Properties

Label 4830.2.a.bn.1.1
Level $4830$
Weight $2$
Character 4830.1
Self dual yes
Analytic conductor $38.568$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4830.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.5677441763\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 4830.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} -2.60555 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +4.60555 q^{17} -1.00000 q^{18} -2.60555 q^{19} -1.00000 q^{20} -1.00000 q^{21} +2.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.60555 q^{26} +1.00000 q^{27} -1.00000 q^{28} +5.21110 q^{29} +1.00000 q^{30} +2.60555 q^{31} -1.00000 q^{32} -2.00000 q^{33} -4.60555 q^{34} +1.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} +2.60555 q^{38} -2.60555 q^{39} +1.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} +11.2111 q^{43} -2.00000 q^{44} -1.00000 q^{45} -1.00000 q^{46} -1.39445 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} +4.60555 q^{51} -2.60555 q^{52} -3.21110 q^{53} -1.00000 q^{54} +2.00000 q^{55} +1.00000 q^{56} -2.60555 q^{57} -5.21110 q^{58} -5.21110 q^{59} -1.00000 q^{60} +10.0000 q^{61} -2.60555 q^{62} -1.00000 q^{63} +1.00000 q^{64} +2.60555 q^{65} +2.00000 q^{66} +3.21110 q^{67} +4.60555 q^{68} +1.00000 q^{69} -1.00000 q^{70} -6.00000 q^{71} -1.00000 q^{72} -2.00000 q^{73} +6.00000 q^{74} +1.00000 q^{75} -2.60555 q^{76} +2.00000 q^{77} +2.60555 q^{78} -14.4222 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -10.6056 q^{83} -1.00000 q^{84} -4.60555 q^{85} -11.2111 q^{86} +5.21110 q^{87} +2.00000 q^{88} -5.39445 q^{89} +1.00000 q^{90} +2.60555 q^{91} +1.00000 q^{92} +2.60555 q^{93} +1.39445 q^{94} +2.60555 q^{95} -1.00000 q^{96} -11.8167 q^{97} -1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + 2q^{10} - 4q^{11} + 2q^{12} + 2q^{13} + 2q^{14} - 2q^{15} + 2q^{16} + 2q^{17} - 2q^{18} + 2q^{19} - 2q^{20} - 2q^{21} + 4q^{22} + 2q^{23} - 2q^{24} + 2q^{25} - 2q^{26} + 2q^{27} - 2q^{28} - 4q^{29} + 2q^{30} - 2q^{31} - 2q^{32} - 4q^{33} - 2q^{34} + 2q^{35} + 2q^{36} - 12q^{37} - 2q^{38} + 2q^{39} + 2q^{40} - 12q^{41} + 2q^{42} + 8q^{43} - 4q^{44} - 2q^{45} - 2q^{46} - 10q^{47} + 2q^{48} + 2q^{49} - 2q^{50} + 2q^{51} + 2q^{52} + 8q^{53} - 2q^{54} + 4q^{55} + 2q^{56} + 2q^{57} + 4q^{58} + 4q^{59} - 2q^{60} + 20q^{61} + 2q^{62} - 2q^{63} + 2q^{64} - 2q^{65} + 4q^{66} - 8q^{67} + 2q^{68} + 2q^{69} - 2q^{70} - 12q^{71} - 2q^{72} - 4q^{73} + 12q^{74} + 2q^{75} + 2q^{76} + 4q^{77} - 2q^{78} - 2q^{80} + 2q^{81} + 12q^{82} - 14q^{83} - 2q^{84} - 2q^{85} - 8q^{86} - 4q^{87} + 4q^{88} - 18q^{89} + 2q^{90} - 2q^{91} + 2q^{92} - 2q^{93} + 10q^{94} - 2q^{95} - 2q^{96} - 2q^{97} - 2q^{98} - 4q^{99} + O(q^{100}) \)

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
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.60555 −0.722650 −0.361325 0.932440i \(-0.617675\pi\)
−0.361325 + 0.932440i \(0.617675\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 4.60555 1.11701 0.558505 0.829501i \(-0.311375\pi\)
0.558505 + 0.829501i \(0.311375\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.60555 −0.597754 −0.298877 0.954292i \(-0.596612\pi\)
−0.298877 + 0.954292i \(0.596612\pi\)
\(20\) −1.00000 −0.223607
\(21\) −1.00000 −0.218218
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 2.60555 0.510991
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 5.21110 0.967677 0.483839 0.875157i \(-0.339242\pi\)
0.483839 + 0.875157i \(0.339242\pi\)
\(30\) 1.00000 0.182574
\(31\) 2.60555 0.467971 0.233985 0.972240i \(-0.424823\pi\)
0.233985 + 0.972240i \(0.424823\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) −4.60555 −0.789846
\(35\) 1.00000 0.169031
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 2.60555 0.422676
\(39\) −2.60555 −0.417222
\(40\) 1.00000 0.158114
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 1.00000 0.154303
\(43\) 11.2111 1.70968 0.854839 0.518894i \(-0.173656\pi\)
0.854839 + 0.518894i \(0.173656\pi\)
\(44\) −2.00000 −0.301511
\(45\) −1.00000 −0.149071
\(46\) −1.00000 −0.147442
\(47\) −1.39445 −0.203401 −0.101701 0.994815i \(-0.532428\pi\)
−0.101701 + 0.994815i \(0.532428\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 4.60555 0.644906
\(52\) −2.60555 −0.361325
\(53\) −3.21110 −0.441079 −0.220539 0.975378i \(-0.570782\pi\)
−0.220539 + 0.975378i \(0.570782\pi\)
\(54\) −1.00000 −0.136083
\(55\) 2.00000 0.269680
\(56\) 1.00000 0.133631
\(57\) −2.60555 −0.345114
\(58\) −5.21110 −0.684251
\(59\) −5.21110 −0.678428 −0.339214 0.940709i \(-0.610161\pi\)
−0.339214 + 0.940709i \(0.610161\pi\)
\(60\) −1.00000 −0.129099
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) −2.60555 −0.330905
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 2.60555 0.323179
\(66\) 2.00000 0.246183
\(67\) 3.21110 0.392299 0.196149 0.980574i \(-0.437156\pi\)
0.196149 + 0.980574i \(0.437156\pi\)
\(68\) 4.60555 0.558505
\(69\) 1.00000 0.120386
