Properties

Label 1850.2.b.c.149.1
Level $1850$
Weight $2$
Character 1850.149
Analytic conductor $14.772$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 370)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.c.149.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -2.00000i q^{3} -1.00000 q^{4} -2.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +3.00000 q^{11} +2.00000i q^{12} +1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +1.00000i q^{18} +6.00000 q^{19} +2.00000 q^{21} -3.00000i q^{22} -2.00000i q^{23} +2.00000 q^{24} -4.00000i q^{27} -1.00000i q^{28} +3.00000 q^{29} +3.00000 q^{31} -1.00000i q^{32} -6.00000i q^{33} +3.00000 q^{34} +1.00000 q^{36} -1.00000i q^{37} -6.00000i q^{38} +3.00000 q^{41} -2.00000i q^{42} +1.00000i q^{43} -3.00000 q^{44} -2.00000 q^{46} +4.00000i q^{47} -2.00000i q^{48} +6.00000 q^{49} +6.00000 q^{51} -13.0000i q^{53} -4.00000 q^{54} -1.00000 q^{56} -12.0000i q^{57} -3.00000i q^{58} -15.0000 q^{61} -3.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} -3.00000i q^{68} -4.00000 q^{69} -2.00000 q^{71} -1.00000i q^{72} -1.00000 q^{74} -6.00000 q^{76} +3.00000i q^{77} +8.00000 q^{79} -11.0000 q^{81} -3.00000i q^{82} +4.00000i q^{83} -2.00000 q^{84} +1.00000 q^{86} -6.00000i q^{87} +3.00000i q^{88} +18.0000 q^{89} +2.00000i q^{92} -6.00000i q^{93} +4.00000 q^{94} -2.00000 q^{96} -7.00000i q^{97} -6.00000i q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} + 6 q^{11} + 2 q^{14} + 2 q^{16} + 12 q^{19} + 4 q^{21} + 4 q^{24} + 6 q^{29} + 6 q^{31} + 6 q^{34} + 2 q^{36} + 6 q^{41} - 6 q^{44} - 4 q^{46} + 12 q^{49} + 12 q^{51} - 8 q^{54} - 2 q^{56} - 30 q^{61} - 2 q^{64} - 12 q^{66} - 8 q^{69} - 4 q^{71} - 2 q^{74} - 12 q^{76} + 16 q^{79} - 22 q^{81} - 4 q^{84} + 2 q^{86} + 36 q^{89} + 8 q^{94} - 4 q^{96} - 6 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −2.00000 −0.816497
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 2.00000i 0.577350i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) − 3.00000i − 0.639602i
\(23\) − 2.00000i − 0.417029i −0.978019 0.208514i \(-0.933137\pi\)
0.978019 0.208514i \(-0.0668628\pi\)
\(24\) 2.00000 0.408248
\(25\) 0 0
\(26\) 0 0
\(27\) − 4.00000i − 0.769800i
\(28\) − 1.00000i − 0.188982i
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 6.00000i − 1.04447i
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 1.00000i − 0.164399i
\(38\) − 6.00000i − 0.973329i
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) 1.00000i 0.152499i 0.997089 + 0.0762493i \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) − 2.00000i − 0.288675i
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) − 13.0000i − 1.78569i −0.450367 0.892844i \(-0.648707\pi\)
0.450367 0.892844i \(-0.351293\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) − 12.0000i − 1.58944i
\(58\) − 3.00000i − 0.393919i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −15.0000 −1.92055 −0.960277 0.279050i \(-0.909981\pi\)
−0.960277 + 0.279050i \(0.909981\pi\)
\(62\) − 3.00000i − 0.381000i
\(63\) − 1.00000i − 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) − 3.00000i − 0.363803i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 3.00000i 0.341882i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) − 3.00000i − 0.331295i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 1.00000 0.107833
\(87\) − 6.00000i − 0.643268i
\(88\) 3.00000i 0.319801i
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.00000i 0.208514i
\(93\) − 6.00000i − 0.622171i
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) −2.00000 −0.204124
\(97\) − 7.00000i − 0.710742i −0.934725 0.355371i \(-0.884354\pi\)
0.934725 0.355371i \(-0.115646\pi\)
\(98\) − 6.00000i − 0.606092i
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) − 6.00000i − 0.594089i
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −13.0000 −1.26267
\(107\) 2.00000i 0.193347i 0.995316 + 0.0966736i \(0.0308203\pi\)
−0.995316 + 0.0966736i \(0.969180\pi\)
\(108\) 4.00000i 0.384900i
\(109\) 3.00000 0.287348 0.143674 0.989625i \(-0.454108\pi\)
