Properties

Label 8034.2.a.bb.1.4
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $14$
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: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 6 x^{13} - 29 x^{12} + 207 x^{11} + 269 x^{10} - 2601 x^{9} - 847 x^{8} + 14851 x^{7} + 678 x^{6} - 39390 x^{5} - 3280 x^{4} + 42456 x^{3} + 10816 x^{2} - 7296 x - 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.57230\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.57230 q^{5} -1.00000 q^{6} -4.81484 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.57230 q^{5} -1.00000 q^{6} -4.81484 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.57230 q^{10} +1.68147 q^{11} +1.00000 q^{12} -1.00000 q^{13} +4.81484 q^{14} -2.57230 q^{15} +1.00000 q^{16} +6.18217 q^{17} -1.00000 q^{18} +0.771079 q^{19} -2.57230 q^{20} -4.81484 q^{21} -1.68147 q^{22} -2.46234 q^{23} -1.00000 q^{24} +1.61675 q^{25} +1.00000 q^{26} +1.00000 q^{27} -4.81484 q^{28} -4.09908 q^{29} +2.57230 q^{30} -7.36712 q^{31} -1.00000 q^{32} +1.68147 q^{33} -6.18217 q^{34} +12.3852 q^{35} +1.00000 q^{36} +7.10946 q^{37} -0.771079 q^{38} -1.00000 q^{39} +2.57230 q^{40} +11.9672 q^{41} +4.81484 q^{42} +4.87285 q^{43} +1.68147 q^{44} -2.57230 q^{45} +2.46234 q^{46} -7.45264 q^{47} +1.00000 q^{48} +16.1827 q^{49} -1.61675 q^{50} +6.18217 q^{51} -1.00000 q^{52} +4.54055 q^{53} -1.00000 q^{54} -4.32526 q^{55} +4.81484 q^{56} +0.771079 q^{57} +4.09908 q^{58} -5.91864 q^{59} -2.57230 q^{60} -0.0881773 q^{61} +7.36712 q^{62} -4.81484 q^{63} +1.00000 q^{64} +2.57230 q^{65} -1.68147 q^{66} -9.81599 q^{67} +6.18217 q^{68} -2.46234 q^{69} -12.3852 q^{70} -6.15722 q^{71} -1.00000 q^{72} +16.5077 q^{73} -7.10946 q^{74} +1.61675 q^{75} +0.771079 q^{76} -8.09602 q^{77} +1.00000 q^{78} -3.92812 q^{79} -2.57230 q^{80} +1.00000 q^{81} -11.9672 q^{82} -1.22361 q^{83} -4.81484 q^{84} -15.9024 q^{85} -4.87285 q^{86} -4.09908 q^{87} -1.68147 q^{88} +6.45570 q^{89} +2.57230 q^{90} +4.81484 q^{91} -2.46234 q^{92} -7.36712 q^{93} +7.45264 q^{94} -1.98345 q^{95} -1.00000 q^{96} -8.09177 q^{97} -16.1827 q^{98} +1.68147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + 6q^{10} - 8q^{11} + 14q^{12} - 14q^{13} + 4q^{14} - 6q^{15} + 14q^{16} - 4q^{17} - 14q^{18} - q^{19} - 6q^{20} - 4q^{21} + 8q^{22} - 9q^{23} - 14q^{24} + 24q^{25} + 14q^{26} + 14q^{27} - 4q^{28} - 10q^{29} + 6q^{30} - 5q^{31} - 14q^{32} - 8q^{33} + 4q^{34} - 16q^{35} + 14q^{36} - 4q^{37} + q^{38} - 14q^{39} + 6q^{40} - 24q^{41} + 4q^{42} - 8q^{44} - 6q^{45} + 9q^{46} - 32q^{47} + 14q^{48} + 24q^{49} - 24q^{50} - 4q^{51} - 14q^{52} - 5q^{53} - 14q^{54} - 8q^{55} + 4q^{56} - q^{57} + 10q^{58} - 13q^{59} - 6q^{60} + 2q^{61} + 5q^{62} - 4q^{63} + 14q^{64} + 6q^{65} + 8q^{66} - 16q^{67} - 4q^{68} - 9q^{69} + 16q^{70} - 29q^{71} - 14q^{72} + 4q^{74} + 24q^{75} - q^{76} - 9q^{77} + 14q^{78} - 21q^{79} - 6q^{80} + 14q^{81} + 24q^{82} - 40q^{83} - 4q^{84} - 7q^{85} - 10q^{87} + 8q^{88} - 48q^{89} + 6q^{90} + 4q^{91} - 9q^{92} - 5q^{93} + 32q^{94} - 26q^{95} - 14q^{96} + 18q^{97} - 24q^{98} - 8q^{99} + O(q^{100}) \)

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
\(5\) −2.57230 −1.15037 −0.575185 0.818024i \(-0.695070\pi\)
−0.575185 + 0.818024i \(0.695070\pi\)
\(6\) −1.00000 −0.408248
\(7\) −4.81484 −1.81984 −0.909919 0.414786i \(-0.863857\pi\)
−0.909919 + 0.414786i \(0.863857\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.57230 0.813434
\(11\) 1.68147 0.506983 0.253491 0.967338i \(-0.418421\pi\)
0.253491 + 0.967338i \(0.418421\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 4.81484 1.28682
\(15\) −2.57230 −0.664166
\(16\) 1.00000 0.250000
\(17\) 6.18217 1.49940 0.749698 0.661780i \(-0.230199\pi\)
0.749698 + 0.661780i \(0.230199\pi\)
\(18\) −1.00000 −0.235702
\(19\) 0.771079 0.176898 0.0884488 0.996081i \(-0.471809\pi\)
0.0884488 + 0.996081i \(0.471809\pi\)
\(20\) −2.57230 −0.575185
\(21\) −4.81484 −1.05068
\(22\) −1.68147 −0.358491
\(23\) −2.46234 −0.513432 −0.256716 0.966487i \(-0.582641\pi\)
−0.256716 + 0.966487i \(0.582641\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.61675 0.323350
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) −4.81484 −0.909919
\(29\) −4.09908 −0.761179 −0.380590 0.924744i \(-0.624279\pi\)
−0.380590 + 0.924744i \(0.624279\pi\)
\(30\) 2.57230 0.469636
\(31\) −7.36712 −1.32317 −0.661587 0.749868i \(-0.730117\pi\)
−0.661587 + 0.749868i \(0.730117\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.68147 0.292707
\(34\) −6.18217 −1.06023
\(35\) 12.3852 2.09349
\(36\) 1.00000 0.166667
\(37\) 7.10946 1.16879 0.584394 0.811470i \(-0.301332\pi\)
0.584394 + 0.811470i \(0.301332\pi\)
\(38\) −0.771079 −0.125085
\(39\) −1.00000 −0.160128
\(40\) 2.57230 0.406717
\(41\) 11.9672 1.86897 0.934484 0.356005i \(-0.115861\pi\)
0.934484 + 0.356005i \(0.115861\pi\)
\(42\) 4.81484 0.742946
\(43\) 4.87285 0.743103 0.371551 0.928412i \(-0.378826\pi\)
0.371551 + 0.928412i \(0.378826\pi\)
\(44\) 1.68147 0.253491
\(45\) −2.57230 −0.383456
\(46\) 2.46234 0.363052
\(47\) −7.45264 −1.08708 −0.543539 0.839384i \(-0.682916\pi\)
−0.543539 + 0.839384i \(0.682916\pi\)
\(48\) 1.00000 0.144338
\(49\) 16.1827 2.31181
\(50\) −1.61675 −0.228643
\(51\) 6.18217 0.865677
