Properties

Label 2667.2.a.l.1.5
Level 2667
Weight 2
Character 2667.1
Self dual yes
Analytic conductor 21.296
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2667 = 3 \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2667.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.2961022191\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 4 x^{12} - 10 x^{11} + 53 x^{10} + 19 x^{9} - 242 x^{8} + 61 x^{7} + 467 x^{6} - 211 x^{5} - 372 x^{4} + 146 x^{3} + 116 x^{2} - 12 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.423652\) of defining polynomial
Character \(\chi\) \(=\) 2667.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.423652 q^{2} -1.00000 q^{3} -1.82052 q^{4} +2.66196 q^{5} +0.423652 q^{6} +1.00000 q^{7} +1.61857 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.423652 q^{2} -1.00000 q^{3} -1.82052 q^{4} +2.66196 q^{5} +0.423652 q^{6} +1.00000 q^{7} +1.61857 q^{8} +1.00000 q^{9} -1.12774 q^{10} +3.97552 q^{11} +1.82052 q^{12} +4.95058 q^{13} -0.423652 q^{14} -2.66196 q^{15} +2.95533 q^{16} +5.63178 q^{17} -0.423652 q^{18} +3.72073 q^{19} -4.84615 q^{20} -1.00000 q^{21} -1.68424 q^{22} -4.13626 q^{23} -1.61857 q^{24} +2.08603 q^{25} -2.09732 q^{26} -1.00000 q^{27} -1.82052 q^{28} +7.04468 q^{29} +1.12774 q^{30} -2.57265 q^{31} -4.48917 q^{32} -3.97552 q^{33} -2.38592 q^{34} +2.66196 q^{35} -1.82052 q^{36} +2.65212 q^{37} -1.57630 q^{38} -4.95058 q^{39} +4.30857 q^{40} +2.05217 q^{41} +0.423652 q^{42} +4.03075 q^{43} -7.23751 q^{44} +2.66196 q^{45} +1.75233 q^{46} -7.52083 q^{47} -2.95533 q^{48} +1.00000 q^{49} -0.883750 q^{50} -5.63178 q^{51} -9.01263 q^{52} +0.214398 q^{53} +0.423652 q^{54} +10.5827 q^{55} +1.61857 q^{56} -3.72073 q^{57} -2.98449 q^{58} -12.5754 q^{59} +4.84615 q^{60} -2.37431 q^{61} +1.08991 q^{62} +1.00000 q^{63} -4.00881 q^{64} +13.1783 q^{65} +1.68424 q^{66} +4.26602 q^{67} -10.2528 q^{68} +4.13626 q^{69} -1.12774 q^{70} -13.7119 q^{71} +1.61857 q^{72} -10.8892 q^{73} -1.12358 q^{74} -2.08603 q^{75} -6.77367 q^{76} +3.97552 q^{77} +2.09732 q^{78} +2.69299 q^{79} +7.86696 q^{80} +1.00000 q^{81} -0.869404 q^{82} +4.22143 q^{83} +1.82052 q^{84} +14.9916 q^{85} -1.70763 q^{86} -7.04468 q^{87} +6.43466 q^{88} +4.08362 q^{89} -1.12774 q^{90} +4.95058 q^{91} +7.53014 q^{92} +2.57265 q^{93} +3.18621 q^{94} +9.90444 q^{95} +4.48917 q^{96} +0.135110 q^{97} -0.423652 q^{98} +3.97552 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 4q^{2} - 13q^{3} + 10q^{4} + 12q^{5} - 4q^{6} + 13q^{7} + 9q^{8} + 13q^{9} + O(q^{10}) \) \( 13q + 4q^{2} - 13q^{3} + 10q^{4} + 12q^{5} - 4q^{6} + 13q^{7} + 9q^{8} + 13q^{9} + 6q^{10} + 3q^{11} - 10q^{12} + 21q^{13} + 4q^{14} - 12q^{15} + 8q^{16} + 17q^{17} + 4q^{18} + 5q^{19} + 29q^{20} - 13q^{21} + q^{22} + 4q^{23} - 9q^{24} + q^{25} + 22q^{26} - 13q^{27} + 10q^{28} + 21q^{29} - 6q^{30} - 7q^{31} + 12q^{32} - 3q^{33} + 2q^{34} + 12q^{35} + 10q^{36} + 7q^{37} - 9q^{38} - 21q^{39} + 29q^{40} + 21q^{41} - 4q^{42} - 9q^{43} - 2q^{44} + 12q^{45} - 28q^{46} + 23q^{47} - 8q^{48} + 13q^{49} + 15q^{50} - 17q^{51} + 15q^{52} + 31q^{53} - 4q^{54} - 8q^{55} + 9q^{56} - 5q^{57} - 25q^{58} + 28q^{59} - 29q^{60} + 29q^{61} - 3q^{62} + 13q^{63} + 9q^{64} + 30q^{65} - q^{66} - 18q^{67} + 34q^{68} - 4q^{69} + 6q^{70} + 10q^{71} + 9q^{72} + 24q^{73} - 19q^{74} - q^{75} + 3q^{77} - 22q^{78} - 28q^{79} + 26q^{80} + 13q^{81} + 18q^{82} + 26q^{83} - 10q^{84} + 20q^{85} - 2q^{86} - 21q^{87} - 17q^{88} + 44q^{89} + 6q^{90} + 21q^{91} + 6q^{92} + 7q^{93} - 9q^{94} - 2q^{95} - 12q^{96} + 17q^{97} + 4q^{98} + 3q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.423652 −0.299567 −0.149784 0.988719i \(-0.547858\pi\)
−0.149784 + 0.988719i \(0.547858\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.82052 −0.910259
\(5\) 2.66196 1.19046 0.595232 0.803554i \(-0.297060\pi\)
0.595232 + 0.803554i \(0.297060\pi\)
\(6\) 0.423652 0.172955
\(7\) 1.00000 0.377964
\(8\) 1.61857 0.572251
\(9\) 1.00000 0.333333
\(10\) −1.12774 −0.356624
\(11\) 3.97552 1.19866 0.599332 0.800500i \(-0.295433\pi\)
0.599332 + 0.800500i \(0.295433\pi\)
\(12\) 1.82052 0.525539
\(13\) 4.95058 1.37304 0.686522 0.727109i \(-0.259136\pi\)
0.686522 + 0.727109i \(0.259136\pi\)
\(14\) −0.423652 −0.113226
\(15\) −2.66196 −0.687315
\(16\) 2.95533 0.738832
\(17\) 5.63178 1.36591 0.682954 0.730462i \(-0.260695\pi\)
0.682954 + 0.730462i \(0.260695\pi\)
\(18\) −0.423652 −0.0998558
\(19\) 3.72073 0.853595 0.426797 0.904347i \(-0.359642\pi\)
0.426797 + 0.904347i \(0.359642\pi\)
\(20\) −4.84615 −1.08363
\(21\) −1.00000 −0.218218
\(22\) −1.68424 −0.359081
\(23\) −4.13626 −0.862469 −0.431235 0.902240i \(-0.641922\pi\)
−0.431235 + 0.902240i \(0.641922\pi\)
\(24\) −1.61857 −0.330389
\(25\) 2.08603 0.417205
\(26\) −2.09732 −0.411319
\(27\) −1.00000 −0.192450
\(28\) −1.82052 −0.344046
\(29\) 7.04468 1.30816 0.654082 0.756423i \(-0.273055\pi\)
0.654082 + 0.756423i \(0.273055\pi\)
\(30\) 1.12774 0.205897
\(31\) −2.57265 −0.462062 −0.231031 0.972946i \(-0.574210\pi\)
−0.231031 + 0.972946i \(0.574210\pi\)
\(32\) −4.48917 −0.793581
\(33\) −3.97552 −0.692049
\(34\) −2.38592 −0.409181
\(35\) 2.66196 0.449953
\(36\) −1.82052 −0.303420
\(37\) 2.65212 0.436006 0.218003 0.975948i \(-0.430046\pi\)
0.218003 + 0.975948i \(0.430046\pi\)
\(38\) −1.57630 −0.255709
