Properties

Label 4598.2.a.by.1.4
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 16x^{6} - 4x^{5} + 75x^{4} + 32x^{3} - 90x^{2} - 28x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.02984\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.55594 q^{3} +1.00000 q^{4} +2.24033 q^{5} -1.55594 q^{6} +2.27752 q^{7} -1.00000 q^{8} -0.579065 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.55594 q^{3} +1.00000 q^{4} +2.24033 q^{5} -1.55594 q^{6} +2.27752 q^{7} -1.00000 q^{8} -0.579065 q^{9} -2.24033 q^{10} +1.55594 q^{12} -2.75611 q^{13} -2.27752 q^{14} +3.48581 q^{15} +1.00000 q^{16} +0.571514 q^{17} +0.579065 q^{18} +1.00000 q^{19} +2.24033 q^{20} +3.54367 q^{21} +9.47493 q^{23} -1.55594 q^{24} +0.0190711 q^{25} +2.75611 q^{26} -5.56879 q^{27} +2.27752 q^{28} -1.99533 q^{29} -3.48581 q^{30} +8.92659 q^{31} -1.00000 q^{32} -0.571514 q^{34} +5.10239 q^{35} -0.579065 q^{36} +7.48207 q^{37} -1.00000 q^{38} -4.28832 q^{39} -2.24033 q^{40} +0.0862369 q^{41} -3.54367 q^{42} -5.53342 q^{43} -1.29730 q^{45} -9.47493 q^{46} -9.68313 q^{47} +1.55594 q^{48} -1.81292 q^{49} -0.0190711 q^{50} +0.889238 q^{51} -2.75611 q^{52} +4.40072 q^{53} +5.56879 q^{54} -2.27752 q^{56} +1.55594 q^{57} +1.99533 q^{58} +8.02502 q^{59} +3.48581 q^{60} -3.73883 q^{61} -8.92659 q^{62} -1.31883 q^{63} +1.00000 q^{64} -6.17458 q^{65} +6.91634 q^{67} +0.571514 q^{68} +14.7424 q^{69} -5.10239 q^{70} +4.91454 q^{71} +0.579065 q^{72} +10.7414 q^{73} -7.48207 q^{74} +0.0296734 q^{75} +1.00000 q^{76} +4.28832 q^{78} +5.92179 q^{79} +2.24033 q^{80} -6.92749 q^{81} -0.0862369 q^{82} +16.8470 q^{83} +3.54367 q^{84} +1.28038 q^{85} +5.53342 q^{86} -3.10460 q^{87} +5.14468 q^{89} +1.29730 q^{90} -6.27708 q^{91} +9.47493 q^{92} +13.8892 q^{93} +9.68313 q^{94} +2.24033 q^{95} -1.55594 q^{96} +3.98358 q^{97} +1.81292 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9} + 8 q^{12} + 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} + 4 q^{17} - 22 q^{18} + 8 q^{19} + 20 q^{21} + 14 q^{23} - 8 q^{24} + 36 q^{25} - 12 q^{26} + 32 q^{27} - 4 q^{28} + 2 q^{29} - 4 q^{30} - 8 q^{32} - 4 q^{34} - 36 q^{35} + 22 q^{36} + 24 q^{37} - 8 q^{38} - 16 q^{39} - 8 q^{41} - 20 q^{42} - 8 q^{43} + 16 q^{45} - 14 q^{46} - 16 q^{47} + 8 q^{48} + 34 q^{49} - 36 q^{50} - 18 q^{51} + 12 q^{52} + 36 q^{53} - 32 q^{54} + 4 q^{56} + 8 q^{57} - 2 q^{58} - 24 q^{59} + 4 q^{60} - 12 q^{61} - 24 q^{63} + 8 q^{64} - 16 q^{65} + 16 q^{67} + 4 q^{68} + 4 q^{69} + 36 q^{70} + 4 q^{71} - 22 q^{72} + 20 q^{73} - 24 q^{74} + 40 q^{75} + 8 q^{76} + 16 q^{78} + 12 q^{79} + 40 q^{81} + 8 q^{82} - 20 q^{83} + 20 q^{84} - 12 q^{85} + 8 q^{86} + 36 q^{87} + 8 q^{89} - 16 q^{90} - 24 q^{91} + 14 q^{92} + 12 q^{93} + 16 q^{94} - 8 q^{96} + 4 q^{97} - 34 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.55594 0.898320 0.449160 0.893451i \(-0.351723\pi\)
0.449160 + 0.893451i \(0.351723\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.24033 1.00191 0.500953 0.865475i \(-0.332983\pi\)
0.500953 + 0.865475i \(0.332983\pi\)
\(6\) −1.55594 −0.635208
\(7\) 2.27752 0.860821 0.430410 0.902633i \(-0.358369\pi\)
0.430410 + 0.902633i \(0.358369\pi\)
\(8\) −1.00000 −0.353553
\(9\) −0.579065 −0.193022
\(10\) −2.24033 −0.708454
\(11\) 0 0
\(12\) 1.55594 0.449160
\(13\) −2.75611 −0.764406 −0.382203 0.924078i \(-0.624835\pi\)
−0.382203 + 0.924078i \(0.624835\pi\)
\(14\) −2.27752 −0.608692
\(15\) 3.48581 0.900031
\(16\) 1.00000 0.250000
\(17\) 0.571514 0.138612 0.0693062 0.997595i \(-0.477921\pi\)
0.0693062 + 0.997595i \(0.477921\pi\)
\(18\) 0.579065 0.136487
\(19\) 1.00000 0.229416
\(20\) 2.24033 0.500953
\(21\) 3.54367 0.773292
\(22\) 0 0
\(23\) 9.47493 1.97566 0.987830 0.155537i \(-0.0497107\pi\)
0.987830 + 0.155537i \(0.0497107\pi\)
\(24\) −1.55594 −0.317604
\(25\) 0.0190711 0.00381423
\(26\) 2.75611 0.540517
\(27\) −5.56879 −1.07171
\(28\) 2.27752 0.430410
\(29\) −1.99533 −0.370523 −0.185261 0.982689i \(-0.559313\pi\)
−0.185261 + 0.982689i \(0.559313\pi\)
\(30\) −3.48581 −0.636418
\(31\) 8.92659 1.60326 0.801631 0.597819i \(-0.203966\pi\)
0.801631 + 0.597819i \(0.203966\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −0.571514 −0.0980138
\(35\) 5.10239 0.862461
\(36\) −0.579065 −0.0965108
\(37\) 7.48207 1.23004 0.615022 0.788510i \(-0.289147\pi\)
0.615022 + 0.788510i \(0.289147\pi\)
\(38\) −1.00000 −0.162221
\(39\) −4.28832 −0.686681
\(40\) −2.24033 −0.354227
\(41\) 0.0862369 0.0134679 0.00673397 0.999977i \(-0.497856\pi\)
0.00673397 + 0.999977i \(0.497856\pi\)
\(42\) −3.54367 −0.546800
\(43\) −5.53342 −0.843839 −0.421920 0.906633i \(-0.638644\pi\)
−0.421920 + 0.906633i \(0.638644\pi\)
\(44\) 0 0
\(45\) −1.29730 −0.193389
\(46\) −9.47493 −1.39700
\(47\) −9.68313 −1.41243 −0.706215 0.707997i \(-0.749599\pi\)
−0.706215 + 0.707997i \(0.749599\pi\)
\(48\) 1.55594 0.224580
\(49\) −1.81292 −0.258988
\(50\) −0.0190711 −0.00269706
\(51\) 0.889238 0.124518
\(52\) −2.75611 −0.382203
\(53\) 4.40072 0.604485 0.302243 0.953231i \(-0.402265\pi\)
0.302243 + 0.953231i \(0.402265\pi\)
\(54\) 5.56879 0.757817
