Properties

Label 6003.2.a.j.1.7
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 3 x^{6} - 5 x^{5} + 18 x^{4} + 4 x^{3} - 26 x^{2} + x + 8\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2001)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(2.37349\) of defining polynomial
Character \(\chi\) \(=\) 6003.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.37349 q^{2} +3.63344 q^{4} +1.31800 q^{5} -0.859291 q^{7} +3.87694 q^{8} +O(q^{10})\) \(q+2.37349 q^{2} +3.63344 q^{4} +1.31800 q^{5} -0.859291 q^{7} +3.87694 q^{8} +3.12824 q^{10} -4.30847 q^{11} -4.15542 q^{13} -2.03951 q^{14} +1.93498 q^{16} -2.08920 q^{17} -2.45615 q^{19} +4.78885 q^{20} -10.2261 q^{22} -1.00000 q^{23} -3.26289 q^{25} -9.86283 q^{26} -3.12218 q^{28} +1.00000 q^{29} -4.70746 q^{31} -3.16122 q^{32} -4.95870 q^{34} -1.13254 q^{35} -3.94232 q^{37} -5.82963 q^{38} +5.10979 q^{40} +3.17442 q^{41} +11.0153 q^{43} -15.6545 q^{44} -2.37349 q^{46} -11.1103 q^{47} -6.26162 q^{49} -7.74442 q^{50} -15.0984 q^{52} +14.0165 q^{53} -5.67854 q^{55} -3.33142 q^{56} +2.37349 q^{58} -8.47222 q^{59} -6.39249 q^{61} -11.1731 q^{62} -11.3731 q^{64} -5.47682 q^{65} +4.74715 q^{67} -7.59099 q^{68} -2.68807 q^{70} +12.4909 q^{71} +8.54106 q^{73} -9.35703 q^{74} -8.92426 q^{76} +3.70223 q^{77} -13.0092 q^{79} +2.55030 q^{80} +7.53445 q^{82} +14.2415 q^{83} -2.75356 q^{85} +26.1447 q^{86} -16.7037 q^{88} -0.795808 q^{89} +3.57071 q^{91} -3.63344 q^{92} -26.3702 q^{94} -3.23719 q^{95} +3.92369 q^{97} -14.8619 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 3q^{2} + 5q^{4} + 3q^{5} - 5q^{7} + 6q^{8} + O(q^{10}) \) \( 7q + 3q^{2} + 5q^{4} + 3q^{5} - 5q^{7} + 6q^{8} + 3q^{10} + 4q^{11} - 18q^{13} + 2q^{14} - 7q^{16} + 3q^{17} - 4q^{19} + 2q^{20} - 26q^{22} - 7q^{23} - 8q^{25} + 7q^{26} - 6q^{28} + 7q^{29} - 22q^{31} - 5q^{32} + 9q^{34} - 3q^{35} - 25q^{37} - 14q^{38} - 10q^{40} + 13q^{41} - 2q^{43} - 4q^{44} - 3q^{46} + 25q^{47} - 8q^{49} - 19q^{50} - 12q^{52} + 5q^{53} - 15q^{55} - 18q^{56} + 3q^{58} - 11q^{59} - 33q^{61} - 28q^{62} - 14q^{64} + 2q^{65} + 8q^{67} - 12q^{68} - 22q^{70} + 6q^{71} + 15q^{73} - 34q^{74} - 28q^{76} + q^{77} - 15q^{79} + 12q^{80} - 14q^{82} - 21q^{83} - 28q^{85} + 12q^{86} - 13q^{88} - 8q^{89} + 6q^{91} - 5q^{92} - 35q^{94} + 25q^{95} + 13q^{97} - q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.37349 1.67831 0.839154 0.543894i \(-0.183051\pi\)
0.839154 + 0.543894i \(0.183051\pi\)
\(3\) 0 0
\(4\) 3.63344 1.81672
\(5\) 1.31800 0.589426 0.294713 0.955586i \(-0.404776\pi\)
0.294713 + 0.955586i \(0.404776\pi\)
\(6\) 0 0
\(7\) −0.859291 −0.324781 −0.162391 0.986727i \(-0.551920\pi\)
−0.162391 + 0.986727i \(0.551920\pi\)
\(8\) 3.87694 1.37070
\(9\) 0 0
\(10\) 3.12824 0.989238
\(11\) −4.30847 −1.29905 −0.649526 0.760339i \(-0.725033\pi\)
−0.649526 + 0.760339i \(0.725033\pi\)
\(12\) 0 0
\(13\) −4.15542 −1.15251 −0.576253 0.817272i \(-0.695486\pi\)
−0.576253 + 0.817272i \(0.695486\pi\)
\(14\) −2.03951 −0.545083
\(15\) 0 0
\(16\) 1.93498 0.483746
\(17\) −2.08920 −0.506706 −0.253353 0.967374i \(-0.581533\pi\)
−0.253353 + 0.967374i \(0.581533\pi\)
\(18\) 0 0
\(19\) −2.45615 −0.563479 −0.281740 0.959491i \(-0.590911\pi\)
−0.281740 + 0.959491i \(0.590911\pi\)
\(20\) 4.78885 1.07082
\(21\) 0 0
\(22\) −10.2261 −2.18021
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −3.26289 −0.652577
\(26\) −9.86283 −1.93426
\(27\) 0 0
\(28\) −3.12218 −0.590036
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −4.70746 −0.845484 −0.422742 0.906250i \(-0.638932\pi\)
−0.422742 + 0.906250i \(0.638932\pi\)
\(32\) −3.16122 −0.558830
\(33\) 0 0
\(34\) −4.95870 −0.850409
\(35\) −1.13254 −0.191434
\(36\) 0 0
\(37\) −3.94232 −0.648113 −0.324056 0.946038i \(-0.605047\pi\)
−0.324056 + 0.946038i \(0.605047\pi\)
\(38\) −5.82963 −0.945692
\(39\) 0 0
\(40\) 5.10979 0.807928
\(41\) 3.17442 0.495762 0.247881 0.968791i \(-0.420266\pi\)
0.247881 + 0.968791i \(0.420266\pi\)
\(42\) 0 0
\(43\) 11.0153 1.67982 0.839909 0.542727i \(-0.182608\pi\)
0.839909 + 0.542727i \(0.182608\pi\)
\(44\) −15.6545 −2.36001
\(45\) 0 0
\(46\) −2.37349 −0.349951
\(47\) −11.1103 −1.62061 −0.810303 0.586012i \(-0.800697\pi\)
−0.810303 + 0.586012i \(0.800697\pi\)
\(48\) 0 0
\(49\) −6.26162 −0.894517
\(50\) −7.74442 −1.09523
\(51\) 0 0
\(52\) −15.0984 −2.09378
\(53\) 14.0165 1.92531 0.962657 0.270723i \(-0.0872626\pi\)
0.962657 + 0.270723i \(0.0872626\pi\)
\(54\) 0 0
\(55\) −5.67854 −0.765695
\(56\) −3.33142 −0.445179
\(57\) 0 0
\(58\) 2.37349 0.311654