0.143674 + 0.989625i \(0.454108\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 1.00000i 0.0944911i
\(113\) 7.00000i 0.658505i 0.944242 + 0.329252i \(0.106797\pi\)
−0.944242 + 0.329252i \(0.893203\pi\)
\(114\) −12.0000 −1.12390
\(115\) 0 0
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 15.0000i 1.35804i
\(123\) − 6.00000i − 0.541002i
\(124\) −3.00000 −0.269408
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) − 4.00000i − 0.354943i −0.984126 0.177471i \(-0.943208\pi\)
0.984126 0.177471i \(-0.0567917\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 6.00000i 0.522233i
\(133\) 6.00000i 0.520266i
\(134\) 0 0
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 8.00000i 0.683486i 0.939793 + 0.341743i \(0.111017\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) 4.00000i 0.340503i
\(139\) −3.00000 −0.254457 −0.127228 0.991873i \(-0.540608\pi\)
−0.127228 + 0.991873i \(0.540608\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 2.00000i 0.167836i
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 0 0
\(147\) − 12.0000i − 0.989743i
\(148\) 1.00000i 0.0821995i
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 6.00000i 0.486664i
\(153\) − 3.00000i − 0.242536i
\(154\) 3.00000 0.241747
\(155\) 0 0
\(156\) 0 0
\(157\) − 3.00000i − 0.239426i −0.992809 0.119713i \(-0.961803\pi\)
0.992809 0.119713i \(-0.0381975\pi\)
\(158\) − 8.00000i − 0.636446i
\(159\) −26.0000 −2.06193
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 11.0000i 0.864242i
\(163\) 5.00000i 0.391630i 0.980641 + 0.195815i \(0.0627352\pi\)
−0.980641 + 0.195815i \(0.937265\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 2.00000i 0.154303i
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) − 1.00000i − 0.0762493i
\(173\) − 9.00000i − 0.684257i −0.939653 0.342129i \(-0.888852\pi\)
0.939653 0.342129i \(-0.111148\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) − 18.0000i − 1.34916i
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 30.0000i 2.21766i
\(184\) 2.00000 0.147442
\(185\) 0 0
\(186\) −6.00000 −0.439941
\(187\) 9.00000i 0.658145i
\(188\) − 4.00000i − 0.291730i
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) 21.0000 1.51951 0.759753 0.650211i \(-0.225320\pi\)
0.759753 + 0.650211i \(0.225320\pi\)
\(192\) 2.00000i 0.144338i
\(193\) 10.0000i 0.719816i 0.932988 + 0.359908i \(0.117192\pi\)
−0.932988 + 0.359908i \(0.882808\pi\)
\(194\) −7.00000 −0.502571
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) − 18.0000i − 1.28245i −0.767354 0.641223i \(-0.778427\pi\)
0.767354 0.641223i \(-0.221573\pi\)
\(198\) 3.00000i 0.213201i
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 10.0000i 0.703598i
\(203\) 3.00000i 0.210559i
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 2.00000i 0.139010i
\(208\) 0 0
\(209\) 18.0000 1.24509
\(210\) 0 0
\(211\) −3.00000 −0.206529 −0.103264 0.994654i \(-0.532929\pi\)
−0.103264 + 0.994654i \(0.532929\pi\)
\(212\) 13.0000i 0.892844i
\(213\) 4.00000i 0.274075i
\(214\) 2.00000 0.136717
\(215\) 0 0
\(216\) 4.00000 0.272166
\(217\) 3.00000i 0.203653i
\(218\) − 3.00000i − 0.203186i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 2.00000i 0.134231i
\(223\) − 23.0000i − 1.54019i −0.637927 0.770097i \(-0.720208\pi\)
0.637927 0.770097i \(-0.279792\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 7.00000 0.465633
\(227\) 13.0000i 0.862840i 0.902151 + 0.431420i \(0.141987\pi\)
−0.902151 + 0.431420i \(0.858013\pi\)
\(228\) 12.0000i 0.794719i
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) 3.00000i 0.196960i
\(233\) − 4.00000i − 0.262049i −0.991379 0.131024i \(-0.958173\pi\)
0.991379 0.131024i \(-0.0418266\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 16.0000i − 1.03931i
\(238\) 3.00000i 0.194461i
\(239\) −9.00000 −0.582162 −0.291081 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) 0 0
\(241\) 24.0000 1.54598 0.772988 0.634421i \(-0.218761\pi\)
0.772988 + 0.634421i \(0.218761\pi\)
\(242\) 2.00000i 0.128565i
\(243\) 10.0000i 0.641500i
\(244\) 15.0000 0.960277
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 0 0
\(248\) 3.00000i 0.190500i
\(249\) 8.00000 0.506979
