Properties

Label 8470.2.a.cr.1.2
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.517638\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.41421 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.82843 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.41421 q^{6} -1.00000 q^{7} +1.00000 q^{8} +2.82843 q^{9} +1.00000 q^{10} -2.41421 q^{12} +2.76733 q^{13} -1.00000 q^{14} -2.41421 q^{15} +1.00000 q^{16} +4.44949 q^{17} +2.82843 q^{18} -2.62106 q^{19} +1.00000 q^{20} +2.41421 q^{21} +1.54587 q^{23} -2.41421 q^{24} +1.00000 q^{25} +2.76733 q^{26} +0.414214 q^{27} -1.00000 q^{28} -9.32780 q^{29} -2.41421 q^{30} -10.1769 q^{31} +1.00000 q^{32} +4.44949 q^{34} -1.00000 q^{35} +2.82843 q^{36} -4.39960 q^{37} -2.62106 q^{38} -6.68092 q^{39} +1.00000 q^{40} +4.80776 q^{41} +2.41421 q^{42} -11.1416 q^{43} +2.82843 q^{45} +1.54587 q^{46} -12.3424 q^{47} -2.41421 q^{48} +1.00000 q^{49} +1.00000 q^{50} -10.7420 q^{51} +2.76733 q^{52} +9.77729 q^{53} +0.414214 q^{54} -1.00000 q^{56} +6.32780 q^{57} -9.32780 q^{58} +0.510590 q^{59} -2.41421 q^{60} -2.63567 q^{61} -10.1769 q^{62} -2.82843 q^{63} +1.00000 q^{64} +2.76733 q^{65} +2.68556 q^{67} +4.44949 q^{68} -3.73205 q^{69} -1.00000 q^{70} +14.8484 q^{71} +2.82843 q^{72} +3.41421 q^{73} -4.39960 q^{74} -2.41421 q^{75} -2.62106 q^{76} -6.68092 q^{78} +2.04524 q^{79} +1.00000 q^{80} -9.48528 q^{81} +4.80776 q^{82} +4.27452 q^{83} +2.41421 q^{84} +4.44949 q^{85} -11.1416 q^{86} +22.5193 q^{87} +11.9700 q^{89} +2.82843 q^{90} -2.76733 q^{91} +1.54587 q^{92} +24.5692 q^{93} -12.3424 q^{94} -2.62106 q^{95} -2.41421 q^{96} -2.35827 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} - 4q^{7} + 4q^{8} + O(q^{10}) \) \( 4q + 4q^{2} - 4q^{3} + 4q^{4} + 4q^{5} - 4q^{6} - 4q^{7} + 4q^{8} + 4q^{10} - 4q^{12} - 4q^{14} - 4q^{15} + 4q^{16} + 8q^{17} - 12q^{19} + 4q^{20} + 4q^{21} - 8q^{23} - 4q^{24} + 4q^{25} - 4q^{27} - 4q^{28} - 8q^{29} - 4q^{30} + 4q^{32} + 8q^{34} - 4q^{35} - 16q^{37} - 12q^{38} + 8q^{39} + 4q^{40} + 8q^{41} + 4q^{42} - 8q^{43} - 8q^{46} - 16q^{47} - 4q^{48} + 4q^{49} + 4q^{50} - 8q^{51} - 4q^{54} - 4q^{56} - 4q^{57} - 8q^{58} - 8q^{59} - 4q^{60} - 8q^{61} + 4q^{64} + 8q^{68} - 8q^{69} - 4q^{70} - 8q^{71} + 8q^{73} - 16q^{74} - 4q^{75} - 12q^{76} + 8q^{78} - 24q^{79} + 4q^{80} - 4q^{81} + 8q^{82} + 8q^{83} + 4q^{84} + 8q^{85} - 8q^{86} + 16q^{87} - 8q^{92} + 16q^{93} - 16q^{94} - 12q^{95} - 4q^{96} - 8q^{97} + 4q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.41421 −1.39385 −0.696923 0.717146i \(-0.745448\pi\)
−0.696923 + 0.717146i \(0.745448\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) −2.41421 −0.985599
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 2.82843 0.942809
\(10\) 1.00000 0.316228
\(11\) 0 0
\(12\) −2.41421 −0.696923
\(13\) 2.76733 0.767518 0.383759 0.923433i \(-0.374629\pi\)
0.383759 + 0.923433i \(0.374629\pi\)
\(14\) −1.00000 −0.267261
\(15\) −2.41421 −0.623347
\(16\) 1.00000 0.250000
\(17\) 4.44949 1.07916 0.539580 0.841934i \(-0.318583\pi\)
0.539580 + 0.841934i \(0.318583\pi\)
\(18\) 2.82843 0.666667
\(19\) −2.62106 −0.601313 −0.300657 0.953732i \(-0.597206\pi\)
−0.300657 + 0.953732i \(0.597206\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.41421 0.526825
\(22\) 0 0
\(23\) 1.54587 0.322335 0.161168 0.986927i \(-0.448474\pi\)
0.161168 + 0.986927i \(0.448474\pi\)
\(24\) −2.41421 −0.492799
\(25\) 1.00000 0.200000
\(26\) 2.76733 0.542717
\(27\) 0.414214 0.0797154
\(28\) −1.00000 −0.188982
\(29\) −9.32780 −1.73213 −0.866065 0.499932i \(-0.833359\pi\)
−0.866065 + 0.499932i \(0.833359\pi\)
\(30\) −2.41421 −0.440773
\(31\) −10.1769 −1.82782 −0.913912 0.405912i \(-0.866954\pi\)
−0.913912 + 0.405912i \(0.866954\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.44949 0.763081
\(35\) −1.00000 −0.169031
\(36\) 2.82843 0.471405
\(37\) −4.39960 −0.723290 −0.361645 0.932316i \(-0.617785\pi\)
−0.361645 + 0.932316i \(0.617785\pi\)
\(38\) −2.62106 −0.425193
\(39\) −6.68092 −1.06980
\(40\) 1.00000 0.158114
\(41\) 4.80776 0.750846 0.375423 0.926854i \(-0.377497\pi\)
0.375423 + 0.926854i \(0.377497\pi\)
\(42\) 2.41421 0.372521
\(43\) −11.1416 −1.69908 −0.849541 0.527523i \(-0.823121\pi\)
−0.849541 + 0.527523i \(0.823121\pi\)
\(44\) 0 0
\(45\) 2.82843 0.421637
\(46\) 1.54587 0.227926
\(47\) −12.3424 −1.80033 −0.900163 0.435553i \(-0.856553\pi\)
−0.900163 + 0.435553i \(0.856553\pi\)
\(48\) −2.41421 −0.348462
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −10.7420 −1.50418
\(52\) 2.76733 0.383759
\(53\) 9.77729 1.34301 0.671507 0.740998i \(-0.265647\pi\)
0.671507 + 0.740998i \(0.265647\pi\)
\(54\) 0.414214 0.0563673
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 6.32780 0.838138
\(58\) −9.32780 −1.22480
\(59\) 0.510590 0.0664731 0.0332366 0.999448i \(-0.489419\pi\)
0.0332366 + 0.999448i \(0.489419\pi\)
\(60\) −2.41421 −0.311674
\(61\) −2.63567 −0.337464 −0.168732 0.985662i \(-0.553967\pi\)
−0.168732 + 0.985662i \(0.553967\pi\)
\(62\) −10.1769 −1.29247
