Properties

Label 1950.2.a.bc.1.2
Level $1950$
Weight $2$
Character 1950.1
Self dual yes
Analytic conductor $15.571$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
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 1.2
Root \(-2.70156\) of defining polynomial
Character \(\chi\) \(=\) 1950.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.70156 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.70156 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.70156 q^{11} -1.00000 q^{12} +1.00000 q^{13} -1.70156 q^{14} +1.00000 q^{16} -5.70156 q^{17} -1.00000 q^{18} +4.70156 q^{19} -1.70156 q^{21} +1.70156 q^{22} +1.00000 q^{24} -1.00000 q^{26} -1.00000 q^{27} +1.70156 q^{28} +6.40312 q^{29} +9.10469 q^{31} -1.00000 q^{32} +1.70156 q^{33} +5.70156 q^{34} +1.00000 q^{36} -4.70156 q^{37} -4.70156 q^{38} -1.00000 q^{39} +2.70156 q^{41} +1.70156 q^{42} -1.40312 q^{43} -1.70156 q^{44} -7.00000 q^{47} -1.00000 q^{48} -4.10469 q^{49} +5.70156 q^{51} +1.00000 q^{52} +10.4031 q^{53} +1.00000 q^{54} -1.70156 q^{56} -4.70156 q^{57} -6.40312 q^{58} -3.70156 q^{59} -5.10469 q^{61} -9.10469 q^{62} +1.70156 q^{63} +1.00000 q^{64} -1.70156 q^{66} -6.40312 q^{67} -5.70156 q^{68} +4.70156 q^{71} -1.00000 q^{72} +12.0000 q^{73} +4.70156 q^{74} +4.70156 q^{76} -2.89531 q^{77} +1.00000 q^{78} +0.701562 q^{79} +1.00000 q^{81} -2.70156 q^{82} +4.29844 q^{83} -1.70156 q^{84} +1.40312 q^{86} -6.40312 q^{87} +1.70156 q^{88} -1.40312 q^{89} +1.70156 q^{91} -9.10469 q^{93} +7.00000 q^{94} +1.00000 q^{96} +15.4031 q^{97} +4.10469 q^{98} -1.70156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 3q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 3q^{7} - 2q^{8} + 2q^{9} + 3q^{11} - 2q^{12} + 2q^{13} + 3q^{14} + 2q^{16} - 5q^{17} - 2q^{18} + 3q^{19} + 3q^{21} - 3q^{22} + 2q^{24} - 2q^{26} - 2q^{27} - 3q^{28} - q^{31} - 2q^{32} - 3q^{33} + 5q^{34} + 2q^{36} - 3q^{37} - 3q^{38} - 2q^{39} - q^{41} - 3q^{42} + 10q^{43} + 3q^{44} - 14q^{47} - 2q^{48} + 11q^{49} + 5q^{51} + 2q^{52} + 8q^{53} + 2q^{54} + 3q^{56} - 3q^{57} - q^{59} + 9q^{61} + q^{62} - 3q^{63} + 2q^{64} + 3q^{66} - 5q^{68} + 3q^{71} - 2q^{72} + 24q^{73} + 3q^{74} + 3q^{76} - 25q^{77} + 2q^{78} - 5q^{79} + 2q^{81} + q^{82} + 15q^{83} + 3q^{84} - 10q^{86} - 3q^{88} + 10q^{89} - 3q^{91} + q^{93} + 14q^{94} + 2q^{96} + 18q^{97} - 11q^{98} + 3q^{99} + O(q^{100}) \)

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
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 1.70156 0.643130 0.321565 0.946888i \(-0.395791\pi\)
0.321565 + 0.946888i \(0.395791\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.70156 −0.513040 −0.256520 0.966539i \(-0.582576\pi\)
−0.256520 + 0.966539i \(0.582576\pi\)
\(12\) −1.00000 −0.288675
\(13\) 1.00000 0.277350
\(14\) −1.70156 −0.454762
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.70156 −1.38283 −0.691416 0.722457i \(-0.743013\pi\)
−0.691416 + 0.722457i \(0.743013\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.70156 1.07861 0.539306 0.842110i \(-0.318687\pi\)
0.539306 + 0.842110i \(0.318687\pi\)
\(20\) 0 0
\(21\) −1.70156 −0.371311
\(22\) 1.70156 0.362774
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) −1.00000 −0.192450
\(28\) 1.70156 0.321565
\(29\) 6.40312 1.18903 0.594515 0.804084i \(-0.297344\pi\)
0.594515 + 0.804084i \(0.297344\pi\)
\(30\) 0 0
\(31\) 9.10469 1.63525 0.817625 0.575751i \(-0.195290\pi\)
0.817625 + 0.575751i \(0.195290\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.70156 0.296204
\(34\) 5.70156 0.977810
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.70156 −0.772932 −0.386466 0.922304i \(-0.626304\pi\)
−0.386466 + 0.922304i \(0.626304\pi\)
\(38\) −4.70156 −0.762694
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 2.70156 0.421913 0.210957 0.977495i \(-0.432342\pi\)
0.210957 + 0.977495i \(0.432342\pi\)
\(42\) 1.70156 0.262557
\(43\) −1.40312 −0.213974 −0.106987 0.994260i \(-0.534120\pi\)
−0.106987 + 0.994260i \(0.534120\pi\)
\(44\) −1.70156 −0.256520
\(45\) 0 0
\(46\) 0 0
\(47\) −7.00000 −1.02105 −0.510527 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) −1.00000 −0.144338
\(49\) −4.10469 −0.586384
\(50\) 0 0
\(51\) 5.70156 0.798378
\(52\) 1.00000 0.138675
\(53\) 10.4031 1.42898 0.714490 0.699646i \(-0.246659\pi\)
0.714490 + 0.699646i \(0.246659\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.70156 −0.227381
\(57\) −4.70156 −0.622737
\(58\) −6.40312 −0.840771
\(59\) −3.70156 −0.481902 −0.240951 0.970537i \(-0.577459\pi\)
−0.240951 + 0.970537i \(0.577459\pi\)
\(60\) 0 0
\(61\) −5.10469 −0.653588 −0.326794 0.945096i \(-0.605968\pi\)
−0.326794 + 0.945096i \(0.605968\pi\)
\(62\) −9.10469 −1.15630
\(63\) 1.70156 0.214377
