Properties

Label 3381.2.a.bc.1.2
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.7997584.1
Defining polynomial: \(x^{6} - x^{5} - 7 x^{4} + 5 x^{3} + 12 x^{2} - 4 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.83264\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.83264 q^{2} -1.00000 q^{3} +1.35859 q^{4} -0.943490 q^{5} +1.83264 q^{6} +1.17548 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.83264 q^{2} -1.00000 q^{3} +1.35859 q^{4} -0.943490 q^{5} +1.83264 q^{6} +1.17548 q^{8} +1.00000 q^{9} +1.72908 q^{10} +2.08680 q^{11} -1.35859 q^{12} +3.64405 q^{13} +0.943490 q^{15} -4.87142 q^{16} +0.299440 q^{17} -1.83264 q^{18} -0.579464 q^{19} -1.28181 q^{20} -3.82436 q^{22} -1.00000 q^{23} -1.17548 q^{24} -4.10983 q^{25} -6.67825 q^{26} -1.00000 q^{27} -5.89363 q^{29} -1.72908 q^{30} -6.02387 q^{31} +6.57660 q^{32} -2.08680 q^{33} -0.548767 q^{34} +1.35859 q^{36} -1.23744 q^{37} +1.06195 q^{38} -3.64405 q^{39} -1.10906 q^{40} +8.42024 q^{41} -9.08583 q^{43} +2.83510 q^{44} -0.943490 q^{45} +1.83264 q^{46} +5.06246 q^{47} +4.87142 q^{48} +7.53185 q^{50} -0.299440 q^{51} +4.95075 q^{52} +2.32577 q^{53} +1.83264 q^{54} -1.96887 q^{55} +0.579464 q^{57} +10.8009 q^{58} +2.08015 q^{59} +1.28181 q^{60} +12.4540 q^{61} +11.0396 q^{62} -2.30974 q^{64} -3.43812 q^{65} +3.82436 q^{66} -4.13763 q^{67} +0.406815 q^{68} +1.00000 q^{69} -11.6439 q^{71} +1.17548 q^{72} -11.5700 q^{73} +2.26778 q^{74} +4.10983 q^{75} -0.787251 q^{76} +6.67825 q^{78} +17.1584 q^{79} +4.59613 q^{80} +1.00000 q^{81} -15.4313 q^{82} +10.9740 q^{83} -0.282519 q^{85} +16.6511 q^{86} +5.89363 q^{87} +2.45300 q^{88} +2.67561 q^{89} +1.72908 q^{90} -1.35859 q^{92} +6.02387 q^{93} -9.27770 q^{94} +0.546718 q^{95} -6.57660 q^{96} +0.589843 q^{97} +2.08680 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{2} - 6q^{3} + 3q^{4} + 3q^{5} + q^{6} - 3q^{8} + 6q^{9} + O(q^{10}) \) \( 6q - q^{2} - 6q^{3} + 3q^{4} + 3q^{5} + q^{6} - 3q^{8} + 6q^{9} - 3q^{10} - 14q^{11} - 3q^{12} - 3q^{15} - 7q^{16} + 15q^{17} - q^{18} - q^{19} + 17q^{20} - 6q^{22} - 6q^{23} + 3q^{24} + 9q^{25} - 15q^{26} - 6q^{27} - 6q^{29} + 3q^{30} - 11q^{31} + 3q^{32} + 14q^{33} + 15q^{34} + 3q^{36} - 5q^{37} + 14q^{38} - 17q^{40} + 18q^{41} - 37q^{43} - 10q^{44} + 3q^{45} + q^{46} + 3q^{47} + 7q^{48} - 30q^{50} - 15q^{51} - 7q^{52} - 15q^{53} + q^{54} - 2q^{55} + q^{57} + 4q^{58} - 2q^{59} - 17q^{60} - 12q^{61} + 36q^{62} - 23q^{64} - 17q^{65} + 6q^{66} - 10q^{67} - q^{68} + 6q^{69} - 21q^{71} - 3q^{72} - 8q^{73} - 16q^{74} - 9q^{75} + 18q^{76} + 15q^{78} - 17q^{79} + 3q^{80} + 6q^{81} - 48q^{82} + 12q^{83} - 13q^{85} + 22q^{86} + 6q^{87} - 2q^{88} + 18q^{89} - 3q^{90} - 3q^{92} + 11q^{93} + 3q^{94} - 16q^{95} - 3q^{96} - 2q^{97} - 14q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.83264 −1.29588 −0.647938 0.761693i \(-0.724368\pi\)
−0.647938 + 0.761693i \(0.724368\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.35859 0.679293
\(5\) −0.943490 −0.421941 −0.210971 0.977492i \(-0.567662\pi\)
−0.210971 + 0.977492i \(0.567662\pi\)
\(6\) 1.83264 0.748174
\(7\) 0 0
\(8\) 1.17548 0.415597
\(9\) 1.00000 0.333333
\(10\) 1.72908 0.546783
\(11\) 2.08680 0.629194 0.314597 0.949225i \(-0.398131\pi\)
0.314597 + 0.949225i \(0.398131\pi\)
\(12\) −1.35859 −0.392190
\(13\) 3.64405 1.01068 0.505339 0.862921i \(-0.331368\pi\)
0.505339 + 0.862921i \(0.331368\pi\)
\(14\) 0 0
\(15\) 0.943490 0.243608
\(16\) −4.87142 −1.21785
\(17\) 0.299440 0.0726249 0.0363124 0.999340i \(-0.488439\pi\)
0.0363124 + 0.999340i \(0.488439\pi\)
\(18\) −1.83264 −0.431958
\(19\) −0.579464 −0.132938 −0.0664690 0.997788i \(-0.521173\pi\)
−0.0664690 + 0.997788i \(0.521173\pi\)
\(20\) −1.28181 −0.286622
\(21\) 0 0
\(22\) −3.82436 −0.815357
\(23\) −1.00000 −0.208514
\(24\) −1.17548 −0.239945
\(25\) −4.10983 −0.821965
\(26\) −6.67825 −1.30971
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.89363 −1.09442 −0.547210 0.836995i \(-0.684310\pi\)
−0.547210 + 0.836995i \(0.684310\pi\)
\(30\) −1.72908 −0.315686
\(31\) −6.02387 −1.08192 −0.540960 0.841048i \(-0.681939\pi\)
−0.540960 + 0.841048i \(0.681939\pi\)
\(32\) 6.57660 1.16259
\(33\) −2.08680 −0.363265
\(34\) −0.548767 −0.0941128
\(35\) 0 0
\(36\) 1.35859 0.226431
\(37\) −1.23744 −0.203433 −0.101717 0.994813i \(-0.532433\pi\)
−0.101717 + 0.994813i \(0.532433\pi\)
\(38\) 1.06195 0.172271
\(39\) −3.64405 −0.583515
\(40\) −1.10906 −0.175357
\(41\) 8.42024 1.31502 0.657510 0.753446i \(-0.271610\pi\)
0.657510 + 0.753446i \(0.271610\pi\)
\(42\) 0 0
\(43\) −9.08583 −1.38558 −0.692788 0.721142i \(-0.743618\pi\)
−0.692788 + 0.721142i \(0.743618\pi\)
\(44\) 2.83510 0.427407
\(45\) −0.943490 −0.140647
\(46\) 1.83264 0.270209
\(47\) 5.06246 0.738436 0.369218 0.929343i \(-0.379626\pi\)
0.369218 + 0.929343i \(0.379626\pi\)
\(48\) 4.87142 0.703128
\(49\) 0 0
\(50\) 7.53185 1.06516
\(51\) −0.299440 −0.0419300
\(52\) 4.95075 0.686546
\(53\) 2.32577 0.319469 0.159734 0.987160i \(-0.448936\pi\)
