Properties

Label 7623.2.a.ck.1.2
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 4
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{7})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.16372\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421 q^{2} +2.64575 q^{5} +1.00000 q^{7} +2.82843 q^{8} +O(q^{10})\) \(q-1.41421 q^{2} +2.64575 q^{5} +1.00000 q^{7} +2.82843 q^{8} -3.74166 q^{10} -5.74166 q^{13} -1.41421 q^{14} -4.00000 q^{16} -5.47418 q^{17} +5.74166 q^{19} -3.87729 q^{23} +2.00000 q^{25} +8.11993 q^{26} -3.87729 q^{29} +5.48331 q^{31} +7.74166 q^{34} +2.64575 q^{35} +3.48331 q^{37} -8.11993 q^{38} +7.48331 q^{40} +5.65685 q^{41} -1.00000 q^{43} +5.48331 q^{46} -5.47418 q^{47} +1.00000 q^{49} -2.82843 q^{50} +13.4114 q^{53} +2.82843 q^{56} +5.48331 q^{58} +8.30261 q^{59} +1.74166 q^{61} -7.75458 q^{62} +8.00000 q^{64} -15.1910 q^{65} -6.48331 q^{67} -3.74166 q^{70} -1.41421 q^{71} -15.7417 q^{73} -4.92615 q^{74} -13.4833 q^{79} -10.5830 q^{80} -8.00000 q^{82} -11.1310 q^{83} -14.4833 q^{85} +1.41421 q^{86} +5.83953 q^{89} -5.74166 q^{91} +7.74166 q^{94} +15.1910 q^{95} -7.22497 q^{97} -1.41421 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} + O(q^{10}) \) \( 4q + 4q^{7} - 8q^{13} - 16q^{16} + 8q^{19} + 8q^{25} - 8q^{31} + 16q^{34} - 16q^{37} - 4q^{43} - 8q^{46} + 4q^{49} - 8q^{58} - 8q^{61} + 32q^{64} + 4q^{67} - 48q^{73} - 24q^{79} - 32q^{82} - 28q^{85} - 8q^{91} + 16q^{94} + 16q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 0 0
\(4\) 0 0
\(5\) 2.64575 1.18322 0.591608 0.806226i \(-0.298493\pi\)
0.591608 + 0.806226i \(0.298493\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 2.82843 1.00000
\(9\) 0 0
\(10\) −3.74166 −1.18322
\(11\) 0 0
\(12\) 0 0
\(13\) −5.74166 −1.59245 −0.796225 0.605001i \(-0.793173\pi\)
−0.796225 + 0.605001i \(0.793173\pi\)
\(14\) −1.41421 −0.377964
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −5.47418 −1.32768 −0.663842 0.747873i \(-0.731075\pi\)
−0.663842 + 0.747873i \(0.731075\pi\)
\(18\) 0 0
\(19\) 5.74166 1.31723 0.658613 0.752482i \(-0.271143\pi\)
0.658613 + 0.752482i \(0.271143\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.87729 −0.808471 −0.404235 0.914655i \(-0.632462\pi\)
−0.404235 + 0.914655i \(0.632462\pi\)
\(24\) 0 0
\(25\) 2.00000 0.400000
\(26\) 8.11993 1.59245
\(27\) 0 0
\(28\) 0 0
\(29\) −3.87729 −0.719995 −0.359997 0.932953i \(-0.617222\pi\)
−0.359997 + 0.932953i \(0.617222\pi\)
\(30\) 0 0
\(31\) 5.48331 0.984832 0.492416 0.870360i \(-0.336114\pi\)
0.492416 + 0.870360i \(0.336114\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 7.74166 1.32768
\(35\) 2.64575 0.447214
\(36\) 0 0
\(37\) 3.48331 0.572653 0.286327 0.958132i \(-0.407566\pi\)
0.286327 + 0.958132i \(0.407566\pi\)
\(38\) −8.11993 −1.31723
\(39\) 0 0
\(40\) 7.48331 1.18322
\(41\) 5.65685 0.883452 0.441726 0.897150i \(-0.354366\pi\)
0.441726 + 0.897150i \(0.354366\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 5.48331 0.808471
\(47\) −5.47418 −0.798491 −0.399245 0.916844i \(-0.630728\pi\)
−0.399245 + 0.916844i \(0.630728\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −2.82843 −0.400000
\(51\) 0 0
\(52\) 0 0
\(53\) 13.4114 1.84220 0.921101 0.389324i \(-0.127291\pi\)
0.921101 + 0.389324i \(0.127291\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.82843 0.377964
\(57\) 0 0
\(58\) 5.48331 0.719995
\(59\) 8.30261 1.08091 0.540454 0.841374i \(-0.318253\pi\)
0.540454 + 0.841374i \(0.318253\pi\)
\(60\) 0 0
\(61\) 1.74166 0.222996 0.111498 0.993765i \(-0.464435\pi\)
0.111498 + 0.993765i \(0.464435\pi\)
\(62\) −7.75458 −0.984832
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) −15.1910 −1.88421
\(66\) 0 0
\(67\) −6.48331 −0.792063 −0.396031 0.918237i \(-0.629613\pi\)
−0.396031 + 0.918237i \(0.629613\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −3.74166 −0.447214
\(71\) −1.41421 −0.167836 −0.0839181 0.996473i \(-0.526743\pi\)
−0.0839181 + 0.996473i \(0.526743\pi\)
\(72\) 0 0
\(73\) −15.7417 −1.84242 −0.921211 0.389064i \(-0.872799\pi\)
−0.921211 + 0.389064i \(0.872799\pi\)
\(74\) −4.92615 −0.572653
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −13.4833 −1.51699 −0.758496 0.651678i \(-0.774065\pi\)
