Properties

Label 390.2.a.h.1.1
Level $390$
Weight $2$
Character 390.1
Self dual yes
Analytic conductor $3.114$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 390.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.82843 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.82843 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +5.65685 q^{11} +1.00000 q^{12} -1.00000 q^{13} -2.82843 q^{14} +1.00000 q^{15} +1.00000 q^{16} +0.828427 q^{17} +1.00000 q^{18} +2.82843 q^{19} +1.00000 q^{20} -2.82843 q^{21} +5.65685 q^{22} -8.48528 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} -2.82843 q^{28} -8.82843 q^{29} +1.00000 q^{30} +4.00000 q^{31} +1.00000 q^{32} +5.65685 q^{33} +0.828427 q^{34} -2.82843 q^{35} +1.00000 q^{36} -11.6569 q^{37} +2.82843 q^{38} -1.00000 q^{39} +1.00000 q^{40} -7.65685 q^{41} -2.82843 q^{42} +9.65685 q^{43} +5.65685 q^{44} +1.00000 q^{45} -8.48528 q^{46} -8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +0.828427 q^{51} -1.00000 q^{52} +13.3137 q^{53} +1.00000 q^{54} +5.65685 q^{55} -2.82843 q^{56} +2.82843 q^{57} -8.82843 q^{58} -2.34315 q^{59} +1.00000 q^{60} +6.00000 q^{61} +4.00000 q^{62} -2.82843 q^{63} +1.00000 q^{64} -1.00000 q^{65} +5.65685 q^{66} +5.65685 q^{67} +0.828427 q^{68} -8.48528 q^{69} -2.82843 q^{70} -5.65685 q^{71} +1.00000 q^{72} -14.4853 q^{73} -11.6569 q^{74} +1.00000 q^{75} +2.82843 q^{76} -16.0000 q^{77} -1.00000 q^{78} +2.34315 q^{79} +1.00000 q^{80} +1.00000 q^{81} -7.65685 q^{82} +6.34315 q^{83} -2.82843 q^{84} +0.828427 q^{85} +9.65685 q^{86} -8.82843 q^{87} +5.65685 q^{88} -15.6569 q^{89} +1.00000 q^{90} +2.82843 q^{91} -8.48528 q^{92} +4.00000 q^{93} -8.00000 q^{94} +2.82843 q^{95} +1.00000 q^{96} -3.17157 q^{97} +1.00000 q^{98} +5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{8} + 2q^{9} + 2q^{10} + 2q^{12} - 2q^{13} + 2q^{15} + 2q^{16} - 4q^{17} + 2q^{18} + 2q^{20} + 2q^{24} + 2q^{25} - 2q^{26} + 2q^{27} - 12q^{29} + 2q^{30} + 8q^{31} + 2q^{32} - 4q^{34} + 2q^{36} - 12q^{37} - 2q^{39} + 2q^{40} - 4q^{41} + 8q^{43} + 2q^{45} - 16q^{47} + 2q^{48} + 2q^{49} + 2q^{50} - 4q^{51} - 2q^{52} + 4q^{53} + 2q^{54} - 12q^{58} - 16q^{59} + 2q^{60} + 12q^{61} + 8q^{62} + 2q^{64} - 2q^{65} - 4q^{68} + 2q^{72} - 12q^{73} - 12q^{74} + 2q^{75} - 32q^{77} - 2q^{78} + 16q^{79} + 2q^{80} + 2q^{81} - 4q^{82} + 24q^{83} - 4q^{85} + 8q^{86} - 12q^{87} - 20q^{89} + 2q^{90} + 8q^{93} - 16q^{94} + 2q^{96} - 12q^{97} + 2q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 5.65685 1.70561 0.852803 0.522233i \(-0.174901\pi\)
0.852803 + 0.522233i \(0.174901\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) −2.82843 −0.755929
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 0.828427 0.200923 0.100462 0.994941i \(-0.467968\pi\)
0.100462 + 0.994941i \(0.467968\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.82843 −0.617213
\(22\) 5.65685 1.20605
\(23\) −8.48528 −1.76930 −0.884652 0.466252i \(-0.845604\pi\)
−0.884652 + 0.466252i \(0.845604\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) −2.82843 −0.534522
\(29\) −8.82843 −1.63940 −0.819699 0.572795i \(-0.805859\pi\)
−0.819699 + 0.572795i \(0.805859\pi\)
\(30\) 1.00000 0.182574
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.65685 0.984732
\(34\) 0.828427 0.142074
\(35\) −2.82843 −0.478091
\(36\) 1.00000 0.166667
\(37\) −11.6569 −1.91638 −0.958188 0.286141i \(-0.907627\pi\)
−0.958188 + 0.286141i \(0.907627\pi\)
\(38\) 2.82843 0.458831
\(39\) −1.00000 −0.160128
\(40\) 1.00000 0.158114
\(41\) −7.65685 −1.19580 −0.597900 0.801571i \(-0.703998\pi\)
−0.597900 + 0.801571i \(0.703998\pi\)
\(42\) −2.82843 −0.436436
\(43\) 9.65685 1.47266 0.736328 0.676625i \(-0.236558\pi\)
0.736328 + 0.676625i \(0.236558\pi\)
\(44\) 5.65685 0.852803
\(45\) 1.00000 0.149071
\(46\) −8.48528 −1.25109
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0.828427 0.116003
\(52\) −1.00000 −0.138675
\(53\) 13.3137 1.82878 0.914389 0.404836i \(-0.132671\pi\)
0.914389 + 0.404836i \(0.132671\pi\)
\(54\) 1.00000 0.136083
\(55\) 5.65685 0.762770
\(56\) −2.82843 −0.377964
\(57\) 2.82843 0.374634
\(58\) −8.82843 −1.15923
\(59\) −2.34315 −0.305052 −0.152526 0.988299i \(-0.548741\pi\)
−0.152526 + 0.988299i \(0.548741\pi\)
\(60\) 1.00000 0.129099
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 4.00000 0.508001
\(63\) −2.82843 −0.356348
\(64\) 1.00000 0.125000
\(65\) −1.00000 −0.124035
\(66\) 5.65685 0.696311
\(67\) 5.65685 0.691095 0.345547 0.938401i \(-0.387693\pi\)
0.345547 + 0.938401i \(0.387693\pi\)
\(68\) 0.828427 0.100462
\(69\) −8.48528 −1.02151
\(70\) −2.82843 −0.338062
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.4853 −1.69537 −0.847687 0.530497i \(-0.822005\pi\)
−0.847687 + 0.530497i \(0.822005\pi\)
\(74\) −11.6569 −1.35508
\(75\) 1.00000 0.115470
\(76\) 2.82843 0.324443
\(77\) −16.0000 −1.82337
\(78\) −1.00000 −0.113228
\(79\) 2.34315 0.263624 0.131812 0.991275i \(-0.457920\pi\)
0.131812 + 0.991275i \(0.457920\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −7.65685 −0.845558
\(83\) 6.34315 0.696251 0.348125 0.937448i \(-0.386818\pi\)
