Properties

Label 2169.2.a.e.1.5
Level $2169$
Weight $2$
Character 2169.1
Self dual yes
Analytic conductor $17.320$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2169 = 3^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2169.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3195521984\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: 7.7.31056073.1
Defining polynomial: \(x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 241)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.356270\) of defining polynomial
Character \(\chi\) \(=\) 2169.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.35627 q^{2} -0.160532 q^{4} -2.74184 q^{5} -0.283608 q^{7} -2.93026 q^{8} +O(q^{10})\) \(q+1.35627 q^{2} -0.160532 q^{4} -2.74184 q^{5} -0.283608 q^{7} -2.93026 q^{8} -3.71867 q^{10} +4.12582 q^{11} +0.0271909 q^{13} -0.384649 q^{14} -3.65317 q^{16} +1.28740 q^{17} -5.72717 q^{19} +0.440153 q^{20} +5.59572 q^{22} +5.97702 q^{23} +2.51768 q^{25} +0.0368782 q^{26} +0.0455281 q^{28} +2.55610 q^{29} -2.02967 q^{31} +0.905851 q^{32} +1.74607 q^{34} +0.777607 q^{35} +2.42844 q^{37} -7.76759 q^{38} +8.03431 q^{40} +11.0324 q^{41} +10.4984 q^{43} -0.662326 q^{44} +8.10645 q^{46} -4.54349 q^{47} -6.91957 q^{49} +3.41465 q^{50} -0.00436502 q^{52} +9.30751 q^{53} -11.3123 q^{55} +0.831046 q^{56} +3.46676 q^{58} +9.94762 q^{59} +8.17350 q^{61} -2.75279 q^{62} +8.53491 q^{64} -0.0745531 q^{65} +4.40964 q^{67} -0.206670 q^{68} +1.05464 q^{70} +3.80954 q^{71} -15.6571 q^{73} +3.29362 q^{74} +0.919395 q^{76} -1.17011 q^{77} +6.69229 q^{79} +10.0164 q^{80} +14.9629 q^{82} +4.32880 q^{83} -3.52985 q^{85} +14.2386 q^{86} -12.0897 q^{88} -0.746861 q^{89} -0.00771155 q^{91} -0.959503 q^{92} -6.16219 q^{94} +15.7030 q^{95} +11.9245 q^{97} -9.38480 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 2q^{4} + 8q^{5} - 7q^{7} + 6q^{8} + 3q^{10} + 18q^{11} - q^{13} + 6q^{14} + 4q^{16} + 2q^{17} - 6q^{19} + 8q^{20} + 10q^{22} + 22q^{23} + 5q^{25} - 8q^{26} + 9q^{28} + 16q^{29} - 18q^{31} + 6q^{32} + 11q^{34} - 7q^{35} + 8q^{37} - 16q^{38} + 14q^{40} + 15q^{41} + 14q^{43} + 4q^{44} + 11q^{46} + 10q^{47} + 6q^{49} + 4q^{50} + 27q^{52} - 15q^{53} + 29q^{55} - 13q^{56} + 17q^{58} + 18q^{59} + 4q^{61} - 13q^{62} + 2q^{64} + 7q^{65} + 18q^{67} + 15q^{68} + 8q^{70} + 50q^{71} - 10q^{74} - 20q^{76} - 17q^{77} - 15q^{79} + 11q^{80} + 45q^{82} + 24q^{83} - 2q^{85} + 23q^{86} + 8q^{88} + 13q^{89} - 12q^{91} + 10q^{92} - 32q^{94} + 41q^{95} + q^{97} - 9q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.35627 0.959028 0.479514 0.877534i \(-0.340813\pi\)
0.479514 + 0.877534i \(0.340813\pi\)
\(3\) 0 0
\(4\) −0.160532 −0.0802661
\(5\) −2.74184 −1.22619 −0.613094 0.790010i \(-0.710075\pi\)
−0.613094 + 0.790010i \(0.710075\pi\)
\(6\) 0 0
\(7\) −0.283608 −0.107194 −0.0535968 0.998563i \(-0.517069\pi\)
−0.0535968 + 0.998563i \(0.517069\pi\)
\(8\) −2.93026 −1.03600
\(9\) 0 0
\(10\) −3.71867 −1.17595
\(11\) 4.12582 1.24398 0.621990 0.783025i \(-0.286324\pi\)
0.621990 + 0.783025i \(0.286324\pi\)
\(12\) 0 0
\(13\) 0.0271909 0.00754140 0.00377070 0.999993i \(-0.498800\pi\)
0.00377070 + 0.999993i \(0.498800\pi\)
\(14\) −0.384649 −0.102802
\(15\) 0 0
\(16\) −3.65317 −0.913291
\(17\) 1.28740 0.312241 0.156121 0.987738i \(-0.450101\pi\)
0.156121 + 0.987738i \(0.450101\pi\)
\(18\) 0 0
\(19\) −5.72717 −1.31390 −0.656952 0.753933i \(-0.728154\pi\)
−0.656952 + 0.753933i \(0.728154\pi\)
\(20\) 0.440153 0.0984212
\(21\) 0 0
\(22\) 5.59572 1.19301
\(23\) 5.97702 1.24629 0.623147 0.782104i \(-0.285854\pi\)
0.623147 + 0.782104i \(0.285854\pi\)
\(24\) 0 0
\(25\) 2.51768 0.503536
\(26\) 0.0368782 0.00723241
\(27\) 0 0
\(28\) 0.0455281 0.00860401
\(29\) 2.55610 0.474656 0.237328 0.971430i \(-0.423728\pi\)
0.237328 + 0.971430i \(0.423728\pi\)
\(30\) 0 0
\(31\) −2.02967 −0.364540 −0.182270 0.983248i \(-0.558345\pi\)
−0.182270 + 0.983248i \(0.558345\pi\)
\(32\) 0.905851 0.160133
\(33\) 0 0
\(34\) 1.74607 0.299448
\(35\) 0.777607 0.131440
\(36\) 0 0
\(37\) 2.42844 0.399233 0.199617 0.979874i \(-0.436030\pi\)
0.199617 + 0.979874i \(0.436030\pi\)
\(38\) −7.76759 −1.26007
\(39\) 0 0
\(40\) 8.03431 1.27034
\(41\) 11.0324 1.72297 0.861483 0.507786i \(-0.169536\pi\)
0.861483 + 0.507786i \(0.169536\pi\)
\(42\) 0 0
\(43\) 10.4984 1.60099 0.800493 0.599342i \(-0.204571\pi\)
0.800493 + 0.599342i \(0.204571\pi\)
\(44\) −0.662326 −0.0998494
\(45\) 0 0
\(46\) 8.10645 1.19523
\(47\) −4.54349 −0.662736 −0.331368 0.943502i \(-0.607510\pi\)
−0.331368 + 0.943502i \(0.607510\pi\)
\(48\) 0 0
\(49\) −6.91957 −0.988510
\(50\) 3.41465 0.482904
\(51\) 0 0
