Properties

Label 1840.2.a.v.1.5
Level $1840$
Weight $2$
Character 1840.1
Self dual yes
Analytic conductor $14.692$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.13955077.1
Defining polynomial: \(x^{5} - 14 x^{3} - x^{2} + 32 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 1.5
Root \(-3.36002\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.36002 q^{3} -1.00000 q^{5} +1.90754 q^{7} +8.28974 q^{9} +O(q^{10})\) \(q+3.36002 q^{3} -1.00000 q^{5} +1.90754 q^{7} +8.28974 q^{9} +5.48021 q^{11} -1.04937 q^{13} -3.36002 q^{15} -6.74222 q^{17} +1.55049 q^{19} +6.40939 q^{21} +1.00000 q^{23} +1.00000 q^{25} +17.7736 q^{27} -3.38219 q^{29} -10.9327 q^{31} +18.4136 q^{33} -1.90754 q^{35} +5.26201 q^{37} -3.52589 q^{39} +6.09801 q^{41} -8.28974 q^{45} -0.403830 q^{47} -3.36128 q^{49} -22.6540 q^{51} +5.88332 q^{53} -5.48021 q^{55} +5.20968 q^{57} -9.60111 q^{59} -7.09927 q^{61} +15.8130 q^{63} +1.04937 q^{65} -13.7971 q^{67} +3.36002 q^{69} -0.478950 q^{71} +2.40383 q^{73} +3.36002 q^{75} +10.4537 q^{77} +4.24037 q^{79} +34.8505 q^{81} +11.2620 q^{83} +6.74222 q^{85} -11.3642 q^{87} +4.90495 q^{89} -2.00171 q^{91} -36.7340 q^{93} -1.55049 q^{95} -12.3433 q^{97} +45.4295 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{5} + 2q^{7} + 13q^{9} + O(q^{10}) \) \( 5q - 5q^{5} + 2q^{7} + 13q^{9} + q^{11} + 4q^{13} + 4q^{17} - 7q^{19} + 6q^{21} + 5q^{23} + 5q^{25} - 3q^{27} + 4q^{29} - 19q^{31} + 17q^{33} - 2q^{35} + 15q^{37} - 19q^{39} + 25q^{41} - 13q^{45} + 11q^{47} + 25q^{49} - 19q^{51} + 3q^{53} - q^{55} + 48q^{57} + q^{59} - 5q^{61} + 41q^{63} - 4q^{65} - 9q^{67} - q^{71} - q^{73} + 18q^{77} + 2q^{79} + 57q^{81} + 45q^{83} - 4q^{85} + 9q^{87} + 6q^{89} - 11q^{91} - 39q^{93} + 7q^{95} + 25q^{97} + 65q^{99} + O(q^{100}) \)

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
\(3\) 3.36002 1.93991 0.969954 0.243287i \(-0.0782256\pi\)
0.969954 + 0.243287i \(0.0782256\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.90754 0.720984 0.360492 0.932762i \(-0.382609\pi\)
0.360492 + 0.932762i \(0.382609\pi\)
\(8\) 0 0
\(9\) 8.28974 2.76325
\(10\) 0 0
\(11\) 5.48021 1.65234 0.826172 0.563418i \(-0.190514\pi\)
0.826172 + 0.563418i \(0.190514\pi\)
\(12\) 0 0
\(13\) −1.04937 −0.291041 −0.145521 0.989355i \(-0.546486\pi\)
−0.145521 + 0.989355i \(0.546486\pi\)
\(14\) 0 0
\(15\) −3.36002 −0.867554
\(16\) 0 0
\(17\) −6.74222 −1.63523 −0.817614 0.575767i \(-0.804703\pi\)
−0.817614 + 0.575767i \(0.804703\pi\)
\(18\) 0 0
\(19\) 1.55049 0.355707 0.177853 0.984057i \(-0.443085\pi\)
0.177853 + 0.984057i \(0.443085\pi\)
\(20\) 0 0
\(21\) 6.40939 1.39864
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 17.7736 3.42054
\(28\) 0 0
\(29\) −3.38219 −0.628058 −0.314029 0.949413i \(-0.601679\pi\)
−0.314029 + 0.949413i \(0.601679\pi\)
\(30\) 0 0
\(31\) −10.9327 −1.96357 −0.981784 0.190000i \(-0.939151\pi\)
−0.981784 + 0.190000i \(0.939151\pi\)
\(32\) 0 0
\(33\) 18.4136 3.20540
\(34\) 0 0
\(35\) −1.90754 −0.322434
\(36\) 0 0
\(37\) 5.26201 0.865069 0.432534 0.901617i \(-0.357619\pi\)
0.432534 + 0.901617i \(0.357619\pi\)
\(38\) 0 0
\(39\) −3.52589 −0.564594
\(40\) 0 0
\(41\) 6.09801 0.952350 0.476175 0.879351i \(-0.342023\pi\)
0.476175 + 0.879351i \(0.342023\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −8.28974 −1.23576
\(46\) 0 0
\(47\) −0.403830 −0.0589046 −0.0294523 0.999566i \(-0.509376\pi\)
−0.0294523 + 0.999566i \(0.509376\pi\)
\(48\) 0 0
\(49\) −3.36128 −0.480183
\(50\) 0 0
\(51\) −22.6540 −3.17219
\(52\) 0 0
\(53\) 5.88332 0.808136 0.404068 0.914729i \(-0.367596\pi\)
0.404068 + 0.914729i \(0.367596\pi\)
\(54\) 0 0
\(55\) −5.48021 −0.738951
\(56\) 0 0
\(57\) 5.20968 0.690039
\(58\) 0 0
\(59\) −9.60111 −1.24996 −0.624979 0.780641i \(-0.714893\pi\)
−0.624979 + 0.780641i \(0.714893\pi\)
\(60\) 0 0
\(61\) −7.09927 −0.908968 −0.454484 0.890755i \(-0.650176\pi\)
−0.454484 + 0.890755i \(0.650176\pi\)
\(62\) 0 0
\(63\) 15.8130 1.99226
\(64\) 0 0
\(65\) 1.04937 0.130158
\(66\) 0 0
\(67\) −13.7971 −1.68559 −0.842794 0.538236i \(-0.819091\pi\)
−0.842794 + 0.538236i \(0.819091\pi\)
\(68\) 0 0
\(69\) 3.36002 0.404499
\(70\) 0 0
\(71\) −0.478950 −0.0568409 −0.0284205 0.999596i \(-0.509048\pi\)
