Properties

Label 731.2.a.b.1.2
Level 731
Weight 2
Character 731.1
Self dual yes
Analytic conductor 5.837
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.83706438776\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\)
Character \(\chi\) = 731.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.56155 q^{3} -1.00000 q^{4} -2.56155 q^{5} -2.56155 q^{6} +3.00000 q^{8} +3.56155 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.56155 q^{3} -1.00000 q^{4} -2.56155 q^{5} -2.56155 q^{6} +3.00000 q^{8} +3.56155 q^{9} +2.56155 q^{10} -2.00000 q^{11} -2.56155 q^{12} -4.56155 q^{13} -6.56155 q^{15} -1.00000 q^{16} -1.00000 q^{17} -3.56155 q^{18} -1.12311 q^{19} +2.56155 q^{20} +2.00000 q^{22} +2.00000 q^{23} +7.68466 q^{24} +1.56155 q^{25} +4.56155 q^{26} +1.43845 q^{27} -5.12311 q^{29} +6.56155 q^{30} -6.00000 q^{31} -5.00000 q^{32} -5.12311 q^{33} +1.00000 q^{34} -3.56155 q^{36} -1.43845 q^{37} +1.12311 q^{38} -11.6847 q^{39} -7.68466 q^{40} +3.12311 q^{41} -1.00000 q^{43} +2.00000 q^{44} -9.12311 q^{45} -2.00000 q^{46} -5.43845 q^{47} -2.56155 q^{48} -7.00000 q^{49} -1.56155 q^{50} -2.56155 q^{51} +4.56155 q^{52} +1.68466 q^{53} -1.43845 q^{54} +5.12311 q^{55} -2.87689 q^{57} +5.12311 q^{58} +8.80776 q^{59} +6.56155 q^{60} -11.6847 q^{61} +6.00000 q^{62} +7.00000 q^{64} +11.6847 q^{65} +5.12311 q^{66} +1.43845 q^{67} +1.00000 q^{68} +5.12311 q^{69} +1.43845 q^{71} +10.6847 q^{72} +8.80776 q^{73} +1.43845 q^{74} +4.00000 q^{75} +1.12311 q^{76} +11.6847 q^{78} -0.876894 q^{79} +2.56155 q^{80} -7.00000 q^{81} -3.12311 q^{82} -4.80776 q^{83} +2.56155 q^{85} +1.00000 q^{86} -13.1231 q^{87} -6.00000 q^{88} +16.2462 q^{89} +9.12311 q^{90} -2.00000 q^{92} -15.3693 q^{93} +5.43845 q^{94} +2.87689 q^{95} -12.8078 q^{96} -0.246211 q^{97} +7.00000 q^{98} -7.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + q^{3} - 2q^{4} - q^{5} - q^{6} + 6q^{8} + 3q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + q^{3} - 2q^{4} - q^{5} - q^{6} + 6q^{8} + 3q^{9} + q^{10} - 4q^{11} - q^{12} - 5q^{13} - 9q^{15} - 2q^{16} - 2q^{17} - 3q^{18} + 6q^{19} + q^{20} + 4q^{22} + 4q^{23} + 3q^{24} - q^{25} + 5q^{26} + 7q^{27} - 2q^{29} + 9q^{30} - 12q^{31} - 10q^{32} - 2q^{33} + 2q^{34} - 3q^{36} - 7q^{37} - 6q^{38} - 11q^{39} - 3q^{40} - 2q^{41} - 2q^{43} + 4q^{44} - 10q^{45} - 4q^{46} - 15q^{47} - q^{48} - 14q^{49} + q^{50} - q^{51} + 5q^{52} - 9q^{53} - 7q^{54} + 2q^{55} - 14q^{57} + 2q^{58} - 3q^{59} + 9q^{60} - 11q^{61} + 12q^{62} + 14q^{64} + 11q^{65} + 2q^{66} + 7q^{67} + 2q^{68} + 2q^{69} + 7q^{71} + 9q^{72} - 3q^{73} + 7q^{74} + 8q^{75} - 6q^{76} + 11q^{78} - 10q^{79} + q^{80} - 14q^{81} + 2q^{82} + 11q^{83} + q^{85} + 2q^{86} - 18q^{87} - 12q^{88} + 16q^{89} + 10q^{90} - 4q^{92} - 6q^{93} + 15q^{94} + 14q^{95} - 5q^{96} + 16q^{97} + 14q^{98} - 6q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 2.56155 1.47891 0.739457 0.673204i \(-0.235083\pi\)
0.739457 + 0.673204i \(0.235083\pi\)
\(4\) −1.00000 −0.500000
\(5\) −2.56155 −1.14556 −0.572781 0.819709i \(-0.694135\pi\)
−0.572781 + 0.819709i \(0.694135\pi\)
\(6\) −2.56155 −1.04575
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 3.00000 1.06066
\(9\) 3.56155 1.18718
\(10\) 2.56155 0.810034
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −2.56155 −0.739457
\(13\) −4.56155 −1.26515 −0.632574 0.774500i \(-0.718001\pi\)
−0.632574 + 0.774500i \(0.718001\pi\)
\(14\) 0 0
\(15\) −6.56155 −1.69419
\(16\) −1.00000 −0.250000
\(17\) −1.00000 −0.242536
\(18\) −3.56155 −0.839466
\(19\) −1.12311 −0.257658 −0.128829 0.991667i \(-0.541122\pi\)
−0.128829 + 0.991667i \(0.541122\pi\)
\(20\) 2.56155 0.572781
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 7.68466 1.56862
\(25\) 1.56155 0.312311
\(26\) 4.56155 0.894594
\(27\) 1.43845 0.276829
\(28\) 0 0
\(29\) −5.12311 −0.951337 −0.475668 0.879625i \(-0.657794\pi\)
−0.475668 + 0.879625i \(0.657794\pi\)
\(30\) 6.56155 1.19797
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −5.00000 −0.883883
\(33\) −5.12311 −0.891818
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) −3.56155 −0.593592
\(37\) −1.43845 −0.236479 −0.118240 0.992985i \(-0.537725\pi\)
−0.118240 + 0.992985i \(0.537725\pi\)
\(38\) 1.12311 0.182192
\(39\) −11.6847 −1.87104
\(40\) −7.68466 −1.21505
\(41\) 3.12311 0.487747 0.243874 0.969807i \(-0.421582\pi\)
0.243874 + 0.969807i \(0.421582\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499
\(44\) 2.00000 0.301511
\(45\) −9.12311 −1.35999
\(46\) −2.00000 −0.294884
\(47\) −5.43845 −0.793279 −0.396640 0.917974i \(-0.629824\pi\)
−0.396640 + 0.917974i \(0.629824\pi\)
\(48\) −2.56155 −0.369728
\(49\) −7.00000 −1.00000
\(50\) −1.56155 −0.220837
\(51\) −2.56155 −0.358689
\(52\) 4.56155 0.632574
\(53\) 1.68466 0.231406 0.115703 0.993284i \(-0.463088\pi\)
0.115703 + 0.993284i \(0.463088\pi\)
\(54\) −1.43845 −0.195748