−0.758496 + 0.651678i \(0.774065\pi\)
\(80\) −10.5830 −1.18322
\(81\) 0 0
\(82\) −8.00000 −0.883452
\(83\) −11.1310 −1.22179 −0.610895 0.791712i \(-0.709190\pi\)
−0.610895 + 0.791712i \(0.709190\pi\)
\(84\) 0 0
\(85\) −14.4833 −1.57094
\(86\) 1.41421 0.152499
\(87\) 0 0
\(88\) 0 0
\(89\) 5.83953 0.618989 0.309494 0.950901i \(-0.399840\pi\)
0.309494 + 0.950901i \(0.399840\pi\)
\(90\) 0 0
\(91\) −5.74166 −0.601889
\(92\) 0 0
\(93\) 0 0
\(94\) 7.74166 0.798491
\(95\) 15.1910 1.55856
\(96\) 0 0
\(97\) −7.22497 −0.733585 −0.366792 0.930303i \(-0.619544\pi\)
−0.366792 + 0.930303i \(0.619544\pi\)
\(98\) −1.41421 −0.142857
\(99\) 0 0
\(100\) 0 0
\(101\) 8.66796 0.862494 0.431247 0.902234i \(-0.358074\pi\)
0.431247 + 0.902234i \(0.358074\pi\)
\(102\) 0 0
\(103\) −13.2250 −1.30310 −0.651548 0.758608i \(-0.725880\pi\)
−0.651548 + 0.758608i \(0.725880\pi\)
\(104\) −16.2399 −1.59245
\(105\) 0 0
\(106\) −18.9666 −1.84220
\(107\) 9.53414 0.921700 0.460850 0.887478i \(-0.347545\pi\)
0.460850 + 0.887478i \(0.347545\pi\)
\(108\) 0 0
\(109\) 16.4833 1.57882 0.789408 0.613869i \(-0.210388\pi\)
0.789408 + 0.613869i \(0.210388\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) 5.29150 0.497783 0.248891 0.968531i \(-0.419934\pi\)
0.248891 + 0.968531i \(0.419934\pi\)
\(114\) 0 0
\(115\) −10.2583 −0.956595
\(116\) 0 0
\(117\) 0 0
\(118\) −11.7417 −1.08091
\(119\) −5.47418 −0.501817
\(120\) 0 0
\(121\) 0 0
\(122\) −2.46308 −0.222996
\(123\) 0 0
\(124\) 0 0
\(125\) −7.93725 −0.709930
\(126\) 0 0
\(127\) −16.4833 −1.46266 −0.731329 0.682025i \(-0.761100\pi\)
−0.731329 + 0.682025i \(0.761100\pi\)
\(128\) −11.3137 −1.00000
\(129\) 0 0
\(130\) 21.4833 1.88421
\(131\) −8.66796 −0.757323 −0.378661 0.925535i \(-0.623616\pi\)
−0.378661 + 0.925535i \(0.623616\pi\)
\(132\) 0 0
\(133\) 5.74166 0.497865
\(134\) 9.16879 0.792063
\(135\) 0 0
\(136\) −15.4833 −1.32768
\(137\) 8.80344 0.752129 0.376064 0.926594i \(-0.377277\pi\)
0.376064 + 0.926594i \(0.377277\pi\)
\(138\) 0 0
\(139\) 3.74166 0.317363 0.158682 0.987330i \(-0.449276\pi\)
0.158682 + 0.987330i \(0.449276\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.00000 0.167836
\(143\) 0 0
\(144\) 0 0
\(145\) −10.2583 −0.851909
\(146\) 22.2621 1.84242
\(147\) 0 0
\(148\) 0 0
\(149\) −23.3109 −1.90971 −0.954853 0.297079i \(-0.903987\pi\)
−0.954853 + 0.297079i \(0.903987\pi\)
\(150\) 0 0
\(151\) −7.51669 −0.611699 −0.305850 0.952080i \(-0.598940\pi\)
−0.305850 + 0.952080i \(0.598940\pi\)
\(152\) 16.2399 1.31723
\(153\) 0 0
\(154\) 0 0
\(155\) 14.5075 1.16527
\(156\) 0 0
\(157\) −12.9666 −1.03485 −0.517425 0.855729i \(-0.673109\pi\)
−0.517425 + 0.855729i \(0.673109\pi\)
\(158\) 19.0683 1.51699
\(159\) 0 0
\(160\) 0 0
\(161\) −3.87729 −0.305573
\(162\) 0 0
\(163\) −24.4499 −1.91507 −0.957534 0.288321i \(-0.906903\pi\)
−0.957534 + 0.288321i \(0.906903\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 15.7417 1.22179
\(167\) 8.66796 0.670747 0.335373 0.942085i \(-0.391138\pi\)
0.335373 + 0.942085i \(0.391138\pi\)
\(168\) 0 0
\(169\) 19.9666 1.53589
\(170\) 20.4825 1.57094
\(171\) 0 0
\(172\) 0 0
\(173\) −5.10883 −0.388417 −0.194208 0.980960i \(-0.562214\pi\)
−0.194208 + 0.980960i \(0.562214\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) 0 0
\(178\) −8.25834 −0.618989
\(179\) 14.1421 1.05703 0.528516 0.848923i \(-0.322748\pi\)
0.528516 + 0.848923i \(0.322748\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 8.11993 0.601889
\(183\) 0 0
\(184\) −10.9666 −0.808471
\(185\) 9.21598 0.677573
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) −21.4833 −1.55856
\(191\) −1.41421 −0.102329 −0.0511645 0.998690i \(-0.516293\pi\)
−0.0511645 + 0.998690i \(0.516293\pi\)
\(192\) 0 0
\(193\) −16.4833 −1.18649 −0.593247 0.805020i \(-0.702154\pi\)
−0.593247 + 0.805020i \(0.702154\pi\)
\(194\) 10.2177 0.733585
\(195\) 0 0
\(196\) 0 0
\(197\) −22.2621 −1.58611 −0.793053 0.609152i \(-0.791510\pi\)
−0.793053 + 0.609152i \(0.791510\pi\)
\(198\) 0 0
\(199\) 2.25834 0.160090 0.0800448 0.996791i \(-0.474494\pi\)
0.0800448 + 0.996791i \(0.474494\pi\)
\(200\) 5.65685 0.400000
\(201\) 0 0
\(202\) −12.2583 −0.862494
\(203\) −3.87729 −0.272132
\(204\) 0 0
\(205\) 14.9666 1.04531
\(206\) 18.7029 1.30310