\(52\) −1.00000 −0.138675
\(53\) 4.54055 0.623692 0.311846 0.950133i \(-0.399053\pi\)
0.311846 + 0.950133i \(0.399053\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.32526 −0.583217
\(56\) 4.81484 0.643410
\(57\) 0.771079 0.102132
\(58\) 4.09908 0.538235
\(59\) −5.91864 −0.770541 −0.385271 0.922804i \(-0.625892\pi\)
−0.385271 + 0.922804i \(0.625892\pi\)
\(60\) −2.57230 −0.332083
\(61\) −0.0881773 −0.0112900 −0.00564498 0.999984i \(-0.501797\pi\)
−0.00564498 + 0.999984i \(0.501797\pi\)
\(62\) 7.36712 0.935626
\(63\) −4.81484 −0.606613
\(64\) 1.00000 0.125000
\(65\) 2.57230 0.319055
\(66\) −1.68147 −0.206975
\(67\) −9.81599 −1.19921 −0.599607 0.800295i \(-0.704676\pi\)
−0.599607 + 0.800295i \(0.704676\pi\)
\(68\) 6.18217 0.749698
\(69\) −2.46234 −0.296430
\(70\) −12.3852 −1.48032
\(71\) −6.15722 −0.730727 −0.365364 0.930865i \(-0.619055\pi\)
−0.365364 + 0.930865i \(0.619055\pi\)
\(72\) −1.00000 −0.117851
\(73\) 16.5077 1.93207 0.966037 0.258402i \(-0.0831959\pi\)
0.966037 + 0.258402i \(0.0831959\pi\)
\(74\) −7.10946 −0.826458
\(75\) 1.61675 0.186686
\(76\) 0.771079 0.0884488
\(77\) −8.09602 −0.922627
\(78\) 1.00000 0.113228
\(79\) −3.92812 −0.441948 −0.220974 0.975280i \(-0.570924\pi\)
−0.220974 + 0.975280i \(0.570924\pi\)
\(80\) −2.57230 −0.287592
\(81\) 1.00000 0.111111
\(82\) −11.9672 −1.32156
\(83\) −1.22361 −0.134309 −0.0671545 0.997743i \(-0.521392\pi\)
−0.0671545 + 0.997743i \(0.521392\pi\)
\(84\) −4.81484 −0.525342
\(85\) −15.9024 −1.72486
\(86\) −4.87285 −0.525453
\(87\) −4.09908 −0.439467
\(88\) −1.68147 −0.179245
\(89\) 6.45570 0.684303 0.342152 0.939645i \(-0.388844\pi\)
0.342152 + 0.939645i \(0.388844\pi\)
\(90\) 2.57230 0.271145
\(91\) 4.81484 0.504732
\(92\) −2.46234 −0.256716
\(93\) −7.36712 −0.763935
\(94\) 7.45264 0.768681
\(95\) −1.98345 −0.203498
\(96\) −1.00000 −0.102062
\(97\) −8.09177 −0.821595 −0.410797 0.911727i \(-0.634750\pi\)
−0.410797 + 0.911727i \(0.634750\pi\)
\(98\) −16.1827 −1.63470
\(99\) 1.68147 0.168994
\(100\) 1.61675 0.161675
\(101\) 6.68754 0.665435 0.332718 0.943027i \(-0.392034\pi\)
0.332718 + 0.943027i \(0.392034\pi\)
\(102\) −6.18217 −0.612126
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) 12.3852 1.20867
\(106\) −4.54055 −0.441017
\(107\) 9.93787 0.960730 0.480365 0.877069i \(-0.340504\pi\)
0.480365 + 0.877069i \(0.340504\pi\)
\(108\) 1.00000 0.0962250
\(109\) 10.9229 1.04623 0.523113 0.852263i \(-0.324771\pi\)
0.523113 + 0.852263i \(0.324771\pi\)
\(110\) 4.32526 0.412397
\(111\) 7.10946 0.674800
\(112\) −4.81484 −0.454960
\(113\) 2.41897 0.227557 0.113779 0.993506i \(-0.463705\pi\)
0.113779 + 0.993506i \(0.463705\pi\)
\(114\) −0.771079 −0.0722181
\(115\) 6.33388 0.590637
\(116\) −4.09908 −0.380590
\(117\) −1.00000 −0.0924500
\(118\) 5.91864 0.544855
\(119\) −29.7661 −2.72866
\(120\) 2.57230 0.234818
\(121\) −8.17265 −0.742969
\(122\) 0.0881773 0.00798320
\(123\) 11.9672 1.07905
\(124\) −7.36712 −0.661587
\(125\) 8.70275 0.778398
\(126\) 4.81484 0.428940
\(127\) 18.6336 1.65346 0.826731 0.562598i \(-0.190198\pi\)
0.826731 + 0.562598i \(0.190198\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.87285 0.429031
\(130\) −2.57230 −0.225606
\(131\) −15.8374 −1.38372 −0.691861 0.722030i \(-0.743209\pi\)
−0.691861 + 0.722030i \(0.743209\pi\)
\(132\) 1.68147 0.146353
\(133\) −3.71262 −0.321925
\(134\) 9.81599 0.847973
\(135\) −2.57230 −0.221389
\(136\) −6.18217 −0.530116
\(137\) −2.03436 −0.173807 −0.0869034 0.996217i \(-0.527697\pi\)
−0.0869034 + 0.996217i \(0.527697\pi\)
\(138\) 2.46234 0.209608
\(139\) 1.34700 0.114251 0.0571254 0.998367i \(-0.481807\pi\)
0.0571254 + 0.998367i \(0.481807\pi\)
\(140\) 12.3852 1.04674
\(141\) −7.45264 −0.627625
\(142\) 6.15722 0.516702
\(143\) −1.68147 −0.140612
\(144\) 1.00000 0.0833333
\(145\) 10.5441 0.875638
\(146\) −16.5077 −1.36618
\(147\) 16.1827 1.33472
\(148\) 7.10946 0.584394
\(149\) −16.3859 −1.34239 −0.671194 0.741282i \(-0.734218\pi\)
−0.671194 + 0.741282i \(0.734218\pi\)
\(150\) −1.61675 −0.132007
\(151\) 19.8000 1.61130 0.805650 0.592392i \(-0.201816\pi\)
0.805650 + 0.592392i \(0.201816\pi\)
\(152\) −0.771079 −0.0625427
\(153\) 6.18217 0.499799
\(154\) 8.09602 0.652396
\(155\) 18.9505 1.52214
\(156\) −1.00000 −0.0800641
\(157\) 16.4161 1.31015 0.655073 0.755566i \(-0.272638\pi\)
0.655073 + 0.755566i \(0.272638\pi\)
\(158\) 3.92812 0.312504
\(159\) 4.54055 0.360089
\(160\) 2.57230 0.203359
\(161\) 11.8557 0.934364
\(162\) −1.00000 −0.0785674
\(163\) −17.7158 −1.38761 −0.693803 0.720165i \(-0.744066\pi\)
−0.693803 + 0.720165i \(0.744066\pi\)
\(164\) 11.9672 0.934484
\(165\) −4.32526 −0.336721
\(166\) 1.22361 0.0949708
\(167\) −15.7136 −1.21595 −0.607977 0.793954i \(-0.708019\pi\)
−0.607977 + 0.793954i \(0.708019\pi\)
\(168\) 4.81484 0.371473
\(169\) 1.00000 0.0769231
\(170\) 15.9024 1.21966
\(171\) 0.771079 0.0589659
\(172\) 4.87285 0.371551
\(173\) 8.65865 0.658305 0.329153 0.944277i \(-0.393237\pi\)
0.329153 + 0.944277i \(0.393237\pi\)
\(174\) 4.09908 0.310750
\(175\) −7.78439 −0.588444
\(176\) 1.68147 0.126746
\(177\) −5.91864 −0.444872
\(178\) −6.45570 −0.483875
\(179\) −16.8361 −1.25839 −0.629196 0.777247i \(-0.716616\pi\)
−0.629196 + 0.777247i \(0.716616\pi\)
