Properties

Label 7623.2.a.cc.1.1
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 3
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
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 \(-1.81361\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.81361 q^{2} +1.28917 q^{4} +2.10278 q^{5} -1.00000 q^{7} +1.28917 q^{8} +O(q^{10})\) \(q-1.81361 q^{2} +1.28917 q^{4} +2.10278 q^{5} -1.00000 q^{7} +1.28917 q^{8} -3.81361 q^{10} -0.186393 q^{13} +1.81361 q^{14} -4.91638 q^{16} -2.10278 q^{17} -4.52444 q^{19} +2.71083 q^{20} +1.94610 q^{23} -0.578337 q^{25} +0.338044 q^{26} -1.28917 q^{28} +0.186393 q^{29} +4.20555 q^{31} +6.33804 q^{32} +3.81361 q^{34} -2.10278 q^{35} -1.28917 q^{37} +8.20555 q^{38} +2.71083 q^{40} +8.91638 q^{41} -10.9653 q^{43} -3.52946 q^{46} +9.01916 q^{47} +1.00000 q^{49} +1.04888 q^{50} -0.240293 q^{52} +6.71083 q^{53} -1.28917 q^{56} -0.338044 q^{58} +4.44082 q^{59} -11.3083 q^{61} -7.62721 q^{62} -1.66196 q^{64} -0.391944 q^{65} -5.07306 q^{67} -2.71083 q^{68} +3.81361 q^{70} -6.72999 q^{71} +11.2005 q^{73} +2.33804 q^{74} -5.83276 q^{76} +7.45998 q^{79} -10.3380 q^{80} -16.1708 q^{82} -12.1758 q^{83} -4.42166 q^{85} +19.8867 q^{86} -0.946101 q^{89} +0.186393 q^{91} +2.50885 q^{92} -16.3572 q^{94} -9.51388 q^{95} +7.27358 q^{97} -1.81361 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} + 3q^{4} - q^{5} - 3q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + q^{2} + 3q^{4} - q^{5} - 3q^{7} + 3q^{8} - 5q^{10} - 7q^{13} - q^{14} - q^{16} + q^{17} - 8q^{19} + 9q^{20} + 2q^{23} - 11q^{26} - 3q^{28} + 7q^{29} - 2q^{31} + 7q^{32} + 5q^{34} + q^{35} - 3q^{37} + 10q^{38} + 9q^{40} + 13q^{41} - 8q^{43} - 20q^{46} + 6q^{47} + 3q^{49} - 8q^{50} - 11q^{52} + 21q^{53} - 3q^{56} + 11q^{58} - 6q^{59} - 12q^{61} - 10q^{62} - 17q^{64} + 7q^{65} + 2q^{67} - 9q^{68} + 5q^{70} + 4q^{73} - 5q^{74} + 10q^{76} - 18q^{79} - 19q^{80} - 9q^{82} - 12q^{83} - 15q^{85} + 36q^{86} + q^{89} + 7q^{91} - 44q^{92} - 16q^{94} + 8q^{95} - 25q^{97} + q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.81361 −1.28241 −0.641207 0.767368i \(-0.721566\pi\)
−0.641207 + 0.767368i \(0.721566\pi\)
\(3\) 0 0
\(4\) 1.28917 0.644584
\(5\) 2.10278 0.940390 0.470195 0.882563i \(-0.344184\pi\)
0.470195 + 0.882563i \(0.344184\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.28917 0.455790
\(9\) 0 0
\(10\) −3.81361 −1.20597
\(11\) 0 0
\(12\) 0 0
\(13\) −0.186393 −0.0516963 −0.0258481 0.999666i \(-0.508229\pi\)
−0.0258481 + 0.999666i \(0.508229\pi\)
\(14\) 1.81361 0.484707
\(15\) 0 0
\(16\) −4.91638 −1.22910
\(17\) −2.10278 −0.509998 −0.254999 0.966941i \(-0.582075\pi\)
−0.254999 + 0.966941i \(0.582075\pi\)
\(18\) 0 0
\(19\) −4.52444 −1.03798 −0.518989 0.854781i \(-0.673691\pi\)
−0.518989 + 0.854781i \(0.673691\pi\)
\(20\) 2.71083 0.606160
\(21\) 0 0
\(22\) 0 0
\(23\) 1.94610 0.405790 0.202895 0.979200i \(-0.434965\pi\)
0.202895 + 0.979200i \(0.434965\pi\)
\(24\) 0 0
\(25\) −0.578337 −0.115667
\(26\) 0.338044 0.0662960
\(27\) 0 0
\(28\) −1.28917 −0.243630
\(29\) 0.186393 0.0346124 0.0173062 0.999850i \(-0.494491\pi\)
0.0173062 + 0.999850i \(0.494491\pi\)
\(30\) 0 0
\(31\) 4.20555 0.755339 0.377670 0.925940i \(-0.376726\pi\)
0.377670 + 0.925940i \(0.376726\pi\)
\(32\) 6.33804 1.12042
\(33\) 0 0
\(34\) 3.81361 0.654028
\(35\) −2.10278 −0.355434
\(36\) 0 0
\(37\) −1.28917 −0.211938 −0.105969 0.994369i \(-0.533794\pi\)
−0.105969 + 0.994369i \(0.533794\pi\)
\(38\) 8.20555 1.33112
\(39\) 0 0
\(40\) 2.71083 0.428620
\(41\) 8.91638 1.39250 0.696252 0.717797i \(-0.254849\pi\)
0.696252 + 0.717797i \(0.254849\pi\)
\(42\) 0 0
\(43\) −10.9653 −1.67219 −0.836093 0.548588i \(-0.815166\pi\)
−0.836093 + 0.548588i \(0.815166\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −3.52946 −0.520391
\(47\) 9.01916 1.31558 0.657790 0.753202i \(-0.271492\pi\)
0.657790 + 0.753202i \(0.271492\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.04888 0.148333
\(51\) 0 0
\(52\) −0.240293 −0.0333226
\(53\) 6.71083 0.921804 0.460902 0.887451i \(-0.347526\pi\)
0.460902 + 0.887451i \(0.347526\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.28917 −0.172272
\(57\) 0 0
\(58\) −0.338044 −0.0443874
\(59\) 4.44082 0.578145 0.289073 0.957307i \(-0.406653\pi\)
0.289073 + 0.957307i \(0.406653\pi\)
\(60\) 0 0
\(61\) −11.3083 −1.44788 −0.723941 0.689862i \(-0.757671\pi\)
−0.723941 + 0.689862i \(0.757671\pi\)
\(62\) −7.62721 −0.968657
\(63\) 0 0
\(64\) −1.66196 −0.207744
