Properties

Label 8034.2.a.w.1.3
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 6 x^{11} - 8 x^{10} + 94 x^{9} - 7 x^{8} - 580 x^{7} + 180 x^{6} + 1787 x^{5} - 308 x^{4} - 2790 x^{3} - 352 x^{2} + 1768 x + 768\)
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 1.3
Root \(1.78941\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.74714 q^{5} -1.00000 q^{6} +3.39290 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.74714 q^{5} -1.00000 q^{6} +3.39290 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.74714 q^{10} -0.860848 q^{11} +1.00000 q^{12} +1.00000 q^{13} -3.39290 q^{14} -2.74714 q^{15} +1.00000 q^{16} -4.14293 q^{17} -1.00000 q^{18} -2.70287 q^{19} -2.74714 q^{20} +3.39290 q^{21} +0.860848 q^{22} +0.937524 q^{23} -1.00000 q^{24} +2.54676 q^{25} -1.00000 q^{26} +1.00000 q^{27} +3.39290 q^{28} +0.521486 q^{29} +2.74714 q^{30} +7.56769 q^{31} -1.00000 q^{32} -0.860848 q^{33} +4.14293 q^{34} -9.32077 q^{35} +1.00000 q^{36} -0.373450 q^{37} +2.70287 q^{38} +1.00000 q^{39} +2.74714 q^{40} -5.16599 q^{41} -3.39290 q^{42} -9.15820 q^{43} -0.860848 q^{44} -2.74714 q^{45} -0.937524 q^{46} +3.73475 q^{47} +1.00000 q^{48} +4.51179 q^{49} -2.54676 q^{50} -4.14293 q^{51} +1.00000 q^{52} +6.82448 q^{53} -1.00000 q^{54} +2.36487 q^{55} -3.39290 q^{56} -2.70287 q^{57} -0.521486 q^{58} -10.7140 q^{59} -2.74714 q^{60} +4.99993 q^{61} -7.56769 q^{62} +3.39290 q^{63} +1.00000 q^{64} -2.74714 q^{65} +0.860848 q^{66} +7.00412 q^{67} -4.14293 q^{68} +0.937524 q^{69} +9.32077 q^{70} -9.02958 q^{71} -1.00000 q^{72} -8.63216 q^{73} +0.373450 q^{74} +2.54676 q^{75} -2.70287 q^{76} -2.92077 q^{77} -1.00000 q^{78} +1.21230 q^{79} -2.74714 q^{80} +1.00000 q^{81} +5.16599 q^{82} +15.5907 q^{83} +3.39290 q^{84} +11.3812 q^{85} +9.15820 q^{86} +0.521486 q^{87} +0.860848 q^{88} -4.60576 q^{89} +2.74714 q^{90} +3.39290 q^{91} +0.937524 q^{92} +7.56769 q^{93} -3.73475 q^{94} +7.42515 q^{95} -1.00000 q^{96} -8.14027 q^{97} -4.51179 q^{98} -0.860848 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + O(q^{10}) \) \( 12q - 12q^{2} + 12q^{3} + 12q^{4} - 4q^{5} - 12q^{6} - 12q^{8} + 12q^{9} + 4q^{10} - 9q^{11} + 12q^{12} + 12q^{13} - 4q^{15} + 12q^{16} - 20q^{17} - 12q^{18} + 4q^{19} - 4q^{20} + 9q^{22} - 30q^{23} - 12q^{24} + 14q^{25} - 12q^{26} + 12q^{27} - 29q^{29} + 4q^{30} + 6q^{31} - 12q^{32} - 9q^{33} + 20q^{34} - 22q^{35} + 12q^{36} + 7q^{37} - 4q^{38} + 12q^{39} + 4q^{40} - 8q^{41} - 8q^{43} - 9q^{44} - 4q^{45} + 30q^{46} - 16q^{47} + 12q^{48} + 10q^{49} - 14q^{50} - 20q^{51} + 12q^{52} - 9q^{53} - 12q^{54} - 20q^{55} + 4q^{57} + 29q^{58} - 29q^{59} - 4q^{60} - 26q^{61} - 6q^{62} + 12q^{64} - 4q^{65} + 9q^{66} + 12q^{67} - 20q^{68} - 30q^{69} + 22q^{70} - 35q^{71} - 12q^{72} + 18q^{73} - 7q^{74} + 14q^{75} + 4q^{76} - 25q^{77} - 12q^{78} - 37q^{79} - 4q^{80} + 12q^{81} + 8q^{82} - 24q^{83} - 17q^{85} + 8q^{86} - 29q^{87} + 9q^{88} + 15q^{89} + 4q^{90} - 30q^{92} + 6q^{93} + 16q^{94} - 54q^{95} - 12q^{96} - 11q^{97} - 10q^{98} - 9q^{99} + O(q^{100}) \)

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
\(5\) −2.74714 −1.22856 −0.614278 0.789089i \(-0.710553\pi\)
−0.614278 + 0.789089i \(0.710553\pi\)
\(6\) −1.00000 −0.408248
\(7\) 3.39290 1.28240 0.641198 0.767375i \(-0.278438\pi\)
0.641198 + 0.767375i \(0.278438\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.74714 0.868721
\(11\) −0.860848 −0.259556 −0.129778 0.991543i \(-0.541426\pi\)
−0.129778 + 0.991543i \(0.541426\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) −3.39290 −0.906791
\(15\) −2.74714 −0.709308
\(16\) 1.00000 0.250000
\(17\) −4.14293 −1.00481 −0.502404 0.864633i \(-0.667551\pi\)
−0.502404 + 0.864633i \(0.667551\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.70287 −0.620081 −0.310040 0.950723i \(-0.600343\pi\)
−0.310040 + 0.950723i \(0.600343\pi\)
\(20\) −2.74714 −0.614278
\(21\) 3.39290 0.740392
\(22\) 0.860848 0.183533
\(23\) 0.937524 0.195487 0.0977436 0.995212i \(-0.468837\pi\)
0.0977436 + 0.995212i \(0.468837\pi\)
\(24\) −1.00000 −0.204124
\(25\) 2.54676 0.509352
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 3.39290 0.641198
\(29\) 0.521486 0.0968375 0.0484187 0.998827i \(-0.484582\pi\)
0.0484187 + 0.998827i \(0.484582\pi\)
\(30\) 2.74714 0.501556
\(31\) 7.56769 1.35920 0.679598 0.733584i \(-0.262154\pi\)
0.679598 + 0.733584i \(0.262154\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.860848 −0.149854
\(34\) 4.14293 0.710507
\(35\) −9.32077 −1.57550
\(36\) 1.00000 0.166667
\(37\) −0.373450 −0.0613947 −0.0306974 0.999529i \(-0.509773\pi\)
−0.0306974 + 0.999529i \(0.509773\pi\)
\(38\) 2.70287 0.438463
\(39\) 1.00000 0.160128
\(40\) 2.74714 0.434360
\(41\) −5.16599 −0.806792 −0.403396 0.915025i \(-0.632170\pi\)
−0.403396 + 0.915025i \(0.632170\pi\)
\(42\) −3.39290 −0.523536
\(43\) −9.15820 −1.39661 −0.698306 0.715799i \(-0.746063\pi\)
−0.698306 + 0.715799i \(0.746063\pi\)
\(44\) −0.860848 −0.129778
\(45\) −2.74714 −0.409519
\(46\) −0.937524 −0.138230
\(47\) 3.73475 0.544770 0.272385 0.962188i \(-0.412188\pi\)
0.272385 + 0.962188i \(0.412188\pi\)
\(48\) 1.00000 0.144338
\(49\) 4.51179 0.644541
\(50\) −2.54676 −0.360166
\(51\) −4.14293 −0.580126
\(52\) 1.00000 0.138675