\(180\) −2.57230 −0.191728
\(181\) 0.312325 0.0232149 0.0116075 0.999933i \(-0.496305\pi\)
0.0116075 + 0.999933i \(0.496305\pi\)
\(182\) −4.81484 −0.356900
\(183\) −0.0881773 −0.00651826
\(184\) 2.46234 0.181526
\(185\) −18.2877 −1.34454
\(186\) 7.36712 0.540184
\(187\) 10.3951 0.760168
\(188\) −7.45264 −0.543539
\(189\) −4.81484 −0.350228
\(190\) 1.98345 0.143895
\(191\) 12.0110 0.869084 0.434542 0.900652i \(-0.356910\pi\)
0.434542 + 0.900652i \(0.356910\pi\)
\(192\) 1.00000 0.0721688
\(193\) −12.2193 −0.879564 −0.439782 0.898104i \(-0.644944\pi\)
−0.439782 + 0.898104i \(0.644944\pi\)
\(194\) 8.09177 0.580955
\(195\) 2.57230 0.184207
\(196\) 16.1827 1.15591
\(197\) 5.75541 0.410056 0.205028 0.978756i \(-0.434271\pi\)
0.205028 + 0.978756i \(0.434271\pi\)
\(198\) −1.68147 −0.119497
\(199\) −24.5893 −1.74309 −0.871544 0.490317i \(-0.836881\pi\)
−0.871544 + 0.490317i \(0.836881\pi\)
\(200\) −1.61675 −0.114321
\(201\) −9.81599 −0.692367
\(202\) −6.68754 −0.470534
\(203\) 19.7364 1.38522
\(204\) 6.18217 0.432838
\(205\) −30.7834 −2.15000
\(206\) −1.00000 −0.0696733
\(207\) −2.46234 −0.171144
\(208\) −1.00000 −0.0693375
\(209\) 1.29655 0.0896840
\(210\) −12.3852 −0.854662
\(211\) 7.63512 0.525623 0.262812 0.964847i \(-0.415350\pi\)
0.262812 + 0.964847i \(0.415350\pi\)
\(212\) 4.54055 0.311846
\(213\) −6.15722 −0.421886
\(214\) −9.93787 −0.679339
\(215\) −12.5345 −0.854843
\(216\) −1.00000 −0.0680414
\(217\) 35.4715 2.40796
\(218\) −10.9229 −0.739793
\(219\) 16.5077 1.11548
\(220\) −4.32526 −0.291609
\(221\) −6.18217 −0.415858
\(222\) −7.10946 −0.477156
\(223\) 3.09337 0.207148 0.103574 0.994622i \(-0.466972\pi\)
0.103574 + 0.994622i \(0.466972\pi\)
\(224\) 4.81484 0.321705
\(225\) 1.61675 0.107783
\(226\) −2.41897 −0.160907
\(227\) −24.1516 −1.60300 −0.801499 0.597996i \(-0.795964\pi\)
−0.801499 + 0.597996i \(0.795964\pi\)
\(228\) 0.771079 0.0510659
\(229\) −9.02543 −0.596417 −0.298209 0.954501i \(-0.596389\pi\)
−0.298209 + 0.954501i \(0.596389\pi\)
\(230\) −6.33388 −0.417643
\(231\) −8.09602 −0.532679
\(232\) 4.09908 0.269118
\(233\) −19.8825 −1.30255 −0.651274 0.758843i \(-0.725765\pi\)
−0.651274 + 0.758843i \(0.725765\pi\)
\(234\) 1.00000 0.0653720
\(235\) 19.1704 1.25054
\(236\) −5.91864 −0.385271
\(237\) −3.92812 −0.255159
\(238\) 29.7661 1.92945
\(239\) −21.7455 −1.40660 −0.703301 0.710893i \(-0.748291\pi\)
−0.703301 + 0.710893i \(0.748291\pi\)
\(240\) −2.57230 −0.166042
\(241\) −29.9056 −1.92639 −0.963195 0.268805i \(-0.913371\pi\)
−0.963195 + 0.268805i \(0.913371\pi\)
\(242\) 8.17265 0.525358
\(243\) 1.00000 0.0641500
\(244\) −0.0881773 −0.00564498
\(245\) −41.6268 −2.65944
\(246\) −11.9672 −0.763003
\(247\) −0.771079 −0.0490626
\(248\) 7.36712 0.467813
\(249\) −1.22361 −0.0775433
\(250\) −8.70275 −0.550410
\(251\) −10.3576 −0.653769 −0.326884 0.945064i \(-0.605999\pi\)
−0.326884 + 0.945064i \(0.605999\pi\)
\(252\) −4.81484 −0.303306
\(253\) −4.14035 −0.260301
\(254\) −18.6336 −1.16917
\(255\) −15.9024 −0.995848
\(256\) 1.00000 0.0625000
\(257\) −2.58228 −0.161078 −0.0805392 0.996751i \(-0.525664\pi\)
−0.0805392 + 0.996751i \(0.525664\pi\)
\(258\) −4.87285 −0.303370
\(259\) −34.2309 −2.12700
\(260\) 2.57230 0.159528
\(261\) −4.09908 −0.253726
\(262\) 15.8374 0.978440
\(263\) −19.9903 −1.23265 −0.616326 0.787491i \(-0.711380\pi\)
−0.616326 + 0.787491i \(0.711380\pi\)
\(264\) −1.68147 −0.103487
\(265\) −11.6797 −0.717476
\(266\) 3.71262 0.227635
\(267\) 6.45570 0.395083
\(268\) −9.81599 −0.599607
\(269\) −1.69478 −0.103333 −0.0516663 0.998664i \(-0.516453\pi\)
−0.0516663 + 0.998664i \(0.516453\pi\)
\(270\) 2.57230 0.156545
\(271\) 24.7894 1.50585 0.752924 0.658107i \(-0.228643\pi\)
0.752924 + 0.658107i \(0.228643\pi\)
\(272\) 6.18217 0.374849
\(273\) 4.81484 0.291407
\(274\) 2.03436 0.122900
\(275\) 2.71852 0.163933
\(276\) −2.46234 −0.148215
\(277\) −3.08089 −0.185113 −0.0925565 0.995707i \(-0.529504\pi\)
−0.0925565 + 0.995707i \(0.529504\pi\)
\(278\) −1.34700 −0.0807875
\(279\) −7.36712 −0.441058
\(280\) −12.3852 −0.740159
\(281\) −3.29557 −0.196597 −0.0982986 0.995157i \(-0.531340\pi\)
−0.0982986 + 0.995157i \(0.531340\pi\)
\(282\) 7.45264 0.443798
\(283\) 24.9865 1.48529 0.742646 0.669684i \(-0.233570\pi\)
0.742646 + 0.669684i \(0.233570\pi\)
\(284\) −6.15722 −0.365364
\(285\) −1.98345 −0.117489
\(286\) 1.68147 0.0994275
\(287\) −57.6203 −3.40122
\(288\) −1.00000 −0.0589256
\(289\) 21.2192 1.24819
\(290\) −10.5441 −0.619169
\(291\) −8.09177 −0.474348
\(292\) 16.5077 0.966037
\(293\) −9.62707 −0.562419 −0.281210 0.959646i \(-0.590736\pi\)
−0.281210 + 0.959646i \(0.590736\pi\)
\(294\) −16.1827 −0.943793
\(295\) 15.2245 0.886407
\(296\) −7.10946 −0.413229
\(297\) 1.68147 0.0975689
\(298\) 16.3859 0.949211
\(299\) 2.46234 0.142401
\(300\) 1.61675 0.0933431
\(301\) −23.4620 −1.35233
\(302\) −19.8000 −1.13936
\(303\) 6.68754 0.384189
\(304\) 0.771079 0.0442244
\(305\) 0.226819 0.0129876
\(306\) −6.18217 −0.353411
\(307\) −20.2738 −1.15709 −0.578545 0.815651i \(-0.696379\pi\)
−0.578545 + 0.815651i \(0.696379\pi\)
\(308\) −8.09602 −0.461313
\(309\) 1.00000 0.0568880
\(310\) −18.9505 −1.07632
\(311\) −32.4947 −1.84260 −0.921302 0.388847i \(-0.872873\pi\)
−0.921302 + 0.388847i \(0.872873\pi\)