\(63\) −2.82843 −0.356348
\(64\) 1.00000 0.125000
\(65\) 2.76733 0.343245
\(66\) 0 0
\(67\) 2.68556 0.328094 0.164047 0.986453i \(-0.447545\pi\)
0.164047 + 0.986453i \(0.447545\pi\)
\(68\) 4.44949 0.539580
\(69\) −3.73205 −0.449286
\(70\) −1.00000 −0.119523
\(71\) 14.8484 1.76218 0.881088 0.472952i \(-0.156812\pi\)
0.881088 + 0.472952i \(0.156812\pi\)
\(72\) 2.82843 0.333333
\(73\) 3.41421 0.399603 0.199802 0.979836i \(-0.435970\pi\)
0.199802 + 0.979836i \(0.435970\pi\)
\(74\) −4.39960 −0.511443
\(75\) −2.41421 −0.278769
\(76\) −2.62106 −0.300657
\(77\) 0 0
\(78\) −6.68092 −0.756465
\(79\) 2.04524 0.230108 0.115054 0.993359i \(-0.463296\pi\)
0.115054 + 0.993359i \(0.463296\pi\)
\(80\) 1.00000 0.111803
\(81\) −9.48528 −1.05392
\(82\) 4.80776 0.530929
\(83\) 4.27452 0.469189 0.234595 0.972093i \(-0.424624\pi\)
0.234595 + 0.972093i \(0.424624\pi\)
\(84\) 2.41421 0.263412
\(85\) 4.44949 0.482615
\(86\) −11.1416 −1.20143
\(87\) 22.5193 2.41432
\(88\) 0 0
\(89\) 11.9700 1.26882 0.634411 0.772996i \(-0.281243\pi\)
0.634411 + 0.772996i \(0.281243\pi\)
\(90\) 2.82843 0.298142
\(91\) −2.76733 −0.290095
\(92\) 1.54587 0.161168
\(93\) 24.5692 2.54771
\(94\) −12.3424 −1.27302
\(95\) −2.62106 −0.268915
\(96\) −2.41421 −0.246400
\(97\) −2.35827 −0.239446 −0.119723 0.992807i \(-0.538201\pi\)
−0.119723 + 0.992807i \(0.538201\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −0.464102 −0.0461798 −0.0230899 0.999733i \(-0.507350\pi\)
−0.0230899 + 0.999733i \(0.507350\pi\)
\(102\) −10.7420 −1.06362
\(103\) −12.4495 −1.22668 −0.613342 0.789817i \(-0.710175\pi\)
−0.613342 + 0.789817i \(0.710175\pi\)
\(104\) 2.76733 0.271359
\(105\) 2.41421 0.235603
\(106\) 9.77729 0.949655
\(107\) 18.4901 1.78750 0.893752 0.448561i \(-0.148063\pi\)
0.893752 + 0.448561i \(0.148063\pi\)
\(108\) 0.414214 0.0398577
\(109\) 1.01461 0.0971822 0.0485911 0.998819i \(-0.484527\pi\)
0.0485911 + 0.998819i \(0.484527\pi\)
\(110\) 0 0
\(111\) 10.6216 1.00816
\(112\) −1.00000 −0.0944911
\(113\) −11.3324 −1.06607 −0.533034 0.846094i \(-0.678948\pi\)
−0.533034 + 0.846094i \(0.678948\pi\)
\(114\) 6.32780 0.592653
\(115\) 1.54587 0.144153
\(116\) −9.32780 −0.866065
\(117\) 7.82718 0.723623
\(118\) 0.510590 0.0470036
\(119\) −4.44949 −0.407884
\(120\) −2.41421 −0.220387
\(121\) 0 0
\(122\) −2.63567 −0.238623
\(123\) −11.6070 −1.04656
\(124\) −10.1769 −0.913912
\(125\) 1.00000 0.0894427
\(126\) −2.82843 −0.251976
\(127\) −18.1616 −1.61158 −0.805789 0.592203i \(-0.798258\pi\)
−0.805789 + 0.592203i \(0.798258\pi\)
\(128\) 1.00000 0.0883883
\(129\) 26.8983 2.36826
\(130\) 2.76733 0.242711
\(131\) 1.25674 0.109802 0.0549008 0.998492i \(-0.482516\pi\)
0.0549008 + 0.998492i \(0.482516\pi\)
\(132\) 0 0
\(133\) 2.62106 0.227275
\(134\) 2.68556 0.231997
\(135\) 0.414214 0.0356498
\(136\) 4.44949 0.381541
\(137\) −19.2526 −1.64486 −0.822431 0.568865i \(-0.807383\pi\)
−0.822431 + 0.568865i \(0.807383\pi\)
\(138\) −3.73205 −0.317693
\(139\) 3.20612 0.271940 0.135970 0.990713i \(-0.456585\pi\)
0.135970 + 0.990713i \(0.456585\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 29.7972 2.50938
\(142\) 14.8484 1.24605
\(143\) 0 0
\(144\) 2.82843 0.235702
\(145\) −9.32780 −0.774632
\(146\) 3.41421 0.282562
\(147\) −2.41421 −0.199121
\(148\) −4.39960 −0.361645
\(149\) −2.56993 −0.210537 −0.105268 0.994444i \(-0.533570\pi\)
−0.105268 + 0.994444i \(0.533570\pi\)
\(150\) −2.41421 −0.197120
\(151\) −10.0969 −0.821673 −0.410837 0.911709i \(-0.634763\pi\)
−0.410837 + 0.911709i \(0.634763\pi\)
\(152\) −2.62106 −0.212596
\(153\) 12.5851 1.01744
\(154\) 0 0
\(155\) −10.1769 −0.817428
\(156\) −6.68092 −0.534902
\(157\) 7.83324 0.625160 0.312580 0.949891i \(-0.398807\pi\)
0.312580 + 0.949891i \(0.398807\pi\)
\(158\) 2.04524 0.162711
\(159\) −23.6045 −1.87196
\(160\) 1.00000 0.0790569
\(161\) −1.54587 −0.121831
\(162\) −9.48528 −0.745234
\(163\) −14.7062 −1.15188 −0.575940 0.817492i \(-0.695364\pi\)
−0.575940 + 0.817492i \(0.695364\pi\)
\(164\) 4.80776 0.375423
\(165\) 0 0
\(166\) 4.27452 0.331767
\(167\) −4.87351 −0.377123 −0.188562 0.982061i \(-0.560383\pi\)
−0.188562 + 0.982061i \(0.560383\pi\)
\(168\) 2.41421 0.186261
\(169\) −5.34190 −0.410915
\(170\) 4.44949 0.341260
\(171\) −7.41348 −0.566923
\(172\) −11.1416 −0.849541
\(173\) 6.38551 0.485481 0.242740 0.970091i \(-0.421954\pi\)
0.242740 + 0.970091i \(0.421954\pi\)
\(174\) 22.5193 1.70718
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −1.23267 −0.0926534
\(178\) 11.9700 0.897193
\(179\) −6.51523 −0.486971 −0.243486 0.969904i \(-0.578291\pi\)
−0.243486 + 0.969904i \(0.578291\pi\)
\(180\) 2.82843 0.210819
\(181\) −6.06591 −0.450875 −0.225438 0.974258i \(-0.572381\pi\)
−0.225438 + 0.974258i \(0.572381\pi\)
\(182\) −2.76733 −0.205128
\(183\) 6.36308 0.470372
\(184\) 1.54587 0.113963
\(185\) −4.39960 −0.323465
\(186\) 24.5692 1.80150
\(187\) 0 0
\(188\) −12.3424 −0.900163
\(189\) −0.414214 −0.0301296
\(190\) −2.62106 −0.190152
\(191\) −7.72208 −0.558750 −0.279375 0.960182i \(-0.590127\pi\)