\(207\) 0 0
\(208\) 22.9666 1.59245
\(209\) 0 0
\(210\) 0 0
\(211\) −9.51669 −0.655156 −0.327578 0.944824i \(-0.606232\pi\)
−0.327578 + 0.944824i \(0.606232\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −13.4833 −0.921700
\(215\) −2.64575 −0.180439
\(216\) 0 0
\(217\) 5.48331 0.372232
\(218\) −23.3109 −1.57882
\(219\) 0 0
\(220\) 0 0
\(221\) 31.4309 2.11427
\(222\) 0 0
\(223\) 17.4833 1.17077 0.585385 0.810756i \(-0.300943\pi\)
0.585385 + 0.810756i \(0.300943\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −7.48331 −0.497783
\(227\) −27.7362 −1.84092 −0.920460 0.390838i \(-0.872185\pi\)
−0.920460 + 0.390838i \(0.872185\pi\)
\(228\) 0 0
\(229\) 13.4833 0.891003 0.445501 0.895281i \(-0.353025\pi\)
0.445501 + 0.895281i \(0.353025\pi\)
\(230\) 14.5075 0.956595
\(231\) 0 0
\(232\) −10.9666 −0.719995
\(233\) −6.34036 −0.415371 −0.207686 0.978196i \(-0.566593\pi\)
−0.207686 + 0.978196i \(0.566593\pi\)
\(234\) 0 0
\(235\) −14.4833 −0.944787
\(236\) 0 0
\(237\) 0 0
\(238\) 7.74166 0.501817
\(239\) 8.11993 0.525235 0.262617 0.964900i \(-0.415414\pi\)
0.262617 + 0.964900i \(0.415414\pi\)
\(240\) 0 0
\(241\) 3.22497 0.207739 0.103869 0.994591i \(-0.466878\pi\)
0.103869 + 0.994591i \(0.466878\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.64575 0.169031
\(246\) 0 0
\(247\) −32.9666 −2.09762
\(248\) 15.5092 0.984832
\(249\) 0 0
\(250\) 11.2250 0.709930
\(251\) −2.09772 −0.132407 −0.0662036 0.997806i \(-0.521089\pi\)
−0.0662036 + 0.997806i \(0.521089\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 23.3109 1.46266
\(255\) 0 0
\(256\) 0 0
\(257\) 0.548027 0.0341850 0.0170925 0.999854i \(-0.494559\pi\)
0.0170925 + 0.999854i \(0.494559\pi\)
\(258\) 0 0
\(259\) 3.48331 0.216443
\(260\) 0 0
\(261\) 0 0
\(262\) 12.2583 0.757323
\(263\) 20.1171 1.24048 0.620238 0.784413i \(-0.287036\pi\)
0.620238 + 0.784413i \(0.287036\pi\)
\(264\) 0 0
\(265\) 35.4833 2.17972
\(266\) −8.11993 −0.497865
\(267\) 0 0
\(268\) 0 0
\(269\) −24.3598 −1.48524 −0.742621 0.669712i \(-0.766418\pi\)
−0.742621 + 0.669712i \(0.766418\pi\)
\(270\) 0 0
\(271\) −8.96663 −0.544684 −0.272342 0.962201i \(-0.587798\pi\)
−0.272342 + 0.962201i \(0.587798\pi\)
\(272\) 21.8967 1.32768
\(273\) 0 0
\(274\) −12.4499 −0.752129
\(275\) 0 0
\(276\) 0 0
\(277\) −19.9666 −1.19968 −0.599839 0.800121i \(-0.704769\pi\)
−0.599839 + 0.800121i \(0.704769\pi\)
\(278\) −5.29150 −0.317363
\(279\) 0 0
\(280\) 7.48331 0.447214
\(281\) 16.2399 0.968789 0.484394 0.874850i \(-0.339040\pi\)
0.484394 + 0.874850i \(0.339040\pi\)
\(282\) 0 0
\(283\) 8.96663 0.533011 0.266505 0.963833i \(-0.414131\pi\)
0.266505 + 0.963833i \(0.414131\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.65685 0.333914
\(288\) 0 0
\(289\) 12.9666 0.762743
\(290\) 14.5075 0.851909
\(291\) 0 0
\(292\) 0 0
\(293\) 28.1016 1.64171 0.820856 0.571135i \(-0.193497\pi\)
0.820856 + 0.571135i \(0.193497\pi\)
\(294\) 0 0
\(295\) 21.9666 1.27895
\(296\) 9.85230 0.572653
\(297\) 0 0
\(298\) 32.9666 1.90971
\(299\) 22.2621 1.28745
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) 10.6302 0.611699
\(303\) 0 0
\(304\) −22.9666 −1.31723
\(305\) 4.60799 0.263853
\(306\) 0 0
\(307\) −6.70829 −0.382862 −0.191431 0.981506i \(-0.561313\pi\)
−0.191431 + 0.981506i \(0.561313\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −20.5167 −1.16527
\(311\) 15.3265 0.869085 0.434542 0.900651i \(-0.356910\pi\)
0.434542 + 0.900651i \(0.356910\pi\)
\(312\) 0 0
\(313\) −4.51669 −0.255298 −0.127649 0.991819i \(-0.540743\pi\)
−0.127649 + 0.991819i \(0.540743\pi\)
\(314\) 18.3376 1.03485
\(315\) 0 0
\(316\) 0 0
\(317\) 9.53414 0.535491 0.267745 0.963490i \(-0.413721\pi\)
0.267745 + 0.963490i \(0.413721\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 21.1660 1.18322
\(321\) 0 0
\(322\) 5.48331 0.305573
\(323\) −31.4309 −1.74886
\(324\) 0 0
\(325\) −11.4833 −0.636980
\(326\) 34.5774 1.91507
\(327\) 0 0
\(328\) 16.0000 0.883452
\(329\) −5.47418 −0.301801
\(330\) 0 0
\(331\) 20.4833 1.12586 0.562932 0.826503i \(-0.309673\pi\)
0.562932 + 0.826503i \(0.309673\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −12.2583 −0.670747
\(335\) −17.1532 −0.937182
\(336\) 0 0