\(65\) −0.391944 −0.0486146
\(66\) 0 0
\(67\) −5.07306 −0.619772 −0.309886 0.950774i \(-0.600291\pi\)
−0.309886 + 0.950774i \(0.600291\pi\)
\(68\) −2.71083 −0.328737
\(69\) 0 0
\(70\) 3.81361 0.455813
\(71\) −6.72999 −0.798703 −0.399351 0.916798i \(-0.630765\pi\)
−0.399351 + 0.916798i \(0.630765\pi\)
\(72\) 0 0
\(73\) 11.2005 1.31092 0.655461 0.755229i \(-0.272474\pi\)
0.655461 + 0.755229i \(0.272474\pi\)
\(74\) 2.33804 0.271792
\(75\) 0 0
\(76\) −5.83276 −0.669064
\(77\) 0 0
\(78\) 0 0
\(79\) 7.45998 0.839313 0.419656 0.907683i \(-0.362151\pi\)
0.419656 + 0.907683i \(0.362151\pi\)
\(80\) −10.3380 −1.15583
\(81\) 0 0
\(82\) −16.1708 −1.78577
\(83\) −12.1758 −1.33647 −0.668236 0.743950i \(-0.732950\pi\)
−0.668236 + 0.743950i \(0.732950\pi\)
\(84\) 0 0
\(85\) −4.42166 −0.479597
\(86\) 19.8867 2.14443
\(87\) 0 0
\(88\) 0 0
\(89\) −0.946101 −0.100286 −0.0501432 0.998742i \(-0.515968\pi\)
−0.0501432 + 0.998742i \(0.515968\pi\)
\(90\) 0 0
\(91\) 0.186393 0.0195393
\(92\) 2.50885 0.261566
\(93\) 0 0
\(94\) −16.3572 −1.68712
\(95\) −9.51388 −0.976103
\(96\) 0 0
\(97\) 7.27358 0.738520 0.369260 0.929326i \(-0.379611\pi\)
0.369260 + 0.929326i \(0.379611\pi\)
\(98\) −1.81361 −0.183202
\(99\) 0 0
\(100\) −0.745574 −0.0745574
\(101\) −15.3919 −1.53156 −0.765778 0.643105i \(-0.777646\pi\)
−0.765778 + 0.643105i \(0.777646\pi\)
\(102\) 0 0
\(103\) 10.5244 1.03700 0.518502 0.855077i \(-0.326490\pi\)
0.518502 + 0.855077i \(0.326490\pi\)
\(104\) −0.240293 −0.0235626
\(105\) 0 0
\(106\) −12.1708 −1.18213
\(107\) 13.5733 1.31218 0.656091 0.754682i \(-0.272209\pi\)
0.656091 + 0.754682i \(0.272209\pi\)
\(108\) 0 0
\(109\) −17.6167 −1.68737 −0.843685 0.536839i \(-0.819618\pi\)
−0.843685 + 0.536839i \(0.819618\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.91638 0.464554
\(113\) −10.5436 −0.991858 −0.495929 0.868363i \(-0.665172\pi\)
−0.495929 + 0.868363i \(0.665172\pi\)
\(114\) 0 0
\(115\) 4.09221 0.381601
\(116\) 0.240293 0.0223106
\(117\) 0 0
\(118\) −8.05390 −0.741422
\(119\) 2.10278 0.192761
\(120\) 0 0
\(121\) 0 0
\(122\) 20.5089 1.85678
\(123\) 0 0
\(124\) 5.42166 0.486880
\(125\) −11.7300 −1.04916
\(126\) 0 0
\(127\) −21.2091 −1.88201 −0.941003 0.338399i \(-0.890115\pi\)
−0.941003 + 0.338399i \(0.890115\pi\)
\(128\) −9.66196 −0.854004
\(129\) 0 0
\(130\) 0.710831 0.0623440
\(131\) 5.55918 0.485708 0.242854 0.970063i \(-0.421916\pi\)
0.242854 + 0.970063i \(0.421916\pi\)
\(132\) 0 0
\(133\) 4.52444 0.392319
\(134\) 9.20053 0.794804
\(135\) 0 0
\(136\) −2.71083 −0.232452
\(137\) −22.4550 −1.91846 −0.959228 0.282633i \(-0.908792\pi\)
−0.959228 + 0.282633i \(0.908792\pi\)
\(138\) 0 0
\(139\) −3.27001 −0.277359 −0.138679 0.990337i \(-0.544286\pi\)
−0.138679 + 0.990337i \(0.544286\pi\)
\(140\) −2.71083 −0.229107
\(141\) 0 0
\(142\) 12.2056 1.02427
\(143\) 0 0
\(144\) 0 0
\(145\) 0.391944 0.0325491
\(146\) −20.3133 −1.68114
\(147\) 0 0
\(148\) −1.66196 −0.136612
\(149\) −2.65693 −0.217664 −0.108832 0.994060i \(-0.534711\pi\)
−0.108832 + 0.994060i \(0.534711\pi\)
\(150\) 0 0
\(151\) 16.5330 1.34544 0.672720 0.739898i \(-0.265126\pi\)
0.672720 + 0.739898i \(0.265126\pi\)
\(152\) −5.83276 −0.473100
\(153\) 0 0
\(154\) 0 0
\(155\) 8.84333 0.710313
\(156\) 0 0
\(157\) 17.1950 1.37231 0.686155 0.727456i \(-0.259297\pi\)
0.686155 + 0.727456i \(0.259297\pi\)
\(158\) −13.5295 −1.07635
\(159\) 0 0
\(160\) 13.3275 1.05363
\(161\) −1.94610 −0.153374
\(162\) 0 0
\(163\) −4.20555 −0.329404 −0.164702 0.986343i \(-0.552666\pi\)
−0.164702 + 0.986343i \(0.552666\pi\)
\(164\) 11.4947 0.897587
\(165\) 0 0
\(166\) 22.0822 1.71391
\(167\) 0.646370 0.0500176 0.0250088 0.999687i \(-0.492039\pi\)
0.0250088 + 0.999687i \(0.492039\pi\)
\(168\) 0 0
\(169\) −12.9653 −0.997327
\(170\) 8.01916 0.615041
\(171\) 0 0
\(172\) −14.1361 −1.07786
\(173\) −1.35363 −0.102915 −0.0514573 0.998675i \(-0.516387\pi\)
−0.0514573 + 0.998675i \(0.516387\pi\)
\(174\) 0 0
\(175\) 0.578337 0.0437182
\(176\) 0 0
\(177\) 0 0
\(178\) 1.71585 0.128609
\(179\) 13.9406 1.04197 0.520983 0.853567i \(-0.325565\pi\)
0.520983 + 0.853567i \(0.325565\pi\)
\(180\) 0 0
\(181\) 1.39697 0.103836 0.0519179 0.998651i \(-0.483467\pi\)
0.0519179 + 0.998651i \(0.483467\pi\)
\(182\) −0.338044 −0.0250575
\(183\) 0 0
\(184\) 2.50885 0.184955
\(185\) −2.71083 −0.199304
\(186\) 0 0
\(187\) 0 0
\(188\) 11.6272 0.848002
\(189\) 0 0
\(190\) 17.2544 1.25177
\(191\) −13.4061 −0.970030 −0.485015 0.874506i \(-0.661186\pi\)
