Properties

Label 7623.2.a.dc.1.15
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 16
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 693)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Root \(2.60390\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.60390 q^{2} +4.78032 q^{4} -4.31860 q^{5} +1.00000 q^{7} +7.23969 q^{8} +O(q^{10})\) \(q+2.60390 q^{2} +4.78032 q^{4} -4.31860 q^{5} +1.00000 q^{7} +7.23969 q^{8} -11.2452 q^{10} -2.46429 q^{13} +2.60390 q^{14} +9.29081 q^{16} -1.18654 q^{17} -3.42691 q^{19} -20.6443 q^{20} +8.32084 q^{23} +13.6503 q^{25} -6.41676 q^{26} +4.78032 q^{28} +1.45371 q^{29} +2.98816 q^{31} +9.71302 q^{32} -3.08964 q^{34} -4.31860 q^{35} +6.34373 q^{37} -8.92335 q^{38} -31.2653 q^{40} -6.16490 q^{41} +3.15289 q^{43} +21.6667 q^{46} +8.73499 q^{47} +1.00000 q^{49} +35.5440 q^{50} -11.7801 q^{52} -6.68159 q^{53} +7.23969 q^{56} +3.78533 q^{58} +11.7548 q^{59} +7.76909 q^{61} +7.78087 q^{62} +6.71015 q^{64} +10.6423 q^{65} +3.11080 q^{67} -5.67204 q^{68} -11.2452 q^{70} +5.48760 q^{71} +8.66003 q^{73} +16.5185 q^{74} -16.3817 q^{76} -8.83563 q^{79} -40.1233 q^{80} -16.0528 q^{82} -9.66363 q^{83} +5.12419 q^{85} +8.20982 q^{86} +0.985558 q^{89} -2.46429 q^{91} +39.7763 q^{92} +22.7451 q^{94} +14.7995 q^{95} +6.97238 q^{97} +2.60390 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 20q^{4} + 16q^{7} + O(q^{10}) \) \( 16q + 20q^{4} + 16q^{7} - 6q^{10} + 32q^{16} + 10q^{19} + 44q^{25} + 20q^{28} + 30q^{31} + 12q^{34} + 38q^{37} - 68q^{40} + 16q^{43} + 80q^{46} + 16q^{49} + 2q^{52} + 18q^{58} - 28q^{61} + 34q^{64} + 52q^{67} - 6q^{70} - 14q^{73} - 14q^{76} + 54q^{79} + 64q^{82} + 30q^{85} - 60q^{94} + 108q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.60390 1.84124 0.920619 0.390461i \(-0.127685\pi\)
0.920619 + 0.390461i \(0.127685\pi\)
\(3\) 0 0
\(4\) 4.78032 2.39016
\(5\) −4.31860 −1.93134 −0.965668 0.259781i \(-0.916350\pi\)
−0.965668 + 0.259781i \(0.916350\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 7.23969 2.55962
\(9\) 0 0
\(10\) −11.2452 −3.55605
\(11\) 0 0
\(12\) 0 0
\(13\) −2.46429 −0.683470 −0.341735 0.939796i \(-0.611014\pi\)
−0.341735 + 0.939796i \(0.611014\pi\)
\(14\) 2.60390 0.695923
\(15\) 0 0
\(16\) 9.29081 2.32270
\(17\) −1.18654 −0.287778 −0.143889 0.989594i \(-0.545961\pi\)
−0.143889 + 0.989594i \(0.545961\pi\)
\(18\) 0 0
\(19\) −3.42691 −0.786188 −0.393094 0.919498i \(-0.628595\pi\)
−0.393094 + 0.919498i \(0.628595\pi\)
\(20\) −20.6443 −4.61620
\(21\) 0 0
\(22\) 0 0
\(23\) 8.32084 1.73501 0.867507 0.497425i \(-0.165721\pi\)
0.867507 + 0.497425i \(0.165721\pi\)
\(24\) 0 0
\(25\) 13.6503 2.73006
\(26\) −6.41676 −1.25843
\(27\) 0 0
\(28\) 4.78032 0.903395
\(29\) 1.45371 0.269947 0.134974 0.990849i \(-0.456905\pi\)
0.134974 + 0.990849i \(0.456905\pi\)
\(30\) 0 0
\(31\) 2.98816 0.536689 0.268344 0.963323i \(-0.413523\pi\)
0.268344 + 0.963323i \(0.413523\pi\)
\(32\) 9.71302 1.71704
\(33\) 0 0
\(34\) −3.08964 −0.529869
\(35\) −4.31860 −0.729976
\(36\) 0 0
\(37\) 6.34373 1.04290 0.521452 0.853281i \(-0.325391\pi\)
0.521452 + 0.853281i \(0.325391\pi\)
\(38\) −8.92335 −1.44756
\(39\) 0 0
\(40\) −31.2653 −4.94348
\(41\) −6.16490 −0.962795 −0.481398 0.876502i \(-0.659871\pi\)
−0.481398 + 0.876502i \(0.659871\pi\)
\(42\) 0 0
\(43\) 3.15289 0.480811 0.240406 0.970673i \(-0.422720\pi\)
0.240406 + 0.970673i \(0.422720\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 21.6667 3.19458
\(47\) 8.73499 1.27413 0.637064 0.770811i \(-0.280149\pi\)
0.637064 + 0.770811i \(0.280149\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 35.5440 5.02668
\(51\) 0 0
\(52\) −11.7801 −1.63360
\(53\) −6.68159 −0.917787 −0.458894 0.888491i \(-0.651754\pi\)
−0.458894 + 0.888491i \(0.651754\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 7.23969 0.967444
\(57\) 0 0
\(58\) 3.78533 0.497038
\(59\) 11.7548 1.53035 0.765175 0.643822i \(-0.222652\pi\)
0.765175 + 0.643822i \(0.222652\pi\)
\(60\) 0 0
\(61\) 7.76909 0.994729 0.497365 0.867542i \(-0.334301\pi\)
0.497365 + 0.867542i \(0.334301\pi\)
\(62\) 7.78087 0.988172
\(63\) 0 0
\(64\) 6.71015 0.838769
\(65\) 10.6423 1.32001
\(66\) 0 0
\(67\) 3.11080 0.380045 0.190022 0.981780i \(-0.439144\pi\)
0.190022 + 0.981780i \(0.439144\pi\)
\(68\) −5.67204 −0.687836
\(69\) 0 0
\(70\) −11.2452 −1.34406
\(71\) 5.48760 0.651258 0.325629 0.945498i \(-0.394424\pi\)
0.325629 + 0.945498i \(0.394424\pi\)
