Properties

Label 7623.2.a.dc.1.11
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 1.11
Root \(0.899827\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.899827 q^{2} -1.19031 q^{4} -0.803961 q^{5} +1.00000 q^{7} -2.87073 q^{8} +O(q^{10})\) \(q+0.899827 q^{2} -1.19031 q^{4} -0.803961 q^{5} +1.00000 q^{7} -2.87073 q^{8} -0.723426 q^{10} +2.94705 q^{13} +0.899827 q^{14} -0.202537 q^{16} +3.30399 q^{17} +7.93896 q^{19} +0.956964 q^{20} -7.53481 q^{23} -4.35365 q^{25} +2.65184 q^{26} -1.19031 q^{28} -0.234450 q^{29} +3.98395 q^{31} +5.55921 q^{32} +2.97302 q^{34} -0.803961 q^{35} -0.536441 q^{37} +7.14369 q^{38} +2.30795 q^{40} -4.78449 q^{41} +3.16645 q^{43} -6.78003 q^{46} -11.0114 q^{47} +1.00000 q^{49} -3.91753 q^{50} -3.50791 q^{52} +11.7317 q^{53} -2.87073 q^{56} -0.210965 q^{58} -5.25351 q^{59} +3.65442 q^{61} +3.58486 q^{62} +5.40740 q^{64} -2.36931 q^{65} -13.6736 q^{67} -3.93278 q^{68} -0.723426 q^{70} -10.3371 q^{71} +8.90937 q^{73} -0.482705 q^{74} -9.44983 q^{76} -8.37651 q^{79} +0.162832 q^{80} -4.30522 q^{82} -7.26234 q^{83} -2.65628 q^{85} +2.84925 q^{86} +8.03460 q^{89} +2.94705 q^{91} +8.96877 q^{92} -9.90831 q^{94} -6.38261 q^{95} +19.2711 q^{97} +0.899827 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 20q^{4} + 16q^{7} + O(q^{10}) \) \( 16q + 20q^{4} + 16q^{7} - 6q^{10} + 32q^{16} + 10q^{19} + 44q^{25} + 20q^{28} + 30q^{31} + 12q^{34} + 38q^{37} - 68q^{40} + 16q^{43} + 80q^{46} + 16q^{49} + 2q^{52} + 18q^{58} - 28q^{61} + 34q^{64} + 52q^{67} - 6q^{70} - 14q^{73} - 14q^{76} + 54q^{79} + 64q^{82} + 30q^{85} - 60q^{94} + 108q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.899827 0.636274 0.318137 0.948045i \(-0.396943\pi\)
0.318137 + 0.948045i \(0.396943\pi\)
\(3\) 0 0
\(4\) −1.19031 −0.595156
\(5\) −0.803961 −0.359542 −0.179771 0.983708i \(-0.557536\pi\)
−0.179771 + 0.983708i \(0.557536\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −2.87073 −1.01496
\(9\) 0 0
\(10\) −0.723426 −0.228767
\(11\) 0 0
\(12\) 0 0
\(13\) 2.94705 0.817365 0.408682 0.912677i \(-0.365988\pi\)
0.408682 + 0.912677i \(0.365988\pi\)
\(14\) 0.899827 0.240489
\(15\) 0 0
\(16\) −0.202537 −0.0506344
\(17\) 3.30399 0.801336 0.400668 0.916223i \(-0.368778\pi\)
0.400668 + 0.916223i \(0.368778\pi\)
\(18\) 0 0
\(19\) 7.93896 1.82132 0.910661 0.413155i \(-0.135573\pi\)
0.910661 + 0.413155i \(0.135573\pi\)
\(20\) 0.956964 0.213984
\(21\) 0 0
\(22\) 0 0
\(23\) −7.53481 −1.57112 −0.785558 0.618788i \(-0.787624\pi\)
−0.785558 + 0.618788i \(0.787624\pi\)
\(24\) 0 0
\(25\) −4.35365 −0.870729
\(26\) 2.65184 0.520068
\(27\) 0 0
\(28\) −1.19031 −0.224948
\(29\) −0.234450 −0.0435364 −0.0217682 0.999763i \(-0.506930\pi\)
−0.0217682 + 0.999763i \(0.506930\pi\)
\(30\) 0 0
\(31\) 3.98395 0.715538 0.357769 0.933810i \(-0.383538\pi\)
0.357769 + 0.933810i \(0.383538\pi\)
\(32\) 5.55921 0.982738
\(33\) 0 0
\(34\) 2.97302 0.509869
\(35\) −0.803961 −0.135894
\(36\) 0 0
\(37\) −0.536441 −0.0881904 −0.0440952 0.999027i \(-0.514040\pi\)
−0.0440952 + 0.999027i \(0.514040\pi\)
\(38\) 7.14369 1.15886
\(39\) 0 0
\(40\) 2.30795 0.364920
\(41\) −4.78449 −0.747212 −0.373606 0.927587i \(-0.621879\pi\)
−0.373606 + 0.927587i \(0.621879\pi\)
\(42\) 0 0
\(43\) 3.16645 0.482878 0.241439 0.970416i \(-0.422381\pi\)
0.241439 + 0.970416i \(0.422381\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −6.78003 −0.999660
\(47\) −11.0114 −1.60617 −0.803085 0.595864i \(-0.796810\pi\)
−0.803085 + 0.595864i \(0.796810\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −3.91753 −0.554022
\(51\) 0 0
\(52\) −3.50791 −0.486459
\(53\) 11.7317 1.61148 0.805738 0.592273i \(-0.201769\pi\)
0.805738 + 0.592273i \(0.201769\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.87073 −0.383617
\(57\) 0 0
\(58\) −0.210965 −0.0277011
\(59\) −5.25351 −0.683949 −0.341974 0.939709i \(-0.611096\pi\)
−0.341974 + 0.939709i \(0.611096\pi\)
\(60\) 0 0
\(61\) 3.65442 0.467901 0.233950 0.972249i \(-0.424835\pi\)
0.233950 + 0.972249i \(0.424835\pi\)
\(62\) 3.58486 0.455278
\(63\) 0 0
\(64\) 5.40740 0.675925
\(65\) −2.36931 −0.293877
\(66\) 0 0
\(67\) −13.6736 −1.67050 −0.835250 0.549871i \(-0.814677\pi\)
−0.835250 + 0.549871i \(0.814677\pi\)
\(68\) −3.93278 −0.476920
\(69\) 0 0
\(70\) −0.723426 −0.0864660
\(71\) −10.3371 −1.22678 −0.613392 0.789779i \(-0.710195\pi\)
−0.613392 + 0.789779i \(0.710195\pi\)
\(72\) 0 0
\(73\) 8.90937 1.04276 0.521382 0.853324i \(-0.325417\pi\)
