Properties

Label 1840.2.a.s.1.1
Level $1840$
Weight $2$
Character 1840.1
Self dual yes
Analytic conductor $14.692$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2597.1
Defining polynomial: \(x^{3} - x^{2} - 9 x + 8\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.07912\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.07912 q^{3} +1.00000 q^{5} +2.07912 q^{7} +6.48097 q^{9} +O(q^{10})\) \(q-3.07912 q^{3} +1.00000 q^{5} +2.07912 q^{7} +6.48097 q^{9} -5.07912 q^{11} -3.48097 q^{13} -3.07912 q^{15} +6.48097 q^{17} +7.48097 q^{19} -6.40185 q^{21} -1.00000 q^{23} +1.00000 q^{25} -10.7183 q^{27} -1.56009 q^{29} -0.0791189 q^{31} +15.6392 q^{33} +2.07912 q^{35} -9.71833 q^{37} +10.7183 q^{39} -0.480973 q^{41} -8.00000 q^{43} +6.48097 q^{45} -6.96195 q^{47} -2.67726 q^{49} -19.9557 q^{51} +11.7183 q^{53} -5.07912 q^{55} -23.0348 q^{57} +11.5601 q^{59} +7.88283 q^{61} +13.4747 q^{63} -3.48097 q^{65} +9.71833 q^{67} +3.07912 q^{69} +9.67726 q^{71} -13.2784 q^{73} -3.07912 q^{75} -10.5601 q^{77} +12.3165 q^{79} +13.5601 q^{81} +4.59815 q^{83} +6.48097 q^{85} +4.80371 q^{87} +8.31648 q^{89} -7.23736 q^{91} +0.243616 q^{93} +7.48097 q^{95} +7.23736 q^{97} -32.9176 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - q^{3} + 3q^{5} - 2q^{7} + 10q^{9} + O(q^{10}) \) \( 3q - q^{3} + 3q^{5} - 2q^{7} + 10q^{9} - 7q^{11} - q^{13} - q^{15} + 10q^{17} + 13q^{19} - 18q^{21} - 3q^{23} + 3q^{25} + 2q^{27} + 13q^{29} + 8q^{31} + 21q^{33} - 2q^{35} + 5q^{37} - 2q^{39} + 8q^{41} - 24q^{43} + 10q^{45} - 2q^{47} - q^{49} - q^{51} + q^{53} - 7q^{55} - 2q^{57} + 17q^{59} + 13q^{61} - 9q^{63} - q^{65} - 5q^{67} + q^{69} + 22q^{71} + 12q^{73} - q^{75} - 14q^{77} + 4q^{79} + 23q^{81} + 15q^{83} + 10q^{85} + 12q^{87} - 8q^{89} + 3q^{91} + 16q^{93} + 13q^{95} - 3q^{97} - 21q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.07912 −1.77773 −0.888865 0.458169i \(-0.848505\pi\)
−0.888865 + 0.458169i \(0.848505\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.07912 0.785833 0.392917 0.919574i \(-0.371466\pi\)
0.392917 + 0.919574i \(0.371466\pi\)
\(8\) 0 0
\(9\) 6.48097 2.16032
\(10\) 0 0
\(11\) −5.07912 −1.53141 −0.765706 0.643191i \(-0.777610\pi\)
−0.765706 + 0.643191i \(0.777610\pi\)
\(12\) 0 0
\(13\) −3.48097 −0.965448 −0.482724 0.875772i \(-0.660353\pi\)
−0.482724 + 0.875772i \(0.660353\pi\)
\(14\) 0 0
\(15\) −3.07912 −0.795025
\(16\) 0 0
\(17\) 6.48097 1.57187 0.785933 0.618311i \(-0.212183\pi\)
0.785933 + 0.618311i \(0.212183\pi\)
\(18\) 0 0
\(19\) 7.48097 1.71625 0.858126 0.513438i \(-0.171629\pi\)
0.858126 + 0.513438i \(0.171629\pi\)
\(20\) 0 0
\(21\) −6.40185 −1.39700
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −10.7183 −2.06274
\(28\) 0 0
\(29\) −1.56009 −0.289702 −0.144851 0.989453i \(-0.546270\pi\)
−0.144851 + 0.989453i \(0.546270\pi\)
\(30\) 0 0
\(31\) −0.0791189 −0.0142102 −0.00710508 0.999975i \(-0.502262\pi\)
−0.00710508 + 0.999975i \(0.502262\pi\)
\(32\) 0 0
\(33\) 15.6392 2.72244
\(34\) 0 0
\(35\) 2.07912 0.351435
\(36\) 0 0
\(37\) −9.71833 −1.59768 −0.798842 0.601541i \(-0.794554\pi\)
−0.798842 + 0.601541i \(0.794554\pi\)
\(38\) 0 0
\(39\) 10.7183 1.71631
\(40\) 0 0
\(41\) −0.480973 −0.0751154 −0.0375577 0.999294i \(-0.511958\pi\)
−0.0375577 + 0.999294i \(0.511958\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 6.48097 0.966126
\(46\) 0 0
\(47\) −6.96195 −1.01550 −0.507752 0.861503i \(-0.669523\pi\)
−0.507752 + 0.861503i \(0.669523\pi\)
\(48\) 0 0
\(49\) −2.67726 −0.382466
\(50\) 0 0
\(51\) −19.9557 −2.79435
\(52\) 0 0
\(53\) 11.7183 1.60964 0.804818 0.593521i \(-0.202263\pi\)
0.804818 + 0.593521i \(0.202263\pi\)
\(54\) 0 0
\(55\) −5.07912 −0.684868
\(56\) 0 0
\(57\) −23.0348 −3.05103
\(58\) 0 0
\(59\) 11.5601 1.50500 0.752498 0.658595i \(-0.228849\pi\)
0.752498 + 0.658595i \(0.228849\pi\)
\(60\) 0 0
\(61\) 7.88283 1.00929 0.504646 0.863326i \(-0.331623\pi\)
0.504646 + 0.863326i \(0.331623\pi\)
\(62\) 0 0
\(63\) 13.4747 1.69765
\(64\) 0 0
\(65\) −3.48097 −0.431762
\(66\) 0 0
\(67\) 9.71833 1.18728 0.593641 0.804730i \(-0.297690\pi\)
0.593641 + 0.804730i \(0.297690\pi\)
\(68\) 0 0
\(69\) 3.07912 0.370682
\(70\) 0 0