\(39\) −4.95058 −0.792728
\(40\) 4.30857 0.681245
\(41\) 2.05217 0.320494 0.160247 0.987077i \(-0.448771\pi\)
0.160247 + 0.987077i \(0.448771\pi\)
\(42\) 0.423652 0.0653709
\(43\) 4.03075 0.614683 0.307342 0.951599i \(-0.400561\pi\)
0.307342 + 0.951599i \(0.400561\pi\)
\(44\) −7.23751 −1.09110
\(45\) 2.66196 0.396821
\(46\) 1.75233 0.258368
\(47\) −7.52083 −1.09703 −0.548513 0.836142i \(-0.684806\pi\)
−0.548513 + 0.836142i \(0.684806\pi\)
\(48\) −2.95533 −0.426565
\(49\) 1.00000 0.142857
\(50\) −0.883750 −0.124981
\(51\) −5.63178 −0.788607
\(52\) −9.01263 −1.24983
\(53\) 0.214398 0.0294499 0.0147249 0.999892i \(-0.495313\pi\)
0.0147249 + 0.999892i \(0.495313\pi\)
\(54\) 0.423652 0.0576517
\(55\) 10.5827 1.42697
\(56\) 1.61857 0.216291
\(57\) −3.72073 −0.492823
\(58\) −2.98449 −0.391883
\(59\) −12.5754 −1.63718 −0.818591 0.574376i \(-0.805245\pi\)
−0.818591 + 0.574376i \(0.805245\pi\)
\(60\) 4.84615 0.625635
\(61\) −2.37431 −0.303999 −0.151999 0.988381i \(-0.548571\pi\)
−0.151999 + 0.988381i \(0.548571\pi\)
\(62\) 1.08991 0.138419
\(63\) 1.00000 0.125988
\(64\) −4.00881 −0.501101
\(65\) 13.1783 1.63456
\(66\) 1.68424 0.207315
\(67\) 4.26602 0.521177 0.260588 0.965450i \(-0.416083\pi\)
0.260588 + 0.965450i \(0.416083\pi\)
\(68\) −10.2528 −1.24333
\(69\) 4.13626 0.497947
\(70\) −1.12774 −0.134791
\(71\) −13.7119 −1.62731 −0.813653 0.581351i \(-0.802524\pi\)
−0.813653 + 0.581351i \(0.802524\pi\)
\(72\) 1.61857 0.190750
\(73\) −10.8892 −1.27449 −0.637244 0.770662i \(-0.719926\pi\)
−0.637244 + 0.770662i \(0.719926\pi\)
\(74\) −1.12358 −0.130613
\(75\) −2.08603 −0.240874
\(76\) −6.77367 −0.776993
\(77\) 3.97552 0.453053
\(78\) 2.09732 0.237475
\(79\) 2.69299 0.302985 0.151492 0.988458i \(-0.451592\pi\)
0.151492 + 0.988458i \(0.451592\pi\)
\(80\) 7.86696 0.879553
\(81\) 1.00000 0.111111
\(82\) −0.869404 −0.0960096
\(83\) 4.22143 0.463362 0.231681 0.972792i \(-0.425577\pi\)
0.231681 + 0.972792i \(0.425577\pi\)
\(84\) 1.82052 0.198635
\(85\) 14.9916 1.62606
\(86\) −1.70763 −0.184139
\(87\) −7.04468 −0.755269
\(88\) 6.43466 0.685937
\(89\) 4.08362 0.432863 0.216431 0.976298i \(-0.430558\pi\)
0.216431 + 0.976298i \(0.430558\pi\)
\(90\) −1.12774 −0.118875
\(91\) 4.95058 0.518962
\(92\) 7.53014 0.785071
\(93\) 2.57265 0.266772
\(94\) 3.18621 0.328633
\(95\) 9.90444 1.01617
\(96\) 4.48917 0.458174
\(97\) 0.135110 0.0137183 0.00685917 0.999976i \(-0.497817\pi\)
0.00685917 + 0.999976i \(0.497817\pi\)
\(98\) −0.423652 −0.0427953
\(99\) 3.97552 0.399555
\(100\) −3.79765 −0.379765
\(101\) 0.725428 0.0721828 0.0360914 0.999348i \(-0.488509\pi\)
0.0360914 + 0.999348i \(0.488509\pi\)
\(102\) 2.38592 0.236241
\(103\) 14.2476 1.40386 0.701930 0.712245i \(-0.252322\pi\)
0.701930 + 0.712245i \(0.252322\pi\)
\(104\) 8.01287 0.785726
\(105\) −2.66196 −0.259781
\(106\) −0.0908303 −0.00882222
\(107\) −18.0628 −1.74620 −0.873098 0.487544i \(-0.837893\pi\)
−0.873098 + 0.487544i \(0.837893\pi\)
\(108\) 1.82052 0.175180
\(109\) −7.40505 −0.709276 −0.354638 0.935004i \(-0.615396\pi\)
−0.354638 + 0.935004i \(0.615396\pi\)
\(110\) −4.48337 −0.427473
\(111\) −2.65212 −0.251728
\(112\) 2.95533 0.279252
\(113\) 18.6046 1.75017 0.875087 0.483966i \(-0.160804\pi\)
0.875087 + 0.483966i \(0.160804\pi\)
\(114\) 1.57630 0.147634
\(115\) −11.0105 −1.02674
\(116\) −12.8250 −1.19077
\(117\) 4.95058 0.457682
\(118\) 5.32761 0.490446
\(119\) 5.63178 0.516264
\(120\) −4.30857 −0.393317
\(121\) 4.80477 0.436797
\(122\) 1.00588 0.0910681
\(123\) −2.05217 −0.185038
\(124\) 4.68357 0.420597
\(125\) −7.75688 −0.693796
\(126\) −0.423652 −0.0377419
\(127\) −1.00000 −0.0887357
\(128\) 10.6767 0.943694
\(129\) −4.03075 −0.354888
\(130\) −5.58299 −0.489661
\(131\) 11.7101 1.02312 0.511559 0.859248i \(-0.329068\pi\)
0.511559 + 0.859248i \(0.329068\pi\)
\(132\) 7.23751 0.629945
\(133\) 3.72073 0.322629
\(134\) −1.80731 −0.156127
\(135\) −2.66196 −0.229105
\(136\) 9.11543 0.781642
\(137\) 2.39147 0.204317 0.102159 0.994768i \(-0.467425\pi\)
0.102159 + 0.994768i \(0.467425\pi\)
\(138\) −1.75233 −0.149169
\(139\) −2.14800 −0.182191 −0.0910953 0.995842i \(-0.529037\pi\)
−0.0910953 + 0.995842i \(0.529037\pi\)
\(140\) −4.84615 −0.409574
\(141\) 7.52083 0.633368
\(142\) 5.80909 0.487488
\(143\) 19.6811 1.64582
\(144\) 2.95533 0.246277
\(145\) 18.7527 1.55732
\(146\) 4.61324 0.381795
\(147\) −1.00000 −0.0824786
\(148\) −4.82823 −0.396878
\(149\) −2.05953 −0.168723 −0.0843615 0.996435i \(-0.526885\pi\)
−0.0843615 + 0.996435i \(0.526885\pi\)
\(150\) 0.883750 0.0721579
\(151\) 12.3000 1.00096 0.500478 0.865749i \(-0.333157\pi\)
0.500478 + 0.865749i \(0.333157\pi\)
\(152\) 6.02227 0.488471
\(153\) 5.63178 0.455302
\(154\) −1.68424 −0.135720
\(155\) −6.84830 −0.550069
\(156\) 9.01263 0.721588
\(157\) −19.3425 −1.54370 −0.771850 0.635804i \(-0.780669\pi\)
−0.771850 + 0.635804i \(0.780669\pi\)
\(158\) −1.14089 −0.0907643
\(159\) −0.214398 −0.0170029
\(160\) −11.9500 −0.944730
\(161\) −4.13626 −0.325983
\(162\) −0.423652 −0.0332853
\(163\) −12.4805 −0.977551 −0.488775 0.872410i \(-0.662556\pi\)
−0.488775 + 0.872410i \(0.662556\pi\)
\(164\) −3.73601 −0.291733
\(165\) −10.5827 −0.823860
\(166\) −1.78842 −0.138808
\(167\) 2.16315 0.167390 0.0836949 0.996491i \(-0.473328\pi\)
0.0836949 + 0.996491i \(0.473328\pi\)
\(168\) −1.61857 −0.124875