−0.0284205 + 0.999596i \(0.509048\pi\)
\(72\) 0 0
\(73\) 2.40383 0.281347 0.140673 0.990056i \(-0.455073\pi\)
0.140673 + 0.990056i \(0.455073\pi\)
\(74\) 0 0
\(75\) 3.36002 0.387982
\(76\) 0 0
\(77\) 10.4537 1.19131
\(78\) 0 0
\(79\) 4.24037 0.477079 0.238540 0.971133i \(-0.423331\pi\)
0.238540 + 0.971133i \(0.423331\pi\)
\(80\) 0 0
\(81\) 34.8505 3.87228
\(82\) 0 0
\(83\) 11.2620 1.23617 0.618083 0.786113i \(-0.287910\pi\)
0.618083 + 0.786113i \(0.287910\pi\)
\(84\) 0 0
\(85\) 6.74222 0.731296
\(86\) 0 0
\(87\) −11.3642 −1.21837
\(88\) 0 0
\(89\) 4.90495 0.519924 0.259962 0.965619i \(-0.416290\pi\)
0.259962 + 0.965619i \(0.416290\pi\)
\(90\) 0 0
\(91\) −2.00171 −0.209836
\(92\) 0 0
\(93\) −36.7340 −3.80914
\(94\) 0 0
\(95\) −1.55049 −0.159077
\(96\) 0 0
\(97\) −12.3433 −1.25327 −0.626637 0.779311i \(-0.715569\pi\)
−0.626637 + 0.779311i \(0.715569\pi\)
\(98\) 0 0
\(99\) 45.4295 4.56583
\(100\) 0 0
\(101\) −1.44692 −0.143974 −0.0719870 0.997406i \(-0.522934\pi\)
−0.0719870 + 0.997406i \(0.522934\pi\)
\(102\) 0 0
\(103\) 7.76385 0.764995 0.382497 0.923957i \(-0.375064\pi\)
0.382497 + 0.923957i \(0.375064\pi\)
\(104\) 0 0
\(105\) −6.40939 −0.625492
\(106\) 0 0
\(107\) 4.54197 0.439089 0.219544 0.975603i \(-0.429543\pi\)
0.219544 + 0.975603i \(0.429543\pi\)
\(108\) 0 0
\(109\) 12.4136 1.18901 0.594504 0.804093i \(-0.297348\pi\)
0.594504 + 0.804093i \(0.297348\pi\)
\(110\) 0 0
\(111\) 17.6805 1.67815
\(112\) 0 0
\(113\) 9.27312 0.872342 0.436171 0.899864i \(-0.356334\pi\)
0.436171 + 0.899864i \(0.356334\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −8.69896 −0.804219
\(118\) 0 0
\(119\) −12.8611 −1.17897
\(120\) 0 0
\(121\) 19.0327 1.73024
\(122\) 0 0
\(123\) 20.4894 1.84747
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −12.4149 −1.10165 −0.550824 0.834621i \(-0.685686\pi\)
−0.550824 + 0.834621i \(0.685686\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9.55434 −0.834766 −0.417383 0.908731i \(-0.637053\pi\)
−0.417383 + 0.908731i \(0.637053\pi\)
\(132\) 0 0
\(133\) 2.95763 0.256459
\(134\) 0 0
\(135\) −17.7736 −1.52971
\(136\) 0 0
\(137\) −3.36558 −0.287541 −0.143770 0.989611i \(-0.545923\pi\)
−0.143770 + 0.989611i \(0.545923\pi\)
\(138\) 0 0
\(139\) 20.7361 1.75881 0.879406 0.476072i \(-0.157940\pi\)
0.879406 + 0.476072i \(0.157940\pi\)
\(140\) 0 0
\(141\) −1.35688 −0.114270
\(142\) 0 0
\(143\) −5.75074 −0.480901
\(144\) 0 0
\(145\) 3.38219 0.280876
\(146\) 0 0
\(147\) −11.2940 −0.931510
\(148\) 0 0
\(149\) −19.9399 −1.63354 −0.816769 0.576965i \(-0.804237\pi\)
−0.816769 + 0.576965i \(0.804237\pi\)
\(150\) 0 0
\(151\) −23.7979 −1.93664 −0.968321 0.249709i \(-0.919665\pi\)
−0.968321 + 0.249709i \(0.919665\pi\)
\(152\) 0 0
\(153\) −55.8912 −4.51854
\(154\) 0 0
\(155\) 10.9327 0.878134
\(156\) 0 0
\(157\) 13.3175 1.06285 0.531425 0.847105i \(-0.321657\pi\)
0.531425 + 0.847105i \(0.321657\pi\)
\(158\) 0 0
\(159\) 19.7681 1.56771
\(160\) 0 0
\(161\) 1.90754 0.150335
\(162\) 0 0
\(163\) −16.1529 −1.26519 −0.632595 0.774483i \(-0.718010\pi\)
−0.632595 + 0.774483i \(0.718010\pi\)
\(164\) 0 0
\(165\) −18.4136 −1.43350
\(166\) 0 0
\(167\) 20.6782 1.60013 0.800064 0.599915i \(-0.204799\pi\)
0.800064 + 0.599915i \(0.204799\pi\)
\(168\) 0 0
\(169\) −11.8988 −0.915295
\(170\) 0 0
\(171\) 12.8532 0.982905
\(172\) 0 0
\(173\) 7.72058 0.586985 0.293492 0.955961i \(-0.405182\pi\)
0.293492 + 0.955961i \(0.405182\pi\)
\(174\) 0 0
\(175\) 1.90754 0.144197
\(176\) 0 0
\(177\) −32.2599 −2.42480
\(178\) 0 0
\(179\) −9.88683 −0.738976 −0.369488 0.929236i \(-0.620467\pi\)
−0.369488 + 0.929236i \(0.620467\pi\)
\(180\) 0 0
\(181\) −23.9883 −1.78304 −0.891519 0.452983i \(-0.850360\pi\)
−0.891519 + 0.452983i \(0.850360\pi\)
\(182\) 0 0
\(183\) −23.8537 −1.76332
\(184\) 0 0
\(185\) −5.26201 −0.386871
\(186\) 0 0
\(187\) −36.9487 −2.70196
\(188\) 0 0
\(189\) 33.9040 2.46615
\(190\) 0 0
\(191\) −6.72004 −0.486245 −0.243123 0.969996i \(-0.578172\pi\)
−0.243123 + 0.969996i \(0.578172\pi\)
\(192\) 0 0
\(193\) 8.95007 0.644240 0.322120 0.946699i \(-0.395605\pi\)
0.322120 + 0.946699i \(0.395605\pi\)
\(194\) 0 0
\(195\) 3.52589 0.252494
\(196\) 0 0
\(197\) −6.65109 −0.473871 −0.236935 0.971525i \(-0.576143\pi\)
