Properties

Label 8034.2.a.z.1.1
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
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: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 3 x^{13} - 36 x^{12} + 108 x^{11} + 434 x^{10} - 1239 x^{9} - 2404 x^{8} + 6204 x^{7} + 6361 x^{6} - 14618 x^{5} - 7015 x^{4} + 15763 x^{3} + 1118 x^{2} - 6316 x + 1552\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.68342\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.68342 q^{5} +1.00000 q^{6} +5.03243 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.68342 q^{5} +1.00000 q^{6} +5.03243 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.68342 q^{10} -6.07783 q^{11} -1.00000 q^{12} +1.00000 q^{13} -5.03243 q^{14} +3.68342 q^{15} +1.00000 q^{16} +0.687836 q^{17} -1.00000 q^{18} +4.84475 q^{19} -3.68342 q^{20} -5.03243 q^{21} +6.07783 q^{22} +5.54263 q^{23} +1.00000 q^{24} +8.56755 q^{25} -1.00000 q^{26} -1.00000 q^{27} +5.03243 q^{28} +6.66499 q^{29} -3.68342 q^{30} -3.81844 q^{31} -1.00000 q^{32} +6.07783 q^{33} -0.687836 q^{34} -18.5365 q^{35} +1.00000 q^{36} +0.460154 q^{37} -4.84475 q^{38} -1.00000 q^{39} +3.68342 q^{40} -7.54316 q^{41} +5.03243 q^{42} -8.38067 q^{43} -6.07783 q^{44} -3.68342 q^{45} -5.54263 q^{46} +5.56288 q^{47} -1.00000 q^{48} +18.3254 q^{49} -8.56755 q^{50} -0.687836 q^{51} +1.00000 q^{52} +7.78611 q^{53} +1.00000 q^{54} +22.3872 q^{55} -5.03243 q^{56} -4.84475 q^{57} -6.66499 q^{58} -5.42438 q^{59} +3.68342 q^{60} -6.72933 q^{61} +3.81844 q^{62} +5.03243 q^{63} +1.00000 q^{64} -3.68342 q^{65} -6.07783 q^{66} +10.8290 q^{67} +0.687836 q^{68} -5.54263 q^{69} +18.5365 q^{70} +14.1374 q^{71} -1.00000 q^{72} -11.0586 q^{73} -0.460154 q^{74} -8.56755 q^{75} +4.84475 q^{76} -30.5863 q^{77} +1.00000 q^{78} +1.41867 q^{79} -3.68342 q^{80} +1.00000 q^{81} +7.54316 q^{82} -5.18575 q^{83} -5.03243 q^{84} -2.53359 q^{85} +8.38067 q^{86} -6.66499 q^{87} +6.07783 q^{88} +4.44024 q^{89} +3.68342 q^{90} +5.03243 q^{91} +5.54263 q^{92} +3.81844 q^{93} -5.56288 q^{94} -17.8452 q^{95} +1.00000 q^{96} -14.0901 q^{97} -18.3254 q^{98} -6.07783 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - 3q^{5} + 14q^{6} + 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} - 14q^{3} + 14q^{4} - 3q^{5} + 14q^{6} + 4q^{7} - 14q^{8} + 14q^{9} + 3q^{10} + 5q^{11} - 14q^{12} + 14q^{13} - 4q^{14} + 3q^{15} + 14q^{16} - 7q^{17} - 14q^{18} + 31q^{19} - 3q^{20} - 4q^{21} - 5q^{22} + 14q^{23} + 14q^{24} + 11q^{25} - 14q^{26} - 14q^{27} + 4q^{28} + 9q^{29} - 3q^{30} + 23q^{31} - 14q^{32} - 5q^{33} + 7q^{34} - 2q^{35} + 14q^{36} + 12q^{37} - 31q^{38} - 14q^{39} + 3q^{40} - 5q^{41} + 4q^{42} - 2q^{43} + 5q^{44} - 3q^{45} - 14q^{46} - 11q^{47} - 14q^{48} + 52q^{49} - 11q^{50} + 7q^{51} + 14q^{52} + 6q^{53} + 14q^{54} + 30q^{55} - 4q^{56} - 31q^{57} - 9q^{58} - 4q^{59} + 3q^{60} + 12q^{61} - 23q^{62} + 4q^{63} + 14q^{64} - 3q^{65} + 5q^{66} + 24q^{67} - 7q^{68} - 14q^{69} + 2q^{70} + 20q^{71} - 14q^{72} + 2q^{73} - 12q^{74} - 11q^{75} + 31q^{76} - 28q^{77} + 14q^{78} + 59q^{79} - 3q^{80} + 14q^{81} + 5q^{82} + q^{83} - 4q^{84} - 29q^{85} + 2q^{86} - 9q^{87} - 5q^{88} + 6q^{89} + 3q^{90} + 4q^{91} + 14q^{92} - 23q^{93} + 11q^{94} - 58q^{95} + 14q^{96} - 6q^{97} - 52q^{98} + 5q^{99} + O(q^{100}) \)

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
\(5\) −3.68342 −1.64727 −0.823637 0.567118i \(-0.808059\pi\)
−0.823637 + 0.567118i \(0.808059\pi\)
\(6\) 1.00000 0.408248
\(7\) 5.03243 1.90208 0.951040 0.309067i \(-0.100017\pi\)
0.951040 + 0.309067i \(0.100017\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.68342 1.16480
\(11\) −6.07783 −1.83254 −0.916268 0.400567i \(-0.868813\pi\)
−0.916268 + 0.400567i \(0.868813\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) −5.03243 −1.34497
\(15\) 3.68342 0.951054
\(16\) 1.00000 0.250000
\(17\) 0.687836 0.166825 0.0834124 0.996515i \(-0.473418\pi\)
0.0834124 + 0.996515i \(0.473418\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.84475 1.11146 0.555731 0.831362i \(-0.312438\pi\)
0.555731 + 0.831362i \(0.312438\pi\)
\(20\) −3.68342 −0.823637
\(21\) −5.03243 −1.09817
\(22\) 6.07783 1.29580
\(23\) 5.54263 1.15572 0.577859 0.816136i \(-0.303888\pi\)
0.577859 + 0.816136i \(0.303888\pi\)
\(24\) 1.00000 0.204124
\(25\) 8.56755 1.71351
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 5.03243 0.951040
\(29\) 6.66499 1.23766 0.618829 0.785526i \(-0.287607\pi\)
0.618829 + 0.785526i \(0.287607\pi\)
\(30\) −3.68342 −0.672497
\(31\) −3.81844 −0.685813 −0.342906 0.939370i \(-0.611411\pi\)
−0.342906 + 0.939370i \(0.611411\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.07783 1.05801
\(34\) −0.687836 −0.117963
\(35\) −18.5365 −3.13325
\(36\) 1.00000 0.166667
\(37\) 0.460154 0.0756489 0.0378244 0.999284i \(-0.487957\pi\)
0.0378244 + 0.999284i \(0.487957\pi\)
\(38\) −4.84475 −0.785923
\(39\) −1.00000 −0.160128
\(40\) 3.68342 0.582399
\(41\) −7.54316 −1.17804 −0.589022 0.808117i \(-0.700487\pi\)
−0.589022 + 0.808117i \(0.700487\pi\)
\(42\) 5.03243 0.776521
\(43\) −8.38067 −1.27804 −0.639020 0.769190i \(-0.720660\pi\)
−0.639020 + 0.769190i \(0.720660\pi\)
\(44\) −6.07783 −0.916268
\(45\) −3.68342 −0.549091
\(46\) −5.54263 −0.817217