0.348125 + 0.937448i \(0.386818\pi\)
\(84\) −2.82843 −0.308607
\(85\) 0.828427 0.0898555
\(86\) 9.65685 1.04133
\(87\) −8.82843 −0.946507
\(88\) 5.65685 0.603023
\(89\) −15.6569 −1.65962 −0.829812 0.558044i \(-0.811552\pi\)
−0.829812 + 0.558044i \(0.811552\pi\)
\(90\) 1.00000 0.105409
\(91\) 2.82843 0.296500
\(92\) −8.48528 −0.884652
\(93\) 4.00000 0.414781
\(94\) −8.00000 −0.825137
\(95\) 2.82843 0.290191
\(96\) 1.00000 0.102062
\(97\) −3.17157 −0.322024 −0.161012 0.986952i \(-0.551476\pi\)
−0.161012 + 0.986952i \(0.551476\pi\)
\(98\) 1.00000 0.101015
\(99\) 5.65685 0.568535
\(100\) 1.00000 0.100000
\(101\) 16.1421 1.60620 0.803101 0.595843i \(-0.203182\pi\)
0.803101 + 0.595843i \(0.203182\pi\)
\(102\) 0.828427 0.0820265
\(103\) −1.65685 −0.163255 −0.0816274 0.996663i \(-0.526012\pi\)
−0.0816274 + 0.996663i \(0.526012\pi\)
\(104\) −1.00000 −0.0980581
\(105\) −2.82843 −0.276026
\(106\) 13.3137 1.29314
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 1.00000 0.0962250
\(109\) 8.82843 0.845610 0.422805 0.906221i \(-0.361046\pi\)
0.422805 + 0.906221i \(0.361046\pi\)
\(110\) 5.65685 0.539360
\(111\) −11.6569 −1.10642
\(112\) −2.82843 −0.267261
\(113\) 6.48528 0.610084 0.305042 0.952339i \(-0.401330\pi\)
0.305042 + 0.952339i \(0.401330\pi\)
\(114\) 2.82843 0.264906
\(115\) −8.48528 −0.791257
\(116\) −8.82843 −0.819699
\(117\) −1.00000 −0.0924500
\(118\) −2.34315 −0.215704
\(119\) −2.34315 −0.214796
\(120\) 1.00000 0.0912871
\(121\) 21.0000 1.90909
\(122\) 6.00000 0.543214
\(123\) −7.65685 −0.690395
\(124\) 4.00000 0.359211
\(125\) 1.00000 0.0894427
\(126\) −2.82843 −0.251976
\(127\) 9.65685 0.856907 0.428454 0.903564i \(-0.359059\pi\)
0.428454 + 0.903564i \(0.359059\pi\)
\(128\) 1.00000 0.0883883
\(129\) 9.65685 0.850239
\(130\) −1.00000 −0.0877058
\(131\) −6.14214 −0.536641 −0.268320 0.963330i \(-0.586469\pi\)
−0.268320 + 0.963330i \(0.586469\pi\)
\(132\) 5.65685 0.492366
\(133\) −8.00000 −0.693688
\(134\) 5.65685 0.488678
\(135\) 1.00000 0.0860663
\(136\) 0.828427 0.0710370
\(137\) −17.3137 −1.47921 −0.739605 0.673041i \(-0.764988\pi\)
−0.739605 + 0.673041i \(0.764988\pi\)
\(138\) −8.48528 −0.722315
\(139\) 6.34315 0.538019 0.269009 0.963138i \(-0.413304\pi\)
0.269009 + 0.963138i \(0.413304\pi\)
\(140\) −2.82843 −0.239046
\(141\) −8.00000 −0.673722
\(142\) −5.65685 −0.474713
\(143\) −5.65685 −0.473050
\(144\) 1.00000 0.0833333
\(145\) −8.82843 −0.733161
\(146\) −14.4853 −1.19881
\(147\) 1.00000 0.0824786
\(148\) −11.6569 −0.958188
\(149\) 3.65685 0.299581 0.149791 0.988718i \(-0.452140\pi\)
0.149791 + 0.988718i \(0.452140\pi\)
\(150\) 1.00000 0.0816497
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 2.82843 0.229416
\(153\) 0.828427 0.0669744
\(154\) −16.0000 −1.28932
\(155\) 4.00000 0.321288
\(156\) −1.00000 −0.0800641
\(157\) 5.31371 0.424080 0.212040 0.977261i \(-0.431989\pi\)
0.212040 + 0.977261i \(0.431989\pi\)
\(158\) 2.34315 0.186411
\(159\) 13.3137 1.05585
\(160\) 1.00000 0.0790569
\(161\) 24.0000 1.89146
\(162\) 1.00000 0.0785674
\(163\) 11.3137 0.886158 0.443079 0.896483i \(-0.353886\pi\)
0.443079 + 0.896483i \(0.353886\pi\)
\(164\) −7.65685 −0.597900
\(165\) 5.65685 0.440386
\(166\) 6.34315 0.492324
\(167\) 8.97056 0.694163 0.347081 0.937835i \(-0.387173\pi\)
0.347081 + 0.937835i \(0.387173\pi\)
\(168\) −2.82843 −0.218218
\(169\) 1.00000 0.0769231
\(170\) 0.828427 0.0635375
\(171\) 2.82843 0.216295
\(172\) 9.65685 0.736328
\(173\) −9.31371 −0.708108 −0.354054 0.935225i \(-0.615197\pi\)
−0.354054 + 0.935225i \(0.615197\pi\)
\(174\) −8.82843 −0.669281
\(175\) −2.82843 −0.213809
\(176\) 5.65685 0.426401
\(177\) −2.34315 −0.176122
\(178\) −15.6569 −1.17353
\(179\) 7.51472 0.561676 0.280838 0.959755i \(-0.409388\pi\)
0.280838 + 0.959755i \(0.409388\pi\)
\(180\) 1.00000 0.0745356
\(181\) −7.65685 −0.569129 −0.284565 0.958657i \(-0.591849\pi\)
−0.284565 + 0.958657i \(0.591849\pi\)
\(182\) 2.82843 0.209657
\(183\) 6.00000 0.443533
\(184\) −8.48528 −0.625543
\(185\) −11.6569 −0.857029
\(186\) 4.00000 0.293294
\(187\) 4.68629 0.342696
\(188\) −8.00000 −0.583460
\(189\) −2.82843 −0.205738
\(190\) 2.82843 0.205196
\(191\) 11.3137 0.818631 0.409316 0.912393i \(-0.365768\pi\)
0.409316 + 0.912393i \(0.365768\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.48528 0.178894 0.0894472 0.995992i \(-0.471490\pi\)
0.0894472 + 0.995992i \(0.471490\pi\)
\(194\) −3.17157 −0.227706
\(195\) −1.00000 −0.0716115
\(196\) 1.00000 0.0714286
\(197\) −13.3137 −0.948562 −0.474281 0.880373i \(-0.657292\pi\)
−0.474281 + 0.880373i \(0.657292\pi\)
\(198\) 5.65685 0.402015
\(199\) 10.3431 0.733206 0.366603 0.930377i \(-0.380521\pi\)
0.366603 + 0.930377i \(0.380521\pi\)
\(200\) 1.00000 0.0707107
\(201\) 5.65685 0.399004
\(202\) 16.1421 1.13576
\(203\) 24.9706 1.75259
\(204\) 0.828427 0.0580015
\(205\) −7.65685 −0.534778
\(206\) −1.65685 −0.115439
\(207\) −8.48528 −0.589768
\(208\) −1.00000 −0.0693375
\(209\) 16.0000 1.10674
\(210\) −2.82843 −0.195180
\(211\) 0.686292 0.0472463 0.0236231 0.999721i \(-0.492480\pi\)