\(337\) 22.9666 1.25107 0.625536 0.780195i \(-0.284880\pi\)
0.625536 + 0.780195i \(0.284880\pi\)
\(338\) −28.2371 −1.53589
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −2.82843 −0.152499
\(345\) 0 0
\(346\) 7.22497 0.388417
\(347\) −33.1632 −1.78030 −0.890148 0.455672i \(-0.849399\pi\)
−0.890148 + 0.455672i \(0.849399\pi\)
\(348\) 0 0
\(349\) −33.4833 −1.79232 −0.896160 0.443730i \(-0.853655\pi\)
−0.896160 + 0.443730i \(0.853655\pi\)
\(350\) −2.82843 −0.151186
\(351\) 0 0
\(352\) 0 0
\(353\) −22.9928 −1.22378 −0.611891 0.790942i \(-0.709591\pi\)
−0.611891 + 0.790942i \(0.709591\pi\)
\(354\) 0 0
\(355\) −3.74166 −0.198587
\(356\) 0 0
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) 14.0949 0.743903 0.371951 0.928252i \(-0.378689\pi\)
0.371951 + 0.928252i \(0.378689\pi\)
\(360\) 0 0
\(361\) 13.9666 0.735086
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −41.6485 −2.17998
\(366\) 0 0
\(367\) 17.2250 0.899136 0.449568 0.893246i \(-0.351578\pi\)
0.449568 + 0.893246i \(0.351578\pi\)
\(368\) 15.5092 0.808471
\(369\) 0 0
\(370\) −13.0334 −0.677573
\(371\) 13.4114 0.696287
\(372\) 0 0
\(373\) −19.4499 −1.00708 −0.503540 0.863972i \(-0.667969\pi\)
−0.503540 + 0.863972i \(0.667969\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −15.4833 −0.798491
\(377\) 22.2621 1.14655
\(378\) 0 0
\(379\) −34.4833 −1.77129 −0.885644 0.464364i \(-0.846283\pi\)
−0.885644 + 0.464364i \(0.846283\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 2.00000 0.102329
\(383\) −21.3487 −1.09087 −0.545433 0.838154i \(-0.683635\pi\)
−0.545433 + 0.838154i \(0.683635\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 23.3109 1.18649
\(387\) 0 0
\(388\) 0 0
\(389\) −33.5758 −1.70236 −0.851180 0.524874i \(-0.824112\pi\)
−0.851180 + 0.524874i \(0.824112\pi\)
\(390\) 0 0
\(391\) 21.2250 1.07339
\(392\) 2.82843 0.142857
\(393\) 0 0
\(394\) 31.4833 1.58611
\(395\) −35.6735 −1.79493
\(396\) 0 0
\(397\) 1.48331 0.0744454 0.0372227 0.999307i \(-0.488149\pi\)
0.0372227 + 0.999307i \(0.488149\pi\)
\(398\) −3.19378 −0.160090
\(399\) 0 0
\(400\) −8.00000 −0.400000
\(401\) 8.11993 0.405490 0.202745 0.979232i \(-0.435014\pi\)
0.202745 + 0.979232i \(0.435014\pi\)
\(402\) 0 0
\(403\) −31.4833 −1.56830
\(404\) 0 0
\(405\) 0 0
\(406\) 5.48331 0.272132
\(407\) 0 0
\(408\) 0 0
\(409\) 9.22497 0.456146 0.228073 0.973644i \(-0.426758\pi\)
0.228073 + 0.973644i \(0.426758\pi\)
\(410\) −21.1660 −1.04531
\(411\) 0 0
\(412\) 0 0
\(413\) 8.30261 0.408545
\(414\) 0 0
\(415\) −29.4499 −1.44564
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −29.4686 −1.43964 −0.719818 0.694163i \(-0.755775\pi\)
−0.719818 + 0.694163i \(0.755775\pi\)
\(420\) 0 0
\(421\) −7.51669 −0.366341 −0.183170 0.983081i \(-0.558636\pi\)
−0.183170 + 0.983081i \(0.558636\pi\)
\(422\) 13.4586 0.655156
\(423\) 0 0
\(424\) 37.9333 1.84220
\(425\) −10.9484 −0.531073
\(426\) 0 0
\(427\) 1.74166 0.0842847
\(428\) 0 0
\(429\) 0 0
\(430\) 3.74166 0.180439
\(431\) 15.8745 0.764648 0.382324 0.924028i \(-0.375124\pi\)
0.382324 + 0.924028i \(0.375124\pi\)
\(432\) 0 0
\(433\) −1.74166 −0.0836987 −0.0418494 0.999124i \(-0.513325\pi\)
−0.0418494 + 0.999124i \(0.513325\pi\)
\(434\) −7.75458 −0.372232
\(435\) 0 0
\(436\) 0 0
\(437\) −22.2621 −1.06494
\(438\) 0 0
\(439\) −19.4833 −0.929888 −0.464944 0.885340i \(-0.653926\pi\)
−0.464944 + 0.885340i \(0.653926\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −44.4499 −2.11427
\(443\) −27.2354 −1.29399 −0.646997 0.762493i \(-0.723975\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(444\) 0 0
\(445\) 15.4499 0.732398
\(446\) −24.7251 −1.17077
\(447\) 0 0
\(448\) 8.00000 0.377964
\(449\) 10.2177 0.482201 0.241100 0.970500i \(-0.422492\pi\)
0.241100 + 0.970500i \(0.422492\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 39.2250 1.84092
\(455\) −15.1910 −0.712165
\(456\) 0 0
\(457\) −23.0000 −1.07589 −0.537947 0.842978i \(-0.680800\pi\)
−0.537947 + 0.842978i \(0.680800\pi\)
\(458\) −19.0683 −0.891003
\(459\) 0 0
\(460\) 0 0
\(461\) −7.20655 −0.335643 −0.167821 0.985817i \(-0.553673\pi\)
−0.167821 + 0.985817i \(0.553673\pi\)
\(462\) 0 0
\(463\) −17.4833 −0.812519 −0.406259 0.913758i \(-0.633167\pi\)
−0.406259 + 0.913758i \(0.633167\pi\)