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.70156 −0.209448
\(67\) −6.40312 −0.782266 −0.391133 0.920334i \(-0.627917\pi\)
−0.391133 + 0.920334i \(0.627917\pi\)
\(68\) −5.70156 −0.691416
\(69\) 0 0
\(70\) 0 0
\(71\) 4.70156 0.557973 0.278986 0.960295i \(-0.410002\pi\)
0.278986 + 0.960295i \(0.410002\pi\)
\(72\) −1.00000 −0.117851
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 4.70156 0.546545
\(75\) 0 0
\(76\) 4.70156 0.539306
\(77\) −2.89531 −0.329952
\(78\) 1.00000 0.113228
\(79\) 0.701562 0.0789319 0.0394660 0.999221i \(-0.487434\pi\)
0.0394660 + 0.999221i \(0.487434\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.70156 −0.298338
\(83\) 4.29844 0.471815 0.235907 0.971776i \(-0.424194\pi\)
0.235907 + 0.971776i \(0.424194\pi\)
\(84\) −1.70156 −0.185656
\(85\) 0 0
\(86\) 1.40312 0.151303
\(87\) −6.40312 −0.686487
\(88\) 1.70156 0.181387
\(89\) −1.40312 −0.148731 −0.0743654 0.997231i \(-0.523693\pi\)
−0.0743654 + 0.997231i \(0.523693\pi\)
\(90\) 0 0
\(91\) 1.70156 0.178372
\(92\) 0 0
\(93\) −9.10469 −0.944112
\(94\) 7.00000 0.721995
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 15.4031 1.56395 0.781975 0.623310i \(-0.214212\pi\)
0.781975 + 0.623310i \(0.214212\pi\)
\(98\) 4.10469 0.414636
\(99\) −1.70156 −0.171013
\(100\) 0 0
\(101\) 0.298438 0.0296957 0.0148478 0.999890i \(-0.495274\pi\)
0.0148478 + 0.999890i \(0.495274\pi\)
\(102\) −5.70156 −0.564539
\(103\) 11.4031 1.12358 0.561792 0.827279i \(-0.310112\pi\)
0.561792 + 0.827279i \(0.310112\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −10.4031 −1.01044
\(107\) 0.104686 0.0101204 0.00506021 0.999987i \(-0.498389\pi\)
0.00506021 + 0.999987i \(0.498389\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 10.7016 1.02502 0.512512 0.858680i \(-0.328715\pi\)
0.512512 + 0.858680i \(0.328715\pi\)
\(110\) 0 0
\(111\) 4.70156 0.446253
\(112\) 1.70156 0.160783
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 4.70156 0.440342
\(115\) 0 0
\(116\) 6.40312 0.594515
\(117\) 1.00000 0.0924500
\(118\) 3.70156 0.340756
\(119\) −9.70156 −0.889341
\(120\) 0 0
\(121\) −8.10469 −0.736790
\(122\) 5.10469 0.462157
\(123\) −2.70156 −0.243592
\(124\) 9.10469 0.817625
\(125\) 0 0
\(126\) −1.70156 −0.151587
\(127\) 12.7016 1.12708 0.563541 0.826088i \(-0.309439\pi\)
0.563541 + 0.826088i \(0.309439\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.40312 0.123538
\(130\) 0 0
\(131\) −9.50781 −0.830701 −0.415351 0.909661i \(-0.636341\pi\)
−0.415351 + 0.909661i \(0.636341\pi\)
\(132\) 1.70156 0.148102
\(133\) 8.00000 0.693688
\(134\) 6.40312 0.553146
\(135\) 0 0
\(136\) 5.70156 0.488905
\(137\) −7.29844 −0.623548 −0.311774 0.950156i \(-0.600923\pi\)
−0.311774 + 0.950156i \(0.600923\pi\)
\(138\) 0 0
\(139\) −3.40312 −0.288649 −0.144325 0.989530i \(-0.546101\pi\)
−0.144325 + 0.989530i \(0.546101\pi\)
\(140\) 0 0
\(141\) 7.00000 0.589506
\(142\) −4.70156 −0.394546
\(143\) −1.70156 −0.142292
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −12.0000 −0.993127
\(147\) 4.10469 0.338549
\(148\) −4.70156 −0.386466
\(149\) 19.4031 1.58957 0.794783 0.606894i \(-0.207585\pi\)
0.794783 + 0.606894i \(0.207585\pi\)
\(150\) 0 0
\(151\) 5.10469 0.415413 0.207707 0.978191i \(-0.433400\pi\)
0.207707 + 0.978191i \(0.433400\pi\)
\(152\) −4.70156 −0.381347
\(153\) −5.70156 −0.460944
\(154\) 2.89531 0.233311
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) 16.2984 1.30076 0.650378 0.759610i \(-0.274610\pi\)
0.650378 + 0.759610i \(0.274610\pi\)
\(158\) −0.701562 −0.0558133
\(159\) −10.4031 −0.825021
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −6.80625 −0.533107 −0.266553 0.963820i \(-0.585885\pi\)
−0.266553 + 0.963820i \(0.585885\pi\)
\(164\) 2.70156 0.210957
\(165\) 0 0
\(166\) −4.29844 −0.333623
\(167\) −18.1047 −1.40098 −0.700491 0.713661i \(-0.747036\pi\)
−0.700491 + 0.713661i \(0.747036\pi\)
\(168\) 1.70156 0.131278
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 4.70156 0.359537
\(172\) −1.40312 −0.106987
\(173\) 25.8062 1.96201 0.981006 0.193976i \(-0.0621382\pi\)
0.981006 + 0.193976i \(0.0621382\pi\)
\(174\) 6.40312 0.485420
\(175\) 0 0
\(176\) −1.70156 −0.128260
\(177\) 3.70156 0.278226
\(178\) 1.40312 0.105169
\(179\) 14.2094 1.06206 0.531029 0.847354i \(-0.321805\pi\)
0.531029 + 0.847354i \(0.321805\pi\)
\(180\) 0 0
\(181\) 17.7016 1.31575 0.657873 0.753129i \(-0.271456\pi\)
0.657873 + 0.753129i \(0.271456\pi\)
\(182\) −1.70156 −0.126128
\(183\) 5.10469 0.377349
\(184\) 0 0
\(185\) 0 0
\(186\) 9.10469 0.667588
\(187\) 9.70156 0.709448
\(188\) −7.00000 −0.510527
\(189\) −1.70156 −0.123770
\(190\) 0 0
\(191\) 12.8062 0.926628 0.463314 0.886194i \(-0.346660\pi\)
0.463314 + 0.886194i \(0.346660\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 22.2094 1.59867 0.799333 0.600889i \(-0.205186\pi\)