\(71\) 9.67726 1.14848 0.574240 0.818687i \(-0.305298\pi\)
0.574240 + 0.818687i \(0.305298\pi\)
\(72\) 0 0
\(73\) −13.2784 −1.55412 −0.777061 0.629425i \(-0.783290\pi\)
−0.777061 + 0.629425i \(0.783290\pi\)
\(74\) 0 0
\(75\) −3.07912 −0.355546
\(76\) 0 0
\(77\) −10.5601 −1.20343
\(78\) 0 0
\(79\) 12.3165 1.38571 0.692856 0.721076i \(-0.256352\pi\)
0.692856 + 0.721076i \(0.256352\pi\)
\(80\) 0 0
\(81\) 13.5601 1.50668
\(82\) 0 0
\(83\) 4.59815 0.504712 0.252356 0.967634i \(-0.418795\pi\)
0.252356 + 0.967634i \(0.418795\pi\)
\(84\) 0 0
\(85\) 6.48097 0.702960
\(86\) 0 0
\(87\) 4.80371 0.515012
\(88\) 0 0
\(89\) 8.31648 0.881545 0.440772 0.897619i \(-0.354705\pi\)
0.440772 + 0.897619i \(0.354705\pi\)
\(90\) 0 0
\(91\) −7.23736 −0.758681
\(92\) 0 0
\(93\) 0.243616 0.0252618
\(94\) 0 0
\(95\) 7.48097 0.767532
\(96\) 0 0
\(97\) 7.23736 0.734842 0.367421 0.930055i \(-0.380241\pi\)
0.367421 + 0.930055i \(0.380241\pi\)
\(98\) 0 0
\(99\) −32.9176 −3.30835
\(100\) 0 0
\(101\) 2.43991 0.242780 0.121390 0.992605i \(-0.461265\pi\)
0.121390 + 0.992605i \(0.461265\pi\)
\(102\) 0 0
\(103\) −7.88283 −0.776718 −0.388359 0.921508i \(-0.626958\pi\)
−0.388359 + 0.921508i \(0.626958\pi\)
\(104\) 0 0
\(105\) −6.40185 −0.624757
\(106\) 0 0
\(107\) 16.6803 1.61254 0.806272 0.591546i \(-0.201482\pi\)
0.806272 + 0.591546i \(0.201482\pi\)
\(108\) 0 0
\(109\) 5.32274 0.509826 0.254913 0.966964i \(-0.417953\pi\)
0.254913 + 0.966964i \(0.417953\pi\)
\(110\) 0 0
\(111\) 29.9239 2.84025
\(112\) 0 0
\(113\) −2.20556 −0.207482 −0.103741 0.994604i \(-0.533081\pi\)
−0.103741 + 0.994604i \(0.533081\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −22.5601 −2.08568
\(118\) 0 0
\(119\) 13.4747 1.23522
\(120\) 0 0
\(121\) 14.7974 1.34522
\(122\) 0 0
\(123\) 1.48097 0.133535
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −6.15824 −0.546455 −0.273228 0.961949i \(-0.588091\pi\)
−0.273228 + 0.961949i \(0.588091\pi\)
\(128\) 0 0
\(129\) 24.6330 2.16881
\(130\) 0 0
\(131\) 7.19629 0.628743 0.314371 0.949300i \(-0.398206\pi\)
0.314371 + 0.949300i \(0.398206\pi\)
\(132\) 0 0
\(133\) 15.5538 1.34869
\(134\) 0 0
\(135\) −10.7183 −0.922487
\(136\) 0 0
\(137\) 12.6012 1.07659 0.538295 0.842757i \(-0.319069\pi\)
0.538295 + 0.842757i \(0.319069\pi\)
\(138\) 0 0
\(139\) 8.75638 0.742707 0.371353 0.928492i \(-0.378894\pi\)
0.371353 + 0.928492i \(0.378894\pi\)
\(140\) 0 0
\(141\) 21.4367 1.80529
\(142\) 0 0
\(143\) 17.6803 1.47850
\(144\) 0 0
\(145\) −1.56009 −0.129559
\(146\) 0 0
\(147\) 8.24362 0.679922
\(148\) 0 0
\(149\) 1.79745 0.147253 0.0736264 0.997286i \(-0.476543\pi\)
0.0736264 + 0.997286i \(0.476543\pi\)
\(150\) 0 0
\(151\) 19.3956 1.57839 0.789196 0.614142i \(-0.210498\pi\)
0.789196 + 0.614142i \(0.210498\pi\)
\(152\) 0 0
\(153\) 42.0030 3.39574
\(154\) 0 0
\(155\) −0.0791189 −0.00635498
\(156\) 0 0
\(157\) 4.36380 0.348269 0.174135 0.984722i \(-0.444287\pi\)
0.174135 + 0.984722i \(0.444287\pi\)
\(158\) 0 0
\(159\) −36.0821 −2.86150
\(160\) 0 0
\(161\) −2.07912 −0.163858
\(162\) 0 0
\(163\) −5.32274 −0.416909 −0.208454 0.978032i \(-0.566843\pi\)
−0.208454 + 0.978032i \(0.566843\pi\)
\(164\) 0 0
\(165\) 15.6392 1.21751
\(166\) 0 0
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) −0.882827 −0.0679098
\(170\) 0 0
\(171\) 48.4840 3.70766
\(172\) 0 0
\(173\) −9.23736 −0.702303 −0.351152 0.936319i \(-0.614210\pi\)
−0.351152 + 0.936319i \(0.614210\pi\)
\(174\) 0 0
\(175\) 2.07912 0.157167
\(176\) 0 0
\(177\) −35.5949 −2.67548
\(178\) 0 0
\(179\) −2.96195 −0.221386 −0.110693 0.993855i \(-0.535307\pi\)
−0.110693 + 0.993855i \(0.535307\pi\)
\(180\) 0 0
\(181\) 10.8355 0.805397 0.402698 0.915333i \(-0.368072\pi\)
0.402698 + 0.915333i \(0.368072\pi\)
\(182\) 0 0
\(183\) −24.2722 −1.79425
\(184\) 0 0
\(185\) −9.71833 −0.714506
\(186\) 0 0
\(187\) −32.9176 −2.40718
\(188\) 0 0
\(189\) −22.2847 −1.62097
\(190\) 0 0
\(191\) −7.76565 −0.561903 −0.280952 0.959722i \(-0.590650\pi\)
−0.280952 + 0.959722i \(0.590650\pi\)
\(192\) 0 0
\(193\) −2.15824 −0.155353 −0.0776767 0.996979i \(-0.524750\pi\)
−0.0776767 + 0.996979i \(0.524750\pi\)
\(194\) 0 0
\(195\) 10.7183 0.767556
\(196\) 0 0