−0.236935 + 0.971525i \(0.576143\pi\)
\(198\) 0 0
\(199\) 10.9050 0.773032 0.386516 0.922283i \(-0.373678\pi\)
0.386516 + 0.922283i \(0.373678\pi\)
\(200\) 0 0
\(201\) −46.3587 −3.26989
\(202\) 0 0
\(203\) −6.45168 −0.452819
\(204\) 0 0
\(205\) −6.09801 −0.425904
\(206\) 0 0
\(207\) 8.28974 0.576177
\(208\) 0 0
\(209\) 8.49700 0.587750
\(210\) 0 0
\(211\) 1.21874 0.0839014 0.0419507 0.999120i \(-0.486643\pi\)
0.0419507 + 0.999120i \(0.486643\pi\)
\(212\) 0 0
\(213\) −1.60928 −0.110266
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −20.8546 −1.41570
\(218\) 0 0
\(219\) 8.07692 0.545787
\(220\) 0 0
\(221\) 7.07505 0.475919
\(222\) 0 0
\(223\) −14.1542 −0.947835 −0.473917 0.880569i \(-0.657160\pi\)
−0.473917 + 0.880569i \(0.657160\pi\)
\(224\) 0 0
\(225\) 8.28974 0.552649
\(226\) 0 0
\(227\) 4.89902 0.325159 0.162580 0.986695i \(-0.448019\pi\)
0.162580 + 0.986695i \(0.448019\pi\)
\(228\) 0 0
\(229\) 12.3946 0.819056 0.409528 0.912298i \(-0.365693\pi\)
0.409528 + 0.912298i \(0.365693\pi\)
\(230\) 0 0
\(231\) 35.1248 2.31104
\(232\) 0 0
\(233\) 0.882062 0.0577858 0.0288929 0.999583i \(-0.490802\pi\)
0.0288929 + 0.999583i \(0.490802\pi\)
\(234\) 0 0
\(235\) 0.403830 0.0263429
\(236\) 0 0
\(237\) 14.2477 0.925490
\(238\) 0 0
\(239\) 19.9506 1.29049 0.645247 0.763974i \(-0.276754\pi\)
0.645247 + 0.763974i \(0.276754\pi\)
\(240\) 0 0
\(241\) −2.81877 −0.181573 −0.0907865 0.995870i \(-0.528938\pi\)
−0.0907865 + 0.995870i \(0.528938\pi\)
\(242\) 0 0
\(243\) 63.7777 4.09134
\(244\) 0 0
\(245\) 3.36128 0.214744
\(246\) 0 0
\(247\) −1.62703 −0.103525
\(248\) 0 0
\(249\) 37.8406 2.39805
\(250\) 0 0
\(251\) −17.9883 −1.13541 −0.567706 0.823231i \(-0.692169\pi\)
−0.567706 + 0.823231i \(0.692169\pi\)
\(252\) 0 0
\(253\) 5.48021 0.344538
\(254\) 0 0
\(255\) 22.6540 1.41865
\(256\) 0 0
\(257\) −14.1705 −0.883930 −0.441965 0.897032i \(-0.645718\pi\)
−0.441965 + 0.897032i \(0.645718\pi\)
\(258\) 0 0
\(259\) 10.0375 0.623701
\(260\) 0 0
\(261\) −28.0375 −1.73548
\(262\) 0 0
\(263\) 6.88386 0.424477 0.212238 0.977218i \(-0.431925\pi\)
0.212238 + 0.977218i \(0.431925\pi\)
\(264\) 0 0
\(265\) −5.88332 −0.361409
\(266\) 0 0
\(267\) 16.4807 1.00861
\(268\) 0 0
\(269\) −23.3463 −1.42345 −0.711724 0.702459i \(-0.752085\pi\)
−0.711724 + 0.702459i \(0.752085\pi\)
\(270\) 0 0
\(271\) −13.9519 −0.847517 −0.423759 0.905775i \(-0.639289\pi\)
−0.423759 + 0.905775i \(0.639289\pi\)
\(272\) 0 0
\(273\) −6.72579 −0.407063
\(274\) 0 0
\(275\) 5.48021 0.330469
\(276\) 0 0
\(277\) 3.69490 0.222005 0.111003 0.993820i \(-0.464594\pi\)
0.111003 + 0.993820i \(0.464594\pi\)
\(278\) 0 0
\(279\) −90.6291 −5.42582
\(280\) 0 0
\(281\) 15.9582 0.951984 0.475992 0.879450i \(-0.342089\pi\)
0.475992 + 0.879450i \(0.342089\pi\)
\(282\) 0 0
\(283\) 8.21649 0.488420 0.244210 0.969722i \(-0.421471\pi\)
0.244210 + 0.969722i \(0.421471\pi\)
\(284\) 0 0
\(285\) −5.20968 −0.308595
\(286\) 0 0
\(287\) 11.6322 0.686628
\(288\) 0 0
\(289\) 28.4575 1.67397
\(290\) 0 0
\(291\) −41.4738 −2.43124
\(292\) 0 0
\(293\) 22.0878 1.29038 0.645191 0.764021i \(-0.276778\pi\)
0.645191 + 0.764021i \(0.276778\pi\)
\(294\) 0 0
\(295\) 9.60111 0.558998
\(296\) 0 0
\(297\) 97.4032 5.65191
\(298\) 0 0
\(299\) −1.04937 −0.0606863
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −4.86168 −0.279296
\(304\) 0 0
\(305\) 7.09927 0.406503
\(306\) 0 0
\(307\) −15.2091 −0.868030 −0.434015 0.900906i \(-0.642903\pi\)
−0.434015 + 0.900906i \(0.642903\pi\)
\(308\) 0 0
\(309\) 26.0867 1.48402
\(310\) 0 0
\(311\) −0.254818 −0.0144494 −0.00722471 0.999974i \(-0.502300\pi\)
−0.00722471 + 0.999974i \(0.502300\pi\)
\(312\) 0 0
\(313\) 3.57095 0.201842 0.100921 0.994894i \(-0.467821\pi\)
0.100921 + 0.994894i \(0.467821\pi\)
\(314\) 0 0
\(315\) −15.8130 −0.890964
\(316\) 0 0
\(317\) 12.3235 0.692155 0.346078 0.938206i \(-0.387513\pi\)
0.346078 + 0.938206i \(0.387513\pi\)
\(318\) 0 0
\(319\) −18.5351 −1.03777
\(320\) 0 0
\(321\) 15.2611 0.851792
\(322\) 0 0
\(323\) −10.4537 −0.581661
\(324\) 0 0
\(325\) −1.04937 −0.0582083
\(326\) 0 0
\(327\) 41.7100 2.30657
\(328\) 0 0
\(329\) −0.770323 −0.0424693
\(330\) 0 0
