Properties

Label 2850.2.d.s.799.1
Level $2850$
Weight $2$
Character 2850.799
Analytic conductor $22.757$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.7573645761\)
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 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2850.799
Dual form 2850.2.d.s.799.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} +4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -1.00000i q^{12} +4.00000 q^{14} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} -1.00000 q^{19} -4.00000 q^{21} -4.00000i q^{22} +2.00000i q^{23} -1.00000 q^{24} -1.00000i q^{27} -4.00000i q^{28} +6.00000 q^{29} +6.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} -8.00000i q^{37} +1.00000i q^{38} +10.0000 q^{41} +4.00000i q^{42} +12.0000i q^{43} -4.00000 q^{44} +2.00000 q^{46} +10.0000i q^{47} +1.00000i q^{48} -9.00000 q^{49} +2.00000 q^{51} -2.00000i q^{53} -1.00000 q^{54} -4.00000 q^{56} -1.00000i q^{57} -6.00000i q^{58} -4.00000 q^{59} -10.0000 q^{61} -6.00000i q^{62} -4.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} +2.00000i q^{68} -2.00000 q^{69} -16.0000 q^{71} -1.00000i q^{72} +2.00000i q^{73} -8.00000 q^{74} +1.00000 q^{76} +16.0000i q^{77} -10.0000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +16.0000i q^{83} +4.00000 q^{84} +12.0000 q^{86} +6.00000i q^{87} +4.00000i q^{88} +2.00000 q^{89} -2.00000i q^{92} +6.00000i q^{93} +10.0000 q^{94} +1.00000 q^{96} -10.0000i q^{97} +9.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + 8q^{11} + 8q^{14} + 2q^{16} - 2q^{19} - 8q^{21} - 2q^{24} + 12q^{29} + 12q^{31} - 4q^{34} + 2q^{36} + 20q^{41} - 8q^{44} + 4q^{46} - 18q^{49} + 4q^{51} - 2q^{54} - 8q^{56} - 8q^{59} - 20q^{61} - 2q^{64} + 8q^{66} - 4q^{69} - 32q^{71} - 16q^{74} + 2q^{76} - 20q^{79} + 2q^{81} + 8q^{84} + 24q^{86} + 4q^{89} + 20q^{94} + 2q^{96} - 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times\).

\(n\) \(1027\) \(1351\) \(1901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) − 4.00000i − 0.852803i
\(23\) 2.00000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) − 4.00000i − 0.755929i
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 4.00000i 0.696311i
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 1.00000i 0.162221i
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 4.00000i 0.617213i
\(43\) 12.0000i 1.82998i 0.403473 + 0.914991i \(0.367803\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) 10.0000i 1.45865i 0.684167 + 0.729325i \(0.260166\pi\)
−0.684167 + 0.729325i \(0.739834\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) − 1.00000i − 0.132453i
\(58\) − 6.00000i − 0.787839i
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) − 6.00000i − 0.762001i
\(63\) − 4.00000i − 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 2.00000i 0.242536i
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 16.0000i 1.82337i
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 10.0000i − 1.10432i
\(83\) 16.0000i 1.75623i 0.478451 + 0.878114i \(0.341198\pi\)
−0.478451 + 0.878114i \(0.658802\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 6.00000i 0.643268i
\(88\) 4.00000i 0.426401i
\(89\) 2.00000 0.212000 0.106000 0.994366i \(-0.466196\pi\)
0.106000 + 0.994366i \(0.466196\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 2.00000i − 0.208514i
\(93\) 6.00000i 0.622171i
\(94\) 10.0000 1.03142
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 9.00000i 0.909137i
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) − 2.00000i − 0.198030i
\(103\) 6.00000i 0.591198i 0.955312 + 0.295599i \(0.0955191\pi\)