\(53\) 6.82448 0.937414 0.468707 0.883354i \(-0.344720\pi\)
0.468707 + 0.883354i \(0.344720\pi\)
\(54\) −1.00000 −0.136083
\(55\) 2.36487 0.318879
\(56\) −3.39290 −0.453396
\(57\) −2.70287 −0.358004
\(58\) −0.521486 −0.0684744
\(59\) −10.7140 −1.39484 −0.697422 0.716661i \(-0.745670\pi\)
−0.697422 + 0.716661i \(0.745670\pi\)
\(60\) −2.74714 −0.354654
\(61\) 4.99993 0.640175 0.320087 0.947388i \(-0.396288\pi\)
0.320087 + 0.947388i \(0.396288\pi\)
\(62\) −7.56769 −0.961097
\(63\) 3.39290 0.427465
\(64\) 1.00000 0.125000
\(65\) −2.74714 −0.340740
\(66\) 0.860848 0.105963
\(67\) 7.00412 0.855689 0.427845 0.903852i \(-0.359273\pi\)
0.427845 + 0.903852i \(0.359273\pi\)
\(68\) −4.14293 −0.502404
\(69\) 0.937524 0.112865
\(70\) 9.32077 1.11404
\(71\) −9.02958 −1.07161 −0.535807 0.844341i \(-0.679992\pi\)
−0.535807 + 0.844341i \(0.679992\pi\)
\(72\) −1.00000 −0.117851
\(73\) −8.63216 −1.01032 −0.505159 0.863026i \(-0.668566\pi\)
−0.505159 + 0.863026i \(0.668566\pi\)
\(74\) 0.373450 0.0434126
\(75\) 2.54676 0.294075
\(76\) −2.70287 −0.310040
\(77\) −2.92077 −0.332853
\(78\) −1.00000 −0.113228
\(79\) 1.21230 0.136394 0.0681970 0.997672i \(-0.478275\pi\)
0.0681970 + 0.997672i \(0.478275\pi\)
\(80\) −2.74714 −0.307139
\(81\) 1.00000 0.111111
\(82\) 5.16599 0.570488
\(83\) 15.5907 1.71130 0.855649 0.517556i \(-0.173158\pi\)
0.855649 + 0.517556i \(0.173158\pi\)
\(84\) 3.39290 0.370196
\(85\) 11.3812 1.23446
\(86\) 9.15820 0.987554
\(87\) 0.521486 0.0559092
\(88\) 0.860848 0.0917667
\(89\) −4.60576 −0.488209 −0.244105 0.969749i \(-0.578494\pi\)
−0.244105 + 0.969749i \(0.578494\pi\)
\(90\) 2.74714 0.289574
\(91\) 3.39290 0.355673
\(92\) 0.937524 0.0977436
\(93\) 7.56769 0.784732
\(94\) −3.73475 −0.385210
\(95\) 7.42515 0.761805
\(96\) −1.00000 −0.102062
\(97\) −8.14027 −0.826519 −0.413260 0.910613i \(-0.635610\pi\)
−0.413260 + 0.910613i \(0.635610\pi\)
\(98\) −4.51179 −0.455759
\(99\) −0.860848 −0.0865185
\(100\) 2.54676 0.254676
\(101\) −7.07845 −0.704332 −0.352166 0.935938i \(-0.614555\pi\)
−0.352166 + 0.935938i \(0.614555\pi\)
\(102\) 4.14293 0.410211
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) −9.32077 −0.909614
\(106\) −6.82448 −0.662852
\(107\) −17.1864 −1.66148 −0.830738 0.556664i \(-0.812081\pi\)
−0.830738 + 0.556664i \(0.812081\pi\)
\(108\) 1.00000 0.0962250
\(109\) 4.88430 0.467831 0.233916 0.972257i \(-0.424846\pi\)
0.233916 + 0.972257i \(0.424846\pi\)
\(110\) −2.36487 −0.225481
\(111\) −0.373450 −0.0354463
\(112\) 3.39290 0.320599
\(113\) −14.8209 −1.39423 −0.697115 0.716959i \(-0.745533\pi\)
−0.697115 + 0.716959i \(0.745533\pi\)
\(114\) 2.70287 0.253147
\(115\) −2.57551 −0.240167
\(116\) 0.521486 0.0484187
\(117\) 1.00000 0.0924500
\(118\) 10.7140 0.986303
\(119\) −14.0566 −1.28856
\(120\) 2.74714 0.250778
\(121\) −10.2589 −0.932631
\(122\) −4.99993 −0.452672
\(123\) −5.16599 −0.465802
\(124\) 7.56769 0.679598
\(125\) 6.73938 0.602789
\(126\) −3.39290 −0.302264
\(127\) −11.1401 −0.988524 −0.494262 0.869313i \(-0.664562\pi\)
−0.494262 + 0.869313i \(0.664562\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −9.15820 −0.806335
\(130\) 2.74714 0.240940
\(131\) 3.97766 0.347529 0.173765 0.984787i \(-0.444407\pi\)
0.173765 + 0.984787i \(0.444407\pi\)
\(132\) −0.860848 −0.0749272
\(133\) −9.17057 −0.795190
\(134\) −7.00412 −0.605064
\(135\) −2.74714 −0.236436
\(136\) 4.14293 0.355253
\(137\) −3.45070 −0.294813 −0.147406 0.989076i \(-0.547093\pi\)
−0.147406 + 0.989076i \(0.547093\pi\)
\(138\) −0.937524 −0.0798073
\(139\) 1.88476 0.159863 0.0799317 0.996800i \(-0.474530\pi\)
0.0799317 + 0.996800i \(0.474530\pi\)
\(140\) −9.32077 −0.787749
\(141\) 3.73475 0.314523
\(142\) 9.02958 0.757745
\(143\) −0.860848 −0.0719878
\(144\) 1.00000 0.0833333
\(145\) −1.43259 −0.118970
\(146\) 8.63216 0.714402
\(147\) 4.51179 0.372126
\(148\) −0.373450 −0.0306974
\(149\) 3.18177 0.260661 0.130330 0.991471i \(-0.458396\pi\)
0.130330 + 0.991471i \(0.458396\pi\)
\(150\) −2.54676 −0.207942
\(151\) 21.2767 1.73147 0.865736 0.500501i \(-0.166851\pi\)
0.865736 + 0.500501i \(0.166851\pi\)
\(152\) 2.70287 0.219232
\(153\) −4.14293 −0.334936
\(154\) 2.92077 0.235363
\(155\) −20.7895 −1.66985
\(156\) 1.00000 0.0800641
\(157\) 8.89329 0.709762 0.354881 0.934912i \(-0.384521\pi\)
0.354881 + 0.934912i \(0.384521\pi\)
\(158\) −1.21230 −0.0964452
\(159\) 6.82448 0.541216
\(160\) 2.74714 0.217180
\(161\) 3.18093 0.250692
\(162\) −1.00000 −0.0785674
\(163\) −18.6770 −1.46289 −0.731447 0.681898i \(-0.761155\pi\)
−0.731447 + 0.681898i \(0.761155\pi\)
\(164\) −5.16599 −0.403396
\(165\) 2.36487 0.184105
\(166\) −15.5907 −1.21007
\(167\) −12.3917 −0.958896 −0.479448 0.877570i \(-0.659163\pi\)
−0.479448 + 0.877570i \(0.659163\pi\)
\(168\) −3.39290 −0.261768
\(169\) 1.00000 0.0769231
\(170\) −11.3812 −0.872898
\(171\) −2.70287 −0.206694
\(172\) −9.15820 −0.698306
\(173\) −10.7413 −0.816644 −0.408322 0.912838i \(-0.633886\pi\)
−0.408322 + 0.912838i \(0.633886\pi\)
\(174\) −0.521486 −0.0395337
\(175\) 8.64091 0.653191
\(176\) −0.860848 −0.0648889
\(177\) −10.7140 −0.805313
\(178\) 4.60576 0.345216
\(179\) −5.05121 −0.377545 −0.188772 0.982021i \(-0.560451\pi\)
−0.188772 + 0.982021i \(0.560451\pi\)
\(180\) −2.74714 −0.204759
