Properties

Label 8034.2.a.bc.1.5
Level 8034
Weight 2
Character 8034.1
Self dual yes
Analytic conductor 64.152
Analytic rank 0
Dimension 15
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + 33191 x^{7} - 16428 x^{6} - 71361 x^{5} + 21747 x^{4} + 65434 x^{3} - 11840 x^{2} - 17600 x + 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.05101\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.05101 q^{5} -1.00000 q^{6} -1.63593 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.05101 q^{5} -1.00000 q^{6} -1.63593 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.05101 q^{10} -1.96737 q^{11} -1.00000 q^{12} -1.00000 q^{13} -1.63593 q^{14} +2.05101 q^{15} +1.00000 q^{16} -4.75332 q^{17} +1.00000 q^{18} -7.72431 q^{19} -2.05101 q^{20} +1.63593 q^{21} -1.96737 q^{22} -0.952745 q^{23} -1.00000 q^{24} -0.793369 q^{25} -1.00000 q^{26} -1.00000 q^{27} -1.63593 q^{28} -2.09348 q^{29} +2.05101 q^{30} +9.14308 q^{31} +1.00000 q^{32} +1.96737 q^{33} -4.75332 q^{34} +3.35531 q^{35} +1.00000 q^{36} +2.77305 q^{37} -7.72431 q^{38} +1.00000 q^{39} -2.05101 q^{40} -9.56781 q^{41} +1.63593 q^{42} +3.74993 q^{43} -1.96737 q^{44} -2.05101 q^{45} -0.952745 q^{46} +5.58312 q^{47} -1.00000 q^{48} -4.32373 q^{49} -0.793369 q^{50} +4.75332 q^{51} -1.00000 q^{52} -1.37438 q^{53} -1.00000 q^{54} +4.03509 q^{55} -1.63593 q^{56} +7.72431 q^{57} -2.09348 q^{58} -9.37885 q^{59} +2.05101 q^{60} +2.60018 q^{61} +9.14308 q^{62} -1.63593 q^{63} +1.00000 q^{64} +2.05101 q^{65} +1.96737 q^{66} +5.60782 q^{67} -4.75332 q^{68} +0.952745 q^{69} +3.35531 q^{70} -0.768551 q^{71} +1.00000 q^{72} +9.15376 q^{73} +2.77305 q^{74} +0.793369 q^{75} -7.72431 q^{76} +3.21848 q^{77} +1.00000 q^{78} +3.23726 q^{79} -2.05101 q^{80} +1.00000 q^{81} -9.56781 q^{82} +12.4302 q^{83} +1.63593 q^{84} +9.74909 q^{85} +3.74993 q^{86} +2.09348 q^{87} -1.96737 q^{88} -6.28143 q^{89} -2.05101 q^{90} +1.63593 q^{91} -0.952745 q^{92} -9.14308 q^{93} +5.58312 q^{94} +15.8426 q^{95} -1.00000 q^{96} -7.59891 q^{97} -4.32373 q^{98} -1.96737 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 15q^{2} - 15q^{3} + 15q^{4} - q^{5} - 15q^{6} + 5q^{7} + 15q^{8} + 15q^{9} - q^{10} + 3q^{11} - 15q^{12} - 15q^{13} + 5q^{14} + q^{15} + 15q^{16} - 2q^{17} + 15q^{18} + 8q^{19} - q^{20} - 5q^{21} + 3q^{22} + 3q^{23} - 15q^{24} + 22q^{25} - 15q^{26} - 15q^{27} + 5q^{28} + 26q^{29} + q^{30} + 15q^{32} - 3q^{33} - 2q^{34} - 8q^{35} + 15q^{36} + 25q^{37} + 8q^{38} + 15q^{39} - q^{40} - q^{41} - 5q^{42} + 10q^{43} + 3q^{44} - q^{45} + 3q^{46} - 3q^{47} - 15q^{48} + 32q^{49} + 22q^{50} + 2q^{51} - 15q^{52} + 13q^{53} - 15q^{54} - 2q^{55} + 5q^{56} - 8q^{57} + 26q^{58} + 28q^{59} + q^{60} + 22q^{61} + 5q^{63} + 15q^{64} + q^{65} - 3q^{66} + 29q^{67} - 2q^{68} - 3q^{69} - 8q^{70} + 18q^{71} + 15q^{72} + 23q^{73} + 25q^{74} - 22q^{75} + 8q^{76} + 17q^{77} + 15q^{78} + 27q^{79} - q^{80} + 15q^{81} - q^{82} + 7q^{83} - 5q^{84} + 43q^{85} + 10q^{86} - 26q^{87} + 3q^{88} + 35q^{89} - q^{90} - 5q^{91} + 3q^{92} - 3q^{94} + 6q^{95} - 15q^{96} + 19q^{97} + 32q^{98} + 3q^{99} + O(q^{100}) \)

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
\(5\) −2.05101 −0.917238 −0.458619 0.888633i \(-0.651656\pi\)
−0.458619 + 0.888633i \(0.651656\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.63593 −0.618324 −0.309162 0.951009i \(-0.600049\pi\)
−0.309162 + 0.951009i \(0.600049\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.05101 −0.648585
\(11\) −1.96737 −0.593185 −0.296592 0.955004i \(-0.595850\pi\)
−0.296592 + 0.955004i \(0.595850\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.00000 −0.277350
\(14\) −1.63593 −0.437221
\(15\) 2.05101 0.529568
\(16\) 1.00000 0.250000
\(17\) −4.75332 −1.15285 −0.576424 0.817151i \(-0.695552\pi\)
−0.576424 + 0.817151i \(0.695552\pi\)
\(18\) 1.00000 0.235702
\(19\) −7.72431 −1.77208 −0.886039 0.463611i \(-0.846553\pi\)
−0.886039 + 0.463611i \(0.846553\pi\)
\(20\) −2.05101 −0.458619
\(21\) 1.63593 0.356990
\(22\) −1.96737 −0.419445
\(23\) −0.952745 −0.198661 −0.0993306 0.995054i \(-0.531670\pi\)
−0.0993306 + 0.995054i \(0.531670\pi\)
\(24\) −1.00000 −0.204124
\(25\) −0.793369 −0.158674
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) −1.63593 −0.309162
\(29\) −2.09348 −0.388749 −0.194374 0.980927i \(-0.562268\pi\)
−0.194374 + 0.980927i \(0.562268\pi\)
\(30\) 2.05101 0.374461
\(31\) 9.14308 1.64215 0.821073 0.570823i \(-0.193376\pi\)
0.821073 + 0.570823i \(0.193376\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.96737 0.342475
\(34\) −4.75332 −0.815187
\(35\) 3.35531 0.567151
\(36\) 1.00000 0.166667
\(37\) 2.77305 0.455887 0.227943 0.973674i \(-0.426800\pi\)
0.227943 + 0.973674i \(0.426800\pi\)
\(38\) −7.72431 −1.25305
\(39\) 1.00000 0.160128
\(40\) −2.05101 −0.324293
\(41\) −9.56781 −1.49424 −0.747121 0.664689i \(-0.768564\pi\)
−0.747121 + 0.664689i \(0.768564\pi\)
\(42\) 1.63593 0.252430
\(43\) 3.74993 0.571859 0.285929 0.958251i \(-0.407698\pi\)
0.285929 + 0.958251i \(0.407698\pi\)
\(44\) −1.96737 −0.296592
\(45\) −2.05101 −0.305746
\(46\) −0.952745 −0.140475
\(47\) 5.58312 0.814382 0.407191 0.913343i \(-0.366508\pi\)
0.407191 + 0.913343i \(0.366508\pi\)
\(48\) −1.00000 −0.144338
\(49\) −4.32373 −0.617675