−0.485015 + 0.874506i \(0.661186\pi\)
\(192\) 0 0
\(193\) −15.0978 −1.08676 −0.543380 0.839487i \(-0.682856\pi\)
−0.543380 + 0.839487i \(0.682856\pi\)
\(194\) −13.1914 −0.947089
\(195\) 0 0
\(196\) 1.28917 0.0920835
\(197\) −3.96526 −0.282513 −0.141256 0.989973i \(-0.545114\pi\)
−0.141256 + 0.989973i \(0.545114\pi\)
\(198\) 0 0
\(199\) 22.5628 1.59943 0.799716 0.600379i \(-0.204984\pi\)
0.799716 + 0.600379i \(0.204984\pi\)
\(200\) −0.745574 −0.0527200
\(201\) 0 0
\(202\) 27.9149 1.96409
\(203\) −0.186393 −0.0130823
\(204\) 0 0
\(205\) 18.7491 1.30950
\(206\) −19.0872 −1.32987
\(207\) 0 0
\(208\) 0.916382 0.0635396
\(209\) 0 0
\(210\) 0 0
\(211\) −6.18137 −0.425543 −0.212772 0.977102i \(-0.568249\pi\)
−0.212772 + 0.977102i \(0.568249\pi\)
\(212\) 8.65139 0.594180
\(213\) 0 0
\(214\) −24.6167 −1.68276
\(215\) −23.0575 −1.57251
\(216\) 0 0
\(217\) −4.20555 −0.285491
\(218\) 31.9497 2.16390
\(219\) 0 0
\(220\) 0 0
\(221\) 0.391944 0.0263650
\(222\) 0 0
\(223\) −11.2544 −0.753652 −0.376826 0.926284i \(-0.622985\pi\)
−0.376826 + 0.926284i \(0.622985\pi\)
\(224\) −6.33804 −0.423478
\(225\) 0 0
\(226\) 19.1219 1.27197
\(227\) 21.2630 1.41128 0.705638 0.708572i \(-0.250660\pi\)
0.705638 + 0.708572i \(0.250660\pi\)
\(228\) 0 0
\(229\) −6.07306 −0.401319 −0.200659 0.979661i \(-0.564308\pi\)
−0.200659 + 0.979661i \(0.564308\pi\)
\(230\) −7.42166 −0.489370
\(231\) 0 0
\(232\) 0.240293 0.0157760
\(233\) 1.60806 0.105347 0.0526736 0.998612i \(-0.483226\pi\)
0.0526736 + 0.998612i \(0.483226\pi\)
\(234\) 0 0
\(235\) 18.9653 1.23716
\(236\) 5.72496 0.372663
\(237\) 0 0
\(238\) −3.81361 −0.247199
\(239\) 0.578337 0.0374095 0.0187048 0.999825i \(-0.494046\pi\)
0.0187048 + 0.999825i \(0.494046\pi\)
\(240\) 0 0
\(241\) −20.5244 −1.32210 −0.661048 0.750344i \(-0.729888\pi\)
−0.661048 + 0.750344i \(0.729888\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −14.5783 −0.933282
\(245\) 2.10278 0.134341
\(246\) 0 0
\(247\) 0.843326 0.0536595
\(248\) 5.42166 0.344276
\(249\) 0 0
\(250\) 21.2736 1.34546
\(251\) −12.8222 −0.809330 −0.404665 0.914465i \(-0.632612\pi\)
−0.404665 + 0.914465i \(0.632612\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 38.4650 2.41351
\(255\) 0 0
\(256\) 20.8469 1.30293
\(257\) −24.4444 −1.52480 −0.762400 0.647107i \(-0.775979\pi\)
−0.762400 + 0.647107i \(0.775979\pi\)
\(258\) 0 0
\(259\) 1.28917 0.0801050
\(260\) −0.505281 −0.0313362
\(261\) 0 0
\(262\) −10.0822 −0.622878
\(263\) −22.7683 −1.40395 −0.701977 0.712200i \(-0.747699\pi\)
−0.701977 + 0.712200i \(0.747699\pi\)
\(264\) 0 0
\(265\) 14.1114 0.866855
\(266\) −8.20555 −0.503115
\(267\) 0 0
\(268\) −6.54002 −0.399496
\(269\) −19.5925 −1.19457 −0.597287 0.802028i \(-0.703755\pi\)
−0.597287 + 0.802028i \(0.703755\pi\)
\(270\) 0 0
\(271\) −10.9511 −0.665233 −0.332617 0.943062i \(-0.607932\pi\)
−0.332617 + 0.943062i \(0.607932\pi\)
\(272\) 10.3380 0.626836
\(273\) 0 0
\(274\) 40.7244 2.46025
\(275\) 0 0
\(276\) 0 0
\(277\) −9.89169 −0.594334 −0.297167 0.954826i \(-0.596042\pi\)
−0.297167 + 0.954826i \(0.596042\pi\)
\(278\) 5.93051 0.355689
\(279\) 0 0
\(280\) −2.71083 −0.162003
\(281\) 31.5960 1.88486 0.942431 0.334401i \(-0.108534\pi\)
0.942431 + 0.334401i \(0.108534\pi\)
\(282\) 0 0
\(283\) −5.52946 −0.328692 −0.164346 0.986403i \(-0.552551\pi\)
−0.164346 + 0.986403i \(0.552551\pi\)
\(284\) −8.67609 −0.514831
\(285\) 0 0
\(286\) 0 0
\(287\) −8.91638 −0.526317
\(288\) 0 0
\(289\) −12.5783 −0.739902
\(290\) −0.710831 −0.0417415
\(291\) 0 0
\(292\) 14.4394 0.845000
\(293\) 8.94610 0.522637 0.261318 0.965253i \(-0.415843\pi\)
0.261318 + 0.965253i \(0.415843\pi\)
\(294\) 0 0
\(295\) 9.33804 0.543682
\(296\) −1.66196 −0.0965992
\(297\) 0 0
\(298\) 4.81863 0.279136
\(299\) −0.362741 −0.0209778
\(300\) 0 0
\(301\) 10.9653 0.632027
\(302\) −29.9844 −1.72541
\(303\) 0 0
\(304\) 22.2439 1.27577
\(305\) −23.7789 −1.36157
\(306\) 0 0
\(307\) 10.2494 0.584964 0.292482 0.956271i \(-0.405519\pi\)
0.292482 + 0.956271i \(0.405519\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −16.0383 −0.910915
\(311\) 5.82417 0.330258 0.165129 0.986272i \(-0.447196\pi\)
0.165129 + 0.986272i \(0.447196\pi\)
\(312\) 0 0
\(313\) −18.8569 −1.06586 −0.532929 0.846160i \(-0.678909\pi\)
−0.532929 + 0.846160i \(0.678909\pi\)
\(314\) −31.1849 −1.75987
\(315\) 0 0
\(316\) 9.61717 0.541008
\(317\) 33.9844 1.90875 0.954377 0.298603i \(-0.0965206\pi\)
0.954377 + 0.298603i \(0.0965206\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −3.49472 −0.195361
