Properties

Label 5070.2.a.bs.1.3
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
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 1.3
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.19806 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.19806 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +3.93900 q^{11} -1.00000 q^{12} -1.19806 q^{14} +1.00000 q^{15} +1.00000 q^{16} +1.74094 q^{17} +1.00000 q^{18} +0.911854 q^{19} -1.00000 q^{20} +1.19806 q^{21} +3.93900 q^{22} +1.24698 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.19806 q^{28} -3.47219 q^{29} +1.00000 q^{30} -2.15883 q^{31} +1.00000 q^{32} -3.93900 q^{33} +1.74094 q^{34} +1.19806 q^{35} +1.00000 q^{36} +4.80194 q^{37} +0.911854 q^{38} -1.00000 q^{40} +0.198062 q^{41} +1.19806 q^{42} +3.43296 q^{43} +3.93900 q^{44} -1.00000 q^{45} +1.24698 q^{46} +0.902165 q^{47} -1.00000 q^{48} -5.56465 q^{49} +1.00000 q^{50} -1.74094 q^{51} -6.91185 q^{53} -1.00000 q^{54} -3.93900 q^{55} -1.19806 q^{56} -0.911854 q^{57} -3.47219 q^{58} -10.2446 q^{59} +1.00000 q^{60} +9.92692 q^{61} -2.15883 q^{62} -1.19806 q^{63} +1.00000 q^{64} -3.93900 q^{66} -9.16852 q^{67} +1.74094 q^{68} -1.24698 q^{69} +1.19806 q^{70} +11.0707 q^{71} +1.00000 q^{72} +4.13706 q^{73} +4.80194 q^{74} -1.00000 q^{75} +0.911854 q^{76} -4.71917 q^{77} +2.73556 q^{79} -1.00000 q^{80} +1.00000 q^{81} +0.198062 q^{82} +15.5864 q^{83} +1.19806 q^{84} -1.74094 q^{85} +3.43296 q^{86} +3.47219 q^{87} +3.93900 q^{88} +8.57673 q^{89} -1.00000 q^{90} +1.24698 q^{92} +2.15883 q^{93} +0.902165 q^{94} -0.911854 q^{95} -1.00000 q^{96} +16.9976 q^{97} -5.56465 q^{98} +3.93900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} - 8q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} - 8q^{7} + 3q^{8} + 3q^{9} - 3q^{10} + 2q^{11} - 3q^{12} - 8q^{14} + 3q^{15} + 3q^{16} - 9q^{17} + 3q^{18} - q^{19} - 3q^{20} + 8q^{21} + 2q^{22} - q^{23} - 3q^{24} + 3q^{25} - 3q^{27} - 8q^{28} - 4q^{29} + 3q^{30} + 2q^{31} + 3q^{32} - 2q^{33} - 9q^{34} + 8q^{35} + 3q^{36} + 10q^{37} - q^{38} - 3q^{40} + 5q^{41} + 8q^{42} - 9q^{43} + 2q^{44} - 3q^{45} - q^{46} + 21q^{47} - 3q^{48} + 5q^{49} + 3q^{50} + 9q^{51} - 17q^{53} - 3q^{54} - 2q^{55} - 8q^{56} + q^{57} - 4q^{58} + 15q^{59} + 3q^{60} + q^{61} + 2q^{62} - 8q^{63} + 3q^{64} - 2q^{66} + 3q^{67} - 9q^{68} + q^{69} + 8q^{70} + 21q^{71} + 3q^{72} + 7q^{73} + 10q^{74} - 3q^{75} - q^{76} - 3q^{77} - 3q^{79} - 3q^{80} + 3q^{81} + 5q^{82} + 22q^{83} + 8q^{84} + 9q^{85} - 9q^{86} + 4q^{87} + 2q^{88} + 23q^{89} - 3q^{90} - q^{92} - 2q^{93} + 21q^{94} + q^{95} - 3q^{96} + 10q^{97} + 5q^{98} + 2q^{99} + O(q^{100}) \)

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
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) −1.19806 −0.452825 −0.226412 0.974032i \(-0.572700\pi\)
−0.226412 + 0.974032i \(0.572700\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 3.93900 1.18765 0.593827 0.804593i \(-0.297616\pi\)
0.593827 + 0.804593i \(0.297616\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.19806 −0.320196
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 1.74094 0.422240 0.211120 0.977460i \(-0.432289\pi\)
0.211120 + 0.977460i \(0.432289\pi\)
\(18\) 1.00000 0.235702
\(19\) 0.911854 0.209194 0.104597 0.994515i \(-0.466645\pi\)
0.104597 + 0.994515i \(0.466645\pi\)
\(20\) −1.00000 −0.223607
\(21\) 1.19806 0.261439
\(22\) 3.93900 0.839798
\(23\) 1.24698 0.260013 0.130007 0.991513i \(-0.458500\pi\)
0.130007 + 0.991513i \(0.458500\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −1.19806 −0.226412
\(29\) −3.47219 −0.644769 −0.322385 0.946609i \(-0.604484\pi\)
−0.322385 + 0.946609i \(0.604484\pi\)
\(30\) 1.00000 0.182574
\(31\) −2.15883 −0.387738 −0.193869 0.981027i \(-0.562104\pi\)
−0.193869 + 0.981027i \(0.562104\pi\)
\(32\) 1.00000 0.176777
\(33\) −3.93900 −0.685692
\(34\) 1.74094 0.298569
\(35\) 1.19806 0.202509
\(36\) 1.00000 0.166667
\(37\) 4.80194 0.789434 0.394717 0.918803i \(-0.370843\pi\)
0.394717 + 0.918803i \(0.370843\pi\)
\(38\) 0.911854 0.147922
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 0.198062 0.0309321 0.0154661 0.999880i \(-0.495077\pi\)
0.0154661 + 0.999880i \(0.495077\pi\)
\(42\) 1.19806 0.184865
\(43\) 3.43296 0.523522 0.261761 0.965133i \(-0.415697\pi\)
0.261761 + 0.965133i \(0.415697\pi\)
\(44\) 3.93900 0.593827
\(45\) −1.00000 −0.149071
\(46\) 1.24698 0.183857
\(47\) 0.902165 0.131594 0.0657972 0.997833i \(-0.479041\pi\)
0.0657972 + 0.997833i \(0.479041\pi\)
\(48\) −1.00000 −0.144338
\(49\) −5.56465 −0.794950
\(50\) 1.00000 0.141421
\(51\) −1.74094 −0.243780
\(52\) 0 0
\(53\) −6.91185 −0.949416 −0.474708 0.880143i \(-0.657446\pi\)
−0.474708 + 0.880143i \(0.657446\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.93900 −0.531135
\(56\) −1.19806 −0.160098
\(57\) −0.911854 −0.120778
\(58\) −3.47219 −0.455921
\(59\) −10.2446 −1.33373 −0.666866 0.745178i \(-0.732365\pi\)
−0.666866 + 0.745178i \(0.732365\pi\)
\(60\) 1.00000 0.129099
\(61\) 9.92692 1.27101 0.635506 0.772096i \(-0.280792\pi\)
0.635506 + 0.772096i \(0.280792\pi\)
\(62\) −2.15883 −0.274172