0.159734 + 0.987160i \(0.448936\pi\)
\(54\) 1.83264 0.249391
\(55\) −1.96887 −0.265483
\(56\) 0 0
\(57\) 0.579464 0.0767518
\(58\) 10.8009 1.41823
\(59\) 2.08015 0.270812 0.135406 0.990790i \(-0.456766\pi\)
0.135406 + 0.990790i \(0.456766\pi\)
\(60\) 1.28181 0.165481
\(61\) 12.4540 1.59457 0.797287 0.603600i \(-0.206268\pi\)
0.797287 + 0.603600i \(0.206268\pi\)
\(62\) 11.0396 1.40203
\(63\) 0 0
\(64\) −2.30974 −0.288718
\(65\) −3.43812 −0.426447
\(66\) 3.82436 0.470747
\(67\) −4.13763 −0.505493 −0.252746 0.967533i \(-0.581334\pi\)
−0.252746 + 0.967533i \(0.581334\pi\)
\(68\) 0.406815 0.0493335
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −11.6439 −1.38188 −0.690938 0.722914i \(-0.742802\pi\)
−0.690938 + 0.722914i \(0.742802\pi\)
\(72\) 1.17548 0.138532
\(73\) −11.5700 −1.35416 −0.677082 0.735908i \(-0.736756\pi\)
−0.677082 + 0.735908i \(0.736756\pi\)
\(74\) 2.26778 0.263624
\(75\) 4.10983 0.474562
\(76\) −0.787251 −0.0903039
\(77\) 0 0
\(78\) 6.67825 0.756163
\(79\) 17.1584 1.93047 0.965235 0.261382i \(-0.0841783\pi\)
0.965235 + 0.261382i \(0.0841783\pi\)
\(80\) 4.59613 0.513863
\(81\) 1.00000 0.111111
\(82\) −15.4313 −1.70410
\(83\) 10.9740 1.20455 0.602274 0.798289i \(-0.294261\pi\)
0.602274 + 0.798289i \(0.294261\pi\)
\(84\) 0 0
\(85\) −0.282519 −0.0306434
\(86\) 16.6511 1.79553
\(87\) 5.89363 0.631864
\(88\) 2.45300 0.261491
\(89\) 2.67561 0.283614 0.141807 0.989894i \(-0.454709\pi\)
0.141807 + 0.989894i \(0.454709\pi\)
\(90\) 1.72908 0.182261
\(91\) 0 0
\(92\) −1.35859 −0.141642
\(93\) 6.02387 0.624647
\(94\) −9.27770 −0.956921
\(95\) 0.546718 0.0560921
\(96\) −6.57660 −0.671222
\(97\) 0.589843 0.0598895 0.0299447 0.999552i \(-0.490467\pi\)
0.0299447 + 0.999552i \(0.490467\pi\)
\(98\) 0 0
\(99\) 2.08680 0.209731
\(100\) −5.58355 −0.558355
\(101\) 6.22834 0.619743 0.309872 0.950778i \(-0.399714\pi\)
0.309872 + 0.950778i \(0.399714\pi\)
\(102\) 0.548767 0.0543360
\(103\) −18.6455 −1.83719 −0.918597 0.395195i \(-0.870677\pi\)
−0.918597 + 0.395195i \(0.870677\pi\)
\(104\) 4.28352 0.420034
\(105\) 0 0
\(106\) −4.26230 −0.413992
\(107\) −0.648883 −0.0627299 −0.0313650 0.999508i \(-0.509985\pi\)
−0.0313650 + 0.999508i \(0.509985\pi\)
\(108\) −1.35859 −0.130730
\(109\) −13.6533 −1.30775 −0.653876 0.756602i \(-0.726858\pi\)
−0.653876 + 0.756602i \(0.726858\pi\)
\(110\) 3.60825 0.344033
\(111\) 1.23744 0.117452
\(112\) 0 0
\(113\) 0.547279 0.0514836 0.0257418 0.999669i \(-0.491805\pi\)
0.0257418 + 0.999669i \(0.491805\pi\)
\(114\) −1.06195 −0.0994608
\(115\) 0.943490 0.0879809
\(116\) −8.00700 −0.743431
\(117\) 3.64405 0.336893
\(118\) −3.81217 −0.350939
\(119\) 0 0
\(120\) 1.10906 0.101243
\(121\) −6.64526 −0.604115
\(122\) −22.8238 −2.06637
\(123\) −8.42024 −0.759227
\(124\) −8.18395 −0.734940
\(125\) 8.59503 0.768763
\(126\) 0 0
\(127\) −1.01278 −0.0898693 −0.0449347 0.998990i \(-0.514308\pi\)
−0.0449347 + 0.998990i \(0.514308\pi\)
\(128\) −8.92027 −0.788448
\(129\) 9.08583 0.799962
\(130\) 6.30086 0.552622
\(131\) −15.0311 −1.31327 −0.656635 0.754209i \(-0.728021\pi\)
−0.656635 + 0.754209i \(0.728021\pi\)
\(132\) −2.83510 −0.246764
\(133\) 0 0
\(134\) 7.58281 0.655055
\(135\) 0.943490 0.0812027
\(136\) 0.351987 0.0301827
\(137\) 9.27867 0.792730 0.396365 0.918093i \(-0.370271\pi\)
0.396365 + 0.918093i \(0.370271\pi\)
\(138\) −1.83264 −0.156005
\(139\) 6.58750 0.558744 0.279372 0.960183i \(-0.409874\pi\)
0.279372 + 0.960183i \(0.409874\pi\)
\(140\) 0 0
\(141\) −5.06246 −0.426336
\(142\) 21.3391 1.79074
\(143\) 7.60440 0.635912
\(144\) −4.87142 −0.405951
\(145\) 5.56058 0.461781
\(146\) 21.2037 1.75483
\(147\) 0 0
\(148\) −1.68116 −0.138191
\(149\) −5.88903 −0.482448 −0.241224 0.970469i \(-0.577549\pi\)
−0.241224 + 0.970469i \(0.577549\pi\)
\(150\) −7.53185 −0.614973
\(151\) 13.1539 1.07045 0.535223 0.844711i \(-0.320228\pi\)
0.535223 + 0.844711i \(0.320228\pi\)
\(152\) −0.681151 −0.0552486
\(153\) 0.299440 0.0242083
\(154\) 0 0
\(155\) 5.68346 0.456507
\(156\) −4.95075 −0.396377
\(157\) −8.42558 −0.672434 −0.336217 0.941784i \(-0.609148\pi\)
−0.336217 + 0.941784i \(0.609148\pi\)
\(158\) −31.4452 −2.50165
\(159\) −2.32577 −0.184445
\(160\) −6.20496 −0.490545
\(161\) 0 0
\(162\) −1.83264 −0.143986
\(163\) 4.15736 0.325630 0.162815 0.986657i \(-0.447943\pi\)
0.162815 + 0.986657i \(0.447943\pi\)
\(164\) 11.4396 0.893284
\(165\) 1.96887 0.153277
\(166\) −20.1114 −1.56095
\(167\) −3.04445 −0.235587 −0.117793 0.993038i \(-0.537582\pi\)
−0.117793 + 0.993038i \(0.537582\pi\)
\(168\) 0 0
\(169\) 0.279097 0.0214690
\(170\) 0.517756 0.0397101
\(171\) −0.579464 −0.0443127
\(172\) −12.3439 −0.941211
\(173\) 16.0543 1.22059 0.610293 0.792176i \(-0.291052\pi\)
0.610293 + 0.792176i \(0.291052\pi\)
\(174\) −10.8009 −0.818816
\(175\) 0 0
\(176\) −10.1657 −0.766267
\(177\) −2.08015 −0.156354
\(178\) −4.90345 −0.367529
\(179\) −5.34509 −0.399511 −0.199755 0.979846i \(-0.564015\pi\)
−0.199755 + 0.979846i \(0.564015\pi\)