\(331\) −26.1082 −1.43503 −0.717517 0.696541i \(-0.754722\pi\)
−0.717517 + 0.696541i \(0.754722\pi\)
\(332\) 0 0
\(333\) 43.6207 2.39040
\(334\) 0 0
\(335\) 13.7971 0.753818
\(336\) 0 0
\(337\) 9.68246 0.527437 0.263718 0.964600i \(-0.415051\pi\)
0.263718 + 0.964600i \(0.415051\pi\)
\(338\) 0 0
\(339\) 31.1579 1.69226
\(340\) 0 0
\(341\) −59.9134 −3.24449
\(342\) 0 0
\(343\) −19.7646 −1.06719
\(344\) 0 0
\(345\) −3.36002 −0.180897
\(346\) 0 0
\(347\) −24.9747 −1.34071 −0.670355 0.742040i \(-0.733858\pi\)
−0.670355 + 0.742040i \(0.733858\pi\)
\(348\) 0 0
\(349\) −35.3019 −1.88967 −0.944835 0.327547i \(-0.893778\pi\)
−0.944835 + 0.327547i \(0.893778\pi\)
\(350\) 0 0
\(351\) −18.6510 −0.995518
\(352\) 0 0
\(353\) −10.0326 −0.533980 −0.266990 0.963699i \(-0.586029\pi\)
−0.266990 + 0.963699i \(0.586029\pi\)
\(354\) 0 0
\(355\) 0.478950 0.0254200
\(356\) 0 0
\(357\) −43.2135 −2.28710
\(358\) 0 0
\(359\) 30.8691 1.62921 0.814603 0.580019i \(-0.196955\pi\)
0.814603 + 0.580019i \(0.196955\pi\)
\(360\) 0 0
\(361\) −16.5960 −0.873473
\(362\) 0 0
\(363\) 63.9502 3.35651
\(364\) 0 0
\(365\) −2.40383 −0.125822
\(366\) 0 0
\(367\) −33.3234 −1.73947 −0.869734 0.493521i \(-0.835709\pi\)
−0.869734 + 0.493521i \(0.835709\pi\)
\(368\) 0 0
\(369\) 50.5509 2.63158
\(370\) 0 0
\(371\) 11.2227 0.582653
\(372\) 0 0
\(373\) 4.62023 0.239227 0.119613 0.992821i \(-0.461835\pi\)
0.119613 + 0.992821i \(0.461835\pi\)
\(374\) 0 0
\(375\) −3.36002 −0.173511
\(376\) 0 0
\(377\) 3.54916 0.182791
\(378\) 0 0
\(379\) −15.0020 −0.770600 −0.385300 0.922791i \(-0.625902\pi\)
−0.385300 + 0.922791i \(0.625902\pi\)
\(380\) 0 0
\(381\) −41.7145 −2.13710
\(382\) 0 0
\(383\) −1.87882 −0.0960033 −0.0480016 0.998847i \(-0.515285\pi\)
−0.0480016 + 0.998847i \(0.515285\pi\)
\(384\) 0 0
\(385\) −10.4537 −0.532772
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.48539 −0.0753121 −0.0376560 0.999291i \(-0.511989\pi\)
−0.0376560 + 0.999291i \(0.511989\pi\)
\(390\) 0 0
\(391\) −6.74222 −0.340968
\(392\) 0 0
\(393\) −32.1028 −1.61937
\(394\) 0 0
\(395\) −4.24037 −0.213356
\(396\) 0 0
\(397\) 5.26722 0.264354 0.132177 0.991226i \(-0.457803\pi\)
0.132177 + 0.991226i \(0.457803\pi\)
\(398\) 0 0
\(399\) 9.93769 0.497507
\(400\) 0 0
\(401\) 26.1490 1.30582 0.652910 0.757436i \(-0.273548\pi\)
0.652910 + 0.757436i \(0.273548\pi\)
\(402\) 0 0
\(403\) 11.4724 0.571480
\(404\) 0 0
\(405\) −34.8505 −1.73174
\(406\) 0 0
\(407\) 28.8369 1.42939
\(408\) 0 0
\(409\) −14.7503 −0.729355 −0.364677 0.931134i \(-0.618821\pi\)
−0.364677 + 0.931134i \(0.618821\pi\)
\(410\) 0 0
\(411\) −11.3084 −0.557803
\(412\) 0 0
\(413\) −18.3145 −0.901200
\(414\) 0 0
\(415\) −11.2620 −0.552830
\(416\) 0 0
\(417\) 69.6737 3.41194
\(418\) 0 0
\(419\) −1.10616 −0.0540394 −0.0270197 0.999635i \(-0.508602\pi\)
−0.0270197 + 0.999635i \(0.508602\pi\)
\(420\) 0 0
\(421\) 9.16362 0.446607 0.223304 0.974749i \(-0.428316\pi\)
0.223304 + 0.974749i \(0.428316\pi\)
\(422\) 0 0
\(423\) −3.34764 −0.162768
\(424\) 0 0
\(425\) −6.74222 −0.327045
\(426\) 0 0
\(427\) −13.5422 −0.655351
\(428\) 0 0
\(429\) −19.3226 −0.932904
\(430\) 0 0
\(431\) 32.1712 1.54963 0.774817 0.632186i \(-0.217842\pi\)
0.774817 + 0.632186i \(0.217842\pi\)
\(432\) 0 0
\(433\) −8.37861 −0.402650 −0.201325 0.979524i \(-0.564525\pi\)
−0.201325 + 0.979524i \(0.564525\pi\)
\(434\) 0 0
\(435\) 11.3642 0.544874
\(436\) 0 0
\(437\) 1.55049 0.0741700
\(438\) 0 0
\(439\) 15.3093 0.730673 0.365336 0.930876i \(-0.380954\pi\)
0.365336 + 0.930876i \(0.380954\pi\)
\(440\) 0 0
\(441\) −27.8641 −1.32686
\(442\) 0 0
\(443\) 3.24129 0.153998 0.0769992 0.997031i \(-0.475466\pi\)
0.0769992 + 0.997031i \(0.475466\pi\)
\(444\) 0 0
\(445\) −4.90495 −0.232517
\(446\) 0 0
\(447\) −66.9984 −3.16892
\(448\) 0 0
\(449\) −9.02215 −0.425782 −0.212891 0.977076i \(-0.568288\pi\)
−0.212891 + 0.977076i \(0.568288\pi\)
\(450\) 0 0
\(451\) 33.4184 1.57361
\(452\) 0 0
\(453\) −79.9613 −3.75691
\(454\) 0 0
\(455\) 2.00171 0.0938416
\(456\) 0 0
\(457\) 7.03526 0.329096 0.164548 0.986369i \(-0.447384\pi\)
0.164548 + 0.986369i \(0.447384\pi\)
\(458\) 0 0
\(459\) −119.834 −5.59336
\(460\) 0 0
\(461\) −1.59617 −0.0743411 −0.0371705 0.999309i \(-0.511834\pi\)
