Properties

Label 7623.2.a.cg.1.1
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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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: \(4\)
Coefficient field: 4.4.7488.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2541)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.698857\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.43091 q^{2} +3.90931 q^{4} -1.43091 q^{5} -1.00000 q^{7} -4.64136 q^{8} +O(q^{10})\) \(q-2.43091 q^{2} +3.90931 q^{4} -1.43091 q^{5} -1.00000 q^{7} -4.64136 q^{8} +3.47841 q^{10} +2.30114 q^{13} +2.43091 q^{14} +3.46410 q^{16} -1.14407 q^{17} +5.47841 q^{19} -5.59387 q^{20} -3.00588 q^{23} -2.95250 q^{25} -5.59387 q^{26} -3.90931 q^{28} +3.16727 q^{29} -6.99589 q^{31} +0.861816 q^{32} +2.78112 q^{34} +1.43091 q^{35} -8.16884 q^{37} -13.3175 q^{38} +6.64136 q^{40} -5.59387 q^{41} +10.5939 q^{43} +7.30703 q^{46} +12.9384 q^{47} +1.00000 q^{49} +7.17726 q^{50} +8.99589 q^{52} -9.28273 q^{53} +4.64136 q^{56} -7.69933 q^{58} +6.89501 q^{59} -8.50160 q^{61} +17.0064 q^{62} -9.02320 q^{64} -3.29272 q^{65} +7.61405 q^{67} -4.47252 q^{68} -3.47841 q^{70} +1.92362 q^{71} +4.83704 q^{73} +19.8577 q^{74} +21.4168 q^{76} +13.5348 q^{79} -4.95681 q^{80} +13.5982 q^{82} -9.40661 q^{83} +1.63706 q^{85} -25.7527 q^{86} +4.35911 q^{89} -2.30114 q^{91} -11.7509 q^{92} -31.4520 q^{94} -7.83909 q^{95} -6.56067 q^{97} -2.43091 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} + 4q^{4} + 2q^{5} - 4q^{7} + O(q^{10}) \) \( 4q - 2q^{2} + 4q^{4} + 2q^{5} - 4q^{7} + 10q^{10} + 10q^{13} + 2q^{14} - 6q^{17} + 18q^{19} + 2q^{23} - 8q^{25} - 4q^{28} - 6q^{29} - 12q^{32} - 2q^{34} - 2q^{35} - 4q^{37} + 8q^{40} + 20q^{43} + 16q^{46} + 6q^{47} + 4q^{49} + 24q^{50} + 8q^{52} + 24q^{58} + 6q^{59} - 10q^{61} - 16q^{64} + 10q^{65} + 4q^{67} - 28q^{68} - 10q^{70} + 6q^{71} + 34q^{73} + 36q^{74} + 36q^{76} + 24q^{79} - 12q^{80} + 28q^{82} - 6q^{83} - 8q^{85} - 38q^{86} - 18q^{89} - 10q^{91} - 24q^{92} - 6q^{94} + 18q^{95} - 10q^{97} - 2q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.43091 −1.71891 −0.859456 0.511210i \(-0.829197\pi\)
−0.859456 + 0.511210i \(0.829197\pi\)
\(3\) 0 0
\(4\) 3.90931 1.95466
\(5\) −1.43091 −0.639921 −0.319961 0.947431i \(-0.603670\pi\)
−0.319961 + 0.947431i \(0.603670\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −4.64136 −1.64097
\(9\) 0 0
\(10\) 3.47841 1.09997
\(11\) 0 0
\(12\) 0 0
\(13\) 2.30114 0.638222 0.319111 0.947717i \(-0.396616\pi\)
0.319111 + 0.947717i \(0.396616\pi\)
\(14\) 2.43091 0.649687
\(15\) 0 0
\(16\) 3.46410 0.866025
\(17\) −1.14407 −0.277477 −0.138739 0.990329i \(-0.544305\pi\)
−0.138739 + 0.990329i \(0.544305\pi\)
\(18\) 0 0
\(19\) 5.47841 1.25683 0.628416 0.777877i \(-0.283704\pi\)
0.628416 + 0.777877i \(0.283704\pi\)
\(20\) −5.59387 −1.25083
\(21\) 0 0
\(22\) 0 0
\(23\) −3.00588 −0.626770 −0.313385 0.949626i \(-0.601463\pi\)
−0.313385 + 0.949626i \(0.601463\pi\)
\(24\) 0 0
\(25\) −2.95250 −0.590501
\(26\) −5.59387 −1.09705
\(27\) 0 0
\(28\) −3.90931 −0.738791
\(29\) 3.16727 0.588147 0.294073 0.955783i \(-0.404989\pi\)
0.294073 + 0.955783i \(0.404989\pi\)
\(30\) 0 0
\(31\) −6.99589 −1.25650 −0.628249 0.778012i \(-0.716228\pi\)
−0.628249 + 0.778012i \(0.716228\pi\)
\(32\) 0.861816 0.152349
\(33\) 0 0
\(34\) 2.78112 0.476959
\(35\) 1.43091 0.241868
\(36\) 0 0
\(37\) −8.16884 −1.34295 −0.671475 0.741027i \(-0.734339\pi\)
−0.671475 + 0.741027i \(0.734339\pi\)
\(38\) −13.3175 −2.16038
\(39\) 0 0
\(40\) 6.64136 1.05009
\(41\) −5.59387 −0.873615 −0.436808 0.899555i \(-0.643891\pi\)
−0.436808 + 0.899555i \(0.643891\pi\)
\(42\) 0 0
\(43\) 10.5939 1.61555 0.807775 0.589491i \(-0.200672\pi\)
0.807775 + 0.589491i \(0.200672\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 7.30703 1.07736
\(47\) 12.9384 1.88726 0.943629 0.331004i \(-0.107387\pi\)
0.943629 + 0.331004i \(0.107387\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 7.17726 1.01502
\(51\) 0 0
\(52\) 8.99589 1.24751
\(53\) −9.28273 −1.27508 −0.637540 0.770417i \(-0.720048\pi\)
−0.637540 + 0.770417i \(0.720048\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.64136 0.620228
\(57\) 0 0
\(58\) −7.69933 −1.01097
\(59\) 6.89501 0.897654 0.448827 0.893619i \(-0.351842\pi\)
0.448827 + 0.893619i \(0.351842\pi\)
\(60\) 0 0
\(61\) −8.50160 −1.08852 −0.544259 0.838917i \(-0.683189\pi\)
−0.544259 + 0.838917i \(0.683189\pi\)