\(180\) −1.28181 −0.0955406
\(181\) −11.6201 −0.863716 −0.431858 0.901942i \(-0.642142\pi\)
−0.431858 + 0.901942i \(0.642142\pi\)
\(182\) 0 0
\(183\) −12.4540 −0.920628
\(184\) −1.17548 −0.0866579
\(185\) 1.16751 0.0858369
\(186\) −11.0396 −0.809464
\(187\) 0.624872 0.0456951
\(188\) 6.87779 0.501614
\(189\) 0 0
\(190\) −1.00194 −0.0726883
\(191\) −17.0751 −1.23551 −0.617757 0.786369i \(-0.711958\pi\)
−0.617757 + 0.786369i \(0.711958\pi\)
\(192\) 2.30974 0.166691
\(193\) −4.65052 −0.334752 −0.167376 0.985893i \(-0.553529\pi\)
−0.167376 + 0.985893i \(0.553529\pi\)
\(194\) −1.08097 −0.0776093
\(195\) 3.43812 0.246209
\(196\) 0 0
\(197\) 0.254912 0.0181618 0.00908088 0.999959i \(-0.497109\pi\)
0.00908088 + 0.999959i \(0.497109\pi\)
\(198\) −3.82436 −0.271786
\(199\) 5.47597 0.388181 0.194091 0.980984i \(-0.437824\pi\)
0.194091 + 0.980984i \(0.437824\pi\)
\(200\) −4.83104 −0.341606
\(201\) 4.13763 0.291846
\(202\) −11.4143 −0.803110
\(203\) 0 0
\(204\) −0.406815 −0.0284827
\(205\) −7.94441 −0.554861
\(206\) 34.1705 2.38077
\(207\) −1.00000 −0.0695048
\(208\) −17.7517 −1.23086
\(209\) −1.20923 −0.0836438
\(210\) 0 0
\(211\) −3.18599 −0.219332 −0.109666 0.993968i \(-0.534978\pi\)
−0.109666 + 0.993968i \(0.534978\pi\)
\(212\) 3.15975 0.217013
\(213\) 11.6439 0.797827
\(214\) 1.18917 0.0812902
\(215\) 8.57238 0.584632
\(216\) −1.17548 −0.0799816
\(217\) 0 0
\(218\) 25.0217 1.69468
\(219\) 11.5700 0.781827
\(220\) −2.67488 −0.180341
\(221\) 1.09117 0.0734003
\(222\) −2.26778 −0.152203
\(223\) −22.7674 −1.52462 −0.762309 0.647213i \(-0.775934\pi\)
−0.762309 + 0.647213i \(0.775934\pi\)
\(224\) 0 0
\(225\) −4.10983 −0.273988
\(226\) −1.00297 −0.0667164
\(227\) 10.4521 0.693729 0.346864 0.937915i \(-0.387246\pi\)
0.346864 + 0.937915i \(0.387246\pi\)
\(228\) 0.787251 0.0521370
\(229\) −10.7678 −0.711559 −0.355779 0.934570i \(-0.615785\pi\)
−0.355779 + 0.934570i \(0.615785\pi\)
\(230\) −1.72908 −0.114012
\(231\) 0 0
\(232\) −6.92787 −0.454837
\(233\) 11.7508 0.769818 0.384909 0.922955i \(-0.374233\pi\)
0.384909 + 0.922955i \(0.374233\pi\)
\(234\) −6.67825 −0.436571
\(235\) −4.77638 −0.311577
\(236\) 2.82606 0.183961
\(237\) −17.1584 −1.11456
\(238\) 0 0
\(239\) −12.5491 −0.811734 −0.405867 0.913932i \(-0.633030\pi\)
−0.405867 + 0.913932i \(0.633030\pi\)
\(240\) −4.59613 −0.296679
\(241\) −10.6305 −0.684768 −0.342384 0.939560i \(-0.611234\pi\)
−0.342384 + 0.939560i \(0.611234\pi\)
\(242\) 12.1784 0.782858
\(243\) −1.00000 −0.0641500
\(244\) 16.9199 1.08318
\(245\) 0 0
\(246\) 15.4313 0.983864
\(247\) −2.11159 −0.134358
\(248\) −7.08097 −0.449642
\(249\) −10.9740 −0.695447
\(250\) −15.7516 −0.996220
\(251\) −17.4415 −1.10090 −0.550450 0.834868i \(-0.685544\pi\)
−0.550450 + 0.834868i \(0.685544\pi\)
\(252\) 0 0
\(253\) −2.08680 −0.131196
\(254\) 1.85606 0.116459
\(255\) 0.282519 0.0176920
\(256\) 20.9672 1.31045
\(257\) 18.7261 1.16810 0.584050 0.811718i \(-0.301467\pi\)
0.584050 + 0.811718i \(0.301467\pi\)
\(258\) −16.6511 −1.03665
\(259\) 0 0
\(260\) −4.67098 −0.289682
\(261\) −5.89363 −0.364807
\(262\) 27.5466 1.70183
\(263\) 29.9086 1.84424 0.922121 0.386902i \(-0.126455\pi\)
0.922121 + 0.386902i \(0.126455\pi\)
\(264\) −2.45300 −0.150972
\(265\) −2.19434 −0.134797
\(266\) 0 0
\(267\) −2.67561 −0.163745
\(268\) −5.62133 −0.343377
\(269\) 22.6461 1.38076 0.690378 0.723449i \(-0.257444\pi\)
0.690378 + 0.723449i \(0.257444\pi\)
\(270\) −1.72908 −0.105229
\(271\) −12.7364 −0.773679 −0.386839 0.922147i \(-0.626433\pi\)
−0.386839 + 0.922147i \(0.626433\pi\)
\(272\) −1.45870 −0.0884465
\(273\) 0 0
\(274\) −17.0045 −1.02728
\(275\) −8.57639 −0.517176
\(276\) 1.35859 0.0817772
\(277\) −29.0127 −1.74320 −0.871601 0.490216i \(-0.836918\pi\)
−0.871601 + 0.490216i \(0.836918\pi\)
\(278\) −12.0725 −0.724063
\(279\) −6.02387 −0.360640
\(280\) 0 0
\(281\) −12.4466 −0.742504 −0.371252 0.928532i \(-0.621071\pi\)
−0.371252 + 0.928532i \(0.621071\pi\)
\(282\) 9.27770 0.552479
\(283\) 9.36397 0.556630 0.278315 0.960490i \(-0.410224\pi\)
0.278315 + 0.960490i \(0.410224\pi\)
\(284\) −15.8192 −0.938699
\(285\) −0.546718 −0.0323848
\(286\) −13.9362 −0.824063
\(287\) 0 0
\(288\) 6.57660 0.387530
\(289\) −16.9103 −0.994726
\(290\) −10.1906 −0.598411
\(291\) −0.589843 −0.0345772
\(292\) −15.7188 −0.919874
\(293\) 5.69521 0.332717 0.166359 0.986065i \(-0.446799\pi\)
0.166359 + 0.986065i \(0.446799\pi\)
\(294\) 0 0
\(295\) −1.96260 −0.114267
\(296\) −1.45459 −0.0845461
\(297\) −2.08680 −0.121088
\(298\) 10.7925 0.625192
\(299\) −3.64405 −0.210741
\(300\) 5.58355 0.322367
\(301\) 0 0
\(302\) −24.1063 −1.38716
\(303\) −6.22834 −0.357809
\(304\) 2.82281 0.161899
\(305\) −11.7502 −0.672817
\(306\) −0.548767 −0.0313709
\(307\) 17.0596 0.973642 0.486821 0.873502i \(-0.338156\pi\)
0.486821 + 0.873502i \(0.338156\pi\)
\(308\) 0 0
\(309\) 18.6455 1.06070
\(310\) −10.4158 −0.591576
\(311\) −16.5243 −0.937005 −0.468503 0.883462i \(-0.655206\pi\)
−0.468503 + 0.883462i \(0.655206\pi\)