\(63\) −1.19806 −0.150942
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.93900 −0.484858
\(67\) −9.16852 −1.12011 −0.560057 0.828454i \(-0.689221\pi\)
−0.560057 + 0.828454i \(0.689221\pi\)
\(68\) 1.74094 0.211120
\(69\) −1.24698 −0.150119
\(70\) 1.19806 0.143196
\(71\) 11.0707 1.31385 0.656924 0.753956i \(-0.271857\pi\)
0.656924 + 0.753956i \(0.271857\pi\)
\(72\) 1.00000 0.117851
\(73\) 4.13706 0.484207 0.242103 0.970250i \(-0.422163\pi\)
0.242103 + 0.970250i \(0.422163\pi\)
\(74\) 4.80194 0.558214
\(75\) −1.00000 −0.115470
\(76\) 0.911854 0.104597
\(77\) −4.71917 −0.537799
\(78\) 0 0
\(79\) 2.73556 0.307775 0.153887 0.988088i \(-0.450821\pi\)
0.153887 + 0.988088i \(0.450821\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) 0.198062 0.0218723
\(83\) 15.5864 1.71083 0.855416 0.517942i \(-0.173302\pi\)
0.855416 + 0.517942i \(0.173302\pi\)
\(84\) 1.19806 0.130719
\(85\) −1.74094 −0.188831
\(86\) 3.43296 0.370186
\(87\) 3.47219 0.372258
\(88\) 3.93900 0.419899
\(89\) 8.57673 0.909131 0.454566 0.890713i \(-0.349794\pi\)
0.454566 + 0.890713i \(0.349794\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 1.24698 0.130007
\(93\) 2.15883 0.223861
\(94\) 0.902165 0.0930512
\(95\) −0.911854 −0.0935542
\(96\) −1.00000 −0.102062
\(97\) 16.9976 1.72585 0.862923 0.505336i \(-0.168631\pi\)
0.862923 + 0.505336i \(0.168631\pi\)
\(98\) −5.56465 −0.562114
\(99\) 3.93900 0.395885
\(100\) 1.00000 0.100000
\(101\) 6.67025 0.663715 0.331857 0.943330i \(-0.392325\pi\)
0.331857 + 0.943330i \(0.392325\pi\)
\(102\) −1.74094 −0.172379
\(103\) 10.0315 0.988429 0.494215 0.869340i \(-0.335456\pi\)
0.494215 + 0.869340i \(0.335456\pi\)
\(104\) 0 0
\(105\) −1.19806 −0.116919
\(106\) −6.91185 −0.671339
\(107\) −10.1032 −0.976714 −0.488357 0.872644i \(-0.662404\pi\)
−0.488357 + 0.872644i \(0.662404\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.5211 1.00774 0.503870 0.863780i \(-0.331909\pi\)
0.503870 + 0.863780i \(0.331909\pi\)
\(110\) −3.93900 −0.375569
\(111\) −4.80194 −0.455780
\(112\) −1.19806 −0.113206
\(113\) −11.2131 −1.05484 −0.527421 0.849604i \(-0.676841\pi\)
−0.527421 + 0.849604i \(0.676841\pi\)
\(114\) −0.911854 −0.0854030
\(115\) −1.24698 −0.116281
\(116\) −3.47219 −0.322385
\(117\) 0 0
\(118\) −10.2446 −0.943091
\(119\) −2.08575 −0.191201
\(120\) 1.00000 0.0912871
\(121\) 4.51573 0.410521
\(122\) 9.92692 0.898741
\(123\) −0.198062 −0.0178587
\(124\) −2.15883 −0.193869
\(125\) −1.00000 −0.0894427
\(126\) −1.19806 −0.106732
\(127\) −9.32736 −0.827669 −0.413834 0.910352i \(-0.635811\pi\)
−0.413834 + 0.910352i \(0.635811\pi\)
\(128\) 1.00000 0.0883883
\(129\) −3.43296 −0.302255
\(130\) 0 0
\(131\) 4.62133 0.403768 0.201884 0.979409i \(-0.435294\pi\)
0.201884 + 0.979409i \(0.435294\pi\)
\(132\) −3.93900 −0.342846
\(133\) −1.09246 −0.0947281
\(134\) −9.16852 −0.792040
\(135\) 1.00000 0.0860663
\(136\) 1.74094 0.149284
\(137\) 0.0760644 0.00649862 0.00324931 0.999995i \(-0.498966\pi\)
0.00324931 + 0.999995i \(0.498966\pi\)
\(138\) −1.24698 −0.106150
\(139\) 18.7017 1.58626 0.793129 0.609053i \(-0.208451\pi\)
0.793129 + 0.609053i \(0.208451\pi\)
\(140\) 1.19806 0.101255
\(141\) −0.902165 −0.0759760
\(142\) 11.0707 0.929031
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 3.47219 0.288350
\(146\) 4.13706 0.342386
\(147\) 5.56465 0.458964
\(148\) 4.80194 0.394717
\(149\) 13.6799 1.12070 0.560352 0.828254i \(-0.310666\pi\)
0.560352 + 0.828254i \(0.310666\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −1.15883 −0.0943045 −0.0471523 0.998888i \(-0.515015\pi\)
−0.0471523 + 0.998888i \(0.515015\pi\)
\(152\) 0.911854 0.0739611
\(153\) 1.74094 0.140747
\(154\) −4.71917 −0.380281
\(155\) 2.15883 0.173402
\(156\) 0 0
\(157\) −12.2664 −0.978962 −0.489481 0.872014i \(-0.662814\pi\)
−0.489481 + 0.872014i \(0.662814\pi\)
\(158\) 2.73556 0.217630
\(159\) 6.91185 0.548146
\(160\) −1.00000 −0.0790569
\(161\) −1.49396 −0.117740
\(162\) 1.00000 0.0785674
\(163\) −2.48858 −0.194921 −0.0974604 0.995239i \(-0.531072\pi\)
−0.0974604 + 0.995239i \(0.531072\pi\)
\(164\) 0.198062 0.0154661
\(165\) 3.93900 0.306651
\(166\) 15.5864 1.20974
\(167\) 7.27950 0.563305 0.281652 0.959516i \(-0.409117\pi\)
0.281652 + 0.959516i \(0.409117\pi\)
\(168\) 1.19806 0.0924325
\(169\) 0 0
\(170\) −1.74094 −0.133524
\(171\) 0.911854 0.0697312
\(172\) 3.43296 0.261761
\(173\) 15.8485 1.20494 0.602468 0.798143i \(-0.294184\pi\)
0.602468 + 0.798143i \(0.294184\pi\)
\(174\) 3.47219 0.263226
\(175\) −1.19806 −0.0905650
\(176\) 3.93900 0.296913
\(177\) 10.2446 0.770030
\(178\) 8.57673 0.642853
\(179\) 18.4819 1.38140 0.690700 0.723141i \(-0.257302\pi\)
0.690700 + 0.723141i \(0.257302\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 10.7041 0.795630 0.397815 0.917466i \(-0.369769\pi\)
0.397815 + 0.917466i \(0.369769\pi\)
\(182\) 0 0
\(183\) −9.92692 −0.733819
\(184\) 1.24698 0.0919286
\(185\) −4.80194 −0.353045
\(186\) 2.15883 0.158293
\(187\) 6.85756 0.501474
\(188\) 0.902165 0.0657972
\(189\) 1.19806 0.0871462
\(190\) −0.911854 −0.0661528
\(191\) 15.3623 1.11158 0.555788 0.831324i \(-0.312417\pi\)