0.521382 + 0.853324i \(0.325417\pi\)
\(74\) −0.482705 −0.0561133
\(75\) 0 0
\(76\) −9.44983 −1.08397
\(77\) 0 0
\(78\) 0 0
\(79\) −8.37651 −0.942431 −0.471215 0.882018i \(-0.656185\pi\)
−0.471215 + 0.882018i \(0.656185\pi\)
\(80\) 0.162832 0.0182052
\(81\) 0 0
\(82\) −4.30522 −0.475432
\(83\) −7.26234 −0.797145 −0.398573 0.917137i \(-0.630494\pi\)
−0.398573 + 0.917137i \(0.630494\pi\)
\(84\) 0 0
\(85\) −2.65628 −0.288114
\(86\) 2.84925 0.307243
\(87\) 0 0
\(88\) 0 0
\(89\) 8.03460 0.851666 0.425833 0.904802i \(-0.359981\pi\)
0.425833 + 0.904802i \(0.359981\pi\)
\(90\) 0 0
\(91\) 2.94705 0.308935
\(92\) 8.96877 0.935058
\(93\) 0 0
\(94\) −9.90831 −1.02196
\(95\) −6.38261 −0.654842
\(96\) 0 0
\(97\) 19.2711 1.95669 0.978343 0.206992i \(-0.0663675\pi\)
0.978343 + 0.206992i \(0.0663675\pi\)
\(98\) 0.899827 0.0908963
\(99\) 0 0
\(100\) 5.18219 0.518219
\(101\) 17.2794 1.71936 0.859682 0.510830i \(-0.170662\pi\)
0.859682 + 0.510830i \(0.170662\pi\)
\(102\) 0 0
\(103\) 14.3573 1.41467 0.707334 0.706880i \(-0.249898\pi\)
0.707334 + 0.706880i \(0.249898\pi\)
\(104\) −8.46018 −0.829589
\(105\) 0 0
\(106\) 10.5565 1.02534
\(107\) 15.3202 1.48106 0.740529 0.672024i \(-0.234575\pi\)
0.740529 + 0.672024i \(0.234575\pi\)
\(108\) 0 0
\(109\) −5.50237 −0.527032 −0.263516 0.964655i \(-0.584882\pi\)
−0.263516 + 0.964655i \(0.584882\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.202537 −0.0191380
\(113\) −6.69477 −0.629791 −0.314895 0.949126i \(-0.601969\pi\)
−0.314895 + 0.949126i \(0.601969\pi\)
\(114\) 0 0
\(115\) 6.05769 0.564883
\(116\) 0.279069 0.0259109
\(117\) 0 0
\(118\) −4.72725 −0.435179
\(119\) 3.30399 0.302877
\(120\) 0 0
\(121\) 0 0
\(122\) 3.28835 0.297713
\(123\) 0 0
\(124\) −4.74214 −0.425856
\(125\) 7.51997 0.672607
\(126\) 0 0
\(127\) −11.5436 −1.02433 −0.512165 0.858887i \(-0.671156\pi\)
−0.512165 + 0.858887i \(0.671156\pi\)
\(128\) −6.25269 −0.552665
\(129\) 0 0
\(130\) −2.13197 −0.186986
\(131\) 21.2008 1.85232 0.926161 0.377128i \(-0.123088\pi\)
0.926161 + 0.377128i \(0.123088\pi\)
\(132\) 0 0
\(133\) 7.93896 0.688395
\(134\) −12.3039 −1.06290
\(135\) 0 0
\(136\) −9.48487 −0.813321
\(137\) 0.815166 0.0696443 0.0348222 0.999394i \(-0.488914\pi\)
0.0348222 + 0.999394i \(0.488914\pi\)
\(138\) 0 0
\(139\) 9.27263 0.786495 0.393247 0.919433i \(-0.371352\pi\)
0.393247 + 0.919433i \(0.371352\pi\)
\(140\) 0.956964 0.0808782
\(141\) 0 0
\(142\) −9.30157 −0.780570
\(143\) 0 0
\(144\) 0 0
\(145\) 0.188489 0.0156532
\(146\) 8.01690 0.663483
\(147\) 0 0
\(148\) 0.638532 0.0524870
\(149\) 6.94847 0.569241 0.284621 0.958640i \(-0.408132\pi\)
0.284621 + 0.958640i \(0.408132\pi\)
\(150\) 0 0
\(151\) 14.0283 1.14161 0.570803 0.821087i \(-0.306632\pi\)
0.570803 + 0.821087i \(0.306632\pi\)
\(152\) −22.7906 −1.84856
\(153\) 0 0
\(154\) 0 0
\(155\) −3.20294 −0.257266
\(156\) 0 0
\(157\) 0.733382 0.0585302 0.0292651 0.999572i \(-0.490683\pi\)
0.0292651 + 0.999572i \(0.490683\pi\)
\(158\) −7.53741 −0.599644
\(159\) 0 0
\(160\) −4.46939 −0.353336
\(161\) −7.53481 −0.593826
\(162\) 0 0
\(163\) 2.96358 0.232125 0.116063 0.993242i \(-0.462973\pi\)
0.116063 + 0.993242i \(0.462973\pi\)
\(164\) 5.69504 0.444708
\(165\) 0 0
\(166\) −6.53485 −0.507203
\(167\) 8.04008 0.622160 0.311080 0.950384i \(-0.399309\pi\)
0.311080 + 0.950384i \(0.399309\pi\)
\(168\) 0 0
\(169\) −4.31490 −0.331915
\(170\) −2.39020 −0.183320
\(171\) 0 0
\(172\) −3.76906 −0.287388
\(173\) 5.15560 0.391973 0.195986 0.980607i \(-0.437209\pi\)
0.195986 + 0.980607i \(0.437209\pi\)
\(174\) 0 0
\(175\) −4.35365 −0.329105
\(176\) 0 0
\(177\) 0 0
\(178\) 7.22975 0.541893
\(179\) 6.01273 0.449412 0.224706 0.974427i \(-0.427858\pi\)
0.224706 + 0.974427i \(0.427858\pi\)
\(180\) 0 0
\(181\) 2.45046 0.182141 0.0910707 0.995844i \(-0.470971\pi\)
0.0910707 + 0.995844i \(0.470971\pi\)
\(182\) 2.65184 0.196567
\(183\) 0 0
\(184\) 21.6304 1.59461
\(185\) 0.431278 0.0317082
\(186\) 0 0
\(187\) 0 0
\(188\) 13.1069 0.955922
\(189\) 0 0
\(190\) −5.74325 −0.416659
\(191\) −21.2746 −1.53938 −0.769689 0.638419i \(-0.779589\pi\)
−0.769689 + 0.638419i \(0.779589\pi\)
\(192\) 0 0
\(193\) −8.39924 −0.604590 −0.302295 0.953214i \(-0.597753\pi\)
−0.302295 + 0.953214i \(0.597753\pi\)
\(194\) 17.3407 1.24499
\(195\) 0 0
\(196\) −1.19031 −0.0850222
\(197\) 10.9646 0.781194 0.390597 0.920562i \(-0.372269\pi\)
0.390597 + 0.920562i \(0.372269\pi\)
\(198\) 0 0
\(199\) 6.26130 0.443852 0.221926 0.975064i \(-0.428766\pi\)
