Properties

Label 4029.2.a.k.1.6
Level 4029
Weight 2
Character 4029.1
Self dual yes
Analytic conductor 32.172
Analytic rank 0
Dimension 31
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4029 = 3 \cdot 17 \cdot 79 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4029.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.1717269744\)
Analytic rank: \(0\)
Dimension: \(31\)
Coefficient ring index: multiple of None
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Character \(\chi\) = 4029.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.88694 q^{2} +1.00000 q^{3} +1.56054 q^{4} -0.846288 q^{5} -1.88694 q^{6} +3.01609 q^{7} +0.829236 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.88694 q^{2} +1.00000 q^{3} +1.56054 q^{4} -0.846288 q^{5} -1.88694 q^{6} +3.01609 q^{7} +0.829236 q^{8} +1.00000 q^{9} +1.59689 q^{10} -0.920932 q^{11} +1.56054 q^{12} -3.85540 q^{13} -5.69118 q^{14} -0.846288 q^{15} -4.68580 q^{16} +1.00000 q^{17} -1.88694 q^{18} +1.27900 q^{19} -1.32067 q^{20} +3.01609 q^{21} +1.73774 q^{22} -0.718821 q^{23} +0.829236 q^{24} -4.28380 q^{25} +7.27490 q^{26} +1.00000 q^{27} +4.70673 q^{28} +4.88556 q^{29} +1.59689 q^{30} -5.70052 q^{31} +7.18334 q^{32} -0.920932 q^{33} -1.88694 q^{34} -2.55248 q^{35} +1.56054 q^{36} +2.37427 q^{37} -2.41340 q^{38} -3.85540 q^{39} -0.701773 q^{40} +10.5478 q^{41} -5.69118 q^{42} -4.21626 q^{43} -1.43715 q^{44} -0.846288 q^{45} +1.35637 q^{46} +5.34685 q^{47} -4.68580 q^{48} +2.09681 q^{49} +8.08326 q^{50} +1.00000 q^{51} -6.01650 q^{52} -4.27035 q^{53} -1.88694 q^{54} +0.779374 q^{55} +2.50105 q^{56} +1.27900 q^{57} -9.21876 q^{58} +0.996670 q^{59} -1.32067 q^{60} -2.60886 q^{61} +10.7565 q^{62} +3.01609 q^{63} -4.18293 q^{64} +3.26278 q^{65} +1.73774 q^{66} +10.0505 q^{67} +1.56054 q^{68} -0.718821 q^{69} +4.81638 q^{70} +13.2458 q^{71} +0.829236 q^{72} +9.80450 q^{73} -4.48011 q^{74} -4.28380 q^{75} +1.99593 q^{76} -2.77762 q^{77} +7.27490 q^{78} +1.00000 q^{79} +3.96553 q^{80} +1.00000 q^{81} -19.9030 q^{82} +14.1014 q^{83} +4.70673 q^{84} -0.846288 q^{85} +7.95583 q^{86} +4.88556 q^{87} -0.763670 q^{88} +3.60832 q^{89} +1.59689 q^{90} -11.6282 q^{91} -1.12175 q^{92} -5.70052 q^{93} -10.0892 q^{94} -1.08240 q^{95} +7.18334 q^{96} +14.0403 q^{97} -3.95656 q^{98} -0.920932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 31q + 4q^{2} + 31q^{3} + 34q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + 12q^{8} + 31q^{9} + O(q^{10}) \) \( 31q + 4q^{2} + 31q^{3} + 34q^{4} + 11q^{5} + 4q^{6} + 4q^{7} + 12q^{8} + 31q^{9} + 5q^{10} + 26q^{11} + 34q^{12} + 7q^{13} + 19q^{14} + 11q^{15} + 40q^{16} + 31q^{17} + 4q^{18} + 32q^{19} + 23q^{20} + 4q^{21} + 2q^{22} + 29q^{23} + 12q^{24} + 32q^{25} + 13q^{26} + 31q^{27} - 13q^{28} + 25q^{29} + 5q^{30} + 22q^{31} + 28q^{32} + 26q^{33} + 4q^{34} + 20q^{35} + 34q^{36} - 4q^{37} + 19q^{38} + 7q^{39} - 3q^{40} + 33q^{41} + 19q^{42} + 6q^{43} + 30q^{44} + 11q^{45} - 11q^{46} + 23q^{47} + 40q^{48} + 31q^{49} + 6q^{50} + 31q^{51} - 7q^{52} + 12q^{53} + 4q^{54} + 40q^{56} + 32q^{57} + 9q^{58} + 27q^{59} + 23q^{60} - 4q^{61} + 25q^{62} + 4q^{63} + 10q^{64} + 54q^{65} + 2q^{66} + 34q^{68} + 29q^{69} - 59q^{70} + 35q^{71} + 12q^{72} + 5q^{73} + 48q^{74} + 32q^{75} + 32q^{76} + 42q^{77} + 13q^{78} + 31q^{79} + 24q^{80} + 31q^{81} + 5q^{82} + 67q^{83} - 13q^{84} + 11q^{85} - 20q^{86} + 25q^{87} - 7q^{88} + 22q^{89} + 5q^{90} + 16q^{91} + 57q^{92} + 22q^{93} + 45q^{94} + 73q^{95} + 28q^{96} - 13q^{97} - 19q^{98} + 26q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.88694 −1.33427 −0.667134 0.744938i \(-0.732479\pi\)
−0.667134 + 0.744938i \(0.732479\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.56054 0.780270
\(5\) −0.846288 −0.378472 −0.189236 0.981932i \(-0.560601\pi\)
−0.189236 + 0.981932i \(0.560601\pi\)
\(6\) −1.88694 −0.770340
\(7\) 3.01609 1.13998 0.569988 0.821653i \(-0.306948\pi\)
0.569988 + 0.821653i \(0.306948\pi\)
\(8\) 0.829236 0.293179
\(9\) 1.00000 0.333333
\(10\) 1.59689 0.504982
\(11\) −0.920932 −0.277672 −0.138836 0.990315i \(-0.544336\pi\)
−0.138836 + 0.990315i \(0.544336\pi\)
\(12\) 1.56054 0.450489
\(13\) −3.85540 −1.06930 −0.534648 0.845075i \(-0.679556\pi\)
−0.534648 + 0.845075i \(0.679556\pi\)
\(14\) −5.69118 −1.52103
\(15\) −0.846288 −0.218511
\(16\) −4.68580 −1.17145
\(17\) 1.00000 0.242536
\(18\) −1.88694 −0.444756
\(19\) 1.27900 0.293423 0.146711 0.989179i \(-0.453131\pi\)
0.146711 + 0.989179i \(0.453131\pi\)
\(20\) −1.32067 −0.295310
\(21\) 3.01609 0.658165
\(22\) 1.73774 0.370488
\(23\) −0.718821 −0.149885 −0.0749423 0.997188i \(-0.523877\pi\)
−0.0749423 + 0.997188i \(0.523877\pi\)
\(24\) 0.829236 0.169267
\(25\) −4.28380 −0.856759
\(26\) 7.27490 1.42673
\(27\) 1.00000 0.192450
\(28\) 4.70673 0.889488
\(29\) 4.88556 0.907226 0.453613 0.891199i \(-0.350135\pi\)
0.453613 + 0.891199i \(0.350135\pi\)
\(30\) 1.59689 0.291552
\(31\) −5.70052 −1.02384 −0.511922 0.859032i \(-0.671066\pi\)
−0.511922 + 0.859032i \(0.671066\pi\)
\(32\) 7.18334 1.26985
\(33\) −0.920932 −0.160314
\(34\) −1.88694 −0.323607
\(35\) −2.55248 −0.431448
\(36\) 1.56054 0.260090
\(37\) 2.37427 0.390328 0.195164 0.980771i \(-0.437476\pi\)
0.195164 + 0.980771i \(0.437476\pi\)
\(38\) −2.41340 −0.391505
\(39\) −3.85540 −0.617358
