Properties

Label 503.2.a.e.1.9
Level $503$
Weight $2$
Character 503.1
Self dual yes
Analytic conductor $4.016$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 503.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.01647522167\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} - 9 x^{8} + 14 x^{7} + 27 x^{6} - 27 x^{5} - 34 x^{4} + 14 x^{3} + 17 x^{2} + x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-1.40552\) of defining polynomial
Character \(\chi\) \(=\) 503.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.36113 q^{2} -2.40552 q^{3} -0.147314 q^{4} +0.590303 q^{5} -3.27424 q^{6} +1.95900 q^{7} -2.92278 q^{8} +2.78655 q^{9} +O(q^{10})\) \(q+1.36113 q^{2} -2.40552 q^{3} -0.147314 q^{4} +0.590303 q^{5} -3.27424 q^{6} +1.95900 q^{7} -2.92278 q^{8} +2.78655 q^{9} +0.803481 q^{10} -1.52746 q^{11} +0.354366 q^{12} -4.67738 q^{13} +2.66646 q^{14} -1.41999 q^{15} -3.68367 q^{16} -3.04913 q^{17} +3.79286 q^{18} +0.338159 q^{19} -0.0869597 q^{20} -4.71242 q^{21} -2.07908 q^{22} -7.98874 q^{23} +7.03082 q^{24} -4.65154 q^{25} -6.36654 q^{26} +0.513468 q^{27} -0.288587 q^{28} +6.01794 q^{29} -1.93279 q^{30} -4.17447 q^{31} +0.831593 q^{32} +3.67434 q^{33} -4.15028 q^{34} +1.15640 q^{35} -0.410496 q^{36} -11.0654 q^{37} +0.460279 q^{38} +11.2515 q^{39} -1.72533 q^{40} -4.82359 q^{41} -6.41423 q^{42} +12.4475 q^{43} +0.225016 q^{44} +1.64491 q^{45} -10.8737 q^{46} +11.0094 q^{47} +8.86116 q^{48} -3.16233 q^{49} -6.33137 q^{50} +7.33476 q^{51} +0.689042 q^{52} -3.43684 q^{53} +0.698899 q^{54} -0.901664 q^{55} -5.72572 q^{56} -0.813449 q^{57} +8.19122 q^{58} +2.63490 q^{59} +0.209184 q^{60} +11.4908 q^{61} -5.68202 q^{62} +5.45884 q^{63} +8.49925 q^{64} -2.76107 q^{65} +5.00127 q^{66} -10.2554 q^{67} +0.449179 q^{68} +19.2171 q^{69} +1.57402 q^{70} +9.66931 q^{71} -8.14447 q^{72} -6.33688 q^{73} -15.0615 q^{74} +11.1894 q^{75} -0.0498154 q^{76} -2.99229 q^{77} +15.3149 q^{78} -0.667323 q^{79} -2.17448 q^{80} -9.59480 q^{81} -6.56556 q^{82} -2.26231 q^{83} +0.694203 q^{84} -1.79991 q^{85} +16.9427 q^{86} -14.4763 q^{87} +4.46443 q^{88} +2.21979 q^{89} +2.23894 q^{90} -9.16297 q^{91} +1.17685 q^{92} +10.0418 q^{93} +14.9853 q^{94} +0.199616 q^{95} -2.00042 q^{96} +14.2878 q^{97} -4.30435 q^{98} -4.25634 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} + O(q^{10}) \) \( 10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} - 3q^{8} - 2q^{9} - 4q^{10} - 3q^{11} - 7q^{12} - 18q^{13} + q^{14} - 2q^{15} - 4q^{16} - 11q^{17} - q^{18} - 3q^{20} + q^{21} - 18q^{22} - 2q^{23} + 10q^{24} - 27q^{25} + 11q^{26} - 2q^{27} - 22q^{28} - 9q^{29} + 12q^{30} - 22q^{31} - 10q^{32} - 10q^{33} - 10q^{34} - 6q^{35} + 2q^{36} - 35q^{37} + 2q^{38} + 8q^{39} - 19q^{40} - 4q^{41} + 4q^{42} - 20q^{43} + 9q^{44} + 2q^{45} - q^{46} + 7q^{47} - 27q^{49} + 16q^{50} + 9q^{51} - 7q^{52} - 24q^{53} + 17q^{54} - 11q^{55} + 12q^{56} - 23q^{57} + 2q^{58} + 17q^{59} - 4q^{61} + 8q^{62} + 10q^{63} + 3q^{64} - 16q^{65} + 46q^{66} - 6q^{67} + 28q^{68} - 2q^{69} + 26q^{70} - q^{71} - q^{72} - 31q^{73} + 11q^{74} + 30q^{75} + 20q^{76} + 3q^{77} + 11q^{78} - 10q^{79} + 24q^{80} - 6q^{81} - 9q^{82} + 22q^{83} + 22q^{84} - 6q^{85} + 38q^{86} + 25q^{87} - 3q^{88} + q^{89} + 2q^{90} + 10q^{91} + 27q^{92} - 6q^{93} + 33q^{94} + 39q^{95} + 46q^{96} - 57q^{97} + 40q^{98} + 35q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.36113 0.962467 0.481234 0.876592i \(-0.340189\pi\)
0.481234 + 0.876592i \(0.340189\pi\)
\(3\) −2.40552 −1.38883 −0.694415 0.719575i \(-0.744337\pi\)
−0.694415 + 0.719575i \(0.744337\pi\)
\(4\) −0.147314 −0.0736568
\(5\) 0.590303 0.263991 0.131996 0.991250i \(-0.457861\pi\)
0.131996 + 0.991250i \(0.457861\pi\)
\(6\) −3.27424 −1.33670
\(7\) 1.95900 0.740431 0.370216 0.928946i \(-0.379284\pi\)
0.370216 + 0.928946i \(0.379284\pi\)
\(8\) −2.92278 −1.03336
\(9\) 2.78655 0.928849
\(10\) 0.803481 0.254083
\(11\) −1.52746 −0.460546 −0.230273 0.973126i \(-0.573962\pi\)
−0.230273 + 0.973126i \(0.573962\pi\)
\(12\) 0.354366 0.102297
\(13\) −4.67738 −1.29727 −0.648636 0.761099i \(-0.724660\pi\)
−0.648636 + 0.761099i \(0.724660\pi\)
\(14\) 2.66646 0.712641
\(15\) −1.41999 −0.366639
\(16\) −3.68367 −0.920918
\(17\) −3.04913 −0.739523 −0.369761 0.929127i \(-0.620561\pi\)
−0.369761 + 0.929127i \(0.620561\pi\)
\(18\) 3.79286 0.893986
\(19\) 0.338159 0.0775789 0.0387895 0.999247i \(-0.487650\pi\)
0.0387895 + 0.999247i \(0.487650\pi\)
\(20\) −0.0869597 −0.0194448
\(21\) −4.71242 −1.02833
\(22\) −2.07908 −0.443261
\(23\) −7.98874 −1.66577 −0.832884 0.553448i \(-0.813312\pi\)
−0.832884 + 0.553448i \(0.813312\pi\)
\(24\) 7.03082 1.43516
\(25\) −4.65154 −0.930309
\(26\) −6.36654 −1.24858
\(27\) 0.513468 0.0988170
\(28\) −0.288587 −0.0545378
\(29\) 6.01794 1.11750 0.558751 0.829335i \(-0.311281\pi\)
0.558751 + 0.829335i \(0.311281\pi\)
\(30\) −1.93279 −0.352878
\(31\) −4.17447 −0.749758 −0.374879 0.927074i \(-0.622316\pi\)
−0.374879 + 0.927074i \(0.622316\pi\)
\(32\) 0.831593 0.147006
\(33\) 3.67434 0.639621
\(34\) −4.15028 −0.711767
\(35\) 1.15640 0.195468
\(36\) −0.410496 −0.0684160
\(37\) −11.0654 −1.81914 −0.909569 0.415553i \(-0.863588\pi\)
−0.909569 + 0.415553i \(0.863588\pi\)
\(38\) 0.460279 0.0746672
\(39\) 11.2515 1.80169
\(40\) −1.72533 −0.272798
