Properties

Label 7098.2.a.cg.1.2
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 1.2
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.55496 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.55496 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.55496 q^{10} +3.24698 q^{11} +1.00000 q^{12} +1.00000 q^{14} +2.55496 q^{15} +1.00000 q^{16} +3.40581 q^{17} -1.00000 q^{18} -2.85086 q^{19} +2.55496 q^{20} -1.00000 q^{21} -3.24698 q^{22} -8.54288 q^{23} -1.00000 q^{24} +1.52781 q^{25} +1.00000 q^{27} -1.00000 q^{28} -6.18598 q^{29} -2.55496 q^{30} -0.423272 q^{31} -1.00000 q^{32} +3.24698 q^{33} -3.40581 q^{34} -2.55496 q^{35} +1.00000 q^{36} +5.54288 q^{37} +2.85086 q^{38} -2.55496 q^{40} -3.14675 q^{41} +1.00000 q^{42} +4.93900 q^{43} +3.24698 q^{44} +2.55496 q^{45} +8.54288 q^{46} +4.50604 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.52781 q^{50} +3.40581 q^{51} +4.12498 q^{53} -1.00000 q^{54} +8.29590 q^{55} +1.00000 q^{56} -2.85086 q^{57} +6.18598 q^{58} +14.9269 q^{59} +2.55496 q^{60} +3.17629 q^{61} +0.423272 q^{62} -1.00000 q^{63} +1.00000 q^{64} -3.24698 q^{66} +5.47219 q^{67} +3.40581 q^{68} -8.54288 q^{69} +2.55496 q^{70} -4.21983 q^{71} -1.00000 q^{72} +12.2959 q^{73} -5.54288 q^{74} +1.52781 q^{75} -2.85086 q^{76} -3.24698 q^{77} +10.2567 q^{79} +2.55496 q^{80} +1.00000 q^{81} +3.14675 q^{82} +16.8509 q^{83} -1.00000 q^{84} +8.70171 q^{85} -4.93900 q^{86} -6.18598 q^{87} -3.24698 q^{88} +1.96615 q^{89} -2.55496 q^{90} -8.54288 q^{92} -0.423272 q^{93} -4.50604 q^{94} -7.28382 q^{95} -1.00000 q^{96} +1.12737 q^{97} -1.00000 q^{98} +3.24698 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} + 8q^{5} - 3q^{6} - 3q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} + 8q^{5} - 3q^{6} - 3q^{7} - 3q^{8} + 3q^{9} - 8q^{10} + 5q^{11} + 3q^{12} + 3q^{14} + 8q^{15} + 3q^{16} - 3q^{17} - 3q^{18} + 5q^{19} + 8q^{20} - 3q^{21} - 5q^{22} - 7q^{23} - 3q^{24} + 11q^{25} + 3q^{27} - 3q^{28} - 4q^{29} - 8q^{30} - 4q^{31} - 3q^{32} + 5q^{33} + 3q^{34} - 8q^{35} + 3q^{36} - 2q^{37} - 5q^{38} - 8q^{40} + 18q^{41} + 3q^{42} + 5q^{43} + 5q^{44} + 8q^{45} + 7q^{46} + 23q^{47} + 3q^{48} + 3q^{49} - 11q^{50} - 3q^{51} - 12q^{53} - 3q^{54} + 11q^{55} + 3q^{56} + 5q^{57} + 4q^{58} + 16q^{59} + 8q^{60} + 17q^{61} + 4q^{62} - 3q^{63} + 3q^{64} - 5q^{66} + 10q^{67} - 3q^{68} - 7q^{69} + 8q^{70} - 14q^{71} - 3q^{72} + 23q^{73} + 2q^{74} + 11q^{75} + 5q^{76} - 5q^{77} + 4q^{79} + 8q^{80} + 3q^{81} - 18q^{82} + 37q^{83} - 3q^{84} - q^{85} - 5q^{86} - 4q^{87} - 5q^{88} - 10q^{89} - 8q^{90} - 7q^{92} - 4q^{93} - 23q^{94} + 11q^{95} - 3q^{96} + 20q^{97} - 3q^{98} + 5q^{99} + O(q^{100}) \)

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
\(5\) 2.55496 1.14261 0.571306 0.820737i \(-0.306437\pi\)
0.571306 + 0.820737i \(0.306437\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.55496 −0.807949
\(11\) 3.24698 0.979001 0.489501 0.872003i \(-0.337179\pi\)
0.489501 + 0.872003i \(0.337179\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) 2.55496 0.659687
\(16\) 1.00000 0.250000
\(17\) 3.40581 0.826031 0.413016 0.910724i \(-0.364476\pi\)
0.413016 + 0.910724i \(0.364476\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.85086 −0.654031 −0.327016 0.945019i \(-0.606043\pi\)
−0.327016 + 0.945019i \(0.606043\pi\)
\(20\) 2.55496 0.571306
\(21\) −1.00000 −0.218218
\(22\) −3.24698 −0.692258
\(23\) −8.54288 −1.78131 −0.890656 0.454677i \(-0.849755\pi\)
−0.890656 + 0.454677i \(0.849755\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.52781 0.305562
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −6.18598 −1.14871 −0.574354 0.818607i \(-0.694747\pi\)
−0.574354 + 0.818607i \(0.694747\pi\)
\(30\) −2.55496 −0.466469
\(31\) −0.423272 −0.0760218 −0.0380109 0.999277i \(-0.512102\pi\)
−0.0380109 + 0.999277i \(0.512102\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.24698 0.565227
\(34\) −3.40581 −0.584092
\(35\) −2.55496 −0.431867
\(36\) 1.00000 0.166667
\(37\) 5.54288 0.911243 0.455622 0.890174i \(-0.349417\pi\)
0.455622 + 0.890174i \(0.349417\pi\)
\(38\) 2.85086 0.462470
\(39\) 0 0
\(40\) −2.55496 −0.403974
\(41\) −3.14675 −0.491440 −0.245720 0.969341i \(-0.579024\pi\)
−0.245720 + 0.969341i \(0.579024\pi\)
\(42\) 1.00000 0.154303
\(43\) 4.93900 0.753191 0.376595 0.926378i \(-0.377095\pi\)
0.376595 + 0.926378i \(0.377095\pi\)
\(44\) 3.24698 0.489501
\(45\) 2.55496 0.380871
\(46\) 8.54288 1.25958
\(47\) 4.50604 0.657274 0.328637 0.944456i \(-0.393411\pi\)
0.328637 + 0.944456i \(0.393411\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.52781 −0.216065
\(51\) 3.40581 0.476909
\(52\) 0 0
\(53\) 4.12498 0.566610 0.283305 0.959030i \(-0.408569\pi\)
0.283305 + 0.959030i \(0.408569\pi\)
\(54\) −1.00000 −0.136083
\(55\) 8.29590 1.11862
\(56\) 1.00000 0.133631
\(57\) −2.85086 −0.377605
\(58\) 6.18598 0.812259
\(59\) 14.9269 1.94332 0.971660 0.236384i \(-0.0759624\pi\)
0.971660 + 0.236384i \(0.0759624\pi\)
\(60\) 2.55496 0.329844
\(61\) 3.17629 0.406683 0.203341 0.979108i \(-0.434820\pi\)