0.0236231 + 0.999721i \(0.492480\pi\)
\(212\) 13.3137 0.914389
\(213\) −5.65685 −0.387601
\(214\) −4.00000 −0.273434
\(215\) 9.65685 0.658592
\(216\) 1.00000 0.0680414
\(217\) −11.3137 −0.768025
\(218\) 8.82843 0.597937
\(219\) −14.4853 −0.978825
\(220\) 5.65685 0.381385
\(221\) −0.828427 −0.0557260
\(222\) −11.6569 −0.782357
\(223\) 10.8284 0.725125 0.362563 0.931959i \(-0.381902\pi\)
0.362563 + 0.931959i \(0.381902\pi\)
\(224\) −2.82843 −0.188982
\(225\) 1.00000 0.0666667
\(226\) 6.48528 0.431394
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 2.82843 0.187317
\(229\) 4.14214 0.273720 0.136860 0.990590i \(-0.456299\pi\)
0.136860 + 0.990590i \(0.456299\pi\)
\(230\) −8.48528 −0.559503
\(231\) −16.0000 −1.05272
\(232\) −8.82843 −0.579615
\(233\) 5.51472 0.361281 0.180641 0.983549i \(-0.442183\pi\)
0.180641 + 0.983549i \(0.442183\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −8.00000 −0.521862
\(236\) −2.34315 −0.152526
\(237\) 2.34315 0.152204
\(238\) −2.34315 −0.151884
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 1.00000 0.0645497
\(241\) 5.31371 0.342286 0.171143 0.985246i \(-0.445254\pi\)
0.171143 + 0.985246i \(0.445254\pi\)
\(242\) 21.0000 1.34993
\(243\) 1.00000 0.0641500
\(244\) 6.00000 0.384111
\(245\) 1.00000 0.0638877
\(246\) −7.65685 −0.488183
\(247\) −2.82843 −0.179969
\(248\) 4.00000 0.254000
\(249\) 6.34315 0.401981
\(250\) 1.00000 0.0632456
\(251\) 10.8284 0.683484 0.341742 0.939794i \(-0.388983\pi\)
0.341742 + 0.939794i \(0.388983\pi\)
\(252\) −2.82843 −0.178174
\(253\) −48.0000 −3.01773
\(254\) 9.65685 0.605925
\(255\) 0.828427 0.0518781
\(256\) 1.00000 0.0625000
\(257\) −4.82843 −0.301189 −0.150595 0.988596i \(-0.548119\pi\)
−0.150595 + 0.988596i \(0.548119\pi\)
\(258\) 9.65685 0.601209
\(259\) 32.9706 2.04869
\(260\) −1.00000 −0.0620174
\(261\) −8.82843 −0.546466
\(262\) −6.14214 −0.379462
\(263\) 16.4853 1.01653 0.508263 0.861202i \(-0.330288\pi\)
0.508263 + 0.861202i \(0.330288\pi\)
\(264\) 5.65685 0.348155
\(265\) 13.3137 0.817855
\(266\) −8.00000 −0.490511
\(267\) −15.6569 −0.958184
\(268\) 5.65685 0.345547
\(269\) −14.4853 −0.883183 −0.441592 0.897216i \(-0.645586\pi\)
−0.441592 + 0.897216i \(0.645586\pi\)
\(270\) 1.00000 0.0608581
\(271\) 7.31371 0.444276 0.222138 0.975015i \(-0.428696\pi\)
0.222138 + 0.975015i \(0.428696\pi\)
\(272\) 0.828427 0.0502308
\(273\) 2.82843 0.171184
\(274\) −17.3137 −1.04596
\(275\) 5.65685 0.341121
\(276\) −8.48528 −0.510754
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) 6.34315 0.380437
\(279\) 4.00000 0.239474
\(280\) −2.82843 −0.169031
\(281\) 8.34315 0.497710 0.248855 0.968541i \(-0.419946\pi\)
0.248855 + 0.968541i \(0.419946\pi\)
\(282\) −8.00000 −0.476393
\(283\) −17.6569 −1.04959 −0.524796 0.851228i \(-0.675858\pi\)
−0.524796 + 0.851228i \(0.675858\pi\)
\(284\) −5.65685 −0.335673
\(285\) 2.82843 0.167542
\(286\) −5.65685 −0.334497
\(287\) 21.6569 1.27836
\(288\) 1.00000 0.0589256
\(289\) −16.3137 −0.959630
\(290\) −8.82843 −0.518423
\(291\) −3.17157 −0.185921
\(292\) −14.4853 −0.847687
\(293\) −16.6274 −0.971384 −0.485692 0.874130i \(-0.661432\pi\)
−0.485692 + 0.874130i \(0.661432\pi\)
\(294\) 1.00000 0.0583212
\(295\) −2.34315 −0.136423
\(296\) −11.6569 −0.677541
\(297\) 5.65685 0.328244
\(298\) 3.65685 0.211836
\(299\) 8.48528 0.490716
\(300\) 1.00000 0.0577350
\(301\) −27.3137 −1.57434
\(302\) 12.0000 0.690522
\(303\) 16.1421 0.927341
\(304\) 2.82843 0.162221
\(305\) 6.00000 0.343559
\(306\) 0.828427 0.0473580
\(307\) −21.6569 −1.23602 −0.618011 0.786169i \(-0.712061\pi\)
−0.618011 + 0.786169i \(0.712061\pi\)
\(308\) −16.0000 −0.911685
\(309\) −1.65685 −0.0942551
\(310\) 4.00000 0.227185
\(311\) −24.0000 −1.36092 −0.680458 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −30.9706 −1.75056 −0.875280 0.483617i \(-0.839323\pi\)
−0.875280 + 0.483617i \(0.839323\pi\)
\(314\) 5.31371 0.299870
\(315\) −2.82843 −0.159364
\(316\) 2.34315 0.131812
\(317\) 25.3137 1.42176 0.710880 0.703314i \(-0.248297\pi\)
0.710880 + 0.703314i \(0.248297\pi\)
\(318\) 13.3137 0.746596
\(319\) −49.9411 −2.79617
\(320\) 1.00000 0.0559017
\(321\) −4.00000 −0.223258
\(322\) 24.0000 1.33747
\(323\) 2.34315 0.130376
\(324\) 1.00000 0.0555556
\(325\) −1.00000 −0.0554700
\(326\) 11.3137 0.626608
\(327\) 8.82843 0.488213
\(328\) −7.65685 −0.422779
\(329\) 22.6274 1.24749
\(330\) 5.65685 0.311400
\(331\) −8.48528 −0.466393 −0.233197 0.972430i \(-0.574919\pi\)
−0.233197 + 0.972430i \(0.574919\pi\)
\(332\) 6.34315 0.348125
\(333\) −11.6569 −0.638792
\(334\) 8.97056 0.490847
\(335\) 5.65685 0.309067
\(336\) −2.82843 −0.154303
\(337\) 10.9706 0.597605 0.298802 0.954315i \(-0.403413\pi\)
0.298802 + 0.954315i \(0.403413\pi\)
\(338\) 1.00000 0.0543928
\(339\) 6.48528 0.352232
\(340\) 0.828427 0.0449278
\(341\) 22.6274 1.22534
\(342\) 2.82843 0.152944
\(343\) 16.9706 0.916324
\(344\) 9.65685 0.520663
\(345\) −8.48528 −0.456832
\(346\) −9.31371 −0.500708
\(347\) −9.65685 −0.518407 −0.259204 0.965823i \(-0.583460\pi\)
−0.259204 + 0.965823i \(0.583460\pi\)