\(169\) 11.5083 0.885252
\(170\) −6.35121 −0.487115
\(171\) 3.72073 0.284532
\(172\) −7.33805 −0.559521
\(173\) −15.5879 −1.18513 −0.592563 0.805524i \(-0.701884\pi\)
−0.592563 + 0.805524i \(0.701884\pi\)
\(174\) 2.98449 0.226254
\(175\) 2.08603 0.157689
\(176\) 11.7490 0.885612
\(177\) 12.5754 0.945228
\(178\) −1.73003 −0.129671
\(179\) −11.1995 −0.837090 −0.418545 0.908196i \(-0.637460\pi\)
−0.418545 + 0.908196i \(0.637460\pi\)
\(180\) −4.84615 −0.361210
\(181\) −6.72881 −0.500148 −0.250074 0.968227i \(-0.580455\pi\)
−0.250074 + 0.968227i \(0.580455\pi\)
\(182\) −2.09732 −0.155464
\(183\) 2.37431 0.175514
\(184\) −6.69483 −0.493549
\(185\) 7.05983 0.519049
\(186\) −1.08991 −0.0799161
\(187\) 22.3893 1.63726
\(188\) 13.6918 0.998578
\(189\) −1.00000 −0.0727393
\(190\) −4.19604 −0.304413
\(191\) 23.3444 1.68914 0.844570 0.535446i \(-0.179856\pi\)
0.844570 + 0.535446i \(0.179856\pi\)
\(192\) 4.00881 0.289311
\(193\) 2.13924 0.153986 0.0769928 0.997032i \(-0.475468\pi\)
0.0769928 + 0.997032i \(0.475468\pi\)
\(194\) −0.0572397 −0.00410957
\(195\) −13.1783 −0.943714
\(196\) −1.82052 −0.130037
\(197\) 7.01767 0.499988 0.249994 0.968247i \(-0.419571\pi\)
0.249994 + 0.968247i \(0.419571\pi\)
\(198\) −1.68424 −0.119694
\(199\) −18.7133 −1.32655 −0.663277 0.748374i \(-0.730835\pi\)
−0.663277 + 0.748374i \(0.730835\pi\)
\(200\) 3.37638 0.238746
\(201\) −4.26602 −0.300902
\(202\) −0.307329 −0.0216236
\(203\) 7.04468 0.494440
\(204\) 10.2528 0.717837
\(205\) 5.46278 0.381537
\(206\) −6.03604 −0.420551
\(207\) −4.13626 −0.287490
\(208\) 14.6306 1.01445
\(209\) 14.7919 1.02317
\(210\) 1.12774 0.0778218
\(211\) −11.4113 −0.785585 −0.392792 0.919627i \(-0.628491\pi\)
−0.392792 + 0.919627i \(0.628491\pi\)
\(212\) −0.390316 −0.0268070
\(213\) 13.7119 0.939526
\(214\) 7.65234 0.523103
\(215\) 10.7297 0.731759
\(216\) −1.61857 −0.110130
\(217\) −2.57265 −0.174643
\(218\) 3.13717 0.212476
\(219\) 10.8892 0.735826
\(220\) −19.2660 −1.29891
\(221\) 27.8806 1.87545
\(222\) 1.12358 0.0754095
\(223\) 12.7903 0.856505 0.428252 0.903659i \(-0.359129\pi\)
0.428252 + 0.903659i \(0.359129\pi\)
\(224\) −4.48917 −0.299945
\(225\) 2.08603 0.139068
\(226\) −7.88188 −0.524295
\(227\) −23.6675 −1.57087 −0.785434 0.618945i \(-0.787560\pi\)
−0.785434 + 0.618945i \(0.787560\pi\)
\(228\) 6.77367 0.448597
\(229\) −16.7152 −1.10457 −0.552287 0.833654i \(-0.686245\pi\)
−0.552287 + 0.833654i \(0.686245\pi\)
\(230\) 4.66464 0.307577
\(231\) −3.97552 −0.261570
\(232\) 11.4023 0.748599
\(233\) 27.5835 1.80706 0.903528 0.428529i \(-0.140968\pi\)
0.903528 + 0.428529i \(0.140968\pi\)
\(234\) −2.09732 −0.137106
\(235\) −20.0201 −1.30597
\(236\) 22.8938 1.49026
\(237\) −2.69299 −0.174928
\(238\) −2.38592 −0.154656
\(239\) −6.10076 −0.394625 −0.197313 0.980341i \(-0.563221\pi\)
−0.197313 + 0.980341i \(0.563221\pi\)
\(240\) −7.86696 −0.507810
\(241\) −16.4442 −1.05926 −0.529632 0.848228i \(-0.677670\pi\)
−0.529632 + 0.848228i \(0.677670\pi\)
\(242\) −2.03555 −0.130850
\(243\) −1.00000 −0.0641500
\(244\) 4.32247 0.276718
\(245\) 2.66196 0.170066
\(246\) 0.869404 0.0554312
\(247\) 18.4198 1.17202
\(248\) −4.16402 −0.264416
\(249\) −4.22143 −0.267522
\(250\) 3.28622 0.207839
\(251\) 31.2771 1.97420 0.987098 0.160118i \(-0.0511875\pi\)
0.987098 + 0.160118i \(0.0511875\pi\)
\(252\) −1.82052 −0.114682
\(253\) −16.4438 −1.03381
\(254\) 0.423652 0.0265823
\(255\) −14.9916 −0.938808
\(256\) 3.49442 0.218401
\(257\) 15.4673 0.964824 0.482412 0.875944i \(-0.339761\pi\)
0.482412 + 0.875944i \(0.339761\pi\)
\(258\) 1.70763 0.106313
\(259\) 2.65212 0.164795
\(260\) −23.9913 −1.48787
\(261\) 7.04468 0.436055
\(262\) −4.96102 −0.306493
\(263\) −5.45766 −0.336534 −0.168267 0.985741i \(-0.553817\pi\)
−0.168267 + 0.985741i \(0.553817\pi\)
\(264\) −6.43466 −0.396026
\(265\) 0.570720 0.0350590
\(266\) −1.57630 −0.0966489
\(267\) −4.08362 −0.249913
\(268\) −7.76636 −0.474406
\(269\) 24.3057 1.48194 0.740971 0.671536i \(-0.234365\pi\)
0.740971 + 0.671536i \(0.234365\pi\)
\(270\) 1.12774 0.0686323
\(271\) 26.4187 1.60482 0.802410 0.596774i \(-0.203551\pi\)
0.802410 + 0.596774i \(0.203551\pi\)
\(272\) 16.6438 1.00918
\(273\) −4.95058 −0.299623
\(274\) −1.01315 −0.0612068
\(275\) 8.29304 0.500089
\(276\) −7.53014 −0.453261
\(277\) −13.7390 −0.825494 −0.412747 0.910846i \(-0.635431\pi\)
−0.412747 + 0.910846i \(0.635431\pi\)
\(278\) 0.910003 0.0545784
\(279\) −2.57265 −0.154021
\(280\) 4.30857 0.257486
\(281\) −5.06765 −0.302311 −0.151155 0.988510i \(-0.548299\pi\)
−0.151155 + 0.988510i \(0.548299\pi\)
\(282\) −3.18621 −0.189736
\(283\) −27.6362 −1.64280 −0.821400 0.570353i \(-0.806806\pi\)
−0.821400 + 0.570353i \(0.806806\pi\)
\(284\) 24.9628 1.48127
\(285\) −9.90444 −0.586688
\(286\) −8.33796 −0.493034
\(287\) 2.05217 0.121135
\(288\) −4.48917 −0.264527
\(289\) 14.7169 0.865702
\(290\) −7.94460 −0.466523
\(291\) −0.135110 −0.00792029
\(292\) 19.8240 1.16011
\(293\) 26.9116 1.57219 0.786097 0.618104i \(-0.212099\pi\)
0.786097 + 0.618104i \(0.212099\pi\)
\(294\) 0.423652 0.0247079
\(295\) −33.4753 −1.94901
\(296\) 4.29264 0.249505
\(297\) −3.97552 −0.230683
\(298\) 0.872522 0.0505439
\(299\) −20.4769 −1.18421
\(300\) 3.79765 0.219258
\(301\) 4.03075 0.232328
\(302\) −5.21091 −0.299854
\(303\) −0.725428 −0.0416747
