Properties

Label 1950.2.f.m.649.4
Level $1950$
Weight $2$
Character 1950.649
Analytic conductor $15.571$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 390)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.4
Root \(-2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 1950.649
Dual form 1950.2.f.m.649.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} +4.60555 q^{7} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000i q^{6} +4.60555 q^{7} -1.00000 q^{8} -1.00000 q^{9} +1.00000i q^{12} -3.60555i q^{13} -4.60555 q^{14} +1.00000 q^{16} +4.60555i q^{17} +1.00000 q^{18} +4.60555i q^{19} +4.60555i q^{21} +1.39445i q^{23} -1.00000i q^{24} +3.60555i q^{26} -1.00000i q^{27} +4.60555 q^{28} -4.60555 q^{29} +6.00000i q^{31} -1.00000 q^{32} -4.60555i q^{34} -1.00000 q^{36} +9.21110 q^{37} -4.60555i q^{38} +3.60555 q^{39} +3.21110i q^{41} -4.60555i q^{42} -8.00000i q^{43} -1.39445i q^{46} +9.21110 q^{47} +1.00000i q^{48} +14.2111 q^{49} -4.60555 q^{51} -3.60555i q^{52} -6.00000i q^{53} +1.00000i q^{54} -4.60555 q^{56} -4.60555 q^{57} +4.60555 q^{58} +9.21110i q^{59} -11.2111 q^{61} -6.00000i q^{62} -4.60555 q^{63} +1.00000 q^{64} +3.21110 q^{67} +4.60555i q^{68} -1.39445 q^{69} +9.21110i q^{71} +1.00000 q^{72} -1.39445 q^{73} -9.21110 q^{74} +4.60555i q^{76} -3.60555 q^{78} +14.4222 q^{79} +1.00000 q^{81} -3.21110i q^{82} +2.78890 q^{83} +4.60555i q^{84} +8.00000i q^{86} -4.60555i q^{87} -15.2111i q^{89} -16.6056i q^{91} +1.39445i q^{92} -6.00000 q^{93} -9.21110 q^{94} -1.00000i q^{96} -1.39445 q^{97} -14.2111 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{2} + 4q^{4} + 4q^{7} - 4q^{8} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{2} + 4q^{4} + 4q^{7} - 4q^{8} - 4q^{9} - 4q^{14} + 4q^{16} + 4q^{18} + 4q^{28} - 4q^{29} - 4q^{32} - 4q^{36} + 8q^{37} + 8q^{47} + 28q^{49} - 4q^{51} - 4q^{56} - 4q^{57} + 4q^{58} - 16q^{61} - 4q^{63} + 4q^{64} - 16q^{67} - 20q^{69} + 4q^{72} - 20q^{73} - 8q^{74} + 4q^{81} + 40q^{83} - 24q^{93} - 8q^{94} - 20q^{97} - 28q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000i 0.577350i
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) − 1.00000i − 0.408248i
\(7\) 4.60555 1.74073 0.870367 0.492403i \(-0.163881\pi\)
0.870367 + 0.492403i \(0.163881\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 3.60555i − 1.00000i
\(14\) −4.60555 −1.23089
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.60555i 1.11701i 0.829501 + 0.558505i \(0.188625\pi\)
−0.829501 + 0.558505i \(0.811375\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.60555i 1.05659i 0.849062 + 0.528293i \(0.177168\pi\)
−0.849062 + 0.528293i \(0.822832\pi\)
\(20\) 0 0
\(21\) 4.60555i 1.00501i
\(22\) 0 0
\(23\) 1.39445i 0.290763i 0.989376 + 0.145381i \(0.0464409\pi\)
−0.989376 + 0.145381i \(0.953559\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) 0 0
\(26\) 3.60555i 0.707107i
\(27\) − 1.00000i − 0.192450i
\(28\) 4.60555 0.870367
\(29\) −4.60555 −0.855229 −0.427615 0.903961i \(-0.640646\pi\)
−0.427615 + 0.903961i \(0.640646\pi\)
\(30\) 0 0