\(348\) −8.82843 −0.473253
\(349\) 12.1421 0.649954 0.324977 0.945722i \(-0.394644\pi\)
0.324977 + 0.945722i \(0.394644\pi\)
\(350\) −2.82843 −0.151186
\(351\) −1.00000 −0.0533761
\(352\) 5.65685 0.301511
\(353\) 5.31371 0.282820 0.141410 0.989951i \(-0.454836\pi\)
0.141410 + 0.989951i \(0.454836\pi\)
\(354\) −2.34315 −0.124537
\(355\) −5.65685 −0.300235
\(356\) −15.6569 −0.829812
\(357\) −2.34315 −0.124012
\(358\) 7.51472 0.397165
\(359\) −28.2843 −1.49279 −0.746393 0.665505i \(-0.768216\pi\)
−0.746393 + 0.665505i \(0.768216\pi\)
\(360\) 1.00000 0.0527046
\(361\) −11.0000 −0.578947
\(362\) −7.65685 −0.402435
\(363\) 21.0000 1.10221
\(364\) 2.82843 0.148250
\(365\) −14.4853 −0.758194
\(366\) 6.00000 0.313625
\(367\) 25.6569 1.33928 0.669638 0.742687i \(-0.266449\pi\)
0.669638 + 0.742687i \(0.266449\pi\)
\(368\) −8.48528 −0.442326
\(369\) −7.65685 −0.398600
\(370\) −11.6569 −0.606011
\(371\) −37.6569 −1.95505
\(372\) 4.00000 0.207390
\(373\) −2.68629 −0.139091 −0.0695455 0.997579i \(-0.522155\pi\)
−0.0695455 + 0.997579i \(0.522155\pi\)
\(374\) 4.68629 0.242322
\(375\) 1.00000 0.0516398
\(376\) −8.00000 −0.412568
\(377\) 8.82843 0.454687
\(378\) −2.82843 −0.145479
\(379\) −7.51472 −0.386005 −0.193003 0.981198i \(-0.561823\pi\)
−0.193003 + 0.981198i \(0.561823\pi\)
\(380\) 2.82843 0.145095
\(381\) 9.65685 0.494736
\(382\) 11.3137 0.578860
\(383\) −29.6569 −1.51539 −0.757697 0.652606i \(-0.773676\pi\)
−0.757697 + 0.652606i \(0.773676\pi\)
\(384\) 1.00000 0.0510310
\(385\) −16.0000 −0.815436
\(386\) 2.48528 0.126497
\(387\) 9.65685 0.490885
\(388\) −3.17157 −0.161012
\(389\) −6.48528 −0.328817 −0.164408 0.986392i \(-0.552571\pi\)
−0.164408 + 0.986392i \(0.552571\pi\)
\(390\) −1.00000 −0.0506370
\(391\) −7.02944 −0.355494
\(392\) 1.00000 0.0505076
\(393\) −6.14214 −0.309830
\(394\) −13.3137 −0.670735
\(395\) 2.34315 0.117896
\(396\) 5.65685 0.284268
\(397\) 30.2843 1.51992 0.759962 0.649968i \(-0.225218\pi\)
0.759962 + 0.649968i \(0.225218\pi\)
\(398\) 10.3431 0.518455
\(399\) −8.00000 −0.400501
\(400\) 1.00000 0.0500000
\(401\) −26.9706 −1.34685 −0.673423 0.739258i \(-0.735177\pi\)
−0.673423 + 0.739258i \(0.735177\pi\)
\(402\) 5.65685 0.282138
\(403\) −4.00000 −0.199254
\(404\) 16.1421 0.803101
\(405\) 1.00000 0.0496904
\(406\) 24.9706 1.23927
\(407\) −65.9411 −3.26858
\(408\) 0.828427 0.0410133
\(409\) −3.65685 −0.180820 −0.0904099 0.995905i \(-0.528818\pi\)
−0.0904099 + 0.995905i \(0.528818\pi\)
\(410\) −7.65685 −0.378145
\(411\) −17.3137 −0.854022
\(412\) −1.65685 −0.0816274
\(413\) 6.62742 0.326114
\(414\) −8.48528 −0.417029
\(415\) 6.34315 0.311373
\(416\) −1.00000 −0.0490290
\(417\) 6.34315 0.310625
\(418\) 16.0000 0.782586
\(419\) −10.8284 −0.529003 −0.264502 0.964385i \(-0.585207\pi\)
−0.264502 + 0.964385i \(0.585207\pi\)
\(420\) −2.82843 −0.138013
\(421\) −24.1421 −1.17662 −0.588308 0.808637i \(-0.700206\pi\)
−0.588308 + 0.808637i \(0.700206\pi\)
\(422\) 0.686292 0.0334081
\(423\) −8.00000 −0.388973
\(424\) 13.3137 0.646571
\(425\) 0.828427 0.0401846
\(426\) −5.65685 −0.274075
\(427\) −16.9706 −0.821263
\(428\) −4.00000 −0.193347
\(429\) −5.65685 −0.273115
\(430\) 9.65685 0.465695
\(431\) −16.0000 −0.770693 −0.385346 0.922772i \(-0.625918\pi\)
−0.385346 + 0.922772i \(0.625918\pi\)
\(432\) 1.00000 0.0481125
\(433\) −22.9706 −1.10389 −0.551947 0.833879i \(-0.686115\pi\)
−0.551947 + 0.833879i \(0.686115\pi\)
\(434\) −11.3137 −0.543075
\(435\) −8.82843 −0.423291
\(436\) 8.82843 0.422805
\(437\) −24.0000 −1.14808
\(438\) −14.4853 −0.692134
\(439\) 22.6274 1.07995 0.539974 0.841682i \(-0.318434\pi\)
0.539974 + 0.841682i \(0.318434\pi\)
\(440\) 5.65685 0.269680
\(441\) 1.00000 0.0476190
\(442\) −0.828427 −0.0394043
\(443\) 30.3431 1.44165 0.720823 0.693119i \(-0.243764\pi\)
0.720823 + 0.693119i \(0.243764\pi\)
\(444\) −11.6569 −0.553210
\(445\) −15.6569 −0.742206
\(446\) 10.8284 0.512741
\(447\) 3.65685 0.172963
\(448\) −2.82843 −0.133631
\(449\) 26.2843 1.24043 0.620216 0.784431i \(-0.287045\pi\)
0.620216 + 0.784431i \(0.287045\pi\)
\(450\) 1.00000 0.0471405
\(451\) −43.3137 −2.03956
\(452\) 6.48528 0.305042
\(453\) 12.0000 0.563809
\(454\) 4.00000 0.187729
\(455\) 2.82843 0.132599
\(456\) 2.82843 0.132453
\(457\) 20.8284 0.974313 0.487156 0.873315i \(-0.338034\pi\)
0.487156 + 0.873315i \(0.338034\pi\)
\(458\) 4.14214 0.193549
\(459\) 0.828427 0.0386677
\(460\) −8.48528 −0.395628
\(461\) 14.0000 0.652045 0.326023 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) −16.0000 −0.744387
\(463\) −3.79899 −0.176554 −0.0882770 0.996096i \(-0.528136\pi\)
−0.0882770 + 0.996096i \(0.528136\pi\)
\(464\) −8.82843 −0.409849
\(465\) 4.00000 0.185496
\(466\) 5.51472 0.255464
\(467\) 7.31371 0.338438 0.169219 0.985578i \(-0.445875\pi\)
0.169219 + 0.985578i \(0.445875\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −16.0000 −0.738811
\(470\) −8.00000 −0.369012
\(471\) 5.31371 0.244843
\(472\) −2.34315 −0.107852
\(473\) 54.6274 2.51177
\(474\) 2.34315 0.107624
\(475\) 2.82843 0.129777
\(476\) −2.34315 −0.107398
\(477\) 13.3137 0.609593