\(55\) 5.12311 0.690799
\(56\) 0 0
\(57\) −2.87689 −0.381054
\(58\) 5.12311 0.672697
\(59\) 8.80776 1.14667 0.573337 0.819320i \(-0.305649\pi\)
0.573337 + 0.819320i \(0.305649\pi\)
\(60\) 6.56155 0.847093
\(61\) −11.6847 −1.49607 −0.748034 0.663661i \(-0.769002\pi\)
−0.748034 + 0.663661i \(0.769002\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 11.6847 1.44930
\(66\) 5.12311 0.630611
\(67\) 1.43845 0.175734 0.0878671 0.996132i \(-0.471995\pi\)
0.0878671 + 0.996132i \(0.471995\pi\)
\(68\) 1.00000 0.121268
\(69\) 5.12311 0.616749
\(70\) 0 0
\(71\) 1.43845 0.170712 0.0853561 0.996351i \(-0.472797\pi\)
0.0853561 + 0.996351i \(0.472797\pi\)
\(72\) 10.6847 1.25920
\(73\) 8.80776 1.03087 0.515435 0.856928i \(-0.327630\pi\)
0.515435 + 0.856928i \(0.327630\pi\)
\(74\) 1.43845 0.167216
\(75\) 4.00000 0.461880
\(76\) 1.12311 0.128829
\(77\) 0 0
\(78\) 11.6847 1.32303
\(79\) −0.876894 −0.0986583 −0.0493292 0.998783i \(-0.515708\pi\)
−0.0493292 + 0.998783i \(0.515708\pi\)
\(80\) 2.56155 0.286390
\(81\) −7.00000 −0.777778
\(82\) −3.12311 −0.344889
\(83\) −4.80776 −0.527721 −0.263860 0.964561i \(-0.584996\pi\)
−0.263860 + 0.964561i \(0.584996\pi\)
\(84\) 0 0
\(85\) 2.56155 0.277839
\(86\) 1.00000 0.107833
\(87\) −13.1231 −1.40694
\(88\) −6.00000 −0.639602
\(89\) 16.2462 1.72209 0.861047 0.508525i \(-0.169809\pi\)
0.861047 + 0.508525i \(0.169809\pi\)
\(90\) 9.12311 0.961660
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) −15.3693 −1.59372
\(94\) 5.43845 0.560933
\(95\) 2.87689 0.295163
\(96\) −12.8078 −1.30719
\(97\) −0.246211 −0.0249990 −0.0124995 0.999922i \(-0.503979\pi\)
−0.0124995 + 0.999922i \(0.503979\pi\)
\(98\) 7.00000 0.707107
\(99\) −7.12311 −0.715899
\(100\) −1.56155 −0.156155
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 2.56155 0.253632
\(103\) −13.4384 −1.32413 −0.662065 0.749447i \(-0.730320\pi\)
−0.662065 + 0.749447i \(0.730320\pi\)
\(104\) −13.6847 −1.34189
\(105\) 0 0
\(106\) −1.68466 −0.163628
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) −1.43845 −0.138415
\(109\) 12.2462 1.17297 0.586487 0.809959i \(-0.300510\pi\)
0.586487 + 0.809959i \(0.300510\pi\)
\(110\) −5.12311 −0.488469
\(111\) −3.68466 −0.349732
\(112\) 0 0
\(113\) 8.80776 0.828565 0.414282 0.910148i \(-0.364033\pi\)
0.414282 + 0.910148i \(0.364033\pi\)
\(114\) 2.87689 0.269446
\(115\) −5.12311 −0.477732
\(116\) 5.12311 0.475668
\(117\) −16.2462 −1.50196
\(118\) −8.80776 −0.810820
\(119\) 0 0
\(120\) −19.6847 −1.79696
\(121\) −7.00000 −0.636364
\(122\) 11.6847 1.05788
\(123\) 8.00000 0.721336
\(124\) 6.00000 0.538816
\(125\) 8.80776 0.787790
\(126\) 0 0
\(127\) −15.0540 −1.33582 −0.667912 0.744240i \(-0.732812\pi\)
−0.667912 + 0.744240i \(0.732812\pi\)
\(128\) 3.00000 0.265165
\(129\) −2.56155 −0.225532
\(130\) −11.6847 −1.02481
\(131\) −8.80776 −0.769538 −0.384769 0.923013i \(-0.625719\pi\)
−0.384769 + 0.923013i \(0.625719\pi\)
\(132\) 5.12311 0.445909
\(133\) 0 0
\(134\) −1.43845 −0.124263
\(135\) −3.68466 −0.317125
\(136\) −3.00000 −0.257248
\(137\) −4.24621 −0.362778 −0.181389 0.983411i \(-0.558059\pi\)
−0.181389 + 0.983411i \(0.558059\pi\)
\(138\) −5.12311 −0.436108
\(139\) 5.36932 0.455420 0.227710 0.973729i \(-0.426876\pi\)
0.227710 + 0.973729i \(0.426876\pi\)
\(140\) 0 0
\(141\) −13.9309 −1.17319
\(142\) −1.43845 −0.120712
\(143\) 9.12311 0.762912
\(144\) −3.56155 −0.296796
\(145\) 13.1231 1.08981
\(146\) −8.80776 −0.728936
\(147\) −17.9309 −1.47891
\(148\) 1.43845 0.118240
\(149\) 8.24621 0.675556 0.337778 0.941226i \(-0.390325\pi\)
0.337778 + 0.941226i \(0.390325\pi\)
\(150\) −4.00000 −0.326599
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) −3.36932 −0.273288
\(153\) −3.56155 −0.287934
\(154\) 0 0
\(155\) 15.3693 1.23449
\(156\) 11.6847 0.935521
\(157\) 10.4924 0.837386 0.418693 0.908128i \(-0.362488\pi\)
0.418693 + 0.908128i \(0.362488\pi\)
\(158\) 0.876894 0.0697620
\(159\) 4.31534 0.342229
\(160\) 12.8078 1.01254
\(161\) 0 0
\(162\) 7.00000 0.549972
\(163\) −17.4384 −1.36588 −0.682942 0.730472i \(-0.739300\pi\)
−0.682942 + 0.730472i \(0.739300\pi\)
\(164\) −3.12311 −0.243874
\(165\) 13.1231 1.02163
\(166\) 4.80776 0.373155
\(167\) 22.0000 1.70241 0.851206 0.524832i \(-0.175872\pi\)
0.851206 + 0.524832i \(0.175872\pi\)
\(168\) 0 0
\(169\) 7.80776 0.600597
\(170\) −2.56155 −0.196462
\(171\) −4.00000 −0.305888
\(172\) 1.00000 0.0762493
\(173\) 3.12311 0.237445 0.118723 0.992927i \(-0.462120\pi\)
0.118723 + 0.992927i \(0.462120\pi\)
\(174\) 13.1231 0.994860
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 22.5616 1.69583
\(178\) −16.2462 −1.21771
\(179\) 17.1231 1.27984 0.639921 0.768441i \(-0.278967\pi\)
0.639921 + 0.768441i \(0.278967\pi\)
\(180\) 9.12311 0.679996
\(181\) 5.36932 0.399098 0.199549 0.979888i \(-0.436052\pi\)
0.199549 + 0.979888i \(0.436052\pi\)
\(182\) 0 0
\(183\) −29.9309 −2.21255