\(197\) 12.0411 0.857890 0.428945 0.903331i \(-0.358885\pi\)
0.428945 + 0.903331i \(0.358885\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) −29.9239 −2.11067
\(202\) 0 0
\(203\) −3.24362 −0.227657
\(204\) 0 0
\(205\) −0.480973 −0.0335926
\(206\) 0 0
\(207\) −6.48097 −0.450459
\(208\) 0 0
\(209\) −37.9968 −2.62829
\(210\) 0 0
\(211\) −3.40185 −0.234193 −0.117097 0.993121i \(-0.537359\pi\)
−0.117097 + 0.993121i \(0.537359\pi\)
\(212\) 0 0
\(213\) −29.7974 −2.04169
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −0.164498 −0.0111668
\(218\) 0 0
\(219\) 40.8858 2.76281
\(220\) 0 0
\(221\) −22.5601 −1.51756
\(222\) 0 0
\(223\) −5.19629 −0.347969 −0.173985 0.984748i \(-0.555664\pi\)
−0.173985 + 0.984748i \(0.555664\pi\)
\(224\) 0 0
\(225\) 6.48097 0.432065
\(226\) 0 0
\(227\) −5.35453 −0.355393 −0.177696 0.984085i \(-0.556864\pi\)
−0.177696 + 0.984085i \(0.556864\pi\)
\(228\) 0 0
\(229\) −0.803708 −0.0531105 −0.0265553 0.999647i \(-0.508454\pi\)
−0.0265553 + 0.999647i \(0.508454\pi\)
\(230\) 0 0
\(231\) 32.5158 2.13938
\(232\) 0 0
\(233\) 14.1582 0.927537 0.463768 0.885956i \(-0.346497\pi\)
0.463768 + 0.885956i \(0.346497\pi\)
\(234\) 0 0
\(235\) −6.96195 −0.454147
\(236\) 0 0
\(237\) −37.9239 −2.46342
\(238\) 0 0
\(239\) −10.9146 −0.706008 −0.353004 0.935622i \(-0.614840\pi\)
−0.353004 + 0.935622i \(0.614840\pi\)
\(240\) 0 0
\(241\) −27.4367 −1.76735 −0.883675 0.468100i \(-0.844939\pi\)
−0.883675 + 0.468100i \(0.844939\pi\)
\(242\) 0 0
\(243\) −9.59815 −0.615721
\(244\) 0 0
\(245\) −2.67726 −0.171044
\(246\) 0 0
\(247\) −26.0411 −1.65695
\(248\) 0 0
\(249\) −14.1582 −0.897242
\(250\) 0 0
\(251\) 16.5190 1.04267 0.521336 0.853352i \(-0.325434\pi\)
0.521336 + 0.853352i \(0.325434\pi\)
\(252\) 0 0
\(253\) 5.07912 0.319321
\(254\) 0 0
\(255\) −19.9557 −1.24967
\(256\) 0 0
\(257\) 19.7657 1.23295 0.616474 0.787375i \(-0.288561\pi\)
0.616474 + 0.787375i \(0.288561\pi\)
\(258\) 0 0
\(259\) −20.2056 −1.25551
\(260\) 0 0
\(261\) −10.1109 −0.625850
\(262\) 0 0
\(263\) −7.44292 −0.458950 −0.229475 0.973315i \(-0.573701\pi\)
−0.229475 + 0.973315i \(0.573701\pi\)
\(264\) 0 0
\(265\) 11.7183 0.719851
\(266\) 0 0
\(267\) −25.6074 −1.56715
\(268\) 0 0
\(269\) −22.7564 −1.38748 −0.693741 0.720225i \(-0.744039\pi\)
−0.693741 + 0.720225i \(0.744039\pi\)
\(270\) 0 0
\(271\) 19.0411 1.15666 0.578331 0.815802i \(-0.303704\pi\)
0.578331 + 0.815802i \(0.303704\pi\)
\(272\) 0 0
\(273\) 22.2847 1.34873
\(274\) 0 0
\(275\) −5.07912 −0.306282
\(276\) 0 0
\(277\) −18.3165 −1.10053 −0.550265 0.834990i \(-0.685473\pi\)
−0.550265 + 0.834990i \(0.685473\pi\)
\(278\) 0 0
\(279\) −0.512767 −0.0306986
\(280\) 0 0
\(281\) 26.0821 1.55593 0.777965 0.628308i \(-0.216252\pi\)
0.777965 + 0.628308i \(0.216252\pi\)
\(282\) 0 0
\(283\) 25.6422 1.52427 0.762136 0.647417i \(-0.224151\pi\)
0.762136 + 0.647417i \(0.224151\pi\)
\(284\) 0 0
\(285\) −23.0348 −1.36446
\(286\) 0 0
\(287\) −1.00000 −0.0590281
\(288\) 0 0
\(289\) 25.0030 1.47077
\(290\) 0 0
\(291\) −22.2847 −1.30635
\(292\) 0 0
\(293\) 30.8385 1.80161 0.900803 0.434229i \(-0.142979\pi\)
0.900803 + 0.434229i \(0.142979\pi\)
\(294\) 0 0
\(295\) 11.5601 0.673055
\(296\) 0 0
\(297\) 54.4397 3.15891
\(298\) 0 0
\(299\) 3.48097 0.201310
\(300\) 0 0
\(301\) −16.6330 −0.958707
\(302\) 0 0
\(303\) −7.51277 −0.431597
\(304\) 0 0
\(305\) 7.88283 0.451369
\(306\) 0 0
\(307\) 1.88283 0.107459 0.0537293 0.998556i \(-0.482889\pi\)
0.0537293 + 0.998556i \(0.482889\pi\)
\(308\) 0 0
\(309\) 24.2722 1.38080
\(310\) 0 0
\(311\) −1.60742 −0.0911482 −0.0455741 0.998961i \(-0.514512\pi\)
−0.0455741 + 0.998961i \(0.514512\pi\)
\(312\) 0 0
\(313\) −1.68654 −0.0953286 −0.0476643 0.998863i \(-0.515178\pi\)
−0.0476643 + 0.998863i \(0.515178\pi\)
\(314\) 0 0
\(315\) 13.4747 0.759214
\(316\) 0 0
\(317\) −12.6773 −0.712026 −0.356013 0.934481i \(-0.615864\pi\)
−0.356013 + 0.934481i \(0.615864\pi\)
\(318\) 0 0
\(319\) 7.92389 0.443653
\(320\) 0 0
\(321\) −51.3606 −2.86667
\(322\) 0 0
\(323\) 48.4840 2.69772
\(324\) 0 0
\(325\) −3.48097 −0.193090
\(326\) 0 0
\(327\) −16.3893 −0.906332
\(328\) 0 0
\(329\) −14.4747 −0.798017
\(330\) 0 0