\(41\) −4.82359 −0.753319 −0.376659 0.926352i \(-0.622927\pi\)
−0.376659 + 0.926352i \(0.622927\pi\)
\(42\) −6.41423 −0.989737
\(43\) 12.4475 1.89822 0.949111 0.314941i \(-0.101985\pi\)
0.949111 + 0.314941i \(0.101985\pi\)
\(44\) 0.225016 0.0339224
\(45\) 1.64491 0.245208
\(46\) −10.8737 −1.60325
\(47\) 11.0094 1.60589 0.802945 0.596052i \(-0.203265\pi\)
0.802945 + 0.596052i \(0.203265\pi\)
\(48\) 8.86116 1.27900
\(49\) −3.16233 −0.451761
\(50\) −6.33137 −0.895391
\(51\) 7.33476 1.02707
\(52\) 0.689042 0.0955529
\(53\) −3.43684 −0.472087 −0.236043 0.971743i \(-0.575851\pi\)
−0.236043 + 0.971743i \(0.575851\pi\)
\(54\) 0.698899 0.0951081
\(55\) −0.901664 −0.121580
\(56\) −5.72572 −0.765132
\(57\) −0.813449 −0.107744
\(58\) 8.19122 1.07556
\(59\) 2.63490 0.343034 0.171517 0.985181i \(-0.445133\pi\)
0.171517 + 0.985181i \(0.445133\pi\)
\(60\) 0.209184 0.0270055
\(61\) 11.4908 1.47125 0.735625 0.677389i \(-0.236889\pi\)
0.735625 + 0.677389i \(0.236889\pi\)
\(62\) −5.68202 −0.721617
\(63\) 5.45884 0.687749
\(64\) 8.49925 1.06241
\(65\) −2.76107 −0.342469
\(66\) 5.00127 0.615614
\(67\) −10.2554 −1.25289 −0.626445 0.779465i \(-0.715491\pi\)
−0.626445 + 0.779465i \(0.715491\pi\)
\(68\) 0.449179 0.0544709
\(69\) 19.2171 2.31347
\(70\) 1.57402 0.188131
\(71\) 9.66931 1.14754 0.573768 0.819018i \(-0.305481\pi\)
0.573768 + 0.819018i \(0.305481\pi\)
\(72\) −8.14447 −0.959835
\(73\) −6.33688 −0.741676 −0.370838 0.928698i \(-0.620929\pi\)
−0.370838 + 0.928698i \(0.620929\pi\)
\(74\) −15.0615 −1.75086
\(75\) 11.1894 1.29204
\(76\) −0.0498154 −0.00571422
\(77\) −2.99229 −0.341003
\(78\) 15.3149 1.73407
\(79\) −0.667323 −0.0750798 −0.0375399 0.999295i \(-0.511952\pi\)
−0.0375399 + 0.999295i \(0.511952\pi\)
\(80\) −2.17448 −0.243114
\(81\) −9.59480 −1.06609
\(82\) −6.56556 −0.725045
\(83\) −2.26231 −0.248321 −0.124161 0.992262i \(-0.539624\pi\)
−0.124161 + 0.992262i \(0.539624\pi\)
\(84\) 0.694203 0.0757438
\(85\) −1.79991 −0.195228
\(86\) 16.9427 1.82698
\(87\) −14.4763 −1.55202
\(88\) 4.46443 0.475910
\(89\) 2.21979 0.235297 0.117649 0.993055i \(-0.462464\pi\)
0.117649 + 0.993055i \(0.462464\pi\)
\(90\) 2.23894 0.236005
\(91\) −9.16297 −0.960541
\(92\) 1.17685 0.122695
\(93\) 10.0418 1.04129
\(94\) 14.9853 1.54562
\(95\) 0.199616 0.0204802
\(96\) −2.00042 −0.204167
\(97\) 14.2878 1.45070 0.725352 0.688378i \(-0.241677\pi\)
0.725352 + 0.688378i \(0.241677\pi\)
\(98\) −4.30435 −0.434806
\(99\) −4.25634 −0.427778
\(100\) 0.685236 0.0685236
\(101\) −0.629834 −0.0626708 −0.0313354 0.999509i \(-0.509976\pi\)
−0.0313354 + 0.999509i \(0.509976\pi\)
\(102\) 9.98359 0.988523
\(103\) −4.51410 −0.444788 −0.222394 0.974957i \(-0.571387\pi\)
−0.222394 + 0.974957i \(0.571387\pi\)
\(104\) 13.6710 1.34055
\(105\) −2.78175 −0.271471
\(106\) −4.67801 −0.454368
\(107\) 3.43562 0.332134 0.166067 0.986115i \(-0.446893\pi\)
0.166067 + 0.986115i \(0.446893\pi\)
\(108\) −0.0756409 −0.00727855
\(109\) 9.90039 0.948285 0.474142 0.880448i \(-0.342758\pi\)
0.474142 + 0.880448i \(0.342758\pi\)
\(110\) −1.22729 −0.117017
\(111\) 26.6180 2.52647
\(112\) −7.21630 −0.681876
\(113\) −9.47684 −0.891506 −0.445753 0.895156i \(-0.647064\pi\)
−0.445753 + 0.895156i \(0.647064\pi\)
\(114\) −1.10721 −0.103700
\(115\) −4.71578 −0.439748
\(116\) −0.886524 −0.0823117
\(117\) −13.0337 −1.20497
\(118\) 3.58645 0.330159
\(119\) −5.97324 −0.547566
\(120\) 4.15031 0.378870
\(121\) −8.66687 −0.787897
\(122\) 15.6406 1.41603
\(123\) 11.6033 1.04623
\(124\) 0.614957 0.0552248
\(125\) −5.69733 −0.509585
\(126\) 7.43021 0.661936
\(127\) −11.3574 −1.00781 −0.503904 0.863760i \(-0.668104\pi\)
−0.503904 + 0.863760i \(0.668104\pi\)
\(128\) 9.90544 0.875525
\(129\) −29.9427 −2.63631
\(130\) −3.75819 −0.329615
\(131\) 15.9119 1.39023 0.695113 0.718900i \(-0.255354\pi\)
0.695113 + 0.718900i \(0.255354\pi\)
\(132\) −0.541281 −0.0471124
\(133\) 0.662452 0.0574419
\(134\) −13.9589 −1.20587
\(135\) 0.303102 0.0260868
\(136\) 8.91195 0.764193
\(137\) −15.0390 −1.28486 −0.642432 0.766343i \(-0.722075\pi\)
−0.642432 + 0.766343i \(0.722075\pi\)
\(138\) 26.1571 2.22664
\(139\) −16.6486 −1.41212 −0.706059 0.708153i \(-0.749529\pi\)
−0.706059 + 0.708153i \(0.749529\pi\)
\(140\) −0.170354 −0.0143975
\(141\) −26.4835 −2.23031
\(142\) 13.1612 1.10447
\(143\) 7.14451 0.597454
\(144\) −10.2647 −0.855393
\(145\) 3.55241 0.295011
\(146\) −8.62534 −0.713838
\(147\) 7.60706 0.627420
\(148\) 1.63008 0.133992
\(149\) −12.0657 −0.988463 −0.494232 0.869330i \(-0.664550\pi\)
−0.494232 + 0.869330i \(0.664550\pi\)
\(150\) 15.2303 1.24355
\(151\) −16.4897 −1.34192 −0.670958 0.741495i \(-0.734117\pi\)
−0.670958 + 0.741495i \(0.734117\pi\)
\(152\) −0.988364 −0.0801669
\(153\) −8.49655 −0.686905
\(154\) −4.07291 −0.328204
\(155\) −2.46420 −0.197930
\(156\) −1.65751 −0.132707
\(157\) 4.10607 0.327700 0.163850 0.986485i \(-0.447609\pi\)
0.163850 + 0.986485i \(0.447609\pi\)
\(158\) −0.908317 −0.0722618
\(159\) 8.26741 0.655648
\(160\) 0.490892 0.0388084
\(161\) −15.6499 −1.23339
\(162\) −13.0598 −1.02608
\(163\) 21.2459 1.66411 0.832053 0.554696i \(-0.187165\pi\)
0.832053 + 0.554696i \(0.187165\pi\)
\(164\) 0.710581 0.0554871
\(165\) 2.16897 0.168854
\(166\) −3.07931 −0.239001
\(167\) −5.96545 −0.461620 −0.230810 0.972999i \(-0.574138\pi\)
−0.230810 + 0.972999i \(0.574138\pi\)