\(70\) −1.00000 −0.119523
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) −1.00000 −0.117851
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 6.00000 0.697486
\(75\) 1.00000 0.115470
\(76\) −2.60555 −0.298877
\(77\) 2.00000 0.227921
\(78\) 2.60555 0.295021
\(79\) −14.4222 −1.62262 −0.811312 0.584613i \(-0.801246\pi\)
−0.811312 + 0.584613i \(0.801246\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −10.6056 −1.16411 −0.582055 0.813149i \(-0.697751\pi\)
−0.582055 + 0.813149i \(0.697751\pi\)
\(84\) −1.00000 −0.109109
\(85\) −4.60555 −0.499542
\(86\) −11.2111 −1.20892
\(87\) 5.21110 0.558689
\(88\) 2.00000 0.213201
\(89\) −5.39445 −0.571810 −0.285905 0.958258i \(-0.592294\pi\)
−0.285905 + 0.958258i \(0.592294\pi\)
\(90\) 1.00000 0.105409
\(91\) 2.60555 0.273136
\(92\) 1.00000 0.104257
\(93\) 2.60555 0.270183
\(94\) 1.39445 0.143826
\(95\) 2.60555 0.267324
\(96\) −1.00000 −0.102062
\(97\) −11.8167 −1.19980 −0.599900 0.800075i \(-0.704793\pi\)
−0.599900 + 0.800075i \(0.704793\pi\)
\(98\) −1.00000 −0.101015
\(99\) −2.00000 −0.201008
\(100\) 1.00000 0.100000
\(101\) −15.8167 −1.57382 −0.786908 0.617070i \(-0.788319\pi\)
−0.786908 + 0.617070i \(0.788319\pi\)
\(102\) −4.60555 −0.456018
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 2.60555 0.255495
\(105\) 1.00000 0.0975900
\(106\) 3.21110 0.311890
\(107\) −9.21110 −0.890471 −0.445235 0.895414i \(-0.646880\pi\)
−0.445235 + 0.895414i \(0.646880\pi\)
\(108\) 1.00000 0.0962250
\(109\) 16.4222 1.57296 0.786481 0.617614i \(-0.211901\pi\)
0.786481 + 0.617614i \(0.211901\pi\)
\(110\) −2.00000 −0.190693
\(111\) −6.00000 −0.569495
\(112\) −1.00000 −0.0944911
\(113\) 0.788897 0.0742132 0.0371066 0.999311i \(-0.488186\pi\)
0.0371066 + 0.999311i \(0.488186\pi\)
\(114\) 2.60555 0.244032
\(115\) −1.00000 −0.0932505
\(116\) 5.21110 0.483839
\(117\) −2.60555 −0.240883
\(118\) 5.21110 0.479721
\(119\) −4.60555 −0.422190
\(120\) 1.00000 0.0912871
\(121\) −7.00000 −0.636364
\(122\) −10.0000 −0.905357
\(123\) −6.00000 −0.541002
\(124\) 2.60555 0.233985
\(125\) −1.00000 −0.0894427
\(126\) 1.00000 0.0890871
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 11.2111 0.987083
\(130\) −2.60555 −0.228522
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −2.00000 −0.174078
\(133\) 2.60555 0.225930
\(134\) −3.21110 −0.277397
\(135\) −1.00000 −0.0860663
\(136\) −4.60555 −0.394923
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 1.00000 0.0845154
\(141\) −1.39445 −0.117434
\(142\) 6.00000 0.503509
\(143\) 5.21110 0.435774
\(144\) 1.00000 0.0833333
\(145\) −5.21110 −0.432759
\(146\) 2.00000 0.165521
\(147\) 1.00000 0.0824786
\(148\) −6.00000 −0.493197
\(149\) −14.0000 −1.14692 −0.573462 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 17.2111 1.40062 0.700310 0.713839i \(-0.253045\pi\)
0.700310 + 0.713839i \(0.253045\pi\)
\(152\) 2.60555 0.211338
\(153\) 4.60555 0.372337
\(154\) −2.00000 −0.161165
\(155\) −2.60555 −0.209283
\(156\) −2.60555 −0.208611
\(157\) 8.42221 0.672165 0.336083 0.941833i \(-0.390898\pi\)
0.336083 + 0.941833i \(0.390898\pi\)
\(158\) 14.4222 1.14737
\(159\) −3.21110 −0.254657
\(160\) 1.00000 0.0790569
\(161\) −1.00000 −0.0788110
\(162\) −1.00000 −0.0785674
\(163\) 19.6333 1.53780 0.768900 0.639369i \(-0.220805\pi\)
0.768900 + 0.639369i \(0.220805\pi\)
\(164\) −6.00000 −0.468521
\(165\) 2.00000 0.155700
\(166\) 10.6056 0.823150
\(167\) 13.3944 1.03649 0.518247 0.855231i \(-0.326585\pi\)
0.518247 + 0.855231i \(0.326585\pi\)
\(168\) 1.00000 0.0771517
\(169\) −6.21110 −0.477777
\(170\) 4.60555 0.353230
\(171\) −2.60555 −0.199251
\(172\) 11.2111 0.854839
\(173\) −19.0278 −1.44665 −0.723327 0.690506i \(-0.757388\pi\)
−0.723327 + 0.690506i \(0.757388\pi\)
\(174\) −5.21110 −0.395053
\(175\) −1.00000 −0.0755929
\(176\) −2.00000 −0.150756
\(177\) −5.21110 −0.391690
\(178\) 5.39445 0.404331
\(179\) −18.4222 −1.37694 −0.688470 0.725265i \(-0.741717\pi\)
−0.688470 + 0.725265i \(0.741717\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) −2.60555 −0.193136
\(183\) 10.0000 0.739221
\(184\) −1.00000 −0.0737210
\(185\) 6.00000 0.441129
\(186\) −2.60555 −0.191048
\(187\) −9.21110 −0.673583
\(188\) −1.39445 −0.101701
\(189\) −1.00000 −0.0727393
\(190\) −2.60555 −0.189027
\(191\) −6.78890 −0.491227 −0.245614 0.969368i \(-0.578989\pi\)
−0.245614 + 0.969368i \(0.578989\pi\)
\(192\) 1.00000 0.0721688
\(193\) −12.4222 −0.894170 −0.447085 0.894492i \(-0.647538\pi\)
−0.447085 + 0.894492i \(0.647538\pi\)
\(194\) 11.8167 0.848386
\(195\) 2.60555 0.186587
\(196\) 1.00000 0.0714286
\(197\) −8.78890 −0.626183 −0.313092 0.949723i \(-0.601365\pi\)
−0.313092 + 0.949723i \(0.601365\pi\)
\(198\) 2.00000 0.142134
\(199\) 26.4222 1.87302 0.936510 0.350640i \(-0.114036\pi\)
0.936510 + 0.350640i \(0.114036\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 3.21110 0.226494
\(202\) 15.8167 1.11286
\(203\) −5.21110 −0.365748
\(204\) 4.60555 0.322453