\(304\) 10.9960 0.630663
\(305\) −6.32031 −0.361900
\(306\) −2.38592 −0.136394
\(307\) −10.2148 −0.582989 −0.291494 0.956573i \(-0.594152\pi\)
−0.291494 + 0.956573i \(0.594152\pi\)
\(308\) −7.23751 −0.412395
\(309\) −14.2476 −0.810519
\(310\) 2.90130 0.164783
\(311\) 9.69180 0.549572 0.274786 0.961505i \(-0.411393\pi\)
0.274786 + 0.961505i \(0.411393\pi\)
\(312\) −8.01287 −0.453639
\(313\) 13.6848 0.773509 0.386755 0.922183i \(-0.373596\pi\)
0.386755 + 0.922183i \(0.373596\pi\)
\(314\) 8.19450 0.462442
\(315\) 2.66196 0.149984
\(316\) −4.90263 −0.275795
\(317\) 14.1092 0.792451 0.396226 0.918153i \(-0.370320\pi\)
0.396226 + 0.918153i \(0.370320\pi\)
\(318\) 0.0908303 0.00509351
\(319\) 28.0063 1.56805
\(320\) −10.6713 −0.596543
\(321\) 18.0628 1.00817
\(322\) 1.75233 0.0976538
\(323\) 20.9544 1.16593
\(324\) −1.82052 −0.101140
\(325\) 10.3271 0.572842
\(326\) 5.28740 0.292842
\(327\) 7.40505 0.409500
\(328\) 3.32157 0.183403
\(329\) −7.52083 −0.414637
\(330\) 4.48337 0.246802
\(331\) 1.11891 0.0615011 0.0307505 0.999527i \(-0.490210\pi\)
0.0307505 + 0.999527i \(0.490210\pi\)
\(332\) −7.68520 −0.421780
\(333\) 2.65212 0.145335
\(334\) −0.916424 −0.0501445
\(335\) 11.3560 0.620442
\(336\) −2.95533 −0.161226
\(337\) −23.4974 −1.27999 −0.639993 0.768381i \(-0.721063\pi\)
−0.639993 + 0.768381i \(0.721063\pi\)
\(338\) −4.87550 −0.265192
\(339\) −18.6046 −1.01046
\(340\) −27.2924 −1.48014
\(341\) −10.2276 −0.553858
\(342\) −1.57630 −0.0852364
\(343\) 1.00000 0.0539949
\(344\) 6.52405 0.351753
\(345\) 11.0105 0.592788
\(346\) 6.60384 0.355025
\(347\) 6.87719 0.369187 0.184593 0.982815i \(-0.440903\pi\)
0.184593 + 0.982815i \(0.440903\pi\)
\(348\) 12.8250 0.687491
\(349\) −3.13512 −0.167819 −0.0839094 0.996473i \(-0.526741\pi\)
−0.0839094 + 0.996473i \(0.526741\pi\)
\(350\) −0.883750 −0.0472384
\(351\) −4.95058 −0.264243
\(352\) −17.8468 −0.951238
\(353\) −29.3356 −1.56138 −0.780689 0.624920i \(-0.785131\pi\)
−0.780689 + 0.624920i \(0.785131\pi\)
\(354\) −5.32761 −0.283159
\(355\) −36.5006 −1.93725
\(356\) −7.43430 −0.394017
\(357\) −5.63178 −0.298065
\(358\) 4.74469 0.250765
\(359\) −6.92912 −0.365705 −0.182852 0.983140i \(-0.558533\pi\)
−0.182852 + 0.983140i \(0.558533\pi\)
\(360\) 4.30857 0.227082
\(361\) −5.15614 −0.271376
\(362\) 2.85067 0.149828
\(363\) −4.80477 −0.252185
\(364\) −9.01263 −0.472390
\(365\) −28.9867 −1.51723
\(366\) −1.00588 −0.0525782
\(367\) 20.4454 1.06724 0.533621 0.845724i \(-0.320831\pi\)
0.533621 + 0.845724i \(0.320831\pi\)
\(368\) −12.2240 −0.637220
\(369\) 2.05217 0.106831
\(370\) −2.99091 −0.155490
\(371\) 0.214398 0.0111310
\(372\) −4.68357 −0.242832
\(373\) −1.24610 −0.0645209 −0.0322604 0.999479i \(-0.510271\pi\)
−0.0322604 + 0.999479i \(0.510271\pi\)
\(374\) −9.48526 −0.490471
\(375\) 7.75688 0.400563
\(376\) −12.1730 −0.627774
\(377\) 34.8753 1.79617
\(378\) 0.423652 0.0217903
\(379\) −11.8870 −0.610592 −0.305296 0.952258i \(-0.598755\pi\)
−0.305296 + 0.952258i \(0.598755\pi\)
\(380\) −18.0312 −0.924982
\(381\) 1.00000 0.0512316
\(382\) −9.88989 −0.506011
\(383\) −14.6285 −0.747481 −0.373741 0.927533i \(-0.621925\pi\)
−0.373741 + 0.927533i \(0.621925\pi\)
\(384\) −10.6767 −0.544842
\(385\) 10.5827 0.539343
\(386\) −0.906292 −0.0461290
\(387\) 4.03075 0.204894
\(388\) −0.245970 −0.0124873
\(389\) −21.5646 −1.09337 −0.546685 0.837338i \(-0.684110\pi\)
−0.546685 + 0.837338i \(0.684110\pi\)
\(390\) 5.58299 0.282706
\(391\) −23.2945 −1.17805
\(392\) 1.61857 0.0817502
\(393\) −11.7101 −0.590698
\(394\) −2.97305 −0.149780
\(395\) 7.16862 0.360693
\(396\) −7.23751 −0.363699
\(397\) 8.65271 0.434267 0.217133 0.976142i \(-0.430329\pi\)
0.217133 + 0.976142i \(0.430329\pi\)
\(398\) 7.92794 0.397392
\(399\) −3.72073 −0.186270
\(400\) 6.16489 0.308245
\(401\) −25.4873 −1.27277 −0.636387 0.771370i \(-0.719572\pi\)
−0.636387 + 0.771370i \(0.719572\pi\)
\(402\) 1.80731 0.0901402
\(403\) −12.7361 −0.634432
\(404\) −1.32065 −0.0657050
\(405\) 2.66196 0.132274
\(406\) −2.98449 −0.148118
\(407\) 10.5436 0.522625
\(408\) −9.11543 −0.451281
\(409\) 38.0003 1.87900 0.939498 0.342555i \(-0.111292\pi\)
0.939498 + 0.342555i \(0.111292\pi\)
\(410\) −2.31432 −0.114296
\(411\) −2.39147 −0.117963
\(412\) −25.9381 −1.27788
\(413\) −12.5754 −0.618797
\(414\) 1.75233 0.0861225
\(415\) 11.2373 0.551616
\(416\) −22.2240 −1.08962
\(417\) 2.14800 0.105188
\(418\) −6.26660 −0.306509
\(419\) −4.34686 −0.212358 −0.106179 0.994347i \(-0.533862\pi\)
−0.106179 + 0.994347i \(0.533862\pi\)
\(420\) 4.84615 0.236468
\(421\) 29.4508 1.43535 0.717673 0.696380i \(-0.245207\pi\)
0.717673 + 0.696380i \(0.245207\pi\)
\(422\) 4.83441 0.235335
\(423\) −7.52083 −0.365675
\(424\) 0.347019 0.0168527
\(425\) 11.7480 0.569864
\(426\) −5.80909 −0.281451
\(427\) −2.37431 −0.114901
\(428\) 32.8837 1.58949
\(429\) −19.6811 −0.950215
\(430\) −4.54565 −0.219211
\(431\) 29.6562 1.42849 0.714245 0.699895i \(-0.246770\pi\)
0.714245 + 0.699895i \(0.246770\pi\)
\(432\) −2.95533 −0.142188
\(433\) −9.59705 −0.461205 −0.230602 0.973048i \(-0.574070\pi\)
−0.230602 + 0.973048i \(0.574070\pi\)
\(434\) 1.08991 0.0523174
\(435\) −18.7527 −0.899121
\(436\) 13.4810 0.645625
\(437\) −15.3899 −0.736199
\(438\) −4.61324 −0.220429
\(439\) −29.8729 −1.42576 −0.712878 0.701288i \(-0.752609\pi\)
−0.712878 + 0.701288i \(0.752609\pi\)