\(464\) 15.5092 0.719995
\(465\) 0 0
\(466\) 8.96663 0.415371
\(467\) 20.8007 0.962540 0.481270 0.876572i \(-0.340176\pi\)
0.481270 + 0.876572i \(0.340176\pi\)
\(468\) 0 0
\(469\) −6.48331 −0.299372
\(470\) 20.4825 0.944787
\(471\) 0 0
\(472\) 23.4833 1.08091
\(473\) 0 0
\(474\) 0 0
\(475\) 11.4833 0.526891
\(476\) 0 0
\(477\) 0 0
\(478\) −11.4833 −0.525235
\(479\) −23.1754 −1.05891 −0.529457 0.848337i \(-0.677604\pi\)
−0.529457 + 0.848337i \(0.677604\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) −4.56080 −0.207739
\(483\) 0 0
\(484\) 0 0
\(485\) −19.1155 −0.867989
\(486\) 0 0
\(487\) 5.51669 0.249985 0.124992 0.992158i \(-0.460109\pi\)
0.124992 + 0.992158i \(0.460109\pi\)
\(488\) 4.92615 0.222996
\(489\) 0 0
\(490\) −3.74166 −0.169031
\(491\) 29.6985 1.34027 0.670137 0.742237i \(-0.266235\pi\)
0.670137 + 0.742237i \(0.266235\pi\)
\(492\) 0 0
\(493\) 21.2250 0.955925
\(494\) 46.6219 2.09762
\(495\) 0 0
\(496\) −21.9333 −0.984832
\(497\) −1.41421 −0.0634361
\(498\) 0 0
\(499\) −17.9666 −0.804297 −0.402148 0.915574i \(-0.631736\pi\)
−0.402148 + 0.915574i \(0.631736\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 2.96663 0.132407
\(503\) 34.3948 1.53359 0.766793 0.641894i \(-0.221851\pi\)
0.766793 + 0.641894i \(0.221851\pi\)
\(504\) 0 0
\(505\) 22.9333 1.02052
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −11.8617 −0.525762 −0.262881 0.964828i \(-0.584673\pi\)
−0.262881 + 0.964828i \(0.584673\pi\)
\(510\) 0 0
\(511\) −15.7417 −0.696370
\(512\) 22.6274 1.00000
\(513\) 0 0
\(514\) −0.775028 −0.0341850
\(515\) −34.9900 −1.54184
\(516\) 0 0
\(517\) 0 0
\(518\) −4.92615 −0.216443
\(519\) 0 0
\(520\) −42.9666 −1.88421
\(521\) −7.20655 −0.315725 −0.157862 0.987461i \(-0.550460\pi\)
−0.157862 + 0.987461i \(0.550460\pi\)
\(522\) 0 0
\(523\) 7.22497 0.315926 0.157963 0.987445i \(-0.449507\pi\)
0.157963 + 0.987445i \(0.449507\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −28.4499 −1.24048
\(527\) −30.0166 −1.30755
\(528\) 0 0
\(529\) −7.96663 −0.346375
\(530\) −50.1810 −2.17972
\(531\) 0 0
\(532\) 0 0
\(533\) −32.4797 −1.40685
\(534\) 0 0
\(535\) 25.2250 1.09057
\(536\) −18.3376 −0.792063
\(537\) 0 0
\(538\) 34.4499 1.48524
\(539\) 0 0
\(540\) 0 0
\(541\) −15.0000 −0.644900 −0.322450 0.946586i \(-0.604506\pi\)
−0.322450 + 0.946586i \(0.604506\pi\)
\(542\) 12.6807 0.544684
\(543\) 0 0
\(544\) 0 0
\(545\) 43.6108 1.86808
\(546\) 0 0
\(547\) 10.0000 0.427569 0.213785 0.976881i \(-0.431421\pi\)
0.213785 + 0.976881i \(0.431421\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −22.2621 −0.948396
\(552\) 0 0
\(553\) −13.4833 −0.573369
\(554\) 28.2371 1.19968
\(555\) 0 0
\(556\) 0 0
\(557\) 46.9872 1.99091 0.995456 0.0952236i \(-0.0303566\pi\)
0.995456 + 0.0952236i \(0.0303566\pi\)
\(558\) 0 0
\(559\) 5.74166 0.242846
\(560\) −10.5830 −0.447214
\(561\) 0 0
\(562\) −22.9666 −0.968789
\(563\) 13.2288 0.557526 0.278763 0.960360i \(-0.410076\pi\)
0.278763 + 0.960360i \(0.410076\pi\)
\(564\) 0 0
\(565\) 14.0000 0.588984
\(566\) −12.6807 −0.533011
\(567\) 0 0
\(568\) −4.00000 −0.167836
\(569\) −18.3848 −0.770730 −0.385365 0.922764i \(-0.625924\pi\)
−0.385365 + 0.922764i \(0.625924\pi\)
\(570\) 0 0
\(571\) −26.9666 −1.12852 −0.564259 0.825598i \(-0.690838\pi\)
−0.564259 + 0.825598i \(0.690838\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −8.00000 −0.333914
\(575\) −7.75458 −0.323388
\(576\) 0 0
\(577\) −18.7083 −0.778836 −0.389418 0.921061i \(-0.627324\pi\)
−0.389418 + 0.921061i \(0.627324\pi\)
\(578\) −18.3376 −0.762743
\(579\) 0 0
\(580\) 0 0
\(581\) −11.1310 −0.461793
\(582\) 0 0
\(583\) 0 0
\(584\) −44.5241 −1.84242
\(585\) 0 0
\(586\) −39.7417 −1.64171
\(587\) 1.54970 0.0639628 0.0319814 0.999488i \(-0.489818\pi\)
0.0319814 + 0.999488i \(0.489818\pi\)
\(588\) 0 0
\(589\) 31.4833 1.29725
\(590\) −31.0655 −1.27895
\(591\) 0 0
\(592\) −13.9333 −0.572653
\(593\) 39.6863 1.62972 0.814860 0.579658i \(-0.196814\pi\)
0.814860 + 0.579658i \(0.196814\pi\)
\(594\) 0 0
\(595\) −14.4833 −0.593758
\(596\) 0 0
\(597\) 0 0
\(598\) −31.4833 −1.28745
\(599\) 29.6513 1.21152 0.605759 0.795648i \(-0.292869\pi\)
0.605759 + 0.795648i \(0.292869\pi\)
\(600\) 0 0
\(601\) −37.2250 −1.51844 −0.759219 0.650835i \(-0.774419\pi\)