\(205\) 6.00000 0.419058
\(206\) −8.00000 −0.557386
\(207\) 1.00000 0.0695048
\(208\) −2.60555 −0.180662
\(209\) 5.21110 0.360460
\(210\) −1.00000 −0.0690066
\(211\) 10.4222 0.717494 0.358747 0.933435i \(-0.383204\pi\)
0.358747 + 0.933435i \(0.383204\pi\)
\(212\) −3.21110 −0.220539
\(213\) −6.00000 −0.411113
\(214\) 9.21110 0.629658
\(215\) −11.2111 −0.764591
\(216\) −1.00000 −0.0680414
\(217\) −2.60555 −0.176876
\(218\) −16.4222 −1.11225
\(219\) −2.00000 −0.135147
\(220\) 2.00000 0.134840
\(221\) −12.0000 −0.807207
\(222\) 6.00000 0.402694
\(223\) −11.3944 −0.763029 −0.381514 0.924363i \(-0.624597\pi\)
−0.381514 + 0.924363i \(0.624597\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) −0.788897 −0.0524767
\(227\) 22.6056 1.50038 0.750192 0.661221i \(-0.229961\pi\)
0.750192 + 0.661221i \(0.229961\pi\)
\(228\) −2.60555 −0.172557
\(229\) −23.2111 −1.53383 −0.766916 0.641747i \(-0.778210\pi\)
−0.766916 + 0.641747i \(0.778210\pi\)
\(230\) 1.00000 0.0659380
\(231\) 2.00000 0.131590
\(232\) −5.21110 −0.342126
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 2.60555 0.170330
\(235\) 1.39445 0.0909638
\(236\) −5.21110 −0.339214
\(237\) −14.4222 −0.936823
\(238\) 4.60555 0.298534
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −21.8167 −1.40533 −0.702667 0.711519i \(-0.748008\pi\)
−0.702667 + 0.711519i \(0.748008\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) −1.00000 −0.0638877
\(246\) 6.00000 0.382546
\(247\) 6.78890 0.431967
\(248\) −2.60555 −0.165453
\(249\) −10.6056 −0.672100
\(250\) 1.00000 0.0632456
\(251\) −11.3944 −0.719211 −0.359606 0.933104i \(-0.617089\pi\)
−0.359606 + 0.933104i \(0.617089\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −2.00000 −0.125739
\(254\) 2.00000 0.125491
\(255\) −4.60555 −0.288411
\(256\) 1.00000 0.0625000
\(257\) 25.6333 1.59896 0.799481 0.600692i \(-0.205108\pi\)
0.799481 + 0.600692i \(0.205108\pi\)
\(258\) −11.2111 −0.697973
\(259\) 6.00000 0.372822
\(260\) 2.60555 0.161589
\(261\) 5.21110 0.322559
\(262\) 12.0000 0.741362
\(263\) 10.4222 0.642661 0.321330 0.946967i \(-0.395870\pi\)
0.321330 + 0.946967i \(0.395870\pi\)
\(264\) 2.00000 0.123091
\(265\) 3.21110 0.197256
\(266\) −2.60555 −0.159757
\(267\) −5.39445 −0.330135
\(268\) 3.21110 0.196149
\(269\) 25.0278 1.52597 0.762985 0.646417i \(-0.223733\pi\)
0.762985 + 0.646417i \(0.223733\pi\)
\(270\) 1.00000 0.0608581
\(271\) −23.8167 −1.44676 −0.723379 0.690451i \(-0.757412\pi\)
−0.723379 + 0.690451i \(0.757412\pi\)
\(272\) 4.60555 0.279253
\(273\) 2.60555 0.157695
\(274\) 18.0000 1.08742
\(275\) −2.00000 −0.120605
\(276\) 1.00000 0.0601929
\(277\) 13.2111 0.793778 0.396889 0.917867i \(-0.370090\pi\)
0.396889 + 0.917867i \(0.370090\pi\)
\(278\) 12.0000 0.719712
\(279\) 2.60555 0.155990
\(280\) −1.00000 −0.0597614
\(281\) −10.7889 −0.643612 −0.321806 0.946806i \(-0.604290\pi\)
−0.321806 + 0.946806i \(0.604290\pi\)
\(282\) 1.39445 0.0830382
\(283\) −24.2389 −1.44085 −0.720425 0.693533i \(-0.756053\pi\)
−0.720425 + 0.693533i \(0.756053\pi\)
\(284\) −6.00000 −0.356034
\(285\) 2.60555 0.154340
\(286\) −5.21110 −0.308139
\(287\) 6.00000 0.354169
\(288\) −1.00000 −0.0589256
\(289\) 4.21110 0.247712
\(290\) 5.21110 0.306006
\(291\) −11.8167 −0.692705
\(292\) −2.00000 −0.117041
\(293\) 21.6333 1.26383 0.631916 0.775037i \(-0.282269\pi\)
0.631916 + 0.775037i \(0.282269\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 5.21110 0.303402
\(296\) 6.00000 0.348743
\(297\) −2.00000 −0.116052
\(298\) 14.0000 0.810998
\(299\) −2.60555 −0.150683
\(300\) 1.00000 0.0577350
\(301\) −11.2111 −0.646197
\(302\) −17.2111 −0.990388
\(303\) −15.8167 −0.908643
\(304\) −2.60555 −0.149439
\(305\) −10.0000 −0.572598
\(306\) −4.60555 −0.263282
\(307\) −1.57779 −0.0900495 −0.0450248 0.998986i \(-0.514337\pi\)
−0.0450248 + 0.998986i \(0.514337\pi\)
\(308\) 2.00000 0.113961
\(309\) 8.00000 0.455104
\(310\) 2.60555 0.147985
\(311\) −11.0278 −0.625327 −0.312663 0.949864i \(-0.601221\pi\)
−0.312663 + 0.949864i \(0.601221\pi\)
\(312\) 2.60555 0.147510
\(313\) 13.3944 0.757099 0.378550 0.925581i \(-0.376423\pi\)
0.378550 + 0.925581i \(0.376423\pi\)
\(314\) −8.42221 −0.475293
\(315\) 1.00000 0.0563436
\(316\) −14.4222 −0.811312
\(317\) 8.78890 0.493634 0.246817 0.969062i \(-0.420615\pi\)
0.246817 + 0.969062i \(0.420615\pi\)
\(318\) 3.21110 0.180070
\(319\) −10.4222 −0.583531
\(320\) −1.00000 −0.0559017
\(321\) −9.21110 −0.514114
\(322\) 1.00000 0.0557278
\(323\) −12.0000 −0.667698
\(324\) 1.00000 0.0555556
\(325\) −2.60555 −0.144530
\(326\) −19.6333 −1.08739
\(327\) 16.4222 0.908150
\(328\) 6.00000 0.331295
\(329\) 1.39445 0.0768784
\(330\) −2.00000 −0.110096
\(331\) −14.4222 −0.792716 −0.396358 0.918096i \(-0.629726\pi\)
−0.396358 + 0.918096i \(0.629726\pi\)
\(332\) −10.6056 −0.582055
\(333\) −6.00000 −0.328798
\(334\) −13.3944 −0.732912
\(335\) −3.21110 −0.175441
\(336\) −1.00000 −0.0545545
\(337\) −10.4222 −0.567734 −0.283867 0.958864i \(-0.591617\pi\)