\(40\) −0.701773 −0.110960
\(41\) 10.5478 1.64729 0.823644 0.567107i \(-0.191937\pi\)
0.823644 + 0.567107i \(0.191937\pi\)
\(42\) −5.69118 −0.878169
\(43\) −4.21626 −0.642974 −0.321487 0.946914i \(-0.604183\pi\)
−0.321487 + 0.946914i \(0.604183\pi\)
\(44\) −1.43715 −0.216659
\(45\) −0.846288 −0.126157
\(46\) 1.35637 0.199986
\(47\) 5.34685 0.779918 0.389959 0.920832i \(-0.372489\pi\)
0.389959 + 0.920832i \(0.372489\pi\)
\(48\) −4.68580 −0.676336
\(49\) 2.09681 0.299545
\(50\) 8.08326 1.14315
\(51\) 1.00000 0.140028
\(52\) −6.01650 −0.834339
\(53\) −4.27035 −0.586577 −0.293289 0.956024i \(-0.594750\pi\)
−0.293289 + 0.956024i \(0.594750\pi\)
\(54\) −1.88694 −0.256780
\(55\) 0.779374 0.105091
\(56\) 2.50105 0.334217
\(57\) 1.27900 0.169408
\(58\) −9.21876 −1.21048
\(59\) 0.996670 0.129755 0.0648777 0.997893i \(-0.479334\pi\)
0.0648777 + 0.997893i \(0.479334\pi\)
\(60\) −1.32067 −0.170497
\(61\) −2.60886 −0.334031 −0.167015 0.985954i \(-0.553413\pi\)
−0.167015 + 0.985954i \(0.553413\pi\)
\(62\) 10.7565 1.36608
\(63\) 3.01609 0.379992
\(64\) −4.18293 −0.522866
\(65\) 3.26278 0.404698
\(66\) 1.73774 0.213901
\(67\) 10.0505 1.22786 0.613931 0.789360i \(-0.289587\pi\)
0.613931 + 0.789360i \(0.289587\pi\)
\(68\) 1.56054 0.189243
\(69\) −0.718821 −0.0865359
\(70\) 4.81638 0.575668
\(71\) 13.2458 1.57199 0.785996 0.618231i \(-0.212151\pi\)
0.785996 + 0.618231i \(0.212151\pi\)
\(72\) 0.829236 0.0977264
\(73\) 9.80450 1.14753 0.573765 0.819020i \(-0.305482\pi\)
0.573765 + 0.819020i \(0.305482\pi\)
\(74\) −4.48011 −0.520802
\(75\) −4.28380 −0.494650
\(76\) 1.99593 0.228949
\(77\) −2.77762 −0.316539
\(78\) 7.27490 0.823721
\(79\) 1.00000 0.112509
\(80\) 3.96553 0.443360
\(81\) 1.00000 0.111111
\(82\) −19.9030 −2.19792
\(83\) 14.1014 1.54783 0.773913 0.633292i \(-0.218297\pi\)
0.773913 + 0.633292i \(0.218297\pi\)
\(84\) 4.70673 0.513546
\(85\) −0.846288 −0.0917928
\(86\) 7.95583 0.857899
\(87\) 4.88556 0.523787
\(88\) −0.763670 −0.0814075
\(89\) 3.60832 0.382481 0.191240 0.981543i \(-0.438749\pi\)
0.191240 + 0.981543i \(0.438749\pi\)
\(90\) 1.59689 0.168327
\(91\) −11.6282 −1.21897
\(92\) −1.12175 −0.116950
\(93\) −5.70052 −0.591117
\(94\) −10.0892 −1.04062
\(95\) −1.08240 −0.111052
\(96\) 7.18334 0.733146
\(97\) 14.0403 1.42558 0.712788 0.701380i \(-0.247432\pi\)
0.712788 + 0.701380i \(0.247432\pi\)
\(98\) −3.95656 −0.399673
\(99\) −0.920932 −0.0925572
\(100\) −6.68503 −0.668503
\(101\) −0.778050 −0.0774188 −0.0387094 0.999251i \(-0.512325\pi\)
−0.0387094 + 0.999251i \(0.512325\pi\)
\(102\) −1.88694 −0.186835
\(103\) −0.390840 −0.0385106 −0.0192553 0.999815i \(-0.506130\pi\)
−0.0192553 + 0.999815i \(0.506130\pi\)
\(104\) −3.19704 −0.313495
\(105\) −2.55248 −0.249097
\(106\) 8.05789 0.782651
\(107\) 2.33288 0.225528 0.112764 0.993622i \(-0.464030\pi\)
0.112764 + 0.993622i \(0.464030\pi\)
\(108\) 1.56054 0.150163
\(109\) −17.8437 −1.70912 −0.854558 0.519356i \(-0.826172\pi\)
−0.854558 + 0.519356i \(0.826172\pi\)
\(110\) −1.47063 −0.140219
\(111\) 2.37427 0.225356
\(112\) −14.1328 −1.33542
\(113\) −18.2810 −1.71973 −0.859867 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(114\) −2.41340 −0.226035
\(115\) 0.608330 0.0567270
\(116\) 7.62411 0.707881
\(117\) −3.85540 −0.356432
\(118\) −1.88066 −0.173128
\(119\) 3.01609 0.276485
\(120\) −0.701773 −0.0640628
\(121\) −10.1519 −0.922899
\(122\) 4.92277 0.445686
\(123\) 10.5478 0.951062
\(124\) −8.89589 −0.798875
\(125\) 7.85677 0.702731
\(126\) −5.69118 −0.507011
\(127\) −12.4803 −1.10745 −0.553725 0.832700i \(-0.686794\pi\)
−0.553725 + 0.832700i \(0.686794\pi\)
\(128\) −6.47374 −0.572203
\(129\) −4.21626 −0.371221
\(130\) −6.15666 −0.539975
\(131\) −2.16959 −0.189558 −0.0947791 0.995498i \(-0.530214\pi\)
−0.0947791 + 0.995498i \(0.530214\pi\)
\(132\) −1.43715 −0.125088
\(133\) 3.85758 0.334495
\(134\) −18.9646 −1.63830
\(135\) −0.846288 −0.0728369
\(136\) 0.829236 0.0711064
\(137\) 17.0324 1.45517 0.727587 0.686015i \(-0.240642\pi\)
0.727587 + 0.686015i \(0.240642\pi\)
\(138\) 1.35637 0.115462
\(139\) 3.52916 0.299340 0.149670 0.988736i \(-0.452179\pi\)
0.149670 + 0.988736i \(0.452179\pi\)
\(140\) −3.98325 −0.336646
\(141\) 5.34685 0.450286
\(142\) −24.9941 −2.09746
\(143\) 3.55056 0.296913
\(144\) −4.68580 −0.390483
\(145\) −4.13459 −0.343359
\(146\) −18.5005 −1.53111
\(147\) 2.09681 0.172942
\(148\) 3.70514 0.304561
\(149\) 10.4477 0.855912 0.427956 0.903800i \(-0.359234\pi\)
0.427956 + 0.903800i \(0.359234\pi\)
\(150\) 8.08326 0.659996
\(151\) 20.4933 1.66772 0.833862 0.551972i \(-0.186125\pi\)
0.833862 + 0.551972i \(0.186125\pi\)
\(152\) 1.06059 0.0860255
\(153\) 1.00000 0.0808452
\(154\) 5.24119 0.422347
\(155\) 4.82429 0.387496
\(156\) −6.01650 −0.481706
\(157\) −5.04399 −0.402554 −0.201277 0.979534i \(-0.564509\pi\)
−0.201277 + 0.979534i \(0.564509\pi\)
\(158\) −1.88694 −0.150117
\(159\) −4.27035 −0.338661
\(160\) −6.07917 −0.480601
\(161\) −2.16803 −0.170865
\(162\) −1.88694 −0.148252
\(163\) 19.3163 1.51297 0.756486 0.654010i \(-0.226915\pi\)
0.756486 + 0.654010i \(0.226915\pi\)
\(164\) 16.4602 1.28533
\(165\) 0.779374 0.0606742
\(166\) −26.6084 −2.06521
\(167\) −22.6179 −1.75023 −0.875113 0.483919i \(-0.839213\pi\)
−0.875113 + 0.483919i \(0.839213\pi\)
\(168\) 2.50105 0.192960
\(169\) 1.86410 0.143392