\(184\) 6.00000 0.442326
\(185\) 3.68466 0.270901
\(186\) 15.3693 1.12693
\(187\) 2.00000 0.146254
\(188\) 5.43845 0.396640
\(189\) 0 0
\(190\) −2.87689 −0.208712
\(191\) 2.87689 0.208165 0.104082 0.994569i \(-0.466809\pi\)
0.104082 + 0.994569i \(0.466809\pi\)
\(192\) 17.9309 1.29405
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0.246211 0.0176769
\(195\) 29.9309 2.14339
\(196\) 7.00000 0.500000
\(197\) −23.6155 −1.68254 −0.841268 0.540618i \(-0.818191\pi\)
−0.841268 + 0.540618i \(0.818191\pi\)
\(198\) 7.12311 0.506217
\(199\) 25.9309 1.83819 0.919095 0.394035i \(-0.128921\pi\)
0.919095 + 0.394035i \(0.128921\pi\)
\(200\) 4.68466 0.331255
\(201\) 3.68466 0.259896
\(202\) 2.00000 0.140720
\(203\) 0 0
\(204\) 2.56155 0.179345
\(205\) −8.00000 −0.558744
\(206\) 13.4384 0.936301
\(207\) 7.12311 0.495090
\(208\) 4.56155 0.316287
\(209\) 2.24621 0.155374
\(210\) 0 0
\(211\) −24.8078 −1.70784 −0.853918 0.520407i \(-0.825780\pi\)
−0.853918 + 0.520407i \(0.825780\pi\)
\(212\) −1.68466 −0.115703
\(213\) 3.68466 0.252469
\(214\) −2.00000 −0.136717
\(215\) 2.56155 0.174696
\(216\) 4.31534 0.293622
\(217\) 0 0
\(218\) −12.2462 −0.829418
\(219\) 22.5616 1.52457
\(220\) −5.12311 −0.345400
\(221\) 4.56155 0.306843
\(222\) 3.68466 0.247298
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 5.56155 0.370770
\(226\) −8.80776 −0.585884
\(227\) −12.4924 −0.829151 −0.414576 0.910015i \(-0.636070\pi\)
−0.414576 + 0.910015i \(0.636070\pi\)
\(228\) 2.87689 0.190527
\(229\) −14.8078 −0.978525 −0.489262 0.872137i \(-0.662734\pi\)
−0.489262 + 0.872137i \(0.662734\pi\)
\(230\) 5.12311 0.337808
\(231\) 0 0
\(232\) −15.3693 −1.00905
\(233\) 10.8769 0.712569 0.356285 0.934378i \(-0.384043\pi\)
0.356285 + 0.934378i \(0.384043\pi\)
\(234\) 16.2462 1.06205
\(235\) 13.9309 0.908750
\(236\) −8.80776 −0.573337
\(237\) −2.24621 −0.145907
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 6.56155 0.423546
\(241\) 19.3693 1.24769 0.623844 0.781549i \(-0.285570\pi\)
0.623844 + 0.781549i \(0.285570\pi\)
\(242\) 7.00000 0.449977
\(243\) −22.2462 −1.42710
\(244\) 11.6847 0.748034
\(245\) 17.9309 1.14556
\(246\) −8.00000 −0.510061
\(247\) 5.12311 0.325975
\(248\) −18.0000 −1.14300
\(249\) −12.3153 −0.780453
\(250\) −8.80776 −0.557052
\(251\) −10.5616 −0.666639 −0.333320 0.942814i \(-0.608169\pi\)
−0.333320 + 0.942814i \(0.608169\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 15.0540 0.944570
\(255\) 6.56155 0.410900
\(256\) −17.0000 −1.06250
\(257\) 26.0000 1.62184 0.810918 0.585160i \(-0.198968\pi\)
0.810918 + 0.585160i \(0.198968\pi\)
\(258\) 2.56155 0.159475
\(259\) 0 0
\(260\) −11.6847 −0.724652
\(261\) −18.2462 −1.12941
\(262\) 8.80776 0.544145
\(263\) −20.4924 −1.26362 −0.631808 0.775125i \(-0.717687\pi\)
−0.631808 + 0.775125i \(0.717687\pi\)
\(264\) −15.3693 −0.945916
\(265\) −4.31534 −0.265089
\(266\) 0 0
\(267\) 41.6155 2.54683
\(268\) −1.43845 −0.0878671
\(269\) −8.24621 −0.502780 −0.251390 0.967886i \(-0.580888\pi\)
−0.251390 + 0.967886i \(0.580888\pi\)
\(270\) 3.68466 0.224241
\(271\) 9.93087 0.603257 0.301629 0.953425i \(-0.402470\pi\)
0.301629 + 0.953425i \(0.402470\pi\)
\(272\) 1.00000 0.0606339
\(273\) 0 0
\(274\) 4.24621 0.256523
\(275\) −3.12311 −0.188330
\(276\) −5.12311 −0.308375
\(277\) 2.06913 0.124322 0.0621610 0.998066i \(-0.480201\pi\)
0.0621610 + 0.998066i \(0.480201\pi\)
\(278\) −5.36932 −0.322030
\(279\) −21.3693 −1.27935
\(280\) 0 0
\(281\) 7.93087 0.473116 0.236558 0.971617i \(-0.423981\pi\)
0.236558 + 0.971617i \(0.423981\pi\)
\(282\) 13.9309 0.829571
\(283\) −4.24621 −0.252411 −0.126206 0.992004i \(-0.540280\pi\)
−0.126206 + 0.992004i \(0.540280\pi\)
\(284\) −1.43845 −0.0853561
\(285\) 7.36932 0.436521
\(286\) −9.12311 −0.539461
\(287\) 0 0
\(288\) −17.8078 −1.04933
\(289\) 1.00000 0.0588235
\(290\) −13.1231 −0.770615
\(291\) −0.630683 −0.0369713
\(292\) −8.80776 −0.515435
\(293\) 16.2462 0.949114 0.474557 0.880225i \(-0.342608\pi\)
0.474557 + 0.880225i \(0.342608\pi\)
\(294\) 17.9309 1.04575
\(295\) −22.5616 −1.31358
\(296\) −4.31534 −0.250824
\(297\) −2.87689 −0.166934
\(298\) −8.24621 −0.477690
\(299\) −9.12311 −0.527603
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) −8.00000 −0.460348
\(303\) −5.12311 −0.294315
\(304\) 1.12311 0.0644145
\(305\) 29.9309 1.71384
\(306\) 3.56155 0.203600
\(307\) 2.24621 0.128198 0.0640990 0.997944i \(-0.479583\pi\)
0.0640990 + 0.997944i \(0.479583\pi\)
\(308\) 0 0
\(309\) −34.4233 −1.95827
\(310\) −15.3693 −0.872919
\(311\) −9.36932 −0.531285 −0.265643 0.964072i \(-0.585584\pi\)
−0.265643 + 0.964072i \(0.585584\pi\)
\(312\) −35.0540 −1.98454
\(313\) −4.80776 −0.271751 −0.135875 0.990726i \(-0.543385\pi\)
−0.135875 + 0.990726i \(0.543385\pi\)
\(314\) −10.4924 −0.592122
\(315\) 0 0
\(316\) 0.876894 0.0493292
\(317\) 5.36932 0.301571 0.150785 0.988567i \(-0.451820\pi\)