−0.283867 + 0.958864i \(0.591617\pi\)
\(338\) 6.21110 0.337839
\(339\) 0.788897 0.0428470
\(340\) −4.60555 −0.249771
\(341\) −5.21110 −0.282197
\(342\) 2.60555 0.140892
\(343\) −1.00000 −0.0539949
\(344\) −11.2111 −0.604462
\(345\) −1.00000 −0.0538382
\(346\) 19.0278 1.02294
\(347\) 18.4222 0.988956 0.494478 0.869190i \(-0.335359\pi\)
0.494478 + 0.869190i \(0.335359\pi\)
\(348\) 5.21110 0.279344
\(349\) 28.2389 1.51159 0.755796 0.654807i \(-0.227250\pi\)
0.755796 + 0.654807i \(0.227250\pi\)
\(350\) 1.00000 0.0534522
\(351\) −2.60555 −0.139074
\(352\) 2.00000 0.106600
\(353\) −7.21110 −0.383808 −0.191904 0.981414i \(-0.561466\pi\)
−0.191904 + 0.981414i \(0.561466\pi\)
\(354\) 5.21110 0.276967
\(355\) 6.00000 0.318447
\(356\) −5.39445 −0.285905
\(357\) −4.60555 −0.243752
\(358\) 18.4222 0.973644
\(359\) 14.7889 0.780528 0.390264 0.920703i \(-0.372384\pi\)
0.390264 + 0.920703i \(0.372384\pi\)
\(360\) 1.00000 0.0527046
\(361\) −12.2111 −0.642690
\(362\) 14.0000 0.735824
\(363\) −7.00000 −0.367405
\(364\) 2.60555 0.136568
\(365\) 2.00000 0.104685
\(366\) −10.0000 −0.522708
\(367\) 5.57779 0.291159 0.145579 0.989347i \(-0.453495\pi\)
0.145579 + 0.989347i \(0.453495\pi\)
\(368\) 1.00000 0.0521286
\(369\) −6.00000 −0.312348
\(370\) −6.00000 −0.311925
\(371\) 3.21110 0.166712
\(372\) 2.60555 0.135092
\(373\) 20.4222 1.05742 0.528711 0.848802i \(-0.322676\pi\)
0.528711 + 0.848802i \(0.322676\pi\)
\(374\) 9.21110 0.476295
\(375\) −1.00000 −0.0516398
\(376\) 1.39445 0.0719132
\(377\) −13.5778 −0.699292
\(378\) 1.00000 0.0514344
\(379\) 35.6333 1.83036 0.915180 0.403045i \(-0.132048\pi\)
0.915180 + 0.403045i \(0.132048\pi\)
\(380\) 2.60555 0.133662
\(381\) −2.00000 −0.102463
\(382\) 6.78890 0.347350
\(383\) −5.21110 −0.266275 −0.133137 0.991098i \(-0.542505\pi\)
−0.133137 + 0.991098i \(0.542505\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −2.00000 −0.101929
\(386\) 12.4222 0.632274
\(387\) 11.2111 0.569892
\(388\) −11.8167 −0.599900
\(389\) 28.4222 1.44106 0.720532 0.693422i \(-0.243898\pi\)
0.720532 + 0.693422i \(0.243898\pi\)
\(390\) −2.60555 −0.131937
\(391\) 4.60555 0.232913
\(392\) −1.00000 −0.0505076
\(393\) −12.0000 −0.605320
\(394\) 8.78890 0.442778
\(395\) 14.4222 0.725660
\(396\) −2.00000 −0.100504
\(397\) 13.3944 0.672248 0.336124 0.941818i \(-0.390884\pi\)
0.336124 + 0.941818i \(0.390884\pi\)
\(398\) −26.4222 −1.32443
\(399\) 2.60555 0.130441
\(400\) 1.00000 0.0500000
\(401\) 1.21110 0.0604796 0.0302398 0.999543i \(-0.490373\pi\)
0.0302398 + 0.999543i \(0.490373\pi\)
\(402\) −3.21110 −0.160155
\(403\) −6.78890 −0.338179
\(404\) −15.8167 −0.786908
\(405\) −1.00000 −0.0496904
\(406\) 5.21110 0.258623
\(407\) 12.0000 0.594818
\(408\) −4.60555 −0.228009
\(409\) −17.6333 −0.871911 −0.435955 0.899968i \(-0.643589\pi\)
−0.435955 + 0.899968i \(0.643589\pi\)
\(410\) −6.00000 −0.296319
\(411\) −18.0000 −0.887875
\(412\) 8.00000 0.394132
\(413\) 5.21110 0.256422
\(414\) −1.00000 −0.0491473
\(415\) 10.6056 0.520606
\(416\) 2.60555 0.127748
\(417\) −12.0000 −0.587643
\(418\) −5.21110 −0.254883
\(419\) −31.3944 −1.53372 −0.766860 0.641815i \(-0.778182\pi\)
−0.766860 + 0.641815i \(0.778182\pi\)
\(420\) 1.00000 0.0487950
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) −10.4222 −0.507345
\(423\) −1.39445 −0.0678004
\(424\) 3.21110 0.155945
\(425\) 4.60555 0.223402
\(426\) 6.00000 0.290701
\(427\) −10.0000 −0.483934
\(428\) −9.21110 −0.445235
\(429\) 5.21110 0.251594
\(430\) 11.2111 0.540647
\(431\) −10.4222 −0.502020 −0.251010 0.967984i \(-0.580763\pi\)
−0.251010 + 0.967984i \(0.580763\pi\)
\(432\) 1.00000 0.0481125
\(433\) −35.8167 −1.72124 −0.860619 0.509249i \(-0.829923\pi\)
−0.860619 + 0.509249i \(0.829923\pi\)
\(434\) 2.60555 0.125070
\(435\) −5.21110 −0.249853
\(436\) 16.4222 0.786481
\(437\) −2.60555 −0.124640
\(438\) 2.00000 0.0955637
\(439\) 14.2389 0.679584 0.339792 0.940501i \(-0.389643\pi\)
0.339792 + 0.940501i \(0.389643\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) 5.57779 0.265009 0.132505 0.991182i \(-0.457698\pi\)
0.132505 + 0.991182i \(0.457698\pi\)
\(444\) −6.00000 −0.284747
\(445\) 5.39445 0.255721
\(446\) 11.3944 0.539543
\(447\) −14.0000 −0.662177
\(448\) −1.00000 −0.0472456
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 12.0000 0.565058
\(452\) 0.788897 0.0371066
\(453\) 17.2111 0.808648
\(454\) −22.6056 −1.06093
\(455\) −2.60555 −0.122150
\(456\) 2.60555 0.122016
\(457\) 27.6333 1.29263 0.646316 0.763070i \(-0.276309\pi\)
0.646316 + 0.763070i \(0.276309\pi\)
\(458\) 23.2111 1.08458
\(459\) 4.60555 0.214969
\(460\) −1.00000 −0.0466252
\(461\) −23.8167 −1.10925 −0.554626 0.832100i \(-0.687139\pi\)
−0.554626 + 0.832100i \(0.687139\pi\)
\(462\) −2.00000 −0.0930484
\(463\) −29.6333 −1.37718 −0.688588 0.725153i \(-0.741769\pi\)
−0.688588 + 0.725153i \(0.741769\pi\)
\(464\) 5.21110 0.241919
\(465\) −2.60555 −0.120830