\(62\) 17.0064 2.15981
\(63\) 0 0
\(64\) −9.02320 −1.12790
\(65\) −3.29272 −0.408412
\(66\) 0 0
\(67\) 7.61405 0.930205 0.465102 0.885257i \(-0.346018\pi\)
0.465102 + 0.885257i \(0.346018\pi\)
\(68\) −4.47252 −0.542373
\(69\) 0 0
\(70\) −3.47841 −0.415749
\(71\) 1.92362 0.228291 0.114146 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(72\) 0 0
\(73\) 4.83704 0.566133 0.283066 0.959100i \(-0.408648\pi\)
0.283066 + 0.959100i \(0.408648\pi\)
\(74\) 19.8577 2.30841
\(75\) 0 0
\(76\) 21.4168 2.45668
\(77\) 0 0
\(78\) 0 0
\(79\) 13.5348 1.52278 0.761392 0.648292i \(-0.224516\pi\)
0.761392 + 0.648292i \(0.224516\pi\)
\(80\) −4.95681 −0.554188
\(81\) 0 0
\(82\) 13.5982 1.50167
\(83\) −9.40661 −1.03251 −0.516255 0.856435i \(-0.672674\pi\)
−0.516255 + 0.856435i \(0.672674\pi\)
\(84\) 0 0
\(85\) 1.63706 0.177564
\(86\) −25.7527 −2.77699
\(87\) 0 0
\(88\) 0 0
\(89\) 4.35911 0.462065 0.231032 0.972946i \(-0.425790\pi\)
0.231032 + 0.972946i \(0.425790\pi\)
\(90\) 0 0
\(91\) −2.30114 −0.241225
\(92\) −11.7509 −1.22512
\(93\) 0 0
\(94\) −31.4520 −3.24403
\(95\) −7.83909 −0.804274
\(96\) 0 0
\(97\) −6.56067 −0.666135 −0.333068 0.942903i \(-0.608084\pi\)
−0.333068 + 0.942903i \(0.608084\pi\)
\(98\) −2.43091 −0.245559
\(99\) 0 0
\(100\) −11.5423 −1.15423
\(101\) −10.1239 −1.00736 −0.503682 0.863889i \(-0.668022\pi\)
−0.503682 + 0.863889i \(0.668022\pi\)
\(102\) 0 0
\(103\) −5.00158 −0.492820 −0.246410 0.969166i \(-0.579251\pi\)
−0.246410 + 0.969166i \(0.579251\pi\)
\(104\) −10.6804 −1.04730
\(105\) 0 0
\(106\) 22.5655 2.19175
\(107\) 17.6186 1.70326 0.851629 0.524145i \(-0.175615\pi\)
0.851629 + 0.524145i \(0.175615\pi\)
\(108\) 0 0
\(109\) −12.5813 −1.20507 −0.602537 0.798091i \(-0.705843\pi\)
−0.602537 + 0.798091i \(0.705843\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.46410 −0.327327
\(113\) −3.20457 −0.301461 −0.150730 0.988575i \(-0.548163\pi\)
−0.150730 + 0.988575i \(0.548163\pi\)
\(114\) 0 0
\(115\) 4.30114 0.401084
\(116\) 12.3818 1.14962
\(117\) 0 0
\(118\) −16.7611 −1.54299
\(119\) 1.14407 0.104877
\(120\) 0 0
\(121\) 0 0
\(122\) 20.6666 1.87107
\(123\) 0 0
\(124\) −27.3491 −2.45602
\(125\) 11.3793 1.01780
\(126\) 0 0
\(127\) 4.77524 0.423734 0.211867 0.977298i \(-0.432046\pi\)
0.211867 + 0.977298i \(0.432046\pi\)
\(128\) 20.2109 1.78641
\(129\) 0 0
\(130\) 8.00431 0.702024
\(131\) 16.5713 1.44784 0.723922 0.689882i \(-0.242337\pi\)
0.723922 + 0.689882i \(0.242337\pi\)
\(132\) 0 0
\(133\) −5.47841 −0.475038
\(134\) −18.5091 −1.59894
\(135\) 0 0
\(136\) 5.31004 0.455332
\(137\) −9.64725 −0.824220 −0.412110 0.911134i \(-0.635208\pi\)
−0.412110 + 0.911134i \(0.635208\pi\)
\(138\) 0 0
\(139\) −20.9647 −1.77821 −0.889103 0.457707i \(-0.848671\pi\)
−0.889103 + 0.457707i \(0.848671\pi\)
\(140\) 5.59387 0.472768
\(141\) 0 0
\(142\) −4.67613 −0.392412
\(143\) 0 0
\(144\) 0 0
\(145\) −4.53207 −0.376368
\(146\) −11.7584 −0.973132
\(147\) 0 0
\(148\) −31.9346 −2.62500
\(149\) −12.3823 −1.01440 −0.507199 0.861829i \(-0.669319\pi\)
−0.507199 + 0.861829i \(0.669319\pi\)
\(150\) 0 0
\(151\) −19.6952 −1.60277 −0.801387 0.598146i \(-0.795904\pi\)
−0.801387 + 0.598146i \(0.795904\pi\)
\(152\) −25.4273 −2.06242
\(153\) 0 0
\(154\) 0 0
\(155\) 10.0105 0.804060
\(156\) 0 0
\(157\) −1.54001 −0.122906 −0.0614531 0.998110i \(-0.519573\pi\)
−0.0614531 + 0.998110i \(0.519573\pi\)
\(158\) −32.9018 −2.61753
\(159\) 0 0
\(160\) −1.23318 −0.0974913
\(161\) 3.00588 0.236897
\(162\) 0 0
\(163\) 24.7343 1.93734 0.968670 0.248352i \(-0.0798890\pi\)
0.968670 + 0.248352i \(0.0798890\pi\)
\(164\) −21.8682 −1.70762
\(165\) 0 0
\(166\) 22.8666 1.77479
\(167\) 12.2020 0.944222 0.472111 0.881539i \(-0.343492\pi\)
0.472111 + 0.881539i \(0.343492\pi\)
\(168\) 0 0
\(169\) −7.70474 −0.592672
\(170\) −3.97953 −0.305216
\(171\) 0 0
\(172\) 41.4147 3.15784
\(173\) 12.3486 0.938850 0.469425 0.882972i \(-0.344461\pi\)
0.469425 + 0.882972i \(0.344461\pi\)
\(174\) 0 0
\(175\) 2.95250 0.223188
\(176\) 0 0
\(177\) 0 0
\(178\) −10.5966 −0.794249
\(179\) −17.7036 −1.32323 −0.661616 0.749843i \(-0.730129\pi\)
−0.661616 + 0.749843i \(0.730129\pi\)
\(180\) 0 0
\(181\) −5.64677 −0.419721 −0.209861 0.977731i \(-0.567301\pi\)
−0.209861 + 0.977731i \(0.567301\pi\)
\(182\) 5.59387 0.414645
\(183\) 0 0
\(184\) 13.9514 1.02851
\(185\) 11.6889 0.859382
\(186\) 0 0
\(187\) 0 0
\(188\) 50.5802 3.68894
\(189\) 0 0
\(190\) 19.0561 1.38248
