Properties

Label 4830.2.a.bn.1.2
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 1.2
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 4830.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{11} +1.00000 q^{12} +4.60555 q^{13} +1.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} -2.60555 q^{17} -1.00000 q^{18} +4.60555 q^{19} -1.00000 q^{20} -1.00000 q^{21} +2.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} -4.60555 q^{26} +1.00000 q^{27} -1.00000 q^{28} -9.21110 q^{29} +1.00000 q^{30} -4.60555 q^{31} -1.00000 q^{32} -2.00000 q^{33} +2.60555 q^{34} +1.00000 q^{35} +1.00000 q^{36} -6.00000 q^{37} -4.60555 q^{38} +4.60555 q^{39} +1.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} -3.21110 q^{43} -2.00000 q^{44} -1.00000 q^{45} -1.00000 q^{46} -8.60555 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.60555 q^{51} +4.60555 q^{52} +11.2111 q^{53} -1.00000 q^{54} +2.00000 q^{55} +1.00000 q^{56} +4.60555 q^{57} +9.21110 q^{58} +9.21110 q^{59} -1.00000 q^{60} +10.0000 q^{61} +4.60555 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.60555 q^{65} +2.00000 q^{66} -11.2111 q^{67} -2.60555 q^{68} +1.00000 q^{69} -1.00000 q^{70} -6.00000 q^{71} -1.00000 q^{72} -2.00000 q^{73} +6.00000 q^{74} +1.00000 q^{75} +4.60555 q^{76} +2.00000 q^{77} -4.60555 q^{78} +14.4222 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -3.39445 q^{83} -1.00000 q^{84} +2.60555 q^{85} +3.21110 q^{86} -9.21110 q^{87} +2.00000 q^{88} -12.6056 q^{89} +1.00000 q^{90} -4.60555 q^{91} +1.00000 q^{92} -4.60555 q^{93} +8.60555 q^{94} -4.60555 q^{95} -1.00000 q^{96} +9.81665 q^{97} -1.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} - 2q^{8} + 2q^{9} + 2q^{10} - 4q^{11} + 2q^{12} + 2q^{13} + 2q^{14} - 2q^{15} + 2q^{16} + 2q^{17} - 2q^{18} + 2q^{19} - 2q^{20} - 2q^{21} + 4q^{22} + 2q^{23} - 2q^{24} + 2q^{25} - 2q^{26} + 2q^{27} - 2q^{28} - 4q^{29} + 2q^{30} - 2q^{31} - 2q^{32} - 4q^{33} - 2q^{34} + 2q^{35} + 2q^{36} - 12q^{37} - 2q^{38} + 2q^{39} + 2q^{40} - 12q^{41} + 2q^{42} + 8q^{43} - 4q^{44} - 2q^{45} - 2q^{46} - 10q^{47} + 2q^{48} + 2q^{49} - 2q^{50} + 2q^{51} + 2q^{52} + 8q^{53} - 2q^{54} + 4q^{55} + 2q^{56} + 2q^{57} + 4q^{58} + 4q^{59} - 2q^{60} + 20q^{61} + 2q^{62} - 2q^{63} + 2q^{64} - 2q^{65} + 4q^{66} - 8q^{67} + 2q^{68} + 2q^{69} - 2q^{70} - 12q^{71} - 2q^{72} - 4q^{73} + 12q^{74} + 2q^{75} + 2q^{76} + 4q^{77} - 2q^{78} - 2q^{80} + 2q^{81} + 12q^{82} - 14q^{83} - 2q^{84} - 2q^{85} - 8q^{86} - 4q^{87} + 4q^{88} - 18q^{89} + 2q^{90} - 2q^{91} + 2q^{92} - 2q^{93} + 10q^{94} - 2q^{95} - 2q^{96} - 2q^{97} - 2q^{98} - 4q^{99} + O(q^{100}) \)

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
\(13\) 4.60555 1.27735 0.638675 0.769477i \(-0.279483\pi\)
0.638675 + 0.769477i \(0.279483\pi\)
\(14\) 1.00000 0.267261
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −2.60555 −0.631939 −0.315970 0.948769i \(-0.602330\pi\)
−0.315970 + 0.948769i \(0.602330\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.60555 1.05659 0.528293 0.849062i \(-0.322832\pi\)
0.528293 + 0.849062i \(0.322832\pi\)
\(20\) −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
\(26\) −4.60555 −0.903223
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −9.21110 −1.71046 −0.855229 0.518250i \(-0.826584\pi\)
−0.855229 + 0.518250i \(0.826584\pi\)
\(30\) 1.00000 0.182574
\(31\) −4.60555 −0.827181 −0.413591 0.910463i \(-0.635726\pi\)
−0.413591 + 0.910463i \(0.635726\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) 2.60555 0.446848
\(35\) 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 \(0.664172\pi\)
\(38\) −4.60555 −0.747119
\(39\) 4.60555 0.737478
\(40\) 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
\(43\) −3.21110 −0.489689 −0.244844 0.969562i \(-0.578737\pi\)
−0.244844 + 0.969562i \(0.578737\pi\)
\(44\) −2.00000 −0.301511
\(45\) −1.00000 −0.149071
\(46\) −1.00000 −0.147442
\(47\) −8.60555 −1.25525 −0.627624 0.778516i \(-0.715973\pi\)
−0.627624 + 0.778516i \(0.715973\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) −2.60555 −0.364850
\(52\) 4.60555 0.638675
\(53\) 11.2111 1.53996 0.769982 0.638066i \(-0.220265\pi\)
0.769982 + 0.638066i \(0.220265\pi\)
\(54\) −1.00000 −0.136083
\(55\) 2.00000 0.269680
\(56\) 1.00000 0.133631
\(57\) 4.60555 0.610020
\(58\) 9.21110 1.20948
\(59\) 9.21110 1.19918 0.599592 0.800306i \(-0.295330\pi\)
0.599592 + 0.800306i \(0.295330\pi\)
\(60\) −1.00000 −0.129099
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 4.60555 0.584906
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −4.60555 −0.571248
\(66\) 2.00000 0.246183
\(67\) −11.2111 −1.36965 −0.684827 0.728706i \(-0.740122\pi\)
−0.684827 + 0.728706i \(0.740122\pi\)
\(68\) −2.60555 −0.315970
\(69\) 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
\(76\) 4.60555 0.528293
\(77\) 2.00000 0.227921