0.221926 + 0.975064i \(0.428766\pi\)
\(200\) 12.4981 0.883752
\(201\) 0 0
\(202\) 15.5485 1.09399
\(203\) −0.234450 −0.0164552
\(204\) 0 0
\(205\) 3.84655 0.268655
\(206\) 12.9191 0.900116
\(207\) 0 0
\(208\) −0.596888 −0.0413867
\(209\) 0 0
\(210\) 0 0
\(211\) 16.7203 1.15107 0.575537 0.817776i \(-0.304793\pi\)
0.575537 + 0.817776i \(0.304793\pi\)
\(212\) −13.9644 −0.959078
\(213\) 0 0
\(214\) 13.7855 0.942359
\(215\) −2.54570 −0.173615
\(216\) 0 0
\(217\) 3.98395 0.270448
\(218\) −4.95118 −0.335337
\(219\) 0 0
\(220\) 0 0
\(221\) 9.73704 0.654984
\(222\) 0 0
\(223\) 13.0968 0.877026 0.438513 0.898725i \(-0.355505\pi\)
0.438513 + 0.898725i \(0.355505\pi\)
\(224\) 5.55921 0.371440
\(225\) 0 0
\(226\) −6.02413 −0.400719
\(227\) −15.9025 −1.05548 −0.527742 0.849405i \(-0.676961\pi\)
−0.527742 + 0.849405i \(0.676961\pi\)
\(228\) 0 0
\(229\) 9.01594 0.595790 0.297895 0.954599i \(-0.403715\pi\)
0.297895 + 0.954599i \(0.403715\pi\)
\(230\) 5.45088 0.359420
\(231\) 0 0
\(232\) 0.673044 0.0441875
\(233\) 2.54200 0.166532 0.0832659 0.996527i \(-0.473465\pi\)
0.0832659 + 0.996527i \(0.473465\pi\)
\(234\) 0 0
\(235\) 8.85270 0.577487
\(236\) 6.25331 0.407056
\(237\) 0 0
\(238\) 2.97302 0.192713
\(239\) 19.1749 1.24032 0.620161 0.784475i \(-0.287067\pi\)
0.620161 + 0.784475i \(0.287067\pi\)
\(240\) 0 0
\(241\) −7.50063 −0.483158 −0.241579 0.970381i \(-0.577665\pi\)
−0.241579 + 0.970381i \(0.577665\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −4.34990 −0.278474
\(245\) −0.803961 −0.0513632
\(246\) 0 0
\(247\) 23.3965 1.48868
\(248\) −11.4368 −0.726239
\(249\) 0 0
\(250\) 6.76667 0.427962
\(251\) 14.6903 0.927245 0.463622 0.886033i \(-0.346549\pi\)
0.463622 + 0.886033i \(0.346549\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −10.3872 −0.651754
\(255\) 0 0
\(256\) −16.4411 −1.02757
\(257\) 20.7501 1.29436 0.647178 0.762339i \(-0.275949\pi\)
0.647178 + 0.762339i \(0.275949\pi\)
\(258\) 0 0
\(259\) −0.536441 −0.0333328
\(260\) 2.82022 0.174903
\(261\) 0 0
\(262\) 19.0771 1.17858
\(263\) −27.3462 −1.68624 −0.843118 0.537729i \(-0.819283\pi\)
−0.843118 + 0.537729i \(0.819283\pi\)
\(264\) 0 0
\(265\) −9.43185 −0.579394
\(266\) 7.14369 0.438008
\(267\) 0 0
\(268\) 16.2759 0.994207
\(269\) −1.57905 −0.0962763 −0.0481381 0.998841i \(-0.515329\pi\)
−0.0481381 + 0.998841i \(0.515329\pi\)
\(270\) 0 0
\(271\) 15.1014 0.917342 0.458671 0.888606i \(-0.348326\pi\)
0.458671 + 0.888606i \(0.348326\pi\)
\(272\) −0.669183 −0.0405752
\(273\) 0 0
\(274\) 0.733509 0.0443129
\(275\) 0 0
\(276\) 0 0
\(277\) −16.2348 −0.975453 −0.487727 0.872996i \(-0.662174\pi\)
−0.487727 + 0.872996i \(0.662174\pi\)
\(278\) 8.34377 0.500426
\(279\) 0 0
\(280\) 2.30795 0.137927
\(281\) 17.0866 1.01930 0.509651 0.860381i \(-0.329774\pi\)
0.509651 + 0.860381i \(0.329774\pi\)
\(282\) 0 0
\(283\) −0.115027 −0.00683762 −0.00341881 0.999994i \(-0.501088\pi\)
−0.00341881 + 0.999994i \(0.501088\pi\)
\(284\) 12.3043 0.730127
\(285\) 0 0
\(286\) 0 0
\(287\) −4.78449 −0.282420
\(288\) 0 0
\(289\) −6.08362 −0.357860
\(290\) 0.169608 0.00995970
\(291\) 0 0
\(292\) −10.6049 −0.620606
\(293\) 18.6601 1.09013 0.545067 0.838392i \(-0.316504\pi\)
0.545067 + 0.838392i \(0.316504\pi\)
\(294\) 0 0
\(295\) 4.22362 0.245909
\(296\) 1.53998 0.0895094
\(297\) 0 0
\(298\) 6.25243 0.362193
\(299\) −22.2055 −1.28417
\(300\) 0 0
\(301\) 3.16645 0.182511
\(302\) 12.6230 0.726374
\(303\) 0 0
\(304\) −1.60794 −0.0922215
\(305\) −2.93802 −0.168230
\(306\) 0 0
\(307\) 13.9263 0.794817 0.397408 0.917642i \(-0.369910\pi\)
0.397408 + 0.917642i \(0.369910\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.88209 −0.163692
\(311\) −10.1568 −0.575939 −0.287970 0.957640i \(-0.592980\pi\)
−0.287970 + 0.957640i \(0.592980\pi\)
\(312\) 0 0
\(313\) 4.81885 0.272377 0.136189 0.990683i \(-0.456515\pi\)
0.136189 + 0.990683i \(0.456515\pi\)
\(314\) 0.659917 0.0372413
\(315\) 0 0
\(316\) 9.97065 0.560893
\(317\) 8.07318 0.453435 0.226718 0.973961i \(-0.427201\pi\)
0.226718 + 0.973961i \(0.427201\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −4.34734 −0.243024
\(321\) 0 0
\(322\) −6.78003 −0.377836
\(323\) 26.2303 1.45949
\(324\) 0 0
\(325\) −12.8304 −0.711703
\(326\) 2.66671 0.147695
\(327\) 0 0
\(328\) 13.7350 0.758387
\(329\) −11.0114 −0.607076
\(330\) 0 0
\(331\) 0.776780 0.0426957 0.0213479 0.999772i \(-0.493204\pi\)
0.0213479 + 0.999772i \(0.493204\pi\)
\(332\) 8.64444 0.474425
\(333\) 0 0