0.799333 + 0.600889i \(0.205186\pi\)
\(194\) −15.4031 −1.10588
\(195\) 0 0
\(196\) −4.10469 −0.293192
\(197\) −7.40312 −0.527451 −0.263725 0.964598i \(-0.584951\pi\)
−0.263725 + 0.964598i \(0.584951\pi\)
\(198\) 1.70156 0.120925
\(199\) 14.7016 1.04217 0.521083 0.853506i \(-0.325528\pi\)
0.521083 + 0.853506i \(0.325528\pi\)
\(200\) 0 0
\(201\) 6.40312 0.451642
\(202\) −0.298438 −0.0209980
\(203\) 10.8953 0.764701
\(204\) 5.70156 0.399189
\(205\) 0 0
\(206\) −11.4031 −0.794493
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) −18.8062 −1.29468 −0.647338 0.762203i \(-0.724118\pi\)
−0.647338 + 0.762203i \(0.724118\pi\)
\(212\) 10.4031 0.714490
\(213\) −4.70156 −0.322146
\(214\) −0.104686 −0.00715621
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 15.4922 1.05168
\(218\) −10.7016 −0.724801
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) −5.70156 −0.383529
\(222\) −4.70156 −0.315548
\(223\) 1.40312 0.0939601 0.0469801 0.998896i \(-0.485040\pi\)
0.0469801 + 0.998896i \(0.485040\pi\)
\(224\) −1.70156 −0.113690
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) −3.10469 −0.206065 −0.103033 0.994678i \(-0.532855\pi\)
−0.103033 + 0.994678i \(0.532855\pi\)
\(228\) −4.70156 −0.311369
\(229\) 9.29844 0.614458 0.307229 0.951636i \(-0.400598\pi\)
0.307229 + 0.951636i \(0.400598\pi\)
\(230\) 0 0
\(231\) 2.89531 0.190498
\(232\) −6.40312 −0.420386
\(233\) −18.2094 −1.19294 −0.596468 0.802637i \(-0.703430\pi\)
−0.596468 + 0.802637i \(0.703430\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) −3.70156 −0.240951
\(237\) −0.701562 −0.0455714
\(238\) 9.70156 0.628859
\(239\) −13.1047 −0.847672 −0.423836 0.905739i \(-0.639317\pi\)
−0.423836 + 0.905739i \(0.639317\pi\)
\(240\) 0 0
\(241\) −15.4031 −0.992202 −0.496101 0.868265i \(-0.665236\pi\)
−0.496101 + 0.868265i \(0.665236\pi\)
\(242\) 8.10469 0.520989
\(243\) −1.00000 −0.0641500
\(244\) −5.10469 −0.326794
\(245\) 0 0
\(246\) 2.70156 0.172245
\(247\) 4.70156 0.299153
\(248\) −9.10469 −0.578148
\(249\) −4.29844 −0.272402
\(250\) 0 0
\(251\) 6.70156 0.422999 0.211499 0.977378i \(-0.432165\pi\)
0.211499 + 0.977378i \(0.432165\pi\)
\(252\) 1.70156 0.107188
\(253\) 0 0
\(254\) −12.7016 −0.796967
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 21.7016 1.35371 0.676853 0.736118i \(-0.263343\pi\)
0.676853 + 0.736118i \(0.263343\pi\)
\(258\) −1.40312 −0.0873547
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 6.40312 0.396343
\(262\) 9.50781 0.587395
\(263\) −2.00000 −0.123325 −0.0616626 0.998097i \(-0.519640\pi\)
−0.0616626 + 0.998097i \(0.519640\pi\)
\(264\) −1.70156 −0.104724
\(265\) 0 0
\(266\) −8.00000 −0.490511
\(267\) 1.40312 0.0858698
\(268\) −6.40312 −0.391133
\(269\) −7.20937 −0.439563 −0.219782 0.975549i \(-0.570534\pi\)
−0.219782 + 0.975549i \(0.570534\pi\)
\(270\) 0 0
\(271\) −12.5078 −0.759795 −0.379898 0.925029i \(-0.624041\pi\)
−0.379898 + 0.925029i \(0.624041\pi\)
\(272\) −5.70156 −0.345708
\(273\) −1.70156 −0.102983
\(274\) 7.29844 0.440915
\(275\) 0 0
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 3.40312 0.204106
\(279\) 9.10469 0.545083
\(280\) 0 0
\(281\) 22.9109 1.36675 0.683376 0.730067i \(-0.260511\pi\)
0.683376 + 0.730067i \(0.260511\pi\)
\(282\) −7.00000 −0.416844
\(283\) −8.59688 −0.511031 −0.255516 0.966805i \(-0.582245\pi\)
−0.255516 + 0.966805i \(0.582245\pi\)
\(284\) 4.70156 0.278986
\(285\) 0 0
\(286\) 1.70156 0.100615
\(287\) 4.59688 0.271345
\(288\) −1.00000 −0.0589256
\(289\) 15.5078 0.912224
\(290\) 0 0
\(291\) −15.4031 −0.902947
\(292\) 12.0000 0.702247
\(293\) −8.59688 −0.502235 −0.251117 0.967957i \(-0.580798\pi\)
−0.251117 + 0.967957i \(0.580798\pi\)
\(294\) −4.10469 −0.239390
\(295\) 0 0
\(296\) 4.70156 0.273273
\(297\) 1.70156 0.0987346
\(298\) −19.4031 −1.12399
\(299\) 0 0
\(300\) 0 0
\(301\) −2.38750 −0.137613
\(302\) −5.10469 −0.293742
\(303\) −0.298438 −0.0171448
\(304\) 4.70156 0.269653
\(305\) 0 0
\(306\) 5.70156 0.325937
\(307\) −0.701562 −0.0400403 −0.0200201 0.999800i \(-0.506373\pi\)
−0.0200201 + 0.999800i \(0.506373\pi\)
\(308\) −2.89531 −0.164976
\(309\) −11.4031 −0.648701
\(310\) 0 0
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 1.00000 0.0566139
\(313\) −17.2094 −0.972731 −0.486366 0.873755i \(-0.661678\pi\)
−0.486366 + 0.873755i \(0.661678\pi\)
\(314\) −16.2984 −0.919774
\(315\) 0 0
\(316\) 0.701562 0.0394660
\(317\) −26.8062 −1.50559 −0.752794 0.658256i \(-0.771295\pi\)
−0.752794 + 0.658256i \(0.771295\pi\)
\(318\) 10.4031 0.583378
\(319\) −10.8953 −0.610020
\(320\) 0 0
\(321\) −0.104686 −0.00584302
\(322\) 0 0
\(323\) −26.8062 −1.49154
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 6.80625 0.376963