−0.279375 + 0.960182i \(0.590127\pi\)
\(192\) −2.41421 −0.174231
\(193\) −3.22678 −0.232269 −0.116135 0.993233i \(-0.537050\pi\)
−0.116135 + 0.993233i \(0.537050\pi\)
\(194\) −2.35827 −0.169314
\(195\) −6.68092 −0.478430
\(196\) 1.00000 0.0714286
\(197\) −13.2066 −0.940934 −0.470467 0.882418i \(-0.655914\pi\)
−0.470467 + 0.882418i \(0.655914\pi\)
\(198\) 0 0
\(199\) 5.99270 0.424811 0.212406 0.977182i \(-0.431870\pi\)
0.212406 + 0.977182i \(0.431870\pi\)
\(200\) 1.00000 0.0707107
\(201\) −6.48352 −0.457312
\(202\) −0.464102 −0.0326541
\(203\) 9.32780 0.654684
\(204\) −10.7420 −0.752092
\(205\) 4.80776 0.335789
\(206\) −12.4495 −0.867397
\(207\) 4.37237 0.303901
\(208\) 2.76733 0.191880
\(209\) 0 0
\(210\) 2.41421 0.166597
\(211\) −13.9343 −0.959274 −0.479637 0.877467i \(-0.659232\pi\)
−0.479637 + 0.877467i \(0.659232\pi\)
\(212\) 9.77729 0.671507
\(213\) −35.8471 −2.45620
\(214\) 18.4901 1.26396
\(215\) −11.1416 −0.759852
\(216\) 0.414214 0.0281837
\(217\) 10.1769 0.690853
\(218\) 1.01461 0.0687182
\(219\) −8.24264 −0.556986
\(220\) 0 0
\(221\) 12.3132 0.828275
\(222\) 10.6216 0.712874
\(223\) −5.69161 −0.381139 −0.190569 0.981674i \(-0.561033\pi\)
−0.190569 + 0.981674i \(0.561033\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 2.82843 0.188562
\(226\) −11.3324 −0.753823
\(227\) −15.3326 −1.01766 −0.508831 0.860867i \(-0.669922\pi\)
−0.508831 + 0.860867i \(0.669922\pi\)
\(228\) 6.32780 0.419069
\(229\) 14.3734 0.949821 0.474911 0.880034i \(-0.342480\pi\)
0.474911 + 0.880034i \(0.342480\pi\)
\(230\) 1.54587 0.101931
\(231\) 0 0
\(232\) −9.32780 −0.612400
\(233\) 20.7201 1.35742 0.678710 0.734407i \(-0.262539\pi\)
0.678710 + 0.734407i \(0.262539\pi\)
\(234\) 7.82718 0.511679
\(235\) −12.3424 −0.805130
\(236\) 0.510590 0.0332366
\(237\) −4.93766 −0.320735
\(238\) −4.44949 −0.288418
\(239\) −22.0876 −1.42873 −0.714364 0.699774i \(-0.753284\pi\)
−0.714364 + 0.699774i \(0.753284\pi\)
\(240\) −2.41421 −0.155837
\(241\) 0.236073 0.0152068 0.00760339 0.999971i \(-0.497580\pi\)
0.00760339 + 0.999971i \(0.497580\pi\)
\(242\) 0 0
\(243\) 21.6569 1.38929
\(244\) −2.63567 −0.168732
\(245\) 1.00000 0.0638877
\(246\) −11.6070 −0.740033
\(247\) −7.25334 −0.461519
\(248\) −10.1769 −0.646234
\(249\) −10.3196 −0.653978
\(250\) 1.00000 0.0632456
\(251\) 14.5851 0.920601 0.460300 0.887763i \(-0.347742\pi\)
0.460300 + 0.887763i \(0.347742\pi\)
\(252\) −2.82843 −0.178174
\(253\) 0 0
\(254\) −18.1616 −1.13956
\(255\) −10.7420 −0.672691
\(256\) 1.00000 0.0625000
\(257\) 9.57973 0.597567 0.298784 0.954321i \(-0.403419\pi\)
0.298784 + 0.954321i \(0.403419\pi\)
\(258\) 26.8983 1.67461
\(259\) 4.39960 0.273378
\(260\) 2.76733 0.171622
\(261\) −26.3830 −1.63307
\(262\) 1.25674 0.0776415
\(263\) 3.02066 0.186262 0.0931311 0.995654i \(-0.470312\pi\)
0.0931311 + 0.995654i \(0.470312\pi\)
\(264\) 0 0
\(265\) 9.77729 0.600614
\(266\) 2.62106 0.160708
\(267\) −28.8983 −1.76854
\(268\) 2.68556 0.164047
\(269\) −28.6510 −1.74688 −0.873440 0.486932i \(-0.838116\pi\)
−0.873440 + 0.486932i \(0.838116\pi\)
\(270\) 0.414214 0.0252082
\(271\) −0.406899 −0.0247173 −0.0123587 0.999924i \(-0.503934\pi\)
−0.0123587 + 0.999924i \(0.503934\pi\)
\(272\) 4.44949 0.269790
\(273\) 6.68092 0.404348
\(274\) −19.2526 −1.16309
\(275\) 0 0
\(276\) −3.73205 −0.224643
\(277\) −17.9993 −1.08147 −0.540736 0.841193i \(-0.681854\pi\)
−0.540736 + 0.841193i \(0.681854\pi\)
\(278\) 3.20612 0.192290
\(279\) −28.7846 −1.72329
\(280\) −1.00000 −0.0597614
\(281\) −13.9833 −0.834171 −0.417085 0.908867i \(-0.636948\pi\)
−0.417085 + 0.908867i \(0.636948\pi\)
\(282\) 29.7972 1.77440
\(283\) −16.1309 −0.958884 −0.479442 0.877574i \(-0.659161\pi\)
−0.479442 + 0.877574i \(0.659161\pi\)
\(284\) 14.8484 0.881088
\(285\) 6.32780 0.374827
\(286\) 0 0
\(287\) −4.80776 −0.283793
\(288\) 2.82843 0.166667
\(289\) 2.79796 0.164586
\(290\) −9.32780 −0.547748
\(291\) 5.69337 0.333752
\(292\) 3.41421 0.199802
\(293\) 16.1317 0.942421 0.471211 0.882021i \(-0.343817\pi\)
0.471211 + 0.882021i \(0.343817\pi\)
\(294\) −2.41421 −0.140800
\(295\) 0.510590 0.0297277
\(296\) −4.39960 −0.255722
\(297\) 0 0
\(298\) −2.56993 −0.148872
\(299\) 4.27792 0.247398
\(300\) −2.41421 −0.139385
\(301\) 11.1416 0.642192
\(302\) −10.0969 −0.581011
\(303\) 1.12044 0.0643676
\(304\) −2.62106 −0.150328
\(305\) −2.63567 −0.150918
\(306\) 12.5851 0.719440
\(307\) −27.0119 −1.54165 −0.770825 0.637047i \(-0.780156\pi\)
−0.770825 + 0.637047i \(0.780156\pi\)
\(308\) 0 0
\(309\) 30.0557 1.70981
\(310\) −10.1769 −0.578009
\(311\) 24.9182 1.41298 0.706490 0.707723i \(-0.250277\pi\)
0.706490 + 0.707723i \(0.250277\pi\)
\(312\) −6.68092 −0.378233
\(313\) 12.4044 0.701139 0.350569 0.936537i \(-0.385988\pi\)
0.350569 + 0.936537i \(0.385988\pi\)
\(314\) 7.83324 0.442055
\(315\) −2.82843 −0.159364
\(316\) 2.04524 0.115054
\(317\) −30.0333 −1.68684 −0.843419 0.537256i \(-0.819461\pi\)
−0.843419 + 0.537256i \(0.819461\pi\)
\(318\) −23.6045 −1.32367