\(191\) 6.86229 0.496538 0.248269 0.968691i \(-0.420138\pi\)
0.248269 + 0.968691i \(0.420138\pi\)
\(192\) 0 0
\(193\) 0.803848 0.0578622 0.0289311 0.999581i \(-0.490790\pi\)
0.0289311 + 0.999581i \(0.490790\pi\)
\(194\) 15.9484 1.14503
\(195\) 0 0
\(196\) 3.90931 0.279237
\(197\) 16.4482 1.17189 0.585944 0.810352i \(-0.300724\pi\)
0.585944 + 0.810352i \(0.300724\pi\)
\(198\) 0 0
\(199\) −3.65156 −0.258852 −0.129426 0.991589i \(-0.541313\pi\)
−0.129426 + 0.991589i \(0.541313\pi\)
\(200\) 13.7036 0.968994
\(201\) 0 0
\(202\) 24.6102 1.73157
\(203\) −3.16727 −0.222298
\(204\) 0 0
\(205\) 8.00431 0.559045
\(206\) 12.1584 0.847114
\(207\) 0 0
\(208\) 7.97139 0.552717
\(209\) 0 0
\(210\) 0 0
\(211\) 18.2468 1.25616 0.628081 0.778148i \(-0.283841\pi\)
0.628081 + 0.778148i \(0.283841\pi\)
\(212\) −36.2891 −2.49234
\(213\) 0 0
\(214\) −42.8293 −2.92775
\(215\) −15.1588 −1.03382
\(216\) 0 0
\(217\) 6.99589 0.474912
\(218\) 30.5841 2.07141
\(219\) 0 0
\(220\) 0 0
\(221\) −2.63266 −0.177092
\(222\) 0 0
\(223\) −24.3164 −1.62835 −0.814173 0.580622i \(-0.802809\pi\)
−0.814173 + 0.580622i \(0.802809\pi\)
\(224\) −0.861816 −0.0575825
\(225\) 0 0
\(226\) 7.79002 0.518184
\(227\) −9.08069 −0.602707 −0.301353 0.953513i \(-0.597438\pi\)
−0.301353 + 0.953513i \(0.597438\pi\)
\(228\) 0 0
\(229\) 1.23318 0.0814907 0.0407454 0.999170i \(-0.487027\pi\)
0.0407454 + 0.999170i \(0.487027\pi\)
\(230\) −10.4557 −0.689427
\(231\) 0 0
\(232\) −14.7004 −0.965131
\(233\) 9.13996 0.598778 0.299389 0.954131i \(-0.403217\pi\)
0.299389 + 0.954131i \(0.403217\pi\)
\(234\) 0 0
\(235\) −18.5137 −1.20770
\(236\) 26.9547 1.75460
\(237\) 0 0
\(238\) −2.78112 −0.180274
\(239\) −25.7745 −1.66721 −0.833606 0.552359i \(-0.813728\pi\)
−0.833606 + 0.552359i \(0.813728\pi\)
\(240\) 0 0
\(241\) 3.70617 0.238736 0.119368 0.992850i \(-0.461913\pi\)
0.119368 + 0.992850i \(0.461913\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −33.2354 −2.12768
\(245\) −1.43091 −0.0914173
\(246\) 0 0
\(247\) 12.6066 0.802138
\(248\) 32.4705 2.06188
\(249\) 0 0
\(250\) −27.6620 −1.74950
\(251\) 17.7354 1.11945 0.559724 0.828679i \(-0.310907\pi\)
0.559724 + 0.828679i \(0.310907\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −11.6082 −0.728361
\(255\) 0 0
\(256\) −31.0845 −1.94278
\(257\) −8.43426 −0.526114 −0.263057 0.964780i \(-0.584731\pi\)
−0.263057 + 0.964780i \(0.584731\pi\)
\(258\) 0 0
\(259\) 8.16884 0.507587
\(260\) −12.8723 −0.798305
\(261\) 0 0
\(262\) −40.2834 −2.48872
\(263\) 3.09226 0.190677 0.0953386 0.995445i \(-0.469607\pi\)
0.0953386 + 0.995445i \(0.469607\pi\)
\(264\) 0 0
\(265\) 13.2827 0.815951
\(266\) 13.3175 0.816548
\(267\) 0 0
\(268\) 29.7657 1.81823
\(269\) 12.7468 0.777188 0.388594 0.921409i \(-0.372961\pi\)
0.388594 + 0.921409i \(0.372961\pi\)
\(270\) 0 0
\(271\) 22.6995 1.37890 0.689449 0.724334i \(-0.257853\pi\)
0.689449 + 0.724334i \(0.257853\pi\)
\(272\) −3.96317 −0.240302
\(273\) 0 0
\(274\) 23.4516 1.41676
\(275\) 0 0
\(276\) 0 0
\(277\) 6.43679 0.386749 0.193375 0.981125i \(-0.438057\pi\)
0.193375 + 0.981125i \(0.438057\pi\)
\(278\) 50.9634 3.05658
\(279\) 0 0
\(280\) −6.64136 −0.396897
\(281\) 1.70455 0.101685 0.0508423 0.998707i \(-0.483809\pi\)
0.0508423 + 0.998707i \(0.483809\pi\)
\(282\) 0 0
\(283\) −0.00861477 −0.000512095 0 −0.000256048 1.00000i \(-0.500082\pi\)
−0.000256048 1.00000i \(0.500082\pi\)
\(284\) 7.52002 0.446231
\(285\) 0 0
\(286\) 0 0
\(287\) 5.59387 0.330195
\(288\) 0 0
\(289\) −15.6911 −0.923006
\(290\) 11.0170 0.646943
\(291\) 0 0
\(292\) 18.9095 1.10660
\(293\) −2.99130 −0.174754 −0.0873768 0.996175i \(-0.527848\pi\)
−0.0873768 + 0.996175i \(0.527848\pi\)
\(294\) 0 0
\(295\) −9.86612 −0.574428
\(296\) 37.9146 2.20374
\(297\) 0 0
\(298\) 30.1003 1.74366
\(299\) −6.91697 −0.400019
\(300\) 0 0
\(301\) −10.5939 −0.610620
\(302\) 47.8773 2.75503
\(303\) 0 0
\(304\) 18.9778 1.08845
\(305\) 12.1650 0.696566
\(306\) 0 0
\(307\) 16.6115 0.948069 0.474035 0.880506i \(-0.342797\pi\)
0.474035 + 0.880506i \(0.342797\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −24.3345 −1.38211
\(311\) −18.0280 −1.02227 −0.511136 0.859500i \(-0.670775\pi\)
−0.511136 + 0.859500i \(0.670775\pi\)
\(312\) 0 0
\(313\) 32.1350 1.81638 0.908189 0.418559i \(-0.137465\pi\)
0.908189 + 0.418559i \(0.137465\pi\)
\(314\) 3.74362 0.211265
\(315\) 0 0
\(316\) 52.9118 2.97652
\(317\) 25.9290 1.45632 0.728159 0.685408i \(-0.240376\pi\)
