Properties

Label 4598.2.a.cb.1.5
Level $4598$
Weight $2$
Character 4598.1
Self dual yes
Analytic conductor $36.715$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4598.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7152148494\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 16 x^{6} - 4 x^{5} + 75 x^{4} + 32 x^{3} - 90 x^{2} - 28 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.84109\) of defining polynomial
Character \(\chi\) \(=\) 4598.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.67352 q^{3} +1.00000 q^{4} -0.566489 q^{5} +1.67352 q^{6} +3.40033 q^{7} +1.00000 q^{8} -0.199325 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.67352 q^{3} +1.00000 q^{4} -0.566489 q^{5} +1.67352 q^{6} +3.40033 q^{7} +1.00000 q^{8} -0.199325 q^{9} -0.566489 q^{10} +1.67352 q^{12} +1.35442 q^{13} +3.40033 q^{14} -0.948032 q^{15} +1.00000 q^{16} +4.30453 q^{17} -0.199325 q^{18} -1.00000 q^{19} -0.566489 q^{20} +5.69053 q^{21} -2.71065 q^{23} +1.67352 q^{24} -4.67909 q^{25} +1.35442 q^{26} -5.35414 q^{27} +3.40033 q^{28} +5.64462 q^{29} -0.948032 q^{30} +9.62449 q^{31} +1.00000 q^{32} +4.30453 q^{34} -1.92625 q^{35} -0.199325 q^{36} +3.63304 q^{37} -1.00000 q^{38} +2.26664 q^{39} -0.566489 q^{40} -3.64782 q^{41} +5.69053 q^{42} -7.94488 q^{43} +0.112916 q^{45} -2.71065 q^{46} +13.5117 q^{47} +1.67352 q^{48} +4.56226 q^{49} -4.67909 q^{50} +7.20373 q^{51} +1.35442 q^{52} +4.25789 q^{53} -5.35414 q^{54} +3.40033 q^{56} -1.67352 q^{57} +5.64462 q^{58} +7.97498 q^{59} -0.948032 q^{60} +9.37992 q^{61} +9.62449 q^{62} -0.677772 q^{63} +1.00000 q^{64} -0.767262 q^{65} +1.01003 q^{67} +4.30453 q^{68} -4.53633 q^{69} -1.92625 q^{70} +6.30982 q^{71} -0.199325 q^{72} -14.0378 q^{73} +3.63304 q^{74} -7.83056 q^{75} -1.00000 q^{76} +2.26664 q^{78} -7.11185 q^{79} -0.566489 q^{80} -8.36229 q^{81} -3.64782 q^{82} -1.02452 q^{83} +5.69053 q^{84} -2.43847 q^{85} -7.94488 q^{86} +9.44639 q^{87} -15.1613 q^{89} +0.112916 q^{90} +4.60546 q^{91} -2.71065 q^{92} +16.1068 q^{93} +13.5117 q^{94} +0.566489 q^{95} +1.67352 q^{96} +7.04118 q^{97} +4.56226 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 8 q^{6} + 4 q^{7} + 8 q^{8} + 22 q^{9} + O(q^{10}) \) \( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 8 q^{6} + 4 q^{7} + 8 q^{8} + 22 q^{9} + 8 q^{12} - 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} - 4 q^{17} + 22 q^{18} - 8 q^{19} - 20 q^{21} + 14 q^{23} + 8 q^{24} + 36 q^{25} - 12 q^{26} + 32 q^{27} + 4 q^{28} - 2 q^{29} + 4 q^{30} + 8 q^{32} - 4 q^{34} + 36 q^{35} + 22 q^{36} + 24 q^{37} - 8 q^{38} + 16 q^{39} + 8 q^{41} - 20 q^{42} + 8 q^{43} + 16 q^{45} + 14 q^{46} - 16 q^{47} + 8 q^{48} + 34 q^{49} + 36 q^{50} + 18 q^{51} - 12 q^{52} + 36 q^{53} + 32 q^{54} + 4 q^{56} - 8 q^{57} - 2 q^{58} - 24 q^{59} + 4 q^{60} + 12 q^{61} + 24 q^{63} + 8 q^{64} + 16 q^{65} + 16 q^{67} - 4 q^{68} + 4 q^{69} + 36 q^{70} + 4 q^{71} + 22 q^{72} - 20 q^{73} + 24 q^{74} + 40 q^{75} - 8 q^{76} + 16 q^{78} - 12 q^{79} + 40 q^{81} + 8 q^{82} + 20 q^{83} - 20 q^{84} + 12 q^{85} + 8 q^{86} - 36 q^{87} + 8 q^{89} + 16 q^{90} - 24 q^{91} + 14 q^{92} + 12 q^{93} - 16 q^{94} + 8 q^{96} + 4 q^{97} + 34 q^{98} + O(q^{100}) \)

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
\(3\) 1.67352 0.966208 0.483104 0.875563i \(-0.339509\pi\)
0.483104 + 0.875563i \(0.339509\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.566489 −0.253342 −0.126671 0.991945i \(-0.540429\pi\)
−0.126671 + 0.991945i \(0.540429\pi\)
\(6\) 1.67352 0.683212
\(7\) 3.40033 1.28520 0.642602 0.766200i \(-0.277855\pi\)
0.642602 + 0.766200i \(0.277855\pi\)
\(8\) 1.00000 0.353553
\(9\) −0.199325 −0.0664418
\(10\) −0.566489 −0.179140
\(11\) 0 0
\(12\) 1.67352 0.483104
\(13\) 1.35442 0.375647 0.187824 0.982203i \(-0.439857\pi\)
0.187824 + 0.982203i \(0.439857\pi\)
\(14\) 3.40033 0.908777
\(15\) −0.948032 −0.244781
\(16\) 1.00000 0.250000
\(17\) 4.30453 1.04400 0.522002 0.852945i \(-0.325186\pi\)
0.522002 + 0.852945i \(0.325186\pi\)
\(18\) −0.199325 −0.0469814
\(19\) −1.00000 −0.229416
\(20\) −0.566489 −0.126671
\(21\) 5.69053 1.24178
\(22\) 0 0
\(23\) −2.71065 −0.565210 −0.282605 0.959236i \(-0.591199\pi\)
−0.282605 + 0.959236i \(0.591199\pi\)
\(24\) 1.67352 0.341606
\(25\) −4.67909 −0.935818
\(26\) 1.35442 0.265623
\(27\) −5.35414 −1.03040
\(28\) 3.40033 0.642602
\(29\) 5.64462 1.04818 0.524089 0.851663i \(-0.324406\pi\)
0.524089 + 0.851663i \(0.324406\pi\)
\(30\) −0.948032 −0.173086
\(31\) 9.62449 1.72861 0.864305 0.502968i \(-0.167759\pi\)
0.864305 + 0.502968i \(0.167759\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.30453 0.738222
\(35\) −1.92625 −0.325596
\(36\) −0.199325 −0.0332209
\(37\) 3.63304 0.597269 0.298634 0.954368i \(-0.403469\pi\)
0.298634 + 0.954368i \(0.403469\pi\)
\(38\) −1.00000 −0.162221
\(39\) 2.26664 0.362953
\(40\) −0.566489 −0.0895698
\(41\) −3.64782 −0.569694 −0.284847 0.958573i \(-0.591943\pi\)
−0.284847 + 0.958573i \(0.591943\pi\)
\(42\) 5.69053 0.878068
\(43\) −7.94488 −1.21158 −0.605791 0.795624i \(-0.707143\pi\)
−0.605791 + 0.795624i \(0.707143\pi\)
\(44\) 0 0
\(45\) 0.112916 0.0168325