0.150785 + 0.988567i \(0.451820\pi\)
\(318\) −4.31534 −0.241992
\(319\) 10.2462 0.573678
\(320\) −17.9309 −1.00237
\(321\) 5.12311 0.285944
\(322\) 0 0
\(323\) 1.12311 0.0624913
\(324\) 7.00000 0.388889
\(325\) −7.12311 −0.395119
\(326\) 17.4384 0.965826
\(327\) 31.3693 1.73473
\(328\) 9.36932 0.517334
\(329\) 0 0
\(330\) −13.1231 −0.722403
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 4.80776 0.263860
\(333\) −5.12311 −0.280744
\(334\) −22.0000 −1.20379
\(335\) −3.68466 −0.201314
\(336\) 0 0
\(337\) −20.2462 −1.10288 −0.551441 0.834214i \(-0.685922\pi\)
−0.551441 + 0.834214i \(0.685922\pi\)
\(338\) −7.80776 −0.424686
\(339\) 22.5616 1.22538
\(340\) −2.56155 −0.138920
\(341\) 12.0000 0.649836
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) −3.00000 −0.161749
\(345\) −13.1231 −0.706524
\(346\) −3.12311 −0.167899
\(347\) −22.5616 −1.21117 −0.605584 0.795782i \(-0.707060\pi\)
−0.605584 + 0.795782i \(0.707060\pi\)
\(348\) 13.1231 0.703472
\(349\) −19.1231 −1.02364 −0.511818 0.859094i \(-0.671028\pi\)
−0.511818 + 0.859094i \(0.671028\pi\)
\(350\) 0 0
\(351\) −6.56155 −0.350230
\(352\) 10.0000 0.533002
\(353\) 26.1771 1.39327 0.696633 0.717428i \(-0.254681\pi\)
0.696633 + 0.717428i \(0.254681\pi\)
\(354\) −22.5616 −1.19913
\(355\) −3.68466 −0.195561
\(356\) −16.2462 −0.861047
\(357\) 0 0
\(358\) −17.1231 −0.904984
\(359\) 16.8078 0.887080 0.443540 0.896255i \(-0.353722\pi\)
0.443540 + 0.896255i \(0.353722\pi\)
\(360\) −27.3693 −1.44249
\(361\) −17.7386 −0.933612
\(362\) −5.36932 −0.282205
\(363\) −17.9309 −0.941127
\(364\) 0 0
\(365\) −22.5616 −1.18093
\(366\) 29.9309 1.56451
\(367\) −14.4924 −0.756498 −0.378249 0.925704i \(-0.623474\pi\)
−0.378249 + 0.925704i \(0.623474\pi\)
\(368\) −2.00000 −0.104257
\(369\) 11.1231 0.579046
\(370\) −3.68466 −0.191556
\(371\) 0 0
\(372\) 15.3693 0.796862
\(373\) −37.3693 −1.93491 −0.967455 0.253043i \(-0.918568\pi\)
−0.967455 + 0.253043i \(0.918568\pi\)
\(374\) −2.00000 −0.103418
\(375\) 22.5616 1.16507
\(376\) −16.3153 −0.841399
\(377\) 23.3693 1.20358
\(378\) 0 0
\(379\) −0.876894 −0.0450430 −0.0225215 0.999746i \(-0.507169\pi\)
−0.0225215 + 0.999746i \(0.507169\pi\)
\(380\) −2.87689 −0.147582
\(381\) −38.5616 −1.97557
\(382\) −2.87689 −0.147195
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 7.68466 0.392156
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) −3.56155 −0.181044
\(388\) 0.246211 0.0124995
\(389\) −30.4924 −1.54603 −0.773014 0.634389i \(-0.781252\pi\)
−0.773014 + 0.634389i \(0.781252\pi\)
\(390\) −29.9309 −1.51561
\(391\) −2.00000 −0.101144
\(392\) −21.0000 −1.06066
\(393\) −22.5616 −1.13808
\(394\) 23.6155 1.18973
\(395\) 2.24621 0.113019
\(396\) 7.12311 0.357950
\(397\) 15.1231 0.759007 0.379503 0.925190i \(-0.376095\pi\)
0.379503 + 0.925190i \(0.376095\pi\)
\(398\) −25.9309 −1.29980
\(399\) 0 0
\(400\) −1.56155 −0.0780776
\(401\) −20.2462 −1.01105 −0.505524 0.862813i \(-0.668701\pi\)
−0.505524 + 0.862813i \(0.668701\pi\)
\(402\) −3.68466 −0.183774
\(403\) 27.3693 1.36336
\(404\) 2.00000 0.0995037
\(405\) 17.9309 0.890992
\(406\) 0 0
\(407\) 2.87689 0.142602
\(408\) −7.68466 −0.380447
\(409\) 21.8617 1.08099 0.540497 0.841346i \(-0.318236\pi\)
0.540497 + 0.841346i \(0.318236\pi\)
\(410\) 8.00000 0.395092
\(411\) −10.8769 −0.536518
\(412\) 13.4384 0.662065
\(413\) 0 0
\(414\) −7.12311 −0.350082
\(415\) 12.3153 0.604536
\(416\) 22.8078 1.11824
\(417\) 13.7538 0.673526
\(418\) −2.24621 −0.109866
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) −36.7386 −1.79053 −0.895266 0.445533i \(-0.853014\pi\)
−0.895266 + 0.445533i \(0.853014\pi\)
\(422\) 24.8078 1.20762
\(423\) −19.3693 −0.941768
\(424\) 5.05398 0.245443
\(425\) −1.56155 −0.0757464
\(426\) −3.68466 −0.178522
\(427\) 0 0
\(428\) −2.00000 −0.0966736
\(429\) 23.3693 1.12828
\(430\) −2.56155 −0.123529
\(431\) −4.87689 −0.234912 −0.117456 0.993078i \(-0.537474\pi\)
−0.117456 + 0.993078i \(0.537474\pi\)
\(432\) −1.43845 −0.0692073
\(433\) −13.3693 −0.642488 −0.321244 0.946996i \(-0.604101\pi\)
−0.321244 + 0.946996i \(0.604101\pi\)
\(434\) 0 0
\(435\) 33.6155 1.61174
\(436\) −12.2462 −0.586487
\(437\) −2.24621 −0.107451
\(438\) −22.5616 −1.07803
\(439\) −9.36932 −0.447173 −0.223587 0.974684i \(-0.571777\pi\)
−0.223587 + 0.974684i \(0.571777\pi\)
\(440\) 15.3693 0.732703
\(441\) −24.9309 −1.18718
\(442\) −4.56155 −0.216971
\(443\) 30.4233 1.44545 0.722727 0.691133i \(-0.242888\pi\)
0.722727 + 0.691133i \(0.242888\pi\)
\(444\) 3.68466 0.174866
\(445\) −41.6155 −1.97277
\(446\) 16.0000 0.757622
\(447\) 21.1231 0.999089
\(448\) 0 0
\(449\) −34.5616 −1.63106 −0.815530 0.578714i \(-0.803555\pi\)
−0.815530 + 0.578714i \(0.803555\pi\)
\(450\) −5.56155 −0.262174
\(451\) −6.24621 −0.294123
\(452\) −8.80776 −0.414282
\(453\) 20.4924 0.962818
\(454\) 12.4924 0.586298
\(455\) 0 0
\(456\) −8.63068 −0.404169