\(72\) 0 0
\(73\) 8.66003 1.01358 0.506790 0.862070i \(-0.330832\pi\)
0.506790 + 0.862070i \(0.330832\pi\)
\(74\) 16.5185 1.92023
\(75\) 0 0
\(76\) −16.3817 −1.87911
\(77\) 0 0
\(78\) 0 0
\(79\) −8.83563 −0.994086 −0.497043 0.867726i \(-0.665581\pi\)
−0.497043 + 0.867726i \(0.665581\pi\)
\(80\) −40.1233 −4.48592
\(81\) 0 0
\(82\) −16.0528 −1.77274
\(83\) −9.66363 −1.06072 −0.530360 0.847772i \(-0.677943\pi\)
−0.530360 + 0.847772i \(0.677943\pi\)
\(84\) 0 0
\(85\) 5.12419 0.555797
\(86\) 8.20982 0.885288
\(87\) 0 0
\(88\) 0 0
\(89\) 0.985558 0.104469 0.0522345 0.998635i \(-0.483366\pi\)
0.0522345 + 0.998635i \(0.483366\pi\)
\(90\) 0 0
\(91\) −2.46429 −0.258327
\(92\) 39.7763 4.14696
\(93\) 0 0
\(94\) 22.7451 2.34597
\(95\) 14.7995 1.51839
\(96\) 0 0
\(97\) 6.97238 0.707938 0.353969 0.935257i \(-0.384832\pi\)
0.353969 + 0.935257i \(0.384832\pi\)
\(98\) 2.60390 0.263034
\(99\) 0 0
\(100\) 65.2527 6.52527
\(101\) 16.6907 1.66079 0.830394 0.557177i \(-0.188115\pi\)
0.830394 + 0.557177i \(0.188115\pi\)
\(102\) 0 0
\(103\) 11.5782 1.14084 0.570419 0.821354i \(-0.306781\pi\)
0.570419 + 0.821354i \(0.306781\pi\)
\(104\) −17.8407 −1.74942
\(105\) 0 0
\(106\) −17.3982 −1.68987
\(107\) −6.01586 −0.581575 −0.290788 0.956788i \(-0.593917\pi\)
−0.290788 + 0.956788i \(0.593917\pi\)
\(108\) 0 0
\(109\) 6.68101 0.639924 0.319962 0.947430i \(-0.396330\pi\)
0.319962 + 0.947430i \(0.396330\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 9.29081 0.877899
\(113\) −18.7558 −1.76440 −0.882199 0.470877i \(-0.843937\pi\)
−0.882199 + 0.470877i \(0.843937\pi\)
\(114\) 0 0
\(115\) −35.9343 −3.35089
\(116\) 6.94920 0.645217
\(117\) 0 0
\(118\) 30.6085 2.81774
\(119\) −1.18654 −0.108770
\(120\) 0 0
\(121\) 0 0
\(122\) 20.2300 1.83153
\(123\) 0 0
\(124\) 14.2843 1.28277
\(125\) −37.3571 −3.34132
\(126\) 0 0
\(127\) 3.17002 0.281294 0.140647 0.990060i \(-0.455082\pi\)
0.140647 + 0.990060i \(0.455082\pi\)
\(128\) −1.95344 −0.172662
\(129\) 0 0
\(130\) 27.7114 2.43045
\(131\) 7.83376 0.684439 0.342220 0.939620i \(-0.388821\pi\)
0.342220 + 0.939620i \(0.388821\pi\)
\(132\) 0 0
\(133\) −3.42691 −0.297151
\(134\) 8.10023 0.699753
\(135\) 0 0
\(136\) −8.59018 −0.736602
\(137\) 7.18722 0.614045 0.307023 0.951702i \(-0.400667\pi\)
0.307023 + 0.951702i \(0.400667\pi\)
\(138\) 0 0
\(139\) 3.36801 0.285671 0.142836 0.989746i \(-0.454378\pi\)
0.142836 + 0.989746i \(0.454378\pi\)
\(140\) −20.6443 −1.74476
\(141\) 0 0
\(142\) 14.2892 1.19912
\(143\) 0 0
\(144\) 0 0
\(145\) −6.27799 −0.521359
\(146\) 22.5499 1.86624
\(147\) 0 0
\(148\) 30.3251 2.49271
\(149\) −18.6235 −1.52569 −0.762846 0.646580i \(-0.776199\pi\)
−0.762846 + 0.646580i \(0.776199\pi\)
\(150\) 0 0
\(151\) 14.4245 1.17385 0.586924 0.809642i \(-0.300339\pi\)
0.586924 + 0.809642i \(0.300339\pi\)
\(152\) −24.8098 −2.01234
\(153\) 0 0
\(154\) 0 0
\(155\) −12.9046 −1.03653
\(156\) 0 0
\(157\) 5.31457 0.424148 0.212074 0.977254i \(-0.431978\pi\)
0.212074 + 0.977254i \(0.431978\pi\)
\(158\) −23.0071 −1.83035
\(159\) 0 0
\(160\) −41.9466 −3.31617
\(161\) 8.32084 0.655774
\(162\) 0 0
\(163\) 0.906332 0.0709894 0.0354947 0.999370i \(-0.488699\pi\)
0.0354947 + 0.999370i \(0.488699\pi\)
\(164\) −29.4702 −2.30123
\(165\) 0 0
\(166\) −25.1632 −1.95304
\(167\) 11.6434 0.900994 0.450497 0.892778i \(-0.351247\pi\)
0.450497 + 0.892778i \(0.351247\pi\)
\(168\) 0 0
\(169\) −6.92730 −0.532869
\(170\) 13.3429 1.02335
\(171\) 0 0
\(172\) 15.0718 1.14922
\(173\) −6.85042 −0.520828 −0.260414 0.965497i \(-0.583859\pi\)
−0.260414 + 0.965497i \(0.583859\pi\)
\(174\) 0 0
\(175\) 13.6503 1.03186
\(176\) 0 0
\(177\) 0 0
\(178\) 2.56630 0.192352
\(179\) −8.41259 −0.628787 −0.314393 0.949293i \(-0.601801\pi\)
−0.314393 + 0.949293i \(0.601801\pi\)
\(180\) 0 0
\(181\) 13.0288 0.968426 0.484213 0.874950i \(-0.339106\pi\)
0.484213 + 0.874950i \(0.339106\pi\)
\(182\) −6.41676 −0.475642
\(183\) 0 0
\(184\) 60.2402 4.44097
\(185\) −27.3960 −2.01420
\(186\) 0 0
\(187\) 0 0
\(188\) 41.7560 3.04537
\(189\) 0 0
\(190\) 38.5364 2.79572
\(191\) 3.45753 0.250178 0.125089 0.992145i \(-0.460078\pi\)
0.125089 + 0.992145i \(0.460078\pi\)
\(192\) 0 0
\(193\) 17.0472 1.22709 0.613544 0.789661i \(-0.289743\pi\)
0.613544 + 0.789661i \(0.289743\pi\)
\(194\) 18.1554 1.30348
\(195\) 0 0
\(196\) 4.78032 0.341451
\(197\) −3.47970 −0.247918 −0.123959 0.992287i \(-0.539559\pi\)
−0.123959 + 0.992287i \(0.539559\pi\)
\(198\) 0 0