\(321\) 0 0
\(322\) 3.52946 0.196689
\(323\) 9.51388 0.529366
\(324\) 0 0
\(325\) 0.107798 0.00597957
\(326\) 7.62721 0.422432
\(327\) 0 0
\(328\) 11.4947 0.634690
\(329\) −9.01916 −0.497242
\(330\) 0 0
\(331\) 17.4741 0.960464 0.480232 0.877142i \(-0.340552\pi\)
0.480232 + 0.877142i \(0.340552\pi\)
\(332\) −15.6967 −0.861468
\(333\) 0 0
\(334\) −1.17226 −0.0641432
\(335\) −10.6675 −0.582828
\(336\) 0 0
\(337\) −19.0141 −1.03577 −0.517883 0.855452i \(-0.673280\pi\)
−0.517883 + 0.855452i \(0.673280\pi\)
\(338\) 23.5139 1.27899
\(339\) 0 0
\(340\) −5.70027 −0.309140
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −14.1361 −0.762166
\(345\) 0 0
\(346\) 2.45495 0.131979
\(347\) 19.8272 1.06438 0.532191 0.846625i \(-0.321369\pi\)
0.532191 + 0.846625i \(0.321369\pi\)
\(348\) 0 0
\(349\) −10.9547 −0.586391 −0.293196 0.956052i \(-0.594719\pi\)
−0.293196 + 0.956052i \(0.594719\pi\)
\(350\) −1.04888 −0.0560648
\(351\) 0 0
\(352\) 0 0
\(353\) −22.9164 −1.21972 −0.609858 0.792511i \(-0.708774\pi\)
−0.609858 + 0.792511i \(0.708774\pi\)
\(354\) 0 0
\(355\) −14.1517 −0.751092
\(356\) −1.21968 −0.0646431
\(357\) 0 0
\(358\) −25.2827 −1.33623
\(359\) 13.4161 0.708076 0.354038 0.935231i \(-0.384808\pi\)
0.354038 + 0.935231i \(0.384808\pi\)
\(360\) 0 0
\(361\) 1.47054 0.0773968
\(362\) −2.53355 −0.133160
\(363\) 0 0
\(364\) 0.240293 0.0125948
\(365\) 23.5522 1.23278
\(366\) 0 0
\(367\) −33.6499 −1.75651 −0.878256 0.478190i \(-0.841293\pi\)
−0.878256 + 0.478190i \(0.841293\pi\)
\(368\) −9.56777 −0.498755
\(369\) 0 0
\(370\) 4.91638 0.255591
\(371\) −6.71083 −0.348409
\(372\) 0 0
\(373\) 14.4947 0.750508 0.375254 0.926922i \(-0.377555\pi\)
0.375254 + 0.926922i \(0.377555\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 11.6272 0.599628
\(377\) −0.0347425 −0.00178933
\(378\) 0 0
\(379\) 28.0524 1.44096 0.720479 0.693477i \(-0.243922\pi\)
0.720479 + 0.693477i \(0.243922\pi\)
\(380\) −12.2650 −0.629181
\(381\) 0 0
\(382\) 24.3133 1.24398
\(383\) −11.7053 −0.598112 −0.299056 0.954235i \(-0.596672\pi\)
−0.299056 + 0.954235i \(0.596672\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 27.3814 1.39368
\(387\) 0 0
\(388\) 9.37687 0.476039
\(389\) 23.1219 1.17233 0.586164 0.810192i \(-0.300637\pi\)
0.586164 + 0.810192i \(0.300637\pi\)
\(390\) 0 0
\(391\) −4.09221 −0.206952
\(392\) 1.28917 0.0651128
\(393\) 0 0
\(394\) 7.19142 0.362298
\(395\) 15.6867 0.789281
\(396\) 0 0
\(397\) 17.8675 0.896744 0.448372 0.893847i \(-0.352004\pi\)
0.448372 + 0.893847i \(0.352004\pi\)
\(398\) −40.9200 −2.05113
\(399\) 0 0
\(400\) 2.84333 0.142166
\(401\) −13.6620 −0.682246 −0.341123 0.940019i \(-0.610807\pi\)
−0.341123 + 0.940019i \(0.610807\pi\)
\(402\) 0 0
\(403\) −0.783887 −0.0390482
\(404\) −19.8428 −0.987217
\(405\) 0 0
\(406\) 0.338044 0.0167769
\(407\) 0 0
\(408\) 0 0
\(409\) −12.8030 −0.633070 −0.316535 0.948581i \(-0.602519\pi\)
−0.316535 + 0.948581i \(0.602519\pi\)
\(410\) −34.0036 −1.67932
\(411\) 0 0
\(412\) 13.5678 0.668436
\(413\) −4.44082 −0.218518
\(414\) 0 0
\(415\) −25.6030 −1.25680
\(416\) −1.18137 −0.0579214
\(417\) 0 0
\(418\) 0 0
\(419\) −6.91136 −0.337642 −0.168821 0.985647i \(-0.553996\pi\)
−0.168821 + 0.985647i \(0.553996\pi\)
\(420\) 0 0
\(421\) 5.41110 0.263721 0.131860 0.991268i \(-0.457905\pi\)
0.131860 + 0.991268i \(0.457905\pi\)
\(422\) 11.2106 0.545722
\(423\) 0 0
\(424\) 8.65139 0.420149
\(425\) 1.21611 0.0589901
\(426\) 0 0
\(427\) 11.3083 0.547248
\(428\) 17.4983 0.845812
\(429\) 0 0
\(430\) 41.8172 2.01660
\(431\) −18.4705 −0.889695 −0.444847 0.895606i \(-0.646742\pi\)
−0.444847 + 0.895606i \(0.646742\pi\)
\(432\) 0 0
\(433\) −13.3330 −0.640744 −0.320372 0.947292i \(-0.603808\pi\)
−0.320372 + 0.947292i \(0.603808\pi\)
\(434\) 7.62721 0.366118
\(435\) 0 0
\(436\) −22.7108 −1.08765
\(437\) −8.80501 −0.421201
\(438\) 0 0
\(439\) −9.51941 −0.454337 −0.227168 0.973855i \(-0.572947\pi\)
−0.227168 + 0.973855i \(0.572947\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −0.710831 −0.0338108
\(443\) 37.2005 1.76745 0.883725 0.468006i \(-0.155028\pi\)
0.883725 + 0.468006i \(0.155028\pi\)
\(444\) 0 0
\(445\) −1.98944 −0.0943084
\(446\) 20.4111 0.966494
\(447\) 0 0
\(448\) 1.66196 0.0785200
\(449\) −15.7003 −0.740941 −0.370471 0.928844i \(-0.620804\pi\)
−0.370471 + 0.928844i \(0.620804\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −13.5925 −0.639336
\(453\) 0 0
\(454\) −38.5628 −1.80984
