Properties

Label 7098.2.a.ck.1.1
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.692021 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.692021 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.692021 q^{10} -0.664874 q^{11} -1.00000 q^{12} -1.00000 q^{14} +0.692021 q^{15} +1.00000 q^{16} -7.74094 q^{17} +1.00000 q^{18} -3.89977 q^{19} -0.692021 q^{20} +1.00000 q^{21} -0.664874 q^{22} +0.841166 q^{23} -1.00000 q^{24} -4.52111 q^{25} -1.00000 q^{27} -1.00000 q^{28} +8.98792 q^{29} +0.692021 q^{30} +6.26875 q^{31} +1.00000 q^{32} +0.664874 q^{33} -7.74094 q^{34} +0.692021 q^{35} +1.00000 q^{36} +11.1685 q^{37} -3.89977 q^{38} -0.692021 q^{40} -11.7409 q^{41} +1.00000 q^{42} +9.87800 q^{43} -0.664874 q^{44} -0.692021 q^{45} +0.841166 q^{46} -10.3177 q^{47} -1.00000 q^{48} +1.00000 q^{49} -4.52111 q^{50} +7.74094 q^{51} -7.92154 q^{53} -1.00000 q^{54} +0.460107 q^{55} -1.00000 q^{56} +3.89977 q^{57} +8.98792 q^{58} +10.8901 q^{59} +0.692021 q^{60} -6.05429 q^{61} +6.26875 q^{62} -1.00000 q^{63} +1.00000 q^{64} +0.664874 q^{66} -2.98792 q^{67} -7.74094 q^{68} -0.841166 q^{69} +0.692021 q^{70} +9.93661 q^{71} +1.00000 q^{72} +14.5918 q^{73} +11.1685 q^{74} +4.52111 q^{75} -3.89977 q^{76} +0.664874 q^{77} -1.60388 q^{79} -0.692021 q^{80} +1.00000 q^{81} -11.7409 q^{82} +4.39612 q^{83} +1.00000 q^{84} +5.35690 q^{85} +9.87800 q^{86} -8.98792 q^{87} -0.664874 q^{88} +0.454731 q^{89} -0.692021 q^{90} +0.841166 q^{92} -6.26875 q^{93} -10.3177 q^{94} +2.69873 q^{95} -1.00000 q^{96} +8.61596 q^{97} +1.00000 q^{98} -0.664874 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} - 3q^{7} + 3q^{8} + 3q^{9} + 3q^{10} - 3q^{11} - 3q^{12} - 3q^{14} - 3q^{15} + 3q^{16} - 9q^{17} + 3q^{18} + 11q^{19} + 3q^{20} + 3q^{21} - 3q^{22} + 11q^{23} - 3q^{24} + 2q^{25} - 3q^{27} - 3q^{28} + 8q^{29} - 3q^{30} + 11q^{31} + 3q^{32} + 3q^{33} - 9q^{34} - 3q^{35} + 3q^{36} + 3q^{37} + 11q^{38} + 3q^{40} - 21q^{41} + 3q^{42} + 10q^{43} - 3q^{44} + 3q^{45} + 11q^{46} - 14q^{47} - 3q^{48} + 3q^{49} + 2q^{50} + 9q^{51} + 2q^{53} - 3q^{54} - 24q^{55} - 3q^{56} - 11q^{57} + 8q^{58} + 32q^{59} - 3q^{60} - 6q^{61} + 11q^{62} - 3q^{63} + 3q^{64} + 3q^{66} + 10q^{67} - 9q^{68} - 11q^{69} - 3q^{70} - 21q^{71} + 3q^{72} + 16q^{73} + 3q^{74} - 2q^{75} + 11q^{76} + 3q^{77} + 4q^{79} + 3q^{80} + 3q^{81} - 21q^{82} + 22q^{83} + 3q^{84} + 12q^{85} + 10q^{86} - 8q^{87} - 3q^{88} - 21q^{89} + 3q^{90} + 11q^{92} - 11q^{93} - 14q^{94} + 25q^{95} - 3q^{96} + 36q^{97} + 3q^{98} - 3q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −0.692021 −0.309481 −0.154741 0.987955i \(-0.549454\pi\)
−0.154741 + 0.987955i \(0.549454\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −0.692021 −0.218836
\(11\) −0.664874 −0.200467 −0.100234 0.994964i \(-0.531959\pi\)
−0.100234 + 0.994964i \(0.531959\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) 0.692021 0.178679
\(16\) 1.00000 0.250000
\(17\) −7.74094 −1.87745 −0.938727 0.344662i \(-0.887993\pi\)
−0.938727 + 0.344662i \(0.887993\pi\)
\(18\) 1.00000 0.235702
\(19\) −3.89977 −0.894669 −0.447335 0.894367i \(-0.647627\pi\)
−0.447335 + 0.894367i \(0.647627\pi\)
\(20\) −0.692021 −0.154741
\(21\) 1.00000 0.218218
\(22\) −0.664874 −0.141752
\(23\) 0.841166 0.175395 0.0876977 0.996147i \(-0.472049\pi\)
0.0876977 + 0.996147i \(0.472049\pi\)
\(24\) −1.00000 −0.204124
\(25\) −4.52111 −0.904221
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) 8.98792 1.66901 0.834507 0.550997i \(-0.185752\pi\)
0.834507 + 0.550997i \(0.185752\pi\)
\(30\) 0.692021 0.126345
\(31\) 6.26875 1.12590 0.562950 0.826491i \(-0.309666\pi\)
0.562950 + 0.826491i \(0.309666\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.664874 0.115740
\(34\) −7.74094 −1.32756
\(35\) 0.692021 0.116973
\(36\) 1.00000 0.166667
\(37\) 11.1685 1.83609 0.918047 0.396472i \(-0.129766\pi\)
0.918047 + 0.396472i \(0.129766\pi\)
\(38\) −3.89977 −0.632627
\(39\) 0 0
\(40\) −0.692021 −0.109418
\(41\) −11.7409 −1.83363 −0.916813 0.399316i \(-0.869248\pi\)
−0.916813 + 0.399316i \(0.869248\pi\)
\(42\) 1.00000 0.154303
\(43\) 9.87800 1.50638 0.753191 0.657802i \(-0.228514\pi\)
0.753191 + 0.657802i \(0.228514\pi\)
\(44\) −0.664874 −0.100234
\(45\) −0.692021 −0.103160
\(46\) 0.841166 0.124023
\(47\) −10.3177 −1.50499 −0.752493 0.658600i \(-0.771149\pi\)
−0.752493 + 0.658600i \(0.771149\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −4.52111 −0.639381
\(51\) 7.74094 1.08395
\(52\) 0 0
\(53\) −7.92154 −1.08811 −0.544054 0.839050i \(-0.683111\pi\)
−0.544054 + 0.839050i \(0.683111\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0.460107 0.0620409
\(56\) −1.00000 −0.133631
\(57\) 3.89977 0.516537
\(58\) 8.98792 1.18017
\(59\) 10.8901 1.41777 0.708884 0.705325i \(-0.249199\pi\)
0.708884 + 0.705325i \(0.249199\pi\)
\(60\) 0.692021 0.0893396
\(61\) −6.05429 −0.775173 −0.387586 0.921833i \(-0.626691\pi\)
−0.387586 + 0.921833i \(0.626691\pi\)
\(62\) 6.26875 0.796132
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0.664874 0.0818404