0.555788 + 0.831324i \(0.312417\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −18.3207 −1.31875 −0.659375 0.751815i \(-0.729179\pi\)
−0.659375 + 0.751815i \(0.729179\pi\)
\(194\) 16.9976 1.22036
\(195\) 0 0
\(196\) −5.56465 −0.397475
\(197\) −2.31336 −0.164820 −0.0824099 0.996599i \(-0.526262\pi\)
−0.0824099 + 0.996599i \(0.526262\pi\)
\(198\) 3.93900 0.279933
\(199\) 6.36765 0.451391 0.225695 0.974198i \(-0.427535\pi\)
0.225695 + 0.974198i \(0.427535\pi\)
\(200\) 1.00000 0.0707107
\(201\) 9.16852 0.646698
\(202\) 6.67025 0.469317
\(203\) 4.15990 0.291968
\(204\) −1.74094 −0.121890
\(205\) −0.198062 −0.0138333
\(206\) 10.0315 0.698925
\(207\) 1.24698 0.0866711
\(208\) 0 0
\(209\) 3.59179 0.248450
\(210\) −1.19806 −0.0826742
\(211\) −11.6775 −0.803915 −0.401958 0.915658i \(-0.631670\pi\)
−0.401958 + 0.915658i \(0.631670\pi\)
\(212\) −6.91185 −0.474708
\(213\) −11.0707 −0.758551
\(214\) −10.1032 −0.690641
\(215\) −3.43296 −0.234126
\(216\) −1.00000 −0.0680414
\(217\) 2.58642 0.175577
\(218\) 10.5211 0.712579
\(219\) −4.13706 −0.279557
\(220\) −3.93900 −0.265567
\(221\) 0 0
\(222\) −4.80194 −0.322285
\(223\) 11.4004 0.763430 0.381715 0.924280i \(-0.375334\pi\)
0.381715 + 0.924280i \(0.375334\pi\)
\(224\) −1.19806 −0.0800489
\(225\) 1.00000 0.0666667
\(226\) −11.2131 −0.745886
\(227\) 8.86725 0.588540 0.294270 0.955722i \(-0.404924\pi\)
0.294270 + 0.955722i \(0.404924\pi\)
\(228\) −0.911854 −0.0603890
\(229\) 12.6136 0.833528 0.416764 0.909015i \(-0.363164\pi\)
0.416764 + 0.909015i \(0.363164\pi\)
\(230\) −1.24698 −0.0822234
\(231\) 4.71917 0.310498
\(232\) −3.47219 −0.227960
\(233\) −23.4155 −1.53400 −0.767000 0.641647i \(-0.778252\pi\)
−0.767000 + 0.641647i \(0.778252\pi\)
\(234\) 0 0
\(235\) −0.902165 −0.0588508
\(236\) −10.2446 −0.666866
\(237\) −2.73556 −0.177694
\(238\) −2.08575 −0.135199
\(239\) 8.87263 0.573922 0.286961 0.957942i \(-0.407355\pi\)
0.286961 + 0.957942i \(0.407355\pi\)
\(240\) 1.00000 0.0645497
\(241\) 9.72886 0.626691 0.313345 0.949639i \(-0.398550\pi\)
0.313345 + 0.949639i \(0.398550\pi\)
\(242\) 4.51573 0.290282
\(243\) −1.00000 −0.0641500
\(244\) 9.92692 0.635506
\(245\) 5.56465 0.355512
\(246\) −0.198062 −0.0126280
\(247\) 0 0
\(248\) −2.15883 −0.137086
\(249\) −15.5864 −0.987749
\(250\) −1.00000 −0.0632456
\(251\) 29.8823 1.88615 0.943077 0.332573i \(-0.107917\pi\)
0.943077 + 0.332573i \(0.107917\pi\)
\(252\) −1.19806 −0.0754708
\(253\) 4.91185 0.308806
\(254\) −9.32736 −0.585250
\(255\) 1.74094 0.109022
\(256\) 1.00000 0.0625000
\(257\) −8.52111 −0.531532 −0.265766 0.964038i \(-0.585625\pi\)
−0.265766 + 0.964038i \(0.585625\pi\)
\(258\) −3.43296 −0.213727
\(259\) −5.75302 −0.357475
\(260\) 0 0
\(261\) −3.47219 −0.214923
\(262\) 4.62133 0.285507
\(263\) −22.6450 −1.39635 −0.698176 0.715926i \(-0.746005\pi\)
−0.698176 + 0.715926i \(0.746005\pi\)
\(264\) −3.93900 −0.242429
\(265\) 6.91185 0.424592
\(266\) −1.09246 −0.0669829
\(267\) −8.57673 −0.524887
\(268\) −9.16852 −0.560057
\(269\) −9.02608 −0.550330 −0.275165 0.961397i \(-0.588732\pi\)
−0.275165 + 0.961397i \(0.588732\pi\)
\(270\) 1.00000 0.0608581
\(271\) −8.59611 −0.522176 −0.261088 0.965315i \(-0.584081\pi\)
−0.261088 + 0.965315i \(0.584081\pi\)
\(272\) 1.74094 0.105560
\(273\) 0 0
\(274\) 0.0760644 0.00459522
\(275\) 3.93900 0.237531
\(276\) −1.24698 −0.0750594
\(277\) 6.31527 0.379448 0.189724 0.981837i \(-0.439241\pi\)
0.189724 + 0.981837i \(0.439241\pi\)
\(278\) 18.7017 1.12165
\(279\) −2.15883 −0.129246
\(280\) 1.19806 0.0715979
\(281\) 21.7409 1.29696 0.648478 0.761234i \(-0.275406\pi\)
0.648478 + 0.761234i \(0.275406\pi\)
\(282\) −0.902165 −0.0537232
\(283\) 12.9487 0.769720 0.384860 0.922975i \(-0.374250\pi\)
0.384860 + 0.922975i \(0.374250\pi\)
\(284\) 11.0707 0.656924
\(285\) 0.911854 0.0540136
\(286\) 0 0
\(287\) −0.237291 −0.0140068
\(288\) 1.00000 0.0589256
\(289\) −13.9691 −0.821714
\(290\) 3.47219 0.203894
\(291\) −16.9976 −0.996417
\(292\) 4.13706 0.242103
\(293\) 27.3056 1.59521 0.797605 0.603181i \(-0.206100\pi\)
0.797605 + 0.603181i \(0.206100\pi\)
\(294\) 5.56465 0.324537
\(295\) 10.2446 0.596463
\(296\) 4.80194 0.279107
\(297\) −3.93900 −0.228564
\(298\) 13.6799 0.792458
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) −4.11290 −0.237064
\(302\) −1.15883 −0.0666834
\(303\) −6.67025 −0.383196
\(304\) 0.911854 0.0522984
\(305\) −9.92692 −0.568414
\(306\) 1.74094 0.0995228
\(307\) −24.9124 −1.42183 −0.710914 0.703279i \(-0.751719\pi\)
−0.710914 + 0.703279i \(0.751719\pi\)
\(308\) −4.71917 −0.268900
\(309\) −10.0315 −0.570670
\(310\) 2.15883 0.122614
\(311\) 21.3787 1.21227 0.606136 0.795361i \(-0.292719\pi\)
0.606136 + 0.795361i \(0.292719\pi\)
\(312\) 0 0
\(313\) −23.8713 −1.34929 −0.674643 0.738144i \(-0.735702\pi\)
−0.674643 + 0.738144i \(0.735702\pi\)
\(314\) −12.2664 −0.692231
\(315\) 1.19806 0.0675032
\(316\) 2.73556 0.153887
\(317\) −12.6407 −0.709973 −0.354987 0.934871i \(-0.615515\pi\)
−0.354987 + 0.934871i \(0.615515\pi\)
\(318\) 6.91185 0.387598
\(319\) −13.6770 −0.765763
\(320\) −1.00000 −0.0559017
