.511480i \(0.170903\pi\)
\(314\) 3.05675e8 0.557194
\(315\) 2.48722e8 0.448360
\(316\) −6.20745e8 −1.10664
\(317\) 8.93740e8 1.57581 0.787905 0.615797i \(-0.211166\pi\)
0.787905 + 0.615797i \(0.211166\pi\)
\(318\) 7.31274e8 1.27522
\(319\) −7.87503e7 −0.135827
\(320\) 5.03725e8 0.859348
\(321\) −5.87281e8 −0.991011
\(322\) −1.86465e8 −0.311245
\(323\) −9.04100e8 −1.49282
\(324\) 5.45844e7 0.0891581
\(325\) 1.03822e9 1.67764
\(326\) −1.02682e9 −1.64147
\(327\) 2.51787e8 0.398214
\(328\) −1.40867e8 −0.220419
\(329\) 6.72104e8 1.04052
\(330\) −4.16157e8 −0.637468
\(331\) 9.07658e8 1.37570 0.687851 0.725852i \(-0.258554\pi\)
0.687851 + 0.725852i \(0.258554\pi\)
\(332\) −4.84914e8 −0.727246
\(333\) 3.62893e8 0.538548
\(334\) 1.68283e9 2.47131
\(335\) −7.78348e8 −1.13114
\(336\) −4.17510e8 −0.600454
\(337\) 7.89412e8 1.12357 0.561784 0.827284i \(-0.310115\pi\)
0.561784 + 0.827284i \(0.310115\pi\)
\(338\) 7.77198e8 1.09477
\(339\) −4.06071e8 −0.566113
\(340\) 1.14875e9 1.58507
\(341\) −6.41099e7 −0.0875557
\(342\) 3.74868e8 0.506742
\(343\) 8.01151e8 1.07198
\(344\) −1.89707e8 −0.251264
\(345\) 1.70400e8 0.223410
\(346\) 1.73690e9 2.25429
\(347\) −5.28902e8 −0.679551 −0.339775 0.940507i \(-0.610351\pi\)
−0.339775 + 0.940507i \(0.610351\pi\)
\(348\) −9.01328e7 −0.114645
\(349\) 2.00046e8 0.251908 0.125954 0.992036i \(-0.459801\pi\)
0.125954 + 0.992036i \(0.459801\pi\)
\(350\) −1.20366e9 −1.50060
\(351\) −2.10080e8 −0.259304
\(352\) 5.79438e8 0.708121
\(353\) 2.52531e8 0.305565 0.152782 0.988260i \(-0.451177\pi\)
0.152782 + 0.988260i \(0.451177\pi\)
\(354\) 8.42274e7 0.100912
\(355\) −1.52794e9 −1.81262
\(356\) −8.99072e7 −0.105613
\(357\) −5.87402e8 −0.683277
\(358\) −2.87448e8 −0.331107
\(359\) −6.83126e8 −0.779238 −0.389619 0.920976i \(-0.627393\pi\)
−0.389619 + 0.920976i \(0.627393\pi\)
\(360\) 1.17279e8 0.132484
\(361\) 2.52261e8 0.282212
\(362\) 1.41345e9 1.56603
\(363\) 3.67643e8 0.403416
\(364\) −8.93057e8 −0.970565
\(365\) −2.30748e9 −2.48378
\(366\) −2.57795e8 −0.274847
\(367\) 5.09383e8 0.537914 0.268957 0.963152i \(-0.413321\pi\)
0.268957 + 0.963152i \(0.413321\pi\)
\(368\) −2.86037e8 −0.299196
\(369\) 2.67335e8 0.276990
\(370\) −3.16664e9 −3.25007
\(371\) 1.45263e9 1.47689
\(372\) −7.33762e7 −0.0739018
\(373\) 7.78305e8 0.776549 0.388274 0.921544i \(-0.373071\pi\)
0.388274 + 0.921544i \(0.373071\pi\)
\(374\) 9.82832e8 0.971468
\(375\) 2.16533e8 0.212038
\(376\) 3.16915e8 0.307458
\(377\) 3.46896e8 0.333430
\(378\) 2.43555e8 0.231940
\(379\) 1.51777e9 1.43209 0.716043 0.698056i \(-0.245951\pi\)
0.716043 + 0.698056i \(0.245951\pi\)
\(380\) −1.45628e9 −1.36145
\(381\) −4.81715e8 −0.446224
\(382\) 1.94830e9 1.78827
\(383\) 1.35344e9 1.23096 0.615481 0.788152i \(-0.288962\pi\)
0.615481 + 0.788152i \(0.288962\pi\)
\(384\) −3.33221e8 −0.300312
\(385\) −8.26671e8 −0.738279
\(386\) −1.13824e9 −1.00735
\(387\) 3.60025e8 0.315750
\(388\) −1.29559e9 −1.12605
\(389\) −2.16650e7 −0.0186610 −0.00933052 0.999956i \(-0.502970\pi\)
−0.00933052 + 0.999956i \(0.502970\pi\)
\(390\) 1.83318e9 1.56487
\(391\) −4.02430e8 −0.340465
\(392\) 6.14164e7 0.0514971
\(393\) −1.02154e8 −0.0848951
\(394\) −2.17031e9 −1.78766
\(395\) 2.53112e9 2.06644
\(396\) −1.81421e8 −0.146810
\(397\) 9.27708e8 0.744123 0.372061 0.928208i \(-0.378651\pi\)
0.372061 + 0.928208i \(0.378651\pi\)
\(398\) −1.15317e9 −0.916862
\(399\) 7.44653e8 0.586879
\(400\) −1.84641e9 −1.44251