\(455\) 0.391944 0.0183746
\(456\) 0 0
\(457\) 31.7738 1.48632 0.743159 0.669115i \(-0.233327\pi\)
0.743159 + 0.669115i \(0.233327\pi\)
\(458\) 11.0141 0.514657
\(459\) 0 0
\(460\) 5.27555 0.245974
\(461\) −9.04385 −0.421214 −0.210607 0.977571i \(-0.567544\pi\)
−0.210607 + 0.977571i \(0.567544\pi\)
\(462\) 0 0
\(463\) −38.3799 −1.78367 −0.891833 0.452364i \(-0.850581\pi\)
−0.891833 + 0.452364i \(0.850581\pi\)
\(464\) −0.916382 −0.0425419
\(465\) 0 0
\(466\) −2.91638 −0.135099
\(467\) −31.1950 −1.44353 −0.721766 0.692137i \(-0.756669\pi\)
−0.721766 + 0.692137i \(0.756669\pi\)
\(468\) 0 0
\(469\) 5.07306 0.234252
\(470\) −34.3955 −1.58655
\(471\) 0 0
\(472\) 5.72496 0.263513
\(473\) 0 0
\(474\) 0 0
\(475\) 2.61665 0.120060
\(476\) 2.71083 0.124251
\(477\) 0 0
\(478\) −1.04888 −0.0479745
\(479\) 31.9391 1.45934 0.729668 0.683802i \(-0.239675\pi\)
0.729668 + 0.683802i \(0.239675\pi\)
\(480\) 0 0
\(481\) 0.240293 0.0109564
\(482\) 37.2233 1.69547
\(483\) 0 0
\(484\) 0 0
\(485\) 15.2947 0.694497
\(486\) 0 0
\(487\) −26.3869 −1.19571 −0.597853 0.801606i \(-0.703979\pi\)
−0.597853 + 0.801606i \(0.703979\pi\)
\(488\) −14.5783 −0.659930
\(489\) 0 0
\(490\) −3.81361 −0.172281
\(491\) −14.3572 −0.647931 −0.323966 0.946069i \(-0.605016\pi\)
−0.323966 + 0.946069i \(0.605016\pi\)
\(492\) 0 0
\(493\) −0.391944 −0.0176523
\(494\) −1.52946 −0.0688137
\(495\) 0 0
\(496\) −20.6761 −0.928384
\(497\) 6.72999 0.301881
\(498\) 0 0
\(499\) −30.3174 −1.35719 −0.678597 0.734510i \(-0.737412\pi\)
−0.678597 + 0.734510i \(0.737412\pi\)
\(500\) −15.1219 −0.676273
\(501\) 0 0
\(502\) 23.2544 1.03790
\(503\) −12.0086 −0.535437 −0.267718 0.963497i \(-0.586270\pi\)
−0.267718 + 0.963497i \(0.586270\pi\)
\(504\) 0 0
\(505\) −32.3658 −1.44026
\(506\) 0 0
\(507\) 0 0
\(508\) −27.3421 −1.21311
\(509\) 9.54913 0.423258 0.211629 0.977350i \(-0.432123\pi\)
0.211629 + 0.977350i \(0.432123\pi\)
\(510\) 0 0
\(511\) −11.2005 −0.495482
\(512\) −18.4842 −0.816892
\(513\) 0 0
\(514\) 44.3325 1.95542
\(515\) 22.1305 0.975187
\(516\) 0 0
\(517\) 0 0
\(518\) −2.33804 −0.102728
\(519\) 0 0
\(520\) −0.505281 −0.0221581
\(521\) 26.7542 1.17212 0.586061 0.810267i \(-0.300678\pi\)
0.586061 + 0.810267i \(0.300678\pi\)
\(522\) 0 0
\(523\) 24.9739 1.09203 0.546015 0.837775i \(-0.316144\pi\)
0.546015 + 0.837775i \(0.316144\pi\)
\(524\) 7.16672 0.313080
\(525\) 0 0
\(526\) 41.2927 1.80045
\(527\) −8.84333 −0.385221
\(528\) 0 0
\(529\) −19.2127 −0.835334
\(530\) −25.5925 −1.11167
\(531\) 0 0
\(532\) 5.83276 0.252882
\(533\) −1.66196 −0.0719873
\(534\) 0 0
\(535\) 28.5416 1.23396
\(536\) −6.54002 −0.282486
\(537\) 0 0
\(538\) 35.5330 1.53194
\(539\) 0 0
\(540\) 0 0
\(541\) −9.03474 −0.388434 −0.194217 0.980959i \(-0.562217\pi\)
−0.194217 + 0.980959i \(0.562217\pi\)
\(542\) 19.8610 0.853104
\(543\) 0 0
\(544\) −13.3275 −0.571411
\(545\) −37.0439 −1.58678
\(546\) 0 0
\(547\) −11.3522 −0.485384 −0.242692 0.970103i \(-0.578030\pi\)
−0.242692 + 0.970103i \(0.578030\pi\)
\(548\) −28.9482 −1.23661
\(549\) 0 0
\(550\) 0 0
\(551\) −0.843326 −0.0359269
\(552\) 0 0
\(553\) −7.45998 −0.317230
\(554\) 17.9396 0.762182
\(555\) 0 0
\(556\) −4.21560 −0.178781
\(557\) −16.5089 −0.699503 −0.349751 0.936843i \(-0.613734\pi\)
−0.349751 + 0.936843i \(0.613734\pi\)
\(558\) 0 0
\(559\) 2.04385 0.0864458
\(560\) 10.3380 0.436862
\(561\) 0 0
\(562\) −57.3028 −2.41717
\(563\) −2.51030 −0.105797 −0.0528984 0.998600i \(-0.516846\pi\)
−0.0528984 + 0.998600i \(0.516846\pi\)
\(564\) 0 0
\(565\) −22.1708 −0.932733
\(566\) 10.0283 0.421519
\(567\) 0 0
\(568\) −8.67609 −0.364041
\(569\) 5.92498 0.248388 0.124194 0.992258i \(-0.460366\pi\)
0.124194 + 0.992258i \(0.460366\pi\)
\(570\) 0 0
\(571\) −25.7633 −1.07816 −0.539080 0.842255i \(-0.681228\pi\)
−0.539080 + 0.842255i \(0.681228\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 16.1708 0.674956
\(575\) −1.12550 −0.0469367
\(576\) 0 0
\(577\) 22.4997 0.936677 0.468338 0.883549i \(-0.344853\pi\)
0.468338 + 0.883549i \(0.344853\pi\)
\(578\) 22.8122 0.948861
\(579\) 0 0
\(580\) 0.505281 0.0209807
\(581\) 12.1758 0.505139
\(582\) 0 0
\(583\) 0 0
\(584\) 14.4394 0.597505
\(585\) 0 0
\(586\) −16.2247 −0.670236
\(587\) 16.2736 0.671683 0.335841 0.941919i \(-0.390979\pi\)
0.335841 + 0.941919i \(0.390979\pi\)
\(588\) 0 0
\(589\) −19.0278 −0.784025
\(590\) −16.9355 −0.697225
\(591\) 0 0
\(592\) 6.33804 0.260492
\(593\) 36.1794 1.48571 0.742855 0.669452i \(-0.233471\pi\)