\(312\) −4.28352 −0.242507
\(313\) −1.49388 −0.0844389 −0.0422195 0.999108i \(-0.513443\pi\)
−0.0422195 + 0.999108i \(0.513443\pi\)
\(314\) 15.4411 0.871391
\(315\) 0 0
\(316\) 23.3112 1.31135
\(317\) −18.2234 −1.02353 −0.511765 0.859126i \(-0.671008\pi\)
−0.511765 + 0.859126i \(0.671008\pi\)
\(318\) 4.26230 0.239018
\(319\) −12.2988 −0.688602
\(320\) 2.17922 0.121822
\(321\) 0.648883 0.0362171
\(322\) 0 0
\(323\) −0.173515 −0.00965461
\(324\) 1.35859 0.0754770
\(325\) −14.9764 −0.830742
\(326\) −7.61897 −0.421976
\(327\) 13.6533 0.755031
\(328\) 9.89786 0.546518
\(329\) 0 0
\(330\) −3.60825 −0.198627
\(331\) −22.9549 −1.26171 −0.630857 0.775899i \(-0.717297\pi\)
−0.630857 + 0.775899i \(0.717297\pi\)
\(332\) 14.9091 0.818241
\(333\) −1.23744 −0.0678111
\(334\) 5.57939 0.305291
\(335\) 3.90381 0.213288
\(336\) 0 0
\(337\) 22.2266 1.21076 0.605381 0.795936i \(-0.293021\pi\)
0.605381 + 0.795936i \(0.293021\pi\)
\(338\) −0.511486 −0.0278212
\(339\) −0.547279 −0.0297241
\(340\) −0.383826 −0.0208159
\(341\) −12.5706 −0.680737
\(342\) 1.06195 0.0574237
\(343\) 0 0
\(344\) −10.6802 −0.575840
\(345\) −0.943490 −0.0507958
\(346\) −29.4218 −1.58173
\(347\) −5.40842 −0.290339 −0.145170 0.989407i \(-0.546373\pi\)
−0.145170 + 0.989407i \(0.546373\pi\)
\(348\) 8.00700 0.429220
\(349\) 10.3938 0.556369 0.278184 0.960528i \(-0.410267\pi\)
0.278184 + 0.960528i \(0.410267\pi\)
\(350\) 0 0
\(351\) −3.64405 −0.194505
\(352\) 13.7241 0.731495
\(353\) −13.0691 −0.695596 −0.347798 0.937570i \(-0.613070\pi\)
−0.347798 + 0.937570i \(0.613070\pi\)
\(354\) 3.81217 0.202615
\(355\) 10.9859 0.583071
\(356\) 3.63505 0.192657
\(357\) 0 0
\(358\) 9.79565 0.517716
\(359\) −19.2232 −1.01456 −0.507281 0.861781i \(-0.669349\pi\)
−0.507281 + 0.861781i \(0.669349\pi\)
\(360\) −1.10906 −0.0584525
\(361\) −18.6642 −0.982327
\(362\) 21.2955 1.11927
\(363\) 6.64526 0.348786
\(364\) 0 0
\(365\) 10.9162 0.571378
\(366\) 22.8238 1.19302
\(367\) −1.77510 −0.0926592 −0.0463296 0.998926i \(-0.514752\pi\)
−0.0463296 + 0.998926i \(0.514752\pi\)
\(368\) 4.87142 0.253940
\(369\) 8.42024 0.438340
\(370\) −2.13963 −0.111234
\(371\) 0 0
\(372\) 8.18395 0.424318
\(373\) −7.44021 −0.385240 −0.192620 0.981273i \(-0.561698\pi\)
−0.192620 + 0.981273i \(0.561698\pi\)
\(374\) −1.14517 −0.0592152
\(375\) −8.59503 −0.443845
\(376\) 5.95085 0.306892
\(377\) −21.4767 −1.10611
\(378\) 0 0
\(379\) −4.31050 −0.221415 −0.110708 0.993853i \(-0.535312\pi\)
−0.110708 + 0.993853i \(0.535312\pi\)
\(380\) 0.742763 0.0381029
\(381\) 1.01278 0.0518861
\(382\) 31.2927 1.60107
\(383\) 34.6915 1.77265 0.886326 0.463062i \(-0.153249\pi\)
0.886326 + 0.463062i \(0.153249\pi\)
\(384\) 8.92027 0.455211
\(385\) 0 0
\(386\) 8.52275 0.433796
\(387\) −9.08583 −0.461858
\(388\) 0.801352 0.0406825
\(389\) 33.1439 1.68046 0.840230 0.542230i \(-0.182420\pi\)
0.840230 + 0.542230i \(0.182420\pi\)
\(390\) −6.30086 −0.319056
\(391\) −0.299440 −0.0151433
\(392\) 0 0
\(393\) 15.0311 0.758217
\(394\) −0.467164 −0.0235354
\(395\) −16.1888 −0.814545
\(396\) 2.83510 0.142469
\(397\) −28.4564 −1.42818 −0.714092 0.700052i \(-0.753160\pi\)
−0.714092 + 0.700052i \(0.753160\pi\)
\(398\) −10.0355 −0.503035
\(399\) 0 0
\(400\) 20.0207 1.00103
\(401\) −37.8811 −1.89169 −0.945846 0.324615i \(-0.894765\pi\)
−0.945846 + 0.324615i \(0.894765\pi\)
\(402\) −7.58281 −0.378196
\(403\) −21.9513 −1.09347
\(404\) 8.46174 0.420987
\(405\) −0.943490 −0.0468824
\(406\) 0 0
\(407\) −2.58228 −0.127999
\(408\) −0.351987 −0.0174260
\(409\) 0.957146 0.0473278 0.0236639 0.999720i \(-0.492467\pi\)
0.0236639 + 0.999720i \(0.492467\pi\)
\(410\) 14.5593 0.719031
\(411\) −9.27867 −0.457683
\(412\) −25.3315 −1.24799
\(413\) 0 0
\(414\) 1.83264 0.0900696
\(415\) −10.3538 −0.508249
\(416\) 23.9655 1.17500
\(417\) −6.58750 −0.322591
\(418\) 2.21608 0.108392
\(419\) −40.7600 −1.99126 −0.995628 0.0934066i \(-0.970224\pi\)
−0.995628 + 0.0934066i \(0.970224\pi\)
\(420\) 0 0
\(421\) −26.0184 −1.26806 −0.634031 0.773308i \(-0.718601\pi\)
−0.634031 + 0.773308i \(0.718601\pi\)
\(422\) 5.83878 0.284227
\(423\) 5.06246 0.246145
\(424\) 2.73390 0.132770
\(425\) −1.23065 −0.0596951
\(426\) −21.3391 −1.03388
\(427\) 0 0
\(428\) −0.881564 −0.0426120
\(429\) −7.60440 −0.367144
\(430\) −15.7101 −0.757610
\(431\) −31.4441 −1.51461 −0.757305 0.653061i \(-0.773485\pi\)
−0.757305 + 0.653061i \(0.773485\pi\)
\(432\) 4.87142 0.234376
\(433\) 3.61281 0.173621 0.0868103 0.996225i \(-0.472333\pi\)
0.0868103 + 0.996225i \(0.472333\pi\)
\(434\) 0 0
\(435\) −5.56058 −0.266609
\(436\) −18.5492 −0.888347
\(437\) 0.579464 0.0277195
\(438\) −21.2037 −1.01315
\(439\) 18.0511 0.861530 0.430765 0.902464i \(-0.358244\pi\)
0.430765 + 0.902464i \(0.358244\pi\)
\(440\) −2.31438 −0.110334
\(441\) 0 0
\(442\) −1.99973 −0.0951177
\(443\) −35.3301 −1.67858 −0.839291 0.543683i \(-0.817030\pi\)
−0.839291 + 0.543683i \(0.817030\pi\)
\(444\) 1.68116 0.0797844
\(445\) −2.52441 −0.119669
\(446\) 41.7246 1.97571