−0.759219 + 0.650835i \(0.774419\pi\)
\(602\) 1.41421 0.0576390
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −35.7417 −1.45071 −0.725355 0.688375i \(-0.758324\pi\)
−0.725355 + 0.688375i \(0.758324\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −6.51669 −0.263853
\(611\) 31.4309 1.27156
\(612\) 0 0
\(613\) −26.4833 −1.06965 −0.534826 0.844963i \(-0.679623\pi\)
−0.534826 + 0.844963i \(0.679623\pi\)
\(614\) 9.48695 0.382862
\(615\) 0 0
\(616\) 0 0
\(617\) −2.09772 −0.0844512 −0.0422256 0.999108i \(-0.513445\pi\)
−0.0422256 + 0.999108i \(0.513445\pi\)
\(618\) 0 0
\(619\) 34.4499 1.38466 0.692330 0.721581i \(-0.256584\pi\)
0.692330 + 0.721581i \(0.256584\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −21.6749 −0.869085
\(623\) 5.83953 0.233956
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 6.38756 0.255298
\(627\) 0 0
\(628\) 0 0
\(629\) −19.0683 −0.760302
\(630\) 0 0
\(631\) −25.4499 −1.01315 −0.506573 0.862197i \(-0.669088\pi\)
−0.506573 + 0.862197i \(0.669088\pi\)
\(632\) −38.1366 −1.51699
\(633\) 0 0
\(634\) −13.4833 −0.535491
\(635\) −43.6108 −1.73064
\(636\) 0 0
\(637\) −5.74166 −0.227493
\(638\) 0 0
\(639\) 0 0
\(640\) −29.9333 −1.18322
\(641\) −11.2665 −0.445001 −0.222500 0.974933i \(-0.571422\pi\)
−0.222500 + 0.974933i \(0.571422\pi\)
\(642\) 0 0
\(643\) −40.4499 −1.59519 −0.797595 0.603193i \(-0.793895\pi\)
−0.797595 + 0.603193i \(0.793895\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 44.4499 1.74886
\(647\) 47.9006 1.88317 0.941583 0.336781i \(-0.109338\pi\)
0.941583 + 0.336781i \(0.109338\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 16.2399 0.636980
\(651\) 0 0
\(652\) 0 0
\(653\) 27.8717 1.09070 0.545352 0.838207i \(-0.316396\pi\)
0.545352 + 0.838207i \(0.316396\pi\)
\(654\) 0 0
\(655\) −22.9333 −0.896077
\(656\) −22.6274 −0.883452
\(657\) 0 0
\(658\) 7.74166 0.301801
\(659\) 32.5269 1.26707 0.633534 0.773715i \(-0.281604\pi\)
0.633534 + 0.773715i \(0.281604\pi\)
\(660\) 0 0
\(661\) 40.0000 1.55582 0.777910 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) −28.9678 −1.12586
\(663\) 0 0
\(664\) −31.4833 −1.22179
\(665\) 15.1910 0.589082
\(666\) 0 0
\(667\) 15.0334 0.582094
\(668\) 0 0
\(669\) 0 0
\(670\) 24.2583 0.937182
\(671\) 0 0
\(672\) 0 0
\(673\) 5.44994 0.210080 0.105040 0.994468i \(-0.466503\pi\)
0.105040 + 0.994468i \(0.466503\pi\)
\(674\) −32.4797 −1.25107
\(675\) 0 0
\(676\) 0 0
\(677\) −15.3265 −0.589044 −0.294522 0.955645i \(-0.595161\pi\)
−0.294522 + 0.955645i \(0.595161\pi\)
\(678\) 0 0
\(679\) −7.22497 −0.277269
\(680\) −40.9650 −1.57094
\(681\) 0 0
\(682\) 0 0
\(683\) −14.1421 −0.541134 −0.270567 0.962701i \(-0.587211\pi\)
−0.270567 + 0.962701i \(0.587211\pi\)
\(684\) 0 0
\(685\) 23.2917 0.889931
\(686\) −1.41421 −0.0539949
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −77.0038 −2.93361
\(690\) 0 0
\(691\) −10.4499 −0.397535 −0.198767 0.980047i \(-0.563694\pi\)
−0.198767 + 0.980047i \(0.563694\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 46.8999 1.78030
\(695\) 9.89949 0.375509
\(696\) 0 0
\(697\) −30.9666 −1.17294
\(698\) 47.3526 1.79232
\(699\) 0 0
\(700\) 0 0
\(701\) −4.97334 −0.187841 −0.0939203 0.995580i \(-0.529940\pi\)
−0.0939203 + 0.995580i \(0.529940\pi\)
\(702\) 0 0
\(703\) 20.0000 0.754314
\(704\) 0 0
\(705\) 0 0
\(706\) 32.5167 1.22378
\(707\) 8.66796 0.325992
\(708\) 0 0
\(709\) 40.4166 1.51788 0.758938 0.651163i \(-0.225718\pi\)
0.758938 + 0.651163i \(0.225718\pi\)
\(710\) 5.29150 0.198587
\(711\) 0 0
\(712\) 16.5167 0.618989
\(713\) −21.2604 −0.796208
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) −19.9333 −0.743903
\(719\) −2.82843 −0.105483 −0.0527413 0.998608i \(-0.516796\pi\)
−0.0527413 + 0.998608i \(0.516796\pi\)
\(720\) 0 0
\(721\) −13.2250 −0.492524
\(722\) −19.7518 −0.735086
\(723\) 0 0
\(724\) 0 0
\(725\) −7.75458 −0.287998
\(726\) 0 0
\(727\) 10.1916 0.377986 0.188993 0.981978i \(-0.439478\pi\)
0.188993 + 0.981978i \(0.439478\pi\)
\(728\) −16.2399 −0.601889
\(729\) 0 0
\(730\) 58.8999 2.17998
\(731\) 5.47418 0.202470
\(732\) 0 0
\(733\) 1.29171 0.0477105 0.0238553 0.999715i \(-0.492406\pi\)
0.0238553 + 0.999715i \(0.492406\pi\)
\(734\) −24.3598 −0.899136