\(457\) −29.3693 −1.37384 −0.686919 0.726734i \(-0.741037\pi\)
−0.686919 + 0.726734i \(0.741037\pi\)
\(458\) 14.8078 0.691921
\(459\) −1.43845 −0.0671410
\(460\) 5.12311 0.238866
\(461\) 2.80776 0.130771 0.0653853 0.997860i \(-0.479172\pi\)
0.0653853 + 0.997860i \(0.479172\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 5.12311 0.237834
\(465\) 39.3693 1.82571
\(466\) −10.8769 −0.503862
\(467\) 0.492423 0.0227866 0.0113933 0.999935i \(-0.496373\pi\)
0.0113933 + 0.999935i \(0.496373\pi\)
\(468\) 16.2462 0.750981
\(469\) 0 0
\(470\) −13.9309 −0.642583
\(471\) 26.8769 1.23842
\(472\) 26.4233 1.21623
\(473\) 2.00000 0.0919601
\(474\) 2.24621 0.103172
\(475\) −1.75379 −0.0804693
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 8.00000 0.365911
\(479\) 6.63068 0.302964 0.151482 0.988460i \(-0.451595\pi\)
0.151482 + 0.988460i \(0.451595\pi\)
\(480\) 32.8078 1.49746
\(481\) 6.56155 0.299181
\(482\) −19.3693 −0.882248
\(483\) 0 0
\(484\) 7.00000 0.318182
\(485\) 0.630683 0.0286378
\(486\) 22.2462 1.00911
\(487\) 6.49242 0.294200 0.147100 0.989122i \(-0.453006\pi\)
0.147100 + 0.989122i \(0.453006\pi\)
\(488\) −35.0540 −1.58682
\(489\) −44.6695 −2.02002
\(490\) −17.9309 −0.810034
\(491\) 19.3693 0.874125 0.437063 0.899431i \(-0.356019\pi\)
0.437063 + 0.899431i \(0.356019\pi\)
\(492\) −8.00000 −0.360668
\(493\) 5.12311 0.230733
\(494\) −5.12311 −0.230499
\(495\) 18.2462 0.820106
\(496\) 6.00000 0.269408
\(497\) 0 0
\(498\) 12.3153 0.551864
\(499\) −7.68466 −0.344013 −0.172006 0.985096i \(-0.555025\pi\)
−0.172006 + 0.985096i \(0.555025\pi\)
\(500\) −8.80776 −0.393895
\(501\) 56.3542 2.51772
\(502\) 10.5616 0.471385
\(503\) −22.4233 −0.999805 −0.499903 0.866082i \(-0.666631\pi\)
−0.499903 + 0.866082i \(0.666631\pi\)
\(504\) 0 0
\(505\) 5.12311 0.227975
\(506\) 4.00000 0.177822
\(507\) 20.0000 0.888231
\(508\) 15.0540 0.667912
\(509\) −33.0540 −1.46509 −0.732546 0.680718i \(-0.761668\pi\)
−0.732546 + 0.680718i \(0.761668\pi\)
\(510\) −6.56155 −0.290550
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) −1.61553 −0.0713273
\(514\) −26.0000 −1.14681
\(515\) 34.4233 1.51687
\(516\) 2.56155 0.112766
\(517\) 10.8769 0.478365
\(518\) 0 0
\(519\) 8.00000 0.351161
\(520\) 35.0540 1.53722
\(521\) −27.3693 −1.19907 −0.599536 0.800348i \(-0.704648\pi\)
−0.599536 + 0.800348i \(0.704648\pi\)
\(522\) 18.2462 0.798615
\(523\) 30.8769 1.35015 0.675076 0.737748i \(-0.264111\pi\)
0.675076 + 0.737748i \(0.264111\pi\)
\(524\) 8.80776 0.384769
\(525\) 0 0
\(526\) 20.4924 0.893512
\(527\) 6.00000 0.261364
\(528\) 5.12311 0.222955
\(529\) −19.0000 −0.826087
\(530\) 4.31534 0.187446
\(531\) 31.3693 1.36131
\(532\) 0 0
\(533\) −14.2462 −0.617072
\(534\) −41.6155 −1.80088
\(535\) −5.12311 −0.221491
\(536\) 4.31534 0.186394
\(537\) 43.8617 1.89277
\(538\) 8.24621 0.355519
\(539\) 14.0000 0.603023
\(540\) 3.68466 0.158562
\(541\) −19.6155 −0.843337 −0.421669 0.906750i \(-0.638555\pi\)
−0.421669 + 0.906750i \(0.638555\pi\)
\(542\) −9.93087 −0.426567
\(543\) 13.7538 0.590232
\(544\) 5.00000 0.214373
\(545\) −31.3693 −1.34371
\(546\) 0 0
\(547\) 1.50758 0.0644594 0.0322297 0.999480i \(-0.489739\pi\)
0.0322297 + 0.999480i \(0.489739\pi\)
\(548\) 4.24621 0.181389
\(549\) −41.6155 −1.77611
\(550\) 3.12311 0.133170
\(551\) 5.75379 0.245120
\(552\) 15.3693 0.654162
\(553\) 0 0
\(554\) −2.06913 −0.0879089
\(555\) 9.43845 0.400640
\(556\) −5.36932 −0.227710
\(557\) −26.1771 −1.10916 −0.554579 0.832131i \(-0.687121\pi\)
−0.554579 + 0.832131i \(0.687121\pi\)
\(558\) 21.3693 0.904635
\(559\) 4.56155 0.192933
\(560\) 0 0
\(561\) 5.12311 0.216298
\(562\) −7.93087 −0.334544
\(563\) −18.2462 −0.768986 −0.384493 0.923128i \(-0.625624\pi\)
−0.384493 + 0.923128i \(0.625624\pi\)
\(564\) 13.9309 0.586595
\(565\) −22.5616 −0.949172
\(566\) 4.24621 0.178482
\(567\) 0 0
\(568\) 4.31534 0.181068
\(569\) −0.561553 −0.0235415 −0.0117708 0.999931i \(-0.503747\pi\)
−0.0117708 + 0.999931i \(0.503747\pi\)
\(570\) −7.36932 −0.308667
\(571\) −13.7538 −0.575578 −0.287789 0.957694i \(-0.592920\pi\)
−0.287789 + 0.957694i \(0.592920\pi\)
\(572\) −9.12311 −0.381456
\(573\) 7.36932 0.307858
\(574\) 0 0
\(575\) 3.12311 0.130243
\(576\) 24.9309 1.03879
\(577\) 44.7386 1.86249 0.931247 0.364389i \(-0.118722\pi\)
0.931247 + 0.364389i \(0.118722\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 5.12311 0.212909
\(580\) −13.1231 −0.544907
\(581\) 0 0
\(582\) 0.630683 0.0261427
\(583\) −3.36932 −0.139543
\(584\) 26.4233 1.09340
\(585\) 41.6155 1.72059
\(586\) −16.2462 −0.671125
\(587\) −27.3693 −1.12965 −0.564826 0.825210i \(-0.691057\pi\)
−0.564826 + 0.825210i \(0.691057\pi\)
\(588\) 17.9309 0.739457
\(589\) 6.73863 0.277661
\(590\) 22.5616 0.928844
\(591\) −60.4924 −2.48833
\(592\) 1.43845 0.0591198
\(593\) −4.24621 −0.174371 −0.0871855 0.996192i \(-0.527787\pi\)
−0.0871855 + 0.996192i \(0.527787\pi\)