\(312\) 1.00000 0.0566139
\(313\) 7.50820 0.424389 0.212194 0.977227i \(-0.431939\pi\)
0.212194 + 0.977227i \(0.431939\pi\)
\(314\) −16.4161 −0.926413
\(315\) 12.3852 0.697829
\(316\) −3.92812 −0.220974
\(317\) −2.86805 −0.161086 −0.0805429 0.996751i \(-0.525665\pi\)
−0.0805429 + 0.996751i \(0.525665\pi\)
\(318\) −4.54055 −0.254621
\(319\) −6.89248 −0.385905
\(320\) −2.57230 −0.143796
\(321\) 9.93787 0.554678
\(322\) −11.8557 −0.660695
\(323\) 4.76694 0.265240
\(324\) 1.00000 0.0555556
\(325\) −1.61675 −0.0896811
\(326\) 17.7158 0.981186
\(327\) 10.9229 0.604039
\(328\) −11.9672 −0.660780
\(329\) 35.8832 1.97831
\(330\) 4.32526 0.238098
\(331\) 26.5172 1.45752 0.728758 0.684772i \(-0.240098\pi\)
0.728758 + 0.684772i \(0.240098\pi\)
\(332\) −1.22361 −0.0671545
\(333\) 7.10946 0.389596
\(334\) 15.7136 0.859810
\(335\) 25.2497 1.37954
\(336\) −4.81484 −0.262671
\(337\) −14.6616 −0.798668 −0.399334 0.916806i \(-0.630759\pi\)
−0.399334 + 0.916806i \(0.630759\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 2.41897 0.131380
\(340\) −15.9024 −0.862430
\(341\) −12.3876 −0.670827
\(342\) −0.771079 −0.0416952
\(343\) −44.2131 −2.38728
\(344\) −4.87285 −0.262727
\(345\) 6.33388 0.341004
\(346\) −8.65865 −0.465492
\(347\) 2.56359 0.137621 0.0688103 0.997630i \(-0.478080\pi\)
0.0688103 + 0.997630i \(0.478080\pi\)
\(348\) −4.09908 −0.219734
\(349\) −24.6663 −1.32035 −0.660177 0.751110i \(-0.729519\pi\)
−0.660177 + 0.751110i \(0.729519\pi\)
\(350\) 7.78439 0.416093
\(351\) −1.00000 −0.0533761
\(352\) −1.68147 −0.0896227
\(353\) 11.7074 0.623122 0.311561 0.950226i \(-0.399148\pi\)
0.311561 + 0.950226i \(0.399148\pi\)
\(354\) 5.91864 0.314572
\(355\) 15.8382 0.840607
\(356\) 6.45570 0.342152
\(357\) −29.7661 −1.57539
\(358\) 16.8361 0.889817
\(359\) −2.53080 −0.133570 −0.0667851 0.997767i \(-0.521274\pi\)
−0.0667851 + 0.997767i \(0.521274\pi\)
\(360\) 2.57230 0.135572
\(361\) −18.4054 −0.968707
\(362\) −0.312325 −0.0164154
\(363\) −8.17265 −0.428953
\(364\) 4.81484 0.252366
\(365\) −42.4627 −2.22260
\(366\) 0.0881773 0.00460910
\(367\) 5.13485 0.268037 0.134018 0.990979i \(-0.457212\pi\)
0.134018 + 0.990979i \(0.457212\pi\)
\(368\) −2.46234 −0.128358
\(369\) 11.9672 0.622989
\(370\) 18.2877 0.950732
\(371\) −21.8620 −1.13502
\(372\) −7.36712 −0.381968
\(373\) −33.0765 −1.71263 −0.856317 0.516450i \(-0.827253\pi\)
−0.856317 + 0.516450i \(0.827253\pi\)
\(374\) −10.3951 −0.537520
\(375\) 8.70275 0.449408
\(376\) 7.45264 0.384340
\(377\) 4.09908 0.211113
\(378\) 4.81484 0.247649
\(379\) −15.1628 −0.778859 −0.389430 0.921056i \(-0.627328\pi\)
−0.389430 + 0.921056i \(0.627328\pi\)
\(380\) −1.98345 −0.101749
\(381\) 18.6336 0.954627
\(382\) −12.0110 −0.614535
\(383\) −31.7073 −1.62017 −0.810085 0.586313i \(-0.800579\pi\)
−0.810085 + 0.586313i \(0.800579\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 20.8254 1.06136
\(386\) 12.2193 0.621946
\(387\) 4.87285 0.247701
\(388\) −8.09177 −0.410797
\(389\) −23.6651 −1.19987 −0.599934 0.800050i \(-0.704806\pi\)
−0.599934 + 0.800050i \(0.704806\pi\)
\(390\) −2.57230 −0.130254
\(391\) −15.2226 −0.769838
\(392\) −16.1827 −0.817349
\(393\) −15.8374 −0.798893
\(394\) −5.75541 −0.289953
\(395\) 10.1043 0.508403
\(396\) 1.68147 0.0844971
\(397\) 25.1755 1.26352 0.631762 0.775163i \(-0.282332\pi\)
0.631762 + 0.775163i \(0.282332\pi\)
\(398\) 24.5893 1.23255
\(399\) −3.71262 −0.185863
\(400\) 1.61675 0.0808375
\(401\) −15.0419 −0.751158 −0.375579 0.926790i \(-0.622556\pi\)
−0.375579 + 0.926790i \(0.622556\pi\)
\(402\) 9.81599 0.489577
\(403\) 7.36712 0.366983
\(404\) 6.68754 0.332718
\(405\) −2.57230 −0.127819
\(406\) −19.7364 −0.979501
\(407\) 11.9544 0.592555
\(408\) −6.18217 −0.306063
\(409\) 20.1039 0.994072 0.497036 0.867730i \(-0.334422\pi\)
0.497036 + 0.867730i \(0.334422\pi\)
\(410\) 30.7834 1.52028
\(411\) −2.03436 −0.100347
\(412\) 1.00000 0.0492665
\(413\) 28.4973 1.40226
\(414\) 2.46234 0.121017
\(415\) 3.14750 0.154505
\(416\) 1.00000 0.0490290
\(417\) 1.34700 0.0659627
\(418\) −1.29655 −0.0634162
\(419\) −34.5445 −1.68761 −0.843805 0.536650i \(-0.819690\pi\)
−0.843805 + 0.536650i \(0.819690\pi\)
\(420\) 12.3852 0.604337
\(421\) 4.53726 0.221132 0.110566 0.993869i \(-0.464734\pi\)
0.110566 + 0.993869i \(0.464734\pi\)
\(422\) −7.63512 −0.371672
\(423\) −7.45264 −0.362360
\(424\) −4.54055 −0.220508
\(425\) 9.99501 0.484829
\(426\) 6.15722 0.298318
\(427\) 0.424560 0.0205459
\(428\) 9.93787 0.480365
\(429\) −1.68147 −0.0811822
\(430\) 12.5345 0.604465
\(431\) 8.15170 0.392653 0.196327 0.980539i \(-0.437099\pi\)
0.196327 + 0.980539i \(0.437099\pi\)
\(432\) 1.00000 0.0481125
\(433\) −24.1238 −1.15931 −0.579657 0.814860i \(-0.696814\pi\)
−0.579657 + 0.814860i \(0.696814\pi\)
\(434\) −35.4715 −1.70269
\(435\) 10.5441 0.505550
\(436\) 10.9229 0.523113
\(437\) −1.89865 −0.0908249
\(438\) −16.5077 −0.788766
\(439\) 34.3120 1.63762 0.818810 0.574065i \(-0.194634\pi\)
0.818810 + 0.574065i \(0.194634\pi\)
\(440\) 4.32526 0.206198
\(441\) 16.1827 0.770604
\(442\) 6.18217 0.294056
\(443\) 13.8306 0.657112 0.328556 0.944484i \(-0.393438\pi\)
0.328556 + 0.944484i \(0.393438\pi\)
\(444\) 7.10946 0.337400
\(445\) −16.6060 −0.787201
\(446\) −3.09337 −0.146476