\(59\) −8.47222 −1.10299 −0.551495 0.834178i \(-0.685942\pi\)
−0.551495 + 0.834178i \(0.685942\pi\)
\(60\) 0 0
\(61\) −6.39249 −0.818475 −0.409237 0.912428i \(-0.634205\pi\)
−0.409237 + 0.912428i \(0.634205\pi\)
\(62\) −11.1731 −1.41898
\(63\) 0 0
\(64\) −11.3731 −1.42163
\(65\) −5.47682 −0.679316
\(66\) 0 0
\(67\) 4.74715 0.579956 0.289978 0.957033i \(-0.406352\pi\)
0.289978 + 0.957033i \(0.406352\pi\)
\(68\) −7.59099 −0.920542
\(69\) 0 0
\(70\) −2.68807 −0.321286
\(71\) 12.4909 1.48240 0.741201 0.671283i \(-0.234257\pi\)
0.741201 + 0.671283i \(0.234257\pi\)
\(72\) 0 0
\(73\) 8.54106 0.999655 0.499827 0.866125i \(-0.333397\pi\)
0.499827 + 0.866125i \(0.333397\pi\)
\(74\) −9.35703 −1.08773
\(75\) 0 0
\(76\) −8.92426 −1.02368
\(77\) 3.70223 0.421908
\(78\) 0 0
\(79\) −13.0092 −1.46365 −0.731827 0.681491i \(-0.761332\pi\)
−0.731827 + 0.681491i \(0.761332\pi\)
\(80\) 2.55030 0.285132
\(81\) 0 0
\(82\) 7.53445 0.832041
\(83\) 14.2415 1.56321 0.781606 0.623772i \(-0.214401\pi\)
0.781606 + 0.623772i \(0.214401\pi\)
\(84\) 0 0
\(85\) −2.75356 −0.298666
\(86\) 26.1447 2.81925
\(87\) 0 0
\(88\) −16.7037 −1.78062
\(89\) −0.795808 −0.0843554 −0.0421777 0.999110i \(-0.513430\pi\)
−0.0421777 + 0.999110i \(0.513430\pi\)
\(90\) 0 0
\(91\) 3.57071 0.374312
\(92\) −3.63344 −0.378812
\(93\) 0 0
\(94\) −26.3702 −2.71988
\(95\) −3.23719 −0.332129
\(96\) 0 0
\(97\) 3.92369 0.398390 0.199195 0.979960i \(-0.436167\pi\)
0.199195 + 0.979960i \(0.436167\pi\)
\(98\) −14.8619 −1.50128
\(99\) 0 0
\(100\) −11.8555 −1.18555
\(101\) −3.90652 −0.388713 −0.194357 0.980931i \(-0.562262\pi\)
−0.194357 + 0.980931i \(0.562262\pi\)
\(102\) 0 0
\(103\) 1.68054 0.165589 0.0827944 0.996567i \(-0.473616\pi\)
0.0827944 + 0.996567i \(0.473616\pi\)
\(104\) −16.1103 −1.57974
\(105\) 0 0
\(106\) 33.2680 3.23127
\(107\) −11.2977 −1.09219 −0.546096 0.837722i \(-0.683887\pi\)
−0.546096 + 0.837722i \(0.683887\pi\)
\(108\) 0 0
\(109\) −17.5897 −1.68479 −0.842394 0.538863i \(-0.818854\pi\)
−0.842394 + 0.538863i \(0.818854\pi\)
\(110\) −13.4779 −1.28507
\(111\) 0 0
\(112\) −1.66271 −0.157112
\(113\) 3.98253 0.374645 0.187323 0.982298i \(-0.440019\pi\)
0.187323 + 0.982298i \(0.440019\pi\)
\(114\) 0 0
\(115\) −1.31800 −0.122904
\(116\) 3.63344 0.337356
\(117\) 0 0
\(118\) −20.1087 −1.85116
\(119\) 1.79523 0.164569
\(120\) 0 0
\(121\) 7.56290 0.687537
\(122\) −15.1725 −1.37365
\(123\) 0 0
\(124\) −17.1042 −1.53601
\(125\) −10.8905 −0.974072
\(126\) 0 0
\(127\) 16.7386 1.48531 0.742654 0.669675i \(-0.233567\pi\)
0.742654 + 0.669675i \(0.233567\pi\)
\(128\) −20.6714 −1.82711
\(129\) 0 0
\(130\) −12.9992 −1.14010
\(131\) 22.3630 1.95386 0.976930 0.213559i \(-0.0685055\pi\)
0.976930 + 0.213559i \(0.0685055\pi\)
\(132\) 0 0
\(133\) 2.11055 0.183008
\(134\) 11.2673 0.973345
\(135\) 0 0
\(136\) −8.09971 −0.694544
\(137\) 12.8745 1.09995 0.549973 0.835182i \(-0.314638\pi\)
0.549973 + 0.835182i \(0.314638\pi\)
\(138\) 0 0
\(139\) 12.8067 1.08625 0.543123 0.839653i \(-0.317242\pi\)
0.543123 + 0.839653i \(0.317242\pi\)
\(140\) −4.11502 −0.347782
\(141\) 0 0
\(142\) 29.6471 2.48793
\(143\) 17.9035 1.49716
\(144\) 0 0
\(145\) 1.31800 0.109454
\(146\) 20.2721 1.67773
\(147\) 0 0
\(148\) −14.3242 −1.17744
\(149\) −11.7517 −0.962740 −0.481370 0.876518i \(-0.659861\pi\)
−0.481370 + 0.876518i \(0.659861\pi\)
\(150\) 0 0
\(151\) −11.1096 −0.904086 −0.452043 0.891996i \(-0.649305\pi\)
−0.452043 + 0.891996i \(0.649305\pi\)
\(152\) −9.52233 −0.772363
\(153\) 0 0
\(154\) 8.78718 0.708091
\(155\) −6.20441 −0.498350
\(156\) 0 0
\(157\) −19.4984 −1.55614 −0.778072 0.628175i \(-0.783802\pi\)
−0.778072 + 0.628175i \(0.783802\pi\)
\(158\) −30.8772 −2.45646
\(159\) 0 0
\(160\) −4.16647 −0.329389
\(161\) 0.859291 0.0677216
\(162\) 0 0
\(163\) 21.4125 1.67716 0.838579 0.544779i \(-0.183387\pi\)
0.838579 + 0.544779i \(0.183387\pi\)
\(164\) 11.5341 0.900659
\(165\) 0 0
\(166\) 33.8021 2.62355
\(167\) 3.76831 0.291600 0.145800 0.989314i \(-0.453424\pi\)
0.145800 + 0.989314i \(0.453424\pi\)
\(168\) 0 0
\(169\) 4.26750 0.328269
\(170\) −6.53554 −0.501253
\(171\) 0 0
\(172\) 40.0234 3.05176
\(173\) −18.5637 −1.41137 −0.705686 0.708525i \(-0.749361\pi\)
−0.705686 + 0.708525i \(0.749361\pi\)
\(174\) 0 0
\(175\) 2.80377 0.211945
\(176\) −8.33681 −0.628411
\(177\) 0 0
\(178\) −1.88884 −0.141574
\(179\) 21.5082 1.60760 0.803798 0.594902i \(-0.202809\pi\)
0.803798 + 0.594902i \(0.202809\pi\)
\(180\) 0 0
\(181\) −12.8583 −0.955754 −0.477877 0.878427i \(-0.658593\pi\)
−0.477877 + 0.878427i \(0.658593\pi\)