\(401\) −2.89379e8 −0.224110 −0.112055 0.993702i \(-0.535743\pi\)
−0.112055 + 0.993702i \(0.535743\pi\)
\(402\) −7.62179e8 −0.585148
\(403\) 2.82404e8 0.214933
\(404\) 1.46960e8 0.110883
\(405\) −2.22571e8 −0.166485
\(406\) −4.02172e8 −0.298243
\(407\) −1.20614e9 −0.886784
\(408\) −2.76976e8 −0.201898
\(409\) 1.85339e9 1.33947 0.669737 0.742598i \(-0.266407\pi\)
0.669737 + 0.742598i \(0.266407\pi\)
\(410\) −2.33279e9 −1.67160
\(411\) 7.71229e8 0.547945
\(412\) 5.76100e8 0.405843
\(413\) 1.67313e8 0.116870
\(414\) 1.66860e8 0.115572
\(415\) 1.97727e9 1.35799
\(416\) −2.55243e9 −1.73831
\(417\) −9.29671e8 −0.627846
\(418\) −1.24594e9 −0.834412
\(419\) 2.29514e9 1.52427 0.762133 0.647420i \(-0.224152\pi\)
0.762133 + 0.647420i \(0.224152\pi\)
\(420\) −9.46157e8 −0.623147
\(421\) 1.32746e9 0.867028 0.433514 0.901147i \(-0.357273\pi\)
0.433514 + 0.901147i \(0.357273\pi\)
\(422\) −1.04945e9 −0.679782
\(423\) −6.01439e8 −0.386367
\(424\) 6.84956e8 0.436397
\(425\) −2.59774e9 −1.64148
\(426\) −1.49620e9 −0.937683
\(427\) −5.12094e8 −0.318312
\(428\) 2.23406e9 1.37734
\(429\) 6.98239e8 0.426976
\(430\) −3.14161e9 −1.90551
\(431\) 4.95028e8 0.297823 0.148912 0.988850i \(-0.452423\pi\)
0.148912 + 0.988850i \(0.452423\pi\)
\(432\) 3.73613e8 0.222961
\(433\) −1.81935e9 −1.07698 −0.538491 0.842631i \(-0.681005\pi\)
−0.538491 + 0.842631i \(0.681005\pi\)
\(434\) −3.27404e8 −0.192252
\(435\) 3.67522e8 0.214077
\(436\) −9.57818e8 −0.553453
\(437\) 5.10163e8 0.292431
\(438\) −2.25954e9 −1.28488
\(439\) −3.28149e9 −1.85117 −0.925583 0.378546i \(-0.876424\pi\)
−0.925583 + 0.378546i \(0.876424\pi\)
\(440\) −3.89798e8 −0.218150
\(441\) −1.16555e8 −0.0647139
\(442\) −4.32938e9 −2.38478
\(443\) −2.09880e9 −1.14699 −0.573493 0.819210i \(-0.694412\pi\)
−0.573493 + 0.819210i \(0.694412\pi\)
\(444\) −1.38048e9 −0.748494
\(445\) 3.66602e8 0.197212
\(446\) −3.39716e9 −1.81320
\(447\) −1.29779e8 −0.0687274
\(448\) 9.79835e8 0.514849
\(449\) 8.92700e7 0.0465418 0.0232709 0.999729i \(-0.492592\pi\)
0.0232709 + 0.999729i \(0.492592\pi\)
\(450\) 1.07710e9 0.557203
\(451\) −8.88537e8 −0.456098
\(452\) 1.54473e9 0.786805
\(453\) 1.37526e8 0.0695088
\(454\) −3.98821e8 −0.200024
\(455\) 3.64149e9 1.81234
\(456\) 3.51124e8 0.173414
\(457\) −1.68348e9 −0.825092 −0.412546 0.910937i \(-0.635360\pi\)
−0.412546 + 0.910937i \(0.635360\pi\)
\(458\) 4.22833e9 2.05655
\(459\) 5.25642e8 0.253715
\(460\) −6.48214e8 −0.310503
\(461\) −1.92435e9 −0.914811 −0.457406 0.889258i \(-0.651221\pi\)
−0.457406 + 0.889258i \(0.651221\pi\)
\(462\) −8.09499e8 −0.381917
\(463\) −1.42542e9 −0.667437 −0.333718 0.942673i \(-0.608303\pi\)
−0.333718 + 0.942673i \(0.608303\pi\)
\(464\) −6.16931e8 −0.286697
\(465\) 2.99196e8 0.137997
\(466\) 6.58691e8 0.301530
\(467\) 1.67879e9 0.762759 0.381380 0.924418i \(-0.375449\pi\)
0.381380 + 0.924418i \(0.375449\pi\)
\(468\) 7.99161e8 0.360391
\(469\) −1.51402e9 −0.677685
\(470\) 5.24820e9 2.33168
\(471\) −5.43363e8 −0.239617
\(472\) 7.88925e7 0.0345333
\(473\) −1.19661e9 −0.519921
\(474\) 2.47854e9 1.06899
\(475\) 3.29317e9 1.40989
\(476\) 2.23452e9 0.949644
\(477\) −1.29990e9 −0.548398
\(478\) −4.99528e9 −2.09200
\(479\) 2.45408e9 1.02027 0.510135 0.860094i \(-0.329595\pi\)
0.510135 + 0.860094i \(0.329595\pi\)
\(480\) −2.70419e9 −1.11607
\(481\) 5.31306e9 2.17689
\(482\) 5.03617e8 0.204850
\(483\) 3.31458e8 0.133848
\(484\) −1.39854e9 −0.560683