\(47\) 5.56288 0.811429 0.405714 0.914000i \(-0.367023\pi\)
0.405714 + 0.914000i \(0.367023\pi\)
\(48\) −1.00000 −0.144338
\(49\) 18.3254 2.61791
\(50\) −8.56755 −1.21163
\(51\) −0.687836 −0.0963163
\(52\) 1.00000 0.138675
\(53\) 7.78611 1.06950 0.534752 0.845009i \(-0.320405\pi\)
0.534752 + 0.845009i \(0.320405\pi\)
\(54\) 1.00000 0.136083
\(55\) 22.3872 3.01869
\(56\) −5.03243 −0.672487
\(57\) −4.84475 −0.641703
\(58\) −6.66499 −0.875157
\(59\) −5.42438 −0.706194 −0.353097 0.935587i \(-0.614871\pi\)
−0.353097 + 0.935587i \(0.614871\pi\)
\(60\) 3.68342 0.475527
\(61\) −6.72933 −0.861603 −0.430801 0.902447i \(-0.641769\pi\)
−0.430801 + 0.902447i \(0.641769\pi\)
\(62\) 3.81844 0.484943
\(63\) 5.03243 0.634027
\(64\) 1.00000 0.125000
\(65\) −3.68342 −0.456872
\(66\) −6.07783 −0.748129
\(67\) 10.8290 1.32298 0.661488 0.749956i \(-0.269925\pi\)
0.661488 + 0.749956i \(0.269925\pi\)
\(68\) 0.687836 0.0834124
\(69\) −5.54263 −0.667254
\(70\) 18.5365 2.21554
\(71\) 14.1374 1.67780 0.838902 0.544282i \(-0.183198\pi\)
0.838902 + 0.544282i \(0.183198\pi\)
\(72\) −1.00000 −0.117851
\(73\) −11.0586 −1.29431 −0.647157 0.762357i \(-0.724042\pi\)
−0.647157 + 0.762357i \(0.724042\pi\)
\(74\) −0.460154 −0.0534918
\(75\) −8.56755 −0.989296
\(76\) 4.84475 0.555731
\(77\) −30.5863 −3.48563
\(78\) 1.00000 0.113228
\(79\) 1.41867 0.159613 0.0798067 0.996810i \(-0.474570\pi\)
0.0798067 + 0.996810i \(0.474570\pi\)
\(80\) −3.68342 −0.411818
\(81\) 1.00000 0.111111
\(82\) 7.54316 0.833003
\(83\) −5.18575 −0.569210 −0.284605 0.958645i \(-0.591862\pi\)
−0.284605 + 0.958645i \(0.591862\pi\)
\(84\) −5.03243 −0.549083
\(85\) −2.53359 −0.274806
\(86\) 8.38067 0.903711
\(87\) −6.66499 −0.714562
\(88\) 6.07783 0.647899
\(89\) 4.44024 0.470665 0.235332 0.971915i \(-0.424382\pi\)
0.235332 + 0.971915i \(0.424382\pi\)
\(90\) 3.68342 0.388266
\(91\) 5.03243 0.527542
\(92\) 5.54263 0.577859
\(93\) 3.81844 0.395954
\(94\) −5.56288 −0.573767
\(95\) −17.8452 −1.83088
\(96\) 1.00000 0.102062
\(97\) −14.0901 −1.43063 −0.715315 0.698802i \(-0.753717\pi\)
−0.715315 + 0.698802i \(0.753717\pi\)
\(98\) −18.3254 −1.85114
\(99\) −6.07783 −0.610845
\(100\) 8.56755 0.856755
\(101\) −6.16928 −0.613866 −0.306933 0.951731i \(-0.599303\pi\)
−0.306933 + 0.951731i \(0.599303\pi\)
\(102\) 0.687836 0.0681059
\(103\) −1.00000 −0.0985329
\(104\) −1.00000 −0.0980581
\(105\) 18.5365 1.80898
\(106\) −7.78611 −0.756254
\(107\) 20.0623 1.93950 0.969748 0.244108i \(-0.0784951\pi\)
0.969748 + 0.244108i \(0.0784951\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.7098 −1.02581 −0.512907 0.858444i \(-0.671431\pi\)
−0.512907 + 0.858444i \(0.671431\pi\)
\(110\) −22.3872 −2.13453
\(111\) −0.460154 −0.0436759
\(112\) 5.03243 0.475520
\(113\) −0.229504 −0.0215899 −0.0107950 0.999942i \(-0.503436\pi\)
−0.0107950 + 0.999942i \(0.503436\pi\)
\(114\) 4.84475 0.453753
\(115\) −20.4158 −1.90378
\(116\) 6.66499 0.618829
\(117\) 1.00000 0.0924500
\(118\) 5.42438 0.499355
\(119\) 3.46149 0.317314
\(120\) −3.68342 −0.336248
\(121\) 25.9400 2.35818
\(122\) 6.72933 0.609245
\(123\) 7.54316 0.680144
\(124\) −3.81844 −0.342906
\(125\) −13.1408 −1.17535
\(126\) −5.03243 −0.448325
\(127\) −0.988933 −0.0877536 −0.0438768 0.999037i \(-0.513971\pi\)
−0.0438768 + 0.999037i \(0.513971\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.38067 0.737877
\(130\) 3.68342 0.323057
\(131\) −12.7607 −1.11491 −0.557455 0.830207i \(-0.688222\pi\)
−0.557455 + 0.830207i \(0.688222\pi\)
\(132\) 6.07783 0.529007
\(133\) 24.3809 2.11409
\(134\) −10.8290 −0.935485
\(135\) 3.68342 0.317018
\(136\) −0.687836 −0.0589815
\(137\) 8.89751 0.760166 0.380083 0.924952i \(-0.375895\pi\)
0.380083 + 0.924952i \(0.375895\pi\)
\(138\) 5.54263 0.471820
\(139\) −8.11255 −0.688097 −0.344049 0.938952i \(-0.611799\pi\)
−0.344049 + 0.938952i \(0.611799\pi\)
\(140\) −18.5365 −1.56662
\(141\) −5.56288 −0.468479
\(142\) −14.1374 −1.18639
\(143\) −6.07783 −0.508254
\(144\) 1.00000 0.0833333
\(145\) −24.5499 −2.03876
\(146\) 11.0586 0.915219
\(147\) −18.3254 −1.51145
\(148\) 0.460154 0.0378244
\(149\) −15.4097 −1.26241 −0.631204 0.775616i \(-0.717439\pi\)
−0.631204 + 0.775616i \(0.717439\pi\)
\(150\) 8.56755 0.699538
\(151\) −14.6840 −1.19496 −0.597482 0.801882i \(-0.703832\pi\)
−0.597482 + 0.801882i \(0.703832\pi\)
\(152\) −4.84475 −0.392962
\(153\) 0.687836 0.0556083
\(154\) 30.5863 2.46471
\(155\) 14.0649 1.12972
\(156\) −1.00000 −0.0800641
\(157\) 12.6431 1.00903 0.504515 0.863403i \(-0.331671\pi\)
0.504515 + 0.863403i \(0.331671\pi\)
\(158\) −1.41867 −0.112864
\(159\) −7.78611 −0.617479
\(160\) 3.68342 0.291200
\(161\) 27.8929 2.19827
\(162\) −1.00000 −0.0785674
\(163\) 22.7600 1.78270 0.891349 0.453319i \(-0.149760\pi\)
0.891349 + 0.453319i \(0.149760\pi\)
\(164\) −7.54316 −0.589022
\(165\) −22.3872 −1.74284
\(166\) 5.18575 0.402492
\(167\) −9.07055 −0.701900 −0.350950 0.936394i \(-0.614141\pi\)
−0.350950 + 0.936394i \(0.614141\pi\)
\(168\) 5.03243 0.388261
\(169\) 1.00000 0.0769231
\(170\) 2.53359 0.194317
\(171\) 4.84475 0.370488
\(172\) −8.38067 −0.639020
\(173\) 20.7522 1.57776 0.788878 0.614549i \(-0.210662\pi\)
0.788878 + 0.614549i \(0.210662\pi\)
\(174\) 6.66499 0.505272
\(175\) 43.1156 3.25924