\(334\) 7.23468 0.395864
\(335\) 10.9931 0.600615
\(336\) 0 0
\(337\) 0.209600 0.0114176 0.00570882 0.999984i \(-0.498183\pi\)
0.00570882 + 0.999984i \(0.498183\pi\)
\(338\) −3.88266 −0.211189
\(339\) 0 0
\(340\) 3.16180 0.171473
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −9.09001 −0.490100
\(345\) 0 0
\(346\) 4.63915 0.249402
\(347\) 22.7970 1.22380 0.611902 0.790933i \(-0.290405\pi\)
0.611902 + 0.790933i \(0.290405\pi\)
\(348\) 0 0
\(349\) −14.0294 −0.750976 −0.375488 0.926827i \(-0.622525\pi\)
−0.375488 + 0.926827i \(0.622525\pi\)
\(350\) −3.91753 −0.209401
\(351\) 0 0
\(352\) 0 0
\(353\) −22.3703 −1.19065 −0.595327 0.803484i \(-0.702977\pi\)
−0.595327 + 0.803484i \(0.702977\pi\)
\(354\) 0 0
\(355\) 8.31060 0.441081
\(356\) −9.56367 −0.506874
\(357\) 0 0
\(358\) 5.41042 0.285949
\(359\) −2.70211 −0.142612 −0.0713060 0.997454i \(-0.522717\pi\)
−0.0713060 + 0.997454i \(0.522717\pi\)
\(360\) 0 0
\(361\) 44.0270 2.31721
\(362\) 2.20499 0.115892
\(363\) 0 0
\(364\) −3.50791 −0.183864
\(365\) −7.16279 −0.374918
\(366\) 0 0
\(367\) 28.2820 1.47631 0.738155 0.674631i \(-0.235697\pi\)
0.738155 + 0.674631i \(0.235697\pi\)
\(368\) 1.52608 0.0795525
\(369\) 0 0
\(370\) 0.388076 0.0201751
\(371\) 11.7317 0.609080
\(372\) 0 0
\(373\) 8.53857 0.442110 0.221055 0.975261i \(-0.429050\pi\)
0.221055 + 0.975261i \(0.429050\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 31.6106 1.63019
\(377\) −0.690937 −0.0355851
\(378\) 0 0
\(379\) 34.6825 1.78152 0.890759 0.454475i \(-0.150173\pi\)
0.890759 + 0.454475i \(0.150173\pi\)
\(380\) 7.59729 0.389733
\(381\) 0 0
\(382\) −19.1435 −0.979466
\(383\) 12.7173 0.649826 0.324913 0.945744i \(-0.394665\pi\)
0.324913 + 0.945744i \(0.394665\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −7.55786 −0.384685
\(387\) 0 0
\(388\) −22.9386 −1.16453
\(389\) 2.48749 0.126121 0.0630605 0.998010i \(-0.479914\pi\)
0.0630605 + 0.998010i \(0.479914\pi\)
\(390\) 0 0
\(391\) −24.8950 −1.25899
\(392\) −2.87073 −0.144994
\(393\) 0 0
\(394\) 9.86622 0.497053
\(395\) 6.73439 0.338844
\(396\) 0 0
\(397\) 19.1634 0.961781 0.480891 0.876781i \(-0.340313\pi\)
0.480891 + 0.876781i \(0.340313\pi\)
\(398\) 5.63409 0.282411
\(399\) 0 0
\(400\) 0.881777 0.0440888
\(401\) −9.17005 −0.457930 −0.228965 0.973435i \(-0.573534\pi\)
−0.228965 + 0.973435i \(0.573534\pi\)
\(402\) 0 0
\(403\) 11.7409 0.584855
\(404\) −20.5679 −1.02329
\(405\) 0 0
\(406\) −0.210965 −0.0104700
\(407\) 0 0
\(408\) 0 0
\(409\) −29.6366 −1.46544 −0.732718 0.680533i \(-0.761748\pi\)
−0.732718 + 0.680533i \(0.761748\pi\)
\(410\) 3.46123 0.170938
\(411\) 0 0
\(412\) −17.0897 −0.841947
\(413\) −5.25351 −0.258508
\(414\) 0 0
\(415\) 5.83864 0.286607
\(416\) 16.3833 0.803256
\(417\) 0 0
\(418\) 0 0
\(419\) 31.7616 1.55166 0.775828 0.630944i \(-0.217332\pi\)
0.775828 + 0.630944i \(0.217332\pi\)
\(420\) 0 0
\(421\) 5.94862 0.289918 0.144959 0.989438i \(-0.453695\pi\)
0.144959 + 0.989438i \(0.453695\pi\)
\(422\) 15.0454 0.732399
\(423\) 0 0
\(424\) −33.6786 −1.63558
\(425\) −14.3844 −0.697747
\(426\) 0 0
\(427\) 3.65442 0.176850
\(428\) −18.2358 −0.881460
\(429\) 0 0
\(430\) −2.29069 −0.110467
\(431\) 11.2906 0.543847 0.271923 0.962319i \(-0.412340\pi\)
0.271923 + 0.962319i \(0.412340\pi\)
\(432\) 0 0
\(433\) 17.1930 0.826243 0.413122 0.910676i \(-0.364438\pi\)
0.413122 + 0.910676i \(0.364438\pi\)
\(434\) 3.58486 0.172079
\(435\) 0 0
\(436\) 6.54953 0.313666
\(437\) −59.8185 −2.86151
\(438\) 0 0
\(439\) −33.6983 −1.60833 −0.804167 0.594404i \(-0.797388\pi\)
−0.804167 + 0.594404i \(0.797388\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 8.76165 0.416749
\(443\) −1.38268 −0.0656930 −0.0328465 0.999460i \(-0.510457\pi\)
−0.0328465 + 0.999460i \(0.510457\pi\)
\(444\) 0 0
\(445\) −6.45951 −0.306210
\(446\) 11.7848 0.558029
\(447\) 0 0
\(448\) 5.40740 0.255476
\(449\) −21.4531 −1.01243 −0.506217 0.862406i \(-0.668956\pi\)
−0.506217 + 0.862406i \(0.668956\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 7.96886 0.374823
\(453\) 0 0
\(454\) −14.3095 −0.671577
\(455\) −2.36931 −0.111075
\(456\) 0 0
\(457\) −3.54337 −0.165752 −0.0828760 0.996560i \(-0.526411\pi\)
−0.0828760 + 0.996560i \(0.526411\pi\)
\(458\) 8.11279 0.379086
\(459\) 0 0
\(460\) −7.21054 −0.336193
\(461\) −20.0540 −0.934006 −0.467003 0.884256i \(-0.654666\pi\)
−0.467003 + 0.884256i \(0.654666\pi\)
\(462\) 0 0
\(463\) −9.80547 −0.455699 −0.227849 0.973696i \(-0.573169\pi\)