\(321\) 10.1032 0.563906
\(322\) −1.49396 −0.0832551
\(323\) 1.58748 0.0883299
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −2.48858 −0.137830
\(327\) −10.5211 −0.581819
\(328\) 0.198062 0.0109362
\(329\) −1.08085 −0.0595892
\(330\) 3.93900 0.216835
\(331\) 20.5109 1.12738 0.563692 0.825985i \(-0.309381\pi\)
0.563692 + 0.825985i \(0.309381\pi\)
\(332\) 15.5864 0.855416
\(333\) 4.80194 0.263145
\(334\) 7.27950 0.398317
\(335\) 9.16852 0.500930
\(336\) 1.19806 0.0653597
\(337\) 6.82371 0.371711 0.185856 0.982577i \(-0.440494\pi\)
0.185856 + 0.982577i \(0.440494\pi\)
\(338\) 0 0
\(339\) 11.2131 0.609014
\(340\) −1.74094 −0.0944157
\(341\) −8.50365 −0.460498
\(342\) 0.911854 0.0493074
\(343\) 15.0532 0.812798
\(344\) 3.43296 0.185093
\(345\) 1.24698 0.0671351
\(346\) 15.8485 0.852019
\(347\) −29.4741 −1.58225 −0.791127 0.611653i \(-0.790505\pi\)
−0.791127 + 0.611653i \(0.790505\pi\)
\(348\) 3.47219 0.186129
\(349\) −34.5394 −1.84885 −0.924426 0.381361i \(-0.875456\pi\)
−0.924426 + 0.381361i \(0.875456\pi\)
\(350\) −1.19806 −0.0640391
\(351\) 0 0
\(352\) 3.93900 0.209949
\(353\) 31.2295 1.66218 0.831090 0.556138i \(-0.187717\pi\)
0.831090 + 0.556138i \(0.187717\pi\)
\(354\) 10.2446 0.544494
\(355\) −11.0707 −0.587571
\(356\) 8.57673 0.454566
\(357\) 2.08575 0.110390
\(358\) 18.4819 0.976798
\(359\) 18.2524 0.963323 0.481661 0.876357i \(-0.340034\pi\)
0.481661 + 0.876357i \(0.340034\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −18.1685 −0.956238
\(362\) 10.7041 0.562595
\(363\) −4.51573 −0.237014
\(364\) 0 0
\(365\) −4.13706 −0.216544
\(366\) −9.92692 −0.518888
\(367\) −19.3773 −1.01149 −0.505744 0.862683i \(-0.668782\pi\)
−0.505744 + 0.862683i \(0.668782\pi\)
\(368\) 1.24698 0.0650033
\(369\) 0.198062 0.0103107
\(370\) −4.80194 −0.249641
\(371\) 8.28083 0.429919
\(372\) 2.15883 0.111930
\(373\) −13.6280 −0.705633 −0.352817 0.935693i \(-0.614776\pi\)
−0.352817 + 0.935693i \(0.614776\pi\)
\(374\) 6.85756 0.354596
\(375\) 1.00000 0.0516398
\(376\) 0.902165 0.0465256
\(377\) 0 0
\(378\) 1.19806 0.0616217
\(379\) −1.32006 −0.0678069 −0.0339035 0.999425i \(-0.510794\pi\)
−0.0339035 + 0.999425i \(0.510794\pi\)
\(380\) −0.911854 −0.0467771
\(381\) 9.32736 0.477855
\(382\) 15.3623 0.786002
\(383\) 25.5308 1.30456 0.652281 0.757977i \(-0.273812\pi\)
0.652281 + 0.757977i \(0.273812\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 4.71917 0.240511
\(386\) −18.3207 −0.932497
\(387\) 3.43296 0.174507
\(388\) 16.9976 0.862923
\(389\) −6.05967 −0.307238 −0.153619 0.988130i \(-0.549093\pi\)
−0.153619 + 0.988130i \(0.549093\pi\)
\(390\) 0 0
\(391\) 2.17092 0.109788
\(392\) −5.56465 −0.281057
\(393\) −4.62133 −0.233115
\(394\) −2.31336 −0.116545
\(395\) −2.73556 −0.137641
\(396\) 3.93900 0.197942
\(397\) 5.99330 0.300795 0.150397 0.988626i \(-0.451945\pi\)
0.150397 + 0.988626i \(0.451945\pi\)
\(398\) 6.36765 0.319181
\(399\) 1.09246 0.0546913
\(400\) 1.00000 0.0500000
\(401\) 17.3773 0.867783 0.433891 0.900965i \(-0.357140\pi\)
0.433891 + 0.900965i \(0.357140\pi\)
\(402\) 9.16852 0.457284
\(403\) 0 0
\(404\) 6.67025 0.331857
\(405\) −1.00000 −0.0496904
\(406\) 4.15990 0.206452
\(407\) 18.9148 0.937574
\(408\) −1.74094 −0.0861893
\(409\) 29.3672 1.45211 0.726057 0.687635i \(-0.241351\pi\)
0.726057 + 0.687635i \(0.241351\pi\)
\(410\) −0.198062 −0.00978160
\(411\) −0.0760644 −0.00375198
\(412\) 10.0315 0.494215
\(413\) 12.2737 0.603947
\(414\) 1.24698 0.0612857
\(415\) −15.5864 −0.765107
\(416\) 0 0
\(417\) −18.7017 −0.915827
\(418\) 3.59179 0.175680
\(419\) −26.5472 −1.29692 −0.648458 0.761251i \(-0.724586\pi\)
−0.648458 + 0.761251i \(0.724586\pi\)
\(420\) −1.19806 −0.0584595
\(421\) −31.0834 −1.51491 −0.757455 0.652887i \(-0.773558\pi\)
−0.757455 + 0.652887i \(0.773558\pi\)
\(422\) −11.6775 −0.568454
\(423\) 0.902165 0.0438648
\(424\) −6.91185 −0.335669
\(425\) 1.74094 0.0844479
\(426\) −11.0707 −0.536377
\(427\) −11.8931 −0.575546
\(428\) −10.1032 −0.488357
\(429\) 0 0
\(430\) −3.43296 −0.165552
\(431\) 36.1226 1.73996 0.869982 0.493084i \(-0.164130\pi\)
0.869982 + 0.493084i \(0.164130\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −35.2868 −1.69578 −0.847888 0.530176i \(-0.822126\pi\)
−0.847888 + 0.530176i \(0.822126\pi\)
\(434\) 2.58642 0.124152
\(435\) −3.47219 −0.166479
\(436\) 10.5211 0.503870
\(437\) 1.13706 0.0543931
\(438\) −4.13706 −0.197677
\(439\) −32.0170 −1.52809 −0.764044 0.645165i \(-0.776789\pi\)
−0.764044 + 0.645165i \(0.776789\pi\)
\(440\) −3.93900 −0.187785
\(441\) −5.56465 −0.264983
\(442\) 0 0
\(443\) −23.5948 −1.12102 −0.560511 0.828147i \(-0.689395\pi\)
−0.560511 + 0.828147i \(0.689395\pi\)
\(444\) −4.80194 −0.227890
\(445\) −8.57673 −0.406576
\(446\) 11.4004 0.539826
\(447\) −13.6799 −0.647039
\(448\) −1.19806 −0.0566031
\(449\) −19.8374 −0.936187 −0.468093 0.883679i \(-0.655059\pi\)
−0.468093 + 0.883679i \(0.655059\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0.780167 0.0367367
\(452\) −11.2131 −0.527421
\(453\) 1.15883 0.0544468
\(454\) 8.86725 0.416161
\(455\) 0 0
\(456\) −0.911854 −0.0427015