\(55\) 0 0
\(56\) −2.27752 −0.304346
\(57\) 1.55594 0.206089
\(58\) 1.99533 0.261999
\(59\) 8.02502 1.04477 0.522384 0.852710i \(-0.325043\pi\)
0.522384 + 0.852710i \(0.325043\pi\)
\(60\) 3.48581 0.450016
\(61\) −3.73883 −0.478709 −0.239354 0.970932i \(-0.576936\pi\)
−0.239354 + 0.970932i \(0.576936\pi\)
\(62\) −8.92659 −1.13368
\(63\) −1.31883 −0.166157
\(64\) 1.00000 0.125000
\(65\) −6.17458 −0.765863
\(66\) 0 0
\(67\) 6.91634 0.844965 0.422483 0.906371i \(-0.361159\pi\)
0.422483 + 0.906371i \(0.361159\pi\)
\(68\) 0.571514 0.0693062
\(69\) 14.7424 1.77477
\(70\) −5.10239 −0.609852
\(71\) 4.91454 0.583248 0.291624 0.956533i \(-0.405804\pi\)
0.291624 + 0.956533i \(0.405804\pi\)
\(72\) 0.579065 0.0682435
\(73\) 10.7414 1.25719 0.628593 0.777734i \(-0.283631\pi\)
0.628593 + 0.777734i \(0.283631\pi\)
\(74\) −7.48207 −0.869773
\(75\) 0.0296734 0.00342639
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 4.28832 0.485557
\(79\) 5.92179 0.666254 0.333127 0.942882i \(-0.391896\pi\)
0.333127 + 0.942882i \(0.391896\pi\)
\(80\) 2.24033 0.250476
\(81\) −6.92749 −0.769721
\(82\) −0.0862369 −0.00952328
\(83\) 16.8470 1.84920 0.924601 0.380938i \(-0.124399\pi\)
0.924601 + 0.380938i \(0.124399\pi\)
\(84\) 3.54367 0.386646
\(85\) 1.28038 0.138877
\(86\) 5.53342 0.596684
\(87\) −3.10460 −0.332848
\(88\) 0 0
\(89\) 5.14468 0.545335 0.272668 0.962108i \(-0.412094\pi\)
0.272668 + 0.962108i \(0.412094\pi\)
\(90\) 1.29730 0.136747
\(91\) −6.27708 −0.658017
\(92\) 9.47493 0.987830
\(93\) 13.8892 1.44024
\(94\) 9.68313 0.998739
\(95\) 2.24033 0.229853
\(96\) −1.55594 −0.158802
\(97\) 3.98358 0.404471 0.202236 0.979337i \(-0.435179\pi\)
0.202236 + 0.979337i \(0.435179\pi\)
\(98\) 1.81292 0.183132
\(99\) 0 0
\(100\) 0.0190711 0.00190711
\(101\) −1.22629 −0.122021 −0.0610104 0.998137i \(-0.519432\pi\)
−0.0610104 + 0.998137i \(0.519432\pi\)
\(102\) −0.889238 −0.0880477
\(103\) 6.31633 0.622367 0.311183 0.950350i \(-0.399275\pi\)
0.311183 + 0.950350i \(0.399275\pi\)
\(104\) 2.75611 0.270258
\(105\) 7.93898 0.774765
\(106\) −4.40072 −0.427436
\(107\) −15.9287 −1.53988 −0.769940 0.638116i \(-0.779714\pi\)
−0.769940 + 0.638116i \(0.779714\pi\)
\(108\) −5.56879 −0.535857
\(109\) −8.67279 −0.830703 −0.415352 0.909661i \(-0.636341\pi\)
−0.415352 + 0.909661i \(0.636341\pi\)
\(110\) 0 0
\(111\) 11.6416 1.10497
\(112\) 2.27752 0.215205
\(113\) −0.297807 −0.0280153 −0.0140077 0.999902i \(-0.504459\pi\)
−0.0140077 + 0.999902i \(0.504459\pi\)
\(114\) −1.55594 −0.145727
\(115\) 21.2270 1.97942
\(116\) −1.99533 −0.185261
\(117\) 1.59596 0.147547
\(118\) −8.02502 −0.738763
\(119\) 1.30163 0.119320
\(120\) −3.48581 −0.318209
\(121\) 0 0
\(122\) 3.73883 0.338498
\(123\) 0.134179 0.0120985
\(124\) 8.92659 0.801631
\(125\) −11.1589 −0.998084
\(126\) 1.31883 0.117491
\(127\) 0.637264 0.0565480 0.0282740 0.999600i \(-0.490999\pi\)
0.0282740 + 0.999600i \(0.490999\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.60965 −0.758037
\(130\) 6.17458 0.541547
\(131\) 21.8110 1.90564 0.952818 0.303542i \(-0.0981693\pi\)
0.952818 + 0.303542i \(0.0981693\pi\)
\(132\) 0 0
\(133\) 2.27752 0.197486
\(134\) −6.91634 −0.597481
\(135\) −12.4759 −1.07376
\(136\) −0.571514 −0.0490069
\(137\) −1.60089 −0.136773 −0.0683866 0.997659i \(-0.521785\pi\)
−0.0683866 + 0.997659i \(0.521785\pi\)
\(138\) −14.7424 −1.25496
\(139\) 15.9213 1.35043 0.675214 0.737621i \(-0.264051\pi\)
0.675214 + 0.737621i \(0.264051\pi\)
\(140\) 5.10239 0.431230
\(141\) −15.0663 −1.26881
\(142\) −4.91454 −0.412419
\(143\) 0 0
\(144\) −0.579065 −0.0482554
\(145\) −4.47019 −0.371229
\(146\) −10.7414 −0.888965
\(147\) −2.82078 −0.232654
\(148\) 7.48207 0.615022
\(149\) −4.19884 −0.343982 −0.171991 0.985098i \(-0.555020\pi\)
−0.171991 + 0.985098i \(0.555020\pi\)
\(150\) −0.0296734 −0.00242283
\(151\) 0.918503 0.0747467 0.0373733 0.999301i \(-0.488101\pi\)
0.0373733 + 0.999301i \(0.488101\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −0.330944 −0.0267552
\(154\) 0 0
\(155\) 19.9985 1.60632
\(156\) −4.28832 −0.343341
\(157\) −11.8346 −0.944505 −0.472252 0.881463i \(-0.656559\pi\)
−0.472252 + 0.881463i \(0.656559\pi\)
\(158\) −5.92179 −0.471113
\(159\) 6.84723 0.543021
\(160\) −2.24033 −0.177114
\(161\) 21.5793 1.70069
\(162\) 6.92749 0.544275
\(163\) −7.54426 −0.590912 −0.295456 0.955356i \(-0.595472\pi\)
−0.295456 + 0.955356i \(0.595472\pi\)
\(164\) 0.0862369 0.00673397
\(165\) 0 0
\(166\) −16.8470 −1.30758
\(167\) 9.75174 0.754612 0.377306 0.926089i \(-0.376850\pi\)
0.377306 + 0.926089i \(0.376850\pi\)
\(168\) −3.54367 −0.273400
\(169\) −5.40388 −0.415683
\(170\) −1.28038 −0.0982005
\(171\) −0.579065 −0.0442822
\(172\) −5.53342 −0.421920
\(173\) −9.90137 −0.752787 −0.376393 0.926460i \(-0.622836\pi\)
−0.376393 + 0.926460i \(0.622836\pi\)
\(174\) 3.10460 0.235359
\(175\) 0.0434348 0.00328336
\(176\) 0 0
\(177\) 12.4864 0.938536
\(178\) −5.14468 −0.385610
\(179\) 15.8720 1.18633 0.593164 0.805081i \(-0.297878\pi\)
0.593164 + 0.805081i \(0.297878\pi\)
\(180\) −1.29730 −0.0966947