\(485\) 5.28283e9 2.10267
\(486\) −2.17948e8 −0.0861242
\(487\) 1.22859e9 0.482008 0.241004 0.970524i \(-0.422523\pi\)
0.241004 + 0.970524i \(0.422523\pi\)
\(488\) −2.41467e8 −0.0940563
\(489\) 1.82526e9 0.705901
\(490\) 1.01707e9 0.390540
\(491\) −4.00300e9 −1.52616 −0.763081 0.646303i \(-0.776314\pi\)
−0.763081 + 0.646303i \(0.776314\pi\)
\(492\) −1.01697e9 −0.384971
\(493\) −8.67970e8 −0.326242
\(494\) 5.48838e9 2.04833
\(495\) 7.39755e8 0.274138
\(496\) −5.02237e8 −0.184809
\(497\) −2.97211e9 −1.08597
\(498\) 1.93619e9 0.702499
\(499\) −6.37493e8 −0.229680 −0.114840 0.993384i \(-0.536636\pi\)
−0.114840 + 0.993384i \(0.536636\pi\)
\(500\) −8.23707e8 −0.294698
\(501\) −2.99137e9 −1.06277
\(502\) 4.47806e9 1.57989
\(503\) 3.52591e9 1.23533 0.617666 0.786440i \(-0.288078\pi\)
0.617666 + 0.786440i \(0.288078\pi\)
\(504\) 2.28128e8 0.0793730
\(505\) −5.99239e8 −0.207052
\(506\) −5.54590e8 −0.190303
\(507\) −1.38154e9 −0.470798
\(508\) 1.83248e9 0.620179
\(509\) −1.03941e9 −0.349360 −0.174680 0.984625i \(-0.555889\pi\)
−0.174680 + 0.984625i \(0.555889\pi\)
\(510\) −4.58680e9 −1.53114
\(511\) −4.48845e9 −1.48807
\(512\) 3.60603e9 1.18736
\(513\) −6.66360e8 −0.217920
\(514\) −3.90243e9 −1.26755
\(515\) −2.34908e9 −0.757832
\(516\) −1.36956e9 −0.438842
\(517\) 1.99899e9 0.636200
\(518\) −6.15967e9 −1.94717
\(519\) −3.08749e9 −0.969439
\(520\) 1.71706e9 0.535519
\(521\) −1.04135e9 −0.322600 −0.161300 0.986905i \(-0.551569\pi\)
−0.161300 + 0.986905i \(0.551569\pi\)
\(522\) 3.59887e8 0.110744
\(523\) 4.18130e9 1.27807 0.639036 0.769177i \(-0.279334\pi\)
0.639036 + 0.769177i \(0.279334\pi\)
\(524\) 3.88603e8 0.117990
\(525\) 2.13960e9 0.645320
\(526\) 4.58190e9 1.37276
\(527\) −7.06606e8 −0.210300
\(528\) −1.24177e9 −0.367133
\(529\) −3.17774e9 −0.933306
\(530\) 1.13431e10 3.30952
\(531\) −1.49721e8 −0.0433963
\(532\) −2.83272e9 −0.815667
\(533\) 3.91401e9 1.11964
\(534\) 3.58986e8 0.102020
\(535\) −9.10953e9 −2.57192
\(536\) −7.13904e8 −0.200246
\(537\) 5.10964e8 0.142390
\(538\) −3.97300e9 −1.09997
\(539\) 3.87393e8 0.106559
\(540\) 8.46678e8 0.231388
\(541\) −5.65879e9 −1.53650 −0.768251 0.640149i \(-0.778873\pi\)
−0.768251 + 0.640149i \(0.778873\pi\)
\(542\) −2.19497e9 −0.592148
\(543\) −2.51252e9 −0.673458
\(544\) 6.38645e9 1.70084
\(545\) 3.90556e9 1.03346
\(546\) 3.56585e9 0.937538
\(547\) 5.02175e8 0.131190 0.0655948 0.997846i \(-0.479106\pi\)
0.0655948 + 0.997846i \(0.479106\pi\)
\(548\) −2.93382e9 −0.761555
\(549\) 4.58253e8 0.118196
\(550\) −3.57995e9 −0.917503
\(551\) 1.10033e9 0.280216
\(552\) 1.56291e8 0.0395501
\(553\) 4.92348e9 1.23804
\(554\) −7.08533e8 −0.177042
\(555\) 5.62897e9 1.39767
\(556\) 3.53654e9 0.872604
\(557\) −2.86842e9 −0.703314 −0.351657 0.936129i \(-0.614382\pi\)
−0.351657 + 0.936129i \(0.614382\pi\)
\(558\) 2.92981e8 0.0713870
\(559\) 5.27106e9 1.27631
\(560\) −6.47615e9 −1.55833
\(561\) −1.74707e9 −0.417772
\(562\) 1.03027e10 2.44835
\(563\) 3.72076e9 0.878723 0.439361 0.898310i \(-0.355205\pi\)
0.439361 + 0.898310i \(0.355205\pi\)
\(564\) 2.28792e9 0.536987
\(565\) −6.29871e9 −1.46920
\(566\) 3.55191e9 0.823388
\(567\) −4.32940e8 −0.0997440
\(568\) −1.40143e9 −0.320888
\(569\) −5.12299e9 −1.16582 −0.582909 0.812538i \(-0.698085\pi\)
−0.582909 + 0.812538i \(0.698085\pi\)
\(570\) 5.81471e9 1.31512
\(571\) −3.51571e9 −0.790289 −0.395145 0.918619i \(-0.629306\pi\)
−0.395145 + 0.918619i \(0.629306\pi\)