\(46\) −2.71065 −0.399664
\(47\) 13.5117 1.97088 0.985440 0.170023i \(-0.0543841\pi\)
0.985440 + 0.170023i \(0.0543841\pi\)
\(48\) 1.67352 0.241552
\(49\) 4.56226 0.651752
\(50\) −4.67909 −0.661723
\(51\) 7.20373 1.00872
\(52\) 1.35442 0.187824
\(53\) 4.25789 0.584867 0.292433 0.956286i \(-0.405535\pi\)
0.292433 + 0.956286i \(0.405535\pi\)
\(54\) −5.35414 −0.728606
\(55\) 0 0
\(56\) 3.40033 0.454389
\(57\) −1.67352 −0.221663
\(58\) 5.64462 0.741174
\(59\) 7.97498 1.03825 0.519127 0.854697i \(-0.326257\pi\)
0.519127 + 0.854697i \(0.326257\pi\)
\(60\) −0.948032 −0.122390
\(61\) 9.37992 1.20098 0.600488 0.799634i \(-0.294973\pi\)
0.600488 + 0.799634i \(0.294973\pi\)
\(62\) 9.62449 1.22231
\(63\) −0.677772 −0.0853913
\(64\) 1.00000 0.125000
\(65\) −0.767262 −0.0951671
\(66\) 0 0
\(67\) 1.01003 0.123395 0.0616973 0.998095i \(-0.480349\pi\)
0.0616973 + 0.998095i \(0.480349\pi\)
\(68\) 4.30453 0.522002
\(69\) −4.53633 −0.546110
\(70\) −1.92625 −0.230231
\(71\) 6.30982 0.748838 0.374419 0.927260i \(-0.377842\pi\)
0.374419 + 0.927260i \(0.377842\pi\)
\(72\) −0.199325 −0.0234907
\(73\) −14.0378 −1.64300 −0.821498 0.570211i \(-0.806861\pi\)
−0.821498 + 0.570211i \(0.806861\pi\)
\(74\) 3.63304 0.422333
\(75\) −7.83056 −0.904195
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 2.26664 0.256647
\(79\) −7.11185 −0.800145 −0.400073 0.916483i \(-0.631015\pi\)
−0.400073 + 0.916483i \(0.631015\pi\)
\(80\) −0.566489 −0.0633354
\(81\) −8.36229 −0.929144
\(82\) −3.64782 −0.402835
\(83\) −1.02452 −0.112456 −0.0562280 0.998418i \(-0.517907\pi\)
−0.0562280 + 0.998418i \(0.517907\pi\)
\(84\) 5.69053 0.620888
\(85\) −2.43847 −0.264489
\(86\) −7.94488 −0.856718
\(87\) 9.44639 1.01276
\(88\) 0 0
\(89\) −15.1613 −1.60710 −0.803548 0.595239i \(-0.797057\pi\)
−0.803548 + 0.595239i \(0.797057\pi\)
\(90\) 0.112916 0.0119024
\(91\) 4.60546 0.482784
\(92\) −2.71065 −0.282605
\(93\) 16.1068 1.67020
\(94\) 13.5117 1.39362
\(95\) 0.566489 0.0581206
\(96\) 1.67352 0.170803
\(97\) 7.04118 0.714923 0.357462 0.933928i \(-0.383642\pi\)
0.357462 + 0.933928i \(0.383642\pi\)
\(98\) 4.56226 0.460858
\(99\) 0 0
\(100\) −4.67909 −0.467909
\(101\) −8.12037 −0.808007 −0.404003 0.914758i \(-0.632382\pi\)
−0.404003 + 0.914758i \(0.632382\pi\)
\(102\) 7.20373 0.713276
\(103\) −3.00499 −0.296091 −0.148045 0.988981i \(-0.547298\pi\)
−0.148045 + 0.988981i \(0.547298\pi\)
\(104\) 1.35442 0.132811
\(105\) −3.22362 −0.314593
\(106\) 4.25789 0.413563
\(107\) −7.33334 −0.708941 −0.354471 0.935067i \(-0.615339\pi\)
−0.354471 + 0.935067i \(0.615339\pi\)
\(108\) −5.35414 −0.515202
\(109\) 3.90465 0.373998 0.186999 0.982360i \(-0.440124\pi\)
0.186999 + 0.982360i \(0.440124\pi\)
\(110\) 0 0
\(111\) 6.07998 0.577086
\(112\) 3.40033 0.321301
\(113\) −6.36327 −0.598606 −0.299303 0.954158i \(-0.596754\pi\)
−0.299303 + 0.954158i \(0.596754\pi\)
\(114\) −1.67352 −0.156740
\(115\) 1.53555 0.143191
\(116\) 5.64462 0.524089
\(117\) −0.269969 −0.0249587
\(118\) 7.97498 0.734156
\(119\) 14.6369 1.34176
\(120\) −0.948032 −0.0865431
\(121\) 0 0
\(122\) 9.37992 0.849218
\(123\) −6.10471 −0.550443
\(124\) 9.62449 0.864305
\(125\) 5.48310 0.490423
\(126\) −0.677772 −0.0603808
\(127\) 17.3575 1.54023 0.770115 0.637905i \(-0.220199\pi\)
0.770115 + 0.637905i \(0.220199\pi\)
\(128\) 1.00000 0.0883883
\(129\) −13.2959 −1.17064
\(130\) −0.767262 −0.0672933
\(131\) −3.39115 −0.296286 −0.148143 0.988966i \(-0.547330\pi\)
−0.148143 + 0.988966i \(0.547330\pi\)
\(132\) 0 0
\(133\) −3.40033 −0.294846
\(134\) 1.01003 0.0872531
\(135\) 3.03306 0.261044
\(136\) 4.30453 0.369111
\(137\) 0.0612122 0.00522971 0.00261486 0.999997i \(-0.499168\pi\)
0.00261486 + 0.999997i \(0.499168\pi\)
\(138\) −4.53633 −0.386158
\(139\) −16.7756 −1.42289 −0.711443 0.702744i \(-0.751958\pi\)
−0.711443 + 0.702744i \(0.751958\pi\)
\(140\) −1.92625 −0.162798
\(141\) 22.6121 1.90428
\(142\) 6.30982 0.529508
\(143\) 0 0
\(144\) −0.199325 −0.0166104
\(145\) −3.19761 −0.265547
\(146\) −14.0378 −1.16177
\(147\) 7.63504 0.629728
\(148\) 3.63304 0.298634
\(149\) 8.95206 0.733382 0.366691 0.930343i \(-0.380491\pi\)
0.366691 + 0.930343i \(0.380491\pi\)
\(150\) −7.83056 −0.639362
\(151\) −8.51404 −0.692862 −0.346431 0.938075i \(-0.612607\pi\)
−0.346431 + 0.938075i \(0.612607\pi\)
\(152\) −1.00000 −0.0811107
\(153\) −0.858003 −0.0693654
\(154\) 0 0
\(155\) −5.45217 −0.437929
\(156\) 2.26664 0.181477
\(157\) 12.0876 0.964698 0.482349 0.875979i \(-0.339784\pi\)
0.482349 + 0.875979i \(0.339784\pi\)
\(158\) −7.11185 −0.565788
\(159\) 7.12568 0.565103
\(160\) −0.566489 −0.0447849
\(161\) −9.21711 −0.726410
\(162\) −8.36229 −0.657004
\(163\) 13.6320 1.06774 0.533870 0.845566i \(-0.320737\pi\)
0.533870 + 0.845566i \(0.320737\pi\)
\(164\) −3.64782 −0.284847
\(165\) 0 0
\(166\) −1.02452 −0.0795184
\(167\) −3.77640 −0.292226 −0.146113 0.989268i \(-0.546676\pi\)
−0.146113 + 0.989268i \(0.546676\pi\)
\(168\) 5.69053 0.439034
\(169\) −11.1656 −0.858889
\(170\) −2.43847 −0.187022
\(171\) 0.199325 0.0152428
\(172\) −7.94488 −0.605791
\(173\) −1.92088 −0.146042 −0.0730210 0.997330i \(-0.523264\pi\)