\(199\) 8.20279 0.581481 0.290740 0.956802i \(-0.406098\pi\)
0.290740 + 0.956802i \(0.406098\pi\)
\(200\) 98.8237 6.98789
\(201\) 0 0
\(202\) 43.4610 3.05791
\(203\) 1.45371 0.102031
\(204\) 0 0
\(205\) 26.6237 1.85948
\(206\) 30.1486 2.10055
\(207\) 0 0
\(208\) −22.8952 −1.58750
\(209\) 0 0
\(210\) 0 0
\(211\) 22.2192 1.52963 0.764816 0.644248i \(-0.222830\pi\)
0.764816 + 0.644248i \(0.222830\pi\)
\(212\) −31.9401 −2.19366
\(213\) 0 0
\(214\) −15.6647 −1.07082
\(215\) −13.6161 −0.928607
\(216\) 0 0
\(217\) 2.98816 0.202849
\(218\) 17.3967 1.17825
\(219\) 0 0
\(220\) 0 0
\(221\) 2.92397 0.196688
\(222\) 0 0
\(223\) 21.8353 1.46220 0.731100 0.682270i \(-0.239007\pi\)
0.731100 + 0.682270i \(0.239007\pi\)
\(224\) 9.71302 0.648979
\(225\) 0 0
\(226\) −48.8383 −3.24868
\(227\) −24.3708 −1.61755 −0.808774 0.588120i \(-0.799868\pi\)
−0.808774 + 0.588120i \(0.799868\pi\)
\(228\) 0 0
\(229\) −2.58137 −0.170582 −0.0852910 0.996356i \(-0.527182\pi\)
−0.0852910 + 0.996356i \(0.527182\pi\)
\(230\) −93.5696 −6.16980
\(231\) 0 0
\(232\) 10.5244 0.690962
\(233\) −14.2545 −0.933845 −0.466923 0.884298i \(-0.654637\pi\)
−0.466923 + 0.884298i \(0.654637\pi\)
\(234\) 0 0
\(235\) −37.7229 −2.46077
\(236\) 56.1919 3.65778
\(237\) 0 0
\(238\) −3.08964 −0.200272
\(239\) −11.0487 −0.714683 −0.357341 0.933974i \(-0.616317\pi\)
−0.357341 + 0.933974i \(0.616317\pi\)
\(240\) 0 0
\(241\) −11.0131 −0.709419 −0.354709 0.934977i \(-0.615420\pi\)
−0.354709 + 0.934977i \(0.615420\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 37.1387 2.37756
\(245\) −4.31860 −0.275905
\(246\) 0 0
\(247\) 8.44489 0.537335
\(248\) 21.6333 1.37372
\(249\) 0 0
\(250\) −97.2742 −6.15216
\(251\) −11.2971 −0.713065 −0.356532 0.934283i \(-0.616041\pi\)
−0.356532 + 0.934283i \(0.616041\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 8.25443 0.517929
\(255\) 0 0
\(256\) −18.5069 −1.15668
\(257\) 5.10986 0.318745 0.159372 0.987219i \(-0.449053\pi\)
0.159372 + 0.987219i \(0.449053\pi\)
\(258\) 0 0
\(259\) 6.34373 0.394180
\(260\) 50.8734 3.15503
\(261\) 0 0
\(262\) 20.3984 1.26022
\(263\) −6.56589 −0.404870 −0.202435 0.979296i \(-0.564885\pi\)
−0.202435 + 0.979296i \(0.564885\pi\)
\(264\) 0 0
\(265\) 28.8551 1.77256
\(266\) −8.92335 −0.547126
\(267\) 0 0
\(268\) 14.8706 0.908367
\(269\) −11.6093 −0.707833 −0.353916 0.935277i \(-0.615150\pi\)
−0.353916 + 0.935277i \(0.615150\pi\)
\(270\) 0 0
\(271\) −14.6728 −0.891307 −0.445654 0.895205i \(-0.647029\pi\)
−0.445654 + 0.895205i \(0.647029\pi\)
\(272\) −11.0239 −0.668424
\(273\) 0 0
\(274\) 18.7148 1.13060
\(275\) 0 0
\(276\) 0 0
\(277\) −23.2754 −1.39848 −0.699241 0.714886i \(-0.746478\pi\)
−0.699241 + 0.714886i \(0.746478\pi\)
\(278\) 8.76999 0.525989
\(279\) 0 0
\(280\) −31.2653 −1.86846
\(281\) −10.2114 −0.609160 −0.304580 0.952487i \(-0.598516\pi\)
−0.304580 + 0.952487i \(0.598516\pi\)
\(282\) 0 0
\(283\) −6.41587 −0.381384 −0.190692 0.981650i \(-0.561073\pi\)
−0.190692 + 0.981650i \(0.561073\pi\)
\(284\) 26.2325 1.55661
\(285\) 0 0
\(286\) 0 0
\(287\) −6.16490 −0.363902
\(288\) 0 0
\(289\) −15.5921 −0.917184
\(290\) −16.3473 −0.959946
\(291\) 0 0
\(292\) 41.3977 2.42262
\(293\) 15.7851 0.922174 0.461087 0.887355i \(-0.347460\pi\)
0.461087 + 0.887355i \(0.347460\pi\)
\(294\) 0 0
\(295\) −50.7644 −2.95562
\(296\) 45.9266 2.66943
\(297\) 0 0
\(298\) −48.4937 −2.80916
\(299\) −20.5049 −1.18583
\(300\) 0 0
\(301\) 3.15289 0.181730
\(302\) 37.5600 2.16134
\(303\) 0 0
\(304\) −31.8388 −1.82608
\(305\) −33.5515 −1.92116
\(306\) 0 0
\(307\) 17.4204 0.994235 0.497117 0.867683i \(-0.334392\pi\)
0.497117 + 0.867683i \(0.334392\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −33.6025 −1.90849
\(311\) 13.1950 0.748221 0.374110 0.927384i \(-0.377948\pi\)
0.374110 + 0.927384i \(0.377948\pi\)
\(312\) 0 0
\(313\) 18.0139 1.01821 0.509104 0.860705i \(-0.329977\pi\)
0.509104 + 0.860705i \(0.329977\pi\)
\(314\) 13.8386 0.780959
\(315\) 0 0
\(316\) −42.2371 −2.37602
\(317\) −6.70435 −0.376554 −0.188277 0.982116i \(-0.560290\pi\)
−0.188277 + 0.982116i \(0.560290\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −28.9784 −1.61994
\(321\) 0 0
\(322\) 21.6667 1.20744
\(323\) 4.06617 0.226248
\(324\) 0 0
\(325\) −33.6382 −1.86591
\(326\) 2.36000 0.130708
\(327\) 0 0
\(328\) −44.6319 −2.46439
\(329\) 8.73499 0.481575
\(330\) 0 0
\(331\) 3.96252 0.217800 0.108900 0.994053i \(-0.465267\pi\)
