Properties

Label 5070.2.a.bh.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 390)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -5.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} -5.12311 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -3.12311 q^{11} +1.00000 q^{12} -5.12311 q^{14} -1.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} -1.00000 q^{20} -5.12311 q^{21} -3.12311 q^{22} -3.12311 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -5.12311 q^{28} +2.00000 q^{29} -1.00000 q^{30} +5.12311 q^{31} +1.00000 q^{32} -3.12311 q^{33} -2.00000 q^{34} +5.12311 q^{35} +1.00000 q^{36} +3.12311 q^{37} +6.00000 q^{38} -1.00000 q^{40} -9.12311 q^{41} -5.12311 q^{42} +10.2462 q^{43} -3.12311 q^{44} -1.00000 q^{45} -3.12311 q^{46} +10.2462 q^{47} +1.00000 q^{48} +19.2462 q^{49} +1.00000 q^{50} -2.00000 q^{51} +11.3693 q^{53} +1.00000 q^{54} +3.12311 q^{55} -5.12311 q^{56} +6.00000 q^{57} +2.00000 q^{58} +7.12311 q^{59} -1.00000 q^{60} +10.0000 q^{61} +5.12311 q^{62} -5.12311 q^{63} +1.00000 q^{64} -3.12311 q^{66} -13.1231 q^{67} -2.00000 q^{68} -3.12311 q^{69} +5.12311 q^{70} +6.24621 q^{71} +1.00000 q^{72} -4.87689 q^{73} +3.12311 q^{74} +1.00000 q^{75} +6.00000 q^{76} +16.0000 q^{77} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -9.12311 q^{82} -10.2462 q^{83} -5.12311 q^{84} +2.00000 q^{85} +10.2462 q^{86} +2.00000 q^{87} -3.12311 q^{88} -5.12311 q^{89} -1.00000 q^{90} -3.12311 q^{92} +5.12311 q^{93} +10.2462 q^{94} -6.00000 q^{95} +1.00000 q^{96} +4.87689 q^{97} +19.2462 q^{98} -3.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} - 2q^{7} + 2q^{8} + 2q^{9} - 2q^{10} + 2q^{11} + 2q^{12} - 2q^{14} - 2q^{15} + 2q^{16} - 4q^{17} + 2q^{18} + 12q^{19} - 2q^{20} - 2q^{21} + 2q^{22} + 2q^{23} + 2q^{24} + 2q^{25} + 2q^{27} - 2q^{28} + 4q^{29} - 2q^{30} + 2q^{31} + 2q^{32} + 2q^{33} - 4q^{34} + 2q^{35} + 2q^{36} - 2q^{37} + 12q^{38} - 2q^{40} - 10q^{41} - 2q^{42} + 4q^{43} + 2q^{44} - 2q^{45} + 2q^{46} + 4q^{47} + 2q^{48} + 22q^{49} + 2q^{50} - 4q^{51} - 2q^{53} + 2q^{54} - 2q^{55} - 2q^{56} + 12q^{57} + 4q^{58} + 6q^{59} - 2q^{60} + 20q^{61} + 2q^{62} - 2q^{63} + 2q^{64} + 2q^{66} - 18q^{67} - 4q^{68} + 2q^{69} + 2q^{70} - 4q^{71} + 2q^{72} - 18q^{73} - 2q^{74} + 2q^{75} + 12q^{76} + 32q^{77} + 16q^{79} - 2q^{80} + 2q^{81} - 10q^{82} - 4q^{83} - 2q^{84} + 4q^{85} + 4q^{86} + 4q^{87} + 2q^{88} - 2q^{89} - 2q^{90} + 2q^{92} + 2q^{93} + 4q^{94} - 12q^{95} + 2q^{96} + 18q^{97} + 22q^{98} + 2q^{99} + O(q^{100}) \)

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
\(5\) −1.00000 −0.447214
\(6\) 1.00000 0.408248
\(7\) −5.12311 −1.93635 −0.968176 0.250270i \(-0.919480\pi\)
−0.968176 + 0.250270i \(0.919480\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) −3.12311 −0.941652 −0.470826 0.882226i \(-0.656044\pi\)
−0.470826 + 0.882226i \(0.656044\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −5.12311 −1.36921
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) −1.00000 −0.223607
\(21\) −5.12311 −1.11795
\(22\) −3.12311 −0.665848
\(23\) −3.12311 −0.651213 −0.325606 0.945505i \(-0.605568\pi\)
−0.325606 + 0.945505i \(0.605568\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −5.12311 −0.968176
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −1.00000 −0.182574
\(31\) 5.12311 0.920137 0.460068 0.887883i \(-0.347825\pi\)
0.460068 + 0.887883i \(0.347825\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.12311 −0.543663
\(34\) −2.00000 −0.342997
\(35\) 5.12311 0.865963
\(36\) 1.00000 0.166667
\(37\) 3.12311 0.513435 0.256718 0.966486i \(-0.417359\pi\)
0.256718 + 0.966486i \(0.417359\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −9.12311 −1.42479 −0.712395 0.701779i \(-0.752389\pi\)
−0.712395 + 0.701779i \(0.752389\pi\)
\(42\) −5.12311 −0.790512
\(43\) 10.2462 1.56253 0.781266 0.624198i \(-0.214574\pi\)
0.781266 + 0.624198i \(0.214574\pi\)
\(44\) −3.12311 −0.470826
\(45\) −1.00000 −0.149071
\(46\) −3.12311 −0.460477
\(47\) 10.2462 1.49456 0.747282 0.664507i \(-0.231359\pi\)
0.747282 + 0.664507i \(0.231359\pi\)
\(48\) 1.00000 0.144338
\(49\) 19.2462 2.74946
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 11.3693 1.56170 0.780848 0.624721i \(-0.214787\pi\)
0.780848 + 0.624721i \(0.214787\pi\)
\(54\) 1.00000 0.136083
\(55\) 3.12311 0.421119
\(56\) −5.12311 −0.684604
\(57\) 6.00000 0.794719
\(58\) 2.00000 0.262613
\(59\) 7.12311 0.927349 0.463675 0.886006i \(-0.346531\pi\)
0.463675 + 0.886006i \(0.346531\pi\)
\(60\) −1.00000 −0.129099
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 5.12311 0.650635
\(63\) −5.12311 −0.645451
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.12311 −0.384428
\(67\) −13.1231 −1.60324 −0.801621 0.597832i \(-0.796029\pi\)
−0.801621 + 0.597832i \(0.796029\pi\)
\(68\) −2.00000 −0.242536
\(69\) −3.12311 −0.375978
\(70\) 5.12311 0.612328
\(71\) 6.24621 0.741289 0.370644 0.928775i \(-0.379137\pi\)
0.370644 + 0.928775i \(0.379137\pi\)