0.742855 + 0.669452i \(0.233471\pi\)
\(594\) 0 0
\(595\) 4.42166 0.181271
\(596\) −3.42523 −0.140303
\(597\) 0 0
\(598\) 0.657869 0.0269022
\(599\) −1.69670 −0.0693252 −0.0346626 0.999399i \(-0.511036\pi\)
−0.0346626 + 0.999399i \(0.511036\pi\)
\(600\) 0 0
\(601\) 2.61862 0.106816 0.0534079 0.998573i \(-0.482992\pi\)
0.0534079 + 0.998573i \(0.482992\pi\)
\(602\) −19.8867 −0.810520
\(603\) 0 0
\(604\) 21.3139 0.867249
\(605\) 0 0
\(606\) 0 0
\(607\) 37.0333 1.50313 0.751567 0.659656i \(-0.229298\pi\)
0.751567 + 0.659656i \(0.229298\pi\)
\(608\) −28.6761 −1.16297
\(609\) 0 0
\(610\) 43.1255 1.74610
\(611\) −1.68111 −0.0680105
\(612\) 0 0
\(613\) −18.7733 −0.758247 −0.379124 0.925346i \(-0.623775\pi\)
−0.379124 + 0.925346i \(0.623775\pi\)
\(614\) −18.5884 −0.750166
\(615\) 0 0
\(616\) 0 0
\(617\) 28.3764 1.14239 0.571195 0.820815i \(-0.306480\pi\)
0.571195 + 0.820815i \(0.306480\pi\)
\(618\) 0 0
\(619\) 8.09775 0.325476 0.162738 0.986669i \(-0.447967\pi\)
0.162738 + 0.986669i \(0.447967\pi\)
\(620\) 11.4005 0.457857
\(621\) 0 0
\(622\) −10.5628 −0.423528
\(623\) 0.946101 0.0379047
\(624\) 0 0
\(625\) −21.7738 −0.870954
\(626\) 34.1991 1.36687
\(627\) 0 0
\(628\) 22.1672 0.884569
\(629\) 2.71083 0.108088
\(630\) 0 0
\(631\) −20.5925 −0.819773 −0.409887 0.912136i \(-0.634432\pi\)
−0.409887 + 0.912136i \(0.634432\pi\)
\(632\) 9.61717 0.382550
\(633\) 0 0
\(634\) −61.6344 −2.44781
\(635\) −44.5980 −1.76982
\(636\) 0 0
\(637\) −0.186393 −0.00738518
\(638\) 0 0
\(639\) 0 0
\(640\) −20.3169 −0.803097
\(641\) −16.8242 −0.664515 −0.332257 0.943189i \(-0.607810\pi\)
−0.332257 + 0.943189i \(0.607810\pi\)
\(642\) 0 0
\(643\) −27.6655 −1.09102 −0.545511 0.838104i \(-0.683664\pi\)
−0.545511 + 0.838104i \(0.683664\pi\)
\(644\) −2.50885 −0.0988626
\(645\) 0 0
\(646\) −17.2544 −0.678866
\(647\) −37.2530 −1.46457 −0.732283 0.681001i \(-0.761545\pi\)
−0.732283 + 0.681001i \(0.761545\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −0.195504 −0.00766828
\(651\) 0 0
\(652\) −5.42166 −0.212329
\(653\) 0.280575 0.0109797 0.00548987 0.999985i \(-0.498253\pi\)
0.00548987 + 0.999985i \(0.498253\pi\)
\(654\) 0 0
\(655\) 11.6897 0.456755
\(656\) −43.8363 −1.71152
\(657\) 0 0
\(658\) 16.3572 0.637670
\(659\) 25.9250 1.00989 0.504947 0.863150i \(-0.331512\pi\)
0.504947 + 0.863150i \(0.331512\pi\)
\(660\) 0 0
\(661\) 30.9446 1.20361 0.601804 0.798644i \(-0.294449\pi\)
0.601804 + 0.798644i \(0.294449\pi\)
\(662\) −31.6912 −1.23171
\(663\) 0 0
\(664\) −15.6967 −0.609150
\(665\) 9.51388 0.368932
\(666\) 0 0
\(667\) 0.362741 0.0140454
\(668\) 0.833279 0.0322405
\(669\) 0 0
\(670\) 19.3466 0.747426
\(671\) 0 0
\(672\) 0 0
\(673\) −10.9653 −0.422680 −0.211340 0.977413i \(-0.567783\pi\)
−0.211340 + 0.977413i \(0.567783\pi\)
\(674\) 34.4842 1.32828
\(675\) 0 0
\(676\) −16.7144 −0.642862
\(677\) −32.4544 −1.24733 −0.623663 0.781694i \(-0.714356\pi\)
−0.623663 + 0.781694i \(0.714356\pi\)
\(678\) 0 0
\(679\) −7.27358 −0.279134
\(680\) −5.70027 −0.218595
\(681\) 0 0
\(682\) 0 0
\(683\) −17.0972 −0.654208 −0.327104 0.944988i \(-0.606073\pi\)
−0.327104 + 0.944988i \(0.606073\pi\)
\(684\) 0 0
\(685\) −47.2177 −1.80410
\(686\) 1.81361 0.0692438
\(687\) 0 0
\(688\) 53.9094 2.05528
\(689\) −1.25086 −0.0476538
\(690\) 0 0
\(691\) −22.8816 −0.870459 −0.435229 0.900320i \(-0.643333\pi\)
−0.435229 + 0.900320i \(0.643333\pi\)
\(692\) −1.74506 −0.0663371
\(693\) 0 0
\(694\) −35.9588 −1.36498
\(695\) −6.87610 −0.260825
\(696\) 0 0
\(697\) −18.7491 −0.710174
\(698\) 19.8675 0.751996
\(699\) 0 0
\(700\) 0.745574 0.0281800
\(701\) −34.8414 −1.31594 −0.657970 0.753044i \(-0.728585\pi\)
−0.657970 + 0.753044i \(0.728585\pi\)
\(702\) 0 0
\(703\) 5.83276 0.219987
\(704\) 0 0
\(705\) 0 0
\(706\) 41.5613 1.56418
\(707\) 15.3919 0.578874
\(708\) 0 0
\(709\) −7.78746 −0.292464 −0.146232 0.989250i \(-0.546715\pi\)
−0.146232 + 0.989250i \(0.546715\pi\)
\(710\) 25.6655 0.963210
\(711\) 0 0
\(712\) −1.21968 −0.0457096
\(713\) 8.18442 0.306509
\(714\) 0 0
\(715\) 0 0
\(716\) 17.9717 0.671635
\(717\) 0 0
\(718\) −24.3316 −0.908046
\(719\) −36.0383 −1.34400 −0.672001 0.740550i \(-0.734565\pi\)
−0.672001 + 0.740550i \(0.734565\pi\)
\(720\) 0 0
\(721\) −10.5244 −0.391951
\(722\) −2.66698 −0.0992547
\(723\) 0 0
\(724\) 1.80093 0.0669309
\(725\) −0.107798 −0.00400353
\(726\) 0 0
\(727\) −46.6988 −1.73196 −0.865982 0.500076i \(-0.833305\pi\)
−0.865982 + 0.500076i \(0.833305\pi\)
\(728\) 0.240293 0.00890584
\(729\) 0 0