\(176\) −6.07783 −0.458134
\(177\) 5.42438 0.407721
\(178\) −4.44024 −0.332810
\(179\) −14.8788 −1.11209 −0.556046 0.831151i \(-0.687682\pi\)
−0.556046 + 0.831151i \(0.687682\pi\)
\(180\) −3.68342 −0.274546
\(181\) 1.77972 0.132285 0.0661426 0.997810i \(-0.478931\pi\)
0.0661426 + 0.997810i \(0.478931\pi\)
\(182\) −5.03243 −0.373029
\(183\) 6.72933 0.497447
\(184\) −5.54263 −0.408608
\(185\) −1.69494 −0.124614
\(186\) −3.81844 −0.279982
\(187\) −4.18055 −0.305712
\(188\) 5.56288 0.405714
\(189\) −5.03243 −0.366056
\(190\) 17.8452 1.29463
\(191\) −27.1288 −1.96297 −0.981485 0.191541i \(-0.938651\pi\)
−0.981485 + 0.191541i \(0.938651\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 25.0012 1.79963 0.899814 0.436274i \(-0.143702\pi\)
0.899814 + 0.436274i \(0.143702\pi\)
\(194\) 14.0901 1.01161
\(195\) 3.68342 0.263775
\(196\) 18.3254 1.30896
\(197\) −16.4004 −1.16848 −0.584238 0.811582i \(-0.698607\pi\)
−0.584238 + 0.811582i \(0.698607\pi\)
\(198\) 6.07783 0.431933
\(199\) −1.59261 −0.112897 −0.0564485 0.998406i \(-0.517978\pi\)
−0.0564485 + 0.998406i \(0.517978\pi\)
\(200\) −8.56755 −0.605817
\(201\) −10.8290 −0.763821
\(202\) 6.16928 0.434069
\(203\) 33.5411 2.35413
\(204\) −0.687836 −0.0481582
\(205\) 27.7846 1.94056
\(206\) 1.00000 0.0696733
\(207\) 5.54263 0.385240
\(208\) 1.00000 0.0693375
\(209\) −29.4456 −2.03679
\(210\) −18.5365 −1.27914
\(211\) 12.4299 0.855709 0.427855 0.903848i \(-0.359269\pi\)
0.427855 + 0.903848i \(0.359269\pi\)
\(212\) 7.78611 0.534752
\(213\) −14.1374 −0.968681
\(214\) −20.0623 −1.37143
\(215\) 30.8695 2.10528
\(216\) 1.00000 0.0680414
\(217\) −19.2161 −1.30447
\(218\) 10.7098 0.725360
\(219\) 11.0586 0.747273
\(220\) 22.3872 1.50934
\(221\) 0.687836 0.0462689
\(222\) 0.460154 0.0308835
\(223\) −12.2201 −0.818317 −0.409159 0.912463i \(-0.634178\pi\)
−0.409159 + 0.912463i \(0.634178\pi\)
\(224\) −5.03243 −0.336244
\(225\) 8.56755 0.571170
\(226\) 0.229504 0.0152664
\(227\) −6.31622 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(228\) −4.84475 −0.320852
\(229\) 9.72002 0.642317 0.321158 0.947025i \(-0.395928\pi\)
0.321158 + 0.947025i \(0.395928\pi\)
\(230\) 20.4158 1.34618
\(231\) 30.5863 2.01243
\(232\) −6.66499 −0.437578
\(233\) 18.4155 1.20644 0.603220 0.797575i \(-0.293884\pi\)
0.603220 + 0.797575i \(0.293884\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −20.4904 −1.33665
\(236\) −5.42438 −0.353097
\(237\) −1.41867 −0.0921528
\(238\) −3.46149 −0.224375
\(239\) 20.0784 1.29876 0.649382 0.760462i \(-0.275027\pi\)
0.649382 + 0.760462i \(0.275027\pi\)
\(240\) 3.68342 0.237763
\(241\) 20.9951 1.35241 0.676207 0.736711i \(-0.263622\pi\)
0.676207 + 0.736711i \(0.263622\pi\)
\(242\) −25.9400 −1.66749
\(243\) −1.00000 −0.0641500
\(244\) −6.72933 −0.430801
\(245\) −67.5000 −4.31242
\(246\) −7.54316 −0.480934
\(247\) 4.84475 0.308264
\(248\) 3.81844 0.242471
\(249\) 5.18575 0.328634
\(250\) 13.1408 0.831096
\(251\) −5.21065 −0.328893 −0.164447 0.986386i \(-0.552584\pi\)
−0.164447 + 0.986386i \(0.552584\pi\)
\(252\) 5.03243 0.317013
\(253\) −33.6872 −2.11790
\(254\) 0.988933 0.0620512
\(255\) 2.53359 0.158659
\(256\) 1.00000 0.0625000
\(257\) 0.904912 0.0564468 0.0282234 0.999602i \(-0.491015\pi\)
0.0282234 + 0.999602i \(0.491015\pi\)
\(258\) −8.38067 −0.521758
\(259\) 2.31569 0.143890
\(260\) −3.68342 −0.228436
\(261\) 6.66499 0.412553
\(262\) 12.7607 0.788361
\(263\) 8.88779 0.548045 0.274022 0.961723i \(-0.411646\pi\)
0.274022 + 0.961723i \(0.411646\pi\)
\(264\) −6.07783 −0.374065
\(265\) −28.6795 −1.76177
\(266\) −24.3809 −1.49489
\(267\) −4.44024 −0.271738
\(268\) 10.8290 0.661488
\(269\) −6.00072 −0.365870 −0.182935 0.983125i \(-0.558560\pi\)
−0.182935 + 0.983125i \(0.558560\pi\)
\(270\) −3.68342 −0.224166
\(271\) −23.0789 −1.40194 −0.700972 0.713188i \(-0.747250\pi\)
−0.700972 + 0.713188i \(0.747250\pi\)
\(272\) 0.687836 0.0417062
\(273\) −5.03243 −0.304577
\(274\) −8.89751 −0.537518
\(275\) −52.0721 −3.14007
\(276\) −5.54263 −0.333627
\(277\) −2.17618 −0.130754 −0.0653770 0.997861i \(-0.520825\pi\)
−0.0653770 + 0.997861i \(0.520825\pi\)
\(278\) 8.11255 0.486558
\(279\) −3.81844 −0.228604
\(280\) 18.5365 1.10777
\(281\) 1.74352 0.104010 0.0520049 0.998647i \(-0.483439\pi\)
0.0520049 + 0.998647i \(0.483439\pi\)
\(282\) 5.56288 0.331264
\(283\) 12.7346 0.756991 0.378496 0.925603i \(-0.376442\pi\)
0.378496 + 0.925603i \(0.376442\pi\)
\(284\) 14.1374 0.838902
\(285\) 17.8452 1.05706
\(286\) 6.07783 0.359390
\(287\) −37.9605 −2.24074
\(288\) −1.00000 −0.0589256
\(289\) −16.5269 −0.972170
\(290\) 24.5499 1.44162
\(291\) 14.0901 0.825975
\(292\) −11.0586 −0.647157
\(293\) −21.2582 −1.24191 −0.620957 0.783844i \(-0.713256\pi\)
−0.620957 + 0.783844i \(0.713256\pi\)
\(294\) 18.3254 1.06876
\(295\) 19.9802 1.16329
\(296\) −0.460154 −0.0267459
\(297\) 6.07783 0.352672
\(298\) 15.4097 0.892658
\(299\) 5.54263 0.320539
\(300\) −8.56755 −0.494648
\(301\) −42.1752 −2.43094
\(302\) 14.6840 0.844967
\(303\) 6.16928 0.354416
\(304\) 4.84475 0.277866
\(305\) 24.7869 1.41930
\(306\) −0.687836 −0.0393210
\(307\) −23.5910 −1.34641 −0.673204 0.739457i \(-0.735082\pi\)
−0.673204 + 0.739457i \(0.735082\pi\)
\(308\) −30.5863 −1.74281
\(309\) 1.00000 0.0568880