−0.227849 + 0.973696i \(0.573169\pi\)
\(464\) 0.0474850 0.00220444
\(465\) 0 0
\(466\) 2.28736 0.105960
\(467\) −18.1244 −0.838697 −0.419349 0.907825i \(-0.637742\pi\)
−0.419349 + 0.907825i \(0.637742\pi\)
\(468\) 0 0
\(469\) −13.6736 −0.631389
\(470\) 7.96590 0.367440
\(471\) 0 0
\(472\) 15.0814 0.694178
\(473\) 0 0
\(474\) 0 0
\(475\) −34.5634 −1.58588
\(476\) −3.93278 −0.180259
\(477\) 0 0
\(478\) 17.2541 0.789185
\(479\) −38.6169 −1.76445 −0.882227 0.470825i \(-0.843956\pi\)
−0.882227 + 0.470825i \(0.843956\pi\)
\(480\) 0 0
\(481\) −1.58092 −0.0720837
\(482\) −6.74927 −0.307421
\(483\) 0 0
\(484\) 0 0
\(485\) −15.4932 −0.703511
\(486\) 0 0
\(487\) 16.4315 0.744581 0.372290 0.928116i \(-0.378573\pi\)
0.372290 + 0.928116i \(0.378573\pi\)
\(488\) −10.4909 −0.474899
\(489\) 0 0
\(490\) −0.723426 −0.0326811
\(491\) −26.0846 −1.17718 −0.588590 0.808431i \(-0.700317\pi\)
−0.588590 + 0.808431i \(0.700317\pi\)
\(492\) 0 0
\(493\) −0.774623 −0.0348873
\(494\) 21.0528 0.947210
\(495\) 0 0
\(496\) −0.806899 −0.0362308
\(497\) −10.3371 −0.463680
\(498\) 0 0
\(499\) −32.7320 −1.46528 −0.732642 0.680615i \(-0.761713\pi\)
−0.732642 + 0.680615i \(0.761713\pi\)
\(500\) −8.95110 −0.400306
\(501\) 0 0
\(502\) 13.2187 0.589982
\(503\) −18.6428 −0.831241 −0.415621 0.909538i \(-0.636436\pi\)
−0.415621 + 0.909538i \(0.636436\pi\)
\(504\) 0 0
\(505\) −13.8920 −0.618184
\(506\) 0 0
\(507\) 0 0
\(508\) 13.7405 0.609635
\(509\) −23.8913 −1.05896 −0.529482 0.848321i \(-0.677614\pi\)
−0.529482 + 0.848321i \(0.677614\pi\)
\(510\) 0 0
\(511\) 8.90937 0.394127
\(512\) −2.28881 −0.101152
\(513\) 0 0
\(514\) 18.6715 0.823565
\(515\) −11.5427 −0.508633
\(516\) 0 0
\(517\) 0 0
\(518\) −0.482705 −0.0212088
\(519\) 0 0
\(520\) 6.80166 0.298272
\(521\) −36.0222 −1.57816 −0.789080 0.614290i \(-0.789443\pi\)
−0.789080 + 0.614290i \(0.789443\pi\)
\(522\) 0 0
\(523\) 28.3402 1.23923 0.619615 0.784906i \(-0.287289\pi\)
0.619615 + 0.784906i \(0.287289\pi\)
\(524\) −25.2355 −1.10242
\(525\) 0 0
\(526\) −24.6068 −1.07291
\(527\) 13.1629 0.573387
\(528\) 0 0
\(529\) 33.7733 1.46841
\(530\) −8.48703 −0.368653
\(531\) 0 0
\(532\) −9.44983 −0.409702
\(533\) −14.1001 −0.610745
\(534\) 0 0
\(535\) −12.3168 −0.532503
\(536\) 39.2533 1.69548
\(537\) 0 0
\(538\) −1.42087 −0.0612581
\(539\) 0 0
\(540\) 0 0
\(541\) 25.7194 1.10576 0.552882 0.833259i \(-0.313528\pi\)
0.552882 + 0.833259i \(0.313528\pi\)
\(542\) 13.5886 0.583681
\(543\) 0 0
\(544\) 18.3676 0.787504
\(545\) 4.42369 0.189490
\(546\) 0 0
\(547\) 33.1803 1.41869 0.709343 0.704864i \(-0.248992\pi\)
0.709343 + 0.704864i \(0.248992\pi\)
\(548\) −0.970301 −0.0414492
\(549\) 0 0
\(550\) 0 0
\(551\) −1.86129 −0.0792937
\(552\) 0 0
\(553\) −8.37651 −0.356205
\(554\) −14.6085 −0.620656
\(555\) 0 0
\(556\) −11.0373 −0.468087
\(557\) 32.3919 1.37249 0.686245 0.727370i \(-0.259258\pi\)
0.686245 + 0.727370i \(0.259258\pi\)
\(558\) 0 0
\(559\) 9.33167 0.394688
\(560\) 0.162832 0.00688092
\(561\) 0 0
\(562\) 15.3750 0.648555
\(563\) −34.6650 −1.46096 −0.730478 0.682937i \(-0.760702\pi\)
−0.730478 + 0.682937i \(0.760702\pi\)
\(564\) 0 0
\(565\) 5.38233 0.226437
\(566\) −0.103504 −0.00435060
\(567\) 0 0
\(568\) 29.6749 1.24513
\(569\) −18.4658 −0.774127 −0.387064 0.922053i \(-0.626511\pi\)
−0.387064 + 0.922053i \(0.626511\pi\)
\(570\) 0 0
\(571\) 14.2034 0.594394 0.297197 0.954816i \(-0.403948\pi\)
0.297197 + 0.954816i \(0.403948\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −4.30522 −0.179696
\(575\) 32.8039 1.36802
\(576\) 0 0
\(577\) 16.9012 0.703606 0.351803 0.936074i \(-0.385569\pi\)
0.351803 + 0.936074i \(0.385569\pi\)
\(578\) −5.47421 −0.227697
\(579\) 0 0
\(580\) −0.224361 −0.00931607
\(581\) −7.26234 −0.301292
\(582\) 0 0
\(583\) 0 0
\(584\) −25.5764 −1.05836
\(585\) 0 0
\(586\) 16.7909 0.693624
\(587\) 34.7412 1.43392 0.716962 0.697113i \(-0.245532\pi\)
0.716962 + 0.697113i \(0.245532\pi\)
\(588\) 0 0
\(589\) 31.6284 1.30322
\(590\) 3.80053 0.156465
\(591\) 0 0
\(592\) 0.108649 0.00446547
\(593\) 2.43045 0.0998068 0.0499034 0.998754i \(-0.484109\pi\)
0.0499034 + 0.998754i \(0.484109\pi\)
\(594\) 0 0
\(595\) −2.65628 −0.108897
\(596\) −8.27085 −0.338787
\(597\) 0 0
\(598\) −19.9811 −0.817087
\(599\) −16.7195 −0.683142 −0.341571 0.939856i \(-0.610959\pi\)
−0.341571 + 0.939856i \(0.610959\pi\)
\(600\) 0 0
\(601\) 0.429964 0.0175386 0.00876929 0.999962i \(-0.497209\pi\)
0.00876929 + 0.999962i \(0.497209\pi\)