\(181\) −17.7160 −1.31682 −0.658410 0.752660i \(-0.728771\pi\)
−0.658410 + 0.752660i \(0.728771\pi\)
\(182\) 6.27708 0.465288
\(183\) −5.81738 −0.430033
\(184\) −9.47493 −0.698501
\(185\) 16.7623 1.23239
\(186\) −13.8892 −1.01841
\(187\) 0 0
\(188\) −9.68313 −0.706215
\(189\) −12.6830 −0.922554
\(190\) −2.24033 −0.162531
\(191\) −17.5478 −1.26971 −0.634857 0.772629i \(-0.718941\pi\)
−0.634857 + 0.772629i \(0.718941\pi\)
\(192\) 1.55594 0.112290
\(193\) 13.1066 0.943437 0.471719 0.881749i \(-0.343634\pi\)
0.471719 + 0.881749i \(0.343634\pi\)
\(194\) −3.98358 −0.286004
\(195\) −9.60725 −0.687990
\(196\) −1.81292 −0.129494
\(197\) −15.2492 −1.08646 −0.543230 0.839584i \(-0.682799\pi\)
−0.543230 + 0.839584i \(0.682799\pi\)
\(198\) 0 0
\(199\) 13.4862 0.956009 0.478005 0.878357i \(-0.341360\pi\)
0.478005 + 0.878357i \(0.341360\pi\)
\(200\) −0.0190711 −0.00134853
\(201\) 10.7614 0.759049
\(202\) 1.22629 0.0862818
\(203\) −4.54439 −0.318954
\(204\) 0.889238 0.0622591
\(205\) 0.193199 0.0134936
\(206\) −6.31633 −0.440080
\(207\) −5.48660 −0.381345
\(208\) −2.75611 −0.191102
\(209\) 0 0
\(210\) −7.93898 −0.547842
\(211\) −2.15813 −0.148572 −0.0742858 0.997237i \(-0.523668\pi\)
−0.0742858 + 0.997237i \(0.523668\pi\)
\(212\) 4.40072 0.302243
\(213\) 7.64670 0.523943
\(214\) 15.9287 1.08886
\(215\) −12.3967 −0.845447
\(216\) 5.56879 0.378908
\(217\) 20.3305 1.38012
\(218\) 8.67279 0.587396
\(219\) 16.7129 1.12936
\(220\) 0 0
\(221\) −1.57515 −0.105956
\(222\) −11.6416 −0.781334
\(223\) 6.76418 0.452963 0.226482 0.974015i \(-0.427278\pi\)
0.226482 + 0.974015i \(0.427278\pi\)
\(224\) −2.27752 −0.152173
\(225\) −0.0110434 −0.000736228 0
\(226\) 0.297807 0.0198098
\(227\) 6.68912 0.443973 0.221986 0.975050i \(-0.428746\pi\)
0.221986 + 0.975050i \(0.428746\pi\)
\(228\) 1.55594 0.103044
\(229\) 0.645439 0.0426518 0.0213259 0.999773i \(-0.493211\pi\)
0.0213259 + 0.999773i \(0.493211\pi\)
\(230\) −21.2270 −1.39966
\(231\) 0 0
\(232\) 1.99533 0.131000
\(233\) −5.52617 −0.362031 −0.181016 0.983480i \(-0.557938\pi\)
−0.181016 + 0.983480i \(0.557938\pi\)
\(234\) −1.59596 −0.104331
\(235\) −21.6934 −1.41512
\(236\) 8.02502 0.522384
\(237\) 9.21393 0.598509
\(238\) −1.30163 −0.0843723
\(239\) −24.0112 −1.55316 −0.776578 0.630021i \(-0.783046\pi\)
−0.776578 + 0.630021i \(0.783046\pi\)
\(240\) 3.48581 0.225008
\(241\) 12.5817 0.810461 0.405231 0.914214i \(-0.367191\pi\)
0.405231 + 0.914214i \(0.367191\pi\)
\(242\) 0 0
\(243\) 5.92766 0.380259
\(244\) −3.73883 −0.239354
\(245\) −4.06153 −0.259481
\(246\) −0.134179 −0.00855495
\(247\) −2.75611 −0.175367
\(248\) −8.92659 −0.566839
\(249\) 26.2129 1.66117
\(250\) 11.1589 0.705752
\(251\) −27.2335 −1.71896 −0.859480 0.511169i \(-0.829213\pi\)
−0.859480 + 0.511169i \(0.829213\pi\)
\(252\) −1.31883 −0.0830785
\(253\) 0 0
\(254\) −0.637264 −0.0399855
\(255\) 1.99219 0.124756
\(256\) 1.00000 0.0625000
\(257\) −27.8288 −1.73591 −0.867956 0.496641i \(-0.834567\pi\)
−0.867956 + 0.496641i \(0.834567\pi\)
\(258\) 8.60965 0.536013
\(259\) 17.0405 1.05885
\(260\) −6.17458 −0.382931
\(261\) 1.15542 0.0715189
\(262\) −21.8110 −1.34749
\(263\) −29.9986 −1.84979 −0.924895 0.380222i \(-0.875848\pi\)
−0.924895 + 0.380222i \(0.875848\pi\)
\(264\) 0 0
\(265\) 9.85905 0.605637
\(266\) −2.27752 −0.139644
\(267\) 8.00479 0.489885
\(268\) 6.91634 0.422483
\(269\) −1.89146 −0.115325 −0.0576623 0.998336i \(-0.518365\pi\)
−0.0576623 + 0.998336i \(0.518365\pi\)
\(270\) 12.4759 0.759261
\(271\) −18.2983 −1.11154 −0.555772 0.831335i \(-0.687577\pi\)
−0.555772 + 0.831335i \(0.687577\pi\)
\(272\) 0.571514 0.0346531
\(273\) −9.76673 −0.591109
\(274\) 1.60089 0.0967132
\(275\) 0 0
\(276\) 14.7424 0.887387
\(277\) 3.69005 0.221714 0.110857 0.993836i \(-0.464640\pi\)
0.110857 + 0.993836i \(0.464640\pi\)
\(278\) −15.9213 −0.954897
\(279\) −5.16908 −0.309464
\(280\) −5.10239 −0.304926
\(281\) 23.5460 1.40463 0.702317 0.711864i \(-0.252149\pi\)
0.702317 + 0.711864i \(0.252149\pi\)
\(282\) 15.0663 0.897187
\(283\) 15.4242 0.916871 0.458435 0.888728i \(-0.348410\pi\)
0.458435 + 0.888728i \(0.348410\pi\)
\(284\) 4.91454 0.291624
\(285\) 3.48581 0.206481
\(286\) 0 0
\(287\) 0.196406 0.0115935
\(288\) 0.579065 0.0341217
\(289\) −16.6734 −0.980787
\(290\) 4.47019 0.262498
\(291\) 6.19819 0.363344
\(292\) 10.7414 0.628593
\(293\) 15.3414 0.896251 0.448126 0.893971i \(-0.352092\pi\)
0.448126 + 0.893971i \(0.352092\pi\)
\(294\) 2.82078 0.164511
\(295\) 17.9787 1.04676
\(296\) −7.48207 −0.434887
\(297\) 0 0
\(298\) 4.19884 0.243232
\(299\) −26.1139 −1.51021
\(300\) 0.0296734 0.00171320
\(301\) −12.6025 −0.726394
\(302\) −0.918503 −0.0528539
\(303\) −1.90803 −0.109614
\(304\) 1.00000 0.0573539
\(305\) −8.37622 −0.479621
\(306\) 0.330944 0.0189188
\(307\) 23.5750 1.34550 0.672749 0.739870i \(-0.265113\pi\)
0.672749 + 0.739870i \(0.265113\pi\)
\(308\) 0 0
\(309\) 9.82780 0.559084
\(310\) −19.9985 −1.13584
\(311\) −4.17781 −0.236902 −0.118451 0.992960i \(-0.537793\pi\)
−0.118451 + 0.992960i \(0.537793\pi\)
\(312\) 4.28832 0.242779
\(313\) −30.2227 −1.70829 −0.854144 0.520036i \(-0.825919\pi\)