\(52\) −0.00436502 −0.000605319 0
\(53\) 9.30751 1.27848 0.639242 0.769005i \(-0.279248\pi\)
0.639242 + 0.769005i \(0.279248\pi\)
\(54\) 0 0
\(55\) −11.3123 −1.52535
\(56\) 0.831046 0.111053
\(57\) 0 0
\(58\) 3.46676 0.455208
\(59\) 9.94762 1.29507 0.647535 0.762036i \(-0.275800\pi\)
0.647535 + 0.762036i \(0.275800\pi\)
\(60\) 0 0
\(61\) 8.17350 1.04651 0.523255 0.852176i \(-0.324718\pi\)
0.523255 + 0.852176i \(0.324718\pi\)
\(62\) −2.75279 −0.349604
\(63\) 0 0
\(64\) 8.53491 1.06686
\(65\) −0.0745531 −0.00924717
\(66\) 0 0
\(67\) 4.40964 0.538723 0.269361 0.963039i \(-0.413187\pi\)
0.269361 + 0.963039i \(0.413187\pi\)
\(68\) −0.206670 −0.0250624
\(69\) 0 0
\(70\) 1.05464 0.126054
\(71\) 3.80954 0.452109 0.226054 0.974115i \(-0.427417\pi\)
0.226054 + 0.974115i \(0.427417\pi\)
\(72\) 0 0
\(73\) −15.6571 −1.83252 −0.916261 0.400583i \(-0.868808\pi\)
−0.916261 + 0.400583i \(0.868808\pi\)
\(74\) 3.29362 0.382876
\(75\) 0 0
\(76\) 0.919395 0.105462
\(77\) −1.17011 −0.133347
\(78\) 0 0
\(79\) 6.69229 0.752942 0.376471 0.926428i \(-0.377138\pi\)
0.376471 + 0.926428i \(0.377138\pi\)
\(80\) 10.0164 1.11987
\(81\) 0 0
\(82\) 14.9629 1.65237
\(83\) 4.32880 0.475147 0.237574 0.971370i \(-0.423648\pi\)
0.237574 + 0.971370i \(0.423648\pi\)
\(84\) 0 0
\(85\) −3.52985 −0.382866
\(86\) 14.2386 1.53539
\(87\) 0 0
\(88\) −12.0897 −1.28877
\(89\) −0.746861 −0.0791671 −0.0395835 0.999216i \(-0.512603\pi\)
−0.0395835 + 0.999216i \(0.512603\pi\)
\(90\) 0 0
\(91\) −0.00771155 −0.000808391 0
\(92\) −0.959503 −0.100035
\(93\) 0 0
\(94\) −6.16219 −0.635582
\(95\) 15.7030 1.61109
\(96\) 0 0
\(97\) 11.9245 1.21075 0.605373 0.795942i \(-0.293024\pi\)
0.605373 + 0.795942i \(0.293024\pi\)
\(98\) −9.38480 −0.948008
\(99\) 0 0
\(100\) −0.404168 −0.0404168
\(101\) −17.7456 −1.76576 −0.882878 0.469603i \(-0.844397\pi\)
−0.882878 + 0.469603i \(0.844397\pi\)
\(102\) 0 0
\(103\) 5.10848 0.503354 0.251677 0.967811i \(-0.419018\pi\)
0.251677 + 0.967811i \(0.419018\pi\)
\(104\) −0.0796766 −0.00781293
\(105\) 0 0
\(106\) 12.6235 1.22610
\(107\) 0.619213 0.0598615 0.0299308 0.999552i \(-0.490471\pi\)
0.0299308 + 0.999552i \(0.490471\pi\)
\(108\) 0 0
\(109\) 3.34125 0.320034 0.160017 0.987114i \(-0.448845\pi\)
0.160017 + 0.987114i \(0.448845\pi\)
\(110\) −15.3426 −1.46286
\(111\) 0 0
\(112\) 1.03607 0.0978990
\(113\) 3.42957 0.322627 0.161313 0.986903i \(-0.448427\pi\)
0.161313 + 0.986903i \(0.448427\pi\)
\(114\) 0 0
\(115\) −16.3880 −1.52819
\(116\) −0.410336 −0.0380988
\(117\) 0 0
\(118\) 13.4917 1.24201
\(119\) −0.365118 −0.0334703
\(120\) 0 0
\(121\) 6.02237 0.547488
\(122\) 11.0855 1.00363
\(123\) 0 0
\(124\) 0.325828 0.0292602
\(125\) 6.80613 0.608758
\(126\) 0 0
\(127\) 8.11791 0.720348 0.360174 0.932885i \(-0.382717\pi\)
0.360174 + 0.932885i \(0.382717\pi\)
\(128\) 9.76394 0.863018
\(129\) 0 0
\(130\) −0.101114 −0.00886830
\(131\) −2.73258 −0.238747 −0.119373 0.992849i \(-0.538089\pi\)
−0.119373 + 0.992849i \(0.538089\pi\)
\(132\) 0 0
\(133\) 1.62427 0.140842
\(134\) 5.98066 0.516650
\(135\) 0 0
\(136\) −3.77243 −0.323483
\(137\) −5.47355 −0.467637 −0.233819 0.972280i \(-0.575122\pi\)
−0.233819 + 0.972280i \(0.575122\pi\)
\(138\) 0 0
\(139\) −16.3761 −1.38901 −0.694503 0.719490i \(-0.744376\pi\)
−0.694503 + 0.719490i \(0.744376\pi\)
\(140\) −0.124831 −0.0105501
\(141\) 0 0
\(142\) 5.16676 0.433585
\(143\) 0.112185 0.00938136
\(144\) 0 0
\(145\) −7.00842 −0.582017
\(146\) −21.2352 −1.75744
\(147\) 0 0
\(148\) −0.389843 −0.0320449
\(149\) 15.2495 1.24929 0.624645 0.780909i \(-0.285244\pi\)
0.624645 + 0.780909i \(0.285244\pi\)
\(150\) 0 0
\(151\) −3.19011 −0.259607 −0.129804 0.991540i \(-0.541435\pi\)
−0.129804 + 0.991540i \(0.541435\pi\)
\(152\) 16.7821 1.36121
\(153\) 0 0
\(154\) −1.58699 −0.127883
\(155\) 5.56504 0.446995
\(156\) 0 0
\(157\) 12.7755 1.01959 0.509797 0.860295i \(-0.329721\pi\)
0.509797 + 0.860295i \(0.329721\pi\)
\(158\) 9.07655 0.722092
\(159\) 0 0
\(160\) −2.48370 −0.196354
\(161\) −1.69513 −0.133595
\(162\) 0 0
\(163\) −14.1420 −1.10769 −0.553844 0.832620i \(-0.686840\pi\)
−0.553844 + 0.832620i \(0.686840\pi\)
\(164\) −1.77105 −0.138296
\(165\) 0 0
\(166\) 5.87102 0.455679
\(167\) −9.14541 −0.707693 −0.353846 0.935304i \(-0.615126\pi\)
−0.353846 + 0.935304i \(0.615126\pi\)
\(168\) 0 0
\(169\) −12.9993 −0.999943
\(170\) −4.78743 −0.367179
\(171\) 0 0
\(172\) −1.68533 −0.128505
\(173\) −14.0992 −1.07194 −0.535970 0.844237i \(-0.680054\pi\)
−0.535970 + 0.844237i \(0.680054\pi\)
\(174\) 0 0
\(175\) −0.714033 −0.0539758
\(176\) −15.0723 −1.13612
\(177\) 0 0
\(178\) −1.01294 −0.0759234