\(67\) −2.98792 −0.365032 −0.182516 0.983203i \(-0.558424\pi\)
−0.182516 + 0.983203i \(0.558424\pi\)
\(68\) −7.74094 −0.938727
\(69\) −0.841166 −0.101265
\(70\) 0.692021 0.0827124
\(71\) 9.93661 1.17926 0.589629 0.807674i \(-0.299274\pi\)
0.589629 + 0.807674i \(0.299274\pi\)
\(72\) 1.00000 0.117851
\(73\) 14.5918 1.70784 0.853920 0.520404i \(-0.174219\pi\)
0.853920 + 0.520404i \(0.174219\pi\)
\(74\) 11.1685 1.29831
\(75\) 4.52111 0.522052
\(76\) −3.89977 −0.447335
\(77\) 0.664874 0.0757695
\(78\) 0 0
\(79\) −1.60388 −0.180450 −0.0902250 0.995921i \(-0.528759\pi\)
−0.0902250 + 0.995921i \(0.528759\pi\)
\(80\) −0.692021 −0.0773704
\(81\) 1.00000 0.111111
\(82\) −11.7409 −1.29657
\(83\) 4.39612 0.482537 0.241269 0.970458i \(-0.422436\pi\)
0.241269 + 0.970458i \(0.422436\pi\)
\(84\) 1.00000 0.109109
\(85\) 5.35690 0.581037
\(86\) 9.87800 1.06517
\(87\) −8.98792 −0.963606
\(88\) −0.664874 −0.0708758
\(89\) 0.454731 0.0482013 0.0241007 0.999710i \(-0.492328\pi\)
0.0241007 + 0.999710i \(0.492328\pi\)
\(90\) −0.692021 −0.0729455
\(91\) 0 0
\(92\) 0.841166 0.0876977
\(93\) −6.26875 −0.650039
\(94\) −10.3177 −1.06419
\(95\) 2.69873 0.276883
\(96\) −1.00000 −0.102062
\(97\) 8.61596 0.874818 0.437409 0.899263i \(-0.355896\pi\)
0.437409 + 0.899263i \(0.355896\pi\)
\(98\) 1.00000 0.101015
\(99\) −0.664874 −0.0668224
\(100\) −4.52111 −0.452111
\(101\) 17.7235 1.76355 0.881776 0.471668i \(-0.156348\pi\)
0.881776 + 0.471668i \(0.156348\pi\)
\(102\) 7.74094 0.766467
\(103\) −12.5972 −1.24124 −0.620618 0.784113i \(-0.713118\pi\)
−0.620618 + 0.784113i \(0.713118\pi\)
\(104\) 0 0
\(105\) −0.692021 −0.0675344
\(106\) −7.92154 −0.769408
\(107\) −0.692021 −0.0669002 −0.0334501 0.999440i \(-0.510649\pi\)
−0.0334501 + 0.999440i \(0.510649\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 8.03923 0.770018 0.385009 0.922913i \(-0.374198\pi\)
0.385009 + 0.922913i \(0.374198\pi\)
\(110\) 0.460107 0.0438695
\(111\) −11.1685 −1.06007
\(112\) −1.00000 −0.0944911
\(113\) 12.5918 1.18454 0.592268 0.805741i \(-0.298233\pi\)
0.592268 + 0.805741i \(0.298233\pi\)
\(114\) 3.89977 0.365247
\(115\) −0.582105 −0.0542816
\(116\) 8.98792 0.834507
\(117\) 0 0
\(118\) 10.8901 1.00251
\(119\) 7.74094 0.709611
\(120\) 0.692021 0.0631726
\(121\) −10.5579 −0.959813
\(122\) −6.05429 −0.548130
\(123\) 11.7409 1.05864
\(124\) 6.26875 0.562950
\(125\) 6.58881 0.589321
\(126\) −1.00000 −0.0890871
\(127\) 6.76809 0.600571 0.300285 0.953849i \(-0.402918\pi\)
0.300285 + 0.953849i \(0.402918\pi\)
\(128\) 1.00000 0.0883883
\(129\) −9.87800 −0.869710
\(130\) 0 0
\(131\) 0.987918 0.0863148 0.0431574 0.999068i \(-0.486258\pi\)
0.0431574 + 0.999068i \(0.486258\pi\)
\(132\) 0.664874 0.0578699
\(133\) 3.89977 0.338153
\(134\) −2.98792 −0.258117
\(135\) 0.692021 0.0595597
\(136\) −7.74094 −0.663780
\(137\) −0.572417 −0.0489048 −0.0244524 0.999701i \(-0.507784\pi\)
−0.0244524 + 0.999701i \(0.507784\pi\)
\(138\) −0.841166 −0.0716048
\(139\) 16.7627 1.42179 0.710897 0.703296i \(-0.248289\pi\)
0.710897 + 0.703296i \(0.248289\pi\)
\(140\) 0.692021 0.0584865
\(141\) 10.3177 0.868904
\(142\) 9.93661 0.833862
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −6.21983 −0.516529
\(146\) 14.5918 1.20763
\(147\) −1.00000 −0.0824786
\(148\) 11.1685 0.918047
\(149\) −12.0194 −0.984666 −0.492333 0.870407i \(-0.663856\pi\)
−0.492333 + 0.870407i \(0.663856\pi\)
\(150\) 4.52111 0.369147
\(151\) −19.3491 −1.57461 −0.787305 0.616564i \(-0.788524\pi\)
−0.787305 + 0.616564i \(0.788524\pi\)
\(152\) −3.89977 −0.316313
\(153\) −7.74094 −0.625818
\(154\) 0.664874 0.0535771
\(155\) −4.33811 −0.348445
\(156\) 0 0
\(157\) −2.19567 −0.175233 −0.0876167 0.996154i \(-0.527925\pi\)
−0.0876167 + 0.996154i \(0.527925\pi\)
\(158\) −1.60388 −0.127597
\(159\) 7.92154 0.628219
\(160\) −0.692021 −0.0547091
\(161\) −0.841166 −0.0662932
\(162\) 1.00000 0.0785674
\(163\) 19.3599 1.51638 0.758191 0.652032i \(-0.226083\pi\)
0.758191 + 0.652032i \(0.226083\pi\)
\(164\) −11.7409 −0.916813
\(165\) −0.460107 −0.0358193
\(166\) 4.39612 0.341205
\(167\) 20.5133 1.58737 0.793685 0.608329i \(-0.208160\pi\)
0.793685 + 0.608329i \(0.208160\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) 5.35690 0.410855
\(171\) −3.89977 −0.298223
\(172\) 9.87800 0.753191
\(173\) 9.08815 0.690959 0.345479 0.938426i \(-0.387716\pi\)
0.345479 + 0.938426i \(0.387716\pi\)
\(174\) −8.98792 −0.681372
\(175\) 4.52111 0.341764
\(176\) −0.664874 −0.0501168
\(177\) −10.8901 −0.818549
\(178\) 0.454731 0.0340835
\(179\) 8.98792 0.671789 0.335894 0.941900i \(-0.390962\pi\)
0.335894 + 0.941900i \(0.390962\pi\)
\(180\) −0.692021 −0.0515802
\(181\) −1.76941 −0.131519 −0.0657597 0.997835i \(-0.520947\pi\)
−0.0657597 + 0.997835i \(0.520947\pi\)
\(182\) 0 0
\(183\) 6.05429 0.447546
\(184\) 0.841166 0.0620116
\(185\) −7.72886 −0.568237
\(186\) −6.26875 −0.459647
\(187\) 5.14675 0.376368
\(188\) −10.3177 −0.752493
\(189\) 1.00000 0.0727393
\(190\) 2.69873 0.195786
\(191\) −16.7657 −1.21312 −0.606561 0.795037i \(-0.707452\pi\)