\(331\) −33.3893 −1.83524 −0.917622 0.397454i \(-0.869894\pi\)
−0.917622 + 0.397454i \(0.869894\pi\)
\(332\) 0 0
\(333\) −62.9842 −3.45151
\(334\) 0 0
\(335\) 9.71833 0.530969
\(336\) 0 0
\(337\) −17.6392 −0.960869 −0.480435 0.877031i \(-0.659521\pi\)
−0.480435 + 0.877031i \(0.659521\pi\)
\(338\) 0 0
\(339\) 6.79119 0.368847
\(340\) 0 0
\(341\) 0.401854 0.0217616
\(342\) 0 0
\(343\) −20.1202 −1.08639
\(344\) 0 0
\(345\) 3.07912 0.165774
\(346\) 0 0
\(347\) −19.7121 −1.05820 −0.529100 0.848560i \(-0.677470\pi\)
−0.529100 + 0.848560i \(0.677470\pi\)
\(348\) 0 0
\(349\) −16.0348 −0.858323 −0.429162 0.903228i \(-0.641191\pi\)
−0.429162 + 0.903228i \(0.641191\pi\)
\(350\) 0 0
\(351\) 37.3102 1.99147
\(352\) 0 0
\(353\) −8.00000 −0.425797 −0.212899 0.977074i \(-0.568290\pi\)
−0.212899 + 0.977074i \(0.568290\pi\)
\(354\) 0 0
\(355\) 9.67726 0.513616
\(356\) 0 0
\(357\) −41.4902 −2.19590
\(358\) 0 0
\(359\) 13.1202 0.692457 0.346228 0.938150i \(-0.387462\pi\)
0.346228 + 0.938150i \(0.387462\pi\)
\(360\) 0 0
\(361\) 36.9650 1.94552
\(362\) 0 0
\(363\) −45.5631 −2.39144
\(364\) 0 0
\(365\) −13.2784 −0.695024
\(366\) 0 0
\(367\) −3.47796 −0.181548 −0.0907741 0.995872i \(-0.528934\pi\)
−0.0907741 + 0.995872i \(0.528934\pi\)
\(368\) 0 0
\(369\) −3.11717 −0.162274
\(370\) 0 0
\(371\) 24.3638 1.26491
\(372\) 0 0
\(373\) 21.5949 1.11814 0.559071 0.829120i \(-0.311158\pi\)
0.559071 + 0.829120i \(0.311158\pi\)
\(374\) 0 0
\(375\) −3.07912 −0.159005
\(376\) 0 0
\(377\) 5.43064 0.279692
\(378\) 0 0
\(379\) −7.80070 −0.400695 −0.200347 0.979725i \(-0.564207\pi\)
−0.200347 + 0.979725i \(0.564207\pi\)
\(380\) 0 0
\(381\) 18.9619 0.971450
\(382\) 0 0
\(383\) −32.4274 −1.65696 −0.828481 0.560017i \(-0.810795\pi\)
−0.828481 + 0.560017i \(0.810795\pi\)
\(384\) 0 0
\(385\) −10.5601 −0.538192
\(386\) 0 0
\(387\) −51.8478 −2.63557
\(388\) 0 0
\(389\) 23.5538 1.19423 0.597113 0.802157i \(-0.296314\pi\)
0.597113 + 0.802157i \(0.296314\pi\)
\(390\) 0 0
\(391\) −6.48097 −0.327757
\(392\) 0 0
\(393\) −22.1582 −1.11774
\(394\) 0 0
\(395\) 12.3165 0.619709
\(396\) 0 0
\(397\) −0.370060 −0.0185728 −0.00928639 0.999957i \(-0.502956\pi\)
−0.00928639 + 0.999957i \(0.502956\pi\)
\(398\) 0 0
\(399\) −47.8921 −2.39760
\(400\) 0 0
\(401\) 11.3545 0.567018 0.283509 0.958970i \(-0.408501\pi\)
0.283509 + 0.958970i \(0.408501\pi\)
\(402\) 0 0
\(403\) 0.275411 0.0137192
\(404\) 0 0
\(405\) 13.5601 0.673806
\(406\) 0 0
\(407\) 49.3606 2.44671
\(408\) 0 0
\(409\) 4.88283 0.241440 0.120720 0.992687i \(-0.461480\pi\)
0.120720 + 0.992687i \(0.461480\pi\)
\(410\) 0 0
\(411\) −38.8005 −1.91389
\(412\) 0 0
\(413\) 24.0348 1.18268
\(414\) 0 0
\(415\) 4.59815 0.225714
\(416\) 0 0
\(417\) −26.9619 −1.32033
\(418\) 0 0
\(419\) −1.51277 −0.0739035 −0.0369518 0.999317i \(-0.511765\pi\)
−0.0369518 + 0.999317i \(0.511765\pi\)
\(420\) 0 0
\(421\) 22.2722 1.08548 0.542739 0.839901i \(-0.317387\pi\)
0.542739 + 0.839901i \(0.317387\pi\)
\(422\) 0 0
\(423\) −45.1202 −2.19382
\(424\) 0 0
\(425\) 6.48097 0.314373
\(426\) 0 0
\(427\) 16.3893 0.793135
\(428\) 0 0
\(429\) −54.4397 −2.62837
\(430\) 0 0
\(431\) 16.8858 0.813362 0.406681 0.913570i \(-0.366686\pi\)
0.406681 + 0.913570i \(0.366686\pi\)
\(432\) 0 0
\(433\) 24.3956 1.17238 0.586189 0.810175i \(-0.300628\pi\)
0.586189 + 0.810175i \(0.300628\pi\)
\(434\) 0 0
\(435\) 4.80371 0.230320
\(436\) 0 0
\(437\) −7.48097 −0.357863
\(438\) 0 0
\(439\) −40.0378 −1.91090 −0.955450 0.295152i \(-0.904630\pi\)
−0.955450 + 0.295152i \(0.904630\pi\)
\(440\) 0 0
\(441\) −17.3513 −0.826251
\(442\) 0 0
\(443\) 5.22809 0.248394 0.124197 0.992258i \(-0.460365\pi\)
0.124197 + 0.992258i \(0.460365\pi\)
\(444\) 0 0
\(445\) 8.31648 0.394239
\(446\) 0 0
\(447\) −5.53456 −0.261776
\(448\) 0 0
\(449\) 4.72459 0.222967 0.111484 0.993766i \(-0.464440\pi\)
0.111484 + 0.993766i \(0.464440\pi\)
\(450\) 0 0
\(451\) 2.44292 0.115033
\(452\) 0 0
\(453\) −59.7213 −2.80595
\(454\) 0 0
\(455\) −7.23736 −0.339293
\(456\) 0 0
\(457\) 33.2311 1.55449 0.777243 0.629201i \(-0.216618\pi\)
0.777243 + 0.629201i \(0.216618\pi\)
\(458\) 0 0
\(459\) −69.4652 −3.24236
\(460\) 0 0