−0.955312 + 0.295599i \(0.904481\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) − 4.00000i − 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 4.00000 0.383131 0.191565 0.981480i \(-0.438644\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 4.00000i 0.377964i
\(113\) 14.0000i 1.31701i 0.752577 + 0.658505i \(0.228811\pi\)
−0.752577 + 0.658505i \(0.771189\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 4.00000i 0.368230i
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000i 0.905357i
\(123\) 10.0000i 0.901670i
\(124\) −6.00000 −0.538816
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) 22.0000i 1.95218i 0.217357 + 0.976092i \(0.430256\pi\)
−0.217357 + 0.976092i \(0.569744\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −12.0000 −1.05654
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) − 4.00000i − 0.348155i
\(133\) − 4.00000i − 0.346844i
\(134\) 0 0
\(135\) 0 0
\(136\) 2.00000 0.171499
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 2.00000i 0.170251i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −10.0000 −0.842152
\(142\) 16.0000i 1.34269i
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) − 9.00000i − 0.742307i
\(148\) 8.00000i 0.657596i
\(149\) −20.0000 −1.63846 −0.819232 0.573462i \(-0.805600\pi\)
−0.819232 + 0.573462i \(0.805600\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) − 1.00000i − 0.0811107i
\(153\) 2.00000i 0.161690i
\(154\) 16.0000 1.28932
\(155\) 0 0
\(156\) 0 0
\(157\) 18.0000i 1.43656i 0.695756 + 0.718278i \(0.255069\pi\)
−0.695756 + 0.718278i \(0.744931\pi\)
\(158\) 10.0000i 0.795557i
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) −8.00000 −0.630488
\(162\) − 1.00000i − 0.0785674i
\(163\) − 20.0000i − 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) 16.0000 1.24184
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) − 4.00000i − 0.308607i
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) − 12.0000i − 0.914991i
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) − 4.00000i − 0.300658i
\(178\) − 2.00000i − 0.149906i
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 12.0000 0.891953 0.445976 0.895045i \(-0.352856\pi\)
0.445976 + 0.895045i \(0.352856\pi\)
\(182\) 0 0
\(183\) − 10.0000i − 0.739221i
\(184\) −2.00000 −0.147442
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) − 8.00000i − 0.585018i
\(188\) − 10.0000i − 0.729325i
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −2.00000 −0.144715 −0.0723575 0.997379i \(-0.523052\pi\)
−0.0723575 + 0.997379i \(0.523052\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) − 20.0000i − 1.42494i −0.701702 0.712470i \(-0.747576\pi\)
0.701702 0.712470i \(-0.252424\pi\)
\(198\) 4.00000i 0.284268i
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 8.00000i − 0.562878i
\(203\) 24.0000i 1.68447i
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) 6.00000 0.418040
\(207\) − 2.00000i − 0.139010i
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 2.00000i 0.137361i
\(213\) − 16.0000i − 1.09630i
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 24.0000i 1.62923i
\(218\) − 4.00000i − 0.270914i
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) − 8.00000i − 0.536925i
\(223\) − 6.00000i − 0.401790i −0.979613 0.200895i \(-0.935615\pi\)
0.979613 0.200895i \(-0.0643850\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) 20.0000i 1.32745i 0.747978 + 0.663723i \(0.231025\pi\)
−0.747978 + 0.663723i \(0.768975\pi\)
\(228\) 1.00000i 0.0662266i
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) 6.00000i 0.393919i
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) − 10.0000i − 0.649570i
\(238\) − 8.00000i − 0.518563i
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) 1.00000i 0.0641500i
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) 0 0
\(248\) 6.00000i 0.381000i