−0.606561 + 0.795037i \(0.707452\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 16.9933 1.22320 0.611602 0.791166i \(-0.290525\pi\)
0.611602 + 0.791166i \(0.290525\pi\)
\(194\) 8.61596 0.618590
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −4.90946 −0.349785 −0.174892 0.984588i \(-0.555958\pi\)
−0.174892 + 0.984588i \(0.555958\pi\)
\(198\) −0.664874 −0.0472506
\(199\) −14.9729 −1.06140 −0.530699 0.847561i \(-0.678070\pi\)
−0.530699 + 0.847561i \(0.678070\pi\)
\(200\) −4.52111 −0.319690
\(201\) 2.98792 0.210752
\(202\) 17.7235 1.24702
\(203\) −8.98792 −0.630828
\(204\) 7.74094 0.541974
\(205\) 8.12498 0.567473
\(206\) −12.5972 −0.877686
\(207\) 0.841166 0.0584651
\(208\) 0 0
\(209\) 2.59286 0.179352
\(210\) −0.692021 −0.0477540
\(211\) −6.98792 −0.481068 −0.240534 0.970641i \(-0.577323\pi\)
−0.240534 + 0.970641i \(0.577323\pi\)
\(212\) −7.92154 −0.544054
\(213\) −9.93661 −0.680845
\(214\) −0.692021 −0.0473056
\(215\) −6.83579 −0.466197
\(216\) −1.00000 −0.0680414
\(217\) −6.26875 −0.425550
\(218\) 8.03923 0.544485
\(219\) −14.5918 −0.986022
\(220\) 0.460107 0.0310204
\(221\) 0 0
\(222\) −11.1685 −0.749582
\(223\) −1.13467 −0.0759832 −0.0379916 0.999278i \(-0.512096\pi\)
−0.0379916 + 0.999278i \(0.512096\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −4.52111 −0.301407
\(226\) 12.5918 0.837594
\(227\) 5.26205 0.349254 0.174627 0.984635i \(-0.444128\pi\)
0.174627 + 0.984635i \(0.444128\pi\)
\(228\) 3.89977 0.258269
\(229\) 3.40821 0.225221 0.112610 0.993639i \(-0.464079\pi\)
0.112610 + 0.993639i \(0.464079\pi\)
\(230\) −0.582105 −0.0383829
\(231\) −0.664874 −0.0437455
\(232\) 8.98792 0.590086
\(233\) −23.6775 −1.55117 −0.775584 0.631245i \(-0.782544\pi\)
−0.775584 + 0.631245i \(0.782544\pi\)
\(234\) 0 0
\(235\) 7.14005 0.465765
\(236\) 10.8901 0.708884
\(237\) 1.60388 0.104183
\(238\) 7.74094 0.501771
\(239\) −20.2610 −1.31057 −0.655287 0.755380i \(-0.727452\pi\)
−0.655287 + 0.755380i \(0.727452\pi\)
\(240\) 0.692021 0.0446698
\(241\) 0.0435405 0.00280469 0.00140235 0.999999i \(-0.499554\pi\)
0.00140235 + 0.999999i \(0.499554\pi\)
\(242\) −10.5579 −0.678690
\(243\) −1.00000 −0.0641500
\(244\) −6.05429 −0.387586
\(245\) −0.692021 −0.0442116
\(246\) 11.7409 0.748575
\(247\) 0 0
\(248\) 6.26875 0.398066
\(249\) −4.39612 −0.278593
\(250\) 6.58881 0.416713
\(251\) −27.3056 −1.72351 −0.861757 0.507322i \(-0.830636\pi\)
−0.861757 + 0.507322i \(0.830636\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −0.559270 −0.0351610
\(254\) 6.76809 0.424667
\(255\) −5.35690 −0.335462
\(256\) 1.00000 0.0625000
\(257\) 7.97584 0.497519 0.248760 0.968565i \(-0.419977\pi\)
0.248760 + 0.968565i \(0.419977\pi\)
\(258\) −9.87800 −0.614978
\(259\) −11.1685 −0.693978
\(260\) 0 0
\(261\) 8.98792 0.556338
\(262\) 0.987918 0.0610338
\(263\) 9.37867 0.578313 0.289157 0.957282i \(-0.406625\pi\)
0.289157 + 0.957282i \(0.406625\pi\)
\(264\) 0.664874 0.0409202
\(265\) 5.48188 0.336749
\(266\) 3.89977 0.239110
\(267\) −0.454731 −0.0278291
\(268\) −2.98792 −0.182516
\(269\) 7.96077 0.485377 0.242688 0.970104i \(-0.421971\pi\)
0.242688 + 0.970104i \(0.421971\pi\)
\(270\) 0.692021 0.0421151
\(271\) 10.6377 0.646196 0.323098 0.946366i \(-0.395276\pi\)
0.323098 + 0.946366i \(0.395276\pi\)
\(272\) −7.74094 −0.469363
\(273\) 0 0
\(274\) −0.572417 −0.0345809
\(275\) 3.00597 0.181267
\(276\) −0.841166 −0.0506323
\(277\) 26.4674 1.59027 0.795136 0.606431i \(-0.207399\pi\)
0.795136 + 0.606431i \(0.207399\pi\)
\(278\) 16.7627 1.00536
\(279\) 6.26875 0.375300
\(280\) 0.692021 0.0413562
\(281\) 5.60388 0.334299 0.167150 0.985932i \(-0.446544\pi\)
0.167150 + 0.985932i \(0.446544\pi\)
\(282\) 10.3177 0.614408
\(283\) −11.3351 −0.673803 −0.336902 0.941540i \(-0.609379\pi\)
−0.336902 + 0.941540i \(0.609379\pi\)
\(284\) 9.93661 0.589629
\(285\) −2.69873 −0.159859
\(286\) 0 0
\(287\) 11.7409 0.693046
\(288\) 1.00000 0.0589256
\(289\) 42.9221 2.52483
\(290\) −6.21983 −0.365241
\(291\) −8.61596 −0.505076
\(292\) 14.5918 0.853920
\(293\) 20.7530 1.21240 0.606202 0.795311i \(-0.292692\pi\)
0.606202 + 0.795311i \(0.292692\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −7.53617 −0.438773
\(296\) 11.1685 0.649157
\(297\) 0.664874 0.0385799
\(298\) −12.0194 −0.696264
\(299\) 0 0
\(300\) 4.52111 0.261026
\(301\) −9.87800 −0.569359
\(302\) −19.3491 −1.11342
\(303\) −17.7235 −1.01819
\(304\) −3.89977 −0.223667
\(305\) 4.18970 0.239902
\(306\) −7.74094 −0.442520
\(307\) 14.5961 0.833044 0.416522 0.909126i \(-0.363249\pi\)
0.416522 + 0.909126i \(0.363249\pi\)
\(308\) 0.664874 0.0378847
\(309\) 12.5972 0.716628
\(310\) −4.33811 −0.246388
\(311\) −1.82371 −0.103413 −0.0517065 0.998662i \(-0.516466\pi\)
−0.0517065 + 0.998662i \(0.516466\pi\)
\(312\) 0 0
\(313\) 34.6112 1.95634 0.978170 0.207808i \(-0.0666331\pi\)
0.978170 + 0.207808i \(0.0666331\pi\)
\(314\) −2.19567 −0.123909
\(315\) 0.692021 0.0389910
\(316\) −1.60388 −0.0902250
\(317\) −4.49396 −0.252406 −0.126203 0.992004i \(-0.540279\pi\)
−0.126203 + 0.992004i \(0.540279\pi\)
\(318\) 7.92154 0.444218
\(319\) −5.97584 −0.334583
\(320\) −0.692021 −0.0386852
