Properties

Label 2850.2.d.g.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: yes
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.g.799.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} +1.00000i q^{12} -4.00000i q^{13} +1.00000 q^{16} +4.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -1.00000i q^{22} -5.00000i q^{23} +1.00000 q^{24} -4.00000 q^{26} +1.00000i q^{27} -3.00000 q^{29} -5.00000 q^{31} -1.00000i q^{32} -1.00000i q^{33} +4.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} -1.00000i q^{38} -4.00000 q^{39} -2.00000 q^{41} -4.00000i q^{43} -1.00000 q^{44} -5.00000 q^{46} -1.00000i q^{48} +7.00000 q^{49} +4.00000 q^{51} +4.00000i q^{52} -9.00000i q^{53} +1.00000 q^{54} -1.00000i q^{57} +3.00000i q^{58} -11.0000 q^{61} +5.00000i q^{62} -1.00000 q^{64} -1.00000 q^{66} -1.00000i q^{67} -4.00000i q^{68} -5.00000 q^{69} +2.00000 q^{71} -1.00000i q^{72} +3.00000i q^{73} -6.00000 q^{74} -1.00000 q^{76} +4.00000i q^{78} -17.0000 q^{79} +1.00000 q^{81} +2.00000i q^{82} -3.00000i q^{83} -4.00000 q^{86} +3.00000i q^{87} +1.00000i q^{88} -7.00000 q^{89} +5.00000i q^{92} +5.00000i q^{93} -1.00000 q^{96} +10.0000i q^{97} -7.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + 2q^{11} + 2q^{16} + 2q^{19} + 2q^{24} - 8q^{26} - 6q^{29} - 10q^{31} + 8q^{34} + 2q^{36} - 8q^{39} - 4q^{41} - 2q^{44} - 10q^{46} + 14q^{49} + 8q^{51} + 2q^{54} - 22q^{61} - 2q^{64} - 2q^{66} - 10q^{69} + 4q^{71} - 12q^{74} - 2q^{76} - 34q^{79} + 2q^{81} - 8q^{86} - 14q^{89} - 2q^{96} - 2q^{99} + O(q^{100}) \)

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
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 4.00000i − 1.10940i −0.832050 0.554700i \(-0.812833\pi\)
0.832050 0.554700i \(-0.187167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000i 0.970143i 0.874475 + 0.485071i \(0.161206\pi\)
−0.874475 + 0.485071i \(0.838794\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) − 1.00000i − 0.213201i
\(23\) − 5.00000i − 1.04257i −0.853382 0.521286i \(-0.825452\pi\)
0.853382 0.521286i \(-0.174548\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 1.00000i − 0.174078i
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −5.00000 −0.737210
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 4.00000i 0.554700i
\(53\) − 9.00000i − 1.23625i −0.786082 0.618123i \(-0.787894\pi\)
0.786082 0.618123i \(-0.212106\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.00000i − 0.132453i
\(58\) 3.00000i 0.393919i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −11.0000 −1.40841 −0.704203 0.709999i \(-0.748695\pi\)
−0.704203 + 0.709999i \(0.748695\pi\)
\(62\) 5.00000i 0.635001i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −1.00000 −0.123091
\(67\) − 1.00000i − 0.122169i −0.998133 0.0610847i \(-0.980544\pi\)
0.998133 0.0610847i \(-0.0194560\pi\)
\(68\) − 4.00000i − 0.485071i
\(69\) −5.00000 −0.601929
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 3.00000i 0.351123i 0.984468 + 0.175562i \(0.0561742\pi\)
−0.984468 + 0.175562i \(0.943826\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 0 0
\(78\) 4.00000i 0.452911i
\(79\) −17.0000 −1.91265 −0.956325 0.292306i \(-0.905577\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000i 0.220863i
\(83\) − 3.00000i − 0.329293i −0.986353 0.164646i \(-0.947352\pi\)
0.986353 0.164646i \(-0.0526483\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 3.00000i 0.321634i
\(88\) 1.00000i 0.106600i
\(89\) −7.00000 −0.741999 −0.370999 0.928633i \(-0.620985\pi\)
−0.370999 + 0.928633i \(0.620985\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 5.00000i 0.521286i
\(93\) 5.00000i 0.518476i
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) − 7.00000i − 0.707107i
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) − 4.00000i − 0.396059i
\(103\) − 13.0000i − 1.28093i −0.767988 0.640464i \(-0.778742\pi\)
0.767988 0.640464i \(-0.221258\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 10.0000i 0.966736i 0.875417 + 0.483368i \(0.160587\pi\)