\(457\) −26.5284 −1.24095 −0.620473 0.784228i \(-0.713059\pi\)
−0.620473 + 0.784228i \(0.713059\pi\)
\(458\) 12.6136 0.589393
\(459\) −1.74094 −0.0812601
\(460\) −1.24698 −0.0581407
\(461\) −12.2717 −0.571552 −0.285776 0.958297i \(-0.592251\pi\)
−0.285776 + 0.958297i \(0.592251\pi\)
\(462\) 4.71917 0.219556
\(463\) −12.6455 −0.587686 −0.293843 0.955854i \(-0.594934\pi\)
−0.293843 + 0.955854i \(0.594934\pi\)
\(464\) −3.47219 −0.161192
\(465\) −2.15883 −0.100114
\(466\) −23.4155 −1.08470
\(467\) 11.3870 0.526929 0.263464 0.964669i \(-0.415135\pi\)
0.263464 + 0.964669i \(0.415135\pi\)
\(468\) 0 0
\(469\) 10.9845 0.507215
\(470\) −0.902165 −0.0416138
\(471\) 12.2664 0.565204
\(472\) −10.2446 −0.471545
\(473\) 13.5224 0.621762
\(474\) −2.73556 −0.125649
\(475\) 0.911854 0.0418387
\(476\) −2.08575 −0.0956003
\(477\) −6.91185 −0.316472
\(478\) 8.87263 0.405824
\(479\) 18.8431 0.860963 0.430481 0.902600i \(-0.358344\pi\)
0.430481 + 0.902600i \(0.358344\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 9.72886 0.443137
\(483\) 1.49396 0.0679775
\(484\) 4.51573 0.205260
\(485\) −16.9976 −0.771822
\(486\) −1.00000 −0.0453609
\(487\) 27.6383 1.25241 0.626206 0.779658i \(-0.284607\pi\)
0.626206 + 0.779658i \(0.284607\pi\)
\(488\) 9.92692 0.449371
\(489\) 2.48858 0.112538
\(490\) 5.56465 0.251385
\(491\) −13.9866 −0.631206 −0.315603 0.948891i \(-0.602207\pi\)
−0.315603 + 0.948891i \(0.602207\pi\)
\(492\) −0.198062 −0.00892934
\(493\) −6.04487 −0.272247
\(494\) 0 0
\(495\) −3.93900 −0.177045
\(496\) −2.15883 −0.0969345
\(497\) −13.2634 −0.594944
\(498\) −15.5864 −0.698444
\(499\) −8.87071 −0.397107 −0.198554 0.980090i \(-0.563624\pi\)
−0.198554 + 0.980090i \(0.563624\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −7.27950 −0.325224
\(502\) 29.8823 1.33371
\(503\) 1.99761 0.0890689 0.0445344 0.999008i \(-0.485820\pi\)
0.0445344 + 0.999008i \(0.485820\pi\)
\(504\) −1.19806 −0.0533659
\(505\) −6.67025 −0.296822
\(506\) 4.91185 0.218359
\(507\) 0 0
\(508\) −9.32736 −0.413834
\(509\) −21.7023 −0.961938 −0.480969 0.876738i \(-0.659715\pi\)
−0.480969 + 0.876738i \(0.659715\pi\)
\(510\) 1.74094 0.0770901
\(511\) −4.95646 −0.219261
\(512\) 1.00000 0.0441942
\(513\) −0.911854 −0.0402593
\(514\) −8.52111 −0.375850
\(515\) −10.0315 −0.442039
\(516\) −3.43296 −0.151128
\(517\) 3.55363 0.156288
\(518\) −5.75302 −0.252773
\(519\) −15.8485 −0.695670
\(520\) 0 0
\(521\) 11.8364 0.518561 0.259281 0.965802i \(-0.416515\pi\)
0.259281 + 0.965802i \(0.416515\pi\)
\(522\) −3.47219 −0.151974
\(523\) 24.6112 1.07617 0.538086 0.842890i \(-0.319148\pi\)
0.538086 + 0.842890i \(0.319148\pi\)
\(524\) 4.62133 0.201884
\(525\) 1.19806 0.0522877
\(526\) −22.6450 −0.987370
\(527\) −3.75840 −0.163718
\(528\) −3.93900 −0.171423
\(529\) −21.4450 −0.932393
\(530\) 6.91185 0.300232
\(531\) −10.2446 −0.444577
\(532\) −1.09246 −0.0473641
\(533\) 0 0
\(534\) −8.57673 −0.371151
\(535\) 10.1032 0.436800
\(536\) −9.16852 −0.396020
\(537\) −18.4819 −0.797552
\(538\) −9.02608 −0.389142
\(539\) −21.9191 −0.944125
\(540\) 1.00000 0.0430331
\(541\) 29.1282 1.25232 0.626160 0.779694i \(-0.284626\pi\)
0.626160 + 0.779694i \(0.284626\pi\)
\(542\) −8.59611 −0.369234
\(543\) −10.7041 −0.459357
\(544\) 1.74094 0.0746421
\(545\) −10.5211 −0.450675
\(546\) 0 0
\(547\) −10.1752 −0.435061 −0.217531 0.976053i \(-0.569800\pi\)
−0.217531 + 0.976053i \(0.569800\pi\)
\(548\) 0.0760644 0.00324931
\(549\) 9.92692 0.423671
\(550\) 3.93900 0.167960
\(551\) −3.16613 −0.134882
\(552\) −1.24698 −0.0530750
\(553\) −3.27737 −0.139368
\(554\) 6.31527 0.268310
\(555\) 4.80194 0.203831
\(556\) 18.7017 0.793129
\(557\) 13.0398 0.552515 0.276257 0.961084i \(-0.410906\pi\)
0.276257 + 0.961084i \(0.410906\pi\)
\(558\) −2.15883 −0.0913907
\(559\) 0 0
\(560\) 1.19806 0.0506274
\(561\) −6.85756 −0.289526
\(562\) 21.7409 0.917086
\(563\) −32.9221 −1.38750 −0.693751 0.720215i \(-0.744043\pi\)
−0.693751 + 0.720215i \(0.744043\pi\)
\(564\) −0.902165 −0.0379880
\(565\) 11.2131 0.471740
\(566\) 12.9487 0.544274
\(567\) −1.19806 −0.0503139
\(568\) 11.0707 0.464516
\(569\) 7.53319 0.315808 0.157904 0.987454i \(-0.449526\pi\)
0.157904 + 0.987454i \(0.449526\pi\)
\(570\) 0.911854 0.0381934
\(571\) 36.2422 1.51669 0.758344 0.651854i \(-0.226009\pi\)
0.758344 + 0.651854i \(0.226009\pi\)
\(572\) 0 0
\(573\) −15.3623 −0.641768
\(574\) −0.237291 −0.00990433
\(575\) 1.24698 0.0520026
\(576\) 1.00000 0.0416667
\(577\) −14.2198 −0.591979 −0.295990 0.955191i \(-0.595649\pi\)
−0.295990 + 0.955191i \(0.595649\pi\)
\(578\) −13.9691 −0.581039
\(579\) 18.3207 0.761380
\(580\) 3.47219 0.144175
\(581\) −18.6735 −0.774707
\(582\) −16.9976 −0.704573
\(583\) −27.2258 −1.12758
\(584\) 4.13706 0.171193
\(585\) 0 0
\(586\) 27.3056 1.12798
\(587\) 22.3207 0.921272 0.460636 0.887589i \(-0.347621\pi\)
0.460636 + 0.887589i \(0.347621\pi\)
\(588\) 5.56465 0.229482
\(589\) −1.96854 −0.0811123
\(590\) 10.2446 0.421763
\(591\) 2.31336 0.0951587
\(592\) 4.80194 0.197358
\(593\) −1.64742 −0.0676513 −0.0338256 0.999428i \(-0.510769\pi\)
−0.0338256 + 0.999428i \(0.510769\pi\)