0.203341 + 0.979108i \(0.434820\pi\)
\(62\) 0.423272 0.0537555
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.24698 −0.399676
\(67\) 5.47219 0.668534 0.334267 0.942478i \(-0.391511\pi\)
0.334267 + 0.942478i \(0.391511\pi\)
\(68\) 3.40581 0.413016
\(69\) −8.54288 −1.02844
\(70\) 2.55496 0.305376
\(71\) −4.21983 −0.500802 −0.250401 0.968142i \(-0.580562\pi\)
−0.250401 + 0.968142i \(0.580562\pi\)
\(72\) −1.00000 −0.117851
\(73\) 12.2959 1.43913 0.719563 0.694427i \(-0.244342\pi\)
0.719563 + 0.694427i \(0.244342\pi\)
\(74\) −5.54288 −0.644346
\(75\) 1.52781 0.176416
\(76\) −2.85086 −0.327016
\(77\) −3.24698 −0.370028
\(78\) 0 0
\(79\) 10.2567 1.15397 0.576983 0.816756i \(-0.304230\pi\)
0.576983 + 0.816756i \(0.304230\pi\)
\(80\) 2.55496 0.285653
\(81\) 1.00000 0.111111
\(82\) 3.14675 0.347501
\(83\) 16.8509 1.84962 0.924811 0.380427i \(-0.124223\pi\)
0.924811 + 0.380427i \(0.124223\pi\)
\(84\) −1.00000 −0.109109
\(85\) 8.70171 0.943833
\(86\) −4.93900 −0.532586
\(87\) −6.18598 −0.663207
\(88\) −3.24698 −0.346129
\(89\) 1.96615 0.208411 0.104206 0.994556i \(-0.466770\pi\)
0.104206 + 0.994556i \(0.466770\pi\)
\(90\) −2.55496 −0.269316
\(91\) 0 0
\(92\) −8.54288 −0.890656
\(93\) −0.423272 −0.0438912
\(94\) −4.50604 −0.464763
\(95\) −7.28382 −0.747304
\(96\) −1.00000 −0.102062
\(97\) 1.12737 0.114468 0.0572338 0.998361i \(-0.481772\pi\)
0.0572338 + 0.998361i \(0.481772\pi\)
\(98\) −1.00000 −0.101015
\(99\) 3.24698 0.326334
\(100\) 1.52781 0.152781
\(101\) −1.03385 −0.102872 −0.0514361 0.998676i \(-0.516380\pi\)
−0.0514361 + 0.998676i \(0.516380\pi\)
\(102\) −3.40581 −0.337226
\(103\) 8.18060 0.806059 0.403029 0.915187i \(-0.367957\pi\)
0.403029 + 0.915187i \(0.367957\pi\)
\(104\) 0 0
\(105\) −2.55496 −0.249338
\(106\) −4.12498 −0.400654
\(107\) −0.780167 −0.0754216 −0.0377108 0.999289i \(-0.512007\pi\)
−0.0377108 + 0.999289i \(0.512007\pi\)
\(108\) 1.00000 0.0962250
\(109\) 6.93900 0.664636 0.332318 0.943167i \(-0.392169\pi\)
0.332318 + 0.943167i \(0.392169\pi\)
\(110\) −8.29590 −0.790983
\(111\) 5.54288 0.526107
\(112\) −1.00000 −0.0944911
\(113\) −5.20237 −0.489398 −0.244699 0.969599i \(-0.578689\pi\)
−0.244699 + 0.969599i \(0.578689\pi\)
\(114\) 2.85086 0.267007
\(115\) −21.8267 −2.03535
\(116\) −6.18598 −0.574354
\(117\) 0 0
\(118\) −14.9269 −1.37413
\(119\) −3.40581 −0.312210
\(120\) −2.55496 −0.233235
\(121\) −0.457123 −0.0415567
\(122\) −3.17629 −0.287568
\(123\) −3.14675 −0.283733
\(124\) −0.423272 −0.0380109
\(125\) −8.87130 −0.793473
\(126\) 1.00000 0.0890871
\(127\) −14.6136 −1.29674 −0.648372 0.761324i \(-0.724550\pi\)
−0.648372 + 0.761324i \(0.724550\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.93900 0.434855
\(130\) 0 0
\(131\) 19.7778 1.72799 0.863996 0.503499i \(-0.167954\pi\)
0.863996 + 0.503499i \(0.167954\pi\)
\(132\) 3.24698 0.282613
\(133\) 2.85086 0.247200
\(134\) −5.47219 −0.472725
\(135\) 2.55496 0.219896
\(136\) −3.40581 −0.292046
\(137\) 16.7627 1.43213 0.716067 0.698031i \(-0.245940\pi\)
0.716067 + 0.698031i \(0.245940\pi\)
\(138\) 8.54288 0.727218
\(139\) 12.1685 1.03212 0.516060 0.856552i \(-0.327398\pi\)
0.516060 + 0.856552i \(0.327398\pi\)
\(140\) −2.55496 −0.215933
\(141\) 4.50604 0.379477
\(142\) 4.21983 0.354120
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −15.8049 −1.31253
\(146\) −12.2959 −1.01762
\(147\) 1.00000 0.0824786
\(148\) 5.54288 0.455622
\(149\) 0.753020 0.0616898 0.0308449 0.999524i \(-0.490180\pi\)
0.0308449 + 0.999524i \(0.490180\pi\)
\(150\) −1.52781 −0.124745
\(151\) 15.1021 1.22900 0.614498 0.788919i \(-0.289359\pi\)
0.614498 + 0.788919i \(0.289359\pi\)
\(152\) 2.85086 0.231235
\(153\) 3.40581 0.275344
\(154\) 3.24698 0.261649
\(155\) −1.08144 −0.0868634
\(156\) 0 0
\(157\) −18.5133 −1.47753 −0.738763 0.673966i \(-0.764590\pi\)
−0.738763 + 0.673966i \(0.764590\pi\)
\(158\) −10.2567 −0.815977
\(159\) 4.12498 0.327132
\(160\) −2.55496 −0.201987
\(161\) 8.54288 0.673273
\(162\) −1.00000 −0.0785674
\(163\) −20.1444 −1.57783 −0.788914 0.614504i \(-0.789356\pi\)
−0.788914 + 0.614504i \(0.789356\pi\)
\(164\) −3.14675 −0.245720
\(165\) 8.29590 0.645835
\(166\) −16.8509 −1.30788
\(167\) 18.5157 1.43279 0.716395 0.697695i \(-0.245791\pi\)
0.716395 + 0.697695i \(0.245791\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) −8.70171 −0.667391
\(171\) −2.85086 −0.218010
\(172\) 4.93900 0.376595
\(173\) −22.5676 −1.71579 −0.857893 0.513829i \(-0.828227\pi\)
−0.857893 + 0.513829i \(0.828227\pi\)
\(174\) 6.18598 0.468958
\(175\) −1.52781 −0.115492
\(176\) 3.24698 0.244750
\(177\) 14.9269 1.12198
\(178\) −1.96615 −0.147369
\(179\) 11.8116 0.882842 0.441421 0.897300i \(-0.354474\pi\)
0.441421 + 0.897300i \(0.354474\pi\)
\(180\) 2.55496 0.190435
\(181\) −5.48858 −0.407963 −0.203982 0.978975i \(-0.565388\pi\)
−0.203982 + 0.978975i \(0.565388\pi\)
\(182\) 0 0
\(183\) 3.17629 0.234798
\(184\) 8.54288 0.629789
\(185\) 14.1618 1.04120
\(186\) 0.423272 0.0310358
\(187\) 11.0586 0.808685
\(188\) 4.50604 0.328637
\(189\) −1.00000 −0.0727393