\(478\) 16.0000 0.731823
\(479\) −11.3137 −0.516937 −0.258468 0.966020i \(-0.583218\pi\)
−0.258468 + 0.966020i \(0.583218\pi\)
\(480\) 1.00000 0.0456435
\(481\) 11.6569 0.531507
\(482\) 5.31371 0.242033
\(483\) 24.0000 1.09204
\(484\) 21.0000 0.954545
\(485\) −3.17157 −0.144014
\(486\) 1.00000 0.0453609
\(487\) 16.4853 0.747019 0.373510 0.927626i \(-0.378154\pi\)
0.373510 + 0.927626i \(0.378154\pi\)
\(488\) 6.00000 0.271607
\(489\) 11.3137 0.511624
\(490\) 1.00000 0.0451754
\(491\) −38.1421 −1.72133 −0.860665 0.509171i \(-0.829952\pi\)
−0.860665 + 0.509171i \(0.829952\pi\)
\(492\) −7.65685 −0.345198
\(493\) −7.31371 −0.329393
\(494\) −2.82843 −0.127257
\(495\) 5.65685 0.254257
\(496\) 4.00000 0.179605
\(497\) 16.0000 0.717698
\(498\) 6.34315 0.284243
\(499\) 0.485281 0.0217242 0.0108621 0.999941i \(-0.496542\pi\)
0.0108621 + 0.999941i \(0.496542\pi\)
\(500\) 1.00000 0.0447214
\(501\) 8.97056 0.400775
\(502\) 10.8284 0.483296
\(503\) −23.5147 −1.04847 −0.524235 0.851574i \(-0.675649\pi\)
−0.524235 + 0.851574i \(0.675649\pi\)
\(504\) −2.82843 −0.125988
\(505\) 16.1421 0.718316
\(506\) −48.0000 −2.13386
\(507\) 1.00000 0.0444116
\(508\) 9.65685 0.428454
\(509\) −37.3137 −1.65390 −0.826951 0.562275i \(-0.809926\pi\)
−0.826951 + 0.562275i \(0.809926\pi\)
\(510\) 0.828427 0.0366834
\(511\) 40.9706 1.81243
\(512\) 1.00000 0.0441942
\(513\) 2.82843 0.124878
\(514\) −4.82843 −0.212973
\(515\) −1.65685 −0.0730097
\(516\) 9.65685 0.425119
\(517\) −45.2548 −1.99031
\(518\) 32.9706 1.44864
\(519\) −9.31371 −0.408826
\(520\) −1.00000 −0.0438529
\(521\) 26.9706 1.18160 0.590801 0.806817i \(-0.298812\pi\)
0.590801 + 0.806817i \(0.298812\pi\)
\(522\) −8.82843 −0.386410
\(523\) −10.6274 −0.464704 −0.232352 0.972632i \(-0.574642\pi\)
−0.232352 + 0.972632i \(0.574642\pi\)
\(524\) −6.14214 −0.268320
\(525\) −2.82843 −0.123443
\(526\) 16.4853 0.718792
\(527\) 3.31371 0.144347
\(528\) 5.65685 0.246183
\(529\) 49.0000 2.13043
\(530\) 13.3137 0.578311
\(531\) −2.34315 −0.101684
\(532\) −8.00000 −0.346844
\(533\) 7.65685 0.331655
\(534\) −15.6569 −0.677538
\(535\) −4.00000 −0.172935
\(536\) 5.65685 0.244339
\(537\) 7.51472 0.324284
\(538\) −14.4853 −0.624505
\(539\) 5.65685 0.243658
\(540\) 1.00000 0.0430331
\(541\) 14.4853 0.622771 0.311385 0.950284i \(-0.399207\pi\)
0.311385 + 0.950284i \(0.399207\pi\)
\(542\) 7.31371 0.314151
\(543\) −7.65685 −0.328587
\(544\) 0.828427 0.0355185
\(545\) 8.82843 0.378168
\(546\) 2.82843 0.121046
\(547\) 0.686292 0.0293437 0.0146719 0.999892i \(-0.495330\pi\)
0.0146719 + 0.999892i \(0.495330\pi\)
\(548\) −17.3137 −0.739605
\(549\) 6.00000 0.256074
\(550\) 5.65685 0.241209
\(551\) −24.9706 −1.06378
\(552\) −8.48528 −0.361158
\(553\) −6.62742 −0.281826
\(554\) 26.0000 1.10463
\(555\) −11.6569 −0.494806
\(556\) 6.34315 0.269009
\(557\) 10.6863 0.452793 0.226396 0.974035i \(-0.427306\pi\)
0.226396 + 0.974035i \(0.427306\pi\)
\(558\) 4.00000 0.169334
\(559\) −9.65685 −0.408441
\(560\) −2.82843 −0.119523
\(561\) 4.68629 0.197855
\(562\) 8.34315 0.351934
\(563\) −30.3431 −1.27881 −0.639406 0.768870i \(-0.720819\pi\)
−0.639406 + 0.768870i \(0.720819\pi\)
\(564\) −8.00000 −0.336861
\(565\) 6.48528 0.272838
\(566\) −17.6569 −0.742173
\(567\) −2.82843 −0.118783
\(568\) −5.65685 −0.237356
\(569\) 31.6569 1.32712 0.663562 0.748121i \(-0.269044\pi\)
0.663562 + 0.748121i \(0.269044\pi\)
\(570\) 2.82843 0.118470
\(571\) 20.9706 0.877591 0.438795 0.898587i \(-0.355405\pi\)
0.438795 + 0.898587i \(0.355405\pi\)
\(572\) −5.65685 −0.236525
\(573\) 11.3137 0.472637
\(574\) 21.6569 0.903940
\(575\) −8.48528 −0.353861
\(576\) 1.00000 0.0416667
\(577\) −23.4558 −0.976480 −0.488240 0.872710i \(-0.662361\pi\)
−0.488240 + 0.872710i \(0.662361\pi\)
\(578\) −16.3137 −0.678561
\(579\) 2.48528 0.103285
\(580\) −8.82843 −0.366580
\(581\) −17.9411 −0.744323
\(582\) −3.17157 −0.131466
\(583\) 75.3137 3.11918
\(584\) −14.4853 −0.599405
\(585\) −1.00000 −0.0413449
\(586\) −16.6274 −0.686872
\(587\) 2.62742 0.108445 0.0542226 0.998529i \(-0.482732\pi\)
0.0542226 + 0.998529i \(0.482732\pi\)
\(588\) 1.00000 0.0412393
\(589\) 11.3137 0.466173
\(590\) −2.34315 −0.0964658
\(591\) −13.3137 −0.547653
\(592\) −11.6569 −0.479094
\(593\) −0.343146 −0.0140913 −0.00704565 0.999975i \(-0.502243\pi\)
−0.00704565 + 0.999975i \(0.502243\pi\)
\(594\) 5.65685 0.232104
\(595\) −2.34315 −0.0960596
\(596\) 3.65685 0.149791
\(597\) 10.3431 0.423317
\(598\) 8.48528 0.346989
\(599\) 40.0000 1.63436 0.817178 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) 1.00000 0.0408248
\(601\) 29.3137 1.19573 0.597866 0.801596i \(-0.296016\pi\)
0.597866 + 0.801596i \(0.296016\pi\)
\(602\) −27.3137 −1.11322
\(603\) 5.65685 0.230365
\(604\) 12.0000 0.488273
\(605\) 21.0000 0.853771
\(606\) 16.1421 0.655729
\(607\) 28.9706 1.17588 0.587939 0.808905i \(-0.299939\pi\)
0.587939 + 0.808905i \(0.299939\pi\)
\(608\) 2.82843 0.114708
\(609\) 24.9706 1.01186
\(610\) 6.00000 0.242933
\(611\) 8.00000 0.323645
\(612\) 0.828427 0.0334872
\(613\) 22.2843 0.900053 0.450027 0.893015i \(-0.351414\pi\)