0.728159 + 0.685408i \(0.240376\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 12.9114 0.721767
\(321\) 0 0
\(322\) −7.30703 −0.407205
\(323\) −6.26767 −0.348742
\(324\) 0 0
\(325\) −6.79413 −0.376871
\(326\) −60.1268 −3.33012
\(327\) 0 0
\(328\) 25.9632 1.43358
\(329\) −12.9384 −0.713317
\(330\) 0 0
\(331\) −10.6498 −0.585365 −0.292683 0.956210i \(-0.594548\pi\)
−0.292683 + 0.956210i \(0.594548\pi\)
\(332\) −36.7734 −2.01820
\(333\) 0 0
\(334\) −29.6620 −1.62303
\(335\) −10.8950 −0.595258
\(336\) 0 0
\(337\) 14.5148 0.790669 0.395334 0.918537i \(-0.370629\pi\)
0.395334 + 0.918537i \(0.370629\pi\)
\(338\) 18.7295 1.01875
\(339\) 0 0
\(340\) 6.39977 0.347076
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −49.1700 −2.65107
\(345\) 0 0
\(346\) −30.0184 −1.61380
\(347\) −14.7064 −0.789479 −0.394740 0.918793i \(-0.629165\pi\)
−0.394740 + 0.918793i \(0.629165\pi\)
\(348\) 0 0
\(349\) −1.89548 −0.101463 −0.0507315 0.998712i \(-0.516155\pi\)
−0.0507315 + 0.998712i \(0.516155\pi\)
\(350\) −7.17726 −0.383641
\(351\) 0 0
\(352\) 0 0
\(353\) −23.1014 −1.22956 −0.614780 0.788698i \(-0.710755\pi\)
−0.614780 + 0.788698i \(0.710755\pi\)
\(354\) 0 0
\(355\) −2.75252 −0.146088
\(356\) 17.0411 0.903178
\(357\) 0 0
\(358\) 43.0359 2.27452
\(359\) 4.19869 0.221598 0.110799 0.993843i \(-0.464659\pi\)
0.110799 + 0.993843i \(0.464659\pi\)
\(360\) 0 0
\(361\) 11.0129 0.579627
\(362\) 13.7268 0.721464
\(363\) 0 0
\(364\) −8.99589 −0.471513
\(365\) −6.92136 −0.362281
\(366\) 0 0
\(367\) −2.95073 −0.154027 −0.0770134 0.997030i \(-0.524538\pi\)
−0.0770134 + 0.997030i \(0.524538\pi\)
\(368\) −10.4127 −0.542799
\(369\) 0 0
\(370\) −28.4145 −1.47720
\(371\) 9.28273 0.481935
\(372\) 0 0
\(373\) −21.6666 −1.12185 −0.560927 0.827865i \(-0.689555\pi\)
−0.560927 + 0.827865i \(0.689555\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −60.0518 −3.09693
\(377\) 7.28833 0.375368
\(378\) 0 0
\(379\) 29.3165 1.50589 0.752945 0.658084i \(-0.228633\pi\)
0.752945 + 0.658084i \(0.228633\pi\)
\(380\) −30.6455 −1.57208
\(381\) 0 0
\(382\) −16.6816 −0.853505
\(383\) −37.2326 −1.90249 −0.951247 0.308429i \(-0.900197\pi\)
−0.951247 + 0.308429i \(0.900197\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.95408 −0.0994600
\(387\) 0 0
\(388\) −25.6477 −1.30207
\(389\) 29.3869 1.48997 0.744987 0.667079i \(-0.232455\pi\)
0.744987 + 0.667079i \(0.232455\pi\)
\(390\) 0 0
\(391\) 3.43894 0.173915
\(392\) −4.64136 −0.234424
\(393\) 0 0
\(394\) −39.9841 −2.01437
\(395\) −19.3670 −0.974462
\(396\) 0 0
\(397\) −18.5537 −0.931183 −0.465591 0.885000i \(-0.654158\pi\)
−0.465591 + 0.885000i \(0.654158\pi\)
\(398\) 8.87659 0.444943
\(399\) 0 0
\(400\) −10.2278 −0.511388
\(401\) −3.09911 −0.154762 −0.0773810 0.997002i \(-0.524656\pi\)
−0.0773810 + 0.997002i \(0.524656\pi\)
\(402\) 0 0
\(403\) −16.0985 −0.801925
\(404\) −39.5774 −1.96905
\(405\) 0 0
\(406\) 7.69933 0.382111
\(407\) 0 0
\(408\) 0 0
\(409\) 13.5912 0.672041 0.336020 0.941855i \(-0.390919\pi\)
0.336020 + 0.941855i \(0.390919\pi\)
\(410\) −19.4577 −0.960949
\(411\) 0 0
\(412\) −19.5527 −0.963294
\(413\) −6.89501 −0.339281
\(414\) 0 0
\(415\) 13.4600 0.660725
\(416\) 1.98316 0.0972325
\(417\) 0 0
\(418\) 0 0
\(419\) 1.38136 0.0674838 0.0337419 0.999431i \(-0.489258\pi\)
0.0337419 + 0.999431i \(0.489258\pi\)
\(420\) 0 0
\(421\) 28.1477 1.37184 0.685919 0.727678i \(-0.259401\pi\)
0.685919 + 0.727678i \(0.259401\pi\)
\(422\) −44.3563 −2.15923
\(423\) 0 0
\(424\) 43.0845 2.09237
\(425\) 3.37786 0.163851
\(426\) 0 0
\(427\) 8.50160 0.411421
\(428\) 68.8768 3.32928
\(429\) 0 0
\(430\) 36.8498 1.77705
\(431\) −10.7754 −0.519034 −0.259517 0.965738i \(-0.583563\pi\)
−0.259517 + 0.965738i \(0.583563\pi\)
\(432\) 0 0
\(433\) −14.7516 −0.708917 −0.354459 0.935072i \(-0.615335\pi\)
−0.354459 + 0.935072i \(0.615335\pi\)
\(434\) −17.0064 −0.816331
\(435\) 0 0
\(436\) −49.1844 −2.35550
\(437\) −16.4675 −0.787745
\(438\) 0 0
\(439\) 4.56545 0.217897 0.108949 0.994047i \(-0.465252\pi\)
0.108949 + 0.994047i \(0.465252\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 6.39977 0.304406
\(443\) −19.1300 −0.908896 −0.454448 0.890773i \(-0.650163\pi\)
−0.454448 + 0.890773i \(0.650163\pi\)
\(444\) 0 0
\(445\) −6.23749 −0.295685
\(446\) 59.1109 2.79898
\(447\) 0 0
\(448\) 9.02320 0.426306
\(449\) 27.4346 1.29472 0.647359 0.762185i \(-0.275873\pi\)
0.647359 + 0.762185i \(0.275873\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −12.5277 −0.589252