\(466\) 22.0000 1.01913
\(467\) 29.0278 1.34324 0.671622 0.740894i \(-0.265598\pi\)
0.671622 + 0.740894i \(0.265598\pi\)
\(468\) −2.60555 −0.120442
\(469\) −3.21110 −0.148275
\(470\) −1.39445 −0.0643211
\(471\) 8.42221 0.388075
\(472\) 5.21110 0.239860
\(473\) −22.4222 −1.03097
\(474\) 14.4222 0.662434
\(475\) −2.60555 −0.119551
\(476\) −4.60555 −0.211095
\(477\) −3.21110 −0.147026
\(478\) 6.00000 0.274434
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 1.00000 0.0456435
\(481\) 15.6333 0.712817
\(482\) 21.8167 0.993721
\(483\) −1.00000 −0.0455016
\(484\) −7.00000 −0.318182
\(485\) 11.8167 0.536567
\(486\) −1.00000 −0.0453609
\(487\) −40.0555 −1.81509 −0.907544 0.419956i \(-0.862045\pi\)
−0.907544 + 0.419956i \(0.862045\pi\)
\(488\) −10.0000 −0.452679
\(489\) 19.6333 0.887849
\(490\) 1.00000 0.0451754
\(491\) −18.7889 −0.847931 −0.423966 0.905678i \(-0.639362\pi\)
−0.423966 + 0.905678i \(0.639362\pi\)
\(492\) −6.00000 −0.270501
\(493\) 24.0000 1.08091
\(494\) −6.78890 −0.305447
\(495\) 2.00000 0.0898933
\(496\) 2.60555 0.116993
\(497\) 6.00000 0.269137
\(498\) 10.6056 0.475246
\(499\) −28.8444 −1.29125 −0.645627 0.763653i \(-0.723404\pi\)
−0.645627 + 0.763653i \(0.723404\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 13.3944 0.598420
\(502\) 11.3944 0.508559
\(503\) −34.4222 −1.53481 −0.767405 0.641163i \(-0.778452\pi\)
−0.767405 + 0.641163i \(0.778452\pi\)
\(504\) 1.00000 0.0445435
\(505\) 15.8167 0.703832
\(506\) 2.00000 0.0889108
\(507\) −6.21110 −0.275845
\(508\) −2.00000 −0.0887357
\(509\) −23.8167 −1.05565 −0.527827 0.849352i \(-0.676993\pi\)
−0.527827 + 0.849352i \(0.676993\pi\)
\(510\) 4.60555 0.203937
\(511\) 2.00000 0.0884748
\(512\) −1.00000 −0.0441942
\(513\) −2.60555 −0.115038
\(514\) −25.6333 −1.13064
\(515\) −8.00000 −0.352522
\(516\) 11.2111 0.493541
\(517\) 2.78890 0.122656
\(518\) −6.00000 −0.263625
\(519\) −19.0278 −0.835226
\(520\) −2.60555 −0.114261
\(521\) −7.81665 −0.342454 −0.171227 0.985232i \(-0.554773\pi\)
−0.171227 + 0.985232i \(0.554773\pi\)
\(522\) −5.21110 −0.228084
\(523\) −24.2389 −1.05989 −0.529946 0.848032i \(-0.677788\pi\)
−0.529946 + 0.848032i \(0.677788\pi\)
\(524\) −12.0000 −0.524222
\(525\) −1.00000 −0.0436436
\(526\) −10.4222 −0.454430
\(527\) 12.0000 0.522728
\(528\) −2.00000 −0.0870388
\(529\) 1.00000 0.0434783
\(530\) −3.21110 −0.139481
\(531\) −5.21110 −0.226143
\(532\) 2.60555 0.112965
\(533\) 15.6333 0.677154
\(534\) 5.39445 0.233441
\(535\) 9.21110 0.398231
\(536\) −3.21110 −0.138699
\(537\) −18.4222 −0.794977
\(538\) −25.0278 −1.07902
\(539\) −2.00000 −0.0861461
\(540\) −1.00000 −0.0430331
\(541\) −18.8444 −0.810184 −0.405092 0.914276i \(-0.632761\pi\)
−0.405092 + 0.914276i \(0.632761\pi\)
\(542\) 23.8167 1.02301
\(543\) −14.0000 −0.600798
\(544\) −4.60555 −0.197461
\(545\) −16.4222 −0.703450
\(546\) −2.60555 −0.111507
\(547\) 14.4222 0.616649 0.308324 0.951281i \(-0.400232\pi\)
0.308324 + 0.951281i \(0.400232\pi\)
\(548\) −18.0000 −0.768922
\(549\) 10.0000 0.426790
\(550\) 2.00000 0.0852803
\(551\) −13.5778 −0.578434
\(552\) −1.00000 −0.0425628
\(553\) 14.4222 0.613295
\(554\) −13.2111 −0.561286
\(555\) 6.00000 0.254686
\(556\) −12.0000 −0.508913
\(557\) −16.4222 −0.695831 −0.347916 0.937526i \(-0.613110\pi\)
−0.347916 + 0.937526i \(0.613110\pi\)
\(558\) −2.60555 −0.110302
\(559\) −29.2111 −1.23550
\(560\) 1.00000 0.0422577
\(561\) −9.21110 −0.388893
\(562\) 10.7889 0.455102
\(563\) 31.8167 1.34091 0.670456 0.741949i \(-0.266098\pi\)
0.670456 + 0.741949i \(0.266098\pi\)
\(564\) −1.39445 −0.0587169
\(565\) −0.788897 −0.0331892
\(566\) 24.2389 1.01884
\(567\) −1.00000 −0.0419961
\(568\) 6.00000 0.251754
\(569\) 14.7889 0.619983 0.309991 0.950739i \(-0.399674\pi\)
0.309991 + 0.950739i \(0.399674\pi\)
\(570\) −2.60555 −0.109135
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 5.21110 0.217887
\(573\) −6.78890 −0.283610
\(574\) −6.00000 −0.250435
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) 0.788897 0.0328422 0.0164211 0.999865i \(-0.494773\pi\)
0.0164211 + 0.999865i \(0.494773\pi\)
\(578\) −4.21110 −0.175159
\(579\) −12.4222 −0.516249
\(580\) −5.21110 −0.216379
\(581\) 10.6056 0.439992
\(582\) 11.8167 0.489816
\(583\) 6.42221 0.265981
\(584\) 2.00000 0.0827606
\(585\) 2.60555 0.107726
\(586\) −21.6333 −0.893664
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 1.00000 0.0412393
\(589\) −6.78890 −0.279732
\(590\) −5.21110 −0.214538
\(591\) −8.78890 −0.361527
\(592\) −6.00000 −0.246598
\(593\) 28.0555 1.15210 0.576051 0.817414i \(-0.304593\pi\)
0.576051 + 0.817414i \(0.304593\pi\)
\(594\) 2.00000 0.0820610
\(595\) 4.60555 0.188809
\(596\) −14.0000 −0.573462
\(597\) 26.4222 1.08139
\(598\) 2.60555 0.106549
\(599\) −15.2111 −0.621509 −0.310754 0.950490i \(-0.600582\pi\)
−0.310754 + 0.950490i \(0.600582\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 11.2111 0.456930
\(603\) 3.21110 0.130766
\(604\) 17.2111 0.700310