\(170\) 1.59689 0.122476
\(171\) 1.27900 0.0978076
\(172\) −6.57964 −0.501693
\(173\) −5.37195 −0.408422 −0.204211 0.978927i \(-0.565463\pi\)
−0.204211 + 0.978927i \(0.565463\pi\)
\(174\) −9.21876 −0.698872
\(175\) −12.9203 −0.976685
\(176\) 4.31530 0.325278
\(177\) 0.996670 0.0749143
\(178\) −6.80867 −0.510332
\(179\) 17.5000 1.30801 0.654006 0.756490i \(-0.273087\pi\)
0.654006 + 0.756490i \(0.273087\pi\)
\(180\) −1.32067 −0.0984366
\(181\) 25.4752 1.89356 0.946778 0.321889i \(-0.104318\pi\)
0.946778 + 0.321889i \(0.104318\pi\)
\(182\) 21.9418 1.62643
\(183\) −2.60886 −0.192853
\(184\) −0.596072 −0.0439430
\(185\) −2.00932 −0.147728
\(186\) 10.7565 0.788708
\(187\) −0.920932 −0.0673452
\(188\) 8.34397 0.608546
\(189\) 3.01609 0.219388
\(190\) 2.04243 0.148173
\(191\) −4.97527 −0.359998 −0.179999 0.983667i \(-0.557609\pi\)
−0.179999 + 0.983667i \(0.557609\pi\)
\(192\) −4.18293 −0.301877
\(193\) 8.95833 0.644835 0.322417 0.946598i \(-0.395505\pi\)
0.322417 + 0.946598i \(0.395505\pi\)
\(194\) −26.4932 −1.90210
\(195\) 3.26278 0.233652
\(196\) 3.27216 0.233726
\(197\) 12.5153 0.891678 0.445839 0.895113i \(-0.352905\pi\)
0.445839 + 0.895113i \(0.352905\pi\)
\(198\) 1.73774 0.123496
\(199\) −25.9024 −1.83617 −0.918086 0.396381i \(-0.870266\pi\)
−0.918086 + 0.396381i \(0.870266\pi\)
\(200\) −3.55228 −0.251184
\(201\) 10.0505 0.708906
\(202\) 1.46813 0.103297
\(203\) 14.7353 1.03422
\(204\) 1.56054 0.109260
\(205\) −8.92647 −0.623452
\(206\) 0.737491 0.0513835
\(207\) −0.718821 −0.0499615
\(208\) 18.0656 1.25262
\(209\) −1.17787 −0.0814752
\(210\) 4.81638 0.332362
\(211\) 12.2946 0.846397 0.423199 0.906037i \(-0.360907\pi\)
0.423199 + 0.906037i \(0.360907\pi\)
\(212\) −6.66404 −0.457688
\(213\) 13.2458 0.907590
\(214\) −4.40201 −0.300915
\(215\) 3.56817 0.243347
\(216\) 0.829236 0.0564224
\(217\) −17.1933 −1.16716
\(218\) 33.6700 2.28042
\(219\) 9.80450 0.662527
\(220\) 1.21624 0.0819991
\(221\) −3.85540 −0.259342
\(222\) −4.48011 −0.300685
\(223\) 0.593490 0.0397430 0.0198715 0.999803i \(-0.493674\pi\)
0.0198715 + 0.999803i \(0.493674\pi\)
\(224\) 21.6656 1.44759
\(225\) −4.28380 −0.285586
\(226\) 34.4952 2.29459
\(227\) −18.5422 −1.23069 −0.615344 0.788259i \(-0.710983\pi\)
−0.615344 + 0.788259i \(0.710983\pi\)
\(228\) 1.99593 0.132184
\(229\) 7.01948 0.463860 0.231930 0.972732i \(-0.425496\pi\)
0.231930 + 0.972732i \(0.425496\pi\)
\(230\) −1.14788 −0.0756891
\(231\) −2.77762 −0.182754
\(232\) 4.05128 0.265980
\(233\) −17.8632 −1.17026 −0.585130 0.810940i \(-0.698956\pi\)
−0.585130 + 0.810940i \(0.698956\pi\)
\(234\) 7.27490 0.475575
\(235\) −4.52497 −0.295177
\(236\) 1.55534 0.101244
\(237\) 1.00000 0.0649570
\(238\) −5.69118 −0.368905
\(239\) −6.45389 −0.417467 −0.208734 0.977973i \(-0.566934\pi\)
−0.208734 + 0.977973i \(0.566934\pi\)
\(240\) 3.96553 0.255974
\(241\) −23.6366 −1.52257 −0.761285 0.648418i \(-0.775431\pi\)
−0.761285 + 0.648418i \(0.775431\pi\)
\(242\) 19.1560 1.23139
\(243\) 1.00000 0.0641500
\(244\) −4.07123 −0.260634
\(245\) −1.77451 −0.113369
\(246\) −19.9030 −1.26897
\(247\) −4.93106 −0.313756
\(248\) −4.72708 −0.300170
\(249\) 14.1014 0.893637
\(250\) −14.8252 −0.937630
\(251\) 15.7470 0.993942 0.496971 0.867767i \(-0.334445\pi\)
0.496971 + 0.867767i \(0.334445\pi\)
\(252\) 4.70673 0.296496
\(253\) 0.661986 0.0416187
\(254\) 23.5496 1.47763
\(255\) −0.846288 −0.0529966
\(256\) 20.5814 1.28634
\(257\) 16.7472 1.04466 0.522331 0.852743i \(-0.325063\pi\)
0.522331 + 0.852743i \(0.325063\pi\)
\(258\) 7.95583 0.495308
\(259\) 7.16102 0.444964
\(260\) 5.09169 0.315773
\(261\) 4.88556 0.302409
\(262\) 4.09389 0.252921
\(263\) −9.30111 −0.573531 −0.286766 0.958001i \(-0.592580\pi\)
−0.286766 + 0.958001i \(0.592580\pi\)
\(264\) −0.763670 −0.0470007
\(265\) 3.61394 0.222003
\(266\) −7.27903 −0.446306
\(267\) 3.60832 0.220825
\(268\) 15.6842 0.958063
\(269\) 19.0596 1.16208 0.581041 0.813874i \(-0.302646\pi\)
0.581041 + 0.813874i \(0.302646\pi\)
\(270\) 1.59689 0.0971839
\(271\) 29.3396 1.78225 0.891127 0.453754i \(-0.149915\pi\)
0.891127 + 0.453754i \(0.149915\pi\)
\(272\) −4.68580 −0.284118
\(273\) −11.6282 −0.703773
\(274\) −32.1391 −1.94159
\(275\) 3.94509 0.237898
\(276\) −1.12175 −0.0675213
\(277\) 31.4076 1.88710 0.943550 0.331231i \(-0.107464\pi\)
0.943550 + 0.331231i \(0.107464\pi\)
\(278\) −6.65931 −0.399399
\(279\) −5.70052 −0.341281
\(280\) −2.11661 −0.126492
\(281\) 17.1452 1.02279 0.511397 0.859344i \(-0.329128\pi\)
0.511397 + 0.859344i \(0.329128\pi\)
\(282\) −10.0892 −0.600802
\(283\) −10.0764 −0.598981 −0.299490 0.954099i \(-0.596817\pi\)
−0.299490 + 0.954099i \(0.596817\pi\)
\(284\) 20.6707 1.22658
\(285\) −1.08240 −0.0641160
\(286\) −6.69969 −0.396161
\(287\) 31.8131 1.87787
\(288\) 7.18334 0.423282
\(289\) 1.00000 0.0588235
\(290\) 7.80173 0.458133
\(291\) 14.0403 0.823056
\(292\) 15.3003 0.895383
\(293\) −15.1049 −0.882438 −0.441219 0.897399i \(-0.645454\pi\)
−0.441219 + 0.897399i \(0.645454\pi\)
\(294\) −3.95656 −0.230751
\(295\) −0.843470 −0.0491087
\(296\) 1.96883 0.114436
\(297\) −0.920932 −0.0534379
\(298\) −19.7142 −1.14202
\(299\) 2.77134 0.160271
\(300\) −6.68503 −0.385960
\(301\) −12.7166 −0.732975
\(302\) −38.6697 −2.22519
\(303\) −0.778050 −0.0446978
\(304\) −5.99314 −0.343730