\(72\) 1.00000 0.117851
\(73\) −4.87689 −0.570797 −0.285399 0.958409i \(-0.592126\pi\)
−0.285399 + 0.958409i \(0.592126\pi\)
\(74\) 3.12311 0.363054
\(75\) 1.00000 0.115470
\(76\) 6.00000 0.688247
\(77\) 16.0000 1.82337
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −9.12311 −1.00748
\(83\) −10.2462 −1.12467 −0.562334 0.826910i \(-0.690096\pi\)
−0.562334 + 0.826910i \(0.690096\pi\)
\(84\) −5.12311 −0.558977
\(85\) 2.00000 0.216930
\(86\) 10.2462 1.10488
\(87\) 2.00000 0.214423
\(88\) −3.12311 −0.332924
\(89\) −5.12311 −0.543048 −0.271524 0.962432i \(-0.587528\pi\)
−0.271524 + 0.962432i \(0.587528\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −3.12311 −0.325606
\(93\) 5.12311 0.531241
\(94\) 10.2462 1.05682
\(95\) −6.00000 −0.615587
\(96\) 1.00000 0.102062
\(97\) 4.87689 0.495174 0.247587 0.968866i \(-0.420362\pi\)
0.247587 + 0.968866i \(0.420362\pi\)
\(98\) 19.2462 1.94416
\(99\) −3.12311 −0.313884
\(100\) 1.00000 0.100000
\(101\) −4.24621 −0.422514 −0.211257 0.977431i \(-0.567756\pi\)
−0.211257 + 0.977431i \(0.567756\pi\)
\(102\) −2.00000 −0.198030
\(103\) 4.87689 0.480535 0.240267 0.970707i \(-0.422765\pi\)
0.240267 + 0.970707i \(0.422765\pi\)
\(104\) 0 0
\(105\) 5.12311 0.499964
\(106\) 11.3693 1.10429
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 1.00000 0.0962250
\(109\) −11.1231 −1.06540 −0.532700 0.846304i \(-0.678823\pi\)
−0.532700 + 0.846304i \(0.678823\pi\)
\(110\) 3.12311 0.297776
\(111\) 3.12311 0.296432
\(112\) −5.12311 −0.484088
\(113\) −4.24621 −0.399450 −0.199725 0.979852i \(-0.564005\pi\)
−0.199725 + 0.979852i \(0.564005\pi\)
\(114\) 6.00000 0.561951
\(115\) 3.12311 0.291231
\(116\) 2.00000 0.185695
\(117\) 0 0
\(118\) 7.12311 0.655735
\(119\) 10.2462 0.939269
\(120\) −1.00000 −0.0912871
\(121\) −1.24621 −0.113292
\(122\) 10.0000 0.905357
\(123\) −9.12311 −0.822603
\(124\) 5.12311 0.460068
\(125\) −1.00000 −0.0894427
\(126\) −5.12311 −0.456403
\(127\) 4.87689 0.432754 0.216377 0.976310i \(-0.430576\pi\)
0.216377 + 0.976310i \(0.430576\pi\)
\(128\) 1.00000 0.0883883
\(129\) 10.2462 0.902129
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) −3.12311 −0.271831
\(133\) −30.7386 −2.66538
\(134\) −13.1231 −1.13366
\(135\) −1.00000 −0.0860663
\(136\) −2.00000 −0.171499
\(137\) 22.4924 1.92166 0.960829 0.277143i \(-0.0893876\pi\)
0.960829 + 0.277143i \(0.0893876\pi\)
\(138\) −3.12311 −0.265856
\(139\) 16.4924 1.39887 0.699435 0.714697i \(-0.253435\pi\)
0.699435 + 0.714697i \(0.253435\pi\)
\(140\) 5.12311 0.432981
\(141\) 10.2462 0.862887
\(142\) 6.24621 0.524170
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −2.00000 −0.166091
\(146\) −4.87689 −0.403615
\(147\) 19.2462 1.58740
\(148\) 3.12311 0.256718
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 1.00000 0.0816497
\(151\) −11.3693 −0.925222 −0.462611 0.886561i \(-0.653087\pi\)
−0.462611 + 0.886561i \(0.653087\pi\)
\(152\) 6.00000 0.486664
\(153\) −2.00000 −0.161690
\(154\) 16.0000 1.28932
\(155\) −5.12311 −0.411498
\(156\) 0 0
\(157\) −3.36932 −0.268901 −0.134450 0.990920i \(-0.542927\pi\)
−0.134450 + 0.990920i \(0.542927\pi\)
\(158\) 8.00000 0.636446
\(159\) 11.3693 0.901645
\(160\) −1.00000 −0.0790569
\(161\) 16.0000 1.26098
\(162\) 1.00000 0.0785674
\(163\) 1.12311 0.0879684 0.0439842 0.999032i \(-0.485995\pi\)
0.0439842 + 0.999032i \(0.485995\pi\)
\(164\) −9.12311 −0.712395
\(165\) 3.12311 0.243133
\(166\) −10.2462 −0.795260
\(167\) −5.75379 −0.445242 −0.222621 0.974905i \(-0.571461\pi\)
−0.222621 + 0.974905i \(0.571461\pi\)
\(168\) −5.12311 −0.395256
\(169\) 0 0
\(170\) 2.00000 0.153393
\(171\) 6.00000 0.458831
\(172\) 10.2462 0.781266
\(173\) −14.8769 −1.13107 −0.565535 0.824725i \(-0.691330\pi\)
−0.565535 + 0.824725i \(0.691330\pi\)
\(174\) 2.00000 0.151620
\(175\) −5.12311 −0.387270
\(176\) −3.12311 −0.235413
\(177\) 7.12311 0.535405
\(178\) −5.12311 −0.383993
\(179\) 16.4924 1.23270 0.616351 0.787472i \(-0.288610\pi\)
0.616351 + 0.787472i \(0.288610\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 3.75379 0.279017 0.139508 0.990221i \(-0.455448\pi\)
0.139508 + 0.990221i \(0.455448\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) −3.12311 −0.230238
\(185\) −3.12311 −0.229615
\(186\) 5.12311 0.375644
\(187\) 6.24621 0.456768
\(188\) 10.2462 0.747282
\(189\) −5.12311 −0.372651
\(190\) −6.00000 −0.435286
\(191\) −16.4924 −1.19335 −0.596675 0.802483i \(-0.703512\pi\)
−0.596675 + 0.802483i \(0.703512\pi\)
\(192\) 1.00000 0.0721688
\(193\) −16.8769 −1.21483 −0.607413 0.794386i \(-0.707793\pi\)
−0.607413 + 0.794386i \(0.707793\pi\)
\(194\) 4.87689 0.350141
\(195\) 0 0
\(196\) 19.2462 1.37473
\(197\) 0.246211 0.0175418 0.00877091 0.999962i \(-0.497208\pi\)
0.00877091 + 0.999962i \(0.497208\pi\)
\(198\) −3.12311 −0.221949
\(199\) −1.75379 −0.124323 −0.0621614 0.998066i \(-0.519799\pi\)
−0.0621614 + 0.998066i \(0.519799\pi\)
\(200\) 1.00000 0.0707107
\(201\) −13.1231 −0.925633