\(182\) 8.47503 0.628211
\(183\) 0 0
\(184\) −3.87694 −0.285812
\(185\) −5.19596 −0.382014
\(186\) 0 0
\(187\) 9.00127 0.658238
\(188\) −40.3686 −2.94418
\(189\) 0 0
\(190\) −7.68344 −0.557415
\(191\) −1.67695 −0.121340 −0.0606700 0.998158i \(-0.519324\pi\)
−0.0606700 + 0.998158i \(0.519324\pi\)
\(192\) 0 0
\(193\) −13.8369 −0.996002 −0.498001 0.867177i \(-0.665932\pi\)
−0.498001 + 0.867177i \(0.665932\pi\)
\(194\) 9.31282 0.668622
\(195\) 0 0
\(196\) −22.7512 −1.62509
\(197\) 27.8816 1.98648 0.993241 0.116074i \(-0.0370311\pi\)
0.993241 + 0.116074i \(0.0370311\pi\)
\(198\) 0 0
\(199\) −17.7635 −1.25922 −0.629609 0.776912i \(-0.716785\pi\)
−0.629609 + 0.776912i \(0.716785\pi\)
\(200\) −12.6500 −0.894490
\(201\) 0 0
\(202\) −9.27207 −0.652380
\(203\) −0.859291 −0.0603104
\(204\) 0 0
\(205\) 4.18388 0.292215
\(206\) 3.98875 0.277909
\(207\) 0 0
\(208\) −8.04066 −0.557520
\(209\) 10.5822 0.731989
\(210\) 0 0
\(211\) −21.7285 −1.49585 −0.747926 0.663783i \(-0.768950\pi\)
−0.747926 + 0.663783i \(0.768950\pi\)
\(212\) 50.9281 3.49775
\(213\) 0 0
\(214\) −26.8150 −1.83304
\(215\) 14.5181 0.990128
\(216\) 0 0
\(217\) 4.04507 0.274598
\(218\) −41.7489 −2.82759
\(219\) 0 0
\(220\) −20.6326 −1.39105
\(221\) 8.68152 0.583982
\(222\) 0 0
\(223\) −0.162927 −0.0109104 −0.00545520 0.999985i \(-0.501736\pi\)
−0.00545520 + 0.999985i \(0.501736\pi\)
\(224\) 2.71641 0.181497
\(225\) 0 0
\(226\) 9.45248 0.628770
\(227\) 14.9422 0.991752 0.495876 0.868393i \(-0.334847\pi\)
0.495876 + 0.868393i \(0.334847\pi\)
\(228\) 0 0
\(229\) −9.10145 −0.601441 −0.300720 0.953712i \(-0.597227\pi\)
−0.300720 + 0.953712i \(0.597227\pi\)
\(230\) −3.12824 −0.206270
\(231\) 0 0
\(232\) 3.87694 0.254533
\(233\) 12.6237 0.827007 0.413504 0.910502i \(-0.364305\pi\)
0.413504 + 0.910502i \(0.364305\pi\)
\(234\) 0 0
\(235\) −14.6433 −0.955227
\(236\) −30.7833 −2.00382
\(237\) 0 0
\(238\) 4.26096 0.276197
\(239\) −16.8514 −1.09003 −0.545013 0.838428i \(-0.683475\pi\)
−0.545013 + 0.838428i \(0.683475\pi\)
\(240\) 0 0
\(241\) −17.4282 −1.12265 −0.561323 0.827597i \(-0.689708\pi\)
−0.561323 + 0.827597i \(0.689708\pi\)
\(242\) 17.9504 1.15390
\(243\) 0 0
\(244\) −23.2267 −1.48694
\(245\) −8.25279 −0.527251
\(246\) 0 0
\(247\) 10.2063 0.649413
\(248\) −18.2505 −1.15891
\(249\) 0 0
\(250\) −25.8483 −1.63479
\(251\) −10.2885 −0.649407 −0.324703 0.945816i \(-0.605264\pi\)
−0.324703 + 0.945816i \(0.605264\pi\)
\(252\) 0 0
\(253\) 4.30847 0.270871
\(254\) 39.7288 2.49280
\(255\) 0 0
\(256\) −26.3171 −1.64482
\(257\) 5.27151 0.328827 0.164414 0.986391i \(-0.447427\pi\)
0.164414 + 0.986391i \(0.447427\pi\)
\(258\) 0 0
\(259\) 3.38760 0.210495
\(260\) −19.8997 −1.23413
\(261\) 0 0
\(262\) 53.0782 3.27918
\(263\) 4.72941 0.291628 0.145814 0.989312i \(-0.453420\pi\)
0.145814 + 0.989312i \(0.453420\pi\)
\(264\) 0 0
\(265\) 18.4737 1.13483
\(266\) 5.00935 0.307143
\(267\) 0 0
\(268\) 17.2485 1.05362
\(269\) 21.2129 1.29338 0.646688 0.762754i \(-0.276154\pi\)
0.646688 + 0.762754i \(0.276154\pi\)
\(270\) 0 0
\(271\) −11.0667 −0.672253 −0.336127 0.941817i \(-0.609117\pi\)
−0.336127 + 0.941817i \(0.609117\pi\)
\(272\) −4.04257 −0.245117
\(273\) 0 0
\(274\) 30.5575 1.84605
\(275\) 14.0580 0.847732
\(276\) 0 0
\(277\) −7.98577 −0.479818 −0.239909 0.970795i \(-0.577118\pi\)
−0.239909 + 0.970795i \(0.577118\pi\)
\(278\) 30.3964 1.82306
\(279\) 0 0
\(280\) −4.39079 −0.262400
\(281\) −6.63114 −0.395581 −0.197790 0.980244i \(-0.563377\pi\)
−0.197790 + 0.980244i \(0.563377\pi\)
\(282\) 0 0
\(283\) 2.98274 0.177306 0.0886528 0.996063i \(-0.471744\pi\)
0.0886528 + 0.996063i \(0.471744\pi\)
\(284\) 45.3850 2.69311
\(285\) 0 0
\(286\) 42.4937 2.51270
\(287\) −2.72775 −0.161014
\(288\) 0 0
\(289\) −12.6352 −0.743249
\(290\) 3.12824 0.183697
\(291\) 0 0
\(292\) 31.0334 1.81609
\(293\) −13.4117 −0.783521 −0.391760 0.920067i \(-0.628134\pi\)
−0.391760 + 0.920067i \(0.628134\pi\)
\(294\) 0 0
\(295\) −11.1664 −0.650130
\(296\) −15.2841 −0.888371
\(297\) 0 0
\(298\) −27.8926 −1.61577
\(299\) 4.15542 0.240314
\(300\) 0 0
\(301\) −9.46535 −0.545574
\(302\) −26.3685 −1.51734
\(303\) 0 0
\(304\) −4.75261 −0.272581
\(305\) −8.42528 −0.482430
\(306\) 0 0
\(307\) −7.23125 −0.412709 −0.206355 0.978477i \(-0.566160\pi\)
−0.206355 + 0.978477i \(0.566160\pi\)
\(308\) 13.4518 0.766488
\(309\) 0 0
\(310\) −14.7261 −0.836385
\(311\) −0.648787 −0.0367894 −0.0183947 0.999831i \(-0.505856\pi\)
−0.0183947 + 0.999831i \(0.505856\pi\)
\(312\) 0 0
\(313\) 18.3081 1.03483 0.517416 0.855734i \(-0.326894\pi\)