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 4.00000i 0.251976i
\(253\) 8.00000i 0.502956i
\(254\) 22.0000 1.38040
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 12.0000i 0.747087i
\(259\) 32.0000 1.98838
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 6.00000i 0.369976i 0.982741 + 0.184988i \(0.0592246\pi\)
−0.982741 + 0.184988i \(0.940775\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) 2.00000i 0.122398i
\(268\) 0 0
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) −4.00000 −0.242983 −0.121491 0.992592i \(-0.538768\pi\)
−0.121491 + 0.992592i \(0.538768\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 2.00000 0.120386
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 10.0000i 0.595491i
\(283\) − 28.0000i − 1.66443i −0.554455 0.832214i \(-0.687073\pi\)
0.554455 0.832214i \(-0.312927\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 0 0
\(287\) 40.0000i 2.36113i
\(288\) 1.00000i 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) − 2.00000i − 0.117041i
\(293\) − 14.0000i − 0.817889i −0.912559 0.408944i \(-0.865897\pi\)
0.912559 0.408944i \(-0.134103\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) − 4.00000i − 0.232104i
\(298\) 20.0000i 1.15857i
\(299\) 0 0
\(300\) 0 0
\(301\) −48.0000 −2.76667
\(302\) − 10.0000i − 0.575435i
\(303\) 8.00000i 0.459588i
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 2.00000 0.114332
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) − 16.0000i − 0.911685i
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) 34.0000 1.92796 0.963982 0.265969i \(-0.0856919\pi\)
0.963982 + 0.265969i \(0.0856919\pi\)
\(312\) 0 0
\(313\) − 6.00000i − 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) − 2.00000i − 0.112154i
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 8.00000i 0.445823i
\(323\) 2.00000i 0.111283i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −20.0000 −1.10770
\(327\) 4.00000i 0.221201i
\(328\) 10.0000i 0.552158i
\(329\) −40.0000 −2.20527
\(330\) 0 0
\(331\) 24.0000 1.31916 0.659580 0.751635i \(-0.270734\pi\)
0.659580 + 0.751635i \(0.270734\pi\)
\(332\) − 16.0000i − 0.878114i
\(333\) 8.00000i 0.438397i
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) 26.0000i 1.41631i 0.706057 + 0.708155i \(0.250472\pi\)
−0.706057 + 0.708155i \(0.749528\pi\)
\(338\) − 13.0000i − 0.707107i
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) − 1.00000i − 0.0540738i
\(343\) − 8.00000i − 0.431959i
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 28.0000i 1.50312i 0.659665 + 0.751559i \(0.270698\pi\)
−0.659665 + 0.751559i \(0.729302\pi\)
\(348\) − 6.00000i − 0.321634i
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 4.00000i − 0.213201i
\(353\) 30.0000i 1.59674i 0.602168 + 0.798369i \(0.294304\pi\)
−0.602168 + 0.798369i \(0.705696\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 8.00000i 0.423405i
\(358\) − 20.0000i − 1.05703i
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) − 12.0000i − 0.630706i
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) − 28.0000i − 1.46159i −0.682598 0.730794i \(-0.739150\pi\)
0.682598 0.730794i \(-0.260850\pi\)
\(368\) 2.00000i 0.104257i
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 8.00000 0.415339
\(372\) − 6.00000i − 0.311086i
\(373\) − 8.00000i − 0.414224i −0.978317 0.207112i \(-0.933593\pi\)
0.978317 0.207112i \(-0.0664065\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) −10.0000 −0.515711
\(377\) 0 0
\(378\) − 4.00000i − 0.205738i
\(379\) −8.00000 −0.410932 −0.205466 0.978664i \(-0.565871\pi\)
−0.205466 + 0.978664i \(0.565871\pi\)
\(380\) 0 0
\(381\) −22.0000 −1.12709
\(382\) 2.00000i 0.102329i
\(383\) − 8.00000i − 0.408781i −0.978889 0.204390i \(-0.934479\pi\)