−0.854144 + 0.520036i \(0.825919\pi\)
\(314\) 11.8346 0.667866
\(315\) −2.95461 −0.166474
\(316\) 5.92179 0.333127
\(317\) 17.3175 0.972647 0.486324 0.873779i \(-0.338338\pi\)
0.486324 + 0.873779i \(0.338338\pi\)
\(318\) −6.84723 −0.383974
\(319\) 0 0
\(320\) 2.24033 0.125238
\(321\) −24.7840 −1.38331
\(322\) −21.5793 −1.20257
\(323\) 0.571514 0.0317999
\(324\) −6.92749 −0.384860
\(325\) −0.0525621 −0.00291562
\(326\) 7.54426 0.417838
\(327\) −13.4943 −0.746237
\(328\) −0.0862369 −0.00476164
\(329\) −22.0535 −1.21585
\(330\) 0 0
\(331\) −9.13686 −0.502207 −0.251104 0.967960i \(-0.580793\pi\)
−0.251104 + 0.967960i \(0.580793\pi\)
\(332\) 16.8470 0.924601
\(333\) −4.33261 −0.237425
\(334\) −9.75174 −0.533591
\(335\) 15.4949 0.846575
\(336\) 3.54367 0.193323
\(337\) −9.49487 −0.517219 −0.258609 0.965982i \(-0.583264\pi\)
−0.258609 + 0.965982i \(0.583264\pi\)
\(338\) 5.40388 0.293932
\(339\) −0.463368 −0.0251667
\(340\) 1.28038 0.0694383
\(341\) 0 0
\(342\) 0.579065 0.0313122
\(343\) −20.0716 −1.08376
\(344\) 5.53342 0.298342
\(345\) 33.0278 1.77816
\(346\) 9.90137 0.532301
\(347\) 12.4664 0.669234 0.334617 0.942354i \(-0.391393\pi\)
0.334617 + 0.942354i \(0.391393\pi\)
\(348\) −3.10460 −0.166424
\(349\) 29.2430 1.56534 0.782672 0.622434i \(-0.213856\pi\)
0.782672 + 0.622434i \(0.213856\pi\)
\(350\) −0.0434348 −0.00232169
\(351\) 15.3482 0.819226
\(352\) 0 0
\(353\) −9.95957 −0.530095 −0.265047 0.964235i \(-0.585388\pi\)
−0.265047 + 0.964235i \(0.585388\pi\)
\(354\) −12.4864 −0.663646
\(355\) 11.0102 0.584359
\(356\) 5.14468 0.272668
\(357\) 2.02526 0.107188
\(358\) −15.8720 −0.838861
\(359\) 9.89422 0.522197 0.261099 0.965312i \(-0.415915\pi\)
0.261099 + 0.965312i \(0.415915\pi\)
\(360\) 1.29730 0.0683735
\(361\) 1.00000 0.0526316
\(362\) 17.7160 0.931132
\(363\) 0 0
\(364\) −6.27708 −0.329008
\(365\) 24.0643 1.25958
\(366\) 5.81738 0.304080
\(367\) −1.78588 −0.0932222 −0.0466111 0.998913i \(-0.514842\pi\)
−0.0466111 + 0.998913i \(0.514842\pi\)
\(368\) 9.47493 0.493915
\(369\) −0.0499368 −0.00259961
\(370\) −16.7623 −0.871430
\(371\) 10.0227 0.520353
\(372\) 13.8892 0.720121
\(373\) 11.8730 0.614762 0.307381 0.951586i \(-0.400547\pi\)
0.307381 + 0.951586i \(0.400547\pi\)
\(374\) 0 0
\(375\) −17.3626 −0.896598
\(376\) 9.68313 0.499369
\(377\) 5.49933 0.283230
\(378\) 12.6830 0.652344
\(379\) 14.5469 0.747226 0.373613 0.927585i \(-0.378119\pi\)
0.373613 + 0.927585i \(0.378119\pi\)
\(380\) 2.24033 0.114926
\(381\) 0.991542 0.0507982
\(382\) 17.5478 0.897824
\(383\) 10.9005 0.556988 0.278494 0.960438i \(-0.410165\pi\)
0.278494 + 0.960438i \(0.410165\pi\)
\(384\) −1.55594 −0.0794010
\(385\) 0 0
\(386\) −13.1066 −0.667111
\(387\) 3.20421 0.162879
\(388\) 3.98358 0.202236
\(389\) 2.85935 0.144975 0.0724874 0.997369i \(-0.476906\pi\)
0.0724874 + 0.997369i \(0.476906\pi\)
\(390\) 9.60725 0.486482
\(391\) 5.41505 0.273851
\(392\) 1.81292 0.0915661
\(393\) 33.9365 1.71187
\(394\) 15.2492 0.768244
\(395\) 13.2668 0.667523
\(396\) 0 0
\(397\) −1.34242 −0.0673741 −0.0336870 0.999432i \(-0.510725\pi\)
−0.0336870 + 0.999432i \(0.510725\pi\)
\(398\) −13.4862 −0.676001
\(399\) 3.54367 0.177405
\(400\) 0.0190711 0.000953556 0
\(401\) −25.2632 −1.26159 −0.630793 0.775951i \(-0.717270\pi\)
−0.630793 + 0.775951i \(0.717270\pi\)
\(402\) −10.7614 −0.536729
\(403\) −24.6026 −1.22554
\(404\) −1.22629 −0.0610104
\(405\) −15.5198 −0.771188
\(406\) 4.54439 0.225534
\(407\) 0 0
\(408\) −0.889238 −0.0440239
\(409\) −10.9069 −0.539312 −0.269656 0.962957i \(-0.586910\pi\)
−0.269656 + 0.962957i \(0.586910\pi\)
\(410\) −0.193199 −0.00954142
\(411\) −2.49088 −0.122866
\(412\) 6.31633 0.311183
\(413\) 18.2771 0.899358
\(414\) 5.48660 0.269652
\(415\) 37.7429 1.85272
\(416\) 2.75611 0.135129
\(417\) 24.7725 1.21312
\(418\) 0 0
\(419\) 9.08955 0.444054 0.222027 0.975041i \(-0.428733\pi\)
0.222027 + 0.975041i \(0.428733\pi\)
\(420\) 7.93898 0.387383
\(421\) 34.5218 1.68249 0.841244 0.540655i \(-0.181824\pi\)
0.841244 + 0.540655i \(0.181824\pi\)
\(422\) 2.15813 0.105056
\(423\) 5.60716 0.272630
\(424\) −4.40072 −0.213718
\(425\) 0.0108994 0.000528699 0
\(426\) −7.64670 −0.370484
\(427\) −8.51526 −0.412082
\(428\) −15.9287 −0.769940
\(429\) 0 0
\(430\) 12.3967 0.597821
\(431\) 3.67160 0.176855 0.0884273 0.996083i \(-0.471816\pi\)
0.0884273 + 0.996083i \(0.471816\pi\)
\(432\) −5.56879 −0.267929
\(433\) −9.59701 −0.461203 −0.230602 0.973048i \(-0.574069\pi\)
−0.230602 + 0.973048i \(0.574069\pi\)
\(434\) −20.3305 −0.975893
\(435\) −6.95532 −0.333482
\(436\) −8.67279 −0.415352
\(437\) 9.47493 0.453248
\(438\) −16.7129 −0.798575
\(439\) 5.20790 0.248560 0.124280 0.992247i \(-0.460338\pi\)
0.124280 + 0.992247i \(0.460338\pi\)
\(440\) 0 0
\(441\) 1.04980 0.0499903
\(442\) 1.57515 0.0749224
\(443\) −11.5968 −0.550981 −0.275490 0.961304i \(-0.588840\pi\)
−0.275490 + 0.961304i \(0.588840\pi\)
\(444\) 11.6416 0.552487
\(445\) 11.5258 0.546374
\(446\) −6.76418 −0.320293
\(447\) −6.53312 −0.309006
\(448\) 2.27752 0.107603
\(449\) −19.1493 −0.903711 −0.451856 0.892091i \(-0.649238\pi\)