\(327\) −10.7016 −0.591798
\(328\) −2.70156 −0.149169
\(329\) −11.9109 −0.656671
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 4.29844 0.235907
\(333\) −4.70156 −0.257644
\(334\) 18.1047 0.990644
\(335\) 0 0
\(336\) −1.70156 −0.0928278
\(337\) 11.1047 0.604911 0.302455 0.953164i \(-0.402194\pi\)
0.302455 + 0.953164i \(0.402194\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) −15.4922 −0.838949
\(342\) −4.70156 −0.254231
\(343\) −18.8953 −1.02025
\(344\) 1.40312 0.0756514
\(345\) 0 0
\(346\) −25.8062 −1.38735
\(347\) 21.5078 1.15460 0.577300 0.816532i \(-0.304106\pi\)
0.577300 + 0.816532i \(0.304106\pi\)
\(348\) −6.40312 −0.343243
\(349\) 1.40312 0.0751075 0.0375538 0.999295i \(-0.488043\pi\)
0.0375538 + 0.999295i \(0.488043\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 1.70156 0.0906936
\(353\) 17.5078 0.931847 0.465923 0.884825i \(-0.345722\pi\)
0.465923 + 0.884825i \(0.345722\pi\)
\(354\) −3.70156 −0.196736
\(355\) 0 0
\(356\) −1.40312 −0.0743654
\(357\) 9.70156 0.513461
\(358\) −14.2094 −0.750989
\(359\) 16.6125 0.876774 0.438387 0.898786i \(-0.355550\pi\)
0.438387 + 0.898786i \(0.355550\pi\)
\(360\) 0 0
\(361\) 3.10469 0.163405
\(362\) −17.7016 −0.930373
\(363\) 8.10469 0.425386
\(364\) 1.70156 0.0891861
\(365\) 0 0
\(366\) −5.10469 −0.266826
\(367\) 8.70156 0.454218 0.227109 0.973869i \(-0.427073\pi\)
0.227109 + 0.973869i \(0.427073\pi\)
\(368\) 0 0
\(369\) 2.70156 0.140638
\(370\) 0 0
\(371\) 17.7016 0.919019
\(372\) −9.10469 −0.472056
\(373\) −11.7016 −0.605884 −0.302942 0.953009i \(-0.597969\pi\)
−0.302942 + 0.953009i \(0.597969\pi\)
\(374\) −9.70156 −0.501656
\(375\) 0 0
\(376\) 7.00000 0.360997
\(377\) 6.40312 0.329778
\(378\) 1.70156 0.0875189
\(379\) −33.1047 −1.70047 −0.850237 0.526400i \(-0.823541\pi\)
−0.850237 + 0.526400i \(0.823541\pi\)
\(380\) 0 0
\(381\) −12.7016 −0.650721
\(382\) −12.8062 −0.655225
\(383\) −0.492189 −0.0251497 −0.0125749 0.999921i \(-0.504003\pi\)
−0.0125749 + 0.999921i \(0.504003\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −22.2094 −1.13043
\(387\) −1.40312 −0.0713248
\(388\) 15.4031 0.781975
\(389\) −7.89531 −0.400308 −0.200154 0.979764i \(-0.564144\pi\)
−0.200154 + 0.979764i \(0.564144\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 4.10469 0.207318
\(393\) 9.50781 0.479606
\(394\) 7.40312 0.372964
\(395\) 0 0
\(396\) −1.70156 −0.0855067
\(397\) −22.9109 −1.14987 −0.574933 0.818200i \(-0.694972\pi\)
−0.574933 + 0.818200i \(0.694972\pi\)
\(398\) −14.7016 −0.736923
\(399\) −8.00000 −0.400501
\(400\) 0 0
\(401\) −20.2094 −1.00921 −0.504604 0.863351i \(-0.668361\pi\)
−0.504604 + 0.863351i \(0.668361\pi\)
\(402\) −6.40312 −0.319359
\(403\) 9.10469 0.453537
\(404\) 0.298438 0.0148478
\(405\) 0 0
\(406\) −10.8953 −0.540725
\(407\) 8.00000 0.396545
\(408\) −5.70156 −0.282269
\(409\) 16.5969 0.820663 0.410331 0.911937i \(-0.365413\pi\)
0.410331 + 0.911937i \(0.365413\pi\)
\(410\) 0 0
\(411\) 7.29844 0.360005
\(412\) 11.4031 0.561792
\(413\) −6.29844 −0.309926
\(414\) 0 0
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 3.40312 0.166652
\(418\) 8.00000 0.391293
\(419\) −34.9109 −1.70551 −0.852755 0.522310i \(-0.825070\pi\)
−0.852755 + 0.522310i \(0.825070\pi\)
\(420\) 0 0
\(421\) −13.4031 −0.653228 −0.326614 0.945158i \(-0.605908\pi\)
−0.326614 + 0.945158i \(0.605908\pi\)
\(422\) 18.8062 0.915474
\(423\) −7.00000 −0.340352
\(424\) −10.4031 −0.505220
\(425\) 0 0
\(426\) 4.70156 0.227791
\(427\) −8.68594 −0.420342
\(428\) 0.104686 0.00506021
\(429\) 1.70156 0.0821522
\(430\) 0 0
\(431\) −22.3141 −1.07483 −0.537415 0.843318i \(-0.680599\pi\)
−0.537415 + 0.843318i \(0.680599\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −8.10469 −0.389486 −0.194743 0.980854i \(-0.562387\pi\)
−0.194743 + 0.980854i \(0.562387\pi\)
\(434\) −15.4922 −0.743649
\(435\) 0 0
\(436\) 10.7016 0.512512
\(437\) 0 0
\(438\) 12.0000 0.573382
\(439\) −16.7016 −0.797122 −0.398561 0.917142i \(-0.630490\pi\)
−0.398561 + 0.917142i \(0.630490\pi\)
\(440\) 0 0
\(441\) −4.10469 −0.195461
\(442\) 5.70156 0.271196
\(443\) −0.492189 −0.0233846 −0.0116923 0.999932i \(-0.503722\pi\)
−0.0116923 + 0.999932i \(0.503722\pi\)
\(444\) 4.70156 0.223126
\(445\) 0 0
\(446\) −1.40312 −0.0664399
\(447\) −19.4031 −0.917736
\(448\) 1.70156 0.0803913
\(449\) 8.70156 0.410652 0.205326 0.978694i \(-0.434175\pi\)
0.205326 + 0.978694i \(0.434175\pi\)
\(450\) 0 0
\(451\) −4.59688 −0.216458
\(452\) 14.0000 0.658505
\(453\) −5.10469 −0.239839
\(454\) 3.10469 0.145710
\(455\) 0 0
\(456\) 4.70156 0.220171
\(457\) 15.4031 0.720528 0.360264 0.932850i \(-0.382687\pi\)
0.360264 + 0.932850i \(0.382687\pi\)
\(458\) −9.29844 −0.434487
\(459\) 5.70156 0.266126
\(460\) 0 0