\(181\) −17.9386 −1.33337 −0.666684 0.745341i \(-0.732287\pi\)
−0.666684 + 0.745341i \(0.732287\pi\)
\(182\) −3.39290 −0.251499
\(183\) 4.99993 0.369605
\(184\) −0.937524 −0.0691152
\(185\) 1.02592 0.0754269
\(186\) −7.56769 −0.554890
\(187\) 3.56644 0.260804
\(188\) 3.73475 0.272385
\(189\) 3.39290 0.246797
\(190\) −7.42515 −0.538677
\(191\) 16.2968 1.17919 0.589597 0.807697i \(-0.299286\pi\)
0.589597 + 0.807697i \(0.299286\pi\)
\(192\) 1.00000 0.0721688
\(193\) 25.9923 1.87097 0.935485 0.353366i \(-0.114963\pi\)
0.935485 + 0.353366i \(0.114963\pi\)
\(194\) 8.14027 0.584437
\(195\) −2.74714 −0.196727
\(196\) 4.51179 0.322270
\(197\) 26.0591 1.85663 0.928315 0.371794i \(-0.121257\pi\)
0.928315 + 0.371794i \(0.121257\pi\)
\(198\) 0.860848 0.0611778
\(199\) −12.4115 −0.879829 −0.439915 0.898040i \(-0.644991\pi\)
−0.439915 + 0.898040i \(0.644991\pi\)
\(200\) −2.54676 −0.180083
\(201\) 7.00412 0.494032
\(202\) 7.07845 0.498038
\(203\) 1.76935 0.124184
\(204\) −4.14293 −0.290063
\(205\) 14.1917 0.991190
\(206\) 1.00000 0.0696733
\(207\) 0.937524 0.0651624
\(208\) 1.00000 0.0693375
\(209\) 2.32676 0.160945
\(210\) 9.32077 0.643194
\(211\) −15.7658 −1.08537 −0.542683 0.839938i \(-0.682591\pi\)
−0.542683 + 0.839938i \(0.682591\pi\)
\(212\) 6.82448 0.468707
\(213\) −9.02958 −0.618696
\(214\) 17.1864 1.17484
\(215\) 25.1588 1.71582
\(216\) −1.00000 −0.0680414
\(217\) 25.6764 1.74303
\(218\) −4.88430 −0.330807
\(219\) −8.63216 −0.583307
\(220\) 2.36487 0.159439
\(221\) −4.14293 −0.278684
\(222\) 0.373450 0.0250643
\(223\) 26.3575 1.76503 0.882514 0.470286i \(-0.155849\pi\)
0.882514 + 0.470286i \(0.155849\pi\)
\(224\) −3.39290 −0.226698
\(225\) 2.54676 0.169784
\(226\) 14.8209 0.985870
\(227\) −6.92514 −0.459638 −0.229819 0.973233i \(-0.573813\pi\)
−0.229819 + 0.973233i \(0.573813\pi\)
\(228\) −2.70287 −0.179002
\(229\) 24.9537 1.64899 0.824495 0.565870i \(-0.191459\pi\)
0.824495 + 0.565870i \(0.191459\pi\)
\(230\) 2.57551 0.169824
\(231\) −2.92077 −0.192173
\(232\) −0.521486 −0.0342372
\(233\) −8.28474 −0.542751 −0.271376 0.962474i \(-0.587479\pi\)
−0.271376 + 0.962474i \(0.587479\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −10.2599 −0.669281
\(236\) −10.7140 −0.697422
\(237\) 1.21230 0.0787471
\(238\) 14.0566 0.911152
\(239\) −26.6667 −1.72493 −0.862464 0.506118i \(-0.831080\pi\)
−0.862464 + 0.506118i \(0.831080\pi\)
\(240\) −2.74714 −0.177327
\(241\) −27.2863 −1.75767 −0.878834 0.477128i \(-0.841678\pi\)
−0.878834 + 0.477128i \(0.841678\pi\)
\(242\) 10.2589 0.659470
\(243\) 1.00000 0.0641500
\(244\) 4.99993 0.320087
\(245\) −12.3945 −0.791855
\(246\) 5.16599 0.329372
\(247\) −2.70287 −0.171980
\(248\) −7.56769 −0.480549
\(249\) 15.5907 0.988019
\(250\) −6.73938 −0.426236
\(251\) 19.0037 1.19950 0.599750 0.800187i \(-0.295267\pi\)
0.599750 + 0.800187i \(0.295267\pi\)
\(252\) 3.39290 0.213733
\(253\) −0.807066 −0.0507398
\(254\) 11.1401 0.698992
\(255\) 11.3812 0.712718
\(256\) 1.00000 0.0625000
\(257\) −18.7819 −1.17158 −0.585791 0.810462i \(-0.699216\pi\)
−0.585791 + 0.810462i \(0.699216\pi\)
\(258\) 9.15820 0.570165
\(259\) −1.26708 −0.0787324
\(260\) −2.74714 −0.170370
\(261\) 0.521486 0.0322792
\(262\) −3.97766 −0.245740
\(263\) 7.84505 0.483746 0.241873 0.970308i \(-0.422238\pi\)
0.241873 + 0.970308i \(0.422238\pi\)
\(264\) 0.860848 0.0529816
\(265\) −18.7478 −1.15167
\(266\) 9.17057 0.562284
\(267\) −4.60576 −0.281868
\(268\) 7.00412 0.427845
\(269\) −7.66512 −0.467351 −0.233675 0.972315i \(-0.575075\pi\)
−0.233675 + 0.972315i \(0.575075\pi\)
\(270\) 2.74714 0.167185
\(271\) 0.226570 0.0137631 0.00688157 0.999976i \(-0.497810\pi\)
0.00688157 + 0.999976i \(0.497810\pi\)
\(272\) −4.14293 −0.251202
\(273\) 3.39290 0.205348
\(274\) 3.45070 0.208464
\(275\) −2.19237 −0.132205
\(276\) 0.937524 0.0564323
\(277\) −31.6272 −1.90029 −0.950147 0.311802i \(-0.899067\pi\)
−0.950147 + 0.311802i \(0.899067\pi\)
\(278\) −1.88476 −0.113040
\(279\) 7.56769 0.453066
\(280\) 9.32077 0.557022
\(281\) −5.79701 −0.345820 −0.172910 0.984938i \(-0.555317\pi\)
−0.172910 + 0.984938i \(0.555317\pi\)
\(282\) −3.73475 −0.222401
\(283\) 31.1609 1.85232 0.926161 0.377129i \(-0.123089\pi\)
0.926161 + 0.377129i \(0.123089\pi\)
\(284\) −9.02958 −0.535807
\(285\) 7.42515 0.439828
\(286\) 0.860848 0.0509030
\(287\) −17.5277 −1.03463
\(288\) −1.00000 −0.0589256
\(289\) 0.163881 0.00964004
\(290\) 1.43259 0.0841248
\(291\) −8.14027 −0.477191
\(292\) −8.63216 −0.505159
\(293\) −29.9482 −1.74959 −0.874795 0.484493i \(-0.839004\pi\)
−0.874795 + 0.484493i \(0.839004\pi\)
\(294\) −4.51179 −0.263133
\(295\) 29.4328 1.71364
\(296\) 0.373450 0.0217063
\(297\) −0.860848 −0.0499515
\(298\) −3.18177 −0.184315
\(299\) 0.937524 0.0542184
\(300\) 2.54676 0.147037
\(301\) −31.0729 −1.79101
\(302\) −21.2767 −1.22434
\(303\) −7.07845 −0.406646
\(304\) −2.70287 −0.155020
\(305\) −13.7355 −0.786491
\(306\) 4.14293 0.236836
\(307\) −23.7820 −1.35731 −0.678657 0.734456i \(-0.737438\pi\)
−0.678657 + 0.734456i \(0.737438\pi\)
\(308\) −2.92077 −0.166427
\(309\) −1.00000 −0.0568880
\(310\) 20.7895 1.18076
\(311\) 29.4598 1.67051 0.835256 0.549861i \(-0.185319\pi\)
0.835256 + 0.549861i \(0.185319\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −5.15810 −0.291553 −0.145776 0.989318i \(-0.546568\pi\)