\(50\) −0.793369 −0.112199
\(51\) 4.75332 0.665597
\(52\) −1.00000 −0.138675
\(53\) −1.37438 −0.188785 −0.0943926 0.995535i \(-0.530091\pi\)
−0.0943926 + 0.995535i \(0.530091\pi\)
\(54\) −1.00000 −0.136083
\(55\) 4.03509 0.544092
\(56\) −1.63593 −0.218611
\(57\) 7.72431 1.02311
\(58\) −2.09348 −0.274887
\(59\) −9.37885 −1.22102 −0.610511 0.792008i \(-0.709036\pi\)
−0.610511 + 0.792008i \(0.709036\pi\)
\(60\) 2.05101 0.264784
\(61\) 2.60018 0.332919 0.166459 0.986048i \(-0.446767\pi\)
0.166459 + 0.986048i \(0.446767\pi\)
\(62\) 9.14308 1.16117
\(63\) −1.63593 −0.206108
\(64\) 1.00000 0.125000
\(65\) 2.05101 0.254396
\(66\) 1.96737 0.242167
\(67\) 5.60782 0.685104 0.342552 0.939499i \(-0.388709\pi\)
0.342552 + 0.939499i \(0.388709\pi\)
\(68\) −4.75332 −0.576424
\(69\) 0.952745 0.114697
\(70\) 3.35531 0.401036
\(71\) −0.768551 −0.0912102 −0.0456051 0.998960i \(-0.514522\pi\)
−0.0456051 + 0.998960i \(0.514522\pi\)
\(72\) 1.00000 0.117851
\(73\) 9.15376 1.07137 0.535683 0.844419i \(-0.320054\pi\)
0.535683 + 0.844419i \(0.320054\pi\)
\(74\) 2.77305 0.322360
\(75\) 0.793369 0.0916103
\(76\) −7.72431 −0.886039
\(77\) 3.21848 0.366780
\(78\) 1.00000 0.113228
\(79\) 3.23726 0.364220 0.182110 0.983278i \(-0.441707\pi\)
0.182110 + 0.983278i \(0.441707\pi\)
\(80\) −2.05101 −0.229310
\(81\) 1.00000 0.111111
\(82\) −9.56781 −1.05659
\(83\) 12.4302 1.36439 0.682193 0.731172i \(-0.261026\pi\)
0.682193 + 0.731172i \(0.261026\pi\)
\(84\) 1.63593 0.178495
\(85\) 9.74909 1.05744
\(86\) 3.74993 0.404365
\(87\) 2.09348 0.224444
\(88\) −1.96737 −0.209722
\(89\) −6.28143 −0.665830 −0.332915 0.942957i \(-0.608032\pi\)
−0.332915 + 0.942957i \(0.608032\pi\)
\(90\) −2.05101 −0.216195
\(91\) 1.63593 0.171492
\(92\) −0.952745 −0.0993306
\(93\) −9.14308 −0.948093
\(94\) 5.58312 0.575855
\(95\) 15.8426 1.62542
\(96\) −1.00000 −0.102062
\(97\) −7.59891 −0.771552 −0.385776 0.922592i \(-0.626066\pi\)
−0.385776 + 0.922592i \(0.626066\pi\)
\(98\) −4.32373 −0.436762
\(99\) −1.96737 −0.197728
\(100\) −0.793369 −0.0793369
\(101\) 16.7085 1.66256 0.831280 0.555854i \(-0.187609\pi\)
0.831280 + 0.555854i \(0.187609\pi\)
\(102\) 4.75332 0.470648
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −3.35531 −0.327445
\(106\) −1.37438 −0.133491
\(107\) −15.8026 −1.52769 −0.763847 0.645397i \(-0.776692\pi\)
−0.763847 + 0.645397i \(0.776692\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −7.02817 −0.673176 −0.336588 0.941652i \(-0.609273\pi\)
−0.336588 + 0.941652i \(0.609273\pi\)
\(110\) 4.03509 0.384731
\(111\) −2.77305 −0.263206
\(112\) −1.63593 −0.154581
\(113\) −4.15073 −0.390468 −0.195234 0.980757i \(-0.562547\pi\)
−0.195234 + 0.980757i \(0.562547\pi\)
\(114\) 7.72431 0.723448
\(115\) 1.95409 0.182220
\(116\) −2.09348 −0.194374
\(117\) −1.00000 −0.0924500
\(118\) −9.37885 −0.863393
\(119\) 7.77610 0.712834
\(120\) 2.05101 0.187230
\(121\) −7.12945 −0.648132
\(122\) 2.60018 0.235409
\(123\) 9.56781 0.862701
\(124\) 9.14308 0.821073
\(125\) 11.8822 1.06278
\(126\) −1.63593 −0.145740
\(127\) 2.15288 0.191037 0.0955184 0.995428i \(-0.469549\pi\)
0.0955184 + 0.995428i \(0.469549\pi\)
\(128\) 1.00000 0.0883883
\(129\) −3.74993 −0.330163
\(130\) 2.05101 0.179885
\(131\) 6.40579 0.559676 0.279838 0.960047i \(-0.409719\pi\)
0.279838 + 0.960047i \(0.409719\pi\)
\(132\) 1.96737 0.171238
\(133\) 12.6364 1.09572
\(134\) 5.60782 0.484442
\(135\) 2.05101 0.176523
\(136\) −4.75332 −0.407594
\(137\) 3.49417 0.298527 0.149264 0.988797i \(-0.452310\pi\)
0.149264 + 0.988797i \(0.452310\pi\)
\(138\) 0.952745 0.0811031
\(139\) −8.19183 −0.694822 −0.347411 0.937713i \(-0.612939\pi\)
−0.347411 + 0.937713i \(0.612939\pi\)
\(140\) 3.35531 0.283575
\(141\) −5.58312 −0.470184
\(142\) −0.768551 −0.0644954
\(143\) 1.96737 0.164520
\(144\) 1.00000 0.0833333
\(145\) 4.29373 0.356575
\(146\) 9.15376 0.757570
\(147\) 4.32373 0.356615
\(148\) 2.77305 0.227943
\(149\) 15.2604 1.25018 0.625090 0.780553i \(-0.285062\pi\)
0.625090 + 0.780553i \(0.285062\pi\)
\(150\) 0.793369 0.0647783
\(151\) 3.51375 0.285945 0.142973 0.989727i \(-0.454334\pi\)
0.142973 + 0.989727i \(0.454334\pi\)
\(152\) −7.72431 −0.626524
\(153\) −4.75332 −0.384283
\(154\) 3.21848 0.259353
\(155\) −18.7525 −1.50624
\(156\) 1.00000 0.0800641
\(157\) 13.6112 1.08629 0.543146 0.839638i \(-0.317233\pi\)
0.543146 + 0.839638i \(0.317233\pi\)
\(158\) 3.23726 0.257542
\(159\) 1.37438 0.108995
\(160\) −2.05101 −0.162146
\(161\) 1.55863 0.122837
\(162\) 1.00000 0.0785674
\(163\) −2.48927 −0.194975 −0.0974875 0.995237i \(-0.531081\pi\)
−0.0974875 + 0.995237i \(0.531081\pi\)
\(164\) −9.56781 −0.747121
\(165\) −4.03509 −0.314131
\(166\) 12.4302 0.964767
\(167\) 5.51732 0.426943 0.213471 0.976949i \(-0.431523\pi\)
0.213471 + 0.976949i \(0.431523\pi\)
\(168\) 1.63593 0.126215
\(169\) 1.00000 0.0769231
\(170\) 9.74909 0.747721
\(171\) −7.72431 −0.590692
\(172\) 3.74993 0.285929
\(173\) −3.49076 −0.265398 −0.132699 0.991156i \(-0.542364\pi\)
−0.132699 + 0.991156i \(0.542364\pi\)
\(174\) 2.09348 0.158706
\(175\) 1.29790 0.0981118
\(176\) −1.96737 −0.148296
\(177\) 9.37885 0.704957
\(178\) −6.28143 −0.470813
\(179\) 0.869690 0.0650037 0.0325018 0.999472i \(-0.489653\pi\)
0.0325018 + 0.999472i \(0.489653\pi\)