\(168\) 13.7734 1.06264
\(169\) 8.87787 0.682913
\(170\) −2.44992 −0.187900
\(171\) 0.942295 0.0720591
\(172\) −1.83368 −0.139817
\(173\) −16.2621 −1.23638 −0.618192 0.786027i \(-0.712135\pi\)
−0.618192 + 0.786027i \(0.712135\pi\)
\(174\) −19.7042 −1.49377
\(175\) −9.11236 −0.688830
\(176\) 5.62666 0.424125
\(177\) −6.33831 −0.476416
\(178\) 3.02143 0.226466
\(179\) 9.36724 0.700140 0.350070 0.936723i \(-0.386158\pi\)
0.350070 + 0.936723i \(0.386158\pi\)
\(180\) −0.242317 −0.0180612
\(181\) 1.62885 0.121072 0.0605358 0.998166i \(-0.480719\pi\)
0.0605358 + 0.998166i \(0.480719\pi\)
\(182\) −12.4720 −0.924489
\(183\) −27.6415 −2.04332
\(184\) 23.3493 1.72134
\(185\) −6.53193 −0.480237
\(186\) 13.6682 1.00220
\(187\) 4.65742 0.340585
\(188\) −1.62184 −0.118285
\(189\) 1.00588 0.0731672
\(190\) 0.271704 0.0197115
\(191\) −8.41952 −0.609215 −0.304607 0.952478i \(-0.598525\pi\)
−0.304607 + 0.952478i \(0.598525\pi\)
\(192\) −20.4452 −1.47550
\(193\) −24.9514 −1.79604 −0.898022 0.439950i \(-0.854996\pi\)
−0.898022 + 0.439950i \(0.854996\pi\)
\(194\) 19.4476 1.39625
\(195\) 6.64182 0.475631
\(196\) 0.465854 0.0332753
\(197\) 2.36559 0.168541 0.0842705 0.996443i \(-0.473144\pi\)
0.0842705 + 0.996443i \(0.473144\pi\)
\(198\) −5.79345 −0.411722
\(199\) 11.7817 0.835180 0.417590 0.908636i \(-0.362875\pi\)
0.417590 + 0.908636i \(0.362875\pi\)
\(200\) 13.5954 0.961343
\(201\) 24.6695 1.74005
\(202\) −0.857288 −0.0603186
\(203\) 11.7891 0.827434
\(204\) −1.08051 −0.0756508
\(205\) −2.84738 −0.198870
\(206\) −6.14430 −0.428094
\(207\) −22.2610 −1.54725
\(208\) 17.2299 1.19468
\(209\) −0.516524 −0.0357287
\(210\) −3.78634 −0.261282
\(211\) −16.8124 −1.15742 −0.578708 0.815535i \(-0.696443\pi\)
−0.578708 + 0.815535i \(0.696443\pi\)
\(212\) 0.506294 0.0347724
\(213\) −23.2597 −1.59373
\(214\) 4.67633 0.319668
\(215\) 7.34778 0.501115
\(216\) −1.50076 −0.102114
\(217\) −8.17778 −0.555144
\(218\) 13.4758 0.912693
\(219\) 15.2435 1.03006
\(220\) 0.132827 0.00895522
\(221\) 14.2619 0.959362
\(222\) 36.2307 2.43165
\(223\) 8.32490 0.557476 0.278738 0.960367i \(-0.410084\pi\)
0.278738 + 0.960367i \(0.410084\pi\)
\(224\) 1.62909 0.108848
\(225\) −12.9617 −0.864116
\(226\) −12.8992 −0.858045
\(227\) 6.35919 0.422074 0.211037 0.977478i \(-0.432316\pi\)
0.211037 + 0.977478i \(0.432316\pi\)
\(228\) 0.119832 0.00793607
\(229\) 11.4619 0.757426 0.378713 0.925514i \(-0.376367\pi\)
0.378713 + 0.925514i \(0.376367\pi\)
\(230\) −6.41880 −0.423243
\(231\) 7.19802 0.473595
\(232\) −17.5891 −1.15478
\(233\) 21.5891 1.41435 0.707175 0.707039i \(-0.249969\pi\)
0.707175 + 0.707039i \(0.249969\pi\)
\(234\) −17.7407 −1.15974
\(235\) 6.49890 0.423941
\(236\) −0.388156 −0.0252668
\(237\) 1.60526 0.104273
\(238\) −8.13038 −0.527014
\(239\) −4.52568 −0.292742 −0.146371 0.989230i \(-0.546759\pi\)
−0.146371 + 0.989230i \(0.546759\pi\)
\(240\) 5.23077 0.337645
\(241\) −9.84982 −0.634483 −0.317241 0.948345i \(-0.602757\pi\)
−0.317241 + 0.948345i \(0.602757\pi\)
\(242\) −11.7968 −0.758325
\(243\) 21.5401 1.38180
\(244\) −1.69276 −0.108368
\(245\) −1.86673 −0.119261
\(246\) 15.7936 1.00696
\(247\) −1.58170 −0.100641
\(248\) 12.2011 0.774769
\(249\) 5.44205 0.344876
\(250\) −7.75483 −0.490459
\(251\) 22.0558 1.39215 0.696075 0.717969i \(-0.254928\pi\)
0.696075 + 0.717969i \(0.254928\pi\)
\(252\) −0.804161 −0.0506574
\(253\) 12.2025 0.767163
\(254\) −15.4590 −0.969982
\(255\) 4.32973 0.271138
\(256\) −3.51587 −0.219742
\(257\) 25.3211 1.57948 0.789742 0.613439i \(-0.210214\pi\)
0.789742 + 0.613439i \(0.210214\pi\)
\(258\) −40.7560 −2.53736
\(259\) −21.6771 −1.34695
\(260\) 0.406743 0.0252251
\(261\) 16.7693 1.03799
\(262\) 21.6582 1.33805
\(263\) 10.7243 0.661289 0.330644 0.943755i \(-0.392734\pi\)
0.330644 + 0.943755i \(0.392734\pi\)
\(264\) −10.7393 −0.660958
\(265\) −2.02878 −0.124627
\(266\) 0.901686 0.0552859
\(267\) −5.33975 −0.326788
\(268\) 1.51075 0.0922839
\(269\) −11.9609 −0.729266 −0.364633 0.931151i \(-0.618806\pi\)
−0.364633 + 0.931151i \(0.618806\pi\)
\(270\) 0.412562 0.0251077
\(271\) −11.7258 −0.712295 −0.356147 0.934430i \(-0.615910\pi\)
−0.356147 + 0.934430i \(0.615910\pi\)
\(272\) 11.2320 0.681040
\(273\) 22.0418 1.33403
\(274\) −20.4700 −1.23664
\(275\) 7.10504 0.428450
\(276\) −2.83094 −0.170403
\(277\) −0.916745 −0.0550819 −0.0275409 0.999621i \(-0.508768\pi\)
−0.0275409 + 0.999621i \(0.508768\pi\)
\(278\) −22.6610 −1.35912
\(279\) −11.6324 −0.696411
\(280\) −3.37991 −0.201988
\(281\) 21.7464 1.29728 0.648640 0.761096i \(-0.275338\pi\)
0.648640 + 0.761096i \(0.275338\pi\)
\(282\) −36.0475 −2.14660
\(283\) −8.06684 −0.479524 −0.239762 0.970832i \(-0.577069\pi\)
−0.239762 + 0.970832i \(0.577069\pi\)
\(284\) −1.42442 −0.0845238
\(285\) −0.480181 −0.0284435
\(286\) 9.72463 0.575030
\(287\) −9.44941 −0.557781
\(288\) 2.31727 0.136547
\(289\) −7.70280 −0.453106
\(290\) 4.83530 0.283939
\(291\) −34.3696 −2.01478
\(292\) 0.933509 0.0546295
\(293\) −2.16609 −0.126544 −0.0632722 0.997996i \(-0.520154\pi\)
−0.0632722 + 0.997996i \(0.520154\pi\)
\(294\) 10.3542 0.603871
\(295\) 1.55539 0.0905581
\(296\) 32.3417 1.87982
\(297\) −0.784302 −0.0455098
\(298\) −16.4231 −0.951363
\(299\) 37.3664 2.16095
\(300\) −1.64835 −0.0951676
\(301\) 24.3846 1.40550
\(302\) −22.4448 −1.29155
\(303\) 1.51508 0.0870391