\(179\) −23.3458 −1.74495 −0.872474 0.488660i \(-0.837486\pi\)
−0.872474 + 0.488660i \(0.837486\pi\)
\(180\) 0 0
\(181\) 19.8207 1.47326 0.736631 0.676295i \(-0.236415\pi\)
0.736631 + 0.676295i \(0.236415\pi\)
\(182\) −0.0104589 −0.000775269 0
\(183\) 0 0
\(184\) −17.5142 −1.29117
\(185\) −6.65839 −0.489535
\(186\) 0 0
\(187\) 5.31159 0.388422
\(188\) 0.729375 0.0531952
\(189\) 0 0
\(190\) 21.2975 1.54508
\(191\) −1.89196 −0.136898 −0.0684488 0.997655i \(-0.521805\pi\)
−0.0684488 + 0.997655i \(0.521805\pi\)
\(192\) 0 0
\(193\) 17.2319 1.24038 0.620189 0.784452i \(-0.287056\pi\)
0.620189 + 0.784452i \(0.287056\pi\)
\(194\) 16.1728 1.16114
\(195\) 0 0
\(196\) 1.11081 0.0793438
\(197\) −4.62238 −0.329331 −0.164665 0.986349i \(-0.552654\pi\)
−0.164665 + 0.986349i \(0.552654\pi\)
\(198\) 0 0
\(199\) −17.7065 −1.25518 −0.627589 0.778545i \(-0.715958\pi\)
−0.627589 + 0.778545i \(0.715958\pi\)
\(200\) −7.37746 −0.521665
\(201\) 0 0
\(202\) −24.0679 −1.69341
\(203\) −0.724930 −0.0508801
\(204\) 0 0
\(205\) −30.2490 −2.11268
\(206\) 6.92848 0.482730
\(207\) 0 0
\(208\) −0.0993329 −0.00688750
\(209\) −23.6293 −1.63447
\(210\) 0 0
\(211\) −11.1970 −0.770834 −0.385417 0.922742i \(-0.625942\pi\)
−0.385417 + 0.922742i \(0.625942\pi\)
\(212\) −1.49415 −0.102619
\(213\) 0 0
\(214\) 0.839819 0.0574089
\(215\) −28.7848 −1.96311
\(216\) 0 0
\(217\) 0.575631 0.0390764
\(218\) 4.53164 0.306921
\(219\) 0 0
\(220\) 1.81599 0.122434
\(221\) 0.0350057 0.00235474
\(222\) 0 0
\(223\) −2.90292 −0.194394 −0.0971969 0.995265i \(-0.530988\pi\)
−0.0971969 + 0.995265i \(0.530988\pi\)
\(224\) −0.256906 −0.0171653
\(225\) 0 0
\(226\) 4.65142 0.309408
\(227\) −21.4022 −1.42051 −0.710255 0.703944i \(-0.751421\pi\)
−0.710255 + 0.703944i \(0.751421\pi\)
\(228\) 0 0
\(229\) 17.5833 1.16194 0.580968 0.813926i \(-0.302674\pi\)
0.580968 + 0.813926i \(0.302674\pi\)
\(230\) −22.2266 −1.46558
\(231\) 0 0
\(232\) −7.49005 −0.491746
\(233\) 28.8991 1.89324 0.946621 0.322348i \(-0.104472\pi\)
0.946621 + 0.322348i \(0.104472\pi\)
\(234\) 0 0
\(235\) 12.4575 0.812638
\(236\) −1.59691 −0.103950
\(237\) 0 0
\(238\) −0.495198 −0.0320989
\(239\) −25.6630 −1.66000 −0.830000 0.557763i \(-0.811660\pi\)
−0.830000 + 0.557763i \(0.811660\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157
\(242\) 8.16796 0.525056
\(243\) 0 0
\(244\) −1.31211 −0.0839992
\(245\) 18.9723 1.21210
\(246\) 0 0
\(247\) −0.155727 −0.00990868
\(248\) 5.94748 0.377665
\(249\) 0 0
\(250\) 9.23094 0.583816
\(251\) 28.3138 1.78715 0.893576 0.448913i \(-0.148189\pi\)
0.893576 + 0.448913i \(0.148189\pi\)
\(252\) 0 0
\(253\) 24.6601 1.55037
\(254\) 11.0101 0.690833
\(255\) 0 0
\(256\) −3.82728 −0.239205
\(257\) 17.0359 1.06267 0.531336 0.847161i \(-0.321690\pi\)
0.531336 + 0.847161i \(0.321690\pi\)
\(258\) 0 0
\(259\) −0.688725 −0.0427953
\(260\) 0.0119682 0.000742234 0
\(261\) 0 0
\(262\) −3.70611 −0.228965
\(263\) 21.9895 1.35593 0.677966 0.735094i \(-0.262862\pi\)
0.677966 + 0.735094i \(0.262862\pi\)
\(264\) 0 0
\(265\) −25.5197 −1.56766
\(266\) 2.20295 0.135071
\(267\) 0 0
\(268\) −0.707888 −0.0432412
\(269\) −8.91005 −0.543255 −0.271628 0.962402i \(-0.587562\pi\)
−0.271628 + 0.962402i \(0.587562\pi\)
\(270\) 0 0
\(271\) 29.9968 1.82218 0.911088 0.412213i \(-0.135244\pi\)
0.911088 + 0.412213i \(0.135244\pi\)
\(272\) −4.70310 −0.285167
\(273\) 0 0
\(274\) −7.42362 −0.448477
\(275\) 10.3875 0.626389
\(276\) 0 0
\(277\) −14.9125 −0.896007 −0.448003 0.894032i \(-0.647865\pi\)
−0.448003 + 0.894032i \(0.647865\pi\)
\(278\) −22.2105 −1.33210
\(279\) 0 0
\(280\) −2.27859 −0.136172
\(281\) 28.8871 1.72326 0.861631 0.507535i \(-0.169443\pi\)
0.861631 + 0.507535i \(0.169443\pi\)
\(282\) 0 0
\(283\) −16.5841 −0.985820 −0.492910 0.870080i \(-0.664067\pi\)
−0.492910 + 0.870080i \(0.664067\pi\)
\(284\) −0.611553 −0.0362890
\(285\) 0 0
\(286\) 0.152153 0.00899698
\(287\) −3.12886 −0.184691
\(288\) 0 0
\(289\) −15.3426 −0.902505
\(290\) −9.50531 −0.558171
\(291\) 0 0
\(292\) 2.51346 0.147089
\(293\) 11.2756 0.658729 0.329364 0.944203i \(-0.393166\pi\)
0.329364 + 0.944203i \(0.393166\pi\)
\(294\) 0 0
\(295\) −27.2748 −1.58800
\(296\) −7.11597 −0.413608
\(297\) 0 0
\(298\) 20.6825 1.19810
\(299\) 0.162521 0.00939881
\(300\) 0 0
\(301\) −2.97742 −0.171616
\(302\) −4.32665 −0.248971
\(303\) 0 0
\(304\) 20.9223 1.19998
\(305\) −22.4104 −1.28322
\(306\) 0 0
\(307\) −1.24678 −0.0711575 −0.0355787 0.999367i \(-0.511327\pi\)
−0.0355787 + 0.999367i \(0.511327\pi\)
\(308\) 0.187841 0.0107032
\(309\) 0 0
\(310\) 7.54769 0.428680
\(311\) 17.8941 1.01468 0.507340 0.861746i \(-0.330629\pi\)
0.507340 + 0.861746i \(0.330629\pi\)