−0.451856 + 0.892091i \(0.649238\pi\)
\(450\) 0.0110434 0.000520592 0
\(451\) 0 0
\(452\) −0.297807 −0.0140077
\(453\) 1.42913 0.0671464
\(454\) −6.68912 −0.313936
\(455\) −14.0627 −0.659270
\(456\) −1.55594 −0.0728634
\(457\) −15.2474 −0.713243 −0.356622 0.934249i \(-0.616071\pi\)
−0.356622 + 0.934249i \(0.616071\pi\)
\(458\) −0.645439 −0.0301594
\(459\) −3.18264 −0.148553
\(460\) 21.2270 0.989712
\(461\) −13.1903 −0.614336 −0.307168 0.951655i \(-0.599381\pi\)
−0.307168 + 0.951655i \(0.599381\pi\)
\(462\) 0 0
\(463\) −15.8312 −0.735740 −0.367870 0.929877i \(-0.619913\pi\)
−0.367870 + 0.929877i \(0.619913\pi\)
\(464\) −1.99533 −0.0926307
\(465\) 31.1164 1.44299
\(466\) 5.52617 0.255995
\(467\) −14.8060 −0.685140 −0.342570 0.939492i \(-0.611297\pi\)
−0.342570 + 0.939492i \(0.611297\pi\)
\(468\) 1.59596 0.0737735
\(469\) 15.7521 0.727363
\(470\) 21.6934 1.00064
\(471\) −18.4139 −0.848467
\(472\) −8.02502 −0.369382
\(473\) 0 0
\(474\) −9.21393 −0.423210
\(475\) 0.0190711 0.000875043 0
\(476\) 1.30163 0.0596602
\(477\) −2.54830 −0.116679
\(478\) 24.0112 1.09825
\(479\) −21.4058 −0.978055 −0.489028 0.872268i \(-0.662648\pi\)
−0.489028 + 0.872268i \(0.662648\pi\)
\(480\) −3.48581 −0.159105
\(481\) −20.6214 −0.940254
\(482\) −12.5817 −0.573083
\(483\) 33.5760 1.52776
\(484\) 0 0
\(485\) 8.92453 0.405242
\(486\) −5.92766 −0.268884
\(487\) −18.0587 −0.818320 −0.409160 0.912463i \(-0.634178\pi\)
−0.409160 + 0.912463i \(0.634178\pi\)
\(488\) 3.73883 0.169249
\(489\) −11.7384 −0.530828
\(490\) 4.06153 0.183481
\(491\) −26.9808 −1.21763 −0.608813 0.793314i \(-0.708354\pi\)
−0.608813 + 0.793314i \(0.708354\pi\)
\(492\) 0.134179 0.00604926
\(493\) −1.14036 −0.0513590
\(494\) 2.75611 0.124003
\(495\) 0 0
\(496\) 8.92659 0.400816
\(497\) 11.1929 0.502072
\(498\) −26.2129 −1.17463
\(499\) 41.4218 1.85429 0.927147 0.374699i \(-0.122254\pi\)
0.927147 + 0.374699i \(0.122254\pi\)
\(500\) −11.1589 −0.499042
\(501\) 15.1731 0.677883
\(502\) 27.2335 1.21549
\(503\) −36.6381 −1.63361 −0.816806 0.576912i \(-0.804258\pi\)
−0.816806 + 0.576912i \(0.804258\pi\)
\(504\) 1.31883 0.0587454
\(505\) −2.74730 −0.122253
\(506\) 0 0
\(507\) −8.40808 −0.373416
\(508\) 0.637264 0.0282740
\(509\) −1.94309 −0.0861260 −0.0430630 0.999072i \(-0.513712\pi\)
−0.0430630 + 0.999072i \(0.513712\pi\)
\(510\) −1.99219 −0.0882155
\(511\) 24.4637 1.08221
\(512\) −1.00000 −0.0441942
\(513\) −5.56879 −0.245868
\(514\) 27.8288 1.22748
\(515\) 14.1507 0.623552
\(516\) −8.60965 −0.379019
\(517\) 0 0
\(518\) −17.0405 −0.748719
\(519\) −15.4059 −0.676243
\(520\) 6.17458 0.270773
\(521\) −42.3465 −1.85523 −0.927616 0.373535i \(-0.878145\pi\)
−0.927616 + 0.373535i \(0.878145\pi\)
\(522\) −1.15542 −0.0505715
\(523\) −40.4000 −1.76657 −0.883284 0.468838i \(-0.844673\pi\)
−0.883284 + 0.468838i \(0.844673\pi\)
\(524\) 21.8110 0.952818
\(525\) 0.0675818 0.00294951
\(526\) 29.9986 1.30800
\(527\) 5.10167 0.222232
\(528\) 0 0
\(529\) 66.7744 2.90323
\(530\) −9.85905 −0.428250
\(531\) −4.64701 −0.201663
\(532\) 2.27752 0.0987429
\(533\) −0.237678 −0.0102950
\(534\) −8.00479 −0.346401
\(535\) −35.6854 −1.54281
\(536\) −6.91634 −0.298740
\(537\) 24.6958 1.06570
\(538\) 1.89146 0.0815468
\(539\) 0 0
\(540\) −12.4759 −0.536878
\(541\) −4.72409 −0.203104 −0.101552 0.994830i \(-0.532381\pi\)
−0.101552 + 0.994830i \(0.532381\pi\)
\(542\) 18.2983 0.785981
\(543\) −27.5649 −1.18293
\(544\) −0.571514 −0.0245034
\(545\) −19.4299 −0.832286
\(546\) 9.76673 0.417978
\(547\) −26.5069 −1.13335 −0.566677 0.823940i \(-0.691771\pi\)
−0.566677 + 0.823940i \(0.691771\pi\)
\(548\) −1.60089 −0.0683866
\(549\) 2.16503 0.0924011
\(550\) 0 0
\(551\) −1.99533 −0.0850037
\(552\) −14.7424 −0.627478
\(553\) 13.4870 0.573525
\(554\) −3.69005 −0.156775
\(555\) 26.0811 1.10708
\(556\) 15.9213 0.675214
\(557\) 19.2433 0.815364 0.407682 0.913124i \(-0.366337\pi\)
0.407682 + 0.913124i \(0.366337\pi\)
\(558\) 5.16908 0.218824
\(559\) 15.2507 0.645036
\(560\) 5.10239 0.215615
\(561\) 0 0
\(562\) −23.5460 −0.993226
\(563\) −3.50880 −0.147878 −0.0739391 0.997263i \(-0.523557\pi\)
−0.0739391 + 0.997263i \(0.523557\pi\)
\(564\) −15.0663 −0.634407
\(565\) −0.667185 −0.0280687
\(566\) −15.4242 −0.648326
\(567\) −15.7775 −0.662592
\(568\) −4.91454 −0.206209
\(569\) −34.8083 −1.45924 −0.729620 0.683852i \(-0.760303\pi\)
−0.729620 + 0.683852i \(0.760303\pi\)
\(570\) −3.48581 −0.146004
\(571\) 8.86605 0.371033 0.185516 0.982641i \(-0.440604\pi\)
0.185516 + 0.982641i \(0.440604\pi\)
\(572\) 0 0
\(573\) −27.3033 −1.14061
\(574\) −0.196406 −0.00819783
\(575\) 0.180698 0.00753561
\(576\) −0.579065 −0.0241277
\(577\) 15.5873 0.648910 0.324455 0.945901i \(-0.394819\pi\)
0.324455 + 0.945901i \(0.394819\pi\)
\(578\) 16.6734 0.693521
\(579\) 20.3931 0.847508
\(580\) −4.47019 −0.185614
\(581\) 38.3694 1.59183
\(582\) −6.19819 −0.256923
\(583\) 0 0
\(584\) −10.7414 −0.444483
\(585\) 3.57549 0.147828
\(586\) −15.3414 −0.633745
\(587\) 26.9304 1.11154 0.555769 0.831337i \(-0.312424\pi\)
0.555769 + 0.831337i \(0.312424\pi\)