\(304\) −1.24567 −0.0714438
\(305\) 6.78307 0.388397
\(306\) −11.5649 −0.661124
\(307\) 6.60523 0.376980 0.188490 0.982075i \(-0.439641\pi\)
0.188490 + 0.982075i \(0.439641\pi\)
\(308\) 0.440805 0.0251172
\(309\) 10.8588 0.617735
\(310\) −3.35411 −0.190501
\(311\) 3.84184 0.217851 0.108925 0.994050i \(-0.465259\pi\)
0.108925 + 0.994050i \(0.465259\pi\)
\(312\) −32.8858 −1.86179
\(313\) −25.4617 −1.43918 −0.719591 0.694398i \(-0.755671\pi\)
−0.719591 + 0.694398i \(0.755671\pi\)
\(314\) 5.58891 0.315401
\(315\) 3.22237 0.181560
\(316\) 0.0983058 0.00553014
\(317\) −8.23817 −0.462702 −0.231351 0.972870i \(-0.574314\pi\)
−0.231351 + 0.972870i \(0.574314\pi\)
\(318\) 11.2531 0.631040
\(319\) −9.19216 −0.514662
\(320\) 5.01713 0.280466
\(321\) −8.26446 −0.461277
\(322\) −21.3016 −1.18709
\(323\) −1.03109 −0.0573714
\(324\) 1.41344 0.0785247
\(325\) 21.7570 1.20686
\(326\) 28.9185 1.60165
\(327\) −23.8156 −1.31701
\(328\) 14.0983 0.778449
\(329\) 21.5675 1.18905
\(330\) 2.95226 0.162517
\(331\) −6.68597 −0.367494 −0.183747 0.982974i \(-0.558823\pi\)
−0.183747 + 0.982974i \(0.558823\pi\)
\(332\) 0.333270 0.0182906
\(333\) −30.8342 −1.68970
\(334\) −8.11977 −0.444294
\(335\) −6.05376 −0.330752
\(336\) 17.3590 0.947011
\(337\) −34.9491 −1.90380 −0.951899 0.306411i \(-0.900872\pi\)
−0.951899 + 0.306411i \(0.900872\pi\)
\(338\) 12.0840 0.657282
\(339\) 22.7968 1.23815
\(340\) 0.265151 0.0143799
\(341\) 6.37634 0.345298
\(342\) 1.28259 0.0693545
\(343\) −19.9080 −1.07493
\(344\) −36.3813 −1.96155
\(345\) 11.3439 0.610736
\(346\) −22.1349 −1.18998
\(347\) −16.1317 −0.865998 −0.432999 0.901394i \(-0.642545\pi\)
−0.432999 + 0.901394i \(0.642545\pi\)
\(348\) 2.13256 0.114317
\(349\) −16.2651 −0.870653 −0.435326 0.900273i \(-0.643367\pi\)
−0.435326 + 0.900273i \(0.643367\pi\)
\(350\) −12.4031 −0.662976
\(351\) −2.40169 −0.128192
\(352\) −1.27022 −0.0677032
\(353\) 15.5427 0.827254 0.413627 0.910446i \(-0.364262\pi\)
0.413627 + 0.910446i \(0.364262\pi\)
\(354\) −8.62729 −0.458535
\(355\) 5.70782 0.302940
\(356\) −0.327005 −0.0173312
\(357\) 14.3688 0.760476
\(358\) 12.7501 0.673862
\(359\) −16.5339 −0.872625 −0.436312 0.899795i \(-0.643716\pi\)
−0.436312 + 0.899795i \(0.643716\pi\)
\(360\) −4.80770 −0.253388
\(361\) −18.8856 −0.993982
\(362\) 2.21708 0.116527
\(363\) 20.8484 1.09425
\(364\) 1.34983 0.0707504
\(365\) −3.74068 −0.195796
\(366\) −37.6237 −1.96662
\(367\) 5.62500 0.293623 0.146811 0.989165i \(-0.453099\pi\)
0.146811 + 0.989165i \(0.453099\pi\)
\(368\) 29.4279 1.53403
\(369\) −13.4412 −0.699719
\(370\) −8.89083 −0.462212
\(371\) −6.73277 −0.349548
\(372\) −1.47929 −0.0766978
\(373\) −18.0602 −0.935122 −0.467561 0.883961i \(-0.654867\pi\)
−0.467561 + 0.883961i \(0.654867\pi\)
\(374\) 6.33938 0.327802
\(375\) 13.7051 0.707727
\(376\) −32.1782 −1.65946
\(377\) −28.1482 −1.44970
\(378\) 1.36914 0.0704211
\(379\) −2.42773 −0.124704 −0.0623521 0.998054i \(-0.519860\pi\)
−0.0623521 + 0.998054i \(0.519860\pi\)
\(380\) −0.0294062 −0.00150850
\(381\) 27.3206 1.39967
\(382\) −11.4601 −0.586349
\(383\) −16.2778 −0.831757 −0.415878 0.909420i \(-0.636526\pi\)
−0.415878 + 0.909420i \(0.636526\pi\)
\(384\) −23.8278 −1.21596
\(385\) −1.76636 −0.0900219
\(386\) −33.9623 −1.72863
\(387\) 34.6855 1.76316
\(388\) −2.10478 −0.106854
\(389\) −15.0862 −0.764899 −0.382450 0.923976i \(-0.624919\pi\)
−0.382450 + 0.923976i \(0.624919\pi\)
\(390\) 9.04041 0.457779
\(391\) 24.3587 1.23187
\(392\) 9.24280 0.466832
\(393\) −38.2764 −1.93079
\(394\) 3.21988 0.162215
\(395\) −0.393923 −0.0198204
\(396\) 0.627016 0.0315088
\(397\) −26.7523 −1.34266 −0.671329 0.741160i \(-0.734276\pi\)
−0.671329 + 0.741160i \(0.734276\pi\)
\(398\) 16.0364 0.803834
\(399\) −1.59354 −0.0797770
\(400\) 17.1348 0.856738
\(401\) −21.1930 −1.05833 −0.529165 0.848519i \(-0.677495\pi\)
−0.529165 + 0.848519i \(0.677495\pi\)
\(402\) 33.5785 1.67474
\(403\) 19.5256 0.972639
\(404\) 0.0927831 0.00461613
\(405\) −5.66384 −0.281438
\(406\) 16.0466 0.796378
\(407\) 16.9019 0.837797
\(408\) −21.4379 −1.06133
\(409\) 14.2178 0.703023 0.351511 0.936184i \(-0.385668\pi\)
0.351511 + 0.936184i \(0.385668\pi\)
\(410\) −3.87567 −0.191406
\(411\) 36.1766 1.78446
\(412\) 0.664989 0.0327617
\(413\) 5.16176 0.253993
\(414\) −30.3002 −1.48917
\(415\) −1.33545 −0.0655547
\(416\) −3.88968 −0.190707
\(417\) 40.0487 1.96119
\(418\) −0.703058 −0.0343877
\(419\) 40.7220 1.98940 0.994700 0.102818i \(-0.0327858\pi\)
0.994700 + 0.102818i \(0.0327858\pi\)
\(420\) 0.409790 0.0199957
\(421\) 30.2685 1.47519 0.737597 0.675241i \(-0.235960\pi\)
0.737597 + 0.675241i \(0.235960\pi\)
\(422\) −22.8840 −1.11397
\(423\) 30.6783 1.49163
\(424\) 10.0451 0.487835
\(425\) 14.1832 0.687984
\(426\) −31.6596 −1.53391
\(427\) 22.5105 1.08936
\(428\) −0.506113 −0.0244639
\(429\) −17.1863 −0.829762
\(430\) 10.0013 0.482306
\(431\) 14.3367 0.690572 0.345286 0.938497i \(-0.387782\pi\)
0.345286 + 0.938497i \(0.387782\pi\)
\(432\) −1.89145 −0.0910024
\(433\) −26.4985 −1.27344 −0.636719 0.771096i \(-0.719709\pi\)
−0.636719 + 0.771096i \(0.719709\pi\)
\(434\) −11.1311 −0.534308
\(435\) −8.54540 −0.409720
\(436\) −1.45846 −0.0698477
\(437\) −2.70146 −0.129228
\(438\) 20.7485 0.991400
\(439\) 19.2176 0.917208 0.458604 0.888641i \(-0.348350\pi\)
0.458604 + 0.888641i \(0.348350\pi\)