\(453\) 0 0
\(454\) 22.0743 1.03600
\(455\) 3.29272 0.154365
\(456\) 0 0
\(457\) −5.58244 −0.261135 −0.130568 0.991439i \(-0.541680\pi\)
−0.130568 + 0.991439i \(0.541680\pi\)
\(458\) −2.99774 −0.140075
\(459\) 0 0
\(460\) 16.8145 0.783981
\(461\) 31.2918 1.45740 0.728701 0.684832i \(-0.240124\pi\)
0.728701 + 0.684832i \(0.240124\pi\)
\(462\) 0 0
\(463\) 20.5780 0.956340 0.478170 0.878267i \(-0.341300\pi\)
0.478170 + 0.878267i \(0.341300\pi\)
\(464\) 10.9717 0.509350
\(465\) 0 0
\(466\) −22.2184 −1.02925
\(467\) 40.8220 1.88902 0.944508 0.328489i \(-0.106539\pi\)
0.944508 + 0.328489i \(0.106539\pi\)
\(468\) 0 0
\(469\) −7.61405 −0.351584
\(470\) 45.0050 2.07592
\(471\) 0 0
\(472\) −32.0022 −1.47302
\(473\) 0 0
\(474\) 0 0
\(475\) −16.1750 −0.742160
\(476\) 4.47252 0.204998
\(477\) 0 0
\(478\) 62.6554 2.86579
\(479\) −26.5969 −1.21525 −0.607623 0.794226i \(-0.707877\pi\)
−0.607623 + 0.794226i \(0.707877\pi\)
\(480\) 0 0
\(481\) −18.7977 −0.857100
\(482\) −9.00937 −0.410366
\(483\) 0 0
\(484\) 0 0
\(485\) 9.38772 0.426274
\(486\) 0 0
\(487\) 23.6150 1.07010 0.535049 0.844821i \(-0.320293\pi\)
0.535049 + 0.844821i \(0.320293\pi\)
\(488\) 39.4590 1.78623
\(489\) 0 0
\(490\) 3.47841 0.157138
\(491\) −40.8045 −1.84148 −0.920741 0.390174i \(-0.872415\pi\)
−0.920741 + 0.390174i \(0.872415\pi\)
\(492\) 0 0
\(493\) −3.62357 −0.163197
\(494\) −30.6455 −1.37880
\(495\) 0 0
\(496\) −24.2345 −1.08816
\(497\) −1.92362 −0.0862860
\(498\) 0 0
\(499\) 9.57197 0.428500 0.214250 0.976779i \(-0.431269\pi\)
0.214250 + 0.976779i \(0.431269\pi\)
\(500\) 44.4852 1.98944
\(501\) 0 0
\(502\) −43.1131 −1.92423
\(503\) −20.1131 −0.896797 −0.448399 0.893834i \(-0.648005\pi\)
−0.448399 + 0.893834i \(0.648005\pi\)
\(504\) 0 0
\(505\) 14.4863 0.644634
\(506\) 0 0
\(507\) 0 0
\(508\) 18.6679 0.828255
\(509\) −6.27889 −0.278307 −0.139154 0.990271i \(-0.544438\pi\)
−0.139154 + 0.990271i \(0.544438\pi\)
\(510\) 0 0
\(511\) −4.83704 −0.213978
\(512\) 35.1417 1.55306
\(513\) 0 0
\(514\) 20.5029 0.904344
\(515\) 7.15680 0.315366
\(516\) 0 0
\(517\) 0 0
\(518\) −19.8577 −0.872497
\(519\) 0 0
\(520\) 15.2827 0.670192
\(521\) 23.5173 1.03031 0.515156 0.857097i \(-0.327734\pi\)
0.515156 + 0.857097i \(0.327734\pi\)
\(522\) 0 0
\(523\) −7.38211 −0.322797 −0.161399 0.986889i \(-0.551600\pi\)
−0.161399 + 0.986889i \(0.551600\pi\)
\(524\) 64.7825 2.83004
\(525\) 0 0
\(526\) −7.51701 −0.327757
\(527\) 8.00377 0.348650
\(528\) 0 0
\(529\) −13.9647 −0.607159
\(530\) −32.2891 −1.40255
\(531\) 0 0
\(532\) −21.4168 −0.928536
\(533\) −12.8723 −0.557561
\(534\) 0 0
\(535\) −25.2107 −1.08995
\(536\) −35.3396 −1.52644
\(537\) 0 0
\(538\) −30.9864 −1.33592
\(539\) 0 0
\(540\) 0 0
\(541\) 10.5161 0.452123 0.226061 0.974113i \(-0.427415\pi\)
0.226061 + 0.974113i \(0.427415\pi\)
\(542\) −55.1805 −2.37020
\(543\) 0 0
\(544\) −0.985976 −0.0422734
\(545\) 18.0027 0.771152
\(546\) 0 0
\(547\) 9.42803 0.403114 0.201557 0.979477i \(-0.435400\pi\)
0.201557 + 0.979477i \(0.435400\pi\)
\(548\) −37.7141 −1.61107
\(549\) 0 0
\(550\) 0 0
\(551\) 17.3516 0.739202
\(552\) 0 0
\(553\) −13.5348 −0.575558
\(554\) −15.6472 −0.664788
\(555\) 0 0
\(556\) −81.9578 −3.47578
\(557\) −2.83580 −0.120157 −0.0600784 0.998194i \(-0.519135\pi\)
−0.0600784 + 0.998194i \(0.519135\pi\)
\(558\) 0 0
\(559\) 24.3780 1.03108
\(560\) 4.95681 0.209463
\(561\) 0 0
\(562\) −4.14359 −0.174787
\(563\) 13.8127 0.582138 0.291069 0.956702i \(-0.405989\pi\)
0.291069 + 0.956702i \(0.405989\pi\)
\(564\) 0 0
\(565\) 4.58545 0.192911
\(566\) 0.0209417 0.000880246 0
\(567\) 0 0
\(568\) −8.92820 −0.374619
\(569\) 20.9491 0.878230 0.439115 0.898431i \(-0.355292\pi\)
0.439115 + 0.898431i \(0.355292\pi\)
\(570\) 0 0
\(571\) 30.2109 1.26429 0.632144 0.774851i \(-0.282175\pi\)
0.632144 + 0.774851i \(0.282175\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −13.5982 −0.567577
\(575\) 8.87488 0.370108
\(576\) 0 0
\(577\) 44.1976 1.83997 0.919985 0.391955i \(-0.128201\pi\)
0.919985 + 0.391955i \(0.128201\pi\)
\(578\) 38.1436 1.58657
\(579\) 0 0
\(580\) −17.7173 −0.735669
\(581\) 9.40661 0.390252
\(582\) 0 0
\(583\) 0 0
\(584\) −22.4505 −0.929007
\(585\) 0 0
\(586\) 7.27158 0.300386
\(587\) 7.43693 0.306955 0.153477 0.988152i \(-0.450953\pi\)
0.153477 + 0.988152i \(0.450953\pi\)
\(588\) 0 0
\(589\) −38.3263 −1.57921
\(590\) 23.9836 0.987391
\(591\) 0 0
\(592\) −28.2977 −1.16303
