Properties

Label 7623.2.a.dc.1.6
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 1.6
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 \(-0.603057\pi\)
−0.318137 + 0.948045i \(0.603057\pi\)
\(3\) 0 0
\(4\) −1.19031 −0.595156
\(5\) 0.803961 0.359542 0.179771 0.983708i \(-0.442464\pi\)
0.179771 + 0.983708i \(0.442464\pi\)
\(6\) 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 \(-0.631222\pi\)
−0.400668 + 0.916223i \(0.631222\pi\)
\(18\) 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 \(-0.212376\pi\)
0.785558 + 0.618788i \(0.212376\pi\)
\(24\) 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 \(-0.493070\pi\)
0.0217682 + 0.999763i \(0.493070\pi\)
\(30\) 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 \(-0.378121\pi\)
0.373606 + 0.927587i \(0.378121\pi\)
\(42\) 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 \(-0.203190\pi\)
0.803085 + 0.595864i \(0.203190\pi\)
\(48\) 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 \(-0.798231\pi\)
−0.805738 + 0.592273i \(0.798231\pi\)
\(54\) 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 \(-0.388904\pi\)
0.341974 + 0.939709i \(0.388904\pi\)
\(60\) 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 \(-0.289805\pi\)
0.613392 + 0.789779i \(0.289805\pi\)
\(72\) 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 \(-0.369506\pi\)
0.398573 + 0.917137i \(0.369506\pi\)
\(84\) 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 \(-0.640019\pi\)
−0.425833 + 0.904802i \(0.640019\pi\)
\(90\) 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 \(-0.829338\pi\)
−0.859682 + 0.510830i \(0.829338\pi\)
\(102\) 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 \(-0.765425\pi\)
−0.740529 + 0.672024i \(0.765425\pi\)
\(108\) 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 \(-0.398031\pi\)
0.314895 + 0.949126i \(0.398031\pi\)
\(114\) 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 \(-0.876912\pi\)
−0.926161 + 0.377128i \(0.876912\pi\)
\(132\) 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 \(-0.511086\pi\)
−0.0348222 + 0.999394i \(0.511086\pi\)
\(138\) 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 \(-0.591868\pi\)
−0.284621 + 0.958640i \(0.591868\pi\)
\(150\) 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 \(-0.600691\pi\)
−0.311080 + 0.950384i \(0.600691\pi\)
\(168\) 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 \(-0.562791\pi\)
−0.195986 + 0.980607i \(0.562791\pi\)
\(174\) 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 \(-0.572142\pi\)
−0.224706 + 0.974427i \(0.572142\pi\)
\(180\) 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 \(-0.220411\pi\)
0.769689 + 0.638419i \(0.220411\pi\)
\(192\) 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 \(-0.627731\pi\)
−0.390597 + 0.920562i \(0.627731\pi\)
\(198\) 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 \(-0.323039\pi\)
0.527742 + 0.849405i \(0.323039\pi\)
\(228\) 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 \(-0.526535\pi\)
−0.0832659 + 0.996527i \(0.526535\pi\)
\(234\) 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 \(-0.712933\pi\)
−0.620161 + 0.784475i \(0.712933\pi\)
\(240\) 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 \(-0.653451\pi\)
−0.463622 + 0.886033i \(0.653451\pi\)
\(252\) 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 \(-0.724051\pi\)
−0.647178 + 0.762339i \(0.724051\pi\)
\(258\) 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 \(-0.180717\pi\)
0.843118 + 0.537729i \(0.180717\pi\)
\(264\) 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 \(-0.484671\pi\)
0.0481381 + 0.998841i \(0.484671\pi\)
\(270\) 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 \(-0.670226\pi\)
−0.509651 + 0.860381i \(0.670226\pi\)
\(282\) 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 \(-0.683496\pi\)
−0.545067 + 0.838392i \(0.683496\pi\)
\(294\) 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 \(-0.407020\pi\)
0.287970 + 0.957640i \(0.407020\pi\)
\(312\) 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 \(-0.572799\pi\)
−0.226718 + 0.973961i \(0.572799\pi\)
\(318\) 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 \(-0.709595\pi\)
−0.611902 + 0.790933i \(0.709595\pi\)
\(348\) 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 \(-0.297023\pi\)
0.595327 + 0.803484i \(0.297023\pi\)
\(354\) 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 \(-0.477283\pi\)
0.0713060 + 0.997454i \(0.477283\pi\)
\(360\) 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 \(-0.605335\pi\)
−0.324913 + 0.945744i \(0.605335\pi\)
\(384\) 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 \(-0.520086\pi\)
−0.0630605 + 0.998010i \(0.520086\pi\)
\(390\) 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 \(-0.426466\pi\)
0.228965 + 0.973435i \(0.426466\pi\)
\(402\) 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 \(-0.782668\pi\)
−0.775828 + 0.630944i \(0.782668\pi\)
\(420\) 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 \(-0.587660\pi\)
−0.271923 + 0.962319i \(0.587660\pi\)
\(432\) 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 \(-0.489543\pi\)
0.0328465 + 0.999460i \(0.489543\pi\)
\(444\) 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 \(-0.331044\pi\)
0.506217 + 0.862406i \(0.331044\pi\)
\(450\) 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 \(-0.345334\pi\)
0.467003 + 0.884256i \(0.345334\pi\)