\(78\) −4.60555 −0.521476
\(79\) 14.4222 1.62262 0.811312 0.584613i \(-0.198754\pi\)
0.811312 + 0.584613i \(0.198754\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) −3.39445 −0.372589 −0.186295 0.982494i \(-0.559648\pi\)
−0.186295 + 0.982494i \(0.559648\pi\)
\(84\) −1.00000 −0.109109
\(85\) 2.60555 0.282612
\(86\) 3.21110 0.346262
\(87\) −9.21110 −0.987534
\(88\) 2.00000 0.213201
\(89\) −12.6056 −1.33619 −0.668093 0.744078i \(-0.732889\pi\)
−0.668093 + 0.744078i \(0.732889\pi\)
\(90\) 1.00000 0.105409
\(91\) −4.60555 −0.482793
\(92\) 1.00000 0.104257
\(93\) −4.60555 −0.477573
\(94\) 8.60555 0.887595
\(95\) −4.60555 −0.472520
\(96\) −1.00000 −0.102062
\(97\) 9.81665 0.996730 0.498365 0.866967i \(-0.333934\pi\)
0.498365 + 0.866967i \(0.333934\pi\)
\(98\) −1.00000 −0.101015
\(99\) −2.00000 −0.201008
\(100\) 1.00000 0.100000
\(101\) 5.81665 0.578779 0.289389 0.957211i \(-0.406548\pi\)
0.289389 + 0.957211i \(0.406548\pi\)
\(102\) 2.60555 0.257988
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −4.60555 −0.451611
\(105\) 1.00000 0.0975900
\(106\) −11.2111 −1.08892
\(107\) 5.21110 0.503776 0.251888 0.967756i \(-0.418948\pi\)
0.251888 + 0.967756i \(0.418948\pi\)
\(108\) 1.00000 0.0962250
\(109\) −12.4222 −1.18983 −0.594916 0.803788i \(-0.702815\pi\)
−0.594916 + 0.803788i \(0.702815\pi\)
\(110\) −2.00000 −0.190693
\(111\) −6.00000 −0.569495
\(112\) −1.00000 −0.0944911
\(113\) 15.2111 1.43094 0.715470 0.698643i \(-0.246213\pi\)
0.715470 + 0.698643i \(0.246213\pi\)
\(114\) −4.60555 −0.431349
\(115\) −1.00000 −0.0932505
\(116\) −9.21110 −0.855229
\(117\) 4.60555 0.425783
\(118\) −9.21110 −0.847951
\(119\) 2.60555 0.238850
\(120\) 1.00000 0.0912871
\(121\) −7.00000 −0.636364
\(122\) −10.0000 −0.905357
\(123\) −6.00000 −0.541002
\(124\) −4.60555 −0.413591
\(125\) −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
\(129\) −3.21110 −0.282722
\(130\) 4.60555 0.403934
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) −2.00000 −0.174078
\(133\) −4.60555 −0.399352
\(134\) 11.2111 0.968492
\(135\) −1.00000 −0.0860663
\(136\) 2.60555 0.223424
\(137\) −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
\(141\) −8.60555 −0.724718
\(142\) 6.00000 0.503509
\(143\) −9.21110 −0.770271
\(144\) 1.00000 0.0833333
\(145\) 9.21110 0.764940
\(146\) 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
\(151\) 2.78890 0.226957 0.113479 0.993540i \(-0.463801\pi\)
0.113479 + 0.993540i \(0.463801\pi\)
\(152\) −4.60555 −0.373560
\(153\) −2.60555 −0.210646
\(154\) −2.00000 −0.161165
\(155\) 4.60555 0.369927
\(156\) 4.60555 0.368739
\(157\) −20.4222 −1.62987 −0.814935 0.579553i \(-0.803227\pi\)
−0.814935 + 0.579553i \(0.803227\pi\)
\(158\) −14.4222 −1.14737
\(159\) 11.2111 0.889098
\(160\) 1.00000 0.0790569
\(161\) −1.00000 −0.0788110
\(162\) −1.00000 −0.0785674
\(163\) −23.6333 −1.85110 −0.925552 0.378621i \(-0.876398\pi\)
−0.925552 + 0.378621i \(0.876398\pi\)
\(164\) −6.00000 −0.468521
\(165\) 2.00000 0.155700
\(166\) 3.39445 0.263460
\(167\) 20.6056 1.59451 0.797253 0.603646i \(-0.206286\pi\)
0.797253 + 0.603646i \(0.206286\pi\)
\(168\) 1.00000 0.0771517
\(169\) 8.21110 0.631623
\(170\) −2.60555 −0.199837
\(171\) 4.60555 0.352195
\(172\) −3.21110 −0.244844
\(173\) 17.0278 1.29460 0.647298 0.762237i \(-0.275899\pi\)
0.647298 + 0.762237i \(0.275899\pi\)
\(174\) 9.21110 0.698292
\(175\) −1.00000 −0.0755929
\(176\) −2.00000 −0.150756
\(177\) 9.21110 0.692349
\(178\) 12.6056 0.944826
\(179\) 10.4222 0.778992 0.389496 0.921028i \(-0.372649\pi\)
0.389496 + 0.921028i \(0.372649\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 4.60555 0.341386
\(183\) 10.0000 0.739221
\(184\) −1.00000 −0.0737210
\(185\) 6.00000 0.441129
\(186\) 4.60555 0.337695
\(187\) 5.21110 0.381074
\(188\) −8.60555 −0.627624
\(189\) −1.00000 −0.0727393
\(190\) 4.60555 0.334122
\(191\) −21.2111 −1.53478 −0.767391 0.641180i \(-0.778445\pi\)
−0.767391 + 0.641180i \(0.778445\pi\)
\(192\) 1.00000 0.0721688
\(193\) 16.4222 1.18210 0.591048 0.806636i \(-0.298714\pi\)
0.591048 + 0.806636i \(0.298714\pi\)
\(194\) −9.81665 −0.704795
\(195\) −4.60555 −0.329810
\(196\) 1.00000 0.0714286
\(197\) −23.2111 −1.65372 −0.826861 0.562406i \(-0.809876\pi\)
−0.826861 + 0.562406i \(0.809876\pi\)
\(198\) 2.00000 0.142134
\(199\) −2.42221 −0.171706 −0.0858528 0.996308i \(-0.527361\pi\)
−0.0858528 + 0.996308i \(0.527361\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −11.2111 −0.790770
\(202\) −5.81665 −0.409258
\(203\) 9.21110 0.646493
\(204\) −2.60555 −0.182425
\(205\) 6.00000 0.419058
\(206\) −8.00000 −0.557386
\(207\) 1.00000 0.0695048
\(208\) 4.60555 0.319338
\(209\) −9.21110 −0.637145
\(210\) −1.00000 −0.0690066
\(211\) −18.4222 −1.26824 −0.634118 0.773236i \(-0.718637\pi\)
−0.634118 + 0.773236i \(0.718637\pi\)
\(212\) 11.2111 0.769982
\(213\) −6.00000 −0.411113