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −44.6390 −2.49151
\(322\) −1.54587 −0.0861477
\(323\) −11.6624 −0.648913
\(324\) −9.48528 −0.526960
\(325\) 2.76733 0.153504
\(326\) −14.7062 −0.814503
\(327\) −2.44949 −0.135457
\(328\) 4.80776 0.265464
\(329\) 12.3424 0.680459
\(330\) 0 0
\(331\) 9.67752 0.531925 0.265962 0.963983i \(-0.414310\pi\)
0.265962 + 0.963983i \(0.414310\pi\)
\(332\) 4.27452 0.234595
\(333\) −12.4440 −0.681924
\(334\) −4.87351 −0.266666
\(335\) 2.68556 0.146728
\(336\) 2.41421 0.131706
\(337\) −17.9968 −0.980346 −0.490173 0.871625i \(-0.663066\pi\)
−0.490173 + 0.871625i \(0.663066\pi\)
\(338\) −5.34190 −0.290561
\(339\) 27.3590 1.48593
\(340\) 4.44949 0.241307
\(341\) 0 0
\(342\) −7.41348 −0.400875
\(343\) −1.00000 −0.0539949
\(344\) −11.1416 −0.600716
\(345\) −3.73205 −0.200927
\(346\) 6.38551 0.343287
\(347\) −16.1806 −0.868623 −0.434311 0.900763i \(-0.643008\pi\)
−0.434311 + 0.900763i \(0.643008\pi\)
\(348\) 22.5193 1.20716
\(349\) 9.94031 0.532092 0.266046 0.963960i \(-0.414283\pi\)
0.266046 + 0.963960i \(0.414283\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 1.14626 0.0611831
\(352\) 0 0
\(353\) −24.9547 −1.32821 −0.664103 0.747642i \(-0.731186\pi\)
−0.664103 + 0.747642i \(0.731186\pi\)
\(354\) −1.23267 −0.0655158
\(355\) 14.8484 0.788069
\(356\) 11.9700 0.634411
\(357\) 10.7420 0.568528
\(358\) −6.51523 −0.344341
\(359\) 28.3933 1.49854 0.749271 0.662263i \(-0.230404\pi\)
0.749271 + 0.662263i \(0.230404\pi\)
\(360\) 2.82843 0.149071
\(361\) −12.1300 −0.638423
\(362\) −6.06591 −0.318817
\(363\) 0 0
\(364\) −2.76733 −0.145047
\(365\) 3.41421 0.178708
\(366\) 6.36308 0.332604
\(367\) 19.7164 1.02919 0.514593 0.857435i \(-0.327943\pi\)
0.514593 + 0.857435i \(0.327943\pi\)
\(368\) 1.54587 0.0805838
\(369\) 13.5984 0.707905
\(370\) −4.39960 −0.228724
\(371\) −9.77729 −0.507612
\(372\) 24.5692 1.27385
\(373\) 32.7705 1.69679 0.848396 0.529361i \(-0.177568\pi\)
0.848396 + 0.529361i \(0.177568\pi\)
\(374\) 0 0
\(375\) −2.41421 −0.124669
\(376\) −12.3424 −0.636512
\(377\) −25.8131 −1.32944
\(378\) −0.414214 −0.0213048
\(379\) −12.8332 −0.659199 −0.329600 0.944121i \(-0.606914\pi\)
−0.329600 + 0.944121i \(0.606914\pi\)
\(380\) −2.62106 −0.134458
\(381\) 43.8459 2.24629
\(382\) −7.72208 −0.395096
\(383\) 13.1557 0.672226 0.336113 0.941822i \(-0.390888\pi\)
0.336113 + 0.941822i \(0.390888\pi\)
\(384\) −2.41421 −0.123200
\(385\) 0 0
\(386\) −3.22678 −0.164239
\(387\) −31.5133 −1.60191
\(388\) −2.35827 −0.119723
\(389\) −20.8831 −1.05882 −0.529408 0.848367i \(-0.677586\pi\)
−0.529408 + 0.848367i \(0.677586\pi\)
\(390\) −6.68092 −0.338301
\(391\) 6.87832 0.347851
\(392\) 1.00000 0.0505076
\(393\) −3.03403 −0.153047
\(394\) −13.2066 −0.665341
\(395\) 2.04524 0.102907
\(396\) 0 0
\(397\) −8.01314 −0.402168 −0.201084 0.979574i \(-0.564446\pi\)
−0.201084 + 0.979574i \(0.564446\pi\)
\(398\) 5.99270 0.300387
\(399\) −6.32780 −0.316786
\(400\) 1.00000 0.0500000
\(401\) 4.25120 0.212295 0.106147 0.994350i \(-0.466148\pi\)
0.106147 + 0.994350i \(0.466148\pi\)
\(402\) −6.48352 −0.323369
\(403\) −28.1628 −1.40289
\(404\) −0.464102 −0.0230899
\(405\) −9.48528 −0.471327
\(406\) 9.32780 0.462931
\(407\) 0 0
\(408\) −10.7420 −0.531809
\(409\) 5.90606 0.292036 0.146018 0.989282i \(-0.453354\pi\)
0.146018 + 0.989282i \(0.453354\pi\)
\(410\) 4.80776 0.237438
\(411\) 46.4799 2.29268
\(412\) −12.4495 −0.613342
\(413\) −0.510590 −0.0251245
\(414\) 4.37237 0.214890
\(415\) 4.27452 0.209828
\(416\) 2.76733 0.135679
\(417\) −7.74026 −0.379042
\(418\) 0 0
\(419\) 30.0428 1.46768 0.733842 0.679320i \(-0.237725\pi\)
0.733842 + 0.679320i \(0.237725\pi\)
\(420\) 2.41421 0.117802
\(421\) −10.2288 −0.498520 −0.249260 0.968437i \(-0.580187\pi\)
−0.249260 + 0.968437i \(0.580187\pi\)
\(422\) −13.9343 −0.678309
\(423\) −34.9096 −1.69736
\(424\) 9.77729 0.474827
\(425\) 4.44949 0.215832
\(426\) −35.8471 −1.73680
\(427\) 2.63567 0.127549
\(428\) 18.4901 0.893752
\(429\) 0 0
\(430\) −11.1416 −0.537297
\(431\) −1.14626 −0.0552136 −0.0276068 0.999619i \(-0.508789\pi\)
−0.0276068 + 0.999619i \(0.508789\pi\)
\(432\) 0.414214 0.0199289
\(433\) −9.29931 −0.446896 −0.223448 0.974716i \(-0.571731\pi\)
−0.223448 + 0.974716i \(0.571731\pi\)
\(434\) 10.1769 0.488507
\(435\) 22.5193 1.07972
\(436\) 1.01461 0.0485911
\(437\) −4.05181 −0.193824
\(438\) −8.24264 −0.393849
\(439\) 9.16604 0.437471 0.218736 0.975784i \(-0.429807\pi\)
0.218736 + 0.975784i \(0.429807\pi\)
\(440\) 0 0
\(441\) 2.82843 0.134687
\(442\) 12.3132 0.585679
\(443\) −33.0771 −1.57154 −0.785771 0.618517i \(-0.787734\pi\)
−0.785771 + 0.618517i \(0.787734\pi\)
\(444\) 10.6216 0.504078
\(445\) 11.9700 0.567435
\(446\) −5.69161 −0.269506
\(447\) 6.20436 0.293456
\(448\) −1.00000 −0.0472456
\(449\) −19.3933 −0.915228 −0.457614 0.889151i \(-0.651296\pi\)
−0.457614 + 0.889151i \(0.651296\pi\)
\(450\) 2.82843 0.133333
\(451\) 0 0
\(452\) −11.3324 −0.533034
\(453\) 24.3761 1.14529
\(454\) −15.3326 −0.719595