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 27.9333 1.02754 0.513771 0.857927i \(-0.328248\pi\)
0.513771 + 0.857927i \(0.328248\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −18.9666 −0.696287
\(743\) 32.1616 1.17989 0.589947 0.807442i \(-0.299149\pi\)
0.589947 + 0.807442i \(0.299149\pi\)
\(744\) 0 0
\(745\) −61.6749 −2.25959
\(746\) 27.5064 1.00708
\(747\) 0 0
\(748\) 0 0
\(749\) 9.53414 0.348370
\(750\) 0 0
\(751\) −31.9666 −1.16648 −0.583239 0.812300i \(-0.698215\pi\)
−0.583239 + 0.812300i \(0.698215\pi\)
\(752\) 21.8967 0.798491
\(753\) 0 0
\(754\) −31.4833 −1.14655
\(755\) −19.8873 −0.723772
\(756\) 0 0
\(757\) −24.4833 −0.889861 −0.444931 0.895565i \(-0.646772\pi\)
−0.444931 + 0.895565i \(0.646772\pi\)
\(758\) 48.7668 1.77129
\(759\) 0 0
\(760\) 42.9666 1.55856
\(761\) 41.5130 1.50485 0.752423 0.658680i \(-0.228885\pi\)
0.752423 + 0.658680i \(0.228885\pi\)
\(762\) 0 0
\(763\) 16.4833 0.596736
\(764\) 0 0
\(765\) 0 0
\(766\) 30.1916 1.09087
\(767\) −47.6707 −1.72129
\(768\) 0 0
\(769\) −3.55006 −0.128018 −0.0640091 0.997949i \(-0.520389\pi\)
−0.0640091 + 0.997949i \(0.520389\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −46.8045 −1.68344 −0.841721 0.539913i \(-0.818457\pi\)
−0.841721 + 0.539913i \(0.818457\pi\)
\(774\) 0 0
\(775\) 10.9666 0.393933
\(776\) −20.4353 −0.733585
\(777\) 0 0
\(778\) 47.4833 1.70236
\(779\) 32.4797 1.16371
\(780\) 0 0
\(781\) 0 0
\(782\) −30.0166 −1.07339
\(783\) 0 0
\(784\) −4.00000 −0.142857
\(785\) −34.3065 −1.22445
\(786\) 0 0
\(787\) −7.55006 −0.269130 −0.134565 0.990905i \(-0.542964\pi\)
−0.134565 + 0.990905i \(0.542964\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 50.4499 1.79493
\(791\) 5.29150 0.188144
\(792\) 0 0
\(793\) −10.0000 −0.355110
\(794\) −2.09772 −0.0744454
\(795\) 0 0
\(796\) 0 0
\(797\) −25.5441 −0.904820 −0.452410 0.891810i \(-0.649436\pi\)
−0.452410 + 0.891810i \(0.649436\pi\)
\(798\) 0 0
\(799\) 29.9666 1.06014
\(800\) 0 0
\(801\) 0 0
\(802\) −11.4833 −0.405490
\(803\) 0 0
\(804\) 0 0
\(805\) −10.2583 −0.361559
\(806\) 44.5241 1.56830
\(807\) 0 0
\(808\) 24.5167 0.862494
\(809\) 10.9955 0.386583 0.193291 0.981141i \(-0.438084\pi\)
0.193291 + 0.981141i \(0.438084\pi\)
\(810\) 0 0
\(811\) −26.7750 −0.940198 −0.470099 0.882614i \(-0.655782\pi\)
−0.470099 + 0.882614i \(0.655782\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −64.6885 −2.26594
\(816\) 0 0
\(817\) −5.74166 −0.200875
\(818\) −13.0461 −0.456146
\(819\) 0 0
\(820\) 0 0
\(821\) −33.9411 −1.18455 −0.592277 0.805735i \(-0.701771\pi\)
−0.592277 + 0.805735i \(0.701771\pi\)
\(822\) 0 0
\(823\) −55.9333 −1.94971 −0.974855 0.222838i \(-0.928468\pi\)
−0.974855 + 0.222838i \(0.928468\pi\)
\(824\) −37.4059 −1.30310
\(825\) 0 0
\(826\) −11.7417 −0.408545
\(827\) 21.4842 0.747078 0.373539 0.927615i \(-0.378144\pi\)
0.373539 + 0.927615i \(0.378144\pi\)
\(828\) 0 0
\(829\) 4.77503 0.165844 0.0829218 0.996556i \(-0.473575\pi\)
0.0829218 + 0.996556i \(0.473575\pi\)
\(830\) 41.6485 1.44564
\(831\) 0 0
\(832\) −45.9333 −1.59245
\(833\) −5.47418 −0.189669
\(834\) 0 0
\(835\) 22.9333 0.793638
\(836\) 0 0
\(837\) 0 0
\(838\) 41.6749 1.43964
\(839\) −22.0794 −0.762265 −0.381133 0.924520i \(-0.624466\pi\)
−0.381133 + 0.924520i \(0.624466\pi\)
\(840\) 0 0
\(841\) −13.9666 −0.481608
\(842\) 10.6302 0.366341
\(843\) 0 0
\(844\) 0 0
\(845\) 52.8267 1.81729
\(846\) 0 0
\(847\) 0 0
\(848\) −53.6457 −1.84220
\(849\) 0 0
\(850\) 15.4833 0.531073
\(851\) −13.5058 −0.462973
\(852\) 0 0
\(853\) 38.9666 1.33419 0.667096 0.744972i \(-0.267537\pi\)
0.667096 + 0.744972i \(0.267537\pi\)
\(854\) −2.46308 −0.0842847
\(855\) 0 0
\(856\) 26.9666 0.921700
\(857\) 12.8634 0.439406 0.219703 0.975567i \(-0.429491\pi\)
0.219703 + 0.975567i \(0.429491\pi\)
\(858\) 0 0
\(859\) −30.4499 −1.03894 −0.519469 0.854489i \(-0.673870\pi\)
−0.519469 + 0.854489i \(0.673870\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −22.4499 −0.764648
\(863\) 18.7029 0.636655 0.318328 0.947981i \(-0.396879\pi\)
0.318328 + 0.947981i \(0.396879\pi\)
\(864\) 0 0
\(865\) −13.5167 −0.459581
\(866\) 2.46308 0.0836987
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 37.2250 1.26132
\(872\) 46.6219 1.57882