\(214\) −5.21110 −0.356224
\(215\) 3.21110 0.218995
\(216\) −1.00000 −0.0680414
\(217\) 4.60555 0.312645
\(218\) 12.4222 0.841338
\(219\) −2.00000 −0.135147
\(220\) 2.00000 0.134840
\(221\) −12.0000 −0.807207
\(222\) 6.00000 0.402694
\(223\) −18.6056 −1.24592 −0.622960 0.782254i \(-0.714070\pi\)
−0.622960 + 0.782254i \(0.714070\pi\)
\(224\) 1.00000 0.0668153
\(225\) 1.00000 0.0666667
\(226\) −15.2111 −1.01183
\(227\) 15.3944 1.02177 0.510883 0.859650i \(-0.329319\pi\)
0.510883 + 0.859650i \(0.329319\pi\)
\(228\) 4.60555 0.305010
\(229\) −8.78890 −0.580787 −0.290393 0.956907i \(-0.593786\pi\)
−0.290393 + 0.956907i \(0.593786\pi\)
\(230\) 1.00000 0.0659380
\(231\) 2.00000 0.131590
\(232\) 9.21110 0.604739
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) −4.60555 −0.301074
\(235\) 8.60555 0.561364
\(236\) 9.21110 0.599592
\(237\) 14.4222 0.936823
\(238\) −2.60555 −0.168893
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −0.183346 −0.0118104 −0.00590518 0.999983i \(-0.501880\pi\)
−0.00590518 + 0.999983i \(0.501880\pi\)
\(242\) 7.00000 0.449977
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) −1.00000 −0.0638877
\(246\) 6.00000 0.382546
\(247\) 21.2111 1.34963
\(248\) 4.60555 0.292453
\(249\) −3.39445 −0.215114
\(250\) 1.00000 0.0632456
\(251\) −18.6056 −1.17437 −0.587186 0.809452i \(-0.699764\pi\)
−0.587186 + 0.809452i \(0.699764\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −2.00000 −0.125739
\(254\) 2.00000 0.125491
\(255\) 2.60555 0.163166
\(256\) 1.00000 0.0625000
\(257\) −17.6333 −1.09994 −0.549968 0.835186i \(-0.685360\pi\)
−0.549968 + 0.835186i \(0.685360\pi\)
\(258\) 3.21110 0.199915
\(259\) 6.00000 0.372822
\(260\) −4.60555 −0.285624
\(261\) −9.21110 −0.570153
\(262\) 12.0000 0.741362
\(263\) −18.4222 −1.13596 −0.567981 0.823042i \(-0.692275\pi\)
−0.567981 + 0.823042i \(0.692275\pi\)
\(264\) 2.00000 0.123091
\(265\) −11.2111 −0.688693
\(266\) 4.60555 0.282384
\(267\) −12.6056 −0.771447
\(268\) −11.2111 −0.684827
\(269\) −11.0278 −0.672374 −0.336187 0.941795i \(-0.609137\pi\)
−0.336187 + 0.941795i \(0.609137\pi\)
\(270\) 1.00000 0.0608581
\(271\) −2.18335 −0.132629 −0.0663144 0.997799i \(-0.521124\pi\)
−0.0663144 + 0.997799i \(0.521124\pi\)
\(272\) −2.60555 −0.157985
\(273\) −4.60555 −0.278741
\(274\) 18.0000 1.08742
\(275\) −2.00000 −0.120605
\(276\) 1.00000 0.0601929
\(277\) −1.21110 −0.0727681 −0.0363840 0.999338i \(-0.511584\pi\)
−0.0363840 + 0.999338i \(0.511584\pi\)
\(278\) 12.0000 0.719712
\(279\) −4.60555 −0.275727
\(280\) −1.00000 −0.0597614
\(281\) −25.2111 −1.50397 −0.751984 0.659181i \(-0.770903\pi\)
−0.751984 + 0.659181i \(0.770903\pi\)
\(282\) 8.60555 0.512453
\(283\) 26.2389 1.55974 0.779869 0.625943i \(-0.215286\pi\)
0.779869 + 0.625943i \(0.215286\pi\)
\(284\) −6.00000 −0.356034
\(285\) −4.60555 −0.272809
\(286\) 9.21110 0.544664
\(287\) 6.00000 0.354169
\(288\) −1.00000 −0.0589256
\(289\) −10.2111 −0.600653
\(290\) −9.21110 −0.540895
\(291\) 9.81665 0.575462
\(292\) −2.00000 −0.117041
\(293\) −21.6333 −1.26383 −0.631916 0.775037i \(-0.717731\pi\)
−0.631916 + 0.775037i \(0.717731\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −9.21110 −0.536291
\(296\) 6.00000 0.348743
\(297\) −2.00000 −0.116052
\(298\) 14.0000 0.810998
\(299\) 4.60555 0.266346
\(300\) 1.00000 0.0577350
\(301\) 3.21110 0.185085
\(302\) −2.78890 −0.160483
\(303\) 5.81665 0.334158
\(304\) 4.60555 0.264146
\(305\) −10.0000 −0.572598
\(306\) 2.60555 0.148949
\(307\) −30.4222 −1.73629 −0.868143 0.496313i \(-0.834687\pi\)
−0.868143 + 0.496313i \(0.834687\pi\)
\(308\) 2.00000 0.113961
\(309\) 8.00000 0.455104
\(310\) −4.60555 −0.261578
\(311\) 25.0278 1.41919 0.709597 0.704608i \(-0.248877\pi\)
0.709597 + 0.704608i \(0.248877\pi\)
\(312\) −4.60555 −0.260738
\(313\) 20.6056 1.16469 0.582347 0.812940i \(-0.302134\pi\)
0.582347 + 0.812940i \(0.302134\pi\)
\(314\) 20.4222 1.15249
\(315\) 1.00000 0.0563436
\(316\) 14.4222 0.811312
\(317\) 23.2111 1.30367 0.651833 0.758363i \(-0.274000\pi\)
0.651833 + 0.758363i \(0.274000\pi\)
\(318\) −11.2111 −0.628687
\(319\) 18.4222 1.03145
\(320\) −1.00000 −0.0559017
\(321\) 5.21110 0.290855
\(322\) 1.00000 0.0557278
\(323\) −12.0000 −0.667698
\(324\) 1.00000 0.0555556
\(325\) 4.60555 0.255470
\(326\) 23.6333 1.30893
\(327\) −12.4222 −0.686950
\(328\) 6.00000 0.331295
\(329\) 8.60555 0.474439
\(330\) −2.00000 −0.110096
\(331\) 14.4222 0.792716 0.396358 0.918096i \(-0.370274\pi\)
0.396358 + 0.918096i \(0.370274\pi\)
\(332\) −3.39445 −0.186295
\(333\) −6.00000 −0.328798
\(334\) −20.6056 −1.12749
\(335\) 11.2111 0.612528
\(336\) −1.00000 −0.0545545
\(337\) 18.4222 1.00352 0.501761 0.865006i \(-0.332686\pi\)
0.501761 + 0.865006i \(0.332686\pi\)
\(338\) −8.21110 −0.446625
\(339\) 15.2111 0.826154
\(340\) 2.60555 0.141306
\(341\) 9.21110 0.498809