\(605\) 7.00000 0.284590
\(606\) 15.8167 0.642508
\(607\) 1.81665 0.0737357 0.0368679 0.999320i \(-0.488262\pi\)
0.0368679 + 0.999320i \(0.488262\pi\)
\(608\) 2.60555 0.105669
\(609\) −5.21110 −0.211165
\(610\) 10.0000 0.404888
\(611\) 3.63331 0.146988
\(612\) 4.60555 0.186168
\(613\) 41.2666 1.66674 0.833371 0.552713i \(-0.186408\pi\)
0.833371 + 0.552713i \(0.186408\pi\)
\(614\) 1.57779 0.0636746
\(615\) 6.00000 0.241943
\(616\) −2.00000 −0.0805823
\(617\) −26.8444 −1.08072 −0.540358 0.841435i \(-0.681711\pi\)
−0.540358 + 0.841435i \(0.681711\pi\)
\(618\) −8.00000 −0.321807
\(619\) 31.4500 1.26408 0.632040 0.774935i \(-0.282218\pi\)
0.632040 + 0.774935i \(0.282218\pi\)
\(620\) −2.60555 −0.104641
\(621\) 1.00000 0.0401286
\(622\) 11.0278 0.442173
\(623\) 5.39445 0.216124
\(624\) −2.60555 −0.104306
\(625\) 1.00000 0.0400000
\(626\) −13.3944 −0.535350
\(627\) 5.21110 0.208111
\(628\) 8.42221 0.336083
\(629\) −27.6333 −1.10181
\(630\) −1.00000 −0.0398410
\(631\) −9.57779 −0.381286 −0.190643 0.981659i \(-0.561057\pi\)
−0.190643 + 0.981659i \(0.561057\pi\)
\(632\) 14.4222 0.573685
\(633\) 10.4222 0.414245
\(634\) −8.78890 −0.349052
\(635\) 2.00000 0.0793676
\(636\) −3.21110 −0.127328
\(637\) −2.60555 −0.103236
\(638\) 10.4222 0.412619
\(639\) −6.00000 −0.237356
\(640\) 1.00000 0.0395285
\(641\) 15.6333 0.617479 0.308739 0.951147i \(-0.400093\pi\)
0.308739 + 0.951147i \(0.400093\pi\)
\(642\) 9.21110 0.363533
\(643\) −37.4500 −1.47688 −0.738441 0.674318i \(-0.764438\pi\)
−0.738441 + 0.674318i \(0.764438\pi\)
\(644\) −1.00000 −0.0394055
\(645\) −11.2111 −0.441437
\(646\) 12.0000 0.472134
\(647\) 6.23886 0.245275 0.122637 0.992452i \(-0.460865\pi\)
0.122637 + 0.992452i \(0.460865\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 10.4222 0.409107
\(650\) 2.60555 0.102198
\(651\) −2.60555 −0.102120
\(652\) 19.6333 0.768900
\(653\) −26.0000 −1.01746 −0.508729 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) −16.4222 −0.642159
\(655\) 12.0000 0.468879
\(656\) −6.00000 −0.234261
\(657\) −2.00000 −0.0780274
\(658\) −1.39445 −0.0543613
\(659\) 24.7889 0.965638 0.482819 0.875720i \(-0.339613\pi\)
0.482819 + 0.875720i \(0.339613\pi\)
\(660\) 2.00000 0.0778499
\(661\) −7.21110 −0.280479 −0.140240 0.990118i \(-0.544787\pi\)
−0.140240 + 0.990118i \(0.544787\pi\)
\(662\) 14.4222 0.560535
\(663\) −12.0000 −0.466041
\(664\) 10.6056 0.411575
\(665\) −2.60555 −0.101039
\(666\) 6.00000 0.232495
\(667\) 5.21110 0.201775
\(668\) 13.3944 0.518247
\(669\) −11.3944 −0.440535
\(670\) 3.21110 0.124056
\(671\) −20.0000 −0.772091
\(672\) 1.00000 0.0385758
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 10.4222 0.401448
\(675\) 1.00000 0.0384900
\(676\) −6.21110 −0.238889
\(677\) −0.422205 −0.0162267 −0.00811333 0.999967i \(-0.502583\pi\)
−0.00811333 + 0.999967i \(0.502583\pi\)
\(678\) −0.788897 −0.0302974
\(679\) 11.8167 0.453482
\(680\) 4.60555 0.176615
\(681\) 22.6056 0.866247
\(682\) 5.21110 0.199543
\(683\) 18.4222 0.704906 0.352453 0.935829i \(-0.385348\pi\)
0.352453 + 0.935829i \(0.385348\pi\)
\(684\) −2.60555 −0.0996257
\(685\) 18.0000 0.687745
\(686\) 1.00000 0.0381802
\(687\) −23.2111 −0.885559
\(688\) 11.2111 0.427419
\(689\) 8.36669 0.318746
\(690\) 1.00000 0.0380693
\(691\) 19.6333 0.746886 0.373443 0.927653i \(-0.378177\pi\)
0.373443 + 0.927653i \(0.378177\pi\)
\(692\) −19.0278 −0.723327
\(693\) 2.00000 0.0759737
\(694\) −18.4222 −0.699297
\(695\) 12.0000 0.455186
\(696\) −5.21110 −0.197526
\(697\) −27.6333 −1.04669
\(698\) −28.2389 −1.06886
\(699\) −22.0000 −0.832116
\(700\) −1.00000 −0.0377964
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 2.60555 0.0983402
\(703\) 15.6333 0.589621
\(704\) −2.00000 −0.0753778
\(705\) 1.39445 0.0525180
\(706\) 7.21110 0.271393
\(707\) 15.8167 0.594846
\(708\) −5.21110 −0.195845
\(709\) 20.7889 0.780743 0.390372 0.920657i \(-0.372346\pi\)
0.390372 + 0.920657i \(0.372346\pi\)
\(710\) −6.00000 −0.225176
\(711\) −14.4222 −0.540875
\(712\) 5.39445 0.202166
\(713\) 2.60555 0.0975787
\(714\) 4.60555 0.172358
\(715\) −5.21110 −0.194884
\(716\) −18.4222 −0.688470
\(717\) −6.00000 −0.224074
\(718\) −14.7889 −0.551917
\(719\) −7.39445 −0.275766 −0.137883 0.990448i \(-0.544030\pi\)
−0.137883 + 0.990448i \(0.544030\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −8.00000 −0.297936
\(722\) 12.2111 0.454450
\(723\) −21.8167 −0.811370
\(724\) −14.0000 −0.520306
\(725\) 5.21110 0.193535
\(726\) 7.00000 0.259794
\(727\) 33.2111 1.23173 0.615866 0.787851i \(-0.288806\pi\)
0.615866 + 0.787851i \(0.288806\pi\)
\(728\) −2.60555 −0.0965682
\(729\) 1.00000 0.0370370
\(730\) −2.00000 −0.0740233
\(731\) 51.6333 1.90973
\(732\) 10.0000 0.369611
\(733\) −1.63331 −0.0603276 −0.0301638 0.999545i \(-0.509603\pi\)
−0.0301638 + 0.999545i \(0.509603\pi\)
\(734\) −5.57779 −0.205880
\(735\) −1.00000 −0.0368856
\(736\) −1.00000 −0.0368605
\(737\) −6.42221 −0.236565
\(738\) 6.00000 0.220863
