Properties

Label 1840.2.a.u.1.1
Level $1840$
Weight $2$
Character 1840.1
Self dual yes
Analytic conductor $14.692$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.15317.1
Defining polynomial: \(x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 115)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.32973\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.56155 q^{3} +1.00000 q^{5} -4.06562 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q-1.56155 q^{3} +1.00000 q^{5} -4.06562 q^{7} -0.561553 q^{9} -2.65945 q^{11} -5.91023 q^{13} -1.56155 q^{15} -3.40617 q^{17} +2.65945 q^{19} +6.34868 q^{21} +1.00000 q^{23} +1.00000 q^{25} +5.56155 q^{27} +5.84461 q^{29} +9.31640 q^{31} +4.15288 q^{33} -4.06562 q^{35} -4.18059 q^{37} +9.22914 q^{39} +2.15539 q^{41} -2.34868 q^{43} -0.561553 q^{45} -0.242644 q^{47} +9.52927 q^{49} +5.31891 q^{51} +9.03585 q^{53} -2.65945 q^{55} -4.15288 q^{57} -14.4143 q^{59} +2.46365 q^{61} +2.28306 q^{63} -5.91023 q^{65} +7.75485 q^{67} -1.56155 q^{69} -12.4395 q^{71} -2.08977 q^{73} -1.56155 q^{75} +10.8123 q^{77} +4.15288 q^{79} -7.00000 q^{81} +12.5293 q^{83} -3.40617 q^{85} -9.12667 q^{87} -9.35682 q^{89} +24.0288 q^{91} -14.5481 q^{93} +2.65945 q^{95} +3.47179 q^{97} +1.49342 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{3} + 4q^{5} + 3q^{7} + 6q^{9} + O(q^{10}) \) \( 4q + 2q^{3} + 4q^{5} + 3q^{7} + 6q^{9} - 4q^{11} + 2q^{15} - q^{17} + 4q^{19} + 10q^{21} + 4q^{23} + 4q^{25} + 14q^{27} + 19q^{29} + q^{31} - 2q^{33} + 3q^{35} - 3q^{37} + 13q^{41} + 6q^{43} + 6q^{45} - 6q^{47} + 9q^{49} + 8q^{51} + 19q^{53} - 4q^{55} + 2q^{57} - 23q^{59} + 13q^{63} + 3q^{67} + 2q^{69} + 3q^{71} - 32q^{73} + 2q^{75} + 18q^{77} - 2q^{79} - 28q^{81} + 21q^{83} - q^{85} + 18q^{87} + 40q^{91} - 8q^{93} + 4q^{95} - 18q^{97} - 6q^{99} + O(q^{100}) \)

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
\(3\) −1.56155 −0.901563 −0.450781 0.892634i \(-0.648855\pi\)
−0.450781 + 0.892634i \(0.648855\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −4.06562 −1.53666 −0.768330 0.640054i \(-0.778912\pi\)
−0.768330 + 0.640054i \(0.778912\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) −2.65945 −0.801856 −0.400928 0.916110i \(-0.631312\pi\)
−0.400928 + 0.916110i \(0.631312\pi\)
\(12\) 0 0
\(13\) −5.91023 −1.63920 −0.819602 0.572933i \(-0.805805\pi\)
−0.819602 + 0.572933i \(0.805805\pi\)
\(14\) 0 0
\(15\) −1.56155 −0.403191
\(16\) 0 0
\(17\) −3.40617 −0.826117 −0.413058 0.910705i \(-0.635539\pi\)
−0.413058 + 0.910705i \(0.635539\pi\)
\(18\) 0 0
\(19\) 2.65945 0.610121 0.305060 0.952333i \(-0.401323\pi\)
0.305060 + 0.952333i \(0.401323\pi\)
\(20\) 0 0
\(21\) 6.34868 1.38540
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.56155 1.07032
\(28\) 0 0
\(29\) 5.84461 1.08532 0.542659 0.839953i \(-0.317418\pi\)
0.542659 + 0.839953i \(0.317418\pi\)
\(30\) 0 0
\(31\) 9.31640 1.67327 0.836637 0.547757i \(-0.184518\pi\)
0.836637 + 0.547757i \(0.184518\pi\)
\(32\) 0 0
\(33\) 4.15288 0.722923
\(34\) 0 0
\(35\) −4.06562 −0.687215
\(36\) 0 0
\(37\) −4.18059 −0.687285 −0.343642 0.939101i \(-0.611661\pi\)
−0.343642 + 0.939101i \(0.611661\pi\)
\(38\) 0 0
\(39\) 9.22914 1.47785
\(40\) 0 0
\(41\) 2.15539 0.336615 0.168307 0.985735i \(-0.446170\pi\)
0.168307 + 0.985735i \(0.446170\pi\)
\(42\) 0 0
\(43\) −2.34868 −0.358171 −0.179085 0.983834i \(-0.557314\pi\)
−0.179085 + 0.983834i \(0.557314\pi\)
\(44\) 0 0
\(45\) −0.561553 −0.0837114
\(46\) 0 0
\(47\) −0.242644 −0.0353933 −0.0176966 0.999843i \(-0.505633\pi\)
−0.0176966 + 0.999843i \(0.505633\pi\)
\(48\) 0 0
\(49\) 9.52927 1.36132
\(50\) 0 0
\(51\) 5.31891 0.744796
\(52\) 0 0
\(53\) 9.03585 1.24117 0.620585 0.784140i \(-0.286895\pi\)
0.620585 + 0.784140i \(0.286895\pi\)
\(54\) 0 0
\(55\) −2.65945 −0.358601
\(56\) 0 0
\(57\) −4.15288 −0.550062
\(58\) 0 0
\(59\) −14.4143 −1.87658 −0.938291 0.345846i \(-0.887592\pi\)
−0.938291 + 0.345846i \(0.887592\pi\)
\(60\) 0 0
\(61\) 2.46365 0.315438 0.157719 0.987484i \(-0.449586\pi\)
0.157719 + 0.987484i \(0.449586\pi\)
\(62\) 0 0
\(63\) 2.28306 0.287639
\(64\) 0 0
\(65\) −5.91023 −0.733074
\(66\) 0 0
\(67\) 7.75485 0.947405 0.473703 0.880685i \(-0.342917\pi\)
0.473703 + 0.880685i \(0.342917\pi\)
\(68\) 0 0