\(342\) −4.60555 −0.249040
\(343\) −1.00000 −0.0539949
\(344\) 3.21110 0.173131
\(345\) −1.00000 −0.0538382
\(346\) −17.0278 −0.915418
\(347\) −10.4222 −0.559493 −0.279747 0.960074i \(-0.590250\pi\)
−0.279747 + 0.960074i \(0.590250\pi\)
\(348\) −9.21110 −0.493767
\(349\) −22.2389 −1.19042 −0.595209 0.803571i \(-0.702931\pi\)
−0.595209 + 0.803571i \(0.702931\pi\)
\(350\) 1.00000 0.0534522
\(351\) 4.60555 0.245826
\(352\) 2.00000 0.106600
\(353\) 7.21110 0.383808 0.191904 0.981414i \(-0.438534\pi\)
0.191904 + 0.981414i \(0.438534\pi\)
\(354\) −9.21110 −0.489565
\(355\) 6.00000 0.318447
\(356\) −12.6056 −0.668093
\(357\) 2.60555 0.137900
\(358\) −10.4222 −0.550831
\(359\) 29.2111 1.54170 0.770852 0.637015i \(-0.219831\pi\)
0.770852 + 0.637015i \(0.219831\pi\)
\(360\) 1.00000 0.0527046
\(361\) 2.21110 0.116374
\(362\) 14.0000 0.735824
\(363\) −7.00000 −0.367405
\(364\) −4.60555 −0.241396
\(365\) 2.00000 0.104685
\(366\) −10.0000 −0.522708
\(367\) 34.4222 1.79682 0.898412 0.439153i \(-0.144721\pi\)
0.898412 + 0.439153i \(0.144721\pi\)
\(368\) 1.00000 0.0521286
\(369\) −6.00000 −0.312348
\(370\) −6.00000 −0.311925
\(371\) −11.2111 −0.582051
\(372\) −4.60555 −0.238787
\(373\) −8.42221 −0.436085 −0.218043 0.975939i \(-0.569967\pi\)
−0.218043 + 0.975939i \(0.569967\pi\)
\(374\) −5.21110 −0.269460
\(375\) −1.00000 −0.0516398
\(376\) 8.60555 0.443797
\(377\) −42.4222 −2.18485
\(378\) 1.00000 0.0514344
\(379\) −7.63331 −0.392097 −0.196048 0.980594i \(-0.562811\pi\)
−0.196048 + 0.980594i \(0.562811\pi\)
\(380\) −4.60555 −0.236260
\(381\) −2.00000 −0.102463
\(382\) 21.2111 1.08525
\(383\) 9.21110 0.470665 0.235333 0.971915i \(-0.424382\pi\)
0.235333 + 0.971915i \(0.424382\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −2.00000 −0.101929
\(386\) −16.4222 −0.835868
\(387\) −3.21110 −0.163230
\(388\) 9.81665 0.498365
\(389\) −0.422205 −0.0214066 −0.0107033 0.999943i \(-0.503407\pi\)
−0.0107033 + 0.999943i \(0.503407\pi\)
\(390\) 4.60555 0.233211
\(391\) −2.60555 −0.131768
\(392\) −1.00000 −0.0505076
\(393\) −12.0000 −0.605320
\(394\) 23.2111 1.16936
\(395\) −14.4222 −0.725660
\(396\) −2.00000 −0.100504
\(397\) 20.6056 1.03416 0.517081 0.855936i \(-0.327018\pi\)
0.517081 + 0.855936i \(0.327018\pi\)
\(398\) 2.42221 0.121414
\(399\) −4.60555 −0.230566
\(400\) 1.00000 0.0500000
\(401\) −13.2111 −0.659731 −0.329865 0.944028i \(-0.607003\pi\)
−0.329865 + 0.944028i \(0.607003\pi\)
\(402\) 11.2111 0.559159
\(403\) −21.2111 −1.05660
\(404\) 5.81665 0.289389
\(405\) −1.00000 −0.0496904
\(406\) −9.21110 −0.457139
\(407\) 12.0000 0.594818
\(408\) 2.60555 0.128994
\(409\) 25.6333 1.26749 0.633743 0.773544i \(-0.281518\pi\)
0.633743 + 0.773544i \(0.281518\pi\)
\(410\) −6.00000 −0.296319
\(411\) −18.0000 −0.887875
\(412\) 8.00000 0.394132
\(413\) −9.21110 −0.453249
\(414\) −1.00000 −0.0491473
\(415\) 3.39445 0.166627
\(416\) −4.60555 −0.225806
\(417\) −12.0000 −0.587643
\(418\) 9.21110 0.450530
\(419\) −38.6056 −1.88600 −0.943002 0.332786i \(-0.892011\pi\)
−0.943002 + 0.332786i \(0.892011\pi\)
\(420\) 1.00000 0.0487950
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) 18.4222 0.896779
\(423\) −8.60555 −0.418416
\(424\) −11.2111 −0.544459
\(425\) −2.60555 −0.126388
\(426\) 6.00000 0.290701
\(427\) −10.0000 −0.483934
\(428\) 5.21110 0.251888
\(429\) −9.21110 −0.444716
\(430\) −3.21110 −0.154853
\(431\) 18.4222 0.887366 0.443683 0.896184i \(-0.353672\pi\)
0.443683 + 0.896184i \(0.353672\pi\)
\(432\) 1.00000 0.0481125
\(433\) −14.1833 −0.681608 −0.340804 0.940134i \(-0.610699\pi\)
−0.340804 + 0.940134i \(0.610699\pi\)
\(434\) −4.60555 −0.221074
\(435\) 9.21110 0.441639
\(436\) −12.4222 −0.594916
\(437\) 4.60555 0.220313
\(438\) 2.00000 0.0955637
\(439\) −36.2389 −1.72959 −0.864793 0.502128i \(-0.832551\pi\)
−0.864793 + 0.502128i \(0.832551\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 1.00000 0.0476190
\(442\) 12.0000 0.570782
\(443\) 34.4222 1.63545 0.817724 0.575610i \(-0.195235\pi\)
0.817724 + 0.575610i \(0.195235\pi\)
\(444\) −6.00000 −0.284747
\(445\) 12.6056 0.597560
\(446\) 18.6056 0.880998
\(447\) −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
\(452\) 15.2111 0.715470
\(453\) 2.78890 0.131034
\(454\) −15.3944 −0.722497
\(455\) 4.60555 0.215912
\(456\) −4.60555 −0.215675
\(457\) −15.6333 −0.731295 −0.365648 0.930753i \(-0.619152\pi\)
−0.365648 + 0.930753i \(0.619152\pi\)
\(458\) 8.78890 0.410678
\(459\) −2.60555 −0.121617
\(460\) −1.00000 −0.0466252
\(461\) −2.18335 −0.101689 −0.0508443 0.998707i \(-0.516191\pi\)
−0.0508443 + 0.998707i \(0.516191\pi\)
\(462\) −2.00000 −0.0930484
\(463\) 13.6333 0.633594 0.316797 0.948493i \(-0.397393\pi\)
0.316797 + 0.948493i \(0.397393\pi\)
\(464\) −9.21110 −0.427615
\(465\) 4.60555 0.213577
\(466\) 22.0000 1.01913
\(467\) −7.02776 −0.325206 −0.162603 0.986692i \(-0.551989\pi\)