0.108900 + 0.994053i \(0.465267\pi\)
\(332\) −46.1952 −2.53529
\(333\) 0 0
\(334\) 30.3183 1.65894
\(335\) −13.4343 −0.733994
\(336\) 0 0
\(337\) −12.2604 −0.667866 −0.333933 0.942597i \(-0.608376\pi\)
−0.333933 + 0.942597i \(0.608376\pi\)
\(338\) −18.0380 −0.981139
\(339\) 0 0
\(340\) 24.4953 1.32844
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 22.8259 1.23069
\(345\) 0 0
\(346\) −17.8378 −0.958968
\(347\) −11.4238 −0.613261 −0.306630 0.951829i \(-0.599202\pi\)
−0.306630 + 0.951829i \(0.599202\pi\)
\(348\) 0 0
\(349\) −29.2338 −1.56485 −0.782425 0.622745i \(-0.786017\pi\)
−0.782425 + 0.622745i \(0.786017\pi\)
\(350\) 35.5440 1.89991
\(351\) 0 0
\(352\) 0 0
\(353\) 7.76081 0.413066 0.206533 0.978440i \(-0.433782\pi\)
0.206533 + 0.978440i \(0.433782\pi\)
\(354\) 0 0
\(355\) −23.6987 −1.25780
\(356\) 4.71128 0.249697
\(357\) 0 0
\(358\) −21.9056 −1.15775
\(359\) 29.9693 1.58172 0.790860 0.611997i \(-0.209634\pi\)
0.790860 + 0.611997i \(0.209634\pi\)
\(360\) 0 0
\(361\) −7.25627 −0.381909
\(362\) 33.9259 1.78310
\(363\) 0 0
\(364\) −11.7801 −0.617443
\(365\) −37.3992 −1.95756
\(366\) 0 0
\(367\) −3.08710 −0.161145 −0.0805726 0.996749i \(-0.525675\pi\)
−0.0805726 + 0.996749i \(0.525675\pi\)
\(368\) 77.3073 4.02992
\(369\) 0 0
\(370\) −71.3366 −3.70862
\(371\) −6.68159 −0.346891
\(372\) 0 0
\(373\) 29.2948 1.51683 0.758413 0.651774i \(-0.225975\pi\)
0.758413 + 0.651774i \(0.225975\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 63.2386 3.26128
\(377\) −3.58236 −0.184501
\(378\) 0 0
\(379\) 15.6869 0.805784 0.402892 0.915247i \(-0.368005\pi\)
0.402892 + 0.915247i \(0.368005\pi\)
\(380\) 70.7461 3.62920
\(381\) 0 0
\(382\) 9.00309 0.460638
\(383\) 8.03590 0.410615 0.205308 0.978697i \(-0.434180\pi\)
0.205308 + 0.978697i \(0.434180\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 44.3894 2.25936
\(387\) 0 0
\(388\) 33.3302 1.69208
\(389\) −28.7874 −1.45958 −0.729789 0.683672i \(-0.760382\pi\)
−0.729789 + 0.683672i \(0.760382\pi\)
\(390\) 0 0
\(391\) −9.87301 −0.499300
\(392\) 7.23969 0.365659
\(393\) 0 0
\(394\) −9.06080 −0.456477
\(395\) 38.1575 1.91991
\(396\) 0 0
\(397\) −33.7315 −1.69293 −0.846467 0.532441i \(-0.821275\pi\)
−0.846467 + 0.532441i \(0.821275\pi\)
\(398\) 21.3593 1.07064
\(399\) 0 0
\(400\) 126.822 6.34111
\(401\) 0.924552 0.0461699 0.0230850 0.999734i \(-0.492651\pi\)
0.0230850 + 0.999734i \(0.492651\pi\)
\(402\) 0 0
\(403\) −7.36367 −0.366810
\(404\) 79.7869 3.96955
\(405\) 0 0
\(406\) 3.78533 0.187863
\(407\) 0 0
\(408\) 0 0
\(409\) 25.7923 1.27535 0.637673 0.770307i \(-0.279897\pi\)
0.637673 + 0.770307i \(0.279897\pi\)
\(410\) 69.3256 3.42375
\(411\) 0 0
\(412\) 55.3477 2.72678
\(413\) 11.7548 0.578418
\(414\) 0 0
\(415\) 41.7333 2.04861
\(416\) −23.9357 −1.17354
\(417\) 0 0
\(418\) 0 0
\(419\) 9.41300 0.459855 0.229928 0.973208i \(-0.426151\pi\)
0.229928 + 0.973208i \(0.426151\pi\)
\(420\) 0 0
\(421\) −29.3045 −1.42821 −0.714107 0.700037i \(-0.753167\pi\)
−0.714107 + 0.700037i \(0.753167\pi\)
\(422\) 57.8567 2.81642
\(423\) 0 0
\(424\) −48.3726 −2.34918
\(425\) −16.1966 −0.785651
\(426\) 0 0
\(427\) 7.76909 0.375972
\(428\) −28.7577 −1.39006
\(429\) 0 0
\(430\) −35.4549 −1.70979
\(431\) 6.97670 0.336056 0.168028 0.985782i \(-0.446260\pi\)
0.168028 + 0.985782i \(0.446260\pi\)
\(432\) 0 0
\(433\) 1.32914 0.0638746 0.0319373 0.999490i \(-0.489832\pi\)
0.0319373 + 0.999490i \(0.489832\pi\)
\(434\) 7.78087 0.373494
\(435\) 0 0
\(436\) 31.9374 1.52952
\(437\) −28.5148 −1.36405
\(438\) 0 0
\(439\) −3.63801 −0.173633 −0.0868163 0.996224i \(-0.527669\pi\)
−0.0868163 + 0.996224i \(0.527669\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 7.61375 0.362149
\(443\) 13.4099 0.637125 0.318563 0.947902i \(-0.396800\pi\)
0.318563 + 0.947902i \(0.396800\pi\)
\(444\) 0 0
\(445\) −4.25623 −0.201765
\(446\) 56.8570 2.69226
\(447\) 0 0
\(448\) 6.71015 0.317025
\(449\) 14.9993 0.707859 0.353929 0.935272i \(-0.384845\pi\)
0.353929 + 0.935272i \(0.384845\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −89.6587 −4.21719
\(453\) 0 0
\(454\) −63.4593 −2.97829
\(455\) 10.6423 0.498917
\(456\) 0 0
\(457\) −19.7121 −0.922094 −0.461047 0.887376i \(-0.652526\pi\)
−0.461047 + 0.887376i \(0.652526\pi\)
\(458\) −6.72165 −0.314082
\(459\) 0 0
\(460\) −171.778 −8.00917
\(461\) 16.3627 0.762086 0.381043 0.924557i \(-0.375565\pi\)
0.381043 + 0.924557i \(0.375565\pi\)
\(462\) 0 0