−0.875417 + 0.483368i \(0.839413\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) − 5.00000i − 0.470360i −0.971952 0.235180i \(-0.924432\pi\)
0.971952 0.235180i \(-0.0755680\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) 4.00000i 0.369800i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 11.0000i 0.995893i
\(123\) 2.00000i 0.180334i
\(124\) 5.00000 0.449013
\(125\) 0 0
\(126\) 0 0
\(127\) − 5.00000i − 0.443678i −0.975083 0.221839i \(-0.928794\pi\)
0.975083 0.221839i \(-0.0712060\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 1.00000i 0.0870388i
\(133\) 0 0
\(134\) −1.00000 −0.0863868
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) − 4.00000i − 0.341743i −0.985293 0.170872i \(-0.945342\pi\)
0.985293 0.170872i \(-0.0546583\pi\)
\(138\) 5.00000i 0.425628i
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 2.00000i − 0.167836i
\(143\) − 4.00000i − 0.334497i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 3.00000 0.248282
\(147\) − 7.00000i − 0.577350i
\(148\) 6.00000i 0.493197i
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −24.0000 −1.95309 −0.976546 0.215308i \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) − 4.00000i − 0.323381i
\(154\) 0 0
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) 17.0000i 1.35245i
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) 0 0
\(162\) − 1.00000i − 0.0785674i
\(163\) − 10.0000i − 0.783260i −0.920123 0.391630i \(-0.871911\pi\)
0.920123 0.391630i \(-0.128089\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −3.00000 −0.232845
\(167\) 18.0000i 1.39288i 0.717614 + 0.696441i \(0.245234\pi\)
−0.717614 + 0.696441i \(0.754766\pi\)
\(168\) 0 0
\(169\) −3.00000 −0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 4.00000i 0.304997i
\(173\) − 7.00000i − 0.532200i −0.963945 0.266100i \(-0.914265\pi\)
0.963945 0.266100i \(-0.0857352\pi\)
\(174\) 3.00000 0.227429
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 7.00000i 0.524672i
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 11.0000i 0.813143i
\(184\) 5.00000 0.368605
\(185\) 0 0
\(186\) 5.00000 0.366618
\(187\) 4.00000i 0.292509i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.0000 0.795932 0.397966 0.917400i \(-0.369716\pi\)
0.397966 + 0.917400i \(0.369716\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 10.0000i − 0.719816i −0.932988 0.359908i \(-0.882808\pi\)
0.932988 0.359908i \(-0.117192\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) − 20.0000i − 1.42494i −0.701702 0.712470i \(-0.747576\pi\)
0.701702 0.712470i \(-0.252424\pi\)
\(198\) 1.00000i 0.0710669i
\(199\) −6.00000 −0.425329 −0.212664 0.977125i \(-0.568214\pi\)
−0.212664 + 0.977125i \(0.568214\pi\)
\(200\) 0 0
\(201\) −1.00000 −0.0705346
\(202\) 8.00000i 0.562878i
\(203\) 0 0
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) −13.0000 −0.905753
\(207\) 5.00000i 0.347524i
\(208\) − 4.00000i − 0.277350i
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 9.00000i 0.618123i
\(213\) − 2.00000i − 0.137038i
\(214\) 10.0000 0.683586
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 6.00000i 0.406371i
\(219\) 3.00000 0.202721
\(220\) 0 0
\(221\) 16.0000 1.07628
\(222\) 6.00000i 0.402694i
\(223\) 15.0000i 1.00447i 0.864730 + 0.502237i \(0.167490\pi\)
−0.864730 + 0.502237i \(0.832510\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −5.00000 −0.332595
\(227\) 2.00000i 0.132745i 0.997795 + 0.0663723i \(0.0211425\pi\)
−0.997795 + 0.0663723i \(0.978857\pi\)
\(228\) 1.00000i 0.0662266i
\(229\) −5.00000 −0.330409 −0.165205 0.986259i \(-0.552828\pi\)
−0.165205 + 0.986259i \(0.552828\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 3.00000i − 0.196960i
\(233\) 24.0000i 1.57229i 0.618041 + 0.786146i \(0.287927\pi\)
−0.618041 + 0.786146i \(0.712073\pi\)
\(234\) 4.00000 0.261488
\(235\) 0 0
\(236\) 0 0
\(237\) 17.0000i 1.10427i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −4.00000 −0.257663 −0.128831 0.991667i \(-0.541123\pi\)
−0.128831 + 0.991667i \(0.541123\pi\)
\(242\) 10.0000i 0.642824i
\(243\) − 1.00000i − 0.0641500i
\(244\) 11.0000 0.704203
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) − 4.00000i − 0.254514i