−0.162603 + 0.986692i \(0.551989\pi\)
\(468\) 4.60555 0.212892
\(469\) 11.2111 0.517681
\(470\) −8.60555 −0.396944
\(471\) −20.4222 −0.941006
\(472\) −9.21110 −0.423975
\(473\) 6.42221 0.295293
\(474\) −14.4222 −0.662434
\(475\) 4.60555 0.211317
\(476\) 2.60555 0.119425
\(477\) 11.2111 0.513321
\(478\) 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
\(481\) −27.6333 −1.25997
\(482\) 0.183346 0.00835119
\(483\) −1.00000 −0.0455016
\(484\) −7.00000 −0.318182
\(485\) −9.81665 −0.445751
\(486\) −1.00000 −0.0453609
\(487\) 32.0555 1.45257 0.726287 0.687392i \(-0.241244\pi\)
0.726287 + 0.687392i \(0.241244\pi\)
\(488\) −10.0000 −0.452679
\(489\) −23.6333 −1.06874
\(490\) 1.00000 0.0451754
\(491\) −33.2111 −1.49880 −0.749398 0.662120i \(-0.769657\pi\)
−0.749398 + 0.662120i \(0.769657\pi\)
\(492\) −6.00000 −0.270501
\(493\) 24.0000 1.08091
\(494\) −21.2111 −0.954333
\(495\) 2.00000 0.0898933
\(496\) −4.60555 −0.206795
\(497\) 6.00000 0.269137
\(498\) 3.39445 0.152109
\(499\) 28.8444 1.29125 0.645627 0.763653i \(-0.276596\pi\)
0.645627 + 0.763653i \(0.276596\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 20.6056 0.920588
\(502\) 18.6056 0.830406
\(503\) −5.57779 −0.248702 −0.124351 0.992238i \(-0.539685\pi\)
−0.124351 + 0.992238i \(0.539685\pi\)
\(504\) 1.00000 0.0445435
\(505\) −5.81665 −0.258838
\(506\) 2.00000 0.0889108
\(507\) 8.21110 0.364668
\(508\) −2.00000 −0.0887357
\(509\) −2.18335 −0.0967751 −0.0483876 0.998829i \(-0.515408\pi\)
−0.0483876 + 0.998829i \(0.515408\pi\)
\(510\) −2.60555 −0.115376
\(511\) 2.00000 0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 4.60555 0.203340
\(514\) 17.6333 0.777772
\(515\) −8.00000 −0.352522
\(516\) −3.21110 −0.141361
\(517\) 17.2111 0.756943
\(518\) −6.00000 −0.263625
\(519\) 17.0278 0.747436
\(520\) 4.60555 0.201967
\(521\) 13.8167 0.605319 0.302659 0.953099i \(-0.402126\pi\)
0.302659 + 0.953099i \(0.402126\pi\)
\(522\) 9.21110 0.403159
\(523\) 26.2389 1.14735 0.573673 0.819085i \(-0.305518\pi\)
0.573673 + 0.819085i \(0.305518\pi\)
\(524\) −12.0000 −0.524222
\(525\) −1.00000 −0.0436436
\(526\) 18.4222 0.803246
\(527\) 12.0000 0.522728
\(528\) −2.00000 −0.0870388
\(529\) 1.00000 0.0434783
\(530\) 11.2111 0.486979
\(531\) 9.21110 0.399728
\(532\) −4.60555 −0.199676
\(533\) −27.6333 −1.19693
\(534\) 12.6056 0.545496
\(535\) −5.21110 −0.225296
\(536\) 11.2111 0.484246
\(537\) 10.4222 0.449751
\(538\) 11.0278 0.475440
\(539\) −2.00000 −0.0861461
\(540\) −1.00000 −0.0430331
\(541\) 38.8444 1.67005 0.835026 0.550211i \(-0.185453\pi\)
0.835026 + 0.550211i \(0.185453\pi\)
\(542\) 2.18335 0.0937827
\(543\) −14.0000 −0.600798
\(544\) 2.60555 0.111712
\(545\) 12.4222 0.532109
\(546\) 4.60555 0.197099
\(547\) −14.4222 −0.616649 −0.308324 0.951281i \(-0.599768\pi\)
−0.308324 + 0.951281i \(0.599768\pi\)
\(548\) −18.0000 −0.768922
\(549\) 10.0000 0.426790
\(550\) 2.00000 0.0852803
\(551\) −42.4222 −1.80725
\(552\) −1.00000 −0.0425628
\(553\) −14.4222 −0.613295
\(554\) 1.21110 0.0514548
\(555\) 6.00000 0.254686
\(556\) −12.0000 −0.508913
\(557\) 12.4222 0.526346 0.263173 0.964749i \(-0.415231\pi\)
0.263173 + 0.964749i \(0.415231\pi\)
\(558\) 4.60555 0.194969
\(559\) −14.7889 −0.625504
\(560\) 1.00000 0.0422577
\(561\) 5.21110 0.220013
\(562\) 25.2111 1.06347
\(563\) 10.1833 0.429177 0.214588 0.976705i \(-0.431159\pi\)
0.214588 + 0.976705i \(0.431159\pi\)
\(564\) −8.60555 −0.362359
\(565\) −15.2111 −0.639936
\(566\) −26.2389 −1.10290
\(567\) −1.00000 −0.0419961
\(568\) 6.00000 0.251754
\(569\) 29.2111 1.22459 0.612297 0.790628i \(-0.290246\pi\)
0.612297 + 0.790628i \(0.290246\pi\)
\(570\) 4.60555 0.192905
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) −9.21110 −0.385136
\(573\) −21.2111 −0.886107
\(574\) −6.00000 −0.250435
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) 15.2111 0.633246 0.316623 0.948551i \(-0.397451\pi\)
0.316623 + 0.948551i \(0.397451\pi\)
\(578\) 10.2111 0.424726
\(579\) 16.4222 0.682484
\(580\) 9.21110 0.382470
\(581\) 3.39445 0.140825
\(582\) −9.81665 −0.406913
\(583\) −22.4222 −0.928633
\(584\) 2.00000 0.0827606
\(585\) −4.60555 −0.190416
\(586\) 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
\(589\) −21.2111 −0.873988
\(590\) 9.21110 0.379215
\(591\) −23.2111 −0.954777
\(592\) −6.00000 −0.246598
\(593\) −44.0555 −1.80914 −0.904572 0.426322i \(-0.859809\pi\)
−0.904572 + 0.426322i \(0.859809\pi\)
\(594\) 2.00000 0.0820610
\(595\) −2.60555 −0.106817
\(596\) −14.0000 −0.573462
\(597\) −2.42221 −0.0991343
\(598\) −4.60555 −0.188335
\(599\) −0.788897 −0.0322335 −0.0161167 0.999870i \(-0.505130\pi\)
−0.0161167 + 0.999870i \(0.505130\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) −3.21110 −0.130875
\(603\) −11.2111 −0.456551
\(604\) 2.78890 0.113479
\(605\) 7.00000 0.284590
\(606\) −5.81665 −0.236285