\(461\) −2.20937 −0.102901 −0.0514504 0.998676i \(-0.516384\pi\)
−0.0514504 + 0.998676i \(0.516384\pi\)
\(462\) −2.89531 −0.134702
\(463\) 27.7016 1.28740 0.643700 0.765278i \(-0.277398\pi\)
0.643700 + 0.765278i \(0.277398\pi\)
\(464\) 6.40312 0.297258
\(465\) 0 0
\(466\) 18.2094 0.843533
\(467\) −36.7016 −1.69835 −0.849173 0.528115i \(-0.822899\pi\)
−0.849173 + 0.528115i \(0.822899\pi\)
\(468\) 1.00000 0.0462250
\(469\) −10.8953 −0.503099
\(470\) 0 0
\(471\) −16.2984 −0.750992
\(472\) 3.70156 0.170378
\(473\) 2.38750 0.109778
\(474\) 0.701562 0.0322238
\(475\) 0 0
\(476\) −9.70156 −0.444670
\(477\) 10.4031 0.476326
\(478\) 13.1047 0.599394
\(479\) 10.6125 0.484897 0.242449 0.970164i \(-0.422049\pi\)
0.242449 + 0.970164i \(0.422049\pi\)
\(480\) 0 0
\(481\) −4.70156 −0.214373
\(482\) 15.4031 0.701593
\(483\) 0 0
\(484\) −8.10469 −0.368395
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 27.1047 1.22823 0.614115 0.789216i \(-0.289513\pi\)
0.614115 + 0.789216i \(0.289513\pi\)
\(488\) 5.10469 0.231078
\(489\) 6.80625 0.307789
\(490\) 0 0
\(491\) 20.5969 0.929524 0.464762 0.885436i \(-0.346140\pi\)
0.464762 + 0.885436i \(0.346140\pi\)
\(492\) −2.70156 −0.121796
\(493\) −36.5078 −1.64423
\(494\) −4.70156 −0.211533
\(495\) 0 0
\(496\) 9.10469 0.408812
\(497\) 8.00000 0.358849
\(498\) 4.29844 0.192618
\(499\) −21.2094 −0.949462 −0.474731 0.880131i \(-0.657455\pi\)
−0.474731 + 0.880131i \(0.657455\pi\)
\(500\) 0 0
\(501\) 18.1047 0.808858
\(502\) −6.70156 −0.299105
\(503\) −26.2094 −1.16862 −0.584309 0.811531i \(-0.698634\pi\)
−0.584309 + 0.811531i \(0.698634\pi\)
\(504\) −1.70156 −0.0757936
\(505\) 0 0
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 12.7016 0.563541
\(509\) 7.40312 0.328138 0.164069 0.986449i \(-0.447538\pi\)
0.164069 + 0.986449i \(0.447538\pi\)
\(510\) 0 0
\(511\) 20.4187 0.903272
\(512\) −1.00000 −0.0441942
\(513\) −4.70156 −0.207579
\(514\) −21.7016 −0.957215
\(515\) 0 0
\(516\) 1.40312 0.0617691
\(517\) 11.9109 0.523842
\(518\) 8.00000 0.351500
\(519\) −25.8062 −1.13277
\(520\) 0 0
\(521\) 2.20937 0.0967944 0.0483972 0.998828i \(-0.484589\pi\)
0.0483972 + 0.998828i \(0.484589\pi\)
\(522\) −6.40312 −0.280257
\(523\) −0.806248 −0.0352548 −0.0176274 0.999845i \(-0.505611\pi\)
−0.0176274 + 0.999845i \(0.505611\pi\)
\(524\) −9.50781 −0.415351
\(525\) 0 0
\(526\) 2.00000 0.0872041
\(527\) −51.9109 −2.26128
\(528\) 1.70156 0.0740510
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −3.70156 −0.160634
\(532\) 8.00000 0.346844
\(533\) 2.70156 0.117018
\(534\) −1.40312 −0.0607191
\(535\) 0 0
\(536\) 6.40312 0.276573
\(537\) −14.2094 −0.613180
\(538\) 7.20937 0.310818
\(539\) 6.98438 0.300838
\(540\) 0 0
\(541\) −40.2094 −1.72874 −0.864368 0.502860i \(-0.832281\pi\)
−0.864368 + 0.502860i \(0.832281\pi\)
\(542\) 12.5078 0.537256
\(543\) −17.7016 −0.759647
\(544\) 5.70156 0.244452
\(545\) 0 0
\(546\) 1.70156 0.0728201
\(547\) −19.6125 −0.838570 −0.419285 0.907855i \(-0.637719\pi\)
−0.419285 + 0.907855i \(0.637719\pi\)
\(548\) −7.29844 −0.311774
\(549\) −5.10469 −0.217863
\(550\) 0 0
\(551\) 30.1047 1.28250
\(552\) 0 0
\(553\) 1.19375 0.0507635
\(554\) −22.0000 −0.934690
\(555\) 0 0
\(556\) −3.40312 −0.144325
\(557\) 8.80625 0.373133 0.186566 0.982442i \(-0.440264\pi\)
0.186566 + 0.982442i \(0.440264\pi\)
\(558\) −9.10469 −0.385432
\(559\) −1.40312 −0.0593458
\(560\) 0 0
\(561\) −9.70156 −0.409600
\(562\) −22.9109 −0.966439
\(563\) 0.492189 0.0207433 0.0103717 0.999946i \(-0.496699\pi\)
0.0103717 + 0.999946i \(0.496699\pi\)
\(564\) 7.00000 0.294753
\(565\) 0 0
\(566\) 8.59688 0.361354
\(567\) 1.70156 0.0714589
\(568\) −4.70156 −0.197273
\(569\) −17.7016 −0.742088 −0.371044 0.928615i \(-0.621000\pi\)
−0.371044 + 0.928615i \(0.621000\pi\)
\(570\) 0 0
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) −1.70156 −0.0711459
\(573\) −12.8062 −0.534989
\(574\) −4.59688 −0.191870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 14.2094 0.591544 0.295772 0.955259i \(-0.404423\pi\)
0.295772 + 0.955259i \(0.404423\pi\)
\(578\) −15.5078 −0.645040
\(579\) −22.2094 −0.922990
\(580\) 0 0
\(581\) 7.31406 0.303438
\(582\) 15.4031 0.638480
\(583\) −17.7016 −0.733124
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) 8.59688 0.355134
\(587\) −24.5078 −1.01155 −0.505773 0.862667i \(-0.668793\pi\)
−0.505773 + 0.862667i \(0.668793\pi\)
\(588\) 4.10469 0.169274
\(589\) 42.8062 1.76380
\(590\) 0 0
\(591\) 7.40312 0.304524
\(592\) −4.70156 −0.193233
\(593\) 26.9109 1.10510 0.552550 0.833480i \(-0.313655\pi\)
0.552550 + 0.833480i \(0.313655\pi\)
\(594\) −1.70156 −0.0698159
\(595\) 0 0
\(596\) 19.4031 0.794783
\(597\) −14.7016 −0.601695
\(598\) 0 0
\(599\) −30.2094 −1.23432 −0.617161 0.786837i \(-0.711717\pi\)