\(588\) −2.82078 −0.116327
\(589\) 8.92659 0.367814
\(590\) −17.9787 −0.740171
\(591\) −23.7268 −0.975989
\(592\) 7.48207 0.307511
\(593\) 23.7386 0.974826 0.487413 0.873172i \(-0.337941\pi\)
0.487413 + 0.873172i \(0.337941\pi\)
\(594\) 0 0
\(595\) 2.91608 0.119548
\(596\) −4.19884 −0.171991
\(597\) 20.9836 0.858802
\(598\) 26.1139 1.06788
\(599\) 21.9522 0.896944 0.448472 0.893797i \(-0.351968\pi\)
0.448472 + 0.893797i \(0.351968\pi\)
\(600\) −0.0296734 −0.00121141
\(601\) 9.40878 0.383792 0.191896 0.981415i \(-0.438536\pi\)
0.191896 + 0.981415i \(0.438536\pi\)
\(602\) 12.6025 0.513638
\(603\) −4.00501 −0.163097
\(604\) 0.918503 0.0373733
\(605\) 0 0
\(606\) 1.90803 0.0775086
\(607\) 29.5786 1.20056 0.600278 0.799791i \(-0.295056\pi\)
0.600278 + 0.799791i \(0.295056\pi\)
\(608\) −1.00000 −0.0405554
\(609\) −7.07078 −0.286522
\(610\) 8.37622 0.339143
\(611\) 26.6877 1.07967
\(612\) −0.330944 −0.0133776
\(613\) 43.1876 1.74433 0.872166 0.489211i \(-0.162715\pi\)
0.872166 + 0.489211i \(0.162715\pi\)
\(614\) −23.5750 −0.951411
\(615\) 0.300605 0.0121216
\(616\) 0 0
\(617\) −29.5678 −1.19036 −0.595178 0.803594i \(-0.702919\pi\)
−0.595178 + 0.803594i \(0.702919\pi\)
\(618\) −9.82780 −0.395332
\(619\) 6.02976 0.242357 0.121178 0.992631i \(-0.461333\pi\)
0.121178 + 0.992631i \(0.461333\pi\)
\(620\) 19.9985 0.803159
\(621\) −52.7640 −2.11734
\(622\) 4.17781 0.167515
\(623\) 11.7171 0.469436
\(624\) −4.28832 −0.171670
\(625\) −25.0950 −1.00380
\(626\) 30.2227 1.20794
\(627\) 0 0
\(628\) −11.8346 −0.472252
\(629\) 4.27611 0.170500
\(630\) 2.95461 0.117715
\(631\) −41.8326 −1.66533 −0.832665 0.553777i \(-0.813186\pi\)
−0.832665 + 0.553777i \(0.813186\pi\)
\(632\) −5.92179 −0.235556
\(633\) −3.35791 −0.133465
\(634\) −17.3175 −0.687766
\(635\) 1.42768 0.0566558
\(636\) 6.84723 0.271510
\(637\) 4.99659 0.197972
\(638\) 0 0
\(639\) −2.84584 −0.112580
\(640\) −2.24033 −0.0885568
\(641\) 46.1632 1.82334 0.911668 0.410928i \(-0.134795\pi\)
0.911668 + 0.410928i \(0.134795\pi\)
\(642\) 24.7840 0.978145
\(643\) 15.9119 0.627504 0.313752 0.949505i \(-0.398414\pi\)
0.313752 + 0.949505i \(0.398414\pi\)
\(644\) 21.5793 0.850344
\(645\) −19.2884 −0.759482
\(646\) −0.571514 −0.0224859
\(647\) 4.91972 0.193414 0.0967071 0.995313i \(-0.469169\pi\)
0.0967071 + 0.995313i \(0.469169\pi\)
\(648\) 6.92749 0.272137
\(649\) 0 0
\(650\) 0.0525621 0.00206165
\(651\) 31.6329 1.23979
\(652\) −7.54426 −0.295456
\(653\) 8.25465 0.323029 0.161515 0.986870i \(-0.448362\pi\)
0.161515 + 0.986870i \(0.448362\pi\)
\(654\) 13.4943 0.527669
\(655\) 48.8638 1.90927
\(656\) 0.0862369 0.00336699
\(657\) −6.21997 −0.242664
\(658\) 22.0535 0.859735
\(659\) −49.5250 −1.92922 −0.964611 0.263677i \(-0.915065\pi\)
−0.964611 + 0.263677i \(0.915065\pi\)
\(660\) 0 0
\(661\) 49.1275 1.91084 0.955420 0.295252i \(-0.0954035\pi\)
0.955420 + 0.295252i \(0.0954035\pi\)
\(662\) 9.13686 0.355114
\(663\) −2.45084 −0.0951826
\(664\) −16.8470 −0.653791
\(665\) 5.10239 0.197862
\(666\) 4.33261 0.167885
\(667\) −18.9056 −0.732027
\(668\) 9.75174 0.377306
\(669\) 10.5246 0.406906
\(670\) −15.4949 −0.598619
\(671\) 0 0
\(672\) −3.54367 −0.136700
\(673\) 39.5853 1.52590 0.762951 0.646457i \(-0.223750\pi\)
0.762951 + 0.646457i \(0.223750\pi\)
\(674\) 9.49487 0.365729
\(675\) −0.106203 −0.00408776
\(676\) −5.40388 −0.207841
\(677\) −44.3011 −1.70263 −0.851315 0.524655i \(-0.824194\pi\)
−0.851315 + 0.524655i \(0.824194\pi\)
\(678\) 0.463368 0.0177955
\(679\) 9.07267 0.348177
\(680\) −1.28038 −0.0491003
\(681\) 10.4078 0.398829
\(682\) 0 0
\(683\) 4.18585 0.160167 0.0800835 0.996788i \(-0.474481\pi\)
0.0800835 + 0.996788i \(0.474481\pi\)
\(684\) −0.579065 −0.0221411
\(685\) −3.58652 −0.137034
\(686\) 20.0716 0.766336
\(687\) 1.00426 0.0383149
\(688\) −5.53342 −0.210960
\(689\) −12.1288 −0.462072
\(690\) −33.0278 −1.25735
\(691\) −17.3471 −0.659916 −0.329958 0.943996i \(-0.607035\pi\)
−0.329958 + 0.943996i \(0.607035\pi\)
\(692\) −9.90137 −0.376393
\(693\) 0 0
\(694\) −12.4664 −0.473220
\(695\) 35.6690 1.35300
\(696\) 3.10460 0.117679
\(697\) 0.0492856 0.00186682
\(698\) −29.2430 −1.10687
\(699\) −8.59836 −0.325220
\(700\) 0.0434348 0.00164168
\(701\) 30.9365 1.16845 0.584227 0.811590i \(-0.301398\pi\)
0.584227 + 0.811590i \(0.301398\pi\)
\(702\) −15.3482 −0.579280
\(703\) 7.48207 0.282192
\(704\) 0 0
\(705\) −33.7535 −1.27123
\(706\) 9.95957 0.374833
\(707\) −2.79291 −0.105038
\(708\) 12.4864 0.469268
\(709\) −17.4009 −0.653505 −0.326752 0.945110i \(-0.605954\pi\)
−0.326752 + 0.945110i \(0.605954\pi\)
\(710\) −11.0102 −0.413205
\(711\) −3.42910 −0.128601
\(712\) −5.14468 −0.192805
\(713\) 84.5788 3.16750
\(714\) −2.02526 −0.0757933
\(715\) 0 0
\(716\) 15.8720 0.593164
\(717\) −37.3599 −1.39523
\(718\) −9.89422 −0.369249
\(719\) −15.1001 −0.563138 −0.281569 0.959541i \(-0.590855\pi\)
−0.281569 + 0.959541i \(0.590855\pi\)
\(720\) −1.29730 −0.0483474
\(721\) 14.3856 0.535746
\(722\) −1.00000 −0.0372161
\(723\) 19.5764 0.728053
\(724\) −17.7160 −0.658410
\(725\) −0.0380531 −0.00141326
\(726\) 0 0