0.517416 + 0.855734i \(0.326894\pi\)
\(314\) −46.2793 −2.61169
\(315\) 0 0
\(316\) −47.2682 −2.65905
\(317\) 21.4173 1.20292 0.601458 0.798904i \(-0.294587\pi\)
0.601458 + 0.798904i \(0.294587\pi\)
\(318\) 0 0
\(319\) −4.30847 −0.241228
\(320\) −14.9897 −0.837948
\(321\) 0 0
\(322\) 2.03951 0.113658
\(323\) 5.13140 0.285518
\(324\) 0 0
\(325\) 13.5587 0.752099
\(326\) 50.8223 2.81479
\(327\) 0 0
\(328\) 12.3070 0.679542
\(329\) 9.54699 0.526342
\(330\) 0 0
\(331\) 2.79629 0.153698 0.0768490 0.997043i \(-0.475514\pi\)
0.0768490 + 0.997043i \(0.475514\pi\)
\(332\) 51.7457 2.83992
\(333\) 0 0
\(334\) 8.94403 0.489395
\(335\) 6.25672 0.341841
\(336\) 0 0
\(337\) −31.0998 −1.69411 −0.847057 0.531501i \(-0.821628\pi\)
−0.847057 + 0.531501i \(0.821628\pi\)
\(338\) 10.1288 0.550937
\(339\) 0 0
\(340\) −10.0049 −0.542591
\(341\) 20.2819 1.09833
\(342\) 0 0
\(343\) 11.3956 0.615304
\(344\) 42.7056 2.30253
\(345\) 0 0
\(346\) −44.0607 −2.36872
\(347\) −34.9752 −1.87757 −0.938785 0.344504i \(-0.888047\pi\)
−0.938785 + 0.344504i \(0.888047\pi\)
\(348\) 0 0
\(349\) 28.8052 1.54191 0.770953 0.636892i \(-0.219780\pi\)
0.770953 + 0.636892i \(0.219780\pi\)
\(350\) 6.65470 0.355709
\(351\) 0 0
\(352\) 13.6200 0.725949
\(353\) 11.5890 0.616819 0.308409 0.951254i \(-0.400203\pi\)
0.308409 + 0.951254i \(0.400203\pi\)
\(354\) 0 0
\(355\) 16.4630 0.873766
\(356\) −2.89152 −0.153250
\(357\) 0 0
\(358\) 51.0493 2.69804
\(359\) −35.7439 −1.88649 −0.943246 0.332095i \(-0.892245\pi\)
−0.943246 + 0.332095i \(0.892245\pi\)
\(360\) 0 0
\(361\) −12.9673 −0.682491
\(362\) −30.5191 −1.60405
\(363\) 0 0
\(364\) 12.9740 0.680020
\(365\) 11.2571 0.589222
\(366\) 0 0
\(367\) −27.4350 −1.43209 −0.716046 0.698053i \(-0.754050\pi\)
−0.716046 + 0.698053i \(0.754050\pi\)
\(368\) −1.93498 −0.100868
\(369\) 0 0
\(370\) −12.3325 −0.641138
\(371\) −12.0443 −0.625306
\(372\) 0 0
\(373\) 13.3266 0.690027 0.345013 0.938598i \(-0.387874\pi\)
0.345013 + 0.938598i \(0.387874\pi\)
\(374\) 21.3644 1.10473
\(375\) 0 0
\(376\) −43.0740 −2.22137
\(377\) −4.15542 −0.214015
\(378\) 0 0
\(379\) −5.84883 −0.300434 −0.150217 0.988653i \(-0.547997\pi\)
−0.150217 + 0.988653i \(0.547997\pi\)
\(380\) −11.7621 −0.603385
\(381\) 0 0
\(382\) −3.98022 −0.203646
\(383\) −13.9417 −0.712388 −0.356194 0.934412i \(-0.615926\pi\)
−0.356194 + 0.934412i \(0.615926\pi\)
\(384\) 0 0
\(385\) 4.87952 0.248683
\(386\) −32.8417 −1.67160
\(387\) 0 0
\(388\) 14.2565 0.723763
\(389\) −24.2834 −1.23122 −0.615608 0.788052i \(-0.711090\pi\)
−0.615608 + 0.788052i \(0.711090\pi\)
\(390\) 0 0
\(391\) 2.08920 0.105656
\(392\) −24.2759 −1.22612
\(393\) 0 0
\(394\) 66.1766 3.33393
\(395\) −17.1461 −0.862715
\(396\) 0 0
\(397\) −33.5680 −1.68473 −0.842364 0.538909i \(-0.818837\pi\)
−0.842364 + 0.538909i \(0.818837\pi\)
\(398\) −42.1613 −2.11336
\(399\) 0 0
\(400\) −6.31363 −0.315681
\(401\) −16.7895 −0.838426 −0.419213 0.907888i \(-0.637694\pi\)
−0.419213 + 0.907888i \(0.637694\pi\)
\(402\) 0 0
\(403\) 19.5615 0.974425
\(404\) −14.1941 −0.706182
\(405\) 0 0
\(406\) −2.03951 −0.101219
\(407\) 16.9854 0.841933
\(408\) 0 0
\(409\) 33.9540 1.67892 0.839458 0.543424i \(-0.182872\pi\)
0.839458 + 0.543424i \(0.182872\pi\)
\(410\) 9.93037 0.490426
\(411\) 0 0
\(412\) 6.10615 0.300828
\(413\) 7.28010 0.358230
\(414\) 0 0
\(415\) 18.7703 0.921398
\(416\) 13.1362 0.644054
\(417\) 0 0
\(418\) 25.1168 1.22850
\(419\) −1.25305 −0.0612157 −0.0306079 0.999531i \(-0.509744\pi\)
−0.0306079 + 0.999531i \(0.509744\pi\)
\(420\) 0 0
\(421\) 19.7566 0.962880 0.481440 0.876479i \(-0.340114\pi\)
0.481440 + 0.876479i \(0.340114\pi\)
\(422\) −51.5723 −2.51050
\(423\) 0 0
\(424\) 54.3411 2.63904
\(425\) 6.81684 0.330665
\(426\) 0 0
\(427\) 5.49301 0.265825
\(428\) −41.0496 −1.98421
\(429\) 0 0
\(430\) 34.4586 1.66174
\(431\) 28.0855 1.35283 0.676416 0.736520i \(-0.263532\pi\)
0.676416 + 0.736520i \(0.263532\pi\)
\(432\) 0 0
\(433\) −37.9606 −1.82427 −0.912135 0.409890i \(-0.865567\pi\)
−0.912135 + 0.409890i \(0.865567\pi\)
\(434\) 9.60093 0.460859
\(435\) 0 0
\(436\) −63.9110 −3.06078
\(437\) 2.45615 0.117494
\(438\) 0 0
\(439\) 33.0666 1.57818 0.789091 0.614277i \(-0.210552\pi\)
0.789091 + 0.614277i \(0.210552\pi\)
\(440\) −22.0154 −1.04954
\(441\) 0 0
\(442\) 20.6055 0.980101
\(443\) 4.67818 0.222267 0.111134 0.993805i \(-0.464552\pi\)
0.111134 + 0.993805i \(0.464552\pi\)
\(444\) 0 0
\(445\) −1.04887 −0.0497213
\(446\) −0.386705 −0.0183110
\(447\) 0 0
\(448\) 9.77278 0.461720
\(449\) −0.0578209 −0.00272873 −0.00136437 0.999999i \(-0.500434\pi\)