\(447\) 5.88903 0.278541
\(448\) 0 0
\(449\) 5.95593 0.281078 0.140539 0.990075i \(-0.455116\pi\)
0.140539 + 0.990075i \(0.455116\pi\)
\(450\) 7.53185 0.355055
\(451\) 17.5714 0.827403
\(452\) 0.743525 0.0349725
\(453\) −13.1539 −0.618022
\(454\) −19.1549 −0.898986
\(455\) 0 0
\(456\) 0.681151 0.0318978
\(457\) 31.9674 1.49537 0.747685 0.664053i \(-0.231165\pi\)
0.747685 + 0.664053i \(0.231165\pi\)
\(458\) 19.7336 0.922091
\(459\) −0.299440 −0.0139767
\(460\) 1.28181 0.0597648
\(461\) 7.10536 0.330930 0.165465 0.986216i \(-0.447088\pi\)
0.165465 + 0.986216i \(0.447088\pi\)
\(462\) 0 0
\(463\) −42.0816 −1.95570 −0.977848 0.209314i \(-0.932877\pi\)
−0.977848 + 0.209314i \(0.932877\pi\)
\(464\) 28.7103 1.33284
\(465\) −5.68346 −0.263564
\(466\) −21.5350 −0.997588
\(467\) 26.4507 1.22399 0.611995 0.790861i \(-0.290367\pi\)
0.611995 + 0.790861i \(0.290367\pi\)
\(468\) 4.95075 0.228849
\(469\) 0 0
\(470\) 8.75341 0.403765
\(471\) 8.42558 0.388230
\(472\) 2.44518 0.112549
\(473\) −18.9603 −0.871796
\(474\) 31.4452 1.44433
\(475\) 2.38150 0.109271
\(476\) 0 0
\(477\) 2.32577 0.106490
\(478\) 22.9980 1.05191
\(479\) −25.2828 −1.15520 −0.577601 0.816319i \(-0.696011\pi\)
−0.577601 + 0.816319i \(0.696011\pi\)
\(480\) 6.20496 0.283216
\(481\) −4.50928 −0.205605
\(482\) 19.4818 0.887373
\(483\) 0 0
\(484\) −9.02816 −0.410371
\(485\) −0.556511 −0.0252698
\(486\) 1.83264 0.0831304
\(487\) 19.9765 0.905222 0.452611 0.891708i \(-0.350493\pi\)
0.452611 + 0.891708i \(0.350493\pi\)
\(488\) 14.6395 0.662700
\(489\) −4.15736 −0.188002
\(490\) 0 0
\(491\) −41.0091 −1.85072 −0.925358 0.379095i \(-0.876235\pi\)
−0.925358 + 0.379095i \(0.876235\pi\)
\(492\) −11.4396 −0.515738
\(493\) −1.76479 −0.0794821
\(494\) 3.86980 0.174111
\(495\) −1.96887 −0.0884943
\(496\) 29.3448 1.31762
\(497\) 0 0
\(498\) 20.1114 0.901212
\(499\) 14.0469 0.628825 0.314412 0.949286i \(-0.398193\pi\)
0.314412 + 0.949286i \(0.398193\pi\)
\(500\) 11.6771 0.522215
\(501\) 3.04445 0.136016
\(502\) 31.9641 1.42663
\(503\) 12.8463 0.572787 0.286393 0.958112i \(-0.407544\pi\)
0.286393 + 0.958112i \(0.407544\pi\)
\(504\) 0 0
\(505\) −5.87638 −0.261495
\(506\) 3.82436 0.170014
\(507\) −0.279097 −0.0123951
\(508\) −1.37594 −0.0610476
\(509\) −26.7812 −1.18706 −0.593528 0.804813i \(-0.702266\pi\)
−0.593528 + 0.804813i \(0.702266\pi\)
\(510\) −0.517756 −0.0229266
\(511\) 0 0
\(512\) −20.5848 −0.909730
\(513\) 0.579464 0.0255839
\(514\) −34.3182 −1.51371
\(515\) 17.5918 0.775188
\(516\) 12.3439 0.543409
\(517\) 10.5644 0.464620
\(518\) 0 0
\(519\) −16.0543 −0.704705
\(520\) −4.04146 −0.177230
\(521\) −26.8305 −1.17546 −0.587732 0.809056i \(-0.699979\pi\)
−0.587732 + 0.809056i \(0.699979\pi\)
\(522\) 10.8009 0.472744
\(523\) −16.0946 −0.703768 −0.351884 0.936044i \(-0.614459\pi\)
−0.351884 + 0.936044i \(0.614459\pi\)
\(524\) −20.4210 −0.892095
\(525\) 0 0
\(526\) −54.8118 −2.38991
\(527\) −1.80379 −0.0785743
\(528\) 10.1657 0.442404
\(529\) 1.00000 0.0434783
\(530\) 4.02144 0.174680
\(531\) 2.08015 0.0902708
\(532\) 0 0
\(533\) 30.6838 1.32906
\(534\) 4.90345 0.212193
\(535\) 0.612215 0.0264683
\(536\) −4.86373 −0.210081
\(537\) 5.34509 0.230658
\(538\) −41.5022 −1.78929
\(539\) 0 0
\(540\) 1.28181 0.0551604
\(541\) −0.676959 −0.0291047 −0.0145524 0.999894i \(-0.504632\pi\)
−0.0145524 + 0.999894i \(0.504632\pi\)
\(542\) 23.3412 1.00259
\(543\) 11.6201 0.498667
\(544\) 1.96930 0.0844330
\(545\) 12.8818 0.551795
\(546\) 0 0
\(547\) −8.38777 −0.358635 −0.179318 0.983791i \(-0.557389\pi\)
−0.179318 + 0.983791i \(0.557389\pi\)
\(548\) 12.6059 0.538496
\(549\) 12.4540 0.531525
\(550\) 15.7175 0.670195
\(551\) 3.41515 0.145490
\(552\) 1.17548 0.0500320
\(553\) 0 0
\(554\) 53.1699 2.25897
\(555\) −1.16751 −0.0495579
\(556\) 8.94968 0.379551
\(557\) −2.49888 −0.105881 −0.0529404 0.998598i \(-0.516859\pi\)
−0.0529404 + 0.998598i \(0.516859\pi\)
\(558\) 11.0396 0.467344
\(559\) −33.1092 −1.40037
\(560\) 0 0
\(561\) −0.624872 −0.0263821
\(562\) 22.8103 0.962193
\(563\) −40.4580 −1.70510 −0.852551 0.522645i \(-0.824945\pi\)
−0.852551 + 0.522645i \(0.824945\pi\)
\(564\) −6.87779 −0.289607
\(565\) −0.516352 −0.0217231
\(566\) −17.1608 −0.721323
\(567\) 0 0
\(568\) −13.6872 −0.574303
\(569\) −12.4802 −0.523199 −0.261599 0.965177i \(-0.584250\pi\)
−0.261599 + 0.965177i \(0.584250\pi\)
\(570\) 1.00194 0.0419666
\(571\) −30.9927 −1.29700 −0.648502 0.761213i \(-0.724604\pi\)
−0.648502 + 0.761213i \(0.724604\pi\)
\(572\) 10.3312 0.431971
\(573\) 17.0751 0.713324
\(574\) 0 0
\(575\) 4.10983 0.171392
\(576\) −2.30974 −0.0962394
\(577\) 10.5302 0.438379 0.219190 0.975682i \(-0.429659\pi\)
0.219190 + 0.975682i \(0.429659\pi\)
\(578\) 30.9906 1.28904
\(579\) 4.65052 0.193269
\(580\) 7.55452 0.313684
\(581\) 0 0
\(582\) 1.08097 0.0448077
\(583\) 4.85341 0.201008
\(584\) −13.6003 −0.562786
\(585\) −3.43812 −0.142149
\(586\) −10.4373 −0.431160
\(587\) 7.06244 0.291498 0.145749 0.989322i \(-0.453441\pi\)