\(310\) −14.0649 −0.798834
\(311\) 12.7248 0.721559 0.360780 0.932651i \(-0.382511\pi\)
0.360780 + 0.932651i \(0.382511\pi\)
\(312\) 1.00000 0.0566139
\(313\) 30.6522 1.73256 0.866282 0.499554i \(-0.166503\pi\)
0.866282 + 0.499554i \(0.166503\pi\)
\(314\) −12.6431 −0.713493
\(315\) −18.5365 −1.04442
\(316\) 1.41867 0.0798067
\(317\) −19.2824 −1.08301 −0.541503 0.840699i \(-0.682144\pi\)
−0.541503 + 0.840699i \(0.682144\pi\)
\(318\) 7.78611 0.436624
\(319\) −40.5087 −2.26805
\(320\) −3.68342 −0.205909
\(321\) −20.0623 −1.11977
\(322\) −27.8929 −1.55441
\(323\) 3.33240 0.185420
\(324\) 1.00000 0.0555556
\(325\) 8.56755 0.475242
\(326\) −22.7600 −1.26056
\(327\) 10.7098 0.592254
\(328\) 7.54316 0.416501
\(329\) 27.9948 1.54340
\(330\) 22.3872 1.23237
\(331\) −34.7227 −1.90853 −0.954267 0.298957i \(-0.903361\pi\)
−0.954267 + 0.298957i \(0.903361\pi\)
\(332\) −5.18575 −0.284605
\(333\) 0.460154 0.0252163
\(334\) 9.07055 0.496319
\(335\) −39.8878 −2.17930
\(336\) −5.03243 −0.274542
\(337\) −2.87112 −0.156400 −0.0782000 0.996938i \(-0.524917\pi\)
−0.0782000 + 0.996938i \(0.524917\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0.229504 0.0124650
\(340\) −2.53359 −0.137403
\(341\) 23.2079 1.25678
\(342\) −4.84475 −0.261974
\(343\) 56.9942 3.07740
\(344\) 8.38067 0.451855
\(345\) 20.4158 1.09915
\(346\) −20.7522 −1.11564
\(347\) 17.9238 0.962201 0.481101 0.876665i \(-0.340237\pi\)
0.481101 + 0.876665i \(0.340237\pi\)
\(348\) −6.66499 −0.357281
\(349\) −5.68935 −0.304544 −0.152272 0.988339i \(-0.548659\pi\)
−0.152272 + 0.988339i \(0.548659\pi\)
\(350\) −43.1156 −2.30463
\(351\) −1.00000 −0.0533761
\(352\) 6.07783 0.323949
\(353\) 12.6046 0.670874 0.335437 0.942063i \(-0.391116\pi\)
0.335437 + 0.942063i \(0.391116\pi\)
\(354\) −5.42438 −0.288302
\(355\) −52.0740 −2.76380
\(356\) 4.44024 0.235332
\(357\) −3.46149 −0.183201
\(358\) 14.8788 0.786368
\(359\) 32.9331 1.73814 0.869070 0.494688i \(-0.164718\pi\)
0.869070 + 0.494688i \(0.164718\pi\)
\(360\) 3.68342 0.194133
\(361\) 4.47165 0.235350
\(362\) −1.77972 −0.0935398
\(363\) −25.9400 −1.36150
\(364\) 5.03243 0.263771
\(365\) 40.7335 2.13209
\(366\) −6.72933 −0.351748
\(367\) −31.8850 −1.66438 −0.832191 0.554489i \(-0.812914\pi\)
−0.832191 + 0.554489i \(0.812914\pi\)
\(368\) 5.54263 0.288930
\(369\) −7.54316 −0.392681
\(370\) 1.69494 0.0881157
\(371\) 39.1831 2.03428
\(372\) 3.81844 0.197977
\(373\) 18.7119 0.968867 0.484433 0.874828i \(-0.339026\pi\)
0.484433 + 0.874828i \(0.339026\pi\)
\(374\) 4.18055 0.216171
\(375\) 13.1408 0.678587
\(376\) −5.56288 −0.286883
\(377\) 6.66499 0.343265
\(378\) 5.03243 0.258840
\(379\) 0.754775 0.0387702 0.0193851 0.999812i \(-0.493829\pi\)
0.0193851 + 0.999812i \(0.493829\pi\)
\(380\) −17.8452 −0.915442
\(381\) 0.988933 0.0506646
\(382\) 27.1288 1.38803
\(383\) 2.46500 0.125955 0.0629777 0.998015i \(-0.479940\pi\)
0.0629777 + 0.998015i \(0.479940\pi\)
\(384\) 1.00000 0.0510310
\(385\) 112.662 5.74179
\(386\) −25.0012 −1.27253
\(387\) −8.38067 −0.426013
\(388\) −14.0901 −0.715315
\(389\) 3.11542 0.157958 0.0789789 0.996876i \(-0.474834\pi\)
0.0789789 + 0.996876i \(0.474834\pi\)
\(390\) −3.68342 −0.186517
\(391\) 3.81242 0.192802
\(392\) −18.3254 −0.925571
\(393\) 12.7607 0.643694
\(394\) 16.4004 0.826238
\(395\) −5.22557 −0.262927
\(396\) −6.07783 −0.305423
\(397\) −1.53453 −0.0770156 −0.0385078 0.999258i \(-0.512260\pi\)
−0.0385078 + 0.999258i \(0.512260\pi\)
\(398\) 1.59261 0.0798302
\(399\) −24.3809 −1.22057
\(400\) 8.56755 0.428378
\(401\) 34.0311 1.69943 0.849717 0.527239i \(-0.176773\pi\)
0.849717 + 0.527239i \(0.176773\pi\)
\(402\) 10.8290 0.540103
\(403\) −3.81844 −0.190210
\(404\) −6.16928 −0.306933
\(405\) −3.68342 −0.183030
\(406\) −33.5411 −1.66462
\(407\) −2.79674 −0.138629
\(408\) 0.687836 0.0340530
\(409\) 18.5088 0.915200 0.457600 0.889158i \(-0.348709\pi\)
0.457600 + 0.889158i \(0.348709\pi\)
\(410\) −27.7846 −1.37218
\(411\) −8.89751 −0.438882
\(412\) −1.00000 −0.0492665
\(413\) −27.2978 −1.34324
\(414\) −5.54263 −0.272406
\(415\) 19.1013 0.937645
\(416\) −1.00000 −0.0490290
\(417\) 8.11255 0.397273
\(418\) 29.4456 1.44023
\(419\) 9.52179 0.465170 0.232585 0.972576i \(-0.425282\pi\)
0.232585 + 0.972576i \(0.425282\pi\)
\(420\) 18.5365 0.904491
\(421\) −40.2831 −1.96328 −0.981639 0.190746i \(-0.938909\pi\)
−0.981639 + 0.190746i \(0.938909\pi\)
\(422\) −12.4299 −0.605078
\(423\) 5.56288 0.270476
\(424\) −7.78611 −0.378127
\(425\) 5.89307 0.285856
\(426\) 14.1374 0.684961
\(427\) −33.8649 −1.63884
\(428\) 20.0623 0.969748
\(429\) 6.07783 0.293440
\(430\) −30.8695 −1.48866
\(431\) −8.41932 −0.405544 −0.202772 0.979226i \(-0.564995\pi\)
−0.202772 + 0.979226i \(0.564995\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 26.6609 1.28124 0.640621 0.767857i \(-0.278677\pi\)
0.640621 + 0.767857i \(0.278677\pi\)
\(434\) 19.2161 0.922400
\(435\) 24.5499 1.17708
\(436\) −10.7098 −0.512907
\(437\) 26.8527 1.28454
\(438\) −11.0586 −0.528402
\(439\) −0.393637 −0.0187873 −0.00939363 0.999956i \(-0.502990\pi\)
−0.00939363 + 0.999956i \(0.502990\pi\)
\(440\) −22.3872 −1.06727
\(441\) 18.3254 0.872637
\(442\) −0.687836 −0.0327170
\(443\) −12.8404 −0.610065 −0.305033 0.952342i \(-0.598667\pi\)