\(463\) −7.48587 −0.347898 −0.173949 0.984755i \(-0.555653\pi\)
−0.173949 + 0.984755i \(0.555653\pi\)
\(464\) 13.5062 0.627008
\(465\) 0 0
\(466\) −37.1174 −1.71943
\(467\) −2.91022 −0.134669 −0.0673344 0.997730i \(-0.521449\pi\)
−0.0673344 + 0.997730i \(0.521449\pi\)
\(468\) 0 0
\(469\) 3.11080 0.143643
\(470\) −98.2268 −4.53086
\(471\) 0 0
\(472\) 85.1014 3.91711
\(473\) 0 0
\(474\) 0 0
\(475\) −46.7783 −2.14634
\(476\) −5.67204 −0.259978
\(477\) 0 0
\(478\) −28.7698 −1.31590
\(479\) −26.5272 −1.21206 −0.606030 0.795442i \(-0.707239\pi\)
−0.606030 + 0.795442i \(0.707239\pi\)
\(480\) 0 0
\(481\) −15.6328 −0.712793
\(482\) −28.6772 −1.30621
\(483\) 0 0
\(484\) 0 0
\(485\) −30.1109 −1.36727
\(486\) 0 0
\(487\) −2.81722 −0.127661 −0.0638303 0.997961i \(-0.520332\pi\)
−0.0638303 + 0.997961i \(0.520332\pi\)
\(488\) 56.2457 2.54613
\(489\) 0 0
\(490\) −11.2452 −0.508007
\(491\) 5.67888 0.256284 0.128142 0.991756i \(-0.459099\pi\)
0.128142 + 0.991756i \(0.459099\pi\)
\(492\) 0 0
\(493\) −1.72489 −0.0776850
\(494\) 21.9897 0.989363
\(495\) 0 0
\(496\) 27.7624 1.24657
\(497\) 5.48760 0.246152
\(498\) 0 0
\(499\) 35.2949 1.58002 0.790008 0.613096i \(-0.210076\pi\)
0.790008 + 0.613096i \(0.210076\pi\)
\(500\) −178.579 −7.98628
\(501\) 0 0
\(502\) −29.4165 −1.31292
\(503\) 29.6610 1.32252 0.661259 0.750158i \(-0.270022\pi\)
0.661259 + 0.750158i \(0.270022\pi\)
\(504\) 0 0
\(505\) −72.0805 −3.20754
\(506\) 0 0
\(507\) 0 0
\(508\) 15.1537 0.672337
\(509\) 4.71393 0.208941 0.104471 0.994528i \(-0.466685\pi\)
0.104471 + 0.994528i \(0.466685\pi\)
\(510\) 0 0
\(511\) 8.66003 0.383097
\(512\) −44.2833 −1.95706
\(513\) 0 0
\(514\) 13.3056 0.586885
\(515\) −50.0018 −2.20334
\(516\) 0 0
\(517\) 0 0
\(518\) 16.5185 0.725780
\(519\) 0 0
\(520\) 77.0466 3.37872
\(521\) −22.6541 −0.992495 −0.496247 0.868181i \(-0.665289\pi\)
−0.496247 + 0.868181i \(0.665289\pi\)
\(522\) 0 0
\(523\) −15.0536 −0.658247 −0.329124 0.944287i \(-0.606753\pi\)
−0.329124 + 0.944287i \(0.606753\pi\)
\(524\) 37.4479 1.63592
\(525\) 0 0
\(526\) −17.0969 −0.745462
\(527\) −3.54557 −0.154447
\(528\) 0 0
\(529\) 46.2363 2.01027
\(530\) 75.1359 3.26370
\(531\) 0 0
\(532\) −16.3817 −0.710238
\(533\) 15.1921 0.658041
\(534\) 0 0
\(535\) 25.9801 1.12322
\(536\) 22.5212 0.972768
\(537\) 0 0
\(538\) −30.2296 −1.30329
\(539\) 0 0
\(540\) 0 0
\(541\) 15.5738 0.669570 0.334785 0.942295i \(-0.391336\pi\)
0.334785 + 0.942295i \(0.391336\pi\)
\(542\) −38.2065 −1.64111
\(543\) 0 0
\(544\) −11.5249 −0.494126
\(545\) −28.8526 −1.23591
\(546\) 0 0
\(547\) 16.9696 0.725568 0.362784 0.931873i \(-0.381826\pi\)
0.362784 + 0.931873i \(0.381826\pi\)
\(548\) 34.3572 1.46767
\(549\) 0 0
\(550\) 0 0
\(551\) −4.98174 −0.212229
\(552\) 0 0
\(553\) −8.83563 −0.375729
\(554\) −60.6068 −2.57494
\(555\) 0 0
\(556\) 16.1002 0.682800
\(557\) −2.65932 −0.112679 −0.0563394 0.998412i \(-0.517943\pi\)
−0.0563394 + 0.998412i \(0.517943\pi\)
\(558\) 0 0
\(559\) −7.76962 −0.328620
\(560\) −40.1233 −1.69552
\(561\) 0 0
\(562\) −26.5895 −1.12161
\(563\) −15.0516 −0.634349 −0.317174 0.948367i \(-0.602734\pi\)
−0.317174 + 0.948367i \(0.602734\pi\)
\(564\) 0 0
\(565\) 80.9987 3.40764
\(566\) −16.7063 −0.702219
\(567\) 0 0
\(568\) 39.7285 1.66697
\(569\) 36.7508 1.54067 0.770337 0.637637i \(-0.220088\pi\)
0.770337 + 0.637637i \(0.220088\pi\)
\(570\) 0 0
\(571\) −3.56292 −0.149104 −0.0745519 0.997217i \(-0.523753\pi\)
−0.0745519 + 0.997217i \(0.523753\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −16.0528 −0.670031
\(575\) 113.582 4.73669
\(576\) 0 0
\(577\) 19.7289 0.821325 0.410663 0.911787i \(-0.365297\pi\)
0.410663 + 0.911787i \(0.365297\pi\)
\(578\) −40.6004 −1.68875
\(579\) 0 0
\(580\) −30.0108 −1.24613
\(581\) −9.66363 −0.400915
\(582\) 0 0
\(583\) 0 0
\(584\) 62.6959 2.59437
\(585\) 0 0
\(586\) 41.1028 1.69794
\(587\) −8.34576 −0.344466 −0.172233 0.985056i \(-0.555098\pi\)
−0.172233 + 0.985056i \(0.555098\pi\)
\(588\) 0 0
\(589\) −10.2402 −0.421938
\(590\) −132.186 −5.44200
\(591\) 0 0
\(592\) 58.9385 2.42236
\(593\) −15.7459 −0.646605 −0.323302 0.946296i \(-0.604793\pi\)
−0.323302 + 0.946296i \(0.604793\pi\)
\(594\) 0 0
\(595\) 5.12419 0.210071
\(596\) −89.0261 −3.64665
\(597\) 0 0
\(598\) −53.3928 −2.18340
\(599\) 0.0424211 0.00173328 0.000866640 1.00000i \(-0.499724\pi\)
0.000866640 1.00000i \(0.499724\pi\)
\(600\) 0 0
\(601\) 4.36523 0.178062 0.0890308 0.996029i \(-0.471623\pi\)