\(440\) 2.63537 0.125636
\(441\) −8.81198 −0.419618
\(442\) 19.4124 0.923355
\(443\) −5.97322 −0.283796 −0.141898 0.989881i \(-0.545321\pi\)
−0.141898 + 0.989881i \(0.545321\pi\)
\(444\) −3.92120 −0.186092
\(445\) 1.31035 0.0621164
\(446\) 11.3313 0.536553
\(447\) 29.0244 1.37281
\(448\) 16.6500 0.786639
\(449\) 1.84304 0.0869784 0.0434892 0.999054i \(-0.486153\pi\)
0.0434892 + 0.999054i \(0.486153\pi\)
\(450\) −17.6427 −0.831683
\(451\) 7.36785 0.346938
\(452\) 1.39607 0.0656655
\(453\) 39.6665 1.86369
\(454\) 8.65571 0.406233
\(455\) −5.40893 −0.253575
\(456\) 2.37753 0.111338
\(457\) 27.3399 1.27891 0.639453 0.768830i \(-0.279161\pi\)
0.639453 + 0.768830i \(0.279161\pi\)
\(458\) 15.6012 0.728998
\(459\) −1.56563 −0.0730774
\(460\) 0.694698 0.0323905
\(461\) 35.1434 1.63679 0.818395 0.574656i \(-0.194864\pi\)
0.818395 + 0.574656i \(0.194864\pi\)
\(462\) 9.79748 0.455820
\(463\) −18.6765 −0.867969 −0.433984 0.900920i \(-0.642893\pi\)
−0.433984 + 0.900920i \(0.642893\pi\)
\(464\) −22.1681 −1.02913
\(465\) 5.92770 0.274891
\(466\) 29.3857 1.36126
\(467\) −16.9925 −0.786318 −0.393159 0.919470i \(-0.628618\pi\)
−0.393159 + 0.919470i \(0.628618\pi\)
\(468\) 1.92005 0.0887542
\(469\) −20.0902 −0.927680
\(470\) 8.84588 0.408030
\(471\) −9.87725 −0.455120
\(472\) −7.70123 −0.354478
\(473\) −19.0130 −0.874220
\(474\) 2.18498 0.100359
\(475\) −1.57296 −0.0721723
\(476\) 0.879940 0.0403320
\(477\) −9.57692 −0.438497
\(478\) −6.16006 −0.281755
\(479\) −11.4554 −0.523409 −0.261705 0.965148i \(-0.584285\pi\)
−0.261705 + 0.965148i \(0.584285\pi\)
\(480\) −1.18085 −0.0538983
\(481\) 51.7570 2.35992
\(482\) −13.4069 −0.610669
\(483\) 37.6463 1.71296
\(484\) 1.27675 0.0580340
\(485\) 8.43411 0.382973
\(486\) 29.3190 1.32994
\(487\) −14.2183 −0.644295 −0.322147 0.946690i \(-0.604405\pi\)
−0.322147 + 0.946690i \(0.604405\pi\)
\(488\) −33.5852 −1.52033
\(489\) −51.1075 −2.31116
\(490\) −2.54087 −0.114785
\(491\) −4.53402 −0.204617 −0.102309 0.994753i \(-0.532623\pi\)
−0.102309 + 0.994753i \(0.532623\pi\)
\(492\) −1.70932 −0.0770621
\(493\) −18.3495 −0.826419
\(494\) −2.15290 −0.0968636
\(495\) −2.51253 −0.112930
\(496\) 15.3774 0.690465
\(497\) 18.9421 0.849671
\(498\) 7.40736 0.331932
\(499\) −38.1411 −1.70743 −0.853715 0.520741i \(-0.825656\pi\)
−0.853715 + 0.520741i \(0.825656\pi\)
\(500\) 0.839295 0.0375344
\(501\) 14.3500 0.641112
\(502\) 30.0209 1.33990
\(503\) −1.00000 −0.0445878
\(504\) −15.9550 −0.710692
\(505\) −0.371793 −0.0165446
\(506\) 16.6092 0.738369
\(507\) −21.3559 −0.948451
\(508\) 1.67310 0.0742319
\(509\) 34.3114 1.52083 0.760413 0.649440i \(-0.224997\pi\)
0.760413 + 0.649440i \(0.224997\pi\)
\(510\) 5.89334 0.260962
\(511\) −12.4139 −0.549160
\(512\) −24.5965 −1.08702
\(513\) 0.173634 0.00766612
\(514\) 34.4654 1.52020
\(515\) −2.66469 −0.117420
\(516\) 4.41097 0.194182
\(517\) −16.8165 −0.739587
\(518\) −29.5054 −1.29639
\(519\) 39.1189 1.71713
\(520\) 8.07001 0.353893
\(521\) 23.8175 1.04346 0.521732 0.853110i \(-0.325286\pi\)
0.521732 + 0.853110i \(0.325286\pi\)
\(522\) 22.8252 0.999032
\(523\) −2.16718 −0.0947643 −0.0473821 0.998877i \(-0.515088\pi\)
−0.0473821 + 0.998877i \(0.515088\pi\)
\(524\) −2.34404 −0.102400
\(525\) 21.9200 0.956667
\(526\) 14.5972 0.636469
\(527\) 12.7285 0.554463
\(528\) −13.5351 −0.589038
\(529\) 40.8200 1.77478
\(530\) −2.76144 −0.119949
\(531\) 7.34226 0.318627
\(532\) −0.0975882 −0.00423099
\(533\) 22.5618 0.977259
\(534\) −7.26812 −0.314522
\(535\) 2.02805 0.0876804
\(536\) 29.9742 1.29469
\(537\) −22.5331 −0.972376
\(538\) −16.2803 −0.701895
\(539\) 4.83033 0.208057
\(540\) −0.0446510 −0.00192147
\(541\) 13.5019 0.580493 0.290246 0.956952i \(-0.406263\pi\)
0.290246 + 0.956952i \(0.406263\pi\)
\(542\) −15.9605 −0.685560
\(543\) −3.91824 −0.168148
\(544\) −2.53564 −0.108714
\(545\) 5.84423 0.250339
\(546\) 30.0018 1.28396
\(547\) −1.05880 −0.0452710 −0.0226355 0.999744i \(-0.507206\pi\)
−0.0226355 + 0.999744i \(0.507206\pi\)
\(548\) 2.21544 0.0946390
\(549\) 32.0197 1.36657
\(550\) 9.67092 0.412369
\(551\) 2.03502 0.0866947
\(552\) −56.1674 −2.39064
\(553\) −1.30728 −0.0555914
\(554\) −1.24781 −0.0530145
\(555\) 15.7127 0.666967
\(556\) 2.45257 0.104012
\(557\) −9.20176 −0.389891 −0.194946 0.980814i \(-0.562453\pi\)
−0.194946 + 0.980814i \(0.562453\pi\)
\(558\) −15.8332 −0.670273
\(559\) −58.2216 −2.46251
\(560\) −4.25980 −0.180010
\(561\) −11.2035 −0.473014
\(562\) 29.5997 1.24859
\(563\) −29.6502 −1.24961 −0.624804 0.780782i \(-0.714821\pi\)
−0.624804 + 0.780782i \(0.714821\pi\)
\(564\) 3.90137 0.164277
\(565\) −5.59420 −0.235350
\(566\) −10.9800 −0.461526
\(567\) −18.7962 −0.789366
\(568\) −28.2613 −1.18582
\(569\) 22.6715 0.950439 0.475220 0.879867i \(-0.342369\pi\)
0.475220 + 0.879867i \(0.342369\pi\)
\(570\) −0.653591 −0.0273759
\(571\) −31.2013 −1.30573 −0.652866 0.757473i \(-0.726434\pi\)
−0.652866 + 0.757473i \(0.726434\pi\)
\(572\) −1.05248 −0.0440065
\(573\) 20.2533 0.846096
\(574\) −12.8619 −0.536846
\(575\) 37.1600 1.54968
\(576\) 23.6836 0.986815
\(577\) −3.98191 −0.165769 −0.0828845 0.996559i \(-0.526413\pi\)
−0.0828845 + 0.996559i \(0.526413\pi\)
\(578\) −10.4845 −0.436099
\(579\) 60.0213 2.49440
\(580\) −0.523318 −0.0217296
\(581\) −4.43187 −0.183865
\(582\) −46.7816 −1.93916
\(583\) 5.24964 0.217418
