Properties

Label 7623.2.a.db.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 \(-0.0836504\pi\)
0.965668 + 0.259781i \(0.0836504\pi\)
\(6\) 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 \(-0.388986\pi\)
0.341735 + 0.939796i \(0.388986\pi\)
\(14\) −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 \(-0.371405\pi\)
0.393094 + 0.919498i \(0.371405\pi\)
\(20\) 20.6443 4.61620
\(21\) 0 0
\(22\) 0 0
\(23\) −8.32084 −1.73501 −0.867507 0.497425i \(-0.834279\pi\)
−0.867507 + 0.497425i \(0.834279\pi\)
\(24\) 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 \(-0.577280\pi\)
−0.240406 + 0.970673i \(0.577280\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −21.6667 −3.19458
\(47\) −8.73499 −1.27413 −0.637064 0.770811i \(-0.719851\pi\)
−0.637064 + 0.770811i \(0.719851\pi\)
\(48\) 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 \(-0.348246\pi\)
0.458894 + 0.888491i \(0.348246\pi\)
\(54\) 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 \(-0.777348\pi\)
−0.765175 + 0.643822i \(0.777348\pi\)
\(60\) 0 0
\(61\) −7.76909 −0.994729 −0.497365 0.867542i \(-0.665699\pi\)
−0.497365 + 0.867542i \(0.665699\pi\)
\(62\) 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 \(-0.605576\pi\)
−0.325629 + 0.945498i \(0.605576\pi\)
\(72\) 0 0
\(73\) −8.66003 −1.01358 −0.506790 0.862070i \(-0.669168\pi\)
−0.506790 + 0.862070i \(0.669168\pi\)
\(74\) 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 \(-0.334419\pi\)
0.497043 + 0.867726i \(0.334419\pi\)
\(80\) 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 \(-0.516634\pi\)
−0.0522345 + 0.998635i \(0.516634\pi\)
\(90\) 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 \(-0.603670\pi\)
−0.319962 + 0.947430i \(0.603670\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −9.29081 −0.877899
\(113\) 18.7558 1.76440 0.882199 0.470877i \(-0.156063\pi\)
0.882199 + 0.470877i \(0.156063\pi\)
\(114\) 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 \(-0.544918\pi\)
−0.140647 + 0.990060i \(0.544918\pi\)
\(128\) −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 \(-0.599333\pi\)
−0.307023 + 0.951702i \(0.599333\pi\)
\(138\) 0 0
\(139\) −3.36801 −0.285671 −0.142836 0.989746i \(-0.545622\pi\)
−0.142836 + 0.989746i \(0.545622\pi\)
\(140\) −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 \(-0.699661\pi\)
−0.586924 + 0.809642i \(0.699661\pi\)
\(152\) 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 \(-0.398199\pi\)
0.314393 + 0.949293i \(0.398199\pi\)
\(180\) 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 \(-0.539922\pi\)
−0.125089 + 0.992145i \(0.539922\pi\)
\(192\) 0 0
\(193\) −17.0472 −1.22709 −0.613544 0.789661i \(-0.710257\pi\)
−0.613544 + 0.789661i \(0.710257\pi\)
\(194\) 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 \(-0.777170\pi\)
−0.764816 + 0.644248i \(0.777170\pi\)
\(212\) 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 \(-0.384580\pi\)
0.354709 + 0.934977i \(0.384580\pi\)
\(242\) 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 \(-0.383959\pi\)
0.356532 + 0.934283i \(0.383959\pi\)
\(252\) 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 \(-0.550947\pi\)
−0.159372 + 0.987219i \(0.550947\pi\)
\(258\) 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 \(-0.384850\pi\)
0.353916 + 0.935277i \(0.384850\pi\)
\(270\) 0 0
\(271\) 14.6728 0.891307 0.445654 0.895205i \(-0.352971\pi\)
0.445654 + 0.895205i \(0.352971\pi\)
\(272\) −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 \(-0.253522\pi\)
0.699241 + 0.714886i \(0.253522\pi\)
\(278\) −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 \(-0.438927\pi\)
0.190692 + 0.981650i \(0.438927\pi\)
\(284\) −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 \(-0.665608\pi\)