0.450027 + 0.893015i \(0.351414\pi\)
\(614\) −21.6569 −0.874000
\(615\) −7.65685 −0.308754
\(616\) −16.0000 −0.644658
\(617\) 2.00000 0.0805170 0.0402585 0.999189i \(-0.487182\pi\)
0.0402585 + 0.999189i \(0.487182\pi\)
\(618\) −1.65685 −0.0666485
\(619\) −34.8284 −1.39987 −0.699936 0.714205i \(-0.746788\pi\)
−0.699936 + 0.714205i \(0.746788\pi\)
\(620\) 4.00000 0.160644
\(621\) −8.48528 −0.340503
\(622\) −24.0000 −0.962312
\(623\) 44.2843 1.77421
\(624\) −1.00000 −0.0400320
\(625\) 1.00000 0.0400000
\(626\) −30.9706 −1.23783
\(627\) 16.0000 0.638978
\(628\) 5.31371 0.212040
\(629\) −9.65685 −0.385044
\(630\) −2.82843 −0.112687
\(631\) −33.6569 −1.33986 −0.669929 0.742425i \(-0.733676\pi\)
−0.669929 + 0.742425i \(0.733676\pi\)
\(632\) 2.34315 0.0932053
\(633\) 0.686292 0.0272776
\(634\) 25.3137 1.00534
\(635\) 9.65685 0.383221
\(636\) 13.3137 0.527923
\(637\) −1.00000 −0.0396214
\(638\) −49.9411 −1.97719
\(639\) −5.65685 −0.223782
\(640\) 1.00000 0.0395285
\(641\) −4.62742 −0.182772 −0.0913860 0.995816i \(-0.529130\pi\)
−0.0913860 + 0.995816i \(0.529130\pi\)
\(642\) −4.00000 −0.157867
\(643\) 39.5980 1.56159 0.780796 0.624786i \(-0.214814\pi\)
0.780796 + 0.624786i \(0.214814\pi\)
\(644\) 24.0000 0.945732
\(645\) 9.65685 0.380238
\(646\) 2.34315 0.0921898
\(647\) 8.48528 0.333591 0.166795 0.985992i \(-0.446658\pi\)
0.166795 + 0.985992i \(0.446658\pi\)
\(648\) 1.00000 0.0392837
\(649\) −13.2548 −0.520298
\(650\) −1.00000 −0.0392232
\(651\) −11.3137 −0.443419
\(652\) 11.3137 0.443079
\(653\) −42.2843 −1.65471 −0.827356 0.561678i \(-0.810156\pi\)
−0.827356 + 0.561678i \(0.810156\pi\)
\(654\) 8.82843 0.345219
\(655\) −6.14214 −0.239993
\(656\) −7.65685 −0.298950
\(657\) −14.4853 −0.565125
\(658\) 22.6274 0.882109
\(659\) −7.51472 −0.292732 −0.146366 0.989231i \(-0.546758\pi\)
−0.146366 + 0.989231i \(0.546758\pi\)
\(660\) 5.65685 0.220193
\(661\) −8.14214 −0.316692 −0.158346 0.987384i \(-0.550616\pi\)
−0.158346 + 0.987384i \(0.550616\pi\)
\(662\) −8.48528 −0.329790
\(663\) −0.828427 −0.0321734
\(664\) 6.34315 0.246162
\(665\) −8.00000 −0.310227
\(666\) −11.6569 −0.451694
\(667\) 74.9117 2.90059
\(668\) 8.97056 0.347081
\(669\) 10.8284 0.418651
\(670\) 5.65685 0.218543
\(671\) 33.9411 1.31028
\(672\) −2.82843 −0.109109
\(673\) 32.6274 1.25769 0.628847 0.777529i \(-0.283527\pi\)
0.628847 + 0.777529i \(0.283527\pi\)
\(674\) 10.9706 0.422570
\(675\) 1.00000 0.0384900
\(676\) 1.00000 0.0384615
\(677\) 12.3431 0.474386 0.237193 0.971463i \(-0.423773\pi\)
0.237193 + 0.971463i \(0.423773\pi\)
\(678\) 6.48528 0.249066
\(679\) 8.97056 0.344259
\(680\) 0.828427 0.0317687
\(681\) 4.00000 0.153280
\(682\) 22.6274 0.866449
\(683\) 33.6569 1.28784 0.643922 0.765091i \(-0.277306\pi\)
0.643922 + 0.765091i \(0.277306\pi\)
\(684\) 2.82843 0.108148
\(685\) −17.3137 −0.661523
\(686\) 16.9706 0.647939
\(687\) 4.14214 0.158032
\(688\) 9.65685 0.368164
\(689\) −13.3137 −0.507212
\(690\) −8.48528 −0.323029
\(691\) −27.7990 −1.05752 −0.528762 0.848770i \(-0.677343\pi\)
−0.528762 + 0.848770i \(0.677343\pi\)
\(692\) −9.31371 −0.354054
\(693\) −16.0000 −0.607790
\(694\) −9.65685 −0.366569
\(695\) 6.34315 0.240609
\(696\) −8.82843 −0.334641
\(697\) −6.34315 −0.240264
\(698\) 12.1421 0.459587
\(699\) 5.51472 0.208586
\(700\) −2.82843 −0.106904
\(701\) 0.142136 0.00536839 0.00268419 0.999996i \(-0.499146\pi\)
0.00268419 + 0.999996i \(0.499146\pi\)
\(702\) −1.00000 −0.0377426
\(703\) −32.9706 −1.24351
\(704\) 5.65685 0.213201
\(705\) −8.00000 −0.301297
\(706\) 5.31371 0.199984
\(707\) −45.6569 −1.71710
\(708\) −2.34315 −0.0880608
\(709\) −7.17157 −0.269334 −0.134667 0.990891i \(-0.542996\pi\)
−0.134667 + 0.990891i \(0.542996\pi\)
\(710\) −5.65685 −0.212298
\(711\) 2.34315 0.0878748
\(712\) −15.6569 −0.586765
\(713\) −33.9411 −1.27111
\(714\) −2.34315 −0.0876900
\(715\) −5.65685 −0.211554
\(716\) 7.51472 0.280838
\(717\) 16.0000 0.597531
\(718\) −28.2843 −1.05556
\(719\) −29.6569 −1.10601 −0.553007 0.833177i \(-0.686520\pi\)
−0.553007 + 0.833177i \(0.686520\pi\)
\(720\) 1.00000 0.0372678
\(721\) 4.68629 0.174527
\(722\) −11.0000 −0.409378
\(723\) 5.31371 0.197619
\(724\) −7.65685 −0.284565
\(725\) −8.82843 −0.327880
\(726\) 21.0000 0.779383
\(727\) −45.9411 −1.70386 −0.851931 0.523654i \(-0.824568\pi\)
−0.851931 + 0.523654i \(0.824568\pi\)
\(728\) 2.82843 0.104828
\(729\) 1.00000 0.0370370
\(730\) −14.4853 −0.536124
\(731\) 8.00000 0.295891
\(732\) 6.00000 0.221766
\(733\) −0.343146 −0.0126744 −0.00633719 0.999980i \(-0.502017\pi\)
−0.00633719 + 0.999980i \(0.502017\pi\)
\(734\) 25.6569 0.947012
\(735\) 1.00000 0.0368856
\(736\) −8.48528 −0.312772
\(737\) 32.0000 1.17874
\(738\) −7.65685 −0.281853
\(739\) −14.1421 −0.520227 −0.260113 0.965578i \(-0.583760\pi\)
−0.260113 + 0.965578i \(0.583760\pi\)
\(740\) −11.6569 −0.428514
\(741\) −2.82843 −0.103905
\(742\) −37.6569 −1.38243
\(743\) −36.2843 −1.33114 −0.665570 0.746335i \(-0.731812\pi\)
−0.665570 + 0.746335i \(0.731812\pi\)
\(744\) 4.00000 0.146647
\(745\) 3.65685 0.133977
\(746\) −2.68629 −0.0983521
\(747\) 6.34315 0.232084