\(584\) 18.5213 0.766418
\(585\) −7.69385 −0.318102
\(586\) −2.94834 −0.121795
\(587\) −12.7494 −0.526225 −0.263113 0.964765i \(-0.584749\pi\)
−0.263113 + 0.964765i \(0.584749\pi\)
\(588\) −1.12062 −0.0462137
\(589\) −1.41163 −0.0581654
\(590\) 2.11709 0.0871592
\(591\) −5.69048 −0.234075
\(592\) 40.7612 1.67528
\(593\) −3.27771 −0.134600 −0.0672998 0.997733i \(-0.521438\pi\)
−0.0672998 + 0.997733i \(0.521438\pi\)
\(594\) −1.06754 −0.0438017
\(595\) −3.52602 −0.144553
\(596\) 1.77745 0.0728070
\(597\) −28.3411 −1.15992
\(598\) 50.8606 2.07985
\(599\) 10.9237 0.446332 0.223166 0.974780i \(-0.428361\pi\)
0.223166 + 0.974780i \(0.428361\pi\)
\(600\) −32.7042 −1.33514
\(601\) −29.7446 −1.21331 −0.606653 0.794967i \(-0.707488\pi\)
−0.606653 + 0.794967i \(0.707488\pi\)
\(602\) 33.1907 1.35275
\(603\) −28.5770 −1.16375
\(604\) 2.42916 0.0988413
\(605\) −5.11608 −0.207998
\(606\) 2.06223 0.0837723
\(607\) −43.8414 −1.77947 −0.889735 0.456478i \(-0.849111\pi\)
−0.889735 + 0.456478i \(0.849111\pi\)
\(608\) 0.281210 0.0114046
\(609\) −28.3590 −1.14917
\(610\) 9.23267 0.373820
\(611\) −51.4953 −2.08328
\(612\) 1.25166 0.0505952
\(613\) −32.3195 −1.30537 −0.652687 0.757628i \(-0.726358\pi\)
−0.652687 + 0.757628i \(0.726358\pi\)
\(614\) 8.99061 0.362831
\(615\) 6.84945 0.276196
\(616\) 8.74581 0.352379
\(617\) 40.9696 1.64937 0.824687 0.565589i \(-0.191351\pi\)
0.824687 + 0.565589i \(0.191351\pi\)
\(618\) 14.7803 0.594549
\(619\) 22.5775 0.907466 0.453733 0.891138i \(-0.350092\pi\)
0.453733 + 0.891138i \(0.350092\pi\)
\(620\) 0.363011 0.0145789
\(621\) −4.10196 −0.164606
\(622\) 5.22926 0.209674
\(623\) 4.34856 0.174221
\(624\) −41.4470 −1.65921
\(625\) 19.8946 0.795782
\(626\) −34.6568 −1.38517
\(627\) 1.24251 0.0496211
\(628\) −0.604880 −0.0241373
\(629\) 33.7398 1.34529
\(630\) 4.38607 0.174745
\(631\) 33.9719 1.35240 0.676199 0.736719i \(-0.263626\pi\)
0.676199 + 0.736719i \(0.263626\pi\)
\(632\) 1.95044 0.0775844
\(633\) 40.4427 1.60745
\(634\) −11.2132 −0.445335
\(635\) −6.70432 −0.266053
\(636\) −1.21790 −0.0482930
\(637\) 14.7914 0.586057
\(638\) −12.5118 −0.495345
\(639\) 26.9440 1.06589
\(640\) 5.84721 0.231131
\(641\) −4.53431 −0.179095 −0.0895473 0.995983i \(-0.528542\pi\)
−0.0895473 + 0.995983i \(0.528542\pi\)
\(642\) −11.2490 −0.443964
\(643\) 38.3929 1.51407 0.757033 0.653376i \(-0.226648\pi\)
0.757033 + 0.653376i \(0.226648\pi\)
\(644\) 2.30545 0.0908473
\(645\) −17.6753 −0.695963
\(646\) −1.40345 −0.0552181
\(647\) −0.255028 −0.0100262 −0.00501309 0.999987i \(-0.501596\pi\)
−0.00501309 + 0.999987i \(0.501596\pi\)
\(648\) 28.0435 1.10165
\(649\) −4.02470 −0.157983
\(650\) 29.6142 1.16157
\(651\) 19.6719 0.771001
\(652\) −3.12981 −0.122573
\(653\) −2.04379 −0.0799797 −0.0399898 0.999200i \(-0.512733\pi\)
−0.0399898 + 0.999200i \(0.512733\pi\)
\(654\) −32.4163 −1.26758
\(655\) 9.39282 0.367008
\(656\) 17.7685 0.693745
\(657\) −17.6580 −0.688904
\(658\) 29.3562 1.14442
\(659\) 10.0768 0.392534 0.196267 0.980550i \(-0.437118\pi\)
0.196267 + 0.980550i \(0.437118\pi\)
\(660\) −0.319519 −0.0124373
\(661\) −5.53010 −0.215096 −0.107548 0.994200i \(-0.534300\pi\)
−0.107548 + 0.994200i \(0.534300\pi\)
\(662\) −9.10050 −0.353701
\(663\) −34.3074 −1.33239
\(664\) 6.61225 0.256605
\(665\) 0.391047 0.0151642
\(666\) −41.9695 −1.62628
\(667\) −48.0757 −1.86150
\(668\) 0.878791 0.0340015
\(669\) −20.0257 −0.774240
\(670\) −8.23998 −0.318338
\(671\) −17.5518 −0.677579
\(672\) −3.91881 −0.151171
\(673\) −3.16280 −0.121917 −0.0609585 0.998140i \(-0.519416\pi\)
−0.0609585 + 0.998140i \(0.519416\pi\)
\(674\) −47.5704 −1.83234
\(675\) −2.38842 −0.0919303
\(676\) −1.30783 −0.0503012
\(677\) −47.8219 −1.83794 −0.918972 0.394323i \(-0.870979\pi\)
−0.918972 + 0.394323i \(0.870979\pi\)
\(678\) 31.0295 1.19168
\(679\) 27.9897 1.07415
\(680\) 5.26075 0.201740
\(681\) −15.2972 −0.586190
\(682\) 8.67906 0.332338
\(683\) 13.2468 0.506873 0.253437 0.967352i \(-0.418439\pi\)
0.253437 + 0.967352i \(0.418439\pi\)
\(684\) −0.138813 −0.00530764
\(685\) −8.87754 −0.339193
\(686\) −27.0974 −1.03458
\(687\) −27.5720 −1.05194
\(688\) −45.8524 −1.74811
\(689\) 16.0754 0.612425
\(690\) 15.4406 0.587813
\(691\) −7.80850 −0.297049 −0.148525 0.988909i \(-0.547452\pi\)
−0.148525 + 0.988909i \(0.547452\pi\)
\(692\) 2.39563 0.0910681
\(693\) −8.33815 −0.316740
\(694\) −21.9575 −0.833494
\(695\) −9.82774 −0.372787
\(696\) 42.3111 1.60380
\(697\) 14.7078 0.557097
\(698\) −22.1390 −0.837975
\(699\) −51.9331 −1.96429
\(700\) 1.34237 0.0507370
\(701\) 7.68972 0.290437 0.145218 0.989400i \(-0.453612\pi\)
0.145218 + 0.989400i \(0.453612\pi\)
\(702\) −3.26902 −0.123381
\(703\) −3.74185 −0.141127
\(704\) −12.9823 −0.489288
\(705\) −15.6333 −0.588783
\(706\) 21.1557 0.796205
\(707\) −1.23384 −0.0464034
\(708\) 0.933719 0.0350913
\(709\) 31.0522 1.16619 0.583096 0.812403i \(-0.301841\pi\)
0.583096 + 0.812403i \(0.301841\pi\)
\(710\) 7.76911 0.291569
\(711\) −1.85953 −0.0697377
\(712\) −6.48796 −0.243146
\(713\) 33.3488 1.24892
\(714\) 19.5578 0.731933
\(715\) 4.21742 0.157723
\(716\) −1.37992 −0.0515701
\(717\) 10.8866 0.406569
\(718\) −22.5048 −0.839873
\(719\) −30.4494 −1.13557 −0.567786 0.823176i \(-0.692200\pi\)
−0.567786 + 0.823176i \(0.692200\pi\)
\(720\) −6.05929 −0.225817
\(721\) −8.84312 −0.329335