\(461\) −28.9619 −1.34889 −0.674446 0.738324i \(-0.735618\pi\)
−0.674446 + 0.738324i \(0.735618\pi\)
\(462\) 0 0
\(463\) −4.39258 −0.204141 −0.102070 0.994777i \(-0.532547\pi\)
−0.102070 + 0.994777i \(0.532547\pi\)
\(464\) 0 0
\(465\) 0.243616 0.0112974
\(466\) 0 0
\(467\) −17.0729 −0.790038 −0.395019 0.918673i \(-0.629262\pi\)
−0.395019 + 0.918673i \(0.629262\pi\)
\(468\) 0 0
\(469\) 20.2056 0.933006
\(470\) 0 0
\(471\) −13.4367 −0.619129
\(472\) 0 0
\(473\) 40.6330 1.86831
\(474\) 0 0
\(475\) 7.48097 0.343251
\(476\) 0 0
\(477\) 75.9462 3.47734
\(478\) 0 0
\(479\) −27.0441 −1.23568 −0.617838 0.786306i \(-0.711991\pi\)
−0.617838 + 0.786306i \(0.711991\pi\)
\(480\) 0 0
\(481\) 33.8292 1.54248
\(482\) 0 0
\(483\) 6.40185 0.291294
\(484\) 0 0
\(485\) 7.23736 0.328631
\(486\) 0 0
\(487\) 28.0821 1.27252 0.636261 0.771474i \(-0.280480\pi\)
0.636261 + 0.771474i \(0.280480\pi\)
\(488\) 0 0
\(489\) 16.3893 0.741151
\(490\) 0 0
\(491\) 7.48398 0.337747 0.168874 0.985638i \(-0.445987\pi\)
0.168874 + 0.985638i \(0.445987\pi\)
\(492\) 0 0
\(493\) −10.1109 −0.455373
\(494\) 0 0
\(495\) −32.9176 −1.47954
\(496\) 0 0
\(497\) 20.1202 0.902514
\(498\) 0 0
\(499\) −21.7183 −0.972246 −0.486123 0.873890i \(-0.661589\pi\)
−0.486123 + 0.873890i \(0.661589\pi\)
\(500\) 0 0
\(501\) −24.6330 −1.10052
\(502\) 0 0
\(503\) 7.27541 0.324395 0.162197 0.986758i \(-0.448142\pi\)
0.162197 + 0.986758i \(0.448142\pi\)
\(504\) 0 0
\(505\) 2.43991 0.108574
\(506\) 0 0
\(507\) 2.71833 0.120725
\(508\) 0 0
\(509\) 22.7151 1.00683 0.503414 0.864045i \(-0.332077\pi\)
0.503414 + 0.864045i \(0.332077\pi\)
\(510\) 0 0
\(511\) −27.6074 −1.22128
\(512\) 0 0
\(513\) −80.1835 −3.54019
\(514\) 0 0
\(515\) −7.88283 −0.347359
\(516\) 0 0
\(517\) 35.3606 1.55516
\(518\) 0 0
\(519\) 28.4429 1.24851
\(520\) 0 0
\(521\) −16.5568 −0.725368 −0.362684 0.931912i \(-0.618140\pi\)
−0.362684 + 0.931912i \(0.618140\pi\)
\(522\) 0 0
\(523\) −24.8037 −1.08459 −0.542295 0.840188i \(-0.682445\pi\)
−0.542295 + 0.840188i \(0.682445\pi\)
\(524\) 0 0
\(525\) −6.40185 −0.279400
\(526\) 0 0
\(527\) −0.512767 −0.0223365
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 74.9206 3.25128
\(532\) 0 0
\(533\) 1.67425 0.0725200
\(534\) 0 0
\(535\) 16.6803 0.721151
\(536\) 0 0
\(537\) 9.12018 0.393565
\(538\) 0 0
\(539\) 13.5981 0.585714
\(540\) 0 0
\(541\) −21.8292 −0.938512 −0.469256 0.883062i \(-0.655478\pi\)
−0.469256 + 0.883062i \(0.655478\pi\)
\(542\) 0 0
\(543\) −33.3638 −1.43178
\(544\) 0 0
\(545\) 5.32274 0.228001
\(546\) 0 0
\(547\) 29.0759 1.24319 0.621597 0.783337i \(-0.286484\pi\)
0.621597 + 0.783337i \(0.286484\pi\)
\(548\) 0 0
\(549\) 51.0884 2.18040
\(550\) 0 0
\(551\) −11.6710 −0.497202
\(552\) 0 0
\(553\) 25.6074 1.08894
\(554\) 0 0
\(555\) 29.9239 1.27020
\(556\) 0 0
\(557\) −22.0348 −0.933645 −0.466822 0.884351i \(-0.654601\pi\)
−0.466822 + 0.884351i \(0.654601\pi\)
\(558\) 0 0
\(559\) 27.8478 1.17784
\(560\) 0 0
\(561\) 101.357 4.27931
\(562\) 0 0
\(563\) −1.40185 −0.0590811 −0.0295406 0.999564i \(-0.509404\pi\)
−0.0295406 + 0.999564i \(0.509404\pi\)
\(564\) 0 0
\(565\) −2.20556 −0.0927887
\(566\) 0 0
\(567\) 28.1930 1.18400
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 18.3575 0.768239 0.384120 0.923283i \(-0.374505\pi\)
0.384120 + 0.923283i \(0.374505\pi\)
\(572\) 0 0
\(573\) 23.9114 0.998912
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) −11.5128 −0.479283 −0.239641 0.970861i \(-0.577030\pi\)
−0.239641 + 0.970861i \(0.577030\pi\)
\(578\) 0 0
\(579\) 6.64547 0.276176
\(580\) 0 0
\(581\) 9.56009 0.396619
\(582\) 0 0
\(583\) −59.5188 −2.46502
\(584\) 0 0
\(585\) −22.5601 −0.932745
\(586\) 0 0
\(587\) −40.1232 −1.65606 −0.828031 0.560683i \(-0.810539\pi\)
−0.828031 + 0.560683i \(0.810539\pi\)
\(588\) 0 0
\(589\) −0.591886 −0.0243882
\(590\) 0 0
\(591\) −37.0759 −1.52510
\(592\) 0 0
\(593\) 18.7912 0.771662 0.385831 0.922570i \(-0.373915\pi\)
0.385831 + 0.922570i \(0.373915\pi\)
\(594\) 0 0
\(595\) 13.4747 0.552409
\(596\) 0 0
\(597\) −43.1077 −1.76428
\(598\) 0 0
\(599\) 27.4777 1.12271 0.561355 0.827575i \(-0.310280\pi\)
0.561355 + 0.827575i \(0.310280\pi\)
\(600\) 0 0