0.978889 0.204390i \(-0.0655212\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) − 12.0000i − 0.609994i
\(388\) 10.0000i 0.507673i
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) − 9.00000i − 0.454569i
\(393\) 0 0
\(394\) −20.0000 −1.00759
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) − 30.0000i − 1.50566i −0.658217 0.752828i \(-0.728689\pi\)
0.658217 0.752828i \(-0.271311\pi\)
\(398\) − 4.00000i − 0.200502i
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −8.00000 −0.398015
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) − 32.0000i − 1.58618i
\(408\) 2.00000i 0.0990148i
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) − 6.00000i − 0.295599i
\(413\) − 16.0000i − 0.787309i
\(414\) −2.00000 −0.0982946
\(415\) 0 0
\(416\) 0 0
\(417\) 4.00000i 0.195881i
\(418\) 4.00000i 0.195646i
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) 20.0000i 0.973585i
\(423\) − 10.0000i − 0.486217i
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) −16.0000 −0.775203
\(427\) − 40.0000i − 1.93574i
\(428\) 4.00000i 0.193347i
\(429\) 0 0
\(430\) 0 0
\(431\) −4.00000 −0.192673 −0.0963366 0.995349i \(-0.530713\pi\)
−0.0963366 + 0.995349i \(0.530713\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 18.0000i 0.865025i 0.901628 + 0.432512i \(0.142373\pi\)
−0.901628 + 0.432512i \(0.857627\pi\)
\(434\) 24.0000 1.15204
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) − 2.00000i − 0.0956730i
\(438\) 2.00000i 0.0955637i
\(439\) −22.0000 −1.05000 −0.525001 0.851101i \(-0.675935\pi\)
−0.525001 + 0.851101i \(0.675935\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) 0 0
\(443\) − 12.0000i − 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) −6.00000 −0.284108
\(447\) − 20.0000i − 0.945968i
\(448\) − 4.00000i − 0.188982i
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 40.0000 1.88353
\(452\) − 14.0000i − 0.658505i
\(453\) 10.0000i 0.469841i
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 2.00000i 0.0935561i 0.998905 + 0.0467780i \(0.0148953\pi\)
−0.998905 + 0.0467780i \(0.985105\pi\)
\(458\) − 2.00000i − 0.0934539i
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 16.0000i 0.744387i
\(463\) − 36.0000i − 1.67306i −0.547920 0.836531i \(-0.684580\pi\)
0.547920 0.836531i \(-0.315420\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 8.00000i 0.370196i 0.982720 + 0.185098i \(0.0592602\pi\)
−0.982720 + 0.185098i \(0.940740\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) − 4.00000i − 0.184115i
\(473\) 48.0000i 2.20704i
\(474\) −10.0000 −0.459315
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) 2.00000i 0.0915737i
\(478\) − 6.00000i − 0.274434i
\(479\) −6.00000 −0.274147 −0.137073 0.990561i \(-0.543770\pi\)
−0.137073 + 0.990561i \(0.543770\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 2.00000i 0.0910975i
\(483\) − 8.00000i − 0.364013i
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) − 10.0000i − 0.452679i
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) − 10.0000i − 0.450835i
\(493\) − 12.0000i − 0.540453i
\(494\) 0 0
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) − 64.0000i − 2.87079i
\(498\) 16.0000i 0.716977i
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 24.0000i 1.07117i
\(503\) − 34.0000i − 1.51599i −0.652263 0.757993i \(-0.726180\pi\)
0.652263 0.757993i \(-0.273820\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 8.00000 0.355643
\(507\) 13.0000i 0.577350i
\(508\) − 22.0000i − 0.976092i
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) − 1.00000i − 0.0441942i
\(513\) 1.00000i 0.0441511i
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 12.0000 0.528271
\(517\) 40.0000i 1.75920i
\(518\) − 32.0000i − 1.40600i
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) 6.00000i 0.262613i