\(69\) −1.56155 −0.187989
\(70\) 0 0
\(71\) −12.4395 −1.47630 −0.738149 0.674638i \(-0.764300\pi\)
−0.738149 + 0.674638i \(0.764300\pi\)
\(72\) 0 0
\(73\) −2.08977 −0.244589 −0.122294 0.992494i \(-0.539025\pi\)
−0.122294 + 0.992494i \(0.539025\pi\)
\(74\) 0 0
\(75\) −1.56155 −0.180313
\(76\) 0 0
\(77\) 10.8123 1.23218
\(78\) 0 0
\(79\) 4.15288 0.467235 0.233618 0.972329i \(-0.424944\pi\)
0.233618 + 0.972329i \(0.424944\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 12.5293 1.37527 0.687633 0.726058i \(-0.258650\pi\)
0.687633 + 0.726058i \(0.258650\pi\)
\(84\) 0 0
\(85\) −3.40617 −0.369451
\(86\) 0 0
\(87\) −9.12667 −0.978482
\(88\) 0 0
\(89\) −9.35682 −0.991821 −0.495910 0.868374i \(-0.665166\pi\)
−0.495910 + 0.868374i \(0.665166\pi\)
\(90\) 0 0
\(91\) 24.0288 2.51890
\(92\) 0 0
\(93\) −14.5481 −1.50856
\(94\) 0 0
\(95\) 2.65945 0.272854
\(96\) 0 0
\(97\) 3.47179 0.352507 0.176253 0.984345i \(-0.443602\pi\)
0.176253 + 0.984345i \(0.443602\pi\)
\(98\) 0 0
\(99\) 1.49342 0.150095
\(100\) 0 0
\(101\) 9.21036 0.916465 0.458233 0.888832i \(-0.348483\pi\)
0.458233 + 0.888832i \(0.348483\pi\)
\(102\) 0 0
\(103\) −4.69736 −0.462845 −0.231422 0.972853i \(-0.574338\pi\)
−0.231422 + 0.972853i \(0.574338\pi\)
\(104\) 0 0
\(105\) 6.34868 0.619568
\(106\) 0 0
\(107\) 0.376394 0.0363873 0.0181937 0.999834i \(-0.494208\pi\)
0.0181937 + 0.999834i \(0.494208\pi\)
\(108\) 0 0
\(109\) 19.0369 1.82340 0.911702 0.410851i \(-0.134768\pi\)
0.911702 + 0.410851i \(0.134768\pi\)
\(110\) 0 0
\(111\) 6.52821 0.619631
\(112\) 0 0
\(113\) 2.94252 0.276809 0.138404 0.990376i \(-0.455803\pi\)
0.138404 + 0.990376i \(0.455803\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) 3.31891 0.306833
\(118\) 0 0
\(119\) 13.8482 1.26946
\(120\) 0 0
\(121\) −3.92730 −0.357028
\(122\) 0 0
\(123\) −3.36575 −0.303479
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −6.00357 −0.532730 −0.266365 0.963872i \(-0.585823\pi\)
−0.266365 + 0.963872i \(0.585823\pi\)
\(128\) 0 0
\(129\) 3.66759 0.322913
\(130\) 0 0
\(131\) 19.9606 1.74397 0.871985 0.489533i \(-0.162833\pi\)
0.871985 + 0.489533i \(0.162833\pi\)
\(132\) 0 0
\(133\) −10.8123 −0.937548
\(134\) 0 0
\(135\) 5.56155 0.478662
\(136\) 0 0
\(137\) −11.7609 −1.00480 −0.502402 0.864634i \(-0.667550\pi\)
−0.502402 + 0.864634i \(0.667550\pi\)
\(138\) 0 0
\(139\) −12.3891 −1.05083 −0.525415 0.850846i \(-0.676090\pi\)
−0.525415 + 0.850846i \(0.676090\pi\)
\(140\) 0 0
\(141\) 0.378902 0.0319093
\(142\) 0 0
\(143\) 15.7180 1.31441
\(144\) 0 0
\(145\) 5.84461 0.485369
\(146\) 0 0
\(147\) −14.8805 −1.22732
\(148\) 0 0
\(149\) −2.13124 −0.174598 −0.0872991 0.996182i \(-0.527824\pi\)
−0.0872991 + 0.996182i \(0.527824\pi\)
\(150\) 0 0
\(151\) 0.787129 0.0640556 0.0320278 0.999487i \(-0.489803\pi\)
0.0320278 + 0.999487i \(0.489803\pi\)
\(152\) 0 0
\(153\) 1.91274 0.154636
\(154\) 0 0
\(155\) 9.31640 0.748311
\(156\) 0 0
\(157\) −9.18873 −0.733340 −0.366670 0.930351i \(-0.619502\pi\)
−0.366670 + 0.930351i \(0.619502\pi\)
\(158\) 0 0
\(159\) −14.1100 −1.11899
\(160\) 0 0
\(161\) −4.06562 −0.320416
\(162\) 0 0
\(163\) 17.6928 1.38581 0.692903 0.721031i \(-0.256331\pi\)
0.692903 + 0.721031i \(0.256331\pi\)
\(164\) 0 0
\(165\) 4.15288 0.323301
\(166\) 0 0
\(167\) −14.2462 −1.10240 −0.551202 0.834372i \(-0.685831\pi\)
−0.551202 + 0.834372i \(0.685831\pi\)
\(168\) 0 0
\(169\) 21.9309 1.68699
\(170\) 0 0
\(171\) −1.49342 −0.114205
\(172\) 0 0
\(173\) −1.20394 −0.0915338 −0.0457669 0.998952i \(-0.514573\pi\)
−0.0457669 + 0.998952i \(0.514573\pi\)
\(174\) 0 0
\(175\) −4.06562 −0.307332
\(176\) 0 0
\(177\) 22.5087 1.69186
\(178\) 0 0
\(179\) −7.42495 −0.554967 −0.277483 0.960730i \(-0.589500\pi\)
−0.277483 + 0.960730i \(0.589500\pi\)
\(180\) 0 0
\(181\) 24.0450 1.78725 0.893627 0.448810i \(-0.148152\pi\)
0.893627 + 0.448810i \(0.148152\pi\)
\(182\) 0 0
\(183\) −3.84712 −0.284387
\(184\) 0 0
\(185\) −4.18059 −0.307363
\(186\) 0 0
\(187\) 9.05854 0.662426
\(188\) 0 0
\(189\) −22.6112 −1.64472
\(190\) 0 0
\(191\) 4.87689 0.352880 0.176440 0.984311i \(-0.443542\pi\)
0.176440 + 0.984311i \(0.443542\pi\)
\(192\) 0 0
\(193\) 10.2714 0.739353 0.369676 0.929161i \(-0.379469\pi\)
0.369676 + 0.929161i \(0.379469\pi\)
\(194\) 0 0