0.0890308 + 0.996029i \(0.471623\pi\)
\(602\) 8.20982 0.334607
\(603\) 0 0
\(604\) 68.9537 2.80569
\(605\) 0 0
\(606\) 0 0
\(607\) −3.46475 −0.140630 −0.0703149 0.997525i \(-0.522400\pi\)
−0.0703149 + 0.997525i \(0.522400\pi\)
\(608\) −33.2857 −1.34991
\(609\) 0 0
\(610\) −87.3650 −3.53731
\(611\) −21.5255 −0.870828
\(612\) 0 0
\(613\) 6.89445 0.278464 0.139232 0.990260i \(-0.455537\pi\)
0.139232 + 0.990260i \(0.455537\pi\)
\(614\) 45.3611 1.83062
\(615\) 0 0
\(616\) 0 0
\(617\) 13.8954 0.559409 0.279705 0.960086i \(-0.409763\pi\)
0.279705 + 0.960086i \(0.409763\pi\)
\(618\) 0 0
\(619\) −30.6734 −1.23287 −0.616434 0.787406i \(-0.711423\pi\)
−0.616434 + 0.787406i \(0.711423\pi\)
\(620\) −61.6883 −2.47746
\(621\) 0 0
\(622\) 34.3586 1.37765
\(623\) 0.985558 0.0394855
\(624\) 0 0
\(625\) 93.0787 3.72315
\(626\) 46.9066 1.87476
\(627\) 0 0
\(628\) 25.4053 1.01378
\(629\) −7.52710 −0.300125
\(630\) 0 0
\(631\) −45.2371 −1.80086 −0.900430 0.435000i \(-0.856748\pi\)
−0.900430 + 0.435000i \(0.856748\pi\)
\(632\) −63.9672 −2.54448
\(633\) 0 0
\(634\) −17.4575 −0.693325
\(635\) −13.6900 −0.543273
\(636\) 0 0
\(637\) −2.46429 −0.0976385
\(638\) 0 0
\(639\) 0 0
\(640\) 8.43614 0.333468
\(641\) 16.1033 0.636043 0.318022 0.948083i \(-0.396982\pi\)
0.318022 + 0.948083i \(0.396982\pi\)
\(642\) 0 0
\(643\) 49.3352 1.94559 0.972795 0.231668i \(-0.0744181\pi\)
0.972795 + 0.231668i \(0.0744181\pi\)
\(644\) 39.7763 1.56740
\(645\) 0 0
\(646\) 10.5879 0.416576
\(647\) −35.0628 −1.37846 −0.689231 0.724542i \(-0.742052\pi\)
−0.689231 + 0.724542i \(0.742052\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −87.5906 −3.43559
\(651\) 0 0
\(652\) 4.33256 0.169676
\(653\) −24.8559 −0.972688 −0.486344 0.873767i \(-0.661670\pi\)
−0.486344 + 0.873767i \(0.661670\pi\)
\(654\) 0 0
\(655\) −33.8309 −1.32188
\(656\) −57.2769 −2.23629
\(657\) 0 0
\(658\) 22.7451 0.886695
\(659\) 12.3854 0.482468 0.241234 0.970467i \(-0.422448\pi\)
0.241234 + 0.970467i \(0.422448\pi\)
\(660\) 0 0
\(661\) −39.9739 −1.55480 −0.777402 0.629004i \(-0.783463\pi\)
−0.777402 + 0.629004i \(0.783463\pi\)
\(662\) 10.3180 0.401021
\(663\) 0 0
\(664\) −69.9616 −2.71504
\(665\) 14.7995 0.573898
\(666\) 0 0
\(667\) 12.0961 0.468363
\(668\) 55.6592 2.15352
\(669\) 0 0
\(670\) −34.9816 −1.35146
\(671\) 0 0
\(672\) 0 0
\(673\) −8.31720 −0.320604 −0.160302 0.987068i \(-0.551247\pi\)
−0.160302 + 0.987068i \(0.551247\pi\)
\(674\) −31.9249 −1.22970
\(675\) 0 0
\(676\) −33.1147 −1.27364
\(677\) 30.7173 1.18056 0.590281 0.807198i \(-0.299017\pi\)
0.590281 + 0.807198i \(0.299017\pi\)
\(678\) 0 0
\(679\) 6.97238 0.267575
\(680\) 37.0975 1.42263
\(681\) 0 0
\(682\) 0 0
\(683\) 13.9802 0.534936 0.267468 0.963567i \(-0.413813\pi\)
0.267468 + 0.963567i \(0.413813\pi\)
\(684\) 0 0
\(685\) −31.0387 −1.18593
\(686\) 2.60390 0.0994175
\(687\) 0 0
\(688\) 29.2929 1.11678
\(689\) 16.4653 0.627280
\(690\) 0 0
\(691\) 28.3460 1.07833 0.539167 0.842199i \(-0.318739\pi\)
0.539167 + 0.842199i \(0.318739\pi\)
\(692\) −32.7472 −1.24486
\(693\) 0 0
\(694\) −29.7464 −1.12916
\(695\) −14.5451 −0.551727
\(696\) 0 0
\(697\) 7.31490 0.277072
\(698\) −76.1220 −2.88126
\(699\) 0 0
\(700\) 65.2527 2.46632
\(701\) −13.0144 −0.491546 −0.245773 0.969327i \(-0.579042\pi\)
−0.245773 + 0.969327i \(0.579042\pi\)
\(702\) 0 0
\(703\) −21.7394 −0.819918
\(704\) 0 0
\(705\) 0 0
\(706\) 20.2084 0.760553
\(707\) 16.6907 0.627719
\(708\) 0 0
\(709\) 22.5145 0.845550 0.422775 0.906235i \(-0.361056\pi\)
0.422775 + 0.906235i \(0.361056\pi\)
\(710\) −61.7092 −2.31590
\(711\) 0 0
\(712\) 7.13513 0.267400
\(713\) 24.8640 0.931163
\(714\) 0 0
\(715\) 0 0
\(716\) −40.2149 −1.50290
\(717\) 0 0
\(718\) 78.0372 2.91232
\(719\) 13.4580 0.501899 0.250950 0.968000i \(-0.419257\pi\)
0.250950 + 0.968000i \(0.419257\pi\)
\(720\) 0 0
\(721\) 11.5782 0.431196
\(722\) −18.8946 −0.703186
\(723\) 0 0
\(724\) 62.2820 2.31469
\(725\) 19.8436 0.736971
\(726\) 0 0
\(727\) −25.2856 −0.937793 −0.468896 0.883253i \(-0.655348\pi\)
−0.468896 + 0.883253i \(0.655348\pi\)
\(728\) −17.8407 −0.661219
\(729\) 0 0
\(730\) −97.3839 −3.60434
\(731\) −3.74103 −0.138367
\(732\) 0 0
\(733\) −40.8544 −1.50899 −0.754496 0.656305i \(-0.772119\pi\)
−0.754496 + 0.656305i \(0.772119\pi\)
\(734\) −8.03851 −0.296707
\(735\) 0 0
\(736\) 80.8205 2.97908
\(737\) 0 0
\(738\) 0 0
\(739\) 37.3128 1.37257 0.686287 0.727331i \(-0.259239\pi\)