\(748\) 4.68629 0.171348
\(749\) 11.3137 0.413394
\(750\) 1.00000 0.0365148
\(751\) 11.3137 0.412843 0.206422 0.978463i \(-0.433818\pi\)
0.206422 + 0.978463i \(0.433818\pi\)
\(752\) −8.00000 −0.291730
\(753\) 10.8284 0.394610
\(754\) 8.82843 0.321512
\(755\) 12.0000 0.436725
\(756\) −2.82843 −0.102869
\(757\) 19.9411 0.724773 0.362386 0.932028i \(-0.381962\pi\)
0.362386 + 0.932028i \(0.381962\pi\)
\(758\) −7.51472 −0.272947
\(759\) −48.0000 −1.74229
\(760\) 2.82843 0.102598
\(761\) 27.6569 1.00256 0.501280 0.865285i \(-0.332863\pi\)
0.501280 + 0.865285i \(0.332863\pi\)
\(762\) 9.65685 0.349831
\(763\) −24.9706 −0.903995
\(764\) 11.3137 0.409316
\(765\) 0.828427 0.0299518
\(766\) −29.6569 −1.07155
\(767\) 2.34315 0.0846061
\(768\) 1.00000 0.0360844
\(769\) −14.0000 −0.504853 −0.252426 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) −16.0000 −0.576600
\(771\) −4.82843 −0.173892
\(772\) 2.48528 0.0894472
\(773\) −53.3137 −1.91756 −0.958780 0.284148i \(-0.908289\pi\)
−0.958780 + 0.284148i \(0.908289\pi\)
\(774\) 9.65685 0.347108
\(775\) 4.00000 0.143684
\(776\) −3.17157 −0.113853
\(777\) 32.9706 1.18281
\(778\) −6.48528 −0.232509
\(779\) −21.6569 −0.775937
\(780\) −1.00000 −0.0358057
\(781\) −32.0000 −1.14505
\(782\) −7.02944 −0.251372
\(783\) −8.82843 −0.315502
\(784\) 1.00000 0.0357143
\(785\) 5.31371 0.189654
\(786\) −6.14214 −0.219083
\(787\) −24.0000 −0.855508 −0.427754 0.903895i \(-0.640695\pi\)
−0.427754 + 0.903895i \(0.640695\pi\)
\(788\) −13.3137 −0.474281
\(789\) 16.4853 0.586892
\(790\) 2.34315 0.0833654
\(791\) −18.3431 −0.652207
\(792\) 5.65685 0.201008
\(793\) −6.00000 −0.213066
\(794\) 30.2843 1.07475
\(795\) 13.3137 0.472189
\(796\) 10.3431 0.366603
\(797\) 16.6274 0.588973 0.294487 0.955656i \(-0.404851\pi\)
0.294487 + 0.955656i \(0.404851\pi\)
\(798\) −8.00000 −0.283197
\(799\) −6.62742 −0.234461
\(800\) 1.00000 0.0353553
\(801\) −15.6569 −0.553208
\(802\) −26.9706 −0.952364
\(803\) −81.9411 −2.89164
\(804\) 5.65685 0.199502
\(805\) 24.0000 0.845889
\(806\) −4.00000 −0.140894
\(807\) −14.4853 −0.509906
\(808\) 16.1421 0.567878
\(809\) 13.3137 0.468085 0.234043 0.972226i \(-0.424804\pi\)
0.234043 + 0.972226i \(0.424804\pi\)
\(810\) 1.00000 0.0351364
\(811\) −1.85786 −0.0652384 −0.0326192 0.999468i \(-0.510385\pi\)
−0.0326192 + 0.999468i \(0.510385\pi\)
\(812\) 24.9706 0.876295
\(813\) 7.31371 0.256503
\(814\) −65.9411 −2.31124
\(815\) 11.3137 0.396302
\(816\) 0.828427 0.0290008
\(817\) 27.3137 0.955586
\(818\) −3.65685 −0.127859
\(819\) 2.82843 0.0988332
\(820\) −7.65685 −0.267389
\(821\) 34.2843 1.19653 0.598265 0.801299i \(-0.295857\pi\)
0.598265 + 0.801299i \(0.295857\pi\)
\(822\) −17.3137 −0.603885
\(823\) −52.9706 −1.84644 −0.923219 0.384275i \(-0.874452\pi\)
−0.923219 + 0.384275i \(0.874452\pi\)
\(824\) −1.65685 −0.0577193
\(825\) 5.65685 0.196946
\(826\) 6.62742 0.230597
\(827\) 9.65685 0.335802 0.167901 0.985804i \(-0.446301\pi\)
0.167901 + 0.985804i \(0.446301\pi\)
\(828\) −8.48528 −0.294884
\(829\) −53.3137 −1.85166 −0.925831 0.377938i \(-0.876633\pi\)
−0.925831 + 0.377938i \(0.876633\pi\)
\(830\) 6.34315 0.220174
\(831\) 26.0000 0.901930
\(832\) −1.00000 −0.0346688
\(833\) 0.828427 0.0287033
\(834\) 6.34315 0.219645
\(835\) 8.97056 0.310439
\(836\) 16.0000 0.553372
\(837\) 4.00000 0.138260
\(838\) −10.8284 −0.374062
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) −2.82843 −0.0975900
\(841\) 48.9411 1.68763
\(842\) −24.1421 −0.831993
\(843\) 8.34315 0.287353
\(844\) 0.686292 0.0236231
\(845\) 1.00000 0.0344010
\(846\) −8.00000 −0.275046
\(847\) −59.3970 −2.04090
\(848\) 13.3137 0.457195
\(849\) −17.6569 −0.605982
\(850\) 0.828427 0.0284148
\(851\) 98.9117 3.39065
\(852\) −5.65685 −0.193801
\(853\) 38.2843 1.31083 0.655414 0.755270i \(-0.272494\pi\)
0.655414 + 0.755270i \(0.272494\pi\)
\(854\) −16.9706 −0.580721
\(855\) 2.82843 0.0967302
\(856\) −4.00000 −0.136717
\(857\) −20.8284 −0.711486 −0.355743 0.934584i \(-0.615772\pi\)
−0.355743 + 0.934584i \(0.615772\pi\)
\(858\) −5.65685 −0.193122
\(859\) 37.9411 1.29453 0.647267 0.762263i \(-0.275912\pi\)
0.647267 + 0.762263i \(0.275912\pi\)
\(860\) 9.65685 0.329296
\(861\) 21.6569 0.738064
\(862\) −16.0000 −0.544962
\(863\) −28.2843 −0.962808 −0.481404 0.876499i \(-0.659873\pi\)
−0.481404 + 0.876499i \(0.659873\pi\)
\(864\) 1.00000 0.0340207
\(865\) −9.31371 −0.316676
\(866\) −22.9706 −0.780571
\(867\) −16.3137 −0.554043
\(868\) −11.3137 −0.384012
\(869\) 13.2548 0.449639
\(870\) −8.82843 −0.299312
\(871\) −5.65685 −0.191675
\(872\) 8.82843 0.298968
\(873\) −3.17157 −0.107341
\(874\) −24.0000 −0.811812
\(875\) −2.82843 −0.0956183
\(876\) −14.4853 −0.489412
\(877\) −51.2548 −1.73075 −0.865376 0.501122i \(-0.832921\pi\)
−0.865376 + 0.501122i \(0.832921\pi\)
\(878\) 22.6274 0.763638
\(879\) −16.6274 −0.560829
\(880\) 5.65685 0.190693
\(881\) −10.2843 −0.346486 −0.173243 0.984879i \(-0.555425\pi\)
−0.173243 + 0.984879i \(0.555425\pi\)
\(882\) 1.00000 0.0336718
\(883\) 31.3137 1.05379 0.526895 0.849930i \(-0.323356\pi\)