\(195\) 9.22914 0.660913
\(196\) 0 0
\(197\) 4.28557 0.305334 0.152667 0.988278i \(-0.451214\pi\)
0.152667 + 0.988278i \(0.451214\pi\)
\(198\) 0 0
\(199\) −4.02164 −0.285086 −0.142543 0.989789i \(-0.545528\pi\)
−0.142543 + 0.989789i \(0.545528\pi\)
\(200\) 0 0
\(201\) −12.1096 −0.854146
\(202\) 0 0
\(203\) −23.7620 −1.66776
\(204\) 0 0
\(205\) 2.15539 0.150539
\(206\) 0 0
\(207\) −0.561553 −0.0390306
\(208\) 0 0
\(209\) −7.07270 −0.489229
\(210\) 0 0
\(211\) −15.3416 −1.05616 −0.528080 0.849195i \(-0.677088\pi\)
−0.528080 + 0.849195i \(0.677088\pi\)
\(212\) 0 0
\(213\) 19.4249 1.33098
\(214\) 0 0
\(215\) −2.34868 −0.160179
\(216\) 0 0
\(217\) −37.8770 −2.57126
\(218\) 0 0
\(219\) 3.26328 0.220512
\(220\) 0 0
\(221\) 20.1312 1.35417
\(222\) 0 0
\(223\) 17.6801 1.18395 0.591973 0.805958i \(-0.298349\pi\)
0.591973 + 0.805958i \(0.298349\pi\)
\(224\) 0 0
\(225\) −0.561553 −0.0374369
\(226\) 0 0
\(227\) −8.63817 −0.573335 −0.286668 0.958030i \(-0.592548\pi\)
−0.286668 + 0.958030i \(0.592548\pi\)
\(228\) 0 0
\(229\) −3.34055 −0.220749 −0.110375 0.993890i \(-0.535205\pi\)
−0.110375 + 0.993890i \(0.535205\pi\)
\(230\) 0 0
\(231\) −16.8840 −1.11089
\(232\) 0 0
\(233\) 8.23728 0.539642 0.269821 0.962910i \(-0.413035\pi\)
0.269821 + 0.962910i \(0.413035\pi\)
\(234\) 0 0
\(235\) −0.242644 −0.0158284
\(236\) 0 0
\(237\) −6.48494 −0.421242
\(238\) 0 0
\(239\) 2.86525 0.185338 0.0926688 0.995697i \(-0.470460\pi\)
0.0926688 + 0.995697i \(0.470460\pi\)
\(240\) 0 0
\(241\) 4.68109 0.301536 0.150768 0.988569i \(-0.451825\pi\)
0.150768 + 0.988569i \(0.451825\pi\)
\(242\) 0 0
\(243\) −5.75379 −0.369106
\(244\) 0 0
\(245\) 9.52927 0.608803
\(246\) 0 0
\(247\) −15.7180 −1.00011
\(248\) 0 0
\(249\) −19.5651 −1.23989
\(250\) 0 0
\(251\) 18.5824 1.17291 0.586455 0.809982i \(-0.300523\pi\)
0.586455 + 0.809982i \(0.300523\pi\)
\(252\) 0 0
\(253\) −2.65945 −0.167198
\(254\) 0 0
\(255\) 5.31891 0.333083
\(256\) 0 0
\(257\) −28.7226 −1.79166 −0.895832 0.444392i \(-0.853420\pi\)
−0.895832 + 0.444392i \(0.853420\pi\)
\(258\) 0 0
\(259\) 16.9967 1.05612
\(260\) 0 0
\(261\) −3.28206 −0.203154
\(262\) 0 0
\(263\) 9.20500 0.567604 0.283802 0.958883i \(-0.408404\pi\)
0.283802 + 0.958883i \(0.408404\pi\)
\(264\) 0 0
\(265\) 9.03585 0.555068
\(266\) 0 0
\(267\) 14.6112 0.894189
\(268\) 0 0
\(269\) 16.9194 1.03160 0.515798 0.856710i \(-0.327496\pi\)
0.515798 + 0.856710i \(0.327496\pi\)
\(270\) 0 0
\(271\) −1.15082 −0.0699072 −0.0349536 0.999389i \(-0.511128\pi\)
−0.0349536 + 0.999389i \(0.511128\pi\)
\(272\) 0 0
\(273\) −37.5222 −2.27095
\(274\) 0 0
\(275\) −2.65945 −0.160371
\(276\) 0 0
\(277\) 2.57505 0.154720 0.0773600 0.997003i \(-0.475351\pi\)
0.0773600 + 0.997003i \(0.475351\pi\)
\(278\) 0 0
\(279\) −5.23165 −0.313211
\(280\) 0 0
\(281\) 27.4718 1.63883 0.819415 0.573201i \(-0.194299\pi\)
0.819415 + 0.573201i \(0.194299\pi\)
\(282\) 0 0
\(283\) 2.36389 0.140519 0.0702595 0.997529i \(-0.477617\pi\)
0.0702595 + 0.997529i \(0.477617\pi\)
\(284\) 0 0
\(285\) −4.15288 −0.245995
\(286\) 0 0
\(287\) −8.76298 −0.517263
\(288\) 0 0
\(289\) −5.39803 −0.317531
\(290\) 0 0
\(291\) −5.42138 −0.317807
\(292\) 0 0
\(293\) 34.1198 1.99330 0.996650 0.0817846i \(-0.0260619\pi\)
0.996650 + 0.0817846i \(0.0260619\pi\)
\(294\) 0 0
\(295\) −14.4143 −0.839233
\(296\) 0 0
\(297\) −14.7907 −0.858243
\(298\) 0 0
\(299\) −5.91023 −0.341798
\(300\) 0 0
\(301\) 9.54885 0.550386
\(302\) 0 0
\(303\) −14.3825 −0.826251
\(304\) 0 0
\(305\) 2.46365 0.141068
\(306\) 0 0
\(307\) −1.49342 −0.0852342 −0.0426171 0.999091i \(-0.513570\pi\)
−0.0426171 + 0.999091i \(0.513570\pi\)
\(308\) 0 0
\(309\) 7.33518 0.417284
\(310\) 0 0
\(311\) 14.1060 0.799880 0.399940 0.916541i \(-0.369031\pi\)
0.399940 + 0.916541i \(0.369031\pi\)
\(312\) 0 0
\(313\) −32.8563 −1.85715 −0.928574 0.371146i \(-0.878965\pi\)
−0.928574 + 0.371146i \(0.878965\pi\)
\(314\) 0 0
\(315\) 2.28306 0.128636
\(316\) 0 0
\(317\) 12.5824 0.706698 0.353349 0.935492i \(-0.385043\pi\)
0.353349 + 0.935492i \(0.385043\pi\)
\(318\) 0 0
\(319\) −15.5435 −0.870268
\(320\) 0 0
\(321\) −0.587758 −0.0328055
\(322\) 0 0
\(323\) −9.05854 −0.504031
\(324\) 0 0
\(325\) −5.91023 −0.327841
\(326\) 0 0
\(327\) −29.7271 −1.64391
\(328\) 0 0