\(602\) 2.84925 0.116127
\(603\) 0 0
\(604\) −16.6980 −0.679433
\(605\) 0 0
\(606\) 0 0
\(607\) 11.5440 0.468557 0.234279 0.972169i \(-0.424727\pi\)
0.234279 + 0.972169i \(0.424727\pi\)
\(608\) 44.1343 1.78988
\(609\) 0 0
\(610\) −2.64371 −0.107041
\(611\) −32.4510 −1.31283
\(612\) 0 0
\(613\) −18.0048 −0.727209 −0.363604 0.931554i \(-0.618454\pi\)
−0.363604 + 0.931554i \(0.618454\pi\)
\(614\) 12.5313 0.505721
\(615\) 0 0
\(616\) 0 0
\(617\) 32.1061 1.29254 0.646272 0.763107i \(-0.276327\pi\)
0.646272 + 0.763107i \(0.276327\pi\)
\(618\) 0 0
\(619\) −0.280294 −0.0112660 −0.00563298 0.999984i \(-0.501793\pi\)
−0.00563298 + 0.999984i \(0.501793\pi\)
\(620\) 3.81249 0.153113
\(621\) 0 0
\(622\) −9.13936 −0.366455
\(623\) 8.03460 0.321899
\(624\) 0 0
\(625\) 15.7225 0.628899
\(626\) 4.33613 0.173307
\(627\) 0 0
\(628\) −0.872952 −0.0348346
\(629\) −1.77240 −0.0706702
\(630\) 0 0
\(631\) 30.3354 1.20763 0.603816 0.797124i \(-0.293646\pi\)
0.603816 + 0.797124i \(0.293646\pi\)
\(632\) 24.0467 0.956525
\(633\) 0 0
\(634\) 7.26447 0.288509
\(635\) 9.28061 0.368290
\(636\) 0 0
\(637\) 2.94705 0.116766
\(638\) 0 0
\(639\) 0 0
\(640\) 5.02692 0.198706
\(641\) 3.93047 0.155244 0.0776222 0.996983i \(-0.475267\pi\)
0.0776222 + 0.996983i \(0.475267\pi\)
\(642\) 0 0
\(643\) −13.7243 −0.541234 −0.270617 0.962687i \(-0.587228\pi\)
−0.270617 + 0.962687i \(0.587228\pi\)
\(644\) 8.96877 0.353419
\(645\) 0 0
\(646\) 23.6027 0.928636
\(647\) −2.01034 −0.0790345 −0.0395173 0.999219i \(-0.512582\pi\)
−0.0395173 + 0.999219i \(0.512582\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −11.5452 −0.452838
\(651\) 0 0
\(652\) −3.52758 −0.138151
\(653\) −0.795349 −0.0311244 −0.0155622 0.999879i \(-0.504954\pi\)
−0.0155622 + 0.999879i \(0.504954\pi\)
\(654\) 0 0
\(655\) −17.0446 −0.665988
\(656\) 0.969039 0.0378346
\(657\) 0 0
\(658\) −9.90831 −0.386266
\(659\) −21.9135 −0.853628 −0.426814 0.904340i \(-0.640364\pi\)
−0.426814 + 0.904340i \(0.640364\pi\)
\(660\) 0 0
\(661\) −19.4952 −0.758276 −0.379138 0.925340i \(-0.623779\pi\)
−0.379138 + 0.925340i \(0.623779\pi\)
\(662\) 0.698968 0.0271662
\(663\) 0 0
\(664\) 20.8482 0.809067
\(665\) −6.38261 −0.247507
\(666\) 0 0
\(667\) 1.76654 0.0684007
\(668\) −9.57020 −0.370282
\(669\) 0 0
\(670\) 9.89186 0.382156
\(671\) 0 0
\(672\) 0 0
\(673\) −7.00748 −0.270118 −0.135059 0.990838i \(-0.543122\pi\)
−0.135059 + 0.990838i \(0.543122\pi\)
\(674\) 0.188604 0.00726474
\(675\) 0 0
\(676\) 5.13607 0.197541
\(677\) −5.85627 −0.225075 −0.112537 0.993647i \(-0.535898\pi\)
−0.112537 + 0.993647i \(0.535898\pi\)
\(678\) 0 0
\(679\) 19.2711 0.739557
\(680\) 7.62547 0.292423
\(681\) 0 0
\(682\) 0 0
\(683\) 34.2255 1.30960 0.654802 0.755801i \(-0.272752\pi\)
0.654802 + 0.755801i \(0.272752\pi\)
\(684\) 0 0
\(685\) −0.655362 −0.0250401
\(686\) 0.899827 0.0343556
\(687\) 0 0
\(688\) −0.641324 −0.0244502
\(689\) 34.5740 1.31716
\(690\) 0 0
\(691\) 15.0086 0.570954 0.285477 0.958386i \(-0.407848\pi\)
0.285477 + 0.958386i \(0.407848\pi\)
\(692\) −6.13676 −0.233285
\(693\) 0 0
\(694\) 20.5133 0.778675
\(695\) −7.45484 −0.282778
\(696\) 0 0
\(697\) −15.8079 −0.598768
\(698\) −12.6240 −0.477826
\(699\) 0 0
\(700\) 5.18219 0.195868
\(701\) 11.9907 0.452884 0.226442 0.974025i \(-0.427291\pi\)
0.226442 + 0.974025i \(0.427291\pi\)
\(702\) 0 0
\(703\) −4.25878 −0.160623
\(704\) 0 0
\(705\) 0 0
\(706\) −20.1294 −0.757582
\(707\) 17.2794 0.649858
\(708\) 0 0
\(709\) 39.5551 1.48552 0.742762 0.669556i \(-0.233515\pi\)
0.742762 + 0.669556i \(0.233515\pi\)
\(710\) 7.47810 0.280648
\(711\) 0 0
\(712\) −23.0651 −0.864403
\(713\) −30.0183 −1.12419
\(714\) 0 0
\(715\) 0 0
\(716\) −7.15702 −0.267470
\(717\) 0 0
\(718\) −2.43143 −0.0907403
\(719\) −44.3811 −1.65514 −0.827568 0.561366i \(-0.810276\pi\)
−0.827568 + 0.561366i \(0.810276\pi\)
\(720\) 0 0
\(721\) 14.3573 0.534694
\(722\) 39.6167 1.47438
\(723\) 0 0
\(724\) −2.91681 −0.108402
\(725\) 1.02071 0.0379084
\(726\) 0 0
\(727\) −45.1325 −1.67387 −0.836935 0.547302i \(-0.815655\pi\)
−0.836935 + 0.547302i \(0.815655\pi\)
\(728\) −8.46018 −0.313555
\(729\) 0 0
\(730\) −6.44527 −0.238550
\(731\) 10.4619 0.386948
\(732\) 0 0
\(733\) 13.7971 0.509608 0.254804 0.966993i \(-0.417989\pi\)
0.254804 + 0.966993i \(0.417989\pi\)
\(734\) 25.4490 0.939338
\(735\) 0 0
\(736\) −41.8876 −1.54400
\(737\) 0 0
\(738\) 0 0
\(739\) −16.7453 −0.615985 −0.307993 0.951389i \(-0.599657\pi\)
−0.307993 + 0.951389i \(0.599657\pi\)