\(447\) −16.3859 −0.775028
\(448\) −4.81484 −0.227480
\(449\) 1.18499 0.0559229 0.0279615 0.999609i \(-0.491098\pi\)
0.0279615 + 0.999609i \(0.491098\pi\)
\(450\) −1.61675 −0.0762143
\(451\) 20.1226 0.947535
\(452\) 2.41897 0.113779
\(453\) 19.8000 0.930284
\(454\) 24.1516 1.13349
\(455\) −12.3852 −0.580629
\(456\) −0.771079 −0.0361091
\(457\) 10.8635 0.508171 0.254086 0.967182i \(-0.418225\pi\)
0.254086 + 0.967182i \(0.418225\pi\)
\(458\) 9.02543 0.421731
\(459\) 6.18217 0.288559
\(460\) 6.33388 0.295318
\(461\) −22.7229 −1.05831 −0.529155 0.848525i \(-0.677491\pi\)
−0.529155 + 0.848525i \(0.677491\pi\)
\(462\) 8.09602 0.376661
\(463\) −24.7743 −1.15136 −0.575679 0.817676i \(-0.695262\pi\)
−0.575679 + 0.817676i \(0.695262\pi\)
\(464\) −4.09908 −0.190295
\(465\) 18.9505 0.878808
\(466\) 19.8825 0.921041
\(467\) 10.2797 0.475686 0.237843 0.971304i \(-0.423560\pi\)
0.237843 + 0.971304i \(0.423560\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 47.2624 2.18238
\(470\) −19.1704 −0.884267
\(471\) 16.4161 0.756413
\(472\) 5.91864 0.272427
\(473\) 8.19356 0.376740
\(474\) 3.92812 0.180424
\(475\) 1.24664 0.0571998
\(476\) −29.7661 −1.36433
\(477\) 4.54055 0.207897
\(478\) 21.7455 0.994617
\(479\) −17.4984 −0.799522 −0.399761 0.916619i \(-0.630907\pi\)
−0.399761 + 0.916619i \(0.630907\pi\)
\(480\) 2.57230 0.117409
\(481\) −7.10946 −0.324163
\(482\) 29.9056 1.36216
\(483\) 11.8557 0.539455
\(484\) −8.17265 −0.371484
\(485\) 20.8145 0.945137
\(486\) −1.00000 −0.0453609
\(487\) 31.5809 1.43107 0.715534 0.698578i \(-0.246184\pi\)
0.715534 + 0.698578i \(0.246184\pi\)
\(488\) 0.0881773 0.00399160
\(489\) −17.7158 −0.801135
\(490\) 41.6268 1.88051
\(491\) 15.1300 0.682808 0.341404 0.939917i \(-0.389098\pi\)
0.341404 + 0.939917i \(0.389098\pi\)
\(492\) 11.9672 0.539525
\(493\) −25.3412 −1.14131
\(494\) 0.771079 0.0346925
\(495\) −4.32526 −0.194406
\(496\) −7.36712 −0.330794
\(497\) 29.6460 1.32981
\(498\) 1.22361 0.0548314
\(499\) 6.31066 0.282504 0.141252 0.989974i \(-0.454887\pi\)
0.141252 + 0.989974i \(0.454887\pi\)
\(500\) 8.70275 0.389199
\(501\) −15.7136 −0.702032
\(502\) 10.3576 0.462284
\(503\) −23.6215 −1.05323 −0.526615 0.850104i \(-0.676539\pi\)
−0.526615 + 0.850104i \(0.676539\pi\)
\(504\) 4.81484 0.214470
\(505\) −17.2024 −0.765496
\(506\) 4.14035 0.184061
\(507\) 1.00000 0.0444116
\(508\) 18.6336 0.826731
\(509\) −4.56360 −0.202278 −0.101139 0.994872i \(-0.532249\pi\)
−0.101139 + 0.994872i \(0.532249\pi\)
\(510\) 15.9024 0.704171
\(511\) −79.4817 −3.51606
\(512\) −1.00000 −0.0441942
\(513\) 0.771079 0.0340440
\(514\) 2.58228 0.113900
\(515\) −2.57230 −0.113349
\(516\) 4.87285 0.214515
\(517\) −12.5314 −0.551130
\(518\) 34.2309 1.50402
\(519\) 8.65865 0.380073
\(520\) −2.57230 −0.112803
\(521\) −8.52013 −0.373274 −0.186637 0.982429i \(-0.559759\pi\)
−0.186637 + 0.982429i \(0.559759\pi\)
\(522\) 4.09908 0.179412
\(523\) −21.2892 −0.930913 −0.465457 0.885071i \(-0.654110\pi\)
−0.465457 + 0.885071i \(0.654110\pi\)
\(524\) −15.8374 −0.691861
\(525\) −7.78439 −0.339739
\(526\) 19.9903 0.871617
\(527\) −45.5448 −1.98396
\(528\) 1.68147 0.0731767
\(529\) −16.9369 −0.736387
\(530\) 11.6797 0.507332
\(531\) −5.91864 −0.256847
\(532\) −3.71262 −0.160963
\(533\) −11.9672 −0.518359
\(534\) −6.45570 −0.279366
\(535\) −25.5632 −1.10519
\(536\) 9.81599 0.423986
\(537\) −16.8361 −0.726533
\(538\) 1.69478 0.0730672
\(539\) 27.2107 1.17205
\(540\) −2.57230 −0.110694
\(541\) −4.95665 −0.213103 −0.106552 0.994307i \(-0.533981\pi\)
−0.106552 + 0.994307i \(0.533981\pi\)
\(542\) −24.7894 −1.06480
\(543\) 0.312325 0.0134031
\(544\) −6.18217 −0.265058
\(545\) −28.0971 −1.20355
\(546\) −4.81484 −0.206056
\(547\) 20.1699 0.862404 0.431202 0.902255i \(-0.358090\pi\)
0.431202 + 0.902255i \(0.358090\pi\)
\(548\) −2.03436 −0.0869034
\(549\) −0.0881773 −0.00376332
\(550\) −2.71852 −0.115918
\(551\) −3.16071 −0.134651
\(552\) 2.46234 0.104804
\(553\) 18.9133 0.804274
\(554\) 3.08089 0.130895
\(555\) −18.2877 −0.776269
\(556\) 1.34700 0.0571254
\(557\) 9.97975 0.422856 0.211428 0.977394i \(-0.432189\pi\)
0.211428 + 0.977394i \(0.432189\pi\)
\(558\) 7.36712 0.311875
\(559\) −4.87285 −0.206100
\(560\) 12.3852 0.523372
\(561\) 10.3951 0.438883
\(562\) 3.29557 0.139015
\(563\) 1.13095 0.0476638 0.0238319 0.999716i \(-0.492413\pi\)
0.0238319 + 0.999716i \(0.492413\pi\)
\(564\) −7.45264 −0.313813
\(565\) −6.22232 −0.261775
\(566\) −24.9865 −1.05026
\(567\) −4.81484 −0.202204
\(568\) 6.15722 0.258351
\(569\) 29.9381 1.25507 0.627535 0.778589i \(-0.284064\pi\)
0.627535 + 0.778589i \(0.284064\pi\)
\(570\) 1.98345 0.0830775
\(571\) 1.46445 0.0612854 0.0306427 0.999530i \(-0.490245\pi\)
0.0306427 + 0.999530i \(0.490245\pi\)
\(572\) −1.68147 −0.0703059
\(573\) 12.0110 0.501766
\(574\) 57.6203 2.40503
\(575\) −3.98098 −0.166018
\(576\) 1.00000 0.0416667
\(577\) 42.4814 1.76852 0.884262 0.466990i \(-0.154662\pi\)
0.884262 + 0.466990i \(0.154662\pi\)
\(578\) −21.2192 −0.882602
\(579\) −12.2193 −0.507817
\(580\) 10.5441 0.437819
\(581\) 5.89150 0.244421
\(582\) 8.09177 0.335415
\(583\) 7.63480 0.316201
\(584\) −16.5077 −0.683092
\(585\) 2.57230 0.106352
\(586\) 9.62707 0.397691
\(587\) −30.5500 −1.26093 −0.630466 0.776217i \(-0.717136\pi\)