\(572\) −2.65616e9 −0.593427
\(573\) −3.46326e9 −0.769031
\(574\) −4.53769e9 −1.00148
\(575\) 1.46585e9 0.321552
\(576\) −8.76815e8 −0.191174
\(577\) −9.70425e8 −0.210304 −0.105152 0.994456i \(-0.533533\pi\)
−0.105152 + 0.994456i \(0.533533\pi\)
\(578\) 4.59988e9 0.990829
\(579\) 2.02332e9 0.433202
\(580\) −1.39808e9 −0.297533
\(581\) 3.84613e9 0.813593
\(582\) 5.17310e9 1.08773
\(583\) 4.32046e9 0.903004
\(584\) −2.11643e9 −0.439702
\(585\) −3.25863e9 −0.672959
\(586\) 4.98516e9 1.02338
\(587\) 5.54244e9 1.13101 0.565506 0.824744i \(-0.308681\pi\)
0.565506 + 0.824744i \(0.308681\pi\)
\(588\) 4.43386e8 0.0899418
\(589\) 8.95768e8 0.180631
\(590\) 1.30648e9 0.261891
\(591\) 3.85792e9 0.768771
\(592\) −9.44893e9 −1.87179
\(593\) −5.71802e8 −0.112604 −0.0563021 0.998414i \(-0.517931\pi\)
−0.0563021 + 0.998414i \(0.517931\pi\)
\(594\) 7.24388e8 0.141814
\(595\) −9.11140e9 −1.77327
\(596\) 4.93691e8 0.0955199
\(597\) 2.04986e9 0.394289
\(598\) 2.44297e9 0.467158
\(599\) 1.65083e9 0.313840 0.156920 0.987611i \(-0.449843\pi\)
0.156920 + 0.987611i \(0.449843\pi\)
\(600\) 1.00888e9 0.190682
\(601\) 4.29026e9 0.806163 0.403081 0.915164i \(-0.367939\pi\)
0.403081 + 0.915164i \(0.367939\pi\)
\(602\) −6.11098e9 −1.14162
\(603\) 1.35484e9 0.251638
\(604\) −5.23158e8 −0.0966059
\(605\) 5.70265e9 1.04697
\(606\) −5.86791e8 −0.107110
\(607\) 8.68657e8 0.157648 0.0788239 0.996889i \(-0.474884\pi\)
0.0788239 + 0.996889i \(0.474884\pi\)
\(608\) −8.09613e9 −1.46088
\(609\) 7.14894e8 0.128257
\(610\) −3.99875e9 −0.713296
\(611\) −8.80556e9 −1.56175
\(612\) −1.99959e9 −0.352623
\(613\) 2.12755e9 0.373050 0.186525 0.982450i \(-0.440277\pi\)
0.186525 + 0.982450i \(0.440277\pi\)
\(614\) 2.77333e9 0.483517
\(615\) 4.14673e9 0.718858
\(616\) −7.58226e8 −0.130697
\(617\) −8.95889e9 −1.53552 −0.767761 0.640736i \(-0.778629\pi\)
−0.767761 + 0.640736i \(0.778629\pi\)
\(618\) −2.30029e9 −0.392033
\(619\) −5.17081e9 −0.876277 −0.438138 0.898908i \(-0.644362\pi\)
−0.438138 + 0.898908i \(0.644362\pi\)
\(620\) −1.13816e9 −0.191794
\(621\) −2.96608e8 −0.0497007
\(622\) −8.87683e9 −1.47908
\(623\) 7.13105e8 0.118153
\(624\) 5.47001e9 0.901244
\(625\) −4.24082e9 −0.694815
\(626\) 1.41616e10 2.30728
\(627\) 2.21477e9 0.358832
\(628\) 2.06700e9 0.333028
\(629\) −1.32938e10 −2.12997
\(630\) 3.77787e9 0.601943
\(631\) −2.71332e9 −0.429931 −0.214965 0.976622i \(-0.568964\pi\)
−0.214965 + 0.976622i \(0.568964\pi\)
\(632\) 2.32156e9 0.365822
\(633\) 1.86549e9 0.292335
\(634\) 1.35751e10 2.11559
\(635\) −7.47206e9 −1.15806
\(636\) 4.94493e9 0.762185
\(637\) −1.70647e9 −0.261583
\(638\) −1.19615e9 −0.182353
\(639\) 2.65962e9 0.403243
\(640\) −5.16871e9 −0.779386
\(641\) 3.35162e9 0.502634 0.251317 0.967905i \(-0.419136\pi\)
0.251317 + 0.967905i \(0.419136\pi\)
\(642\) −8.92030e9 −1.33048
\(643\) 7.57020e9 1.12297 0.561486 0.827486i \(-0.310230\pi\)
0.561486 + 0.827486i \(0.310230\pi\)
\(644\) −1.26089e9 −0.186027
\(645\) 5.58447e9 0.819451
\(646\) −1.37325e10 −2.00418
\(647\) −3.82356e9 −0.555013 −0.277507 0.960724i \(-0.589508\pi\)
−0.277507 + 0.960724i \(0.589508\pi\)
\(648\) −2.04143e8 −0.0294728
\(649\) 4.97626e8 0.0714573
\(650\) 1.57697e10 2.25230
\(651\) 5.81988e8 0.0826763
\(652\) −6.94344e9 −0.981088
\(653\) −9.84140e9 −1.38312 −0.691561 0.722318i \(-0.743077\pi\)
−0.691561 + 0.722318i \(0.743077\pi\)
\(654\) 3.82443e9 0.534619
\(655\) −1.58455e9 −0.220324