\(329\) 0.986499 0.0543874
\(330\) 0 0
\(331\) −1.57677 −0.0866669 −0.0433334 0.999061i \(-0.513798\pi\)
−0.0433334 + 0.999061i \(0.513798\pi\)
\(332\) 0 0
\(333\) 2.34762 0.128649
\(334\) 0 0
\(335\) 7.75485 0.423693
\(336\) 0 0
\(337\) −15.4718 −0.842802 −0.421401 0.906874i \(-0.638461\pi\)
−0.421401 + 0.906874i \(0.638461\pi\)
\(338\) 0 0
\(339\) −4.59489 −0.249560
\(340\) 0 0
\(341\) −24.7765 −1.34172
\(342\) 0 0
\(343\) −10.2831 −0.555233
\(344\) 0 0
\(345\) −1.56155 −0.0840712
\(346\) 0 0
\(347\) 9.00312 0.483313 0.241656 0.970362i \(-0.422309\pi\)
0.241656 + 0.970362i \(0.422309\pi\)
\(348\) 0 0
\(349\) 1.10398 0.0590945 0.0295473 0.999563i \(-0.490593\pi\)
0.0295473 + 0.999563i \(0.490593\pi\)
\(350\) 0 0
\(351\) −32.8701 −1.75448
\(352\) 0 0
\(353\) 1.96566 0.104621 0.0523107 0.998631i \(-0.483341\pi\)
0.0523107 + 0.998631i \(0.483341\pi\)
\(354\) 0 0
\(355\) −12.4395 −0.660220
\(356\) 0 0
\(357\) −21.6247 −1.14450
\(358\) 0 0
\(359\) 2.60403 0.137435 0.0687177 0.997636i \(-0.478109\pi\)
0.0687177 + 0.997636i \(0.478109\pi\)
\(360\) 0 0
\(361\) −11.9273 −0.627753
\(362\) 0 0
\(363\) 6.13269 0.321883
\(364\) 0 0
\(365\) −2.08977 −0.109383
\(366\) 0 0
\(367\) −6.98042 −0.364375 −0.182188 0.983264i \(-0.558318\pi\)
−0.182188 + 0.983264i \(0.558318\pi\)
\(368\) 0 0
\(369\) −1.21036 −0.0630090
\(370\) 0 0
\(371\) −36.7363 −1.90726
\(372\) 0 0
\(373\) 24.3220 1.25935 0.629673 0.776860i \(-0.283189\pi\)
0.629673 + 0.776860i \(0.283189\pi\)
\(374\) 0 0
\(375\) −1.56155 −0.0806382
\(376\) 0 0
\(377\) −34.5430 −1.77906
\(378\) 0 0
\(379\) 24.6541 1.26640 0.633198 0.773990i \(-0.281742\pi\)
0.633198 + 0.773990i \(0.281742\pi\)
\(380\) 0 0
\(381\) 9.37489 0.480290
\(382\) 0 0
\(383\) 4.99292 0.255126 0.127563 0.991830i \(-0.459284\pi\)
0.127563 + 0.991830i \(0.459284\pi\)
\(384\) 0 0
\(385\) 10.8123 0.551048
\(386\) 0 0
\(387\) 1.31891 0.0670439
\(388\) 0 0
\(389\) −32.0234 −1.62365 −0.811826 0.583900i \(-0.801526\pi\)
−0.811826 + 0.583900i \(0.801526\pi\)
\(390\) 0 0
\(391\) −3.40617 −0.172257
\(392\) 0 0
\(393\) −31.1696 −1.57230
\(394\) 0 0
\(395\) 4.15288 0.208954
\(396\) 0 0
\(397\) −17.3441 −0.870476 −0.435238 0.900315i \(-0.643336\pi\)
−0.435238 + 0.900315i \(0.643336\pi\)
\(398\) 0 0
\(399\) 16.8840 0.845259
\(400\) 0 0
\(401\) −21.3352 −1.06543 −0.532714 0.846295i \(-0.678828\pi\)
−0.532714 + 0.846295i \(0.678828\pi\)
\(402\) 0 0
\(403\) −55.0621 −2.74284
\(404\) 0 0
\(405\) −7.00000 −0.347833
\(406\) 0 0
\(407\) 11.1181 0.551103
\(408\) 0 0
\(409\) −12.5328 −0.619709 −0.309855 0.950784i \(-0.600280\pi\)
−0.309855 + 0.950784i \(0.600280\pi\)
\(410\) 0 0
\(411\) 18.3653 0.905894
\(412\) 0 0
\(413\) 58.6031 2.88367
\(414\) 0 0
\(415\) 12.5293 0.615038
\(416\) 0 0
\(417\) 19.3462 0.947389
\(418\) 0 0
\(419\) −12.0288 −0.587644 −0.293822 0.955860i \(-0.594927\pi\)
−0.293822 + 0.955860i \(0.594927\pi\)
\(420\) 0 0
\(421\) −38.3721 −1.87014 −0.935071 0.354462i \(-0.884664\pi\)
−0.935071 + 0.354462i \(0.884664\pi\)
\(422\) 0 0
\(423\) 0.136257 0.00662507
\(424\) 0 0
\(425\) −3.40617 −0.165223
\(426\) 0 0
\(427\) −10.0163 −0.484721
\(428\) 0 0
\(429\) −24.5445 −1.18502
\(430\) 0 0
\(431\) 29.4789 1.41995 0.709975 0.704227i \(-0.248706\pi\)
0.709975 + 0.704227i \(0.248706\pi\)
\(432\) 0 0
\(433\) −12.0531 −0.579236 −0.289618 0.957142i \(-0.593528\pi\)
−0.289618 + 0.957142i \(0.593528\pi\)
\(434\) 0 0
\(435\) −9.12667 −0.437590
\(436\) 0 0
\(437\) 2.65945 0.127219
\(438\) 0 0
\(439\) −16.1656 −0.771541 −0.385771 0.922595i \(-0.626064\pi\)
−0.385771 + 0.922595i \(0.626064\pi\)
\(440\) 0 0
\(441\) −5.35119 −0.254819
\(442\) 0 0
\(443\) −19.3729 −0.920433 −0.460217 0.887807i \(-0.652228\pi\)
−0.460217 + 0.887807i \(0.652228\pi\)
\(444\) 0 0
\(445\) −9.35682 −0.443556
\(446\) 0 0
\(447\) 3.32805 0.157411
\(448\) 0 0
\(449\) 21.4728 1.01337 0.506683 0.862132i \(-0.330871\pi\)
0.506683 + 0.862132i \(0.330871\pi\)
\(450\) 0 0
\(451\) −5.73215 −0.269916
\(452\) 0 0
\(453\) −1.22914 −0.0577502
\(454\) 0 0
\(455\) 24.0288 1.12649
\(456\) 0 0
\(457\) −5.95602 −0.278611 −0.139305 0.990249i \(-0.544487\pi\)
−0.139305 + 0.990249i \(0.544487\pi\)
\(458\) 0 0
\(459\) −18.9436 −0.884210
\(460\) 0 0
\(461\) 34.6918 1.61576 0.807879 0.589348i \(-0.200615\pi\)