\(180\) −2.05101 −0.152873
\(181\) 14.1847 1.05434 0.527171 0.849759i \(-0.323253\pi\)
0.527171 + 0.849759i \(0.323253\pi\)
\(182\) 1.63593 0.121263
\(183\) −2.60018 −0.192211
\(184\) −0.952745 −0.0702373
\(185\) −5.68754 −0.418157
\(186\) −9.14308 −0.670403
\(187\) 9.35154 0.683852
\(188\) 5.58312 0.407191
\(189\) 1.63593 0.118997
\(190\) 15.8426 1.14934
\(191\) 5.68390 0.411272 0.205636 0.978628i \(-0.434074\pi\)
0.205636 + 0.978628i \(0.434074\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 13.5512 0.975439 0.487719 0.873000i \(-0.337829\pi\)
0.487719 + 0.873000i \(0.337829\pi\)
\(194\) −7.59891 −0.545570
\(195\) −2.05101 −0.146876
\(196\) −4.32373 −0.308838
\(197\) 0.197262 0.0140543 0.00702716 0.999975i \(-0.497763\pi\)
0.00702716 + 0.999975i \(0.497763\pi\)
\(198\) −1.96737 −0.139815
\(199\) −4.70360 −0.333430 −0.166715 0.986005i \(-0.553316\pi\)
−0.166715 + 0.986005i \(0.553316\pi\)
\(200\) −0.793369 −0.0560997
\(201\) −5.60782 −0.395545
\(202\) 16.7085 1.17561
\(203\) 3.42478 0.240373
\(204\) 4.75332 0.332799
\(205\) 19.6237 1.37058
\(206\) −1.00000 −0.0696733
\(207\) −0.952745 −0.0662204
\(208\) −1.00000 −0.0693375
\(209\) 15.1966 1.05117
\(210\) −3.35531 −0.231538
\(211\) 9.02680 0.621430 0.310715 0.950503i \(-0.399431\pi\)
0.310715 + 0.950503i \(0.399431\pi\)
\(212\) −1.37438 −0.0943926
\(213\) 0.768551 0.0526602
\(214\) −15.8026 −1.08024
\(215\) −7.69113 −0.524531
\(216\) −1.00000 −0.0680414
\(217\) −14.9575 −1.01538
\(218\) −7.02817 −0.476008
\(219\) −9.15376 −0.618554
\(220\) 4.03509 0.272046
\(221\) 4.75332 0.319743
\(222\) −2.77305 −0.186115
\(223\) −27.5351 −1.84389 −0.921944 0.387324i \(-0.873399\pi\)
−0.921944 + 0.387324i \(0.873399\pi\)
\(224\) −1.63593 −0.109305
\(225\) −0.793369 −0.0528913
\(226\) −4.15073 −0.276103
\(227\) 8.37920 0.556147 0.278073 0.960560i \(-0.410304\pi\)
0.278073 + 0.960560i \(0.410304\pi\)
\(228\) 7.72431 0.511555
\(229\) 7.16726 0.473626 0.236813 0.971555i \(-0.423897\pi\)
0.236813 + 0.971555i \(0.423897\pi\)
\(230\) 1.95409 0.128849
\(231\) −3.21848 −0.211761
\(232\) −2.09348 −0.137443
\(233\) 16.8531 1.10408 0.552042 0.833816i \(-0.313849\pi\)
0.552042 + 0.833816i \(0.313849\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −11.4510 −0.746983
\(236\) −9.37885 −0.610511
\(237\) −3.23726 −0.210282
\(238\) 7.77610 0.504050
\(239\) −23.4242 −1.51519 −0.757593 0.652727i \(-0.773625\pi\)
−0.757593 + 0.652727i \(0.773625\pi\)
\(240\) 2.05101 0.132392
\(241\) 1.65066 0.106328 0.0531641 0.998586i \(-0.483069\pi\)
0.0531641 + 0.998586i \(0.483069\pi\)
\(242\) −7.12945 −0.458299
\(243\) −1.00000 −0.0641500
\(244\) 2.60018 0.166459
\(245\) 8.86800 0.566556
\(246\) 9.56781 0.610021
\(247\) 7.72431 0.491486
\(248\) 9.14308 0.580586
\(249\) −12.4302 −0.787729
\(250\) 11.8822 0.751499
\(251\) −11.1260 −0.702270 −0.351135 0.936325i \(-0.614204\pi\)
−0.351135 + 0.936325i \(0.614204\pi\)
\(252\) −1.63593 −0.103054
\(253\) 1.87440 0.117843
\(254\) 2.15288 0.135083
\(255\) −9.74909 −0.610512
\(256\) 1.00000 0.0625000
\(257\) −17.3102 −1.07978 −0.539890 0.841735i \(-0.681534\pi\)
−0.539890 + 0.841735i \(0.681534\pi\)
\(258\) −3.74993 −0.233460
\(259\) −4.53652 −0.281886
\(260\) 2.05101 0.127198
\(261\) −2.09348 −0.129583
\(262\) 6.40579 0.395751
\(263\) −7.79407 −0.480603 −0.240301 0.970698i \(-0.577246\pi\)
−0.240301 + 0.970698i \(0.577246\pi\)
\(264\) 1.96737 0.121083
\(265\) 2.81886 0.173161
\(266\) 12.6364 0.774790
\(267\) 6.28143 0.384417
\(268\) 5.60782 0.342552
\(269\) −10.1302 −0.617652 −0.308826 0.951119i \(-0.599936\pi\)
−0.308826 + 0.951119i \(0.599936\pi\)
\(270\) 2.05101 0.124820
\(271\) 13.4818 0.818961 0.409480 0.912319i \(-0.365710\pi\)
0.409480 + 0.912319i \(0.365710\pi\)
\(272\) −4.75332 −0.288212
\(273\) −1.63593 −0.0990111
\(274\) 3.49417 0.211091
\(275\) 1.56085 0.0941228
\(276\) 0.952745 0.0573485
\(277\) 26.0301 1.56400 0.781998 0.623281i \(-0.214201\pi\)
0.781998 + 0.623281i \(0.214201\pi\)
\(278\) −8.19183 −0.491313
\(279\) 9.14308 0.547382
\(280\) 3.35531 0.200518
\(281\) 3.54102 0.211240 0.105620 0.994407i \(-0.466317\pi\)
0.105620 + 0.994407i \(0.466317\pi\)
\(282\) −5.58312 −0.332470
\(283\) −18.7384 −1.11388 −0.556941 0.830552i \(-0.688025\pi\)
−0.556941 + 0.830552i \(0.688025\pi\)
\(284\) −0.768551 −0.0456051
\(285\) −15.8426 −0.938435
\(286\) 1.96737 0.116333
\(287\) 15.6523 0.923925
\(288\) 1.00000 0.0589256
\(289\) 5.59402 0.329060
\(290\) 4.29373 0.252137
\(291\) 7.59891 0.445456
\(292\) 9.15376 0.535683
\(293\) −14.0709 −0.822033 −0.411016 0.911628i \(-0.634826\pi\)
−0.411016 + 0.911628i \(0.634826\pi\)
\(294\) 4.32373 0.252165
\(295\) 19.2361 1.11997
\(296\) 2.77305 0.161180
\(297\) 1.96737 0.114158
\(298\) 15.2604 0.884010
\(299\) 0.952745 0.0550987
\(300\) 0.793369 0.0458052
\(301\) −6.13463 −0.353594
\(302\) 3.51375 0.202194
\(303\) −16.7085 −0.959879
\(304\) −7.72431 −0.443019
\(305\) −5.33299 −0.305366
\(306\) −4.75332 −0.271729
\(307\) 8.85911 0.505616 0.252808 0.967516i \(-0.418646\pi\)
0.252808 + 0.967516i \(0.418646\pi\)
\(308\) 3.21848 0.183390
\(309\) 1.00000 0.0568880
\(310\) −18.7525 −1.06507
\(311\) 34.3848 1.94978 0.974892 0.222678i \(-0.0714797\pi\)
0.974892 + 0.222678i \(0.0714797\pi\)
\(312\) 1.00000 0.0566139