−0.00136437 + 0.999999i \(0.500434\pi\)
\(450\) 0 0
\(451\) −13.6769 −0.644020
\(452\) 14.4703 0.680624
\(453\) 0 0
\(454\) 35.4652 1.66446
\(455\) 4.70618 0.220629
\(456\) 0 0
\(457\) −9.68884 −0.453225 −0.226612 0.973985i \(-0.572765\pi\)
−0.226612 + 0.973985i \(0.572765\pi\)
\(458\) −21.6022 −1.00940
\(459\) 0 0
\(460\) −4.78885 −0.223281
\(461\) −13.9639 −0.650365 −0.325182 0.945651i \(-0.605426\pi\)
−0.325182 + 0.945651i \(0.605426\pi\)
\(462\) 0 0
\(463\) 31.3995 1.45926 0.729629 0.683844i \(-0.239693\pi\)
0.729629 + 0.683844i \(0.239693\pi\)
\(464\) 1.93498 0.0898293
\(465\) 0 0
\(466\) 29.9622 1.38797
\(467\) 20.9278 0.968422 0.484211 0.874951i \(-0.339107\pi\)
0.484211 + 0.874951i \(0.339107\pi\)
\(468\) 0 0
\(469\) −4.07918 −0.188359
\(470\) −34.7558 −1.60316
\(471\) 0 0
\(472\) −32.8463 −1.51187
\(473\) −47.4591 −2.18217
\(474\) 0 0
\(475\) 8.01414 0.367714
\(476\) 6.52286 0.298975
\(477\) 0 0
\(478\) −39.9965 −1.82940
\(479\) −41.1688 −1.88105 −0.940525 0.339724i \(-0.889666\pi\)
−0.940525 + 0.339724i \(0.889666\pi\)
\(480\) 0 0
\(481\) 16.3820 0.746954
\(482\) −41.3655 −1.88415
\(483\) 0 0
\(484\) 27.4793 1.24906
\(485\) 5.17141 0.234821
\(486\) 0 0
\(487\) −16.7097 −0.757190 −0.378595 0.925562i \(-0.623593\pi\)
−0.378595 + 0.925562i \(0.623593\pi\)
\(488\) −24.7833 −1.12189
\(489\) 0 0
\(490\) −19.5879 −0.884890
\(491\) −18.7578 −0.846526 −0.423263 0.906007i \(-0.639115\pi\)
−0.423263 + 0.906007i \(0.639115\pi\)
\(492\) 0 0
\(493\) −2.08920 −0.0940930
\(494\) 24.2246 1.08991
\(495\) 0 0
\(496\) −9.10885 −0.408999
\(497\) −10.7333 −0.481456
\(498\) 0 0
\(499\) 4.60344 0.206078 0.103039 0.994677i \(-0.467143\pi\)
0.103039 + 0.994677i \(0.467143\pi\)
\(500\) −39.5698 −1.76961
\(501\) 0 0
\(502\) −24.4197 −1.08990
\(503\) 10.6369 0.474276 0.237138 0.971476i \(-0.423791\pi\)
0.237138 + 0.971476i \(0.423791\pi\)
\(504\) 0 0
\(505\) −5.14878 −0.229117
\(506\) 10.2261 0.454605
\(507\) 0 0
\(508\) 60.8185 2.69839
\(509\) −12.0067 −0.532186 −0.266093 0.963947i \(-0.585733\pi\)
−0.266093 + 0.963947i \(0.585733\pi\)
\(510\) 0 0
\(511\) −7.33925 −0.324669
\(512\) −21.1205 −0.933404
\(513\) 0 0
\(514\) 12.5118 0.551874
\(515\) 2.21495 0.0976023
\(516\) 0 0
\(517\) 47.8684 2.10525
\(518\) 8.04041 0.353275
\(519\) 0 0
\(520\) −21.2333 −0.931142
\(521\) −11.2329 −0.492123 −0.246061 0.969254i \(-0.579136\pi\)
−0.246061 + 0.969254i \(0.579136\pi\)
\(522\) 0 0
\(523\) −30.6342 −1.33954 −0.669770 0.742569i \(-0.733607\pi\)
−0.669770 + 0.742569i \(0.733607\pi\)
\(524\) 81.2544 3.54961
\(525\) 0 0
\(526\) 11.2252 0.489442
\(527\) 9.83484 0.428412
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 43.8471 1.90459
\(531\) 0 0
\(532\) 7.66853 0.332473
\(533\) −13.1911 −0.571368
\(534\) 0 0
\(535\) −14.8904 −0.643766
\(536\) 18.4044 0.794948
\(537\) 0 0
\(538\) 50.3486 2.17068
\(539\) 26.9780 1.16202
\(540\) 0 0
\(541\) −24.1023 −1.03624 −0.518119 0.855308i \(-0.673368\pi\)
−0.518119 + 0.855308i \(0.673368\pi\)
\(542\) −26.2666 −1.12825
\(543\) 0 0
\(544\) 6.60443 0.283163
\(545\) −23.1831 −0.993057
\(546\) 0 0
\(547\) 13.8709 0.593075 0.296538 0.955021i \(-0.404168\pi\)
0.296538 + 0.955021i \(0.404168\pi\)
\(548\) 46.7788 1.99829
\(549\) 0 0
\(550\) 33.3666 1.42276
\(551\) −2.45615 −0.104635
\(552\) 0 0
\(553\) 11.1787 0.475367
\(554\) −18.9541 −0.805283
\(555\) 0 0
\(556\) 46.5322 1.97340
\(557\) −33.6906 −1.42752 −0.713758 0.700392i \(-0.753009\pi\)
−0.713758 + 0.700392i \(0.753009\pi\)
\(558\) 0 0
\(559\) −45.7732 −1.93600
\(560\) −2.19145 −0.0926056
\(561\) 0 0
\(562\) −15.7389 −0.663906
\(563\) −41.3582 −1.74304 −0.871520 0.490360i \(-0.836865\pi\)
−0.871520 + 0.490360i \(0.836865\pi\)
\(564\) 0 0
\(565\) 5.24896 0.220825
\(566\) 7.07949 0.297573
\(567\) 0 0
\(568\) 48.4266 2.03193
\(569\) −14.0644 −0.589612 −0.294806 0.955557i \(-0.595255\pi\)
−0.294806 + 0.955557i \(0.595255\pi\)
\(570\) 0 0
\(571\) −5.85298 −0.244940 −0.122470 0.992472i \(-0.539081\pi\)
−0.122470 + 0.992472i \(0.539081\pi\)
\(572\) 65.0512 2.71993
\(573\) 0 0
\(574\) −6.47428 −0.270231
\(575\) 3.26289 0.136072
\(576\) 0 0
\(577\) −27.3870 −1.14014 −0.570069 0.821597i \(-0.693083\pi\)
−0.570069 + 0.821597i \(0.693083\pi\)
\(578\) −29.9895 −1.24740
\(579\) 0 0
\(580\) 4.78885 0.198846
\(581\) −12.2376 −0.507702
\(582\) 0 0
\(583\) −60.3897 −2.50108
\(584\) 33.1131 1.37023
\(585\) 0 0
\(586\) −31.8325 −1.31499
\(587\) 45.3687 1.87257 0.936284 0.351245i \(-0.114242\pi\)
0.936284 + 0.351245i \(0.114242\pi\)