\(727\) 12.5584 0.465764 0.232882 0.972505i \(-0.425184\pi\)
0.232882 + 0.972505i \(0.425184\pi\)
\(728\) 6.27708 0.232644
\(729\) 30.0055 1.11132
\(730\) −24.0643 −0.890659
\(731\) −3.16243 −0.116967
\(732\) −5.81738 −0.215017
\(733\) −8.63342 −0.318883 −0.159441 0.987207i \(-0.550969\pi\)
−0.159441 + 0.987207i \(0.550969\pi\)
\(734\) 1.78588 0.0659180
\(735\) −6.31947 −0.233097
\(736\) −9.47493 −0.349251
\(737\) 0 0
\(738\) 0.0499368 0.00183820
\(739\) 25.8140 0.949584 0.474792 0.880098i \(-0.342523\pi\)
0.474792 + 0.880098i \(0.342523\pi\)
\(740\) 16.7623 0.616194
\(741\) −4.28832 −0.157536
\(742\) −10.0227 −0.367945
\(743\) −27.8252 −1.02081 −0.510404 0.859935i \(-0.670504\pi\)
−0.510404 + 0.859935i \(0.670504\pi\)
\(744\) −13.8892 −0.509203
\(745\) −9.40678 −0.344638
\(746\) −11.8730 −0.434703
\(747\) −9.75552 −0.356936
\(748\) 0 0
\(749\) −36.2778 −1.32556
\(750\) 17.3626 0.633991
\(751\) 16.8060 0.613259 0.306630 0.951829i \(-0.400799\pi\)
0.306630 + 0.951829i \(0.400799\pi\)
\(752\) −9.68313 −0.353107
\(753\) −42.3735 −1.54418
\(754\) −5.49933 −0.200274
\(755\) 2.05775 0.0748891
\(756\) −12.6830 −0.461277
\(757\) −41.6137 −1.51248 −0.756238 0.654296i \(-0.772965\pi\)
−0.756238 + 0.654296i \(0.772965\pi\)
\(758\) −14.5469 −0.528369
\(759\) 0 0
\(760\) −2.24033 −0.0812653
\(761\) 17.3354 0.628409 0.314204 0.949355i \(-0.398262\pi\)
0.314204 + 0.949355i \(0.398262\pi\)
\(762\) −0.991542 −0.0359198
\(763\) −19.7524 −0.715086
\(764\) −17.5478 −0.634857
\(765\) −0.741422 −0.0268062
\(766\) −10.9005 −0.393850
\(767\) −22.1178 −0.798628
\(768\) 1.55594 0.0561450
\(769\) 6.63601 0.239301 0.119650 0.992816i \(-0.461823\pi\)
0.119650 + 0.992816i \(0.461823\pi\)
\(770\) 0 0
\(771\) −43.2998 −1.55940
\(772\) 13.1066 0.471719
\(773\) −23.1698 −0.833360 −0.416680 0.909053i \(-0.636807\pi\)
−0.416680 + 0.909053i \(0.636807\pi\)
\(774\) −3.20421 −0.115173
\(775\) 0.170240 0.00611521
\(776\) −3.98358 −0.143002
\(777\) 26.5140 0.951184
\(778\) −2.85935 −0.102513
\(779\) 0.0862369 0.00308976
\(780\) −9.60725 −0.343995
\(781\) 0 0
\(782\) −5.41505 −0.193642
\(783\) 11.1116 0.397095
\(784\) −1.81292 −0.0647470
\(785\) −26.5134 −0.946305
\(786\) −33.9365 −1.21048
\(787\) −35.8057 −1.27634 −0.638168 0.769897i \(-0.720307\pi\)
−0.638168 + 0.769897i \(0.720307\pi\)
\(788\) −15.2492 −0.543230
\(789\) −46.6758 −1.66170
\(790\) −13.2668 −0.472010
\(791\) −0.678260 −0.0241162
\(792\) 0 0
\(793\) 10.3046 0.365928
\(794\) 1.34242 0.0476407
\(795\) 15.3400 0.544056
\(796\) 13.4862 0.478005
\(797\) 12.9247 0.457816 0.228908 0.973448i \(-0.426485\pi\)
0.228908 + 0.973448i \(0.426485\pi\)
\(798\) −3.54367 −0.125445
\(799\) −5.53404 −0.195780
\(800\) −0.0190711 −0.000674266 0
\(801\) −2.97910 −0.105261
\(802\) 25.2632 0.892076
\(803\) 0 0
\(804\) 10.7614 0.379524
\(805\) 48.3448 1.70393
\(806\) 24.6026 0.866591
\(807\) −2.94300 −0.103598
\(808\) 1.22629 0.0431409
\(809\) 1.69221 0.0594950 0.0297475 0.999557i \(-0.490530\pi\)
0.0297475 + 0.999557i \(0.490530\pi\)
\(810\) 15.5198 0.545312
\(811\) 14.2119 0.499048 0.249524 0.968369i \(-0.419726\pi\)
0.249524 + 0.968369i \(0.419726\pi\)
\(812\) −4.54439 −0.159477
\(813\) −28.4710 −0.998523
\(814\) 0 0
\(815\) −16.9016 −0.592038
\(816\) 0.889238 0.0311296
\(817\) −5.53342 −0.193590
\(818\) 10.9069 0.381351
\(819\) 3.63484 0.127011
\(820\) 0.193199 0.00674680
\(821\) −36.7897 −1.28397 −0.641985 0.766717i \(-0.721889\pi\)
−0.641985 + 0.766717i \(0.721889\pi\)
\(822\) 2.49088 0.0868794
\(823\) 16.9368 0.590379 0.295189 0.955439i \(-0.404617\pi\)
0.295189 + 0.955439i \(0.404617\pi\)
\(824\) −6.31633 −0.220040
\(825\) 0 0
\(826\) −18.2771 −0.635942
\(827\) 28.9883 1.00802 0.504012 0.863697i \(-0.331857\pi\)
0.504012 + 0.863697i \(0.331857\pi\)
\(828\) −5.48660 −0.190673
\(829\) 38.3557 1.33215 0.666074 0.745886i \(-0.267973\pi\)
0.666074 + 0.745886i \(0.267973\pi\)
\(830\) −37.7429 −1.31007
\(831\) 5.74148 0.199170
\(832\) −2.75611 −0.0955508
\(833\) −1.03611 −0.0358990
\(834\) −24.7725 −0.857803
\(835\) 21.8471 0.756050
\(836\) 0 0
\(837\) −49.7103 −1.71824
\(838\) −9.08955 −0.313993
\(839\) 13.1347 0.453462 0.226731 0.973957i \(-0.427196\pi\)
0.226731 + 0.973957i \(0.427196\pi\)
\(840\) −7.93898 −0.273921
\(841\) −25.0187 −0.862713
\(842\) −34.5218 −1.18970
\(843\) 36.6360 1.26181
\(844\) −2.15813 −0.0742858
\(845\) −12.1065 −0.416475
\(846\) −5.60716 −0.192778
\(847\) 0 0
\(848\) 4.40072 0.151121
\(849\) 23.9990 0.823643
\(850\) −0.0108994 −0.000373847 0
\(851\) 70.8921 2.43015
\(852\) 7.64670 0.261972
\(853\) 47.1433 1.61415 0.807077 0.590446i \(-0.201048\pi\)
0.807077 + 0.590446i \(0.201048\pi\)
\(854\) 8.51526 0.291386
\(855\) −1.29730 −0.0443666
\(856\) 15.9287 0.544430
\(857\) −4.63462 −0.158316 −0.0791579 0.996862i \(-0.525223\pi\)
−0.0791579 + 0.996862i \(0.525223\pi\)
\(858\) 0 0
\(859\) −28.6906 −0.978911 −0.489456 0.872028i \(-0.662805\pi\)
−0.489456 + 0.872028i \(0.662805\pi\)
\(860\) −12.3967 −0.422723
\(861\) 0.305595 0.0104147
\(862\) −3.67160 −0.125055
\(863\) −29.7927 −1.01416 −0.507078 0.861900i \(-0.669275\pi\)