−0.630466 + 0.776217i \(0.717136\pi\)
\(588\) 16.1827 0.667362
\(589\) −5.68063 −0.234066
\(590\) −15.2245 −0.626784
\(591\) 5.75541 0.236746
\(592\) 7.10946 0.292197
\(593\) −43.2197 −1.77482 −0.887411 0.460979i \(-0.847498\pi\)
−0.887411 + 0.460979i \(0.847498\pi\)
\(594\) −1.68147 −0.0689916
\(595\) 76.5676 3.13896
\(596\) −16.3859 −0.671194
\(597\) −24.5893 −1.00637
\(598\) −2.46234 −0.100692
\(599\) 36.4711 1.49017 0.745085 0.666970i \(-0.232409\pi\)
0.745085 + 0.666970i \(0.232409\pi\)
\(600\) −1.61675 −0.0660035
\(601\) −22.2340 −0.906944 −0.453472 0.891271i \(-0.649815\pi\)
−0.453472 + 0.891271i \(0.649815\pi\)
\(602\) 23.4620 0.956240
\(603\) −9.81599 −0.399738
\(604\) 19.8000 0.805650
\(605\) 21.0226 0.854688
\(606\) −6.68754 −0.271663
\(607\) 25.0935 1.01851 0.509257 0.860614i \(-0.329920\pi\)
0.509257 + 0.860614i \(0.329920\pi\)
\(608\) −0.771079 −0.0312714
\(609\) 19.7364 0.799759
\(610\) −0.226819 −0.00918363
\(611\) 7.45264 0.301501
\(612\) 6.18217 0.249899
\(613\) 13.6964 0.553193 0.276597 0.960986i \(-0.410793\pi\)
0.276597 + 0.960986i \(0.410793\pi\)
\(614\) 20.2738 0.818186
\(615\) −30.7834 −1.24131
\(616\) 8.09602 0.326198
\(617\) 27.3265 1.10012 0.550061 0.835124i \(-0.314604\pi\)
0.550061 + 0.835124i \(0.314604\pi\)
\(618\) −1.00000 −0.0402259
\(619\) −30.5733 −1.22885 −0.614423 0.788977i \(-0.710611\pi\)
−0.614423 + 0.788977i \(0.710611\pi\)
\(620\) 18.9505 0.761070
\(621\) −2.46234 −0.0988101
\(622\) 32.4947 1.30292
\(623\) −31.0832 −1.24532
\(624\) −1.00000 −0.0400320
\(625\) −30.4699 −1.21879
\(626\) −7.50820 −0.300088
\(627\) 1.29655 0.0517791
\(628\) 16.4161 0.655073
\(629\) 43.9519 1.75248
\(630\) −12.3852 −0.493439
\(631\) −41.5796 −1.65526 −0.827628 0.561277i \(-0.810310\pi\)
−0.827628 + 0.561277i \(0.810310\pi\)
\(632\) 3.92812 0.156252
\(633\) 7.63512 0.303469
\(634\) 2.86805 0.113905
\(635\) −47.9312 −1.90209
\(636\) 4.54055 0.180044
\(637\) −16.1827 −0.641181
\(638\) 6.89248 0.272876
\(639\) −6.15722 −0.243576
\(640\) 2.57230 0.101679
\(641\) 32.9251 1.30046 0.650232 0.759736i \(-0.274672\pi\)
0.650232 + 0.759736i \(0.274672\pi\)
\(642\) −9.93787 −0.392216
\(643\) −40.5674 −1.59982 −0.799911 0.600119i \(-0.795120\pi\)
−0.799911 + 0.600119i \(0.795120\pi\)
\(644\) 11.8557 0.467182
\(645\) −12.5345 −0.493544
\(646\) −4.76694 −0.187553
\(647\) 3.50592 0.137832 0.0689160 0.997622i \(-0.478046\pi\)
0.0689160 + 0.997622i \(0.478046\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −9.95202 −0.390651
\(650\) 1.61675 0.0634141
\(651\) 35.4715 1.39024
\(652\) −17.7158 −0.693803
\(653\) −23.6156 −0.924149 −0.462075 0.886841i \(-0.652895\pi\)
−0.462075 + 0.886841i \(0.652895\pi\)
\(654\) −10.9229 −0.427120
\(655\) 40.7387 1.59179
\(656\) 11.9672 0.467242
\(657\) 16.5077 0.644025
\(658\) −35.8832 −1.39887
\(659\) −10.2901 −0.400846 −0.200423 0.979709i \(-0.564232\pi\)
−0.200423 + 0.979709i \(0.564232\pi\)
\(660\) −4.32526 −0.168360
\(661\) 3.85558 0.149965 0.0749824 0.997185i \(-0.476110\pi\)
0.0749824 + 0.997185i \(0.476110\pi\)
\(662\) −26.5172 −1.03062
\(663\) −6.18217 −0.240095
\(664\) 1.22361 0.0474854
\(665\) 9.54999 0.370333
\(666\) −7.10946 −0.275486
\(667\) 10.0933 0.390814
\(668\) −15.7136 −0.607977
\(669\) 3.09337 0.119597
\(670\) −25.2497 −0.975482
\(671\) −0.148268 −0.00572381
\(672\) 4.81484 0.185736
\(673\) 22.6898 0.874626 0.437313 0.899309i \(-0.355930\pi\)
0.437313 + 0.899309i \(0.355930\pi\)
\(674\) 14.6616 0.564743
\(675\) 1.61675 0.0622287
\(676\) 1.00000 0.0384615
\(677\) 0.215423 0.00827938 0.00413969 0.999991i \(-0.498682\pi\)
0.00413969 + 0.999991i \(0.498682\pi\)
\(678\) −2.41897 −0.0928999
\(679\) 38.9606 1.49517
\(680\) 15.9024 0.609830
\(681\) −24.1516 −0.925492
\(682\) 12.3876 0.474346
\(683\) −7.30937 −0.279685 −0.139843 0.990174i \(-0.544660\pi\)
−0.139843 + 0.990174i \(0.544660\pi\)
\(684\) 0.771079 0.0294829
\(685\) 5.23298 0.199942
\(686\) 44.2131 1.68807
\(687\) −9.02543 −0.344342
\(688\) 4.87285 0.185776
\(689\) −4.54055 −0.172981
\(690\) −6.33388 −0.241127
\(691\) 39.3684 1.49765 0.748823 0.662770i \(-0.230619\pi\)
0.748823 + 0.662770i \(0.230619\pi\)
\(692\) 8.65865 0.329153
\(693\) −8.09602 −0.307542
\(694\) −2.56359 −0.0973125
\(695\) −3.46489 −0.131431
\(696\) 4.09908 0.155375
\(697\) 73.9835 2.80232
\(698\) 24.6663 0.933632
\(699\) −19.8825 −0.752027
\(700\) −7.78439 −0.294222
\(701\) −21.0019 −0.793232 −0.396616 0.917985i \(-0.629816\pi\)
−0.396616 + 0.917985i \(0.629816\pi\)
\(702\) 1.00000 0.0377426
\(703\) 5.48195 0.206756
\(704\) 1.68147 0.0633728
\(705\) 19.1704 0.722001
\(706\) −11.7074 −0.440614
\(707\) −32.1994 −1.21098
\(708\) −5.91864 −0.222436
\(709\) 0.353263 0.0132671 0.00663354 0.999978i \(-0.497888\pi\)
0.00663354 + 0.999978i \(0.497888\pi\)
\(710\) −15.8382 −0.594399
\(711\) −3.92812 −0.147316
\(712\) −6.45570 −0.241938
\(713\) 18.1403 0.679361
\(714\) 29.7661 1.11397
\(715\) 4.32526 0.161755
\(716\) −16.8361 −0.629196
\(717\) −21.7455 −0.812102
\(718\) 2.53080 0.0944484
\(719\) 8.28885 0.309122 0.154561 0.987983i \(-0.450604\pi\)
0.154561 + 0.987983i \(0.450604\pi\)
\(720\) −2.57230 −0.0958641
\(721\) −4.81484 −0.179314
\(722\) 18.4054 0.684979
\(723\) −29.9056 −1.11220