\(190\) 7.28382 0.528424
\(191\) −5.17523 −0.374466 −0.187233 0.982316i \(-0.559952\pi\)
−0.187233 + 0.982316i \(0.559952\pi\)
\(192\) 1.00000 0.0721688
\(193\) 1.67696 0.120710 0.0603550 0.998177i \(-0.480777\pi\)
0.0603550 + 0.998177i \(0.480777\pi\)
\(194\) −1.12737 −0.0809408
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −20.4916 −1.45996 −0.729982 0.683467i \(-0.760472\pi\)
−0.729982 + 0.683467i \(0.760472\pi\)
\(198\) −3.24698 −0.230753
\(199\) −5.36898 −0.380597 −0.190298 0.981726i \(-0.560946\pi\)
−0.190298 + 0.981726i \(0.560946\pi\)
\(200\) −1.52781 −0.108033
\(201\) 5.47219 0.385978
\(202\) 1.03385 0.0727416
\(203\) 6.18598 0.434171
\(204\) 3.40581 0.238455
\(205\) −8.03982 −0.561525
\(206\) −8.18060 −0.569970
\(207\) −8.54288 −0.593771
\(208\) 0 0
\(209\) −9.25667 −0.640297
\(210\) 2.55496 0.176309
\(211\) 6.69202 0.460698 0.230349 0.973108i \(-0.426013\pi\)
0.230349 + 0.973108i \(0.426013\pi\)
\(212\) 4.12498 0.283305
\(213\) −4.21983 −0.289138
\(214\) 0.780167 0.0533312
\(215\) 12.6189 0.860605
\(216\) −1.00000 −0.0680414
\(217\) 0.423272 0.0287335
\(218\) −6.93900 −0.469968
\(219\) 12.2959 0.830880
\(220\) 8.29590 0.559309
\(221\) 0 0
\(222\) −5.54288 −0.372014
\(223\) 18.5864 1.24464 0.622319 0.782764i \(-0.286191\pi\)
0.622319 + 0.782764i \(0.286191\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.52781 0.101854
\(226\) 5.20237 0.346057
\(227\) 6.92394 0.459558 0.229779 0.973243i \(-0.426200\pi\)
0.229779 + 0.973243i \(0.426200\pi\)
\(228\) −2.85086 −0.188802
\(229\) 5.73855 0.379214 0.189607 0.981860i \(-0.439279\pi\)
0.189607 + 0.981860i \(0.439279\pi\)
\(230\) 21.8267 1.43921
\(231\) −3.24698 −0.213636
\(232\) 6.18598 0.406130
\(233\) 11.2892 0.739580 0.369790 0.929115i \(-0.379430\pi\)
0.369790 + 0.929115i \(0.379430\pi\)
\(234\) 0 0
\(235\) 11.5127 0.751009
\(236\) 14.9269 0.971660
\(237\) 10.2567 0.666242
\(238\) 3.40581 0.220766
\(239\) 19.3491 1.25159 0.625795 0.779987i \(-0.284774\pi\)
0.625795 + 0.779987i \(0.284774\pi\)
\(240\) 2.55496 0.164922
\(241\) −5.95108 −0.383343 −0.191671 0.981459i \(-0.561391\pi\)
−0.191671 + 0.981459i \(0.561391\pi\)
\(242\) 0.457123 0.0293850
\(243\) 1.00000 0.0641500
\(244\) 3.17629 0.203341
\(245\) 2.55496 0.163230
\(246\) 3.14675 0.200630
\(247\) 0 0
\(248\) 0.423272 0.0268778
\(249\) 16.8509 1.06788
\(250\) 8.87130 0.561070
\(251\) −8.84415 −0.558238 −0.279119 0.960257i \(-0.590042\pi\)
−0.279119 + 0.960257i \(0.590042\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −27.7385 −1.74391
\(254\) 14.6136 0.916937
\(255\) 8.70171 0.544922
\(256\) 1.00000 0.0625000
\(257\) −5.50902 −0.343644 −0.171822 0.985128i \(-0.554965\pi\)
−0.171822 + 0.985128i \(0.554965\pi\)
\(258\) −4.93900 −0.307489
\(259\) −5.54288 −0.344418
\(260\) 0 0
\(261\) −6.18598 −0.382903
\(262\) −19.7778 −1.22187
\(263\) −2.84548 −0.175460 −0.0877299 0.996144i \(-0.527961\pi\)
−0.0877299 + 0.996144i \(0.527961\pi\)
\(264\) −3.24698 −0.199838
\(265\) 10.5392 0.647415
\(266\) −2.85086 −0.174797
\(267\) 1.96615 0.120326
\(268\) 5.47219 0.334267
\(269\) 26.0291 1.58702 0.793510 0.608557i \(-0.208251\pi\)
0.793510 + 0.608557i \(0.208251\pi\)
\(270\) −2.55496 −0.155490
\(271\) 2.45042 0.148852 0.0744262 0.997227i \(-0.476287\pi\)
0.0744262 + 0.997227i \(0.476287\pi\)
\(272\) 3.40581 0.206508
\(273\) 0 0
\(274\) −16.7627 −1.01267
\(275\) 4.96077 0.299146
\(276\) −8.54288 −0.514221
\(277\) −11.9812 −0.719881 −0.359941 0.932975i \(-0.617203\pi\)
−0.359941 + 0.932975i \(0.617203\pi\)
\(278\) −12.1685 −0.729819
\(279\) −0.423272 −0.0253406
\(280\) 2.55496 0.152688
\(281\) 2.88231 0.171944 0.0859722 0.996298i \(-0.472600\pi\)
0.0859722 + 0.996298i \(0.472600\pi\)
\(282\) −4.50604 −0.268331
\(283\) −3.71917 −0.221082 −0.110541 0.993872i \(-0.535258\pi\)
−0.110541 + 0.993872i \(0.535258\pi\)
\(284\) −4.21983 −0.250401
\(285\) −7.28382 −0.431456
\(286\) 0 0
\(287\) 3.14675 0.185747
\(288\) −1.00000 −0.0589256
\(289\) −5.40044 −0.317673
\(290\) 15.8049 0.928097
\(291\) 1.12737 0.0660879
\(292\) 12.2959 0.719563
\(293\) 16.2862 0.951450 0.475725 0.879594i \(-0.342186\pi\)
0.475725 + 0.879594i \(0.342186\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 38.1377 2.22046
\(296\) −5.54288 −0.322173
\(297\) 3.24698 0.188409
\(298\) −0.753020 −0.0436213
\(299\) 0 0
\(300\) 1.52781 0.0882082
\(301\) −4.93900 −0.284679
\(302\) −15.1021 −0.869031
\(303\) −1.03385 −0.0593932
\(304\) −2.85086 −0.163508
\(305\) 8.11529 0.464680
\(306\) −3.40581 −0.194697
\(307\) −1.91723 −0.109422 −0.0547111 0.998502i \(-0.517424\pi\)
−0.0547111 + 0.998502i \(0.517424\pi\)
\(308\) −3.24698 −0.185014
\(309\) 8.18060 0.465378
\(310\) 1.08144 0.0614217
\(311\) −16.2064 −0.918982 −0.459491 0.888182i \(-0.651968\pi\)
−0.459491 + 0.888182i \(0.651968\pi\)
\(312\) 0 0
\(313\) −24.2543 −1.37093 −0.685466 0.728104i \(-0.740402\pi\)
−0.685466 + 0.728104i \(0.740402\pi\)
\(314\) 18.5133 1.04477
\(315\) −2.55496 −0.143956
\(316\) 10.2567 0.576983
\(317\) −23.2911 −1.30816 −0.654080 0.756426i \(-0.726944\pi\)