−0.0730210 + 0.997330i \(0.523264\pi\)
\(174\) 9.44639 0.716129
\(175\) −15.9105 −1.20272
\(176\) 0 0
\(177\) 13.3463 1.00317
\(178\) −15.1613 −1.13639
\(179\) −12.9182 −0.965548 −0.482774 0.875745i \(-0.660371\pi\)
−0.482774 + 0.875745i \(0.660371\pi\)
\(180\) 0.112916 0.00841623
\(181\) 17.7782 1.32144 0.660722 0.750630i \(-0.270250\pi\)
0.660722 + 0.750630i \(0.270250\pi\)
\(182\) 4.60546 0.341380
\(183\) 15.6975 1.16039
\(184\) −2.71065 −0.199832
\(185\) −2.05808 −0.151313
\(186\) 16.1068 1.18101
\(187\) 0 0
\(188\) 13.5117 0.985440
\(189\) −18.2059 −1.32428
\(190\) 0.566489 0.0410974
\(191\) −20.7230 −1.49947 −0.749733 0.661741i \(-0.769818\pi\)
−0.749733 + 0.661741i \(0.769818\pi\)
\(192\) 1.67352 0.120776
\(193\) 12.1336 0.873396 0.436698 0.899608i \(-0.356148\pi\)
0.436698 + 0.899608i \(0.356148\pi\)
\(194\) 7.04118 0.505527
\(195\) −1.28403 −0.0919512
\(196\) 4.56226 0.325876
\(197\) −14.2097 −1.01240 −0.506199 0.862417i \(-0.668950\pi\)
−0.506199 + 0.862417i \(0.668950\pi\)
\(198\) 0 0
\(199\) −0.976956 −0.0692545 −0.0346273 0.999400i \(-0.511024\pi\)
−0.0346273 + 0.999400i \(0.511024\pi\)
\(200\) −4.67909 −0.330862
\(201\) 1.69030 0.119225
\(202\) −8.12037 −0.571347
\(203\) 19.1936 1.34712
\(204\) 7.20373 0.504362
\(205\) 2.06645 0.144327
\(206\) −3.00499 −0.209368
\(207\) 0.540301 0.0375535
\(208\) 1.35442 0.0939118
\(209\) 0 0
\(210\) −3.22362 −0.222451
\(211\) −8.44427 −0.581327 −0.290664 0.956825i \(-0.593876\pi\)
−0.290664 + 0.956825i \(0.593876\pi\)
\(212\) 4.25789 0.292433
\(213\) 10.5596 0.723533
\(214\) −7.33334 −0.501297
\(215\) 4.50069 0.306944
\(216\) −5.35414 −0.364303
\(217\) 32.7265 2.22162
\(218\) 3.90465 0.264456
\(219\) −23.4925 −1.58748
\(220\) 0 0
\(221\) 5.83013 0.392177
\(222\) 6.07998 0.408061
\(223\) −0.309593 −0.0207319 −0.0103659 0.999946i \(-0.503300\pi\)
−0.0103659 + 0.999946i \(0.503300\pi\)
\(224\) 3.40033 0.227194
\(225\) 0.932661 0.0621774
\(226\) −6.36327 −0.423278
\(227\) 17.8942 1.18768 0.593840 0.804583i \(-0.297611\pi\)
0.593840 + 0.804583i \(0.297611\pi\)
\(228\) −1.67352 −0.110832
\(229\) −4.77162 −0.315318 −0.157659 0.987494i \(-0.550395\pi\)
−0.157659 + 0.987494i \(0.550395\pi\)
\(230\) 1.53555 0.101251
\(231\) 0 0
\(232\) 5.64462 0.370587
\(233\) −12.5120 −0.819686 −0.409843 0.912156i \(-0.634417\pi\)
−0.409843 + 0.912156i \(0.634417\pi\)
\(234\) −0.269969 −0.0176484
\(235\) −7.65422 −0.499306
\(236\) 7.97498 0.519127
\(237\) −11.9018 −0.773107
\(238\) 14.6369 0.948766
\(239\) 11.2492 0.727653 0.363826 0.931467i \(-0.381470\pi\)
0.363826 + 0.931467i \(0.381470\pi\)
\(240\) −0.948032 −0.0611952
\(241\) 23.4105 1.50801 0.754003 0.656871i \(-0.228121\pi\)
0.754003 + 0.656871i \(0.228121\pi\)
\(242\) 0 0
\(243\) 2.06794 0.132658
\(244\) 9.37992 0.600488
\(245\) −2.58447 −0.165116
\(246\) −6.10471 −0.389222
\(247\) −1.35442 −0.0861794
\(248\) 9.62449 0.611156
\(249\) −1.71456 −0.108656
\(250\) 5.48310 0.346782
\(251\) −27.2448 −1.71967 −0.859837 0.510568i \(-0.829435\pi\)
−0.859837 + 0.510568i \(0.829435\pi\)
\(252\) −0.677772 −0.0426956
\(253\) 0 0
\(254\) 17.3575 1.08911
\(255\) −4.08084 −0.255552
\(256\) 1.00000 0.0625000
\(257\) 18.9406 1.18148 0.590742 0.806861i \(-0.298835\pi\)
0.590742 + 0.806861i \(0.298835\pi\)
\(258\) −13.2959 −0.827768
\(259\) 12.3536 0.767613
\(260\) −0.767262 −0.0475835
\(261\) −1.12511 −0.0696428
\(262\) −3.39115 −0.209506
\(263\) −28.1414 −1.73527 −0.867637 0.497198i \(-0.834362\pi\)
−0.867637 + 0.497198i \(0.834362\pi\)
\(264\) 0 0
\(265\) −2.41205 −0.148171
\(266\) −3.40033 −0.208488
\(267\) −25.3728 −1.55279
\(268\) 1.01003 0.0616973
\(269\) −0.144365 −0.00880207 −0.00440104 0.999990i \(-0.501401\pi\)
−0.00440104 + 0.999990i \(0.501401\pi\)
\(270\) 3.03306 0.184586
\(271\) 7.88292 0.478853 0.239427 0.970914i \(-0.423041\pi\)
0.239427 + 0.970914i \(0.423041\pi\)
\(272\) 4.30453 0.261001
\(273\) 7.70734 0.466469
\(274\) 0.0612122 0.00369797
\(275\) 0 0
\(276\) −4.53633 −0.273055
\(277\) −16.4244 −0.986846 −0.493423 0.869790i \(-0.664254\pi\)
−0.493423 + 0.869790i \(0.664254\pi\)
\(278\) −16.7756 −1.00613
\(279\) −1.91841 −0.114852
\(280\) −1.92625 −0.115116
\(281\) 1.71799 0.102487 0.0512435 0.998686i \(-0.483682\pi\)
0.0512435 + 0.998686i \(0.483682\pi\)
\(282\) 22.6121 1.34653
\(283\) 5.05430 0.300447 0.150223 0.988652i \(-0.452001\pi\)
0.150223 + 0.988652i \(0.452001\pi\)
\(284\) 6.30982 0.374419
\(285\) 0.948032 0.0561566
\(286\) 0 0
\(287\) −12.4038 −0.732174
\(288\) −0.199325 −0.0117454
\(289\) 1.52902 0.0899424
\(290\) −3.19761 −0.187770
\(291\) 11.7836 0.690765
\(292\) −14.0378 −0.821498
\(293\) −19.2526 −1.12475 −0.562376 0.826882i \(-0.690112\pi\)
−0.562376 + 0.826882i \(0.690112\pi\)
\(294\) 7.63504 0.445285
\(295\) −4.51774 −0.263033
\(296\) 3.63304 0.211166
\(297\) 0 0
\(298\) 8.95206 0.518579
\(299\) −3.67135 −0.212319
\(300\) −7.83056 −0.452097
\(301\) −27.0152 −1.55713
\(302\) −8.51404 −0.489928
\(303\) −13.5896 −0.780703
\(304\) −1.00000 −0.0573539
\(305\) −5.31362 −0.304257
\(306\) −0.858003 −0.0490487
\(307\) −12.2585 −0.699629 −0.349814 0.936819i \(-0.613755\pi\)