0.807879 + 0.589348i \(0.200615\pi\)
\(462\) 0 0
\(463\) 30.6399 1.42396 0.711979 0.702200i \(-0.247799\pi\)
0.711979 + 0.702200i \(0.247799\pi\)
\(464\) 0 0
\(465\) −14.5481 −0.674650
\(466\) 0 0
\(467\) 8.89547 0.411633 0.205817 0.978591i \(-0.434015\pi\)
0.205817 + 0.978591i \(0.434015\pi\)
\(468\) 0 0
\(469\) −31.5283 −1.45584
\(470\) 0 0
\(471\) 14.3487 0.661152
\(472\) 0 0
\(473\) 6.24621 0.287201
\(474\) 0 0
\(475\) 2.65945 0.122024
\(476\) 0 0
\(477\) −5.07411 −0.232327
\(478\) 0 0
\(479\) 9.13224 0.417263 0.208631 0.977994i \(-0.433099\pi\)
0.208631 + 0.977994i \(0.433099\pi\)
\(480\) 0 0
\(481\) 24.7083 1.12660
\(482\) 0 0
\(483\) 6.34868 0.288875
\(484\) 0 0
\(485\) 3.47179 0.157646
\(486\) 0 0
\(487\) −14.2085 −0.643849 −0.321924 0.946765i \(-0.604330\pi\)
−0.321924 + 0.946765i \(0.604330\pi\)
\(488\) 0 0
\(489\) −27.6282 −1.24939
\(490\) 0 0
\(491\) 23.1760 1.04592 0.522960 0.852357i \(-0.324828\pi\)
0.522960 + 0.852357i \(0.324828\pi\)
\(492\) 0 0
\(493\) −19.9077 −0.896599
\(494\) 0 0
\(495\) 1.49342 0.0671244
\(496\) 0 0
\(497\) 50.5743 2.26857
\(498\) 0 0
\(499\) 2.19831 0.0984099 0.0492050 0.998789i \(-0.484331\pi\)
0.0492050 + 0.998789i \(0.484331\pi\)
\(500\) 0 0
\(501\) 22.2462 0.993887
\(502\) 0 0
\(503\) 44.0461 1.96392 0.981959 0.189092i \(-0.0605545\pi\)
0.981959 + 0.189092i \(0.0605545\pi\)
\(504\) 0 0
\(505\) 9.21036 0.409856
\(506\) 0 0
\(507\) −34.2462 −1.52093
\(508\) 0 0
\(509\) −9.09254 −0.403020 −0.201510 0.979486i \(-0.564585\pi\)
−0.201510 + 0.979486i \(0.564585\pi\)
\(510\) 0 0
\(511\) 8.49619 0.375850
\(512\) 0 0
\(513\) 14.7907 0.653025
\(514\) 0 0
\(515\) −4.69736 −0.206991
\(516\) 0 0
\(517\) 0.645301 0.0283803
\(518\) 0 0
\(519\) 1.88001 0.0825235
\(520\) 0 0
\(521\) −17.1231 −0.750177 −0.375088 0.926989i \(-0.622388\pi\)
−0.375088 + 0.926989i \(0.622388\pi\)
\(522\) 0 0
\(523\) 26.6112 1.16362 0.581812 0.813323i \(-0.302344\pi\)
0.581812 + 0.813323i \(0.302344\pi\)
\(524\) 0 0
\(525\) 6.34868 0.277079
\(526\) 0 0
\(527\) −31.7332 −1.38232
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 8.09439 0.351267
\(532\) 0 0
\(533\) −12.7388 −0.551780
\(534\) 0 0
\(535\) 0.376394 0.0162729
\(536\) 0 0
\(537\) 11.5944 0.500337
\(538\) 0 0
\(539\) −25.3427 −1.09159
\(540\) 0 0
\(541\) 31.5941 1.35834 0.679168 0.733983i \(-0.262341\pi\)
0.679168 + 0.733983i \(0.262341\pi\)
\(542\) 0 0
\(543\) −37.5476 −1.61132
\(544\) 0 0
\(545\) 19.0369 0.815452
\(546\) 0 0
\(547\) −22.7009 −0.970622 −0.485311 0.874342i \(-0.661294\pi\)
−0.485311 + 0.874342i \(0.661294\pi\)
\(548\) 0 0
\(549\) −1.38347 −0.0590451
\(550\) 0 0
\(551\) 15.5435 0.662175
\(552\) 0 0
\(553\) −16.8840 −0.717982
\(554\) 0 0
\(555\) 6.52821 0.277107
\(556\) 0 0
\(557\) 35.1292 1.48847 0.744236 0.667917i \(-0.232814\pi\)
0.744236 + 0.667917i \(0.232814\pi\)
\(558\) 0 0
\(559\) 13.8813 0.587115
\(560\) 0 0
\(561\) −14.1454 −0.597219
\(562\) 0 0
\(563\) −32.0511 −1.35079 −0.675397 0.737455i \(-0.736028\pi\)
−0.675397 + 0.737455i \(0.736028\pi\)
\(564\) 0 0
\(565\) 2.94252 0.123793
\(566\) 0 0
\(567\) 28.4593 1.19518
\(568\) 0 0
\(569\) −22.0575 −0.924700 −0.462350 0.886697i \(-0.652994\pi\)
−0.462350 + 0.886697i \(0.652994\pi\)
\(570\) 0 0
\(571\) −8.49242 −0.355397 −0.177698 0.984085i \(-0.556865\pi\)
−0.177698 + 0.984085i \(0.556865\pi\)
\(572\) 0 0
\(573\) −7.61553 −0.318143
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 27.1554 1.13050 0.565248 0.824921i \(-0.308780\pi\)
0.565248 + 0.824921i \(0.308780\pi\)
\(578\) 0 0
\(579\) −16.0394 −0.666573
\(580\) 0 0
\(581\) −50.9393 −2.11332
\(582\) 0 0
\(583\) −24.0304 −0.995239
\(584\) 0 0
\(585\) 3.31891 0.137220
\(586\) 0 0
\(587\) 30.8322 1.27258 0.636290 0.771450i \(-0.280468\pi\)
0.636290 + 0.771450i \(0.280468\pi\)
\(588\) 0 0
\(589\) 24.7765 1.02090
\(590\) 0 0
\(591\) −6.69214 −0.275278
\(592\) 0 0
\(593\) −27.7559 −1.13980 −0.569899 0.821715i \(-0.693018\pi\)
−0.569899 + 0.821715i \(0.693018\pi\)
\(594\) 0 0
\(595\) 13.8482 0.567720
\(596\) 0 0
\(597\) 6.28000 0.257023
\(598\) 0 0
\(599\) 30.7712 1.25728 0.628638 0.777698i \(-0.283613\pi\)
0.628638 + 0.777698i \(0.283613\pi\)
\(600\) 0 0
\(601\) −17.9830 −0.733541 −0.366771 0.930311i \(-0.619537\pi\)