\(593\) 30.7149 1.26131 0.630656 0.776063i \(-0.282786\pi\)
0.630656 + 0.776063i \(0.282786\pi\)
\(594\) 0 0
\(595\) −1.63706 −0.0671128
\(596\) −48.4063 −1.98280
\(597\) 0 0
\(598\) 16.8145 0.687597
\(599\) −11.3859 −0.465217 −0.232609 0.972570i \(-0.574726\pi\)
−0.232609 + 0.972570i \(0.574726\pi\)
\(600\) 0 0
\(601\) 4.93519 0.201311 0.100655 0.994921i \(-0.467906\pi\)
0.100655 + 0.994921i \(0.467906\pi\)
\(602\) 25.7527 1.04960
\(603\) 0 0
\(604\) −76.9948 −3.13287
\(605\) 0 0
\(606\) 0 0
\(607\) −2.24652 −0.0911836 −0.0455918 0.998960i \(-0.514517\pi\)
−0.0455918 + 0.998960i \(0.514517\pi\)
\(608\) 4.72138 0.191477
\(609\) 0 0
\(610\) −29.5720 −1.19734
\(611\) 29.7731 1.20449
\(612\) 0 0
\(613\) 26.3261 1.06330 0.531651 0.846964i \(-0.321572\pi\)
0.531651 + 0.846964i \(0.321572\pi\)
\(614\) −40.3811 −1.62965
\(615\) 0 0
\(616\) 0 0
\(617\) −6.08323 −0.244901 −0.122451 0.992475i \(-0.539075\pi\)
−0.122451 + 0.992475i \(0.539075\pi\)
\(618\) 0 0
\(619\) −17.6732 −0.710345 −0.355172 0.934801i \(-0.615578\pi\)
−0.355172 + 0.934801i \(0.615578\pi\)
\(620\) 39.1341 1.57166
\(621\) 0 0
\(622\) 43.8244 1.75720
\(623\) −4.35911 −0.174644
\(624\) 0 0
\(625\) −1.52021 −0.0608085
\(626\) −78.1173 −3.12219
\(627\) 0 0
\(628\) −6.02038 −0.240239
\(629\) 9.34571 0.372638
\(630\) 0 0
\(631\) −15.2270 −0.606178 −0.303089 0.952962i \(-0.598018\pi\)
−0.303089 + 0.952962i \(0.598018\pi\)
\(632\) −62.8199 −2.49884
\(633\) 0 0
\(634\) −63.0310 −2.50328
\(635\) −6.83293 −0.271157
\(636\) 0 0
\(637\) 2.30114 0.0911746
\(638\) 0 0
\(639\) 0 0
\(640\) −28.9200 −1.14316
\(641\) −40.3827 −1.59502 −0.797510 0.603305i \(-0.793850\pi\)
−0.797510 + 0.603305i \(0.793850\pi\)
\(642\) 0 0
\(643\) −0.813217 −0.0320701 −0.0160351 0.999871i \(-0.505104\pi\)
−0.0160351 + 0.999871i \(0.505104\pi\)
\(644\) 11.7509 0.463052
\(645\) 0 0
\(646\) 15.2361 0.599457
\(647\) −30.9476 −1.21667 −0.608337 0.793679i \(-0.708163\pi\)
−0.608337 + 0.793679i \(0.708163\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 16.5159 0.647807
\(651\) 0 0
\(652\) 96.6941 3.78683
\(653\) −7.68593 −0.300774 −0.150387 0.988627i \(-0.548052\pi\)
−0.150387 + 0.988627i \(0.548052\pi\)
\(654\) 0 0
\(655\) −23.7121 −0.926507
\(656\) −19.3777 −0.756573
\(657\) 0 0
\(658\) 31.4520 1.22613
\(659\) −8.17999 −0.318647 −0.159324 0.987226i \(-0.550931\pi\)
−0.159324 + 0.987226i \(0.550931\pi\)
\(660\) 0 0
\(661\) 30.5381 1.18780 0.593898 0.804540i \(-0.297588\pi\)
0.593898 + 0.804540i \(0.297588\pi\)
\(662\) 25.8886 1.00619
\(663\) 0 0
\(664\) 43.6595 1.69432
\(665\) 7.83909 0.303987
\(666\) 0 0
\(667\) −9.52043 −0.368633
\(668\) 47.7016 1.84563
\(669\) 0 0
\(670\) 26.4848 1.02320
\(671\) 0 0
\(672\) 0 0
\(673\) 34.4546 1.32813 0.664063 0.747676i \(-0.268831\pi\)
0.664063 + 0.747676i \(0.268831\pi\)
\(674\) −35.2840 −1.35909
\(675\) 0 0
\(676\) −30.1202 −1.15847
\(677\) 40.6807 1.56349 0.781744 0.623600i \(-0.214331\pi\)
0.781744 + 0.623600i \(0.214331\pi\)
\(678\) 0 0
\(679\) 6.56067 0.251776
\(680\) −7.59817 −0.291377
\(681\) 0 0
\(682\) 0 0
\(683\) 38.4349 1.47067 0.735336 0.677703i \(-0.237024\pi\)
0.735336 + 0.677703i \(0.237024\pi\)
\(684\) 0 0
\(685\) 13.8043 0.527436
\(686\) 2.43091 0.0928125
\(687\) 0 0
\(688\) 36.6982 1.39911
\(689\) −21.3609 −0.813785
\(690\) 0 0
\(691\) −13.8850 −0.528211 −0.264105 0.964494i \(-0.585077\pi\)
−0.264105 + 0.964494i \(0.585077\pi\)
\(692\) 48.2747 1.83513
\(693\) 0 0
\(694\) 35.7498 1.35704
\(695\) 29.9986 1.13791
\(696\) 0 0
\(697\) 6.39977 0.242408
\(698\) 4.60775 0.174406
\(699\) 0 0
\(700\) 11.5423 0.436256
\(701\) −44.2982 −1.67312 −0.836560 0.547876i \(-0.815437\pi\)
−0.836560 + 0.547876i \(0.815437\pi\)
\(702\) 0 0
\(703\) −44.7522 −1.68786
\(704\) 0 0
\(705\) 0 0
\(706\) 56.1573 2.11351
\(707\) 10.1239 0.380748
\(708\) 0 0
\(709\) 40.8278 1.53332 0.766660 0.642053i \(-0.221917\pi\)
0.766660 + 0.642053i \(0.221917\pi\)
\(710\) 6.69112 0.251113
\(711\) 0 0
\(712\) −20.2322 −0.758234
\(713\) 21.0288 0.787536
\(714\) 0 0
\(715\) 0 0
\(716\) −69.2091 −2.58646
\(717\) 0 0
\(718\) −10.2066 −0.380908
\(719\) 30.4882 1.13702 0.568510 0.822676i \(-0.307520\pi\)
0.568510 + 0.822676i \(0.307520\pi\)
\(720\) 0 0
\(721\) 5.00158 0.186268
\(722\) −26.7714 −0.996328
\(723\) 0 0
\(724\) −22.0750 −0.820411
\(725\) −9.35136 −0.347301
\(726\) 0 0
\(727\) 26.4661 0.981572 0.490786 0.871280i \(-0.336710\pi\)
0.490786 + 0.871280i \(0.336710\pi\)
\(728\) 10.6804 0.395843