\(739\) −16.0000 −0.588570 −0.294285 0.955718i \(-0.595081\pi\)
−0.294285 + 0.955718i \(0.595081\pi\)
\(740\) 6.00000 0.220564
\(741\) 6.78890 0.249396
\(742\) −3.21110 −0.117883
\(743\) −30.4222 −1.11608 −0.558041 0.829813i \(-0.688447\pi\)
−0.558041 + 0.829813i \(0.688447\pi\)
\(744\) −2.60555 −0.0955241
\(745\) 14.0000 0.512920
\(746\) −20.4222 −0.747710
\(747\) −10.6056 −0.388037
\(748\) −9.21110 −0.336791
\(749\) 9.21110 0.336566
\(750\) 1.00000 0.0365148
\(751\) 21.5778 0.787385 0.393692 0.919242i \(-0.371197\pi\)
0.393692 + 0.919242i \(0.371197\pi\)
\(752\) −1.39445 −0.0508503
\(753\) −11.3944 −0.415237
\(754\) 13.5778 0.494474
\(755\) −17.2111 −0.626376
\(756\) −1.00000 −0.0363696
\(757\) −18.8444 −0.684912 −0.342456 0.939534i \(-0.611259\pi\)
−0.342456 + 0.939534i \(0.611259\pi\)
\(758\) −35.6333 −1.29426
\(759\) −2.00000 −0.0725954
\(760\) −2.60555 −0.0945133
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 2.00000 0.0724524
\(763\) −16.4222 −0.594524
\(764\) −6.78890 −0.245614
\(765\) −4.60555 −0.166514
\(766\) 5.21110 0.188285
\(767\) 13.5778 0.490266
\(768\) 1.00000 0.0360844
\(769\) 47.8722 1.72631 0.863157 0.504935i \(-0.168484\pi\)
0.863157 + 0.504935i \(0.168484\pi\)
\(770\) 2.00000 0.0720750
\(771\) 25.6333 0.923161
\(772\) −12.4222 −0.447085
\(773\) −46.8444 −1.68488 −0.842438 0.538793i \(-0.818880\pi\)
−0.842438 + 0.538793i \(0.818880\pi\)
\(774\) −11.2111 −0.402975
\(775\) 2.60555 0.0935942
\(776\) 11.8167 0.424193
\(777\) 6.00000 0.215249
\(778\) −28.4222 −1.01899
\(779\) 15.6333 0.560121
\(780\) 2.60555 0.0932937
\(781\) 12.0000 0.429394
\(782\) −4.60555 −0.164694
\(783\) 5.21110 0.186230
\(784\) 1.00000 0.0357143
\(785\) −8.42221 −0.300601
\(786\) 12.0000 0.428026
\(787\) −11.0278 −0.393097 −0.196549 0.980494i \(-0.562973\pi\)
−0.196549 + 0.980494i \(0.562973\pi\)
\(788\) −8.78890 −0.313092
\(789\) 10.4222 0.371040
\(790\) −14.4222 −0.513119
\(791\) −0.788897 −0.0280500
\(792\) 2.00000 0.0710669
\(793\) −26.0555 −0.925258
\(794\) −13.3944 −0.475351
\(795\) 3.21110 0.113886
\(796\) 26.4222 0.936510
\(797\) 31.2111 1.10555 0.552777 0.833329i \(-0.313568\pi\)
0.552777 + 0.833329i \(0.313568\pi\)
\(798\) −2.60555 −0.0922355
\(799\) −6.42221 −0.227201
\(800\) −1.00000 −0.0353553
\(801\) −5.39445 −0.190603
\(802\) −1.21110 −0.0427655
\(803\) 4.00000 0.141157
\(804\) 3.21110 0.113247
\(805\) 1.00000 0.0352454
\(806\) 6.78890 0.239129
\(807\) 25.0278 0.881019
\(808\) 15.8167 0.556428
\(809\) −54.8444 −1.92823 −0.964113 0.265491i \(-0.914466\pi\)
−0.964113 + 0.265491i \(0.914466\pi\)
\(810\) 1.00000 0.0351364
\(811\) −46.4222 −1.63010 −0.815052 0.579388i \(-0.803292\pi\)
−0.815052 + 0.579388i \(0.803292\pi\)
\(812\) −5.21110 −0.182874
\(813\) −23.8167 −0.835287
\(814\) −12.0000 −0.420600
\(815\) −19.6333 −0.687725
\(816\) 4.60555 0.161227
\(817\) −29.2111 −1.02197
\(818\) 17.6333 0.616534
\(819\) 2.60555 0.0910453
\(820\) 6.00000 0.209529
\(821\) 8.84441 0.308672 0.154336 0.988018i \(-0.450676\pi\)
0.154336 + 0.988018i \(0.450676\pi\)
\(822\) 18.0000 0.627822
\(823\) 11.5778 0.403577 0.201788 0.979429i \(-0.435325\pi\)
0.201788 + 0.979429i \(0.435325\pi\)
\(824\) −8.00000 −0.278693
\(825\) −2.00000 −0.0696311
\(826\) −5.21110 −0.181317
\(827\) −3.63331 −0.126342 −0.0631712 0.998003i \(-0.520121\pi\)
−0.0631712 + 0.998003i \(0.520121\pi\)
\(828\) 1.00000 0.0347524
\(829\) −50.6611 −1.75953 −0.879766 0.475407i \(-0.842301\pi\)
−0.879766 + 0.475407i \(0.842301\pi\)
\(830\) −10.6056 −0.368124
\(831\) 13.2111 0.458288
\(832\) −2.60555 −0.0903312
\(833\) 4.60555 0.159573
\(834\) 12.0000 0.415526
\(835\) −13.3944 −0.463534
\(836\) 5.21110 0.180230
\(837\) 2.60555 0.0900610
\(838\) 31.3944 1.08450
\(839\) −27.6333 −0.954008 −0.477004 0.878901i \(-0.658277\pi\)
−0.477004 + 0.878901i \(0.658277\pi\)
\(840\) −1.00000 −0.0345033
\(841\) −1.84441 −0.0636004
\(842\) −18.0000 −0.620321
\(843\) −10.7889 −0.371589
\(844\) 10.4222 0.358747
\(845\) 6.21110 0.213668
\(846\) 1.39445 0.0479421
\(847\) 7.00000 0.240523
\(848\) −3.21110 −0.110270
\(849\) −24.2389 −0.831875
\(850\) −4.60555 −0.157969
\(851\) −6.00000 −0.205677
\(852\) −6.00000 −0.205557
\(853\) −21.0278 −0.719977 −0.359988 0.932957i \(-0.617219\pi\)
−0.359988 + 0.932957i \(0.617219\pi\)
\(854\) 10.0000 0.342193
\(855\) 2.60555 0.0891080
\(856\) 9.21110 0.314829
\(857\) 24.7889 0.846773 0.423386 0.905949i \(-0.360841\pi\)
0.423386 + 0.905949i \(0.360841\pi\)
\(858\) −5.21110 −0.177904
\(859\) 43.6333 1.48875 0.744375 0.667762i \(-0.232748\pi\)
0.744375 + 0.667762i \(0.232748\pi\)
\(860\) −11.2111 −0.382295
\(861\) 6.00000 0.204479
\(862\) 10.4222 0.354982
\(863\) 54.0555 1.84007 0.920036 0.391835i \(-0.128160\pi\)
0.920036 + 0.391835i \(0.128160\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 19.0278 0.646963
\(866\) 35.8167 1.21710
\(867\) 4.21110 0.143017
\(868\) −2.60555 −0.0884382
\(869\) 28.8444 0.978480