\(730\) −42.7144 −1.58093
\(731\) 23.0575 0.852811
\(732\) 0 0
\(733\) 0.116908 0.00431811 0.00215906 0.999998i \(-0.499313\pi\)
0.00215906 + 0.999998i \(0.499313\pi\)
\(734\) 61.0278 2.25258
\(735\) 0 0
\(736\) 12.3345 0.454655
\(737\) 0 0
\(738\) 0 0
\(739\) 46.3799 1.70611 0.853057 0.521818i \(-0.174746\pi\)
0.853057 + 0.521818i \(0.174746\pi\)
\(740\) −3.49472 −0.128468
\(741\) 0 0
\(742\) 12.1708 0.446804
\(743\) −45.3466 −1.66361 −0.831803 0.555070i \(-0.812691\pi\)
−0.831803 + 0.555070i \(0.812691\pi\)
\(744\) 0 0
\(745\) −5.58693 −0.204689
\(746\) −26.2877 −0.962462
\(747\) 0 0
\(748\) 0 0
\(749\) −13.5733 −0.495958
\(750\) 0 0
\(751\) 34.2680 1.25046 0.625229 0.780441i \(-0.285005\pi\)
0.625229 + 0.780441i \(0.285005\pi\)
\(752\) −44.3416 −1.61697
\(753\) 0 0
\(754\) 0.0630093 0.00229466
\(755\) 34.7652 1.26524
\(756\) 0 0
\(757\) −22.0106 −0.799988 −0.399994 0.916518i \(-0.630988\pi\)
−0.399994 + 0.916518i \(0.630988\pi\)
\(758\) −50.8761 −1.84790
\(759\) 0 0
\(760\) −12.2650 −0.444898
\(761\) −11.2877 −0.409179 −0.204590 0.978848i \(-0.565586\pi\)
−0.204590 + 0.978848i \(0.565586\pi\)
\(762\) 0 0
\(763\) 17.6167 0.637766
\(764\) −17.2827 −0.625266
\(765\) 0 0
\(766\) 21.2288 0.767027
\(767\) −0.827740 −0.0298880
\(768\) 0 0
\(769\) 12.3864 0.446665 0.223333 0.974742i \(-0.428306\pi\)
0.223333 + 0.974742i \(0.428306\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −19.4635 −0.700508
\(773\) −26.9013 −0.967573 −0.483786 0.875186i \(-0.660739\pi\)
−0.483786 + 0.875186i \(0.660739\pi\)
\(774\) 0 0
\(775\) −2.43223 −0.0873681
\(776\) 9.37687 0.336610
\(777\) 0 0
\(778\) −41.9341 −1.50341
\(779\) −40.3416 −1.44539
\(780\) 0 0
\(781\) 0 0
\(782\) 7.42166 0.265398
\(783\) 0 0
\(784\) −4.91638 −0.175585
\(785\) 36.1572 1.29051
\(786\) 0 0
\(787\) 16.9583 0.604497 0.302248 0.953229i \(-0.402263\pi\)
0.302248 + 0.953229i \(0.402263\pi\)
\(788\) −5.11189 −0.182103
\(789\) 0 0
\(790\) −28.4494 −1.01218
\(791\) 10.5436 0.374887
\(792\) 0 0
\(793\) 2.10780 0.0748501
\(794\) −32.4046 −1.15000
\(795\) 0 0
\(796\) 29.0872 1.03097
\(797\) −5.99141 −0.212226 −0.106113 0.994354i \(-0.533841\pi\)
−0.106113 + 0.994354i \(0.533841\pi\)
\(798\) 0 0
\(799\) −18.9653 −0.670943
\(800\) −3.66553 −0.129596
\(801\) 0 0
\(802\) 24.7774 0.874921
\(803\) 0 0
\(804\) 0 0
\(805\) −4.09221 −0.144232
\(806\) 1.42166 0.0500759
\(807\) 0 0
\(808\) −19.8428 −0.698068
\(809\) −30.2877 −1.06486 −0.532430 0.846474i \(-0.678721\pi\)
−0.532430 + 0.846474i \(0.678721\pi\)
\(810\) 0 0
\(811\) −25.2388 −0.886256 −0.443128 0.896458i \(-0.646131\pi\)
−0.443128 + 0.896458i \(0.646131\pi\)
\(812\) −0.240293 −0.00843262
\(813\) 0 0
\(814\) 0 0
\(815\) −8.84333 −0.309768
\(816\) 0 0
\(817\) 49.6116 1.73569
\(818\) 23.2197 0.811857
\(819\) 0 0
\(820\) 24.1708 0.844081
\(821\) 0.0977518 0.00341156 0.00170578 0.999999i \(-0.499457\pi\)
0.00170578 + 0.999999i \(0.499457\pi\)
\(822\) 0 0
\(823\) 38.7839 1.35192 0.675961 0.736938i \(-0.263729\pi\)
0.675961 + 0.736938i \(0.263729\pi\)
\(824\) 13.5678 0.472656
\(825\) 0 0
\(826\) 8.05390 0.280231
\(827\) 27.8555 0.968630 0.484315 0.874894i \(-0.339069\pi\)
0.484315 + 0.874894i \(0.339069\pi\)
\(828\) 0 0
\(829\) −40.4585 −1.40518 −0.702591 0.711594i \(-0.747974\pi\)
−0.702591 + 0.711594i \(0.747974\pi\)
\(830\) 46.4338 1.61174
\(831\) 0 0
\(832\) 0.309778 0.0107396
\(833\) −2.10278 −0.0728568
\(834\) 0 0
\(835\) 1.35917 0.0470360
\(836\) 0 0
\(837\) 0 0
\(838\) 12.5345 0.432997
\(839\) 53.6741 1.85304 0.926518 0.376250i \(-0.122787\pi\)
0.926518 + 0.376250i \(0.122787\pi\)
\(840\) 0 0
\(841\) −28.9653 −0.998802
\(842\) −9.81361 −0.338199
\(843\) 0 0
\(844\) −7.96883 −0.274298
\(845\) −27.2630 −0.937876
\(846\) 0 0
\(847\) 0 0
\(848\) −32.9930 −1.13298
\(849\) 0 0
\(850\) −2.20555 −0.0756497
\(851\) −2.50885 −0.0860023
\(852\) 0 0
\(853\) −11.6408 −0.398574 −0.199287 0.979941i \(-0.563863\pi\)
−0.199287 + 0.979941i \(0.563863\pi\)
\(854\) −20.5089 −0.701798
\(855\) 0 0
\(856\) 17.4983 0.598079
\(857\) −34.9497 −1.19386 −0.596929 0.802294i \(-0.703613\pi\)
−0.596929 + 0.802294i \(0.703613\pi\)
\(858\) 0 0
\(859\) −48.1049 −1.64132 −0.820659 0.571418i \(-0.806393\pi\)
−0.820659 + 0.571418i \(0.806393\pi\)
\(860\) −29.7250 −1.01361
\(861\) 0 0
\(862\) 33.4983 1.14096
\(863\) −33.4499 −1.13865 −0.569324 0.822113i \(-0.692795\pi\)
−0.569324 + 0.822113i \(0.692795\pi\)
\(864\) 0 0
\(865\) −2.84638 −0.0967798
\(866\) 24.1809 0.821699