\(312\) 0 0
\(313\) −30.0820 −1.70033 −0.850166 0.526514i \(-0.823499\pi\)
−0.850166 + 0.526514i \(0.823499\pi\)
\(314\) 17.3270 0.977818
\(315\) 0 0
\(316\) −1.07433 −0.0604357
\(317\) 8.17835 0.459342 0.229671 0.973268i \(-0.426235\pi\)
0.229671 + 0.973268i \(0.426235\pi\)
\(318\) 0 0
\(319\) 10.5460 0.590463
\(320\) −23.4013 −1.30817
\(321\) 0 0
\(322\) −2.29905 −0.128121
\(323\) −7.37318 −0.410255
\(324\) 0 0
\(325\) 0.0684580 0.00379736
\(326\) −19.1804 −1.06230
\(327\) 0 0
\(328\) −32.3278 −1.78500
\(329\) 1.28857 0.0710410
\(330\) 0 0
\(331\) 13.4643 0.740067 0.370034 0.929018i \(-0.379346\pi\)
0.370034 + 0.929018i \(0.379346\pi\)
\(332\) −0.694911 −0.0381382
\(333\) 0 0
\(334\) −12.4036 −0.678697
\(335\) −12.0905 −0.660575
\(336\) 0 0
\(337\) −6.06342 −0.330295 −0.165148 0.986269i \(-0.552810\pi\)
−0.165148 + 0.986269i \(0.552810\pi\)
\(338\) −17.6305 −0.958973
\(339\) 0 0
\(340\) 0.566655 0.0307312
\(341\) −8.37407 −0.453481
\(342\) 0 0
\(343\) 3.94770 0.213156
\(344\) −30.7630 −1.65863
\(345\) 0 0
\(346\) −19.1223 −1.02802
\(347\) −23.7297 −1.27388 −0.636938 0.770915i \(-0.719799\pi\)
−0.636938 + 0.770915i \(0.719799\pi\)
\(348\) 0 0
\(349\) 8.46091 0.452902 0.226451 0.974023i \(-0.427288\pi\)
0.226451 + 0.974023i \(0.427288\pi\)
\(350\) −0.968421 −0.0517643
\(351\) 0 0
\(352\) 3.73738 0.199203
\(353\) −6.66847 −0.354927 −0.177463 0.984127i \(-0.556789\pi\)
−0.177463 + 0.984127i \(0.556789\pi\)
\(354\) 0 0
\(355\) −10.4451 −0.554370
\(356\) 0.119895 0.00635443
\(357\) 0 0
\(358\) −31.6632 −1.67345
\(359\) 28.3212 1.49474 0.747369 0.664409i \(-0.231317\pi\)
0.747369 + 0.664409i \(0.231317\pi\)
\(360\) 0 0
\(361\) 13.8005 0.726342
\(362\) 26.8822 1.41290
\(363\) 0 0
\(364\) 0.00123795 6.48863e−5 0
\(365\) 42.9292 2.24701
\(366\) 0 0
\(367\) 7.48573 0.390752 0.195376 0.980728i \(-0.437407\pi\)
0.195376 + 0.980728i \(0.437407\pi\)
\(368\) −21.8350 −1.13823
\(369\) 0 0
\(370\) −9.03058 −0.469477
\(371\) −2.63968 −0.137045
\(372\) 0 0
\(373\) 17.6125 0.911943 0.455972 0.889994i \(-0.349292\pi\)
0.455972 + 0.889994i \(0.349292\pi\)
\(374\) 7.20395 0.372507
\(375\) 0 0
\(376\) 13.3136 0.686597
\(377\) 0.0695028 0.00357957
\(378\) 0 0
\(379\) 7.61632 0.391224 0.195612 0.980681i \(-0.437331\pi\)
0.195612 + 0.980681i \(0.437331\pi\)
\(380\) −2.52083 −0.129316
\(381\) 0 0
\(382\) −2.56601 −0.131289
\(383\) 27.2242 1.39109 0.695546 0.718482i \(-0.255163\pi\)
0.695546 + 0.718482i \(0.255163\pi\)
\(384\) 0 0
\(385\) 3.20826 0.163508
\(386\) 23.3711 1.18956
\(387\) 0 0
\(388\) −1.91426 −0.0971818
\(389\) −23.0380 −1.16807 −0.584036 0.811728i \(-0.698527\pi\)
−0.584036 + 0.811728i \(0.698527\pi\)
\(390\) 0 0
\(391\) 7.69483 0.389145
\(392\) 20.2762 1.02410
\(393\) 0 0
\(394\) −6.26920 −0.315837
\(395\) −18.3492 −0.923248
\(396\) 0 0
\(397\) −17.0475 −0.855587 −0.427794 0.903876i \(-0.640709\pi\)
−0.427794 + 0.903876i \(0.640709\pi\)
\(398\) −24.0147 −1.20375
\(399\) 0 0
\(400\) −9.19749 −0.459875
\(401\) −6.19034 −0.309131 −0.154565 0.987983i \(-0.549398\pi\)
−0.154565 + 0.987983i \(0.549398\pi\)
\(402\) 0 0
\(403\) −0.0551887 −0.00274914
\(404\) 2.84874 0.141730
\(405\) 0 0
\(406\) −0.983201 −0.0487954
\(407\) 10.0193 0.496638
\(408\) 0 0
\(409\) 11.3675 0.562088 0.281044 0.959695i \(-0.409319\pi\)
0.281044 + 0.959695i \(0.409319\pi\)
\(410\) −41.0258 −2.02612
\(411\) 0 0
\(412\) −0.820076 −0.0404022
\(413\) −2.82122 −0.138823
\(414\) 0 0
\(415\) −11.8689 −0.582620
\(416\) 0.0246309 0.00120763
\(417\) 0 0
\(418\) −32.0477 −1.56750
\(419\) −4.14954 −0.202718 −0.101359 0.994850i \(-0.532319\pi\)
−0.101359 + 0.994850i \(0.532319\pi\)
\(420\) 0 0
\(421\) −1.44805 −0.0705734 −0.0352867 0.999377i \(-0.511234\pi\)
−0.0352867 + 0.999377i \(0.511234\pi\)
\(422\) −15.1862 −0.739251
\(423\) 0 0
\(424\) −27.2735 −1.32452
\(425\) 3.24127 0.157225
\(426\) 0 0
\(427\) −2.31807 −0.112179
\(428\) −0.0994035 −0.00480485
\(429\) 0 0
\(430\) −39.0400 −1.88268
\(431\) 22.1450 1.06669 0.533344 0.845898i \(-0.320935\pi\)
0.533344 + 0.845898i \(0.320935\pi\)
\(432\) 0 0
\(433\) 10.3894 0.499284 0.249642 0.968338i \(-0.419687\pi\)
0.249642 + 0.968338i \(0.419687\pi\)
\(434\) 0.780711 0.0374753
\(435\) 0 0
\(436\) −0.536378 −0.0256879
\(437\) −34.2314 −1.63751
\(438\) 0 0
\(439\) 39.5672 1.88844 0.944220 0.329316i \(-0.106818\pi\)
0.944220 + 0.329316i \(0.106818\pi\)
\(440\) 33.1481 1.58027
\(441\) 0 0
\(442\) 0.0474771 0.00225826
\(443\) 2.60234 0.123641 0.0618203 0.998087i \(-0.480309\pi\)
0.0618203 + 0.998087i \(0.480309\pi\)
\(444\) 0 0
\(445\) 2.04777 0.0970737
\(446\) −3.93714 −0.186429