\(31\) 6.00000i 1.07763i 0.842424 + 0.538816i \(0.181128\pi\)
−0.842424 + 0.538816i \(0.818872\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) − 4.60555i − 0.789846i
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 9.21110 1.51430 0.757148 0.653243i \(-0.226592\pi\)
0.757148 + 0.653243i \(0.226592\pi\)
\(38\) − 4.60555i − 0.747119i
\(39\) 3.60555 0.577350
\(40\) 0 0
\(41\) 3.21110i 0.501490i 0.968053 + 0.250745i \(0.0806756\pi\)
−0.968053 + 0.250745i \(0.919324\pi\)
\(42\) − 4.60555i − 0.710652i
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) − 1.39445i − 0.205600i
\(47\) 9.21110 1.34358 0.671789 0.740743i \(-0.265526\pi\)
0.671789 + 0.740743i \(0.265526\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 14.2111 2.03016
\(50\) 0 0
\(51\) −4.60555 −0.644906
\(52\) − 3.60555i − 0.500000i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 0 0
\(56\) −4.60555 −0.615443
\(57\) −4.60555 −0.610020
\(58\) 4.60555 0.604739
\(59\) 9.21110i 1.19918i 0.800306 + 0.599592i \(0.204670\pi\)
−0.800306 + 0.599592i \(0.795330\pi\)
\(60\) 0 0
\(61\) −11.2111 −1.43543 −0.717717 0.696335i \(-0.754813\pi\)
−0.717717 + 0.696335i \(0.754813\pi\)
\(62\) − 6.00000i − 0.762001i
\(63\) −4.60555 −0.580245
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 3.21110 0.392299 0.196149 0.980574i \(-0.437156\pi\)
0.196149 + 0.980574i \(0.437156\pi\)
\(68\) 4.60555i 0.558505i
\(69\) −1.39445 −0.167872
\(70\) 0 0
\(71\) 9.21110i 1.09316i 0.837408 + 0.546578i \(0.184070\pi\)
−0.837408 + 0.546578i \(0.815930\pi\)
\(72\) 1.00000 0.117851
\(73\) −1.39445 −0.163208 −0.0816039 0.996665i \(-0.526004\pi\)
−0.0816039 + 0.996665i \(0.526004\pi\)
\(74\) −9.21110 −1.07077
\(75\) 0 0
\(76\) 4.60555i 0.528293i
\(77\) 0 0
\(78\) −3.60555 −0.408248
\(79\) 14.4222 1.62262 0.811312 0.584613i \(-0.198754\pi\)
0.811312 + 0.584613i \(0.198754\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 3.21110i − 0.354607i
\(83\) 2.78890 0.306121 0.153061 0.988217i \(-0.451087\pi\)
0.153061 + 0.988217i \(0.451087\pi\)
\(84\) 4.60555i 0.502507i
\(85\) 0 0
\(86\) 8.00000i 0.862662i
\(87\) − 4.60555i − 0.493767i
\(88\) 0 0
\(89\) − 15.2111i − 1.61237i −0.591661 0.806187i \(-0.701528\pi\)
0.591661 0.806187i \(-0.298472\pi\)
\(90\) 0 0
\(91\) − 16.6056i − 1.74073i
\(92\) 1.39445i 0.145381i
\(93\) −6.00000 −0.622171
\(94\) −9.21110 −0.950053
\(95\) 0 0
\(96\) − 1.00000i − 0.102062i
\(97\) −1.39445 −0.141585 −0.0707924 0.997491i \(-0.522553\pi\)
−0.0707924 + 0.997491i \(0.522553\pi\)
\(98\) −14.2111 −1.43554
\(99\) 0 0
\(100\) 0 0
\(101\) −7.39445 −0.735775 −0.367888 0.929870i \(-0.619919\pi\)
−0.367888 + 0.929870i \(0.619919\pi\)
\(102\) 4.60555 0.456018
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 3.60555i 0.353553i
\(105\) 0 0
\(106\) 6.00000i 0.582772i
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) − 1.39445i − 0.133564i −0.997768 0.0667820i \(-0.978727\pi\)
0.997768 0.0667820i \(-0.0212732\pi\)
\(110\) 0 0
\(111\) 9.21110i 0.874279i
\(112\) 4.60555 0.435184
\(113\) 13.8167i 1.29976i 0.760036 + 0.649881i \(0.225181\pi\)