\(202\) −4.24621 −0.298762
\(203\) −10.2462 −0.719143
\(204\) −2.00000 −0.140028
\(205\) 9.12311 0.637185
\(206\) 4.87689 0.339789
\(207\) −3.12311 −0.217071
\(208\) 0 0
\(209\) −18.7386 −1.29618
\(210\) 5.12311 0.353528
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 11.3693 0.780848
\(213\) 6.24621 0.427983
\(214\) −8.00000 −0.546869
\(215\) −10.2462 −0.698786
\(216\) 1.00000 0.0680414
\(217\) −26.2462 −1.78171
\(218\) −11.1231 −0.753352
\(219\) −4.87689 −0.329550
\(220\) 3.12311 0.210560
\(221\) 0 0
\(222\) 3.12311 0.209609
\(223\) 15.3693 1.02921 0.514603 0.857429i \(-0.327939\pi\)
0.514603 + 0.857429i \(0.327939\pi\)
\(224\) −5.12311 −0.342302
\(225\) 1.00000 0.0666667
\(226\) −4.24621 −0.282454
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 6.00000 0.397360
\(229\) 3.12311 0.206381 0.103190 0.994662i \(-0.467095\pi\)
0.103190 + 0.994662i \(0.467095\pi\)
\(230\) 3.12311 0.205931
\(231\) 16.0000 1.05272
\(232\) 2.00000 0.131306
\(233\) 24.2462 1.58842 0.794211 0.607642i \(-0.207884\pi\)
0.794211 + 0.607642i \(0.207884\pi\)
\(234\) 0 0
\(235\) −10.2462 −0.668389
\(236\) 7.12311 0.463675
\(237\) 8.00000 0.519656
\(238\) 10.2462 0.664163
\(239\) 28.4924 1.84302 0.921511 0.388353i \(-0.126956\pi\)
0.921511 + 0.388353i \(0.126956\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 2.24621 0.144691 0.0723456 0.997380i \(-0.476952\pi\)
0.0723456 + 0.997380i \(0.476952\pi\)
\(242\) −1.24621 −0.0801095
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) −19.2462 −1.22960
\(246\) −9.12311 −0.581668
\(247\) 0 0
\(248\) 5.12311 0.325318
\(249\) −10.2462 −0.649327
\(250\) −1.00000 −0.0632456
\(251\) −9.75379 −0.615654 −0.307827 0.951442i \(-0.599602\pi\)
−0.307827 + 0.951442i \(0.599602\pi\)
\(252\) −5.12311 −0.322725
\(253\) 9.75379 0.613215
\(254\) 4.87689 0.306004
\(255\) 2.00000 0.125245
\(256\) 1.00000 0.0625000
\(257\) −4.24621 −0.264871 −0.132436 0.991192i \(-0.542280\pi\)
−0.132436 + 0.991192i \(0.542280\pi\)
\(258\) 10.2462 0.637901
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 4.00000 0.247121
\(263\) −2.63068 −0.162215 −0.0811074 0.996705i \(-0.525846\pi\)
−0.0811074 + 0.996705i \(0.525846\pi\)
\(264\) −3.12311 −0.192214
\(265\) −11.3693 −0.698412
\(266\) −30.7386 −1.88471
\(267\) −5.12311 −0.313529
\(268\) −13.1231 −0.801621
\(269\) 0.246211 0.0150118 0.00750588 0.999972i \(-0.497611\pi\)
0.00750588 + 0.999972i \(0.497611\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 14.8769 0.903707 0.451853 0.892092i \(-0.350763\pi\)
0.451853 + 0.892092i \(0.350763\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 22.4924 1.35882
\(275\) −3.12311 −0.188330
\(276\) −3.12311 −0.187989
\(277\) −27.8617 −1.67405 −0.837025 0.547165i \(-0.815707\pi\)
−0.837025 + 0.547165i \(0.815707\pi\)
\(278\) 16.4924 0.989150
\(279\) 5.12311 0.306712
\(280\) 5.12311 0.306164
\(281\) 5.12311 0.305619 0.152809 0.988256i \(-0.451168\pi\)
0.152809 + 0.988256i \(0.451168\pi\)
\(282\) 10.2462 0.610153
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 6.24621 0.370644
\(285\) −6.00000 −0.355409
\(286\) 0 0
\(287\) 46.7386 2.75889
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) −2.00000 −0.117444
\(291\) 4.87689 0.285889
\(292\) −4.87689 −0.285399
\(293\) −20.7386 −1.21156 −0.605782 0.795631i \(-0.707140\pi\)
−0.605782 + 0.795631i \(0.707140\pi\)
\(294\) 19.2462 1.12246
\(295\) −7.12311 −0.414723
\(296\) 3.12311 0.181527
\(297\) −3.12311 −0.181221
\(298\) 14.0000 0.810998
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) −52.4924 −3.02561
\(302\) −11.3693 −0.654231
\(303\) −4.24621 −0.243938
\(304\) 6.00000 0.344124
\(305\) −10.0000 −0.572598
\(306\) −2.00000 −0.114332
\(307\) 22.8769 1.30565 0.652827 0.757507i \(-0.273583\pi\)
0.652827 + 0.757507i \(0.273583\pi\)
\(308\) 16.0000 0.911685
\(309\) 4.87689 0.277437
\(310\) −5.12311 −0.290973
\(311\) 24.4924 1.38884 0.694419 0.719571i \(-0.255661\pi\)
0.694419 + 0.719571i \(0.255661\pi\)
\(312\) 0 0
\(313\) −0.246211 −0.0139167 −0.00695834 0.999976i \(-0.502215\pi\)
−0.00695834 + 0.999976i \(0.502215\pi\)
\(314\) −3.36932 −0.190142
\(315\) 5.12311 0.288654
\(316\) 8.00000 0.450035
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 11.3693 0.637560
\(319\) −6.24621 −0.349721
\(320\) −1.00000 −0.0559017
\(321\) −8.00000 −0.446516
\(322\) 16.0000 0.891645
\(323\) −12.0000 −0.667698
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 1.12311 0.0622031
\(327\) −11.1231 −0.615109
\(328\) −9.12311 −0.503739
\(329\) −52.4924 −2.89400
\(330\) 3.12311 0.171921
\(331\) 24.2462 1.33269 0.666346 0.745643i \(-0.267857\pi\)
0.666346 + 0.745643i \(0.267857\pi\)
\(332\) −10.2462 −0.562334
\(333\) 3.12311 0.171145
\(334\) −5.75379 −0.314833
\(335\) 13.1231 0.716992
\(336\) −5.12311 −0.279488
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 0 0
\(339\) −4.24621 −0.230623
\(340\) 2.00000 0.108465
\(341\) −16.0000 −0.866449
\(342\) 6.00000 0.324443