\(729\) 0 0
\(730\) 16.8252 0.622728
\(731\) −12.1201 −0.448278
\(732\) 0 0
\(733\) 1.68895 0.0623826 0.0311913 0.999513i \(-0.490070\pi\)
0.0311913 + 0.999513i \(0.490070\pi\)
\(734\) 7.17295 0.264759
\(735\) 0 0
\(736\) −2.59052 −0.0954878
\(737\) 0 0
\(738\) 0 0
\(739\) 43.5644 1.60254 0.801270 0.598302i \(-0.204158\pi\)
0.801270 + 0.598302i \(0.204158\pi\)
\(740\) 45.6954 1.67980
\(741\) 0 0
\(742\) −22.5655 −0.828404
\(743\) 9.89726 0.363095 0.181548 0.983382i \(-0.441889\pi\)
0.181548 + 0.983382i \(0.441889\pi\)
\(744\) 0 0
\(745\) 17.7179 0.649135
\(746\) 52.6695 1.92837
\(747\) 0 0
\(748\) 0 0
\(749\) −17.6186 −0.643771
\(750\) 0 0
\(751\) −26.8450 −0.979589 −0.489795 0.871838i \(-0.662928\pi\)
−0.489795 + 0.871838i \(0.662928\pi\)
\(752\) 44.8199 1.63441
\(753\) 0 0
\(754\) −17.7173 −0.645225
\(755\) 28.1820 1.02565
\(756\) 0 0
\(757\) 21.5277 0.782437 0.391218 0.920298i \(-0.372054\pi\)
0.391218 + 0.920298i \(0.372054\pi\)
\(758\) −71.2658 −2.58849
\(759\) 0 0
\(760\) 36.3841 1.31979
\(761\) 29.3264 1.06308 0.531540 0.847033i \(-0.321613\pi\)
0.531540 + 0.847033i \(0.321613\pi\)
\(762\) 0 0
\(763\) 12.5813 0.455475
\(764\) 26.8268 0.970561
\(765\) 0 0
\(766\) 90.5089 3.27022
\(767\) 15.8664 0.572903
\(768\) 0 0
\(769\) −30.6982 −1.10701 −0.553503 0.832847i \(-0.686709\pi\)
−0.553503 + 0.832847i \(0.686709\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 3.14249 0.113101
\(773\) 23.7252 0.853336 0.426668 0.904408i \(-0.359687\pi\)
0.426668 + 0.904408i \(0.359687\pi\)
\(774\) 0 0
\(775\) 20.6554 0.741963
\(776\) 30.4505 1.09311
\(777\) 0 0
\(778\) −71.4368 −2.56113
\(779\) −30.6455 −1.09799
\(780\) 0 0
\(781\) 0 0
\(782\) −8.35974 −0.298944
\(783\) 0 0
\(784\) 3.46410 0.123718
\(785\) 2.20361 0.0786503
\(786\) 0 0
\(787\) 1.87454 0.0668202 0.0334101 0.999442i \(-0.489363\pi\)
0.0334101 + 0.999442i \(0.489363\pi\)
\(788\) 64.3012 2.29064
\(789\) 0 0
\(790\) 47.0795 1.67501
\(791\) 3.20457 0.113941
\(792\) 0 0
\(793\) −19.5634 −0.694717
\(794\) 45.1023 1.60062
\(795\) 0 0
\(796\) −14.2751 −0.505966
\(797\) 9.58066 0.339365 0.169682 0.985499i \(-0.445726\pi\)
0.169682 + 0.985499i \(0.445726\pi\)
\(798\) 0 0
\(799\) −14.8024 −0.523672
\(800\) −2.54451 −0.0899621
\(801\) 0 0
\(802\) 7.53364 0.266022
\(803\) 0 0
\(804\) 0 0
\(805\) −4.30114 −0.151595
\(806\) 39.1341 1.37844
\(807\) 0 0
\(808\) 46.9886 1.65305
\(809\) 38.1237 1.34036 0.670179 0.742199i \(-0.266217\pi\)
0.670179 + 0.742199i \(0.266217\pi\)
\(810\) 0 0
\(811\) 39.4724 1.38606 0.693032 0.720907i \(-0.256274\pi\)
0.693032 + 0.720907i \(0.256274\pi\)
\(812\) −12.3818 −0.434517
\(813\) 0 0
\(814\) 0 0
\(815\) −35.3925 −1.23975
\(816\) 0 0
\(817\) 58.0375 2.03047
\(818\) −33.0389 −1.15518
\(819\) 0 0
\(820\) 31.2913 1.09274
\(821\) −28.2469 −0.985822 −0.492911 0.870080i \(-0.664067\pi\)
−0.492911 + 0.870080i \(0.664067\pi\)
\(822\) 0 0
\(823\) 37.4954 1.30701 0.653503 0.756924i \(-0.273299\pi\)
0.653503 + 0.756924i \(0.273299\pi\)
\(824\) 23.2141 0.808703
\(825\) 0 0
\(826\) 16.7611 0.583194
\(827\) −19.9210 −0.692722 −0.346361 0.938101i \(-0.612583\pi\)
−0.346361 + 0.938101i \(0.612583\pi\)
\(828\) 0 0
\(829\) 49.7189 1.72681 0.863404 0.504513i \(-0.168328\pi\)
0.863404 + 0.504513i \(0.168328\pi\)
\(830\) −32.7200 −1.13573
\(831\) 0 0
\(832\) −20.7637 −0.719851
\(833\) −1.14407 −0.0396396
\(834\) 0 0
\(835\) −17.4600 −0.604228
\(836\) 0 0
\(837\) 0 0
\(838\) −3.35796 −0.115999
\(839\) −2.57431 −0.0888749 −0.0444375 0.999012i \(-0.514150\pi\)
−0.0444375 + 0.999012i \(0.514150\pi\)
\(840\) 0 0
\(841\) −18.9684 −0.654084
\(842\) −68.4246 −2.35807
\(843\) 0 0
\(844\) 71.3325 2.45536
\(845\) 11.0248 0.379264
\(846\) 0 0
\(847\) 0 0
\(848\) −32.1563 −1.10425
\(849\) 0 0
\(850\) −8.21128 −0.281645
\(851\) 24.5546 0.841721
\(852\) 0 0
\(853\) 40.3968 1.38316 0.691580 0.722300i \(-0.256915\pi\)
0.691580 + 0.722300i \(0.256915\pi\)
\(854\) −20.6666 −0.707197
\(855\) 0 0
\(856\) −81.7745 −2.79500
\(857\) 3.24467 0.110836 0.0554179 0.998463i \(-0.482351\pi\)
0.0554179 + 0.998463i \(0.482351\pi\)
\(858\) 0 0
\(859\) 6.75730 0.230556 0.115278 0.993333i \(-0.463224\pi\)
0.115278 + 0.993333i \(0.463224\pi\)
\(860\) −59.2607 −2.02077
\(861\) 0 0
\(862\) 26.1941 0.892174
\(863\) −13.7259 −0.467234 −0.233617 0.972329i \(-0.575056\pi\)
−0.233617 + 0.972329i \(0.575056\pi\)
\(864\) 0 0
\(865\) −17.6698 −0.600790
\(866\) 35.8598 1.21857
\(867\) 0 0