−0.145776 + 0.989318i \(0.546568\pi\)
\(314\) −8.89329 −0.501877
\(315\) −9.32077 −0.525166
\(316\) 1.21230 0.0681970
\(317\) 16.2036 0.910085 0.455043 0.890470i \(-0.349624\pi\)
0.455043 + 0.890470i \(0.349624\pi\)
\(318\) −6.82448 −0.382698
\(319\) −0.448920 −0.0251347
\(320\) −2.74714 −0.153570
\(321\) −17.1864 −0.959254
\(322\) −3.18093 −0.177266
\(323\) 11.1978 0.623063
\(324\) 1.00000 0.0555556
\(325\) 2.54676 0.141269
\(326\) 18.6770 1.03442
\(327\) 4.88430 0.270103
\(328\) 5.16599 0.285244
\(329\) 12.6717 0.698611
\(330\) −2.36487 −0.130182
\(331\) 32.8982 1.80825 0.904125 0.427269i \(-0.140524\pi\)
0.904125 + 0.427269i \(0.140524\pi\)
\(332\) 15.5907 0.855649
\(333\) −0.373450 −0.0204649
\(334\) 12.3917 0.678042
\(335\) −19.2413 −1.05126
\(336\) 3.39290 0.185098
\(337\) −11.5902 −0.631357 −0.315678 0.948866i \(-0.602232\pi\)
−0.315678 + 0.948866i \(0.602232\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −14.8209 −0.804959
\(340\) 11.3812 0.617232
\(341\) −6.51463 −0.352787
\(342\) 2.70287 0.146154
\(343\) −8.44227 −0.455840
\(344\) 9.15820 0.493777
\(345\) −2.57551 −0.138661
\(346\) 10.7413 0.577455
\(347\) 14.5311 0.780072 0.390036 0.920800i \(-0.372463\pi\)
0.390036 + 0.920800i \(0.372463\pi\)
\(348\) 0.521486 0.0279546
\(349\) 2.57425 0.137796 0.0688981 0.997624i \(-0.478052\pi\)
0.0688981 + 0.997624i \(0.478052\pi\)
\(350\) −8.64091 −0.461876
\(351\) 1.00000 0.0533761
\(352\) 0.860848 0.0458834
\(353\) 4.27233 0.227393 0.113697 0.993516i \(-0.463731\pi\)
0.113697 + 0.993516i \(0.463731\pi\)
\(354\) 10.7140 0.569442
\(355\) 24.8055 1.31654
\(356\) −4.60576 −0.244105
\(357\) −14.0566 −0.743952
\(358\) 5.05121 0.266965
\(359\) −12.0179 −0.634279 −0.317139 0.948379i \(-0.602722\pi\)
−0.317139 + 0.948379i \(0.602722\pi\)
\(360\) 2.74714 0.144787
\(361\) −11.6945 −0.615500
\(362\) 17.9386 0.942833
\(363\) −10.2589 −0.538455
\(364\) 3.39290 0.177836
\(365\) 23.7137 1.24123
\(366\) −4.99993 −0.261350
\(367\) 3.95835 0.206624 0.103312 0.994649i \(-0.467056\pi\)
0.103312 + 0.994649i \(0.467056\pi\)
\(368\) 0.937524 0.0488718
\(369\) −5.16599 −0.268931
\(370\) −1.02592 −0.0533349
\(371\) 23.1548 1.20214
\(372\) 7.56769 0.392366
\(373\) 6.22306 0.322218 0.161109 0.986937i \(-0.448493\pi\)
0.161109 + 0.986937i \(0.448493\pi\)
\(374\) −3.56644 −0.184416
\(375\) 6.73938 0.348020
\(376\) −3.73475 −0.192605
\(377\) 0.521486 0.0268579
\(378\) −3.39290 −0.174512
\(379\) −10.0976 −0.518678 −0.259339 0.965786i \(-0.583505\pi\)
−0.259339 + 0.965786i \(0.583505\pi\)
\(380\) 7.42515 0.380902
\(381\) −11.1401 −0.570725
\(382\) −16.2968 −0.833816
\(383\) −38.4445 −1.96442 −0.982211 0.187779i \(-0.939871\pi\)
−0.982211 + 0.187779i \(0.939871\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 8.02377 0.408929
\(386\) −25.9923 −1.32298
\(387\) −9.15820 −0.465537
\(388\) −8.14027 −0.413260
\(389\) −23.4430 −1.18861 −0.594304 0.804240i \(-0.702573\pi\)
−0.594304 + 0.804240i \(0.702573\pi\)
\(390\) 2.74714 0.139107
\(391\) −3.88410 −0.196427
\(392\) −4.51179 −0.227880
\(393\) 3.97766 0.200646
\(394\) −26.0591 −1.31284
\(395\) −3.33035 −0.167568
\(396\) −0.860848 −0.0432593
\(397\) −16.4848 −0.827350 −0.413675 0.910425i \(-0.635755\pi\)
−0.413675 + 0.910425i \(0.635755\pi\)
\(398\) 12.4115 0.622133
\(399\) −9.17057 −0.459103
\(400\) 2.54676 0.127338
\(401\) 0.0156239 0.000780218 0 0.000390109 1.00000i \(-0.499876\pi\)
0.000390109 1.00000i \(0.499876\pi\)
\(402\) −7.00412 −0.349334
\(403\) 7.56769 0.376973
\(404\) −7.07845 −0.352166
\(405\) −2.74714 −0.136506
\(406\) −1.76935 −0.0878114
\(407\) 0.321483 0.0159353
\(408\) 4.14293 0.205106
\(409\) −38.0942 −1.88364 −0.941818 0.336124i \(-0.890884\pi\)
−0.941818 + 0.336124i \(0.890884\pi\)
\(410\) −14.1917 −0.700877
\(411\) −3.45070 −0.170210
\(412\) −1.00000 −0.0492665
\(413\) −36.3515 −1.78874
\(414\) −0.937524 −0.0460768
\(415\) −42.8297 −2.10243
\(416\) −1.00000 −0.0490290
\(417\) 1.88476 0.0922972
\(418\) −2.32676 −0.113806
\(419\) 31.0704 1.51789 0.758943 0.651157i \(-0.225716\pi\)
0.758943 + 0.651157i \(0.225716\pi\)
\(420\) −9.32077 −0.454807
\(421\) −15.5883 −0.759729 −0.379864 0.925042i \(-0.624029\pi\)
−0.379864 + 0.925042i \(0.624029\pi\)
\(422\) 15.7658 0.767470
\(423\) 3.73475 0.181590
\(424\) −6.82448 −0.331426
\(425\) −10.5511 −0.511801
\(426\) 9.02958 0.437484
\(427\) 16.9643 0.820958
\(428\) −17.1864 −0.830738
\(429\) −0.860848 −0.0415621
\(430\) −25.1588 −1.21327
\(431\) −17.0616 −0.821830 −0.410915 0.911674i \(-0.634791\pi\)
−0.410915 + 0.911674i \(0.634791\pi\)
\(432\) 1.00000 0.0481125
\(433\) 5.18891 0.249363 0.124682 0.992197i \(-0.460209\pi\)
0.124682 + 0.992197i \(0.460209\pi\)
\(434\) −25.6764 −1.23251
\(435\) −1.43259 −0.0686876
\(436\) 4.88430 0.233916
\(437\) −2.53401 −0.121218
\(438\) 8.63216 0.412460
\(439\) −41.6551 −1.98809 −0.994045 0.108975i \(-0.965243\pi\)
−0.994045 + 0.108975i \(0.965243\pi\)
\(440\) −2.36487 −0.112741
\(441\) 4.51179 0.214847
\(442\) 4.14293 0.197059
\(443\) −23.0790 −1.09652 −0.548258 0.836309i \(-0.684709\pi\)
−0.548258 + 0.836309i \(0.684709\pi\)
\(444\) −0.373450 −0.0177231
\(445\) 12.6526 0.599793
\(446\) −26.3575 −1.24806
\(447\) 3.18177 0.150492