\(601\) −19.5283 −0.796576 −0.398288 0.917260i \(-0.630396\pi\)
−0.398288 + 0.917260i \(0.630396\pi\)
\(602\) 0 0
\(603\) 62.9842 2.56492
\(604\) 0 0
\(605\) 14.7974 0.601602
\(606\) 0 0
\(607\) −45.9239 −1.86399 −0.931997 0.362467i \(-0.881935\pi\)
−0.931997 + 0.362467i \(0.881935\pi\)
\(608\) 0 0
\(609\) 9.98748 0.404713
\(610\) 0 0
\(611\) 24.2343 0.980417
\(612\) 0 0
\(613\) 12.7091 0.513314 0.256657 0.966503i \(-0.417379\pi\)
0.256657 + 0.966503i \(0.417379\pi\)
\(614\) 0 0
\(615\) 1.48097 0.0597186
\(616\) 0 0
\(617\) 48.3102 1.94490 0.972448 0.233120i \(-0.0748934\pi\)
0.972448 + 0.233120i \(0.0748934\pi\)
\(618\) 0 0
\(619\) −0.677265 −0.0272216 −0.0136108 0.999907i \(-0.504333\pi\)
−0.0136108 + 0.999907i \(0.504333\pi\)
\(620\) 0 0
\(621\) 10.7183 0.430112
\(622\) 0 0
\(623\) 17.2909 0.692747
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 116.997 4.67239
\(628\) 0 0
\(629\) −62.9842 −2.51135
\(630\) 0 0
\(631\) −44.3986 −1.76748 −0.883740 0.467978i \(-0.844983\pi\)
−0.883740 + 0.467978i \(0.844983\pi\)
\(632\) 0 0
\(633\) 10.4747 0.416332
\(634\) 0 0
\(635\) −6.15824 −0.244382
\(636\) 0 0
\(637\) 9.31949 0.369251
\(638\) 0 0
\(639\) 62.7181 2.48109
\(640\) 0 0
\(641\) 16.8798 0.666713 0.333356 0.942801i \(-0.391819\pi\)
0.333356 + 0.942801i \(0.391819\pi\)
\(642\) 0 0
\(643\) 26.7564 1.05517 0.527584 0.849503i \(-0.323098\pi\)
0.527584 + 0.849503i \(0.323098\pi\)
\(644\) 0 0
\(645\) 24.6330 0.969921
\(646\) 0 0
\(647\) −32.5568 −1.27994 −0.639971 0.768399i \(-0.721054\pi\)
−0.639971 + 0.768399i \(0.721054\pi\)
\(648\) 0 0
\(649\) −58.7151 −2.30477
\(650\) 0 0
\(651\) 0.506507 0.0198516
\(652\) 0 0
\(653\) −9.87356 −0.386382 −0.193191 0.981161i \(-0.561884\pi\)
−0.193191 + 0.981161i \(0.561884\pi\)
\(654\) 0 0
\(655\) 7.19629 0.281182
\(656\) 0 0
\(657\) −86.0571 −3.35741
\(658\) 0 0
\(659\) 13.5128 0.526383 0.263191 0.964744i \(-0.415225\pi\)
0.263191 + 0.964744i \(0.415225\pi\)
\(660\) 0 0
\(661\) −26.7687 −1.04118 −0.520590 0.853807i \(-0.674288\pi\)
−0.520590 + 0.853807i \(0.674288\pi\)
\(662\) 0 0
\(663\) 69.4652 2.69780
\(664\) 0 0
\(665\) 15.5538 0.603152
\(666\) 0 0
\(667\) 1.56009 0.0604070
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −40.0378 −1.54564
\(672\) 0 0
\(673\) −0.803708 −0.0309807 −0.0154903 0.999880i \(-0.504931\pi\)
−0.0154903 + 0.999880i \(0.504931\pi\)
\(674\) 0 0
\(675\) −10.7183 −0.412549
\(676\) 0 0
\(677\) −33.6422 −1.29298 −0.646488 0.762924i \(-0.723763\pi\)
−0.646488 + 0.762924i \(0.723763\pi\)
\(678\) 0 0
\(679\) 15.0473 0.577463
\(680\) 0 0
\(681\) 16.4872 0.631792
\(682\) 0 0
\(683\) 11.8736 0.454329 0.227165 0.973856i \(-0.427054\pi\)
0.227165 + 0.973856i \(0.427054\pi\)
\(684\) 0 0
\(685\) 12.6012 0.481465
\(686\) 0 0
\(687\) 2.47471 0.0944162
\(688\) 0 0
\(689\) −40.7912 −1.55402
\(690\) 0 0
\(691\) −27.9114 −1.06180 −0.530899 0.847435i \(-0.678146\pi\)
−0.530899 + 0.847435i \(0.678146\pi\)
\(692\) 0 0
\(693\) −68.4397 −2.59981
\(694\) 0 0
\(695\) 8.75638 0.332149
\(696\) 0 0
\(697\) −3.11717 −0.118071
\(698\) 0 0
\(699\) −43.5949 −1.64891
\(700\) 0 0
\(701\) −26.3668 −0.995861 −0.497930 0.867217i \(-0.665906\pi\)
−0.497930 + 0.867217i \(0.665906\pi\)
\(702\) 0 0
\(703\) −72.7026 −2.74203
\(704\) 0 0
\(705\) 21.4367 0.807351
\(706\) 0 0
\(707\) 5.07286 0.190785
\(708\) 0 0
\(709\) 11.8007 0.443184 0.221592 0.975139i \(-0.428875\pi\)
0.221592 + 0.975139i \(0.428875\pi\)
\(710\) 0 0
\(711\) 79.8227 2.99359
\(712\) 0 0
\(713\) 0.0791189 0.00296302
\(714\) 0 0
\(715\) 17.6803 0.661205
\(716\) 0 0
\(717\) 33.6074 1.25509
\(718\) 0 0
\(719\) 12.9589 0.483287 0.241643 0.970365i \(-0.422314\pi\)
0.241643 + 0.970365i \(0.422314\pi\)
\(720\) 0 0
\(721\) −16.3893 −0.610371
\(722\) 0 0
\(723\) 84.4807 3.14187
\(724\) 0 0
\(725\) −1.56009 −0.0579404
\(726\) 0 0
\(727\) 19.0318 0.705850 0.352925 0.935652i \(-0.385187\pi\)
0.352925 + 0.935652i \(0.385187\pi\)
\(728\) 0 0
\(729\) −11.1264 −0.412090
\(730\) 0 0
\(731\) −51.8478 −1.91766
\(732\) 0 0
\(733\) −30.5220 −1.12736 −0.563679 0.825994i \(-0.690614\pi\)
−0.563679 + 0.825994i \(0.690614\pi\)
\(734\) 0 0
\(735\) 8.24362 0.304070
\(736\) 0 0
\(737\) −49.3606 −1.81822