0.686287 + 0.727331i \(0.259239\pi\)
\(740\) −130.962 −4.81425
\(741\) 0 0
\(742\) −17.3982 −0.638709
\(743\) −39.9831 −1.46684 −0.733418 0.679778i \(-0.762076\pi\)
−0.733418 + 0.679778i \(0.762076\pi\)
\(744\) 0 0
\(745\) 80.4272 2.94662
\(746\) 76.2808 2.79284
\(747\) 0 0
\(748\) 0 0
\(749\) −6.01586 −0.219815
\(750\) 0 0
\(751\) 29.2931 1.06892 0.534460 0.845194i \(-0.320515\pi\)
0.534460 + 0.845194i \(0.320515\pi\)
\(752\) 81.1551 2.95942
\(753\) 0 0
\(754\) −9.32812 −0.339710
\(755\) −62.2936 −2.26709
\(756\) 0 0
\(757\) −32.8416 −1.19365 −0.596824 0.802372i \(-0.703571\pi\)
−0.596824 + 0.802372i \(0.703571\pi\)
\(758\) 40.8473 1.48364
\(759\) 0 0
\(760\) 107.143 3.88650
\(761\) −43.3133 −1.57011 −0.785053 0.619429i \(-0.787364\pi\)
−0.785053 + 0.619429i \(0.787364\pi\)
\(762\) 0 0
\(763\) 6.68101 0.241869
\(764\) 16.5281 0.597966
\(765\) 0 0
\(766\) 20.9247 0.756041
\(767\) −28.9673 −1.04595
\(768\) 0 0
\(769\) 31.4543 1.13427 0.567135 0.823625i \(-0.308052\pi\)
0.567135 + 0.823625i \(0.308052\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 81.4913 2.93294
\(773\) −41.8131 −1.50391 −0.751957 0.659212i \(-0.770890\pi\)
−0.751957 + 0.659212i \(0.770890\pi\)
\(774\) 0 0
\(775\) 40.7892 1.46519
\(776\) 50.4778 1.81205
\(777\) 0 0
\(778\) −74.9596 −2.68743
\(779\) 21.1266 0.756938
\(780\) 0 0
\(781\) 0 0
\(782\) −25.7084 −0.919330
\(783\) 0 0
\(784\) 9.29081 0.331815
\(785\) −22.9515 −0.819173
\(786\) 0 0
\(787\) −46.4054 −1.65418 −0.827088 0.562073i \(-0.810004\pi\)
−0.827088 + 0.562073i \(0.810004\pi\)
\(788\) −16.6341 −0.592564
\(789\) 0 0
\(790\) 99.3585 3.53502
\(791\) −18.7558 −0.666880
\(792\) 0 0
\(793\) −19.1452 −0.679867
\(794\) −87.8336 −3.11710
\(795\) 0 0
\(796\) 39.2120 1.38983
\(797\) 35.3958 1.25378 0.626891 0.779107i \(-0.284327\pi\)
0.626891 + 0.779107i \(0.284327\pi\)
\(798\) 0 0
\(799\) −10.3644 −0.366667
\(800\) 132.585 4.68760
\(801\) 0 0
\(802\) 2.40744 0.0850098
\(803\) 0 0
\(804\) 0 0
\(805\) −35.9343 −1.26652
\(806\) −19.1743 −0.675386
\(807\) 0 0
\(808\) 120.836 4.25098
\(809\) −23.7838 −0.836193 −0.418097 0.908403i \(-0.637303\pi\)
−0.418097 + 0.908403i \(0.637303\pi\)
\(810\) 0 0
\(811\) −7.21740 −0.253437 −0.126719 0.991939i \(-0.540445\pi\)
−0.126719 + 0.991939i \(0.540445\pi\)
\(812\) 6.94920 0.243869
\(813\) 0 0
\(814\) 0 0
\(815\) −3.91408 −0.137104
\(816\) 0 0
\(817\) −10.8047 −0.378008
\(818\) 67.1606 2.34822
\(819\) 0 0
\(820\) 127.270 4.44446
\(821\) 46.4367 1.62065 0.810327 0.585978i \(-0.199290\pi\)
0.810327 + 0.585978i \(0.199290\pi\)
\(822\) 0 0
\(823\) 22.4195 0.781496 0.390748 0.920498i \(-0.372216\pi\)
0.390748 + 0.920498i \(0.372216\pi\)
\(824\) 83.8228 2.92011
\(825\) 0 0
\(826\) 30.6085 1.06501
\(827\) 30.4556 1.05904 0.529522 0.848296i \(-0.322371\pi\)
0.529522 + 0.848296i \(0.322371\pi\)
\(828\) 0 0
\(829\) −43.9403 −1.52611 −0.763055 0.646333i \(-0.776302\pi\)
−0.763055 + 0.646333i \(0.776302\pi\)
\(830\) 108.670 3.77197
\(831\) 0 0
\(832\) −16.5357 −0.573273
\(833\) −1.18654 −0.0411112
\(834\) 0 0
\(835\) −50.2832 −1.74012
\(836\) 0 0
\(837\) 0 0
\(838\) 24.5105 0.846703
\(839\) 2.80149 0.0967181 0.0483591 0.998830i \(-0.484601\pi\)
0.0483591 + 0.998830i \(0.484601\pi\)
\(840\) 0 0
\(841\) −26.8867 −0.927128
\(842\) −76.3061 −2.62968
\(843\) 0 0
\(844\) 106.215 3.65607
\(845\) 29.9162 1.02915
\(846\) 0 0
\(847\) 0 0
\(848\) −62.0774 −2.13175
\(849\) 0 0
\(850\) −42.1744 −1.44657
\(851\) 52.7852 1.80945
\(852\) 0 0
\(853\) −8.38258 −0.287014 −0.143507 0.989649i \(-0.545838\pi\)
−0.143507 + 0.989649i \(0.545838\pi\)
\(854\) 20.2300 0.692255
\(855\) 0 0
\(856\) −43.5529 −1.48861
\(857\) −53.1740 −1.81639 −0.908194 0.418550i \(-0.862539\pi\)
−0.908194 + 0.418550i \(0.862539\pi\)
\(858\) 0 0
\(859\) 39.7762 1.35715 0.678573 0.734533i \(-0.262599\pi\)
0.678573 + 0.734533i \(0.262599\pi\)
\(860\) −65.0891 −2.21952
\(861\) 0 0
\(862\) 18.1667 0.618759
\(863\) −9.11801 −0.310381 −0.155190 0.987885i \(-0.549599\pi\)
−0.155190 + 0.987885i \(0.549599\pi\)
\(864\) 0 0
\(865\) 29.5842 1.00589
\(866\) 3.46096 0.117608
\(867\) 0 0
\(868\) 14.2843 0.484842
\(869\) 0 0
\(870\) 0 0
\(871\) −7.66590 −0.259749
\(872\) 48.3684 1.63796
\(873\) 0 0
\(874\) −74.2498 −2.51154
\(875\) −37.3571 −1.26290
\(876\) 0 0
\(877\) −23.3441 −0.788274 −0.394137 0.919052i \(-0.628956\pi\)
−0.394137 + 0.919052i \(0.628956\pi\)
\(878\) −9.47303 −0.319699
\(879\) 0 0
\(880\) 0 0