\(321\) 0.692021 0.0386249
\(322\) −0.841166 −0.0468764
\(323\) 30.1879 1.67970
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 19.3599 1.07224
\(327\) −8.03923 −0.444570
\(328\) −11.7409 −0.648285
\(329\) 10.3177 0.568831
\(330\) −0.460107 −0.0253281
\(331\) −8.75733 −0.481347 −0.240673 0.970606i \(-0.577368\pi\)
−0.240673 + 0.970606i \(0.577368\pi\)
\(332\) 4.39612 0.241269
\(333\) 11.1685 0.612031
\(334\) 20.5133 1.12244
\(335\) 2.06770 0.112971
\(336\) 1.00000 0.0545545
\(337\) 12.6136 0.687105 0.343552 0.939134i \(-0.388370\pi\)
0.343552 + 0.939134i \(0.388370\pi\)
\(338\) 0 0
\(339\) −12.5918 −0.683892
\(340\) 5.35690 0.290518
\(341\) −4.16793 −0.225706
\(342\) −3.89977 −0.210876
\(343\) −1.00000 −0.0539949
\(344\) 9.87800 0.532586
\(345\) 0.582105 0.0313395
\(346\) 9.08815 0.488582
\(347\) 19.4034 1.04163 0.520815 0.853670i \(-0.325628\pi\)
0.520815 + 0.853670i \(0.325628\pi\)
\(348\) −8.98792 −0.481803
\(349\) 0.415502 0.0222413 0.0111207 0.999938i \(-0.496460\pi\)
0.0111207 + 0.999938i \(0.496460\pi\)
\(350\) 4.52111 0.241663
\(351\) 0 0
\(352\) −0.664874 −0.0354379
\(353\) 16.4450 0.875281 0.437641 0.899150i \(-0.355814\pi\)
0.437641 + 0.899150i \(0.355814\pi\)
\(354\) −10.8901 −0.578801
\(355\) −6.87635 −0.364959
\(356\) 0.454731 0.0241007
\(357\) −7.74094 −0.409694
\(358\) 8.98792 0.475026
\(359\) 6.61058 0.348893 0.174447 0.984667i \(-0.444186\pi\)
0.174447 + 0.984667i \(0.444186\pi\)
\(360\) −0.692021 −0.0364727
\(361\) −3.79178 −0.199567
\(362\) −1.76941 −0.0929983
\(363\) 10.5579 0.554148
\(364\) 0 0
\(365\) −10.0978 −0.528545
\(366\) 6.05429 0.316463
\(367\) 15.5773 0.813129 0.406565 0.913622i \(-0.366727\pi\)
0.406565 + 0.913622i \(0.366727\pi\)
\(368\) 0.841166 0.0438488
\(369\) −11.7409 −0.611209
\(370\) −7.72886 −0.401804
\(371\) 7.92154 0.411266
\(372\) −6.26875 −0.325020
\(373\) −1.81940 −0.0942048 −0.0471024 0.998890i \(-0.514999\pi\)
−0.0471024 + 0.998890i \(0.514999\pi\)
\(374\) 5.14675 0.266132
\(375\) −6.58881 −0.340245
\(376\) −10.3177 −0.532093
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) −7.53617 −0.387107 −0.193554 0.981090i \(-0.562001\pi\)
−0.193554 + 0.981090i \(0.562001\pi\)
\(380\) 2.69873 0.138442
\(381\) −6.76809 −0.346740
\(382\) −16.7657 −0.857807
\(383\) 3.78017 0.193158 0.0965788 0.995325i \(-0.469210\pi\)
0.0965788 + 0.995325i \(0.469210\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −0.460107 −0.0234492
\(386\) 16.9933 0.864936
\(387\) 9.87800 0.502127
\(388\) 8.61596 0.437409
\(389\) 5.01208 0.254123 0.127061 0.991895i \(-0.459446\pi\)
0.127061 + 0.991895i \(0.459446\pi\)
\(390\) 0 0
\(391\) −6.51142 −0.329297
\(392\) 1.00000 0.0505076
\(393\) −0.987918 −0.0498339
\(394\) −4.90946 −0.247335
\(395\) 1.10992 0.0558459
\(396\) −0.664874 −0.0334112
\(397\) −5.05562 −0.253734 −0.126867 0.991920i \(-0.540492\pi\)
−0.126867 + 0.991920i \(0.540492\pi\)
\(398\) −14.9729 −0.750521
\(399\) −3.89977 −0.195233
\(400\) −4.52111 −0.226055
\(401\) 38.8745 1.94130 0.970651 0.240492i \(-0.0773089\pi\)
0.970651 + 0.240492i \(0.0773089\pi\)
\(402\) 2.98792 0.149024
\(403\) 0 0
\(404\) 17.7235 0.881776
\(405\) −0.692021 −0.0343868
\(406\) −8.98792 −0.446063
\(407\) −7.42566 −0.368077
\(408\) 7.74094 0.383234
\(409\) 32.6461 1.61425 0.807123 0.590384i \(-0.201024\pi\)
0.807123 + 0.590384i \(0.201024\pi\)
\(410\) 8.12498 0.401264
\(411\) 0.572417 0.0282352
\(412\) −12.5972 −0.620618
\(413\) −10.8901 −0.535866
\(414\) 0.841166 0.0413411
\(415\) −3.04221 −0.149336
\(416\) 0 0
\(417\) −16.7627 −0.820873
\(418\) 2.59286 0.126821
\(419\) 23.3250 1.13950 0.569750 0.821818i \(-0.307040\pi\)
0.569750 + 0.821818i \(0.307040\pi\)
\(420\) −0.692021 −0.0337672
\(421\) 18.2282 0.888388 0.444194 0.895931i \(-0.353490\pi\)
0.444194 + 0.895931i \(0.353490\pi\)
\(422\) −6.98792 −0.340167
\(423\) −10.3177 −0.501662
\(424\) −7.92154 −0.384704
\(425\) 34.9976 1.69763
\(426\) −9.93661 −0.481430
\(427\) 6.05429 0.292988
\(428\) −0.692021 −0.0334501
\(429\) 0 0
\(430\) −6.83579 −0.329651
\(431\) −4.02177 −0.193722 −0.0968609 0.995298i \(-0.530880\pi\)
−0.0968609 + 0.995298i \(0.530880\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −11.4470 −0.550106 −0.275053 0.961429i \(-0.588695\pi\)
−0.275053 + 0.961429i \(0.588695\pi\)
\(434\) −6.26875 −0.300910
\(435\) 6.21983 0.298218
\(436\) 8.03923 0.385009
\(437\) −3.28036 −0.156921
\(438\) −14.5918 −0.697223
\(439\) −37.6450 −1.79670 −0.898349 0.439282i \(-0.855233\pi\)
−0.898349 + 0.439282i \(0.855233\pi\)
\(440\) 0.460107 0.0219348
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −31.7797 −1.50990 −0.754949 0.655783i \(-0.772339\pi\)
−0.754949 + 0.655783i \(0.772339\pi\)
\(444\) −11.1685 −0.530035
\(445\) −0.314683 −0.0149174
\(446\) −1.13467 −0.0537282
\(447\) 12.0194 0.568497
\(448\) −1.00000 −0.0472456
\(449\) 1.82371 0.0860661 0.0430331 0.999074i \(-0.486298\pi\)
0.0430331 + 0.999074i \(0.486298\pi\)
\(450\) −4.52111 −0.213127
\(451\) 7.80625 0.367582
\(452\) 12.5918 0.592268
\(453\) 19.3491 0.909101
\(454\) 5.26205 0.246960
\(455\) 0 0