\(607\) −19.8167 −0.804333 −0.402167 0.915567i \(-0.631743\pi\)
−0.402167 + 0.915567i \(0.631743\pi\)
\(608\) −4.60555 −0.186780
\(609\) 9.21110 0.373253
\(610\) 10.0000 0.404888
\(611\) −39.6333 −1.60339
\(612\) −2.60555 −0.105323
\(613\) −45.2666 −1.82830 −0.914151 0.405375i \(-0.867141\pi\)
−0.914151 + 0.405375i \(0.867141\pi\)
\(614\) 30.4222 1.22774
\(615\) 6.00000 0.241943
\(616\) −2.00000 −0.0805823
\(617\) 30.8444 1.24175 0.620875 0.783910i \(-0.286778\pi\)
0.620875 + 0.783910i \(0.286778\pi\)
\(618\) −8.00000 −0.321807
\(619\) −33.4500 −1.34447 −0.672234 0.740339i \(-0.734665\pi\)
−0.672234 + 0.740339i \(0.734665\pi\)
\(620\) 4.60555 0.184963
\(621\) 1.00000 0.0401286
\(622\) −25.0278 −1.00352
\(623\) 12.6056 0.505031
\(624\) 4.60555 0.184370
\(625\) 1.00000 0.0400000
\(626\) −20.6056 −0.823563
\(627\) −9.21110 −0.367856
\(628\) −20.4222 −0.814935
\(629\) 15.6333 0.623341
\(630\) −1.00000 −0.0398410
\(631\) −38.4222 −1.52956 −0.764782 0.644289i \(-0.777153\pi\)
−0.764782 + 0.644289i \(0.777153\pi\)
\(632\) −14.4222 −0.573685
\(633\) −18.4222 −0.732217
\(634\) −23.2111 −0.921831
\(635\) 2.00000 0.0793676
\(636\) 11.2111 0.444549
\(637\) 4.60555 0.182479
\(638\) −18.4222 −0.729342
\(639\) −6.00000 −0.237356
\(640\) 1.00000 0.0395285
\(641\) −27.6333 −1.09145 −0.545725 0.837964i \(-0.683746\pi\)
−0.545725 + 0.837964i \(0.683746\pi\)
\(642\) −5.21110 −0.205666
\(643\) 27.4500 1.08252 0.541260 0.840855i \(-0.317947\pi\)
0.541260 + 0.840855i \(0.317947\pi\)
\(644\) −1.00000 −0.0394055
\(645\) 3.21110 0.126437
\(646\) 12.0000 0.472134
\(647\) −44.2389 −1.73921 −0.869605 0.493749i \(-0.835626\pi\)
−0.869605 + 0.493749i \(0.835626\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −18.4222 −0.723135
\(650\) −4.60555 −0.180645
\(651\) 4.60555 0.180506
\(652\) −23.6333 −0.925552
\(653\) −26.0000 −1.01746 −0.508729 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) 12.4222 0.485747
\(655\) 12.0000 0.468879
\(656\) −6.00000 −0.234261
\(657\) −2.00000 −0.0780274
\(658\) −8.60555 −0.335479
\(659\) 39.2111 1.52745 0.763724 0.645543i \(-0.223369\pi\)
0.763724 + 0.645543i \(0.223369\pi\)
\(660\) 2.00000 0.0778499
\(661\) 7.21110 0.280479 0.140240 0.990118i \(-0.455213\pi\)
0.140240 + 0.990118i \(0.455213\pi\)
\(662\) −14.4222 −0.560535
\(663\) −12.0000 −0.466041
\(664\) 3.39445 0.131730
\(665\) 4.60555 0.178596
\(666\) 6.00000 0.232495
\(667\) −9.21110 −0.356655
\(668\) 20.6056 0.797253
\(669\) −18.6056 −0.719332
\(670\) −11.2111 −0.433123
\(671\) −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 \(0.586968\pi\)
\(674\) −18.4222 −0.709597
\(675\) 1.00000 0.0384900
\(676\) 8.21110 0.315812
\(677\) 28.4222 1.09235 0.546177 0.837670i \(-0.316083\pi\)
0.546177 + 0.837670i \(0.316083\pi\)
\(678\) −15.2111 −0.584179
\(679\) −9.81665 −0.376729
\(680\) −2.60555 −0.0999183
\(681\) 15.3944 0.589917
\(682\) −9.21110 −0.352711
\(683\) −10.4222 −0.398795 −0.199397 0.979919i \(-0.563898\pi\)
−0.199397 + 0.979919i \(0.563898\pi\)
\(684\) 4.60555 0.176098
\(685\) 18.0000 0.687745
\(686\) 1.00000 0.0381802
\(687\) −8.78890 −0.335317
\(688\) −3.21110 −0.122422
\(689\) 51.6333 1.96707
\(690\) 1.00000 0.0380693
\(691\) −23.6333 −0.899053 −0.449527 0.893267i \(-0.648407\pi\)
−0.449527 + 0.893267i \(0.648407\pi\)
\(692\) 17.0278 0.647298
\(693\) 2.00000 0.0759737
\(694\) 10.4222 0.395621
\(695\) 12.0000 0.455186
\(696\) 9.21110 0.349146
\(697\) 15.6333 0.592154
\(698\) 22.2389 0.841753
\(699\) −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 \(0.463855\pi\)
\(702\) −4.60555 −0.173825
\(703\) −27.6333 −1.04221
\(704\) −2.00000 −0.0753778
\(705\) 8.60555 0.324104
\(706\) −7.21110 −0.271393
\(707\) −5.81665 −0.218758
\(708\) 9.21110 0.346174
\(709\) 35.2111 1.32238 0.661190 0.750218i \(-0.270052\pi\)
0.661190 + 0.750218i \(0.270052\pi\)
\(710\) −6.00000 −0.225176
\(711\) 14.4222 0.540875
\(712\) 12.6056 0.472413
\(713\) −4.60555 −0.172479
\(714\) −2.60555 −0.0975103
\(715\) 9.21110 0.344476
\(716\) 10.4222 0.389496
\(717\) −6.00000 −0.224074
\(718\) −29.2111 −1.09015
\(719\) −14.6056 −0.544695 −0.272348 0.962199i \(-0.587800\pi\)
−0.272348 + 0.962199i \(0.587800\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −8.00000 −0.297936
\(722\) −2.21110 −0.0822887
\(723\) −0.183346 −0.00681872
\(724\) −14.0000 −0.520306
\(725\) −9.21110 −0.342092
\(726\) 7.00000 0.259794
\(727\) 18.7889 0.696842 0.348421 0.937338i \(-0.386718\pi\)
0.348421 + 0.937338i \(0.386718\pi\)
\(728\) 4.60555 0.170693
\(729\) 1.00000 0.0370370
\(730\) −2.00000 −0.0740233
\(731\) 8.36669 0.309453
\(732\) 10.0000 0.369611
\(733\) 41.6333 1.53776 0.768881 0.639392i \(-0.220814\pi\)
0.768881 + 0.639392i \(0.220814\pi\)
\(734\) −34.4222 −1.27055
\(735\) −1.00000 −0.0368856
\(736\) −1.00000 −0.0368605
\(737\) 22.4222 0.825933
\(738\) 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