\(868\) 27.3491 0.928289
\(869\) 0 0
\(870\) 0 0
\(871\) 17.5210 0.593677
\(872\) 58.3946 1.97749
\(873\) 0 0
\(874\) 40.0309 1.35406
\(875\) −11.3793 −0.384691
\(876\) 0 0
\(877\) −9.63610 −0.325388 −0.162694 0.986677i \(-0.552018\pi\)
−0.162694 + 0.986677i \(0.552018\pi\)
\(878\) −11.0982 −0.374546
\(879\) 0 0
\(880\) 0 0
\(881\) −24.2777 −0.817937 −0.408968 0.912549i \(-0.634111\pi\)
−0.408968 + 0.912549i \(0.634111\pi\)
\(882\) 0 0
\(883\) −12.8564 −0.432653 −0.216326 0.976321i \(-0.569407\pi\)
−0.216326 + 0.976321i \(0.569407\pi\)
\(884\) −10.2919 −0.346154
\(885\) 0 0
\(886\) 46.5034 1.56231
\(887\) 47.1594 1.58346 0.791729 0.610873i \(-0.209181\pi\)
0.791729 + 0.610873i \(0.209181\pi\)
\(888\) 0 0
\(889\) −4.77524 −0.160156
\(890\) 15.1628 0.508257
\(891\) 0 0
\(892\) −95.0604 −3.18286
\(893\) 70.8818 2.37197
\(894\) 0 0
\(895\) 25.3323 0.846765
\(896\) −20.2109 −0.675200
\(897\) 0 0
\(898\) −66.6910 −2.22551
\(899\) −22.1578 −0.739005
\(900\) 0 0
\(901\) 10.6201 0.353806
\(902\) 0 0
\(903\) 0 0
\(904\) 14.8736 0.494688
\(905\) 8.08001 0.268589
\(906\) 0 0
\(907\) −4.99890 −0.165986 −0.0829928 0.996550i \(-0.526448\pi\)
−0.0829928 + 0.996550i \(0.526448\pi\)
\(908\) −35.4993 −1.17808
\(909\) 0 0
\(910\) −8.00431 −0.265340
\(911\) −24.2373 −0.803017 −0.401509 0.915855i \(-0.631514\pi\)
−0.401509 + 0.915855i \(0.631514\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 13.5704 0.448869
\(915\) 0 0
\(916\) 4.82088 0.159286
\(917\) −16.5713 −0.547234
\(918\) 0 0
\(919\) 37.5240 1.23780 0.618901 0.785469i \(-0.287578\pi\)
0.618901 + 0.785469i \(0.287578\pi\)
\(920\) −19.9632 −0.658166
\(921\) 0 0
\(922\) −76.0674 −2.50515
\(923\) 4.42652 0.145701
\(924\) 0 0
\(925\) 24.1185 0.793012
\(926\) −50.0232 −1.64386
\(927\) 0 0
\(928\) 2.72960 0.0896035
\(929\) 20.8638 0.684519 0.342259 0.939606i \(-0.388808\pi\)
0.342259 + 0.939606i \(0.388808\pi\)
\(930\) 0 0
\(931\) 5.47841 0.179547
\(932\) 35.7309 1.17041
\(933\) 0 0
\(934\) −99.2345 −3.24705
\(935\) 0 0
\(936\) 0 0
\(937\) −0.463822 −0.0151524 −0.00757620 0.999971i \(-0.502412\pi\)
−0.00757620 + 0.999971i \(0.502412\pi\)
\(938\) 18.5091 0.604342
\(939\) 0 0
\(940\) −72.3757 −2.36063
\(941\) −23.6522 −0.771039 −0.385519 0.922700i \(-0.625978\pi\)
−0.385519 + 0.922700i \(0.625978\pi\)
\(942\) 0 0
\(943\) 16.8145 0.547556
\(944\) 23.8850 0.777391
\(945\) 0 0
\(946\) 0 0
\(947\) 28.3269 0.920499 0.460250 0.887789i \(-0.347760\pi\)
0.460250 + 0.887789i \(0.347760\pi\)
\(948\) 0 0
\(949\) 11.1307 0.361319
\(950\) 39.3199 1.27571
\(951\) 0 0
\(952\) −5.31004 −0.172099
\(953\) 48.2418 1.56270 0.781352 0.624090i \(-0.214530\pi\)
0.781352 + 0.624090i \(0.214530\pi\)
\(954\) 0 0
\(955\) −9.81931 −0.317745
\(956\) −100.760 −3.25883
\(957\) 0 0
\(958\) 64.6547 2.08890
\(959\) 9.64725 0.311526
\(960\) 0 0
\(961\) 17.9424 0.578789
\(962\) 45.6954 1.47328
\(963\) 0 0
\(964\) 14.4886 0.466646
\(965\) −1.15023 −0.0370273
\(966\) 0 0
\(967\) 35.1632 1.13077 0.565387 0.824826i \(-0.308727\pi\)
0.565387 + 0.824826i \(0.308727\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −22.8207 −0.732728
\(971\) 54.0629 1.73496 0.867480 0.497471i \(-0.165738\pi\)
0.867480 + 0.497471i \(0.165738\pi\)
\(972\) 0 0
\(973\) 20.9647 0.672099
\(974\) −57.4059 −1.83940
\(975\) 0 0
\(976\) −29.4504 −0.942685
\(977\) 42.2318 1.35111 0.675557 0.737307i \(-0.263903\pi\)
0.675557 + 0.737307i \(0.263903\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −5.59387 −0.178690
\(981\) 0 0
\(982\) 99.1920 3.16534
\(983\) 0.876060 0.0279420 0.0139710 0.999902i \(-0.495553\pi\)
0.0139710 + 0.999902i \(0.495553\pi\)
\(984\) 0 0
\(985\) −23.5359 −0.749916
\(986\) 8.80856 0.280522
\(987\) 0 0
\(988\) 49.2831 1.56790
\(989\) −31.8439 −1.01258
\(990\) 0 0
\(991\) −41.7171 −1.32519 −0.662594 0.748979i \(-0.730544\pi\)
−0.662594 + 0.748979i \(0.730544\pi\)
\(992\) −6.02917 −0.191426
\(993\) 0 0
\(994\) 4.67613 0.148318
\(995\) 5.22504 0.165645
\(996\) 0 0
\(997\) 7.75731 0.245676 0.122838 0.992427i \(-0.460800\pi\)
0.122838 + 0.992427i \(0.460800\pi\)
\(998\) −23.2686 −0.736554
\(999\) 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.cg.1.1 4
3.2 odd 2 2541.2.a.bp.1.4 yes 4
11.10 odd 2 7623.2.a.cn.1.4 4
33.32 even 2 2541.2.a.bl.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.bl.1.1 4 33.32 even 2
2541.2.a.bp.1.4 yes 4 3.2 odd 2
7623.2.a.cg.1.1 4 1.1 even 1 trivial
7623.2.a.cn.1.4 4 11.10 odd 2