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 1.00000i 0.0629941i
\(253\) − 6.00000i − 0.377217i
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 22.0000i − 1.37232i −0.727450 0.686161i \(-0.759294\pi\)
0.727450 0.686161i \(-0.240706\pi\)
\(258\) − 2.00000i − 0.124515i
\(259\) 1.00000 0.0621370
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) 12.0000i 0.741362i
\(263\) 1.00000i 0.0616626i 0.999525 + 0.0308313i \(0.00981547\pi\)
−0.999525 + 0.0308313i \(0.990185\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) 6.00000 0.367884
\(267\) − 36.0000i − 2.20316i
\(268\) 0 0
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 0 0
\(271\) −10.0000 −0.607457 −0.303728 0.952759i \(-0.598232\pi\)
−0.303728 + 0.952759i \(0.598232\pi\)
\(272\) 3.00000i 0.181902i
\(273\) 0 0
\(274\) 8.00000 0.483298
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) − 20.0000i − 1.20168i −0.799368 0.600842i \(-0.794832\pi\)
0.799368 0.600842i \(-0.205168\pi\)
\(278\) 3.00000i 0.179928i
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) −24.0000 −1.43172 −0.715860 0.698244i \(-0.753965\pi\)
−0.715860 + 0.698244i \(0.753965\pi\)
\(282\) − 8.00000i − 0.476393i
\(283\) 20.0000i 1.18888i 0.804141 + 0.594438i \(0.202626\pi\)
−0.804141 + 0.594438i \(0.797374\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) 3.00000i 0.177084i
\(288\) 1.00000i 0.0589256i
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) − 3.00000i − 0.175262i −0.996153 0.0876309i \(-0.972070\pi\)
0.996153 0.0876309i \(-0.0279296\pi\)
\(294\) −12.0000 −0.699854
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) − 12.0000i − 0.696311i
\(298\) 18.0000i 1.04271i
\(299\) 0 0
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) − 16.0000i − 0.920697i
\(303\) 20.0000i 1.14897i
\(304\) 6.00000 0.344124
\(305\) 0 0
\(306\) −3.00000 −0.171499
\(307\) 34.0000i 1.94048i 0.242140 + 0.970241i \(0.422151\pi\)
−0.242140 + 0.970241i \(0.577849\pi\)
\(308\) − 3.00000i − 0.170941i
\(309\) −16.0000 −0.910208
\(310\) 0 0
\(311\) 25.0000 1.41762 0.708810 0.705399i \(-0.249232\pi\)
0.708810 + 0.705399i \(0.249232\pi\)
\(312\) 0 0
\(313\) − 26.0000i − 1.46961i −0.678280 0.734803i \(-0.737274\pi\)
0.678280 0.734803i \(-0.262726\pi\)
\(314\) −3.00000 −0.169300
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 21.0000i 1.17948i 0.807594 + 0.589739i \(0.200769\pi\)
−0.807594 + 0.589739i \(0.799231\pi\)
\(318\) 26.0000i 1.45801i
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) − 2.00000i − 0.111456i
\(323\) 18.0000i 1.00155i
\(324\) 11.0000 0.611111
\(325\) 0 0
\(326\) 5.00000 0.276924
\(327\) − 6.00000i − 0.331801i
\(328\) 3.00000i 0.165647i
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) 30.0000 1.64895 0.824475 0.565899i \(-0.191471\pi\)
0.824475 + 0.565899i \(0.191471\pi\)
\(332\) − 4.00000i − 0.219529i
\(333\) 1.00000i 0.0547997i
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 12.0000i 0.653682i 0.945079 + 0.326841i \(0.105984\pi\)
−0.945079 + 0.326841i \(0.894016\pi\)
\(338\) − 13.0000i − 0.707107i
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) 9.00000 0.487377
\(342\) 6.00000i 0.324443i
\(343\) 13.0000i 0.701934i
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) −9.00000 −0.483843
\(347\) − 28.0000i − 1.50312i −0.659665 0.751559i \(-0.729302\pi\)
0.659665 0.751559i \(-0.270698\pi\)
\(348\) 6.00000i 0.321634i
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 3.00000i − 0.159901i
\(353\) − 19.0000i − 1.01127i −0.862748 0.505634i \(-0.831259\pi\)
0.862748 0.505634i \(-0.168741\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) 6.00000i 0.317554i
\(358\) 24.0000i 1.26844i
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) − 2.00000i − 0.105118i
\(363\) 4.00000i 0.209946i
\(364\) 0 0
\(365\) 0 0
\(366\) 30.0000 1.56813
\(367\) 13.0000i 0.678594i 0.940679 + 0.339297i \(0.110189\pi\)
−0.940679 + 0.339297i \(0.889811\pi\)
\(368\) − 2.00000i − 0.104257i
\(369\) −3.00000 −0.156174
\(370\) 0 0
\(371\) 13.0000 0.674926
\(372\) 6.00000i 0.311086i
\(373\) − 22.0000i − 1.13912i −0.821951 0.569558i \(-0.807114\pi\)