−0.654080 + 0.756426i \(0.726944\pi\)
\(318\) −4.12498 −0.231317
\(319\) −20.0858 −1.12459
\(320\) 2.55496 0.142827
\(321\) −0.780167 −0.0435447
\(322\) −8.54288 −0.476076
\(323\) −9.70948 −0.540250
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 20.1444 1.11569
\(327\) 6.93900 0.383728
\(328\) 3.14675 0.173750
\(329\) −4.50604 −0.248426
\(330\) −8.29590 −0.456674
\(331\) −34.3957 −1.89056 −0.945278 0.326266i \(-0.894209\pi\)
−0.945278 + 0.326266i \(0.894209\pi\)
\(332\) 16.8509 0.924811
\(333\) 5.54288 0.303748
\(334\) −18.5157 −1.01314
\(335\) 13.9812 0.763875
\(336\) −1.00000 −0.0545545
\(337\) −14.8049 −0.806475 −0.403238 0.915095i \(-0.632115\pi\)
−0.403238 + 0.915095i \(0.632115\pi\)
\(338\) 0 0
\(339\) −5.20237 −0.282554
\(340\) 8.70171 0.471916
\(341\) −1.37435 −0.0744255
\(342\) 2.85086 0.154157
\(343\) −1.00000 −0.0539949
\(344\) −4.93900 −0.266293
\(345\) −21.8267 −1.17511
\(346\) 22.5676 1.21324
\(347\) 26.6122 1.42862 0.714310 0.699830i \(-0.246741\pi\)
0.714310 + 0.699830i \(0.246741\pi\)
\(348\) −6.18598 −0.331603
\(349\) −13.2567 −0.709613 −0.354807 0.934940i \(-0.615453\pi\)
−0.354807 + 0.934940i \(0.615453\pi\)
\(350\) 1.52781 0.0816649
\(351\) 0 0
\(352\) −3.24698 −0.173065
\(353\) 22.5483 1.20012 0.600061 0.799954i \(-0.295143\pi\)
0.600061 + 0.799954i \(0.295143\pi\)
\(354\) −14.9269 −0.793357
\(355\) −10.7815 −0.572222
\(356\) 1.96615 0.104206
\(357\) −3.40581 −0.180255
\(358\) −11.8116 −0.624264
\(359\) 31.8025 1.67847 0.839237 0.543766i \(-0.183002\pi\)
0.839237 + 0.543766i \(0.183002\pi\)
\(360\) −2.55496 −0.134658
\(361\) −10.8726 −0.572243
\(362\) 5.48858 0.288473
\(363\) −0.457123 −0.0239928
\(364\) 0 0
\(365\) 31.4155 1.64436
\(366\) −3.17629 −0.166027
\(367\) −3.31037 −0.172800 −0.0864000 0.996261i \(-0.527536\pi\)
−0.0864000 + 0.996261i \(0.527536\pi\)
\(368\) −8.54288 −0.445328
\(369\) −3.14675 −0.163813
\(370\) −14.1618 −0.736238
\(371\) −4.12498 −0.214158
\(372\) −0.423272 −0.0219456
\(373\) 34.8950 1.80679 0.903397 0.428805i \(-0.141065\pi\)
0.903397 + 0.428805i \(0.141065\pi\)
\(374\) −11.0586 −0.571827
\(375\) −8.87130 −0.458112
\(376\) −4.50604 −0.232381
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) −33.3545 −1.71331 −0.856653 0.515893i \(-0.827460\pi\)
−0.856653 + 0.515893i \(0.827460\pi\)
\(380\) −7.28382 −0.373652
\(381\) −14.6136 −0.748676
\(382\) 5.17523 0.264788
\(383\) −23.3980 −1.19558 −0.597792 0.801651i \(-0.703955\pi\)
−0.597792 + 0.801651i \(0.703955\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −8.29590 −0.422798
\(386\) −1.67696 −0.0853548
\(387\) 4.93900 0.251064
\(388\) 1.12737 0.0572338
\(389\) −9.38644 −0.475911 −0.237956 0.971276i \(-0.576477\pi\)
−0.237956 + 0.971276i \(0.576477\pi\)
\(390\) 0 0
\(391\) −29.0954 −1.47142
\(392\) −1.00000 −0.0505076
\(393\) 19.7778 0.997657
\(394\) 20.4916 1.03235
\(395\) 26.2054 1.31853
\(396\) 3.24698 0.163167
\(397\) 7.96376 0.399689 0.199845 0.979828i \(-0.435956\pi\)
0.199845 + 0.979828i \(0.435956\pi\)
\(398\) 5.36898 0.269123
\(399\) 2.85086 0.142721
\(400\) 1.52781 0.0763906
\(401\) 25.4359 1.27021 0.635105 0.772426i \(-0.280957\pi\)
0.635105 + 0.772426i \(0.280957\pi\)
\(402\) −5.47219 −0.272928
\(403\) 0 0
\(404\) −1.03385 −0.0514361
\(405\) 2.55496 0.126957
\(406\) −6.18598 −0.307005
\(407\) 17.9976 0.892108
\(408\) −3.40581 −0.168613
\(409\) −26.9801 −1.33408 −0.667041 0.745021i \(-0.732440\pi\)
−0.667041 + 0.745021i \(0.732440\pi\)
\(410\) 8.03982 0.397058
\(411\) 16.7627 0.826843
\(412\) 8.18060 0.403029
\(413\) −14.9269 −0.734506
\(414\) 8.54288 0.419859
\(415\) 43.0532 2.11340
\(416\) 0 0
\(417\) 12.1685 0.595895
\(418\) 9.25667 0.452758
\(419\) −38.1594 −1.86421 −0.932105 0.362188i \(-0.882030\pi\)
−0.932105 + 0.362188i \(0.882030\pi\)
\(420\) −2.55496 −0.124669
\(421\) −12.0828 −0.588878 −0.294439 0.955670i \(-0.595133\pi\)
−0.294439 + 0.955670i \(0.595133\pi\)
\(422\) −6.69202 −0.325763
\(423\) 4.50604 0.219091
\(424\) −4.12498 −0.200327
\(425\) 5.20344 0.252404
\(426\) 4.21983 0.204452
\(427\) −3.17629 −0.153712
\(428\) −0.780167 −0.0377108
\(429\) 0 0
\(430\) −12.6189 −0.608539
\(431\) −28.7023 −1.38254 −0.691271 0.722596i \(-0.742949\pi\)
−0.691271 + 0.722596i \(0.742949\pi\)
\(432\) 1.00000 0.0481125
\(433\) 9.33406 0.448566 0.224283 0.974524i \(-0.427996\pi\)
0.224283 + 0.974524i \(0.427996\pi\)
\(434\) −0.423272 −0.0203177
\(435\) −15.8049 −0.757788
\(436\) 6.93900 0.332318
\(437\) 24.3545 1.16503
\(438\) −12.2959 −0.587521
\(439\) −15.3123 −0.730816 −0.365408 0.930848i \(-0.619070\pi\)
−0.365408 + 0.930848i \(0.619070\pi\)
\(440\) −8.29590 −0.395491
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 9.89679 0.470211 0.235105 0.971970i \(-0.424456\pi\)
0.235105 + 0.971970i \(0.424456\pi\)
\(444\) 5.54288 0.263053
\(445\) 5.02343 0.238133
\(446\) −18.5864 −0.880092
\(447\) 0.753020 0.0356166
\(448\) −1.00000 −0.0472456
\(449\) −26.9288 −1.27085 −0.635425 0.772162i \(-0.719175\pi\)
−0.635425 + 0.772162i \(0.719175\pi\)
\(450\) −1.52781 −0.0720217
\(451\) −10.2174 −0.481120