−0.305033 + 0.952342i \(0.598667\pi\)
\(444\) −0.460154 −0.0218379
\(445\) −16.3553 −0.775314
\(446\) 12.2201 0.578638
\(447\) 15.4097 0.728852
\(448\) 5.03243 0.237760
\(449\) 3.61376 0.170544 0.0852719 0.996358i \(-0.472824\pi\)
0.0852719 + 0.996358i \(0.472824\pi\)
\(450\) −8.56755 −0.403878
\(451\) 45.8461 2.15881
\(452\) −0.229504 −0.0107950
\(453\) 14.6840 0.689913
\(454\) 6.31622 0.296435
\(455\) −18.5365 −0.869007
\(456\) 4.84475 0.226876
\(457\) 33.7785 1.58009 0.790045 0.613048i \(-0.210057\pi\)
0.790045 + 0.613048i \(0.210057\pi\)
\(458\) −9.72002 −0.454187
\(459\) −0.687836 −0.0321054
\(460\) −20.4158 −0.951892
\(461\) 39.8441 1.85572 0.927862 0.372923i \(-0.121644\pi\)
0.927862 + 0.372923i \(0.121644\pi\)
\(462\) −30.5863 −1.42300
\(463\) 15.5460 0.722484 0.361242 0.932472i \(-0.382353\pi\)
0.361242 + 0.932472i \(0.382353\pi\)
\(464\) 6.66499 0.309415
\(465\) −14.0649 −0.652245
\(466\) −18.4155 −0.853082
\(467\) −3.24823 −0.150310 −0.0751550 0.997172i \(-0.523945\pi\)
−0.0751550 + 0.997172i \(0.523945\pi\)
\(468\) 1.00000 0.0462250
\(469\) 54.4963 2.51641
\(470\) 20.4904 0.945151
\(471\) −12.6431 −0.582564
\(472\) 5.42438 0.249677
\(473\) 50.9363 2.34205
\(474\) 1.41867 0.0651619
\(475\) 41.5077 1.90450
\(476\) 3.46149 0.158657
\(477\) 7.78611 0.356502
\(478\) −20.0784 −0.918365
\(479\) 29.4666 1.34636 0.673181 0.739478i \(-0.264928\pi\)
0.673181 + 0.739478i \(0.264928\pi\)
\(480\) −3.68342 −0.168124
\(481\) 0.460154 0.0209812
\(482\) −20.9951 −0.956302
\(483\) −27.8929 −1.26917
\(484\) 25.9400 1.17909
\(485\) 51.8996 2.35664
\(486\) 1.00000 0.0453609
\(487\) 39.0218 1.76825 0.884123 0.467254i \(-0.154757\pi\)
0.884123 + 0.467254i \(0.154757\pi\)
\(488\) 6.72933 0.304623
\(489\) −22.7600 −1.02924
\(490\) 67.5000 3.04934
\(491\) 2.37906 0.107365 0.0536827 0.998558i \(-0.482904\pi\)
0.0536827 + 0.998558i \(0.482904\pi\)
\(492\) 7.54316 0.340072
\(493\) 4.58442 0.206472
\(494\) −4.84475 −0.217976
\(495\) 22.3872 1.00623
\(496\) −3.81844 −0.171453
\(497\) 71.1457 3.19132
\(498\) −5.18575 −0.232379
\(499\) 6.26699 0.280549 0.140275 0.990113i \(-0.455201\pi\)
0.140275 + 0.990113i \(0.455201\pi\)
\(500\) −13.1408 −0.587674
\(501\) 9.07055 0.405242
\(502\) 5.21065 0.232563
\(503\) 24.3898 1.08749 0.543744 0.839251i \(-0.317006\pi\)
0.543744 + 0.839251i \(0.317006\pi\)
\(504\) −5.03243 −0.224162
\(505\) 22.7240 1.01121
\(506\) 33.6872 1.49758
\(507\) −1.00000 −0.0444116
\(508\) −0.988933 −0.0438768
\(509\) 11.5495 0.511924 0.255962 0.966687i \(-0.417608\pi\)
0.255962 + 0.966687i \(0.417608\pi\)
\(510\) −2.53359 −0.112189
\(511\) −55.6518 −2.46189
\(512\) −1.00000 −0.0441942
\(513\) −4.84475 −0.213901
\(514\) −0.904912 −0.0399140
\(515\) 3.68342 0.162311
\(516\) 8.38067 0.368938
\(517\) −33.8102 −1.48697
\(518\) −2.31569 −0.101746
\(519\) −20.7522 −0.910918
\(520\) 3.68342 0.161528
\(521\) 24.9889 1.09478 0.547391 0.836877i \(-0.315621\pi\)
0.547391 + 0.836877i \(0.315621\pi\)
\(522\) −6.66499 −0.291719
\(523\) 2.47928 0.108411 0.0542057 0.998530i \(-0.482737\pi\)
0.0542057 + 0.998530i \(0.482737\pi\)
\(524\) −12.7607 −0.557455
\(525\) −43.1156 −1.88172
\(526\) −8.88779 −0.387526
\(527\) −2.62646 −0.114411
\(528\) 6.07783 0.264504
\(529\) 7.72077 0.335686
\(530\) 28.6795 1.24576
\(531\) −5.42438 −0.235398
\(532\) 24.3809 1.05705
\(533\) −7.54316 −0.326731
\(534\) 4.44024 0.192148
\(535\) −73.8978 −3.19488
\(536\) −10.8290 −0.467743
\(537\) 14.8788 0.642067
\(538\) 6.00072 0.258709
\(539\) −111.379 −4.79741
\(540\) 3.68342 0.158509
\(541\) −20.5295 −0.882632 −0.441316 0.897352i \(-0.645488\pi\)
−0.441316 + 0.897352i \(0.645488\pi\)
\(542\) 23.0789 0.991325
\(543\) −1.77972 −0.0763749
\(544\) −0.687836 −0.0294907
\(545\) 39.4487 1.68980
\(546\) 5.03243 0.215368
\(547\) 35.3566 1.51174 0.755869 0.654722i \(-0.227214\pi\)
0.755869 + 0.654722i \(0.227214\pi\)
\(548\) 8.89751 0.380083
\(549\) −6.72933 −0.287201
\(550\) 52.0721 2.22036
\(551\) 32.2903 1.37561
\(552\) 5.54263 0.235910
\(553\) 7.13939 0.303598
\(554\) 2.17618 0.0924571
\(555\) 1.69494 0.0719461
\(556\) −8.11255 −0.344049
\(557\) 25.7429 1.09076 0.545380 0.838189i \(-0.316385\pi\)
0.545380 + 0.838189i \(0.316385\pi\)
\(558\) 3.81844 0.161648
\(559\) −8.38067 −0.354465
\(560\) −18.5365 −0.783312
\(561\) 4.18055 0.176503
\(562\) −1.74352 −0.0735461
\(563\) 18.7924 0.792005 0.396003 0.918249i \(-0.370397\pi\)
0.396003 + 0.918249i \(0.370397\pi\)
\(564\) −5.56288 −0.234239
\(565\) 0.845360 0.0355646
\(566\) −12.7346 −0.535274
\(567\) 5.03243 0.211342
\(568\) −14.1374 −0.593193
\(569\) −34.8935 −1.46281 −0.731406 0.681942i \(-0.761136\pi\)
−0.731406 + 0.681942i \(0.761136\pi\)
\(570\) −17.8452 −0.747455
\(571\) 25.9802 1.08724 0.543619 0.839332i \(-0.317054\pi\)
0.543619 + 0.839332i \(0.317054\pi\)
\(572\) −6.07783 −0.254127
\(573\) 27.1288 1.13332
\(574\) 37.9605 1.58444
\(575\) 47.4868 1.98034
\(576\) 1.00000 0.0416667
\(577\) −24.0007 −0.999162 −0.499581 0.866267i \(-0.666513\pi\)
−0.499581 + 0.866267i \(0.666513\pi\)
\(578\) 16.5269 0.687428
\(579\) −25.0012 −1.03902
\(580\) −24.5499 −1.01938
\(581\) −26.0969 −1.08268
\(582\) −14.0901 −0.584053
\(583\) −47.3227 −1.95991
\(584\) 11.0586 0.457609
\(585\) −3.68342 −0.152291