0.145749 + 0.989322i \(0.453441\pi\)
\(588\) 0 0
\(589\) 3.49062 0.143828
\(590\) 3.59675 0.148076
\(591\) −0.254912 −0.0104857
\(592\) 6.02806 0.247752
\(593\) 39.8925 1.63819 0.819094 0.573660i \(-0.194477\pi\)
0.819094 + 0.573660i \(0.194477\pi\)
\(594\) 3.82436 0.156916
\(595\) 0 0
\(596\) −8.00075 −0.327723
\(597\) −5.47597 −0.224117
\(598\) 6.67825 0.273094
\(599\) 11.8646 0.484773 0.242386 0.970180i \(-0.422070\pi\)
0.242386 + 0.970180i \(0.422070\pi\)
\(600\) 4.83104 0.197226
\(601\) 15.0105 0.612290 0.306145 0.951985i \(-0.400961\pi\)
0.306145 + 0.951985i \(0.400961\pi\)
\(602\) 0 0
\(603\) −4.13763 −0.168498
\(604\) 17.8706 0.727146
\(605\) 6.26974 0.254901
\(606\) 11.4143 0.463676
\(607\) −4.19990 −0.170469 −0.0852343 0.996361i \(-0.527164\pi\)
−0.0852343 + 0.996361i \(0.527164\pi\)
\(608\) −3.81090 −0.154553
\(609\) 0 0
\(610\) 21.5340 0.871887
\(611\) 18.4479 0.746321
\(612\) 0.406815 0.0164445
\(613\) 30.9865 1.25153 0.625766 0.780011i \(-0.284786\pi\)
0.625766 + 0.780011i \(0.284786\pi\)
\(614\) −31.2641 −1.26172
\(615\) 7.94441 0.320349
\(616\) 0 0
\(617\) 1.93041 0.0777153 0.0388577 0.999245i \(-0.487628\pi\)
0.0388577 + 0.999245i \(0.487628\pi\)
\(618\) −34.1705 −1.37454
\(619\) 33.2844 1.33781 0.668907 0.743346i \(-0.266763\pi\)
0.668907 + 0.743346i \(0.266763\pi\)
\(620\) 7.72147 0.310102
\(621\) 1.00000 0.0401286
\(622\) 30.2831 1.21424
\(623\) 0 0
\(624\) 17.7517 0.710636
\(625\) 12.4398 0.497593
\(626\) 2.73775 0.109422
\(627\) 1.20923 0.0482918
\(628\) −11.4469 −0.456780
\(629\) −0.370538 −0.0147743
\(630\) 0 0
\(631\) 4.49815 0.179068 0.0895342 0.995984i \(-0.471462\pi\)
0.0895342 + 0.995984i \(0.471462\pi\)
\(632\) 20.1694 0.802297
\(633\) 3.18599 0.126632
\(634\) 33.3971 1.32637
\(635\) 0.955543 0.0379196
\(636\) −3.15975 −0.125292
\(637\) 0 0
\(638\) 22.5394 0.892343
\(639\) −11.6439 −0.460625
\(640\) 8.41618 0.332679
\(641\) −26.8209 −1.05936 −0.529681 0.848197i \(-0.677688\pi\)
−0.529681 + 0.848197i \(0.677688\pi\)
\(642\) −1.18917 −0.0469329
\(643\) −21.1912 −0.835701 −0.417850 0.908516i \(-0.637216\pi\)
−0.417850 + 0.908516i \(0.637216\pi\)
\(644\) 0 0
\(645\) −8.57238 −0.337537
\(646\) 0.317991 0.0125112
\(647\) 3.28238 0.129044 0.0645218 0.997916i \(-0.479448\pi\)
0.0645218 + 0.997916i \(0.479448\pi\)
\(648\) 1.17548 0.0461774
\(649\) 4.34086 0.170393
\(650\) 27.4464 1.07654
\(651\) 0 0
\(652\) 5.64813 0.221198
\(653\) 32.7912 1.28322 0.641610 0.767031i \(-0.278267\pi\)
0.641610 + 0.767031i \(0.278267\pi\)
\(654\) −25.0217 −0.978426
\(655\) 14.1816 0.554123
\(656\) −41.0185 −1.60150
\(657\) −11.5700 −0.451388
\(658\) 0 0
\(659\) 11.1625 0.434828 0.217414 0.976079i \(-0.430238\pi\)
0.217414 + 0.976079i \(0.430238\pi\)
\(660\) 2.67488 0.104120
\(661\) 8.39461 0.326513 0.163256 0.986584i \(-0.447800\pi\)
0.163256 + 0.986584i \(0.447800\pi\)
\(662\) 42.0681 1.63502
\(663\) −1.09117 −0.0423777
\(664\) 12.8997 0.500606
\(665\) 0 0
\(666\) 2.26778 0.0878747
\(667\) 5.89363 0.228202
\(668\) −4.13614 −0.160032
\(669\) 22.7674 0.880239
\(670\) −7.15430 −0.276395
\(671\) 25.9891 1.00330
\(672\) 0 0
\(673\) 30.3183 1.16869 0.584343 0.811507i \(-0.301352\pi\)
0.584343 + 0.811507i \(0.301352\pi\)
\(674\) −40.7335 −1.56900
\(675\) 4.10983 0.158187
\(676\) 0.379177 0.0145837
\(677\) −10.3013 −0.395910 −0.197955 0.980211i \(-0.563430\pi\)
−0.197955 + 0.980211i \(0.563430\pi\)
\(678\) 1.00297 0.0385187
\(679\) 0 0
\(680\) −0.332096 −0.0127353
\(681\) −10.4521 −0.400524
\(682\) 23.0375 0.882151
\(683\) −48.1741 −1.84333 −0.921666 0.387984i \(-0.873172\pi\)
−0.921666 + 0.387984i \(0.873172\pi\)
\(684\) −0.787251 −0.0301013
\(685\) −8.75433 −0.334486
\(686\) 0 0
\(687\) 10.7678 0.410819
\(688\) 44.2608 1.68743
\(689\) 8.47521 0.322880
\(690\) 1.72908 0.0658250
\(691\) −36.7195 −1.39687 −0.698437 0.715671i \(-0.746121\pi\)
−0.698437 + 0.715671i \(0.746121\pi\)
\(692\) 21.8111 0.829135
\(693\) 0 0
\(694\) 9.91171 0.376243
\(695\) −6.21523 −0.235757
\(696\) 6.92787 0.262600
\(697\) 2.52136 0.0955032
\(698\) −19.0482 −0.720985
\(699\) −11.7508 −0.444455
\(700\) 0 0
\(701\) 35.3581 1.33546 0.667728 0.744406i \(-0.267267\pi\)
0.667728 + 0.744406i \(0.267267\pi\)
\(702\) 6.67825 0.252054
\(703\) 0.717049 0.0270440
\(704\) −4.81998 −0.181660
\(705\) 4.77638 0.179889
\(706\) 23.9509 0.901405
\(707\) 0 0
\(708\) −2.82606 −0.106210
\(709\) −10.9962 −0.412969 −0.206485 0.978450i \(-0.566202\pi\)
−0.206485 + 0.978450i \(0.566202\pi\)
\(710\) −20.1332 −0.755587
\(711\) 17.1584 0.643490
\(712\) 3.14514 0.117869
\(713\) 6.02387 0.225596
\(714\) 0 0
\(715\) −7.17468 −0.268318
\(716\) −7.26176 −0.271385
\(717\) 12.5491 0.468655
\(718\) 35.2293 1.31475
\(719\) −36.5754 −1.36403 −0.682016 0.731337i \(-0.738897\pi\)
−0.682016 + 0.731337i \(0.738897\pi\)
\(720\) 4.59613 0.171288
\(721\) 0 0
\(722\) 34.2049 1.27297
\(723\) 10.6305 0.395351
\(724\) −15.7869 −0.586716
\(725\) 24.2218 0.899575
\(726\) −12.1784 −0.451983