−0.617161 + 0.786837i \(0.711717\pi\)
\(600\) 0 0
\(601\) −40.6125 −1.65662 −0.828309 0.560271i \(-0.810697\pi\)
−0.828309 + 0.560271i \(0.810697\pi\)
\(602\) 2.38750 0.0973074
\(603\) −6.40312 −0.260755
\(604\) 5.10469 0.207707
\(605\) 0 0
\(606\) 0.298438 0.0121232
\(607\) 15.8953 0.645171 0.322585 0.946540i \(-0.395448\pi\)
0.322585 + 0.946540i \(0.395448\pi\)
\(608\) −4.70156 −0.190674
\(609\) −10.8953 −0.441500
\(610\) 0 0
\(611\) −7.00000 −0.283190
\(612\) −5.70156 −0.230472
\(613\) −12.8062 −0.517240 −0.258620 0.965979i \(-0.583268\pi\)
−0.258620 + 0.965979i \(0.583268\pi\)
\(614\) 0.701562 0.0283127
\(615\) 0 0
\(616\) 2.89531 0.116656
\(617\) 7.89531 0.317853 0.158927 0.987290i \(-0.449197\pi\)
0.158927 + 0.987290i \(0.449197\pi\)
\(618\) 11.4031 0.458701
\(619\) −38.8062 −1.55975 −0.779877 0.625932i \(-0.784719\pi\)
−0.779877 + 0.625932i \(0.784719\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −30.0000 −1.20289
\(623\) −2.38750 −0.0956533
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) 17.2094 0.687825
\(627\) 8.00000 0.319489
\(628\) 16.2984 0.650378
\(629\) 26.8062 1.06884
\(630\) 0 0
\(631\) −17.6125 −0.701142 −0.350571 0.936536i \(-0.614013\pi\)
−0.350571 + 0.936536i \(0.614013\pi\)
\(632\) −0.701562 −0.0279066
\(633\) 18.8062 0.747481
\(634\) 26.8062 1.06461
\(635\) 0 0
\(636\) −10.4031 −0.412511
\(637\) −4.10469 −0.162634
\(638\) 10.8953 0.431350
\(639\) 4.70156 0.185991
\(640\) 0 0
\(641\) −26.5078 −1.04700 −0.523498 0.852027i \(-0.675373\pi\)
−0.523498 + 0.852027i \(0.675373\pi\)
\(642\) 0.104686 0.00413164
\(643\) 16.7016 0.658645 0.329323 0.944217i \(-0.393180\pi\)
0.329323 + 0.944217i \(0.393180\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 26.8062 1.05468
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 6.29844 0.247235
\(650\) 0 0
\(651\) −15.4922 −0.607187
\(652\) −6.80625 −0.266553
\(653\) 27.7016 1.08405 0.542023 0.840364i \(-0.317659\pi\)
0.542023 + 0.840364i \(0.317659\pi\)
\(654\) 10.7016 0.418464
\(655\) 0 0
\(656\) 2.70156 0.105478
\(657\) 12.0000 0.468165
\(658\) 11.9109 0.464337
\(659\) 38.3141 1.49250 0.746252 0.665664i \(-0.231851\pi\)
0.746252 + 0.665664i \(0.231851\pi\)
\(660\) 0 0
\(661\) 43.1203 1.67719 0.838593 0.544759i \(-0.183379\pi\)
0.838593 + 0.544759i \(0.183379\pi\)
\(662\) 12.0000 0.466393
\(663\) 5.70156 0.221430
\(664\) −4.29844 −0.166812
\(665\) 0 0
\(666\) 4.70156 0.182182
\(667\) 0 0
\(668\) −18.1047 −0.700491
\(669\) −1.40312 −0.0542479
\(670\) 0 0
\(671\) 8.68594 0.335317
\(672\) 1.70156 0.0656392
\(673\) 28.0156 1.07992 0.539961 0.841690i \(-0.318439\pi\)
0.539961 + 0.841690i \(0.318439\pi\)
\(674\) −11.1047 −0.427737
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 19.6125 0.753769 0.376885 0.926260i \(-0.376995\pi\)
0.376885 + 0.926260i \(0.376995\pi\)
\(678\) 14.0000 0.537667
\(679\) 26.2094 1.00582
\(680\) 0 0
\(681\) 3.10469 0.118972
\(682\) 15.4922 0.593227
\(683\) −38.2984 −1.46545 −0.732724 0.680525i \(-0.761752\pi\)
−0.732724 + 0.680525i \(0.761752\pi\)
\(684\) 4.70156 0.179769
\(685\) 0 0
\(686\) 18.8953 0.721426
\(687\) −9.29844 −0.354758
\(688\) −1.40312 −0.0534936
\(689\) 10.4031 0.396327
\(690\) 0 0
\(691\) −47.8062 −1.81864 −0.909318 0.416103i \(-0.863396\pi\)
−0.909318 + 0.416103i \(0.863396\pi\)
\(692\) 25.8062 0.981006
\(693\) −2.89531 −0.109984
\(694\) −21.5078 −0.816425
\(695\) 0 0
\(696\) 6.40312 0.242710
\(697\) −15.4031 −0.583435
\(698\) −1.40312 −0.0531090
\(699\) 18.2094 0.688742
\(700\) 0 0
\(701\) −13.9109 −0.525409 −0.262704 0.964876i \(-0.584614\pi\)
−0.262704 + 0.964876i \(0.584614\pi\)
\(702\) 1.00000 0.0377426
\(703\) −22.1047 −0.833694
\(704\) −1.70156 −0.0641300
\(705\) 0 0
\(706\) −17.5078 −0.658915
\(707\) 0.507811 0.0190982
\(708\) 3.70156 0.139113
\(709\) −35.6125 −1.33746 −0.668728 0.743507i \(-0.733161\pi\)
−0.668728 + 0.743507i \(0.733161\pi\)
\(710\) 0 0
\(711\) 0.701562 0.0263106
\(712\) 1.40312 0.0525843
\(713\) 0 0
\(714\) −9.70156 −0.363072
\(715\) 0 0
\(716\) 14.2094 0.531029
\(717\) 13.1047 0.489403
\(718\) −16.6125 −0.619973
\(719\) 53.0156 1.97715 0.988575 0.150733i \(-0.0481633\pi\)
0.988575 + 0.150733i \(0.0481633\pi\)
\(720\) 0 0
\(721\) 19.4031 0.722610
\(722\) −3.10469 −0.115544
\(723\) 15.4031 0.572848
\(724\) 17.7016 0.657873
\(725\) 0 0
\(726\) −8.10469 −0.300793
\(727\) −46.2094 −1.71381 −0.856905 0.515474i \(-0.827616\pi\)
−0.856905 + 0.515474i \(0.827616\pi\)
\(728\) −1.70156 −0.0630641
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) 5.10469 0.188675
\(733\) −3.08907 −0.114097 −0.0570486 0.998371i \(-0.518169\pi\)
−0.0570486 + 0.998371i \(0.518169\pi\)
\(734\) −8.70156 −0.321181
\(735\) 0 0
\(736\) 0 0
\(737\) 10.8953 0.401334