\(248\) − 5.00000i − 0.317500i
\(249\) −3.00000 −0.190117
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) − 5.00000i − 0.314347i
\(254\) −5.00000 −0.313728
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 7.00000i 0.436648i 0.975876 + 0.218324i \(0.0700590\pi\)
−0.975876 + 0.218324i \(0.929941\pi\)
\(258\) 4.00000i 0.249029i
\(259\) 0 0
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) − 15.0000i − 0.926703i
\(263\) 11.0000i 0.678289i 0.940734 + 0.339145i \(0.110138\pi\)
−0.940734 + 0.339145i \(0.889862\pi\)
\(264\) 1.00000 0.0615457
\(265\) 0 0
\(266\) 0 0
\(267\) 7.00000i 0.428393i
\(268\) 1.00000i 0.0610847i
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 4.00000i 0.242536i
\(273\) 0 0
\(274\) −4.00000 −0.241649
\(275\) 0 0
\(276\) 5.00000 0.300965
\(277\) − 1.00000i − 0.0600842i −0.999549 0.0300421i \(-0.990436\pi\)
0.999549 0.0300421i \(-0.00956413\pi\)
\(278\) − 2.00000i − 0.119952i
\(279\) 5.00000 0.299342
\(280\) 0 0
\(281\) −3.00000 −0.178965 −0.0894825 0.995988i \(-0.528521\pi\)
−0.0894825 + 0.995988i \(0.528521\pi\)
\(282\) 0 0
\(283\) 8.00000i 0.475551i 0.971320 + 0.237775i \(0.0764182\pi\)
−0.971320 + 0.237775i \(0.923582\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) − 3.00000i − 0.175562i
\(293\) − 11.0000i − 0.642627i −0.946973 0.321313i \(-0.895876\pi\)
0.946973 0.321313i \(-0.104124\pi\)
\(294\) −7.00000 −0.408248
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 1.00000i 0.0580259i
\(298\) 0 0
\(299\) −20.0000 −1.15663
\(300\) 0 0
\(301\) 0 0
\(302\) 24.0000i 1.38104i
\(303\) 8.00000i 0.459588i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) − 25.0000i − 1.42683i −0.700744 0.713413i \(-0.747149\pi\)
0.700744 0.713413i \(-0.252851\pi\)
\(308\) 0 0
\(309\) −13.0000 −0.739544
\(310\) 0 0
\(311\) 32.0000 1.81455 0.907277 0.420534i \(-0.138157\pi\)
0.907277 + 0.420534i \(0.138157\pi\)
\(312\) − 4.00000i − 0.226455i
\(313\) − 21.0000i − 1.18699i −0.804838 0.593495i \(-0.797748\pi\)
0.804838 0.593495i \(-0.202252\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) 17.0000 0.956325
\(317\) − 13.0000i − 0.730153i −0.930978 0.365076i \(-0.881043\pi\)
0.930978 0.365076i \(-0.118957\pi\)
\(318\) 9.00000i 0.504695i
\(319\) −3.00000 −0.167968
\(320\) 0 0
\(321\) 10.0000 0.558146
\(322\) 0 0
\(323\) 4.00000i 0.222566i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −10.0000 −0.553849
\(327\) 6.00000i 0.331801i
\(328\) − 2.00000i − 0.110432i
\(329\) 0 0
\(330\) 0 0
\(331\) 13.0000 0.714545 0.357272 0.934000i \(-0.383707\pi\)
0.357272 + 0.934000i \(0.383707\pi\)
\(332\) 3.00000i 0.164646i
\(333\) 6.00000i 0.328798i
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) 0 0
\(337\) − 8.00000i − 0.435788i −0.975972 0.217894i \(-0.930081\pi\)
0.975972 0.217894i \(-0.0699187\pi\)
\(338\) 3.00000i 0.163178i
\(339\) −5.00000 −0.271563
\(340\) 0 0
\(341\) −5.00000 −0.270765
\(342\) 1.00000i 0.0540738i
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) −7.00000 −0.376322
\(347\) − 12.0000i − 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) − 3.00000i − 0.160817i
\(349\) 1.00000 0.0535288 0.0267644 0.999642i \(-0.491480\pi\)
0.0267644 + 0.999642i \(0.491480\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) − 1.00000i − 0.0533002i
\(353\) − 28.0000i − 1.49029i −0.666903 0.745145i \(-0.732380\pi\)
0.666903 0.745145i \(-0.267620\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 7.00000 0.370999
\(357\) 0 0
\(358\) − 4.00000i − 0.211407i
\(359\) −24.0000 −1.26667 −0.633336 0.773877i \(-0.718315\pi\)
−0.633336 + 0.773877i \(0.718315\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 10.0000i 0.524864i
\(364\) 0 0
\(365\) 0 0
\(366\) 11.0000 0.574979
\(367\) − 18.0000i − 0.939592i −0.882775 0.469796i \(-0.844327\pi\)
0.882775 0.469796i \(-0.155673\pi\)
\(368\) − 5.00000i − 0.260643i
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) − 5.00000i − 0.259238i
\(373\) − 28.0000i − 1.44979i −0.688862 0.724893i \(-0.741889\pi\)