\(727\) −12.9513 −0.480337 −0.240169 0.970731i \(-0.577203\pi\)
−0.240169 + 0.970731i \(0.577203\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −20.0054 −0.740434
\(731\) −2.72066 −0.100627
\(732\) −16.9199 −0.625376
\(733\) 6.26355 0.231350 0.115675 0.993287i \(-0.463097\pi\)
0.115675 + 0.993287i \(0.463097\pi\)
\(734\) 3.25312 0.120075
\(735\) 0 0
\(736\) −6.57660 −0.242417
\(737\) −8.63442 −0.318053
\(738\) −15.4313 −0.568034
\(739\) 32.4215 1.19264 0.596322 0.802745i \(-0.296628\pi\)
0.596322 + 0.802745i \(0.296628\pi\)
\(740\) 1.58616 0.0583084
\(741\) 2.11159 0.0775714
\(742\) 0 0
\(743\) −4.32354 −0.158615 −0.0793076 0.996850i \(-0.525271\pi\)
−0.0793076 + 0.996850i \(0.525271\pi\)
\(744\) 7.08097 0.259601
\(745\) 5.55624 0.203565
\(746\) 13.6353 0.499222
\(747\) 10.9740 0.401516
\(748\) 0.848942 0.0310404
\(749\) 0 0
\(750\) 15.7516 0.575168
\(751\) −12.0432 −0.439461 −0.219730 0.975561i \(-0.570518\pi\)
−0.219730 + 0.975561i \(0.570518\pi\)
\(752\) −24.6614 −0.899308
\(753\) 17.4415 0.635605
\(754\) 39.3591 1.43337
\(755\) −12.4105 −0.451665
\(756\) 0 0
\(757\) 17.8806 0.649880 0.324940 0.945735i \(-0.394656\pi\)
0.324940 + 0.945735i \(0.394656\pi\)
\(758\) 7.89961 0.286927
\(759\) 2.08680 0.0757461
\(760\) 0.642659 0.0233117
\(761\) 26.2960 0.953231 0.476615 0.879112i \(-0.341863\pi\)
0.476615 + 0.879112i \(0.341863\pi\)
\(762\) −1.85606 −0.0672379
\(763\) 0 0
\(764\) −23.1980 −0.839276
\(765\) −0.282519 −0.0102145
\(766\) −63.5772 −2.29714
\(767\) 7.58016 0.273704
\(768\) −20.9672 −0.756588
\(769\) −32.2632 −1.16344 −0.581720 0.813389i \(-0.697620\pi\)
−0.581720 + 0.813389i \(0.697620\pi\)
\(770\) 0 0
\(771\) −18.7261 −0.674403
\(772\) −6.31813 −0.227394
\(773\) −10.5757 −0.380380 −0.190190 0.981747i \(-0.560910\pi\)
−0.190190 + 0.981747i \(0.560910\pi\)
\(774\) 16.6511 0.598511
\(775\) 24.7571 0.889301
\(776\) 0.693351 0.0248899
\(777\) 0 0
\(778\) −60.7409 −2.17767
\(779\) −4.87922 −0.174816
\(780\) 4.67098 0.167248
\(781\) −24.2985 −0.869468
\(782\) 0.548767 0.0196239
\(783\) 5.89363 0.210621
\(784\) 0 0
\(785\) 7.94945 0.283728
\(786\) −27.5466 −0.982554
\(787\) −27.5234 −0.981103 −0.490552 0.871412i \(-0.663205\pi\)
−0.490552 + 0.871412i \(0.663205\pi\)
\(788\) 0.346320 0.0123371
\(789\) −29.9086 −1.06477
\(790\) 29.6683 1.05555
\(791\) 0 0
\(792\) 2.45300 0.0871636
\(793\) 45.3831 1.61160
\(794\) 52.1504 1.85075
\(795\) 2.19434 0.0778251
\(796\) 7.43958 0.263689
\(797\) −43.4629 −1.53953 −0.769767 0.638324i \(-0.779628\pi\)
−0.769767 + 0.638324i \(0.779628\pi\)
\(798\) 0 0
\(799\) 1.51590 0.0536288
\(800\) −27.0287 −0.955609
\(801\) 2.67561 0.0945381
\(802\) 69.4226 2.45140
\(803\) −24.1442 −0.852032
\(804\) 5.62133 0.198249
\(805\) 0 0
\(806\) 40.2289 1.41700
\(807\) −22.6461 −0.797180
\(808\) 7.32132 0.257563
\(809\) 42.8145 1.50528 0.752639 0.658434i \(-0.228781\pi\)
0.752639 + 0.658434i \(0.228781\pi\)
\(810\) 1.72908 0.0607537
\(811\) 45.2610 1.58933 0.794664 0.607049i \(-0.207647\pi\)
0.794664 + 0.607049i \(0.207647\pi\)
\(812\) 0 0
\(813\) 12.7364 0.446684
\(814\) 4.73240 0.165871
\(815\) −3.92243 −0.137397
\(816\) 1.45870 0.0510646
\(817\) 5.26491 0.184196
\(818\) −1.75411 −0.0613310
\(819\) 0 0
\(820\) −10.7932 −0.376913
\(821\) −36.3213 −1.26762 −0.633812 0.773487i \(-0.718511\pi\)
−0.633812 + 0.773487i \(0.718511\pi\)
\(822\) 17.0045 0.593100
\(823\) 52.1001 1.81610 0.908048 0.418865i \(-0.137572\pi\)
0.908048 + 0.418865i \(0.137572\pi\)
\(824\) −21.9175 −0.763532
\(825\) 8.57639 0.298592
\(826\) 0 0
\(827\) −21.6989 −0.754545 −0.377273 0.926102i \(-0.623138\pi\)
−0.377273 + 0.926102i \(0.623138\pi\)
\(828\) −1.35859 −0.0472141
\(829\) −13.5755 −0.471497 −0.235749 0.971814i \(-0.575754\pi\)
−0.235749 + 0.971814i \(0.575754\pi\)
\(830\) 18.9749 0.658627
\(831\) 29.0127 1.00644
\(832\) −8.41682 −0.291801
\(833\) 0 0
\(834\) 12.0725 0.418038
\(835\) 2.87241 0.0994037
\(836\) −1.64284 −0.0568187
\(837\) 6.02387 0.208216
\(838\) 74.6986 2.58042
\(839\) −40.6915 −1.40483 −0.702413 0.711770i \(-0.747894\pi\)
−0.702413 + 0.711770i \(0.747894\pi\)
\(840\) 0 0
\(841\) 5.73489 0.197755
\(842\) 47.6826 1.64325
\(843\) 12.4466 0.428685
\(844\) −4.32843 −0.148991
\(845\) −0.263325 −0.00905866
\(846\) −9.27770 −0.318974
\(847\) 0 0
\(848\) −11.3298 −0.389066
\(849\) −9.36397 −0.321370
\(850\) 2.25534 0.0773575
\(851\) 1.23744 0.0424187
\(852\) 15.8192 0.541958
\(853\) −53.5882 −1.83482 −0.917412 0.397938i \(-0.869726\pi\)
−0.917412 + 0.397938i \(0.869726\pi\)
\(854\) 0 0
\(855\) 0.546718 0.0186974
\(856\) −0.762752 −0.0260703
\(857\) −15.1853 −0.518719 −0.259360 0.965781i \(-0.583511\pi\)
−0.259360 + 0.965781i \(0.583511\pi\)
\(858\) 13.9362 0.475773
\(859\) −37.9385 −1.29444 −0.647222 0.762302i \(-0.724069\pi\)
−0.647222 + 0.762302i \(0.724069\pi\)
\(860\) 11.6463 0.397136
\(861\) 0 0
\(862\) 57.6259 1.96275
\(863\) 34.9640 1.19019 0.595094 0.803656i \(-0.297115\pi\)
0.595094 + 0.803656i \(0.297115\pi\)
\(864\) −6.57660 −0.223741