\(594\) −3.93900 −0.161619
\(595\) 2.08575 0.0855075
\(596\) 13.6799 0.560352
\(597\) −6.36765 −0.260611
\(598\) 0 0
\(599\) 9.41119 0.384531 0.192265 0.981343i \(-0.438417\pi\)
0.192265 + 0.981343i \(0.438417\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 1.62565 0.0663115 0.0331557 0.999450i \(-0.489444\pi\)
0.0331557 + 0.999450i \(0.489444\pi\)
\(602\) −4.11290 −0.167629
\(603\) −9.16852 −0.373371
\(604\) −1.15883 −0.0471523
\(605\) −4.51573 −0.183591
\(606\) −6.67025 −0.270960
\(607\) 14.9282 0.605919 0.302959 0.953003i \(-0.402025\pi\)
0.302959 + 0.953003i \(0.402025\pi\)
\(608\) 0.911854 0.0369806
\(609\) −4.15990 −0.168568
\(610\) −9.92692 −0.401929
\(611\) 0 0
\(612\) 1.74094 0.0703733
\(613\) 28.8437 1.16499 0.582493 0.812836i \(-0.302078\pi\)
0.582493 + 0.812836i \(0.302078\pi\)
\(614\) −24.9124 −1.00538
\(615\) 0.198062 0.00798664
\(616\) −4.71917 −0.190141
\(617\) −39.7985 −1.60223 −0.801113 0.598513i \(-0.795759\pi\)
−0.801113 + 0.598513i \(0.795759\pi\)
\(618\) −10.0315 −0.403524
\(619\) −38.0465 −1.52922 −0.764609 0.644494i \(-0.777068\pi\)
−0.764609 + 0.644494i \(0.777068\pi\)
\(620\) 2.15883 0.0867008
\(621\) −1.24698 −0.0500396
\(622\) 21.3787 0.857206
\(623\) −10.2755 −0.411677
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −23.8713 −0.954089
\(627\) −3.59179 −0.143442
\(628\) −12.2664 −0.489481
\(629\) 8.35988 0.333330
\(630\) 1.19806 0.0477319
\(631\) 23.0901 0.919201 0.459600 0.888126i \(-0.347993\pi\)
0.459600 + 0.888126i \(0.347993\pi\)
\(632\) 2.73556 0.108815
\(633\) 11.6775 0.464141
\(634\) −12.6407 −0.502027
\(635\) 9.32736 0.370145
\(636\) 6.91185 0.274073
\(637\) 0 0
\(638\) −13.6770 −0.541476
\(639\) 11.0707 0.437950
\(640\) −1.00000 −0.0395285
\(641\) 23.3948 0.924039 0.462019 0.886870i \(-0.347125\pi\)
0.462019 + 0.886870i \(0.347125\pi\)
\(642\) 10.1032 0.398742
\(643\) −6.00836 −0.236947 −0.118473 0.992957i \(-0.537800\pi\)
−0.118473 + 0.992957i \(0.537800\pi\)
\(644\) −1.49396 −0.0588702
\(645\) 3.43296 0.135173
\(646\) 1.58748 0.0624586
\(647\) −4.38511 −0.172396 −0.0861982 0.996278i \(-0.527472\pi\)
−0.0861982 + 0.996278i \(0.527472\pi\)
\(648\) 1.00000 0.0392837
\(649\) −40.3534 −1.58401
\(650\) 0 0
\(651\) −2.58642 −0.101370
\(652\) −2.48858 −0.0974604
\(653\) −20.6160 −0.806765 −0.403382 0.915032i \(-0.632166\pi\)
−0.403382 + 0.915032i \(0.632166\pi\)
\(654\) −10.5211 −0.411408
\(655\) −4.62133 −0.180570
\(656\) 0.198062 0.00773303
\(657\) 4.13706 0.161402
\(658\) −1.08085 −0.0421359
\(659\) 18.4910 0.720306 0.360153 0.932893i \(-0.382724\pi\)
0.360153 + 0.932893i \(0.382724\pi\)
\(660\) 3.93900 0.153325
\(661\) 17.6420 0.686196 0.343098 0.939300i \(-0.388524\pi\)
0.343098 + 0.939300i \(0.388524\pi\)
\(662\) 20.5109 0.797180
\(663\) 0 0
\(664\) 15.5864 0.604870
\(665\) 1.09246 0.0423637
\(666\) 4.80194 0.186071
\(667\) −4.32975 −0.167649
\(668\) 7.27950 0.281652
\(669\) −11.4004 −0.440766
\(670\) 9.16852 0.354211
\(671\) 39.1021 1.50952
\(672\) 1.19806 0.0462163
\(673\) 17.1473 0.660981 0.330491 0.943809i \(-0.392786\pi\)
0.330491 + 0.943809i \(0.392786\pi\)
\(674\) 6.82371 0.262839
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 31.7982 1.22210 0.611052 0.791590i \(-0.290746\pi\)
0.611052 + 0.791590i \(0.290746\pi\)
\(678\) 11.2131 0.430638
\(679\) −20.3642 −0.781506
\(680\) −1.74094 −0.0667620
\(681\) −8.86725 −0.339794
\(682\) −8.50365 −0.325622
\(683\) 6.32783 0.242128 0.121064 0.992645i \(-0.461369\pi\)
0.121064 + 0.992645i \(0.461369\pi\)
\(684\) 0.911854 0.0348656
\(685\) −0.0760644 −0.00290627
\(686\) 15.0532 0.574735
\(687\) −12.6136 −0.481237
\(688\) 3.43296 0.130880
\(689\) 0 0
\(690\) 1.24698 0.0474717
\(691\) −15.1825 −0.577570 −0.288785 0.957394i \(-0.593251\pi\)
−0.288785 + 0.957394i \(0.593251\pi\)
\(692\) 15.8485 0.602468
\(693\) −4.71917 −0.179266
\(694\) −29.4741 −1.11882
\(695\) −18.7017 −0.709396
\(696\) 3.47219 0.131613
\(697\) 0.344814 0.0130608
\(698\) −34.5394 −1.30734
\(699\) 23.4155 0.885656
\(700\) −1.19806 −0.0452825
\(701\) −23.6528 −0.893354 −0.446677 0.894695i \(-0.647393\pi\)
−0.446677 + 0.894695i \(0.647393\pi\)
\(702\) 0 0
\(703\) 4.37867 0.165145
\(704\) 3.93900 0.148457
\(705\) 0.902165 0.0339775
\(706\) 31.2295 1.17534
\(707\) −7.99138 −0.300547
\(708\) 10.2446 0.385015
\(709\) −40.9353 −1.53736 −0.768678 0.639636i \(-0.779085\pi\)
−0.768678 + 0.639636i \(0.779085\pi\)
\(710\) −11.0707 −0.415476
\(711\) 2.73556 0.102592
\(712\) 8.57673 0.321426
\(713\) −2.69202 −0.100817
\(714\) 2.08575 0.0780573
\(715\) 0 0
\(716\) 18.4819 0.690700
\(717\) −8.87263 −0.331354
\(718\) 18.2524 0.681172
\(719\) 4.55363 0.169822 0.0849109 0.996389i \(-0.472939\pi\)
0.0849109 + 0.996389i \(0.472939\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −12.0183 −0.447585
\(722\) −18.1685 −0.676162
\(723\) −9.72886 −0.361820
\(724\) 10.7041 0.397815
\(725\) −3.47219 −0.128954
\(726\) −4.51573 −0.167594
\(727\) 18.1612 0.673563 0.336781 0.941583i \(-0.390662\pi\)
0.336781 + 0.941583i \(0.390662\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −4.13706 −0.153120