0.688862 0.724893i \(-0.258111\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000i 0.618031i
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −5.00000 −0.256158
\(382\) − 11.0000i − 0.562809i
\(383\) 6.00000i 0.306586i 0.988181 + 0.153293i \(0.0489878\pi\)
−0.988181 + 0.153293i \(0.951012\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 4.00000i 0.203331i
\(388\) − 10.0000i − 0.507673i
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 20.0000 1.01144
\(392\) 7.00000i 0.353553i
\(393\) − 15.0000i − 0.756650i
\(394\) −20.0000 −1.00759
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) − 3.00000i − 0.150566i −0.997162 0.0752828i \(-0.976014\pi\)
0.997162 0.0752828i \(-0.0239860\pi\)
\(398\) 6.00000i 0.300753i
\(399\) 0 0
\(400\) 0 0
\(401\) −17.0000 −0.848939 −0.424470 0.905442i \(-0.639539\pi\)
−0.424470 + 0.905442i \(0.639539\pi\)
\(402\) 1.00000i 0.0498755i
\(403\) 20.0000i 0.996271i
\(404\) 8.00000 0.398015
\(405\) 0 0
\(406\) 0 0
\(407\) − 6.00000i − 0.297409i
\(408\) 4.00000i 0.198030i
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) −4.00000 −0.197305
\(412\) 13.0000i 0.640464i
\(413\) 0 0
\(414\) 5.00000 0.245737
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) − 2.00000i − 0.0979404i
\(418\) − 1.00000i − 0.0489116i
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 13.0000i 0.632830i
\(423\) 0 0
\(424\) 9.00000 0.437079
\(425\) 0 0
\(426\) −2.00000 −0.0969003
\(427\) 0 0
\(428\) − 10.0000i − 0.483368i
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 38.0000 1.83040 0.915198 0.403005i \(-0.132034\pi\)
0.915198 + 0.403005i \(0.132034\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 22.0000i 1.05725i 0.848855 + 0.528626i \(0.177293\pi\)
−0.848855 + 0.528626i \(0.822707\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) − 5.00000i − 0.239182i
\(438\) − 3.00000i − 0.143346i
\(439\) −21.0000 −1.00228 −0.501138 0.865368i \(-0.667085\pi\)
−0.501138 + 0.865368i \(0.667085\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) − 16.0000i − 0.761042i
\(443\) 13.0000i 0.617649i 0.951119 + 0.308824i \(0.0999355\pi\)
−0.951119 + 0.308824i \(0.900064\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) 15.0000 0.710271
\(447\) 0 0
\(448\) 0 0
\(449\) 27.0000 1.27421 0.637104 0.770778i \(-0.280132\pi\)
0.637104 + 0.770778i \(0.280132\pi\)
\(450\) 0 0
\(451\) −2.00000 −0.0941763
\(452\) 5.00000i 0.235180i
\(453\) 24.0000i 1.12762i
\(454\) 2.00000 0.0938647
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) − 18.0000i − 0.842004i −0.907060 0.421002i \(-0.861678\pi\)
0.907060 0.421002i \(-0.138322\pi\)
\(458\) 5.00000i 0.233635i
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 4.00000 0.186299 0.0931493 0.995652i \(-0.470307\pi\)
0.0931493 + 0.995652i \(0.470307\pi\)
\(462\) 0 0
\(463\) − 4.00000i − 0.185896i −0.995671 0.0929479i \(-0.970371\pi\)
0.995671 0.0929479i \(-0.0296290\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) 24.0000 1.11178
\(467\) − 11.0000i − 0.509019i −0.967070 0.254510i \(-0.918086\pi\)
0.967070 0.254510i \(-0.0819141\pi\)
\(468\) − 4.00000i − 0.184900i
\(469\) 0 0
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 0 0
\(473\) − 4.00000i − 0.183920i
\(474\) 17.0000 0.780836
\(475\) 0 0
\(476\) 0 0
\(477\) 9.00000i 0.412082i
\(478\) 0 0
\(479\) 37.0000 1.69057 0.845287 0.534313i \(-0.179430\pi\)
0.845287 + 0.534313i \(0.179430\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) 4.00000i 0.182195i
\(483\) 0 0
\(484\) 10.0000 0.454545
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 8.00000i 0.362515i 0.983436 + 0.181257i \(0.0580167\pi\)
−0.983436 + 0.181257i \(0.941983\pi\)
\(488\) − 11.0000i − 0.497947i
\(489\) −10.0000 −0.452216
\(490\) 0 0
\(491\) 36.0000 1.62466 0.812329 0.583200i \(-0.198200\pi\)
0.812329 + 0.583200i \(0.198200\pi\)
\(492\) − 2.00000i − 0.0901670i
\(493\) − 12.0000i − 0.540453i
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −5.00000 −0.224507
\(497\) 0 0
\(498\) 3.00000i 0.134433i
\(499\) 24.0000 1.07439 0.537194 0.843459i \(-0.319484\pi\)