\(456\) 3.89977 0.182624
\(457\) 11.0774 0.518179 0.259089 0.965853i \(-0.416578\pi\)
0.259089 + 0.965853i \(0.416578\pi\)
\(458\) 3.40821 0.159255
\(459\) 7.74094 0.361316
\(460\) −0.582105 −0.0271408
\(461\) −3.82908 −0.178338 −0.0891691 0.996016i \(-0.528421\pi\)
−0.0891691 + 0.996016i \(0.528421\pi\)
\(462\) −0.664874 −0.0309328
\(463\) 36.0737 1.67649 0.838243 0.545297i \(-0.183583\pi\)
0.838243 + 0.545297i \(0.183583\pi\)
\(464\) 8.98792 0.417254
\(465\) 4.33811 0.201175
\(466\) −23.6775 −1.09684
\(467\) −38.3806 −1.77604 −0.888021 0.459803i \(-0.847920\pi\)
−0.888021 + 0.459803i \(0.847920\pi\)
\(468\) 0 0
\(469\) 2.98792 0.137969
\(470\) 7.14005 0.329346
\(471\) 2.19567 0.101171
\(472\) 10.8901 0.501257
\(473\) −6.56763 −0.301980
\(474\) 1.60388 0.0736684
\(475\) 17.6313 0.808979
\(476\) 7.74094 0.354805
\(477\) −7.92154 −0.362703
\(478\) −20.2610 −0.926716
\(479\) 11.3599 0.519046 0.259523 0.965737i \(-0.416435\pi\)
0.259523 + 0.965737i \(0.416435\pi\)
\(480\) 0.692021 0.0315863
\(481\) 0 0
\(482\) 0.0435405 0.00198322
\(483\) 0.841166 0.0382744
\(484\) −10.5579 −0.479906
\(485\) −5.96243 −0.270740
\(486\) −1.00000 −0.0453609
\(487\) 37.4577 1.69737 0.848686 0.528898i \(-0.177395\pi\)
0.848686 + 0.528898i \(0.177395\pi\)
\(488\) −6.05429 −0.274065
\(489\) −19.3599 −0.875484
\(490\) −0.692021 −0.0312623
\(491\) 24.0200 1.08401 0.542003 0.840377i \(-0.317666\pi\)
0.542003 + 0.840377i \(0.317666\pi\)
\(492\) 11.7409 0.529322
\(493\) −69.5749 −3.13350
\(494\) 0 0
\(495\) 0.460107 0.0206803
\(496\) 6.26875 0.281475
\(497\) −9.93661 −0.445718
\(498\) −4.39612 −0.196995
\(499\) 36.3043 1.62520 0.812601 0.582821i \(-0.198051\pi\)
0.812601 + 0.582821i \(0.198051\pi\)
\(500\) 6.58881 0.294661
\(501\) −20.5133 −0.916468
\(502\) −27.3056 −1.21871
\(503\) −15.6340 −0.697086 −0.348543 0.937293i \(-0.613323\pi\)
−0.348543 + 0.937293i \(0.613323\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −12.2650 −0.545787
\(506\) −0.559270 −0.0248626
\(507\) 0 0
\(508\) 6.76809 0.300285
\(509\) 12.0248 0.532988 0.266494 0.963837i \(-0.414135\pi\)
0.266494 + 0.963837i \(0.414135\pi\)
\(510\) −5.35690 −0.237207
\(511\) −14.5918 −0.645503
\(512\) 1.00000 0.0441942
\(513\) 3.89977 0.172179
\(514\) 7.97584 0.351799
\(515\) 8.71751 0.384140
\(516\) −9.87800 −0.434855
\(517\) 6.85995 0.301700
\(518\) −11.1685 −0.490717
\(519\) −9.08815 −0.398925
\(520\) 0 0
\(521\) −16.0054 −0.701208 −0.350604 0.936524i \(-0.614024\pi\)
−0.350604 + 0.936524i \(0.614024\pi\)
\(522\) 8.98792 0.393391
\(523\) 0.143768 0.00628654 0.00314327 0.999995i \(-0.498999\pi\)
0.00314327 + 0.999995i \(0.498999\pi\)
\(524\) 0.987918 0.0431574
\(525\) −4.52111 −0.197317
\(526\) 9.37867 0.408929
\(527\) −48.5260 −2.11383
\(528\) 0.664874 0.0289349
\(529\) −22.2924 −0.969236
\(530\) 5.48188 0.238118
\(531\) 10.8901 0.472589
\(532\) 3.89977 0.169077
\(533\) 0 0
\(534\) −0.454731 −0.0196781
\(535\) 0.478894 0.0207044
\(536\) −2.98792 −0.129058
\(537\) −8.98792 −0.387857
\(538\) 7.96077 0.343213
\(539\) −0.664874 −0.0286382
\(540\) 0.692021 0.0297799
\(541\) −30.2717 −1.30148 −0.650742 0.759299i \(-0.725542\pi\)
−0.650742 + 0.759299i \(0.725542\pi\)
\(542\) 10.6377 0.456930
\(543\) 1.76941 0.0759328
\(544\) −7.74094 −0.331890
\(545\) −5.56332 −0.238306
\(546\) 0 0
\(547\) 30.7681 1.31555 0.657774 0.753215i \(-0.271498\pi\)
0.657774 + 0.753215i \(0.271498\pi\)
\(548\) −0.572417 −0.0244524
\(549\) −6.05429 −0.258391
\(550\) 3.00597 0.128175
\(551\) −35.0508 −1.49322
\(552\) −0.841166 −0.0358024
\(553\) 1.60388 0.0682037
\(554\) 26.4674 1.12449
\(555\) 7.72886 0.328072
\(556\) 16.7627 0.710897
\(557\) −5.01208 −0.212369 −0.106184 0.994346i \(-0.533863\pi\)
−0.106184 + 0.994346i \(0.533863\pi\)
\(558\) 6.26875 0.265377
\(559\) 0 0
\(560\) 0.692021 0.0292432
\(561\) −5.14675 −0.217296
\(562\) 5.60388 0.236385
\(563\) 23.3793 0.985318 0.492659 0.870222i \(-0.336025\pi\)
0.492659 + 0.870222i \(0.336025\pi\)
\(564\) 10.3177 0.434452
\(565\) −8.71379 −0.366592
\(566\) −11.3351 −0.476451
\(567\) −1.00000 −0.0419961
\(568\) 9.93661 0.416931
\(569\) 25.9168 1.08649 0.543243 0.839575i \(-0.317196\pi\)
0.543243 + 0.839575i \(0.317196\pi\)
\(570\) −2.69873 −0.113037
\(571\) −19.3163 −0.808364 −0.404182 0.914679i \(-0.632444\pi\)
−0.404182 + 0.914679i \(0.632444\pi\)
\(572\) 0 0
\(573\) 16.7657 0.700397
\(574\) 11.7409 0.490057
\(575\) −3.80300 −0.158596
\(576\) 1.00000 0.0416667
\(577\) 17.7802 0.740198 0.370099 0.928992i \(-0.379324\pi\)
0.370099 + 0.928992i \(0.379324\pi\)
\(578\) 42.9221 1.78533
\(579\) −16.9933 −0.706217
\(580\) −6.21983 −0.258264
\(581\) −4.39612 −0.182382
\(582\) −8.61596 −0.357143
\(583\) 5.26683 0.218130
\(584\) 14.5918 0.603813
\(585\) 0 0
\(586\) 20.7530 0.857299
\(587\) −2.95300 −0.121883 −0.0609417 0.998141i \(-0.519410\pi\)
−0.0609417 + 0.998141i \(0.519410\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −24.4467 −1.00731
\(590\) −7.53617 −0.310259
\(591\) 4.90946 0.201948
\(592\) 11.1685 0.459023
\(593\) −27.9758 −1.14883 −0.574415 0.818564i \(-0.694770\pi\)