\(741\) 21.2111 0.779209
\(742\) 11.2111 0.411573
\(743\) −1.57779 −0.0578837 −0.0289418 0.999581i \(-0.509214\pi\)
−0.0289418 + 0.999581i \(0.509214\pi\)
\(744\) 4.60555 0.168848
\(745\) 14.0000 0.512920
\(746\) 8.42221 0.308359
\(747\) −3.39445 −0.124196
\(748\) 5.21110 0.190537
\(749\) −5.21110 −0.190410
\(750\) 1.00000 0.0365148
\(751\) 50.4222 1.83993 0.919966 0.391998i \(-0.128216\pi\)
0.919966 + 0.391998i \(0.128216\pi\)
\(752\) −8.60555 −0.313812
\(753\) −18.6056 −0.678024
\(754\) 42.4222 1.54493
\(755\) −2.78890 −0.101498
\(756\) −1.00000 −0.0363696
\(757\) 38.8444 1.41182 0.705912 0.708299i \(-0.250537\pi\)
0.705912 + 0.708299i \(0.250537\pi\)
\(758\) 7.63331 0.277254
\(759\) −2.00000 −0.0725954
\(760\) 4.60555 0.167061
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 2.00000 0.0724524
\(763\) 12.4222 0.449714
\(764\) −21.2111 −0.767391
\(765\) 2.60555 0.0942039
\(766\) −9.21110 −0.332811
\(767\) 42.4222 1.53178
\(768\) 1.00000 0.0360844
\(769\) −45.8722 −1.65419 −0.827096 0.562060i \(-0.810009\pi\)
−0.827096 + 0.562060i \(0.810009\pi\)
\(770\) 2.00000 0.0720750
\(771\) −17.6333 −0.635048
\(772\) 16.4222 0.591048
\(773\) 10.8444 0.390046 0.195023 0.980799i \(-0.437522\pi\)
0.195023 + 0.980799i \(0.437522\pi\)
\(774\) 3.21110 0.115421
\(775\) −4.60555 −0.165436
\(776\) −9.81665 −0.352397
\(777\) 6.00000 0.215249
\(778\) 0.422205 0.0151368
\(779\) −27.6333 −0.990066
\(780\) −4.60555 −0.164905
\(781\) 12.0000 0.429394
\(782\) 2.60555 0.0931743
\(783\) −9.21110 −0.329178
\(784\) 1.00000 0.0357143
\(785\) 20.4222 0.728900
\(786\) 12.0000 0.428026
\(787\) 25.0278 0.892143 0.446072 0.894997i \(-0.352823\pi\)
0.446072 + 0.894997i \(0.352823\pi\)
\(788\) −23.2111 −0.826861
\(789\) −18.4222 −0.655848
\(790\) 14.4222 0.513119
\(791\) −15.2111 −0.540845
\(792\) 2.00000 0.0710669
\(793\) 46.0555 1.63548
\(794\) −20.6056 −0.731264
\(795\) −11.2111 −0.397617
\(796\) −2.42221 −0.0858528
\(797\) 16.7889 0.594693 0.297347 0.954770i \(-0.403898\pi\)
0.297347 + 0.954770i \(0.403898\pi\)
\(798\) 4.60555 0.163035
\(799\) 22.4222 0.793241
\(800\) −1.00000 −0.0353553
\(801\) −12.6056 −0.445395
\(802\) 13.2111 0.466500
\(803\) 4.00000 0.141157
\(804\) −11.2111 −0.395385
\(805\) 1.00000 0.0352454
\(806\) 21.2111 0.747129
\(807\) −11.0278 −0.388195
\(808\) −5.81665 −0.204629
\(809\) 2.84441 0.100004 0.0500021 0.998749i \(-0.484077\pi\)
0.0500021 + 0.998749i \(0.484077\pi\)
\(810\) 1.00000 0.0351364
\(811\) −17.5778 −0.617240 −0.308620 0.951185i \(-0.599867\pi\)
−0.308620 + 0.951185i \(0.599867\pi\)
\(812\) 9.21110 0.323246
\(813\) −2.18335 −0.0765733
\(814\) −12.0000 −0.420600
\(815\) 23.6333 0.827839
\(816\) −2.60555 −0.0912125
\(817\) −14.7889 −0.517398
\(818\) −25.6333 −0.896248
\(819\) −4.60555 −0.160931
\(820\) 6.00000 0.209529
\(821\) −48.8444 −1.70468 −0.852341 0.522987i \(-0.824818\pi\)
−0.852341 + 0.522987i \(0.824818\pi\)
\(822\) 18.0000 0.627822
\(823\) 40.4222 1.40903 0.704515 0.709689i \(-0.251165\pi\)
0.704515 + 0.709689i \(0.251165\pi\)
\(824\) −8.00000 −0.278693
\(825\) −2.00000 −0.0696311
\(826\) 9.21110 0.320495
\(827\) 39.6333 1.37819 0.689093 0.724673i \(-0.258009\pi\)
0.689093 + 0.724673i \(0.258009\pi\)
\(828\) 1.00000 0.0347524
\(829\) 28.6611 0.995440 0.497720 0.867338i \(-0.334171\pi\)
0.497720 + 0.867338i \(0.334171\pi\)
\(830\) −3.39445 −0.117823
\(831\) −1.21110 −0.0420127
\(832\) 4.60555 0.159669
\(833\) −2.60555 −0.0902770
\(834\) 12.0000 0.415526
\(835\) −20.6056 −0.713085
\(836\) −9.21110 −0.318573
\(837\) −4.60555 −0.159191
\(838\) 38.6056 1.33361
\(839\) 15.6333 0.539722 0.269861 0.962899i \(-0.413022\pi\)
0.269861 + 0.962899i \(0.413022\pi\)
\(840\) −1.00000 −0.0345033
\(841\) 55.8444 1.92567
\(842\) −18.0000 −0.620321
\(843\) −25.2111 −0.868316
\(844\) −18.4222 −0.634118
\(845\) −8.21110 −0.282471
\(846\) 8.60555 0.295865
\(847\) 7.00000 0.240523
\(848\) 11.2111 0.384991
\(849\) 26.2389 0.900515
\(850\) 2.60555 0.0893697
\(851\) −6.00000 −0.205677
\(852\) −6.00000 −0.205557
\(853\) 15.0278 0.514541 0.257270 0.966339i \(-0.417177\pi\)
0.257270 + 0.966339i \(0.417177\pi\)
\(854\) 10.0000 0.342193
\(855\) −4.60555 −0.157507
\(856\) −5.21110 −0.178112
\(857\) 39.2111 1.33943 0.669713 0.742620i \(-0.266417\pi\)
0.669713 + 0.742620i \(0.266417\pi\)
\(858\) 9.21110 0.314462
\(859\) 0.366692 0.0125114 0.00625569 0.999980i \(-0.498009\pi\)
0.00625569 + 0.999980i \(0.498009\pi\)
\(860\) 3.21110 0.109498
\(861\) 6.00000 0.204479
\(862\) −18.4222 −0.627463
\(863\) −18.0555 −0.614617 −0.307308 0.951610i \(-0.599428\pi\)
−0.307308 + 0.951610i \(0.599428\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −17.0278 −0.578961
\(866\) 14.1833 0.481970
\(867\) −10.2111 −0.346787
\(868\) 4.60555 0.156323
\(869\) −28.8444 −0.978480
\(870\) −9.21110 −0.312286