\(738\) 0 0
\(739\) 6.18702 0.227593 0.113797 0.993504i \(-0.463699\pi\)
0.113797 + 0.993504i \(0.463699\pi\)
\(740\) 0 0
\(741\) 80.1835 2.94562
\(742\) 0 0
\(743\) −16.2722 −0.596968 −0.298484 0.954415i \(-0.596481\pi\)
−0.298484 + 0.954415i \(0.596481\pi\)
\(744\) 0 0
\(745\) 1.79745 0.0658534
\(746\) 0 0
\(747\) 29.8005 1.09034
\(748\) 0 0
\(749\) 34.6803 1.26719
\(750\) 0 0
\(751\) 17.5949 0.642047 0.321023 0.947071i \(-0.395973\pi\)
0.321023 + 0.947071i \(0.395973\pi\)
\(752\) 0 0
\(753\) −50.8640 −1.85359
\(754\) 0 0
\(755\) 19.3956 0.705878
\(756\) 0 0
\(757\) 41.2496 1.49924 0.749622 0.661866i \(-0.230235\pi\)
0.749622 + 0.661866i \(0.230235\pi\)
\(758\) 0 0
\(759\) −15.6392 −0.567667
\(760\) 0 0
\(761\) 26.8067 0.971743 0.485871 0.874030i \(-0.338502\pi\)
0.485871 + 0.874030i \(0.338502\pi\)
\(762\) 0 0
\(763\) 11.0666 0.400638
\(764\) 0 0
\(765\) 42.0030 1.51862
\(766\) 0 0
\(767\) −40.2404 −1.45300
\(768\) 0 0
\(769\) −26.8798 −0.969311 −0.484655 0.874705i \(-0.661055\pi\)
−0.484655 + 0.874705i \(0.661055\pi\)
\(770\) 0 0
\(771\) −60.8608 −2.19185
\(772\) 0 0
\(773\) 41.0441 1.47625 0.738126 0.674662i \(-0.235711\pi\)
0.738126 + 0.674662i \(0.235711\pi\)
\(774\) 0 0
\(775\) −0.0791189 −0.00284203
\(776\) 0 0
\(777\) 62.2153 2.23196
\(778\) 0 0
\(779\) −3.59815 −0.128917
\(780\) 0 0
\(781\) −49.1520 −1.75880
\(782\) 0 0
\(783\) 16.7216 0.597580
\(784\) 0 0
\(785\) 4.36380 0.155751
\(786\) 0 0
\(787\) 21.4840 0.765821 0.382911 0.923785i \(-0.374922\pi\)
0.382911 + 0.923785i \(0.374922\pi\)
\(788\) 0 0
\(789\) 22.9176 0.815889
\(790\) 0 0
\(791\) −4.58563 −0.163046
\(792\) 0 0
\(793\) −27.4399 −0.974420
\(794\) 0 0
\(795\) −36.0821 −1.27970
\(796\) 0 0
\(797\) 38.2692 1.35556 0.677781 0.735263i \(-0.262942\pi\)
0.677781 + 0.735263i \(0.262942\pi\)
\(798\) 0 0
\(799\) −45.1202 −1.59624
\(800\) 0 0
\(801\) 53.8989 1.90442
\(802\) 0 0
\(803\) 67.4427 2.38000
\(804\) 0 0
\(805\) −2.07912 −0.0732793
\(806\) 0 0
\(807\) 70.0696 2.46657
\(808\) 0 0
\(809\) −27.2026 −0.956391 −0.478195 0.878253i \(-0.658709\pi\)
−0.478195 + 0.878253i \(0.658709\pi\)
\(810\) 0 0
\(811\) −38.5095 −1.35225 −0.676126 0.736786i \(-0.736343\pi\)
−0.676126 + 0.736786i \(0.736343\pi\)
\(812\) 0 0
\(813\) −58.6297 −2.05623
\(814\) 0 0
\(815\) −5.32274 −0.186447
\(816\) 0 0
\(817\) −59.8478 −2.09381
\(818\) 0 0
\(819\) −46.9051 −1.63900
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) −40.1457 −1.39939 −0.699696 0.714441i \(-0.746681\pi\)
−0.699696 + 0.714441i \(0.746681\pi\)
\(824\) 0 0
\(825\) 15.6392 0.544487
\(826\) 0 0
\(827\) −0.851033 −0.0295933 −0.0147967 0.999891i \(-0.504710\pi\)
−0.0147967 + 0.999891i \(0.504710\pi\)
\(828\) 0 0
\(829\) −2.83851 −0.0985856 −0.0492928 0.998784i \(-0.515697\pi\)
−0.0492928 + 0.998784i \(0.515697\pi\)
\(830\) 0 0
\(831\) 56.3986 1.95645
\(832\) 0 0
\(833\) −17.3513 −0.601186
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 0.848022 0.0293119
\(838\) 0 0
\(839\) 10.2343 0.353329 0.176664 0.984271i \(-0.443469\pi\)
0.176664 + 0.984271i \(0.443469\pi\)
\(840\) 0 0
\(841\) −26.5661 −0.916073
\(842\) 0 0
\(843\) −80.3100 −2.76602
\(844\) 0 0
\(845\) −0.882827 −0.0303702
\(846\) 0 0
\(847\) 30.7657 1.05712
\(848\) 0 0
\(849\) −78.9554 −2.70974
\(850\) 0 0
\(851\) 9.71833 0.333140
\(852\) 0 0
\(853\) 17.7213 0.606767 0.303384 0.952869i \(-0.401884\pi\)
0.303384 + 0.952869i \(0.401884\pi\)
\(854\) 0 0
\(855\) 48.4840 1.65812
\(856\) 0 0
\(857\) 17.6835 0.604058 0.302029 0.953299i \(-0.402336\pi\)
0.302029 + 0.953299i \(0.402336\pi\)
\(858\) 0 0
\(859\) 26.6042 0.907722 0.453861 0.891072i \(-0.350046\pi\)
0.453861 + 0.891072i \(0.350046\pi\)
\(860\) 0 0
\(861\) 3.07912 0.104936
\(862\) 0 0
\(863\) 53.5949 1.82439 0.912196 0.409755i \(-0.134386\pi\)
0.912196 + 0.409755i \(0.134386\pi\)
\(864\) 0 0
\(865\) −9.23736 −0.314080
\(866\) 0 0
\(867\) −76.9872 −2.61462
\(868\) 0 0
\(869\) −62.5568 −2.12210
\(870\) 0 0
\(871\) −33.8292 −1.14626
\(872\) 0 0
\(873\) 46.9051 1.58750
\(874\) 0 0
\(875\) 2.07912 0.0702870
\(876\) 0 0
\(877\) 25.9650 0.876774 0.438387 0.898786i \(-0.355550\pi\)
0.438387 + 0.898786i \(0.355550\pi\)