−0.349814 + 0.936819i \(0.613755\pi\)
\(308\) 0 0
\(309\) −5.02892 −0.286085
\(310\) −5.45217 −0.309663
\(311\) −5.48085 −0.310791 −0.155395 0.987852i \(-0.549665\pi\)
−0.155395 + 0.987852i \(0.549665\pi\)
\(312\) 2.26664 0.128323
\(313\) 8.75492 0.494858 0.247429 0.968906i \(-0.420414\pi\)
0.247429 + 0.968906i \(0.420414\pi\)
\(314\) 12.0876 0.682144
\(315\) 0.383951 0.0216332
\(316\) −7.11185 −0.400073
\(317\) −18.4394 −1.03566 −0.517829 0.855484i \(-0.673260\pi\)
−0.517829 + 0.855484i \(0.673260\pi\)
\(318\) 7.12568 0.399588
\(319\) 0 0
\(320\) −0.566489 −0.0316677
\(321\) −12.2725 −0.684985
\(322\) −9.21711 −0.513650
\(323\) −4.30453 −0.239511
\(324\) −8.36229 −0.464572
\(325\) −6.33743 −0.351537
\(326\) 13.6320 0.755007
\(327\) 6.53452 0.361360
\(328\) −3.64782 −0.201417
\(329\) 45.9442 2.53298
\(330\) 0 0
\(331\) 6.57096 0.361172 0.180586 0.983559i \(-0.442201\pi\)
0.180586 + 0.983559i \(0.442201\pi\)
\(332\) −1.02452 −0.0562280
\(333\) −0.724158 −0.0396836
\(334\) −3.77640 −0.206635
\(335\) −0.572170 −0.0312610
\(336\) 5.69053 0.310444
\(337\) −23.5185 −1.28114 −0.640568 0.767902i \(-0.721301\pi\)
−0.640568 + 0.767902i \(0.721301\pi\)
\(338\) −11.1656 −0.607326
\(339\) −10.6491 −0.578378
\(340\) −2.43847 −0.132245
\(341\) 0 0
\(342\) 0.199325 0.0107783
\(343\) −8.28912 −0.447571
\(344\) −7.94488 −0.428359
\(345\) 2.56978 0.138352
\(346\) −1.92088 −0.103267
\(347\) −6.22613 −0.334236 −0.167118 0.985937i \(-0.553446\pi\)
−0.167118 + 0.985937i \(0.553446\pi\)
\(348\) 9.44639 0.506379
\(349\) 22.3550 1.19664 0.598319 0.801258i \(-0.295836\pi\)
0.598319 + 0.801258i \(0.295836\pi\)
\(350\) −15.9105 −0.850450
\(351\) −7.25173 −0.387069
\(352\) 0 0
\(353\) 36.4241 1.93866 0.969329 0.245768i \(-0.0790402\pi\)
0.969329 + 0.245768i \(0.0790402\pi\)
\(354\) 13.3463 0.709348
\(355\) −3.57444 −0.189712
\(356\) −15.1613 −0.803548
\(357\) 24.4951 1.29642
\(358\) −12.9182 −0.682746
\(359\) 7.54684 0.398307 0.199153 0.979968i \(-0.436181\pi\)
0.199153 + 0.979968i \(0.436181\pi\)
\(360\) 0.112916 0.00595118
\(361\) 1.00000 0.0526316
\(362\) 17.7782 0.934403
\(363\) 0 0
\(364\) 4.60546 0.241392
\(365\) 7.95224 0.416240
\(366\) 15.6975 0.820521
\(367\) 25.7229 1.34273 0.671363 0.741129i \(-0.265709\pi\)
0.671363 + 0.741129i \(0.265709\pi\)
\(368\) −2.71065 −0.141302
\(369\) 0.727103 0.0378515
\(370\) −2.05808 −0.106995
\(371\) 14.4783 0.751674
\(372\) 16.1068 0.835099
\(373\) −7.13854 −0.369619 −0.184810 0.982774i \(-0.559167\pi\)
−0.184810 + 0.982774i \(0.559167\pi\)
\(374\) 0 0
\(375\) 9.17609 0.473851
\(376\) 13.5117 0.696811
\(377\) 7.64515 0.393745
\(378\) −18.2059 −0.936408
\(379\) −27.1836 −1.39633 −0.698165 0.715937i \(-0.746000\pi\)
−0.698165 + 0.715937i \(0.746000\pi\)
\(380\) 0.566489 0.0290603
\(381\) 29.0482 1.48818
\(382\) −20.7230 −1.06028
\(383\) −30.3425 −1.55043 −0.775215 0.631698i \(-0.782358\pi\)
−0.775215 + 0.631698i \(0.782358\pi\)
\(384\) 1.67352 0.0854015
\(385\) 0 0
\(386\) 12.1336 0.617584
\(387\) 1.58362 0.0804997
\(388\) 7.04118 0.357462
\(389\) 5.82380 0.295278 0.147639 0.989041i \(-0.452833\pi\)
0.147639 + 0.989041i \(0.452833\pi\)
\(390\) −1.28403 −0.0650193
\(391\) −11.6681 −0.590081
\(392\) 4.56226 0.230429
\(393\) −5.67516 −0.286274
\(394\) −14.2097 −0.715874
\(395\) 4.02878 0.202710
\(396\) 0 0
\(397\) 2.13644 0.107225 0.0536124 0.998562i \(-0.482926\pi\)
0.0536124 + 0.998562i \(0.482926\pi\)
\(398\) −0.976956 −0.0489704
\(399\) −5.69053 −0.284883
\(400\) −4.67909 −0.233954
\(401\) 3.52506 0.176033 0.0880164 0.996119i \(-0.471947\pi\)
0.0880164 + 0.996119i \(0.471947\pi\)
\(402\) 1.69030 0.0843047
\(403\) 13.0356 0.649347
\(404\) −8.12037 −0.404003
\(405\) 4.73715 0.235391
\(406\) 19.1936 0.952561
\(407\) 0 0
\(408\) 7.20373 0.356638
\(409\) 13.0620 0.645874 0.322937 0.946421i \(-0.395330\pi\)
0.322937 + 0.946421i \(0.395330\pi\)
\(410\) 2.06645 0.102055
\(411\) 0.102440 0.00505299
\(412\) −3.00499 −0.148045
\(413\) 27.1176 1.33437
\(414\) 0.540301 0.0265544
\(415\) 0.580381 0.0284898
\(416\) 1.35442 0.0664057
\(417\) −28.0743 −1.37480
\(418\) 0 0
\(419\) 24.4288 1.19342 0.596712 0.802456i \(-0.296474\pi\)
0.596712 + 0.802456i \(0.296474\pi\)
\(420\) −3.22362 −0.157297
\(421\) −11.0083 −0.536514 −0.268257 0.963347i \(-0.586448\pi\)
−0.268257 + 0.963347i \(0.586448\pi\)
\(422\) −8.44427 −0.411061
\(423\) −2.69322 −0.130949
\(424\) 4.25789 0.206782
\(425\) −20.1413 −0.976997
\(426\) 10.5596 0.511615
\(427\) 31.8948 1.54350
\(428\) −7.33334 −0.354471
\(429\) 0 0
\(430\) 4.50069 0.217042
\(431\) −13.5147 −0.650978 −0.325489 0.945546i \(-0.605529\pi\)
−0.325489 + 0.945546i \(0.605529\pi\)
\(432\) −5.35414 −0.257601
\(433\) 22.8978 1.10040 0.550199 0.835033i \(-0.314552\pi\)
0.550199 + 0.835033i \(0.314552\pi\)
\(434\) 32.7265 1.57092
\(435\) −5.35128 −0.256574
\(436\) 3.90465 0.186999
\(437\) 2.71065 0.129668
\(438\) −23.4925 −1.12252
\(439\) −4.19717 −0.200320 −0.100160 0.994971i \(-0.531935\pi\)
−0.100160 + 0.994971i \(0.531935\pi\)
\(440\) 0 0
\(441\) −0.909374 −0.0433035
\(442\) 5.83013 0.277311
\(443\) −31.0222 −1.47391 −0.736954 0.675943i \(-0.763737\pi\)