\(313\) −22.8294 −1.29039 −0.645197 0.764016i \(-0.723225\pi\)
−0.645197 + 0.764016i \(0.723225\pi\)
\(314\) 13.6112 0.768125
\(315\) 3.35531 0.189050
\(316\) 3.23726 0.182110
\(317\) 26.9585 1.51414 0.757071 0.653332i \(-0.226630\pi\)
0.757071 + 0.653332i \(0.226630\pi\)
\(318\) 1.37438 0.0770712
\(319\) 4.11864 0.230600
\(320\) −2.05101 −0.114655
\(321\) 15.8026 0.882015
\(322\) 1.55863 0.0868588
\(323\) 36.7161 2.04294
\(324\) 1.00000 0.0555556
\(325\) 0.793369 0.0440082
\(326\) −2.48927 −0.137868
\(327\) 7.02817 0.388659
\(328\) −9.56781 −0.528294
\(329\) −9.13361 −0.503552
\(330\) −4.03509 −0.222124
\(331\) −14.0868 −0.774280 −0.387140 0.922021i \(-0.626537\pi\)
−0.387140 + 0.922021i \(0.626537\pi\)
\(332\) 12.4302 0.682193
\(333\) 2.77305 0.151962
\(334\) 5.51732 0.301894
\(335\) −11.5017 −0.628404
\(336\) 1.63593 0.0892474
\(337\) 24.1632 1.31625 0.658126 0.752908i \(-0.271349\pi\)
0.658126 + 0.752908i \(0.271349\pi\)
\(338\) 1.00000 0.0543928
\(339\) 4.15073 0.225437
\(340\) 9.74909 0.528719
\(341\) −17.9878 −0.974096
\(342\) −7.72431 −0.417683
\(343\) 18.5248 1.00025
\(344\) 3.74993 0.202183
\(345\) −1.95409 −0.105205
\(346\) −3.49076 −0.187664
\(347\) 22.5146 1.20865 0.604324 0.796739i \(-0.293443\pi\)
0.604324 + 0.796739i \(0.293443\pi\)
\(348\) 2.09348 0.112222
\(349\) −9.14497 −0.489519 −0.244760 0.969584i \(-0.578709\pi\)
−0.244760 + 0.969584i \(0.578709\pi\)
\(350\) 1.29790 0.0693755
\(351\) 1.00000 0.0533761
\(352\) −1.96737 −0.104861
\(353\) −29.2329 −1.55591 −0.777954 0.628321i \(-0.783742\pi\)
−0.777954 + 0.628321i \(0.783742\pi\)
\(354\) 9.37885 0.498480
\(355\) 1.57630 0.0836615
\(356\) −6.28143 −0.332915
\(357\) −7.77610 −0.411555
\(358\) 0.869690 0.0459645
\(359\) −14.2120 −0.750083 −0.375041 0.927008i \(-0.622372\pi\)
−0.375041 + 0.927008i \(0.622372\pi\)
\(360\) −2.05101 −0.108098
\(361\) 40.6649 2.14026
\(362\) 14.1847 0.745532
\(363\) 7.12945 0.374199
\(364\) 1.63593 0.0857461
\(365\) −18.7744 −0.982698
\(366\) −2.60018 −0.135914
\(367\) −33.2324 −1.73472 −0.867358 0.497685i \(-0.834184\pi\)
−0.867358 + 0.497685i \(0.834184\pi\)
\(368\) −0.952745 −0.0496653
\(369\) −9.56781 −0.498080
\(370\) −5.68754 −0.295681
\(371\) 2.24839 0.116730
\(372\) −9.14308 −0.474047
\(373\) 28.9657 1.49979 0.749893 0.661559i \(-0.230105\pi\)
0.749893 + 0.661559i \(0.230105\pi\)
\(374\) 9.35154 0.483556
\(375\) −11.8822 −0.613596
\(376\) 5.58312 0.287928
\(377\) 2.09348 0.107820
\(378\) 1.63593 0.0841432
\(379\) 9.51928 0.488973 0.244486 0.969653i \(-0.421381\pi\)
0.244486 + 0.969653i \(0.421381\pi\)
\(380\) 15.8426 0.812709
\(381\) −2.15288 −0.110295
\(382\) 5.68390 0.290814
\(383\) 4.91506 0.251148 0.125574 0.992084i \(-0.459923\pi\)
0.125574 + 0.992084i \(0.459923\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −6.60113 −0.336425
\(386\) 13.5512 0.689739
\(387\) 3.74993 0.190620
\(388\) −7.59891 −0.385776
\(389\) 13.5847 0.688772 0.344386 0.938828i \(-0.388087\pi\)
0.344386 + 0.938828i \(0.388087\pi\)
\(390\) −2.05101 −0.103857
\(391\) 4.52870 0.229026
\(392\) −4.32373 −0.218381
\(393\) −6.40579 −0.323129
\(394\) 0.197262 0.00993790
\(395\) −6.63964 −0.334076
\(396\) −1.96737 −0.0988641
\(397\) 26.5233 1.33117 0.665583 0.746324i \(-0.268183\pi\)
0.665583 + 0.746324i \(0.268183\pi\)
\(398\) −4.70360 −0.235770
\(399\) −12.6364 −0.632613
\(400\) −0.793369 −0.0396684
\(401\) 11.4425 0.571413 0.285706 0.958317i \(-0.407772\pi\)
0.285706 + 0.958317i \(0.407772\pi\)
\(402\) −5.60782 −0.279692
\(403\) −9.14308 −0.455449
\(404\) 16.7085 0.831280
\(405\) −2.05101 −0.101915
\(406\) 3.42478 0.169969
\(407\) −5.45562 −0.270425
\(408\) 4.75332 0.235324
\(409\) 30.8632 1.52609 0.763043 0.646348i \(-0.223705\pi\)
0.763043 + 0.646348i \(0.223705\pi\)
\(410\) 19.6237 0.969143
\(411\) −3.49417 −0.172355
\(412\) −1.00000 −0.0492665
\(413\) 15.3432 0.754987
\(414\) −0.952745 −0.0468249
\(415\) −25.4943 −1.25147
\(416\) −1.00000 −0.0490290
\(417\) 8.19183 0.401155
\(418\) 15.1966 0.743289
\(419\) −9.88812 −0.483066 −0.241533 0.970393i \(-0.577650\pi\)
−0.241533 + 0.970393i \(0.577650\pi\)
\(420\) −3.35531 −0.163722
\(421\) 37.1616 1.81115 0.905573 0.424191i \(-0.139441\pi\)
0.905573 + 0.424191i \(0.139441\pi\)
\(422\) 9.02680 0.439418
\(423\) 5.58312 0.271461
\(424\) −1.37438 −0.0667457
\(425\) 3.77113 0.182927
\(426\) 0.768551 0.0372364
\(427\) −4.25372 −0.205852
\(428\) −15.8026 −0.763847
\(429\) −1.96737 −0.0949855
\(430\) −7.69113 −0.370899
\(431\) −26.6743 −1.28486 −0.642428 0.766346i \(-0.722073\pi\)
−0.642428 + 0.766346i \(0.722073\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 31.0568 1.49249 0.746247 0.665669i \(-0.231854\pi\)
0.746247 + 0.665669i \(0.231854\pi\)
\(434\) −14.9575 −0.717981
\(435\) −4.29373 −0.205869
\(436\) −7.02817 −0.336588
\(437\) 7.35930 0.352043
\(438\) −9.15376 −0.437384
\(439\) 19.0081 0.907208 0.453604 0.891203i \(-0.350138\pi\)
0.453604 + 0.891203i \(0.350138\pi\)
\(440\) 4.03509 0.192365
\(441\) −4.32373 −0.205892
\(442\) 4.75332 0.226092
\(443\) 27.5936 1.31101 0.655507 0.755189i \(-0.272455\pi\)
0.655507 + 0.755189i \(0.272455\pi\)
\(444\) −2.77305 −0.131603
\(445\) 12.8833 0.610725
\(446\) −27.5351 −1.30383
\(447\) −15.2604 −0.721791