−0.574415 + 0.818564i \(0.694770\pi\)
\(594\) 0.664874 0.0272801
\(595\) −5.35690 −0.219611
\(596\) −12.0194 −0.492333
\(597\) 14.9729 0.612798
\(598\) 0 0
\(599\) −9.53617 −0.389637 −0.194819 0.980839i \(-0.562412\pi\)
−0.194819 + 0.980839i \(0.562412\pi\)
\(600\) 4.52111 0.184573
\(601\) 21.9215 0.894198 0.447099 0.894484i \(-0.352457\pi\)
0.447099 + 0.894484i \(0.352457\pi\)
\(602\) −9.87800 −0.402597
\(603\) −2.98792 −0.121677
\(604\) −19.3491 −0.787305
\(605\) 7.30632 0.297044
\(606\) −17.7235 −0.719967
\(607\) −45.6625 −1.85338 −0.926691 0.375823i \(-0.877360\pi\)
−0.926691 + 0.375823i \(0.877360\pi\)
\(608\) −3.89977 −0.158157
\(609\) 8.98792 0.364209
\(610\) 4.18970 0.169636
\(611\) 0 0
\(612\) −7.74094 −0.312909
\(613\) 29.7071 1.19986 0.599929 0.800053i \(-0.295195\pi\)
0.599929 + 0.800053i \(0.295195\pi\)
\(614\) 14.5961 0.589051
\(615\) −8.12498 −0.327631
\(616\) 0.664874 0.0267886
\(617\) −37.4034 −1.50581 −0.752903 0.658132i \(-0.771347\pi\)
−0.752903 + 0.658132i \(0.771347\pi\)
\(618\) 12.5972 0.506733
\(619\) −12.0000 −0.482321 −0.241160 0.970485i \(-0.577528\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(620\) −4.33811 −0.174223
\(621\) −0.841166 −0.0337548
\(622\) −1.82371 −0.0731240
\(623\) −0.454731 −0.0182184
\(624\) 0 0
\(625\) 18.0459 0.721837
\(626\) 34.6112 1.38334
\(627\) −2.59286 −0.103549
\(628\) −2.19567 −0.0876167
\(629\) −86.4548 −3.44718
\(630\) 0.692021 0.0275708
\(631\) 17.3163 0.689353 0.344676 0.938722i \(-0.387989\pi\)
0.344676 + 0.938722i \(0.387989\pi\)
\(632\) −1.60388 −0.0637987
\(633\) 6.98792 0.277745
\(634\) −4.49396 −0.178478
\(635\) −4.68366 −0.185865
\(636\) 7.92154 0.314110
\(637\) 0 0
\(638\) −5.97584 −0.236586
\(639\) 9.93661 0.393086
\(640\) −0.692021 −0.0273546
\(641\) −27.3599 −1.08065 −0.540325 0.841456i \(-0.681699\pi\)
−0.540325 + 0.841456i \(0.681699\pi\)
\(642\) 0.692021 0.0273119
\(643\) −19.1675 −0.755891 −0.377945 0.925828i \(-0.623369\pi\)
−0.377945 + 0.925828i \(0.623369\pi\)
\(644\) −0.841166 −0.0331466
\(645\) 6.83579 0.269159
\(646\) 30.1879 1.18773
\(647\) −27.4711 −1.08000 −0.540001 0.841665i \(-0.681576\pi\)
−0.540001 + 0.841665i \(0.681576\pi\)
\(648\) 1.00000 0.0392837
\(649\) −7.24054 −0.284216
\(650\) 0 0
\(651\) 6.26875 0.245692
\(652\) 19.3599 0.758191
\(653\) 43.2680 1.69321 0.846604 0.532223i \(-0.178643\pi\)
0.846604 + 0.532223i \(0.178643\pi\)
\(654\) −8.03923 −0.314359
\(655\) −0.683661 −0.0267128
\(656\) −11.7409 −0.458407
\(657\) 14.5918 0.569280
\(658\) 10.3177 0.402225
\(659\) −19.8619 −0.773709 −0.386854 0.922141i \(-0.626438\pi\)
−0.386854 + 0.922141i \(0.626438\pi\)
\(660\) −0.460107 −0.0179097
\(661\) −13.4625 −0.523631 −0.261815 0.965118i \(-0.584321\pi\)
−0.261815 + 0.965118i \(0.584321\pi\)
\(662\) −8.75733 −0.340363
\(663\) 0 0
\(664\) 4.39612 0.170603
\(665\) −2.69873 −0.104652
\(666\) 11.1685 0.432771
\(667\) 7.56033 0.292737
\(668\) 20.5133 0.793685
\(669\) 1.13467 0.0438689
\(670\) 2.06770 0.0798824
\(671\) 4.02535 0.155397
\(672\) 1.00000 0.0385758
\(673\) −19.2731 −0.742922 −0.371461 0.928449i \(-0.621143\pi\)
−0.371461 + 0.928449i \(0.621143\pi\)
\(674\) 12.6136 0.485856
\(675\) 4.52111 0.174017
\(676\) 0 0
\(677\) −20.5241 −0.788805 −0.394402 0.918938i \(-0.629048\pi\)
−0.394402 + 0.918938i \(0.629048\pi\)
\(678\) −12.5918 −0.483585
\(679\) −8.61596 −0.330650
\(680\) 5.35690 0.205428
\(681\) −5.26205 −0.201642
\(682\) −4.16793 −0.159598
\(683\) 4.07202 0.155811 0.0779057 0.996961i \(-0.475177\pi\)
0.0779057 + 0.996961i \(0.475177\pi\)
\(684\) −3.89977 −0.149112
\(685\) 0.396125 0.0151351
\(686\) −1.00000 −0.0381802
\(687\) −3.40821 −0.130031
\(688\) 9.87800 0.376595
\(689\) 0 0
\(690\) 0.582105 0.0221604
\(691\) 35.3739 1.34569 0.672843 0.739785i \(-0.265073\pi\)
0.672843 + 0.739785i \(0.265073\pi\)
\(692\) 9.08815 0.345479
\(693\) 0.664874 0.0252565
\(694\) 19.4034 0.736544
\(695\) −11.6002 −0.440019
\(696\) −8.98792 −0.340686
\(697\) 90.8859 3.44255
\(698\) 0.415502 0.0157270
\(699\) 23.6775 0.895567
\(700\) 4.52111 0.170882
\(701\) −9.27280 −0.350229 −0.175114 0.984548i \(-0.556030\pi\)
−0.175114 + 0.984548i \(0.556030\pi\)
\(702\) 0 0
\(703\) −43.5547 −1.64270
\(704\) −0.664874 −0.0250584
\(705\) −7.14005 −0.268910
\(706\) 16.4450 0.618917
\(707\) −17.7235 −0.666560
\(708\) −10.8901 −0.409274
\(709\) −2.31527 −0.0869520 −0.0434760 0.999054i \(-0.513843\pi\)
−0.0434760 + 0.999054i \(0.513843\pi\)
\(710\) −6.87635 −0.258065
\(711\) −1.60388 −0.0601500
\(712\) 0.454731 0.0170417
\(713\) 5.27306 0.197478
\(714\) −7.74094 −0.289697
\(715\) 0 0
\(716\) 8.98792 0.335894
\(717\) 20.2610 0.756660
\(718\) 6.61058 0.246705
\(719\) 0.987918 0.0368431 0.0184216 0.999830i \(-0.494136\pi\)
0.0184216 + 0.999830i \(0.494136\pi\)
\(720\) −0.692021 −0.0257901
\(721\) 12.5972 0.469143
\(722\) −3.79178 −0.141115
\(723\) −0.0435405 −0.00161929
\(724\) −1.76941 −0.0657597
\(725\) −40.6353 −1.50916
\(726\) 10.5579 0.391842
\(727\) −23.1860 −0.859920 −0.429960 0.902848i \(-0.641472\pi\)