−0.0371705 + 0.999309i \(0.511834\pi\)
\(462\) 0 0
\(463\) −9.61655 −0.446919 −0.223459 0.974713i \(-0.571735\pi\)
−0.223459 + 0.974713i \(0.571735\pi\)
\(464\) 0 0
\(465\) 36.7340 1.70350
\(466\) 0 0
\(467\) −24.4764 −1.13263 −0.566317 0.824188i \(-0.691632\pi\)
−0.566317 + 0.824188i \(0.691632\pi\)
\(468\) 0 0
\(469\) −26.3186 −1.21528
\(470\) 0 0
\(471\) 44.7470 2.06183
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.55049 0.0711413
\(476\) 0 0
\(477\) 48.7712 2.23308
\(478\) 0 0
\(479\) 13.2108 0.603618 0.301809 0.953368i \(-0.402410\pi\)
0.301809 + 0.953368i \(0.402410\pi\)
\(480\) 0 0
\(481\) −5.52177 −0.251771
\(482\) 0 0
\(483\) 6.40939 0.291637
\(484\) 0 0
\(485\) 12.3433 0.560482
\(486\) 0 0
\(487\) 23.7078 1.07431 0.537153 0.843485i \(-0.319500\pi\)
0.537153 + 0.843485i \(0.319500\pi\)
\(488\) 0 0
\(489\) −54.2739 −2.45435
\(490\) 0 0
\(491\) −18.5930 −0.839091 −0.419546 0.907734i \(-0.637811\pi\)
−0.419546 + 0.907734i \(0.637811\pi\)
\(492\) 0 0
\(493\) 22.8035 1.02702
\(494\) 0 0
\(495\) −45.4295 −2.04190
\(496\) 0 0
\(497\) −0.913618 −0.0409814
\(498\) 0 0
\(499\) −17.9121 −0.801858 −0.400929 0.916109i \(-0.631313\pi\)
−0.400929 + 0.916109i \(0.631313\pi\)
\(500\) 0 0
\(501\) 69.4792 3.10410
\(502\) 0 0
\(503\) −1.24890 −0.0556855 −0.0278428 0.999612i \(-0.508864\pi\)
−0.0278428 + 0.999612i \(0.508864\pi\)
\(504\) 0 0
\(505\) 1.44692 0.0643871
\(506\) 0 0
\(507\) −39.9803 −1.77559
\(508\) 0 0
\(509\) 15.8757 0.703679 0.351839 0.936060i \(-0.385556\pi\)
0.351839 + 0.936060i \(0.385556\pi\)
\(510\) 0 0
\(511\) 4.58541 0.202847
\(512\) 0 0
\(513\) 27.5578 1.21671
\(514\) 0 0
\(515\) −7.76385 −0.342116
\(516\) 0 0
\(517\) −2.21307 −0.0973307
\(518\) 0 0
\(519\) 25.9413 1.13870
\(520\) 0 0
\(521\) 26.5488 1.16312 0.581561 0.813503i \(-0.302442\pi\)
0.581561 + 0.813503i \(0.302442\pi\)
\(522\) 0 0
\(523\) −31.0258 −1.35666 −0.678331 0.734757i \(-0.737296\pi\)
−0.678331 + 0.734757i \(0.737296\pi\)
\(524\) 0 0
\(525\) 6.40939 0.279729
\(526\) 0 0
\(527\) 73.7105 3.21088
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −79.5907 −3.45394
\(532\) 0 0
\(533\) −6.39904 −0.277173
\(534\) 0 0
\(535\) −4.54197 −0.196366
\(536\) 0 0
\(537\) −33.2199 −1.43355
\(538\) 0 0
\(539\) −18.4205 −0.793427
\(540\) 0 0
\(541\) 22.4123 0.963579 0.481790 0.876287i \(-0.339987\pi\)
0.481790 + 0.876287i \(0.339987\pi\)
\(542\) 0 0
\(543\) −80.6013 −3.45893
\(544\) 0 0
\(545\) −12.4136 −0.531741
\(546\) 0 0
\(547\) 43.3980 1.85556 0.927782 0.373122i \(-0.121713\pi\)
0.927782 + 0.373122i \(0.121713\pi\)
\(548\) 0 0
\(549\) −58.8511 −2.51170
\(550\) 0 0
\(551\) −5.24406 −0.223404
\(552\) 0 0
\(553\) 8.08870 0.343966
\(554\) 0 0
\(555\) −17.6805 −0.750494
\(556\) 0 0
\(557\) −21.2690 −0.901197 −0.450599 0.892727i \(-0.648789\pi\)
−0.450599 + 0.892727i \(0.648789\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −124.149 −5.24155
\(562\) 0 0
\(563\) −10.6629 −0.449389 −0.224694 0.974429i \(-0.572138\pi\)
−0.224694 + 0.974429i \(0.572138\pi\)
\(564\) 0 0
\(565\) −9.27312 −0.390123
\(566\) 0 0
\(567\) 66.4789 2.79185
\(568\) 0 0
\(569\) 31.8986 1.33726 0.668630 0.743596i \(-0.266881\pi\)
0.668630 + 0.743596i \(0.266881\pi\)
\(570\) 0 0
\(571\) 7.87477 0.329549 0.164774 0.986331i \(-0.447310\pi\)
0.164774 + 0.986331i \(0.447310\pi\)
\(572\) 0 0
\(573\) −22.5795 −0.943271
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 40.7070 1.69466 0.847328 0.531070i \(-0.178210\pi\)
0.847328 + 0.531070i \(0.178210\pi\)
\(578\) 0 0
\(579\) 30.0724 1.24977
\(580\) 0 0
\(581\) 21.4828 0.891256
\(582\) 0 0
\(583\) 32.2418 1.33532
\(584\) 0 0
\(585\) 8.69896 0.359658
\(586\) 0 0
\(587\) 22.9321 0.946508 0.473254 0.880926i \(-0.343079\pi\)
0.473254 + 0.880926i \(0.343079\pi\)
\(588\) 0 0
\(589\) −16.9510 −0.698454
\(590\) 0 0
\(591\) −22.3478 −0.919266
\(592\) 0 0
\(593\) 27.6250 1.13442 0.567211 0.823572i \(-0.308022\pi\)
0.567211 + 0.823572i \(0.308022\pi\)
\(594\) 0 0
\(595\) 12.8611 0.527252
\(596\) 0 0
\(597\) 36.6409 1.49961
\(598\) 0 0
\(599\) 18.2139 0.744199 0.372099 0.928193i \(-0.378638\pi\)
0.372099 + 0.928193i \(0.378638\pi\)
\(600\) 0 0
\(601\) −24.0407 −0.980640 −0.490320 0.871543i \(-0.663120\pi\)