\(440\) 17.1288 0.816584
\(441\) 1.00000 0.0476190
\(442\) −11.8117 −0.561824
\(443\) 1.94573 0.0924444 0.0462222 0.998931i \(-0.485282\pi\)
0.0462222 + 0.998931i \(0.485282\pi\)
\(444\) 4.82823 0.229138
\(445\) 10.8704 0.515308
\(446\) −5.41865 −0.256581
\(447\) 2.05953 0.0974122
\(448\) −4.00881 −0.189398
\(449\) −17.6859 −0.834650 −0.417325 0.908757i \(-0.637032\pi\)
−0.417325 + 0.908757i \(0.637032\pi\)
\(450\) −0.883750 −0.0416604
\(451\) 8.15843 0.384165
\(452\) −33.8700 −1.59311
\(453\) −12.3000 −0.577903
\(454\) 10.0268 0.470581
\(455\) 13.1783 0.617806
\(456\) −6.02227 −0.282019
\(457\) −38.7242 −1.81144 −0.905721 0.423874i \(-0.860670\pi\)
−0.905721 + 0.423874i \(0.860670\pi\)
\(458\) 7.08144 0.330894
\(459\) −5.63178 −0.262869
\(460\) 20.0449 0.934599
\(461\) 2.11009 0.0982766 0.0491383 0.998792i \(-0.484352\pi\)
0.0491383 + 0.998792i \(0.484352\pi\)
\(462\) 1.68424 0.0783578
\(463\) −16.3478 −0.759747 −0.379873 0.925038i \(-0.624032\pi\)
−0.379873 + 0.925038i \(0.624032\pi\)
\(464\) 20.8193 0.966514
\(465\) 6.84830 0.317582
\(466\) −11.6858 −0.541335
\(467\) 15.8333 0.732679 0.366340 0.930481i \(-0.380611\pi\)
0.366340 + 0.930481i \(0.380611\pi\)
\(468\) −9.01263 −0.416609
\(469\) 4.26602 0.196986
\(470\) 8.48157 0.391226
\(471\) 19.3425 0.891256
\(472\) −20.3542 −0.936880
\(473\) 16.0243 0.736799
\(474\) 1.14089 0.0524028
\(475\) 7.76155 0.356124
\(476\) −10.2528 −0.469935
\(477\) 0.214398 0.00981663
\(478\) 2.58460 0.118217
\(479\) −22.1361 −1.01142 −0.505712 0.862702i \(-0.668770\pi\)
−0.505712 + 0.862702i \(0.668770\pi\)
\(480\) 11.9500 0.545440
\(481\) 13.1295 0.598655
\(482\) 6.96661 0.317321
\(483\) 4.13626 0.188206
\(484\) −8.74717 −0.397599
\(485\) 0.359657 0.0163312
\(486\) 0.423652 0.0192172
\(487\) −0.348140 −0.0157757 −0.00788787 0.999969i \(-0.502511\pi\)
−0.00788787 + 0.999969i \(0.502511\pi\)
\(488\) −3.84298 −0.173964
\(489\) 12.4805 0.564389
\(490\) −1.12774 −0.0509463
\(491\) −7.28753 −0.328882 −0.164441 0.986387i \(-0.552582\pi\)
−0.164441 + 0.986387i \(0.552582\pi\)
\(492\) 3.73601 0.168432
\(493\) 39.6741 1.78683
\(494\) −7.80359 −0.351100
\(495\) 10.5827 0.475656
\(496\) −7.60304 −0.341386
\(497\) −13.7119 −0.615064
\(498\) 1.78842 0.0801410
\(499\) −16.5620 −0.741416 −0.370708 0.928749i \(-0.620885\pi\)
−0.370708 + 0.928749i \(0.620885\pi\)
\(500\) 14.1215 0.631535
\(501\) −2.16315 −0.0966425
\(502\) −13.2506 −0.591404
\(503\) −19.6862 −0.877766 −0.438883 0.898544i \(-0.644626\pi\)
−0.438883 + 0.898544i \(0.644626\pi\)
\(504\) 1.61857 0.0720969
\(505\) 1.93106 0.0859310
\(506\) 6.96644 0.309696
\(507\) −11.5083 −0.511100
\(508\) 1.82052 0.0807725
\(509\) 0.855626 0.0379249 0.0189625 0.999820i \(-0.493964\pi\)
0.0189625 + 0.999820i \(0.493964\pi\)
\(510\) 6.35121 0.281236
\(511\) −10.8892 −0.481711
\(512\) −22.8338 −1.00912
\(513\) −3.72073 −0.164274
\(514\) −6.55276 −0.289030
\(515\) 37.9266 1.67125
\(516\) 7.33805 0.323040
\(517\) −29.8992 −1.31497
\(518\) −1.12358 −0.0493671
\(519\) 15.5879 0.684233
\(520\) 21.3299 0.935379
\(521\) −19.7053 −0.863303 −0.431652 0.902040i \(-0.642069\pi\)
−0.431652 + 0.902040i \(0.642069\pi\)
\(522\) −2.98449 −0.130628
\(523\) −45.5771 −1.99295 −0.996473 0.0839120i \(-0.973259\pi\)
−0.996473 + 0.0839120i \(0.973259\pi\)
\(524\) −21.3185 −0.931303
\(525\) −2.08603 −0.0910417
\(526\) 2.31215 0.100814
\(527\) −14.4886 −0.631134
\(528\) −11.7490 −0.511308
\(529\) −5.89137 −0.256147
\(530\) −0.241787 −0.0105025
\(531\) −12.5754 −0.545728
\(532\) −6.77367 −0.293676
\(533\) 10.1594 0.440053
\(534\) 1.73003 0.0748659
\(535\) −48.0824 −2.07878
\(536\) 6.90485 0.298244
\(537\) 11.1995 0.483294
\(538\) −10.2971 −0.443942
\(539\) 3.97552 0.171238
\(540\) 4.84615 0.208545
\(541\) 26.4890 1.13885 0.569426 0.822043i \(-0.307166\pi\)
0.569426 + 0.822043i \(0.307166\pi\)
\(542\) −11.1923 −0.480751
\(543\) 6.72881 0.288761
\(544\) −25.2820 −1.08396
\(545\) −19.7120 −0.844367
\(546\) 2.09732 0.0897572
\(547\) −31.6604 −1.35370 −0.676850 0.736121i \(-0.736655\pi\)
−0.676850 + 0.736121i \(0.736655\pi\)
\(548\) −4.35372 −0.185982
\(549\) −2.37431 −0.101333
\(550\) −3.51337 −0.149810
\(551\) 26.2114 1.11664
\(552\) 6.69483 0.284951
\(553\) 2.69299 0.114517
\(554\) 5.82054 0.247291
\(555\) −7.05983 −0.299673
\(556\) 3.91047 0.165841
\(557\) 9.95634 0.421864 0.210932 0.977501i \(-0.432350\pi\)
0.210932 + 0.977501i \(0.432350\pi\)
\(558\) 1.08991 0.0461396
\(559\) 19.9546 0.843988
\(560\) 7.86696 0.332440
\(561\) −22.3893 −0.945275
\(562\) 2.14692 0.0905624
\(563\) 13.7616 0.579983 0.289992 0.957029i \(-0.406347\pi\)
0.289992 + 0.957029i \(0.406347\pi\)
\(564\) −13.6918 −0.576529
\(565\) 49.5247 2.08352
\(566\) 11.7081 0.492129
\(567\) 1.00000 0.0419961
\(568\) −22.1937 −0.931228
\(569\) −7.58374 −0.317927 −0.158963 0.987284i \(-0.550815\pi\)
−0.158963 + 0.987284i \(0.550815\pi\)
\(570\) 4.19604 0.175753
\(571\) 0.400150 0.0167458 0.00837288 0.999965i \(-0.497335\pi\)
0.00837288 + 0.999965i \(0.497335\pi\)
\(572\) −35.8299 −1.49812
\(573\) −23.3444 −0.975225
\(574\) −0.869404 −0.0362882
\(575\) −8.62835 −0.359827
\(576\) −4.00881 −0.167034
\(577\) −18.6822 −0.777748 −0.388874 0.921291i \(-0.627136\pi\)
−0.388874 + 0.921291i \(0.627136\pi\)
\(578\) −6.23486 −0.259336
\(579\) −2.13924 −0.0889036