\(452\) −5.20237 −0.244699
\(453\) 15.1021 0.709561
\(454\) −6.92394 −0.324956
\(455\) 0 0
\(456\) 2.85086 0.133504
\(457\) −21.3351 −0.998015 −0.499008 0.866598i \(-0.666302\pi\)
−0.499008 + 0.866598i \(0.666302\pi\)
\(458\) −5.73855 −0.268145
\(459\) 3.40581 0.158970
\(460\) −21.8267 −1.01767
\(461\) 24.3045 1.13197 0.565987 0.824414i \(-0.308495\pi\)
0.565987 + 0.824414i \(0.308495\pi\)
\(462\) 3.24698 0.151063
\(463\) −37.0465 −1.72170 −0.860849 0.508860i \(-0.830067\pi\)
−0.860849 + 0.508860i \(0.830067\pi\)
\(464\) −6.18598 −0.287177
\(465\) −1.08144 −0.0501506
\(466\) −11.2892 −0.522962
\(467\) 32.8582 1.52049 0.760247 0.649634i \(-0.225078\pi\)
0.760247 + 0.649634i \(0.225078\pi\)
\(468\) 0 0
\(469\) −5.47219 −0.252682
\(470\) −11.5127 −0.531043
\(471\) −18.5133 −0.853050
\(472\) −14.9269 −0.687067
\(473\) 16.0368 0.737374
\(474\) −10.2567 −0.471104
\(475\) −4.35557 −0.199847
\(476\) −3.40581 −0.156105
\(477\) 4.12498 0.188870
\(478\) −19.3491 −0.885008
\(479\) −20.8829 −0.954164 −0.477082 0.878859i \(-0.658306\pi\)
−0.477082 + 0.878859i \(0.658306\pi\)
\(480\) −2.55496 −0.116617
\(481\) 0 0
\(482\) 5.95108 0.271064
\(483\) 8.54288 0.388714
\(484\) −0.457123 −0.0207783
\(485\) 2.88040 0.130792
\(486\) −1.00000 −0.0453609
\(487\) 33.0224 1.49639 0.748193 0.663481i \(-0.230922\pi\)
0.748193 + 0.663481i \(0.230922\pi\)
\(488\) −3.17629 −0.143784
\(489\) −20.1444 −0.910959
\(490\) −2.55496 −0.115421
\(491\) 1.14244 0.0515576 0.0257788 0.999668i \(-0.491793\pi\)
0.0257788 + 0.999668i \(0.491793\pi\)
\(492\) −3.14675 −0.141867
\(493\) −21.0683 −0.948868
\(494\) 0 0
\(495\) 8.29590 0.372873
\(496\) −0.423272 −0.0190055
\(497\) 4.21983 0.189285
\(498\) −16.8509 −0.755105
\(499\) 16.4239 0.735233 0.367617 0.929977i \(-0.380174\pi\)
0.367617 + 0.929977i \(0.380174\pi\)
\(500\) −8.87130 −0.396736
\(501\) 18.5157 0.827222
\(502\) 8.84415 0.394734
\(503\) −27.7797 −1.23864 −0.619318 0.785141i \(-0.712591\pi\)
−0.619318 + 0.785141i \(0.712591\pi\)
\(504\) 1.00000 0.0445435
\(505\) −2.64145 −0.117543
\(506\) 27.7385 1.23313
\(507\) 0 0
\(508\) −14.6136 −0.648372
\(509\) −16.3827 −0.726151 −0.363076 0.931760i \(-0.618273\pi\)
−0.363076 + 0.931760i \(0.618273\pi\)
\(510\) −8.70171 −0.385318
\(511\) −12.2959 −0.543938
\(512\) −1.00000 −0.0441942
\(513\) −2.85086 −0.125868
\(514\) 5.50902 0.242993
\(515\) 20.9011 0.921013
\(516\) 4.93900 0.217427
\(517\) 14.6310 0.643472
\(518\) 5.54288 0.243540
\(519\) −22.5676 −0.990609
\(520\) 0 0
\(521\) 3.65519 0.160137 0.0800683 0.996789i \(-0.474486\pi\)
0.0800683 + 0.996789i \(0.474486\pi\)
\(522\) 6.18598 0.270753
\(523\) −4.77884 −0.208964 −0.104482 0.994527i \(-0.533318\pi\)
−0.104482 + 0.994527i \(0.533318\pi\)
\(524\) 19.7778 0.863996
\(525\) −1.52781 −0.0666791
\(526\) 2.84548 0.124069
\(527\) −1.44158 −0.0627964
\(528\) 3.24698 0.141307
\(529\) 49.9807 2.17308
\(530\) −10.5392 −0.457792
\(531\) 14.9269 0.647773
\(532\) 2.85086 0.123600
\(533\) 0 0
\(534\) −1.96615 −0.0850836
\(535\) −1.99330 −0.0861777
\(536\) −5.47219 −0.236363
\(537\) 11.8116 0.509709
\(538\) −26.0291 −1.12219
\(539\) 3.24698 0.139857
\(540\) 2.55496 0.109948
\(541\) −26.4450 −1.13696 −0.568481 0.822697i \(-0.692469\pi\)
−0.568481 + 0.822697i \(0.692469\pi\)
\(542\) −2.45042 −0.105254
\(543\) −5.48858 −0.235538
\(544\) −3.40581 −0.146023
\(545\) 17.7289 0.759421
\(546\) 0 0
\(547\) 26.0073 1.11199 0.555996 0.831185i \(-0.312337\pi\)
0.555996 + 0.831185i \(0.312337\pi\)
\(548\) 16.7627 0.716067
\(549\) 3.17629 0.135561
\(550\) −4.96077 −0.211528
\(551\) 17.6353 0.751291
\(552\) 8.54288 0.363609
\(553\) −10.2567 −0.436158
\(554\) 11.9812 0.509033
\(555\) 14.1618 0.601136
\(556\) 12.1685 0.516060
\(557\) 27.9135 1.18273 0.591367 0.806403i \(-0.298589\pi\)
0.591367 + 0.806403i \(0.298589\pi\)
\(558\) 0.423272 0.0179185
\(559\) 0 0
\(560\) −2.55496 −0.107967
\(561\) 11.0586 0.466895
\(562\) −2.88231 −0.121583
\(563\) 35.9323 1.51437 0.757183 0.653203i \(-0.226575\pi\)
0.757183 + 0.653203i \(0.226575\pi\)
\(564\) 4.50604 0.189739
\(565\) −13.2918 −0.559192
\(566\) 3.71917 0.156328
\(567\) −1.00000 −0.0419961
\(568\) 4.21983 0.177060
\(569\) 37.1226 1.55626 0.778130 0.628103i \(-0.216168\pi\)
0.778130 + 0.628103i \(0.216168\pi\)
\(570\) 7.28382 0.305085
\(571\) −14.8280 −0.620533 −0.310267 0.950650i \(-0.600418\pi\)
−0.310267 + 0.950650i \(0.600418\pi\)
\(572\) 0 0
\(573\) −5.17523 −0.216198
\(574\) −3.14675 −0.131343
\(575\) −13.0519 −0.544302
\(576\) 1.00000 0.0416667
\(577\) 27.4161 1.14135 0.570673 0.821177i \(-0.306682\pi\)
0.570673 + 0.821177i \(0.306682\pi\)
\(578\) 5.40044 0.224629
\(579\) 1.67696 0.0696919
\(580\) −15.8049 −0.656264
\(581\) −16.8509 −0.699091
\(582\) −1.12737 −0.0467312
\(583\) 13.3937 0.554712
\(584\) −12.2959 −0.508808
\(585\) 0 0
\(586\) −16.2862 −0.672777
\(587\) −1.54288 −0.0636813 −0.0318407 0.999493i \(-0.510137\pi\)
−0.0318407 + 0.999493i \(0.510137\pi\)
\(588\) 1.00000 0.0412393
\(589\) 1.20669 0.0497206
\(590\) −38.1377 −1.57010
\(591\) −20.4916 −0.842910