\(523\) − 16.0000i − 0.699631i −0.936819 0.349816i \(-0.886244\pi\)
0.936819 0.349816i \(-0.113756\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 6.00000 0.261612
\(527\) − 12.0000i − 0.522728i
\(528\) 4.00000i 0.174078i
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 4.00000i 0.173422i
\(533\) 0 0
\(534\) 2.00000 0.0865485
\(535\) 0 0
\(536\) 0 0
\(537\) 20.0000i 0.863064i
\(538\) 14.0000i 0.603583i
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 4.00000i 0.171815i
\(543\) 12.0000i 0.514969i
\(544\) −2.00000 −0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) − 16.0000i − 0.684111i −0.939680 0.342055i \(-0.888877\pi\)
0.939680 0.342055i \(-0.111123\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) − 2.00000i − 0.0851257i
\(553\) − 40.0000i − 1.70097i
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 4.00000i 0.169485i 0.996403 + 0.0847427i \(0.0270068\pi\)
−0.996403 + 0.0847427i \(0.972993\pi\)
\(558\) 6.00000i 0.254000i
\(559\) 0 0
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) − 10.0000i − 0.421825i
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) 10.0000 0.421076
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) 4.00000i 0.167984i
\(568\) − 16.0000i − 0.671345i
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) − 2.00000i − 0.0835512i
\(574\) 40.0000 1.66957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 14.0000i − 0.582828i −0.956597 0.291414i \(-0.905874\pi\)
0.956597 0.291414i \(-0.0941257\pi\)
\(578\) − 13.0000i − 0.540729i
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) −64.0000 −2.65517
\(582\) − 10.0000i − 0.414513i
\(583\) − 8.00000i − 0.331326i
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 28.0000i 1.15568i 0.816149 + 0.577842i \(0.196105\pi\)
−0.816149 + 0.577842i \(0.803895\pi\)
\(588\) 9.00000i 0.371154i
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) 20.0000 0.822690
\(592\) − 8.00000i − 0.328798i
\(593\) − 10.0000i − 0.410651i −0.978694 0.205325i \(-0.934175\pi\)
0.978694 0.205325i \(-0.0658253\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) 20.0000 0.819232
\(597\) 4.00000i 0.163709i
\(598\) 0 0
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) 42.0000 1.71322 0.856608 0.515968i \(-0.172568\pi\)
0.856608 + 0.515968i \(0.172568\pi\)
\(602\) 48.0000i 1.95633i
\(603\) 0 0
\(604\) −10.0000 −0.406894
\(605\) 0 0
\(606\) 8.00000 0.324978
\(607\) − 22.0000i − 0.892952i −0.894795 0.446476i \(-0.852679\pi\)
0.894795 0.446476i \(-0.147321\pi\)
\(608\) 1.00000i 0.0405554i
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) 0 0
\(612\) − 2.00000i − 0.0808452i
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −16.0000 −0.644658
\(617\) − 6.00000i − 0.241551i −0.992680 0.120775i \(-0.961462\pi\)
0.992680 0.120775i \(-0.0385381\pi\)
\(618\) 6.00000i 0.241355i
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 2.00000 0.0802572
\(622\) − 34.0000i − 1.36328i
\(623\) 8.00000i 0.320513i
\(624\) 0 0
\(625\) 0 0
\(626\) −6.00000 −0.239808
\(627\) − 4.00000i − 0.159745i
\(628\) − 18.0000i − 0.718278i
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) − 10.0000i − 0.397779i
\(633\) − 20.0000i − 0.794929i
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) 0 0
\(638\) − 24.0000i − 0.950169i
\(639\) 16.0000 0.632950
\(640\) 0 0
\(641\) 46.0000 1.81689 0.908445 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) − 4.00000i − 0.157867i
\(643\) − 28.0000i − 1.10421i −0.833774 0.552106i \(-0.813824\pi\)
0.833774 0.552106i \(-0.186176\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) 2.00000 0.0786889
\(647\) − 18.0000i − 0.707653i −0.935311 0.353827i \(-0.884880\pi\)