\(588\) 0 0
\(589\) 11.5622 0.476413
\(590\) −26.5032 −1.09112
\(591\) 0 0
\(592\) −7.62832 −0.313522
\(593\) −17.9412 −0.736756 −0.368378 0.929676i \(-0.620087\pi\)
−0.368378 + 0.929676i \(0.620087\pi\)
\(594\) 0 0
\(595\) 2.36611 0.0970011
\(596\) −42.6992 −1.74903
\(597\) 0 0
\(598\) 9.86283 0.403321
\(599\) 38.0747 1.55569 0.777845 0.628456i \(-0.216313\pi\)
0.777845 + 0.628456i \(0.216313\pi\)
\(600\) 0 0
\(601\) −25.1262 −1.02492 −0.512461 0.858711i \(-0.671266\pi\)
−0.512461 + 0.858711i \(0.671266\pi\)
\(602\) −22.4659 −0.915641
\(603\) 0 0
\(604\) −40.3660 −1.64247
\(605\) 9.96787 0.405252
\(606\) 0 0
\(607\) 19.6470 0.797447 0.398723 0.917071i \(-0.369453\pi\)
0.398723 + 0.917071i \(0.369453\pi\)
\(608\) 7.76442 0.314889
\(609\) 0 0
\(610\) −19.9973 −0.809666
\(611\) 46.1680 1.86776
\(612\) 0 0
\(613\) 17.5314 0.708086 0.354043 0.935229i \(-0.384807\pi\)
0.354043 + 0.935229i \(0.384807\pi\)
\(614\) −17.1633 −0.692653
\(615\) 0 0
\(616\) 14.3533 0.578311
\(617\) −28.1894 −1.13486 −0.567430 0.823421i \(-0.692062\pi\)
−0.567430 + 0.823421i \(0.692062\pi\)
\(618\) 0 0
\(619\) 12.6807 0.509680 0.254840 0.966983i \(-0.417977\pi\)
0.254840 + 0.966983i \(0.417977\pi\)
\(620\) −22.5433 −0.905362
\(621\) 0 0
\(622\) −1.53989 −0.0617439
\(623\) 0.683830 0.0273971
\(624\) 0 0
\(625\) 1.96086 0.0784345
\(626\) 43.4539 1.73677
\(627\) 0 0
\(628\) −70.8463 −2.82708
\(629\) 8.23630 0.328403
\(630\) 0 0
\(631\) 24.4176 0.972050 0.486025 0.873945i \(-0.338446\pi\)
0.486025 + 0.873945i \(0.338446\pi\)
\(632\) −50.4360 −2.00624
\(633\) 0 0
\(634\) 50.8337 2.01886
\(635\) 22.0614 0.875479
\(636\) 0 0
\(637\) 26.0196 1.03094
\(638\) −10.2261 −0.404855
\(639\) 0 0
\(640\) −27.2448 −1.07695
\(641\) 39.9966 1.57977 0.789886 0.613254i \(-0.210140\pi\)
0.789886 + 0.613254i \(0.210140\pi\)
\(642\) 0 0
\(643\) −21.9596 −0.866002 −0.433001 0.901393i \(-0.642545\pi\)
−0.433001 + 0.901393i \(0.642545\pi\)
\(644\) 3.12218 0.123031
\(645\) 0 0
\(646\) 12.1793 0.479188
\(647\) −5.75236 −0.226149 −0.113074 0.993587i \(-0.536070\pi\)
−0.113074 + 0.993587i \(0.536070\pi\)
\(648\) 0 0
\(649\) 36.5023 1.43284
\(650\) 32.1813 1.26225
\(651\) 0 0
\(652\) 77.8010 3.04692
\(653\) 16.3800 0.640998 0.320499 0.947249i \(-0.396149\pi\)
0.320499 + 0.947249i \(0.396149\pi\)
\(654\) 0 0
\(655\) 29.4743 1.15166
\(656\) 6.14245 0.239823
\(657\) 0 0
\(658\) 22.6596 0.883365
\(659\) 7.33501 0.285731 0.142866 0.989742i \(-0.454368\pi\)
0.142866 + 0.989742i \(0.454368\pi\)
\(660\) 0 0
\(661\) 23.2593 0.904681 0.452340 0.891845i \(-0.350589\pi\)
0.452340 + 0.891845i \(0.350589\pi\)
\(662\) 6.63695 0.257953
\(663\) 0 0
\(664\) 55.2136 2.14270
\(665\) 2.78169 0.107869
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) 13.6919 0.529756
\(669\) 0 0
\(670\) 14.8502 0.573715
\(671\) 27.5418 1.06324
\(672\) 0 0
\(673\) 28.8875 1.11353 0.556766 0.830670i \(-0.312042\pi\)
0.556766 + 0.830670i \(0.312042\pi\)
\(674\) −73.8150 −2.84325
\(675\) 0 0
\(676\) 15.5057 0.596372
\(677\) −27.0586 −1.03995 −0.519974 0.854182i \(-0.674058\pi\)
−0.519974 + 0.854182i \(0.674058\pi\)
\(678\) 0 0
\(679\) −3.37159 −0.129390
\(680\) −10.6754 −0.409382
\(681\) 0 0
\(682\) 48.1389 1.84333
\(683\) −33.2775 −1.27333 −0.636664 0.771141i \(-0.719686\pi\)
−0.636664 + 0.771141i \(0.719686\pi\)
\(684\) 0 0
\(685\) 16.9686 0.648337
\(686\) 27.0473 1.03267
\(687\) 0 0
\(688\) 21.3144 0.812605
\(689\) −58.2444 −2.21894
\(690\) 0 0
\(691\) 13.0933 0.498093 0.249046 0.968492i \(-0.419883\pi\)
0.249046 + 0.968492i \(0.419883\pi\)
\(692\) −67.4500 −2.56406
\(693\) 0 0
\(694\) −83.0133 −3.15114
\(695\) 16.8791 0.640262
\(696\) 0 0
\(697\) −6.63202 −0.251206
\(698\) 68.3687 2.58779
\(699\) 0 0
\(700\) 10.1873 0.385044
\(701\) 5.48506 0.207168 0.103584 0.994621i \(-0.466969\pi\)
0.103584 + 0.994621i \(0.466969\pi\)
\(702\) 0 0
\(703\) 9.68292 0.365198
\(704\) 49.0005 1.84678
\(705\) 0 0
\(706\) 27.5063 1.03521
\(707\) 3.35683 0.126247
\(708\) 0 0
\(709\) 9.81311 0.368539 0.184270 0.982876i \(-0.441008\pi\)
0.184270 + 0.982876i \(0.441008\pi\)
\(710\) 39.0747 1.46645
\(711\) 0 0
\(712\) −3.08530 −0.115626
\(713\) 4.70746 0.176296
\(714\) 0 0
\(715\) 23.5967 0.882467
\(716\) 78.1485 2.92055
\(717\) 0 0
\(718\) −84.8377 −3.16611
\(719\) 24.9155 0.929193 0.464596 0.885522i \(-0.346199\pi\)
0.464596 + 0.885522i \(0.346199\pi\)
\(720\) 0 0
\(721\) −1.44408 −0.0537802
\(722\) −30.7778 −1.14543
\(723\) 0 0
\(724\) −46.7200 −1.73633
\(725\) −3.26289 −0.121181
\(726\) 0 0