0.537194 + 0.843459i \(0.319484\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) − 16.0000i − 0.714115i
\(503\) 36.0000i 1.60516i 0.596544 + 0.802580i \(0.296540\pi\)
−0.596544 + 0.802580i \(0.703460\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −5.00000 −0.222277
\(507\) 3.00000i 0.133235i
\(508\) 5.00000i 0.221839i
\(509\) 27.0000 1.19675 0.598377 0.801215i \(-0.295813\pi\)
0.598377 + 0.801215i \(0.295813\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 1.00000i − 0.0441942i
\(513\) 1.00000i 0.0441511i
\(514\) 7.00000 0.308757
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) −7.00000 −0.307266
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) − 3.00000i − 0.131306i
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) −15.0000 −0.655278
\(525\) 0 0
\(526\) 11.0000 0.479623
\(527\) − 20.0000i − 0.871214i
\(528\) − 1.00000i − 0.0435194i
\(529\) −2.00000 −0.0869565
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.00000i 0.346518i
\(534\) 7.00000 0.302920
\(535\) 0 0
\(536\) 1.00000 0.0431934
\(537\) − 4.00000i − 0.172613i
\(538\) − 2.00000i − 0.0862261i
\(539\) 7.00000 0.301511
\(540\) 0 0
\(541\) −13.0000 −0.558914 −0.279457 0.960158i \(-0.590154\pi\)
−0.279457 + 0.960158i \(0.590154\pi\)
\(542\) − 2.00000i − 0.0859074i
\(543\) 0 0
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) 1.00000i 0.0427569i 0.999771 + 0.0213785i \(0.00680549\pi\)
−0.999771 + 0.0213785i \(0.993195\pi\)
\(548\) 4.00000i 0.170872i
\(549\) 11.0000 0.469469
\(550\) 0 0
\(551\) −3.00000 −0.127804
\(552\) − 5.00000i − 0.212814i
\(553\) 0 0
\(554\) −1.00000 −0.0424859
\(555\) 0 0
\(556\) −2.00000 −0.0848189
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) − 5.00000i − 0.211667i
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) 3.00000i 0.126547i
\(563\) − 26.0000i − 1.09577i −0.836554 0.547885i \(-0.815433\pi\)
0.836554 0.547885i \(-0.184567\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 8.00000 0.336265
\(567\) 0 0
\(568\) 2.00000i 0.0839181i
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) −26.0000 −1.08807 −0.544033 0.839064i \(-0.683103\pi\)
−0.544033 + 0.839064i \(0.683103\pi\)
\(572\) 4.00000i 0.167248i
\(573\) − 11.0000i − 0.459532i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 27.0000i − 1.12402i −0.827129 0.562012i \(-0.810027\pi\)
0.827129 0.562012i \(-0.189973\pi\)
\(578\) − 1.00000i − 0.0415945i
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) 0 0
\(582\) − 10.0000i − 0.414513i
\(583\) − 9.00000i − 0.372742i
\(584\) −3.00000 −0.124141
\(585\) 0 0
\(586\) −11.0000 −0.454406
\(587\) − 17.0000i − 0.701665i −0.936438 0.350833i \(-0.885899\pi\)
0.936438 0.350833i \(-0.114101\pi\)
\(588\) 7.00000i 0.288675i
\(589\) −5.00000 −0.206021
\(590\) 0 0
\(591\) −20.0000 −0.822690
\(592\) − 6.00000i − 0.246598i
\(593\) 28.0000i 1.14982i 0.818216 + 0.574911i \(0.194963\pi\)
−0.818216 + 0.574911i \(0.805037\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) 0 0
\(597\) 6.00000i 0.245564i
\(598\) 20.0000i 0.817861i
\(599\) −28.0000 −1.14405 −0.572024 0.820237i \(-0.693842\pi\)
−0.572024 + 0.820237i \(0.693842\pi\)
\(600\) 0 0
\(601\) −2.00000 −0.0815817 −0.0407909 0.999168i \(-0.512988\pi\)
−0.0407909 + 0.999168i \(0.512988\pi\)
\(602\) 0 0
\(603\) 1.00000i 0.0407231i
\(604\) 24.0000 0.976546
\(605\) 0 0
\(606\) 8.00000 0.324978
\(607\) − 25.0000i − 1.01472i −0.861735 0.507359i \(-0.830622\pi\)
0.861735 0.507359i \(-0.169378\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 4.00000i 0.161690i
\(613\) − 26.0000i − 1.05013i −0.851062 0.525065i \(-0.824041\pi\)
0.851062 0.525065i \(-0.175959\pi\)
\(614\) −25.0000 −1.00892
\(615\) 0 0
\(616\) 0 0
\(617\) − 46.0000i − 1.85189i −0.377658 0.925945i \(-0.623271\pi\)
0.377658 0.925945i \(-0.376729\pi\)
\(618\) 13.0000i 0.522937i
\(619\) 8.00000 0.321547 0.160774 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 0 0
\(621\) 5.00000 0.200643
\(622\) − 32.0000i − 1.28308i
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) −21.0000 −0.839329
\(627\) − 1.00000i − 0.0399362i
\(628\) − 14.0000i − 0.558661i