\(455\) −2.76733 −0.129734
\(456\) 6.32780 0.296327
\(457\) −32.5206 −1.52125 −0.760624 0.649193i \(-0.775107\pi\)
−0.760624 + 0.649193i \(0.775107\pi\)
\(458\) 14.3734 0.671625
\(459\) 1.84304 0.0860257
\(460\) 1.54587 0.0720764
\(461\) 42.2301 1.96685 0.983427 0.181306i \(-0.0580325\pi\)
0.983427 + 0.181306i \(0.0580325\pi\)
\(462\) 0 0
\(463\) −22.0593 −1.02518 −0.512592 0.858632i \(-0.671315\pi\)
−0.512592 + 0.858632i \(0.671315\pi\)
\(464\) −9.32780 −0.433032
\(465\) 24.5692 1.13937
\(466\) 20.7201 0.959841
\(467\) −21.3716 −0.988962 −0.494481 0.869189i \(-0.664642\pi\)
−0.494481 + 0.869189i \(0.664642\pi\)
\(468\) 7.82718 0.361812
\(469\) −2.68556 −0.124008
\(470\) −12.3424 −0.569313
\(471\) −18.9111 −0.871378
\(472\) 0.510590 0.0235018
\(473\) 0 0
\(474\) −4.93766 −0.226794
\(475\) −2.62106 −0.120263
\(476\) −4.44949 −0.203942
\(477\) 27.6544 1.26621
\(478\) −22.0876 −1.01026
\(479\) −9.84888 −0.450007 −0.225003 0.974358i \(-0.572239\pi\)
−0.225003 + 0.974358i \(0.572239\pi\)
\(480\) −2.41421 −0.110193
\(481\) −12.1751 −0.555138
\(482\) 0.236073 0.0107528
\(483\) 3.73205 0.169814
\(484\) 0 0
\(485\) −2.35827 −0.107084
\(486\) 21.6569 0.982375
\(487\) −35.0712 −1.58923 −0.794615 0.607114i \(-0.792327\pi\)
−0.794615 + 0.607114i \(0.792327\pi\)
\(488\) −2.63567 −0.119311
\(489\) 35.5040 1.60555
\(490\) 1.00000 0.0451754
\(491\) 1.40389 0.0633569 0.0316784 0.999498i \(-0.489915\pi\)
0.0316784 + 0.999498i \(0.489915\pi\)
\(492\) −11.6070 −0.523282
\(493\) −41.5040 −1.86924
\(494\) −7.25334 −0.326343
\(495\) 0 0
\(496\) −10.1769 −0.456956
\(497\) −14.8484 −0.666040
\(498\) −10.3196 −0.462432
\(499\) −5.73573 −0.256767 −0.128383 0.991725i \(-0.540979\pi\)
−0.128383 + 0.991725i \(0.540979\pi\)
\(500\) 1.00000 0.0447214
\(501\) 11.7657 0.525652
\(502\) 14.5851 0.650963
\(503\) 10.7493 0.479288 0.239644 0.970861i \(-0.422969\pi\)
0.239644 + 0.970861i \(0.422969\pi\)
\(504\) −2.82843 −0.125988
\(505\) −0.464102 −0.0206523
\(506\) 0 0
\(507\) 12.8965 0.572753
\(508\) −18.1616 −0.805789
\(509\) 5.44632 0.241404 0.120702 0.992689i \(-0.461485\pi\)
0.120702 + 0.992689i \(0.461485\pi\)
\(510\) −10.7420 −0.475665
\(511\) −3.41421 −0.151036
\(512\) 1.00000 0.0441942
\(513\) −1.08568 −0.0479339
\(514\) 9.57973 0.422544
\(515\) −12.4495 −0.548590
\(516\) 26.8983 1.18413
\(517\) 0 0
\(518\) 4.39960 0.193307
\(519\) −15.4160 −0.676686
\(520\) 2.76733 0.121355
\(521\) −28.3689 −1.24286 −0.621432 0.783468i \(-0.713449\pi\)
−0.621432 + 0.783468i \(0.713449\pi\)
\(522\) −26.3830 −1.15475
\(523\) 28.2428 1.23497 0.617486 0.786582i \(-0.288151\pi\)
0.617486 + 0.786582i \(0.288151\pi\)
\(524\) 1.25674 0.0549008
\(525\) 2.41421 0.105365
\(526\) 3.02066 0.131707
\(527\) −45.2820 −1.97251
\(528\) 0 0
\(529\) −20.6103 −0.896100
\(530\) 9.77729 0.424699
\(531\) 1.44417 0.0626715
\(532\) 2.62106 0.113637
\(533\) 13.3047 0.576288
\(534\) −28.8983 −1.25055
\(535\) 18.4901 0.799396
\(536\) 2.68556 0.115999
\(537\) 15.7292 0.678764
\(538\) −28.6510 −1.23523
\(539\) 0 0
\(540\) 0.414214 0.0178249
\(541\) −3.15698 −0.135729 −0.0678646 0.997695i \(-0.521619\pi\)
−0.0678646 + 0.997695i \(0.521619\pi\)
\(542\) −0.406899 −0.0174778
\(543\) 14.6444 0.628451
\(544\) 4.44949 0.190770
\(545\) 1.01461 0.0434612
\(546\) 6.68092 0.285917
\(547\) 18.8345 0.805305 0.402652 0.915353i \(-0.368088\pi\)
0.402652 + 0.915353i \(0.368088\pi\)
\(548\) −19.2526 −0.822431
\(549\) −7.45481 −0.318164
\(550\) 0 0
\(551\) 24.4488 1.04155
\(552\) −3.73205 −0.158847
\(553\) −2.04524 −0.0869726
\(554\) −17.9993 −0.764716
\(555\) 10.6216 0.450861
\(556\) 3.20612 0.135970
\(557\) 30.1283 1.27658 0.638288 0.769798i \(-0.279643\pi\)
0.638288 + 0.769798i \(0.279643\pi\)
\(558\) −28.7846 −1.21855
\(559\) −30.8325 −1.30408
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) −13.9833 −0.589848
\(563\) 29.7041 1.25188 0.625939 0.779872i \(-0.284716\pi\)
0.625939 + 0.779872i \(0.284716\pi\)
\(564\) 29.7972 1.25469
\(565\) −11.3324 −0.476760
\(566\) −16.1309 −0.678033
\(567\) 9.48528 0.398344
\(568\) 14.8484 0.623023
\(569\) 12.2387 0.513074 0.256537 0.966534i \(-0.417418\pi\)
0.256537 + 0.966534i \(0.417418\pi\)
\(570\) 6.32780 0.265043
\(571\) 43.9882 1.84085 0.920425 0.390919i \(-0.127843\pi\)
0.920425 + 0.390919i \(0.127843\pi\)
\(572\) 0 0
\(573\) 18.6428 0.778812
\(574\) −4.80776 −0.200672
\(575\) 1.54587 0.0644671
\(576\) 2.82843 0.117851
\(577\) −29.8582 −1.24301 −0.621506 0.783410i \(-0.713479\pi\)
−0.621506 + 0.783410i \(0.713479\pi\)
\(578\) 2.79796 0.116380
\(579\) 7.79015 0.323747
\(580\) −9.32780 −0.387316
\(581\) −4.27452 −0.177337
\(582\) 5.69337 0.235998
\(583\) 0 0
\(584\) 3.41421 0.141281
\(585\) 7.82718 0.323614
\(586\) 16.1317 0.666392
\(587\) −46.1626 −1.90533 −0.952667 0.304016i \(-0.901672\pi\)
−0.952667 + 0.304016i \(0.901672\pi\)
\(588\) −2.41421 −0.0995605
\(589\) 26.6743 1.09909
\(590\) 0.510590 0.0210207
\(591\) 31.8836 1.31152
\(592\) −4.39960 −0.180823