\(873\) 0 0
\(874\) 31.4833 1.06494
\(875\) −7.93725 −0.268328
\(876\) 0 0
\(877\) 48.8999 1.65123 0.825616 0.564232i \(-0.190828\pi\)
0.825616 + 0.564232i \(0.190828\pi\)
\(878\) 27.5536 0.929888
\(879\) 0 0
\(880\) 0 0
\(881\) 11.6791 0.393478 0.196739 0.980456i \(-0.436965\pi\)
0.196739 + 0.980456i \(0.436965\pi\)
\(882\) 0 0
\(883\) 52.4166 1.76396 0.881979 0.471289i \(-0.156211\pi\)
0.881979 + 0.471289i \(0.156211\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 38.5167 1.29399
\(887\) 29.4686 0.989459 0.494730 0.869047i \(-0.335267\pi\)
0.494730 + 0.869047i \(0.335267\pi\)
\(888\) 0 0
\(889\) −16.4833 −0.552833
\(890\) −21.8495 −0.732398
\(891\) 0 0
\(892\) 0 0
\(893\) −31.4309 −1.05179
\(894\) 0 0
\(895\) 37.4166 1.25070
\(896\) −11.3137 −0.377964
\(897\) 0 0
\(898\) −14.4499 −0.482201
\(899\) −21.2604 −0.709074
\(900\) 0 0
\(901\) −73.4166 −2.44586
\(902\) 0 0
\(903\) 0 0
\(904\) 14.9666 0.497783
\(905\) 0 0
\(906\) 0 0
\(907\) −1.48331 −0.0492527 −0.0246263 0.999697i \(-0.507840\pi\)
−0.0246263 + 0.999697i \(0.507840\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 21.4833 0.712165
\(911\) −38.8673 −1.28773 −0.643865 0.765139i \(-0.722670\pi\)
−0.643865 + 0.765139i \(0.722670\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 32.5269 1.07589
\(915\) 0 0
\(916\) 0 0
\(917\) −8.66796 −0.286241
\(918\) 0 0
\(919\) −27.0334 −0.891749 −0.445874 0.895096i \(-0.647107\pi\)
−0.445874 + 0.895096i \(0.647107\pi\)
\(920\) −29.0150 −0.956595
\(921\) 0 0
\(922\) 10.1916 0.335643
\(923\) 8.11993 0.267271
\(924\) 0 0
\(925\) 6.96663 0.229061
\(926\) 24.7251 0.812519
\(927\) 0 0
\(928\) 0 0
\(929\) −16.4225 −0.538806 −0.269403 0.963028i \(-0.586826\pi\)
−0.269403 + 0.963028i \(0.586826\pi\)
\(930\) 0 0
\(931\) 5.74166 0.188175
\(932\) 0 0
\(933\) 0 0
\(934\) −29.4166 −0.962540
\(935\) 0 0
\(936\) 0 0
\(937\) 35.7417 1.16763 0.583815 0.811887i \(-0.301560\pi\)
0.583815 + 0.811887i \(0.301560\pi\)
\(938\) 9.16879 0.299372
\(939\) 0 0
\(940\) 0 0
\(941\) −56.8395 −1.85292 −0.926458 0.376399i \(-0.877162\pi\)
−0.926458 + 0.376399i \(0.877162\pi\)
\(942\) 0 0
\(943\) −21.9333 −0.714245
\(944\) −33.2104 −1.08091
\(945\) 0 0
\(946\) 0 0
\(947\) −55.1071 −1.79074 −0.895371 0.445322i \(-0.853089\pi\)
−0.895371 + 0.445322i \(0.853089\pi\)
\(948\) 0 0
\(949\) 90.3832 2.93396
\(950\) −16.2399 −0.526891
\(951\) 0 0
\(952\) −15.4833 −0.501817
\(953\) 50.5463 1.63736 0.818678 0.574253i \(-0.194707\pi\)
0.818678 + 0.574253i \(0.194707\pi\)
\(954\) 0 0
\(955\) −3.74166 −0.121077
\(956\) 0 0
\(957\) 0 0
\(958\) 32.7750 1.05891
\(959\) 8.80344 0.284278
\(960\) 0 0
\(961\) −0.933259 −0.0301051
\(962\) 28.2843 0.911922
\(963\) 0 0
\(964\) 0 0
\(965\) −43.6108 −1.40388
\(966\) 0 0
\(967\) 30.9333 0.994747 0.497373 0.867537i \(-0.334298\pi\)
0.497373 + 0.867537i \(0.334298\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 27.0334 0.867989
\(971\) −15.3265 −0.491850 −0.245925 0.969289i \(-0.579092\pi\)
−0.245925 + 0.969289i \(0.579092\pi\)
\(972\) 0 0
\(973\) 3.74166 0.119952
\(974\) −7.80177 −0.249985
\(975\) 0 0
\(976\) −6.96663 −0.222996
\(977\) 19.7518 0.631916 0.315958 0.948773i \(-0.397674\pi\)
0.315958 + 0.948773i \(0.397674\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −42.0000 −1.34027
\(983\) 2.82843 0.0902128 0.0451064 0.998982i \(-0.485637\pi\)
0.0451064 + 0.998982i \(0.485637\pi\)
\(984\) 0 0
\(985\) −58.8999 −1.87671
\(986\) −30.0166 −0.955925
\(987\) 0 0
\(988\) 0 0
\(989\) 3.87729 0.123291
\(990\) 0 0
\(991\) 0.449944 0.0142930 0.00714648 0.999974i \(-0.497725\pi\)
0.00714648 + 0.999974i \(0.497725\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 2.00000 0.0634361
\(995\) 5.97501 0.189421
\(996\) 0 0
\(997\) 10.9666 0.347317 0.173658 0.984806i \(-0.444441\pi\)
0.173658 + 0.984806i \(0.444441\pi\)
\(998\) 25.4087 0.804297
\(999\) 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.ck.1.2 yes 4
3.2 odd 2 inner 7623.2.a.ck.1.3 yes 4
11.10 odd 2 7623.2.a.cj.1.4 yes 4
33.32 even 2 7623.2.a.cj.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7623.2.a.cj.1.1 4 33.32 even 2
7623.2.a.cj.1.4 yes 4 11.10 odd 2
7623.2.a.ck.1.2 yes 4 1.1 even 1 trivial
7623.2.a.ck.1.3 yes 4 3.2 odd 2 inner