\(592\) 5.54288 0.227811
\(593\) −35.9124 −1.47475 −0.737374 0.675485i \(-0.763934\pi\)
−0.737374 + 0.675485i \(0.763934\pi\)
\(594\) −3.24698 −0.133225
\(595\) −8.70171 −0.356735
\(596\) 0.753020 0.0308449
\(597\) −5.36898 −0.219738
\(598\) 0 0
\(599\) −25.6209 −1.04684 −0.523420 0.852075i \(-0.675344\pi\)
−0.523420 + 0.852075i \(0.675344\pi\)
\(600\) −1.52781 −0.0623726
\(601\) 9.42519 0.384462 0.192231 0.981350i \(-0.438428\pi\)
0.192231 + 0.981350i \(0.438428\pi\)
\(602\) 4.93900 0.201299
\(603\) 5.47219 0.222845
\(604\) 15.1021 0.614498
\(605\) −1.16793 −0.0474832
\(606\) 1.03385 0.0419974
\(607\) −29.0495 −1.17908 −0.589542 0.807738i \(-0.700692\pi\)
−0.589542 + 0.807738i \(0.700692\pi\)
\(608\) 2.85086 0.115617
\(609\) 6.18598 0.250669
\(610\) −8.11529 −0.328579
\(611\) 0 0
\(612\) 3.40581 0.137672
\(613\) −16.3763 −0.661431 −0.330716 0.943730i \(-0.607290\pi\)
−0.330716 + 0.943730i \(0.607290\pi\)
\(614\) 1.91723 0.0773731
\(615\) −8.03982 −0.324197
\(616\) 3.24698 0.130825
\(617\) −21.7313 −0.874867 −0.437434 0.899251i \(-0.644112\pi\)
−0.437434 + 0.899251i \(0.644112\pi\)
\(618\) −8.18060 −0.329072
\(619\) −29.4437 −1.18344 −0.591721 0.806143i \(-0.701551\pi\)
−0.591721 + 0.806143i \(0.701551\pi\)
\(620\) −1.08144 −0.0434317
\(621\) −8.54288 −0.342814
\(622\) 16.2064 0.649818
\(623\) −1.96615 −0.0787721
\(624\) 0 0
\(625\) −30.3048 −1.21219
\(626\) 24.2543 0.969396
\(627\) −9.25667 −0.369676
\(628\) −18.5133 −0.738763
\(629\) 18.8780 0.752715
\(630\) 2.55496 0.101792
\(631\) 31.9855 1.27332 0.636662 0.771143i \(-0.280315\pi\)
0.636662 + 0.771143i \(0.280315\pi\)
\(632\) −10.2567 −0.407988
\(633\) 6.69202 0.265984
\(634\) 23.2911 0.925008
\(635\) −37.3370 −1.48168
\(636\) 4.12498 0.163566
\(637\) 0 0
\(638\) 20.0858 0.795203
\(639\) −4.21983 −0.166934
\(640\) −2.55496 −0.100994
\(641\) −38.9353 −1.53785 −0.768926 0.639338i \(-0.779208\pi\)
−0.768926 + 0.639338i \(0.779208\pi\)
\(642\) 0.780167 0.0307908
\(643\) 43.4010 1.71157 0.855785 0.517332i \(-0.173075\pi\)
0.855785 + 0.517332i \(0.173075\pi\)
\(644\) 8.54288 0.336636
\(645\) 12.6189 0.496870
\(646\) 9.70948 0.382014
\(647\) 29.4825 1.15908 0.579538 0.814945i \(-0.303233\pi\)
0.579538 + 0.814945i \(0.303233\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 48.4674 1.90251
\(650\) 0 0
\(651\) 0.423272 0.0165893
\(652\) −20.1444 −0.788914
\(653\) −36.3715 −1.42333 −0.711663 0.702521i \(-0.752058\pi\)
−0.711663 + 0.702521i \(0.752058\pi\)
\(654\) −6.93900 −0.271336
\(655\) 50.5314 1.97442
\(656\) −3.14675 −0.122860
\(657\) 12.2959 0.479709
\(658\) 4.50604 0.175664
\(659\) 5.29483 0.206257 0.103129 0.994668i \(-0.467115\pi\)
0.103129 + 0.994668i \(0.467115\pi\)
\(660\) 8.29590 0.322917
\(661\) 4.78209 0.186002 0.0930008 0.995666i \(-0.470354\pi\)
0.0930008 + 0.995666i \(0.470354\pi\)
\(662\) 34.3957 1.33682
\(663\) 0 0
\(664\) −16.8509 −0.653940
\(665\) 7.28382 0.282454
\(666\) −5.54288 −0.214782
\(667\) 52.8461 2.04621
\(668\) 18.5157 0.716395
\(669\) 18.5864 0.718592
\(670\) −13.9812 −0.540141
\(671\) 10.3134 0.398143
\(672\) 1.00000 0.0385758
\(673\) 27.5687 1.06270 0.531348 0.847154i \(-0.321686\pi\)
0.531348 + 0.847154i \(0.321686\pi\)
\(674\) 14.8049 0.570264
\(675\) 1.52781 0.0588055
\(676\) 0 0
\(677\) 6.08038 0.233688 0.116844 0.993150i \(-0.462722\pi\)
0.116844 + 0.993150i \(0.462722\pi\)
\(678\) 5.20237 0.199796
\(679\) −1.12737 −0.0432647
\(680\) −8.70171 −0.333695
\(681\) 6.92394 0.265326
\(682\) 1.37435 0.0526267
\(683\) 50.2941 1.92445 0.962225 0.272255i \(-0.0877695\pi\)
0.962225 + 0.272255i \(0.0877695\pi\)
\(684\) −2.85086 −0.109005
\(685\) 42.8280 1.63637
\(686\) 1.00000 0.0381802
\(687\) 5.73855 0.218939
\(688\) 4.93900 0.188298
\(689\) 0 0
\(690\) 21.8267 0.830928
\(691\) −1.83340 −0.0697457 −0.0348728 0.999392i \(-0.511103\pi\)
−0.0348728 + 0.999392i \(0.511103\pi\)
\(692\) −22.5676 −0.857893
\(693\) −3.24698 −0.123343
\(694\) −26.6122 −1.01019
\(695\) 31.0901 1.17931
\(696\) 6.18598 0.234479
\(697\) −10.7172 −0.405945
\(698\) 13.2567 0.501772
\(699\) 11.2892 0.426996
\(700\) −1.52781 −0.0577458
\(701\) 2.42327 0.0915257 0.0457629 0.998952i \(-0.485428\pi\)
0.0457629 + 0.998952i \(0.485428\pi\)
\(702\) 0 0
\(703\) −15.8019 −0.595981
\(704\) 3.24698 0.122375
\(705\) 11.5127 0.433595
\(706\) −22.5483 −0.848615
\(707\) 1.03385 0.0388820
\(708\) 14.9269 0.560988
\(709\) −21.0355 −0.790005 −0.395003 0.918680i \(-0.629256\pi\)
−0.395003 + 0.918680i \(0.629256\pi\)
\(710\) 10.7815 0.404622
\(711\) 10.2567 0.384655
\(712\) −1.96615 −0.0736845
\(713\) 3.61596 0.135419
\(714\) 3.40581 0.127459
\(715\) 0 0
\(716\) 11.8116 0.441421
\(717\) 19.3491 0.722606
\(718\) −31.8025 −1.18686
\(719\) 10.7584 0.401221 0.200610 0.979671i \(-0.435708\pi\)
0.200610 + 0.979671i \(0.435708\pi\)
\(720\) 2.55496 0.0952177
\(721\) −8.18060 −0.304662
\(722\) 10.8726 0.404637
\(723\) −5.95108 −0.221323
\(724\) −5.48858 −0.203982
\(725\) −9.45101 −0.351002
\(726\) 0.457123 0.0169654
\(727\) −8.15777 −0.302555 −0.151277 0.988491i \(-0.548339\pi\)