\(593\) −32.3523 −1.32855 −0.664276 0.747488i \(-0.731260\pi\)
−0.664276 + 0.747488i \(0.731260\pi\)
\(594\) 0 0
\(595\) −4.44949 −0.182411
\(596\) −2.56993 −0.105268
\(597\) −14.4677 −0.592122
\(598\) 4.27792 0.174937
\(599\) −27.7430 −1.13355 −0.566775 0.823873i \(-0.691809\pi\)
−0.566775 + 0.823873i \(0.691809\pi\)
\(600\) −2.41421 −0.0985599
\(601\) 34.5468 1.40919 0.704596 0.709608i \(-0.251128\pi\)
0.704596 + 0.709608i \(0.251128\pi\)
\(602\) 11.1416 0.454099
\(603\) 7.59592 0.309330
\(604\) −10.0969 −0.410837
\(605\) 0 0
\(606\) 1.12044 0.0455148
\(607\) −42.8562 −1.73948 −0.869740 0.493511i \(-0.835713\pi\)
−0.869740 + 0.493511i \(0.835713\pi\)
\(608\) −2.62106 −0.106298
\(609\) −22.5193 −0.912529
\(610\) −2.63567 −0.106715
\(611\) −34.1555 −1.38178
\(612\) 12.5851 0.508721
\(613\) −22.5795 −0.911978 −0.455989 0.889985i \(-0.650714\pi\)
−0.455989 + 0.889985i \(0.650714\pi\)
\(614\) −27.0119 −1.09011
\(615\) −11.6070 −0.468038
\(616\) 0 0
\(617\) −34.0880 −1.37233 −0.686166 0.727445i \(-0.740707\pi\)
−0.686166 + 0.727445i \(0.740707\pi\)
\(618\) 30.0557 1.20902
\(619\) −2.64208 −0.106194 −0.0530971 0.998589i \(-0.516909\pi\)
−0.0530971 + 0.998589i \(0.516909\pi\)
\(620\) −10.1769 −0.408714
\(621\) 0.640319 0.0256951
\(622\) 24.9182 0.999128
\(623\) −11.9700 −0.479570
\(624\) −6.68092 −0.267451
\(625\) 1.00000 0.0400000
\(626\) 12.4044 0.495780
\(627\) 0 0
\(628\) 7.83324 0.312580
\(629\) −19.5760 −0.780546
\(630\) −2.82843 −0.112687
\(631\) 16.2313 0.646156 0.323078 0.946372i \(-0.395282\pi\)
0.323078 + 0.946372i \(0.395282\pi\)
\(632\) 2.04524 0.0813554
\(633\) 33.6403 1.33708
\(634\) −30.0333 −1.19277
\(635\) −18.1616 −0.720719
\(636\) −23.6045 −0.935978
\(637\) 2.76733 0.109645
\(638\) 0 0
\(639\) 41.9975 1.66140
\(640\) 1.00000 0.0395285
\(641\) −11.5324 −0.455502 −0.227751 0.973719i \(-0.573137\pi\)
−0.227751 + 0.973719i \(0.573137\pi\)
\(642\) −44.6390 −1.76176
\(643\) −13.3722 −0.527346 −0.263673 0.964612i \(-0.584934\pi\)
−0.263673 + 0.964612i \(0.584934\pi\)
\(644\) −1.54587 −0.0609157
\(645\) 26.8983 1.05912
\(646\) −11.6624 −0.458851
\(647\) −49.1639 −1.93283 −0.966416 0.256984i \(-0.917271\pi\)
−0.966416 + 0.256984i \(0.917271\pi\)
\(648\) −9.48528 −0.372617
\(649\) 0 0
\(650\) 2.76733 0.108543
\(651\) −24.5692 −0.962943
\(652\) −14.7062 −0.575940
\(653\) 27.7393 1.08552 0.542762 0.839887i \(-0.317379\pi\)
0.542762 + 0.839887i \(0.317379\pi\)
\(654\) −2.44949 −0.0957826
\(655\) 1.25674 0.0491048
\(656\) 4.80776 0.187712
\(657\) 9.65685 0.376750
\(658\) 12.3424 0.481158
\(659\) −35.7841 −1.39395 −0.696975 0.717095i \(-0.745471\pi\)
−0.696975 + 0.717095i \(0.745471\pi\)
\(660\) 0 0
\(661\) 6.25763 0.243394 0.121697 0.992567i \(-0.461166\pi\)
0.121697 + 0.992567i \(0.461166\pi\)
\(662\) 9.67752 0.376127
\(663\) −29.7267 −1.15449
\(664\) 4.27452 0.165883
\(665\) 2.62106 0.101640
\(666\) −12.4440 −0.482193
\(667\) −14.4195 −0.558327
\(668\) −4.87351 −0.188562
\(669\) 13.7408 0.531249
\(670\) 2.68556 0.103752
\(671\) 0 0
\(672\) 2.41421 0.0931303
\(673\) −34.4710 −1.32876 −0.664379 0.747396i \(-0.731304\pi\)
−0.664379 + 0.747396i \(0.731304\pi\)
\(674\) −17.9968 −0.693209
\(675\) 0.414214 0.0159431
\(676\) −5.34190 −0.205458
\(677\) 39.4297 1.51541 0.757704 0.652598i \(-0.226321\pi\)
0.757704 + 0.652598i \(0.226321\pi\)
\(678\) 27.3590 1.05071
\(679\) 2.35827 0.0905022
\(680\) 4.44949 0.170630
\(681\) 37.0162 1.41846
\(682\) 0 0
\(683\) −7.54298 −0.288624 −0.144312 0.989532i \(-0.546097\pi\)
−0.144312 + 0.989532i \(0.546097\pi\)
\(684\) −7.41348 −0.283462
\(685\) −19.2526 −0.735604
\(686\) −1.00000 −0.0381802
\(687\) −34.7005 −1.32391
\(688\) −11.1416 −0.424770
\(689\) 27.0570 1.03079
\(690\) −3.73205 −0.142077
\(691\) −10.7272 −0.408082 −0.204041 0.978962i \(-0.565408\pi\)
−0.204041 + 0.978962i \(0.565408\pi\)
\(692\) 6.38551 0.242740
\(693\) 0 0
\(694\) −16.1806 −0.614209
\(695\) 3.20612 0.121615
\(696\) 22.5193 0.853592
\(697\) 21.3921 0.810283
\(698\) 9.94031 0.376246
\(699\) −50.0228 −1.89203
\(700\) −1.00000 −0.0377964
\(701\) 32.4865 1.22700 0.613500 0.789695i \(-0.289761\pi\)
0.613500 + 0.789695i \(0.289761\pi\)
\(702\) 1.14626 0.0432630
\(703\) 11.5316 0.434924
\(704\) 0 0
\(705\) 29.7972 1.12223
\(706\) −24.9547 −0.939183
\(707\) 0.464102 0.0174543
\(708\) −1.23267 −0.0463267
\(709\) −4.45805 −0.167426 −0.0837128 0.996490i \(-0.526678\pi\)
−0.0837128 + 0.996490i \(0.526678\pi\)
\(710\) 14.8484 0.557249
\(711\) 5.78482 0.216948
\(712\) 11.9700 0.448596
\(713\) −15.7321 −0.589172
\(714\) 10.7420 0.402010
\(715\) 0 0
\(716\) −6.51523 −0.243486
\(717\) 53.3242 1.99143
\(718\) 28.3933 1.05963
\(719\) 45.1213 1.68274 0.841370 0.540460i \(-0.181750\pi\)
0.841370 + 0.540460i \(0.181750\pi\)
\(720\) 2.82843 0.105409
\(721\) 12.4495 0.463643
\(722\) −12.1300 −0.451433
\(723\) −0.569930 −0.0211959
\(724\) −6.06591 −0.225438
\(725\) −9.32780 −0.346426
\(726\) 0 0
\(727\) 9.01388 0.334306 0.167153 0.985931i \(-0.446543\pi\)
0.167153 + 0.985931i \(0.446543\pi\)