0.821951 0.569558i \(-0.192886\pi\)
\(374\) 9.00000 0.465379
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) 0 0
\(378\) − 4.00000i − 0.205738i
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) − 21.0000i − 1.07445i
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) − 1.00000i − 0.0508329i
\(388\) 7.00000i 0.355371i
\(389\) −15.0000 −0.760530 −0.380265 0.924878i \(-0.624167\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 6.00000i 0.303046i
\(393\) 24.0000i 1.21064i
\(394\) −18.0000 −0.906827
\(395\) 0 0
\(396\) 3.00000 0.150756
\(397\) 18.0000i 0.903394i 0.892171 + 0.451697i \(0.149181\pi\)
−0.892171 + 0.451697i \(0.850819\pi\)
\(398\) 8.00000i 0.401004i
\(399\) 12.0000 0.600751
\(400\) 0 0
\(401\) 34.0000 1.69788 0.848939 0.528490i \(-0.177242\pi\)
0.848939 + 0.528490i \(0.177242\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) 3.00000 0.148888
\(407\) − 3.00000i − 0.148704i
\(408\) 6.00000i 0.297044i
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 16.0000 0.789222
\(412\) 8.00000i 0.394132i
\(413\) 0 0
\(414\) 2.00000 0.0982946
\(415\) 0 0
\(416\) 0 0
\(417\) 6.00000i 0.293821i
\(418\) − 18.0000i − 0.880409i
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 34.0000 1.65706 0.828529 0.559946i \(-0.189178\pi\)
0.828529 + 0.559946i \(0.189178\pi\)
\(422\) 3.00000i 0.146038i
\(423\) − 4.00000i − 0.194487i
\(424\) 13.0000 0.631336
\(425\) 0 0
\(426\) 4.00000 0.193801
\(427\) − 15.0000i − 0.725901i
\(428\) − 2.00000i − 0.0966736i
\(429\) 0 0
\(430\) 0 0
\(431\) −31.0000 −1.49322 −0.746609 0.665263i \(-0.768319\pi\)
−0.746609 + 0.665263i \(0.768319\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) 22.0000i 1.05725i 0.848855 + 0.528626i \(0.177293\pi\)
−0.848855 + 0.528626i \(0.822707\pi\)
\(434\) 3.00000 0.144005
\(435\) 0 0
\(436\) −3.00000 −0.143674
\(437\) − 12.0000i − 0.574038i
\(438\) 0 0
\(439\) −37.0000 −1.76591 −0.882957 0.469454i \(-0.844451\pi\)
−0.882957 + 0.469454i \(0.844451\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 34.0000i 1.61539i 0.589601 + 0.807694i \(0.299285\pi\)
−0.589601 + 0.807694i \(0.700715\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) −23.0000 −1.08908
\(447\) 36.0000i 1.70274i
\(448\) − 1.00000i − 0.0472456i
\(449\) 10.0000 0.471929 0.235965 0.971762i \(-0.424175\pi\)
0.235965 + 0.971762i \(0.424175\pi\)
\(450\) 0 0
\(451\) 9.00000 0.423793
\(452\) − 7.00000i − 0.329252i
\(453\) − 32.0000i − 1.50349i
\(454\) 13.0000 0.610120
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) 9.00000i 0.421002i 0.977594 + 0.210501i \(0.0675096\pi\)
−0.977594 + 0.210501i \(0.932490\pi\)
\(458\) − 6.00000i − 0.280362i
\(459\) 12.0000 0.560112
\(460\) 0 0
\(461\) −1.00000 −0.0465746 −0.0232873 0.999729i \(-0.507413\pi\)
−0.0232873 + 0.999729i \(0.507413\pi\)
\(462\) − 6.00000i − 0.279145i
\(463\) − 12.0000i − 0.557687i −0.960337 0.278844i \(-0.910049\pi\)
0.960337 0.278844i \(-0.0899511\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −4.00000 −0.185296
\(467\) − 37.0000i − 1.71216i −0.516847 0.856078i \(-0.672894\pi\)
0.516847 0.856078i \(-0.327106\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −6.00000 −0.276465
\(472\) 0 0
\(473\) 3.00000i 0.137940i
\(474\) −16.0000 −0.734904
\(475\) 0 0
\(476\) 3.00000 0.137505
\(477\) 13.0000i 0.595229i
\(478\) 9.00000i 0.411650i
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) − 24.0000i − 1.09317i
\(483\) − 4.00000i − 0.182006i
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) 10.0000 0.453609
\(487\) 34.0000i 1.54069i 0.637629 + 0.770344i \(0.279915\pi\)
−0.637629 + 0.770344i \(0.720085\pi\)
\(488\) − 15.0000i − 0.679018i
\(489\) 10.0000 0.452216
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 6.00000i 0.270501i
\(493\) 9.00000i 0.405340i
\(494\) 0 0
\(495\) 0 0
\(496\) 3.00000 0.134704
\(497\) − 2.00000i − 0.0897123i
\(498\) − 8.00000i − 0.358489i
\(499\) 22.0000 0.984855 0.492428 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) − 6.00000i − 0.267793i
\(503\) 24.0000i 1.07011i 0.844818 + 0.535054i \(0.179709\pi\)
−0.844818 + 0.535054i \(0.820291\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) − 26.0000i − 1.15470i