\(447\) 0 0
\(448\) −2.42057 −0.114361
\(449\) 29.9486 1.41336 0.706681 0.707533i \(-0.250192\pi\)
0.706681 + 0.707533i \(0.250192\pi\)
\(450\) 0 0
\(451\) 45.5175 2.14334
\(452\) −0.550556 −0.0258960
\(453\) 0 0
\(454\) −29.0271 −1.36231
\(455\) 0.0211438 0.000991238 0
\(456\) 0 0
\(457\) −14.5052 −0.678526 −0.339263 0.940691i \(-0.610178\pi\)
−0.339263 + 0.940691i \(0.610178\pi\)
\(458\) 23.8477 1.11433
\(459\) 0 0
\(460\) 2.63080 0.122662
\(461\) −38.4334 −1.79002 −0.895011 0.446044i \(-0.852832\pi\)
−0.895011 + 0.446044i \(0.852832\pi\)
\(462\) 0 0
\(463\) 36.2326 1.68387 0.841936 0.539578i \(-0.181416\pi\)
0.841936 + 0.539578i \(0.181416\pi\)
\(464\) −9.33786 −0.433499
\(465\) 0 0
\(466\) 39.1950 1.81567
\(467\) 25.1111 1.16200 0.581001 0.813903i \(-0.302661\pi\)
0.581001 + 0.813903i \(0.302661\pi\)
\(468\) 0 0
\(469\) −1.25061 −0.0577477
\(470\) 16.8957 0.779342
\(471\) 0 0
\(472\) −29.1492 −1.34170
\(473\) 43.3144 1.99160
\(474\) 0 0
\(475\) −14.4192 −0.661597
\(476\) 0.0586131 0.00268653
\(477\) 0 0
\(478\) −34.8059 −1.59199
\(479\) 22.0372 1.00691 0.503454 0.864022i \(-0.332062\pi\)
0.503454 + 0.864022i \(0.332062\pi\)
\(480\) 0 0
\(481\) 0.0660315 0.00301078
\(482\) −1.35627 −0.0617764
\(483\) 0 0
\(484\) −0.966784 −0.0439447
\(485\) −32.6950 −1.48460
\(486\) 0 0
\(487\) −15.1309 −0.685649 −0.342824 0.939400i \(-0.611384\pi\)
−0.342824 + 0.939400i \(0.611384\pi\)
\(488\) −23.9505 −1.08419
\(489\) 0 0
\(490\) 25.7316 1.16244
\(491\) 40.4874 1.82717 0.913585 0.406649i \(-0.133303\pi\)
0.913585 + 0.406649i \(0.133303\pi\)
\(492\) 0 0
\(493\) 3.29073 0.148207
\(494\) −0.211208 −0.00950269
\(495\) 0 0
\(496\) 7.41473 0.332931
\(497\) −1.08041 −0.0484632
\(498\) 0 0
\(499\) −22.5598 −1.00992 −0.504959 0.863144i \(-0.668492\pi\)
−0.504959 + 0.863144i \(0.668492\pi\)
\(500\) −1.09260 −0.0488626
\(501\) 0 0
\(502\) 38.4012 1.71393
\(503\) −25.5277 −1.13823 −0.569113 0.822259i \(-0.692713\pi\)
−0.569113 + 0.822259i \(0.692713\pi\)
\(504\) 0 0
\(505\) 48.6556 2.16515
\(506\) 33.4457 1.48684
\(507\) 0 0
\(508\) −1.30318 −0.0578195
\(509\) 16.9047 0.749287 0.374643 0.927169i \(-0.377765\pi\)
0.374643 + 0.927169i \(0.377765\pi\)
\(510\) 0 0
\(511\) 4.44047 0.196435
\(512\) −24.7187 −1.09242
\(513\) 0 0
\(514\) 23.1053 1.01913
\(515\) −14.0066 −0.617206
\(516\) 0 0
\(517\) −18.7456 −0.824430
\(518\) −0.934096 −0.0410418
\(519\) 0 0
\(520\) 0.218460 0.00958012
\(521\) −28.9409 −1.26792 −0.633961 0.773365i \(-0.718572\pi\)
−0.633961 + 0.773365i \(0.718572\pi\)
\(522\) 0 0
\(523\) 15.3129 0.669586 0.334793 0.942292i \(-0.391334\pi\)
0.334793 + 0.942292i \(0.391334\pi\)
\(524\) 0.438667 0.0191632
\(525\) 0 0
\(526\) 29.8237 1.30038
\(527\) −2.61301 −0.113824
\(528\) 0 0
\(529\) 12.7248 0.553250
\(530\) −34.6116 −1.50343
\(531\) 0 0
\(532\) −0.260748 −0.0113048
\(533\) 0.299980 0.0129936
\(534\) 0 0
\(535\) −1.69778 −0.0734015
\(536\) −12.9214 −0.558120
\(537\) 0 0
\(538\) −12.0844 −0.520997
\(539\) −28.5489 −1.22969
\(540\) 0 0
\(541\) 11.0774 0.476253 0.238127 0.971234i \(-0.423467\pi\)
0.238127 + 0.971234i \(0.423467\pi\)
\(542\) 40.6837 1.74752
\(543\) 0 0
\(544\) 1.16620 0.0500002
\(545\) −9.16118 −0.392422
\(546\) 0 0
\(547\) 4.93535 0.211020 0.105510 0.994418i \(-0.466352\pi\)
0.105510 + 0.994418i \(0.466352\pi\)
\(548\) 0.878681 0.0375354
\(549\) 0 0
\(550\) 14.0882 0.600724
\(551\) −14.6392 −0.623652
\(552\) 0 0
\(553\) −1.89799 −0.0807106
\(554\) −20.2254 −0.859295
\(555\) 0 0
\(556\) 2.62890 0.111490
\(557\) 14.0586 0.595681 0.297840 0.954616i \(-0.403734\pi\)
0.297840 + 0.954616i \(0.403734\pi\)
\(558\) 0 0
\(559\) 0.285460 0.0120737
\(560\) −2.84073 −0.120043
\(561\) 0 0
\(562\) 39.1788 1.65266
\(563\) −44.3164 −1.86771 −0.933856 0.357650i \(-0.883578\pi\)
−0.933856 + 0.357650i \(0.883578\pi\)
\(564\) 0 0
\(565\) −9.40333 −0.395601
\(566\) −22.4925 −0.945429
\(567\) 0 0
\(568\) −11.1629 −0.468387
\(569\) 6.64972 0.278771 0.139385 0.990238i \(-0.455487\pi\)
0.139385 + 0.990238i \(0.455487\pi\)
\(570\) 0 0
\(571\) −22.3573 −0.935625 −0.467812 0.883828i \(-0.654958\pi\)
−0.467812 + 0.883828i \(0.654958\pi\)
\(572\) −0.0180093 −0.000753005 0
\(573\) 0 0
\(574\) −4.24358 −0.177124
\(575\) 15.0482 0.627554
\(576\) 0 0
\(577\) 33.2047 1.38233 0.691165 0.722697i \(-0.257098\pi\)
0.691165 + 0.722697i \(0.257098\pi\)
\(578\) −20.8087 −0.865528
\(579\) 0 0
\(580\) 1.12508 0.0467162
\(581\) −1.22768 −0.0509328
\(582\) 0 0
\(583\) 38.4011 1.59041
\(584\) 45.8794 1.89850
\(585\) 0 0
\(586\) 15.2928 0.631739
\(587\) −28.0737 −1.15873 −0.579363 0.815069i \(-0.696699\pi\)
−0.579363 + 0.815069i \(0.696699\pi\)