−0.366771 + 0.930311i \(0.619537\pi\)
\(602\) 0 0
\(603\) −4.35476 −0.177339
\(604\) 0 0
\(605\) −3.92730 −0.159668
\(606\) 0 0
\(607\) 18.3220 0.743668 0.371834 0.928299i \(-0.378729\pi\)
0.371834 + 0.928299i \(0.378729\pi\)
\(608\) 0 0
\(609\) 37.1056 1.50359
\(610\) 0 0
\(611\) 1.43408 0.0580168
\(612\) 0 0
\(613\) 35.4451 1.43162 0.715808 0.698297i \(-0.246059\pi\)
0.715808 + 0.698297i \(0.246059\pi\)
\(614\) 0 0
\(615\) −3.36575 −0.135720
\(616\) 0 0
\(617\) 43.1567 1.73742 0.868712 0.495318i \(-0.164948\pi\)
0.868712 + 0.495318i \(0.164948\pi\)
\(618\) 0 0
\(619\) −5.81133 −0.233577 −0.116789 0.993157i \(-0.537260\pi\)
−0.116789 + 0.993157i \(0.537260\pi\)
\(620\) 0 0
\(621\) 5.56155 0.223177
\(622\) 0 0
\(623\) 38.0413 1.52409
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 11.0444 0.441070
\(628\) 0 0
\(629\) 14.2398 0.567777
\(630\) 0 0
\(631\) 27.4626 1.09327 0.546635 0.837371i \(-0.315908\pi\)
0.546635 + 0.837371i \(0.315908\pi\)
\(632\) 0 0
\(633\) 23.9567 0.952194
\(634\) 0 0
\(635\) −6.00357 −0.238244
\(636\) 0 0
\(637\) −56.3202 −2.23149
\(638\) 0 0
\(639\) 6.98544 0.276340
\(640\) 0 0
\(641\) 37.7938 1.49277 0.746383 0.665517i \(-0.231789\pi\)
0.746383 + 0.665517i \(0.231789\pi\)
\(642\) 0 0
\(643\) −12.5343 −0.494304 −0.247152 0.968977i \(-0.579495\pi\)
−0.247152 + 0.968977i \(0.579495\pi\)
\(644\) 0 0
\(645\) 3.66759 0.144411
\(646\) 0 0
\(647\) 21.5133 0.845774 0.422887 0.906183i \(-0.361017\pi\)
0.422887 + 0.906183i \(0.361017\pi\)
\(648\) 0 0
\(649\) 38.3342 1.50475
\(650\) 0 0
\(651\) 59.1469 2.31815
\(652\) 0 0
\(653\) 19.9011 0.778790 0.389395 0.921071i \(-0.372684\pi\)
0.389395 + 0.921071i \(0.372684\pi\)
\(654\) 0 0
\(655\) 19.9606 0.779927
\(656\) 0 0
\(657\) 1.17351 0.0457831
\(658\) 0 0
\(659\) 4.95773 0.193126 0.0965628 0.995327i \(-0.469215\pi\)
0.0965628 + 0.995327i \(0.469215\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) −31.4360 −1.22087
\(664\) 0 0
\(665\) −10.8123 −0.419284
\(666\) 0 0
\(667\) 5.84461 0.226304
\(668\) 0 0
\(669\) −27.6084 −1.06740
\(670\) 0 0
\(671\) −6.55197 −0.252936
\(672\) 0 0
\(673\) −28.5176 −1.09927 −0.549637 0.835404i \(-0.685234\pi\)
−0.549637 + 0.835404i \(0.685234\pi\)
\(674\) 0 0
\(675\) 5.56155 0.214064
\(676\) 0 0
\(677\) 28.6214 1.10001 0.550004 0.835162i \(-0.314626\pi\)
0.550004 + 0.835162i \(0.314626\pi\)
\(678\) 0 0
\(679\) −14.1150 −0.541683
\(680\) 0 0
\(681\) 13.4890 0.516898
\(682\) 0 0
\(683\) −11.3983 −0.436144 −0.218072 0.975933i \(-0.569977\pi\)
−0.218072 + 0.975933i \(0.569977\pi\)
\(684\) 0 0
\(685\) −11.7609 −0.449362
\(686\) 0 0
\(687\) 5.21644 0.199020
\(688\) 0 0
\(689\) −53.4040 −2.03453
\(690\) 0 0
\(691\) −45.1898 −1.71910 −0.859550 0.511051i \(-0.829256\pi\)
−0.859550 + 0.511051i \(0.829256\pi\)
\(692\) 0 0
\(693\) −6.07170 −0.230645
\(694\) 0 0
\(695\) −12.3891 −0.469945
\(696\) 0 0
\(697\) −7.34160 −0.278083
\(698\) 0 0
\(699\) −12.8629 −0.486521
\(700\) 0 0
\(701\) 40.4871 1.52918 0.764588 0.644520i \(-0.222943\pi\)
0.764588 + 0.644520i \(0.222943\pi\)
\(702\) 0 0
\(703\) −11.1181 −0.419327
\(704\) 0 0
\(705\) 0.378902 0.0142703
\(706\) 0 0
\(707\) −37.4458 −1.40830
\(708\) 0 0
\(709\) −41.6443 −1.56398 −0.781992 0.623288i \(-0.785796\pi\)
−0.781992 + 0.623288i \(0.785796\pi\)
\(710\) 0 0
\(711\) −2.33206 −0.0874591
\(712\) 0 0
\(713\) 9.31640 0.348902
\(714\) 0 0
\(715\) 15.7180 0.587820
\(716\) 0 0
\(717\) −4.47424 −0.167093
\(718\) 0 0
\(719\) −49.2288 −1.83592 −0.917961 0.396670i \(-0.870166\pi\)
−0.917961 + 0.396670i \(0.870166\pi\)
\(720\) 0 0
\(721\) 19.0977 0.711235
\(722\) 0 0
\(723\) −7.30977 −0.271853
\(724\) 0 0
\(725\) 5.84461 0.217063
\(726\) 0 0
\(727\) −27.9581 −1.03691 −0.518455 0.855105i \(-0.673493\pi\)
−0.518455 + 0.855105i \(0.673493\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 0 0
\(733\) −18.6655 −0.689427 −0.344714 0.938708i \(-0.612024\pi\)
−0.344714 + 0.938708i \(0.612024\pi\)
\(734\) 0 0
\(735\) −14.8805 −0.548874
\(736\) 0 0
\(737\) −20.6237 −0.759682
\(738\) 0 0
\(739\) −2.50407 −0.0921136 −0.0460568 0.998939i \(-0.514666\pi\)
−0.0460568 + 0.998939i \(0.514666\pi\)
\(740\) 0 0
\(741\) 24.5445 0.901664
\(742\) 0 0
\(743\) −4.69736 −0.172330 −0.0861648 0.996281i \(-0.527461\pi\)