\(508\) 4.00000i 0.177471i
\(509\) −36.0000 −1.59567 −0.797836 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) − 24.0000i − 1.05963i
\(514\) −22.0000 −0.970378
\(515\) 0 0
\(516\) −2.00000 −0.0880451
\(517\) 12.0000i 0.527759i
\(518\) − 1.00000i − 0.0439375i
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −23.0000 −1.00765 −0.503824 0.863806i \(-0.668074\pi\)
−0.503824 + 0.863806i \(0.668074\pi\)
\(522\) 3.00000i 0.131306i
\(523\) 44.0000i 1.92399i 0.273075 + 0.961993i \(0.411959\pi\)
−0.273075 + 0.961993i \(0.588041\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 1.00000 0.0436021
\(527\) 9.00000i 0.392046i
\(528\) − 6.00000i − 0.261116i
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) − 6.00000i − 0.260133i
\(533\) 0 0
\(534\) −36.0000 −1.55787
\(535\) 0 0
\(536\) 0 0
\(537\) 48.0000i 2.07135i
\(538\) − 10.0000i − 0.431131i
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 10.0000i 0.429537i
\(543\) − 4.00000i − 0.171656i
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) 0 0
\(547\) 23.0000i 0.983409i 0.870762 + 0.491704i \(0.163626\pi\)
−0.870762 + 0.491704i \(0.836374\pi\)
\(548\) − 8.00000i − 0.341743i
\(549\) 15.0000 0.640184
\(550\) 0 0
\(551\) 18.0000 0.766826
\(552\) − 4.00000i − 0.170251i
\(553\) 8.00000i 0.340195i
\(554\) −20.0000 −0.849719
\(555\) 0 0
\(556\) 3.00000 0.127228
\(557\) − 40.0000i − 1.69485i −0.530912 0.847427i \(-0.678150\pi\)
0.530912 0.847427i \(-0.321850\pi\)
\(558\) 3.00000i 0.127000i
\(559\) 0 0
\(560\) 0 0
\(561\) 18.0000 0.759961
\(562\) 24.0000i 1.01238i
\(563\) 11.0000i 0.463595i 0.972764 + 0.231797i \(0.0744606\pi\)
−0.972764 + 0.231797i \(0.925539\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) − 11.0000i − 0.461957i
\(568\) − 2.00000i − 0.0839181i
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −31.0000 −1.29731 −0.648655 0.761083i \(-0.724668\pi\)
−0.648655 + 0.761083i \(0.724668\pi\)
\(572\) 0 0
\(573\) − 42.0000i − 1.75458i
\(574\) 3.00000 0.125218
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 30.0000i 1.24892i 0.781058 + 0.624458i \(0.214680\pi\)
−0.781058 + 0.624458i \(0.785320\pi\)
\(578\) − 8.00000i − 0.332756i
\(579\) 20.0000 0.831172
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 14.0000i 0.580319i
\(583\) − 39.0000i − 1.61521i
\(584\) 0 0
\(585\) 0 0
\(586\) −3.00000 −0.123929
\(587\) 35.0000i 1.44460i 0.691577 + 0.722302i \(0.256916\pi\)
−0.691577 + 0.722302i \(0.743084\pi\)
\(588\) 12.0000i 0.494872i
\(589\) 18.0000 0.741677
\(590\) 0 0
\(591\) −36.0000 −1.48084
\(592\) − 1.00000i − 0.0410997i
\(593\) − 12.0000i − 0.492781i −0.969171 0.246390i \(-0.920755\pi\)
0.969171 0.246390i \(-0.0792446\pi\)
\(594\) −12.0000 −0.492366
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) 16.0000i 0.654836i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 21.0000 0.856608 0.428304 0.903635i \(-0.359111\pi\)
0.428304 + 0.903635i \(0.359111\pi\)
\(602\) 1.00000i 0.0407570i
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) 20.0000 0.812444
\(607\) 22.0000i 0.892952i 0.894795 + 0.446476i \(0.147321\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(608\) − 6.00000i − 0.243332i
\(609\) 6.00000 0.243132
\(610\) 0 0
\(611\) 0 0
\(612\) 3.00000i 0.121268i
\(613\) − 11.0000i − 0.444286i −0.975014 0.222143i \(-0.928695\pi\)
0.975014 0.222143i \(-0.0713052\pi\)
\(614\) 34.0000 1.37213
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) 36.0000i 1.44931i 0.689114 + 0.724653i \(0.258000\pi\)
−0.689114 + 0.724653i \(0.742000\pi\)
\(618\) 16.0000i 0.643614i
\(619\) 37.0000 1.48716 0.743578 0.668649i \(-0.233127\pi\)
0.743578 + 0.668649i \(0.233127\pi\)
\(620\) 0 0
\(621\) −8.00000 −0.321029
\(622\) − 25.0000i − 1.00241i
\(623\) 18.0000i 0.721155i
\(624\) 0 0
\(625\) 0 0
\(626\) −26.0000 −1.03917
\(627\) − 36.0000i − 1.43770i
\(628\) 3.00000i 0.119713i
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) −41.0000 −1.63218 −0.816092 0.577922i \(-0.803864\pi\)
−0.816092 + 0.577922i \(0.803864\pi\)
\(632\) 8.00000i 0.318223i
\(633\) 6.00000i 0.238479i