\(731\) 5.97657 0.221052
\(732\) −9.92692 −0.366910
\(733\) 9.89115 0.365338 0.182669 0.983174i \(-0.441526\pi\)
0.182669 + 0.983174i \(0.441526\pi\)
\(734\) −19.3773 −0.715231
\(735\) −5.56465 −0.205255
\(736\) 1.24698 0.0459643
\(737\) −36.1148 −1.33031
\(738\) 0.198062 0.00729077
\(739\) −23.9608 −0.881411 −0.440706 0.897652i \(-0.645272\pi\)
−0.440706 + 0.897652i \(0.645272\pi\)
\(740\) −4.80194 −0.176523
\(741\) 0 0
\(742\) 8.28083 0.303999
\(743\) 46.6708 1.71219 0.856094 0.516821i \(-0.172885\pi\)
0.856094 + 0.516821i \(0.172885\pi\)
\(744\) 2.15883 0.0791467
\(745\) −13.6799 −0.501194
\(746\) −13.6280 −0.498958
\(747\) 15.5864 0.570277
\(748\) 6.85756 0.250737
\(749\) 12.1043 0.442281
\(750\) 1.00000 0.0365148
\(751\) −30.9168 −1.12817 −0.564084 0.825717i \(-0.690771\pi\)
−0.564084 + 0.825717i \(0.690771\pi\)
\(752\) 0.902165 0.0328986
\(753\) −29.8823 −1.08897
\(754\) 0 0
\(755\) 1.15883 0.0421743
\(756\) 1.19806 0.0435731
\(757\) 24.4566 0.888892 0.444446 0.895806i \(-0.353401\pi\)
0.444446 + 0.895806i \(0.353401\pi\)
\(758\) −1.32006 −0.0479467
\(759\) −4.91185 −0.178289
\(760\) −0.911854 −0.0330764
\(761\) −20.5536 −0.745069 −0.372534 0.928018i \(-0.621511\pi\)
−0.372534 + 0.928018i \(0.621511\pi\)
\(762\) 9.32736 0.337894
\(763\) −12.6049 −0.456329
\(764\) 15.3623 0.555788
\(765\) −1.74094 −0.0629438
\(766\) 25.5308 0.922465
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −21.0084 −0.757581 −0.378790 0.925482i \(-0.623660\pi\)
−0.378790 + 0.925482i \(0.623660\pi\)
\(770\) 4.71917 0.170067
\(771\) 8.52111 0.306880
\(772\) −18.3207 −0.659375
\(773\) −36.2626 −1.30428 −0.652138 0.758100i \(-0.726128\pi\)
−0.652138 + 0.758100i \(0.726128\pi\)
\(774\) 3.43296 0.123395
\(775\) −2.15883 −0.0775476
\(776\) 16.9976 0.610179
\(777\) 5.75302 0.206388
\(778\) −6.05967 −0.217250
\(779\) 0.180604 0.00647081
\(780\) 0 0
\(781\) 43.6075 1.56040
\(782\) 2.17092 0.0776318
\(783\) 3.47219 0.124086
\(784\) −5.56465 −0.198737
\(785\) 12.2664 0.437805
\(786\) −4.62133 −0.164838
\(787\) −1.55927 −0.0555820 −0.0277910 0.999614i \(-0.508847\pi\)
−0.0277910 + 0.999614i \(0.508847\pi\)
\(788\) −2.31336 −0.0824099
\(789\) 22.6450 0.806184
\(790\) −2.73556 −0.0973269
\(791\) 13.4340 0.477659
\(792\) 3.93900 0.139966
\(793\) 0 0
\(794\) 5.99330 0.212694
\(795\) −6.91185 −0.245138
\(796\) 6.36765 0.225695
\(797\) 5.16793 0.183058 0.0915288 0.995802i \(-0.470825\pi\)
0.0915288 + 0.995802i \(0.470825\pi\)
\(798\) 1.09246 0.0386726
\(799\) 1.57061 0.0555644
\(800\) 1.00000 0.0353553
\(801\) 8.57673 0.303044
\(802\) 17.3773 0.613615
\(803\) 16.2959 0.575070
\(804\) 9.16852 0.323349
\(805\) 1.49396 0.0526551
\(806\) 0 0
\(807\) 9.02608 0.317733
\(808\) 6.67025 0.234659
\(809\) −31.8605 −1.12016 −0.560079 0.828440i \(-0.689229\pi\)
−0.560079 + 0.828440i \(0.689229\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −15.3056 −0.537452 −0.268726 0.963217i \(-0.586603\pi\)
−0.268726 + 0.963217i \(0.586603\pi\)
\(812\) 4.15990 0.145984
\(813\) 8.59611 0.301479
\(814\) 18.9148 0.662965
\(815\) 2.48858 0.0871712
\(816\) −1.74094 −0.0609450
\(817\) 3.13036 0.109517
\(818\) 29.3672 1.02680
\(819\) 0 0
\(820\) −0.198062 −0.00691663
\(821\) 32.0248 1.11767 0.558836 0.829278i \(-0.311248\pi\)
0.558836 + 0.829278i \(0.311248\pi\)
\(822\) −0.0760644 −0.00265305
\(823\) −36.8571 −1.28476 −0.642379 0.766387i \(-0.722052\pi\)
−0.642379 + 0.766387i \(0.722052\pi\)
\(824\) 10.0315 0.349462
\(825\) −3.93900 −0.137138
\(826\) 12.2737 0.427055
\(827\) −37.9269 −1.31885 −0.659424 0.751771i \(-0.729200\pi\)
−0.659424 + 0.751771i \(0.729200\pi\)
\(828\) 1.24698 0.0433355
\(829\) −34.2349 −1.18903 −0.594514 0.804086i \(-0.702655\pi\)
−0.594514 + 0.804086i \(0.702655\pi\)
\(830\) −15.5864 −0.541012
\(831\) −6.31527 −0.219074
\(832\) 0 0
\(833\) −9.68771 −0.335659
\(834\) −18.7017 −0.647587
\(835\) −7.27950 −0.251918
\(836\) 3.59179 0.124225
\(837\) 2.15883 0.0746202
\(838\) −26.5472 −0.917057
\(839\) 18.7318 0.646695 0.323347 0.946280i \(-0.395192\pi\)
0.323347 + 0.946280i \(0.395192\pi\)
\(840\) −1.19806 −0.0413371
\(841\) −16.9439 −0.584273
\(842\) −31.0834 −1.07120
\(843\) −21.7409 −0.748798
\(844\) −11.6775 −0.401958
\(845\) 0 0
\(846\) 0.902165 0.0310171
\(847\) −5.41013 −0.185894
\(848\) −6.91185 −0.237354
\(849\) −12.9487 −0.444398
\(850\) 1.74094 0.0597137
\(851\) 5.98792 0.205263
\(852\) −11.0707 −0.379276
\(853\) −4.64071 −0.158895 −0.0794475 0.996839i \(-0.525316\pi\)
−0.0794475 + 0.996839i \(0.525316\pi\)
\(854\) −11.8931 −0.406972
\(855\) −0.911854 −0.0311847
\(856\) −10.1032 −0.345321
\(857\) 11.1347 0.380353 0.190177 0.981750i \(-0.439094\pi\)
0.190177 + 0.981750i \(0.439094\pi\)
\(858\) 0 0
\(859\) 28.9661 0.988312 0.494156 0.869373i \(-0.335477\pi\)
0.494156 + 0.869373i \(0.335477\pi\)
\(860\) −3.43296 −0.117063
\(861\) 0.237291 0.00808685
\(862\) 36.1226 1.23034
\(863\) 33.1371 1.12800 0.563999 0.825775i \(-0.309262\pi\)
0.563999 + 0.825775i \(0.309262\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −15.8485 −0.538864
\(866\) −35.2868 −1.19909
\(867\) 13.9691 0.474417