\(738\) −2.70156 −0.0994459
\(739\) −46.6125 −1.71467 −0.857334 0.514760i \(-0.827881\pi\)
−0.857334 + 0.514760i \(0.827881\pi\)
\(740\) 0 0
\(741\) −4.70156 −0.172716
\(742\) −17.7016 −0.649845
\(743\) −20.6125 −0.756199 −0.378100 0.925765i \(-0.623422\pi\)
−0.378100 + 0.925765i \(0.623422\pi\)
\(744\) 9.10469 0.333794
\(745\) 0 0
\(746\) 11.7016 0.428425
\(747\) 4.29844 0.157272
\(748\) 9.70156 0.354724
\(749\) 0.178130 0.00650874
\(750\) 0 0
\(751\) −23.5078 −0.857812 −0.428906 0.903349i \(-0.641101\pi\)
−0.428906 + 0.903349i \(0.641101\pi\)
\(752\) −7.00000 −0.255264
\(753\) −6.70156 −0.244218
\(754\) −6.40312 −0.233188
\(755\) 0 0
\(756\) −1.70156 −0.0618852
\(757\) 37.1047 1.34859 0.674296 0.738461i \(-0.264447\pi\)
0.674296 + 0.738461i \(0.264447\pi\)
\(758\) 33.1047 1.20242
\(759\) 0 0
\(760\) 0 0
\(761\) 43.7172 1.58475 0.792373 0.610036i \(-0.208845\pi\)
0.792373 + 0.610036i \(0.208845\pi\)
\(762\) 12.7016 0.460129
\(763\) 18.2094 0.659224
\(764\) 12.8062 0.463314
\(765\) 0 0
\(766\) 0.492189 0.0177835
\(767\) −3.70156 −0.133656
\(768\) −1.00000 −0.0360844
\(769\) −8.00000 −0.288487 −0.144244 0.989542i \(-0.546075\pi\)
−0.144244 + 0.989542i \(0.546075\pi\)
\(770\) 0 0
\(771\) −21.7016 −0.781563
\(772\) 22.2094 0.799333
\(773\) −39.4031 −1.41723 −0.708616 0.705594i \(-0.750680\pi\)
−0.708616 + 0.705594i \(0.750680\pi\)
\(774\) 1.40312 0.0504343
\(775\) 0 0
\(776\) −15.4031 −0.552940
\(777\) 8.00000 0.286998
\(778\) 7.89531 0.283061
\(779\) 12.7016 0.455081
\(780\) 0 0
\(781\) −8.00000 −0.286263
\(782\) 0 0
\(783\) −6.40312 −0.228829
\(784\) −4.10469 −0.146596
\(785\) 0 0
\(786\) −9.50781 −0.339132
\(787\) −53.1047 −1.89298 −0.946489 0.322737i \(-0.895397\pi\)
−0.946489 + 0.322737i \(0.895397\pi\)
\(788\) −7.40312 −0.263725
\(789\) 2.00000 0.0712019
\(790\) 0 0
\(791\) 23.8219 0.847008
\(792\) 1.70156 0.0604624
\(793\) −5.10469 −0.181273
\(794\) 22.9109 0.813079
\(795\) 0 0
\(796\) 14.7016 0.521083
\(797\) 41.9109 1.48456 0.742281 0.670089i \(-0.233744\pi\)
0.742281 + 0.670089i \(0.233744\pi\)
\(798\) 8.00000 0.283197
\(799\) 39.9109 1.41195
\(800\) 0 0
\(801\) −1.40312 −0.0495770
\(802\) 20.2094 0.713618
\(803\) −20.4187 −0.720562
\(804\) 6.40312 0.225821
\(805\) 0 0
\(806\) −9.10469 −0.320699
\(807\) 7.20937 0.253782
\(808\) −0.298438 −0.0104990
\(809\) 32.8062 1.15341 0.576703 0.816954i \(-0.304339\pi\)
0.576703 + 0.816954i \(0.304339\pi\)
\(810\) 0 0
\(811\) 42.2984 1.48530 0.742650 0.669680i \(-0.233569\pi\)
0.742650 + 0.669680i \(0.233569\pi\)
\(812\) 10.8953 0.382351
\(813\) 12.5078 0.438668
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) 5.70156 0.199595
\(817\) −6.59688 −0.230795
\(818\) −16.5969 −0.580296
\(819\) 1.70156 0.0594574
\(820\) 0 0
\(821\) −33.4031 −1.16578 −0.582889 0.812552i \(-0.698078\pi\)
−0.582889 + 0.812552i \(0.698078\pi\)
\(822\) −7.29844 −0.254562
\(823\) 2.10469 0.0733648 0.0366824 0.999327i \(-0.488321\pi\)
0.0366824 + 0.999327i \(0.488321\pi\)
\(824\) −11.4031 −0.397247
\(825\) 0 0
\(826\) 6.29844 0.219151
\(827\) −14.2984 −0.497205 −0.248603 0.968606i \(-0.579971\pi\)
−0.248603 + 0.968606i \(0.579971\pi\)
\(828\) 0 0
\(829\) 26.5078 0.920654 0.460327 0.887749i \(-0.347732\pi\)
0.460327 + 0.887749i \(0.347732\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 1.00000 0.0346688
\(833\) 23.4031 0.810870
\(834\) −3.40312 −0.117841
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) −9.10469 −0.314704
\(838\) 34.9109 1.20598
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 12.0000 0.413793
\(842\) 13.4031 0.461902
\(843\) −22.9109 −0.789095
\(844\) −18.8062 −0.647338
\(845\) 0 0
\(846\) 7.00000 0.240665
\(847\) −13.7906 −0.473852
\(848\) 10.4031 0.357245
\(849\) 8.59688 0.295044
\(850\) 0 0
\(851\) 0 0
\(852\) −4.70156 −0.161073
\(853\) 31.5078 1.07881 0.539403 0.842047i \(-0.318650\pi\)
0.539403 + 0.842047i \(0.318650\pi\)
\(854\) 8.68594 0.297227
\(855\) 0 0
\(856\) −0.104686 −0.00357811
\(857\) −43.6125 −1.48977 −0.744887 0.667190i \(-0.767497\pi\)
−0.744887 + 0.667190i \(0.767497\pi\)
\(858\) −1.70156 −0.0580904
\(859\) 32.2094 1.09897 0.549485 0.835504i \(-0.314824\pi\)
0.549485 + 0.835504i \(0.314824\pi\)
\(860\) 0 0
\(861\) −4.59688 −0.156661
\(862\) 22.3141 0.760020
\(863\) 48.2250 1.64160 0.820799 0.571217i \(-0.193529\pi\)
0.820799 + 0.571217i \(0.193529\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 8.10469 0.275408
\(867\) −15.5078 −0.526673
\(868\) 15.4922 0.525839
\(869\) −1.19375 −0.0404952
\(870\) 0 0
\(871\) −6.40312 −0.216962
\(872\) −10.7016 −0.362401
\(873\) 15.4031 0.521317
\(874\) 0 0
\(875\) 0 0
\(876\) −12.0000 −0.405442
\(877\) −27.7172 −0.935943 −0.467971 0.883744i \(-0.655015\pi\)
−0.467971 + 0.883744i \(0.655015\pi\)