\(634\) 21.0000 0.834017
\(635\) 0 0
\(636\) 26.0000 1.03097
\(637\) 0 0
\(638\) − 9.00000i − 0.356313i
\(639\) 2.00000 0.0791188
\(640\) 0 0
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) − 4.00000i − 0.157867i
\(643\) − 11.0000i − 0.433798i −0.976194 0.216899i \(-0.930406\pi\)
0.976194 0.216899i \(-0.0695942\pi\)
\(644\) −2.00000 −0.0788110
\(645\) 0 0
\(646\) 18.0000 0.708201
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) − 11.0000i − 0.432121i
\(649\) 0 0
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) − 5.00000i − 0.195815i
\(653\) − 30.0000i − 1.17399i −0.809590 0.586995i \(-0.800311\pi\)
0.809590 0.586995i \(-0.199689\pi\)
\(654\) −6.00000 −0.234619
\(655\) 0 0
\(656\) 3.00000 0.117130
\(657\) 0 0
\(658\) 4.00000i 0.155936i
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −13.0000 −0.505641 −0.252821 0.967513i \(-0.581358\pi\)
−0.252821 + 0.967513i \(0.581358\pi\)
\(662\) − 30.0000i − 1.16598i
\(663\) 0 0
\(664\) −4.00000 −0.155230
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) − 6.00000i − 0.232321i
\(668\) − 12.0000i − 0.464294i
\(669\) −46.0000 −1.77846
\(670\) 0 0
\(671\) −45.0000 −1.73721
\(672\) − 2.00000i − 0.0771517i
\(673\) 34.0000i 1.31060i 0.755367 + 0.655302i \(0.227459\pi\)
−0.755367 + 0.655302i \(0.772541\pi\)
\(674\) 12.0000 0.462223
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 34.0000i 1.30673i 0.757045 + 0.653363i \(0.226642\pi\)
−0.757045 + 0.653363i \(0.773358\pi\)
\(678\) − 14.0000i − 0.537667i
\(679\) 7.00000 0.268635
\(680\) 0 0
\(681\) 26.0000 0.996322
\(682\) − 9.00000i − 0.344628i
\(683\) 31.0000i 1.18618i 0.805135 + 0.593091i \(0.202093\pi\)
−0.805135 + 0.593091i \(0.797907\pi\)
\(684\) 6.00000 0.229416
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) − 12.0000i − 0.457829i
\(688\) 1.00000i 0.0381246i
\(689\) 0 0
\(690\) 0 0
\(691\) 5.00000 0.190209 0.0951045 0.995467i \(-0.469681\pi\)
0.0951045 + 0.995467i \(0.469681\pi\)
\(692\) 9.00000i 0.342129i
\(693\) − 3.00000i − 0.113961i
\(694\) −28.0000 −1.06287
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) 9.00000i 0.340899i
\(698\) − 22.0000i − 0.832712i
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) −22.0000 −0.830929 −0.415464 0.909610i \(-0.636381\pi\)
−0.415464 + 0.909610i \(0.636381\pi\)
\(702\) 0 0
\(703\) − 6.00000i − 0.226294i
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) −19.0000 −0.715074
\(707\) − 10.0000i − 0.376089i
\(708\) 0 0
\(709\) 19.0000 0.713560 0.356780 0.934188i \(-0.383875\pi\)
0.356780 + 0.934188i \(0.383875\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 18.0000i 0.674579i
\(713\) − 6.00000i − 0.224702i
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) 18.0000i 0.672222i
\(718\) 16.0000i 0.597115i
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) − 17.0000i − 0.632674i
\(723\) − 48.0000i − 1.78514i
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 4.00000 0.148454
\(727\) 52.0000i 1.92857i 0.264861 + 0.964287i \(0.414674\pi\)
−0.264861 + 0.964287i \(0.585326\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) −3.00000 −0.110959
\(732\) − 30.0000i − 1.10883i
\(733\) 41.0000i 1.51437i 0.653201 + 0.757185i \(0.273426\pi\)
−0.653201 + 0.757185i \(0.726574\pi\)
\(734\) 13.0000 0.479839
\(735\) 0 0
\(736\) −2.00000 −0.0737210
\(737\) 0 0
\(738\) 3.00000i 0.110432i
\(739\) −27.0000 −0.993211 −0.496606 0.867976i \(-0.665420\pi\)
−0.496606 + 0.867976i \(0.665420\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 13.0000i − 0.477245i
\(743\) − 3.00000i − 0.110059i −0.998485 0.0550297i \(-0.982475\pi\)
0.998485 0.0550297i \(-0.0175253\pi\)
\(744\) 6.00000 0.219971
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) − 4.00000i − 0.146352i
\(748\) − 9.00000i − 0.329073i
\(749\) −2.00000 −0.0730784
\(750\) 0 0
\(751\) −22.0000 −0.802791 −0.401396 0.915905i \(-0.631475\pi\)
−0.401396 + 0.915905i \(0.631475\pi\)
\(752\) 4.00000i 0.145865i
\(753\) − 12.0000i − 0.437304i
\(754\) 0 0
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) − 50.0000i − 1.81728i −0.417579 0.908640i \(-0.637121\pi\)