−0.760036 + 0.649881i \(0.774819\pi\)
\(114\) 4.60555 0.431349
\(115\) 0 0
\(116\) −4.60555 −0.427615
\(117\) 3.60555i 0.333333i
\(118\) − 9.21110i − 0.847951i
\(119\) 21.2111i 1.94442i
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 11.2111 1.01501
\(123\) −3.21110 −0.289535
\(124\) 6.00000i 0.538816i
\(125\) 0 0
\(126\) 4.60555 0.410295
\(127\) 1.21110i 0.107468i 0.998555 + 0.0537340i \(0.0171123\pi\)
−0.998555 + 0.0537340i \(0.982888\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 22.6056 1.97506 0.987528 0.157443i \(-0.0503250\pi\)
0.987528 + 0.157443i \(0.0503250\pi\)
\(132\) 0 0
\(133\) 21.2111i 1.83924i
\(134\) −3.21110 −0.277397
\(135\) 0 0
\(136\) − 4.60555i − 0.394923i
\(137\) −3.21110 −0.274343 −0.137172 0.990547i \(-0.543801\pi\)
−0.137172 + 0.990547i \(0.543801\pi\)
\(138\) 1.39445 0.118703
\(139\) −17.2111 −1.45983 −0.729913 0.683540i \(-0.760440\pi\)
−0.729913 + 0.683540i \(0.760440\pi\)
\(140\) 0 0
\(141\) 9.21110i 0.775715i
\(142\) − 9.21110i − 0.772979i
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 1.39445 0.115405
\(147\) 14.2111i 1.17211i
\(148\) 9.21110 0.757148
\(149\) − 15.2111i − 1.24614i −0.782165 0.623071i \(-0.785885\pi\)
0.782165 0.623071i \(-0.214115\pi\)
\(150\) 0 0
\(151\) 6.00000i 0.488273i 0.969741 + 0.244137i \(0.0785045\pi\)
−0.969741 + 0.244137i \(0.921495\pi\)
\(152\) − 4.60555i − 0.373560i
\(153\) − 4.60555i − 0.372337i
\(154\) 0 0
\(155\) 0 0
\(156\) 3.60555 0.288675
\(157\) 20.4222i 1.62987i 0.579553 + 0.814935i \(0.303227\pi\)
−0.579553 + 0.814935i \(0.696773\pi\)
\(158\) −14.4222 −1.14737
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 6.42221i 0.506141i
\(162\) −1.00000 −0.0785674
\(163\) 24.4222 1.91289 0.956447 0.291905i \(-0.0942891\pi\)
0.956447 + 0.291905i \(0.0942891\pi\)
\(164\) 3.21110i 0.250745i
\(165\) 0 0
\(166\) −2.78890 −0.216460
\(167\) −9.21110 −0.712777 −0.356388 0.934338i \(-0.615992\pi\)
−0.356388 + 0.934338i \(0.615992\pi\)
\(168\) − 4.60555i − 0.355326i
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) − 4.60555i − 0.352195i
\(172\) − 8.00000i − 0.609994i
\(173\) 12.4222i 0.944443i 0.881480 + 0.472221i \(0.156548\pi\)
−0.881480 + 0.472221i \(0.843452\pi\)
\(174\) 4.60555i 0.349146i
\(175\) 0 0
\(176\) 0 0
\(177\) −9.21110 −0.692349
\(178\) 15.2111i 1.14012i
\(179\) 19.8167 1.48117 0.740583 0.671965i \(-0.234549\pi\)
0.740583 + 0.671965i \(0.234549\pi\)
\(180\) 0 0
\(181\) 8.42221 0.626018 0.313009 0.949750i \(-0.398663\pi\)
0.313009 + 0.949750i \(0.398663\pi\)
\(182\) 16.6056i 1.23089i
\(183\) − 11.2111i − 0.828749i
\(184\) − 1.39445i − 0.102800i
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) 0 0
\(188\) 9.21110 0.671789
\(189\) − 4.60555i − 0.335005i
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 7.81665 0.562655 0.281328 0.959612i \(-0.409225\pi\)
0.281328 + 0.959612i \(0.409225\pi\)
\(194\) 1.39445 0.100116
\(195\) 0 0
\(196\) 14.2111 1.01508
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −22.4222 −1.58947 −0.794734 0.606958i \(-0.792390\pi\)
−0.794734 + 0.606958i \(0.792390\pi\)
\(200\) 0 0