\(878\) 16.7016 0.563650
\(879\) 8.59688 0.289965
\(880\) 0 0
\(881\) 30.7172 1.03489 0.517444 0.855717i \(-0.326884\pi\)
0.517444 + 0.855717i \(0.326884\pi\)
\(882\) 4.10469 0.138212
\(883\) 5.61250 0.188876 0.0944378 0.995531i \(-0.469895\pi\)
0.0944378 + 0.995531i \(0.469895\pi\)
\(884\) −5.70156 −0.191764
\(885\) 0 0
\(886\) 0.492189 0.0165354
\(887\) 20.0000 0.671534 0.335767 0.941945i \(-0.391004\pi\)
0.335767 + 0.941945i \(0.391004\pi\)
\(888\) −4.70156 −0.157774
\(889\) 21.6125 0.724860
\(890\) 0 0
\(891\) −1.70156 −0.0570045
\(892\) 1.40312 0.0469801
\(893\) −32.9109 −1.10132
\(894\) 19.4031 0.648938
\(895\) 0 0
\(896\) −1.70156 −0.0568452
\(897\) 0 0
\(898\) −8.70156 −0.290375
\(899\) 58.2984 1.94436
\(900\) 0 0
\(901\) −59.3141 −1.97604
\(902\) 4.59688 0.153059
\(903\) 2.38750 0.0794511
\(904\) −14.0000 −0.465633
\(905\) 0 0
\(906\) 5.10469 0.169592
\(907\) −39.0156 −1.29549 −0.647746 0.761856i \(-0.724288\pi\)
−0.647746 + 0.761856i \(0.724288\pi\)
\(908\) −3.10469 −0.103033
\(909\) 0.298438 0.00989856
\(910\) 0 0
\(911\) 3.79063 0.125589 0.0627945 0.998026i \(-0.479999\pi\)
0.0627945 + 0.998026i \(0.479999\pi\)
\(912\) −4.70156 −0.155684
\(913\) −7.31406 −0.242060
\(914\) −15.4031 −0.509490
\(915\) 0 0
\(916\) 9.29844 0.307229
\(917\) −16.1781 −0.534249
\(918\) −5.70156 −0.188180
\(919\) −24.1047 −0.795140 −0.397570 0.917572i \(-0.630146\pi\)
−0.397570 + 0.917572i \(0.630146\pi\)
\(920\) 0 0
\(921\) 0.701562 0.0231173
\(922\) 2.20937 0.0727618
\(923\) 4.70156 0.154754
\(924\) 2.89531 0.0952488
\(925\) 0 0
\(926\) −27.7016 −0.910330
\(927\) 11.4031 0.374528
\(928\) −6.40312 −0.210193
\(929\) 58.7016 1.92594 0.962968 0.269616i \(-0.0868968\pi\)
0.962968 + 0.269616i \(0.0868968\pi\)
\(930\) 0 0
\(931\) −19.2984 −0.632481
\(932\) −18.2094 −0.596468
\(933\) −30.0000 −0.982156
\(934\) 36.7016 1.20091
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) 29.9109 0.977148 0.488574 0.872523i \(-0.337517\pi\)
0.488574 + 0.872523i \(0.337517\pi\)
\(938\) 10.8953 0.355745
\(939\) 17.2094 0.561607
\(940\) 0 0
\(941\) −48.4187 −1.57841 −0.789203 0.614132i \(-0.789506\pi\)
−0.789203 + 0.614132i \(0.789506\pi\)
\(942\) 16.2984 0.531032
\(943\) 0 0
\(944\) −3.70156 −0.120476
\(945\) 0 0
\(946\) −2.38750 −0.0776244
\(947\) −15.4922 −0.503429 −0.251714 0.967802i \(-0.580994\pi\)
−0.251714 + 0.967802i \(0.580994\pi\)
\(948\) −0.701562 −0.0227857
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 26.8062 0.869252
\(952\) 9.70156 0.314429
\(953\) 37.7016 1.22127 0.610637 0.791911i \(-0.290914\pi\)
0.610637 + 0.791911i \(0.290914\pi\)
\(954\) −10.4031 −0.336814
\(955\) 0 0
\(956\) −13.1047 −0.423836
\(957\) 10.8953 0.352195
\(958\) −10.6125 −0.342874
\(959\) −12.4187 −0.401022
\(960\) 0 0
\(961\) 51.8953 1.67404
\(962\) 4.70156 0.151584
\(963\) 0.104686 0.00337347
\(964\) −15.4031 −0.496101
\(965\) 0 0
\(966\) 0 0
\(967\) −49.1047 −1.57910 −0.789550 0.613686i \(-0.789686\pi\)
−0.789550 + 0.613686i \(0.789686\pi\)
\(968\) 8.10469 0.260494
\(969\) 26.8062 0.861141
\(970\) 0 0
\(971\) −56.1047 −1.80049 −0.900243 0.435389i \(-0.856611\pi\)
−0.900243 + 0.435389i \(0.856611\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −5.79063 −0.185639
\(974\) −27.1047 −0.868490
\(975\) 0 0
\(976\) −5.10469 −0.163397
\(977\) 10.5969 0.339024 0.169512 0.985528i \(-0.445781\pi\)
0.169512 + 0.985528i \(0.445781\pi\)
\(978\) −6.80625 −0.217640
\(979\) 2.38750 0.0763049
\(980\) 0 0
\(981\) 10.7016 0.341675
\(982\) −20.5969 −0.657273
\(983\) −24.5078 −0.781678 −0.390839 0.920459i \(-0.627815\pi\)
−0.390839 + 0.920459i \(0.627815\pi\)
\(984\) 2.70156 0.0861227
\(985\) 0 0
\(986\) 36.5078 1.16265
\(987\) 11.9109 0.379129
\(988\) 4.70156 0.149577
\(989\) 0 0
\(990\) 0 0
\(991\) 10.1047 0.320986 0.160493 0.987037i \(-0.448692\pi\)
0.160493 + 0.987037i \(0.448692\pi\)
\(992\) −9.10469 −0.289074
\(993\) 12.0000 0.380808
\(994\) −8.00000 −0.253745
\(995\) 0 0
\(996\) −4.29844 −0.136201
\(997\) 42.5078 1.34624 0.673118 0.739535i \(-0.264955\pi\)
0.673118 + 0.739535i \(0.264955\pi\)
\(998\) 21.2094 0.671371
\(999\) 4.70156 0.148751
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 1950.2.a.bc.1.2 2
3.2 odd 2 5850.2.a.cj.1.2 2
5.2 odd 4 1950.2.e.p.1249.2 4
5.3 odd 4 1950.2.e.p.1249.3 4
5.4 even 2 1950.2.a.bg.1.1 yes 2
15.2 even 4 5850.2.e.bi.5149.4 4
15.8 even 4 5850.2.e.bi.5149.1 4
15.14 odd 2 5850.2.a.cg.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1950.2.a.bc.1.2 2 1.1 even 1 trivial
1950.2.a.bg.1.1 yes 2 5.4 even 2
1950.2.e.p.1249.2 4 5.2 odd 4
1950.2.e.p.1249.3 4 5.3 odd 4
5850.2.a.cg.1.1 2 15.14 odd 2
5850.2.a.cj.1.2 2 3.2 odd 2
5850.2.e.bi.5149.1 4 15.8 even 4
5850.2.e.bi.5149.4 4 15.2 even 4