−0.507078 + 0.861900i \(0.669275\pi\)
\(864\) 5.56879 0.189454
\(865\) −22.1823 −0.754221
\(866\) 9.59701 0.326120
\(867\) −25.9427 −0.881060
\(868\) 20.3305 0.690061
\(869\) 0 0
\(870\) 6.95532 0.235807
\(871\) −19.0622 −0.645897
\(872\) 8.67279 0.293698
\(873\) −2.30675 −0.0780717
\(874\) −9.47493 −0.320494
\(875\) −25.4146 −0.859171
\(876\) 16.7129 0.564678
\(877\) −35.2763 −1.19120 −0.595598 0.803283i \(-0.703085\pi\)
−0.595598 + 0.803283i \(0.703085\pi\)
\(878\) −5.20790 −0.175758
\(879\) 23.8702 0.805120
\(880\) 0 0
\(881\) −0.536228 −0.0180660 −0.00903300 0.999959i \(-0.502875\pi\)
−0.00903300 + 0.999959i \(0.502875\pi\)
\(882\) −1.04980 −0.0353485
\(883\) 9.43044 0.317360 0.158680 0.987330i \(-0.449276\pi\)
0.158680 + 0.987330i \(0.449276\pi\)
\(884\) −1.57515 −0.0529781
\(885\) 27.9737 0.940325
\(886\) 11.5968 0.389602
\(887\) 15.7866 0.530062 0.265031 0.964240i \(-0.414618\pi\)
0.265031 + 0.964240i \(0.414618\pi\)
\(888\) −11.6416 −0.390667
\(889\) 1.45138 0.0486777
\(890\) −11.5258 −0.386345
\(891\) 0 0
\(892\) 6.76418 0.226482
\(893\) −9.68313 −0.324034
\(894\) 6.53312 0.218500
\(895\) 35.5585 1.18859
\(896\) −2.27752 −0.0760865
\(897\) −40.6316 −1.35665
\(898\) 19.1493 0.639020
\(899\) −17.8115 −0.594045
\(900\) −0.0110434 −0.000368114 0
\(901\) 2.51507 0.0837891
\(902\) 0 0
\(903\) −19.6086 −0.652534
\(904\) 0.297807 0.00990491
\(905\) −39.6896 −1.31933
\(906\) −1.42913 −0.0474797
\(907\) −44.7761 −1.48677 −0.743383 0.668866i \(-0.766780\pi\)
−0.743383 + 0.668866i \(0.766780\pi\)
\(908\) 6.68912 0.221986
\(909\) 0.710104 0.0235527
\(910\) 14.0627 0.466175
\(911\) 3.05935 0.101361 0.0506804 0.998715i \(-0.483861\pi\)
0.0506804 + 0.998715i \(0.483861\pi\)
\(912\) 1.55594 0.0515222
\(913\) 0 0
\(914\) 15.2474 0.504339
\(915\) −13.0329 −0.430853
\(916\) 0.645439 0.0213259
\(917\) 49.6749 1.64041
\(918\) 3.18264 0.105043
\(919\) 55.2774 1.82343 0.911717 0.410818i \(-0.134757\pi\)
0.911717 + 0.410818i \(0.134757\pi\)
\(920\) −21.2270 −0.699832
\(921\) 36.6812 1.20869
\(922\) 13.1903 0.434401
\(923\) −13.5450 −0.445839
\(924\) 0 0
\(925\) 0.142692 0.00469167
\(926\) 15.8312 0.520247
\(927\) −3.65757 −0.120130
\(928\) 1.99533 0.0654998
\(929\) −29.1942 −0.957832 −0.478916 0.877861i \(-0.658970\pi\)
−0.478916 + 0.877861i \(0.658970\pi\)
\(930\) −31.1164 −1.02035
\(931\) −1.81292 −0.0594159
\(932\) −5.52617 −0.181016
\(933\) −6.50041 −0.212814
\(934\) 14.8060 0.484467
\(935\) 0 0
\(936\) −1.59596 −0.0521657
\(937\) 18.6969 0.610801 0.305401 0.952224i \(-0.401210\pi\)
0.305401 + 0.952224i \(0.401210\pi\)
\(938\) −15.7521 −0.514324
\(939\) −47.0246 −1.53459
\(940\) −21.6934 −0.707561
\(941\) −4.47168 −0.145773 −0.0728863 0.997340i \(-0.523221\pi\)
−0.0728863 + 0.997340i \(0.523221\pi\)
\(942\) 18.4139 0.599957
\(943\) 0.817089 0.0266081
\(944\) 8.02502 0.261192
\(945\) −28.4141 −0.924312
\(946\) 0 0
\(947\) 8.78679 0.285532 0.142766 0.989756i \(-0.454400\pi\)
0.142766 + 0.989756i \(0.454400\pi\)
\(948\) 9.21393 0.299254
\(949\) −29.6045 −0.961002
\(950\) −0.0190711 −0.000618749 0
\(951\) 26.9449 0.873748
\(952\) −1.30163 −0.0421861
\(953\) 33.1114 1.07258 0.536292 0.844033i \(-0.319825\pi\)
0.536292 + 0.844033i \(0.319825\pi\)
\(954\) 2.54830 0.0825043
\(955\) −39.3129 −1.27213
\(956\) −24.0112 −0.776578
\(957\) 0 0
\(958\) 21.4058 0.691590
\(959\) −3.64605 −0.117737
\(960\) 3.48581 0.112504
\(961\) 48.6840 1.57045
\(962\) 20.6214 0.664860
\(963\) 9.22372 0.297230
\(964\) 12.5817 0.405231
\(965\) 29.3632 0.945235
\(966\) −33.5760 −1.08029
\(967\) 39.9996 1.28630 0.643150 0.765740i \(-0.277627\pi\)
0.643150 + 0.765740i \(0.277627\pi\)
\(968\) 0 0
\(969\) 0.889238 0.0285665
\(970\) −8.92453 −0.286549
\(971\) 4.06604 0.130486 0.0652428 0.997869i \(-0.479218\pi\)
0.0652428 + 0.997869i \(0.479218\pi\)
\(972\) 5.92766 0.190130
\(973\) 36.2611 1.16248
\(974\) 18.0587 0.578639
\(975\) −0.0817832 −0.00261916
\(976\) −3.73883 −0.119677
\(977\) −37.4002 −1.19654 −0.598270 0.801295i \(-0.704145\pi\)
−0.598270 + 0.801295i \(0.704145\pi\)
\(978\) 11.7384 0.375352
\(979\) 0 0
\(980\) −4.06153 −0.129741
\(981\) 5.02211 0.160344
\(982\) 26.9808 0.860992
\(983\) −52.9556 −1.68902 −0.844511 0.535538i \(-0.820109\pi\)
−0.844511 + 0.535538i \(0.820109\pi\)
\(984\) −0.134179 −0.00427747
\(985\) −34.1632 −1.08853
\(986\) 1.14036 0.0363163
\(987\) −34.3138 −1.09222
\(988\) −2.75611 −0.0876834
\(989\) −52.4288 −1.66714
\(990\) 0 0
\(991\) −3.98829 −0.126692 −0.0633461 0.997992i \(-0.520177\pi\)
−0.0633461 + 0.997992i \(0.520177\pi\)
\(992\) −8.92659 −0.283420
\(993\) −14.2164 −0.451143
\(994\) −11.1929 −0.355019
\(995\) 30.2135 0.957831
\(996\) 26.2129 0.830587
\(997\) −32.7194 −1.03623 −0.518117 0.855309i \(-0.673367\pi\)
−0.518117 + 0.855309i \(0.673367\pi\)
\(998\) −41.4218 −1.31118
\(999\) −41.6661 −1.31826
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 4598.2.a.by.1.4 8
11.10 odd 2 4598.2.a.cb.1.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.by.1.4 8 1.1 even 1 trivial
4598.2.a.cb.1.4 yes 8 11.10 odd 2