\(594\) 2.87689 0.118040
\(595\) 0 0
\(596\) −8.24621 −0.337778
\(597\) 66.4233 2.71852
\(598\) 9.12311 0.373072
\(599\) 12.3153 0.503191 0.251596 0.967832i \(-0.419045\pi\)
0.251596 + 0.967832i \(0.419045\pi\)
\(600\) 12.0000 0.489898
\(601\) 6.56155 0.267651 0.133826 0.991005i \(-0.457274\pi\)
0.133826 + 0.991005i \(0.457274\pi\)
\(602\) 0 0
\(603\) 5.12311 0.208629
\(604\) −8.00000 −0.325515
\(605\) 17.9309 0.728994
\(606\) 5.12311 0.208112
\(607\) 31.6847 1.28604 0.643020 0.765849i \(-0.277681\pi\)
0.643020 + 0.765849i \(0.277681\pi\)
\(608\) 5.61553 0.227740
\(609\) 0 0
\(610\) −29.9309 −1.21187
\(611\) 24.8078 1.00361
\(612\) 3.56155 0.143967
\(613\) −34.4924 −1.39314 −0.696568 0.717491i \(-0.745291\pi\)
−0.696568 + 0.717491i \(0.745291\pi\)
\(614\) −2.24621 −0.0906497
\(615\) −20.4924 −0.826334
\(616\) 0 0
\(617\) 23.7538 0.956292 0.478146 0.878280i \(-0.341309\pi\)
0.478146 + 0.878280i \(0.341309\pi\)
\(618\) 34.4233 1.38471
\(619\) −0.738634 −0.0296882 −0.0148441 0.999890i \(-0.504725\pi\)
−0.0148441 + 0.999890i \(0.504725\pi\)
\(620\) −15.3693 −0.617247
\(621\) 2.87689 0.115446
\(622\) 9.36932 0.375675
\(623\) 0 0
\(624\) 11.6847 0.467761
\(625\) −30.3693 −1.21477
\(626\) 4.80776 0.192157
\(627\) 5.75379 0.229784
\(628\) −10.4924 −0.418693
\(629\) 1.43845 0.0573546
\(630\) 0 0
\(631\) −41.6155 −1.65669 −0.828344 0.560220i \(-0.810717\pi\)
−0.828344 + 0.560220i \(0.810717\pi\)
\(632\) −2.63068 −0.104643
\(633\) −63.5464 −2.52574
\(634\) −5.36932 −0.213243
\(635\) 38.5616 1.53027
\(636\) −4.31534 −0.171114
\(637\) 31.9309 1.26515
\(638\) −10.2462 −0.405651
\(639\) 5.12311 0.202667
\(640\) −7.68466 −0.303763
\(641\) 46.4233 1.83361 0.916805 0.399335i \(-0.130759\pi\)
0.916805 + 0.399335i \(0.130759\pi\)
\(642\) −5.12311 −0.202193
\(643\) 27.7538 1.09450 0.547251 0.836968i \(-0.315674\pi\)
0.547251 + 0.836968i \(0.315674\pi\)
\(644\) 0 0
\(645\) 6.56155 0.258361
\(646\) −1.12311 −0.0441880
\(647\) 38.1080 1.49818 0.749089 0.662469i \(-0.230491\pi\)
0.749089 + 0.662469i \(0.230491\pi\)
\(648\) −21.0000 −0.824958
\(649\) −17.6155 −0.691470
\(650\) 7.12311 0.279391
\(651\) 0 0
\(652\) 17.4384 0.682942
\(653\) −2.87689 −0.112582 −0.0562908 0.998414i \(-0.517927\pi\)
−0.0562908 + 0.998414i \(0.517927\pi\)
\(654\) −31.3693 −1.22664
\(655\) 22.5616 0.881553
\(656\) −3.12311 −0.121937
\(657\) 31.3693 1.22383
\(658\) 0 0
\(659\) −29.7538 −1.15904 −0.579522 0.814957i \(-0.696761\pi\)
−0.579522 + 0.814957i \(0.696761\pi\)
\(660\) −13.1231 −0.510816
\(661\) −44.5616 −1.73324 −0.866622 0.498966i \(-0.833713\pi\)
−0.866622 + 0.498966i \(0.833713\pi\)
\(662\) −12.0000 −0.466393
\(663\) 11.6847 0.453795
\(664\) −14.4233 −0.559732
\(665\) 0 0
\(666\) 5.12311 0.198516
\(667\) −10.2462 −0.396735
\(668\) −22.0000 −0.851206
\(669\) −40.9848 −1.58457
\(670\) 3.68466 0.142351
\(671\) 23.3693 0.902162
\(672\) 0 0
\(673\) 48.8078 1.88140 0.940701 0.339238i \(-0.110169\pi\)
0.940701 + 0.339238i \(0.110169\pi\)
\(674\) 20.2462 0.779855
\(675\) 2.24621 0.0864567
\(676\) −7.80776 −0.300299
\(677\) −41.6155 −1.59941 −0.799707 0.600390i \(-0.795012\pi\)
−0.799707 + 0.600390i \(0.795012\pi\)
\(678\) −22.5616 −0.866471
\(679\) 0 0
\(680\) 7.68466 0.294693
\(681\) −32.0000 −1.22624
\(682\) −12.0000 −0.459504
\(683\) 18.6307 0.712883 0.356442 0.934318i \(-0.383990\pi\)
0.356442 + 0.934318i \(0.383990\pi\)
\(684\) 4.00000 0.152944
\(685\) 10.8769 0.415585
\(686\) 0 0
\(687\) −37.9309 −1.44715
\(688\) 1.00000 0.0381246
\(689\) −7.68466 −0.292762
\(690\) 13.1231 0.499588
\(691\) 32.4924 1.23607 0.618035 0.786151i \(-0.287929\pi\)
0.618035 + 0.786151i \(0.287929\pi\)
\(692\) −3.12311 −0.118723
\(693\) 0 0
\(694\) 22.5616 0.856425
\(695\) −13.7538 −0.521711
\(696\) −39.3693 −1.49229
\(697\) −3.12311 −0.118296
\(698\) 19.1231 0.723820
\(699\) 27.8617 1.05383
\(700\) 0 0
\(701\) −9.05398 −0.341964 −0.170982 0.985274i \(-0.554694\pi\)
−0.170982 + 0.985274i \(0.554694\pi\)
\(702\) 6.56155 0.247650
\(703\) 1.61553 0.0609308
\(704\) −14.0000 −0.527645
\(705\) 35.6847 1.34396
\(706\) −26.1771 −0.985187
\(707\) 0 0
\(708\) −22.5616 −0.847915
\(709\) 38.4924 1.44561 0.722807 0.691050i \(-0.242852\pi\)
0.722807 + 0.691050i \(0.242852\pi\)
\(710\) 3.68466 0.138283
\(711\) −3.12311 −0.117126
\(712\) 48.7386 1.82656
\(713\) −12.0000 −0.449404
\(714\) 0 0
\(715\) −23.3693 −0.873963
\(716\) −17.1231 −0.639921
\(717\) −20.4924 −0.765304
\(718\) −16.8078 −0.627260
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 9.12311 0.339998
\(721\) 0 0
\(722\) 17.7386 0.660164
\(723\) 49.6155 1.84522
\(724\) −5.36932 −0.199549
\(725\) −8.00000 −0.297113
\(726\) 17.9309 0.665477
\(727\) −1.26137 −0.0467815 −0.0233907 0.999726i \(-0.507446\pi\)
−0.0233907 + 0.999726i \(0.507446\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 22.5616 0.835041
\(731\) 1.00000 0.0369863