−0.490320 + 0.871543i \(0.663120\pi\)
\(602\) 0 0
\(603\) −114.375 −4.65770
\(604\) 0 0
\(605\) −19.0327 −0.773788
\(606\) 0 0
\(607\) −5.52878 −0.224406 −0.112203 0.993685i \(-0.535791\pi\)
−0.112203 + 0.993685i \(0.535791\pi\)
\(608\) 0 0
\(609\) −21.6778 −0.878428
\(610\) 0 0
\(611\) 0.423765 0.0171437
\(612\) 0 0
\(613\) −21.1205 −0.853050 −0.426525 0.904476i \(-0.640262\pi\)
−0.426525 + 0.904476i \(0.640262\pi\)
\(614\) 0 0
\(615\) −20.4894 −0.826214
\(616\) 0 0
\(617\) 19.6523 0.791174 0.395587 0.918429i \(-0.370541\pi\)
0.395587 + 0.918429i \(0.370541\pi\)
\(618\) 0 0
\(619\) −40.9931 −1.64765 −0.823826 0.566843i \(-0.808164\pi\)
−0.823826 + 0.566843i \(0.808164\pi\)
\(620\) 0 0
\(621\) 17.7736 0.713231
\(622\) 0 0
\(623\) 9.35641 0.374857
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 28.5501 1.14018
\(628\) 0 0
\(629\) −35.4776 −1.41458
\(630\) 0 0
\(631\) 45.5595 1.81369 0.906847 0.421461i \(-0.138483\pi\)
0.906847 + 0.421461i \(0.138483\pi\)
\(632\) 0 0
\(633\) 4.09499 0.162761
\(634\) 0 0
\(635\) 12.4149 0.492672
\(636\) 0 0
\(637\) 3.52721 0.139753
\(638\) 0 0
\(639\) −3.97037 −0.157065
\(640\) 0 0
\(641\) 8.97997 0.354688 0.177344 0.984149i \(-0.443250\pi\)
0.177344 + 0.984149i \(0.443250\pi\)
\(642\) 0 0
\(643\) −2.69616 −0.106326 −0.0531630 0.998586i \(-0.516930\pi\)
−0.0531630 + 0.998586i \(0.516930\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.12638 −0.358795 −0.179398 0.983777i \(-0.557415\pi\)
−0.179398 + 0.983777i \(0.557415\pi\)
\(648\) 0 0
\(649\) −52.6161 −2.06536
\(650\) 0 0
\(651\) −70.0718 −2.74633
\(652\) 0 0
\(653\) 41.5497 1.62596 0.812982 0.582289i \(-0.197843\pi\)
0.812982 + 0.582289i \(0.197843\pi\)
\(654\) 0 0
\(655\) 9.55434 0.373319
\(656\) 0 0
\(657\) 19.9271 0.777431
\(658\) 0 0
\(659\) −20.9256 −0.815145 −0.407573 0.913173i \(-0.633625\pi\)
−0.407573 + 0.913173i \(0.633625\pi\)
\(660\) 0 0
\(661\) 5.25688 0.204469 0.102234 0.994760i \(-0.467401\pi\)
0.102234 + 0.994760i \(0.467401\pi\)
\(662\) 0 0
\(663\) 23.7723 0.923240
\(664\) 0 0
\(665\) −2.95763 −0.114692
\(666\) 0 0
\(667\) −3.38219 −0.130959
\(668\) 0 0
\(669\) −47.5584 −1.83871
\(670\) 0 0
\(671\) −38.9055 −1.50193
\(672\) 0 0
\(673\) 38.1900 1.47212 0.736059 0.676918i \(-0.236685\pi\)
0.736059 + 0.676918i \(0.236685\pi\)
\(674\) 0 0
\(675\) 17.7736 0.684107
\(676\) 0 0
\(677\) −4.83529 −0.185835 −0.0929176 0.995674i \(-0.529619\pi\)
−0.0929176 + 0.995674i \(0.529619\pi\)
\(678\) 0 0
\(679\) −23.5454 −0.903591
\(680\) 0 0
\(681\) 16.4608 0.630780
\(682\) 0 0
\(683\) 50.3044 1.92484 0.962422 0.271559i \(-0.0875391\pi\)
0.962422 + 0.271559i \(0.0875391\pi\)
\(684\) 0 0
\(685\) 3.36558 0.128592
\(686\) 0 0
\(687\) 41.6460 1.58889
\(688\) 0 0
\(689\) −6.17375 −0.235201
\(690\) 0 0
\(691\) −20.6298 −0.784793 −0.392396 0.919796i \(-0.628354\pi\)
−0.392396 + 0.919796i \(0.628354\pi\)
\(692\) 0 0
\(693\) 86.6587 3.29189
\(694\) 0 0
\(695\) −20.7361 −0.786565
\(696\) 0 0
\(697\) −41.1141 −1.55731
\(698\) 0 0
\(699\) 2.96375 0.112099
\(700\) 0 0
\(701\) −22.9598 −0.867182 −0.433591 0.901110i \(-0.642754\pi\)
−0.433591 + 0.901110i \(0.642754\pi\)
\(702\) 0 0
\(703\) 8.15869 0.307711
\(704\) 0 0
\(705\) 1.35688 0.0511029
\(706\) 0 0
\(707\) −2.76006 −0.103803
\(708\) 0 0
\(709\) 11.9069 0.447174 0.223587 0.974684i \(-0.428223\pi\)
0.223587 + 0.974684i \(0.428223\pi\)
\(710\) 0 0
\(711\) 35.1516 1.31829
\(712\) 0 0
\(713\) −10.9327 −0.409432
\(714\) 0 0
\(715\) 5.75074 0.215065
\(716\) 0 0
\(717\) 67.0343 2.50344
\(718\) 0 0
\(719\) −24.5244 −0.914604 −0.457302 0.889311i \(-0.651184\pi\)
−0.457302 + 0.889311i \(0.651184\pi\)
\(720\) 0 0
\(721\) 14.8099 0.551549
\(722\) 0 0
\(723\) −9.47113 −0.352235
\(724\) 0 0
\(725\) −3.38219 −0.125612
\(726\) 0 0
\(727\) −11.4034 −0.422928 −0.211464 0.977386i \(-0.567823\pi\)
−0.211464 + 0.977386i \(0.567823\pi\)
\(728\) 0 0
\(729\) 109.743 4.06454
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 35.8722 1.32497 0.662484 0.749076i \(-0.269502\pi\)
0.662484 + 0.749076i \(0.269502\pi\)
\(734\) 0 0
\(735\) 11.2940 0.416584
\(736\) 0 0
\(737\) −75.6112 −2.78517
\(738\) 0 0
\(739\) 17.1503 0.630883 0.315441 0.948945i \(-0.397847\pi\)