\(305\) 2.20785 0.126421
\(306\) −1.88694 −0.107869
\(307\) −5.28167 −0.301441 −0.150720 0.988576i \(-0.548159\pi\)
−0.150720 + 0.988576i \(0.548159\pi\)
\(308\) −4.33458 −0.246986
\(309\) −0.390840 −0.0222341
\(310\) −9.10313 −0.517023
\(311\) −19.8699 −1.12672 −0.563359 0.826213i \(-0.690491\pi\)
−0.563359 + 0.826213i \(0.690491\pi\)
\(312\) −3.19704 −0.180997
\(313\) 17.3863 0.982734 0.491367 0.870953i \(-0.336497\pi\)
0.491367 + 0.870953i \(0.336497\pi\)
\(314\) 9.51770 0.537115
\(315\) −2.55248 −0.143816
\(316\) 1.56054 0.0877872
\(317\) 30.2204 1.69735 0.848674 0.528916i \(-0.177401\pi\)
0.848674 + 0.528916i \(0.177401\pi\)
\(318\) 8.05789 0.451864
\(319\) −4.49927 −0.251911
\(320\) 3.53997 0.197890
\(321\) 2.33288 0.130209
\(322\) 4.09094 0.227979
\(323\) 1.27900 0.0711655
\(324\) 1.56054 0.0866966
\(325\) 16.5157 0.916129
\(326\) −36.4487 −2.01871
\(327\) −17.8437 −0.986759
\(328\) 8.74661 0.482951
\(329\) 16.1266 0.889088
\(330\) −1.47063 −0.0809556
\(331\) −24.6484 −1.35480 −0.677400 0.735615i \(-0.736893\pi\)
−0.677400 + 0.735615i \(0.736893\pi\)
\(332\) 22.0057 1.20772
\(333\) 2.37427 0.130109
\(334\) 42.6786 2.33527
\(335\) −8.50560 −0.464711
\(336\) −14.1328 −0.771007
\(337\) 2.38693 0.130024 0.0650121 0.997884i \(-0.479291\pi\)
0.0650121 + 0.997884i \(0.479291\pi\)
\(338\) −3.51745 −0.191324
\(339\) −18.2810 −0.992889
\(340\) −1.32067 −0.0716231
\(341\) 5.24980 0.284292
\(342\) −2.41340 −0.130502
\(343\) −14.7885 −0.798502
\(344\) −3.49628 −0.188507
\(345\) 0.608330 0.0327514
\(346\) 10.1365 0.544944
\(347\) 12.7112 0.682372 0.341186 0.939996i \(-0.389171\pi\)
0.341186 + 0.939996i \(0.389171\pi\)
\(348\) 7.62411 0.408695
\(349\) −13.8757 −0.742747 −0.371374 0.928484i \(-0.621113\pi\)
−0.371374 + 0.928484i \(0.621113\pi\)
\(350\) 24.3799 1.30316
\(351\) −3.85540 −0.205786
\(352\) −6.61537 −0.352600
\(353\) 4.35714 0.231907 0.115954 0.993255i \(-0.463008\pi\)
0.115954 + 0.993255i \(0.463008\pi\)
\(354\) −1.88066 −0.0999558
\(355\) −11.2098 −0.594954
\(356\) 5.63092 0.298438
\(357\) 3.01609 0.159629
\(358\) −33.0214 −1.74524
\(359\) −1.47835 −0.0780242 −0.0390121 0.999239i \(-0.512421\pi\)
−0.0390121 + 0.999239i \(0.512421\pi\)
\(360\) −0.701773 −0.0369867
\(361\) −17.3642 −0.913903
\(362\) −48.0701 −2.52651
\(363\) −10.1519 −0.532836
\(364\) −18.1463 −0.951126
\(365\) −8.29743 −0.434308
\(366\) 4.92277 0.257317
\(367\) 1.22044 0.0637064 0.0318532 0.999493i \(-0.489859\pi\)
0.0318532 + 0.999493i \(0.489859\pi\)
\(368\) 3.36825 0.175582
\(369\) 10.5478 0.549096
\(370\) 3.79146 0.197109
\(371\) −12.8798 −0.668684
\(372\) −8.89589 −0.461230
\(373\) 30.4212 1.57515 0.787574 0.616220i \(-0.211337\pi\)
0.787574 + 0.616220i \(0.211337\pi\)
\(374\) 1.73774 0.0898566
\(375\) 7.85677 0.405722
\(376\) 4.43380 0.228656
\(377\) −18.8358 −0.970093
\(378\) −5.69118 −0.292723
\(379\) −29.8919 −1.53545 −0.767723 0.640782i \(-0.778610\pi\)
−0.767723 + 0.640782i \(0.778610\pi\)
\(380\) −1.68913 −0.0866507
\(381\) −12.4803 −0.639386
\(382\) 9.38803 0.480334
\(383\) 32.0422 1.63728 0.818640 0.574307i \(-0.194728\pi\)
0.818640 + 0.574307i \(0.194728\pi\)
\(384\) −6.47374 −0.330362
\(385\) 2.35066 0.119801
\(386\) −16.9038 −0.860382
\(387\) −4.21626 −0.214325
\(388\) 21.9104 1.11233
\(389\) 35.9615 1.82332 0.911659 0.410947i \(-0.134802\pi\)
0.911659 + 0.410947i \(0.134802\pi\)
\(390\) −6.15666 −0.311755
\(391\) −0.718821 −0.0363523
\(392\) 1.73875 0.0878203
\(393\) −2.16959 −0.109441
\(394\) −23.6156 −1.18974
\(395\) −0.846288 −0.0425814
\(396\) −1.43715 −0.0722195
\(397\) 31.2960 1.57070 0.785351 0.619051i \(-0.212483\pi\)
0.785351 + 0.619051i \(0.212483\pi\)
\(398\) 48.8763 2.44995
\(399\) 3.85758 0.193121
\(400\) 20.0730 1.00365
\(401\) −8.29049 −0.414007 −0.207004 0.978340i \(-0.566371\pi\)
−0.207004 + 0.978340i \(0.566371\pi\)
\(402\) −18.9646 −0.945870
\(403\) 21.9778 1.09479
\(404\) −1.21418 −0.0604076
\(405\) −0.846288 −0.0420524
\(406\) −27.8046 −1.37992
\(407\) −2.18654 −0.108383
\(408\) 0.829236 0.0410533
\(409\) −10.0389 −0.496390 −0.248195 0.968710i \(-0.579837\pi\)
−0.248195 + 0.968710i \(0.579837\pi\)
\(410\) 16.8437 0.831851
\(411\) 17.0324 0.840146
\(412\) −0.609921 −0.0300487
\(413\) 3.00605 0.147918
\(414\) 1.35637 0.0666620
\(415\) −11.9338 −0.585808
\(416\) −27.6946 −1.35784
\(417\) 3.52916 0.172824
\(418\) 2.22257 0.108710
\(419\) 17.1659 0.838611 0.419305 0.907845i \(-0.362274\pi\)
0.419305 + 0.907845i \(0.362274\pi\)
\(420\) −3.98325 −0.194363
\(421\) −7.71328 −0.375922 −0.187961 0.982176i \(-0.560188\pi\)
−0.187961 + 0.982176i \(0.560188\pi\)
\(422\) −23.1992 −1.12932
\(423\) 5.34685 0.259973
\(424\) −3.54113 −0.171972
\(425\) −4.28380 −0.207795
\(426\) −24.9941 −1.21097
\(427\) −7.86857 −0.380787
\(428\) 3.64056 0.175973
\(429\) 3.55056 0.171423
\(430\) −6.73293 −0.324691
\(431\) 21.7270 1.04655 0.523275 0.852164i \(-0.324710\pi\)
0.523275 + 0.852164i \(0.324710\pi\)
\(432\) −4.68580 −0.225445
\(433\) 28.5630 1.37265 0.686326 0.727294i \(-0.259222\pi\)
0.686326 + 0.727294i \(0.259222\pi\)
\(434\) 32.4427 1.55730
\(435\) −4.13459 −0.198239
\(436\) −27.8458 −1.33357
\(437\) −0.919373 −0.0439796
\(438\) −18.5005 −0.883988
\(439\) −16.0063 −0.763940 −0.381970 0.924175i \(-0.624754\pi\)
−0.381970 + 0.924175i \(0.624754\pi\)