0.417579 0.908640i \(-0.362879\pi\)
\(758\) 16.0000i 0.581146i
\(759\) −12.0000 −0.435572
\(760\) 0 0
\(761\) −21.0000 −0.761249 −0.380625 0.924730i \(-0.624291\pi\)
−0.380625 + 0.924730i \(0.624291\pi\)
\(762\) 8.00000i 0.289809i
\(763\) 3.00000i 0.108607i
\(764\) −21.0000 −0.759753
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) 0 0
\(768\) − 2.00000i − 0.0721688i
\(769\) −8.00000 −0.288487 −0.144244 0.989542i \(-0.546075\pi\)
−0.144244 + 0.989542i \(0.546075\pi\)
\(770\) 0 0
\(771\) −44.0000 −1.58462
\(772\) − 10.0000i − 0.359908i
\(773\) 41.0000i 1.47467i 0.675529 + 0.737334i \(0.263915\pi\)
−0.675529 + 0.737334i \(0.736085\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 0 0
\(776\) 7.00000 0.251285
\(777\) − 2.00000i − 0.0717496i
\(778\) 15.0000i 0.537776i
\(779\) 18.0000 0.644917
\(780\) 0 0
\(781\) −6.00000 −0.214697
\(782\) − 6.00000i − 0.214560i
\(783\) − 12.0000i − 0.428845i
\(784\) 6.00000 0.214286
\(785\) 0 0
\(786\) 24.0000 0.856052
\(787\) − 10.0000i − 0.356462i −0.983989 0.178231i \(-0.942963\pi\)
0.983989 0.178231i \(-0.0570374\pi\)
\(788\) 18.0000i 0.641223i
\(789\) 2.00000 0.0712019
\(790\) 0 0
\(791\) −7.00000 −0.248891
\(792\) − 3.00000i − 0.106600i
\(793\) 0 0
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) − 26.0000i − 0.920967i −0.887668 0.460484i \(-0.847676\pi\)
0.887668 0.460484i \(-0.152324\pi\)
\(798\) − 12.0000i − 0.424795i
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) − 34.0000i − 1.20058i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 20.0000i − 0.704033i
\(808\) − 10.0000i − 0.351799i
\(809\) −34.0000 −1.19538 −0.597688 0.801729i \(-0.703914\pi\)
−0.597688 + 0.801729i \(0.703914\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) − 3.00000i − 0.105279i
\(813\) 20.0000i 0.701431i
\(814\) −3.00000 −0.105150
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 6.00000i 0.209913i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.00000 0.279202 0.139601 0.990208i \(-0.455418\pi\)
0.139601 + 0.990208i \(0.455418\pi\)
\(822\) − 16.0000i − 0.558064i
\(823\) − 24.0000i − 0.836587i −0.908312 0.418294i \(-0.862628\pi\)
0.908312 0.418294i \(-0.137372\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) − 11.0000i − 0.382507i −0.981541 0.191254i \(-0.938745\pi\)
0.981541 0.191254i \(-0.0612553\pi\)
\(828\) − 2.00000i − 0.0695048i
\(829\) 39.0000 1.35453 0.677263 0.735741i \(-0.263166\pi\)
0.677263 + 0.735741i \(0.263166\pi\)
\(830\) 0 0
\(831\) −40.0000 −1.38758
\(832\) 0 0
\(833\) 18.0000i 0.623663i
\(834\) 6.00000 0.207763
\(835\) 0 0
\(836\) −18.0000 −0.622543
\(837\) − 12.0000i − 0.414781i
\(838\) 12.0000i 0.414533i
\(839\) −36.0000 −1.24286 −0.621429 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) − 34.0000i − 1.17172i
\(843\) 48.0000i 1.65321i
\(844\) 3.00000 0.103264
\(845\) 0 0
\(846\) −4.00000 −0.137523
\(847\) − 2.00000i − 0.0687208i
\(848\) − 13.0000i − 0.446422i
\(849\) 40.0000 1.37280
\(850\) 0 0
\(851\) −2.00000 −0.0685591
\(852\) − 4.00000i − 0.137038i
\(853\) − 22.0000i − 0.753266i −0.926363 0.376633i \(-0.877082\pi\)
0.926363 0.376633i \(-0.122918\pi\)
\(854\) −15.0000 −0.513289
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) 33.0000i 1.12726i 0.826028 + 0.563629i \(0.190595\pi\)
−0.826028 + 0.563629i \(0.809405\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) 6.00000 0.204479
\(862\) 31.0000i 1.05586i
\(863\) − 33.0000i − 1.12333i −0.827364 0.561667i \(-0.810160\pi\)
0.827364 0.561667i \(-0.189840\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) 22.0000 0.747590
\(867\) − 16.0000i − 0.543388i
\(868\) − 3.00000i − 0.101827i
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 0 0
\(872\) 3.00000i 0.101593i
\(873\) 7.00000i 0.236914i
\(874\) −12.0000 −0.405906
\(875\) 0 0
\(876\) 0 0
\(877\) − 43.0000i − 1.45201i −0.687691 0.726003i \(-0.741376\pi\)
0.687691 0.726003i \(-0.258624\pi\)
\(878\) 37.0000i 1.24869i
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −9.00000 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) 6.00000i 0.202031i
\(883\) − 29.0000i − 0.975928i −0.872864 0.487964i \(-0.837740\pi\)