\(448\) −1.63593 −0.0772905
\(449\) 32.0669 1.51333 0.756664 0.653804i \(-0.226828\pi\)
0.756664 + 0.653804i \(0.226828\pi\)
\(450\) −0.793369 −0.0373998
\(451\) 18.8234 0.886361
\(452\) −4.15073 −0.195234
\(453\) −3.51375 −0.165091
\(454\) 8.37920 0.393255
\(455\) −3.35531 −0.157299
\(456\) 7.72431 0.361724
\(457\) 13.9628 0.653151 0.326576 0.945171i \(-0.394105\pi\)
0.326576 + 0.945171i \(0.394105\pi\)
\(458\) 7.16726 0.334904
\(459\) 4.75332 0.221866
\(460\) 1.95409 0.0911098
\(461\) 21.1741 0.986178 0.493089 0.869979i \(-0.335868\pi\)
0.493089 + 0.869979i \(0.335868\pi\)
\(462\) −3.21848 −0.149737
\(463\) −3.10076 −0.144105 −0.0720523 0.997401i \(-0.522955\pi\)
−0.0720523 + 0.997401i \(0.522955\pi\)
\(464\) −2.09348 −0.0971872
\(465\) 18.7525 0.869628
\(466\) 16.8531 0.780705
\(467\) −11.1953 −0.518057 −0.259028 0.965870i \(-0.583402\pi\)
−0.259028 + 0.965870i \(0.583402\pi\)
\(468\) −1.00000 −0.0462250
\(469\) −9.17401 −0.423616
\(470\) −11.4510 −0.528197
\(471\) −13.6112 −0.627171
\(472\) −9.37885 −0.431696
\(473\) −7.37750 −0.339218
\(474\) −3.23726 −0.148692
\(475\) 6.12822 0.281182
\(476\) 7.77610 0.356417
\(477\) −1.37438 −0.0629284
\(478\) −23.4242 −1.07140
\(479\) −20.8766 −0.953876 −0.476938 0.878937i \(-0.658253\pi\)
−0.476938 + 0.878937i \(0.658253\pi\)
\(480\) 2.05101 0.0936152
\(481\) −2.77305 −0.126440
\(482\) 1.65066 0.0751854
\(483\) −1.55863 −0.0709199
\(484\) −7.12945 −0.324066
\(485\) 15.5854 0.707697
\(486\) −1.00000 −0.0453609
\(487\) 37.8999 1.71741 0.858703 0.512473i \(-0.171271\pi\)
0.858703 + 0.512473i \(0.171271\pi\)
\(488\) 2.60018 0.117705
\(489\) 2.48927 0.112569
\(490\) 8.86800 0.400615
\(491\) 15.4924 0.699160 0.349580 0.936906i \(-0.386324\pi\)
0.349580 + 0.936906i \(0.386324\pi\)
\(492\) 9.56781 0.431350
\(493\) 9.95096 0.448168
\(494\) 7.72431 0.347533
\(495\) 4.03509 0.181364
\(496\) 9.14308 0.410536
\(497\) 1.25730 0.0563975
\(498\) −12.4302 −0.557008
\(499\) −33.2335 −1.48774 −0.743868 0.668326i \(-0.767011\pi\)
−0.743868 + 0.668326i \(0.767011\pi\)
\(500\) 11.8822 0.531390
\(501\) −5.51732 −0.246496
\(502\) −11.1260 −0.496580
\(503\) −20.5650 −0.916947 −0.458474 0.888708i \(-0.651604\pi\)
−0.458474 + 0.888708i \(0.651604\pi\)
\(504\) −1.63593 −0.0728702
\(505\) −34.2693 −1.52496
\(506\) 1.87440 0.0833274
\(507\) −1.00000 −0.0444116
\(508\) 2.15288 0.0955184
\(509\) −30.1255 −1.33529 −0.667645 0.744480i \(-0.732697\pi\)
−0.667645 + 0.744480i \(0.732697\pi\)
\(510\) −9.74909 −0.431697
\(511\) −14.9749 −0.662452
\(512\) 1.00000 0.0441942
\(513\) 7.72431 0.341036
\(514\) −17.3102 −0.763520
\(515\) 2.05101 0.0903782
\(516\) −3.74993 −0.165081
\(517\) −10.9841 −0.483079
\(518\) −4.53652 −0.199323
\(519\) 3.49076 0.153227
\(520\) 2.05101 0.0899426
\(521\) −42.4093 −1.85799 −0.928993 0.370097i \(-0.879324\pi\)
−0.928993 + 0.370097i \(0.879324\pi\)
\(522\) −2.09348 −0.0916290
\(523\) −0.665979 −0.0291212 −0.0145606 0.999894i \(-0.504635\pi\)
−0.0145606 + 0.999894i \(0.504635\pi\)
\(524\) 6.40579 0.279838
\(525\) −1.29790 −0.0566449
\(526\) −7.79407 −0.339837
\(527\) −43.4600 −1.89315
\(528\) 1.96737 0.0856188
\(529\) −22.0923 −0.960534
\(530\) 2.81886 0.122443
\(531\) −9.37885 −0.407007
\(532\) 12.6364 0.547859
\(533\) 9.56781 0.414428
\(534\) 6.28143 0.271824
\(535\) 32.4112 1.40126
\(536\) 5.60782 0.242221
\(537\) −0.869690 −0.0375299
\(538\) −10.1302 −0.436746
\(539\) 8.50637 0.366395
\(540\) 2.05101 0.0882613
\(541\) 26.9992 1.16079 0.580393 0.814336i \(-0.302899\pi\)
0.580393 + 0.814336i \(0.302899\pi\)
\(542\) 13.4818 0.579093
\(543\) −14.1847 −0.608724
\(544\) −4.75332 −0.203797
\(545\) 14.4148 0.617463
\(546\) −1.63593 −0.0700114
\(547\) −7.96155 −0.340411 −0.170206 0.985409i \(-0.554443\pi\)
−0.170206 + 0.985409i \(0.554443\pi\)
\(548\) 3.49417 0.149264
\(549\) 2.60018 0.110973
\(550\) 1.56085 0.0665549
\(551\) 16.1706 0.688893
\(552\) 0.952745 0.0405515
\(553\) −5.29593 −0.225206
\(554\) 26.0301 1.10591
\(555\) 5.68754 0.241423
\(556\) −8.19183 −0.347411
\(557\) 15.3251 0.649347 0.324674 0.945826i \(-0.394746\pi\)
0.324674 + 0.945826i \(0.394746\pi\)
\(558\) 9.14308 0.387057
\(559\) −3.74993 −0.158605
\(560\) 3.35531 0.141788
\(561\) −9.35154 −0.394822
\(562\) 3.54102 0.149369
\(563\) −41.2756 −1.73956 −0.869779 0.493441i \(-0.835739\pi\)
−0.869779 + 0.493441i \(0.835739\pi\)
\(564\) −5.58312 −0.235092
\(565\) 8.51318 0.358152
\(566\) −18.7384 −0.787633
\(567\) −1.63593 −0.0687027
\(568\) −0.768551 −0.0322477
\(569\) 27.1026 1.13620 0.568099 0.822960i \(-0.307679\pi\)
0.568099 + 0.822960i \(0.307679\pi\)
\(570\) −15.8426 −0.663574
\(571\) 7.89439 0.330370 0.165185 0.986263i \(-0.447178\pi\)
0.165185 + 0.986263i \(0.447178\pi\)
\(572\) 1.96737 0.0822599
\(573\) −5.68390 −0.237448
\(574\) 15.6523 0.653314
\(575\) 0.755878 0.0315223
\(576\) 1.00000 0.0416667
\(577\) −21.9953 −0.915678 −0.457839 0.889035i \(-0.651376\pi\)
−0.457839 + 0.889035i \(0.651376\pi\)
\(578\) 5.59402 0.232681
\(579\) −13.5512 −0.563170
\(580\) 4.29373 0.178288
\(581\) −20.3349 −0.843633
\(582\) 7.59891 0.314985
\(583\) 2.70391 0.111984
\(584\) 9.15376 0.378785
\(585\) 2.05101 0.0847987
\(586\) −14.0709 −0.581265
\(587\) −10.1600 −0.419346 −0.209673 0.977772i \(-0.567240\pi\)