0.935311 0.353827i \(-0.115120\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) −24.0000 −0.940634
\(652\) 20.0000i 0.783260i
\(653\) 36.0000i 1.40879i 0.709809 + 0.704394i \(0.248781\pi\)
−0.709809 + 0.704394i \(0.751219\pi\)
\(654\) 4.00000 0.156412
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) − 2.00000i − 0.0780274i
\(658\) 40.0000i 1.55936i
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) −8.00000 −0.311164 −0.155582 0.987823i \(-0.549725\pi\)
−0.155582 + 0.987823i \(0.549725\pi\)
\(662\) − 24.0000i − 0.932786i
\(663\) 0 0
\(664\) −16.0000 −0.620920
\(665\) 0 0
\(666\) 8.00000 0.309994
\(667\) 12.0000i 0.464642i
\(668\) − 12.0000i − 0.464294i
\(669\) 6.00000 0.231973
\(670\) 0 0
\(671\) −40.0000 −1.54418
\(672\) 4.00000i 0.154303i
\(673\) − 14.0000i − 0.539660i −0.962908 0.269830i \(-0.913032\pi\)
0.962908 0.269830i \(-0.0869676\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 42.0000i 1.61419i 0.590421 + 0.807096i \(0.298962\pi\)
−0.590421 + 0.807096i \(0.701038\pi\)
\(678\) 14.0000i 0.537667i
\(679\) 40.0000 1.53506
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) − 24.0000i − 0.919007i
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) 2.00000i 0.0763048i
\(688\) 12.0000i 0.457496i
\(689\) 0 0
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 6.00000i 0.228086i
\(693\) − 16.0000i − 0.607790i
\(694\) 28.0000 1.06287
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) − 20.0000i − 0.757554i
\(698\) 10.0000i 0.378506i
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) −8.00000 −0.302156 −0.151078 0.988522i \(-0.548274\pi\)
−0.151078 + 0.988522i \(0.548274\pi\)
\(702\) 0 0
\(703\) 8.00000i 0.301726i
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 32.0000i 1.20348i
\(708\) 4.00000i 0.150329i
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 10.0000 0.375029
\(712\) 2.00000i 0.0749532i
\(713\) 12.0000i 0.449404i
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) 6.00000i 0.224074i
\(718\) 6.00000i 0.223918i
\(719\) −34.0000 −1.26799 −0.633993 0.773339i \(-0.718585\pi\)
−0.633993 + 0.773339i \(0.718585\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) − 1.00000i − 0.0372161i
\(723\) − 2.00000i − 0.0743808i
\(724\) −12.0000 −0.445976
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) 8.00000i 0.296704i 0.988935 + 0.148352i \(0.0473968\pi\)
−0.988935 + 0.148352i \(0.952603\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 10.0000i 0.369611i
\(733\) 38.0000i 1.40356i 0.712393 + 0.701781i \(0.247612\pi\)
−0.712393 + 0.701781i \(0.752388\pi\)
\(734\) −28.0000 −1.03350
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) 0 0
\(738\) 10.0000i 0.368105i
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 8.00000i − 0.293689i
\(743\) 36.0000i 1.32071i 0.750953 + 0.660356i \(0.229595\pi\)
−0.750953 + 0.660356i \(0.770405\pi\)
\(744\) −6.00000 −0.219971
\(745\) 0 0
\(746\) −8.00000 −0.292901
\(747\) − 16.0000i − 0.585409i
\(748\) 8.00000i 0.292509i
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) 10.0000 0.364905 0.182453 0.983215i \(-0.441596\pi\)
0.182453 + 0.983215i \(0.441596\pi\)
\(752\) 10.0000i 0.364662i
\(753\) − 24.0000i − 0.874609i
\(754\) 0 0
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) 8.00000i 0.290573i
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) 14.0000 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(762\) 22.0000i 0.796976i
\(763\) 16.0000i 0.579239i
\(764\) 2.00000 0.0723575
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) 0 0
\(768\) 1.00000i 0.0360844i
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 14.0000i 0.503871i
\(773\) − 6.00000i − 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) −12.0000 −0.431331