\(588\) 0 0
\(589\) 11.6243 0.478971
\(590\) −36.9920 −1.52293
\(591\) 0 0
\(592\) −8.87150 −0.364616
\(593\) 0.499968 0.0205312 0.0102656 0.999947i \(-0.496732\pi\)
0.0102656 + 0.999947i \(0.496732\pi\)
\(594\) 0 0
\(595\) 1.00109 0.0410408
\(596\) −2.44804 −0.100276
\(597\) 0 0
\(598\) 0.220422 0.00901372
\(599\) 5.66864 0.231614 0.115807 0.993272i \(-0.463055\pi\)
0.115807 + 0.993272i \(0.463055\pi\)
\(600\) 0 0
\(601\) 22.5848 0.921252 0.460626 0.887594i \(-0.347625\pi\)
0.460626 + 0.887594i \(0.347625\pi\)
\(602\) −4.03818 −0.164584
\(603\) 0 0
\(604\) 0.512115 0.0208377
\(605\) −16.5124 −0.671323
\(606\) 0 0
\(607\) −4.01260 −0.162867 −0.0814333 0.996679i \(-0.525950\pi\)
−0.0814333 + 0.996679i \(0.525950\pi\)
\(608\) −5.18797 −0.210400
\(609\) 0 0
\(610\) −30.3946 −1.23064
\(611\) −0.123542 −0.00499796
\(612\) 0 0
\(613\) −31.5947 −1.27610 −0.638048 0.769996i \(-0.720258\pi\)
−0.638048 + 0.769996i \(0.720258\pi\)
\(614\) −1.69097 −0.0682420
\(615\) 0 0
\(616\) 3.42874 0.138148
\(617\) 2.64125 0.106333 0.0531664 0.998586i \(-0.483069\pi\)
0.0531664 + 0.998586i \(0.483069\pi\)
\(618\) 0 0
\(619\) 13.8609 0.557115 0.278558 0.960419i \(-0.410144\pi\)
0.278558 + 0.960419i \(0.410144\pi\)
\(620\) −0.893367 −0.0358785
\(621\) 0 0
\(622\) 24.2692 0.973106
\(623\) 0.211815 0.00848621
\(624\) 0 0
\(625\) −31.2497 −1.24999
\(626\) −40.7992 −1.63067
\(627\) 0 0
\(628\) −2.05087 −0.0818387
\(629\) 3.12638 0.124657
\(630\) 0 0
\(631\) −28.9779 −1.15359 −0.576795 0.816889i \(-0.695697\pi\)
−0.576795 + 0.816889i \(0.695697\pi\)
\(632\) −19.6102 −0.780051
\(633\) 0 0
\(634\) 11.0921 0.440522
\(635\) −22.2580 −0.883281
\(636\) 0 0
\(637\) −0.188149 −0.00745475
\(638\) 14.3032 0.566271
\(639\) 0 0
\(640\) −26.7711 −1.05822
\(641\) 36.8202 1.45431 0.727154 0.686474i \(-0.240842\pi\)
0.727154 + 0.686474i \(0.240842\pi\)
\(642\) 0 0
\(643\) 14.9327 0.588890 0.294445 0.955668i \(-0.404865\pi\)
0.294445 + 0.955668i \(0.404865\pi\)
\(644\) 0.272123 0.0107231
\(645\) 0 0
\(646\) −10.0000 −0.393446
\(647\) −1.09720 −0.0431356 −0.0215678 0.999767i \(-0.506866\pi\)
−0.0215678 + 0.999767i \(0.506866\pi\)
\(648\) 0 0
\(649\) 41.0421 1.61104
\(650\) 0.0928475 0.00364178
\(651\) 0 0
\(652\) 2.27025 0.0889097
\(653\) −9.51269 −0.372260 −0.186130 0.982525i \(-0.559595\pi\)
−0.186130 + 0.982525i \(0.559595\pi\)
\(654\) 0 0
\(655\) 7.49229 0.292748
\(656\) −40.3031 −1.57357
\(657\) 0 0
\(658\) 1.74765 0.0681303
\(659\) −3.98928 −0.155400 −0.0777001 0.996977i \(-0.524758\pi\)
−0.0777001 + 0.996977i \(0.524758\pi\)
\(660\) 0 0
\(661\) −25.6014 −0.995780 −0.497890 0.867240i \(-0.665892\pi\)
−0.497890 + 0.867240i \(0.665892\pi\)
\(662\) 18.2613 0.709745
\(663\) 0 0
\(664\) −12.6845 −0.492255
\(665\) −4.45349 −0.172699
\(666\) 0 0
\(667\) 15.2779 0.591561
\(668\) 1.46813 0.0568037
\(669\) 0 0
\(670\) −16.3980 −0.633510
\(671\) 33.7224 1.30184
\(672\) 0 0
\(673\) 0.273027 0.0105244 0.00526221 0.999986i \(-0.498325\pi\)
0.00526221 + 0.999986i \(0.498325\pi\)
\(674\) −8.22363 −0.316762
\(675\) 0 0
\(676\) 2.08680 0.0802615
\(677\) 5.73279 0.220329 0.110165 0.993913i \(-0.464862\pi\)
0.110165 + 0.993913i \(0.464862\pi\)
\(678\) 0 0
\(679\) −3.38187 −0.129784
\(680\) 10.3434 0.396651
\(681\) 0 0
\(682\) −11.3575 −0.434901
\(683\) 18.8046 0.719538 0.359769 0.933041i \(-0.382855\pi\)
0.359769 + 0.933041i \(0.382855\pi\)
\(684\) 0 0
\(685\) 15.0076 0.573411
\(686\) 5.35414 0.204422
\(687\) 0 0
\(688\) −38.3523 −1.46217
\(689\) 0.253080 0.00964157
\(690\) 0 0
\(691\) 15.8329 0.602310 0.301155 0.953575i \(-0.402628\pi\)
0.301155 + 0.953575i \(0.402628\pi\)
\(692\) 2.26337 0.0860403
\(693\) 0 0
\(694\) −32.1838 −1.22168
\(695\) 44.9007 1.70318
\(696\) 0 0
\(697\) 14.2031 0.537981
\(698\) 11.4753 0.434346
\(699\) 0 0
\(700\) 0.114625 0.00433243
\(701\) −40.3523 −1.52409 −0.762043 0.647527i \(-0.775803\pi\)
−0.762043 + 0.647527i \(0.775803\pi\)
\(702\) 0 0
\(703\) −13.9081 −0.524554
\(704\) 35.2135 1.32716
\(705\) 0 0
\(706\) −9.04425 −0.340385
\(707\) 5.03280 0.189278
\(708\) 0 0
\(709\) −41.5401 −1.56007 −0.780036 0.625734i \(-0.784800\pi\)
−0.780036 + 0.625734i \(0.784800\pi\)
\(710\) −14.1664 −0.531656
\(711\) 0 0
\(712\) 2.18850 0.0820175
\(713\) −12.1314 −0.454325
\(714\) 0 0
\(715\) −0.307592 −0.0115033
\(716\) 3.74775 0.140060
\(717\) 0 0
\(718\) 38.4112 1.43349
\(719\) 10.2324 0.381603 0.190801 0.981629i \(-0.438891\pi\)
0.190801 + 0.981629i \(0.438891\pi\)
\(720\) 0 0
\(721\) −1.44881 −0.0539563
\(722\) 18.7172 0.696582
\(723\) 0 0
\(724\) −3.18186 −0.118253
\(725\) 6.43544 0.239006
\(726\) 0 0