\(580\) −34.1396 −1.41757
\(581\) 4.22143 0.175135
\(582\) 0.0572397 0.00237266
\(583\) 0.852345 0.0353005
\(584\) −17.6250 −0.729327
\(585\) 13.1783 0.544854
\(586\) −11.4012 −0.470978
\(587\) 35.7593 1.47595 0.737973 0.674830i \(-0.235783\pi\)
0.737973 + 0.674830i \(0.235783\pi\)
\(588\) 1.82052 0.0750769
\(589\) −9.57216 −0.394414
\(590\) 14.1819 0.583859
\(591\) −7.01767 −0.288668
\(592\) 7.83788 0.322135
\(593\) 27.4101 1.12560 0.562799 0.826594i \(-0.309724\pi\)
0.562799 + 0.826594i \(0.309724\pi\)
\(594\) 1.68424 0.0691051
\(595\) 14.9916 0.614594
\(596\) 3.74941 0.153582
\(597\) 18.7133 0.765886
\(598\) 8.67508 0.354750
\(599\) −13.9434 −0.569711 −0.284856 0.958570i \(-0.591946\pi\)
−0.284856 + 0.958570i \(0.591946\pi\)
\(600\) −3.37638 −0.137840
\(601\) −26.8951 −1.09708 −0.548538 0.836126i \(-0.684815\pi\)
−0.548538 + 0.836126i \(0.684815\pi\)
\(602\) −1.70763 −0.0695980
\(603\) 4.26602 0.173726
\(604\) −22.3923 −0.911131
\(605\) 12.7901 0.519991
\(606\) 0.307329 0.0124844
\(607\) 18.7939 0.762819 0.381410 0.924406i \(-0.375439\pi\)
0.381410 + 0.924406i \(0.375439\pi\)
\(608\) −16.7030 −0.677397
\(609\) −7.04468 −0.285465
\(610\) 2.67761 0.108413
\(611\) −37.2325 −1.50627
\(612\) −10.2528 −0.414443
\(613\) 6.13877 0.247943 0.123971 0.992286i \(-0.460437\pi\)
0.123971 + 0.992286i \(0.460437\pi\)
\(614\) 4.32752 0.174644
\(615\) −5.46278 −0.220281
\(616\) 6.43466 0.259260
\(617\) −16.3314 −0.657477 −0.328739 0.944421i \(-0.606624\pi\)
−0.328739 + 0.944421i \(0.606624\pi\)
\(618\) 6.03604 0.242805
\(619\) 18.1738 0.730465 0.365233 0.930916i \(-0.380989\pi\)
0.365233 + 0.930916i \(0.380989\pi\)
\(620\) 12.4675 0.500705
\(621\) 4.13626 0.165982
\(622\) −4.10595 −0.164634
\(623\) 4.08362 0.163607
\(624\) −14.6306 −0.585692
\(625\) −31.0786 −1.24315
\(626\) −5.79759 −0.231718
\(627\) −14.7919 −0.590730
\(628\) 35.2134 1.40517
\(629\) 14.9362 0.595543
\(630\) −1.12774 −0.0449304
\(631\) −22.8576 −0.909945 −0.454972 0.890506i \(-0.650351\pi\)
−0.454972 + 0.890506i \(0.650351\pi\)
\(632\) 4.35879 0.173383
\(633\) 11.4113 0.453558
\(634\) −5.97739 −0.237393
\(635\) −2.66196 −0.105637
\(636\) 0.390316 0.0154770
\(637\) 4.95058 0.196149
\(638\) −11.8649 −0.469737
\(639\) −13.7119 −0.542435
\(640\) 28.4209 1.12343
\(641\) 34.5503 1.36465 0.682327 0.731047i \(-0.260968\pi\)
0.682327 + 0.731047i \(0.260968\pi\)
\(642\) −7.65234 −0.302014
\(643\) 4.29860 0.169520 0.0847601 0.996401i \(-0.472988\pi\)
0.0847601 + 0.996401i \(0.472988\pi\)
\(644\) 7.53014 0.296729
\(645\) −10.7297 −0.422481
\(646\) −8.87735 −0.349275
\(647\) −17.3258 −0.681147 −0.340573 0.940218i \(-0.610621\pi\)
−0.340573 + 0.940218i \(0.610621\pi\)
\(648\) 1.61857 0.0635835
\(649\) −49.9939 −1.96243
\(650\) −4.37508 −0.171605
\(651\) 2.57265 0.100830
\(652\) 22.7210 0.889825
\(653\) 29.1488 1.14068 0.570341 0.821408i \(-0.306811\pi\)
0.570341 + 0.821408i \(0.306811\pi\)
\(654\) −3.13717 −0.122673
\(655\) 31.1719 1.21799
\(656\) 6.06482 0.236791
\(657\) −10.8892 −0.424829
\(658\) 3.18621 0.124212
\(659\) −35.0106 −1.36382 −0.681910 0.731436i \(-0.738850\pi\)
−0.681910 + 0.731436i \(0.738850\pi\)
\(660\) 19.2660 0.749926
\(661\) 41.6401 1.61961 0.809806 0.586697i \(-0.199572\pi\)
0.809806 + 0.586697i \(0.199572\pi\)
\(662\) −0.474030 −0.0184237
\(663\) −27.8806 −1.08279
\(664\) 6.83269 0.265160
\(665\) 9.90444 0.384078
\(666\) −1.12358 −0.0435377
\(667\) −29.1386 −1.12825
\(668\) −3.93806 −0.152368
\(669\) −12.7903 −0.494503
\(670\) −4.81098 −0.185864
\(671\) −9.43910 −0.364393
\(672\) 4.48917 0.173174
\(673\) 6.26678 0.241567 0.120783 0.992679i \(-0.461459\pi\)
0.120783 + 0.992679i \(0.461459\pi\)
\(674\) 9.95473 0.383442
\(675\) −2.08603 −0.0802912
\(676\) −20.9510 −0.805809
\(677\) −35.5065 −1.36463 −0.682313 0.731060i \(-0.739026\pi\)
−0.682313 + 0.731060i \(0.739026\pi\)
\(678\) 7.88188 0.302702
\(679\) 0.135110 0.00518505
\(680\) 24.2649 0.930517
\(681\) 23.6675 0.906941
\(682\) 4.33296 0.165918
\(683\) 30.5511 1.16900 0.584502 0.811392i \(-0.301290\pi\)
0.584502 + 0.811392i \(0.301290\pi\)
\(684\) −6.77367 −0.258998
\(685\) 6.36601 0.243233
\(686\) −0.423652 −0.0161751
\(687\) 16.7152 0.637726
\(688\) 11.9122 0.454148
\(689\) 1.06140 0.0404360
\(690\) −4.66464 −0.177580
\(691\) 21.9818 0.836229 0.418114 0.908394i \(-0.362691\pi\)
0.418114 + 0.908394i \(0.362691\pi\)
\(692\) 28.3781 1.07877
\(693\) 3.97552 0.151018
\(694\) −2.91354 −0.110596
\(695\) −5.71788 −0.216892
\(696\) −11.4023 −0.432204
\(697\) 11.5573 0.437766
\(698\) 1.32820 0.0502730
\(699\) −27.5835 −1.04330
\(700\) −3.79765 −0.143538
\(701\) −5.79388 −0.218832 −0.109416 0.993996i \(-0.534898\pi\)
−0.109416 + 0.993996i \(0.534898\pi\)
\(702\) 2.09732 0.0791584
\(703\) 9.86783 0.372172
\(704\) −15.9371 −0.600652
\(705\) 20.0201 0.754002
\(706\) 12.4281 0.467737
\(707\) 0.725428 0.0272825
\(708\) −22.8938 −0.860403
\(709\) −12.2837 −0.461324 −0.230662 0.973034i \(-0.574089\pi\)
−0.230662 + 0.973034i \(0.574089\pi\)
\(710\) 15.4636 0.580337
\(711\) 2.69299 0.100995
\(712\) 6.60962 0.247706
\(713\) 10.6412 0.398515
\(714\) 2.38592 0.0892906
\(715\) 52.3904 1.95929
\(716\) 20.3889 0.761969
\(717\) 6.10076 0.227837
\(718\) 2.93554 0.109553
\(719\) −25.6425 −0.956302 −0.478151 0.878278i \(-0.658693\pi\)
−0.478151 + 0.878278i \(0.658693\pi\)