\(656\) −6.96081e9 −0.962712
\(657\) 4.01653e9 0.552551
\(658\) 1.02087e10 1.39694
\(659\) 1.04889e10 1.42768 0.713842 0.700307i \(-0.246954\pi\)
0.713842 + 0.700307i \(0.246954\pi\)
\(660\) −2.81409e9 −0.381008
\(661\) −5.13152e8 −0.0691100 −0.0345550 0.999403i \(-0.511001\pi\)
−0.0345550 + 0.999403i \(0.511001\pi\)
\(662\) 1.37866e10 1.84694
\(663\) 7.69584e9 1.02555
\(664\) 1.81356e9 0.240404
\(665\) 1.15506e10 1.52310
\(666\) 5.51204e9 0.723024
\(667\) 4.89776e8 0.0639082
\(668\) 1.13794e10 1.47707
\(669\) 6.03875e9 0.779750
\(670\) −1.18224e10 −1.51861
\(671\) −1.52309e9 −0.194624
\(672\) −5.26013e9 −0.668658
\(673\) 4.59671e8 0.0581292 0.0290646 0.999578i \(-0.490747\pi\)
0.0290646 + 0.999578i \(0.490747\pi\)
\(674\) 1.19905e10 1.50844
\(675\) −1.91464e9 −0.239621
\(676\) 5.25547e9 0.654333
\(677\) −1.30244e10 −1.61324 −0.806619 0.591072i \(-0.798705\pi\)
−0.806619 + 0.591072i \(0.798705\pi\)
\(678\) −6.16787e9 −0.760031
\(679\) 1.02760e10 1.25974
\(680\) −4.29628e9 −0.523975
\(681\) 7.08939e8 0.0860189
\(682\) −9.73774e8 −0.117547
\(683\) −1.27703e9 −0.153366 −0.0766830 0.997056i \(-0.524433\pi\)
−0.0766830 + 0.997056i \(0.524433\pi\)
\(684\) 2.53489e9 0.302874
\(685\) 1.19628e10 1.42205
\(686\) 1.21688e10 1.43917
\(687\) −7.51622e9 −0.884404
\(688\) −9.37423e9 −1.09743
\(689\) −1.90316e10 −2.21671
\(690\) 2.58823e9 0.299937
\(691\) 1.46413e10 1.68813 0.844063 0.536243i \(-0.180157\pi\)
0.844063 + 0.536243i \(0.180157\pi\)
\(692\) 1.17451e10 1.34736
\(693\) 1.43895e9 0.164241
\(694\) −8.03356e9 −0.912326
\(695\) −1.44205e10 −1.62942
\(696\) 3.37092e8 0.0378980
\(697\) −9.79328e9 −1.09550
\(698\) 3.03853e9 0.338197
\(699\) −1.17088e9 −0.129671
\(700\) −8.13922e9 −0.896891
\(701\) −1.79391e10 −1.96692 −0.983460 0.181127i \(-0.942026\pi\)
−0.983460 + 0.181127i \(0.942026\pi\)
\(702\) −3.19094e9 −0.348127
\(703\) 1.68527e10 1.82947
\(704\) 2.91425e9 0.314791
\(705\) −9.32913e9 −1.00272
\(706\) 3.83573e9 0.410234
\(707\) −1.16563e9 −0.124048
\(708\) 5.69552e8 0.0603138
\(709\) −1.11761e10 −1.17769 −0.588843 0.808248i \(-0.700416\pi\)
−0.588843 + 0.808248i \(0.700416\pi\)
\(710\) −2.32081e10 −2.43352
\(711\) −4.40582e9 −0.459710
\(712\) 3.36248e8 0.0349125
\(713\) 3.98721e8 0.0411961
\(714\) −8.92213e9 −0.917329
\(715\) 1.08306e10 1.10811
\(716\) −1.94375e9 −0.197899
\(717\) 8.87954e9 0.899650
\(718\) −1.03761e10 −1.04616
\(719\) 9.74647e9 0.977904 0.488952 0.872311i \(-0.337379\pi\)
0.488952 + 0.872311i \(0.337379\pi\)
\(720\) 5.79525e9 0.578640
\(721\) −4.56938e9 −0.454029
\(722\) 3.83163e9 0.378881
\(723\) −8.95222e8 −0.0880940
\(724\) 9.55784e9 0.935997
\(725\) 3.16157e9 0.308120
\(726\) 5.58419e9 0.541604
\(727\) 1.00742e10 0.972389 0.486195 0.873851i \(-0.338385\pi\)
0.486195 + 0.873851i \(0.338385\pi\)
\(728\) 3.33999e9 0.320838
\(729\) 3.87420e8 0.0370370
\(730\) −3.50486e10 −3.33458
\(731\) −1.31888e10 −1.24880
\(732\) −1.74323e9 −0.164273
\(733\) 1.97006e10 1.84763 0.923814 0.382841i \(-0.125054\pi\)
0.923814 + 0.382841i \(0.125054\pi\)
\(734\) 7.73709e9 0.722173
\(735\) −1.80793e9 −0.167949
\(736\) −3.60373e9 −0.333180
\(737\) −4.50305e9 −0.414353
\(738\) 4.06060e9 0.371871
\(739\) −9.78856e9 −0.892202 −0.446101 0.894983i \(-0.647188\pi\)
−0.446101 + 0.894983i \(0.647188\pi\)
\(740\) −2.14131e10 −1.94253
\(741\) −9.75606e9 −0.880868
\(742\) 2.20642e10 1.98278
\(743\) 7.25852e9 0.649213 0.324607 0.945849i \(-0.394768\pi\)