\(727\) −45.5533 −1.68948 −0.844739 0.535178i \(-0.820244\pi\)
−0.844739 + 0.535178i \(0.820244\pi\)
\(728\) 0.0225969 0.000837497 0
\(729\) 0 0
\(730\) 58.2235 2.15495
\(731\) 13.5156 0.499894
\(732\) 0 0
\(733\) −20.6241 −0.761770 −0.380885 0.924622i \(-0.624381\pi\)
−0.380885 + 0.924622i \(0.624381\pi\)
\(734\) 10.1527 0.374742
\(735\) 0 0
\(736\) 5.41429 0.199573
\(737\) 18.1934 0.670161
\(738\) 0 0
\(739\) −17.1975 −0.632620 −0.316310 0.948656i \(-0.602444\pi\)
−0.316310 + 0.948656i \(0.602444\pi\)
\(740\) 1.06889 0.0392930
\(741\) 0 0
\(742\) −3.58012 −0.131430
\(743\) 40.8356 1.49811 0.749056 0.662506i \(-0.230507\pi\)
0.749056 + 0.662506i \(0.230507\pi\)
\(744\) 0 0
\(745\) −41.8117 −1.53186
\(746\) 23.8874 0.874579
\(747\) 0 0
\(748\) −0.852681 −0.0311771
\(749\) −0.175613 −0.00641678
\(750\) 0 0
\(751\) 20.2798 0.740021 0.370010 0.929028i \(-0.379354\pi\)
0.370010 + 0.929028i \(0.379354\pi\)
\(752\) 16.5981 0.605271
\(753\) 0 0
\(754\) 0.0942645 0.00343291
\(755\) 8.74676 0.318327
\(756\) 0 0
\(757\) 13.7148 0.498473 0.249237 0.968443i \(-0.419820\pi\)
0.249237 + 0.968443i \(0.419820\pi\)
\(758\) 10.3298 0.375194
\(759\) 0 0
\(760\) −46.0139 −1.66910
\(761\) 6.41706 0.232618 0.116309 0.993213i \(-0.462894\pi\)
0.116309 + 0.993213i \(0.462894\pi\)
\(762\) 0 0
\(763\) −0.947605 −0.0343056
\(764\) 0.303721 0.0109882
\(765\) 0 0
\(766\) 36.9234 1.33410
\(767\) 0.270485 0.00976665
\(768\) 0 0
\(769\) −27.8170 −1.00311 −0.501553 0.865127i \(-0.667238\pi\)
−0.501553 + 0.865127i \(0.667238\pi\)
\(770\) 4.35127 0.156809
\(771\) 0 0
\(772\) −2.76627 −0.0995603
\(773\) −12.8524 −0.462270 −0.231135 0.972922i \(-0.574244\pi\)
−0.231135 + 0.972922i \(0.574244\pi\)
\(774\) 0 0
\(775\) −5.11007 −0.183559
\(776\) −34.9418 −1.25434
\(777\) 0 0
\(778\) −31.2457 −1.12021
\(779\) −63.1843 −2.26381
\(780\) 0 0
\(781\) 15.7174 0.562414
\(782\) 10.4363 0.373200
\(783\) 0 0
\(784\) 25.2783 0.902797
\(785\) −35.0283 −1.25021
\(786\) 0 0
\(787\) 9.53803 0.339994 0.169997 0.985445i \(-0.445624\pi\)
0.169997 + 0.985445i \(0.445624\pi\)
\(788\) 0.742040 0.0264341
\(789\) 0 0
\(790\) −24.8864 −0.885420
\(791\) −0.972653 −0.0345836
\(792\) 0 0
\(793\) 0.222245 0.00789215
\(794\) −23.1209 −0.820532
\(795\) 0 0
\(796\) 2.84246 0.100748
\(797\) 7.72801 0.273740 0.136870 0.990589i \(-0.456296\pi\)
0.136870 + 0.990589i \(0.456296\pi\)
\(798\) 0 0
\(799\) −5.84930 −0.206933
\(800\) 2.28064 0.0806329
\(801\) 0 0
\(802\) −8.39577 −0.296465
\(803\) −64.5982 −2.27962
\(804\) 0 0
\(805\) 4.64777 0.163812
\(806\) −0.0748508 −0.00263651
\(807\) 0 0
\(808\) 51.9994 1.82933
\(809\) 1.99109 0.0700030 0.0350015 0.999387i \(-0.488856\pi\)
0.0350015 + 0.999387i \(0.488856\pi\)
\(810\) 0 0
\(811\) 28.5346 1.00198 0.500992 0.865452i \(-0.332969\pi\)
0.500992 + 0.865452i \(0.332969\pi\)
\(812\) 0.116375 0.00408395
\(813\) 0 0
\(814\) 13.5889 0.476290
\(815\) 38.7751 1.35823
\(816\) 0 0
\(817\) −60.1260 −2.10354
\(818\) 15.4174 0.539058
\(819\) 0 0
\(820\) 4.85593 0.169576
\(821\) −48.3459 −1.68728 −0.843642 0.536907i \(-0.819593\pi\)
−0.843642 + 0.536907i \(0.819593\pi\)
\(822\) 0 0
\(823\) −36.4809 −1.27164 −0.635821 0.771836i \(-0.719338\pi\)
−0.635821 + 0.771836i \(0.719338\pi\)
\(824\) −14.9692 −0.521477
\(825\) 0 0
\(826\) −3.82634 −0.133135
\(827\) −23.2928 −0.809971 −0.404985 0.914323i \(-0.632723\pi\)
−0.404985 + 0.914323i \(0.632723\pi\)
\(828\) 0 0
\(829\) −29.8113 −1.03539 −0.517695 0.855565i \(-0.673210\pi\)
−0.517695 + 0.855565i \(0.673210\pi\)
\(830\) −16.0974 −0.558748
\(831\) 0 0
\(832\) 0.232072 0.00804565
\(833\) −8.90827 −0.308653
\(834\) 0 0
\(835\) 25.0752 0.867764
\(836\) 3.79326 0.131193
\(837\) 0 0
\(838\) −5.62790 −0.194412
\(839\) 48.3508 1.66925 0.834627 0.550815i \(-0.185683\pi\)
0.834627 + 0.550815i \(0.185683\pi\)
\(840\) 0 0
\(841\) −22.4663 −0.774702
\(842\) −1.96394 −0.0676819
\(843\) 0 0
\(844\) 1.79748 0.0618718
\(845\) 35.6419 1.22612
\(846\) 0 0
\(847\) −1.70799 −0.0586873
\(848\) −34.0019 −1.16763
\(849\) 0 0
\(850\) 4.39603 0.150783
\(851\) 14.5148 0.497562
\(852\) 0 0
\(853\) 5.13297 0.175750 0.0878748 0.996132i \(-0.471992\pi\)
0.0878748 + 0.996132i \(0.471992\pi\)
\(854\) −3.14393 −0.107583
\(855\) 0 0
\(856\) −1.81446 −0.0620168
\(857\) −20.8453 −0.712062 −0.356031 0.934474i \(-0.615870\pi\)
−0.356031 + 0.934474i \(0.615870\pi\)
\(858\) 0 0
\(859\) 24.4260 0.833403 0.416701 0.909043i \(-0.363186\pi\)
0.416701 + 0.909043i \(0.363186\pi\)
\(860\) 4.62089 0.157571
\(861\) 0 0
\(862\) 30.0346 1.02298
\(863\) −25.5588 −0.870033 −0.435017 0.900422i \(-0.643257\pi\)
−0.435017 + 0.900422i \(0.643257\pi\)