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 28.0000 1.11466 0.557331 0.830290i \(-0.311825\pi\)
0.557331 + 0.830290i \(0.311825\pi\)
\(632\) − 17.0000i − 0.676224i
\(633\) 13.0000i 0.516704i
\(634\) −13.0000 −0.516296
\(635\) 0 0
\(636\) 9.00000 0.356873
\(637\) − 28.0000i − 1.10940i
\(638\) 3.00000i 0.118771i
\(639\) −2.00000 −0.0791188
\(640\) 0 0
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) − 10.0000i − 0.394669i
\(643\) 14.0000i 0.552106i 0.961142 + 0.276053i \(0.0890266\pi\)
−0.961142 + 0.276053i \(0.910973\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) − 21.0000i − 0.825595i −0.910823 0.412798i \(-0.864552\pi\)
0.910823 0.412798i \(-0.135448\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 10.0000i 0.391630i
\(653\) 48.0000i 1.87839i 0.343391 + 0.939193i \(0.388424\pi\)
−0.343391 + 0.939193i \(0.611576\pi\)
\(654\) 6.00000 0.234619
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) − 3.00000i − 0.117041i
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) − 13.0000i − 0.505259i
\(663\) − 16.0000i − 0.621389i
\(664\) 3.00000 0.116423
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 15.0000i 0.580802i
\(668\) − 18.0000i − 0.696441i
\(669\) 15.0000 0.579934
\(670\) 0 0
\(671\) −11.0000 −0.424650
\(672\) 0 0
\(673\) − 44.0000i − 1.69608i −0.529936 0.848038i \(-0.677784\pi\)
0.529936 0.848038i \(-0.322216\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) − 15.0000i − 0.576497i −0.957556 0.288248i \(-0.906927\pi\)
0.957556 0.288248i \(-0.0930729\pi\)
\(678\) 5.00000i 0.192024i
\(679\) 0 0
\(680\) 0 0
\(681\) 2.00000 0.0766402
\(682\) 5.00000i 0.191460i
\(683\) 18.0000i 0.688751i 0.938832 + 0.344375i \(0.111909\pi\)
−0.938832 + 0.344375i \(0.888091\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) 0 0
\(687\) 5.00000i 0.190762i
\(688\) − 4.00000i − 0.152499i
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 26.0000 0.989087 0.494543 0.869153i \(-0.335335\pi\)
0.494543 + 0.869153i \(0.335335\pi\)
\(692\) 7.00000i 0.266100i
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) − 8.00000i − 0.303022i
\(698\) − 1.00000i − 0.0378506i
\(699\) 24.0000 0.907763
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) − 4.00000i − 0.150970i
\(703\) − 6.00000i − 0.226294i
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −28.0000 −1.05379
\(707\) 0 0
\(708\) 0 0
\(709\) 43.0000 1.61490 0.807449 0.589937i \(-0.200847\pi\)
0.807449 + 0.589937i \(0.200847\pi\)
\(710\) 0 0
\(711\) 17.0000 0.637550
\(712\) − 7.00000i − 0.262336i
\(713\) 25.0000i 0.936257i
\(714\) 0 0
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) 24.0000i 0.895672i
\(719\) 29.0000 1.08152 0.540759 0.841178i \(-0.318137\pi\)
0.540759 + 0.841178i \(0.318137\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 1.00000i − 0.0372161i
\(723\) 4.00000i 0.148762i
\(724\) 0 0
\(725\) 0 0
\(726\) 10.0000 0.371135
\(727\) − 10.0000i − 0.370879i −0.982656 0.185440i \(-0.940629\pi\)
0.982656 0.185440i \(-0.0593710\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 16.0000 0.591781
\(732\) − 11.0000i − 0.406572i
\(733\) − 41.0000i − 1.51437i −0.653201 0.757185i \(-0.726574\pi\)
0.653201 0.757185i \(-0.273426\pi\)
\(734\) −18.0000 −0.664392
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) − 1.00000i − 0.0368355i
\(738\) − 2.00000i − 0.0736210i
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) 16.0000i 0.586983i 0.955962 + 0.293492i \(0.0948173\pi\)
−0.955962 + 0.293492i \(0.905183\pi\)
\(744\) −5.00000 −0.183309
\(745\) 0 0
\(746\) −28.0000 −1.02515
\(747\) 3.00000i 0.109764i
\(748\) − 4.00000i − 0.146254i
\(749\) 0 0
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 0 0
\(753\) − 16.0000i − 0.583072i
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) − 23.0000i − 0.835949i −0.908459 0.417975i \(-0.862740\pi\)
0.908459 0.417975i \(-0.137260\pi\)
\(758\) − 8.00000i − 0.290573i
\(759\) −5.00000 −0.181489
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 5.00000i 0.181131i
\(763\) 0 0
\(764\) −11.0000 −0.397966
\(765\) 0 0
\(766\) 6.00000 0.216789