\(586\) 21.2582 0.878166
\(587\) 26.8579 1.10854 0.554271 0.832336i \(-0.312997\pi\)
0.554271 + 0.832336i \(0.312997\pi\)
\(588\) −18.3254 −0.755726
\(589\) −18.4994 −0.762255
\(590\) −19.9802 −0.822574
\(591\) 16.4004 0.674621
\(592\) 0.460154 0.0189122
\(593\) −5.35277 −0.219812 −0.109906 0.993942i \(-0.535055\pi\)
−0.109906 + 0.993942i \(0.535055\pi\)
\(594\) −6.07783 −0.249376
\(595\) −12.7501 −0.522703
\(596\) −15.4097 −0.631204
\(597\) 1.59261 0.0651811
\(598\) −5.54263 −0.226655
\(599\) −4.91192 −0.200696 −0.100348 0.994952i \(-0.531996\pi\)
−0.100348 + 0.994952i \(0.531996\pi\)
\(600\) 8.56755 0.349769
\(601\) 0.846204 0.0345174 0.0172587 0.999851i \(-0.494506\pi\)
0.0172587 + 0.999851i \(0.494506\pi\)
\(602\) 42.1752 1.71893
\(603\) 10.8290 0.440992
\(604\) −14.6840 −0.597482
\(605\) −95.5479 −3.88458
\(606\) −6.16928 −0.250610
\(607\) −17.1652 −0.696713 −0.348357 0.937362i \(-0.613260\pi\)
−0.348357 + 0.937362i \(0.613260\pi\)
\(608\) −4.84475 −0.196481
\(609\) −33.5411 −1.35916
\(610\) −24.7869 −1.00359
\(611\) 5.56288 0.225050
\(612\) 0.687836 0.0278041
\(613\) 29.9714 1.21054 0.605268 0.796022i \(-0.293066\pi\)
0.605268 + 0.796022i \(0.293066\pi\)
\(614\) 23.5910 0.952054
\(615\) −27.7846 −1.12038
\(616\) 30.5863 1.23236
\(617\) −34.6639 −1.39552 −0.697758 0.716334i \(-0.745819\pi\)
−0.697758 + 0.716334i \(0.745819\pi\)
\(618\) −1.00000 −0.0402259
\(619\) 28.2338 1.13481 0.567406 0.823438i \(-0.307947\pi\)
0.567406 + 0.823438i \(0.307947\pi\)
\(620\) 14.0649 0.564861
\(621\) −5.54263 −0.222418
\(622\) −12.7248 −0.510219
\(623\) 22.3452 0.895243
\(624\) −1.00000 −0.0400320
\(625\) 5.56520 0.222608
\(626\) −30.6522 −1.22511
\(627\) 29.4456 1.17594
\(628\) 12.6431 0.504515
\(629\) 0.316511 0.0126201
\(630\) 18.5365 0.738514
\(631\) −37.9817 −1.51203 −0.756014 0.654555i \(-0.772856\pi\)
−0.756014 + 0.654555i \(0.772856\pi\)
\(632\) −1.41867 −0.0564319
\(633\) −12.4299 −0.494044
\(634\) 19.2824 0.765800
\(635\) 3.64265 0.144554
\(636\) −7.78611 −0.308739
\(637\) 18.3254 0.726078
\(638\) 40.5087 1.60376
\(639\) 14.1374 0.559268
\(640\) 3.68342 0.145600
\(641\) 33.2759 1.31432 0.657159 0.753752i \(-0.271758\pi\)
0.657159 + 0.753752i \(0.271758\pi\)
\(642\) 20.0623 0.791796
\(643\) 31.0624 1.22498 0.612491 0.790478i \(-0.290168\pi\)
0.612491 + 0.790478i \(0.290168\pi\)
\(644\) 27.8929 1.09914
\(645\) −30.8695 −1.21549
\(646\) −3.33240 −0.131111
\(647\) 27.9655 1.09944 0.549719 0.835350i \(-0.314735\pi\)
0.549719 + 0.835350i \(0.314735\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 32.9685 1.29413
\(650\) −8.56755 −0.336047
\(651\) 19.2161 0.753137
\(652\) 22.7600 0.891349
\(653\) 0.814236 0.0318635 0.0159318 0.999873i \(-0.494929\pi\)
0.0159318 + 0.999873i \(0.494929\pi\)
\(654\) −10.7098 −0.418787
\(655\) 47.0031 1.83656
\(656\) −7.54316 −0.294511
\(657\) −11.0586 −0.431438
\(658\) −27.9948 −1.09135
\(659\) −18.3533 −0.714943 −0.357472 0.933924i \(-0.616361\pi\)
−0.357472 + 0.933924i \(0.616361\pi\)
\(660\) −22.3872 −0.871420
\(661\) 48.5449 1.88818 0.944090 0.329689i \(-0.106944\pi\)
0.944090 + 0.329689i \(0.106944\pi\)
\(662\) 34.7227 1.34954
\(663\) −0.687836 −0.0267133
\(664\) 5.18575 0.201246
\(665\) −89.8050 −3.48249
\(666\) −0.460154 −0.0178306
\(667\) 36.9416 1.43038
\(668\) −9.07055 −0.350950
\(669\) 12.2201 0.472456
\(670\) 39.8878 1.54100
\(671\) 40.8998 1.57892
\(672\) 5.03243 0.194130
\(673\) −2.13932 −0.0824647 −0.0412323 0.999150i \(-0.513128\pi\)
−0.0412323 + 0.999150i \(0.513128\pi\)
\(674\) 2.87112 0.110592
\(675\) −8.56755 −0.329765
\(676\) 1.00000 0.0384615
\(677\) −12.4305 −0.477743 −0.238871 0.971051i \(-0.576777\pi\)
−0.238871 + 0.971051i \(0.576777\pi\)
\(678\) −0.229504 −0.00881406
\(679\) −70.9074 −2.72118
\(680\) 2.53359 0.0971586
\(681\) 6.31622 0.242038
\(682\) −23.2079 −0.888675
\(683\) 11.9972 0.459060 0.229530 0.973302i \(-0.426281\pi\)
0.229530 + 0.973302i \(0.426281\pi\)
\(684\) 4.84475 0.185244
\(685\) −32.7732 −1.25220
\(686\) −56.9942 −2.17605
\(687\) −9.72002 −0.370842
\(688\) −8.38067 −0.319510
\(689\) 7.78611 0.296627
\(690\) −20.4158 −0.777217
\(691\) −27.0180 −1.02781 −0.513906 0.857847i \(-0.671802\pi\)
−0.513906 + 0.857847i \(0.671802\pi\)
\(692\) 20.7522 0.788878
\(693\) −30.5863 −1.16188
\(694\) −17.9238 −0.680379
\(695\) 29.8819 1.13348
\(696\) 6.66499 0.252636
\(697\) −5.18846 −0.196527
\(698\) 5.68935 0.215345
\(699\) −18.4155 −0.696538
\(700\) 43.1156 1.62962
\(701\) 11.4617 0.432902 0.216451 0.976293i \(-0.430552\pi\)
0.216451 + 0.976293i \(0.430552\pi\)
\(702\) 1.00000 0.0377426
\(703\) 2.22933 0.0840809
\(704\) −6.07783 −0.229067
\(705\) 20.4904 0.771713
\(706\) −12.6046 −0.474380
\(707\) −31.0465 −1.16762
\(708\) 5.42438 0.203861
\(709\) −4.22477 −0.158665 −0.0793323 0.996848i \(-0.525279\pi\)
−0.0793323 + 0.996848i \(0.525279\pi\)
\(710\) 52.0740 1.95430
\(711\) 1.41867 0.0532045
\(712\) −4.44024 −0.166405
\(713\) −21.1642 −0.792607
\(714\) 3.46149 0.129543
\(715\) 22.3872 0.837233
\(716\) −14.8788 −0.556046
\(717\) −20.0784 −0.749842
\(718\) −32.9331 −1.22905
\(719\) 13.9429 0.519983 0.259992 0.965611i \(-0.416280\pi\)
0.259992 + 0.965611i \(0.416280\pi\)
\(720\) −3.68342 −0.137273
\(721\) −5.03243 −0.187418