\(722\) −25.7059 −0.956675
\(723\) 23.6940 0.881189
\(724\) −0.239952 −0.00891774
\(725\) −27.9927 −1.03962
\(726\) 28.3774 1.05318
\(727\) −6.14587 −0.227938 −0.113969 0.993484i \(-0.536356\pi\)
−0.113969 + 0.993484i \(0.536356\pi\)
\(728\) 26.7814 0.992584
\(729\) −23.0309 −0.852995
\(730\) −5.09156 −0.188447
\(731\) −37.9540 −1.40378
\(732\) 4.07196 0.150504
\(733\) −25.4237 −0.939046 −0.469523 0.882920i \(-0.655574\pi\)
−0.469523 + 0.882920i \(0.655574\pi\)
\(734\) 7.65638 0.282602
\(735\) 4.49047 0.165633
\(736\) −6.64338 −0.244878
\(737\) 15.6646 0.577014
\(738\) −18.2952 −0.673457
\(739\) −19.8464 −0.730063 −0.365032 0.930995i \(-0.618942\pi\)
−0.365032 + 0.930995i \(0.618942\pi\)
\(740\) 0.962242 0.0353727
\(741\) 3.80481 0.139773
\(742\) −9.16420 −0.336428
\(743\) −28.7154 −1.05346 −0.526732 0.850031i \(-0.676583\pi\)
−0.526732 + 0.850031i \(0.676583\pi\)
\(744\) −29.3500 −1.07602
\(745\) −7.12243 −0.260946
\(746\) −24.5824 −0.900025
\(747\) −6.30404 −0.230653
\(748\) −0.686102 −0.0250864
\(749\) 6.73036 0.245922
\(750\) 18.6544 0.681164
\(751\) 24.0374 0.877139 0.438569 0.898697i \(-0.355485\pi\)
0.438569 + 0.898697i \(0.355485\pi\)
\(752\) −40.5551 −1.47889
\(753\) −53.0558 −1.93346
\(754\) −38.3134 −1.39529
\(755\) −9.73394 −0.354254
\(756\) −0.148180 −0.00538926
\(757\) −23.2463 −0.844902 −0.422451 0.906386i \(-0.638830\pi\)
−0.422451 + 0.906386i \(0.638830\pi\)
\(758\) −3.30447 −0.120024
\(759\) −29.3534 −1.06546
\(760\) −0.583434 −0.0211634
\(761\) 44.3555 1.60789 0.803943 0.594706i \(-0.202732\pi\)
0.803943 + 0.594706i \(0.202732\pi\)
\(762\) 37.1869 1.34714
\(763\) 19.3948 0.702140
\(764\) 1.24031 0.0448728
\(765\) −5.01553 −0.181337
\(766\) −22.1563 −0.800539
\(767\) −12.3244 −0.445009
\(768\) 8.45752 0.305185
\(769\) −17.4534 −0.629385 −0.314693 0.949194i \(-0.601901\pi\)
−0.314693 + 0.949194i \(0.601901\pi\)
\(770\) −2.40425 −0.0866431
\(771\) −60.9104 −2.19364
\(772\) 3.67569 0.132291
\(773\) −1.73607 −0.0624422 −0.0312211 0.999513i \(-0.509940\pi\)
−0.0312211 + 0.999513i \(0.509940\pi\)
\(774\) 47.2116 1.69699
\(775\) 19.4177 0.697506
\(776\) −41.7600 −1.49910
\(777\) 52.1447 1.87068
\(778\) −20.5343 −0.736190
\(779\) −1.63114 −0.0584417
\(780\) −0.978431 −0.0350334
\(781\) −14.7695 −0.528493
\(782\) 33.1555 1.18564
\(783\) 3.09002 0.110428
\(784\) 11.6490 0.416035
\(785\) 2.42383 0.0865100
\(786\) −52.0993 −1.85832
\(787\) −16.1636 −0.576169 −0.288085 0.957605i \(-0.593018\pi\)
−0.288085 + 0.957605i \(0.593018\pi\)
\(788\) −0.348483 −0.0124142
\(789\) −25.7976 −0.918418
\(790\) −0.536182 −0.0190765
\(791\) −18.5651 −0.660099
\(792\) 12.4403 0.442048
\(793\) −53.7469 −1.90861
\(794\) −36.4134 −1.29226
\(795\) 4.88028 0.173086
\(796\) −1.73560 −0.0615167
\(797\) −26.8208 −0.950040 −0.475020 0.879975i \(-0.657559\pi\)
−0.475020 + 0.879975i \(0.657559\pi\)
\(798\) −2.16903 −0.0767827
\(799\) −33.5692 −1.18759
\(800\) −3.86819 −0.136761
\(801\) 6.18554 0.218555
\(802\) −28.8466 −1.01861
\(803\) 9.67933 0.341576
\(804\) −3.63415 −0.128167
\(805\) −9.23819 −0.325603
\(806\) 26.5770 0.936133
\(807\) 28.7721 1.01283
\(808\) 1.84087 0.0647615
\(809\) 33.4065 1.17451 0.587255 0.809402i \(-0.300209\pi\)
0.587255 + 0.809402i \(0.300209\pi\)
\(810\) −7.70924 −0.270875
\(811\) 34.8499 1.22375 0.611873 0.790956i \(-0.290416\pi\)
0.611873 + 0.790956i \(0.290416\pi\)
\(812\) −1.73670 −0.0609462
\(813\) 28.2068 0.989256
\(814\) 23.0058 0.806352
\(815\) 12.5415 0.439310
\(816\) −27.0188 −0.945849
\(817\) 4.20922 0.147262
\(818\) 19.3523 0.676636
\(819\) −25.5330 −0.892197
\(820\) 0.419458 0.0146481
\(821\) −24.4182 −0.852201 −0.426101 0.904676i \(-0.640113\pi\)
−0.426101 + 0.904676i \(0.640113\pi\)
\(822\) 49.2412 1.71748
\(823\) 4.51039 0.157222 0.0786112 0.996905i \(-0.474951\pi\)
0.0786112 + 0.996905i \(0.474951\pi\)
\(824\) 13.1937 0.459626
\(825\) −17.0914 −0.595045
\(826\) 7.02584 0.244460
\(827\) −2.91499 −0.101364 −0.0506821 0.998715i \(-0.516140\pi\)
−0.0506821 + 0.998715i \(0.516140\pi\)
\(828\) 3.27935 0.113965
\(829\) 35.2382 1.22387 0.611937 0.790906i \(-0.290390\pi\)
0.611937 + 0.790906i \(0.290390\pi\)
\(830\) −1.81773 −0.0630943
\(831\) 2.20525 0.0764994
\(832\) −39.7542 −1.37823
\(833\) 9.64236 0.334088
\(834\) 54.5116 1.88758
\(835\) −3.52142 −0.121864
\(836\) 0.0760910 0.00263166
\(837\) −2.14346 −0.0740888
\(838\) 55.4281 1.91473
\(839\) −14.4876 −0.500166 −0.250083 0.968224i \(-0.580458\pi\)
−0.250083 + 0.968224i \(0.580458\pi\)
\(840\) 8.13046 0.280527
\(841\) 7.21557 0.248813
\(842\) 41.1994 1.41983
\(843\) −52.3114 −1.80170
\(844\) 2.47670 0.0852515
\(845\) 5.24063 0.180283
\(846\) 41.7573 1.43564
\(847\) −16.9784 −0.583384
\(848\) 12.6602 0.434753
\(849\) 19.4050 0.665977
\(850\) 19.3052 0.662163
\(851\) 88.3985 3.03026
\(852\) 3.42648 0.117389
\(853\) 4.92917 0.168772 0.0843858 0.996433i \(-0.473107\pi\)
0.0843858 + 0.996433i \(0.473107\pi\)
\(854\) 30.6398 1.04847
\(855\) 0.556239 0.0190230
\(856\) −10.0416 −0.343213
\(857\) 52.5344 1.79454 0.897270 0.441482i \(-0.145547\pi\)
0.897270 + 0.441482i \(0.145547\pi\)
\(858\) −23.3928 −0.798618
\(859\) 32.5431 1.11036 0.555179 0.831731i \(-0.312650\pi\)
0.555179 + 0.831731i \(0.312650\pi\)
\(860\) −1.08243 −0.0369105
\(861\) 22.7308 0.774663
\(862\) 19.5141 0.664653