−0.429960 + 0.902848i \(0.641472\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −10.0978 −0.373738
\(731\) −76.4650 −2.82816
\(732\) 6.05429 0.223773
\(733\) 26.5435 0.980405 0.490203 0.871608i \(-0.336923\pi\)
0.490203 + 0.871608i \(0.336923\pi\)
\(734\) 15.5773 0.574969
\(735\) 0.692021 0.0255256
\(736\) 0.841166 0.0310058
\(737\) 1.98659 0.0731770
\(738\) −11.7409 −0.432190
\(739\) 19.3491 0.711769 0.355885 0.934530i \(-0.384180\pi\)
0.355885 + 0.934530i \(0.384180\pi\)
\(740\) −7.72886 −0.284118
\(741\) 0 0
\(742\) 7.92154 0.290809
\(743\) −51.8437 −1.90196 −0.950980 0.309252i \(-0.899921\pi\)
−0.950980 + 0.309252i \(0.899921\pi\)
\(744\) −6.26875 −0.229824
\(745\) 8.31767 0.304736
\(746\) −1.81940 −0.0666128
\(747\) 4.39612 0.160846
\(748\) 5.14675 0.188184
\(749\) 0.692021 0.0252859
\(750\) −6.58881 −0.240589
\(751\) −16.9095 −0.617035 −0.308517 0.951219i \(-0.599833\pi\)
−0.308517 + 0.951219i \(0.599833\pi\)
\(752\) −10.3177 −0.376247
\(753\) 27.3056 0.995071
\(754\) 0 0
\(755\) 13.3900 0.487312
\(756\) 1.00000 0.0363696
\(757\) −23.5297 −0.855203 −0.427601 0.903967i \(-0.640641\pi\)
−0.427601 + 0.903967i \(0.640641\pi\)
\(758\) −7.53617 −0.273726
\(759\) 0.559270 0.0203002
\(760\) 2.69873 0.0978931
\(761\) −9.60627 −0.348227 −0.174113 0.984726i \(-0.555706\pi\)
−0.174113 + 0.984726i \(0.555706\pi\)
\(762\) −6.76809 −0.245182
\(763\) −8.03923 −0.291040
\(764\) −16.7657 −0.606561
\(765\) 5.35690 0.193679
\(766\) 3.78017 0.136583
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −2.29829 −0.0828785 −0.0414392 0.999141i \(-0.513194\pi\)
−0.0414392 + 0.999141i \(0.513194\pi\)
\(770\) −0.460107 −0.0165811
\(771\) −7.97584 −0.287243
\(772\) 16.9933 0.611602
\(773\) 10.2828 0.369845 0.184922 0.982753i \(-0.440797\pi\)
0.184922 + 0.982753i \(0.440797\pi\)
\(774\) 9.87800 0.355057
\(775\) −28.3417 −1.01806
\(776\) 8.61596 0.309295
\(777\) 11.1685 0.400669
\(778\) 5.01208 0.179692
\(779\) 45.7870 1.64049
\(780\) 0 0
\(781\) −6.60660 −0.236403
\(782\) −6.51142 −0.232848
\(783\) −8.98792 −0.321202
\(784\) 1.00000 0.0357143
\(785\) 1.51945 0.0542315
\(786\) −0.987918 −0.0352379
\(787\) −26.2398 −0.935348 −0.467674 0.883901i \(-0.654908\pi\)
−0.467674 + 0.883901i \(0.654908\pi\)
\(788\) −4.90946 −0.174892
\(789\) −9.37867 −0.333889
\(790\) 1.10992 0.0394890
\(791\) −12.5918 −0.447713
\(792\) −0.664874 −0.0236253
\(793\) 0 0
\(794\) −5.05562 −0.179417
\(795\) −5.48188 −0.194422
\(796\) −14.9729 −0.530699
\(797\) −10.6203 −0.376189 −0.188095 0.982151i \(-0.560231\pi\)
−0.188095 + 0.982151i \(0.560231\pi\)
\(798\) −3.89977 −0.138050
\(799\) 79.8684 2.82554
\(800\) −4.52111 −0.159845
\(801\) 0.454731 0.0160671
\(802\) 38.8745 1.37271
\(803\) −9.70171 −0.342366
\(804\) 2.98792 0.105376
\(805\) 0.582105 0.0205165
\(806\) 0 0
\(807\) −7.96077 −0.280232
\(808\) 17.7235 0.623510
\(809\) −32.0435 −1.12659 −0.563295 0.826256i \(-0.690467\pi\)
−0.563295 + 0.826256i \(0.690467\pi\)
\(810\) −0.692021 −0.0243152
\(811\) −25.9909 −0.912664 −0.456332 0.889810i \(-0.650837\pi\)
−0.456332 + 0.889810i \(0.650837\pi\)
\(812\) −8.98792 −0.315414
\(813\) −10.6377 −0.373081
\(814\) −7.42566 −0.260269
\(815\) −13.3975 −0.469292
\(816\) 7.74094 0.270987
\(817\) −38.5220 −1.34771
\(818\) 32.6461 1.14144
\(819\) 0 0
\(820\) 8.12498 0.283737
\(821\) −34.5133 −1.20452 −0.602262 0.798299i \(-0.705734\pi\)
−0.602262 + 0.798299i \(0.705734\pi\)
\(822\) 0.572417 0.0199653
\(823\) 9.84309 0.343108 0.171554 0.985175i \(-0.445121\pi\)
0.171554 + 0.985175i \(0.445121\pi\)
\(824\) −12.5972 −0.438843
\(825\) −3.00597 −0.104654
\(826\) −10.8901 −0.378914
\(827\) 16.2892 0.566431 0.283215 0.959056i \(-0.408599\pi\)
0.283215 + 0.959056i \(0.408599\pi\)
\(828\) 0.841166 0.0292326
\(829\) 13.9323 0.483889 0.241944 0.970290i \(-0.422215\pi\)
0.241944 + 0.970290i \(0.422215\pi\)
\(830\) −3.04221 −0.105597
\(831\) −26.4674 −0.918144
\(832\) 0 0
\(833\) −7.74094 −0.268208
\(834\) −16.7627 −0.580445
\(835\) −14.1957 −0.491261
\(836\) 2.59286 0.0896759
\(837\) −6.26875 −0.216680
\(838\) 23.3250 0.805747
\(839\) −6.49396 −0.224196 −0.112098 0.993697i \(-0.535757\pi\)
−0.112098 + 0.993697i \(0.535757\pi\)
\(840\) −0.692021 −0.0238770
\(841\) 51.7827 1.78561
\(842\) 18.2282 0.628185
\(843\) −5.60388 −0.193008
\(844\) −6.98792 −0.240534
\(845\) 0 0
\(846\) −10.3177 −0.354729
\(847\) 10.5579 0.362775
\(848\) −7.92154 −0.272027
\(849\) 11.3351 0.389021
\(850\) 34.9976 1.20041
\(851\) 9.39459 0.322042
\(852\) −9.93661 −0.340423
\(853\) 45.1099 1.54453 0.772267 0.635298i \(-0.219123\pi\)
0.772267 + 0.635298i \(0.219123\pi\)
\(854\) 6.05429 0.207174
\(855\) 2.69873 0.0922945
\(856\) −0.692021 −0.0236528
\(857\) −4.66189 −0.159247 −0.0796236 0.996825i \(-0.525372\pi\)
−0.0796236 + 0.996825i \(0.525372\pi\)
\(858\) 0 0
\(859\) −37.6364 −1.28414 −0.642069 0.766647i \(-0.721924\pi\)
−0.642069 + 0.766647i \(0.721924\pi\)
\(860\) −6.83579 −0.233098
\(861\) −11.7409 −0.400130
\(862\) −4.02177 −0.136982
\(863\) 14.1021 0.480043 0.240021 0.970768i \(-0.422846\pi\)