\(732\) 29.9309 1.10628
\(733\) −8.24621 −0.304581 −0.152290 0.988336i \(-0.548665\pi\)
−0.152290 + 0.988336i \(0.548665\pi\)
\(734\) 14.4924 0.534925
\(735\) 45.9309 1.69419
\(736\) −10.0000 −0.368605
\(737\) −2.87689 −0.105972
\(738\) −11.1231 −0.409447
\(739\) 34.1080 1.25468 0.627341 0.778745i \(-0.284143\pi\)
0.627341 + 0.778745i \(0.284143\pi\)
\(740\) −3.68466 −0.135451
\(741\) 13.1231 0.482089
\(742\) 0 0
\(743\) −35.5464 −1.30407 −0.652035 0.758188i \(-0.726085\pi\)
−0.652035 + 0.758188i \(0.726085\pi\)
\(744\) −46.1080 −1.69040
\(745\) −21.1231 −0.773891
\(746\) 37.3693 1.36819
\(747\) −17.1231 −0.626502
\(748\) −2.00000 −0.0731272
\(749\) 0 0
\(750\) −22.5616 −0.823831
\(751\) 7.05398 0.257403 0.128702 0.991683i \(-0.458919\pi\)
0.128702 + 0.991683i \(0.458919\pi\)
\(752\) 5.43845 0.198320
\(753\) −27.0540 −0.985902
\(754\) −23.3693 −0.851060
\(755\) −20.4924 −0.745796
\(756\) 0 0
\(757\) −6.49242 −0.235971 −0.117986 0.993015i \(-0.537644\pi\)
−0.117986 + 0.993015i \(0.537644\pi\)
\(758\) 0.876894 0.0318502
\(759\) −10.2462 −0.371914
\(760\) 8.63068 0.313068
\(761\) −52.6004 −1.90676 −0.953381 0.301769i \(-0.902423\pi\)
−0.953381 + 0.301769i \(0.902423\pi\)
\(762\) 38.5616 1.39694
\(763\) 0 0
\(764\) −2.87689 −0.104082
\(765\) 9.12311 0.329847
\(766\) 0 0
\(767\) −40.1771 −1.45071
\(768\) −43.5464 −1.57135
\(769\) −45.5464 −1.64245 −0.821223 0.570608i \(-0.806708\pi\)
−0.821223 + 0.570608i \(0.806708\pi\)
\(770\) 0 0
\(771\) 66.6004 2.39855
\(772\) −2.00000 −0.0719816
\(773\) 10.4924 0.377386 0.188693 0.982036i \(-0.439575\pi\)
0.188693 + 0.982036i \(0.439575\pi\)
\(774\) 3.56155 0.128017
\(775\) −9.36932 −0.336556
\(776\) −0.738634 −0.0265154
\(777\) 0 0
\(778\) 30.4924 1.09321
\(779\) −3.50758 −0.125672
\(780\) −29.9309 −1.07170
\(781\) −2.87689 −0.102943
\(782\) 2.00000 0.0715199
\(783\) −7.36932 −0.263358
\(784\) 7.00000 0.250000
\(785\) −26.8769 −0.959277
\(786\) 22.5616 0.804744
\(787\) 30.9848 1.10449 0.552245 0.833682i \(-0.313771\pi\)
0.552245 + 0.833682i \(0.313771\pi\)
\(788\) 23.6155 0.841268
\(789\) −52.4924 −1.86878
\(790\) −2.24621 −0.0799166
\(791\) 0 0
\(792\) −21.3693 −0.759326
\(793\) 53.3002 1.89275
\(794\) −15.1231 −0.536699
\(795\) −11.0540 −0.392044
\(796\) −25.9309 −0.919095
\(797\) −15.9309 −0.564300 −0.282150 0.959370i \(-0.591048\pi\)
−0.282150 + 0.959370i \(0.591048\pi\)
\(798\) 0 0
\(799\) 5.43845 0.192398
\(800\) −7.80776 −0.276046
\(801\) 57.8617 2.04444
\(802\) 20.2462 0.714919
\(803\) −17.6155 −0.621638
\(804\) −3.68466 −0.129948
\(805\) 0 0
\(806\) −27.3693 −0.964043
\(807\) −21.1231 −0.743569
\(808\) −6.00000 −0.211079
\(809\) 14.6307 0.514387 0.257194 0.966360i \(-0.417202\pi\)
0.257194 + 0.966360i \(0.417202\pi\)
\(810\) −17.9309 −0.630027
\(811\) −40.4924 −1.42188 −0.710941 0.703252i \(-0.751731\pi\)
−0.710941 + 0.703252i \(0.751731\pi\)
\(812\) 0 0
\(813\) 25.4384 0.892165
\(814\) −2.87689 −0.100835
\(815\) 44.6695 1.56470
\(816\) 2.56155 0.0896723
\(817\) 1.12311 0.0392925
\(818\) −21.8617 −0.764378
\(819\) 0 0
\(820\) 8.00000 0.279372
\(821\) 0.246211 0.00859283 0.00429642 0.999991i \(-0.498632\pi\)
0.00429642 + 0.999991i \(0.498632\pi\)
\(822\) 10.8769 0.379375
\(823\) 42.4924 1.48119 0.740596 0.671950i \(-0.234543\pi\)
0.740596 + 0.671950i \(0.234543\pi\)
\(824\) −40.3153 −1.40445
\(825\) −8.00000 −0.278524
\(826\) 0 0
\(827\) 20.7386 0.721153 0.360576 0.932730i \(-0.382580\pi\)
0.360576 + 0.932730i \(0.382580\pi\)
\(828\) −7.12311 −0.247545
\(829\) 32.8769 1.14186 0.570931 0.820998i \(-0.306582\pi\)
0.570931 + 0.820998i \(0.306582\pi\)
\(830\) −12.3153 −0.427472
\(831\) 5.30019 0.183861
\(832\) −31.9309 −1.10700
\(833\) 7.00000 0.242536
\(834\) −13.7538 −0.476255
\(835\) −56.3542 −1.95022
\(836\) −2.24621 −0.0776868
\(837\) −8.63068 −0.298320
\(838\) 20.0000 0.690889
\(839\) −0.807764 −0.0278871 −0.0139436 0.999903i \(-0.504439\pi\)
−0.0139436 + 0.999903i \(0.504439\pi\)
\(840\) 0 0
\(841\) −2.75379 −0.0949582
\(842\) 36.7386 1.26610
\(843\) 20.3153 0.699698
\(844\) 24.8078 0.853918
\(845\) −20.0000 −0.688021
\(846\) 19.3693 0.665931
\(847\) 0 0
\(848\) −1.68466 −0.0578514
\(849\) −10.8769 −0.373294
\(850\) 1.56155 0.0535608
\(851\) −2.87689 −0.0986187
\(852\) −3.68466 −0.126234
\(853\) 10.9848 0.376114 0.188057 0.982158i \(-0.439781\pi\)
0.188057 + 0.982158i \(0.439781\pi\)
\(854\) 0 0
\(855\) 10.2462 0.350413
\(856\) 6.00000 0.205076
\(857\) 28.8769 0.986416 0.493208 0.869911i \(-0.335824\pi\)
0.493208 + 0.869911i \(0.335824\pi\)
\(858\) −23.3693 −0.797815
\(859\) 27.3693 0.933829 0.466915 0.884302i \(-0.345366\pi\)
0.466915 + 0.884302i \(0.345366\pi\)
\(860\) −2.56155 −0.0873482
\(861\) 0 0
\(862\) 4.87689 0.166108
\(863\) 20.4924 0.697570 0.348785 0.937203i \(-0.386594\pi\)
0.348785 + 0.937203i \(0.386594\pi\)
\(864\) −7.19224 −0.244685
\(865\) −8.00000 −0.272008