\(720\) 7.86696 0.293184
\(721\) 14.2476 0.530610
\(722\) 2.18441 0.0812953
\(723\) 16.4442 0.611566
\(724\) 12.2499 0.455265
\(725\) 14.6954 0.545774
\(726\) 2.03555 0.0755463
\(727\) 41.3287 1.53280 0.766399 0.642365i \(-0.222047\pi\)
0.766399 + 0.642365i \(0.222047\pi\)
\(728\) 8.01287 0.296977
\(729\) 1.00000 0.0370370
\(730\) 12.2803 0.454513
\(731\) 22.7003 0.839600
\(732\) −4.32247 −0.159763
\(733\) 27.6156 1.02001 0.510004 0.860172i \(-0.329644\pi\)
0.510004 + 0.860172i \(0.329644\pi\)
\(734\) −8.66174 −0.319711
\(735\) −2.66196 −0.0981878
\(736\) 18.5684 0.684439
\(737\) 16.9596 0.624716
\(738\) −0.869404 −0.0320032
\(739\) −30.8185 −1.13368 −0.566839 0.823828i \(-0.691834\pi\)
−0.566839 + 0.823828i \(0.691834\pi\)
\(740\) −12.8526 −0.472470
\(741\) −18.4198 −0.676668
\(742\) −0.0908303 −0.00333449
\(743\) 24.3137 0.891983 0.445991 0.895037i \(-0.352851\pi\)
0.445991 + 0.895037i \(0.352851\pi\)
\(744\) 4.16402 0.152661
\(745\) −5.48237 −0.200859
\(746\) 0.527915 0.0193283
\(747\) 4.22143 0.154454
\(748\) −40.7601 −1.49034
\(749\) −18.0628 −0.660000
\(750\) −3.28622 −0.119996
\(751\) 13.6195 0.496984 0.248492 0.968634i \(-0.420065\pi\)
0.248492 + 0.968634i \(0.420065\pi\)
\(752\) −22.2265 −0.810517
\(753\) −31.2771 −1.13980
\(754\) −14.7750 −0.538073
\(755\) 32.7420 1.19160
\(756\) 1.82052 0.0662116
\(757\) 33.4512 1.21580 0.607902 0.794012i \(-0.292011\pi\)
0.607902 + 0.794012i \(0.292011\pi\)
\(758\) 5.03593 0.182913
\(759\) 16.4438 0.596871
\(760\) 16.0310 0.581507
\(761\) 18.4399 0.668446 0.334223 0.942494i \(-0.391526\pi\)
0.334223 + 0.942494i \(0.391526\pi\)
\(762\) −0.423652 −0.0153473
\(763\) −7.40505 −0.268081
\(764\) −42.4989 −1.53755
\(765\) 14.9916 0.542021
\(766\) 6.19739 0.223921
\(767\) −62.2558 −2.24793
\(768\) −3.49442 −0.126094
\(769\) 42.6719 1.53879 0.769394 0.638774i \(-0.220558\pi\)
0.769394 + 0.638774i \(0.220558\pi\)
\(770\) −4.48337 −0.161570
\(771\) −15.4673 −0.557041
\(772\) −3.89452 −0.140167
\(773\) 17.6637 0.635319 0.317659 0.948205i \(-0.397103\pi\)
0.317659 + 0.948205i \(0.397103\pi\)
\(774\) −1.70763 −0.0613797
\(775\) −5.36663 −0.192775
\(776\) 0.218685 0.00785034
\(777\) −2.65212 −0.0951443
\(778\) 9.13590 0.327538
\(779\) 7.63556 0.273572
\(780\) 23.9913 0.859025
\(781\) −54.5121 −1.95059
\(782\) 9.86876 0.352906
\(783\) −7.04468 −0.251756
\(784\) 2.95533 0.105547
\(785\) −51.4890 −1.83772
\(786\) 4.96102 0.176954
\(787\) 34.4904 1.22945 0.614724 0.788742i \(-0.289267\pi\)
0.614724 + 0.788742i \(0.289267\pi\)
\(788\) −12.7758 −0.455119
\(789\) 5.45766 0.194298
\(790\) −3.03700 −0.108052
\(791\) 18.6046 0.661503
\(792\) 6.43466 0.228646
\(793\) −11.7542 −0.417404
\(794\) −3.66574 −0.130092
\(795\) −0.570720 −0.0202413
\(796\) 34.0680 1.20751
\(797\) −11.8100 −0.418333 −0.209166 0.977880i \(-0.567075\pi\)
−0.209166 + 0.977880i \(0.567075\pi\)
\(798\) 1.57630 0.0558003
\(799\) −42.3556 −1.49844
\(800\) −9.36453 −0.331086
\(801\) 4.08362 0.144288
\(802\) 10.7977 0.381282
\(803\) −43.2904 −1.52768
\(804\) 7.76636 0.273898
\(805\) −11.0105 −0.388071
\(806\) 5.39569 0.190055
\(807\) −24.3057 −0.855600
\(808\) 1.17416 0.0413067
\(809\) 6.66413 0.234298 0.117149 0.993114i \(-0.462624\pi\)
0.117149 + 0.993114i \(0.462624\pi\)
\(810\) −1.12774 −0.0396249
\(811\) −1.22927 −0.0431654 −0.0215827 0.999767i \(-0.506871\pi\)
−0.0215827 + 0.999767i \(0.506871\pi\)
\(812\) −12.8250 −0.450069
\(813\) −26.4187 −0.926543
\(814\) −4.46680 −0.156561
\(815\) −33.2227 −1.16374
\(816\) −16.6438 −0.582648
\(817\) 14.9973 0.524691
\(818\) −16.0989 −0.562885
\(819\) 4.95058 0.172987
\(820\) −9.94510 −0.347298
\(821\) 28.1827 0.983582 0.491791 0.870713i \(-0.336342\pi\)
0.491791 + 0.870713i \(0.336342\pi\)
\(822\) 1.01315 0.0353378
\(823\) −24.6122 −0.857926 −0.428963 0.903322i \(-0.641121\pi\)
−0.428963 + 0.903322i \(0.641121\pi\)
\(824\) 23.0608 0.803361
\(825\) −8.29304 −0.288727
\(826\) 5.32761 0.185371
\(827\) −1.75039 −0.0608670 −0.0304335 0.999537i \(-0.509689\pi\)
−0.0304335 + 0.999537i \(0.509689\pi\)
\(828\) 7.53014 0.261690
\(829\) −34.2634 −1.19002 −0.595009 0.803719i \(-0.702851\pi\)
−0.595009 + 0.803719i \(0.702851\pi\)
\(830\) −4.76070 −0.165246
\(831\) 13.7390 0.476599
\(832\) −19.8459 −0.688034
\(833\) 5.63178 0.195130
\(834\) −0.910003 −0.0315108
\(835\) 5.75822 0.199271
\(836\) −26.9289 −0.931354
\(837\) 2.57265 0.0889240
\(838\) 1.84156 0.0636155
\(839\) 35.2195 1.21591 0.607955 0.793971i \(-0.291990\pi\)
0.607955 + 0.793971i \(0.291990\pi\)
\(840\) −4.30857 −0.148660
\(841\) 20.6276 0.711296
\(842\) −12.4769 −0.429983
\(843\) 5.06765 0.174539
\(844\) 20.7744 0.715086
\(845\) 30.6346 1.05386
\(846\) 3.18621 0.109544
\(847\) 4.80477 0.165094
\(848\) 0.633617 0.0217585
\(849\) 27.6362 0.948471
\(850\) −4.97708 −0.170713
\(851\) −10.9699 −0.376042
\(852\) −24.9628 −0.855212
\(853\) −0.785201 −0.0268848 −0.0134424 0.999910i \(-0.504279\pi\)
−0.0134424 + 0.999910i \(0.504279\pi\)
\(854\) 1.00588 0.0344205
\(855\) 9.90444 0.338725
\(856\) −29.2359 −0.999263
\(857\) 33.4829 1.14375 0.571876 0.820340i \(-0.306216\pi\)
0.571876 + 0.820340i \(0.306216\pi\)
\(858\) 8.33796 0.284653
\(859\) −8.16211 −0.278488 −0.139244 0.990258i \(-0.544467\pi\)
−0.139244 + 0.990258i \(0.544467\pi\)
\(860\) −19.5336 −0.666090