−0.209673 + 0.977772i \(0.567240\pi\)
\(588\) 4.32373 0.178308
\(589\) −70.6240 −2.91001
\(590\) 19.2361 0.791937
\(591\) −0.197262 −0.00811426
\(592\) 2.77305 0.113972
\(593\) −36.2785 −1.48978 −0.744891 0.667187i \(-0.767498\pi\)
−0.744891 + 0.667187i \(0.767498\pi\)
\(594\) 1.96737 0.0807222
\(595\) −15.9488 −0.653839
\(596\) 15.2604 0.625090
\(597\) 4.70360 0.192506
\(598\) 0.952745 0.0389606
\(599\) 37.9189 1.54932 0.774662 0.632376i \(-0.217920\pi\)
0.774662 + 0.632376i \(0.217920\pi\)
\(600\) 0.793369 0.0323891
\(601\) −28.5638 −1.16514 −0.582571 0.812780i \(-0.697953\pi\)
−0.582571 + 0.812780i \(0.697953\pi\)
\(602\) −6.13463 −0.250029
\(603\) 5.60782 0.228368
\(604\) 3.51375 0.142973
\(605\) 14.6226 0.594492
\(606\) −16.7085 −0.678737
\(607\) 27.2317 1.10530 0.552651 0.833413i \(-0.313616\pi\)
0.552651 + 0.833413i \(0.313616\pi\)
\(608\) −7.72431 −0.313262
\(609\) −3.42478 −0.138779
\(610\) −5.33299 −0.215926
\(611\) −5.58312 −0.225869
\(612\) −4.75332 −0.192141
\(613\) 15.1951 0.613724 0.306862 0.951754i \(-0.400721\pi\)
0.306862 + 0.951754i \(0.400721\pi\)
\(614\) 8.85911 0.357524
\(615\) −19.6237 −0.791302
\(616\) 3.21848 0.129676
\(617\) 0.124161 0.00499852 0.00249926 0.999997i \(-0.499204\pi\)
0.00249926 + 0.999997i \(0.499204\pi\)
\(618\) 1.00000 0.0402259
\(619\) −4.35831 −0.175175 −0.0875876 0.996157i \(-0.527916\pi\)
−0.0875876 + 0.996157i \(0.527916\pi\)
\(620\) −18.7525 −0.753120
\(621\) 0.952745 0.0382323
\(622\) 34.3848 1.37871
\(623\) 10.2760 0.411699
\(624\) 1.00000 0.0400320
\(625\) −20.4037 −0.816149
\(626\) −22.8294 −0.912447
\(627\) −15.1966 −0.606893
\(628\) 13.6112 0.543146
\(629\) −13.1812 −0.525568
\(630\) 3.35531 0.133679
\(631\) −11.7231 −0.466690 −0.233345 0.972394i \(-0.574967\pi\)
−0.233345 + 0.972394i \(0.574967\pi\)
\(632\) 3.23726 0.128771
\(633\) −9.02680 −0.358783
\(634\) 26.9585 1.07066
\(635\) −4.41556 −0.175226
\(636\) 1.37438 0.0544976
\(637\) 4.32373 0.171312
\(638\) 4.11864 0.163059
\(639\) −0.768551 −0.0304034
\(640\) −2.05101 −0.0810732
\(641\) −34.1541 −1.34901 −0.674503 0.738272i \(-0.735642\pi\)
−0.674503 + 0.738272i \(0.735642\pi\)
\(642\) 15.8026 0.623679
\(643\) −5.89952 −0.232654 −0.116327 0.993211i \(-0.537112\pi\)
−0.116327 + 0.993211i \(0.537112\pi\)
\(644\) 1.55863 0.0614185
\(645\) 7.69113 0.302838
\(646\) 36.7161 1.44457
\(647\) −4.52293 −0.177815 −0.0889075 0.996040i \(-0.528338\pi\)
−0.0889075 + 0.996040i \(0.528338\pi\)
\(648\) 1.00000 0.0392837
\(649\) 18.4517 0.724291
\(650\) 0.793369 0.0311185
\(651\) 14.9575 0.586229
\(652\) −2.48927 −0.0974875
\(653\) −1.04118 −0.0407446 −0.0203723 0.999792i \(-0.506485\pi\)
−0.0203723 + 0.999792i \(0.506485\pi\)
\(654\) 7.02817 0.274823
\(655\) −13.1383 −0.513357
\(656\) −9.56781 −0.373560
\(657\) 9.15376 0.357122
\(658\) −9.13361 −0.356065
\(659\) 12.7324 0.495983 0.247991 0.968762i \(-0.420230\pi\)
0.247991 + 0.968762i \(0.420230\pi\)
\(660\) −4.03509 −0.157066
\(661\) 14.5700 0.566709 0.283354 0.959015i \(-0.408553\pi\)
0.283354 + 0.959015i \(0.408553\pi\)
\(662\) −14.0868 −0.547499
\(663\) −4.75332 −0.184604
\(664\) 12.4302 0.482383
\(665\) −25.9174 −1.00503
\(666\) 2.77305 0.107453
\(667\) 1.99455 0.0772293
\(668\) 5.51732 0.213471
\(669\) 27.5351 1.06457
\(670\) −11.5017 −0.444348
\(671\) −5.11552 −0.197482
\(672\) 1.63593 0.0631074
\(673\) −0.563354 −0.0217157 −0.0108579 0.999941i \(-0.503456\pi\)
−0.0108579 + 0.999941i \(0.503456\pi\)
\(674\) 24.1632 0.930731
\(675\) 0.793369 0.0305368
\(676\) 1.00000 0.0384615
\(677\) −50.4341 −1.93834 −0.969169 0.246395i \(-0.920754\pi\)
−0.969169 + 0.246395i \(0.920754\pi\)
\(678\) 4.15073 0.159408
\(679\) 12.4313 0.477069
\(680\) 9.74909 0.373860
\(681\) −8.37920 −0.321092
\(682\) −17.9878 −0.688790
\(683\) 3.47283 0.132884 0.0664420 0.997790i \(-0.478835\pi\)
0.0664420 + 0.997790i \(0.478835\pi\)
\(684\) −7.72431 −0.295346
\(685\) −7.16658 −0.273821
\(686\) 18.5248 0.707282
\(687\) −7.16726 −0.273448
\(688\) 3.74993 0.142965
\(689\) 1.37438 0.0523596
\(690\) −1.95409 −0.0743908
\(691\) 33.3395 1.26829 0.634147 0.773213i \(-0.281351\pi\)
0.634147 + 0.773213i \(0.281351\pi\)
\(692\) −3.49076 −0.132699
\(693\) 3.21848 0.122260
\(694\) 22.5146 0.854643
\(695\) 16.8015 0.637317
\(696\) 2.09348 0.0793530
\(697\) 45.4788 1.72263
\(698\) −9.14497 −0.346142
\(699\) −16.8531 −0.637443
\(700\) 1.29790 0.0490559
\(701\) −15.9443 −0.602207 −0.301103 0.953592i \(-0.597355\pi\)
−0.301103 + 0.953592i \(0.597355\pi\)
\(702\) 1.00000 0.0377426
\(703\) −21.4199 −0.807866
\(704\) −1.96737 −0.0741481
\(705\) 11.4510 0.431271
\(706\) −29.2329 −1.10019
\(707\) −27.3340 −1.02800
\(708\) 9.37885 0.352479
\(709\) 24.4755 0.919196 0.459598 0.888127i \(-0.347993\pi\)
0.459598 + 0.888127i \(0.347993\pi\)
\(710\) 1.57630 0.0591576
\(711\) 3.23726 0.121407
\(712\) −6.28143 −0.235407
\(713\) −8.71103 −0.326231
\(714\) −7.77610 −0.291013
\(715\) −4.03509 −0.150904
\(716\) 0.869690 0.0325018
\(717\) 23.4242 0.874793
\(718\) −14.2120 −0.530389
\(719\) 8.89640 0.331780 0.165890 0.986144i \(-0.446950\pi\)
0.165890 + 0.986144i \(0.446950\pi\)
\(720\) −2.05101 −0.0764365
\(721\) 1.63593 0.0609253
\(722\) 40.6649 1.51339
\(723\) −1.65066 −0.0613886
\(724\) 14.1847 0.527171