0.315441 + 0.948945i \(0.397847\pi\)
\(740\) 0 0
\(741\) −5.46685 −0.200830
\(742\) 0 0
\(743\) −27.7242 −1.01710 −0.508552 0.861031i \(-0.669819\pi\)
−0.508552 + 0.861031i \(0.669819\pi\)
\(744\) 0 0
\(745\) 19.9399 0.730540
\(746\) 0 0
\(747\) 93.3591 3.41583
\(748\) 0 0
\(749\) 8.66400 0.316576
\(750\) 0 0
\(751\) 2.80173 0.102236 0.0511182 0.998693i \(-0.483721\pi\)
0.0511182 + 0.998693i \(0.483721\pi\)
\(752\) 0 0
\(753\) −60.4411 −2.20260
\(754\) 0 0
\(755\) 23.7979 0.866093
\(756\) 0 0
\(757\) −41.6710 −1.51456 −0.757278 0.653092i \(-0.773471\pi\)
−0.757278 + 0.653092i \(0.773471\pi\)
\(758\) 0 0
\(759\) 18.4136 0.668372
\(760\) 0 0
\(761\) −35.4285 −1.28428 −0.642141 0.766587i \(-0.721954\pi\)
−0.642141 + 0.766587i \(0.721954\pi\)
\(762\) 0 0
\(763\) 23.6795 0.857255
\(764\) 0 0
\(765\) 55.8912 2.02075
\(766\) 0 0
\(767\) 10.0751 0.363790
\(768\) 0 0
\(769\) 23.8731 0.860885 0.430442 0.902618i \(-0.358358\pi\)
0.430442 + 0.902618i \(0.358358\pi\)
\(770\) 0 0
\(771\) −47.6131 −1.71474
\(772\) 0 0
\(773\) 11.3403 0.407881 0.203941 0.978983i \(-0.434625\pi\)
0.203941 + 0.978983i \(0.434625\pi\)
\(774\) 0 0
\(775\) −10.9327 −0.392714
\(776\) 0 0
\(777\) 33.7262 1.20992
\(778\) 0 0
\(779\) 9.45490 0.338757
\(780\) 0 0
\(781\) −2.62475 −0.0939208
\(782\) 0 0
\(783\) −60.1139 −2.14829
\(784\) 0 0
\(785\) −13.3175 −0.475321
\(786\) 0 0
\(787\) 26.1906 0.933595 0.466797 0.884364i \(-0.345408\pi\)
0.466797 + 0.884364i \(0.345408\pi\)
\(788\) 0 0
\(789\) 23.1299 0.823446
\(790\) 0 0
\(791\) 17.6889 0.628944
\(792\) 0 0
\(793\) 7.44973 0.264547
\(794\) 0 0
\(795\) −19.7681 −0.701101
\(796\) 0 0
\(797\) −4.48133 −0.158737 −0.0793684 0.996845i \(-0.525290\pi\)
−0.0793684 + 0.996845i \(0.525290\pi\)
\(798\) 0 0
\(799\) 2.72271 0.0963224
\(800\) 0 0
\(801\) 40.6608 1.43668
\(802\) 0 0
\(803\) 13.1735 0.464882
\(804\) 0 0
\(805\) −1.90754 −0.0672321
\(806\) 0 0
\(807\) −78.4440 −2.76136
\(808\) 0 0
\(809\) −32.4608 −1.14126 −0.570630 0.821207i \(-0.693301\pi\)
−0.570630 + 0.821207i \(0.693301\pi\)
\(810\) 0 0
\(811\) −11.3500 −0.398554 −0.199277 0.979943i \(-0.563859\pi\)
−0.199277 + 0.979943i \(0.563859\pi\)
\(812\) 0 0
\(813\) −46.8786 −1.64411
\(814\) 0 0
\(815\) 16.1529 0.565810
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −16.5936 −0.579829
\(820\) 0 0
\(821\) 48.2070 1.68244 0.841218 0.540697i \(-0.181839\pi\)
0.841218 + 0.540697i \(0.181839\pi\)
\(822\) 0 0
\(823\) 37.7179 1.31476 0.657381 0.753558i \(-0.271664\pi\)
0.657381 + 0.753558i \(0.271664\pi\)
\(824\) 0 0
\(825\) 18.4136 0.641080
\(826\) 0 0
\(827\) 8.89736 0.309392 0.154696 0.987962i \(-0.450560\pi\)
0.154696 + 0.987962i \(0.450560\pi\)
\(828\) 0 0
\(829\) 30.9058 1.07340 0.536702 0.843772i \(-0.319670\pi\)
0.536702 + 0.843772i \(0.319670\pi\)
\(830\) 0 0
\(831\) 12.4149 0.430670
\(832\) 0 0
\(833\) 22.6625 0.785208
\(834\) 0 0
\(835\) −20.6782 −0.715599
\(836\) 0 0
\(837\) −194.313 −6.71646
\(838\) 0 0
\(839\) −1.34387 −0.0463954 −0.0231977 0.999731i \(-0.507385\pi\)
−0.0231977 + 0.999731i \(0.507385\pi\)
\(840\) 0 0
\(841\) −17.5608 −0.605543
\(842\) 0 0
\(843\) 53.6198 1.84676
\(844\) 0 0
\(845\) 11.8988 0.409332
\(846\) 0 0
\(847\) 36.3056 1.24748
\(848\) 0 0
\(849\) 27.6076 0.947489
\(850\) 0 0
\(851\) 5.26201 0.180379
\(852\) 0 0
\(853\) −26.3940 −0.903713 −0.451857 0.892091i \(-0.649238\pi\)
−0.451857 + 0.892091i \(0.649238\pi\)
\(854\) 0 0
\(855\) −12.8532 −0.439569
\(856\) 0 0
\(857\) −17.5048 −0.597953 −0.298976 0.954260i \(-0.596645\pi\)
−0.298976 + 0.954260i \(0.596645\pi\)
\(858\) 0 0
\(859\) −11.7199 −0.399877 −0.199938 0.979808i \(-0.564074\pi\)
−0.199938 + 0.979808i \(0.564074\pi\)
\(860\) 0 0
\(861\) 39.0845 1.33200
\(862\) 0 0
\(863\) −26.6578 −0.907443 −0.453722 0.891144i \(-0.649904\pi\)
−0.453722 + 0.891144i \(0.649904\pi\)
\(864\) 0 0
\(865\) −7.72058 −0.262508
\(866\) 0 0
\(867\) 95.6177 3.24735
\(868\) 0 0
\(869\) 23.2381 0.788299
\(870\) 0 0
\(871\) 14.4782 0.490576
\(872\) 0 0
\(873\) −102.323 −3.46311
\(874\) 0 0
\(875\) −1.90754 −0.0644867
\(876\) 0 0
\(877\) 19.9574 0.673912 0.336956 0.941520i \(-0.390603\pi\)
0.336956 + 0.941520i \(0.390603\pi\)
\(878\) 0 0