\(343\) −62.7386 −3.38757
\(344\) 10.2462 0.552439
\(345\) 3.12311 0.168142
\(346\) −14.8769 −0.799787
\(347\) 2.24621 0.120583 0.0602915 0.998181i \(-0.480797\pi\)
0.0602915 + 0.998181i \(0.480797\pi\)
\(348\) 2.00000 0.107211
\(349\) −5.36932 −0.287413 −0.143706 0.989620i \(-0.545902\pi\)
−0.143706 + 0.989620i \(0.545902\pi\)
\(350\) −5.12311 −0.273842
\(351\) 0 0
\(352\) −3.12311 −0.166462
\(353\) −4.24621 −0.226003 −0.113002 0.993595i \(-0.536046\pi\)
−0.113002 + 0.993595i \(0.536046\pi\)
\(354\) 7.12311 0.378589
\(355\) −6.24621 −0.331514
\(356\) −5.12311 −0.271524
\(357\) 10.2462 0.542287
\(358\) 16.4924 0.871652
\(359\) 34.2462 1.80745 0.903723 0.428118i \(-0.140823\pi\)
0.903723 + 0.428118i \(0.140823\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 17.0000 0.894737
\(362\) 3.75379 0.197295
\(363\) −1.24621 −0.0654091
\(364\) 0 0
\(365\) 4.87689 0.255268
\(366\) 10.0000 0.522708
\(367\) 33.3693 1.74186 0.870932 0.491403i \(-0.163516\pi\)
0.870932 + 0.491403i \(0.163516\pi\)
\(368\) −3.12311 −0.162803
\(369\) −9.12311 −0.474930
\(370\) −3.12311 −0.162363
\(371\) −58.2462 −3.02399
\(372\) 5.12311 0.265621
\(373\) −1.12311 −0.0581522 −0.0290761 0.999577i \(-0.509257\pi\)
−0.0290761 + 0.999577i \(0.509257\pi\)
\(374\) 6.24621 0.322984
\(375\) −1.00000 −0.0516398
\(376\) 10.2462 0.528408
\(377\) 0 0
\(378\) −5.12311 −0.263504
\(379\) 6.00000 0.308199 0.154100 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) −6.00000 −0.307794
\(381\) 4.87689 0.249851
\(382\) −16.4924 −0.843826
\(383\) −18.2462 −0.932338 −0.466169 0.884696i \(-0.654366\pi\)
−0.466169 + 0.884696i \(0.654366\pi\)
\(384\) 1.00000 0.0510310
\(385\) −16.0000 −0.815436
\(386\) −16.8769 −0.859011
\(387\) 10.2462 0.520844
\(388\) 4.87689 0.247587
\(389\) −16.2462 −0.823716 −0.411858 0.911248i \(-0.635120\pi\)
−0.411858 + 0.911248i \(0.635120\pi\)
\(390\) 0 0
\(391\) 6.24621 0.315884
\(392\) 19.2462 0.972080
\(393\) 4.00000 0.201773
\(394\) 0.246211 0.0124039
\(395\) −8.00000 −0.402524
\(396\) −3.12311 −0.156942
\(397\) 9.36932 0.470233 0.235116 0.971967i \(-0.424453\pi\)
0.235116 + 0.971967i \(0.424453\pi\)
\(398\) −1.75379 −0.0879095
\(399\) −30.7386 −1.53886
\(400\) 1.00000 0.0500000
\(401\) 23.3693 1.16701 0.583504 0.812110i \(-0.301681\pi\)
0.583504 + 0.812110i \(0.301681\pi\)
\(402\) −13.1231 −0.654521
\(403\) 0 0
\(404\) −4.24621 −0.211257
\(405\) −1.00000 −0.0496904
\(406\) −10.2462 −0.508511
\(407\) −9.75379 −0.483477
\(408\) −2.00000 −0.0990148
\(409\) 24.4924 1.21107 0.605536 0.795818i \(-0.292959\pi\)
0.605536 + 0.795818i \(0.292959\pi\)
\(410\) 9.12311 0.450558
\(411\) 22.4924 1.10947
\(412\) 4.87689 0.240267
\(413\) −36.4924 −1.79567
\(414\) −3.12311 −0.153492
\(415\) 10.2462 0.502967
\(416\) 0 0
\(417\) 16.4924 0.807637
\(418\) −18.7386 −0.916537
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 5.12311 0.249982
\(421\) 25.3693 1.23642 0.618212 0.786011i \(-0.287857\pi\)
0.618212 + 0.786011i \(0.287857\pi\)
\(422\) 4.00000 0.194717
\(423\) 10.2462 0.498188
\(424\) 11.3693 0.552143
\(425\) −2.00000 −0.0970143
\(426\) 6.24621 0.302630
\(427\) −51.2311 −2.47924
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) −10.2462 −0.494116
\(431\) −0.492423 −0.0237192 −0.0118596 0.999930i \(-0.503775\pi\)
−0.0118596 + 0.999930i \(0.503775\pi\)
\(432\) 1.00000 0.0481125
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) −26.2462 −1.25986
\(435\) −2.00000 −0.0958927
\(436\) −11.1231 −0.532700
\(437\) −18.7386 −0.896390
\(438\) −4.87689 −0.233027
\(439\) 3.50758 0.167408 0.0837038 0.996491i \(-0.473325\pi\)
0.0837038 + 0.996491i \(0.473325\pi\)
\(440\) 3.12311 0.148888
\(441\) 19.2462 0.916486
\(442\) 0 0
\(443\) −36.4924 −1.73381 −0.866904 0.498476i \(-0.833893\pi\)
−0.866904 + 0.498476i \(0.833893\pi\)
\(444\) 3.12311 0.148216
\(445\) 5.12311 0.242858
\(446\) 15.3693 0.727758
\(447\) 14.0000 0.662177
\(448\) −5.12311 −0.242044
\(449\) −37.1231 −1.75195 −0.875974 0.482359i \(-0.839780\pi\)
−0.875974 + 0.482359i \(0.839780\pi\)
\(450\) 1.00000 0.0471405
\(451\) 28.4924 1.34166
\(452\) −4.24621 −0.199725
\(453\) −11.3693 −0.534177
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) 6.00000 0.280976
\(457\) −6.63068 −0.310170 −0.155085 0.987901i \(-0.549565\pi\)
−0.155085 + 0.987901i \(0.549565\pi\)
\(458\) 3.12311 0.145933
\(459\) −2.00000 −0.0933520
\(460\) 3.12311 0.145616
\(461\) 14.4924 0.674979 0.337490 0.941329i \(-0.390422\pi\)
0.337490 + 0.941329i \(0.390422\pi\)
\(462\) 16.0000 0.744387
\(463\) 35.8617 1.66664 0.833318 0.552794i \(-0.186438\pi\)
0.833318 + 0.552794i \(0.186438\pi\)
\(464\) 2.00000 0.0928477
\(465\) −5.12311 −0.237578
\(466\) 24.2462 1.12318
\(467\) 5.75379 0.266254 0.133127 0.991099i \(-0.457498\pi\)
0.133127 + 0.991099i \(0.457498\pi\)
\(468\) 0 0
\(469\) 67.2311 3.10444
\(470\) −10.2462 −0.472622
\(471\) −3.36932 −0.155250
\(472\) 7.12311 0.327868
\(473\) −32.0000 −1.47136
\(474\) 8.00000 0.367452