\(881\) 55.3515 1.86484 0.932419 0.361379i \(-0.117694\pi\)
0.932419 + 0.361379i \(0.117694\pi\)
\(882\) 0 0
\(883\) 31.8420 1.07157 0.535784 0.844355i \(-0.320016\pi\)
0.535784 + 0.844355i \(0.320016\pi\)
\(884\) 13.9775 0.470115
\(885\) 0 0
\(886\) 34.9182 1.17310
\(887\) −35.2677 −1.18417 −0.592087 0.805874i \(-0.701696\pi\)
−0.592087 + 0.805874i \(0.701696\pi\)
\(888\) 0 0
\(889\) 3.17002 0.106319
\(890\) −11.0828 −0.371497
\(891\) 0 0
\(892\) 104.380 3.49489
\(893\) −29.9340 −1.00170
\(894\) 0 0
\(895\) 36.3306 1.21440
\(896\) −1.95344 −0.0652600
\(897\) 0 0
\(898\) 39.0566 1.30334
\(899\) 4.34392 0.144878
\(900\) 0 0
\(901\) 7.92798 0.264119
\(902\) 0 0
\(903\) 0 0
\(904\) −135.786 −4.51618
\(905\) −56.2663 −1.87036
\(906\) 0 0
\(907\) −4.60681 −0.152967 −0.0764833 0.997071i \(-0.524369\pi\)
−0.0764833 + 0.997071i \(0.524369\pi\)
\(908\) −116.500 −3.86620
\(909\) 0 0
\(910\) 27.7114 0.918624
\(911\) 6.15440 0.203904 0.101952 0.994789i \(-0.467491\pi\)
0.101952 + 0.994789i \(0.467491\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −51.3285 −1.69779
\(915\) 0 0
\(916\) −12.3398 −0.407718
\(917\) 7.83376 0.258694
\(918\) 0 0
\(919\) 19.9838 0.659204 0.329602 0.944120i \(-0.393086\pi\)
0.329602 + 0.944120i \(0.393086\pi\)
\(920\) −260.153 −8.57700
\(921\) 0 0
\(922\) 42.6069 1.40318
\(923\) −13.5230 −0.445115
\(924\) 0 0
\(925\) 86.5937 2.84718
\(926\) −19.4925 −0.640563
\(927\) 0 0
\(928\) 14.1199 0.463509
\(929\) −7.35186 −0.241207 −0.120603 0.992701i \(-0.538483\pi\)
−0.120603 + 0.992701i \(0.538483\pi\)
\(930\) 0 0
\(931\) −3.42691 −0.112313
\(932\) −68.1412 −2.23204
\(933\) 0 0
\(934\) −7.57793 −0.247957
\(935\) 0 0
\(936\) 0 0
\(937\) −22.4036 −0.731893 −0.365946 0.930636i \(-0.619255\pi\)
−0.365946 + 0.930636i \(0.619255\pi\)
\(938\) 8.10023 0.264482
\(939\) 0 0
\(940\) −180.327 −5.88163
\(941\) −4.86490 −0.158591 −0.0792957 0.996851i \(-0.525267\pi\)
−0.0792957 + 0.996851i \(0.525267\pi\)
\(942\) 0 0
\(943\) −51.2971 −1.67046
\(944\) 109.212 3.55455
\(945\) 0 0
\(946\) 0 0
\(947\) 29.6187 0.962477 0.481238 0.876590i \(-0.340187\pi\)
0.481238 + 0.876590i \(0.340187\pi\)
\(948\) 0 0
\(949\) −21.3408 −0.692751
\(950\) −121.806 −3.95192
\(951\) 0 0
\(952\) −8.59018 −0.278409
\(953\) 5.61143 0.181772 0.0908860 0.995861i \(-0.471030\pi\)
0.0908860 + 0.995861i \(0.471030\pi\)
\(954\) 0 0
\(955\) −14.9317 −0.483178
\(956\) −52.8165 −1.70821
\(957\) 0 0
\(958\) −69.0744 −2.23169
\(959\) 7.18722 0.232087
\(960\) 0 0
\(961\) −22.0709 −0.711965
\(962\) −40.7062 −1.31242
\(963\) 0 0
\(964\) −52.6463 −1.69562
\(965\) −73.6202 −2.36992
\(966\) 0 0
\(967\) −49.9946 −1.60772 −0.803858 0.594821i \(-0.797223\pi\)
−0.803858 + 0.594821i \(0.797223\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −78.4059 −2.51746
\(971\) 59.4504 1.90786 0.953928 0.300037i \(-0.0969990\pi\)
0.953928 + 0.300037i \(0.0969990\pi\)
\(972\) 0 0
\(973\) 3.36801 0.107974
\(974\) −7.33578 −0.235054
\(975\) 0 0
\(976\) 72.1811 2.31046
\(977\) 20.3326 0.650497 0.325249 0.945629i \(-0.394552\pi\)
0.325249 + 0.945629i \(0.394552\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −20.6443 −0.659457
\(981\) 0 0
\(982\) 14.7873 0.471881
\(983\) −25.0460 −0.798843 −0.399422 0.916767i \(-0.630789\pi\)
−0.399422 + 0.916767i \(0.630789\pi\)
\(984\) 0 0
\(985\) 15.0274 0.478813
\(986\) −4.49144 −0.143037
\(987\) 0 0
\(988\) 40.3693 1.28432
\(989\) 26.2347 0.834214
\(990\) 0 0
\(991\) 10.5368 0.334713 0.167356 0.985896i \(-0.446477\pi\)
0.167356 + 0.985896i \(0.446477\pi\)
\(992\) 29.0240 0.921514
\(993\) 0 0
\(994\) 14.2892 0.453225
\(995\) −35.4246 −1.12303
\(996\) 0 0
\(997\) −34.0917 −1.07970 −0.539848 0.841763i \(-0.681518\pi\)
−0.539848 + 0.841763i \(0.681518\pi\)
\(998\) 91.9046 2.90919
\(999\) 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 7623.2.a.dc.1.15 16
3.2 odd 2 inner 7623.2.a.dc.1.2 16
11.3 even 5 693.2.m.k.64.8 yes 32
11.4 even 5 693.2.m.k.379.8 yes 32
11.10 odd 2 7623.2.a.db.1.2 16
33.14 odd 10 693.2.m.k.64.1 32
33.26 odd 10 693.2.m.k.379.1 yes 32
33.32 even 2 7623.2.a.db.1.15 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.2.m.k.64.1 32 33.14 odd 10
693.2.m.k.64.8 yes 32 11.3 even 5
693.2.m.k.379.1 yes 32 33.26 odd 10
693.2.m.k.379.8 yes 32 11.4 even 5
7623.2.a.db.1.2 16 11.10 odd 2
7623.2.a.db.1.15 16 33.32 even 2
7623.2.a.dc.1.2 16 3.2 odd 2 inner
7623.2.a.dc.1.15 16 1.1 even 1 trivial