\(727\) 36.6602 1.35965 0.679825 0.733374i \(-0.262056\pi\)
0.679825 + 0.733374i \(0.262056\pi\)
\(728\) 13.8434 0.513071
\(729\) 0 0
\(730\) 26.7185 0.988897
\(731\) −23.0132 −0.851175
\(732\) 0 0
\(733\) 18.9303 0.699206 0.349603 0.936898i \(-0.386317\pi\)
0.349603 + 0.936898i \(0.386317\pi\)
\(734\) −65.1165 −2.40349
\(735\) 0 0
\(736\) 3.16122 0.116524
\(737\) −20.4529 −0.753393
\(738\) 0 0
\(739\) 8.84818 0.325485 0.162743 0.986669i \(-0.447966\pi\)
0.162743 + 0.986669i \(0.447966\pi\)
\(740\) −18.8792 −0.694012
\(741\) 0 0
\(742\) −28.5869 −1.04946
\(743\) −4.67217 −0.171405 −0.0857026 0.996321i \(-0.527313\pi\)
−0.0857026 + 0.996321i \(0.527313\pi\)
\(744\) 0 0
\(745\) −15.4887 −0.567464
\(746\) 31.6306 1.15808
\(747\) 0 0
\(748\) 32.7055 1.19583
\(749\) 9.70803 0.354724
\(750\) 0 0
\(751\) −23.8022 −0.868555 −0.434277 0.900779i \(-0.642996\pi\)
−0.434277 + 0.900779i \(0.642996\pi\)
\(752\) −21.4983 −0.783961
\(753\) 0 0
\(754\) −9.86283 −0.359183
\(755\) −14.6424 −0.532892
\(756\) 0 0
\(757\) −0.980413 −0.0356337 −0.0178169 0.999841i \(-0.505672\pi\)
−0.0178169 + 0.999841i \(0.505672\pi\)
\(758\) −13.8821 −0.504221
\(759\) 0 0
\(760\) −12.5504 −0.455251
\(761\) −28.6974 −1.04028 −0.520140 0.854081i \(-0.674120\pi\)
−0.520140 + 0.854081i \(0.674120\pi\)
\(762\) 0 0
\(763\) 15.1147 0.547187
\(764\) −6.09309 −0.220440
\(765\) 0 0
\(766\) −33.0904 −1.19561
\(767\) 35.2056 1.27120
\(768\) 0 0
\(769\) 34.6389 1.24911 0.624556 0.780980i \(-0.285280\pi\)
0.624556 + 0.780980i \(0.285280\pi\)
\(770\) 11.5815 0.417367
\(771\) 0 0
\(772\) −50.2755 −1.80945
\(773\) −3.18514 −0.114561 −0.0572807 0.998358i \(-0.518243\pi\)
−0.0572807 + 0.998358i \(0.518243\pi\)
\(774\) 0 0
\(775\) 15.3599 0.551744
\(776\) 15.2119 0.546075
\(777\) 0 0
\(778\) −57.6363 −2.06636
\(779\) −7.79686 −0.279351
\(780\) 0 0
\(781\) −53.8168 −1.92572
\(782\) 4.95870 0.177323
\(783\) 0 0
\(784\) −12.1161 −0.432719
\(785\) −25.6989 −0.917232
\(786\) 0 0
\(787\) 47.6246 1.69764 0.848818 0.528686i \(-0.177315\pi\)
0.848818 + 0.528686i \(0.177315\pi\)
\(788\) 101.306 3.60888
\(789\) 0 0
\(790\) −40.6961 −1.44790
\(791\) −3.42215 −0.121678
\(792\) 0 0
\(793\) 26.5635 0.943297
\(794\) −79.6731 −2.82749
\(795\) 0 0
\(796\) −64.5424 −2.28764
\(797\) −44.5050 −1.57645 −0.788223 0.615389i \(-0.788999\pi\)
−0.788223 + 0.615389i \(0.788999\pi\)
\(798\) 0 0
\(799\) 23.2117 0.821171
\(800\) 10.3147 0.364680
\(801\) 0 0
\(802\) −39.8496 −1.40714
\(803\) −36.7989 −1.29860
\(804\) 0 0
\(805\) 1.13254 0.0399168
\(806\) 46.4288 1.63539
\(807\) 0 0
\(808\) −15.1453 −0.532811
\(809\) 34.0983 1.19883 0.599416 0.800438i \(-0.295400\pi\)
0.599416 + 0.800438i \(0.295400\pi\)
\(810\) 0 0
\(811\) −10.5525 −0.370550 −0.185275 0.982687i \(-0.559318\pi\)
−0.185275 + 0.982687i \(0.559318\pi\)
\(812\) −3.12218 −0.109567
\(813\) 0 0
\(814\) 40.3145 1.41302
\(815\) 28.2216 0.988561
\(816\) 0 0
\(817\) −27.0552 −0.946543
\(818\) 80.5893 2.81774
\(819\) 0 0
\(820\) 15.2018 0.530872
\(821\) −30.2775 −1.05669 −0.528345 0.849030i \(-0.677187\pi\)
−0.528345 + 0.849030i \(0.677187\pi\)
\(822\) 0 0
\(823\) −40.7346 −1.41992 −0.709960 0.704242i \(-0.751287\pi\)
−0.709960 + 0.704242i \(0.751287\pi\)
\(824\) 6.51536 0.226973
\(825\) 0 0
\(826\) 17.2792 0.601221
\(827\) −7.88557 −0.274208 −0.137104 0.990557i \(-0.543779\pi\)
−0.137104 + 0.990557i \(0.543779\pi\)
\(828\) 0 0
\(829\) 8.87941 0.308395 0.154197 0.988040i \(-0.450721\pi\)
0.154197 + 0.988040i \(0.450721\pi\)
\(830\) 44.5510 1.54639
\(831\) 0 0
\(832\) 47.2599 1.63844
\(833\) 13.0818 0.453257
\(834\) 0 0
\(835\) 4.96662 0.171877
\(836\) 38.4499 1.32982
\(837\) 0 0
\(838\) −2.97411 −0.102739
\(839\) 0.405359 0.0139946 0.00699728 0.999976i \(-0.497773\pi\)
0.00699728 + 0.999976i \(0.497773\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 46.8921 1.61601
\(843\) 0 0
\(844\) −78.9491 −2.71754
\(845\) 5.62455 0.193490
\(846\) 0 0
\(847\) −6.49873 −0.223299
\(848\) 27.1217 0.931363
\(849\) 0 0
\(850\) 16.1797 0.554958
\(851\) 3.94232 0.135141
\(852\) 0 0
\(853\) 38.4556 1.31669 0.658347 0.752715i \(-0.271256\pi\)
0.658347 + 0.752715i \(0.271256\pi\)
\(854\) 13.0376 0.446137
\(855\) 0 0
\(856\) −43.8006 −1.49707
\(857\) 4.08648 0.139592 0.0697958 0.997561i \(-0.477765\pi\)
0.0697958 + 0.997561i \(0.477765\pi\)
\(858\) 0 0
\(859\) −26.7695 −0.913362 −0.456681 0.889631i \(-0.650962\pi\)
−0.456681 + 0.889631i \(0.650962\pi\)
\(860\) 52.7507 1.79878
\(861\) 0 0
\(862\) 66.6606 2.27047
\(863\) −26.1278 −0.889402 −0.444701 0.895679i \(-0.646690\pi\)