0.872864 0.487964i \(-0.162260\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 34.0000 1.14225
\(887\) − 33.0000i − 1.10803i −0.832506 0.554016i \(-0.813095\pi\)
0.832506 0.554016i \(-0.186905\pi\)
\(888\) − 2.00000i − 0.0671156i
\(889\) 4.00000 0.134156
\(890\) 0 0
\(891\) −33.0000 −1.10554
\(892\) 23.0000i 0.770097i
\(893\) 24.0000i 0.803129i
\(894\) 36.0000 1.20402
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) − 10.0000i − 0.333704i
\(899\) 9.00000 0.300167
\(900\) 0 0
\(901\) 39.0000 1.29928
\(902\) − 9.00000i − 0.299667i
\(903\) 2.00000i 0.0665558i
\(904\) −7.00000 −0.232817
\(905\) 0 0
\(906\) −32.0000 −1.06313
\(907\) − 12.0000i − 0.398453i −0.979953 0.199227i \(-0.936157\pi\)
0.979953 0.199227i \(-0.0638430\pi\)
\(908\) − 13.0000i − 0.431420i
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) − 12.0000i − 0.397360i
\(913\) 12.0000i 0.397142i
\(914\) 9.00000 0.297694
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) − 12.0000i − 0.396275i
\(918\) − 12.0000i − 0.396059i
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 68.0000 2.24068
\(922\) 1.00000i 0.0329332i
\(923\) 0 0
\(924\) −6.00000 −0.197386
\(925\) 0 0
\(926\) −12.0000 −0.394344
\(927\) 8.00000i 0.262754i
\(928\) − 3.00000i − 0.0984798i
\(929\) 3.00000 0.0984268 0.0492134 0.998788i \(-0.484329\pi\)
0.0492134 + 0.998788i \(0.484329\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) 4.00000i 0.131024i
\(933\) − 50.0000i − 1.63693i
\(934\) −37.0000 −1.21068
\(935\) 0 0
\(936\) 0 0
\(937\) − 14.0000i − 0.457360i −0.973502 0.228680i \(-0.926559\pi\)
0.973502 0.228680i \(-0.0734410\pi\)
\(938\) 0 0
\(939\) −52.0000 −1.69696
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 6.00000i 0.195491i
\(943\) − 6.00000i − 0.195387i
\(944\) 0 0
\(945\) 0 0
\(946\) 3.00000 0.0975384
\(947\) 9.00000i 0.292461i 0.989251 + 0.146230i \(0.0467141\pi\)
−0.989251 + 0.146230i \(0.953286\pi\)
\(948\) 16.0000i 0.519656i
\(949\) 0 0
\(950\) 0 0
\(951\) 42.0000 1.36194
\(952\) − 3.00000i − 0.0972306i
\(953\) − 36.0000i − 1.16615i −0.812417 0.583077i \(-0.801849\pi\)
0.812417 0.583077i \(-0.198151\pi\)
\(954\) 13.0000 0.420891
\(955\) 0 0
\(956\) 9.00000 0.291081
\(957\) − 18.0000i − 0.581857i
\(958\) − 8.00000i − 0.258468i
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) − 2.00000i − 0.0644491i
\(964\) −24.0000 −0.772988
\(965\) 0 0
\(966\) −4.00000 −0.128698
\(967\) 48.0000i 1.54358i 0.635880 + 0.771788i \(0.280637\pi\)
−0.635880 + 0.771788i \(0.719363\pi\)
\(968\) − 2.00000i − 0.0642824i
\(969\) 36.0000 1.15649
\(970\) 0 0
\(971\) 9.00000 0.288824 0.144412 0.989518i \(-0.453871\pi\)
0.144412 + 0.989518i \(0.453871\pi\)
\(972\) − 10.0000i − 0.320750i
\(973\) − 3.00000i − 0.0961756i
\(974\) 34.0000 1.08943
\(975\) 0 0
\(976\) −15.0000 −0.480138
\(977\) 35.0000i 1.11975i 0.828577 + 0.559875i \(0.189151\pi\)
−0.828577 + 0.559875i \(0.810849\pi\)
\(978\) − 10.0000i − 0.319765i
\(979\) 54.0000 1.72585
\(980\) 0 0
\(981\) −3.00000 −0.0957826
\(982\) 20.0000i 0.638226i
\(983\) 27.0000i 0.861166i 0.902551 + 0.430583i \(0.141692\pi\)
−0.902551 + 0.430583i \(0.858308\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 9.00000 0.286618
\(987\) 8.00000i 0.254643i
\(988\) 0 0
\(989\) 2.00000 0.0635963
\(990\) 0 0
\(991\) 33.0000 1.04828 0.524140 0.851632i \(-0.324387\pi\)
0.524140 + 0.851632i \(0.324387\pi\)
\(992\) − 3.00000i − 0.0952501i
\(993\) − 60.0000i − 1.90404i
\(994\) −2.00000 −0.0634361
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) − 12.0000i − 0.380044i −0.981780 0.190022i \(-0.939144\pi\)
0.981780 0.190022i \(-0.0608559\pi\)
\(998\) − 22.0000i − 0.696398i
\(999\) −4.00000 −0.126554
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 1850.2.b.c.149.1 2
5.2 odd 4 1850.2.a.i.1.1 1
5.3 odd 4 370.2.a.c.1.1 1
5.4 even 2 inner 1850.2.b.c.149.2 2
15.8 even 4 3330.2.a.p.1.1 1
20.3 even 4 2960.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.a.c.1.1 1 5.3 odd 4
1850.2.a.i.1.1 1 5.2 odd 4
1850.2.b.c.149.1 2 1.1 even 1 trivial
1850.2.b.c.149.2 2 5.4 even 2 inner
2960.2.a.c.1.1 1 20.3 even 4
3330.2.a.p.1.1 1 15.8 even 4