−0.151277 + 0.988491i \(0.548339\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −31.4155 −1.16274
\(731\) 16.8213 0.622159
\(732\) 3.17629 0.117399
\(733\) 19.3884 0.716125 0.358063 0.933698i \(-0.383437\pi\)
0.358063 + 0.933698i \(0.383437\pi\)
\(734\) 3.31037 0.122188
\(735\) 2.55496 0.0942411
\(736\) 8.54288 0.314895
\(737\) 17.7681 0.654496
\(738\) 3.14675 0.115834
\(739\) −35.7579 −1.31538 −0.657688 0.753290i \(-0.728466\pi\)
−0.657688 + 0.753290i \(0.728466\pi\)
\(740\) 14.1618 0.520599
\(741\) 0 0
\(742\) 4.12498 0.151433
\(743\) −27.8799 −1.02282 −0.511408 0.859338i \(-0.670876\pi\)
−0.511408 + 0.859338i \(0.670876\pi\)
\(744\) 0.423272 0.0155179
\(745\) 1.92394 0.0704875
\(746\) −34.8950 −1.27760
\(747\) 16.8509 0.616541
\(748\) 11.0586 0.404343
\(749\) 0.780167 0.0285067
\(750\) 8.87130 0.323934
\(751\) 20.3526 0.742676 0.371338 0.928498i \(-0.378899\pi\)
0.371338 + 0.928498i \(0.378899\pi\)
\(752\) 4.50604 0.164318
\(753\) −8.84415 −0.322299
\(754\) 0 0
\(755\) 38.5854 1.40426
\(756\) −1.00000 −0.0363696
\(757\) −53.9023 −1.95911 −0.979556 0.201172i \(-0.935525\pi\)
−0.979556 + 0.201172i \(0.935525\pi\)
\(758\) 33.3545 1.21149
\(759\) −27.7385 −1.00685
\(760\) 7.28382 0.264212
\(761\) −20.2446 −0.733866 −0.366933 0.930247i \(-0.619592\pi\)
−0.366933 + 0.930247i \(0.619592\pi\)
\(762\) 14.6136 0.529394
\(763\) −6.93900 −0.251209
\(764\) −5.17523 −0.187233
\(765\) 8.70171 0.314611
\(766\) 23.3980 0.845406
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 10.5351 0.379906 0.189953 0.981793i \(-0.439166\pi\)
0.189953 + 0.981793i \(0.439166\pi\)
\(770\) 8.29590 0.298963
\(771\) −5.50902 −0.198403
\(772\) 1.67696 0.0603550
\(773\) −24.5526 −0.883094 −0.441547 0.897238i \(-0.645570\pi\)
−0.441547 + 0.897238i \(0.645570\pi\)
\(774\) −4.93900 −0.177529
\(775\) −0.646679 −0.0232294
\(776\) −1.12737 −0.0404704
\(777\) −5.54288 −0.198850
\(778\) 9.38644 0.336520
\(779\) 8.97093 0.321417
\(780\) 0 0
\(781\) −13.7017 −0.490286
\(782\) 29.0954 1.04045
\(783\) −6.18598 −0.221069
\(784\) 1.00000 0.0357143
\(785\) −47.3008 −1.68824
\(786\) −19.7778 −0.705450
\(787\) −37.2301 −1.32711 −0.663555 0.748127i \(-0.730953\pi\)
−0.663555 + 0.748127i \(0.730953\pi\)
\(788\) −20.4916 −0.729982
\(789\) −2.84548 −0.101302
\(790\) −26.2054 −0.932345
\(791\) 5.20237 0.184975
\(792\) −3.24698 −0.115376
\(793\) 0 0
\(794\) −7.96376 −0.282623
\(795\) 10.5392 0.373785
\(796\) −5.36898 −0.190298
\(797\) 23.5163 0.832991 0.416495 0.909138i \(-0.363258\pi\)
0.416495 + 0.909138i \(0.363258\pi\)
\(798\) −2.85086 −0.100919
\(799\) 15.3467 0.542928
\(800\) −1.52781 −0.0540163
\(801\) 1.96615 0.0694704
\(802\) −25.4359 −0.898174
\(803\) 39.9245 1.40891
\(804\) 5.47219 0.192989
\(805\) 21.8267 0.769290
\(806\) 0 0
\(807\) 26.0291 0.916267
\(808\) 1.03385 0.0363708
\(809\) 45.8437 1.61178 0.805889 0.592067i \(-0.201688\pi\)
0.805889 + 0.592067i \(0.201688\pi\)
\(810\) −2.55496 −0.0897721
\(811\) 39.7192 1.39473 0.697364 0.716717i \(-0.254356\pi\)
0.697364 + 0.716717i \(0.254356\pi\)
\(812\) 6.18598 0.217085
\(813\) 2.45042 0.0859399
\(814\) −17.9976 −0.630816
\(815\) −51.4680 −1.80285
\(816\) 3.40581 0.119227
\(817\) −14.0804 −0.492610
\(818\) 26.9801 0.943339
\(819\) 0 0
\(820\) −8.03982 −0.280763
\(821\) −21.5319 −0.751467 −0.375734 0.926728i \(-0.622609\pi\)
−0.375734 + 0.926728i \(0.622609\pi\)
\(822\) −16.7627 −0.584667
\(823\) 22.2107 0.774218 0.387109 0.922034i \(-0.373474\pi\)
0.387109 + 0.922034i \(0.373474\pi\)
\(824\) −8.18060 −0.284985
\(825\) 4.96077 0.172712
\(826\) 14.9269 0.519374
\(827\) −15.8793 −0.552178 −0.276089 0.961132i \(-0.589039\pi\)
−0.276089 + 0.961132i \(0.589039\pi\)
\(828\) −8.54288 −0.296885
\(829\) 11.5948 0.402703 0.201352 0.979519i \(-0.435467\pi\)
0.201352 + 0.979519i \(0.435467\pi\)
\(830\) −43.0532 −1.49440
\(831\) −11.9812 −0.415624
\(832\) 0 0
\(833\) 3.40581 0.118004
\(834\) −12.1685 −0.421361
\(835\) 47.3069 1.63712
\(836\) −9.25667 −0.320149
\(837\) −0.423272 −0.0146304
\(838\) 38.1594 1.31820
\(839\) 16.7759 0.579167 0.289583 0.957153i \(-0.406483\pi\)
0.289583 + 0.957153i \(0.406483\pi\)
\(840\) 2.55496 0.0881544
\(841\) 9.26636 0.319530
\(842\) 12.0828 0.416400
\(843\) 2.88231 0.0992722
\(844\) 6.69202 0.230349
\(845\) 0 0
\(846\) −4.50604 −0.154921
\(847\) 0.457123 0.0157069
\(848\) 4.12498 0.141652
\(849\) −3.71917 −0.127642
\(850\) −5.20344 −0.178476
\(851\) −47.3521 −1.62321
\(852\) −4.21983 −0.144569
\(853\) −25.7864 −0.882909 −0.441455 0.897284i \(-0.645537\pi\)
−0.441455 + 0.897284i \(0.645537\pi\)
\(854\) 3.17629 0.108690
\(855\) −7.28382 −0.249101
\(856\) 0.780167 0.0266656
\(857\) −7.85517 −0.268327 −0.134164 0.990959i \(-0.542835\pi\)
−0.134164 + 0.990959i \(0.542835\pi\)
\(858\) 0 0
\(859\) 7.97179 0.271994 0.135997 0.990709i \(-0.456576\pi\)
0.135997 + 0.990709i \(0.456576\pi\)
\(860\) 12.6189 0.430302
\(861\) 3.14675 0.107241
\(862\) 28.7023 0.977604
\(863\) 5.64310 0.192093 0.0960467 0.995377i \(-0.469380\pi\)