\(448\) 3.39290 0.160300
\(449\) 7.77027 0.366702 0.183351 0.983048i \(-0.441306\pi\)
0.183351 + 0.983048i \(0.441306\pi\)
\(450\) −2.54676 −0.120055
\(451\) 4.44713 0.209407
\(452\) −14.8209 −0.697115
\(453\) 21.2767 0.999665
\(454\) 6.92514 0.325013
\(455\) −9.32077 −0.436964
\(456\) 2.70287 0.126574
\(457\) −24.1142 −1.12801 −0.564007 0.825770i \(-0.690741\pi\)
−0.564007 + 0.825770i \(0.690741\pi\)
\(458\) −24.9537 −1.16601
\(459\) −4.14293 −0.193375
\(460\) −2.57551 −0.120084
\(461\) −4.96952 −0.231453 −0.115727 0.993281i \(-0.536920\pi\)
−0.115727 + 0.993281i \(0.536920\pi\)
\(462\) 2.92077 0.135887
\(463\) −0.774424 −0.0359905 −0.0179953 0.999838i \(-0.505728\pi\)
−0.0179953 + 0.999838i \(0.505728\pi\)
\(464\) 0.521486 0.0242094
\(465\) −20.7895 −0.964089
\(466\) 8.28474 0.383783
\(467\) −18.2679 −0.845336 −0.422668 0.906285i \(-0.638906\pi\)
−0.422668 + 0.906285i \(0.638906\pi\)
\(468\) 1.00000 0.0462250
\(469\) 23.7643 1.09733
\(470\) 10.2599 0.473253
\(471\) 8.89329 0.409781
\(472\) 10.7140 0.493152
\(473\) 7.88382 0.362498
\(474\) −1.21230 −0.0556826
\(475\) −6.88357 −0.315840
\(476\) −14.0566 −0.644281
\(477\) 6.82448 0.312471
\(478\) 26.6667 1.21971
\(479\) 5.82877 0.266323 0.133162 0.991094i \(-0.457487\pi\)
0.133162 + 0.991094i \(0.457487\pi\)
\(480\) 2.74714 0.125389
\(481\) −0.373450 −0.0170278
\(482\) 27.2863 1.24286
\(483\) 3.18093 0.144737
\(484\) −10.2589 −0.466315
\(485\) 22.3624 1.01543
\(486\) −1.00000 −0.0453609
\(487\) −7.63754 −0.346090 −0.173045 0.984914i \(-0.555361\pi\)
−0.173045 + 0.984914i \(0.555361\pi\)
\(488\) −4.99993 −0.226336
\(489\) −18.6770 −0.844602
\(490\) 12.3945 0.559926
\(491\) 10.2639 0.463203 0.231602 0.972811i \(-0.425603\pi\)
0.231602 + 0.972811i \(0.425603\pi\)
\(492\) −5.16599 −0.232901
\(493\) −2.16048 −0.0973031
\(494\) 2.70287 0.121608
\(495\) 2.36487 0.106293
\(496\) 7.56769 0.339799
\(497\) −30.6365 −1.37423
\(498\) −15.5907 −0.698635
\(499\) 18.1451 0.812288 0.406144 0.913809i \(-0.366873\pi\)
0.406144 + 0.913809i \(0.366873\pi\)
\(500\) 6.73938 0.301394
\(501\) −12.3917 −0.553619
\(502\) −19.0037 −0.848175
\(503\) −3.82760 −0.170664 −0.0853321 0.996353i \(-0.527195\pi\)
−0.0853321 + 0.996353i \(0.527195\pi\)
\(504\) −3.39290 −0.151132
\(505\) 19.4455 0.865312
\(506\) 0.807066 0.0358785
\(507\) 1.00000 0.0444116
\(508\) −11.1401 −0.494262
\(509\) 23.7607 1.05318 0.526588 0.850121i \(-0.323471\pi\)
0.526588 + 0.850121i \(0.323471\pi\)
\(510\) −11.3812 −0.503968
\(511\) −29.2881 −1.29563
\(512\) −1.00000 −0.0441942
\(513\) −2.70287 −0.119335
\(514\) 18.7819 0.828434
\(515\) 2.74714 0.121053
\(516\) −9.15820 −0.403167
\(517\) −3.21506 −0.141398
\(518\) 1.26708 0.0556722
\(519\) −10.7413 −0.471490
\(520\) 2.74714 0.120470
\(521\) −17.9485 −0.786338 −0.393169 0.919466i \(-0.628621\pi\)
−0.393169 + 0.919466i \(0.628621\pi\)
\(522\) −0.521486 −0.0228248
\(523\) 31.3989 1.37298 0.686490 0.727139i \(-0.259151\pi\)
0.686490 + 0.727139i \(0.259151\pi\)
\(524\) 3.97766 0.173765
\(525\) 8.64091 0.377120
\(526\) −7.84505 −0.342060
\(527\) −31.3524 −1.36573
\(528\) −0.860848 −0.0374636
\(529\) −22.1210 −0.961785
\(530\) 18.7478 0.814351
\(531\) −10.7140 −0.464948
\(532\) −9.17057 −0.397595
\(533\) −5.16599 −0.223764
\(534\) 4.60576 0.199311
\(535\) 47.2135 2.04122
\(536\) −7.00412 −0.302532
\(537\) −5.05121 −0.217976
\(538\) 7.66512 0.330467
\(539\) −3.88396 −0.167294
\(540\) −2.74714 −0.118218
\(541\) −26.4259 −1.13614 −0.568070 0.822980i \(-0.692310\pi\)
−0.568070 + 0.822980i \(0.692310\pi\)
\(542\) −0.226570 −0.00973201
\(543\) −17.9386 −0.769820
\(544\) 4.14293 0.177627
\(545\) −13.4178 −0.574757
\(546\) −3.39290 −0.145203
\(547\) −1.01814 −0.0435325 −0.0217663 0.999763i \(-0.506929\pi\)
−0.0217663 + 0.999763i \(0.506929\pi\)
\(548\) −3.45070 −0.147406
\(549\) 4.99993 0.213392
\(550\) 2.19237 0.0934832
\(551\) −1.40951 −0.0600471
\(552\) −0.937524 −0.0399037
\(553\) 4.11320 0.174911
\(554\) 31.6272 1.34371
\(555\) 1.02592 0.0435478
\(556\) 1.88476 0.0799317
\(557\) 18.7478 0.794371 0.397185 0.917738i \(-0.369987\pi\)
0.397185 + 0.917738i \(0.369987\pi\)
\(558\) −7.56769 −0.320366
\(559\) −9.15820 −0.387351
\(560\) −9.32077 −0.393874
\(561\) 3.56644 0.150575
\(562\) 5.79701 0.244532
\(563\) −29.7944 −1.25568 −0.627841 0.778341i \(-0.716061\pi\)
−0.627841 + 0.778341i \(0.716061\pi\)
\(564\) 3.73475 0.157261
\(565\) 40.7150 1.71289
\(566\) −31.1609 −1.30979
\(567\) 3.39290 0.142488
\(568\) 9.02958 0.378873
\(569\) −29.3063 −1.22858 −0.614292 0.789079i \(-0.710558\pi\)
−0.614292 + 0.789079i \(0.710558\pi\)
\(570\) −7.42515 −0.311006
\(571\) −34.2507 −1.43335 −0.716673 0.697410i \(-0.754336\pi\)
−0.716673 + 0.697410i \(0.754336\pi\)
\(572\) −0.860848 −0.0359939
\(573\) 16.2968 0.680808
\(574\) 17.5277 0.731592
\(575\) 2.38765 0.0995719
\(576\) 1.00000 0.0416667
\(577\) 20.9906 0.873848 0.436924 0.899498i \(-0.356068\pi\)
0.436924 + 0.899498i \(0.356068\pi\)
\(578\) −0.163881 −0.00681654
\(579\) 25.9923 1.08021
\(580\) −1.43259 −0.0594852
\(581\) 52.8976 2.19456
\(582\) 8.14027 0.337425
\(583\) −5.87484 −0.243311
\(584\) 8.63216 0.357201
\(585\) −2.74714 −0.113580
\(586\) 29.9482 1.23715
\(587\) 4.97107 0.205178 0.102589 0.994724i \(-0.467287\pi\)