\(440\) 0.646285 0.0308104
\(441\) 2.09681 0.0998482
\(442\) 7.27490 0.346032
\(443\) 15.3898 0.731191 0.365596 0.930774i \(-0.380865\pi\)
0.365596 + 0.930774i \(0.380865\pi\)
\(444\) 3.70514 0.175838
\(445\) −3.05368 −0.144758
\(446\) −1.11988 −0.0530278
\(447\) 10.4477 0.494161
\(448\) −12.6161 −0.596055
\(449\) −15.7432 −0.742966 −0.371483 0.928440i \(-0.621151\pi\)
−0.371483 + 0.928440i \(0.621151\pi\)
\(450\) 8.08326 0.381049
\(451\) −9.71380 −0.457405
\(452\) −28.5283 −1.34186
\(453\) 20.4933 0.962861
\(454\) 34.9880 1.64207
\(455\) 9.84084 0.461346
\(456\) 1.06059 0.0496668
\(457\) 34.9860 1.63658 0.818289 0.574807i \(-0.194923\pi\)
0.818289 + 0.574807i \(0.194923\pi\)
\(458\) −13.2453 −0.618914
\(459\) 1.00000 0.0466760
\(460\) 0.949322 0.0442624
\(461\) −28.0810 −1.30786 −0.653931 0.756554i \(-0.726882\pi\)
−0.653931 + 0.756554i \(0.726882\pi\)
\(462\) 5.24119 0.243842
\(463\) −31.6142 −1.46923 −0.734617 0.678482i \(-0.762638\pi\)
−0.734617 + 0.678482i \(0.762638\pi\)
\(464\) −22.8927 −1.06277
\(465\) 4.82429 0.223721
\(466\) 33.7068 1.56144
\(467\) −10.2872 −0.476037 −0.238018 0.971261i \(-0.576498\pi\)
−0.238018 + 0.971261i \(0.576498\pi\)
\(468\) −6.01650 −0.278113
\(469\) 30.3132 1.39973
\(470\) 8.53835 0.393845
\(471\) −5.04399 −0.232415
\(472\) 0.826475 0.0380416
\(473\) 3.88289 0.178536
\(474\) −1.88694 −0.0866700
\(475\) −5.47898 −0.251393
\(476\) 4.70673 0.215733
\(477\) −4.27035 −0.195526
\(478\) 12.1781 0.557013
\(479\) −8.35097 −0.381566 −0.190783 0.981632i \(-0.561103\pi\)
−0.190783 + 0.981632i \(0.561103\pi\)
\(480\) −6.07917 −0.277475
\(481\) −9.15377 −0.417376
\(482\) 44.6009 2.03152
\(483\) −2.16803 −0.0986488
\(484\) −15.8424 −0.720110
\(485\) −11.8821 −0.539540
\(486\) −1.88694 −0.0855933
\(487\) −24.0747 −1.09093 −0.545465 0.838134i \(-0.683647\pi\)
−0.545465 + 0.838134i \(0.683647\pi\)
\(488\) −2.16336 −0.0979309
\(489\) 19.3163 0.873515
\(490\) 3.34839 0.151265
\(491\) 19.0373 0.859142 0.429571 0.903033i \(-0.358665\pi\)
0.429571 + 0.903033i \(0.358665\pi\)
\(492\) 16.4602 0.742085
\(493\) 4.88556 0.220035
\(494\) 9.30461 0.418634
\(495\) 0.779374 0.0350303
\(496\) 26.7115 1.19938
\(497\) 39.9507 1.79203
\(498\) −26.6084 −1.19235
\(499\) −12.3272 −0.551840 −0.275920 0.961181i \(-0.588983\pi\)
−0.275920 + 0.961181i \(0.588983\pi\)
\(500\) 12.2608 0.548319
\(501\) −22.6179 −1.01049
\(502\) −29.7136 −1.32619
\(503\) 6.74107 0.300569 0.150285 0.988643i \(-0.451981\pi\)
0.150285 + 0.988643i \(0.451981\pi\)
\(504\) 2.50105 0.111406
\(505\) 0.658454 0.0293008
\(506\) −1.24913 −0.0555304
\(507\) 1.86410 0.0827877
\(508\) −19.4760 −0.864109
\(509\) 25.2221 1.11795 0.558975 0.829184i \(-0.311195\pi\)
0.558975 + 0.829184i \(0.311195\pi\)
\(510\) 1.59689 0.0707117
\(511\) 29.5713 1.30816
\(512\) −25.8884 −1.14412
\(513\) 1.27900 0.0564693
\(514\) −31.6009 −1.39386
\(515\) 0.330763 0.0145752
\(516\) −6.57964 −0.289653
\(517\) −4.92409 −0.216561
\(518\) −13.5124 −0.593701
\(519\) −5.37195 −0.235802
\(520\) 2.70561 0.118649
\(521\) 6.19871 0.271570 0.135785 0.990738i \(-0.456644\pi\)
0.135785 + 0.990738i \(0.456644\pi\)
\(522\) −9.21876 −0.403494
\(523\) −12.1289 −0.530361 −0.265181 0.964199i \(-0.585432\pi\)
−0.265181 + 0.964199i \(0.585432\pi\)
\(524\) −3.38573 −0.147906
\(525\) −12.9203 −0.563889
\(526\) 17.5506 0.765244
\(527\) −5.70052 −0.248319
\(528\) 4.31530 0.187799
\(529\) −22.4833 −0.977535
\(530\) −6.81929 −0.296211
\(531\) 0.996670 0.0432518
\(532\) 6.01991 0.260996
\(533\) −40.6659 −1.76144
\(534\) −6.80867 −0.294640
\(535\) −1.97429 −0.0853561
\(536\) 8.33422 0.359983
\(537\) 17.5000 0.755181
\(538\) −35.9642 −1.55053
\(539\) −1.93102 −0.0831750
\(540\) −1.32067 −0.0568324
\(541\) 37.1875 1.59881 0.799407 0.600789i \(-0.205147\pi\)
0.799407 + 0.600789i \(0.205147\pi\)
\(542\) −55.3621 −2.37800
\(543\) 25.4752 1.09324
\(544\) 7.18334 0.307983
\(545\) 15.1009 0.646852
\(546\) 21.9418 0.939021
\(547\) 42.8321 1.83137 0.915685 0.401897i \(-0.131649\pi\)
0.915685 + 0.401897i \(0.131649\pi\)
\(548\) 26.5797 1.13543
\(549\) −2.60886 −0.111344
\(550\) −7.44414 −0.317419
\(551\) 6.24864 0.266201
\(552\) −0.596072 −0.0253705
\(553\) 3.01609 0.128257
\(554\) −59.2642 −2.51790
\(555\) −2.00932 −0.0852908
\(556\) 5.50740 0.233566
\(557\) 2.89913 0.122840 0.0614201 0.998112i \(-0.480437\pi\)
0.0614201 + 0.998112i \(0.480437\pi\)
\(558\) 10.7565 0.455361
\(559\) 16.2554 0.687529
\(560\) 11.9604 0.505420
\(561\) −0.920932 −0.0388818
\(562\) −32.3519 −1.36468
\(563\) 13.9220 0.586741 0.293370 0.955999i \(-0.405223\pi\)
0.293370 + 0.955999i \(0.405223\pi\)
\(564\) 8.34397 0.351344
\(565\) 15.4710 0.650871
\(566\) 19.0136 0.799201
\(567\) 3.01609 0.126664
\(568\) 10.9839 0.460875
\(569\) −5.20319 −0.218129 −0.109065 0.994035i \(-0.534786\pi\)
−0.109065 + 0.994035i \(0.534786\pi\)
\(570\) 2.04243 0.0855479
\(571\) −19.7899 −0.828180 −0.414090 0.910236i \(-0.635900\pi\)
−0.414090 + 0.910236i \(0.635900\pi\)
\(572\) 5.54079 0.231672
\(573\) −4.97527 −0.207845
\(574\) −60.0294 −2.50558
\(575\) 3.07928 0.128415
\(576\) −4.18293 −0.174289
\(577\) 14.0552 0.585124 0.292562 0.956247i \(-0.405492\pi\)
0.292562 + 0.956247i \(0.405492\pi\)
\(578\) −1.88694 −0.0784863
\(579\) 8.95833 0.372296
\(580\) −6.45219 −0.267913