\(728\) −2.76733 −0.102564
\(729\) −23.8284 −0.882534
\(730\) 3.41421 0.126366
\(731\) −49.5745 −1.83358
\(732\) 6.36308 0.235186
\(733\) 46.3945 1.71362 0.856811 0.515631i \(-0.172443\pi\)
0.856811 + 0.515631i \(0.172443\pi\)
\(734\) 19.7164 0.727744
\(735\) −2.41421 −0.0890496
\(736\) 1.54587 0.0569814
\(737\) 0 0
\(738\) 13.5984 0.500564
\(739\) −26.8879 −0.989089 −0.494544 0.869152i \(-0.664665\pi\)
−0.494544 + 0.869152i \(0.664665\pi\)
\(740\) −4.39960 −0.161733
\(741\) 17.5111 0.643287
\(742\) −9.77729 −0.358936
\(743\) −5.89670 −0.216329 −0.108165 0.994133i \(-0.534497\pi\)
−0.108165 + 0.994133i \(0.534497\pi\)
\(744\) 24.5692 0.900751
\(745\) −2.56993 −0.0941550
\(746\) 32.7705 1.19981
\(747\) 12.0902 0.442356
\(748\) 0 0
\(749\) −18.4901 −0.675613
\(750\) −2.41421 −0.0881546
\(751\) 32.7728 1.19590 0.597948 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(752\) −12.3424 −0.450082
\(753\) −35.2114 −1.28318
\(754\) −25.8131 −0.940057
\(755\) −10.0969 −0.367464
\(756\) −0.414214 −0.0150648
\(757\) 42.6821 1.55131 0.775654 0.631158i \(-0.217420\pi\)
0.775654 + 0.631158i \(0.217420\pi\)
\(758\) −12.8332 −0.466124
\(759\) 0 0
\(760\) −2.62106 −0.0950759
\(761\) −13.3835 −0.485153 −0.242576 0.970132i \(-0.577992\pi\)
−0.242576 + 0.970132i \(0.577992\pi\)
\(762\) 43.8459 1.58837
\(763\) −1.01461 −0.0367314
\(764\) −7.72208 −0.279375
\(765\) 12.5851 0.455014
\(766\) 13.1557 0.475335
\(767\) 1.41297 0.0510194
\(768\) −2.41421 −0.0871154
\(769\) 24.1256 0.869990 0.434995 0.900433i \(-0.356750\pi\)
0.434995 + 0.900433i \(0.356750\pi\)
\(770\) 0 0
\(771\) −23.1275 −0.832917
\(772\) −3.22678 −0.116135
\(773\) 18.1243 0.651886 0.325943 0.945389i \(-0.394318\pi\)
0.325943 + 0.945389i \(0.394318\pi\)
\(774\) −31.5133 −1.13272
\(775\) −10.1769 −0.365565
\(776\) −2.35827 −0.0846571
\(777\) −10.6216 −0.381047
\(778\) −20.8831 −0.748696
\(779\) −12.6014 −0.451494
\(780\) −6.68092 −0.239215
\(781\) 0 0
\(782\) 6.87832 0.245968
\(783\) −3.86370 −0.138077
\(784\) 1.00000 0.0357143
\(785\) 7.83324 0.279580
\(786\) −3.03403 −0.108220
\(787\) −13.1477 −0.468665 −0.234332 0.972157i \(-0.575290\pi\)
−0.234332 + 0.972157i \(0.575290\pi\)
\(788\) −13.2066 −0.470467
\(789\) −7.29253 −0.259621
\(790\) 2.04524 0.0727665
\(791\) 11.3324 0.402936
\(792\) 0 0
\(793\) −7.29377 −0.259009
\(794\) −8.01314 −0.284376
\(795\) −23.6045 −0.837165
\(796\) 5.99270 0.212406
\(797\) 28.3545 1.00437 0.502185 0.864760i \(-0.332530\pi\)
0.502185 + 0.864760i \(0.332530\pi\)
\(798\) −6.32780 −0.224002
\(799\) −54.9175 −1.94284
\(800\) 1.00000 0.0353553
\(801\) 33.8564 1.19626
\(802\) 4.25120 0.150115
\(803\) 0 0
\(804\) −6.48352 −0.228656
\(805\) −1.54587 −0.0544846
\(806\) −28.1628 −0.991992
\(807\) 69.1695 2.43488
\(808\) −0.464102 −0.0163270
\(809\) 20.7073 0.728028 0.364014 0.931393i \(-0.381406\pi\)
0.364014 + 0.931393i \(0.381406\pi\)
\(810\) −9.48528 −0.333279
\(811\) −28.2526 −0.992082 −0.496041 0.868299i \(-0.665213\pi\)
−0.496041 + 0.868299i \(0.665213\pi\)
\(812\) 9.32780 0.327342
\(813\) 0.982340 0.0344522
\(814\) 0 0
\(815\) −14.7062 −0.515137
\(816\) −10.7420 −0.376046
\(817\) 29.2029 1.02168
\(818\) 5.90606 0.206501
\(819\) −7.82718 −0.273504
\(820\) 4.80776 0.167894
\(821\) 22.7165 0.792813 0.396406 0.918075i \(-0.370257\pi\)
0.396406 + 0.918075i \(0.370257\pi\)
\(822\) 46.4799 1.62117
\(823\) −7.38572 −0.257450 −0.128725 0.991680i \(-0.541088\pi\)
−0.128725 + 0.991680i \(0.541088\pi\)
\(824\) −12.4495 −0.433699
\(825\) 0 0
\(826\) −0.510590 −0.0177657
\(827\) 26.2870 0.914088 0.457044 0.889444i \(-0.348908\pi\)
0.457044 + 0.889444i \(0.348908\pi\)
\(828\) 4.37237 0.151950
\(829\) −32.9916 −1.14585 −0.572923 0.819609i \(-0.694191\pi\)
−0.572923 + 0.819609i \(0.694191\pi\)
\(830\) 4.27452 0.148371
\(831\) 43.4541 1.50741
\(832\) 2.76733 0.0959398
\(833\) 4.44949 0.154166
\(834\) −7.74026 −0.268023
\(835\) −4.87351 −0.168655
\(836\) 0 0
\(837\) −4.21541 −0.145706
\(838\) 30.0428 1.03781
\(839\) −16.8466 −0.581609 −0.290805 0.956782i \(-0.593923\pi\)
−0.290805 + 0.956782i \(0.593923\pi\)
\(840\) 2.41421 0.0832983
\(841\) 58.0079 2.00027
\(842\) −10.2288 −0.352507
\(843\) 33.7586 1.16271
\(844\) −13.9343 −0.479637
\(845\) −5.34190 −0.183767
\(846\) −34.9096 −1.20022
\(847\) 0 0
\(848\) 9.77729 0.335754
\(849\) 38.9435 1.33654
\(850\) 4.44949 0.152616
\(851\) −6.80119 −0.233142
\(852\) −35.8471 −1.22810
\(853\) −48.8392 −1.67222 −0.836112 0.548559i \(-0.815177\pi\)
−0.836112 + 0.548559i \(0.815177\pi\)
\(854\) 2.63567 0.0901909
\(855\) −7.41348 −0.253536
\(856\) 18.4901 0.631978
\(857\) −16.4322 −0.561312 −0.280656 0.959808i \(-0.590552\pi\)
−0.280656 + 0.959808i \(0.590552\pi\)
\(858\) 0 0
\(859\) 10.2477 0.349646 0.174823 0.984600i \(-0.444065\pi\)
0.174823 + 0.984600i \(0.444065\pi\)
\(860\) −11.1416 −0.379926
\(861\) 11.6070 0.395564
\(862\) −1.14626 −0.0390419
\(863\) 43.4437 1.47884 0.739421 0.673243i \(-0.235099\pi\)
0.739421 + 0.673243i \(0.235099\pi\)