\(475\) 6.00000 0.275299
\(476\) 10.2462 0.469634
\(477\) 11.3693 0.520565
\(478\) 28.4924 1.30321
\(479\) 20.4924 0.936323 0.468161 0.883643i \(-0.344917\pi\)
0.468161 + 0.883643i \(0.344917\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 2.24621 0.102312
\(483\) 16.0000 0.728025
\(484\) −1.24621 −0.0566460
\(485\) −4.87689 −0.221448
\(486\) 1.00000 0.0453609
\(487\) 7.36932 0.333936 0.166968 0.985962i \(-0.446602\pi\)
0.166968 + 0.985962i \(0.446602\pi\)
\(488\) 10.0000 0.452679
\(489\) 1.12311 0.0507886
\(490\) −19.2462 −0.869455
\(491\) 10.7386 0.484628 0.242314 0.970198i \(-0.422094\pi\)
0.242314 + 0.970198i \(0.422094\pi\)
\(492\) −9.12311 −0.411301
\(493\) −4.00000 −0.180151
\(494\) 0 0
\(495\) 3.12311 0.140373
\(496\) 5.12311 0.230034
\(497\) −32.0000 −1.43540
\(498\) −10.2462 −0.459144
\(499\) −1.50758 −0.0674884 −0.0337442 0.999431i \(-0.510743\pi\)
−0.0337442 + 0.999431i \(0.510743\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −5.75379 −0.257060
\(502\) −9.75379 −0.435333
\(503\) 10.6307 0.473999 0.236999 0.971510i \(-0.423836\pi\)
0.236999 + 0.971510i \(0.423836\pi\)
\(504\) −5.12311 −0.228201
\(505\) 4.24621 0.188954
\(506\) 9.75379 0.433609
\(507\) 0 0
\(508\) 4.87689 0.216377
\(509\) −15.7538 −0.698274 −0.349137 0.937072i \(-0.613525\pi\)
−0.349137 + 0.937072i \(0.613525\pi\)
\(510\) 2.00000 0.0885615
\(511\) 24.9848 1.10526
\(512\) 1.00000 0.0441942
\(513\) 6.00000 0.264906
\(514\) −4.24621 −0.187292
\(515\) −4.87689 −0.214902
\(516\) 10.2462 0.451064
\(517\) −32.0000 −1.40736
\(518\) −16.0000 −0.703000
\(519\) −14.8769 −0.653023
\(520\) 0 0
\(521\) −16.2462 −0.711759 −0.355880 0.934532i \(-0.615819\pi\)
−0.355880 + 0.934532i \(0.615819\pi\)
\(522\) 2.00000 0.0875376
\(523\) 22.7386 0.994291 0.497146 0.867667i \(-0.334382\pi\)
0.497146 + 0.867667i \(0.334382\pi\)
\(524\) 4.00000 0.174741
\(525\) −5.12311 −0.223591
\(526\) −2.63068 −0.114703
\(527\) −10.2462 −0.446332
\(528\) −3.12311 −0.135916
\(529\) −13.2462 −0.575922
\(530\) −11.3693 −0.493852
\(531\) 7.12311 0.309116
\(532\) −30.7386 −1.33269
\(533\) 0 0
\(534\) −5.12311 −0.221698
\(535\) 8.00000 0.345870
\(536\) −13.1231 −0.566832
\(537\) 16.4924 0.711701
\(538\) 0.246211 0.0106149
\(539\) −60.1080 −2.58903
\(540\) −1.00000 −0.0430331
\(541\) 19.1231 0.822167 0.411083 0.911598i \(-0.365151\pi\)
0.411083 + 0.911598i \(0.365151\pi\)
\(542\) 14.8769 0.639017
\(543\) 3.75379 0.161090
\(544\) −2.00000 −0.0857493
\(545\) 11.1231 0.476461
\(546\) 0 0
\(547\) −44.9848 −1.92341 −0.961707 0.274081i \(-0.911626\pi\)
−0.961707 + 0.274081i \(0.911626\pi\)
\(548\) 22.4924 0.960829
\(549\) 10.0000 0.426790
\(550\) −3.12311 −0.133170
\(551\) 12.0000 0.511217
\(552\) −3.12311 −0.132928
\(553\) −40.9848 −1.74285
\(554\) −27.8617 −1.18373
\(555\) −3.12311 −0.132568
\(556\) 16.4924 0.699435
\(557\) 28.2462 1.19683 0.598415 0.801186i \(-0.295797\pi\)
0.598415 + 0.801186i \(0.295797\pi\)
\(558\) 5.12311 0.216878
\(559\) 0 0
\(560\) 5.12311 0.216491
\(561\) 6.24621 0.263715
\(562\) 5.12311 0.216105
\(563\) −32.9848 −1.39015 −0.695073 0.718939i \(-0.744628\pi\)
−0.695073 + 0.718939i \(0.744628\pi\)
\(564\) 10.2462 0.431443
\(565\) 4.24621 0.178639
\(566\) 4.00000 0.168133
\(567\) −5.12311 −0.215150
\(568\) 6.24621 0.262085
\(569\) −36.7386 −1.54016 −0.770082 0.637945i \(-0.779785\pi\)
−0.770082 + 0.637945i \(0.779785\pi\)
\(570\) −6.00000 −0.251312
\(571\) −24.4924 −1.02498 −0.512488 0.858694i \(-0.671276\pi\)
−0.512488 + 0.858694i \(0.671276\pi\)
\(572\) 0 0
\(573\) −16.4924 −0.688981
\(574\) 46.7386 1.95083
\(575\) −3.12311 −0.130243
\(576\) 1.00000 0.0416667
\(577\) −2.63068 −0.109517 −0.0547584 0.998500i \(-0.517439\pi\)
−0.0547584 + 0.998500i \(0.517439\pi\)
\(578\) −13.0000 −0.540729
\(579\) −16.8769 −0.701380
\(580\) −2.00000 −0.0830455
\(581\) 52.4924 2.17775
\(582\) 4.87689 0.202154
\(583\) −35.5076 −1.47057
\(584\) −4.87689 −0.201807
\(585\) 0 0
\(586\) −20.7386 −0.856705
\(587\) −16.4924 −0.680715 −0.340358 0.940296i \(-0.610548\pi\)
−0.340358 + 0.940296i \(0.610548\pi\)
\(588\) 19.2462 0.793700
\(589\) 30.7386 1.26656
\(590\) −7.12311 −0.293254
\(591\) 0.246211 0.0101278
\(592\) 3.12311 0.128359
\(593\) −38.4924 −1.58069 −0.790347 0.612659i \(-0.790100\pi\)
−0.790347 + 0.612659i \(0.790100\pi\)
\(594\) −3.12311 −0.128143
\(595\) −10.2462 −0.420054
\(596\) 14.0000 0.573462
\(597\) −1.75379 −0.0717778
\(598\) 0 0
\(599\) −3.50758 −0.143316 −0.0716579 0.997429i \(-0.522829\pi\)
−0.0716579 + 0.997429i \(0.522829\pi\)
\(600\) 1.00000 0.0408248
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) −52.4924 −2.13943
\(603\) −13.1231 −0.534414
\(604\) −11.3693 −0.462611
\(605\) 1.24621 0.0506657
\(606\) −4.24621 −0.172491
\(607\) −9.36932 −0.380289 −0.190144 0.981756i \(-0.560896\pi\)
−0.190144 + 0.981756i \(0.560896\pi\)
\(608\) 6.00000 0.243332
\(609\) −10.2462 −0.415197
\(610\) −10.0000 −0.404888
\(611\) 0 0
\(612\) −2.00000 −0.0808452