0.526895 + 0.849930i \(0.323356\pi\)
\(884\) −0.828427 −0.0278630
\(885\) −2.34315 −0.0787640
\(886\) 30.3431 1.01940
\(887\) −40.4853 −1.35936 −0.679681 0.733508i \(-0.737882\pi\)
−0.679681 + 0.733508i \(0.737882\pi\)
\(888\) −11.6569 −0.391178
\(889\) −27.3137 −0.916072
\(890\) −15.6569 −0.524819
\(891\) 5.65685 0.189512
\(892\) 10.8284 0.362563
\(893\) −22.6274 −0.757198
\(894\) 3.65685 0.122304
\(895\) 7.51472 0.251189
\(896\) −2.82843 −0.0944911
\(897\) 8.48528 0.283315
\(898\) 26.2843 0.877117
\(899\) −35.3137 −1.17778
\(900\) 1.00000 0.0333333
\(901\) 11.0294 0.367444
\(902\) −43.3137 −1.44219
\(903\) −27.3137 −0.908943
\(904\) 6.48528 0.215697
\(905\) −7.65685 −0.254522
\(906\) 12.0000 0.398673
\(907\) 8.28427 0.275075 0.137537 0.990497i \(-0.456081\pi\)
0.137537 + 0.990497i \(0.456081\pi\)
\(908\) 4.00000 0.132745
\(909\) 16.1421 0.535401
\(910\) 2.82843 0.0937614
\(911\) 24.9706 0.827312 0.413656 0.910433i \(-0.364252\pi\)
0.413656 + 0.910433i \(0.364252\pi\)
\(912\) 2.82843 0.0936586
\(913\) 35.8823 1.18753
\(914\) 20.8284 0.688943
\(915\) 6.00000 0.198354
\(916\) 4.14214 0.136860
\(917\) 17.3726 0.573693
\(918\) 0.828427 0.0273422
\(919\) 41.9411 1.38351 0.691755 0.722132i \(-0.256838\pi\)
0.691755 + 0.722132i \(0.256838\pi\)
\(920\) −8.48528 −0.279751
\(921\) −21.6569 −0.713618
\(922\) 14.0000 0.461065
\(923\) 5.65685 0.186198
\(924\) −16.0000 −0.526361
\(925\) −11.6569 −0.383275
\(926\) −3.79899 −0.124843
\(927\) −1.65685 −0.0544182
\(928\) −8.82843 −0.289807
\(929\) −33.5980 −1.10231 −0.551157 0.834402i \(-0.685813\pi\)
−0.551157 + 0.834402i \(0.685813\pi\)
\(930\) 4.00000 0.131165
\(931\) 2.82843 0.0926980
\(932\) 5.51472 0.180641
\(933\) −24.0000 −0.785725
\(934\) 7.31371 0.239312
\(935\) 4.68629 0.153258
\(936\) −1.00000 −0.0326860
\(937\) 16.6274 0.543194 0.271597 0.962411i \(-0.412448\pi\)
0.271597 + 0.962411i \(0.412448\pi\)
\(938\) −16.0000 −0.522419
\(939\) −30.9706 −1.01069
\(940\) −8.00000 −0.260931
\(941\) 54.9706 1.79199 0.895995 0.444065i \(-0.146464\pi\)
0.895995 + 0.444065i \(0.146464\pi\)
\(942\) 5.31371 0.173130
\(943\) 64.9706 2.11573
\(944\) −2.34315 −0.0762629
\(945\) −2.82843 −0.0920087
\(946\) 54.6274 1.77609
\(947\) 30.3431 0.986020 0.493010 0.870024i \(-0.335897\pi\)
0.493010 + 0.870024i \(0.335897\pi\)
\(948\) 2.34315 0.0761018
\(949\) 14.4853 0.470212
\(950\) 2.82843 0.0917663
\(951\) 25.3137 0.820853
\(952\) −2.34315 −0.0759418
\(953\) −27.8579 −0.902405 −0.451202 0.892422i \(-0.649005\pi\)
−0.451202 + 0.892422i \(0.649005\pi\)
\(954\) 13.3137 0.431047
\(955\) 11.3137 0.366103
\(956\) 16.0000 0.517477
\(957\) −49.9411 −1.61437
\(958\) −11.3137 −0.365529
\(959\) 48.9706 1.58134
\(960\) 1.00000 0.0322749
\(961\) −15.0000 −0.483871
\(962\) 11.6569 0.375832
\(963\) −4.00000 −0.128898
\(964\) 5.31371 0.171143
\(965\) 2.48528 0.0800040
\(966\) 24.0000 0.772187
\(967\) −7.51472 −0.241657 −0.120829 0.992673i \(-0.538555\pi\)
−0.120829 + 0.992673i \(0.538555\pi\)
\(968\) 21.0000 0.674966
\(969\) 2.34315 0.0752727
\(970\) −3.17157 −0.101833
\(971\) 15.5147 0.497891 0.248946 0.968517i \(-0.419916\pi\)
0.248946 + 0.968517i \(0.419916\pi\)
\(972\) 1.00000 0.0320750
\(973\) −17.9411 −0.575166
\(974\) 16.4853 0.528222
\(975\) −1.00000 −0.0320256
\(976\) 6.00000 0.192055
\(977\) −8.34315 −0.266921 −0.133460 0.991054i \(-0.542609\pi\)
−0.133460 + 0.991054i \(0.542609\pi\)
\(978\) 11.3137 0.361773
\(979\) −88.5685 −2.83066
\(980\) 1.00000 0.0319438
\(981\) 8.82843 0.281870
\(982\) −38.1421 −1.21716
\(983\) −2.34315 −0.0747347 −0.0373674 0.999302i \(-0.511897\pi\)
−0.0373674 + 0.999302i \(0.511897\pi\)
\(984\) −7.65685 −0.244092
\(985\) −13.3137 −0.424210
\(986\) −7.31371 −0.232916
\(987\) 22.6274 0.720239
\(988\) −2.82843 −0.0899843
\(989\) −81.9411 −2.60558
\(990\) 5.65685 0.179787
\(991\) −42.9117 −1.36313 −0.681567 0.731755i \(-0.738701\pi\)
−0.681567 + 0.731755i \(0.738701\pi\)
\(992\) 4.00000 0.127000
\(993\) −8.48528 −0.269272
\(994\) 16.0000 0.507489
\(995\) 10.3431 0.327900
\(996\) 6.34315 0.200990
\(997\) 61.3137 1.94182 0.970912 0.239435i \(-0.0769623\pi\)
0.970912 + 0.239435i \(0.0769623\pi\)
\(998\) 0.485281 0.0153613
\(999\) −11.6569 −0.368807
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 390.2.a.h.1.1 2
3.2 odd 2 1170.2.a.o.1.1 2
4.3 odd 2 3120.2.a.bc.1.2 2
5.2 odd 4 1950.2.e.o.1249.3 4
5.3 odd 4 1950.2.e.o.1249.2 4
5.4 even 2 1950.2.a.bd.1.2 2
12.11 even 2 9360.2.a.ch.1.2 2
13.5 odd 4 5070.2.b.q.1351.1 4
13.8 odd 4 5070.2.b.q.1351.4 4
13.12 even 2 5070.2.a.bc.1.2 2
15.2 even 4 5850.2.e.bk.5149.1 4
15.8 even 4 5850.2.e.bk.5149.4 4
15.14 odd 2 5850.2.a.cl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.a.h.1.1 2 1.1 even 1 trivial
1170.2.a.o.1.1 2 3.2 odd 2
1950.2.a.bd.1.2 2 5.4 even 2
1950.2.e.o.1249.2 4 5.3 odd 4
1950.2.e.o.1249.3 4 5.2 odd 4
3120.2.a.bc.1.2 2 4.3 odd 2
5070.2.a.bc.1.2 2 13.12 even 2
5070.2.b.q.1351.1 4 13.5 odd 4
5070.2.b.q.1351.4 4 13.8 odd 4
5850.2.a.cl.1.2 2 15.14 odd 2
5850.2.e.bk.5149.1 4 15.2 even 4
5850.2.e.bk.5149.4 4 15.8 even 4
9360.2.a.ch.1.2 2 12.11 even 2