−0.736954 + 0.675943i \(0.763737\pi\)
\(444\) 6.07998 0.288543
\(445\) 8.58873 0.407145
\(446\) −0.309593 −0.0146596
\(447\) 14.9815 0.708599
\(448\) 3.40033 0.160651
\(449\) −30.0246 −1.41695 −0.708474 0.705737i \(-0.750616\pi\)
−0.708474 + 0.705737i \(0.750616\pi\)
\(450\) 0.932661 0.0439661
\(451\) 0 0
\(452\) −6.36327 −0.299303
\(453\) −14.2484 −0.669449
\(454\) 17.8942 0.839817
\(455\) −2.60894 −0.122309
\(456\) −1.67352 −0.0783698
\(457\) −9.16793 −0.428858 −0.214429 0.976740i \(-0.568789\pi\)
−0.214429 + 0.976740i \(0.568789\pi\)
\(458\) −4.77162 −0.222963
\(459\) −23.0471 −1.07575
\(460\) 1.53555 0.0715956
\(461\) −30.4549 −1.41843 −0.709213 0.704995i \(-0.750949\pi\)
−0.709213 + 0.704995i \(0.750949\pi\)
\(462\) 0 0
\(463\) 16.4686 0.765359 0.382679 0.923881i \(-0.375001\pi\)
0.382679 + 0.923881i \(0.375001\pi\)
\(464\) 5.64462 0.262045
\(465\) −9.12433 −0.423131
\(466\) −12.5120 −0.579606
\(467\) −36.8201 −1.70383 −0.851916 0.523679i \(-0.824559\pi\)
−0.851916 + 0.523679i \(0.824559\pi\)
\(468\) −0.269969 −0.0124793
\(469\) 3.43443 0.158587
\(470\) −7.65422 −0.353063
\(471\) 20.2289 0.932099
\(472\) 7.97498 0.367078
\(473\) 0 0
\(474\) −11.9018 −0.546669
\(475\) 4.67909 0.214691
\(476\) 14.6369 0.670879
\(477\) −0.848706 −0.0388596
\(478\) 11.2492 0.514528
\(479\) −31.1691 −1.42415 −0.712077 0.702102i \(-0.752245\pi\)
−0.712077 + 0.702102i \(0.752245\pi\)
\(480\) −0.948032 −0.0432715
\(481\) 4.92065 0.224362
\(482\) 23.4105 1.06632
\(483\) −15.4250 −0.701864
\(484\) 0 0
\(485\) −3.98875 −0.181120
\(486\) 2.06794 0.0938037
\(487\) 40.4109 1.83119 0.915597 0.402097i \(-0.131718\pi\)
0.915597 + 0.402097i \(0.131718\pi\)
\(488\) 9.37992 0.424609
\(489\) 22.8134 1.03166
\(490\) −2.58447 −0.116755
\(491\) −9.86715 −0.445298 −0.222649 0.974899i \(-0.571470\pi\)
−0.222649 + 0.974899i \(0.571470\pi\)
\(492\) −6.10471 −0.275222
\(493\) 24.2974 1.09430
\(494\) −1.35442 −0.0609380
\(495\) 0 0
\(496\) 9.62449 0.432153
\(497\) 21.4555 0.962410
\(498\) −1.71456 −0.0768313
\(499\) −13.4498 −0.602094 −0.301047 0.953609i \(-0.597336\pi\)
−0.301047 + 0.953609i \(0.597336\pi\)
\(500\) 5.48310 0.245212
\(501\) −6.31988 −0.282352
\(502\) −27.2448 −1.21599
\(503\) −22.5378 −1.00491 −0.502456 0.864603i \(-0.667570\pi\)
−0.502456 + 0.864603i \(0.667570\pi\)
\(504\) −0.677772 −0.0301904
\(505\) 4.60010 0.204702
\(506\) 0 0
\(507\) −18.6858 −0.829866
\(508\) 17.3575 0.770115
\(509\) 4.87076 0.215893 0.107946 0.994157i \(-0.465573\pi\)
0.107946 + 0.994157i \(0.465573\pi\)
\(510\) −4.08084 −0.180702
\(511\) −47.7331 −2.11159
\(512\) 1.00000 0.0441942
\(513\) 5.35414 0.236391
\(514\) 18.9406 0.835435
\(515\) 1.70230 0.0750121
\(516\) −13.2959 −0.585321
\(517\) 0 0
\(518\) 12.3536 0.542784
\(519\) −3.21464 −0.141107
\(520\) −0.767262 −0.0336466
\(521\) 6.08484 0.266582 0.133291 0.991077i \(-0.457446\pi\)
0.133291 + 0.991077i \(0.457446\pi\)
\(522\) −1.12511 −0.0492449
\(523\) 40.9388 1.79013 0.895065 0.445936i \(-0.147129\pi\)
0.895065 + 0.445936i \(0.147129\pi\)
\(524\) −3.39115 −0.148143
\(525\) −26.6265 −1.16208
\(526\) −28.1414 −1.22702
\(527\) 41.4290 1.80467
\(528\) 0 0
\(529\) −15.6524 −0.680538
\(530\) −2.41205 −0.104773
\(531\) −1.58961 −0.0689834
\(532\) −3.40033 −0.147423
\(533\) −4.94067 −0.214004
\(534\) −25.3728 −1.09799
\(535\) 4.15426 0.179604
\(536\) 1.01003 0.0436266
\(537\) −21.6188 −0.932921
\(538\) −0.144365 −0.00622400
\(539\) 0 0
\(540\) 3.03306 0.130522
\(541\) 20.1046 0.864366 0.432183 0.901786i \(-0.357743\pi\)
0.432183 + 0.901786i \(0.357743\pi\)
\(542\) 7.88292 0.338601
\(543\) 29.7522 1.27679
\(544\) 4.30453 0.184555
\(545\) −2.21194 −0.0947493
\(546\) 7.70734 0.329844
\(547\) 30.7418 1.31442 0.657212 0.753706i \(-0.271736\pi\)
0.657212 + 0.753706i \(0.271736\pi\)
\(548\) 0.0612122 0.00261486
\(549\) −1.86965 −0.0797949
\(550\) 0 0
\(551\) −5.64462 −0.240469
\(552\) −4.53633 −0.193079
\(553\) −24.1826 −1.02835
\(554\) −16.4244 −0.697805
\(555\) −3.44424 −0.146200
\(556\) −16.7756 −0.711443
\(557\) 9.53569 0.404040 0.202020 0.979381i \(-0.435249\pi\)
0.202020 + 0.979381i \(0.435249\pi\)
\(558\) −1.91841 −0.0812126
\(559\) −10.7607 −0.455128
\(560\) −1.92625 −0.0813990
\(561\) 0 0
\(562\) 1.71799 0.0724692
\(563\) −38.1065 −1.60600 −0.802998 0.595981i \(-0.796763\pi\)
−0.802998 + 0.595981i \(0.796763\pi\)
\(564\) 22.6121 0.952140
\(565\) 3.60472 0.151652
\(566\) 5.05430 0.212448
\(567\) −28.4346 −1.19414
\(568\) 6.30982 0.264754
\(569\) −22.1572 −0.928879 −0.464439 0.885605i \(-0.653744\pi\)
−0.464439 + 0.885605i \(0.653744\pi\)
\(570\) 0.948032 0.0397087
\(571\) −19.3268 −0.808801 −0.404400 0.914582i \(-0.632520\pi\)
−0.404400 + 0.914582i \(0.632520\pi\)
\(572\) 0 0
\(573\) −34.6804 −1.44880
\(574\) −12.4038 −0.517725
\(575\) 12.6834 0.528933
\(576\) −0.199325 −0.00830522
\(577\) 26.9783 1.12312 0.561560 0.827436i \(-0.310201\pi\)
0.561560 + 0.827436i \(0.310201\pi\)
\(578\) 1.52902 0.0635989
\(579\) 20.3058 0.843882
\(580\) −3.19761 −0.132774
\(581\) −3.48372 −0.144529
\(582\) 11.7836 0.488444
\(583\) 0 0
\(584\) −14.0378 −0.580887
\(585\) 0.152935 0.00632307