\(613\) −14.6307 −0.590928 −0.295464 0.955354i \(-0.595474\pi\)
−0.295464 + 0.955354i \(0.595474\pi\)
\(614\) 22.8769 0.923236
\(615\) 9.12311 0.367879
\(616\) 16.0000 0.644658
\(617\) −8.73863 −0.351804 −0.175902 0.984408i \(-0.556284\pi\)
−0.175902 + 0.984408i \(0.556284\pi\)
\(618\) 4.87689 0.196177
\(619\) −26.9848 −1.08461 −0.542306 0.840181i \(-0.682449\pi\)
−0.542306 + 0.840181i \(0.682449\pi\)
\(620\) −5.12311 −0.205749
\(621\) −3.12311 −0.125326
\(622\) 24.4924 0.982057
\(623\) 26.2462 1.05153
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −0.246211 −0.00984058
\(627\) −18.7386 −0.748349
\(628\) −3.36932 −0.134450
\(629\) −6.24621 −0.249053
\(630\) 5.12311 0.204109
\(631\) 5.61553 0.223551 0.111775 0.993734i \(-0.464346\pi\)
0.111775 + 0.993734i \(0.464346\pi\)
\(632\) 8.00000 0.318223
\(633\) 4.00000 0.158986
\(634\) −6.00000 −0.238290
\(635\) −4.87689 −0.193534
\(636\) 11.3693 0.450823
\(637\) 0 0
\(638\) −6.24621 −0.247290
\(639\) 6.24621 0.247096
\(640\) −1.00000 −0.0395285
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) −8.00000 −0.315735
\(643\) 7.36932 0.290617 0.145309 0.989386i \(-0.453582\pi\)
0.145309 + 0.989386i \(0.453582\pi\)
\(644\) 16.0000 0.630488
\(645\) −10.2462 −0.403444
\(646\) −12.0000 −0.472134
\(647\) 11.6155 0.456654 0.228327 0.973585i \(-0.426675\pi\)
0.228327 + 0.973585i \(0.426675\pi\)
\(648\) 1.00000 0.0392837
\(649\) −22.2462 −0.873240
\(650\) 0 0
\(651\) −26.2462 −1.02867
\(652\) 1.12311 0.0439842
\(653\) 43.8617 1.71644 0.858221 0.513280i \(-0.171570\pi\)
0.858221 + 0.513280i \(0.171570\pi\)
\(654\) −11.1231 −0.434948
\(655\) −4.00000 −0.156293
\(656\) −9.12311 −0.356197
\(657\) −4.87689 −0.190266
\(658\) −52.4924 −2.04637
\(659\) −38.2462 −1.48986 −0.744930 0.667142i \(-0.767517\pi\)
−0.744930 + 0.667142i \(0.767517\pi\)
\(660\) 3.12311 0.121567
\(661\) 0.876894 0.0341072 0.0170536 0.999855i \(-0.494571\pi\)
0.0170536 + 0.999855i \(0.494571\pi\)
\(662\) 24.2462 0.942356
\(663\) 0 0
\(664\) −10.2462 −0.397630
\(665\) 30.7386 1.19199
\(666\) 3.12311 0.121018
\(667\) −6.24621 −0.241854
\(668\) −5.75379 −0.222621
\(669\) 15.3693 0.594212
\(670\) 13.1231 0.506990
\(671\) −31.2311 −1.20566
\(672\) −5.12311 −0.197628
\(673\) 38.9848 1.50276 0.751378 0.659872i \(-0.229390\pi\)
0.751378 + 0.659872i \(0.229390\pi\)
\(674\) −6.00000 −0.231111
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 21.1231 0.811827 0.405913 0.913912i \(-0.366953\pi\)
0.405913 + 0.913912i \(0.366953\pi\)
\(678\) −4.24621 −0.163075
\(679\) −24.9848 −0.958830
\(680\) 2.00000 0.0766965
\(681\) −8.00000 −0.306561
\(682\) −16.0000 −0.612672
\(683\) 48.9848 1.87435 0.937177 0.348856i \(-0.113430\pi\)
0.937177 + 0.348856i \(0.113430\pi\)
\(684\) 6.00000 0.229416
\(685\) −22.4924 −0.859391
\(686\) −62.7386 −2.39537
\(687\) 3.12311 0.119154
\(688\) 10.2462 0.390633
\(689\) 0 0
\(690\) 3.12311 0.118895
\(691\) 20.7386 0.788935 0.394467 0.918910i \(-0.370929\pi\)
0.394467 + 0.918910i \(0.370929\pi\)
\(692\) −14.8769 −0.565535
\(693\) 16.0000 0.607790
\(694\) 2.24621 0.0852650
\(695\) −16.4924 −0.625593
\(696\) 2.00000 0.0758098
\(697\) 18.2462 0.691125
\(698\) −5.36932 −0.203232
\(699\) 24.2462 0.917076
\(700\) −5.12311 −0.193635
\(701\) 14.0000 0.528773 0.264386 0.964417i \(-0.414831\pi\)
0.264386 + 0.964417i \(0.414831\pi\)
\(702\) 0 0
\(703\) 18.7386 0.706741
\(704\) −3.12311 −0.117706
\(705\) −10.2462 −0.385895
\(706\) −4.24621 −0.159808
\(707\) 21.7538 0.818135
\(708\) 7.12311 0.267703
\(709\) 39.6155 1.48779 0.743896 0.668295i \(-0.232976\pi\)
0.743896 + 0.668295i \(0.232976\pi\)
\(710\) −6.24621 −0.234416
\(711\) 8.00000 0.300023
\(712\) −5.12311 −0.191997
\(713\) −16.0000 −0.599205
\(714\) 10.2462 0.383455
\(715\) 0 0
\(716\) 16.4924 0.616351
\(717\) 28.4924 1.06407
\(718\) 34.2462 1.27806
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −24.9848 −0.930484
\(722\) 17.0000 0.632674
\(723\) 2.24621 0.0835375
\(724\) 3.75379 0.139508
\(725\) 2.00000 0.0742781
\(726\) −1.24621 −0.0462512
\(727\) 37.8617 1.40421 0.702107 0.712071i \(-0.252243\pi\)
0.702107 + 0.712071i \(0.252243\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 4.87689 0.180502
\(731\) −20.4924 −0.757940
\(732\) 10.0000 0.369611
\(733\) −10.6307 −0.392653 −0.196327 0.980539i \(-0.562901\pi\)
−0.196327 + 0.980539i \(0.562901\pi\)
\(734\) 33.3693 1.23168
\(735\) −19.2462 −0.709907
\(736\) −3.12311 −0.115119
\(737\) 40.9848 1.50970
\(738\) −9.12311 −0.335826
\(739\) 45.2311 1.66385 0.831926 0.554887i \(-0.187239\pi\)
0.831926 + 0.554887i \(0.187239\pi\)
\(740\) −3.12311 −0.114808
\(741\) 0 0
\(742\) −58.2462 −2.13829
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 5.12311 0.187822
\(745\) −14.0000 −0.512920
\(746\) −1.12311 −0.0411198
\(747\) −10.2462 −0.374889
\(748\) 6.24621 0.228384
\(749\) 40.9848 1.49755
\(750\) −1.00000 −0.0365148
\(751\) 9.75379 0.355921 0.177960 0.984038i \(-0.443050\pi\)