\(767\) 0 0
\(768\) − 1.00000i − 0.0360844i
\(769\) −35.0000 −1.26213 −0.631066 0.775729i \(-0.717382\pi\)
−0.631066 + 0.775729i \(0.717382\pi\)
\(770\) 0 0
\(771\) 7.00000 0.252099
\(772\) 10.0000i 0.359908i
\(773\) − 18.0000i − 0.647415i −0.946157 0.323708i \(-0.895071\pi\)
0.946157 0.323708i \(-0.104929\pi\)
\(774\) 4.00000 0.143777
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) 26.0000i 0.932145i
\(779\) −2.00000 −0.0716574
\(780\) 0 0
\(781\) 2.00000 0.0715656
\(782\) − 20.0000i − 0.715199i
\(783\) − 3.00000i − 0.107211i
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) −15.0000 −0.535032
\(787\) 37.0000i 1.31891i 0.751745 + 0.659454i \(0.229212\pi\)
−0.751745 + 0.659454i \(0.770788\pi\)
\(788\) 20.0000i 0.712470i
\(789\) 11.0000 0.391610
\(790\) 0 0
\(791\) 0 0
\(792\) − 1.00000i − 0.0355335i
\(793\) 44.0000i 1.56249i
\(794\) −3.00000 −0.106466
\(795\) 0 0
\(796\) 6.00000 0.212664
\(797\) − 18.0000i − 0.637593i −0.947823 0.318796i \(-0.896721\pi\)
0.947823 0.318796i \(-0.103279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 7.00000 0.247333
\(802\) 17.0000i 0.600291i
\(803\) 3.00000i 0.105868i
\(804\) 1.00000 0.0352673
\(805\) 0 0
\(806\) 20.0000 0.704470
\(807\) − 2.00000i − 0.0704033i
\(808\) − 8.00000i − 0.281439i
\(809\) 44.0000 1.54696 0.773479 0.633822i \(-0.218515\pi\)
0.773479 + 0.633822i \(0.218515\pi\)
\(810\) 0 0
\(811\) −35.0000 −1.22902 −0.614508 0.788911i \(-0.710645\pi\)
−0.614508 + 0.788911i \(0.710645\pi\)
\(812\) 0 0
\(813\) − 2.00000i − 0.0701431i
\(814\) −6.00000 −0.210300
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) − 4.00000i − 0.139942i
\(818\) 6.00000i 0.209785i
\(819\) 0 0
\(820\) 0 0
\(821\) 26.0000 0.907406 0.453703 0.891153i \(-0.350103\pi\)
0.453703 + 0.891153i \(0.350103\pi\)
\(822\) 4.00000i 0.139516i
\(823\) 26.0000i 0.906303i 0.891434 + 0.453152i \(0.149700\pi\)
−0.891434 + 0.453152i \(0.850300\pi\)
\(824\) 13.0000 0.452876
\(825\) 0 0
\(826\) 0 0
\(827\) 48.0000i 1.66912i 0.550914 + 0.834562i \(0.314279\pi\)
−0.550914 + 0.834562i \(0.685721\pi\)
\(828\) − 5.00000i − 0.173762i
\(829\) 50.0000 1.73657 0.868286 0.496064i \(-0.165222\pi\)
0.868286 + 0.496064i \(0.165222\pi\)
\(830\) 0 0
\(831\) −1.00000 −0.0346896
\(832\) 4.00000i 0.138675i
\(833\) 28.0000i 0.970143i
\(834\) −2.00000 −0.0692543
\(835\) 0 0
\(836\) −1.00000 −0.0345857
\(837\) − 5.00000i − 0.172825i
\(838\) − 12.0000i − 0.414533i
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 34.0000i 1.17172i
\(843\) 3.00000i 0.103325i
\(844\) 13.0000 0.447478
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) − 9.00000i − 0.309061i
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) −30.0000 −1.02839
\(852\) 2.00000i 0.0685189i
\(853\) − 10.0000i − 0.342393i −0.985237 0.171197i \(-0.945237\pi\)
0.985237 0.171197i \(-0.0547634\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −10.0000 −0.341793
\(857\) − 38.0000i − 1.29806i −0.760765 0.649028i \(-0.775176\pi\)
0.760765 0.649028i \(-0.224824\pi\)
\(858\) 4.00000i 0.136558i
\(859\) 18.0000 0.614152 0.307076 0.951685i \(-0.400649\pi\)
0.307076 + 0.951685i \(0.400649\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 38.0000i − 1.29429i
\(863\) − 4.00000i − 0.136162i −0.997680 0.0680808i \(-0.978312\pi\)
0.997680 0.0680808i \(-0.0216876\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 22.0000 0.747590
\(867\) − 1.00000i − 0.0339618i
\(868\) 0 0
\(869\) −17.0000 −0.576686
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) − 6.00000i − 0.203186i
\(873\) − 10.0000i − 0.338449i
\(874\) −5.00000 −0.169128
\(875\) 0 0
\(876\) −3.00000 −0.101361
\(877\) − 2.00000i − 0.0675352i −0.999430 0.0337676i \(-0.989249\pi\)
0.999430 0.0337676i \(-0.0107506\pi\)
\(878\) 21.0000i 0.708716i
\(879\) −11.0000 −0.371021
\(880\) 0 0
\(881\) −48.0000 −1.61716 −0.808581 0.588386i \(-0.799764\pi\)
−0.808581 + 0.588386i \(0.799764\pi\)
\(882\) 7.00000i 0.235702i
\(883\) 38.0000i 1.27880i 0.768874 + 0.639401i \(0.220818\pi\)
−0.768874 + 0.639401i \(0.779182\pi\)
\(884\) −16.0000 −0.538138
\(885\) 0 0