\(581\) 42.5310 1.76448
\(582\) −26.4932 −1.09818
\(583\) 3.93270 0.162876
\(584\) 8.13025 0.336432
\(585\) 3.26278 0.134899
\(586\) 28.5020 1.17741
\(587\) 30.3566 1.25295 0.626475 0.779442i \(-0.284497\pi\)
0.626475 + 0.779442i \(0.284497\pi\)
\(588\) 3.27216 0.134942
\(589\) −7.29098 −0.300419
\(590\) 1.59158 0.0655242
\(591\) 12.5153 0.514811
\(592\) −11.1254 −0.457249
\(593\) −14.4046 −0.591527 −0.295763 0.955261i \(-0.595574\pi\)
−0.295763 + 0.955261i \(0.595574\pi\)
\(594\) 1.73774 0.0713005
\(595\) −2.55248 −0.104642
\(596\) 16.3041 0.667842
\(597\) −25.9024 −1.06011
\(598\) −5.22935 −0.213844
\(599\) 44.1246 1.80288 0.901442 0.432900i \(-0.142510\pi\)
0.901442 + 0.432900i \(0.142510\pi\)
\(600\) −3.55228 −0.145021
\(601\) −12.9118 −0.526685 −0.263342 0.964702i \(-0.584825\pi\)
−0.263342 + 0.964702i \(0.584825\pi\)
\(602\) 23.9955 0.977985
\(603\) 10.0505 0.409287
\(604\) 31.9807 1.30127
\(605\) 8.59142 0.349291
\(606\) 1.46813 0.0596388
\(607\) 7.79738 0.316486 0.158243 0.987400i \(-0.449417\pi\)
0.158243 + 0.987400i \(0.449417\pi\)
\(608\) 9.18750 0.372602
\(609\) 14.7353 0.597105
\(610\) −4.16608 −0.168680
\(611\) −20.6142 −0.833963
\(612\) 1.56054 0.0630811
\(613\) 38.3292 1.54810 0.774050 0.633124i \(-0.218228\pi\)
0.774050 + 0.633124i \(0.218228\pi\)
\(614\) 9.96619 0.402203
\(615\) −8.92647 −0.359950
\(616\) −2.30330 −0.0928026
\(617\) 36.2737 1.46032 0.730162 0.683274i \(-0.239444\pi\)
0.730162 + 0.683274i \(0.239444\pi\)
\(618\) 0.737491 0.0296663
\(619\) 34.8928 1.40246 0.701229 0.712936i \(-0.252635\pi\)
0.701229 + 0.712936i \(0.252635\pi\)
\(620\) 7.52849 0.302351
\(621\) −0.718821 −0.0288453
\(622\) 37.4932 1.50334
\(623\) 10.8830 0.436019
\(624\) 18.0656 0.723203
\(625\) 14.7699 0.590796
\(626\) −32.8070 −1.31123
\(627\) −1.17787 −0.0470397
\(628\) −7.87134 −0.314101
\(629\) 2.37427 0.0946684
\(630\) 4.81638 0.191889
\(631\) 36.1616 1.43957 0.719786 0.694196i \(-0.244240\pi\)
0.719786 + 0.694196i \(0.244240\pi\)
\(632\) 0.829236 0.0329852
\(633\) 12.2946 0.488668
\(634\) −57.0241 −2.26472
\(635\) 10.5619 0.419138
\(636\) −6.66404 −0.264247
\(637\) −8.08405 −0.320302
\(638\) 8.48985 0.336116
\(639\) 13.2458 0.523997
\(640\) 5.47865 0.216563
\(641\) −40.8595 −1.61385 −0.806926 0.590652i \(-0.798871\pi\)
−0.806926 + 0.590652i \(0.798871\pi\)
\(642\) −4.40201 −0.173734
\(643\) −10.3584 −0.408494 −0.204247 0.978919i \(-0.565475\pi\)
−0.204247 + 0.978919i \(0.565475\pi\)
\(644\) −3.38330 −0.133321
\(645\) 3.56817 0.140497
\(646\) −2.41340 −0.0949538
\(647\) −43.4012 −1.70628 −0.853139 0.521683i \(-0.825304\pi\)
−0.853139 + 0.521683i \(0.825304\pi\)
\(648\) 0.829236 0.0325755
\(649\) −0.917866 −0.0360294
\(650\) −31.1642 −1.22236
\(651\) −17.1933 −0.673859
\(652\) 30.1439 1.18053
\(653\) −20.2481 −0.792370 −0.396185 0.918171i \(-0.629666\pi\)
−0.396185 + 0.918171i \(0.629666\pi\)
\(654\) 33.6700 1.31660
\(655\) 1.83610 0.0717424
\(656\) −49.4248 −1.92971
\(657\) 9.80450 0.382510
\(658\) −30.4299 −1.18628
\(659\) 2.17029 0.0845425 0.0422713 0.999106i \(-0.486541\pi\)
0.0422713 + 0.999106i \(0.486541\pi\)
\(660\) 1.21624 0.0473422
\(661\) −43.3422 −1.68582 −0.842908 0.538057i \(-0.819158\pi\)
−0.842908 + 0.538057i \(0.819158\pi\)
\(662\) 46.5101 1.80766
\(663\) −3.85540 −0.149731
\(664\) 11.6934 0.453790
\(665\) −3.26463 −0.126597
\(666\) −4.48011 −0.173601
\(667\) −3.51185 −0.135979
\(668\) −35.2961 −1.36565
\(669\) 0.593490 0.0229456
\(670\) 16.0495 0.620048
\(671\) 2.40259 0.0927508
\(672\) 21.6656 0.835769
\(673\) 32.2301 1.24238 0.621190 0.783660i \(-0.286650\pi\)
0.621190 + 0.783660i \(0.286650\pi\)
\(674\) −4.50399 −0.173487
\(675\) −4.28380 −0.164883
\(676\) 2.90900 0.111885
\(677\) −36.3410 −1.39670 −0.698348 0.715758i \(-0.746081\pi\)
−0.698348 + 0.715758i \(0.746081\pi\)
\(678\) 34.4952 1.32478
\(679\) 42.3468 1.62512
\(680\) −0.701773 −0.0269118
\(681\) −18.5422 −0.710538
\(682\) −9.90605 −0.379322
\(683\) −27.6294 −1.05721 −0.528605 0.848868i \(-0.677285\pi\)
−0.528605 + 0.848868i \(0.677285\pi\)
\(684\) 1.99593 0.0763163
\(685\) −14.4143 −0.550742
\(686\) 27.9049 1.06542
\(687\) 7.01948 0.267810
\(688\) 19.7566 0.753211
\(689\) 16.4639 0.627225
\(690\) −1.14788 −0.0436991
\(691\) 8.03647 0.305722 0.152861 0.988248i \(-0.451151\pi\)
0.152861 + 0.988248i \(0.451151\pi\)
\(692\) −8.38314 −0.318679
\(693\) −2.77762 −0.105513
\(694\) −23.9852 −0.910467
\(695\) −2.98669 −0.113292
\(696\) 4.05128 0.153564
\(697\) 10.5478 0.399526
\(698\) 26.1825 0.991023
\(699\) −17.8632 −0.675649
\(700\) −20.1627 −0.762077
\(701\) 28.6491 1.08206 0.541031 0.841003i \(-0.318034\pi\)
0.541031 + 0.841003i \(0.318034\pi\)
\(702\) 7.27490 0.274574
\(703\) 3.03670 0.114531
\(704\) 3.85220 0.145185
\(705\) −4.52497 −0.170420
\(706\) −8.22166 −0.309426
\(707\) −2.34667 −0.0882556
\(708\) 1.55534 0.0584534
\(709\) 13.2281 0.496793 0.248397 0.968658i \(-0.420096\pi\)
0.248397 + 0.968658i \(0.420096\pi\)
\(710\) 21.1522 0.793828
\(711\) 1.00000 0.0375029
\(712\) 2.99215 0.112135
\(713\) 4.09766 0.153458
\(714\) −5.69118 −0.212987
\(715\) −3.00480 −0.112373
\(716\) 27.3094 1.02060
\(717\) −6.45389 −0.241025
\(718\) 2.78955 0.104105
\(719\) −1.77194 −0.0660821 −0.0330410 0.999454i \(-0.510519\pi\)
−0.0330410 + 0.999454i \(0.510519\pi\)