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) 32.0000i 1.14799i
\(778\) 8.00000i 0.286814i
\(779\) −10.0000 −0.358287
\(780\) 0 0
\(781\) −64.0000 −2.29010
\(782\) − 4.00000i − 0.143040i
\(783\) − 6.00000i − 0.214423i
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) 0 0
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) 20.0000i 0.712470i
\(789\) −6.00000 −0.213606
\(790\) 0 0
\(791\) −56.0000 −1.99113
\(792\) − 4.00000i − 0.142134i
\(793\) 0 0
\(794\) −30.0000 −1.06466
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) − 34.0000i − 1.20434i −0.798367 0.602171i \(-0.794303\pi\)
0.798367 0.602171i \(-0.205697\pi\)
\(798\) − 4.00000i − 0.141598i
\(799\) 20.0000 0.707549
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) 30.0000i 1.05934i
\(803\) 8.00000i 0.282314i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 14.0000i − 0.492823i
\(808\) 8.00000i 0.281439i
\(809\) −54.0000 −1.89854 −0.949269 0.314464i \(-0.898175\pi\)
−0.949269 + 0.314464i \(0.898175\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) − 24.0000i − 0.842235i
\(813\) − 4.00000i − 0.140286i
\(814\) −32.0000 −1.12160
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) − 12.0000i − 0.419827i
\(818\) − 18.0000i − 0.629355i
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 6.00000i 0.209274i
\(823\) − 16.0000i − 0.557725i −0.960331 0.278862i \(-0.910043\pi\)
0.960331 0.278862i \(-0.0899574\pi\)
\(824\) −6.00000 −0.209020
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) − 12.0000i − 0.417281i −0.977992 0.208640i \(-0.933096\pi\)
0.977992 0.208640i \(-0.0669038\pi\)
\(828\) 2.00000i 0.0695048i
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) 18.0000i 0.623663i
\(834\) 4.00000 0.138509
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) − 6.00000i − 0.207390i
\(838\) − 12.0000i − 0.414533i
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 8.00000i 0.275698i
\(843\) 10.0000i 0.344418i
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) −10.0000 −0.343807
\(847\) 20.0000i 0.687208i
\(848\) − 2.00000i − 0.0686803i
\(849\) 28.0000 0.960958
\(850\) 0 0
\(851\) 16.0000 0.548473
\(852\) 16.0000i 0.548151i
\(853\) − 54.0000i − 1.84892i −0.381273 0.924462i \(-0.624514\pi\)
0.381273 0.924462i \(-0.375486\pi\)
\(854\) −40.0000 −1.36877
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) − 42.0000i − 1.43469i −0.696717 0.717346i \(-0.745357\pi\)
0.696717 0.717346i \(-0.254643\pi\)
\(858\) 0 0
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) −40.0000 −1.36320
\(862\) 4.00000i 0.136241i
\(863\) 28.0000i 0.953131i 0.879139 + 0.476566i \(0.158119\pi\)
−0.879139 + 0.476566i \(0.841881\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 18.0000 0.611665
\(867\) 13.0000i 0.441503i
\(868\) − 24.0000i − 0.814613i
\(869\) −40.0000 −1.35691
\(870\) 0 0
\(871\) 0 0
\(872\) 4.00000i 0.135457i
\(873\) 10.0000i 0.338449i
\(874\) −2.00000 −0.0676510
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 8.00000i 0.270141i 0.990836 + 0.135070i \(0.0431261\pi\)
−0.990836 + 0.135070i \(0.956874\pi\)
\(878\) 22.0000i 0.742464i
\(879\) 14.0000 0.472208
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) − 9.00000i − 0.303046i
\(883\) − 36.0000i − 1.21150i −0.795656 0.605748i \(-0.792874\pi\)
0.795656 0.605748i \(-0.207126\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) − 12.0000i − 0.402921i −0.979497 0.201460i \(-0.935431\pi\)
0.979497 0.201460i \(-0.0645687\pi\)
\(888\) 8.00000i 0.268462i
\(889\) −88.0000 −2.95143
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 6.00000i 0.200895i
\(893\) − 10.0000i − 0.334637i
\(894\) −20.0000 −0.668900
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) 0 0
\(898\) − 18.0000i − 0.600668i
\(899\) 36.0000 1.20067