\(886\) 13.0000 0.436744
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) − 6.00000i − 0.201347i
\(889\) 0 0
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) − 15.0000i − 0.502237i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 20.0000i 0.667781i
\(898\) − 27.0000i − 0.901002i
\(899\) 15.0000 0.500278
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 2.00000i 0.0665927i
\(903\) 0 0
\(904\) 5.00000 0.166298
\(905\) 0 0
\(906\) 24.0000 0.797347
\(907\) 8.00000i 0.265636i 0.991140 + 0.132818i \(0.0424025\pi\)
−0.991140 + 0.132818i \(0.957597\pi\)
\(908\) − 2.00000i − 0.0663723i
\(909\) 8.00000 0.265343
\(910\) 0 0
\(911\) −18.0000 −0.596367 −0.298183 0.954509i \(-0.596381\pi\)
−0.298183 + 0.954509i \(0.596381\pi\)
\(912\) − 1.00000i − 0.0331133i
\(913\) − 3.00000i − 0.0992855i
\(914\) −18.0000 −0.595387
\(915\) 0 0
\(916\) 5.00000 0.165205
\(917\) 0 0
\(918\) 4.00000i 0.132020i
\(919\) −14.0000 −0.461817 −0.230909 0.972975i \(-0.574170\pi\)
−0.230909 + 0.972975i \(0.574170\pi\)
\(920\) 0 0
\(921\) −25.0000 −0.823778
\(922\) − 4.00000i − 0.131733i
\(923\) − 8.00000i − 0.263323i
\(924\) 0 0
\(925\) 0 0
\(926\) −4.00000 −0.131448
\(927\) 13.0000i 0.426976i
\(928\) 3.00000i 0.0984798i
\(929\) 12.0000 0.393707 0.196854 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) 0 0
\(931\) 7.00000 0.229416
\(932\) − 24.0000i − 0.786146i
\(933\) − 32.0000i − 1.04763i
\(934\) −11.0000 −0.359931
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) − 10.0000i − 0.326686i −0.986569 0.163343i \(-0.947772\pi\)
0.986569 0.163343i \(-0.0522277\pi\)
\(938\) 0 0
\(939\) −21.0000 −0.685309
\(940\) 0 0
\(941\) 33.0000 1.07577 0.537885 0.843018i \(-0.319224\pi\)
0.537885 + 0.843018i \(0.319224\pi\)
\(942\) − 14.0000i − 0.456145i
\(943\) 10.0000i 0.325645i
\(944\) 0 0
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 20.0000i 0.649913i 0.945729 + 0.324956i \(0.105350\pi\)
−0.945729 + 0.324956i \(0.894650\pi\)
\(948\) − 17.0000i − 0.552134i
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) −13.0000 −0.421554
\(952\) 0 0
\(953\) − 3.00000i − 0.0971795i −0.998819 0.0485898i \(-0.984527\pi\)
0.998819 0.0485898i \(-0.0154727\pi\)
\(954\) 9.00000 0.291386
\(955\) 0 0
\(956\) 0 0
\(957\) 3.00000i 0.0969762i
\(958\) − 37.0000i − 1.19542i
\(959\) 0 0
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 24.0000i 0.773791i
\(963\) − 10.0000i − 0.322245i
\(964\) 4.00000 0.128831
\(965\) 0 0
\(966\) 0 0
\(967\) 14.0000i 0.450210i 0.974335 + 0.225105i \(0.0722725\pi\)
−0.974335 + 0.225105i \(0.927728\pi\)
\(968\) − 10.0000i − 0.321412i
\(969\) 4.00000 0.128499
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 0 0
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) −11.0000 −0.352101
\(977\) 46.0000i 1.47167i 0.677161 + 0.735835i \(0.263210\pi\)
−0.677161 + 0.735835i \(0.736790\pi\)
\(978\) 10.0000i 0.319765i
\(979\) −7.00000 −0.223721
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) − 36.0000i − 1.14881i
\(983\) − 34.0000i − 1.08443i −0.840239 0.542216i \(-0.817586\pi\)
0.840239 0.542216i \(-0.182414\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 0 0
\(986\) −12.0000 −0.382158
\(987\) 0 0
\(988\) 4.00000i 0.127257i
\(989\) −20.0000 −0.635963
\(990\) 0 0
\(991\) 13.0000 0.412959 0.206479 0.978451i \(-0.433799\pi\)
0.206479 + 0.978451i \(0.433799\pi\)
\(992\) 5.00000i 0.158750i
\(993\) − 13.0000i − 0.412543i
\(994\) 0 0
\(995\) 0 0
\(996\) 3.00000 0.0950586
\(997\) 37.0000i 1.17180i 0.810383 + 0.585901i \(0.199259\pi\)
−0.810383 + 0.585901i \(0.800741\pi\)
\(998\) − 24.0000i − 0.759707i
\(999\) 6.00000 0.189832
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.g.799.1 2
5.2 odd 4 2850.2.a.u.1.1 yes 1
5.3 odd 4 2850.2.a.l.1.1 1
5.4 even 2 inner 2850.2.d.g.799.2 2
15.2 even 4 8550.2.a.h.1.1 1
15.8 even 4 8550.2.a.z.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.l.1.1 1 5.3 odd 4
2850.2.a.u.1.1 yes 1 5.2 odd 4
2850.2.d.g.799.1 2 1.1 even 1 trivial
2850.2.d.g.799.2 2 5.4 even 2 inner
8550.2.a.h.1.1 1 15.2 even 4
8550.2.a.z.1.1 1 15.8 even 4