\(722\) −4.47165 −0.166418
\(723\) −20.9951 −0.780817
\(724\) 1.77972 0.0661426
\(725\) 57.1027 2.12074
\(726\) 25.9400 0.962725
\(727\) 33.4359 1.24007 0.620034 0.784575i \(-0.287119\pi\)
0.620034 + 0.784575i \(0.287119\pi\)
\(728\) −5.03243 −0.186514
\(729\) 1.00000 0.0370370
\(730\) −40.7335 −1.50762
\(731\) −5.76453 −0.213209
\(732\) 6.72933 0.248723
\(733\) 35.4827 1.31058 0.655292 0.755376i \(-0.272546\pi\)
0.655292 + 0.755376i \(0.272546\pi\)
\(734\) 31.8850 1.17690
\(735\) 67.5000 2.48977
\(736\) −5.54263 −0.204304
\(737\) −65.8170 −2.42440
\(738\) 7.54316 0.277668
\(739\) 36.8815 1.35671 0.678353 0.734736i \(-0.262694\pi\)
0.678353 + 0.734736i \(0.262694\pi\)
\(740\) −1.69494 −0.0623072
\(741\) −4.84475 −0.177977
\(742\) −39.1831 −1.43846
\(743\) −25.5306 −0.936628 −0.468314 0.883562i \(-0.655138\pi\)
−0.468314 + 0.883562i \(0.655138\pi\)
\(744\) −3.81844 −0.139991
\(745\) 56.7602 2.07953
\(746\) −18.7119 −0.685092
\(747\) −5.18575 −0.189737
\(748\) −4.18055 −0.152856
\(749\) 100.962 3.68908
\(750\) −13.1408 −0.479833
\(751\) −6.00067 −0.218968 −0.109484 0.993989i \(-0.534920\pi\)
−0.109484 + 0.993989i \(0.534920\pi\)
\(752\) 5.56288 0.202857
\(753\) 5.21065 0.189886
\(754\) −6.66499 −0.242725
\(755\) 54.0872 1.96843
\(756\) −5.03243 −0.183028
\(757\) −21.3919 −0.777503 −0.388752 0.921343i \(-0.627094\pi\)
−0.388752 + 0.921343i \(0.627094\pi\)
\(758\) −0.754775 −0.0274147
\(759\) 33.6872 1.22277
\(760\) 17.8452 0.647315
\(761\) 18.5331 0.671823 0.335911 0.941894i \(-0.390956\pi\)
0.335911 + 0.941894i \(0.390956\pi\)
\(762\) −0.988933 −0.0358253
\(763\) −53.8964 −1.95118
\(764\) −27.1288 −0.981485
\(765\) −2.53359 −0.0916020
\(766\) −2.46500 −0.0890639
\(767\) −5.42438 −0.195863
\(768\) −1.00000 −0.0360844
\(769\) 32.4916 1.17168 0.585838 0.810428i \(-0.300766\pi\)
0.585838 + 0.810428i \(0.300766\pi\)
\(770\) −112.662 −4.06006
\(771\) −0.904912 −0.0325896
\(772\) 25.0012 0.899814
\(773\) 44.8293 1.61240 0.806199 0.591644i \(-0.201521\pi\)
0.806199 + 0.591644i \(0.201521\pi\)
\(774\) 8.38067 0.301237
\(775\) −32.7147 −1.17515
\(776\) 14.0901 0.505804
\(777\) −2.31569 −0.0830751
\(778\) −3.11542 −0.111693
\(779\) −36.5448 −1.30935
\(780\) 3.68342 0.131887
\(781\) −85.9249 −3.07463
\(782\) −3.81242 −0.136332
\(783\) −6.66499 −0.238187
\(784\) 18.3254 0.654478
\(785\) −46.5699 −1.66215
\(786\) −12.7607 −0.455160
\(787\) −50.2159 −1.79000 −0.895001 0.446064i \(-0.852825\pi\)
−0.895001 + 0.446064i \(0.852825\pi\)
\(788\) −16.4004 −0.584238
\(789\) −8.88779 −0.316414
\(790\) 5.22557 0.185917
\(791\) −1.15496 −0.0410658
\(792\) 6.07783 0.215966
\(793\) −6.72933 −0.238966
\(794\) 1.53453 0.0544583
\(795\) 28.6795 1.01716
\(796\) −1.59261 −0.0564485
\(797\) 15.1919 0.538124 0.269062 0.963123i \(-0.413286\pi\)
0.269062 + 0.963123i \(0.413286\pi\)
\(798\) 24.3809 0.863075
\(799\) 3.82635 0.135366
\(800\) −8.56755 −0.302909
\(801\) 4.44024 0.156888
\(802\) −34.0311 −1.20168
\(803\) 67.2125 2.37188
\(804\) −10.8290 −0.381910
\(805\) −102.741 −3.62115
\(806\) 3.81844 0.134499
\(807\) 6.00072 0.211235
\(808\) 6.16928 0.217034
\(809\) −21.1652 −0.744128 −0.372064 0.928207i \(-0.621350\pi\)
−0.372064 + 0.928207i \(0.621350\pi\)
\(810\) 3.68342 0.129422
\(811\) 3.68919 0.129545 0.0647725 0.997900i \(-0.479368\pi\)
0.0647725 + 0.997900i \(0.479368\pi\)
\(812\) 33.5411 1.17706
\(813\) 23.0789 0.809413
\(814\) 2.79674 0.0980256
\(815\) −83.8344 −2.93659
\(816\) −0.687836 −0.0240791
\(817\) −40.6023 −1.42049
\(818\) −18.5088 −0.647144
\(819\) 5.03243 0.175847
\(820\) 27.7846 0.970281
\(821\) −25.6266 −0.894374 −0.447187 0.894440i \(-0.647574\pi\)
−0.447187 + 0.894440i \(0.647574\pi\)
\(822\) 8.89751 0.310336
\(823\) 41.7967 1.45694 0.728471 0.685077i \(-0.240231\pi\)
0.728471 + 0.685077i \(0.240231\pi\)
\(824\) 1.00000 0.0348367
\(825\) 52.0721 1.81292
\(826\) 27.2978 0.949813
\(827\) 9.04757 0.314615 0.157307 0.987550i \(-0.449719\pi\)
0.157307 + 0.987550i \(0.449719\pi\)
\(828\) 5.54263 0.192620
\(829\) −11.8349 −0.411042 −0.205521 0.978653i \(-0.565889\pi\)
−0.205521 + 0.978653i \(0.565889\pi\)
\(830\) −19.1013 −0.663015
\(831\) 2.17618 0.0754909
\(832\) 1.00000 0.0346688
\(833\) 12.6049 0.436732
\(834\) −8.11255 −0.280915
\(835\) 33.4106 1.15622
\(836\) −29.4456 −1.01840
\(837\) 3.81844 0.131985
\(838\) −9.52179 −0.328925
\(839\) 7.23430 0.249756 0.124878 0.992172i \(-0.460146\pi\)
0.124878 + 0.992172i \(0.460146\pi\)
\(840\) −18.5365 −0.639572
\(841\) 15.4221 0.531798
\(842\) 40.2831 1.38825
\(843\) −1.74352 −0.0600501
\(844\) 12.4299 0.427855
\(845\) −3.68342 −0.126713
\(846\) −5.56288 −0.191256
\(847\) 130.541 4.48546
\(848\) 7.78611 0.267376
\(849\) −12.7346 −0.437049
\(850\) −5.89307 −0.202131
\(851\) 2.55046 0.0874288
\(852\) −14.1374 −0.484340
\(853\) 19.9288 0.682348 0.341174 0.940000i \(-0.389175\pi\)
0.341174 + 0.940000i \(0.389175\pi\)
\(854\) 33.8649 1.15883
\(855\) −17.8452 −0.610295
\(856\) −20.0623 −0.685715
\(857\) −37.5409 −1.28237 −0.641187 0.767385i \(-0.721558\pi\)
−0.641187 + 0.767385i \(0.721558\pi\)
\(858\) −6.07783 −0.207494
\(859\) 41.5544 1.41782 0.708909 0.705300i \(-0.249188\pi\)
0.708909 + 0.705300i \(0.249188\pi\)
\(860\) 30.8695 1.05264