−0.0861648 + 0.996281i \(0.527461\pi\)
\(744\) 0 0
\(745\) −2.13124 −0.0780826
\(746\) 0 0
\(747\) −7.03585 −0.257428
\(748\) 0 0
\(749\) −1.53027 −0.0559150
\(750\) 0 0
\(751\) 16.0720 0.586477 0.293239 0.956039i \(-0.405267\pi\)
0.293239 + 0.956039i \(0.405267\pi\)
\(752\) 0 0
\(753\) −29.0174 −1.05745
\(754\) 0 0
\(755\) 0.787129 0.0286465
\(756\) 0 0
\(757\) −25.8357 −0.939014 −0.469507 0.882929i \(-0.655568\pi\)
−0.469507 + 0.882929i \(0.655568\pi\)
\(758\) 0 0
\(759\) 4.15288 0.150740
\(760\) 0 0
\(761\) 15.6005 0.565518 0.282759 0.959191i \(-0.408750\pi\)
0.282759 + 0.959191i \(0.408750\pi\)
\(762\) 0 0
\(763\) −77.3968 −2.80195
\(764\) 0 0
\(765\) 1.91274 0.0691553
\(766\) 0 0
\(767\) 85.1919 3.07610
\(768\) 0 0
\(769\) −37.7722 −1.36210 −0.681050 0.732237i \(-0.738476\pi\)
−0.681050 + 0.732237i \(0.738476\pi\)
\(770\) 0 0
\(771\) 44.8518 1.61530
\(772\) 0 0
\(773\) −42.9078 −1.54329 −0.771643 0.636056i \(-0.780565\pi\)
−0.771643 + 0.636056i \(0.780565\pi\)
\(774\) 0 0
\(775\) 9.31640 0.334655
\(776\) 0 0
\(777\) −26.5412 −0.952162
\(778\) 0 0
\(779\) 5.73215 0.205376
\(780\) 0 0
\(781\) 33.0823 1.18378
\(782\) 0 0
\(783\) 32.5051 1.16164
\(784\) 0 0
\(785\) −9.18873 −0.327960
\(786\) 0 0
\(787\) 33.5324 1.19530 0.597650 0.801757i \(-0.296101\pi\)
0.597650 + 0.801757i \(0.296101\pi\)
\(788\) 0 0
\(789\) −14.3741 −0.511731
\(790\) 0 0
\(791\) −11.9632 −0.425361
\(792\) 0 0
\(793\) −14.5608 −0.517068
\(794\) 0 0
\(795\) −14.1100 −0.500428
\(796\) 0 0
\(797\) 32.7488 1.16002 0.580012 0.814608i \(-0.303048\pi\)
0.580012 + 0.814608i \(0.303048\pi\)
\(798\) 0 0
\(799\) 0.826486 0.0292390
\(800\) 0 0
\(801\) 5.25435 0.185653
\(802\) 0 0
\(803\) 5.55764 0.196125
\(804\) 0 0
\(805\) −4.06562 −0.143294
\(806\) 0 0
\(807\) −26.4206 −0.930049
\(808\) 0 0
\(809\) 28.8513 1.01436 0.507179 0.861841i \(-0.330688\pi\)
0.507179 + 0.861841i \(0.330688\pi\)
\(810\) 0 0
\(811\) −36.1791 −1.27042 −0.635211 0.772339i \(-0.719087\pi\)
−0.635211 + 0.772339i \(0.719087\pi\)
\(812\) 0 0
\(813\) 1.79706 0.0630257
\(814\) 0 0
\(815\) 17.6928 0.619752
\(816\) 0 0
\(817\) −6.24621 −0.218527
\(818\) 0 0
\(819\) −13.4934 −0.471498
\(820\) 0 0
\(821\) 5.73752 0.200241 0.100120 0.994975i \(-0.468077\pi\)
0.100120 + 0.994975i \(0.468077\pi\)
\(822\) 0 0
\(823\) 6.87333 0.239589 0.119795 0.992799i \(-0.461776\pi\)
0.119795 + 0.992799i \(0.461776\pi\)
\(824\) 0 0
\(825\) 4.15288 0.144585
\(826\) 0 0
\(827\) 19.7332 0.686191 0.343095 0.939301i \(-0.388525\pi\)
0.343095 + 0.939301i \(0.388525\pi\)
\(828\) 0 0
\(829\) −2.88833 −0.100316 −0.0501580 0.998741i \(-0.515972\pi\)
−0.0501580 + 0.998741i \(0.515972\pi\)
\(830\) 0 0
\(831\) −4.02108 −0.139490
\(832\) 0 0
\(833\) −32.4583 −1.12461
\(834\) 0 0
\(835\) −14.2462 −0.493010
\(836\) 0 0
\(837\) 51.8137 1.79094
\(838\) 0 0
\(839\) −24.1904 −0.835147 −0.417573 0.908643i \(-0.637119\pi\)
−0.417573 + 0.908643i \(0.637119\pi\)
\(840\) 0 0
\(841\) 5.15951 0.177914
\(842\) 0 0
\(843\) −42.8986 −1.47751
\(844\) 0 0
\(845\) 21.9309 0.754445
\(846\) 0 0
\(847\) 15.9669 0.548630
\(848\) 0 0
\(849\) −3.69135 −0.126687
\(850\) 0 0
\(851\) −4.18059 −0.143309
\(852\) 0 0
\(853\) 27.5381 0.942887 0.471444 0.881896i \(-0.343733\pi\)
0.471444 + 0.881896i \(0.343733\pi\)
\(854\) 0 0
\(855\) −1.49342 −0.0510740
\(856\) 0 0
\(857\) −17.3995 −0.594357 −0.297178 0.954822i \(-0.596046\pi\)
−0.297178 + 0.954822i \(0.596046\pi\)
\(858\) 0 0
\(859\) −1.24560 −0.0424993 −0.0212497 0.999774i \(-0.506764\pi\)
−0.0212497 + 0.999774i \(0.506764\pi\)
\(860\) 0 0
\(861\) 13.6839 0.466345
\(862\) 0 0
\(863\) 4.60383 0.156716 0.0783580 0.996925i \(-0.475032\pi\)
0.0783580 + 0.996925i \(0.475032\pi\)
\(864\) 0 0
\(865\) −1.20394 −0.0409352
\(866\) 0 0
\(867\) 8.42931 0.286274
\(868\) 0 0
\(869\) −11.0444 −0.374655
\(870\) 0 0
\(871\) −45.8330 −1.55299
\(872\) 0 0
\(873\) −1.94959 −0.0659837
\(874\) 0 0
\(875\) −4.06562 −0.137443
\(876\) 0 0
\(877\) 33.8276 1.14228 0.571138 0.820854i \(-0.306502\pi\)
0.571138 + 0.820854i \(0.306502\pi\)
\(878\) 0 0
\(879\) −53.2799 −1.79709
\(880\) 0 0
\(881\) 26.1134 0.879782 0.439891 0.898051i \(-0.355017\pi\)
0.439891 + 0.898051i \(0.355017\pi\)
\(882\) 0 0