0.0960467 + 0.995377i \(0.469380\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −57.6594 −1.96048
\(866\) −9.33406 −0.317184
\(867\) −5.40044 −0.183408
\(868\) 0.423272 0.0143668
\(869\) 33.3032 1.12973
\(870\) 15.8049 0.535837
\(871\) 0 0
\(872\) −6.93900 −0.234984
\(873\) 1.12737 0.0381559
\(874\) −24.3545 −0.823803
\(875\) 8.87130 0.299905
\(876\) 12.2959 0.415440
\(877\) 4.71273 0.159137 0.0795687 0.996829i \(-0.474646\pi\)
0.0795687 + 0.996829i \(0.474646\pi\)
\(878\) 15.3123 0.516765
\(879\) 16.2862 0.549320
\(880\) 8.29590 0.279655
\(881\) 24.1400 0.813299 0.406649 0.913584i \(-0.366697\pi\)
0.406649 + 0.913584i \(0.366697\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 9.48858 0.319316 0.159658 0.987172i \(-0.448961\pi\)
0.159658 + 0.987172i \(0.448961\pi\)
\(884\) 0 0
\(885\) 38.1377 1.28198
\(886\) −9.89679 −0.332489
\(887\) 37.4319 1.25684 0.628420 0.777874i \(-0.283702\pi\)
0.628420 + 0.777874i \(0.283702\pi\)
\(888\) −5.54288 −0.186007
\(889\) 14.6136 0.490123
\(890\) −5.02343 −0.168386
\(891\) 3.24698 0.108778
\(892\) 18.5864 0.622319
\(893\) −12.8461 −0.429877
\(894\) −0.753020 −0.0251848
\(895\) 30.1782 1.00875
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) 26.9288 0.898627
\(899\) 2.61835 0.0873269
\(900\) 1.52781 0.0509270
\(901\) 14.0489 0.468037
\(902\) 10.2174 0.340204
\(903\) −4.93900 −0.164360
\(904\) 5.20237 0.173028
\(905\) −14.0231 −0.466144
\(906\) −15.1021 −0.501735
\(907\) −53.1041 −1.76329 −0.881646 0.471912i \(-0.843564\pi\)
−0.881646 + 0.471912i \(0.843564\pi\)
\(908\) 6.92394 0.229779
\(909\) −1.03385 −0.0342907
\(910\) 0 0
\(911\) −4.68127 −0.155097 −0.0775487 0.996989i \(-0.524709\pi\)
−0.0775487 + 0.996989i \(0.524709\pi\)
\(912\) −2.85086 −0.0944012
\(913\) 54.7144 1.81078
\(914\) 21.3351 0.705703
\(915\) 8.11529 0.268283
\(916\) 5.73855 0.189607
\(917\) −19.7778 −0.653120
\(918\) −3.40581 −0.112409
\(919\) 17.1515 0.565777 0.282889 0.959153i \(-0.408707\pi\)
0.282889 + 0.959153i \(0.408707\pi\)
\(920\) 21.8267 0.719605
\(921\) −1.91723 −0.0631749
\(922\) −24.3045 −0.800427
\(923\) 0 0
\(924\) −3.24698 −0.106818
\(925\) 8.46847 0.278442
\(926\) 37.0465 1.21742
\(927\) 8.18060 0.268686
\(928\) 6.18598 0.203065
\(929\) −27.7235 −0.909578 −0.454789 0.890599i \(-0.650285\pi\)
−0.454789 + 0.890599i \(0.650285\pi\)
\(930\) 1.08144 0.0354619
\(931\) −2.85086 −0.0934330
\(932\) 11.2892 0.369790
\(933\) −16.2064 −0.530574
\(934\) −32.8582 −1.07515
\(935\) 28.2543 0.924014
\(936\) 0 0
\(937\) −6.49396 −0.212148 −0.106074 0.994358i \(-0.533828\pi\)
−0.106074 + 0.994358i \(0.533828\pi\)
\(938\) 5.47219 0.178673
\(939\) −24.2543 −0.791508
\(940\) 11.5127 0.375504
\(941\) −7.95779 −0.259416 −0.129708 0.991552i \(-0.541404\pi\)
−0.129708 + 0.991552i \(0.541404\pi\)
\(942\) 18.5133 0.603197
\(943\) 26.8823 0.875409
\(944\) 14.9269 0.485830
\(945\) −2.55496 −0.0831128
\(946\) −16.0368 −0.521403
\(947\) 56.4717 1.83508 0.917542 0.397639i \(-0.130170\pi\)
0.917542 + 0.397639i \(0.130170\pi\)
\(948\) 10.2567 0.333121
\(949\) 0 0
\(950\) 4.35557 0.141313
\(951\) −23.2911 −0.755266
\(952\) 3.40581 0.110383
\(953\) 37.2204 1.20569 0.602844 0.797859i \(-0.294034\pi\)
0.602844 + 0.797859i \(0.294034\pi\)
\(954\) −4.12498 −0.133551
\(955\) −13.2225 −0.427870
\(956\) 19.3491 0.625795
\(957\) −20.0858 −0.649280
\(958\) 20.8829 0.674696
\(959\) −16.7627 −0.541296
\(960\) 2.55496 0.0824609
\(961\) −30.8208 −0.994221
\(962\) 0 0
\(963\) −0.780167 −0.0251405
\(964\) −5.95108 −0.191671
\(965\) 4.28455 0.137925
\(966\) −8.54288 −0.274863
\(967\) −24.0790 −0.774330 −0.387165 0.922010i \(-0.626546\pi\)
−0.387165 + 0.922010i \(0.626546\pi\)
\(968\) 0.457123 0.0146925
\(969\) −9.70948 −0.311913
\(970\) −2.88040 −0.0924839
\(971\) −28.4204 −0.912054 −0.456027 0.889966i \(-0.650728\pi\)
−0.456027 + 0.889966i \(0.650728\pi\)
\(972\) 1.00000 0.0320750
\(973\) −12.1685 −0.390105
\(974\) −33.0224 −1.05810
\(975\) 0 0
\(976\) 3.17629 0.101671
\(977\) −20.6297 −0.660002 −0.330001 0.943981i \(-0.607049\pi\)
−0.330001 + 0.943981i \(0.607049\pi\)
\(978\) 20.1444 0.644146
\(979\) 6.38404 0.204035
\(980\) 2.55496 0.0816151
\(981\) 6.93900 0.221545
\(982\) −1.14244 −0.0364567
\(983\) 48.3749 1.54292 0.771461 0.636277i \(-0.219527\pi\)
0.771461 + 0.636277i \(0.219527\pi\)
\(984\) 3.14675 0.100315
\(985\) −52.3551 −1.66817
\(986\) 21.0683 0.670951
\(987\) −4.50604 −0.143429
\(988\) 0 0
\(989\) −42.1933 −1.34167
\(990\) −8.29590 −0.263661
\(991\) 12.6276 0.401128 0.200564 0.979681i \(-0.435723\pi\)
0.200564 + 0.979681i \(0.435723\pi\)
\(992\) 0.423272 0.0134389
\(993\) −34.3957 −1.09151
\(994\) −4.21983 −0.133845
\(995\) −13.7175 −0.434874
\(996\) 16.8509 0.533940
\(997\) −11.1733 −0.353862 −0.176931 0.984223i \(-0.556617\pi\)
−0.176931 + 0.984223i \(0.556617\pi\)
\(998\) −16.4239 −0.519888
\(999\) 5.54288 0.175369
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.cg.1.2 3
13.12 even 2 7098.2.a.cl.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7098.2.a.cg.1.2 3 1.1 even 1 trivial
7098.2.a.cl.1.2 yes 3 13.12 even 2