\(724\) 0.312325 0.0116075
\(725\) −6.62718 −0.246127
\(726\) 8.17265 0.303316
\(727\) 41.0157 1.52119 0.760594 0.649228i \(-0.224908\pi\)
0.760594 + 0.649228i \(0.224908\pi\)
\(728\) −4.81484 −0.178450
\(729\) 1.00000 0.0370370
\(730\) 42.4627 1.57162
\(731\) 30.1248 1.11421
\(732\) −0.0881773 −0.00325913
\(733\) 2.14522 0.0792354 0.0396177 0.999215i \(-0.487386\pi\)
0.0396177 + 0.999215i \(0.487386\pi\)
\(734\) −5.13485 −0.189531
\(735\) −41.6268 −1.53543
\(736\) 2.46234 0.0907629
\(737\) −16.5053 −0.607981
\(738\) −11.9672 −0.440520
\(739\) −27.8164 −1.02324 −0.511622 0.859211i \(-0.670955\pi\)
−0.511622 + 0.859211i \(0.670955\pi\)
\(740\) −18.2877 −0.672269
\(741\) −0.771079 −0.0283263
\(742\) 21.8620 0.802579
\(743\) 43.1157 1.58176 0.790881 0.611970i \(-0.209623\pi\)
0.790881 + 0.611970i \(0.209623\pi\)
\(744\) 7.36712 0.270092
\(745\) 42.1496 1.54424
\(746\) 33.0765 1.21102
\(747\) −1.22361 −0.0447696
\(748\) 10.3951 0.380084
\(749\) −47.8492 −1.74837
\(750\) −8.70275 −0.317780
\(751\) 6.62649 0.241804 0.120902 0.992664i \(-0.461421\pi\)
0.120902 + 0.992664i \(0.461421\pi\)
\(752\) −7.45264 −0.271770
\(753\) −10.3576 −0.377453
\(754\) −4.09908 −0.149280
\(755\) −50.9316 −1.85359
\(756\) −4.81484 −0.175114
\(757\) −0.588080 −0.0213741 −0.0106871 0.999943i \(-0.503402\pi\)
−0.0106871 + 0.999943i \(0.503402\pi\)
\(758\) 15.1628 0.550737
\(759\) −4.14035 −0.150285
\(760\) 1.98345 0.0719473
\(761\) 27.9768 1.01416 0.507079 0.861899i \(-0.330725\pi\)
0.507079 + 0.861899i \(0.330725\pi\)
\(762\) −18.6336 −0.675023
\(763\) −52.5921 −1.90396
\(764\) 12.0110 0.434542
\(765\) −15.9024 −0.574953
\(766\) 31.7073 1.14563
\(767\) 5.91864 0.213710
\(768\) 1.00000 0.0360844
\(769\) 49.9214 1.80021 0.900106 0.435670i \(-0.143489\pi\)
0.900106 + 0.435670i \(0.143489\pi\)
\(770\) −20.8254 −0.750496
\(771\) −2.58228 −0.0929987
\(772\) −12.2193 −0.439782
\(773\) 0.172234 0.00619482 0.00309741 0.999995i \(-0.499014\pi\)
0.00309741 + 0.999995i \(0.499014\pi\)
\(774\) −4.87285 −0.175151
\(775\) −11.9108 −0.427848
\(776\) 8.09177 0.290478
\(777\) −34.2309 −1.22803
\(778\) 23.6651 0.848434
\(779\) 9.22768 0.330616
\(780\) 2.57230 0.0921033
\(781\) −10.3532 −0.370466
\(782\) 15.2226 0.544358
\(783\) −4.09908 −0.146489
\(784\) 16.1827 0.577953
\(785\) −42.2271 −1.50715
\(786\) 15.8374 0.564902
\(787\) 32.4074 1.15520 0.577599 0.816321i \(-0.303990\pi\)
0.577599 + 0.816321i \(0.303990\pi\)
\(788\) 5.75541 0.205028
\(789\) −19.9903 −0.711672
\(790\) −10.1043 −0.359495
\(791\) −11.6469 −0.414118
\(792\) −1.68147 −0.0597485
\(793\) 0.0881773 0.00313127
\(794\) −25.1755 −0.893446
\(795\) −11.6797 −0.414235
\(796\) −24.5893 −0.871544
\(797\) 15.5656 0.551363 0.275682 0.961249i \(-0.411096\pi\)
0.275682 + 0.961249i \(0.411096\pi\)
\(798\) 3.71262 0.131425
\(799\) −46.0734 −1.62996
\(800\) −1.61675 −0.0571607
\(801\) 6.45570 0.228101
\(802\) 15.0419 0.531149
\(803\) 27.7572 0.979529
\(804\) −9.81599 −0.346183
\(805\) −30.4966 −1.07486
\(806\) −7.36712 −0.259496
\(807\) −1.69478 −0.0596591
\(808\) −6.68754 −0.235267
\(809\) −45.8293 −1.61127 −0.805636 0.592411i \(-0.798176\pi\)
−0.805636 + 0.592411i \(0.798176\pi\)
\(810\) 2.57230 0.0903816
\(811\) 0.288999 0.0101481 0.00507406 0.999987i \(-0.498385\pi\)
0.00507406 + 0.999987i \(0.498385\pi\)
\(812\) 19.7364 0.692612
\(813\) 24.7894 0.869402
\(814\) −11.9544 −0.419000
\(815\) 45.5703 1.59626
\(816\) 6.18217 0.216419
\(817\) 3.75735 0.131453
\(818\) −20.1039 −0.702915
\(819\) 4.81484 0.168244
\(820\) −30.7834 −1.07500
\(821\) −32.2067 −1.12402 −0.562011 0.827130i \(-0.689972\pi\)
−0.562011 + 0.827130i \(0.689972\pi\)
\(822\) 2.03436 0.0709563
\(823\) 42.5356 1.48270 0.741349 0.671120i \(-0.234186\pi\)
0.741349 + 0.671120i \(0.234186\pi\)
\(824\) −1.00000 −0.0348367
\(825\) 2.71852 0.0946466
\(826\) −28.4973 −0.991548
\(827\) −39.6858 −1.38001 −0.690006 0.723804i \(-0.742392\pi\)
−0.690006 + 0.723804i \(0.742392\pi\)
\(828\) −2.46234 −0.0855721
\(829\) −4.07853 −0.141653 −0.0708266 0.997489i \(-0.522564\pi\)
−0.0708266 + 0.997489i \(0.522564\pi\)
\(830\) −3.14750 −0.109251
\(831\) −3.08089 −0.106875
\(832\) −1.00000 −0.0346688
\(833\) 100.044 3.46632
\(834\) −1.34700 −0.0466427
\(835\) 40.4202 1.39880
\(836\) 1.29655 0.0448420
\(837\) −7.36712 −0.254645
\(838\) 34.5445 1.19332
\(839\) 36.1407 1.24772 0.623858 0.781538i \(-0.285564\pi\)
0.623858 + 0.781538i \(0.285564\pi\)
\(840\) −12.3852 −0.427331
\(841\) −12.1976 −0.420606
\(842\) −4.53726 −0.156364
\(843\) −3.29557 −0.113505
\(844\) 7.63512 0.262812
\(845\) −2.57230 −0.0884900
\(846\) 7.45264 0.256227
\(847\) 39.3500 1.35208
\(848\) 4.54055 0.155923
\(849\) 24.9865 0.857534
\(850\) −9.99501 −0.342826
\(851\) −17.5059 −0.600093
\(852\) −6.15722 −0.210943
\(853\) 26.1799 0.896383 0.448192 0.893938i \(-0.352068\pi\)
0.448192 + 0.893938i \(0.352068\pi\)
\(854\) −0.424560 −0.0145281
\(855\) −1.98345 −0.0678325
\(856\) −9.93787 −0.339669
\(857\) −7.27431 −0.248486 −0.124243 0.992252i \(-0.539650\pi\)
−0.124243 + 0.992252i \(0.539650\pi\)
\(858\) 1.68147 0.0574045
\(859\) 9.31821 0.317933 0.158967 0.987284i \(-0.449184\pi\)
0.158967 + 0.987284i \(0.449184\pi\)
\(860\) −12.5345 −0.427421
\(861\) −57.6203 −1.96370