0.324607 + 0.945849i \(0.394768\pi\)
\(744\) 2.74423e8 0.0244296
\(745\) −2.01306e9 −0.178365
\(746\) 1.18218e10 1.04255
\(747\) −3.44175e9 −0.302104
\(748\) 6.64598e9 0.580636
\(749\) −1.77196e10 −1.54088
\(750\) 3.28894e9 0.284670
\(751\) 1.17654e10 1.01360 0.506799 0.862064i \(-0.330829\pi\)
0.506799 + 0.862064i \(0.330829\pi\)
\(752\) 1.56601e10 1.34287
\(753\) −7.96013e9 −0.679419
\(754\) 5.26905e9 0.447644
\(755\) 2.13321e9 0.180393
\(756\) 1.64694e9 0.138628
\(757\) −1.11582e10 −0.934886 −0.467443 0.884023i \(-0.654825\pi\)
−0.467443 + 0.884023i \(0.654825\pi\)
\(758\) 2.30537e10 1.92264
\(759\) 9.85830e8 0.0818382
\(760\) 5.44641e9 0.450052
\(761\) −5.87504e9 −0.483242 −0.241621 0.970371i \(-0.577679\pi\)
−0.241621 + 0.970371i \(0.577679\pi\)
\(762\) −7.31684e9 −0.599075
\(763\) 7.59700e9 0.619165
\(764\) 1.31745e10 1.06883
\(765\) 8.15343e9 0.658454
\(766\) 2.05577e10 1.65262
\(767\) −2.19204e9 −0.175415
\(768\) −9.21810e9 −0.734306
\(769\) 1.13603e10 0.900841 0.450420 0.892817i \(-0.351274\pi\)
0.450420 + 0.892817i \(0.351274\pi\)
\(770\) −1.25564e10 −0.991171
\(771\) 6.93690e9 0.545099
\(772\) −7.69688e9 −0.602080
\(773\) −1.76385e10 −1.37352 −0.686759 0.726885i \(-0.740967\pi\)
−0.686759 + 0.726885i \(0.740967\pi\)
\(774\) 5.46847e9 0.423909
\(775\) 2.57380e9 0.198618
\(776\) 4.84544e9 0.372235
\(777\) 1.09493e10 0.837364
\(778\) −3.29074e8 −0.0250533
\(779\) 1.24150e10 0.940947
\(780\) 1.23961e10 0.935304
\(781\) −8.83974e9 −0.663989
\(782\) −6.11257e9 −0.457089
\(783\) −6.39730e8 −0.0476245
\(784\) 3.03484e9 0.224921
\(785\) −8.42830e9 −0.621865
\(786\) −1.55164e9 −0.113975
\(787\) −2.46188e10 −1.80034 −0.900172 0.435534i \(-0.856560\pi\)
−0.900172 + 0.435534i \(0.856560\pi\)
\(788\) −1.46758e10 −1.06847
\(789\) −8.14472e9 −0.590346
\(790\) 3.84456e10 2.77429
\(791\) −1.22521e10 −0.880224
\(792\) 6.78506e8 0.0485306
\(793\) 6.70920e9 0.477766
\(794\) 1.40911e10 0.999017
\(795\) −2.01632e10 −1.42323
\(796\) −7.79785e9 −0.547998
\(797\) −1.36580e10 −0.955612 −0.477806 0.878465i \(-0.658568\pi\)
−0.477806 + 0.878465i \(0.658568\pi\)
\(798\) 1.13106e10 0.787911
\(799\) 2.20325e10 1.52809
\(800\) −2.32625e10 −1.60636
\(801\) −6.38129e8 −0.0438727
\(802\) −4.39542e9 −0.300878
\(803\) −1.33497e10 −0.909843
\(804\) −5.15392e9 −0.349737
\(805\) 5.14136e9 0.347370
\(806\) 4.28948e9 0.288557
\(807\) 7.06235e9 0.473034
\(808\) −5.49624e8 −0.0366544
\(809\) −4.59136e9 −0.304875 −0.152438 0.988313i \(-0.548712\pi\)
−0.152438 + 0.988313i \(0.548712\pi\)
\(810\) −3.38066e9 −0.223514
\(811\) −2.86469e10 −1.88584 −0.942919 0.333022i \(-0.891931\pi\)
−0.942919 + 0.333022i \(0.891931\pi\)
\(812\) −2.71952e9 −0.178257
\(813\) 3.90174e9 0.254649
\(814\) −1.83203e10 −1.19055
\(815\) 2.83123e10 1.83199
\(816\) −1.36865e10 −0.881817
\(817\) 1.67195e10 1.07262
\(818\) 2.81513e10 1.79830
\(819\) −6.33860e9 −0.403181
\(820\) −1.57745e10 −0.999096
\(821\) −1.33387e10 −0.841225 −0.420613 0.907240i \(-0.638185\pi\)
−0.420613 + 0.907240i \(0.638185\pi\)
\(822\) 1.17143e10 0.735640
\(823\) 1.33719e10 0.836171 0.418085 0.908408i \(-0.362701\pi\)
0.418085 + 0.908408i \(0.362701\pi\)
\(824\) −2.15459e9 −0.134159
\(825\) 6.36366e9 0.394565
\(826\) 2.54133e9 0.156903
\(827\) −1.27792e10 −0.785658 −0.392829 0.919612i \(-0.628503\pi\)
−0.392829 + 0.919612i \(0.628503\pi\)
\(828\) 1.12832e9 0.0690759
\(829\) 3.01708e10 1.83927 0.919635 0.392775i \(-0.128485\pi\)