−0.444701 + 0.895679i \(0.646690\pi\)
\(864\) 0 0
\(865\) −24.4669 −0.831899
\(866\) −90.0989 −3.06169
\(867\) 0 0
\(868\) 14.6975 0.498866
\(869\) 56.0499 1.90136
\(870\) 0 0
\(871\) −19.7264 −0.668403
\(872\) −68.1941 −2.30934
\(873\) 0 0
\(874\) 5.82963 0.197190
\(875\) 9.35806 0.316360
\(876\) 0 0
\(877\) −4.82414 −0.162900 −0.0814498 0.996677i \(-0.525955\pi\)
−0.0814498 + 0.996677i \(0.525955\pi\)
\(878\) 78.4831 2.64867
\(879\) 0 0
\(880\) −10.9879 −0.370402
\(881\) 24.2045 0.815469 0.407734 0.913101i \(-0.366319\pi\)
0.407734 + 0.913101i \(0.366319\pi\)
\(882\) 0 0
\(883\) 18.1602 0.611139 0.305570 0.952170i \(-0.401153\pi\)
0.305570 + 0.952170i \(0.401153\pi\)
\(884\) 31.5437 1.06093
\(885\) 0 0
\(886\) 11.1036 0.373033
\(887\) 40.7965 1.36981 0.684907 0.728630i \(-0.259843\pi\)
0.684907 + 0.728630i \(0.259843\pi\)
\(888\) 0 0
\(889\) −14.3833 −0.482400
\(890\) −2.48948 −0.0834476
\(891\) 0 0
\(892\) −0.591985 −0.0198211
\(893\) 27.2886 0.913177
\(894\) 0 0
\(895\) 28.3477 0.947558
\(896\) 17.7627 0.593411
\(897\) 0 0
\(898\) −0.137237 −0.00457966
\(899\) −4.70746 −0.157002
\(900\) 0 0
\(901\) −29.2833 −0.975569
\(902\) −32.4619 −1.08086
\(903\) 0 0
\(904\) 15.4400 0.513528
\(905\) −16.9473 −0.563346
\(906\) 0 0
\(907\) 20.5919 0.683744 0.341872 0.939747i \(-0.388939\pi\)
0.341872 + 0.939747i \(0.388939\pi\)
\(908\) 54.2917 1.80173
\(909\) 0 0
\(910\) 11.1701 0.370284
\(911\) −0.555205 −0.0183948 −0.00919738 0.999958i \(-0.502928\pi\)
−0.00919738 + 0.999958i \(0.502928\pi\)
\(912\) 0 0
\(913\) −61.3593 −2.03069
\(914\) −22.9963 −0.760651
\(915\) 0 0
\(916\) −33.0695 −1.09265
\(917\) −19.2163 −0.634577
\(918\) 0 0
\(919\) 51.8269 1.70961 0.854805 0.518949i \(-0.173676\pi\)
0.854805 + 0.518949i \(0.173676\pi\)
\(920\) −5.10979 −0.168465
\(921\) 0 0
\(922\) −33.1432 −1.09151
\(923\) −51.9051 −1.70848
\(924\) 0 0
\(925\) 12.8633 0.422944
\(926\) 74.5262 2.44908
\(927\) 0 0
\(928\) −3.16122 −0.103772
\(929\) −24.3448 −0.798727 −0.399364 0.916793i \(-0.630769\pi\)
−0.399364 + 0.916793i \(0.630769\pi\)
\(930\) 0 0
\(931\) 15.3795 0.504042
\(932\) 45.8675 1.50244
\(933\) 0 0
\(934\) 49.6718 1.62531
\(935\) 11.8636 0.387982
\(936\) 0 0
\(937\) 2.62171 0.0856475 0.0428237 0.999083i \(-0.486365\pi\)
0.0428237 + 0.999083i \(0.486365\pi\)
\(938\) −9.68187 −0.316124
\(939\) 0 0
\(940\) −53.2057 −1.73538
\(941\) 27.1040 0.883566 0.441783 0.897122i \(-0.354346\pi\)
0.441783 + 0.897122i \(0.354346\pi\)
\(942\) 0 0
\(943\) −3.17442 −0.103373
\(944\) −16.3936 −0.533566
\(945\) 0 0
\(946\) −112.644 −3.66236
\(947\) −1.67384 −0.0543924 −0.0271962 0.999630i \(-0.508658\pi\)
−0.0271962 + 0.999630i \(0.508658\pi\)
\(948\) 0 0
\(949\) −35.4917 −1.15211
\(950\) 19.0214 0.617137
\(951\) 0 0
\(952\) 6.96001 0.225575
\(953\) 39.1266 1.26743 0.633717 0.773565i \(-0.281528\pi\)
0.633717 + 0.773565i \(0.281528\pi\)
\(954\) 0 0
\(955\) −2.21021 −0.0715209
\(956\) −61.2284 −1.98027
\(957\) 0 0
\(958\) −97.7136 −3.15698
\(959\) −11.0630 −0.357242
\(960\) 0 0
\(961\) −8.83984 −0.285156
\(962\) 38.8824 1.25362
\(963\) 0 0
\(964\) −63.3241 −2.03953
\(965\) −18.2370 −0.587069
\(966\) 0 0
\(967\) −46.2331 −1.48676 −0.743378 0.668872i \(-0.766777\pi\)
−0.743378 + 0.668872i \(0.766777\pi\)
\(968\) 29.3209 0.942409
\(969\) 0 0
\(970\) 12.2743 0.394103
\(971\) −39.1168 −1.25532 −0.627660 0.778488i \(-0.715987\pi\)
−0.627660 + 0.778488i \(0.715987\pi\)
\(972\) 0 0
\(973\) −11.0046 −0.352793
\(974\) −39.6603 −1.27080
\(975\) 0 0
\(976\) −12.3694 −0.395934
\(977\) 11.4002 0.364725 0.182363 0.983231i \(-0.441626\pi\)
0.182363 + 0.983231i \(0.441626\pi\)
\(978\) 0 0
\(979\) 3.42871 0.109582
\(980\) −29.9860 −0.957867
\(981\) 0 0
\(982\) −44.5213 −1.42073
\(983\) −47.4586 −1.51369 −0.756846 0.653593i \(-0.773261\pi\)
−0.756846 + 0.653593i \(0.773261\pi\)
\(984\) 0 0
\(985\) 36.7478 1.17088
\(986\) −4.95870 −0.157917
\(987\) 0 0
\(988\) 37.0840 1.17980
\(989\) −11.0153 −0.350266
\(990\) 0 0
\(991\) 52.3146 1.66183 0.830914 0.556401i \(-0.187818\pi\)
0.830914 + 0.556401i \(0.187818\pi\)
\(992\) 14.8813 0.472482
\(993\) 0 0
\(994\) −25.4754 −0.808032
\(995\) −23.4122 −0.742215
\(996\) 0 0
\(997\) −46.9541 −1.48705 −0.743525 0.668708i \(-0.766848\pi\)
−0.743525 + 0.668708i \(0.766848\pi\)
\(998\) 10.9262 0.345863
\(999\) 0 0
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 6003.2.a.j.1.7 7
3.2 odd 2 2001.2.a.i.1.1 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.i.1.1 7 3.2 odd 2
6003.2.a.j.1.7 7 1.1 even 1 trivial