\(720\) 3.96553 0.147787
\(721\) −1.17881 −0.0439012
\(722\) 32.7651 1.21939
\(723\) −23.6366 −0.879056
\(724\) 39.7550 1.47748
\(725\) −20.9288 −0.777274
\(726\) 19.1560 0.710945
\(727\) −38.2746 −1.41953 −0.709764 0.704440i \(-0.751198\pi\)
−0.709764 + 0.704440i \(0.751198\pi\)
\(728\) −9.64256 −0.357377
\(729\) 1.00000 0.0370370
\(730\) 15.6568 0.579482
\(731\) −4.21626 −0.155944
\(732\) −4.07123 −0.150477
\(733\) 15.2987 0.565069 0.282535 0.959257i \(-0.408825\pi\)
0.282535 + 0.959257i \(0.408825\pi\)
\(734\) −2.30289 −0.0850014
\(735\) −1.77451 −0.0654537
\(736\) −5.16354 −0.190330
\(737\) −9.25581 −0.340942
\(738\) −19.9030 −0.732641
\(739\) −14.4646 −0.532088 −0.266044 0.963961i \(-0.585717\pi\)
−0.266044 + 0.963961i \(0.585717\pi\)
\(740\) −3.13562 −0.115268
\(741\) −4.93106 −0.181147
\(742\) 24.3033 0.892203
\(743\) −17.9639 −0.659032 −0.329516 0.944150i \(-0.606886\pi\)
−0.329516 + 0.944150i \(0.606886\pi\)
\(744\) −4.72708 −0.173303
\(745\) −8.84180 −0.323938
\(746\) −57.4029 −2.10167
\(747\) 14.1014 0.515942
\(748\) −1.43715 −0.0525474
\(749\) 7.03620 0.257097
\(750\) −14.8252 −0.541341
\(751\) −10.3650 −0.378223 −0.189111 0.981956i \(-0.560561\pi\)
−0.189111 + 0.981956i \(0.560561\pi\)
\(752\) −25.0542 −0.913634
\(753\) 15.7470 0.573853
\(754\) 35.5420 1.29436
\(755\) −17.3433 −0.631186
\(756\) 4.70673 0.171182
\(757\) −27.2481 −0.990347 −0.495174 0.868794i \(-0.664896\pi\)
−0.495174 + 0.868794i \(0.664896\pi\)
\(758\) 56.4043 2.04870
\(759\) 0.661986 0.0240286
\(760\) −0.897568 −0.0325582
\(761\) −26.0036 −0.942630 −0.471315 0.881965i \(-0.656221\pi\)
−0.471315 + 0.881965i \(0.656221\pi\)
\(762\) 23.5496 0.853112
\(763\) −53.8182 −1.94835
\(764\) −7.76410 −0.280895
\(765\) −0.846288 −0.0305976
\(766\) −60.4617 −2.18457
\(767\) −3.84256 −0.138747
\(768\) 20.5814 0.742668
\(769\) 24.9618 0.900146 0.450073 0.892992i \(-0.351398\pi\)
0.450073 + 0.892992i \(0.351398\pi\)
\(770\) −4.43556 −0.159846
\(771\) 16.7472 0.603136
\(772\) 13.9798 0.503145
\(773\) −35.7594 −1.28617 −0.643087 0.765793i \(-0.722347\pi\)
−0.643087 + 0.765793i \(0.722347\pi\)
\(774\) 7.95583 0.285966
\(775\) 24.4199 0.877188
\(776\) 11.6427 0.417949
\(777\) 7.16102 0.256900
\(778\) −67.8571 −2.43279
\(779\) 13.4906 0.483352
\(780\) 5.09169 0.182312
\(781\) −12.1985 −0.436497
\(782\) 1.35637 0.0485038
\(783\) 4.88556 0.174596
\(784\) −9.82524 −0.350901
\(785\) 4.26867 0.152355
\(786\) 4.09389 0.146024
\(787\) −50.1097 −1.78622 −0.893109 0.449840i \(-0.851481\pi\)
−0.893109 + 0.449840i \(0.851481\pi\)
\(788\) 19.5306 0.695749
\(789\) −9.30111 −0.331128
\(790\) 1.59689 0.0568149
\(791\) −55.1373 −1.96046
\(792\) −0.763670 −0.0271358
\(793\) 10.0582 0.357178
\(794\) −59.0536 −2.09574
\(795\) 3.61394 0.128173
\(796\) −40.4217 −1.43271
\(797\) −25.3853 −0.899194 −0.449597 0.893232i \(-0.648432\pi\)
−0.449597 + 0.893232i \(0.648432\pi\)
\(798\) −7.27903 −0.257675
\(799\) 5.34685 0.189158
\(800\) −30.7720 −1.08795
\(801\) 3.60832 0.127494
\(802\) 15.6436 0.552396
\(803\) −9.02928 −0.318636
\(804\) 15.6842 0.553138
\(805\) 1.83478 0.0646675
\(806\) −41.4708 −1.46075
\(807\) 19.0596 0.670928
\(808\) −0.645187 −0.0226976
\(809\) 13.9436 0.490232 0.245116 0.969494i \(-0.421174\pi\)
0.245116 + 0.969494i \(0.421174\pi\)
\(810\) 1.59689 0.0561091
\(811\) 6.74545 0.236865 0.118432 0.992962i \(-0.462213\pi\)
0.118432 + 0.992962i \(0.462213\pi\)
\(812\) 22.9950 0.806967
\(813\) 29.3396 1.02898
\(814\) 4.12587 0.144612
\(815\) −16.3472 −0.572617
\(816\) −4.68580 −0.164036
\(817\) −5.39260 −0.188663
\(818\) 18.9427 0.662316
\(819\) −11.6282 −0.406324
\(820\) −13.9301 −0.486460
\(821\) −12.7394 −0.444609 −0.222305 0.974977i \(-0.571358\pi\)
−0.222305 + 0.974977i \(0.571358\pi\)
\(822\) −32.1391 −1.12098
\(823\) −49.2113 −1.71540 −0.857699 0.514151i \(-0.828107\pi\)
−0.857699 + 0.514151i \(0.828107\pi\)
\(824\) −0.324099 −0.0112905
\(825\) 3.94509 0.137350
\(826\) −5.67223 −0.197362
\(827\) 51.9279 1.80571 0.902855 0.429946i \(-0.141467\pi\)
0.902855 + 0.429946i \(0.141467\pi\)
\(828\) −1.12175 −0.0389835
\(829\) −23.6474 −0.821310 −0.410655 0.911791i \(-0.634700\pi\)
−0.410655 + 0.911791i \(0.634700\pi\)
\(830\) 22.5184 0.781624
\(831\) 31.4076 1.08952
\(832\) 16.1269 0.559099
\(833\) 2.09681 0.0726503
\(834\) −6.65931 −0.230593
\(835\) 19.1413 0.662410
\(836\) −1.83812 −0.0635726
\(837\) −5.70052 −0.197039
\(838\) −32.3911 −1.11893
\(839\) 54.2295 1.87221 0.936105 0.351721i \(-0.114403\pi\)
0.936105 + 0.351721i \(0.114403\pi\)
\(840\) −2.11661 −0.0730300
\(841\) −5.13128 −0.176941
\(842\) 14.5545 0.501581
\(843\) 17.1452 0.590511
\(844\) 19.1863 0.660418
\(845\) −1.57757 −0.0542700
\(846\) −10.0892 −0.346873
\(847\) −30.6190 −1.05208
\(848\) 20.0100 0.687146
\(849\) −10.0764 −0.345822
\(850\) 8.08326 0.277254
\(851\) −1.70668 −0.0585041
\(852\) 20.6707 0.708165
\(853\) −32.2549 −1.10439 −0.552193 0.833716i \(-0.686209\pi\)
−0.552193 + 0.833716i \(0.686209\pi\)
\(854\) 14.8475 0.508072
\(855\) −1.08240 −0.0370174
\(856\) 1.93451 0.0661203
\(857\) −15.2326 −0.520337 −0.260168 0.965563i \(-0.583778\pi\)
−0.260168 + 0.965563i \(0.583778\pi\)
\(858\) −6.69969 −0.228724
\(859\) −9.13714 −0.311755 −0.155878 0.987776i \(-0.549821\pi\)
−0.155878 + 0.987776i \(0.549821\pi\)