\(868\) 2.58642 0.0877887
\(869\) 10.7754 0.365530
\(870\) −3.47219 −0.117718
\(871\) 0 0
\(872\) 10.5211 0.356290
\(873\) 16.9976 0.575282
\(874\) 1.13706 0.0384617
\(875\) 1.19806 0.0405019
\(876\) −4.13706 −0.139778
\(877\) 19.1371 0.646213 0.323106 0.946363i \(-0.395273\pi\)
0.323106 + 0.946363i \(0.395273\pi\)
\(878\) −32.0170 −1.08052
\(879\) −27.3056 −0.920995
\(880\) −3.93900 −0.132784
\(881\) −40.3387 −1.35905 −0.679523 0.733655i \(-0.737813\pi\)
−0.679523 + 0.733655i \(0.737813\pi\)
\(882\) −5.56465 −0.187371
\(883\) −4.31336 −0.145156 −0.0725780 0.997363i \(-0.523123\pi\)
−0.0725780 + 0.997363i \(0.523123\pi\)
\(884\) 0 0
\(885\) −10.2446 −0.344368
\(886\) −23.5948 −0.792682
\(887\) −9.38404 −0.315085 −0.157543 0.987512i \(-0.550357\pi\)
−0.157543 + 0.987512i \(0.550357\pi\)
\(888\) −4.80194 −0.161142
\(889\) 11.1748 0.374789
\(890\) −8.57673 −0.287493
\(891\) 3.93900 0.131962
\(892\) 11.4004 0.381715
\(893\) 0.822643 0.0275287
\(894\) −13.6799 −0.457526
\(895\) −18.4819 −0.617781
\(896\) −1.19806 −0.0400245
\(897\) 0 0
\(898\) −19.8374 −0.661984
\(899\) 7.49588 0.250002
\(900\) 1.00000 0.0333333
\(901\) −12.0331 −0.400881
\(902\) 0.780167 0.0259767
\(903\) 4.11290 0.136869
\(904\) −11.2131 −0.372943
\(905\) −10.7041 −0.355816
\(906\) 1.15883 0.0384997
\(907\) 14.2131 0.471939 0.235970 0.971760i \(-0.424173\pi\)
0.235970 + 0.971760i \(0.424173\pi\)
\(908\) 8.86725 0.294270
\(909\) 6.67025 0.221238
\(910\) 0 0
\(911\) 40.2707 1.33423 0.667113 0.744956i \(-0.267530\pi\)
0.667113 + 0.744956i \(0.267530\pi\)
\(912\) −0.911854 −0.0301945
\(913\) 61.3949 2.03188
\(914\) −26.5284 −0.877482
\(915\) 9.92692 0.328174
\(916\) 12.6136 0.416764
\(917\) −5.53665 −0.182836
\(918\) −1.74094 −0.0574595
\(919\) 7.03577 0.232089 0.116044 0.993244i \(-0.462979\pi\)
0.116044 + 0.993244i \(0.462979\pi\)
\(920\) −1.24698 −0.0411117
\(921\) 24.9124 0.820893
\(922\) −12.2717 −0.404148
\(923\) 0 0
\(924\) 4.71917 0.155249
\(925\) 4.80194 0.157887
\(926\) −12.6455 −0.415557
\(927\) 10.0315 0.329476
\(928\) −3.47219 −0.113980
\(929\) −21.3575 −0.700716 −0.350358 0.936616i \(-0.613940\pi\)
−0.350358 + 0.936616i \(0.613940\pi\)
\(930\) −2.15883 −0.0707909
\(931\) −5.07415 −0.166298
\(932\) −23.4155 −0.767000
\(933\) −21.3787 −0.699906
\(934\) 11.3870 0.372595
\(935\) −6.85756 −0.224266
\(936\) 0 0
\(937\) 41.7396 1.36357 0.681787 0.731551i \(-0.261203\pi\)
0.681787 + 0.731551i \(0.261203\pi\)
\(938\) 10.9845 0.358655
\(939\) 23.8713 0.779010
\(940\) −0.902165 −0.0294254
\(941\) −26.2078 −0.854348 −0.427174 0.904169i \(-0.640491\pi\)
−0.427174 + 0.904169i \(0.640491\pi\)
\(942\) 12.2664 0.399660
\(943\) 0.246980 0.00804276
\(944\) −10.2446 −0.333433
\(945\) −1.19806 −0.0389730
\(946\) 13.5224 0.439652
\(947\) −3.15239 −0.102439 −0.0512195 0.998687i \(-0.516311\pi\)
−0.0512195 + 0.998687i \(0.516311\pi\)
\(948\) −2.73556 −0.0888469
\(949\) 0 0
\(950\) 0.911854 0.0295845
\(951\) 12.6407 0.409903
\(952\) −2.08575 −0.0675996
\(953\) −3.56060 −0.115339 −0.0576695 0.998336i \(-0.518367\pi\)
−0.0576695 + 0.998336i \(0.518367\pi\)
\(954\) −6.91185 −0.223780
\(955\) −15.3623 −0.497111
\(956\) 8.87263 0.286961
\(957\) 13.6770 0.442113
\(958\) 18.8431 0.608792
\(959\) −0.0911299 −0.00294274
\(960\) 1.00000 0.0322749
\(961\) −26.3394 −0.849659
\(962\) 0 0
\(963\) −10.1032 −0.325571
\(964\) 9.72886 0.313345
\(965\) 18.3207 0.589763
\(966\) 1.49396 0.0480673
\(967\) −7.26875 −0.233747 −0.116874 0.993147i \(-0.537287\pi\)
−0.116874 + 0.993147i \(0.537287\pi\)
\(968\) 4.51573 0.145141
\(969\) −1.58748 −0.0509973
\(970\) −16.9976 −0.545760
\(971\) −9.53511 −0.305996 −0.152998 0.988226i \(-0.548893\pi\)
−0.152998 + 0.988226i \(0.548893\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −22.4058 −0.718297
\(974\) 27.6383 0.885589
\(975\) 0 0
\(976\) 9.92692 0.317753
\(977\) −17.5700 −0.562115 −0.281057 0.959691i \(-0.590685\pi\)
−0.281057 + 0.959691i \(0.590685\pi\)
\(978\) 2.48858 0.0795761
\(979\) 33.7837 1.07973
\(980\) 5.56465 0.177756
\(981\) 10.5211 0.335913
\(982\) −13.9866 −0.446330
\(983\) −43.4935 −1.38723 −0.693613 0.720347i \(-0.743982\pi\)
−0.693613 + 0.720347i \(0.743982\pi\)
\(984\) −0.198062 −0.00631399
\(985\) 2.31336 0.0737096
\(986\) −6.04487 −0.192508
\(987\) 1.08085 0.0344038
\(988\) 0 0
\(989\) 4.28083 0.136123
\(990\) −3.93900 −0.125190
\(991\) 38.8232 1.23326 0.616630 0.787253i \(-0.288497\pi\)
0.616630 + 0.787253i \(0.288497\pi\)
\(992\) −2.15883 −0.0685430
\(993\) −20.5109 −0.650895
\(994\) −13.2634 −0.420689
\(995\) −6.36765 −0.201868
\(996\) −15.5864 −0.493875
\(997\) −21.9807 −0.696137 −0.348069 0.937469i \(-0.613162\pi\)
−0.348069 + 0.937469i \(0.613162\pi\)
\(998\) −8.87071 −0.280797
\(999\) −4.80194 −0.151927
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 5070.2.a.bs.1.3 yes 3
13.5 odd 4 5070.2.b.v.1351.3 6
13.8 odd 4 5070.2.b.v.1351.4 6
13.12 even 2 5070.2.a.bl.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bl.1.1 3 13.12 even 2
5070.2.a.bs.1.3 yes 3 1.1 even 1 trivial
5070.2.b.v.1351.3 6 13.5 odd 4
5070.2.b.v.1351.4 6 13.8 odd 4