0.177960 + 0.984038i \(0.443050\pi\)
\(752\) 10.2462 0.373641
\(753\) −9.75379 −0.355448
\(754\) 0 0
\(755\) 11.3693 0.413772
\(756\) −5.12311 −0.186326
\(757\) 5.12311 0.186202 0.0931012 0.995657i \(-0.470322\pi\)
0.0931012 + 0.995657i \(0.470322\pi\)
\(758\) 6.00000 0.217930
\(759\) 9.75379 0.354040
\(760\) −6.00000 −0.217643
\(761\) 5.12311 0.185712 0.0928562 0.995680i \(-0.470400\pi\)
0.0928562 + 0.995680i \(0.470400\pi\)
\(762\) 4.87689 0.176671
\(763\) 56.9848 2.06299
\(764\) −16.4924 −0.596675
\(765\) 2.00000 0.0723102
\(766\) −18.2462 −0.659262
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −32.9848 −1.18946 −0.594732 0.803924i \(-0.702742\pi\)
−0.594732 + 0.803924i \(0.702742\pi\)
\(770\) −16.0000 −0.576600
\(771\) −4.24621 −0.152924
\(772\) −16.8769 −0.607413
\(773\) −0.246211 −0.00885560 −0.00442780 0.999990i \(-0.501409\pi\)
−0.00442780 + 0.999990i \(0.501409\pi\)
\(774\) 10.2462 0.368292
\(775\) 5.12311 0.184027
\(776\) 4.87689 0.175070
\(777\) −16.0000 −0.573997
\(778\) −16.2462 −0.582455
\(779\) −54.7386 −1.96122
\(780\) 0 0
\(781\) −19.5076 −0.698036
\(782\) 6.24621 0.223364
\(783\) 2.00000 0.0714742
\(784\) 19.2462 0.687365
\(785\) 3.36932 0.120256
\(786\) 4.00000 0.142675
\(787\) −53.6155 −1.91119 −0.955594 0.294688i \(-0.904784\pi\)
−0.955594 + 0.294688i \(0.904784\pi\)
\(788\) 0.246211 0.00877091
\(789\) −2.63068 −0.0936548
\(790\) −8.00000 −0.284627
\(791\) 21.7538 0.773476
\(792\) −3.12311 −0.110975
\(793\) 0 0
\(794\) 9.36932 0.332505
\(795\) −11.3693 −0.403228
\(796\) −1.75379 −0.0621614
\(797\) 31.8617 1.12860 0.564300 0.825570i \(-0.309146\pi\)
0.564300 + 0.825570i \(0.309146\pi\)
\(798\) −30.7386 −1.08814
\(799\) −20.4924 −0.724970
\(800\) 1.00000 0.0353553
\(801\) −5.12311 −0.181016
\(802\) 23.3693 0.825199
\(803\) 15.2311 0.537492
\(804\) −13.1231 −0.462816
\(805\) −16.0000 −0.563926
\(806\) 0 0
\(807\) 0.246211 0.00866705
\(808\) −4.24621 −0.149381
\(809\) −46.4924 −1.63459 −0.817293 0.576222i \(-0.804526\pi\)
−0.817293 + 0.576222i \(0.804526\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 44.2462 1.55369 0.776847 0.629689i \(-0.216818\pi\)
0.776847 + 0.629689i \(0.216818\pi\)
\(812\) −10.2462 −0.359572
\(813\) 14.8769 0.521755
\(814\) −9.75379 −0.341870
\(815\) −1.12311 −0.0393407
\(816\) −2.00000 −0.0700140
\(817\) 61.4773 2.15082
\(818\) 24.4924 0.856357
\(819\) 0 0
\(820\) 9.12311 0.318593
\(821\) −27.7538 −0.968614 −0.484307 0.874898i \(-0.660928\pi\)
−0.484307 + 0.874898i \(0.660928\pi\)
\(822\) 22.4924 0.784513
\(823\) 51.1231 1.78204 0.891020 0.453965i \(-0.149991\pi\)
0.891020 + 0.453965i \(0.149991\pi\)
\(824\) 4.87689 0.169895
\(825\) −3.12311 −0.108733
\(826\) −36.4924 −1.26973
\(827\) 50.7386 1.76436 0.882178 0.470917i \(-0.156077\pi\)
0.882178 + 0.470917i \(0.156077\pi\)
\(828\) −3.12311 −0.108535
\(829\) 7.75379 0.269300 0.134650 0.990893i \(-0.457009\pi\)
0.134650 + 0.990893i \(0.457009\pi\)
\(830\) 10.2462 0.355651
\(831\) −27.8617 −0.966513
\(832\) 0 0
\(833\) −38.4924 −1.33368
\(834\) 16.4924 0.571086
\(835\) 5.75379 0.199118
\(836\) −18.7386 −0.648089
\(837\) 5.12311 0.177080
\(838\) −28.0000 −0.967244
\(839\) 2.73863 0.0945481 0.0472741 0.998882i \(-0.484947\pi\)
0.0472741 + 0.998882i \(0.484947\pi\)
\(840\) 5.12311 0.176764
\(841\) −25.0000 −0.862069
\(842\) 25.3693 0.874284
\(843\) 5.12311 0.176449
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 10.2462 0.352272
\(847\) 6.38447 0.219373
\(848\) 11.3693 0.390424
\(849\) 4.00000 0.137280
\(850\) −2.00000 −0.0685994
\(851\) −9.75379 −0.334356
\(852\) 6.24621 0.213992
\(853\) −21.8617 −0.748532 −0.374266 0.927321i \(-0.622105\pi\)
−0.374266 + 0.927321i \(0.622105\pi\)
\(854\) −51.2311 −1.75309
\(855\) −6.00000 −0.205196
\(856\) −8.00000 −0.273434
\(857\) −2.49242 −0.0851395 −0.0425698 0.999093i \(-0.513554\pi\)
−0.0425698 + 0.999093i \(0.513554\pi\)
\(858\) 0 0
\(859\) −12.0000 −0.409435 −0.204717 0.978821i \(-0.565628\pi\)
−0.204717 + 0.978821i \(0.565628\pi\)
\(860\) −10.2462 −0.349393
\(861\) 46.7386 1.59285
\(862\) −0.492423 −0.0167720
\(863\) −10.2462 −0.348785 −0.174393 0.984676i \(-0.555796\pi\)
−0.174393 + 0.984676i \(0.555796\pi\)
\(864\) 1.00000 0.0340207
\(865\) 14.8769 0.505830
\(866\) 18.0000 0.611665
\(867\) −13.0000 −0.441503
\(868\) −26.2462 −0.890854
\(869\) −24.9848 −0.847553
\(870\) −2.00000 −0.0678064
\(871\) 0 0
\(872\) −11.1231 −0.376676
\(873\) 4.87689 0.165058
\(874\) −18.7386 −0.633844
\(875\) 5.12311 0.173193
\(876\) −4.87689 −0.164775
\(877\) 27.1231 0.915882 0.457941 0.888983i \(-0.348587\pi\)
0.457941 + 0.888983i \(0.348587\pi\)
\(878\) 3.50758 0.118375
\(879\) −20.7386 −0.699497
\(880\) 3.12311 0.105280
\(881\) −11.7538 −0.395995 −0.197998 0.980203i \(-0.563444\pi\)
−0.197998 + 0.980203i \(0.563444\pi\)
\(882\) 19.2462 0.648054
\(883\) 26.2462 0.883255 0.441628 0.897198i \(-0.354401\pi\)
0.441628 + 0.897198i \(0.354401\pi\)
\(884\) 0 0
\(885\) −7.12311 −0.239441