\(865\) −15.1471 −0.515015
\(866\) −6.62099 −0.224991
\(867\) 16.9103 0.574305
\(868\) 0 0
\(869\) 35.8062 1.21464
\(870\) 10.1906 0.345493
\(871\) −15.0777 −0.510890
\(872\) −16.0493 −0.543497
\(873\) 0.589843 0.0199632
\(874\) −1.06195 −0.0359210
\(875\) 0 0
\(876\) 15.7188 0.531089
\(877\) 39.0428 1.31838 0.659191 0.751976i \(-0.270899\pi\)
0.659191 + 0.751976i \(0.270899\pi\)
\(878\) −33.0812 −1.11644
\(879\) −5.69521 −0.192095
\(880\) 9.59121 0.323320
\(881\) 21.5235 0.725145 0.362572 0.931956i \(-0.381899\pi\)
0.362572 + 0.931956i \(0.381899\pi\)
\(882\) 0 0
\(883\) −17.0630 −0.574217 −0.287109 0.957898i \(-0.592694\pi\)
−0.287109 + 0.957898i \(0.592694\pi\)
\(884\) 1.48245 0.0498603
\(885\) 1.96260 0.0659720
\(886\) 64.7474 2.17523
\(887\) 43.1464 1.44871 0.724357 0.689425i \(-0.242137\pi\)
0.724357 + 0.689425i \(0.242137\pi\)
\(888\) 1.45459 0.0488127
\(889\) 0 0
\(890\) 4.62635 0.155076
\(891\) 2.08680 0.0699104
\(892\) −30.9315 −1.03566
\(893\) −2.93351 −0.0981663
\(894\) −10.7925 −0.360955
\(895\) 5.04304 0.168570
\(896\) 0 0
\(897\) 3.64405 0.121671
\(898\) −10.9151 −0.364242
\(899\) 35.5025 1.18407
\(900\) −5.58355 −0.186118
\(901\) 0.696428 0.0232014
\(902\) −32.2020 −1.07221
\(903\) 0 0
\(904\) 0.643318 0.0213964
\(905\) 10.9635 0.364438
\(906\) 24.1063 0.800879
\(907\) −9.48595 −0.314976 −0.157488 0.987521i \(-0.550340\pi\)
−0.157488 + 0.987521i \(0.550340\pi\)
\(908\) 14.2000 0.471245
\(909\) 6.22834 0.206581
\(910\) 0 0
\(911\) 17.7773 0.588988 0.294494 0.955653i \(-0.404849\pi\)
0.294494 + 0.955653i \(0.404849\pi\)
\(912\) −2.82281 −0.0934725
\(913\) 22.9005 0.757895
\(914\) −58.5848 −1.93781
\(915\) 11.7502 0.388451
\(916\) −14.6290 −0.483357
\(917\) 0 0
\(918\) 0.548767 0.0181120
\(919\) 3.50669 0.115675 0.0578376 0.998326i \(-0.481579\pi\)
0.0578376 + 0.998326i \(0.481579\pi\)
\(920\) 1.10906 0.0365645
\(921\) −17.0596 −0.562132
\(922\) −13.0216 −0.428843
\(923\) −42.4309 −1.39663
\(924\) 0 0
\(925\) 5.08565 0.167215
\(926\) 77.1206 2.53434
\(927\) −18.6455 −0.612398
\(928\) −38.7601 −1.27236
\(929\) 15.2444 0.500153 0.250076 0.968226i \(-0.419544\pi\)
0.250076 + 0.968226i \(0.419544\pi\)
\(930\) 10.4158 0.341546
\(931\) 0 0
\(932\) 15.9644 0.522932
\(933\) 16.5243 0.540980
\(934\) −48.4747 −1.58614
\(935\) −0.589560 −0.0192807
\(936\) 4.28352 0.140011
\(937\) −16.0219 −0.523414 −0.261707 0.965147i \(-0.584285\pi\)
−0.261707 + 0.965147i \(0.584285\pi\)
\(938\) 0 0
\(939\) 1.49388 0.0487508
\(940\) −6.48912 −0.211652
\(941\) −8.29938 −0.270552 −0.135276 0.990808i \(-0.543192\pi\)
−0.135276 + 0.990808i \(0.543192\pi\)
\(942\) −15.4411 −0.503098
\(943\) −8.42024 −0.274201
\(944\) −10.1333 −0.329810
\(945\) 0 0
\(946\) 34.7475 1.12974
\(947\) 48.3210 1.57022 0.785111 0.619355i \(-0.212606\pi\)
0.785111 + 0.619355i \(0.212606\pi\)
\(948\) −23.3112 −0.757111
\(949\) −42.1616 −1.36862
\(950\) −4.36443 −0.141601
\(951\) 18.2234 0.590935
\(952\) 0 0
\(953\) 12.0181 0.389303 0.194652 0.980872i \(-0.437642\pi\)
0.194652 + 0.980872i \(0.437642\pi\)
\(954\) −4.26230 −0.137997
\(955\) 16.1102 0.521314
\(956\) −17.0490 −0.551405
\(957\) 12.2988 0.397565
\(958\) 46.3344 1.49700
\(959\) 0 0
\(960\) −2.17922 −0.0703340
\(961\) 5.28706 0.170550
\(962\) 8.26390 0.266439
\(963\) −0.648883 −0.0209100
\(964\) −14.4424 −0.465158
\(965\) 4.38772 0.141246
\(966\) 0 0
\(967\) 20.9914 0.675036 0.337518 0.941319i \(-0.390413\pi\)
0.337518 + 0.941319i \(0.390413\pi\)
\(968\) −7.81141 −0.251068
\(969\) 0.173515 0.00557409
\(970\) 1.01989 0.0327466
\(971\) −5.29526 −0.169933 −0.0849665 0.996384i \(-0.527078\pi\)
−0.0849665 + 0.996384i \(0.527078\pi\)
\(972\) −1.35859 −0.0435767
\(973\) 0 0
\(974\) −36.6098 −1.17305
\(975\) 14.9764 0.479629
\(976\) −60.6688 −1.94196
\(977\) −26.6768 −0.853466 −0.426733 0.904378i \(-0.640336\pi\)
−0.426733 + 0.904378i \(0.640336\pi\)
\(978\) 7.61897 0.243628
\(979\) 5.58347 0.178448
\(980\) 0 0
\(981\) −13.6533 −0.435917
\(982\) 75.1551 2.39830
\(983\) 6.63941 0.211764 0.105882 0.994379i \(-0.466233\pi\)
0.105882 + 0.994379i \(0.466233\pi\)
\(984\) −9.89786 −0.315532
\(985\) −0.240507 −0.00766320
\(986\) 3.23423 0.102999
\(987\) 0 0
\(988\) −2.86878 −0.0912681
\(989\) 9.08583 0.288912
\(990\) 3.60825 0.114678
\(991\) 30.3443 0.963917 0.481958 0.876194i \(-0.339926\pi\)
0.481958 + 0.876194i \(0.339926\pi\)
\(992\) −39.6166 −1.25783
\(993\) 22.9549 0.728451
\(994\) 0 0
\(995\) −5.16652 −0.163790
\(996\) −14.9091 −0.472412
\(997\) 46.9848 1.48802 0.744011 0.668167i \(-0.232921\pi\)
0.744011 + 0.668167i \(0.232921\pi\)
\(998\) −25.7429 −0.814878
\(999\) 1.23744 0.0391507
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 3381.2.a.bc.1.2 6
7.2 even 3 483.2.i.f.277.5 12
7.4 even 3 483.2.i.f.415.5 yes 12
7.6 odd 2 3381.2.a.bd.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.f.277.5 12 7.2 even 3
483.2.i.f.415.5 yes 12 7.4 even 3
3381.2.a.bc.1.2 6 1.1 even 1 trivial
3381.2.a.bd.1.2 6 7.6 odd 2