0.102589 + 0.994724i \(0.467287\pi\)
\(588\) 4.51179 0.186063
\(589\) −20.4545 −0.842812
\(590\) −29.4328 −1.21173
\(591\) 26.0591 1.07193
\(592\) −0.373450 −0.0153487
\(593\) −18.4742 −0.758643 −0.379321 0.925265i \(-0.623842\pi\)
−0.379321 + 0.925265i \(0.623842\pi\)
\(594\) 0.860848 0.0353210
\(595\) 38.6153 1.58307
\(596\) 3.18177 0.130330
\(597\) −12.4115 −0.507970
\(598\) −0.937524 −0.0383382
\(599\) −10.0889 −0.412222 −0.206111 0.978529i \(-0.566081\pi\)
−0.206111 + 0.978529i \(0.566081\pi\)
\(600\) −2.54676 −0.103971
\(601\) −17.7439 −0.723788 −0.361894 0.932219i \(-0.617870\pi\)
−0.361894 + 0.932219i \(0.617870\pi\)
\(602\) 31.0729 1.26644
\(603\) 7.00412 0.285230
\(604\) 21.2767 0.865736
\(605\) 28.1827 1.14579
\(606\) 7.07845 0.287542
\(607\) 10.5752 0.429233 0.214616 0.976698i \(-0.431150\pi\)
0.214616 + 0.976698i \(0.431150\pi\)
\(608\) 2.70287 0.109616
\(609\) 1.76935 0.0716977
\(610\) 13.7355 0.556133
\(611\) 3.73475 0.151092
\(612\) −4.14293 −0.167468
\(613\) −3.09381 −0.124958 −0.0624789 0.998046i \(-0.519901\pi\)
−0.0624789 + 0.998046i \(0.519901\pi\)
\(614\) 23.7820 0.959765
\(615\) 14.1917 0.572264
\(616\) 2.92077 0.117681
\(617\) 0.671803 0.0270458 0.0135229 0.999909i \(-0.495695\pi\)
0.0135229 + 0.999909i \(0.495695\pi\)
\(618\) 1.00000 0.0402259
\(619\) −0.425707 −0.0171106 −0.00855531 0.999963i \(-0.502723\pi\)
−0.00855531 + 0.999963i \(0.502723\pi\)
\(620\) −20.7895 −0.834925
\(621\) 0.937524 0.0376215
\(622\) −29.4598 −1.18123
\(623\) −15.6269 −0.626078
\(624\) 1.00000 0.0400320
\(625\) −31.2478 −1.24991
\(626\) 5.15810 0.206159
\(627\) 2.32676 0.0929219
\(628\) 8.89329 0.354881
\(629\) 1.54718 0.0616900
\(630\) 9.32077 0.371348
\(631\) 36.4937 1.45279 0.726395 0.687278i \(-0.241194\pi\)
0.726395 + 0.687278i \(0.241194\pi\)
\(632\) −1.21230 −0.0482226
\(633\) −15.7658 −0.626636
\(634\) −16.2036 −0.643528
\(635\) 30.6034 1.21446
\(636\) 6.82448 0.270608
\(637\) 4.51179 0.178763
\(638\) 0.448920 0.0177729
\(639\) −9.02958 −0.357204
\(640\) 2.74714 0.108590
\(641\) −12.5755 −0.496702 −0.248351 0.968670i \(-0.579889\pi\)
−0.248351 + 0.968670i \(0.579889\pi\)
\(642\) 17.1864 0.678295
\(643\) −7.38509 −0.291239 −0.145620 0.989341i \(-0.546518\pi\)
−0.145620 + 0.989341i \(0.546518\pi\)
\(644\) 3.18093 0.125346
\(645\) 25.1588 0.990628
\(646\) −11.1978 −0.440572
\(647\) 42.7103 1.67911 0.839557 0.543271i \(-0.182814\pi\)
0.839557 + 0.543271i \(0.182814\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 9.22313 0.362039
\(650\) −2.54676 −0.0998922
\(651\) 25.6764 1.00634
\(652\) −18.6770 −0.731447
\(653\) −2.17138 −0.0849726 −0.0424863 0.999097i \(-0.513528\pi\)
−0.0424863 + 0.999097i \(0.513528\pi\)
\(654\) −4.88430 −0.190991
\(655\) −10.9272 −0.426960
\(656\) −5.16599 −0.201698
\(657\) −8.63216 −0.336773
\(658\) −12.6717 −0.493992
\(659\) 28.6191 1.11484 0.557421 0.830230i \(-0.311791\pi\)
0.557421 + 0.830230i \(0.311791\pi\)
\(660\) 2.36487 0.0920524
\(661\) −23.2652 −0.904912 −0.452456 0.891787i \(-0.649452\pi\)
−0.452456 + 0.891787i \(0.649452\pi\)
\(662\) −32.8982 −1.27863
\(663\) −4.14293 −0.160898
\(664\) −15.5907 −0.605035
\(665\) 25.1928 0.976936
\(666\) 0.373450 0.0144709
\(667\) 0.488905 0.0189305
\(668\) −12.3917 −0.479448
\(669\) 26.3575 1.01904
\(670\) 19.2413 0.743355
\(671\) −4.30418 −0.166161
\(672\) −3.39290 −0.130884
\(673\) −15.9535 −0.614961 −0.307480 0.951554i \(-0.599486\pi\)
−0.307480 + 0.951554i \(0.599486\pi\)
\(674\) 11.5902 0.446437
\(675\) 2.54676 0.0980249
\(676\) 1.00000 0.0384615
\(677\) 39.7105 1.52620 0.763098 0.646282i \(-0.223677\pi\)
0.763098 + 0.646282i \(0.223677\pi\)
\(678\) 14.8209 0.569192
\(679\) −27.6191 −1.05993
\(680\) −11.3812 −0.436449
\(681\) −6.92514 −0.265372
\(682\) 6.51463 0.249458
\(683\) −4.31372 −0.165060 −0.0825299 0.996589i \(-0.526300\pi\)
−0.0825299 + 0.996589i \(0.526300\pi\)
\(684\) −2.70287 −0.103347
\(685\) 9.47953 0.362194
\(686\) 8.44227 0.322327
\(687\) 24.9537 0.952044
\(688\) −9.15820 −0.349153
\(689\) 6.82448 0.259992
\(690\) 2.57551 0.0980479
\(691\) −16.3101 −0.620464 −0.310232 0.950661i \(-0.600407\pi\)
−0.310232 + 0.950661i \(0.600407\pi\)
\(692\) −10.7413 −0.408322
\(693\) −2.92077 −0.110951
\(694\) −14.5311 −0.551594
\(695\) −5.17770 −0.196401
\(696\) −0.521486 −0.0197669
\(697\) 21.4023 0.810672
\(698\) −2.57425 −0.0974367
\(699\) −8.28474 −0.313358
\(700\) 8.64091 0.326596
\(701\) −46.1044 −1.74134 −0.870670 0.491868i \(-0.836314\pi\)
−0.870670 + 0.491868i \(0.836314\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 1.00939 0.0380697
\(704\) −0.860848 −0.0324444
\(705\) −10.2599 −0.386409
\(706\) −4.27233 −0.160791
\(707\) −24.0165 −0.903233
\(708\) −10.7140 −0.402657
\(709\) 29.9630 1.12529 0.562643 0.826700i \(-0.309785\pi\)
0.562643 + 0.826700i \(0.309785\pi\)
\(710\) −24.8055 −0.930933
\(711\) 1.21230 0.0454647
\(712\) 4.60576 0.172608
\(713\) 7.09489 0.265706
\(714\) 14.0566 0.526054
\(715\) 2.36487 0.0884411
\(716\) −5.05121 −0.188772
\(717\) −26.6667 −0.995888
\(718\) 12.0179 0.448503
\(719\) 28.1540 1.04997 0.524984 0.851112i \(-0.324071\pi\)
0.524984 + 0.851112i \(0.324071\pi\)
\(720\) −2.74714 −0.102380
\(721\) −3.39290 −0.126358
\(722\) 11.6945 0.435224