\(870\) 5.21110 0.176673
\(871\) −8.36669 −0.283495
\(872\) −16.4222 −0.556126
\(873\) −11.8167 −0.399933
\(874\) 2.60555 0.0881341
\(875\) 1.00000 0.0338062
\(876\) −2.00000 −0.0675737
\(877\) 20.8444 0.703866 0.351933 0.936025i \(-0.385525\pi\)
0.351933 + 0.936025i \(0.385525\pi\)
\(878\) −14.2389 −0.480538
\(879\) 21.6333 0.729673
\(880\) 2.00000 0.0674200
\(881\) −16.1833 −0.545231 −0.272615 0.962123i \(-0.587889\pi\)
−0.272615 + 0.962123i \(0.587889\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 38.0555 1.28067 0.640335 0.768096i \(-0.278796\pi\)
0.640335 + 0.768096i \(0.278796\pi\)
\(884\) −12.0000 −0.403604
\(885\) 5.21110 0.175169
\(886\) −5.57779 −0.187390
\(887\) −35.4500 −1.19029 −0.595147 0.803617i \(-0.702906\pi\)
−0.595147 + 0.803617i \(0.702906\pi\)
\(888\) 6.00000 0.201347
\(889\) 2.00000 0.0670778
\(890\) −5.39445 −0.180822
\(891\) −2.00000 −0.0670025
\(892\) −11.3944 −0.381514
\(893\) 3.63331 0.121584
\(894\) 14.0000 0.468230
\(895\) 18.4222 0.615786
\(896\) 1.00000 0.0334077
\(897\) −2.60555 −0.0869968
\(898\) 6.00000 0.200223
\(899\) 13.5778 0.452845
\(900\) 1.00000 0.0333333
\(901\) −14.7889 −0.492690
\(902\) −12.0000 −0.399556
\(903\) −11.2111 −0.373082
\(904\) −0.788897 −0.0262383
\(905\) 14.0000 0.465376
\(906\) −17.2111 −0.571801
\(907\) −2.00000 −0.0664089 −0.0332045 0.999449i \(-0.510571\pi\)
−0.0332045 + 0.999449i \(0.510571\pi\)
\(908\) 22.6056 0.750192
\(909\) −15.8167 −0.524605
\(910\) 2.60555 0.0863732
\(911\) −7.63331 −0.252903 −0.126451 0.991973i \(-0.540359\pi\)
−0.126451 + 0.991973i \(0.540359\pi\)
\(912\) −2.60555 −0.0862784
\(913\) 21.2111 0.701985
\(914\) −27.6333 −0.914029
\(915\) −10.0000 −0.330590
\(916\) −23.2111 −0.766916
\(917\) 12.0000 0.396275
\(918\) −4.60555 −0.152006
\(919\) 46.4222 1.53133 0.765664 0.643241i \(-0.222411\pi\)
0.765664 + 0.643241i \(0.222411\pi\)
\(920\) 1.00000 0.0329690
\(921\) −1.57779 −0.0519901
\(922\) 23.8167 0.784360
\(923\) 15.6333 0.514577
\(924\) 2.00000 0.0657952
\(925\) −6.00000 −0.197279
\(926\) 29.6333 0.973811
\(927\) 8.00000 0.262754
\(928\) −5.21110 −0.171063
\(929\) −21.6333 −0.709766 −0.354883 0.934911i \(-0.615479\pi\)
−0.354883 + 0.934911i \(0.615479\pi\)
\(930\) 2.60555 0.0854394
\(931\) −2.60555 −0.0853935
\(932\) −22.0000 −0.720634
\(933\) −11.0278 −0.361033
\(934\) −29.0278 −0.949817
\(935\) 9.21110 0.301235
\(936\) 2.60555 0.0851651
\(937\) −12.1833 −0.398013 −0.199006 0.979998i \(-0.563771\pi\)
−0.199006 + 0.979998i \(0.563771\pi\)
\(938\) 3.21110 0.104846
\(939\) 13.3944 0.437111
\(940\) 1.39445 0.0454819
\(941\) −37.2666 −1.21486 −0.607428 0.794374i \(-0.707799\pi\)
−0.607428 + 0.794374i \(0.707799\pi\)
\(942\) −8.42221 −0.274410
\(943\) −6.00000 −0.195387
\(944\) −5.21110 −0.169607
\(945\) 1.00000 0.0325300
\(946\) 22.4222 0.729009
\(947\) 21.5778 0.701184 0.350592 0.936528i \(-0.385980\pi\)
0.350592 + 0.936528i \(0.385980\pi\)
\(948\) −14.4222 −0.468411
\(949\) 5.21110 0.169160
\(950\) 2.60555 0.0845352
\(951\) 8.78890 0.285000
\(952\) 4.60555 0.149267
\(953\) −41.6333 −1.34864 −0.674318 0.738441i \(-0.735562\pi\)
−0.674318 + 0.738441i \(0.735562\pi\)
\(954\) 3.21110 0.103963
\(955\) 6.78890 0.219684
\(956\) −6.00000 −0.194054
\(957\) −10.4222 −0.336902
\(958\) 24.0000 0.775405
\(959\) 18.0000 0.581250
\(960\) −1.00000 −0.0322749
\(961\) −24.2111 −0.781003
\(962\) −15.6333 −0.504038
\(963\) −9.21110 −0.296824
\(964\) −21.8167 −0.702667
\(965\) 12.4222 0.399885
\(966\) 1.00000 0.0321745
\(967\) 28.7889 0.925789 0.462894 0.886414i \(-0.346811\pi\)
0.462894 + 0.886414i \(0.346811\pi\)
\(968\) 7.00000 0.224989
\(969\) −12.0000 −0.385496
\(970\) −11.8167 −0.379410
\(971\) 10.1833 0.326799 0.163400 0.986560i \(-0.447754\pi\)
0.163400 + 0.986560i \(0.447754\pi\)
\(972\) 1.00000 0.0320750
\(973\) 12.0000 0.384702
\(974\) 40.0555 1.28346
\(975\) −2.60555 −0.0834444
\(976\) 10.0000 0.320092
\(977\) −24.4222 −0.781336 −0.390668 0.920532i \(-0.627756\pi\)
−0.390668 + 0.920532i \(0.627756\pi\)
\(978\) −19.6333 −0.627804
\(979\) 10.7889 0.344815
\(980\) −1.00000 −0.0319438
\(981\) 16.4222 0.524321
\(982\) 18.7889 0.599578
\(983\) 53.2111 1.69717 0.848585 0.529059i \(-0.177455\pi\)
0.848585 + 0.529059i \(0.177455\pi\)
\(984\) 6.00000 0.191273
\(985\) 8.78890 0.280038
\(986\) −24.0000 −0.764316
\(987\) 1.39445 0.0443858
\(988\) 6.78890 0.215984
\(989\) 11.2111 0.356492
\(990\) −2.00000 −0.0635642
\(991\) 17.2111 0.546729 0.273364 0.961911i \(-0.411864\pi\)
0.273364 + 0.961911i \(0.411864\pi\)
\(992\) −2.60555 −0.0827263
\(993\) −14.4222 −0.457675
\(994\) −6.00000 −0.190308
\(995\) −26.4222 −0.837640
\(996\) −10.6056 −0.336050
\(997\) 36.6611 1.16107 0.580534 0.814236i \(-0.302844\pi\)
0.580534 + 0.814236i \(0.302844\pi\)
\(998\) 28.8444 0.913054
\(999\) −6.00000 −0.189832
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 4830.2.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.bn.1.1 2 1.1 even 1 trivial