\(462\) 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 \(-0.362258\pi\)
0.419349 + 0.907825i \(0.362258\pi\)
\(468\) 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 \(-0.156044\pi\)
0.882227 + 0.470825i \(0.156044\pi\)
\(480\) 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 \(-0.299683\pi\)
0.588590 + 0.808431i \(0.299683\pi\)
\(492\) 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 \(-0.363564\pi\)
0.415621 + 0.909538i \(0.363564\pi\)
\(504\) 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 \(-0.322386\pi\)
0.529482 + 0.848321i \(0.322386\pi\)
\(510\) 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 \(-0.210557\pi\)
0.789080 + 0.614290i \(0.210557\pi\)
\(522\) 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 \(-0.740742\pi\)
−0.686245 + 0.727370i \(0.740742\pi\)
\(558\) 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 \(-0.239298\pi\)
0.730478 + 0.682937i \(0.239298\pi\)
\(564\) 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 \(-0.373489\pi\)
0.387064 + 0.922053i \(0.373489\pi\)
\(570\) 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 \(-0.754468\pi\)
−0.716962 + 0.697113i \(0.754468\pi\)
\(588\) 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 \(-0.515891\pi\)
−0.0499034 + 0.998754i \(0.515891\pi\)
\(594\) 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 \(-0.389041\pi\)
0.341571 + 0.939856i \(0.389041\pi\)
\(600\) 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 \(-0.723673\pi\)
−0.646272 + 0.763107i \(0.723673\pi\)
\(618\) 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 \(-0.524733\pi\)
−0.0776222 + 0.996983i \(0.524733\pi\)
\(642\) 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 \(-0.487418\pi\)
0.0395173 + 0.999219i \(0.487418\pi\)
\(648\) 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 \(-0.495046\pi\)
0.0155622 + 0.999879i \(0.495046\pi\)
\(654\) 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 \(-0.359636\pi\)
0.426814 + 0.904340i \(0.359636\pi\)
\(660\) 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 \(-0.464102\pi\)
0.112537 + 0.993647i \(0.464102\pi\)
\(678\) 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 \(-0.727248\pi\)
−0.654802 + 0.755801i \(0.727248\pi\)
\(684\) 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 \(-0.572709\pi\)
−0.226442 + 0.974025i \(0.572709\pi\)
\(702\) 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 \(-0.189724\pi\)
0.827568 + 0.561366i \(0.189724\pi\)
\(720\) 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 \(-0.435966\pi\)
0.199815 + 0.979834i \(0.435966\pi\)
\(744\) 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 \(-0.746615\pi\)
−0.699547 + 0.714587i \(0.746615\pi\)
\(762\) 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 \(-0.369310\pi\)
0.399135 + 0.916892i \(0.369310\pi\)
\(774\) 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 \(-0.633286\pi\)
−0.406601 + 0.913606i \(0.633286\pi\)
\(798\) 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 \(-0.661277\pi\)
−0.485266 + 0.874367i \(0.661277\pi\)
\(810\) 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 \(-0.342771\pi\)
0.474107 + 0.880467i \(0.342771\pi\)
\(822\) 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 \(-0.677622\pi\)
−0.529503 + 0.848308i \(0.677622\pi\)
\(828\) 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 \(-0.330934\pi\)
0.506514 + 0.862231i \(0.330934\pi\)
\(840\) 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 \(-0.559846\pi\)
−0.186906 + 0.982378i \(0.559846\pi\)
\(858\) 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 \(-0.573583\pi\)
−0.229114 + 0.973400i \(0.573583\pi\)
\(864\) 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 \(-0.394188\pi\)
0.326331 + 0.945256i \(0.394188\pi\)
\(882\) 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 \(-0.541191\pi\)
−0.129043 + 0.991639i \(0.541191\pi\)
\(888\) 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 \(-0.558775\pi\)
−0.183600 + 0.983001i \(0.558775\pi\)
\(912\) 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 \(-0.749354\pi\)
−0.705670 + 0.708541i \(0.749354\pi\)
\(930\) 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 \(-0.587019\pi\)
−0.269986 + 0.962864i \(0.587019\pi\)
\(942\) 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 \(-0.773157\pi\)
−0.756632 + 0.653841i \(0.773157\pi\)
\(948\) 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 \(-0.276006\pi\)
0.647043 + 0.762454i \(0.276006\pi\)
\(954\) 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 \(-0.688778\pi\)
−0.558905 + 0.829232i \(0.688778\pi\)
\(972\) 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 \(-0.378900\pi\)
0.371336 + 0.928499i \(0.378900\pi\)
\(978\) 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 \(-0.198732\pi\)
0.811353 + 0.584557i \(0.198732\pi\)
\(984\) 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.1.6 16
3.2 odd 2 inner 7623.2.a.dc.1.11 16
11.5 even 5 693.2.m.k.190.6 yes 32
11.9 even 5 693.2.m.k.631.6 yes 32
11.10 odd 2 7623.2.a.db.1.11 16
33.5 odd 10 693.2.m.k.190.3 32
33.20 odd 10 693.2.m.k.631.3 yes 32
33.32 even 2 7623.2.a.db.1.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.2.m.k.190.3 32 33.5 odd 10
693.2.m.k.190.6 yes 32 11.5 even 5
693.2.m.k.631.3 yes 32 33.20 odd 10
693.2.m.k.631.6 yes 32 11.9 even 5
7623.2.a.db.1.6 16 33.32 even 2
7623.2.a.db.1.11 16 11.10 odd 2
7623.2.a.dc.1.6 16 1.1 even 1 trivial
7623.2.a.dc.1.11 16 3.2 odd 2 inner