\(725\) 1.66090 0.0616842
\(726\) 7.12945 0.264599
\(727\) −48.6352 −1.80378 −0.901890 0.431967i \(-0.857820\pi\)
−0.901890 + 0.431967i \(0.857820\pi\)
\(728\) 1.63593 0.0606317
\(729\) 1.00000 0.0370370
\(730\) −18.7744 −0.694873
\(731\) −17.8246 −0.659267
\(732\) −2.60018 −0.0961054
\(733\) −25.3578 −0.936611 −0.468305 0.883567i \(-0.655135\pi\)
−0.468305 + 0.883567i \(0.655135\pi\)
\(734\) −33.2324 −1.22663
\(735\) −8.86800 −0.327101
\(736\) −0.952745 −0.0351187
\(737\) −11.0327 −0.406393
\(738\) −9.56781 −0.352196
\(739\) 4.13060 0.151947 0.0759733 0.997110i \(-0.475794\pi\)
0.0759733 + 0.997110i \(0.475794\pi\)
\(740\) −5.68754 −0.209078
\(741\) −7.72431 −0.283759
\(742\) 2.24839 0.0825409
\(743\) −4.36600 −0.160173 −0.0800864 0.996788i \(-0.525520\pi\)
−0.0800864 + 0.996788i \(0.525520\pi\)
\(744\) −9.14308 −0.335202
\(745\) −31.2992 −1.14671
\(746\) 28.9657 1.06051
\(747\) 12.4302 0.454796
\(748\) 9.35154 0.341926
\(749\) 25.8520 0.944611
\(750\) −11.8822 −0.433878
\(751\) −7.68384 −0.280387 −0.140194 0.990124i \(-0.544773\pi\)
−0.140194 + 0.990124i \(0.544773\pi\)
\(752\) 5.58312 0.203596
\(753\) 11.1260 0.405456
\(754\) 2.09348 0.0762399
\(755\) −7.20673 −0.262280
\(756\) 1.63593 0.0594983
\(757\) 37.6196 1.36731 0.683654 0.729807i \(-0.260390\pi\)
0.683654 + 0.729807i \(0.260390\pi\)
\(758\) 9.51928 0.345756
\(759\) −1.87440 −0.0680365
\(760\) 15.8426 0.574672
\(761\) −2.30144 −0.0834272 −0.0417136 0.999130i \(-0.513282\pi\)
−0.0417136 + 0.999130i \(0.513282\pi\)
\(762\) −2.15288 −0.0779905
\(763\) 11.4976 0.416241
\(764\) 5.68390 0.205636
\(765\) 9.74909 0.352479
\(766\) 4.91506 0.177588
\(767\) 9.37885 0.338650
\(768\) −1.00000 −0.0360844
\(769\) 9.62219 0.346985 0.173493 0.984835i \(-0.444495\pi\)
0.173493 + 0.984835i \(0.444495\pi\)
\(770\) −6.60113 −0.237888
\(771\) 17.3102 0.623412
\(772\) 13.5512 0.487719
\(773\) −21.4875 −0.772852 −0.386426 0.922320i \(-0.626291\pi\)
−0.386426 + 0.922320i \(0.626291\pi\)
\(774\) 3.74993 0.134788
\(775\) −7.25384 −0.260565
\(776\) −7.59891 −0.272785
\(777\) 4.53652 0.162747
\(778\) 13.5847 0.487035
\(779\) 73.9047 2.64791
\(780\) −2.05101 −0.0734378
\(781\) 1.51202 0.0541045
\(782\) 4.52870 0.161946
\(783\) 2.09348 0.0748147
\(784\) −4.32373 −0.154419
\(785\) −27.9167 −0.996389
\(786\) −6.40579 −0.228487
\(787\) −52.2357 −1.86200 −0.931002 0.365015i \(-0.881064\pi\)
−0.931002 + 0.365015i \(0.881064\pi\)
\(788\) 0.197262 0.00702716
\(789\) 7.79407 0.277476
\(790\) −6.63964 −0.236228
\(791\) 6.79032 0.241436
\(792\) −1.96737 −0.0699075
\(793\) −2.60018 −0.0923351
\(794\) 26.5233 0.941276
\(795\) −2.81886 −0.0999746
\(796\) −4.70360 −0.166715
\(797\) −38.1373 −1.35089 −0.675446 0.737410i \(-0.736049\pi\)
−0.675446 + 0.737410i \(0.736049\pi\)
\(798\) −12.6364 −0.447325
\(799\) −26.5384 −0.938860
\(800\) −0.793369 −0.0280498
\(801\) −6.28143 −0.221943
\(802\) 11.4425 0.404050
\(803\) −18.0088 −0.635518
\(804\) −5.60782 −0.197772
\(805\) −3.19675 −0.112671
\(806\) −9.14308 −0.322051
\(807\) 10.1302 0.356602
\(808\) 16.7085 0.587804
\(809\) 5.60400 0.197026 0.0985130 0.995136i \(-0.468591\pi\)
0.0985130 + 0.995136i \(0.468591\pi\)
\(810\) −2.05101 −0.0720651
\(811\) 25.2858 0.887906 0.443953 0.896050i \(-0.353576\pi\)
0.443953 + 0.896050i \(0.353576\pi\)
\(812\) 3.42478 0.120186
\(813\) −13.4818 −0.472827
\(814\) −5.45562 −0.191219
\(815\) 5.10552 0.178839
\(816\) 4.75332 0.166399
\(817\) −28.9656 −1.01338
\(818\) 30.8632 1.07911
\(819\) 1.63593 0.0571641
\(820\) 19.6237 0.685288
\(821\) 4.00401 0.139741 0.0698705 0.997556i \(-0.477741\pi\)
0.0698705 + 0.997556i \(0.477741\pi\)
\(822\) −3.49417 −0.121873
\(823\) −52.2745 −1.82217 −0.911087 0.412214i \(-0.864756\pi\)
−0.911087 + 0.412214i \(0.864756\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −1.56085 −0.0543418
\(826\) 15.3432 0.533857
\(827\) −27.6416 −0.961193 −0.480597 0.876942i \(-0.659580\pi\)
−0.480597 + 0.876942i \(0.659580\pi\)
\(828\) −0.952745 −0.0331102
\(829\) 5.56663 0.193337 0.0966685 0.995317i \(-0.469181\pi\)
0.0966685 + 0.995317i \(0.469181\pi\)
\(830\) −25.4943 −0.884921
\(831\) −26.0301 −0.902973
\(832\) −1.00000 −0.0346688
\(833\) 20.5520 0.712086
\(834\) 8.19183 0.283660
\(835\) −11.3161 −0.391608
\(836\) 15.1966 0.525584
\(837\) −9.14308 −0.316031
\(838\) −9.88812 −0.341579
\(839\) 28.6618 0.989516 0.494758 0.869031i \(-0.335257\pi\)
0.494758 + 0.869031i \(0.335257\pi\)
\(840\) −3.35531 −0.115769
\(841\) −24.6174 −0.848874
\(842\) 37.1616 1.28067
\(843\) −3.54102 −0.121959
\(844\) 9.02680 0.310715
\(845\) −2.05101 −0.0705568
\(846\) 5.58312 0.191952
\(847\) 11.6633 0.400756
\(848\) −1.37438 −0.0471963
\(849\) 18.7384 0.643100
\(850\) 3.77113 0.129349
\(851\) −2.64201 −0.0905669
\(852\) 0.768551 0.0263301
\(853\) 53.0347 1.81587 0.907937 0.419106i \(-0.137656\pi\)
0.907937 + 0.419106i \(0.137656\pi\)
\(854\) −4.25372 −0.145559
\(855\) 15.8426 0.541806
\(856\) −15.8026 −0.540122
\(857\) −47.8208 −1.63353 −0.816764 0.576973i \(-0.804234\pi\)
−0.816764 + 0.576973i \(0.804234\pi\)
\(858\) −1.96737 −0.0671649
\(859\) −6.92360 −0.236230 −0.118115 0.993000i \(-0.537685\pi\)
−0.118115 + 0.993000i \(0.537685\pi\)
\(860\) −7.69113 −0.262265
\(861\) −15.6523 −0.533429