\(866\) 13.3693 0.454308
\(867\) 2.56155 0.0869949
\(868\) 0 0
\(869\) 1.75379 0.0594932
\(870\) −33.6155 −1.13967
\(871\) −6.56155 −0.222330
\(872\) 36.7386 1.24413
\(873\) −0.876894 −0.0296784
\(874\) 2.24621 0.0759792
\(875\) 0 0
\(876\) −22.5616 −0.762284
\(877\) −12.8769 −0.434822 −0.217411 0.976080i \(-0.569761\pi\)
−0.217411 + 0.976080i \(0.569761\pi\)
\(878\) 9.36932 0.316199
\(879\) 41.6155 1.40366
\(880\) −5.12311 −0.172700
\(881\) −26.0000 −0.875962 −0.437981 0.898984i \(-0.644306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(882\) 24.9309 0.839466
\(883\) 42.9157 1.44423 0.722114 0.691774i \(-0.243170\pi\)
0.722114 + 0.691774i \(0.243170\pi\)
\(884\) −4.56155 −0.153422
\(885\) −57.7926 −1.94268
\(886\) −30.4233 −1.02209
\(887\) 21.7538 0.730421 0.365210 0.930925i \(-0.380997\pi\)
0.365210 + 0.930925i \(0.380997\pi\)
\(888\) −11.0540 −0.370947
\(889\) 0 0
\(890\) 41.6155 1.39496
\(891\) 14.0000 0.469018
\(892\) 16.0000 0.535720
\(893\) 6.10795 0.204395
\(894\) −21.1231 −0.706462
\(895\) −43.8617 −1.46614
\(896\) 0 0
\(897\) −23.3693 −0.780279
\(898\) 34.5616 1.15333
\(899\) 30.7386 1.02519
\(900\) −5.56155 −0.185385
\(901\) −1.68466 −0.0561241
\(902\) 6.24621 0.207976
\(903\) 0 0
\(904\) 26.4233 0.878826
\(905\) −13.7538 −0.457191
\(906\) −20.4924 −0.680815
\(907\) 22.0000 0.730498 0.365249 0.930910i \(-0.380984\pi\)
0.365249 + 0.930910i \(0.380984\pi\)
\(908\) 12.4924 0.414576
\(909\) −7.12311 −0.236259
\(910\) 0 0
\(911\) 11.1922 0.370815 0.185408 0.982662i \(-0.440639\pi\)
0.185408 + 0.982662i \(0.440639\pi\)
\(912\) 2.87689 0.0952635
\(913\) 9.61553 0.318228
\(914\) 29.3693 0.971451
\(915\) 76.6695 2.53462
\(916\) 14.8078 0.489262
\(917\) 0 0
\(918\) 1.43845 0.0474758
\(919\) 8.80776 0.290541 0.145271 0.989392i \(-0.453595\pi\)
0.145271 + 0.989392i \(0.453595\pi\)
\(920\) −15.3693 −0.506711
\(921\) 5.75379 0.189594
\(922\) −2.80776 −0.0924688
\(923\) −6.56155 −0.215976
\(924\) 0 0
\(925\) −2.24621 −0.0738550
\(926\) 8.00000 0.262896
\(927\) −47.8617 −1.57199
\(928\) 25.6155 0.840871
\(929\) 31.8617 1.04535 0.522675 0.852532i \(-0.324934\pi\)
0.522675 + 0.852532i \(0.324934\pi\)
\(930\) −39.3693 −1.29097
\(931\) 7.86174 0.257658
\(932\) −10.8769 −0.356285
\(933\) −24.0000 −0.785725
\(934\) −0.492423 −0.0161126
\(935\) −5.12311 −0.167543
\(936\) −48.7386 −1.59307
\(937\) −40.1080 −1.31027 −0.655135 0.755512i \(-0.727388\pi\)
−0.655135 + 0.755512i \(0.727388\pi\)
\(938\) 0 0
\(939\) −12.3153 −0.401896
\(940\) −13.9309 −0.454375
\(941\) 53.3693 1.73979 0.869895 0.493237i \(-0.164186\pi\)
0.869895 + 0.493237i \(0.164186\pi\)
\(942\) −26.8769 −0.875697
\(943\) 6.24621 0.203405
\(944\) −8.80776 −0.286668
\(945\) 0 0
\(946\) −2.00000 −0.0650256
\(947\) −16.2462 −0.527931 −0.263965 0.964532i \(-0.585030\pi\)
−0.263965 + 0.964532i \(0.585030\pi\)
\(948\) 2.24621 0.0729535
\(949\) −40.1771 −1.30420
\(950\) 1.75379 0.0569004
\(951\) 13.7538 0.445997
\(952\) 0 0
\(953\) −20.8769 −0.676269 −0.338134 0.941098i \(-0.609796\pi\)
−0.338134 + 0.941098i \(0.609796\pi\)
\(954\) −6.00000 −0.194257
\(955\) −7.36932 −0.238465
\(956\) 8.00000 0.258738
\(957\) 26.2462 0.848420
\(958\) −6.63068 −0.214228
\(959\) 0 0
\(960\) −45.9309 −1.48241
\(961\) 5.00000 0.161290
\(962\) −6.56155 −0.211553
\(963\) 7.12311 0.229539
\(964\) −19.3693 −0.623844
\(965\) −5.12311 −0.164919
\(966\) 0 0
\(967\) 40.9848 1.31798 0.658992 0.752150i \(-0.270983\pi\)
0.658992 + 0.752150i \(0.270983\pi\)
\(968\) −21.0000 −0.674966
\(969\) 2.87689 0.0924192
\(970\) −0.630683 −0.0202500
\(971\) −28.1771 −0.904246 −0.452123 0.891956i \(-0.649333\pi\)
−0.452123 + 0.891956i \(0.649333\pi\)
\(972\) 22.2462 0.713548
\(973\) 0 0
\(974\) −6.49242 −0.208031
\(975\) −18.2462 −0.584346
\(976\) 11.6847 0.374017
\(977\) 40.2462 1.28759 0.643795 0.765198i \(-0.277359\pi\)
0.643795 + 0.765198i \(0.277359\pi\)
\(978\) 44.6695 1.42837
\(979\) −32.4924 −1.03846
\(980\) −17.9309 −0.572781
\(981\) 43.6155 1.39254
\(982\) −19.3693 −0.618100
\(983\) 27.5464 0.878594 0.439297 0.898342i \(-0.355228\pi\)
0.439297 + 0.898342i \(0.355228\pi\)
\(984\) 24.0000 0.765092
\(985\) 60.4924 1.92745
\(986\) −5.12311 −0.163153
\(987\) 0 0
\(988\) −5.12311 −0.162988
\(989\) −2.00000 −0.0635963
\(990\) −18.2462 −0.579903
\(991\) 38.9157 1.23620 0.618099 0.786100i \(-0.287903\pi\)
0.618099 + 0.786100i \(0.287903\pi\)
\(992\) 30.0000 0.952501
\(993\) 30.7386 0.975461
\(994\) 0 0
\(995\) −66.4233 −2.10576
\(996\) 12.3153 0.390227
\(997\) −9.12311 −0.288932 −0.144466 0.989510i \(-0.546146\pi\)
−0.144466 + 0.989510i \(0.546146\pi\)
\(998\) 7.68466 0.243254
\(999\) −2.06913 −0.0654644
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 731.2.a.b.1.2 2
3.2 odd 2 6579.2.a.f.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
731.2.a.b.1.2 2 1.1 even 1 trivial
6579.2.a.f.1.2 2 3.2 odd 2