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) − 40.0000i − 1.33185i
\(903\) − 48.0000i − 1.59734i
\(904\) −14.0000 −0.465633
\(905\) 0 0
\(906\) 10.0000 0.332228
\(907\) − 32.0000i − 1.06254i −0.847202 0.531271i \(-0.821714\pi\)
0.847202 0.531271i \(-0.178286\pi\)
\(908\) − 20.0000i − 0.663723i
\(909\) −8.00000 −0.265343
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) − 1.00000i − 0.0331133i
\(913\) 64.0000i 2.11809i
\(914\) 2.00000 0.0661541
\(915\) 0 0
\(916\) −2.00000 −0.0660819
\(917\) 0 0
\(918\) 2.00000i 0.0660098i
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 12.0000i 0.395199i
\(923\) 0 0
\(924\) 16.0000 0.526361
\(925\) 0 0
\(926\) −36.0000 −1.18303
\(927\) − 6.00000i − 0.197066i
\(928\) − 6.00000i − 0.196960i
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) 9.00000 0.294963
\(932\) − 6.00000i − 0.196537i
\(933\) 34.0000i 1.11311i
\(934\) 8.00000 0.261768
\(935\) 0 0
\(936\) 0 0
\(937\) 38.0000i 1.24141i 0.784046 + 0.620703i \(0.213153\pi\)
−0.784046 + 0.620703i \(0.786847\pi\)
\(938\) 0 0
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 46.0000 1.49956 0.749779 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) 18.0000i 0.586472i
\(943\) 20.0000i 0.651290i
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) − 8.00000i − 0.259965i −0.991516 0.129983i \(-0.958508\pi\)
0.991516 0.129983i \(-0.0414921\pi\)
\(948\) 10.0000i 0.324785i
\(949\) 0 0
\(950\) 0 0
\(951\) −6.00000 −0.194563
\(952\) 8.00000i 0.259281i
\(953\) 30.0000i 0.971795i 0.874016 + 0.485898i \(0.161507\pi\)
−0.874016 + 0.485898i \(0.838493\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) −6.00000 −0.194054
\(957\) 24.0000i 0.775810i
\(958\) 6.00000i 0.193851i
\(959\) −24.0000 −0.775000
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 4.00000i 0.128898i
\(964\) 2.00000 0.0644157
\(965\) 0 0
\(966\) −8.00000 −0.257396
\(967\) 12.0000i 0.385894i 0.981209 + 0.192947i \(0.0618045\pi\)
−0.981209 + 0.192947i \(0.938195\pi\)
\(968\) 5.00000i 0.160706i
\(969\) −2.00000 −0.0642493
\(970\) 0 0
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 16.0000i 0.512936i
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) − 20.0000i − 0.639529i
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) − 24.0000i − 0.765871i
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) −10.0000 −0.318788
\(985\) 0 0
\(986\) −12.0000 −0.382158
\(987\) − 40.0000i − 1.27321i
\(988\) 0 0
\(989\) −24.0000 −0.763156
\(990\) 0 0
\(991\) −50.0000 −1.58830 −0.794151 0.607720i \(-0.792084\pi\)
−0.794151 + 0.607720i \(0.792084\pi\)
\(992\) − 6.00000i − 0.190500i
\(993\) 24.0000i 0.761617i
\(994\) −64.0000 −2.02996
\(995\) 0 0
\(996\) 16.0000 0.506979
\(997\) − 22.0000i − 0.696747i −0.937356 0.348373i \(-0.886734\pi\)
0.937356 0.348373i \(-0.113266\pi\)
\(998\) − 4.00000i − 0.126618i
\(999\) −8.00000 −0.253109
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 2850.2.d.s.799.1 2
5.2 odd 4 2850.2.a.x.1.1 1
5.3 odd 4 114.2.a.a.1.1 1
5.4 even 2 inner 2850.2.d.s.799.2 2
15.2 even 4 8550.2.a.a.1.1 1
15.8 even 4 342.2.a.f.1.1 1
20.3 even 4 912.2.a.h.1.1 1
35.13 even 4 5586.2.a.p.1.1 1
40.3 even 4 3648.2.a.j.1.1 1
40.13 odd 4 3648.2.a.bb.1.1 1
60.23 odd 4 2736.2.a.j.1.1 1
95.18 even 4 2166.2.a.i.1.1 1
285.113 odd 4 6498.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.a.a.1.1 1 5.3 odd 4
342.2.a.f.1.1 1 15.8 even 4
912.2.a.h.1.1 1 20.3 even 4
2166.2.a.i.1.1 1 95.18 even 4
2736.2.a.j.1.1 1 60.23 odd 4
2850.2.a.x.1.1 1 5.2 odd 4
2850.2.d.s.799.1 2 1.1 even 1 trivial
2850.2.d.s.799.2 2 5.4 even 2 inner
3648.2.a.j.1.1 1 40.3 even 4
3648.2.a.bb.1.1 1 40.13 odd 4
5586.2.a.p.1.1 1 35.13 even 4
6498.2.a.h.1.1 1 285.113 odd 4
8550.2.a.a.1.1 1 15.2 even 4