\(878\) 0 0
\(879\) −94.9554 −3.20277
\(880\) 0 0
\(881\) 29.9239 1.00816 0.504081 0.863657i \(-0.331831\pi\)
0.504081 + 0.863657i \(0.331831\pi\)
\(882\) 0 0
\(883\) 4.75939 0.160166 0.0800832 0.996788i \(-0.474481\pi\)
0.0800832 + 0.996788i \(0.474481\pi\)
\(884\) 0 0
\(885\) −35.5949 −1.19651
\(886\) 0 0
\(887\) −38.8037 −1.30290 −0.651451 0.758691i \(-0.725839\pi\)
−0.651451 + 0.758691i \(0.725839\pi\)
\(888\) 0 0
\(889\) −12.8037 −0.429423
\(890\) 0 0
\(891\) −68.8733 −2.30734
\(892\) 0 0
\(893\) −52.0821 −1.74286
\(894\) 0 0
\(895\) −2.96195 −0.0990069
\(896\) 0 0
\(897\) −10.7183 −0.357875
\(898\) 0 0
\(899\) 0.123433 0.00411671
\(900\) 0 0
\(901\) 75.9462 2.53013
\(902\) 0 0
\(903\) 51.2148 1.70432
\(904\) 0 0
\(905\) 10.8355 0.360184
\(906\) 0 0
\(907\) 51.0729 1.69585 0.847923 0.530119i \(-0.177853\pi\)
0.847923 + 0.530119i \(0.177853\pi\)
\(908\) 0 0
\(909\) 15.8130 0.524483
\(910\) 0 0
\(911\) 30.4933 1.01029 0.505143 0.863035i \(-0.331440\pi\)
0.505143 + 0.863035i \(0.331440\pi\)
\(912\) 0 0
\(913\) −23.3545 −0.772922
\(914\) 0 0
\(915\) −24.2722 −0.802413
\(916\) 0 0
\(917\) 14.9619 0.494087
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) −5.79745 −0.191032
\(922\) 0 0
\(923\) −33.6863 −1.10880
\(924\) 0 0
\(925\) −9.71833 −0.319537
\(926\) 0 0
\(927\) −51.0884 −1.67796
\(928\) 0 0
\(929\) −17.3257 −0.568439 −0.284220 0.958759i \(-0.591734\pi\)
−0.284220 + 0.958759i \(0.591734\pi\)
\(930\) 0 0
\(931\) −20.0285 −0.656409
\(932\) 0 0
\(933\) 4.94943 0.162037
\(934\) 0 0
\(935\) −32.9176 −1.07652
\(936\) 0 0
\(937\) −47.5631 −1.55382 −0.776909 0.629612i \(-0.783214\pi\)
−0.776909 + 0.629612i \(0.783214\pi\)
\(938\) 0 0
\(939\) 5.19304 0.169469
\(940\) 0 0
\(941\) 14.2754 0.465365 0.232683 0.972553i \(-0.425250\pi\)
0.232683 + 0.972553i \(0.425250\pi\)
\(942\) 0 0
\(943\) 0.480973 0.0156626
\(944\) 0 0
\(945\) −22.2847 −0.724921
\(946\) 0 0
\(947\) −36.1900 −1.17602 −0.588009 0.808854i \(-0.700088\pi\)
−0.588009 + 0.808854i \(0.700088\pi\)
\(948\) 0 0
\(949\) 46.2218 1.50042
\(950\) 0 0
\(951\) 39.0348 1.26579
\(952\) 0 0
\(953\) 37.7942 1.22427 0.612137 0.790752i \(-0.290310\pi\)
0.612137 + 0.790752i \(0.290310\pi\)
\(954\) 0 0
\(955\) −7.76565 −0.251291
\(956\) 0 0
\(957\) −24.3986 −0.788695
\(958\) 0 0
\(959\) 26.1993 0.846020
\(960\) 0 0
\(961\) −30.9937 −0.999798
\(962\) 0 0
\(963\) 108.104 3.48362
\(964\) 0 0
\(965\) −2.15824 −0.0694761
\(966\) 0 0
\(967\) −50.6390 −1.62844 −0.814220 0.580557i \(-0.802835\pi\)
−0.814220 + 0.580557i \(0.802835\pi\)
\(968\) 0 0
\(969\) −149.288 −4.79582
\(970\) 0 0
\(971\) −2.82623 −0.0906981 −0.0453490 0.998971i \(-0.514440\pi\)
−0.0453490 + 0.998971i \(0.514440\pi\)
\(972\) 0 0
\(973\) 18.2056 0.583644
\(974\) 0 0
\(975\) 10.7183 0.343261
\(976\) 0 0
\(977\) 25.8448 0.826848 0.413424 0.910539i \(-0.364333\pi\)
0.413424 + 0.910539i \(0.364333\pi\)
\(978\) 0 0
\(979\) −42.2404 −1.35001
\(980\) 0 0
\(981\) 34.4965 1.10139
\(982\) 0 0
\(983\) 41.3668 1.31940 0.659698 0.751531i \(-0.270684\pi\)
0.659698 + 0.751531i \(0.270684\pi\)
\(984\) 0 0
\(985\) 12.0411 0.383660
\(986\) 0 0
\(987\) 44.5694 1.41866
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) −54.4745 −1.73044 −0.865219 0.501394i \(-0.832821\pi\)
−0.865219 + 0.501394i \(0.832821\pi\)
\(992\) 0 0
\(993\) 102.810 3.26257
\(994\) 0 0
\(995\) 14.0000 0.443830
\(996\) 0 0
\(997\) −15.5128 −0.491294 −0.245647 0.969359i \(-0.579000\pi\)
−0.245647 + 0.969359i \(0.579000\pi\)
\(998\) 0 0
\(999\) 104.164 3.29561
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 1840.2.a.s.1.1 3
4.3 odd 2 920.2.a.h.1.3 3
5.4 even 2 9200.2.a.ce.1.3 3
8.3 odd 2 7360.2.a.by.1.1 3
8.5 even 2 7360.2.a.cc.1.3 3
12.11 even 2 8280.2.a.bj.1.1 3
20.3 even 4 4600.2.e.p.4049.6 6
20.7 even 4 4600.2.e.p.4049.1 6
20.19 odd 2 4600.2.a.x.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.h.1.3 3 4.3 odd 2
1840.2.a.s.1.1 3 1.1 even 1 trivial
4600.2.a.x.1.1 3 20.19 odd 2
4600.2.e.p.4049.1 6 20.7 even 4
4600.2.e.p.4049.6 6 20.3 even 4
7360.2.a.by.1.1 3 8.3 odd 2
7360.2.a.cc.1.3 3 8.5 even 2
8280.2.a.bj.1.1 3 12.11 even 2
9200.2.a.ce.1.3 3 5.4 even 2