\(586\) −19.2526 −0.795319
\(587\) −27.5285 −1.13622 −0.568112 0.822951i \(-0.692326\pi\)
−0.568112 + 0.822951i \(0.692326\pi\)
\(588\) 7.63504 0.314864
\(589\) −9.62449 −0.396570
\(590\) −4.51774 −0.185992
\(591\) −23.7802 −0.978188
\(592\) 3.63304 0.149317
\(593\) −0.523811 −0.0215103 −0.0107552 0.999942i \(-0.503424\pi\)
−0.0107552 + 0.999942i \(0.503424\pi\)
\(594\) 0 0
\(595\) −8.29162 −0.339923
\(596\) 8.95206 0.366691
\(597\) −1.63496 −0.0669143
\(598\) −3.67135 −0.150133
\(599\) −2.50378 −0.102302 −0.0511508 0.998691i \(-0.516289\pi\)
−0.0511508 + 0.998691i \(0.516289\pi\)
\(600\) −7.83056 −0.319681
\(601\) −19.0143 −0.775608 −0.387804 0.921742i \(-0.626766\pi\)
−0.387804 + 0.921742i \(0.626766\pi\)
\(602\) −27.0152 −1.10106
\(603\) −0.201324 −0.00819855
\(604\) −8.51404 −0.346431
\(605\) 0 0
\(606\) −13.5896 −0.552040
\(607\) −37.9481 −1.54026 −0.770132 0.637884i \(-0.779810\pi\)
−0.770132 + 0.637884i \(0.779810\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 32.1209 1.30160
\(610\) −5.31362 −0.215142
\(611\) 18.3004 0.740356
\(612\) −0.858003 −0.0346827
\(613\) −17.9322 −0.724274 −0.362137 0.932125i \(-0.617953\pi\)
−0.362137 + 0.932125i \(0.617953\pi\)
\(614\) −12.2585 −0.494712
\(615\) 3.45825 0.139450
\(616\) 0 0
\(617\) −40.0578 −1.61267 −0.806333 0.591462i \(-0.798551\pi\)
−0.806333 + 0.591462i \(0.798551\pi\)
\(618\) −5.02892 −0.202293
\(619\) −24.2014 −0.972737 −0.486368 0.873754i \(-0.661679\pi\)
−0.486368 + 0.873754i \(0.661679\pi\)
\(620\) −5.45217 −0.218965
\(621\) 14.5132 0.582395
\(622\) −5.48085 −0.219762
\(623\) −51.5535 −2.06545
\(624\) 2.26664 0.0907383
\(625\) 20.2893 0.811573
\(626\) 8.75492 0.349917
\(627\) 0 0
\(628\) 12.0876 0.482349
\(629\) 15.6386 0.623550
\(630\) 0.383951 0.0152970
\(631\) 48.8748 1.94567 0.972837 0.231489i \(-0.0743599\pi\)
0.972837 + 0.231489i \(0.0743599\pi\)
\(632\) −7.11185 −0.282894
\(633\) −14.1317 −0.561683
\(634\) −18.4394 −0.732320
\(635\) −9.83284 −0.390204
\(636\) 7.12568 0.282552
\(637\) 6.17920 0.244829
\(638\) 0 0
\(639\) −1.25771 −0.0497541
\(640\) −0.566489 −0.0223925
\(641\) −13.6718 −0.540002 −0.270001 0.962860i \(-0.587024\pi\)
−0.270001 + 0.962860i \(0.587024\pi\)
\(642\) −12.2725 −0.484357
\(643\) 35.5605 1.40237 0.701185 0.712980i \(-0.252655\pi\)
0.701185 + 0.712980i \(0.252655\pi\)
\(644\) −9.21711 −0.363205
\(645\) 7.53200 0.296572
\(646\) −4.30453 −0.169360
\(647\) −34.6366 −1.36170 −0.680852 0.732421i \(-0.738391\pi\)
−0.680852 + 0.732421i \(0.738391\pi\)
\(648\) −8.36229 −0.328502
\(649\) 0 0
\(650\) −6.33743 −0.248574
\(651\) 54.7685 2.14655
\(652\) 13.6320 0.533870
\(653\) 7.70803 0.301638 0.150819 0.988561i \(-0.451809\pi\)
0.150819 + 0.988561i \(0.451809\pi\)
\(654\) 6.53452 0.255520
\(655\) 1.92105 0.0750617
\(656\) −3.64782 −0.142424
\(657\) 2.79808 0.109164
\(658\) 45.9442 1.79109
\(659\) −2.66717 −0.103898 −0.0519491 0.998650i \(-0.516543\pi\)
−0.0519491 + 0.998650i \(0.516543\pi\)
\(660\) 0 0
\(661\) 36.9320 1.43649 0.718244 0.695792i \(-0.244946\pi\)
0.718244 + 0.695792i \(0.244946\pi\)
\(662\) 6.57096 0.255387
\(663\) 9.75684 0.378924
\(664\) −1.02452 −0.0397592
\(665\) 1.92625 0.0746968
\(666\) −0.724158 −0.0280605
\(667\) −15.3006 −0.592441
\(668\) −3.77640 −0.146113
\(669\) −0.518110 −0.0200313
\(670\) −0.572170 −0.0221049
\(671\) 0 0
\(672\) 5.69053 0.219517
\(673\) −49.1070 −1.89293 −0.946467 0.322799i \(-0.895376\pi\)
−0.946467 + 0.322799i \(0.895376\pi\)
\(674\) −23.5185 −0.905900
\(675\) 25.0525 0.964271
\(676\) −11.1656 −0.429445
\(677\) −3.66330 −0.140792 −0.0703961 0.997519i \(-0.522426\pi\)
−0.0703961 + 0.997519i \(0.522426\pi\)
\(678\) −10.6491 −0.408975
\(679\) 23.9423 0.918823
\(680\) −2.43847 −0.0935112
\(681\) 29.9463 1.14755
\(682\) 0 0
\(683\) −23.1273 −0.884941 −0.442471 0.896783i \(-0.645898\pi\)
−0.442471 + 0.896783i \(0.645898\pi\)
\(684\) 0.199325 0.00762139
\(685\) −0.0346761 −0.00132490
\(686\) −8.28912 −0.316480
\(687\) −7.98541 −0.304662
\(688\) −7.94488 −0.302896
\(689\) 5.76696 0.219704
\(690\) 2.56978 0.0978300
\(691\) 46.5289 1.77004 0.885021 0.465552i \(-0.154144\pi\)
0.885021 + 0.465552i \(0.154144\pi\)
\(692\) −1.92088 −0.0730210
\(693\) 0 0
\(694\) −6.22613 −0.236341
\(695\) 9.50319 0.360476
\(696\) 9.44639 0.358064
\(697\) −15.7022 −0.594763
\(698\) 22.3550 0.846151
\(699\) −20.9390 −0.791987
\(700\) −15.9105 −0.601359
\(701\) −15.3125 −0.578344 −0.289172 0.957277i \(-0.593380\pi\)
−0.289172 + 0.957277i \(0.593380\pi\)
\(702\) −7.25173 −0.273699
\(703\) −3.63304 −0.137023
\(704\) 0 0
\(705\) −12.8095 −0.482434
\(706\) 36.4241 1.37084
\(707\) −27.6119 −1.03845
\(708\) 13.3463 0.501584
\(709\) 35.5773 1.33613 0.668067 0.744101i \(-0.267122\pi\)
0.668067 + 0.744101i \(0.267122\pi\)
\(710\) −3.57444 −0.134147
\(711\) 1.41757 0.0531631
\(712\) −15.1613 −0.568195
\(713\) −26.0886 −0.977027
\(714\) 24.4951 0.916706
\(715\) 0 0
\(716\) −12.9182 −0.482774
\(717\) 18.8258 0.703064
\(718\) 7.54684 0.281645
\(719\) 41.8647 1.56129 0.780646 0.624974i \(-0.214890\pi\)
0.780646 + 0.624974i \(0.214890\pi\)
\(720\) 0.112916 0.00420812