0.919635 + 0.392775i \(0.128485\pi\)
\(830\) 3.00330e10 1.82316
\(831\) 1.25948e9 0.0761354
\(832\) −1.28373e10 −0.772755
\(833\) 4.26976e9 0.255945
\(834\) −1.41209e10 −0.842911
\(835\) −4.64002e10 −2.75814
\(836\) −8.42515e9 −0.498719
\(837\) −5.20798e8 −0.0306994
\(838\) 3.48613e10 2.04639
\(839\) 1.99280e10 1.16492 0.582460 0.812859i \(-0.302090\pi\)
0.582460 + 0.812859i \(0.302090\pi\)
\(840\) 3.53858e9 0.205993
\(841\) −1.61935e10 −0.938761
\(842\) 2.01629e10 1.16402
\(843\) −1.83139e10 −1.05289
\(844\) −7.09649e9 −0.406298
\(845\) −2.14295e10 −1.22184
\(846\) −9.13534e9 −0.518714
\(847\) 1.10927e10 0.627254
\(848\) 3.38465e10 1.90603
\(849\) −6.31383e9 −0.354092
\(850\) −3.94575e10 −2.20375
\(851\) 7.50142e9 0.417244
\(852\) −1.01174e10 −0.560443
\(853\) −2.14201e10 −1.18168 −0.590840 0.806789i \(-0.701204\pi\)
−0.590840 + 0.806789i \(0.701204\pi\)
\(854\) −7.77828e9 −0.427347
\(855\) −1.03361e10 −0.565558
\(856\) −8.35529e9 −0.455306
\(857\) 3.13832e10 1.70320 0.851599 0.524195i \(-0.175634\pi\)
0.851599 + 0.524195i \(0.175634\pi\)
\(858\) 1.06056e10 0.573234
\(859\) −1.10229e10 −0.593362 −0.296681 0.954977i \(-0.595880\pi\)
−0.296681 + 0.954977i \(0.595880\pi\)
\(860\) −2.12438e10 −1.13890
\(861\) 8.06613e9 0.430679
\(862\) 7.51905e9 0.399841
\(863\) 2.63685e10 1.39652 0.698260 0.715844i \(-0.253958\pi\)
0.698260 + 0.715844i \(0.253958\pi\)
\(864\) 4.70708e9 0.248287
\(865\) −4.78912e10 −2.51594
\(866\) −2.76344e10 −1.44590
\(867\) −8.17667e9 −0.426098
\(868\) −2.21393e9 −0.114907
\(869\) 1.46436e10 0.756967
\(870\) 5.58234e9 0.287408
\(871\) 1.98360e10 1.01716
\(872\) 3.58220e9 0.182954
\(873\) −9.19562e9 −0.467769
\(874\) 7.74894e9 0.392602
\(875\) 6.53329e9 0.329689
\(876\) −1.52792e10 −0.767957
\(877\) −9.58767e9 −0.479970 −0.239985 0.970777i \(-0.577143\pi\)
−0.239985 + 0.970777i \(0.577143\pi\)
\(878\) −4.98430e10 −2.48527
\(879\) −8.86154e9 −0.440097
\(880\) −1.92616e10 −0.952801
\(881\) −2.62769e10 −1.29467 −0.647334 0.762206i \(-0.724116\pi\)
−0.647334 + 0.762206i \(0.724116\pi\)
\(882\) −1.77038e9 −0.0868812
\(883\) −9.16067e9 −0.447780 −0.223890 0.974614i \(-0.571876\pi\)
−0.223890 + 0.974614i \(0.571876\pi\)
\(884\) −2.92756e10 −1.42535
\(885\) −2.32238e9 −0.112624
\(886\) −3.18790e10 −1.53988
\(887\) −2.61104e10 −1.25626 −0.628132 0.778107i \(-0.716180\pi\)
−0.628132 + 0.778107i \(0.716180\pi\)
\(888\) 5.16291e9 0.247428
\(889\) −1.45345e10 −0.693814
\(890\) 5.56837e9 0.264766
\(891\) −1.28766e9 −0.0609860
\(892\) −2.29719e10 −1.08373
\(893\) −2.79307e10 −1.31251
\(894\) −1.97124e9 −0.0922695
\(895\) 7.92574e9 0.369538
\(896\) −1.00541e10 −0.466942
\(897\) −4.34259e9 −0.200898
\(898\) 1.35593e9 0.0624844
\(899\) 8.59971e8 0.0394752
\(900\) 7.28346e9 0.333034
\(901\) 4.76192e10 2.16893
\(902\) −1.34961e10 −0.612331
\(903\) 1.08628e10 0.490946
\(904\) −5.77721e9 −0.260093
\(905\) −3.89726e10 −1.74779
\(906\) 2.08890e9 0.0933185
\(907\) 7.07125e9 0.314681 0.157340 0.987544i \(-0.449708\pi\)
0.157340 + 0.987544i \(0.449708\pi\)
\(908\) −2.69686e9 −0.119552
\(909\) 1.04307e9 0.0460618
\(910\) 5.53112e10 2.43314
\(911\) 2.11411e10 0.926432 0.463216 0.886245i \(-0.346695\pi\)
0.463216 + 0.886245i \(0.346695\pi\)
\(912\) 1.73505e10 0.757409
\(913\) 1.14393e10 0.497451
\(914\) −2.55707e10 −1.10772
\(915\) 7.10812e9 0.306748
\(916\) 2.85923e10 1.22918
\(917\) −3.08223e9 −0.132000
\(918\) 7.98406e9 0.340623