\(864\) 0.414214 0.0140918
\(865\) 6.38551 0.217114
\(866\) −9.29931 −0.316003
\(867\) −6.75487 −0.229407
\(868\) 10.1769 0.345426
\(869\) 0 0
\(870\) 22.5193 0.763476
\(871\) 7.43183 0.251818
\(872\) 1.01461 0.0343591
\(873\) −6.67020 −0.225752
\(874\) −4.05181 −0.137055
\(875\) −1.00000 −0.0338062
\(876\) −8.24264 −0.278493
\(877\) 18.6670 0.630339 0.315170 0.949035i \(-0.397939\pi\)
0.315170 + 0.949035i \(0.397939\pi\)
\(878\) 9.16604 0.309339
\(879\) −38.9453 −1.31359
\(880\) 0 0
\(881\) −53.7108 −1.80956 −0.904782 0.425875i \(-0.859966\pi\)
−0.904782 + 0.425875i \(0.859966\pi\)
\(882\) 2.82843 0.0952381
\(883\) −30.0729 −1.01203 −0.506016 0.862524i \(-0.668882\pi\)
−0.506016 + 0.862524i \(0.668882\pi\)
\(884\) 12.3132 0.414137
\(885\) −1.23267 −0.0414359
\(886\) −33.0771 −1.11125
\(887\) 31.8197 1.06840 0.534200 0.845358i \(-0.320613\pi\)
0.534200 + 0.845358i \(0.320613\pi\)
\(888\) 10.6216 0.356437
\(889\) 18.1616 0.609119
\(890\) 11.9700 0.401237
\(891\) 0 0
\(892\) −5.69161 −0.190569
\(893\) 32.3502 1.08256
\(894\) 6.20436 0.207505
\(895\) −6.51523 −0.217780
\(896\) −1.00000 −0.0334077
\(897\) −10.3278 −0.344835
\(898\) −19.3933 −0.647164
\(899\) 94.9281 3.16603
\(900\) 2.82843 0.0942809
\(901\) 43.5040 1.44933
\(902\) 0 0
\(903\) −26.8983 −0.895118
\(904\) −11.3324 −0.376912
\(905\) −6.06591 −0.201638
\(906\) 24.3761 0.809840
\(907\) −31.8443 −1.05737 −0.528686 0.848817i \(-0.677315\pi\)
−0.528686 + 0.848817i \(0.677315\pi\)
\(908\) −15.3326 −0.508831
\(909\) −1.31268 −0.0435388
\(910\) −2.76733 −0.0917360
\(911\) 0.187429 0.00620980 0.00310490 0.999995i \(-0.499012\pi\)
0.00310490 + 0.999995i \(0.499012\pi\)
\(912\) 6.32780 0.209535
\(913\) 0 0
\(914\) −32.5206 −1.07568
\(915\) 6.36308 0.210357
\(916\) 14.3734 0.474911
\(917\) −1.25674 −0.0415011
\(918\) 1.84304 0.0608294
\(919\) 39.6436 1.30772 0.653861 0.756615i \(-0.273148\pi\)
0.653861 + 0.756615i \(0.273148\pi\)
\(920\) 1.54587 0.0509657
\(921\) 65.2125 2.14882
\(922\) 42.2301 1.39078
\(923\) 41.0903 1.35250
\(924\) 0 0
\(925\) −4.39960 −0.144658
\(926\) −22.0593 −0.724915
\(927\) −35.2125 −1.15653
\(928\) −9.32780 −0.306200
\(929\) −18.4828 −0.606401 −0.303200 0.952927i \(-0.598055\pi\)
−0.303200 + 0.952927i \(0.598055\pi\)
\(930\) 24.5692 0.805656
\(931\) −2.62106 −0.0859019
\(932\) 20.7201 0.678710
\(933\) −60.1578 −1.96948
\(934\) −21.3716 −0.699301
\(935\) 0 0
\(936\) 7.82718 0.255839
\(937\) 1.93426 0.0631894 0.0315947 0.999501i \(-0.489941\pi\)
0.0315947 + 0.999501i \(0.489941\pi\)
\(938\) −2.68556 −0.0876867
\(939\) −29.9469 −0.977280
\(940\) −12.3424 −0.402565
\(941\) 17.7819 0.579672 0.289836 0.957076i \(-0.406399\pi\)
0.289836 + 0.957076i \(0.406399\pi\)
\(942\) −18.9111 −0.616157
\(943\) 7.43216 0.242024
\(944\) 0.510590 0.0166183
\(945\) −0.414214 −0.0134744
\(946\) 0 0
\(947\) 13.6440 0.443371 0.221686 0.975118i \(-0.428844\pi\)
0.221686 + 0.975118i \(0.428844\pi\)
\(948\) −4.93766 −0.160368
\(949\) 9.44825 0.306703
\(950\) −2.62106 −0.0850385
\(951\) 72.5068 2.35119
\(952\) −4.44949 −0.144209
\(953\) −6.71082 −0.217385 −0.108692 0.994075i \(-0.534666\pi\)
−0.108692 + 0.994075i \(0.534666\pi\)
\(954\) 27.6544 0.895343
\(955\) −7.72208 −0.249881
\(956\) −22.0876 −0.714364
\(957\) 0 0
\(958\) −9.84888 −0.318203
\(959\) 19.2526 0.621699
\(960\) −2.41421 −0.0779184
\(961\) 72.5692 2.34094
\(962\) −12.1751 −0.392542
\(963\) 52.2979 1.68528
\(964\) 0.236073 0.00760339
\(965\) −3.22678 −0.103874
\(966\) 3.73205 0.120077
\(967\) 13.8214 0.444467 0.222233 0.974994i \(-0.428665\pi\)
0.222233 + 0.974994i \(0.428665\pi\)
\(968\) 0 0
\(969\) 28.1555 0.904485
\(970\) −2.35827 −0.0757196
\(971\) 52.9417 1.69898 0.849491 0.527604i \(-0.176909\pi\)
0.849491 + 0.527604i \(0.176909\pi\)
\(972\) 21.6569 0.694644
\(973\) −3.20612 −0.102783
\(974\) −35.0712 −1.12375
\(975\) −6.68092 −0.213961
\(976\) −2.63567 −0.0843659
\(977\) 8.77943 0.280879 0.140439 0.990089i \(-0.455148\pi\)
0.140439 + 0.990089i \(0.455148\pi\)
\(978\) 35.5040 1.13529
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 2.86976 0.0916242
\(982\) 1.40389 0.0448001
\(983\) 13.4596 0.429295 0.214648 0.976692i \(-0.431140\pi\)
0.214648 + 0.976692i \(0.431140\pi\)
\(984\) −11.6070 −0.370017
\(985\) −13.2066 −0.420798
\(986\) −41.5040 −1.32176
\(987\) −29.7972 −0.948456
\(988\) −7.25334 −0.230759
\(989\) −17.2235 −0.547674
\(990\) 0 0
\(991\) 60.1845 1.91182 0.955912 0.293653i \(-0.0948711\pi\)
0.955912 + 0.293653i \(0.0948711\pi\)
\(992\) −10.1769 −0.323117
\(993\) −23.3636 −0.741421
\(994\) −14.8484 −0.470961
\(995\) 5.99270 0.189981
\(996\) −10.3196 −0.326989
\(997\) 39.4998 1.25097 0.625486 0.780235i \(-0.284901\pi\)
0.625486 + 0.780235i \(0.284901\pi\)
\(998\) −5.73573 −0.181561
\(999\) −1.82237 −0.0576574
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 8470.2.a.cr.1.2 yes 4
11.10 odd 2 8470.2.a.cp.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8470.2.a.cp.1.1 4 11.10 odd 2
8470.2.a.cr.1.2 yes 4 1.1 even 1 trivial