\(867\) 0 0
\(868\) −5.42166 −0.184023
\(869\) 0 0
\(870\) 0 0
\(871\) 0.945585 0.0320399
\(872\) −22.7108 −0.769086
\(873\) 0 0
\(874\) 15.9688 0.540154
\(875\) 11.7300 0.396546
\(876\) 0 0
\(877\) −27.7980 −0.938672 −0.469336 0.883020i \(-0.655507\pi\)
−0.469336 + 0.883020i \(0.655507\pi\)
\(878\) 17.2645 0.582648
\(879\) 0 0
\(880\) 0 0
\(881\) −33.8363 −1.13998 −0.569988 0.821653i \(-0.693052\pi\)
−0.569988 + 0.821653i \(0.693052\pi\)
\(882\) 0 0
\(883\) −37.4630 −1.26073 −0.630366 0.776298i \(-0.717095\pi\)
−0.630366 + 0.776298i \(0.717095\pi\)
\(884\) 0.505281 0.0169945
\(885\) 0 0
\(886\) −67.4671 −2.26660
\(887\) −3.48970 −0.117173 −0.0585863 0.998282i \(-0.518659\pi\)
−0.0585863 + 0.998282i \(0.518659\pi\)
\(888\) 0 0
\(889\) 21.2091 0.711331
\(890\) 3.60806 0.120942
\(891\) 0 0
\(892\) −14.5089 −0.485792
\(893\) −40.8066 −1.36554
\(894\) 0 0
\(895\) 29.3139 0.979854
\(896\) 9.66196 0.322783
\(897\) 0 0
\(898\) 28.4741 0.950193
\(899\) 0.783887 0.0261441
\(900\) 0 0
\(901\) −14.1114 −0.470118
\(902\) 0 0
\(903\) 0 0
\(904\) −13.5925 −0.452079
\(905\) 2.93751 0.0976460
\(906\) 0 0
\(907\) 25.6655 0.852210 0.426105 0.904674i \(-0.359885\pi\)
0.426105 + 0.904674i \(0.359885\pi\)
\(908\) 27.4116 0.909686
\(909\) 0 0
\(910\) −0.710831 −0.0235638
\(911\) 14.8716 0.492718 0.246359 0.969179i \(-0.420766\pi\)
0.246359 + 0.969179i \(0.420766\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −57.6252 −1.90607
\(915\) 0 0
\(916\) −7.82919 −0.258684
\(917\) −5.55918 −0.183580
\(918\) 0 0
\(919\) 30.7144 1.01317 0.506587 0.862189i \(-0.330907\pi\)
0.506587 + 0.862189i \(0.330907\pi\)
\(920\) 5.27555 0.173930
\(921\) 0 0
\(922\) 16.4020 0.540171
\(923\) 1.25443 0.0412899
\(924\) 0 0
\(925\) 0.745574 0.0245143
\(926\) 69.6061 2.28740
\(927\) 0 0
\(928\) 1.18137 0.0387804
\(929\) 21.2111 0.695913 0.347957 0.937511i \(-0.386876\pi\)
0.347957 + 0.937511i \(0.386876\pi\)
\(930\) 0 0
\(931\) −4.52444 −0.148282
\(932\) 2.07306 0.0679052
\(933\) 0 0
\(934\) 56.5754 1.85120
\(935\) 0 0
\(936\) 0 0
\(937\) −58.9008 −1.92421 −0.962103 0.272688i \(-0.912087\pi\)
−0.962103 + 0.272688i \(0.912087\pi\)
\(938\) −9.20053 −0.300408
\(939\) 0 0
\(940\) 24.4494 0.797452
\(941\) 50.8852 1.65881 0.829405 0.558647i \(-0.188680\pi\)
0.829405 + 0.558647i \(0.188680\pi\)
\(942\) 0 0
\(943\) 17.3522 0.565065
\(944\) −21.8328 −0.710596
\(945\) 0 0
\(946\) 0 0
\(947\) 12.6277 0.410346 0.205173 0.978726i \(-0.434224\pi\)
0.205173 + 0.978726i \(0.434224\pi\)
\(948\) 0 0
\(949\) −2.08771 −0.0677698
\(950\) −4.74557 −0.153967
\(951\) 0 0
\(952\) 2.71083 0.0878586
\(953\) −44.4458 −1.43974 −0.719871 0.694108i \(-0.755799\pi\)
−0.719871 + 0.694108i \(0.755799\pi\)
\(954\) 0 0
\(955\) −28.1900 −0.912206
\(956\) 0.745574 0.0241136
\(957\) 0 0
\(958\) −57.9250 −1.87147
\(959\) 22.4550 0.725108
\(960\) 0 0
\(961\) −13.3133 −0.429463
\(962\) −0.435796 −0.0140506
\(963\) 0 0
\(964\) −26.4595 −0.852202
\(965\) −31.7472 −1.02198
\(966\) 0 0
\(967\) −22.7214 −0.730671 −0.365335 0.930876i \(-0.619046\pi\)
−0.365335 + 0.930876i \(0.619046\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −27.7386 −0.890632
\(971\) −18.2736 −0.586427 −0.293214 0.956047i \(-0.594725\pi\)
−0.293214 + 0.956047i \(0.594725\pi\)
\(972\) 0 0
\(973\) 3.27001 0.104832
\(974\) 47.8555 1.53339
\(975\) 0 0
\(976\) 55.5960 1.77959
\(977\) 31.6152 1.01146 0.505730 0.862692i \(-0.331223\pi\)
0.505730 + 0.862692i \(0.331223\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 2.71083 0.0865943
\(981\) 0 0
\(982\) 26.0383 0.830916
\(983\) −33.8328 −1.07910 −0.539549 0.841954i \(-0.681405\pi\)
−0.539549 + 0.841954i \(0.681405\pi\)
\(984\) 0 0
\(985\) −8.33804 −0.265672
\(986\) 0.710831 0.0226375
\(987\) 0 0
\(988\) 1.08719 0.0345881
\(989\) −21.3395 −0.678557
\(990\) 0 0
\(991\) −2.71440 −0.0862258 −0.0431129 0.999070i \(-0.513728\pi\)
−0.0431129 + 0.999070i \(0.513728\pi\)
\(992\) 26.6550 0.846296
\(993\) 0 0
\(994\) −12.2056 −0.387137
\(995\) 47.4444 1.50409
\(996\) 0 0
\(997\) 43.2197 1.36878 0.684391 0.729116i \(-0.260068\pi\)
0.684391 + 0.729116i \(0.260068\pi\)
\(998\) 54.9839 1.74048
\(999\) 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.cc.1.1 3
3.2 odd 2 2541.2.a.bh.1.3 3
11.10 odd 2 7623.2.a.ca.1.3 3
33.32 even 2 2541.2.a.bj.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.bh.1.3 3 3.2 odd 2
2541.2.a.bj.1.1 yes 3 33.32 even 2
7623.2.a.ca.1.3 3 11.10 odd 2
7623.2.a.cc.1.1 3 1.1 even 1 trivial