\(864\) 0 0
\(865\) 38.6576 1.31440
\(866\) 14.0909 0.478827
\(867\) 0 0
\(868\) −0.0924073 −0.00313651
\(869\) 27.6112 0.936645
\(870\) 0 0
\(871\) 0.119902 0.00406273
\(872\) −9.79076 −0.331557
\(873\) 0 0
\(874\) −46.4270 −1.57042
\(875\) −1.93027 −0.0652550
\(876\) 0 0
\(877\) −7.53456 −0.254424 −0.127212 0.991876i \(-0.540603\pi\)
−0.127212 + 0.991876i \(0.540603\pi\)
\(878\) 53.6638 1.81107
\(879\) 0 0
\(880\) 41.3258 1.39309
\(881\) −21.7361 −0.732308 −0.366154 0.930554i \(-0.619326\pi\)
−0.366154 + 0.930554i \(0.619326\pi\)
\(882\) 0 0
\(883\) −38.6202 −1.29967 −0.649837 0.760074i \(-0.725163\pi\)
−0.649837 + 0.760074i \(0.725163\pi\)
\(884\) −0.00561953 −0.000189005 0
\(885\) 0 0
\(886\) 3.52947 0.118575
\(887\) 14.3165 0.480702 0.240351 0.970686i \(-0.422738\pi\)
0.240351 + 0.970686i \(0.422738\pi\)
\(888\) 0 0
\(889\) −2.30230 −0.0772167
\(890\) 2.77733 0.0930963
\(891\) 0 0
\(892\) 0.466012 0.0156032
\(893\) 26.0213 0.870771
\(894\) 0 0
\(895\) 64.0105 2.13963
\(896\) −2.76913 −0.0925101
\(897\) 0 0
\(898\) 40.6184 1.35545
\(899\) −5.18805 −0.173031
\(900\) 0 0
\(901\) 11.9825 0.399196
\(902\) 61.7341 2.05552
\(903\) 0 0
\(904\) −10.0496 −0.334243
\(905\) −54.3452 −1.80650
\(906\) 0 0
\(907\) −47.6042 −1.58067 −0.790336 0.612674i \(-0.790094\pi\)
−0.790336 + 0.612674i \(0.790094\pi\)
\(908\) 3.43573 0.114019
\(909\) 0 0
\(910\) 0.0286767 0.000950625 0
\(911\) 40.1570 1.33046 0.665230 0.746639i \(-0.268334\pi\)
0.665230 + 0.746639i \(0.268334\pi\)
\(912\) 0 0
\(913\) 17.8598 0.591074
\(914\) −19.6730 −0.650726
\(915\) 0 0
\(916\) −2.82268 −0.0932641
\(917\) 0.774981 0.0255921
\(918\) 0 0
\(919\) −18.0383 −0.595029 −0.297514 0.954717i \(-0.596158\pi\)
−0.297514 + 0.954717i \(0.596158\pi\)
\(920\) 48.0212 1.58321
\(921\) 0 0
\(922\) −52.1261 −1.71668
\(923\) 0.103585 0.00340953
\(924\) 0 0
\(925\) 6.11403 0.201028
\(926\) 49.1412 1.61488
\(927\) 0 0
\(928\) 2.31545 0.0760083
\(929\) 0.679116 0.0222811 0.0111405 0.999938i \(-0.496454\pi\)
0.0111405 + 0.999938i \(0.496454\pi\)
\(930\) 0 0
\(931\) 39.6295 1.29881
\(932\) −4.63923 −0.151963
\(933\) 0 0
\(934\) 34.0574 1.11439
\(935\) −14.5635 −0.476278
\(936\) 0 0
\(937\) −25.7202 −0.840243 −0.420122 0.907468i \(-0.638013\pi\)
−0.420122 + 0.907468i \(0.638013\pi\)
\(938\) −1.69616 −0.0553816
\(939\) 0 0
\(940\) −1.99983 −0.0652272
\(941\) 41.9988 1.36912 0.684560 0.728956i \(-0.259994\pi\)
0.684560 + 0.728956i \(0.259994\pi\)
\(942\) 0 0
\(943\) 65.9407 2.14732
\(944\) −36.3403 −1.18278
\(945\) 0 0
\(946\) 58.7460 1.91000
\(947\) 15.2367 0.495128 0.247564 0.968872i \(-0.420370\pi\)
0.247564 + 0.968872i \(0.420370\pi\)
\(948\) 0 0
\(949\) −0.425730 −0.0138198
\(950\) −19.5563 −0.634490
\(951\) 0 0
\(952\) 1.06989 0.0346754
\(953\) −43.1360 −1.39731 −0.698656 0.715457i \(-0.746218\pi\)
−0.698656 + 0.715457i \(0.746218\pi\)
\(954\) 0 0
\(955\) 5.18745 0.167862
\(956\) 4.11973 0.133242
\(957\) 0 0
\(958\) 29.8885 0.965652
\(959\) 1.55234 0.0501277
\(960\) 0 0
\(961\) −26.8804 −0.867110
\(962\) 0.0895566 0.00288742
\(963\) 0 0
\(964\) 0.160532 0.00517039
\(965\) −47.2471 −1.52094
\(966\) 0 0
\(967\) 45.6699 1.46864 0.734322 0.678801i \(-0.237500\pi\)
0.734322 + 0.678801i \(0.237500\pi\)
\(968\) −17.6471 −0.567201
\(969\) 0 0
\(970\) −44.3432 −1.42377
\(971\) 2.32251 0.0745330 0.0372665 0.999305i \(-0.488135\pi\)
0.0372665 + 0.999305i \(0.488135\pi\)
\(972\) 0 0
\(973\) 4.64440 0.148893
\(974\) −20.5216 −0.657556
\(975\) 0 0
\(976\) −29.8591 −0.955768
\(977\) 50.8644 1.62730 0.813649 0.581357i \(-0.197478\pi\)
0.813649 + 0.581357i \(0.197478\pi\)
\(978\) 0 0
\(979\) −3.08141 −0.0984823
\(980\) −3.04567 −0.0972903
\(981\) 0 0
\(982\) 54.9118 1.75231
\(983\) 21.2587 0.678047 0.339023 0.940778i \(-0.389903\pi\)
0.339023 + 0.940778i \(0.389903\pi\)
\(984\) 0 0
\(985\) 12.6738 0.403821
\(986\) 4.46312 0.142135
\(987\) 0 0
\(988\) 0.0249992 0.000795330 0
\(989\) 62.7490 1.99530
\(990\) 0 0
\(991\) 51.6637 1.64115 0.820575 0.571539i \(-0.193653\pi\)
0.820575 + 0.571539i \(0.193653\pi\)
\(992\) −1.83858 −0.0583751
\(993\) 0 0
\(994\) −1.46533 −0.0464775
\(995\) 48.5483 1.53908
\(996\) 0 0
\(997\) −32.2376 −1.02097 −0.510487 0.859885i \(-0.670535\pi\)
−0.510487 + 0.859885i \(0.670535\pi\)
\(998\) −30.5972 −0.968538
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2169.2.a.e.1.5 7
3.2 odd 2 241.2.a.a.1.3 7
12.11 even 2 3856.2.a.j.1.7 7
15.14 odd 2 6025.2.a.f.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.a.1.3 7 3.2 odd 2
2169.2.a.e.1.5 7 1.1 even 1 trivial
3856.2.a.j.1.7 7 12.11 even 2
6025.2.a.f.1.5 7 15.14 odd 2