\(721\) −10.2180 −0.380537
\(722\) 1.00000 0.0372161
\(723\) 39.1780 1.45705
\(724\) 17.7782 0.660722
\(725\) −26.4117 −0.980904
\(726\) 0 0
\(727\) 17.6492 0.654574 0.327287 0.944925i \(-0.393866\pi\)
0.327287 + 0.944925i \(0.393866\pi\)
\(728\) 4.60546 0.170690
\(729\) 28.5476 1.05732
\(730\) 7.95224 0.294326
\(731\) −34.1990 −1.26490
\(732\) 15.6975 0.580196
\(733\) −42.7110 −1.57756 −0.788782 0.614672i \(-0.789288\pi\)
−0.788782 + 0.614672i \(0.789288\pi\)
\(734\) 25.7229 0.949450
\(735\) −4.32517 −0.159536
\(736\) −2.71065 −0.0999159
\(737\) 0 0
\(738\) 0.727103 0.0267650
\(739\) −5.03814 −0.185331 −0.0926655 0.995697i \(-0.529539\pi\)
−0.0926655 + 0.995697i \(0.529539\pi\)
\(740\) −2.05808 −0.0756565
\(741\) −2.26664 −0.0832672
\(742\) 14.4783 0.531514
\(743\) 22.7649 0.835165 0.417583 0.908639i \(-0.362877\pi\)
0.417583 + 0.908639i \(0.362877\pi\)
\(744\) 16.1068 0.590504
\(745\) −5.07125 −0.185796
\(746\) −7.13854 −0.261360
\(747\) 0.204213 0.00747177
\(748\) 0 0
\(749\) −24.9358 −0.911134
\(750\) 9.17609 0.335063
\(751\) −10.7708 −0.393031 −0.196515 0.980501i \(-0.562963\pi\)
−0.196515 + 0.980501i \(0.562963\pi\)
\(752\) 13.5117 0.492720
\(753\) −45.5947 −1.66156
\(754\) 7.64515 0.278420
\(755\) 4.82311 0.175531
\(756\) −18.2059 −0.662141
\(757\) −39.7209 −1.44368 −0.721841 0.692059i \(-0.756704\pi\)
−0.721841 + 0.692059i \(0.756704\pi\)
\(758\) −27.1836 −0.987355
\(759\) 0 0
\(760\) 0.566489 0.0205487
\(761\) 50.5954 1.83408 0.917041 0.398794i \(-0.130571\pi\)
0.917041 + 0.398794i \(0.130571\pi\)
\(762\) 29.0482 1.05230
\(763\) 13.2771 0.480664
\(764\) −20.7230 −0.749733
\(765\) 0.486049 0.0175731
\(766\) −30.3425 −1.09632
\(767\) 10.8014 0.390017
\(768\) 1.67352 0.0603880
\(769\) −21.0614 −0.759495 −0.379748 0.925090i \(-0.623989\pi\)
−0.379748 + 0.925090i \(0.623989\pi\)
\(770\) 0 0
\(771\) 31.6976 1.14156
\(772\) 12.1336 0.436698
\(773\) −6.82724 −0.245559 −0.122779 0.992434i \(-0.539181\pi\)
−0.122779 + 0.992434i \(0.539181\pi\)
\(774\) 1.58362 0.0569219
\(775\) −45.0339 −1.61766
\(776\) 7.04118 0.252764
\(777\) 20.6739 0.741674
\(778\) 5.82380 0.208793
\(779\) 3.64782 0.130697
\(780\) −1.28403 −0.0459756
\(781\) 0 0
\(782\) −11.6681 −0.417250
\(783\) −30.2221 −1.08005
\(784\) 4.56226 0.162938
\(785\) −6.84751 −0.244398
\(786\) −5.67516 −0.202426
\(787\) −45.8465 −1.63425 −0.817125 0.576461i \(-0.804433\pi\)
−0.817125 + 0.576461i \(0.804433\pi\)
\(788\) −14.2097 −0.506199
\(789\) −47.0953 −1.67664
\(790\) 4.02878 0.143338
\(791\) −21.6372 −0.769331
\(792\) 0 0
\(793\) 12.7043 0.451143
\(794\) 2.13644 0.0758194
\(795\) −4.03662 −0.143164
\(796\) −0.976956 −0.0346273
\(797\) −39.1554 −1.38696 −0.693478 0.720478i \(-0.743923\pi\)
−0.693478 + 0.720478i \(0.743923\pi\)
\(798\) −5.69053 −0.201443
\(799\) 58.1615 2.05761
\(800\) −4.67909 −0.165431
\(801\) 3.02204 0.106778
\(802\) 3.52506 0.124474
\(803\) 0 0
\(804\) 1.69030 0.0596124
\(805\) 5.22140 0.184030
\(806\) 13.0356 0.459158
\(807\) −0.241597 −0.00850463
\(808\) −8.12037 −0.285674
\(809\) 29.0753 1.02223 0.511116 0.859512i \(-0.329232\pi\)
0.511116 + 0.859512i \(0.329232\pi\)
\(810\) 4.73715 0.166446
\(811\) −20.3401 −0.714238 −0.357119 0.934059i \(-0.616241\pi\)
−0.357119 + 0.934059i \(0.616241\pi\)
\(812\) 19.1936 0.673562
\(813\) 13.1922 0.462672
\(814\) 0 0
\(815\) −7.72238 −0.270503
\(816\) 7.20373 0.252181
\(817\) 7.94488 0.277956
\(818\) 13.0620 0.456702
\(819\) −0.917985 −0.0320770
\(820\) 2.06645 0.0721636
\(821\) −10.1130 −0.352947 −0.176474 0.984305i \(-0.556469\pi\)
−0.176474 + 0.984305i \(0.556469\pi\)
\(822\) 0.102440 0.00357301
\(823\) 17.1457 0.597662 0.298831 0.954306i \(-0.403403\pi\)
0.298831 + 0.954306i \(0.403403\pi\)
\(824\) −3.00499 −0.104684
\(825\) 0 0
\(826\) 27.1176 0.943541
\(827\) 46.5796 1.61973 0.809866 0.586615i \(-0.199540\pi\)
0.809866 + 0.586615i \(0.199540\pi\)
\(828\) 0.540301 0.0187768
\(829\) −32.4293 −1.12632 −0.563158 0.826350i \(-0.690414\pi\)
−0.563158 + 0.826350i \(0.690414\pi\)
\(830\) 0.580381 0.0201453
\(831\) −27.4866 −0.953498
\(832\) 1.35442 0.0469559
\(833\) 19.6384 0.680431
\(834\) −28.0743 −0.972134
\(835\) 2.13929 0.0740331
\(836\) 0 0
\(837\) −51.5309 −1.78117
\(838\) 24.4288 0.843878
\(839\) 44.9176 1.55073 0.775363 0.631516i \(-0.217567\pi\)
0.775363 + 0.631516i \(0.217567\pi\)
\(840\) −3.22362 −0.111226
\(841\) 2.86168 0.0986786
\(842\) −11.0083 −0.379373
\(843\) 2.87510 0.0990237
\(844\) −8.44427 −0.290664
\(845\) 6.32517 0.217592
\(846\) −2.69322 −0.0925948
\(847\) 0 0
\(848\) 4.25789 0.146217
\(849\) 8.45847 0.290294
\(850\) −20.1413 −0.690841
\(851\) −9.84791 −0.337582
\(852\) 10.5596 0.361767
\(853\) −41.7985 −1.43115 −0.715577 0.698534i \(-0.753836\pi\)
−0.715577 + 0.698534i \(0.753836\pi\)
\(854\) 31.8948 1.09142
\(855\) −0.112916 −0.00386163
\(856\) −7.33334 −0.250649
\(857\) 20.6115 0.704076 0.352038 0.935986i \(-0.385489\pi\)
0.352038 + 0.935986i \(0.385489\pi\)
\(858\) 0 0
\(859\) 49.0170 1.67244 0.836220 0.548394i \(-0.184761\pi\)
0.836220 + 0.548394i \(0.184761\pi\)
\(860\) 4.50069 0.153472
\(861\) −20.7580 −0.707432
\(862\) −13.5147 −0.460311