\(862\) −26.6743 −0.908530
\(863\) 29.7291 1.01199 0.505996 0.862536i \(-0.331125\pi\)
0.505996 + 0.862536i \(0.331125\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 7.15957 0.243433
\(866\) 31.0568 1.05535
\(867\) −5.59402 −0.189983
\(868\) −14.9575 −0.507689
\(869\) −6.36888 −0.216050
\(870\) −4.29373 −0.145571
\(871\) −5.60782 −0.190014
\(872\) −7.02817 −0.238004
\(873\) −7.59891 −0.257184
\(874\) 7.35930 0.248932
\(875\) −19.4385 −0.657142
\(876\) −9.15376 −0.309277
\(877\) 30.3693 1.02550 0.512750 0.858538i \(-0.328627\pi\)
0.512750 + 0.858538i \(0.328627\pi\)
\(878\) 19.0081 0.641493
\(879\) 14.0709 0.474601
\(880\) 4.03509 0.136023
\(881\) 42.2381 1.42304 0.711519 0.702667i \(-0.248008\pi\)
0.711519 + 0.702667i \(0.248008\pi\)
\(882\) −4.32373 −0.145587
\(883\) 6.78836 0.228447 0.114223 0.993455i \(-0.463562\pi\)
0.114223 + 0.993455i \(0.463562\pi\)
\(884\) 4.75332 0.159871
\(885\) −19.2361 −0.646614
\(886\) 27.5936 0.927027
\(887\) −48.8961 −1.64177 −0.820884 0.571094i \(-0.806519\pi\)
−0.820884 + 0.571094i \(0.806519\pi\)
\(888\) −2.77305 −0.0930574
\(889\) −3.52196 −0.118123
\(890\) 12.8833 0.431848
\(891\) −1.96737 −0.0659094
\(892\) −27.5351 −0.921944
\(893\) −43.1258 −1.44315
\(894\) −15.2604 −0.510384
\(895\) −1.78374 −0.0596239
\(896\) −1.63593 −0.0546526
\(897\) −0.952745 −0.0318112
\(898\) 32.0669 1.07008
\(899\) −19.1408 −0.638382
\(900\) −0.793369 −0.0264456
\(901\) 6.53285 0.217641
\(902\) 18.8234 0.626752
\(903\) 6.13463 0.204148
\(904\) −4.15073 −0.138051
\(905\) −29.0929 −0.967082
\(906\) −3.51375 −0.116737
\(907\) 41.9144 1.39175 0.695873 0.718165i \(-0.255018\pi\)
0.695873 + 0.718165i \(0.255018\pi\)
\(908\) 8.37920 0.278073
\(909\) 16.7085 0.554186
\(910\) −3.35531 −0.111227
\(911\) 5.97789 0.198056 0.0990282 0.995085i \(-0.468427\pi\)
0.0990282 + 0.995085i \(0.468427\pi\)
\(912\) 7.72431 0.255777
\(913\) −24.4547 −0.809333
\(914\) 13.9628 0.461848
\(915\) 5.33299 0.176303
\(916\) 7.16726 0.236813
\(917\) −10.4794 −0.346061
\(918\) 4.75332 0.156883
\(919\) 14.0890 0.464754 0.232377 0.972626i \(-0.425350\pi\)
0.232377 + 0.972626i \(0.425350\pi\)
\(920\) 1.95409 0.0644244
\(921\) −8.85911 −0.291918
\(922\) 21.1741 0.697333
\(923\) 0.768551 0.0252972
\(924\) −3.21848 −0.105880
\(925\) −2.20005 −0.0723372
\(926\) −3.10076 −0.101897
\(927\) −1.00000 −0.0328443
\(928\) −2.09348 −0.0687217
\(929\) 33.7949 1.10878 0.554388 0.832259i \(-0.312953\pi\)
0.554388 + 0.832259i \(0.312953\pi\)
\(930\) 18.7525 0.614920
\(931\) 33.3978 1.09457
\(932\) 16.8531 0.552042
\(933\) −34.3848 −1.12571
\(934\) −11.1953 −0.366322
\(935\) −19.1801 −0.627255
\(936\) −1.00000 −0.0326860
\(937\) −55.7089 −1.81993 −0.909965 0.414684i \(-0.863892\pi\)
−0.909965 + 0.414684i \(0.863892\pi\)
\(938\) −9.17401 −0.299542
\(939\) 22.8294 0.745010
\(940\) −11.4510 −0.373491
\(941\) −28.6326 −0.933396 −0.466698 0.884417i \(-0.654556\pi\)
−0.466698 + 0.884417i \(0.654556\pi\)
\(942\) −13.6112 −0.443477
\(943\) 9.11569 0.296848
\(944\) −9.37885 −0.305255
\(945\) −3.35531 −0.109148
\(946\) −7.37750 −0.239863
\(947\) −38.6602 −1.25629 −0.628143 0.778098i \(-0.716185\pi\)
−0.628143 + 0.778098i \(0.716185\pi\)
\(948\) −3.23726 −0.105141
\(949\) −9.15376 −0.297144
\(950\) 6.12822 0.198826
\(951\) −26.9585 −0.874191
\(952\) 7.77610 0.252025
\(953\) −39.3523 −1.27475 −0.637373 0.770556i \(-0.719979\pi\)
−0.637373 + 0.770556i \(0.719979\pi\)
\(954\) −1.37438 −0.0444971
\(955\) −11.6577 −0.377235
\(956\) −23.4242 −0.757593
\(957\) −4.11864 −0.133137
\(958\) −20.8766 −0.674492
\(959\) −5.71623 −0.184587
\(960\) 2.05101 0.0661960
\(961\) 52.5959 1.69664
\(962\) −2.77305 −0.0894067
\(963\) −15.8026 −0.509232
\(964\) 1.65066 0.0531641
\(965\) −27.7937 −0.894710
\(966\) −1.55863 −0.0501480
\(967\) 7.76103 0.249578 0.124789 0.992183i \(-0.460175\pi\)
0.124789 + 0.992183i \(0.460175\pi\)
\(968\) −7.12945 −0.229149
\(969\) −36.7161 −1.17949
\(970\) 15.5854 0.500417
\(971\) −37.9637 −1.21831 −0.609156 0.793050i \(-0.708492\pi\)
−0.609156 + 0.793050i \(0.708492\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 13.4013 0.429625
\(974\) 37.8999 1.21439
\(975\) −0.793369 −0.0254081
\(976\) 2.60018 0.0832297
\(977\) −44.3722 −1.41959 −0.709797 0.704406i \(-0.751213\pi\)
−0.709797 + 0.704406i \(0.751213\pi\)
\(978\) 2.48927 0.0795982
\(979\) 12.3579 0.394960
\(980\) 8.86800 0.283278
\(981\) −7.02817 −0.224392
\(982\) 15.4924 0.494381
\(983\) 47.9513 1.52941 0.764704 0.644382i \(-0.222885\pi\)
0.764704 + 0.644382i \(0.222885\pi\)
\(984\) 9.56781 0.305011
\(985\) −0.404585 −0.0128912
\(986\) 9.95096 0.316903
\(987\) 9.13361 0.290726
\(988\) 7.72431 0.245743
\(989\) −3.57273 −0.113606
\(990\) 4.03509 0.128244
\(991\) 10.1341 0.321920 0.160960 0.986961i \(-0.448541\pi\)
0.160960 + 0.986961i \(0.448541\pi\)
\(992\) 9.14308 0.290293
\(993\) 14.0868 0.447031
\(994\) 1.25730 0.0398790
\(995\) 9.64713 0.305834
\(996\) −12.4302 −0.393864
\(997\) −23.4023 −0.741157 −0.370578 0.928801i \(-0.620841\pi\)
−0.370578 + 0.928801i \(0.620841\pi\)
\(998\) −33.2335 −1.05199
\(999\) −2.77305 −0.0877354
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 8034.2.a.bc.1.5 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bc.1.5 15 1.1 even 1 trivial