\(871\) −51.6333 −1.74953
\(872\) 12.4222 0.420669
\(873\) 9.81665 0.332243
\(874\) −4.60555 −0.155785
\(875\) 1.00000 0.0338062
\(876\) −2.00000 −0.0675737
\(877\) −36.8444 −1.24415 −0.622074 0.782959i \(-0.713710\pi\)
−0.622074 + 0.782959i \(0.713710\pi\)
\(878\) 36.2389 1.22300
\(879\) −21.6333 −0.729673
\(880\) 2.00000 0.0674200
\(881\) −37.8167 −1.27408 −0.637038 0.770833i \(-0.719840\pi\)
−0.637038 + 0.770833i \(0.719840\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −34.0555 −1.14606 −0.573030 0.819535i \(-0.694232\pi\)
−0.573030 + 0.819535i \(0.694232\pi\)
\(884\) −12.0000 −0.403604
\(885\) −9.21110 −0.309628
\(886\) −34.4222 −1.15644
\(887\) 29.4500 0.988833 0.494416 0.869225i \(-0.335382\pi\)
0.494416 + 0.869225i \(0.335382\pi\)
\(888\) 6.00000 0.201347
\(889\) 2.00000 0.0670778
\(890\) −12.6056 −0.422539
\(891\) −2.00000 −0.0670025
\(892\) −18.6056 −0.622960
\(893\) −39.6333 −1.32628
\(894\) 14.0000 0.468230
\(895\) −10.4222 −0.348376
\(896\) 1.00000 0.0334077
\(897\) 4.60555 0.153775
\(898\) 6.00000 0.200223
\(899\) 42.4222 1.41486
\(900\) 1.00000 0.0333333
\(901\) −29.2111 −0.973163
\(902\) −12.0000 −0.399556
\(903\) 3.21110 0.106859
\(904\) −15.2111 −0.505914
\(905\) 14.0000 0.465376
\(906\) −2.78890 −0.0926549
\(907\) −2.00000 −0.0664089 −0.0332045 0.999449i \(-0.510571\pi\)
−0.0332045 + 0.999449i \(0.510571\pi\)
\(908\) 15.3944 0.510883
\(909\) 5.81665 0.192926
\(910\) −4.60555 −0.152673
\(911\) 35.6333 1.18058 0.590292 0.807190i \(-0.299013\pi\)
0.590292 + 0.807190i \(0.299013\pi\)
\(912\) 4.60555 0.152505
\(913\) 6.78890 0.224680
\(914\) 15.6333 0.517104
\(915\) −10.0000 −0.330590
\(916\) −8.78890 −0.290393
\(917\) 12.0000 0.396275
\(918\) 2.60555 0.0859960
\(919\) 17.5778 0.579838 0.289919 0.957051i \(-0.406372\pi\)
0.289919 + 0.957051i \(0.406372\pi\)
\(920\) 1.00000 0.0329690
\(921\) −30.4222 −1.00245
\(922\) 2.18335 0.0719047
\(923\) −27.6333 −0.909561
\(924\) 2.00000 0.0657952
\(925\) −6.00000 −0.197279
\(926\) −13.6333 −0.448018
\(927\) 8.00000 0.262754
\(928\) 9.21110 0.302369
\(929\) 21.6333 0.709766 0.354883 0.934911i \(-0.384521\pi\)
0.354883 + 0.934911i \(0.384521\pi\)
\(930\) −4.60555 −0.151022
\(931\) 4.60555 0.150941
\(932\) −22.0000 −0.720634
\(933\) 25.0278 0.819372
\(934\) 7.02776 0.229955
\(935\) −5.21110 −0.170421
\(936\) −4.60555 −0.150537
\(937\) −33.8167 −1.10474 −0.552371 0.833598i \(-0.686277\pi\)
−0.552371 + 0.833598i \(0.686277\pi\)
\(938\) −11.2111 −0.366055
\(939\) 20.6056 0.672437
\(940\) 8.60555 0.280682
\(941\) 49.2666 1.60605 0.803023 0.595948i \(-0.203224\pi\)
0.803023 + 0.595948i \(0.203224\pi\)
\(942\) 20.4222 0.665391
\(943\) −6.00000 −0.195387
\(944\) 9.21110 0.299796
\(945\) 1.00000 0.0325300
\(946\) −6.42221 −0.208804
\(947\) 50.4222 1.63850 0.819251 0.573435i \(-0.194390\pi\)
0.819251 + 0.573435i \(0.194390\pi\)
\(948\) 14.4222 0.468411
\(949\) −9.21110 −0.299005
\(950\) −4.60555 −0.149424
\(951\) 23.2111 0.752672
\(952\) −2.60555 −0.0844464
\(953\) 1.63331 0.0529080 0.0264540 0.999650i \(-0.491578\pi\)
0.0264540 + 0.999650i \(0.491578\pi\)
\(954\) −11.2111 −0.362973
\(955\) 21.2111 0.686375
\(956\) −6.00000 −0.194054
\(957\) 18.4222 0.595505
\(958\) 24.0000 0.775405
\(959\) 18.0000 0.581250
\(960\) −1.00000 −0.0322749
\(961\) −9.78890 −0.315771
\(962\) 27.6333 0.890934
\(963\) 5.21110 0.167925
\(964\) −0.183346 −0.00590518
\(965\) −16.4222 −0.528649
\(966\) 1.00000 0.0321745
\(967\) 43.2111 1.38958 0.694788 0.719215i \(-0.255498\pi\)
0.694788 + 0.719215i \(0.255498\pi\)
\(968\) 7.00000 0.224989
\(969\) −12.0000 −0.385496
\(970\) 9.81665 0.315194
\(971\) 31.8167 1.02105 0.510523 0.859864i \(-0.329452\pi\)
0.510523 + 0.859864i \(0.329452\pi\)
\(972\) 1.00000 0.0320750
\(973\) 12.0000 0.384702
\(974\) −32.0555 −1.02712
\(975\) 4.60555 0.147496
\(976\) 10.0000 0.320092
\(977\) 4.42221 0.141479 0.0707394 0.997495i \(-0.477464\pi\)
0.0707394 + 0.997495i \(0.477464\pi\)
\(978\) 23.6333 0.755710
\(979\) 25.2111 0.805750
\(980\) −1.00000 −0.0319438
\(981\) −12.4222 −0.396610
\(982\) 33.2111 1.05981
\(983\) 38.7889 1.23717 0.618587 0.785716i \(-0.287705\pi\)
0.618587 + 0.785716i \(0.287705\pi\)
\(984\) 6.00000 0.191273
\(985\) 23.2111 0.739567
\(986\) −24.0000 −0.764316
\(987\) 8.60555 0.273918
\(988\) 21.2111 0.674815
\(989\) −3.21110 −0.102107
\(990\) −2.00000 −0.0635642
\(991\) 2.78890 0.0885922 0.0442961 0.999018i \(-0.485895\pi\)
0.0442961 + 0.999018i \(0.485895\pi\)
\(992\) 4.60555 0.146226
\(993\) 14.4222 0.457675
\(994\) −6.00000 −0.190308
\(995\) 2.42221 0.0767891
\(996\) −3.39445 −0.107557
\(997\) −42.6611 −1.35109 −0.675545 0.737319i \(-0.736091\pi\)
−0.675545 + 0.737319i \(0.736091\pi\)
\(998\) −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.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4830.2.a.bn.1.2 2 1.1 even 1 trivial