\(863\) 51.5604 1.75514 0.877568 0.479451i \(-0.159164\pi\)
0.877568 + 0.479451i \(0.159164\pi\)
\(864\) 0.426997 0.0145267
\(865\) −9.59956 −0.326395
\(866\) −36.0681 −1.22564
\(867\) 18.5293 0.629287
\(868\) 1.20470 0.0408901
\(869\) 1.01931 0.0345777
\(870\) −11.6314 −0.394342
\(871\) 47.9682 1.62534
\(872\) −28.9367 −0.979919
\(873\) 39.8135 1.34748
\(874\) −3.67705 −0.124378
\(875\) −11.1611 −0.377313
\(876\) −2.24558 −0.0758710
\(877\) 35.8616 1.21096 0.605481 0.795860i \(-0.292981\pi\)
0.605481 + 0.795860i \(0.292981\pi\)
\(878\) 26.1578 0.882783
\(879\) 5.21058 0.175749
\(880\) 3.32143 0.111965
\(881\) −41.2432 −1.38952 −0.694759 0.719243i \(-0.744489\pi\)
−0.694759 + 0.719243i \(0.744489\pi\)
\(882\) −11.9943 −0.403869
\(883\) 24.3041 0.817898 0.408949 0.912557i \(-0.365895\pi\)
0.408949 + 0.912557i \(0.365895\pi\)
\(884\) −2.10098 −0.0706636
\(885\) −3.74152 −0.125770
\(886\) −8.13035 −0.273144
\(887\) 51.7010 1.73595 0.867975 0.496608i \(-0.165421\pi\)
0.867975 + 0.496608i \(0.165421\pi\)
\(888\) −77.7987 −2.61075
\(889\) −22.2492 −0.746213
\(890\) 1.78356 0.0597850
\(891\) 14.6557 0.490983
\(892\) −1.22637 −0.0410619
\(893\) 3.72294 0.124583
\(894\) 39.5061 1.32128
\(895\) 5.52951 0.184831
\(896\) 19.4047 0.648266
\(897\) −89.8857 −3.00120
\(898\) 2.50862 0.0837138
\(899\) −25.1217 −0.837856
\(900\) 1.90944 0.0636480
\(901\) 10.4794 0.349119
\(902\) 10.0286 0.333917
\(903\) −58.6577 −1.95201
\(904\) 27.6987 0.921246
\(905\) 0.961515 0.0319618
\(906\) 53.9914 1.79374
\(907\) −5.06194 −0.168079 −0.0840395 0.996462i \(-0.526782\pi\)
−0.0840395 + 0.996462i \(0.526782\pi\)
\(908\) −0.936796 −0.0310887
\(909\) −1.75506 −0.0582117
\(910\) −7.36228 −0.244057
\(911\) −40.4144 −1.33899 −0.669495 0.742817i \(-0.733489\pi\)
−0.669495 + 0.742817i \(0.733489\pi\)
\(912\) 2.99648 0.0992233
\(913\) 3.45559 0.114363
\(914\) 37.2133 1.23091
\(915\) −16.3168 −0.539418
\(916\) −1.68850 −0.0557896
\(917\) 31.1713 1.02937
\(918\) −2.13104 −0.0703347
\(919\) 31.2276 1.03010 0.515052 0.857159i \(-0.327773\pi\)
0.515052 + 0.857159i \(0.327773\pi\)
\(920\) 13.7832 0.454418
\(921\) −15.8890 −0.523562
\(922\) 47.8349 1.57536
\(923\) −45.2270 −1.48867
\(924\) −1.06037 −0.0348835
\(925\) 51.4711 1.69236
\(926\) −25.4212 −0.835391
\(927\) −12.5788 −0.413141
\(928\) 5.00447 0.164280
\(929\) −51.3773 −1.68563 −0.842817 0.538201i \(-0.819104\pi\)
−0.842817 + 0.538201i \(0.819104\pi\)
\(930\) 8.06840 0.264573
\(931\) −1.06937 −0.0350472
\(932\) −3.18037 −0.104176
\(933\) −9.24164 −0.302558
\(934\) −23.1290 −0.756805
\(935\) 2.74929 0.0899114
\(936\) 38.0948 1.24517
\(937\) 22.6093 0.738614 0.369307 0.929307i \(-0.379595\pi\)
0.369307 + 0.929307i \(0.379595\pi\)
\(938\) −27.3455 −0.892861
\(939\) 61.2488 1.99878
\(940\) −0.957377 −0.0312262
\(941\) −51.2485 −1.67065 −0.835326 0.549754i \(-0.814721\pi\)
−0.835326 + 0.549754i \(0.814721\pi\)
\(942\) −13.4443 −0.438038
\(943\) 38.5344 1.25485
\(944\) −9.70610 −0.315906
\(945\) 0.593776 0.0193155
\(946\) −25.8793 −0.841408
\(947\) −54.1474 −1.75955 −0.879777 0.475387i \(-0.842308\pi\)
−0.879777 + 0.475387i \(0.842308\pi\)
\(948\) −0.236477 −0.00768042
\(949\) 29.6400 0.962155
\(950\) −2.14101 −0.0694635
\(951\) 19.8171 0.642614
\(952\) 17.4585 0.565833
\(953\) −34.0986 −1.10456 −0.552281 0.833658i \(-0.686242\pi\)
−0.552281 + 0.833658i \(0.686242\pi\)
\(954\) −13.0355 −0.422039
\(955\) −4.97006 −0.160827
\(956\) 0.666695 0.0215624
\(957\) 22.1120 0.714778
\(958\) −15.5923 −0.503764
\(959\) −29.4613 −0.951354
\(960\) −12.0688 −0.389520
\(961\) −13.5738 −0.437863
\(962\) 70.4482 2.27134
\(963\) 9.57350 0.308502
\(964\) 1.45101 0.0467340
\(965\) −14.7289 −0.474140
\(966\) 51.2416 1.64867
\(967\) 20.9385 0.673336 0.336668 0.941623i \(-0.390700\pi\)
0.336668 + 0.941623i \(0.390700\pi\)
\(968\) 25.3314 0.814181
\(969\) 2.48031 0.0796791
\(970\) 11.4800 0.368599
\(971\) −5.00722 −0.160689 −0.0803447 0.996767i \(-0.525602\pi\)
−0.0803447 + 0.996767i \(0.525602\pi\)
\(972\) −3.17315 −0.101779
\(973\) −32.6146 −1.04558
\(974\) −19.3531 −0.620113
\(975\) −52.3371 −1.67613
\(976\) −42.3284 −1.35490
\(977\) 12.5547 0.401662 0.200831 0.979626i \(-0.435636\pi\)
0.200831 + 0.979626i \(0.435636\pi\)
\(978\) −69.5642 −2.22442
\(979\) −3.39064 −0.108365
\(980\) 0.274995 0.00878440
\(981\) 27.5879 0.880813
\(982\) −6.17140 −0.196937
\(983\) 1.43810 0.0458683 0.0229341 0.999737i \(-0.492699\pi\)
0.0229341 + 0.999737i \(0.492699\pi\)
\(984\) −33.9138 −1.08113
\(985\) 1.39641 0.0444934
\(986\) −24.9761 −0.795401
\(987\) −51.8810 −1.65139
\(988\) 0.233005 0.00741289
\(989\) −99.4397 −3.16200
\(990\) −3.41989 −0.108691
\(991\) −41.6595 −1.32336 −0.661678 0.749788i \(-0.730155\pi\)
−0.661678 + 0.749788i \(0.730155\pi\)
\(992\) −3.47146 −0.110219
\(993\) 16.0833 0.510387
\(994\) 25.7828 0.817781
\(995\) 6.95475 0.220480
\(996\) −0.801689 −0.0254025
\(997\) 46.5145 1.47313 0.736564 0.676367i \(-0.236447\pi\)
0.736564 + 0.676367i \(0.236447\pi\)
\(998\) −51.9151 −1.64334
\(999\) −5.68172 −0.179762
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 503.2.a.e.1.9 10
3.2 odd 2 4527.2.a.k.1.2 10
4.3 odd 2 8048.2.a.p.1.9 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
503.2.a.e.1.9 10 1.1 even 1 trivial
4527.2.a.k.1.2 10 3.2 odd 2
8048.2.a.p.1.9 10 4.3 odd 2