\(863\) −10.0126 −0.340833 −0.170416 0.985372i \(-0.554511\pi\)
−0.170416 + 0.985372i \(0.554511\pi\)
\(864\) −5.35414 −0.182152
\(865\) 1.08816 0.0369985
\(866\) 22.8978 0.778099
\(867\) 2.55885 0.0869031
\(868\) 32.7265 1.11081
\(869\) 0 0
\(870\) −5.35128 −0.181425
\(871\) 1.36800 0.0463528
\(872\) 3.90465 0.132228
\(873\) −1.40348 −0.0475008
\(874\) 2.71065 0.0916891
\(875\) 18.6444 0.630295
\(876\) −23.4925 −0.793738
\(877\) 40.9542 1.38293 0.691463 0.722412i \(-0.256967\pi\)
0.691463 + 0.722412i \(0.256967\pi\)
\(878\) −4.19717 −0.141648
\(879\) −32.2197 −1.08674
\(880\) 0 0
\(881\) −47.6017 −1.60374 −0.801870 0.597498i \(-0.796161\pi\)
−0.801870 + 0.597498i \(0.796161\pi\)
\(882\) −0.909374 −0.0306202
\(883\) −18.8140 −0.633142 −0.316571 0.948569i \(-0.602532\pi\)
−0.316571 + 0.948569i \(0.602532\pi\)
\(884\) 5.83013 0.196088
\(885\) −7.56053 −0.254145
\(886\) −31.0222 −1.04221
\(887\) −36.2917 −1.21856 −0.609278 0.792957i \(-0.708541\pi\)
−0.609278 + 0.792957i \(0.708541\pi\)
\(888\) 6.07998 0.204031
\(889\) 59.0213 1.97951
\(890\) 8.58873 0.287895
\(891\) 0 0
\(892\) −0.309593 −0.0103659
\(893\) −13.5117 −0.452151
\(894\) 14.9815 0.501055
\(895\) 7.31800 0.244614
\(896\) 3.40033 0.113597
\(897\) −6.14408 −0.205145
\(898\) −30.0246 −1.00193
\(899\) 54.3266 1.81189
\(900\) 0.932661 0.0310887
\(901\) 18.3283 0.610603
\(902\) 0 0
\(903\) −45.2106 −1.50451
\(904\) −6.36327 −0.211639
\(905\) −10.0712 −0.334777
\(906\) −14.2484 −0.473372
\(907\) −44.4140 −1.47474 −0.737371 0.675488i \(-0.763933\pi\)
−0.737371 + 0.675488i \(0.763933\pi\)
\(908\) 17.8942 0.593840
\(909\) 1.61859 0.0536854
\(910\) −2.60894 −0.0864857
\(911\) 14.7495 0.488674 0.244337 0.969690i \(-0.421430\pi\)
0.244337 + 0.969690i \(0.421430\pi\)
\(912\) −1.67352 −0.0554158
\(913\) 0 0
\(914\) −9.16793 −0.303248
\(915\) −8.89246 −0.293976
\(916\) −4.77162 −0.157659
\(917\) −11.5310 −0.380789
\(918\) −23.0471 −0.760667
\(919\) −0.209240 −0.00690220 −0.00345110 0.999994i \(-0.501099\pi\)
−0.00345110 + 0.999994i \(0.501099\pi\)
\(920\) 1.53555 0.0506257
\(921\) −20.5148 −0.675987
\(922\) −30.4549 −1.00298
\(923\) 8.54611 0.281299
\(924\) 0 0
\(925\) −16.9993 −0.558935
\(926\) 16.4686 0.541190
\(927\) 0.598971 0.0196728
\(928\) 5.64462 0.185294
\(929\) 12.5178 0.410695 0.205348 0.978689i \(-0.434168\pi\)
0.205348 + 0.978689i \(0.434168\pi\)
\(930\) −9.12433 −0.299199
\(931\) −4.56226 −0.149522
\(932\) −12.5120 −0.409843
\(933\) −9.17233 −0.300288
\(934\) −36.8201 −1.20479
\(935\) 0 0
\(936\) −0.269969 −0.00882422
\(937\) 38.2958 1.25107 0.625534 0.780197i \(-0.284881\pi\)
0.625534 + 0.780197i \(0.284881\pi\)
\(938\) 3.43443 0.112138
\(939\) 14.6516 0.478135
\(940\) −7.65422 −0.249653
\(941\) 58.5062 1.90725 0.953623 0.301002i \(-0.0973212\pi\)
0.953623 + 0.301002i \(0.0973212\pi\)
\(942\) 20.2289 0.659094
\(943\) 9.88797 0.321997
\(944\) 7.97498 0.259563
\(945\) 10.3134 0.335496
\(946\) 0 0
\(947\) −34.2392 −1.11262 −0.556312 0.830974i \(-0.687784\pi\)
−0.556312 + 0.830974i \(0.687784\pi\)
\(948\) −11.9018 −0.386553
\(949\) −19.0130 −0.617187
\(950\) 4.67909 0.151810
\(951\) −30.8587 −1.00066
\(952\) 14.6369 0.474383
\(953\) −24.3715 −0.789471 −0.394736 0.918795i \(-0.629164\pi\)
−0.394736 + 0.918795i \(0.629164\pi\)
\(954\) −0.848706 −0.0274779
\(955\) 11.7394 0.379877
\(956\) 11.2492 0.363826
\(957\) 0 0
\(958\) −31.1691 −1.00703
\(959\) 0.208142 0.00672126
\(960\) −0.948032 −0.0305976
\(961\) 61.6309 1.98809
\(962\) 4.92065 0.158648
\(963\) 1.46172 0.0471033
\(964\) 23.4105 0.754003
\(965\) −6.87355 −0.221268
\(966\) −15.4250 −0.496293
\(967\) 44.3499 1.42620 0.713099 0.701063i \(-0.247291\pi\)
0.713099 + 0.701063i \(0.247291\pi\)
\(968\) 0 0
\(969\) −7.20373 −0.231417
\(970\) −3.98875 −0.128071
\(971\) −19.7687 −0.634408 −0.317204 0.948357i \(-0.602744\pi\)
−0.317204 + 0.948357i \(0.602744\pi\)
\(972\) 2.06794 0.0663292
\(973\) −57.0426 −1.82870
\(974\) 40.4109 1.29485
\(975\) −10.6058 −0.339658
\(976\) 9.37992 0.300244
\(977\) 18.2277 0.583155 0.291577 0.956547i \(-0.405820\pi\)
0.291577 + 0.956547i \(0.405820\pi\)
\(978\) 22.8134 0.729494
\(979\) 0 0
\(980\) −2.58447 −0.0825579
\(981\) −0.778296 −0.0248491
\(982\) −9.86715 −0.314873
\(983\) 56.2431 1.79388 0.896939 0.442155i \(-0.145786\pi\)
0.896939 + 0.442155i \(0.145786\pi\)
\(984\) −6.10471 −0.194611
\(985\) 8.04964 0.256483
\(986\) 24.2974 0.773788
\(987\) 76.8886 2.44739
\(988\) −1.35442 −0.0430897
\(989\) 21.5358 0.684798
\(990\) 0 0
\(991\) 48.6909 1.54672 0.773358 0.633970i \(-0.218576\pi\)
0.773358 + 0.633970i \(0.218576\pi\)
\(992\) 9.62449 0.305578
\(993\) 10.9966 0.348968
\(994\) 21.4555 0.680527
\(995\) 0.553435 0.0175451
\(996\) −1.71456 −0.0543279
\(997\) −50.2387 −1.59107 −0.795537 0.605905i \(-0.792811\pi\)
−0.795537 + 0.605905i \(0.792811\pi\)
\(998\) −13.4498 −0.425745
\(999\) −19.4518 −0.615429
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 4598.2.a.cb.1.5 yes 8
11.10 odd 2 4598.2.a.by.1.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4598.2.a.by.1.5 8 11.10 odd 2
4598.2.a.cb.1.5 yes 8 1.1 even 1 trivial