−0.497117 + 0.867683i \(0.665608\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 33.6025 1.90849
\(311\) −13.1950 −0.748221 −0.374110 0.927384i \(-0.622052\pi\)
−0.374110 + 0.927384i \(0.622052\pi\)
\(312\) 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 \(-0.439710\pi\)
0.188277 + 0.982116i \(0.439710\pi\)
\(318\) 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 \(-0.391624\pi\)
0.333933 + 0.942597i \(0.391624\pi\)
\(338\) −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 \(-0.213983\pi\)
0.782425 + 0.622745i \(0.213983\pi\)
\(350\) −35.5440 −1.89991
\(351\) 0 0
\(352\) 0 0
\(353\) −7.76081 −0.413066 −0.206533 0.978440i \(-0.566218\pi\)
−0.206533 + 0.978440i \(0.566218\pi\)
\(354\) 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 \(-0.774025\pi\)
−0.758413 + 0.651774i \(0.774025\pi\)
\(374\) 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 \(-0.565820\pi\)
−0.205308 + 0.978697i \(0.565820\pi\)
\(384\) 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 \(-0.239618\pi\)
0.729789 + 0.683672i \(0.239618\pi\)
\(390\) 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 \(-0.507349\pi\)
−0.0230850 + 0.999734i \(0.507349\pi\)
\(402\) 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 \(-0.720103\pi\)
−0.637673 + 0.770307i \(0.720103\pi\)
\(410\) −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 \(-0.573849\pi\)
−0.229928 + 0.973208i \(0.573849\pi\)
\(420\) 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 \(-0.472331\pi\)
0.0868163 + 0.996224i \(0.472331\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −7.61375 −0.362149
\(443\) −13.4099 −0.637125 −0.318563 0.947902i \(-0.603200\pi\)
−0.318563 + 0.947902i \(0.603200\pi\)
\(444\) 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 \(-0.615155\pi\)
−0.353929 + 0.935272i \(0.615155\pi\)
\(450\) 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 \(-0.347474\pi\)
0.461047 + 0.887376i \(0.347474\pi\)
\(458\) −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 \(-0.478551\pi\)
0.0673344 + 0.997730i \(0.478551\pi\)
\(468\) 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 \(-0.533315\pi\)
−0.104471 + 0.994528i \(0.533315\pi\)
\(510\) 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 \(-0.334711\pi\)
0.496247 + 0.868181i \(0.334711\pi\)
\(522\) 0 0
\(523\) 15.0536 0.658247 0.329124 0.944287i \(-0.393247\pi\)
0.329124 + 0.944287i \(0.393247\pi\)
\(524\) 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 \(-0.608664\pi\)
−0.334785 + 0.942295i \(0.608664\pi\)
\(542\) 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 \(-0.618174\pi\)
−0.362784 + 0.931873i \(0.618174\pi\)
\(548\) −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 \(-0.476247\pi\)
0.0745519 + 0.997217i \(0.476247\pi\)
\(572\) 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 \(-0.444902\pi\)
0.172233 + 0.985056i \(0.444902\pi\)
\(588\) 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 \(-0.500276\pi\)
−0.000866640 1.00000i \(0.500276\pi\)
\(600\) 0 0
\(601\) −4.36523 −0.178062 −0.0890308 0.996029i \(-0.528377\pi\)
−0.0890308 + 0.996029i \(0.528377\pi\)
\(602\) 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 \(-0.477600\pi\)
0.0703149 + 0.997525i \(0.477600\pi\)
\(608\) 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 \(-0.544463\pi\)
−0.139232 + 0.990260i \(0.544463\pi\)
\(614\) −45.3611 −1.83062
\(615\) 0 0
\(616\) 0 0
\(617\) −13.8954 −0.559409 −0.279705 0.960086i \(-0.590237\pi\)
−0.279705 + 0.960086i \(0.590237\pi\)
\(618\) 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 \(-0.603018\pi\)
−0.318022 + 0.948083i \(0.603018\pi\)
\(642\) 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 \(-0.257948\pi\)
0.689231 + 0.724542i \(0.257948\pi\)
\(648\) 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 \(-0.338330\pi\)