\(879\) 74.2154 2.50322
\(880\) 0 0
\(881\) −20.7297 −0.698402 −0.349201 0.937048i \(-0.613547\pi\)
−0.349201 + 0.937048i \(0.613547\pi\)
\(882\) 0 0
\(883\) −2.31093 −0.0777690 −0.0388845 0.999244i \(-0.512380\pi\)
−0.0388845 + 0.999244i \(0.512380\pi\)
\(884\) 0 0
\(885\) 32.2599 1.08441
\(886\) 0 0
\(887\) 11.6924 0.392592 0.196296 0.980545i \(-0.437109\pi\)
0.196296 + 0.980545i \(0.437109\pi\)
\(888\) 0 0
\(889\) −23.6820 −0.794270
\(890\) 0 0
\(891\) 190.988 6.39835
\(892\) 0 0
\(893\) −0.626134 −0.0209528
\(894\) 0 0
\(895\) 9.88683 0.330480
\(896\) 0 0
\(897\) −3.52589 −0.117726
\(898\) 0 0
\(899\) 36.9765 1.23323
\(900\) 0 0
\(901\) −39.6666 −1.32149
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 23.9883 0.797399
\(906\) 0 0
\(907\) −32.5061 −1.07935 −0.539673 0.841875i \(-0.681452\pi\)
−0.539673 + 0.841875i \(0.681452\pi\)
\(908\) 0 0
\(909\) −11.9946 −0.397836
\(910\) 0 0
\(911\) −10.7518 −0.356224 −0.178112 0.984010i \(-0.556999\pi\)
−0.178112 + 0.984010i \(0.556999\pi\)
\(912\) 0 0
\(913\) 61.7181 2.04257
\(914\) 0 0
\(915\) 23.8537 0.788579
\(916\) 0 0
\(917\) −18.2253 −0.601853
\(918\) 0 0
\(919\) 24.2758 0.800785 0.400392 0.916344i \(-0.368874\pi\)
0.400392 + 0.916344i \(0.368874\pi\)
\(920\) 0 0
\(921\) −51.1029 −1.68390
\(922\) 0 0
\(923\) 0.502594 0.0165431
\(924\) 0 0
\(925\) 5.26201 0.173014
\(926\) 0 0
\(927\) 64.3603 2.11387
\(928\) 0 0
\(929\) 8.18842 0.268653 0.134327 0.990937i \(-0.457113\pi\)
0.134327 + 0.990937i \(0.457113\pi\)
\(930\) 0 0
\(931\) −5.21163 −0.170804
\(932\) 0 0
\(933\) −0.856194 −0.0280305
\(934\) 0 0
\(935\) 36.9487 1.20835
\(936\) 0 0
\(937\) 26.2312 0.856936 0.428468 0.903557i \(-0.359053\pi\)
0.428468 + 0.903557i \(0.359053\pi\)
\(938\) 0 0
\(939\) 11.9985 0.391556
\(940\) 0 0
\(941\) 23.6292 0.770291 0.385145 0.922856i \(-0.374151\pi\)
0.385145 + 0.922856i \(0.374151\pi\)
\(942\) 0 0
\(943\) 6.09801 0.198579
\(944\) 0 0
\(945\) −33.9040 −1.10290
\(946\) 0 0
\(947\) 15.9026 0.516765 0.258383 0.966043i \(-0.416810\pi\)
0.258383 + 0.966043i \(0.416810\pi\)
\(948\) 0 0
\(949\) −2.52249 −0.0818836
\(950\) 0 0
\(951\) 41.4071 1.34272
\(952\) 0 0
\(953\) 20.9451 0.678478 0.339239 0.940700i \(-0.389831\pi\)
0.339239 + 0.940700i \(0.389831\pi\)
\(954\) 0 0
\(955\) 6.72004 0.217455
\(956\) 0 0
\(957\) −62.2784 −2.01318
\(958\) 0 0
\(959\) −6.41998 −0.207312
\(960\) 0 0
\(961\) 88.5236 2.85560
\(962\) 0 0
\(963\) 37.6517 1.21331
\(964\) 0 0
\(965\) −8.95007 −0.288113
\(966\) 0 0
\(967\) 14.0732 0.452563 0.226281 0.974062i \(-0.427343\pi\)
0.226281 + 0.974062i \(0.427343\pi\)
\(968\) 0 0
\(969\) −35.1248 −1.12837
\(970\) 0 0
\(971\) 11.4404 0.367139 0.183569 0.983007i \(-0.441235\pi\)
0.183569 + 0.983007i \(0.441235\pi\)
\(972\) 0 0
\(973\) 39.5550 1.26808
\(974\) 0 0
\(975\) −3.52589 −0.112919
\(976\) 0 0
\(977\) 43.9119 1.40487 0.702433 0.711750i \(-0.252097\pi\)
0.702433 + 0.711750i \(0.252097\pi\)
\(978\) 0 0
\(979\) 26.8802 0.859094
\(980\) 0 0
\(981\) 102.906 3.28552
\(982\) 0 0
\(983\) 43.0420 1.37283 0.686413 0.727212i \(-0.259184\pi\)
0.686413 + 0.727212i \(0.259184\pi\)
\(984\) 0 0
\(985\) 6.65109 0.211921
\(986\) 0 0
\(987\) −2.58830 −0.0823865
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −43.5288 −1.38274 −0.691369 0.722501i \(-0.742992\pi\)
−0.691369 + 0.722501i \(0.742992\pi\)
\(992\) 0 0
\(993\) −87.7240 −2.78384
\(994\) 0 0
\(995\) −10.9050 −0.345710
\(996\) 0 0
\(997\) 26.4448 0.837517 0.418758 0.908098i \(-0.362465\pi\)
0.418758 + 0.908098i \(0.362465\pi\)
\(998\) 0 0
\(999\) 93.5250 2.95900
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.v.1.5 5
4.3 odd 2 920.2.a.j.1.1 5
5.4 even 2 9200.2.a.cu.1.1 5
8.3 odd 2 7360.2.a.co.1.5 5
8.5 even 2 7360.2.a.cp.1.1 5
12.11 even 2 8280.2.a.bs.1.3 5
20.3 even 4 4600.2.e.u.4049.1 10
20.7 even 4 4600.2.e.u.4049.10 10
20.19 odd 2 4600.2.a.be.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.1 5 4.3 odd 2
1840.2.a.v.1.5 5 1.1 even 1 trivial
4600.2.a.be.1.5 5 20.19 odd 2
4600.2.e.u.4049.1 10 20.3 even 4
4600.2.e.u.4049.10 10 20.7 even 4
7360.2.a.co.1.5 5 8.3 odd 2
7360.2.a.cp.1.1 5 8.5 even 2
8280.2.a.bs.1.3 5 12.11 even 2
9200.2.a.cu.1.1 5 5.4 even 2