\(860\) 5.56827 0.189877
\(861\) 31.8131 1.08419
\(862\) −40.9975 −1.39638
\(863\) 24.4648 0.832793 0.416397 0.909183i \(-0.363293\pi\)
0.416397 + 0.909183i \(0.363293\pi\)
\(864\) 7.18334 0.244382
\(865\) 4.54622 0.154576
\(866\) −53.8967 −1.83148
\(867\) 1.00000 0.0339618
\(868\) −26.8308 −0.910698
\(869\) −0.920932 −0.0312405
\(870\) 7.80173 0.264503
\(871\) −38.7486 −1.31295
\(872\) −14.7966 −0.501077
\(873\) 14.0403 0.475192
\(874\) 1.73480 0.0586805
\(875\) 23.6967 0.801096
\(876\) 15.3003 0.516950
\(877\) −50.8356 −1.71660 −0.858298 0.513152i \(-0.828478\pi\)
−0.858298 + 0.513152i \(0.828478\pi\)
\(878\) 30.2030 1.01930
\(879\) −15.1049 −0.509476
\(880\) −3.65199 −0.123108
\(881\) 42.0380 1.41630 0.708149 0.706063i \(-0.249531\pi\)
0.708149 + 0.706063i \(0.249531\pi\)
\(882\) −3.95656 −0.133224
\(883\) −19.5189 −0.656863 −0.328432 0.944528i \(-0.606520\pi\)
−0.328432 + 0.944528i \(0.606520\pi\)
\(884\) −6.01650 −0.202357
\(885\) −0.843470 −0.0283529
\(886\) −29.0396 −0.975605
\(887\) 4.03208 0.135384 0.0676920 0.997706i \(-0.478436\pi\)
0.0676920 + 0.997706i \(0.478436\pi\)
\(888\) 1.96883 0.0660697
\(889\) −37.6418 −1.26247
\(890\) 5.76210 0.193146
\(891\) −0.920932 −0.0308524
\(892\) 0.926164 0.0310103
\(893\) 6.83862 0.228846
\(894\) −19.7142 −0.659343
\(895\) −14.8100 −0.495045
\(896\) −19.5254 −0.652298
\(897\) 2.77134 0.0925324
\(898\) 29.7064 0.991315
\(899\) −27.8503 −0.928858
\(900\) −6.68503 −0.222834
\(901\) −4.27035 −0.142266
\(902\) 18.3294 0.610301
\(903\) −12.7166 −0.423183
\(904\) −15.1593 −0.504190
\(905\) −21.5593 −0.716657
\(906\) −38.6697 −1.28471
\(907\) 15.0180 0.498664 0.249332 0.968418i \(-0.419789\pi\)
0.249332 + 0.968418i \(0.419789\pi\)
\(908\) −28.9358 −0.960269
\(909\) −0.778050 −0.0258063
\(910\) −18.5691 −0.615559
\(911\) −19.2566 −0.638001 −0.319001 0.947755i \(-0.603347\pi\)
−0.319001 + 0.947755i \(0.603347\pi\)
\(912\) −5.99314 −0.198453
\(913\) −12.9864 −0.429787
\(914\) −66.0165 −2.18363
\(915\) 2.20785 0.0729893
\(916\) 10.9542 0.361936
\(917\) −6.54369 −0.216092
\(918\) −1.88694 −0.0622783
\(919\) 42.4425 1.40005 0.700024 0.714119i \(-0.253173\pi\)
0.700024 + 0.714119i \(0.253173\pi\)
\(920\) 0.504449 0.0166312
\(921\) −5.28167 −0.174037
\(922\) 52.9871 1.74504
\(923\) −51.0680 −1.68092
\(924\) −4.33458 −0.142597
\(925\) −10.1709 −0.334417
\(926\) 59.6540 1.96035
\(927\) −0.390840 −0.0128369
\(928\) 35.0947 1.15204
\(929\) −16.1756 −0.530705 −0.265352 0.964151i \(-0.585488\pi\)
−0.265352 + 0.964151i \(0.585488\pi\)
\(930\) −9.10313 −0.298504
\(931\) 2.68183 0.0878933
\(932\) −27.8763 −0.913117
\(933\) −19.8699 −0.650510
\(934\) 19.4114 0.635160
\(935\) 0.779374 0.0254883
\(936\) −3.19704 −0.104498
\(937\) −41.0799 −1.34202 −0.671011 0.741448i \(-0.734140\pi\)
−0.671011 + 0.741448i \(0.734140\pi\)
\(938\) −57.1991 −1.86762
\(939\) 17.3863 0.567382
\(940\) −7.06140 −0.230317
\(941\) −19.5457 −0.637171 −0.318585 0.947894i \(-0.603208\pi\)
−0.318585 + 0.947894i \(0.603208\pi\)
\(942\) 9.51770 0.310103
\(943\) −7.58198 −0.246903
\(944\) −4.67019 −0.152002
\(945\) −2.55248 −0.0830323
\(946\) −7.32678 −0.238214
\(947\) −31.5035 −1.02372 −0.511862 0.859067i \(-0.671044\pi\)
−0.511862 + 0.859067i \(0.671044\pi\)
\(948\) 1.56054 0.0506840
\(949\) −37.8003 −1.22705
\(950\) 10.3385 0.335425
\(951\) 30.2204 0.979965
\(952\) 2.50105 0.0810596
\(953\) −38.6397 −1.25166 −0.625831 0.779958i \(-0.715240\pi\)
−0.625831 + 0.779958i \(0.715240\pi\)
\(954\) 8.05789 0.260884
\(955\) 4.21051 0.136249
\(956\) −10.0715 −0.325737
\(957\) −4.49927 −0.145441
\(958\) 15.7578 0.509111
\(959\) 51.3712 1.65886
\(960\) 3.53997 0.114252
\(961\) 1.49598 0.0482575
\(962\) 17.2726 0.556891
\(963\) 2.33288 0.0751762
\(964\) −36.8859 −1.18801
\(965\) −7.58133 −0.244052
\(966\) 4.09094 0.131624
\(967\) −38.4574 −1.23671 −0.618353 0.785900i \(-0.712200\pi\)
−0.618353 + 0.785900i \(0.712200\pi\)
\(968\) −8.41831 −0.270575
\(969\) 1.27900 0.0410874
\(970\) 22.4209 0.719890
\(971\) 15.7133 0.504265 0.252132 0.967693i \(-0.418868\pi\)
0.252132 + 0.967693i \(0.418868\pi\)
\(972\) 1.56054 0.0500543
\(973\) 10.6443 0.341240
\(974\) 45.4275 1.45559
\(975\) 16.5157 0.528927
\(976\) 12.2246 0.391300
\(977\) −55.2388 −1.76724 −0.883622 0.468201i \(-0.844902\pi\)
−0.883622 + 0.468201i \(0.844902\pi\)
\(978\) −36.4487 −1.16550
\(979\) −3.32301 −0.106204
\(980\) −2.76919 −0.0884585
\(981\) −17.8437 −0.569705
\(982\) −35.9223 −1.14633
\(983\) −52.6483 −1.67922 −0.839609 0.543191i \(-0.817216\pi\)
−0.839609 + 0.543191i \(0.817216\pi\)
\(984\) 8.74661 0.278832
\(985\) −10.5915 −0.337475
\(986\) −9.21876 −0.293585
\(987\) 16.1266 0.513315
\(988\) −7.69511 −0.244814
\(989\) 3.03074 0.0963719
\(990\) −1.47063 −0.0467397
\(991\) 58.3156 1.85246 0.926228 0.376963i \(-0.123032\pi\)
0.926228 + 0.376963i \(0.123032\pi\)
\(992\) −40.9488 −1.30013
\(993\) −24.6484 −0.782194
\(994\) −75.3845 −2.39105
\(995\) 21.9209 0.694939
\(996\) 22.0057 0.697278
\(997\) 10.1087 0.320147 0.160074 0.987105i \(-0.448827\pi\)
0.160074 + 0.987105i \(0.448827\pi\)
\(998\) 23.2606 0.736302
\(999\) 2.37427 0.0751186
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 4029.2.a.k.1.6 31
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4029.2.a.k.1.6 31 1.1 even 1 trivial