\(886\) −36.4924 −1.22599
\(887\) −27.1231 −0.910705 −0.455352 0.890311i \(-0.650487\pi\)
−0.455352 + 0.890311i \(0.650487\pi\)
\(888\) 3.12311 0.104805
\(889\) −24.9848 −0.837965
\(890\) 5.12311 0.171727
\(891\) −3.12311 −0.104628
\(892\) 15.3693 0.514603
\(893\) 61.4773 2.05726
\(894\) 14.0000 0.468230
\(895\) −16.4924 −0.551281
\(896\) −5.12311 −0.171151
\(897\) 0 0
\(898\) −37.1231 −1.23881
\(899\) 10.2462 0.341730
\(900\) 1.00000 0.0333333
\(901\) −22.7386 −0.757534
\(902\) 28.4924 0.948694
\(903\) −52.4924 −1.74684
\(904\) −4.24621 −0.141227
\(905\) −3.75379 −0.124780
\(906\) −11.3693 −0.377720
\(907\) −42.2462 −1.40276 −0.701381 0.712786i \(-0.747433\pi\)
−0.701381 + 0.712786i \(0.747433\pi\)
\(908\) −8.00000 −0.265489
\(909\) −4.24621 −0.140838
\(910\) 0 0
\(911\) 4.00000 0.132526 0.0662630 0.997802i \(-0.478892\pi\)
0.0662630 + 0.997802i \(0.478892\pi\)
\(912\) 6.00000 0.198680
\(913\) 32.0000 1.05905
\(914\) −6.63068 −0.219324
\(915\) −10.0000 −0.330590
\(916\) 3.12311 0.103190
\(917\) −20.4924 −0.676719
\(918\) −2.00000 −0.0660098
\(919\) −38.2462 −1.26163 −0.630813 0.775935i \(-0.717279\pi\)
−0.630813 + 0.775935i \(0.717279\pi\)
\(920\) 3.12311 0.102966
\(921\) 22.8769 0.753819
\(922\) 14.4924 0.477283
\(923\) 0 0
\(924\) 16.0000 0.526361
\(925\) 3.12311 0.102687
\(926\) 35.8617 1.17849
\(927\) 4.87689 0.160178
\(928\) 2.00000 0.0656532
\(929\) 46.1080 1.51275 0.756376 0.654137i \(-0.226968\pi\)
0.756376 + 0.654137i \(0.226968\pi\)
\(930\) −5.12311 −0.167993
\(931\) 115.477 3.78461
\(932\) 24.2462 0.794211
\(933\) 24.4924 0.801846
\(934\) 5.75379 0.188270
\(935\) −6.24621 −0.204273
\(936\) 0 0
\(937\) −3.75379 −0.122631 −0.0613155 0.998118i \(-0.519530\pi\)
−0.0613155 + 0.998118i \(0.519530\pi\)
\(938\) 67.2311 2.19517
\(939\) −0.246211 −0.00803480
\(940\) −10.2462 −0.334195
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) −3.36932 −0.109778
\(943\) 28.4924 0.927841
\(944\) 7.12311 0.231837
\(945\) 5.12311 0.166655
\(946\) −32.0000 −1.04041
\(947\) −24.4924 −0.795897 −0.397948 0.917408i \(-0.630278\pi\)
−0.397948 + 0.917408i \(0.630278\pi\)
\(948\) 8.00000 0.259828
\(949\) 0 0
\(950\) 6.00000 0.194666
\(951\) −6.00000 −0.194563
\(952\) 10.2462 0.332082
\(953\) 42.9848 1.39242 0.696208 0.717840i \(-0.254869\pi\)
0.696208 + 0.717840i \(0.254869\pi\)
\(954\) 11.3693 0.368095
\(955\) 16.4924 0.533682
\(956\) 28.4924 0.921511
\(957\) −6.24621 −0.201911
\(958\) 20.4924 0.662080
\(959\) −115.231 −3.72100
\(960\) −1.00000 −0.0322749
\(961\) −4.75379 −0.153348
\(962\) 0 0
\(963\) −8.00000 −0.257796
\(964\) 2.24621 0.0723456
\(965\) 16.8769 0.543286
\(966\) 16.0000 0.514792
\(967\) 6.38447 0.205311 0.102655 0.994717i \(-0.467266\pi\)
0.102655 + 0.994717i \(0.467266\pi\)
\(968\) −1.24621 −0.0400547
\(969\) −12.0000 −0.385496
\(970\) −4.87689 −0.156588
\(971\) 54.2462 1.74084 0.870422 0.492307i \(-0.163846\pi\)
0.870422 + 0.492307i \(0.163846\pi\)
\(972\) 1.00000 0.0320750
\(973\) −84.4924 −2.70870
\(974\) 7.36932 0.236128
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −44.7386 −1.43132 −0.715658 0.698451i \(-0.753873\pi\)
−0.715658 + 0.698451i \(0.753873\pi\)
\(978\) 1.12311 0.0359130
\(979\) 16.0000 0.511362
\(980\) −19.2462 −0.614798
\(981\) −11.1231 −0.355133
\(982\) 10.7386 0.342684
\(983\) −27.5076 −0.877355 −0.438678 0.898644i \(-0.644553\pi\)
−0.438678 + 0.898644i \(0.644553\pi\)
\(984\) −9.12311 −0.290834
\(985\) −0.246211 −0.00784494
\(986\) −4.00000 −0.127386
\(987\) −52.4924 −1.67085
\(988\) 0 0
\(989\) −32.0000 −1.01754
\(990\) 3.12311 0.0992588
\(991\) 18.7386 0.595252 0.297626 0.954682i \(-0.403805\pi\)
0.297626 + 0.954682i \(0.403805\pi\)
\(992\) 5.12311 0.162659
\(993\) 24.2462 0.769430
\(994\) −32.0000 −1.01498
\(995\) 1.75379 0.0555988
\(996\) −10.2462 −0.324664
\(997\) −52.3542 −1.65807 −0.829036 0.559195i \(-0.811110\pi\)
−0.829036 + 0.559195i \(0.811110\pi\)
\(998\) −1.50758 −0.0477215
\(999\) 3.12311 0.0988107
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 5070.2.a.bh.1.1 2
13.5 odd 4 390.2.b.d.181.1 4
13.8 odd 4 390.2.b.d.181.4 yes 4
13.12 even 2 5070.2.a.bd.1.2 2
39.5 even 4 1170.2.b.f.181.3 4
39.8 even 4 1170.2.b.f.181.2 4
52.31 even 4 3120.2.g.o.961.4 4
52.47 even 4 3120.2.g.o.961.1 4
65.8 even 4 1950.2.f.o.649.4 4
65.18 even 4 1950.2.f.l.649.3 4
65.34 odd 4 1950.2.b.h.1351.1 4
65.44 odd 4 1950.2.b.h.1351.4 4
65.47 even 4 1950.2.f.l.649.1 4
65.57 even 4 1950.2.f.o.649.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.d.181.1 4 13.5 odd 4
390.2.b.d.181.4 yes 4 13.8 odd 4
1170.2.b.f.181.2 4 39.8 even 4
1170.2.b.f.181.3 4 39.5 even 4
1950.2.b.h.1351.1 4 65.34 odd 4
1950.2.b.h.1351.4 4 65.44 odd 4
1950.2.f.l.649.1 4 65.47 even 4
1950.2.f.l.649.3 4 65.18 even 4
1950.2.f.o.649.2 4 65.57 even 4
1950.2.f.o.649.4 4 65.8 even 4
3120.2.g.o.961.1 4 52.47 even 4
3120.2.g.o.961.4 4 52.31 even 4
5070.2.a.bd.1.2 2 13.12 even 2
5070.2.a.bh.1.1 2 1.1 even 1 trivial