\(919\) 3.00700e9 0.127799 0.0638997 0.997956i \(-0.479646\pi\)
0.0638997 + 0.997956i \(0.479646\pi\)
\(920\) 2.42429e9 0.102642
\(921\) −4.92983e9 −0.207933
\(922\) −2.92293e10 −1.22817
\(923\) 3.89391e10 1.62997
\(924\) −5.47390e9 −0.228268
\(925\) 4.84226e10 2.01165
\(926\) −2.16509e10 −0.896063
\(927\) 4.08895e9 0.168591
\(928\) −7.77259e9 −0.319262
\(929\) −2.87222e9 −0.117534 −0.0587668 0.998272i \(-0.518717\pi\)
−0.0587668 + 0.998272i \(0.518717\pi\)
\(930\) 4.54453e9 0.185267
\(931\) −5.41280e9 −0.219836
\(932\) 4.45412e9 0.180221
\(933\) 1.57793e10 0.636066
\(934\) 2.54994e10 1.02404
\(935\) −2.70994e10 −1.08422
\(936\) −2.98882e9 −0.119134
\(937\) −2.25687e10 −0.896227 −0.448114 0.893977i \(-0.647904\pi\)
−0.448114 + 0.893977i \(0.647904\pi\)
\(938\) −2.29967e10 −0.909821
\(939\) −2.51734e10 −0.992228
\(940\) 3.54887e10 1.39362
\(941\) 1.01256e10 0.396149 0.198075 0.980187i \(-0.436531\pi\)
0.198075 + 0.980187i \(0.436531\pi\)
\(942\) −8.25323e9 −0.321696
\(943\) 5.52612e9 0.214600
\(944\) 3.89841e9 0.150829
\(945\) −6.71548e9 −0.258861
\(946\) −1.81754e10 −0.698017
\(947\) −6.73699e9 −0.257775 −0.128888 0.991659i \(-0.541141\pi\)
−0.128888 + 0.991659i \(0.541141\pi\)
\(948\) 1.67601e10 0.638922
\(949\) 5.88054e10 2.23350
\(950\) 5.00204e10 1.89284
\(951\) −2.41310e10 −0.909794
\(952\) −8.35701e9 −0.313922
\(953\) 2.23147e10 0.835153 0.417577 0.908642i \(-0.362880\pi\)
0.417577 + 0.908642i \(0.362880\pi\)
\(954\) −1.97444e10 −0.736249
\(955\) −5.37199e10 −1.99583
\(956\) −3.37785e10 −1.25037
\(957\) 2.12626e9 0.0784196
\(958\) 3.72754e10 1.36976
\(959\) 2.32698e10 0.851975
\(960\) −1.36006e10 −0.496145
\(961\) −2.68125e10 −0.974554
\(962\) 8.07009e10 2.92257
\(963\) 1.58566e10 0.572160
\(964\) 3.40549e9 0.122436
\(965\) 3.13845e10 1.12427
\(966\) 5.03456e9 0.179697
\(967\) −2.22573e9 −0.0791554 −0.0395777 0.999216i \(-0.512601\pi\)
−0.0395777 + 0.999216i \(0.512601\pi\)
\(968\) 5.23049e9 0.185344
\(969\) 2.44107e10 0.861881
\(970\) 8.02417e10 2.82292
\(971\) −3.19447e9 −0.111978 −0.0559888 0.998431i \(-0.517831\pi\)
−0.0559888 + 0.998431i \(0.517831\pi\)
\(972\) −1.47378e9 −0.0514755
\(973\) −2.80503e10 −0.976210
\(974\) 1.86612e10 0.647116
\(975\) −2.80320e10 −0.968585
\(976\) −1.19319e10 −0.410804
\(977\) −1.60496e10 −0.550598 −0.275299 0.961359i \(-0.588777\pi\)
−0.275299 + 0.961359i \(0.588777\pi\)
\(978\) 2.77242e10 0.947703
\(979\) 2.12094e9 0.0722417
\(980\) 6.87752e9 0.233421
\(981\) −6.79825e9 −0.229909
\(982\) −6.08022e10 −2.04894
\(983\) 3.86768e10 1.29871 0.649357 0.760484i \(-0.275038\pi\)
0.649357 + 0.760484i \(0.275038\pi\)
\(984\) 3.80340e9 0.127259
\(985\) 5.98416e10 1.99515
\(986\) −1.31837e10 −0.437995
\(987\) −1.81468e10 −0.600745
\(988\) 3.71128e10 1.22426
\(989\) 7.44211e9 0.244630
\(990\) 1.12362e10 0.368043
\(991\) 1.63094e10 0.532328 0.266164 0.963928i \(-0.414244\pi\)
0.266164 + 0.963928i \(0.414244\pi\)
\(992\) −6.32759e9 −0.205801
\(993\) −2.45068e10 −0.794262
\(994\) −4.51438e10 −1.45796
\(995\) 3.17962e10 1.02328
\(996\) 1.30927e10 0.419876
\(997\) 2.35819e10 0.753608 0.376804 0.926293i \(-0.377023\pi\)
0.376804 + 0.926293i \(0.377023\pi\)
\(998\) −9.68298e9 −0.308356
\(999\) −9.79812e9 −0.310931
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.1.14 17
3.2 odd 2 531.8.a.c.1.4 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.14 17 1.1 even 1 trivial
531.8.a.c.1.4 17 3.2 odd 2