\(861\) 37.9605 1.29369
\(862\) 8.41932 0.286763
\(863\) 1.79748 0.0611869 0.0305935 0.999532i \(-0.490260\pi\)
0.0305935 + 0.999532i \(0.490260\pi\)
\(864\) 1.00000 0.0340207
\(865\) −76.4388 −2.59900
\(866\) −26.6609 −0.905975
\(867\) 16.5269 0.561282
\(868\) −19.2161 −0.652236
\(869\) −8.62247 −0.292497
\(870\) −24.5499 −0.832321
\(871\) 10.8290 0.366928
\(872\) 10.7098 0.362680
\(873\) −14.0901 −0.476877
\(874\) −26.8527 −0.908306
\(875\) −66.1301 −2.23561
\(876\) 11.0586 0.373636
\(877\) −3.59307 −0.121329 −0.0606647 0.998158i \(-0.519322\pi\)
−0.0606647 + 0.998158i \(0.519322\pi\)
\(878\) 0.393637 0.0132846
\(879\) 21.2582 0.717020
\(880\) 22.3872 0.754672
\(881\) 20.6360 0.695244 0.347622 0.937635i \(-0.386989\pi\)
0.347622 + 0.937635i \(0.386989\pi\)
\(882\) −18.3254 −0.617048
\(883\) 12.6857 0.426909 0.213454 0.976953i \(-0.431529\pi\)
0.213454 + 0.976953i \(0.431529\pi\)
\(884\) 0.687836 0.0231344
\(885\) −19.9802 −0.671629
\(886\) 12.8404 0.431381
\(887\) 48.4068 1.62534 0.812671 0.582723i \(-0.198013\pi\)
0.812671 + 0.582723i \(0.198013\pi\)
\(888\) 0.460154 0.0154418
\(889\) −4.97674 −0.166915
\(890\) 16.3553 0.548230
\(891\) −6.07783 −0.203615
\(892\) −12.2201 −0.409159
\(893\) 26.9508 0.901873
\(894\) −15.4097 −0.515376
\(895\) 54.8047 1.83192
\(896\) −5.03243 −0.168122
\(897\) −5.54263 −0.185063
\(898\) −3.61376 −0.120593
\(899\) −25.4499 −0.848802
\(900\) 8.56755 0.285585
\(901\) 5.35557 0.178420
\(902\) −45.8461 −1.52651
\(903\) 42.1752 1.40350
\(904\) 0.229504 0.00763320
\(905\) −6.55543 −0.217910
\(906\) −14.6840 −0.487842
\(907\) −8.19429 −0.272087 −0.136044 0.990703i \(-0.543439\pi\)
−0.136044 + 0.990703i \(0.543439\pi\)
\(908\) −6.31622 −0.209611
\(909\) −6.16928 −0.204622
\(910\) 18.5365 0.614480
\(911\) 54.0981 1.79235 0.896176 0.443699i \(-0.146334\pi\)
0.896176 + 0.443699i \(0.146334\pi\)
\(912\) −4.84475 −0.160426
\(913\) 31.5181 1.04310
\(914\) −33.7785 −1.11729
\(915\) −24.7869 −0.819431
\(916\) 9.72002 0.321158
\(917\) −64.2176 −2.12065
\(918\) 0.687836 0.0227020
\(919\) −5.38044 −0.177484 −0.0887421 0.996055i \(-0.528285\pi\)
−0.0887421 + 0.996055i \(0.528285\pi\)
\(920\) 20.4158 0.673090
\(921\) 23.5910 0.777349
\(922\) −39.8441 −1.31220
\(923\) 14.1374 0.465339
\(924\) 30.5863 1.00621
\(925\) 3.94239 0.129625
\(926\) −15.5460 −0.510874
\(927\) −1.00000 −0.0328443
\(928\) −6.66499 −0.218789
\(929\) −2.15526 −0.0707117 −0.0353558 0.999375i \(-0.511256\pi\)
−0.0353558 + 0.999375i \(0.511256\pi\)
\(930\) 14.0649 0.461207
\(931\) 88.7820 2.90971
\(932\) 18.4155 0.603220
\(933\) −12.7248 −0.416592
\(934\) 3.24823 0.106285
\(935\) 15.3987 0.503592
\(936\) −1.00000 −0.0326860
\(937\) 3.85420 0.125911 0.0629557 0.998016i \(-0.479947\pi\)
0.0629557 + 0.998016i \(0.479947\pi\)
\(938\) −54.4963 −1.77937
\(939\) −30.6522 −1.00030
\(940\) −20.4904 −0.668323
\(941\) 42.9530 1.40023 0.700114 0.714031i \(-0.253133\pi\)
0.700114 + 0.714031i \(0.253133\pi\)
\(942\) 12.6431 0.411935
\(943\) −41.8090 −1.36149
\(944\) −5.42438 −0.176548
\(945\) 18.5365 0.602994
\(946\) −50.9363 −1.65608
\(947\) −29.8809 −0.970999 −0.485499 0.874237i \(-0.661362\pi\)
−0.485499 + 0.874237i \(0.661362\pi\)
\(948\) −1.41867 −0.0460764
\(949\) −11.0586 −0.358978
\(950\) −41.5077 −1.34669
\(951\) 19.2824 0.625273
\(952\) −3.46149 −0.112188
\(953\) −5.36459 −0.173776 −0.0868880 0.996218i \(-0.527692\pi\)
−0.0868880 + 0.996218i \(0.527692\pi\)
\(954\) −7.78611 −0.252085
\(955\) 99.9266 3.23355
\(956\) 20.0784 0.649382
\(957\) 40.5087 1.30946
\(958\) −29.4666 −0.952021
\(959\) 44.7761 1.44590
\(960\) 3.68342 0.118882
\(961\) −16.4195 −0.529661
\(962\) −0.460154 −0.0148360
\(963\) 20.0623 0.646499
\(964\) 20.9951 0.676207
\(965\) −92.0899 −2.96448
\(966\) 27.8929 0.897440
\(967\) 55.0991 1.77187 0.885935 0.463810i \(-0.153518\pi\)
0.885935 + 0.463810i \(0.153518\pi\)
\(968\) −25.9400 −0.833744
\(969\) −3.33240 −0.107052
\(970\) −51.8996 −1.66640
\(971\) 39.5765 1.27007 0.635035 0.772484i \(-0.280986\pi\)
0.635035 + 0.772484i \(0.280986\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −40.8259 −1.30882
\(974\) −39.0218 −1.25034
\(975\) −8.56755 −0.274381
\(976\) −6.72933 −0.215401
\(977\) 16.4514 0.526327 0.263164 0.964751i \(-0.415234\pi\)
0.263164 + 0.964751i \(0.415234\pi\)
\(978\) 22.7600 0.727783
\(979\) −26.9870 −0.862510
\(980\) −67.5000 −2.15621
\(981\) −10.7098 −0.341938
\(982\) −2.37906 −0.0759188
\(983\) −20.6326 −0.658076 −0.329038 0.944317i \(-0.606725\pi\)
−0.329038 + 0.944317i \(0.606725\pi\)
\(984\) −7.54316 −0.240467
\(985\) 60.4093 1.92480
\(986\) −4.58442 −0.145998
\(987\) −27.9948 −0.891084
\(988\) 4.84475 0.154132
\(989\) −46.4510 −1.47705
\(990\) −22.3872 −0.711511
\(991\) 26.7235 0.848901 0.424451 0.905451i \(-0.360467\pi\)
0.424451 + 0.905451i \(0.360467\pi\)
\(992\) 3.81844 0.121236
\(993\) 34.7227 1.10189
\(994\) −71.1457 −2.25660
\(995\) 5.86624 0.185972
\(996\) 5.18575 0.164317
\(997\) −32.8776 −1.04124 −0.520622 0.853787i \(-0.674300\pi\)
−0.520622 + 0.853787i \(0.674300\pi\)
\(998\) −6.26699 −0.198378
\(999\) −0.460154 −0.0145586
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.z.1.1 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.z.1.1 14 1.1 even 1 trivial