\(740\) −0.513355 −0.0188713
\(741\) 0 0
\(742\) 10.5565 0.387542
\(743\) −10.8932 −0.399631 −0.199815 0.979834i \(-0.564034\pi\)
−0.199815 + 0.979834i \(0.564034\pi\)
\(744\) 0 0
\(745\) −5.58631 −0.204666
\(746\) 7.68323 0.281303
\(747\) 0 0
\(748\) 0 0
\(749\) 15.3202 0.559788
\(750\) 0 0
\(751\) −27.0918 −0.988596 −0.494298 0.869293i \(-0.664575\pi\)
−0.494298 + 0.869293i \(0.664575\pi\)
\(752\) 2.23021 0.0813275
\(753\) 0 0
\(754\) −0.621724 −0.0226419
\(755\) −11.2782 −0.410456
\(756\) 0 0
\(757\) 30.2302 1.09873 0.549367 0.835581i \(-0.314869\pi\)
0.549367 + 0.835581i \(0.314869\pi\)
\(758\) 31.2082 1.13353
\(759\) 0 0
\(760\) 18.3227 0.664636
\(761\) 38.5957 1.39909 0.699547 0.714587i \(-0.253385\pi\)
0.699547 + 0.714587i \(0.253385\pi\)
\(762\) 0 0
\(763\) −5.50237 −0.199199
\(764\) 25.3234 0.916170
\(765\) 0 0
\(766\) 11.4434 0.413467
\(767\) −15.4824 −0.559035
\(768\) 0 0
\(769\) 40.5122 1.46091 0.730454 0.682962i \(-0.239309\pi\)
0.730454 + 0.682962i \(0.239309\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 9.99770 0.359825
\(773\) −22.1942 −0.798270 −0.399135 0.916892i \(-0.630690\pi\)
−0.399135 + 0.916892i \(0.630690\pi\)
\(774\) 0 0
\(775\) −17.3447 −0.623040
\(776\) −55.3221 −1.98595
\(777\) 0 0
\(778\) 2.23831 0.0802474
\(779\) −37.9839 −1.36091
\(780\) 0 0
\(781\) 0 0
\(782\) −22.4012 −0.801064
\(783\) 0 0
\(784\) −0.202537 −0.00723348
\(785\) −0.589610 −0.0210441
\(786\) 0 0
\(787\) 15.0781 0.537478 0.268739 0.963213i \(-0.413393\pi\)
0.268739 + 0.963213i \(0.413393\pi\)
\(788\) −13.0513 −0.464932
\(789\) 0 0
\(790\) 6.05978 0.215597
\(791\) −6.69477 −0.238039
\(792\) 0 0
\(793\) 10.7698 0.382446
\(794\) 17.2437 0.611956
\(795\) 0 0
\(796\) −7.45290 −0.264161
\(797\) 22.9577 0.813202 0.406601 0.913606i \(-0.366714\pi\)
0.406601 + 0.913606i \(0.366714\pi\)
\(798\) 0 0
\(799\) −36.3815 −1.28708
\(800\) −24.2028 −0.855699
\(801\) 0 0
\(802\) −8.25146 −0.291369
\(803\) 0 0
\(804\) 0 0
\(805\) 6.05769 0.213506
\(806\) 10.5648 0.372128
\(807\) 0 0
\(808\) −49.6044 −1.74508
\(809\) 27.6048 0.970531 0.485266 0.874367i \(-0.338723\pi\)
0.485266 + 0.874367i \(0.338723\pi\)
\(810\) 0 0
\(811\) −50.0296 −1.75678 −0.878389 0.477947i \(-0.841381\pi\)
−0.878389 + 0.477947i \(0.841381\pi\)
\(812\) 0.279069 0.00979340
\(813\) 0 0
\(814\) 0 0
\(815\) −2.38260 −0.0834589
\(816\) 0 0
\(817\) 25.1383 0.879477
\(818\) −26.6678 −0.932418
\(819\) 0 0
\(820\) −4.57859 −0.159891
\(821\) −27.1692 −0.948213 −0.474107 0.880467i \(-0.657229\pi\)
−0.474107 + 0.880467i \(0.657229\pi\)
\(822\) 0 0
\(823\) 17.4262 0.607439 0.303720 0.952761i \(-0.401771\pi\)
0.303720 + 0.952761i \(0.401771\pi\)
\(824\) −41.2159 −1.43583
\(825\) 0 0
\(826\) −4.72725 −0.164482
\(827\) 30.4545 1.05901 0.529503 0.848308i \(-0.322378\pi\)
0.529503 + 0.848308i \(0.322378\pi\)
\(828\) 0 0
\(829\) 27.4137 0.952118 0.476059 0.879413i \(-0.342065\pi\)
0.476059 + 0.879413i \(0.342065\pi\)
\(830\) 5.25377 0.182361
\(831\) 0 0
\(832\) 15.9359 0.552477
\(833\) 3.30399 0.114477
\(834\) 0 0
\(835\) −6.46391 −0.223693
\(836\) 0 0
\(837\) 0 0
\(838\) 28.5800 0.987278
\(839\) −29.3429 −1.01303 −0.506514 0.862231i \(-0.669066\pi\)
−0.506514 + 0.862231i \(0.669066\pi\)
\(840\) 0 0
\(841\) −28.9450 −0.998105
\(842\) 5.35273 0.184467
\(843\) 0 0
\(844\) −19.9024 −0.685068
\(845\) 3.46901 0.119338
\(846\) 0 0
\(847\) 0 0
\(848\) −2.37611 −0.0815960
\(849\) 0 0
\(850\) −12.9435 −0.443958
\(851\) 4.04198 0.138557
\(852\) 0 0
\(853\) −19.5073 −0.667916 −0.333958 0.942588i \(-0.608384\pi\)
−0.333958 + 0.942588i \(0.608384\pi\)
\(854\) 3.28835 0.112525
\(855\) 0 0
\(856\) −43.9801 −1.50321
\(857\) 10.9432 0.373811 0.186906 0.982378i \(-0.440154\pi\)
0.186906 + 0.982378i \(0.440154\pi\)
\(858\) 0 0
\(859\) −15.3191 −0.522680 −0.261340 0.965247i \(-0.584164\pi\)
−0.261340 + 0.965247i \(0.584164\pi\)
\(860\) 3.03017 0.103328
\(861\) 0 0
\(862\) 10.1595 0.346036
\(863\) 13.4613 0.458227 0.229114 0.973400i \(-0.426417\pi\)
0.229114 + 0.973400i \(0.426417\pi\)
\(864\) 0 0
\(865\) −4.14490 −0.140931
\(866\) 15.4707 0.525717
\(867\) 0 0
\(868\) −4.74214 −0.160959
\(869\) 0 0
\(870\) 0 0
\(871\) −40.2969 −1.36541
\(872\) 15.7958 0.534914
\(873\) 0 0
\(874\) −53.8263 −1.82070
\(875\) 7.51997 0.254221
\(876\) 0 0
\(877\) 3.22081 0.108759 0.0543796 0.998520i \(-0.482682\pi\)
0.0543796 + 0.998520i \(0.482682\pi\)
\(878\) −30.3227 −1.02334
\(879\) 0 0
\(880\) 0 0