\(862\) −8.15170 −0.277648
\(863\) 53.2521 1.81272 0.906361 0.422504i \(-0.138849\pi\)
0.906361 + 0.422504i \(0.138849\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −22.2727 −0.757294
\(866\) 24.1238 0.819759
\(867\) 21.2192 0.720641
\(868\) 35.4715 1.20398
\(869\) −6.60502 −0.224060
\(870\) −10.5441 −0.357478
\(871\) 9.81599 0.332602
\(872\) −10.9229 −0.369897
\(873\) −8.09177 −0.273865
\(874\) 1.89865 0.0642229
\(875\) −41.9023 −1.41656
\(876\) 16.5077 0.557742
\(877\) 43.1811 1.45812 0.729061 0.684449i \(-0.239957\pi\)
0.729061 + 0.684449i \(0.239957\pi\)
\(878\) −34.3120 −1.15797
\(879\) −9.62707 −0.324713
\(880\) −4.32526 −0.145804
\(881\) −24.2449 −0.816831 −0.408415 0.912796i \(-0.633919\pi\)
−0.408415 + 0.912796i \(0.633919\pi\)
\(882\) −16.1827 −0.544899
\(883\) −32.1646 −1.08243 −0.541213 0.840885i \(-0.682035\pi\)
−0.541213 + 0.840885i \(0.682035\pi\)
\(884\) −6.18217 −0.207929
\(885\) 15.2245 0.511767
\(886\) −13.8306 −0.464649
\(887\) 19.8279 0.665757 0.332878 0.942970i \(-0.391980\pi\)
0.332878 + 0.942970i \(0.391980\pi\)
\(888\) −7.10946 −0.238578
\(889\) −89.7176 −3.00903
\(890\) 16.6060 0.556635
\(891\) 1.68147 0.0563314
\(892\) 3.09337 0.103574
\(893\) −5.74657 −0.192302
\(894\) 16.3859 0.548027
\(895\) 43.3076 1.44761
\(896\) 4.81484 0.160852
\(897\) 2.46234 0.0822150
\(898\) −1.18499 −0.0395435
\(899\) 30.1984 1.00717
\(900\) 1.61675 0.0538916
\(901\) 28.0704 0.935161
\(902\) −20.1226 −0.670008
\(903\) −23.4620 −0.780766
\(904\) −2.41897 −0.0804537
\(905\) −0.803394 −0.0267057
\(906\) −19.8000 −0.657810
\(907\) 15.5207 0.515358 0.257679 0.966231i \(-0.417042\pi\)
0.257679 + 0.966231i \(0.417042\pi\)
\(908\) −24.1516 −0.801499
\(909\) 6.68754 0.221812
\(910\) 12.3852 0.410566
\(911\) −2.59413 −0.0859473 −0.0429737 0.999076i \(-0.513683\pi\)
−0.0429737 + 0.999076i \(0.513683\pi\)
\(912\) 0.771079 0.0255330
\(913\) −2.05747 −0.0680923
\(914\) −10.8635 −0.359331
\(915\) 0.226819 0.00749840
\(916\) −9.02543 −0.298209
\(917\) 76.2547 2.51815
\(918\) −6.18217 −0.204042
\(919\) −41.1902 −1.35874 −0.679370 0.733796i \(-0.737747\pi\)
−0.679370 + 0.733796i \(0.737747\pi\)
\(920\) −6.33388 −0.208822
\(921\) −20.2738 −0.668046
\(922\) 22.7229 0.748339
\(923\) 6.15722 0.202667
\(924\) −8.09602 −0.266339
\(925\) 11.4942 0.377927
\(926\) 24.7743 0.814134
\(927\) 1.00000 0.0328443
\(928\) 4.09908 0.134559
\(929\) −19.9731 −0.655295 −0.327647 0.944800i \(-0.606256\pi\)
−0.327647 + 0.944800i \(0.606256\pi\)
\(930\) −18.9505 −0.621411
\(931\) 12.4781 0.408954
\(932\) −19.8825 −0.651274
\(933\) −32.4947 −1.06383
\(934\) −10.2797 −0.336361
\(935\) −26.7395 −0.874474
\(936\) 1.00000 0.0326860
\(937\) −16.4352 −0.536914 −0.268457 0.963292i \(-0.586514\pi\)
−0.268457 + 0.963292i \(0.586514\pi\)
\(938\) −47.2624 −1.54317
\(939\) 7.50820 0.245021
\(940\) 19.1704 0.625271
\(941\) −7.53303 −0.245570 −0.122785 0.992433i \(-0.539183\pi\)
−0.122785 + 0.992433i \(0.539183\pi\)
\(942\) −16.4161 −0.534865
\(943\) −29.4673 −0.959589
\(944\) −5.91864 −0.192635
\(945\) 12.3852 0.402892
\(946\) −8.19356 −0.266396
\(947\) 55.7249 1.81082 0.905408 0.424543i \(-0.139565\pi\)
0.905408 + 0.424543i \(0.139565\pi\)
\(948\) −3.92812 −0.127579
\(949\) −16.5077 −0.535861
\(950\) −1.24664 −0.0404464
\(951\) −2.86805 −0.0930029
\(952\) 29.7661 0.964726
\(953\) −32.8411 −1.06383 −0.531913 0.846799i \(-0.678527\pi\)
−0.531913 + 0.846799i \(0.678527\pi\)
\(954\) −4.54055 −0.147006
\(955\) −30.8959 −0.999767
\(956\) −21.7455 −0.703301
\(957\) −6.89248 −0.222802
\(958\) 17.4984 0.565348
\(959\) 9.79510 0.316300
\(960\) −2.57230 −0.0830208
\(961\) 23.2745 0.750790
\(962\) 7.10946 0.229218
\(963\) 9.93787 0.320243
\(964\) −29.9056 −0.963195
\(965\) 31.4318 1.01182
\(966\) −11.8557 −0.381452
\(967\) −25.3198 −0.814229 −0.407115 0.913377i \(-0.633465\pi\)
−0.407115 + 0.913377i \(0.633465\pi\)
\(968\) 8.17265 0.262679
\(969\) 4.76694 0.153136
\(970\) −20.8145 −0.668313
\(971\) 10.8294 0.347531 0.173766 0.984787i \(-0.444406\pi\)
0.173766 + 0.984787i \(0.444406\pi\)
\(972\) 1.00000 0.0320750
\(973\) −6.48558 −0.207918
\(974\) −31.5809 −1.01192
\(975\) −1.61675 −0.0517774
\(976\) −0.0881773 −0.00282249
\(977\) 1.64292 0.0525616 0.0262808 0.999655i \(-0.491634\pi\)
0.0262808 + 0.999655i \(0.491634\pi\)
\(978\) 17.7158 0.566488
\(979\) 10.8551 0.346930
\(980\) −41.6268 −1.32972
\(981\) 10.9229 0.348742
\(982\) −15.1300 −0.482818
\(983\) 0.103025 0.00328598 0.00164299 0.999999i \(-0.499477\pi\)
0.00164299 + 0.999999i \(0.499477\pi\)
\(984\) −11.9672 −0.381502
\(985\) −14.8047 −0.471716
\(986\) 25.3412 0.807027
\(987\) 35.8832 1.14218
\(988\) −0.771079 −0.0245313
\(989\) −11.9986 −0.381533
\(990\) 4.32526 0.137466
\(991\) −53.0705 −1.68584 −0.842920 0.538038i \(-0.819166\pi\)
−0.842920 + 0.538038i \(0.819166\pi\)
\(992\) 7.36712 0.233906
\(993\) 26.5172 0.841497
\(994\) −29.6460 −0.940315
\(995\) 63.2511 2.00520
\(996\) −1.22361 −0.0387717
\(997\) −27.8641 −0.882465 −0.441233 0.897393i \(-0.645459\pi\)
−0.441233 + 0.897393i \(0.645459\pi\)
\(998\) −6.31066 −0.199761
\(999\) 7.10946 0.224933
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.bb.1.4 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bb.1.4 14 1.1 even 1 trivial