\(883\) 54.5412 1.83546 0.917729 0.397206i \(-0.130020\pi\)
0.917729 + 0.397206i \(0.130020\pi\)
\(884\) 0 0
\(885\) 22.5087 0.756621
\(886\) 0 0
\(887\) −22.6201 −0.759509 −0.379754 0.925087i \(-0.623991\pi\)
−0.379754 + 0.925087i \(0.623991\pi\)
\(888\) 0 0
\(889\) 24.4082 0.818626
\(890\) 0 0
\(891\) 18.6162 0.623666
\(892\) 0 0
\(893\) −0.645301 −0.0215942
\(894\) 0 0
\(895\) −7.42495 −0.248189
\(896\) 0 0
\(897\) 9.22914 0.308152
\(898\) 0 0
\(899\) 54.4508 1.81603
\(900\) 0 0
\(901\) −30.7776 −1.02535
\(902\) 0 0
\(903\) −14.9110 −0.496208
\(904\) 0 0
\(905\) 24.0450 0.799284
\(906\) 0 0
\(907\) 3.62149 0.120250 0.0601248 0.998191i \(-0.480850\pi\)
0.0601248 + 0.998191i \(0.480850\pi\)
\(908\) 0 0
\(909\) −5.17211 −0.171548
\(910\) 0 0
\(911\) −26.9241 −0.892034 −0.446017 0.895025i \(-0.647158\pi\)
−0.446017 + 0.895025i \(0.647158\pi\)
\(912\) 0 0
\(913\) −33.3210 −1.10277
\(914\) 0 0
\(915\) −3.84712 −0.127182
\(916\) 0 0
\(917\) −81.1524 −2.67989
\(918\) 0 0
\(919\) 19.5059 0.643441 0.321721 0.946835i \(-0.395739\pi\)
0.321721 + 0.946835i \(0.395739\pi\)
\(920\) 0 0
\(921\) 2.33206 0.0768440
\(922\) 0 0
\(923\) 73.5204 2.41995
\(924\) 0 0
\(925\) −4.18059 −0.137457
\(926\) 0 0
\(927\) 2.63782 0.0866373
\(928\) 0 0
\(929\) 27.2139 0.892860 0.446430 0.894819i \(-0.352695\pi\)
0.446430 + 0.894819i \(0.352695\pi\)
\(930\) 0 0
\(931\) 25.3427 0.830572
\(932\) 0 0
\(933\) −22.0273 −0.721142
\(934\) 0 0
\(935\) 9.05854 0.296246
\(936\) 0 0
\(937\) 3.85626 0.125978 0.0629892 0.998014i \(-0.479937\pi\)
0.0629892 + 0.998014i \(0.479937\pi\)
\(938\) 0 0
\(939\) 51.3069 1.67434
\(940\) 0 0
\(941\) −60.6471 −1.97704 −0.988519 0.151097i \(-0.951720\pi\)
−0.988519 + 0.151097i \(0.951720\pi\)
\(942\) 0 0
\(943\) 2.15539 0.0701890
\(944\) 0 0
\(945\) −22.6112 −0.735541
\(946\) 0 0
\(947\) 1.11954 0.0363801 0.0181901 0.999835i \(-0.494210\pi\)
0.0181901 + 0.999835i \(0.494210\pi\)
\(948\) 0 0
\(949\) 12.3510 0.400931
\(950\) 0 0
\(951\) −19.6481 −0.637132
\(952\) 0 0
\(953\) 14.8590 0.481331 0.240666 0.970608i \(-0.422634\pi\)
0.240666 + 0.970608i \(0.422634\pi\)
\(954\) 0 0
\(955\) 4.87689 0.157813
\(956\) 0 0
\(957\) 24.2720 0.784601
\(958\) 0 0
\(959\) 47.8155 1.54404
\(960\) 0 0
\(961\) 55.7953 1.79985
\(962\) 0 0
\(963\) −0.211365 −0.00681114
\(964\) 0 0
\(965\) 10.2714 0.330649
\(966\) 0 0
\(967\) −38.5718 −1.24039 −0.620193 0.784449i \(-0.712946\pi\)
−0.620193 + 0.784449i \(0.712946\pi\)
\(968\) 0 0
\(969\) 14.1454 0.454416
\(970\) 0 0
\(971\) 25.4984 0.818284 0.409142 0.912471i \(-0.365828\pi\)
0.409142 + 0.912471i \(0.365828\pi\)
\(972\) 0 0
\(973\) 50.3694 1.61477
\(974\) 0 0
\(975\) 9.22914 0.295569
\(976\) 0 0
\(977\) −15.8952 −0.508533 −0.254267 0.967134i \(-0.581834\pi\)
−0.254267 + 0.967134i \(0.581834\pi\)
\(978\) 0 0
\(979\) 24.8840 0.795297
\(980\) 0 0
\(981\) −10.6902 −0.341313
\(982\) 0 0
\(983\) 2.50587 0.0799247 0.0399624 0.999201i \(-0.487276\pi\)
0.0399624 + 0.999201i \(0.487276\pi\)
\(984\) 0 0
\(985\) 4.28557 0.136550
\(986\) 0 0
\(987\) −1.54047 −0.0490337
\(988\) 0 0
\(989\) −2.34868 −0.0746837
\(990\) 0 0
\(991\) −12.5272 −0.397938 −0.198969 0.980006i \(-0.563759\pi\)
−0.198969 + 0.980006i \(0.563759\pi\)
\(992\) 0 0
\(993\) 2.46220 0.0781356
\(994\) 0 0
\(995\) −4.02164 −0.127494
\(996\) 0 0
\(997\) −9.34803 −0.296055 −0.148028 0.988983i \(-0.547292\pi\)
−0.148028 + 0.988983i \(0.547292\pi\)
\(998\) 0 0
\(999\) −23.2506 −0.735616
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.u.1.1 4
4.3 odd 2 115.2.a.c.1.2 4
5.4 even 2 9200.2.a.cl.1.4 4
8.3 odd 2 7360.2.a.cj.1.2 4
8.5 even 2 7360.2.a.cg.1.3 4
12.11 even 2 1035.2.a.o.1.3 4
20.3 even 4 575.2.b.e.24.5 8
20.7 even 4 575.2.b.e.24.4 8
20.19 odd 2 575.2.a.h.1.3 4
28.27 even 2 5635.2.a.v.1.2 4
60.59 even 2 5175.2.a.bx.1.2 4
92.91 even 2 2645.2.a.m.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.a.c.1.2 4 4.3 odd 2
575.2.a.h.1.3 4 20.19 odd 2
575.2.b.e.24.4 8 20.7 even 4
575.2.b.e.24.5 8 20.3 even 4
1035.2.a.o.1.3 4 12.11 even 2
1840.2.a.u.1.1 4 1.1 even 1 trivial
2645.2.a.m.1.2 4 92.91 even 2
5175.2.a.bx.1.2 4 60.59 even 2
5635.2.a.v.1.2 4 28.27 even 2
7360.2.a.cg.1.3 4 8.5 even 2
7360.2.a.cj.1.2 4 8.3 odd 2
9200.2.a.cl.1.4 4 5.4 even 2