0.486344 + 0.873767i \(0.338330\pi\)
\(654\) 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 \(-0.448753\pi\)
0.160302 + 0.987068i \(0.448753\pi\)
\(674\) 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 \(-0.586187\pi\)
−0.267468 + 0.963567i \(0.586187\pi\)
\(684\) 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 \(-0.580743\pi\)
−0.250950 + 0.968000i \(0.580743\pi\)
\(720\) 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 \(-0.227881\pi\)
0.754496 + 0.656305i \(0.227881\pi\)
\(734\) −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 \(-0.740761\pi\)
−0.686287 + 0.727331i \(0.740761\pi\)
\(740\) 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 \(-0.691948\pi\)
−0.567135 + 0.823625i \(0.691948\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −81.4913 −2.93294
\(773\) 41.8131 1.50391 0.751957 0.659212i \(-0.229110\pi\)
0.751957 + 0.659212i \(0.229110\pi\)
\(774\) 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 \(-0.189996\pi\)
0.827088 + 0.562073i \(0.189996\pi\)
\(788\) −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 \(-0.715673\pi\)
−0.626891 + 0.779107i \(0.715673\pi\)
\(798\) 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 \(-0.459555\pi\)
0.126719 + 0.991939i \(0.459555\pi\)
\(812\) −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 \(-0.515399\pi\)
−0.0483591 + 0.998830i \(0.515399\pi\)
\(840\) 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 \(-0.454162\pi\)
0.143507 + 0.989649i \(0.454162\pi\)
\(854\) 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 \(-0.450401\pi\)
0.155190 + 0.987885i \(0.450401\pi\)
\(864\) 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 \(-0.371044\pi\)
0.394137 + 0.919052i \(0.371044\pi\)
\(878\) 9.47303 0.319699
\(879\) 0 0
\(880\) 0 0
\(881\) −55.3515 −1.86484 −0.932419 0.361379i \(-0.882306\pi\)
−0.932419 + 0.361379i \(0.882306\pi\)
\(882\) 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 \(-0.532509\pi\)
−0.101952 + 0.994789i \(0.532509\pi\)
\(912\) 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 \(-0.606914\pi\)
−0.329602 + 0.944120i \(0.606914\pi\)
\(920\) −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 \(-0.461517\pi\)
0.120603 + 0.992701i \(0.461517\pi\)
\(930\) 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 \(-0.380745\pi\)
0.365946 + 0.930636i \(0.380745\pi\)
\(938\) −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 \(-0.659813\pi\)
−0.481238 + 0.876590i \(0.659813\pi\)
\(948\) 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 \(-0.202777\pi\)
0.803858 + 0.594821i \(0.202777\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 78.4059 2.51746
\(971\) −59.4504 −1.90786 −0.953928 0.300037i \(-0.903001\pi\)
−0.953928 + 0.300037i \(0.903001\pi\)
\(972\) 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 \(-0.605448\pi\)
−0.325249 + 0.945629i \(0.605448\pi\)
\(978\) 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 \(-0.369211\pi\)
0.399422 + 0.916767i \(0.369211\pi\)
\(984\) 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 \(-0.318482\pi\)
0.539848 + 0.841763i \(0.318482\pi\)
\(998\) 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.db.1.15 16
3.2 odd 2 inner 7623.2.a.db.1.2 16
11.7 odd 10 693.2.m.k.379.1 yes 32
11.8 odd 10 693.2.m.k.64.1 32
11.10 odd 2 7623.2.a.dc.1.2 16
33.8 even 10 693.2.m.k.64.8 yes 32
33.29 even 10 693.2.m.k.379.8 yes 32
33.32 even 2 7623.2.a.dc.1.15 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.2.m.k.64.1 32 11.8 odd 10
693.2.m.k.64.8 yes 32 33.8 even 10
693.2.m.k.379.1 yes 32 11.7 odd 10
693.2.m.k.379.8 yes 32 33.29 even 10
7623.2.a.db.1.2 16 3.2 odd 2 inner
7623.2.a.db.1.15 16 1.1 even 1 trivial
7623.2.a.dc.1.2 16 11.10 odd 2
7623.2.a.dc.1.15 16 33.32 even 2