\(723\) −27.2863 −1.01479
\(724\) −17.9386 −0.666684
\(725\) 1.32810 0.0493244
\(726\) 10.2589 0.380745
\(727\) −41.1786 −1.52723 −0.763614 0.645673i \(-0.776577\pi\)
−0.763614 + 0.645673i \(0.776577\pi\)
\(728\) −3.39290 −0.125749
\(729\) 1.00000 0.0370370
\(730\) −23.7137 −0.877684
\(731\) 37.9418 1.40333
\(732\) 4.99993 0.184803
\(733\) 29.4372 1.08729 0.543645 0.839315i \(-0.317044\pi\)
0.543645 + 0.839315i \(0.317044\pi\)
\(734\) −3.95835 −0.146106
\(735\) −12.3945 −0.457178
\(736\) −0.937524 −0.0345576
\(737\) −6.02948 −0.222099
\(738\) 5.16599 0.190163
\(739\) 39.2150 1.44255 0.721274 0.692650i \(-0.243557\pi\)
0.721274 + 0.692650i \(0.243557\pi\)
\(740\) 1.02592 0.0377135
\(741\) −2.70287 −0.0992924
\(742\) −23.1548 −0.850039
\(743\) 39.5114 1.44953 0.724766 0.688995i \(-0.241948\pi\)
0.724766 + 0.688995i \(0.241948\pi\)
\(744\) −7.56769 −0.277445
\(745\) −8.74075 −0.320236
\(746\) −6.22306 −0.227843
\(747\) 15.5907 0.570433
\(748\) 3.56644 0.130402
\(749\) −58.3119 −2.13067
\(750\) −6.73938 −0.246087
\(751\) 2.11914 0.0773284 0.0386642 0.999252i \(-0.487690\pi\)
0.0386642 + 0.999252i \(0.487690\pi\)
\(752\) 3.73475 0.136192
\(753\) 19.0037 0.692532
\(754\) −0.521486 −0.0189914
\(755\) −58.4499 −2.12721
\(756\) 3.39290 0.123399
\(757\) −51.6443 −1.87704 −0.938521 0.345221i \(-0.887804\pi\)
−0.938521 + 0.345221i \(0.887804\pi\)
\(758\) 10.0976 0.366761
\(759\) −0.807066 −0.0292946
\(760\) −7.42515 −0.269339
\(761\) 14.3106 0.518758 0.259379 0.965776i \(-0.416482\pi\)
0.259379 + 0.965776i \(0.416482\pi\)
\(762\) 11.1401 0.403563
\(763\) 16.5720 0.599945
\(764\) 16.2968 0.589597
\(765\) 11.3812 0.411488
\(766\) 38.4445 1.38906
\(767\) −10.7140 −0.386860
\(768\) 1.00000 0.0360844
\(769\) −23.6577 −0.853118 −0.426559 0.904460i \(-0.640274\pi\)
−0.426559 + 0.904460i \(0.640274\pi\)
\(770\) −8.02377 −0.289156
\(771\) −18.7819 −0.676414
\(772\) 25.9923 0.935485
\(773\) 42.3109 1.52182 0.760908 0.648859i \(-0.224754\pi\)
0.760908 + 0.648859i \(0.224754\pi\)
\(774\) 9.15820 0.329185
\(775\) 19.2731 0.692310
\(776\) 8.14027 0.292219
\(777\) −1.26708 −0.0454562
\(778\) 23.4430 0.840473
\(779\) 13.9630 0.500277
\(780\) −2.74714 −0.0983633
\(781\) 7.77309 0.278143
\(782\) 3.88410 0.138895
\(783\) 0.521486 0.0186364
\(784\) 4.51179 0.161135
\(785\) −24.4311 −0.871983
\(786\) −3.97766 −0.141878
\(787\) −15.9337 −0.567976 −0.283988 0.958828i \(-0.591658\pi\)
−0.283988 + 0.958828i \(0.591658\pi\)
\(788\) 26.0591 0.928315
\(789\) 7.84505 0.279291
\(790\) 3.33035 0.118488
\(791\) −50.2858 −1.78796
\(792\) 0.860848 0.0305889
\(793\) 4.99993 0.177553
\(794\) 16.4848 0.585025
\(795\) −18.7478 −0.664915
\(796\) −12.4115 −0.439915
\(797\) −25.3438 −0.897724 −0.448862 0.893601i \(-0.648171\pi\)
−0.448862 + 0.893601i \(0.648171\pi\)
\(798\) 9.17057 0.324635
\(799\) −15.4728 −0.547389
\(800\) −2.54676 −0.0900416
\(801\) −4.60576 −0.162736
\(802\) −0.0156239 −0.000551698 0
\(803\) 7.43098 0.262234
\(804\) 7.00412 0.247016
\(805\) −8.73844 −0.307990
\(806\) −7.56769 −0.266560
\(807\) −7.66512 −0.269825
\(808\) 7.07845 0.249019
\(809\) 26.9385 0.947108 0.473554 0.880765i \(-0.342971\pi\)
0.473554 + 0.880765i \(0.342971\pi\)
\(810\) 2.74714 0.0965246
\(811\) −37.2117 −1.30668 −0.653341 0.757064i \(-0.726633\pi\)
−0.653341 + 0.757064i \(0.726633\pi\)
\(812\) 1.76935 0.0620920
\(813\) 0.226570 0.00794615
\(814\) −0.321483 −0.0112680
\(815\) 51.3082 1.79725
\(816\) −4.14293 −0.145032
\(817\) 24.7534 0.866013
\(818\) 38.0942 1.33193
\(819\) 3.39290 0.118558
\(820\) 14.1917 0.495595
\(821\) 22.9658 0.801511 0.400756 0.916185i \(-0.368748\pi\)
0.400756 + 0.916185i \(0.368748\pi\)
\(822\) 3.45070 0.120357
\(823\) 2.33619 0.0814343 0.0407172 0.999171i \(-0.487036\pi\)
0.0407172 + 0.999171i \(0.487036\pi\)
\(824\) 1.00000 0.0348367
\(825\) −2.19237 −0.0763287
\(826\) 36.3515 1.26483
\(827\) 11.5453 0.401468 0.200734 0.979646i \(-0.435667\pi\)
0.200734 + 0.979646i \(0.435667\pi\)
\(828\) 0.937524 0.0325812
\(829\) −30.4372 −1.05713 −0.528563 0.848894i \(-0.677269\pi\)
−0.528563 + 0.848894i \(0.677269\pi\)
\(830\) 42.8297 1.48664
\(831\) −31.6272 −1.09714
\(832\) 1.00000 0.0346688
\(833\) −18.6920 −0.647640
\(834\) −1.88476 −0.0652639
\(835\) 34.0416 1.17806
\(836\) 2.32676 0.0804727
\(837\) 7.56769 0.261577
\(838\) −31.0704 −1.07331
\(839\) 45.1025 1.55711 0.778556 0.627575i \(-0.215952\pi\)
0.778556 + 0.627575i \(0.215952\pi\)
\(840\) 9.32077 0.321597
\(841\) −28.7281 −0.990622
\(842\) 15.5883 0.537210
\(843\) −5.79701 −0.199659
\(844\) −15.7658 −0.542683
\(845\) −2.74714 −0.0945044
\(846\) −3.73475 −0.128403
\(847\) −34.8076 −1.19600
\(848\) 6.82448 0.234354
\(849\) 31.1609 1.06944
\(850\) 10.5511 0.361898
\(851\) −0.350118 −0.0120019
\(852\) −9.02958 −0.309348
\(853\) −44.8775 −1.53658 −0.768289 0.640103i \(-0.778891\pi\)
−0.768289 + 0.640103i \(0.778891\pi\)
\(854\) −16.9643 −0.580505
\(855\) 7.42515 0.253935
\(856\) 17.1864 0.587420
\(857\) −37.1959 −1.27059 −0.635294 0.772271i \(-0.719121\pi\)
−0.635294 + 0.772271i \(0.719121\pi\)
\(858\) 0.860848 0.0293889
\(859\) −19.7851 −0.675060 −0.337530 0.941315i \(-0.609591\pi\)
−0.337530 + 0.941315i \(0.609591\pi\)
\(860\) 25.1588 0.857909
\(861\) −17.5277 −0.597342