\(881\) −19.3721 −0.652661 −0.326331 0.945256i \(-0.605812\pi\)
−0.326331 + 0.945256i \(0.605812\pi\)
\(882\) 0 0
\(883\) −2.93248 −0.0986858 −0.0493429 0.998782i \(-0.515713\pi\)
−0.0493429 + 0.998782i \(0.515713\pi\)
\(884\) −11.5901 −0.389817
\(885\) 0 0
\(886\) −1.24417 −0.0417987
\(887\) 7.68647 0.258087 0.129043 0.991639i \(-0.458809\pi\)
0.129043 + 0.991639i \(0.458809\pi\)
\(888\) 0 0
\(889\) −11.5436 −0.387160
\(890\) −5.81244 −0.194833
\(891\) 0 0
\(892\) −15.5893 −0.521967
\(893\) −87.4186 −2.92535
\(894\) 0 0
\(895\) −4.83400 −0.161583
\(896\) −6.25269 −0.208888
\(897\) 0 0
\(898\) −19.3041 −0.644185
\(899\) −0.934038 −0.0311519
\(900\) 0 0
\(901\) 38.7615 1.29133
\(902\) 0 0
\(903\) 0 0
\(904\) 19.2189 0.639210
\(905\) −1.97008 −0.0654875
\(906\) 0 0
\(907\) −12.7746 −0.424172 −0.212086 0.977251i \(-0.568026\pi\)
−0.212086 + 0.977251i \(0.568026\pi\)
\(908\) 18.9289 0.628177
\(909\) 0 0
\(910\) −2.13197 −0.0706742
\(911\) 11.0831 0.367200 0.183600 0.983001i \(-0.441225\pi\)
0.183600 + 0.983001i \(0.441225\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −3.18843 −0.105464
\(915\) 0 0
\(916\) −10.7318 −0.354588
\(917\) 21.2008 0.700112
\(918\) 0 0
\(919\) 13.5522 0.447046 0.223523 0.974699i \(-0.428244\pi\)
0.223523 + 0.974699i \(0.428244\pi\)
\(920\) −17.3900 −0.573331
\(921\) 0 0
\(922\) −18.0451 −0.594284
\(923\) −30.4638 −1.00273
\(924\) 0 0
\(925\) 2.33548 0.0767900
\(926\) −8.82323 −0.289949
\(927\) 0 0
\(928\) −1.30336 −0.0427849
\(929\) 43.0169 1.41134 0.705670 0.708541i \(-0.250646\pi\)
0.705670 + 0.708541i \(0.250646\pi\)
\(930\) 0 0
\(931\) 7.93896 0.260189
\(932\) −3.02577 −0.0991124
\(933\) 0 0
\(934\) −16.3088 −0.533641
\(935\) 0 0
\(936\) 0 0
\(937\) −2.56484 −0.0837896 −0.0418948 0.999122i \(-0.513339\pi\)
−0.0418948 + 0.999122i \(0.513339\pi\)
\(938\) −12.3039 −0.401737
\(939\) 0 0
\(940\) −10.5375 −0.343694
\(941\) 16.5640 0.539972 0.269986 0.962864i \(-0.412981\pi\)
0.269986 + 0.962864i \(0.412981\pi\)
\(942\) 0 0
\(943\) 36.0502 1.17396
\(944\) 1.06403 0.0346313
\(945\) 0 0
\(946\) 0 0
\(947\) 46.5682 1.51326 0.756632 0.653841i \(-0.226843\pi\)
0.756632 + 0.653841i \(0.226843\pi\)
\(948\) 0 0
\(949\) 26.2564 0.852318
\(950\) −31.1011 −1.00905
\(951\) 0 0
\(952\) −9.48487 −0.307406
\(953\) −39.9493 −1.29409 −0.647043 0.762454i \(-0.723994\pi\)
−0.647043 + 0.762454i \(0.723994\pi\)
\(954\) 0 0
\(955\) 17.1040 0.553472
\(956\) −22.8241 −0.738185
\(957\) 0 0
\(958\) −34.7486 −1.12268
\(959\) 0.815166 0.0263231
\(960\) 0 0
\(961\) −15.1282 −0.488005
\(962\) −1.42255 −0.0458650
\(963\) 0 0
\(964\) 8.92809 0.287554
\(965\) 6.75266 0.217376
\(966\) 0 0
\(967\) 35.2572 1.13380 0.566898 0.823788i \(-0.308143\pi\)
0.566898 + 0.823788i \(0.308143\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −13.9412 −0.447626
\(971\) 34.8319 1.11781 0.558905 0.829232i \(-0.311222\pi\)
0.558905 + 0.829232i \(0.311222\pi\)
\(972\) 0 0
\(973\) 9.27263 0.297267
\(974\) 14.7855 0.473757
\(975\) 0 0
\(976\) −0.740158 −0.0236919
\(977\) −23.2137 −0.742672 −0.371336 0.928499i \(-0.621100\pi\)
−0.371336 + 0.928499i \(0.621100\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.956964 0.0305691
\(981\) 0 0
\(982\) −23.4716 −0.749010
\(983\) −50.8764 −1.62271 −0.811353 0.584557i \(-0.801268\pi\)
−0.811353 + 0.584557i \(0.801268\pi\)
\(984\) 0 0
\(985\) −8.81509 −0.280872
\(986\) −0.697027 −0.0221979
\(987\) 0 0
\(988\) −27.8491 −0.885998
\(989\) −23.8586 −0.758658
\(990\) 0 0
\(991\) −49.0338 −1.55761 −0.778805 0.627266i \(-0.784174\pi\)
−0.778805 + 0.627266i \(0.784174\pi\)
\(992\) 22.1476 0.703187
\(993\) 0 0
\(994\) −9.30157 −0.295028
\(995\) −5.03384 −0.159584
\(996\) 0 0
\(997\) −4.64075 −0.146974 −0.0734870 0.997296i \(-0.523413\pi\)
−0.0734870 + 0.997296i \(0.523413\pi\)
\(998\) −29.4531 −0.932322
\(999\) 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.1.11 16
3.2 odd 2 inner 7623.2.a.dc.1.6 16
11.5 even 5 693.2.m.k.190.3 32
11.9 even 5 693.2.m.k.631.3 yes 32
11.10 odd 2 7623.2.a.db.1.6 16
33.5 odd 10 693.2.m.k.190.6 yes 32
33.20 odd 10 693.2.m.k.631.6 yes 32
33.32 even 2 7623.2.a.db.1.11 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.2.m.k.190.3 32 11.5 even 5
693.2.m.k.190.6 yes 32 33.5 odd 10
693.2.m.k.631.3 yes 32 11.9 even 5
693.2.m.k.631.6 yes 32 33.20 odd 10
7623.2.a.db.1.6 16 11.10 odd 2
7623.2.a.db.1.11 16 33.32 even 2
7623.2.a.dc.1.6 16 3.2 odd 2 inner
7623.2.a.dc.1.11 16 1.1 even 1 trivial