0.240021 + 0.970768i \(0.422846\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −6.28919 −0.213839
\(866\) −11.4470 −0.388984
\(867\) −42.9221 −1.45771
\(868\) −6.26875 −0.212775
\(869\) 1.06638 0.0361743
\(870\) 6.21983 0.210872
\(871\) 0 0
\(872\) 8.03923 0.272243
\(873\) 8.61596 0.291606
\(874\) −3.28036 −0.110960
\(875\) −6.58881 −0.222742
\(876\) −14.5918 −0.493011
\(877\) −8.51765 −0.287621 −0.143810 0.989605i \(-0.545936\pi\)
−0.143810 + 0.989605i \(0.545936\pi\)
\(878\) −37.6450 −1.27046
\(879\) −20.7530 −0.699982
\(880\) 0.460107 0.0155102
\(881\) 5.84548 0.196939 0.0984696 0.995140i \(-0.468605\pi\)
0.0984696 + 0.995140i \(0.468605\pi\)
\(882\) 1.00000 0.0336718
\(883\) −26.8358 −0.903096 −0.451548 0.892247i \(-0.649128\pi\)
−0.451548 + 0.892247i \(0.649128\pi\)
\(884\) 0 0
\(885\) 7.53617 0.253326
\(886\) −31.7797 −1.06766
\(887\) −41.9758 −1.40941 −0.704705 0.709500i \(-0.748921\pi\)
−0.704705 + 0.709500i \(0.748921\pi\)
\(888\) −11.1685 −0.374791
\(889\) −6.76809 −0.226994
\(890\) −0.314683 −0.0105482
\(891\) −0.664874 −0.0222741
\(892\) −1.13467 −0.0379916
\(893\) 40.2366 1.34646
\(894\) 12.0194 0.401988
\(895\) −6.21983 −0.207906
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 1.82371 0.0608579
\(899\) 56.3430 1.87914
\(900\) −4.52111 −0.150704
\(901\) 61.3202 2.04287
\(902\) 7.80625 0.259920
\(903\) 9.87800 0.328719
\(904\) 12.5918 0.418797
\(905\) 1.22447 0.0407028
\(906\) 19.3491 0.642832
\(907\) −34.3043 −1.13905 −0.569527 0.821973i \(-0.692874\pi\)
−0.569527 + 0.821973i \(0.692874\pi\)
\(908\) 5.26205 0.174627
\(909\) 17.7235 0.587851
\(910\) 0 0
\(911\) −16.4179 −0.543949 −0.271975 0.962304i \(-0.587677\pi\)
−0.271975 + 0.962304i \(0.587677\pi\)
\(912\) 3.89977 0.129134
\(913\) −2.92287 −0.0967329
\(914\) 11.0774 0.366408
\(915\) −4.18970 −0.138507
\(916\) 3.40821 0.112610
\(917\) −0.987918 −0.0326239
\(918\) 7.74094 0.255489
\(919\) −44.9697 −1.48341 −0.741707 0.670724i \(-0.765984\pi\)
−0.741707 + 0.670724i \(0.765984\pi\)
\(920\) −0.582105 −0.0191914
\(921\) −14.5961 −0.480958
\(922\) −3.82908 −0.126104
\(923\) 0 0
\(924\) −0.664874 −0.0218728
\(925\) −50.4941 −1.66024
\(926\) 36.0737 1.18545
\(927\) −12.5972 −0.413745
\(928\) 8.98792 0.295043
\(929\) −16.5724 −0.543723 −0.271862 0.962336i \(-0.587639\pi\)
−0.271862 + 0.962336i \(0.587639\pi\)
\(930\) 4.33811 0.142252
\(931\) −3.89977 −0.127810
\(932\) −23.6775 −0.775584
\(933\) 1.82371 0.0597055
\(934\) −38.3806 −1.25585
\(935\) −3.56166 −0.116479
\(936\) 0 0
\(937\) 50.5435 1.65118 0.825592 0.564268i \(-0.190842\pi\)
0.825592 + 0.564268i \(0.190842\pi\)
\(938\) 2.98792 0.0975590
\(939\) −34.6112 −1.12949
\(940\) 7.14005 0.232883
\(941\) 37.9191 1.23613 0.618064 0.786127i \(-0.287917\pi\)
0.618064 + 0.786127i \(0.287917\pi\)
\(942\) 2.19567 0.0715388
\(943\) −9.87608 −0.321610
\(944\) 10.8901 0.354442
\(945\) −0.692021 −0.0225115
\(946\) −6.56763 −0.213532
\(947\) 10.2067 0.331673 0.165836 0.986153i \(-0.446968\pi\)
0.165836 + 0.986153i \(0.446968\pi\)
\(948\) 1.60388 0.0520915
\(949\) 0 0
\(950\) 17.6313 0.572034
\(951\) 4.49396 0.145727
\(952\) 7.74094 0.250885
\(953\) 49.6292 1.60765 0.803824 0.594867i \(-0.202795\pi\)
0.803824 + 0.594867i \(0.202795\pi\)
\(954\) −7.92154 −0.256469
\(955\) 11.6022 0.375439
\(956\) −20.2610 −0.655287
\(957\) 5.97584 0.193171
\(958\) 11.3599 0.367021
\(959\) 0.572417 0.0184843
\(960\) 0.692021 0.0223349
\(961\) 8.29722 0.267652
\(962\) 0 0
\(963\) −0.692021 −0.0223001
\(964\) 0.0435405 0.00140235
\(965\) −11.7597 −0.378559
\(966\) 0.841166 0.0270641
\(967\) 38.5461 1.23956 0.619780 0.784776i \(-0.287222\pi\)
0.619780 + 0.784776i \(0.287222\pi\)
\(968\) −10.5579 −0.339345
\(969\) −30.1879 −0.969775
\(970\) −5.96243 −0.191442
\(971\) −6.45904 −0.207281 −0.103640 0.994615i \(-0.533049\pi\)
−0.103640 + 0.994615i \(0.533049\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −16.7627 −0.537388
\(974\) 37.4577 1.20022
\(975\) 0 0
\(976\) −6.05429 −0.193793
\(977\) 9.85649 0.315337 0.157669 0.987492i \(-0.449602\pi\)
0.157669 + 0.987492i \(0.449602\pi\)
\(978\) −19.3599 −0.619061
\(979\) −0.302339 −0.00966279
\(980\) −0.692021 −0.0221058
\(981\) 8.03923 0.256673
\(982\) 24.0200 0.766508
\(983\) 17.2728 0.550917 0.275458 0.961313i \(-0.411170\pi\)
0.275458 + 0.961313i \(0.411170\pi\)
\(984\) 11.7409 0.374287
\(985\) 3.39745 0.108252
\(986\) −69.5749 −2.21572
\(987\) −10.3177 −0.328415
\(988\) 0 0
\(989\) 8.30904 0.264212
\(990\) 0.460107 0.0146232
\(991\) 21.0992 0.670237 0.335118 0.942176i \(-0.391224\pi\)
0.335118 + 0.942176i \(0.391224\pi\)
\(992\) 6.26875 0.199033
\(993\) 8.75733 0.277906
\(994\) −9.93661 −0.315170
\(995\) 10.3615 0.328483
\(996\) −4.39612 −0.139297
\(997\) 4.86113 0.153954 0.0769768 0.997033i \(-0.475473\pi\)
0.0769768 + 0.997033i \(0.475473\pi\)
\(998\) 36.3043 1.14919
\(999\) −11.1685 −0.353356
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 7098.2.a.ck.1.1 yes 3
13.12 even 2 7098.2.a.cb.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7098.2.a.cb.1.3 3 13.12 even 2
7098.2.a.ck.1.1 yes 3 1.1 even 1 trivial