\(861\) −2.05217 −0.0699376
\(862\) −12.5639 −0.427929
\(863\) −5.62343 −0.191424 −0.0957119 0.995409i \(-0.530513\pi\)
−0.0957119 + 0.995409i \(0.530513\pi\)
\(864\) 4.48917 0.152725
\(865\) −41.4943 −1.41085
\(866\) 4.06581 0.138162
\(867\) −14.7169 −0.499814
\(868\) 4.68357 0.158971
\(869\) 10.7060 0.363177
\(870\) 7.94460 0.269347
\(871\) 21.1193 0.715599
\(872\) −11.9856 −0.405884
\(873\) 0.135110 0.00457278
\(874\) 6.51997 0.220541
\(875\) −7.75688 −0.262230
\(876\) −19.8240 −0.669793
\(877\) 2.77495 0.0937033 0.0468516 0.998902i \(-0.485081\pi\)
0.0468516 + 0.998902i \(0.485081\pi\)
\(878\) 12.6557 0.427110
\(879\) −26.9116 −0.907706
\(880\) 31.2753 1.05429
\(881\) −36.0262 −1.21375 −0.606876 0.794797i \(-0.707578\pi\)
−0.606876 + 0.794797i \(0.707578\pi\)
\(882\) −0.423652 −0.0142651
\(883\) 9.10055 0.306258 0.153129 0.988206i \(-0.451065\pi\)
0.153129 + 0.988206i \(0.451065\pi\)
\(884\) −50.7571 −1.70715
\(885\) 33.4753 1.12526
\(886\) −0.824312 −0.0276933
\(887\) 46.7522 1.56979 0.784893 0.619632i \(-0.212718\pi\)
0.784893 + 0.619632i \(0.212718\pi\)
\(888\) −4.29264 −0.144052
\(889\) −1.00000 −0.0335389
\(890\) −4.60528 −0.154369
\(891\) 3.97552 0.133185
\(892\) −23.2851 −0.779641
\(893\) −27.9830 −0.936415
\(894\) −0.872522 −0.0291815
\(895\) −29.8126 −0.996526
\(896\) 10.6767 0.356683
\(897\) 20.4769 0.683703
\(898\) 7.49268 0.250034
\(899\) −18.1235 −0.604454
\(900\) −3.79765 −0.126588
\(901\) 1.20744 0.0402258
\(902\) −3.45633 −0.115083
\(903\) −4.03075 −0.134135
\(904\) 30.1129 1.00154
\(905\) −17.9118 −0.595409
\(906\) 5.21091 0.173121
\(907\) 50.9022 1.69018 0.845090 0.534623i \(-0.179547\pi\)
0.845090 + 0.534623i \(0.179547\pi\)
\(908\) 43.0872 1.42990
\(909\) 0.725428 0.0240609
\(910\) −5.58299 −0.185074
\(911\) −29.0201 −0.961480 −0.480740 0.876863i \(-0.659632\pi\)
−0.480740 + 0.876863i \(0.659632\pi\)
\(912\) −10.9960 −0.364113
\(913\) 16.7824 0.555416
\(914\) 16.4056 0.542649
\(915\) 6.32031 0.208943
\(916\) 30.4304 1.00545
\(917\) 11.7101 0.386702
\(918\) 2.38592 0.0787469
\(919\) −34.4150 −1.13525 −0.567623 0.823288i \(-0.692137\pi\)
−0.567623 + 0.823288i \(0.692137\pi\)
\(920\) −17.8214 −0.587553
\(921\) 10.2148 0.336589
\(922\) −0.893943 −0.0294405
\(923\) −67.8820 −2.23436
\(924\) 7.23751 0.238097
\(925\) 5.53239 0.181904
\(926\) 6.92578 0.227595
\(927\) 14.2476 0.467954
\(928\) −31.6248 −1.03813
\(929\) 41.0030 1.34526 0.672632 0.739977i \(-0.265164\pi\)
0.672632 + 0.739977i \(0.265164\pi\)
\(930\) −2.90130 −0.0951373
\(931\) 3.72073 0.121942
\(932\) −50.2163 −1.64489
\(933\) −9.69180 −0.317295
\(934\) −6.70782 −0.219487
\(935\) 59.5993 1.94911
\(936\) 8.01287 0.261909
\(937\) 45.4052 1.48332 0.741662 0.670774i \(-0.234038\pi\)
0.741662 + 0.670774i \(0.234038\pi\)
\(938\) −1.80731 −0.0590106
\(939\) −13.6848 −0.446586
\(940\) 36.4470 1.18877
\(941\) 6.54913 0.213496 0.106748 0.994286i \(-0.465956\pi\)
0.106748 + 0.994286i \(0.465956\pi\)
\(942\) −8.19450 −0.266991
\(943\) −8.48828 −0.276417
\(944\) −37.1645 −1.20960
\(945\) −2.66196 −0.0865935
\(946\) −6.78874 −0.220721
\(947\) −30.9829 −1.00681 −0.503405 0.864051i \(-0.667919\pi\)
−0.503405 + 0.864051i \(0.667919\pi\)
\(948\) 4.90263 0.159230
\(949\) −53.9080 −1.74993
\(950\) −3.28820 −0.106683
\(951\) −14.1092 −0.457522
\(952\) 9.11543 0.295433
\(953\) 51.1779 1.65782 0.828908 0.559385i \(-0.188963\pi\)
0.828908 + 0.559385i \(0.188963\pi\)
\(954\) −0.0908303 −0.00294074
\(955\) 62.1418 2.01086
\(956\) 11.1065 0.359211
\(957\) −28.0063 −0.905315
\(958\) 9.37801 0.302990
\(959\) 2.39147 0.0772247
\(960\) 10.6713 0.344414
\(961\) −24.3814 −0.786498
\(962\) −5.56236 −0.179338
\(963\) −18.0628 −0.582066
\(964\) 29.9370 0.964204
\(965\) 5.69456 0.183314
\(966\) −1.75233 −0.0563804
\(967\) −38.9551 −1.25271 −0.626356 0.779538i \(-0.715454\pi\)
−0.626356 + 0.779538i \(0.715454\pi\)
\(968\) 7.77686 0.249958
\(969\) −20.9544 −0.673151
\(970\) −0.152370 −0.00489229
\(971\) 42.6229 1.36783 0.683917 0.729559i \(-0.260275\pi\)
0.683917 + 0.729559i \(0.260275\pi\)
\(972\) 1.82052 0.0583932
\(973\) −2.14800 −0.0688616
\(974\) 0.147490 0.00472589
\(975\) −10.3271 −0.330730
\(976\) −7.01685 −0.224604
\(977\) 49.7039 1.59017 0.795084 0.606500i \(-0.207427\pi\)
0.795084 + 0.606500i \(0.207427\pi\)
\(978\) −5.28740 −0.169073
\(979\) 16.2345 0.518857
\(980\) −4.84615 −0.154804
\(981\) −7.40505 −0.236425
\(982\) 3.08738 0.0985222
\(983\) −20.5978 −0.656968 −0.328484 0.944510i \(-0.606538\pi\)
−0.328484 + 0.944510i \(0.606538\pi\)
\(984\) −3.32157 −0.105888
\(985\) 18.6808 0.595218
\(986\) −16.8080 −0.535276
\(987\) 7.52083 0.239391
\(988\) −33.5336 −1.06685
\(989\) −16.6722 −0.530146
\(990\) −4.48337 −0.142491
\(991\) −40.6235 −1.29045 −0.645224 0.763994i \(-0.723236\pi\)
−0.645224 + 0.763994i \(0.723236\pi\)
\(992\) 11.5491 0.366684
\(993\) −1.11891 −0.0355077
\(994\) 5.80909 0.184253
\(995\) −49.8141 −1.57921
\(996\) 7.68520 0.243515
\(997\) −34.6749 −1.09817 −0.549083 0.835768i \(-0.685023\pi\)
−0.549083 + 0.835768i \(0.685023\pi\)
\(998\) 7.01652 0.222104
\(999\) −2.65212 −0.0839094
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 2667.2.a.l.1.5 13
3.2 odd 2 8001.2.a.o.1.9 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.l.1.5 13 1.1 even 1 trivial
8001.2.a.o.1.9 13 3.2 odd 2