\(862\) 17.0616 0.581122
\(863\) −14.5853 −0.496490 −0.248245 0.968697i \(-0.579854\pi\)
−0.248245 + 0.968697i \(0.579854\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 29.5078 1.00329
\(866\) −5.18891 −0.176326
\(867\) 0.163881 0.00556568
\(868\) 25.6764 0.871514
\(869\) −1.04360 −0.0354018
\(870\) 1.43259 0.0485695
\(871\) 7.00412 0.237325
\(872\) −4.88430 −0.165403
\(873\) −8.14027 −0.275506
\(874\) 2.53401 0.0857140
\(875\) 22.8661 0.773014
\(876\) −8.63216 −0.291654
\(877\) 34.7902 1.17478 0.587391 0.809303i \(-0.300155\pi\)
0.587391 + 0.809303i \(0.300155\pi\)
\(878\) 41.6551 1.40579
\(879\) −29.9482 −1.01013
\(880\) 2.36487 0.0797197
\(881\) 25.2054 0.849192 0.424596 0.905383i \(-0.360416\pi\)
0.424596 + 0.905383i \(0.360416\pi\)
\(882\) −4.51179 −0.151920
\(883\) 34.9651 1.17667 0.588335 0.808617i \(-0.299784\pi\)
0.588335 + 0.808617i \(0.299784\pi\)
\(884\) −4.14293 −0.139342
\(885\) 29.4328 0.989373
\(886\) 23.0790 0.775354
\(887\) −41.5902 −1.39646 −0.698231 0.715873i \(-0.746029\pi\)
−0.698231 + 0.715873i \(0.746029\pi\)
\(888\) 0.373450 0.0125321
\(889\) −37.7973 −1.26768
\(890\) −12.6526 −0.424117
\(891\) −0.860848 −0.0288395
\(892\) 26.3575 0.882514
\(893\) −10.0946 −0.337801
\(894\) −3.18177 −0.106414
\(895\) 13.8764 0.463836
\(896\) −3.39290 −0.113349
\(897\) 0.937524 0.0313030
\(898\) −7.77027 −0.259297
\(899\) 3.94644 0.131621
\(900\) 2.54676 0.0848920
\(901\) −28.2733 −0.941922
\(902\) −4.44713 −0.148073
\(903\) −31.0729 −1.03404
\(904\) 14.8209 0.492935
\(905\) 49.2799 1.63812
\(906\) −21.2767 −0.706870
\(907\) −35.7939 −1.18852 −0.594259 0.804274i \(-0.702555\pi\)
−0.594259 + 0.804274i \(0.702555\pi\)
\(908\) −6.92514 −0.229819
\(909\) −7.07845 −0.234777
\(910\) 9.32077 0.308980
\(911\) −49.4881 −1.63961 −0.819806 0.572641i \(-0.805919\pi\)
−0.819806 + 0.572641i \(0.805919\pi\)
\(912\) −2.70287 −0.0895010
\(913\) −13.4212 −0.444177
\(914\) 24.1142 0.797626
\(915\) −13.7355 −0.454081
\(916\) 24.9537 0.824495
\(917\) 13.4958 0.445670
\(918\) 4.14293 0.136737
\(919\) −4.19055 −0.138233 −0.0691167 0.997609i \(-0.522018\pi\)
−0.0691167 + 0.997609i \(0.522018\pi\)
\(920\) 2.57551 0.0849119
\(921\) −23.7820 −0.783645
\(922\) 4.96952 0.163662
\(923\) −9.02958 −0.297212
\(924\) −2.92077 −0.0960864
\(925\) −0.951087 −0.0312715
\(926\) 0.774424 0.0254491
\(927\) −1.00000 −0.0328443
\(928\) −0.521486 −0.0171186
\(929\) −12.6464 −0.414916 −0.207458 0.978244i \(-0.566519\pi\)
−0.207458 + 0.978244i \(0.566519\pi\)
\(930\) 20.7895 0.681714
\(931\) −12.1948 −0.399667
\(932\) −8.28474 −0.271376
\(933\) 29.4598 0.964471
\(934\) 18.2679 0.597743
\(935\) −9.79749 −0.320412
\(936\) −1.00000 −0.0326860
\(937\) −8.42532 −0.275243 −0.137622 0.990485i \(-0.543946\pi\)
−0.137622 + 0.990485i \(0.543946\pi\)
\(938\) −23.7643 −0.775931
\(939\) −5.15810 −0.168328
\(940\) −10.2599 −0.334640
\(941\) 10.7156 0.349319 0.174659 0.984629i \(-0.444118\pi\)
0.174659 + 0.984629i \(0.444118\pi\)
\(942\) −8.89329 −0.289759
\(943\) −4.84324 −0.157718
\(944\) −10.7140 −0.348711
\(945\) −9.32077 −0.303205
\(946\) −7.88382 −0.256325
\(947\) 36.7982 1.19578 0.597891 0.801578i \(-0.296006\pi\)
0.597891 + 0.801578i \(0.296006\pi\)
\(948\) 1.21230 0.0393736
\(949\) −8.63216 −0.280212
\(950\) 6.88357 0.223332
\(951\) 16.2036 0.525438
\(952\) 14.0566 0.455576
\(953\) 12.7429 0.412783 0.206392 0.978469i \(-0.433828\pi\)
0.206392 + 0.978469i \(0.433828\pi\)
\(954\) −6.82448 −0.220951
\(955\) −44.7695 −1.44871
\(956\) −26.6667 −0.862464
\(957\) −0.448920 −0.0145115
\(958\) −5.82877 −0.188319
\(959\) −11.7079 −0.378067
\(960\) −2.74714 −0.0886635
\(961\) 26.2699 0.847415
\(962\) 0.373450 0.0120405
\(963\) −17.1864 −0.553825
\(964\) −27.2863 −0.878834
\(965\) −71.4045 −2.29859
\(966\) −3.18093 −0.102345
\(967\) 19.1336 0.615293 0.307647 0.951501i \(-0.400459\pi\)
0.307647 + 0.951501i \(0.400459\pi\)
\(968\) 10.2589 0.329735
\(969\) 11.1978 0.359725
\(970\) −22.3624 −0.718014
\(971\) 13.9506 0.447697 0.223849 0.974624i \(-0.428138\pi\)
0.223849 + 0.974624i \(0.428138\pi\)
\(972\) 1.00000 0.0320750
\(973\) 6.39481 0.205008
\(974\) 7.63754 0.244722
\(975\) 2.54676 0.0815616
\(976\) 4.99993 0.160044
\(977\) 20.4548 0.654406 0.327203 0.944954i \(-0.393894\pi\)
0.327203 + 0.944954i \(0.393894\pi\)
\(978\) 18.6770 0.597224
\(979\) 3.96486 0.126717
\(980\) −12.3945 −0.395928
\(981\) 4.88430 0.155944
\(982\) −10.2639 −0.327534
\(983\) −38.5517 −1.22961 −0.614805 0.788679i \(-0.710765\pi\)
−0.614805 + 0.788679i \(0.710765\pi\)
\(984\) 5.16599 0.164686
\(985\) −71.5878 −2.28098
\(986\) 2.16048 0.0688037
\(987\) 12.6717 0.403343
\(988\) −2.70287 −0.0859898
\(989\) −8.58603 −0.273020
\(990\) −2.36487 −0.0751604
\(991\) −31.7584 −1.00884 −0.504419 0.863459i \(-0.668293\pi\)
−0.504419 + 0.863459i \(0.668293\pi\)
\(992\) −7.56769 −0.240274
\(993\) 32.8982 1.04399
\(994\) 30.6365 0.971730
\(995\) 34.0961 1.08092
\(996\) 15.5907 0.494009
\(997\) −15.8176 −0.500947 −0.250474 0.968123i \(-0.580586\pi\)
−0.250474 + 0.968123i \(0.580586\pi\)
\(998\) −18.1451 −0.574374
\(999\) −0.373450 −0.0118154
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.w.1.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.w.1.3 12 1.1 even 1 trivial