\(201\) 3.21110i 0.226494i
\(202\) 7.39445 0.520272
\(203\) −21.2111 −1.48873
\(204\) −4.60555 −0.322453
\(205\) 0 0
\(206\) − 4.00000i − 0.278693i
\(207\) − 1.39445i − 0.0969209i
\(208\) − 3.60555i − 0.250000i
\(209\) 0 0
\(210\) 0 0
\(211\) −17.2111 −1.18486 −0.592431 0.805622i \(-0.701832\pi\)
−0.592431 + 0.805622i \(0.701832\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) −9.21110 −0.631134
\(214\) 0 0
\(215\) 0 0
\(216\) 1.00000i 0.0680414i
\(217\) 27.6333i 1.87587i
\(218\) 1.39445i 0.0944440i
\(219\) − 1.39445i − 0.0942281i
\(220\) 0 0
\(221\) 16.6056 1.11701
\(222\) − 9.21110i − 0.618209i
\(223\) 1.81665 0.121652 0.0608261 0.998148i \(-0.480627\pi\)
0.0608261 + 0.998148i \(0.480627\pi\)
\(224\) −4.60555 −0.307721
\(225\) 0 0
\(226\) − 13.8167i − 0.919070i
\(227\) −24.0000 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(228\) −4.60555 −0.305010
\(229\) 19.8167i 1.30952i 0.755836 + 0.654761i \(0.227231\pi\)
−0.755836 + 0.654761i \(0.772769\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 4.60555 0.302369
\(233\) 1.81665i 0.119013i 0.998228 + 0.0595065i \(0.0189527\pi\)
−0.998228 + 0.0595065i \(0.981047\pi\)
\(234\) − 3.60555i − 0.235702i
\(235\) 0 0
\(236\) 9.21110i 0.599592i
\(237\) 14.4222i 0.936823i
\(238\) − 21.2111i − 1.37491i
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) − 6.42221i − 0.413691i −0.978374 0.206845i \(-0.933680\pi\)
0.978374 0.206845i \(-0.0663197\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000i 0.0641500i
\(244\) −11.2111 −0.717717
\(245\) 0 0
\(246\) 3.21110 0.204732
\(247\) 16.6056 1.05659
\(248\) − 6.00000i − 0.381000i
\(249\) 2.78890i 0.176739i
\(250\) 0 0
\(251\) −13.3944 −0.845450 −0.422725 0.906258i \(-0.638926\pi\)
−0.422725 + 0.906258i \(0.638926\pi\)
\(252\) −4.60555 −0.290122
\(253\) 0 0
\(254\) − 1.21110i − 0.0759913i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 28.6056i − 1.78437i −0.451675 0.892183i \(-0.649173\pi\)
0.451675 0.892183i \(-0.350827\pi\)
\(258\) −8.00000 −0.498058
\(259\) 42.4222 2.63599
\(260\) 0 0
\(261\) 4.60555 0.285076
\(262\) −22.6056 −1.39658
\(263\) 7.81665i 0.481996i 0.970526 + 0.240998i \(0.0774746\pi\)
−0.970526 + 0.240998i \(0.922525\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) − 21.2111i − 1.30054i
\(267\) 15.2111 0.930904
\(268\) 3.21110 0.196149
\(269\) −25.8167 −1.57407 −0.787035 0.616909i \(-0.788385\pi\)
−0.787035 + 0.616909i \(0.788385\pi\)
\(270\) 0 0
\(271\) − 0.422205i − 0.0256471i −0.999918 0.0128236i \(-0.995918\pi\)
0.999918 0.0128236i \(-0.00408198\pi\)
\(272\) 4.60555i 0.279253i
\(273\) 16.6056 1.00501
\(274\) 3.21110 0.193990
\(275\) 0 0
\(276\) −1.39445 −0.0839359
\(277\) − 16.4222i − 0.986715i −0.869827 0.493357i \(-0.835770\pi\)
0.869827 0.493357i \(-0.164230\pi\)
\(278\) 17.2111 1.03225
\(279\) − 6.00000i − 0.359211i
\(280\) 0 0
\(281\) 27.2111i 1.62328i 0.584159 + 0.811639i \(0.301424\pi\)
−0.584159 + 0.811639i \(0.698576\pi\)
\(282\) − 9.21110i − 0.548513i
\(283\) − 10.4222i − 0.619536i −0.950812 0.309768i \(-0.899749\pi\)
0.950812 0.309768i \(-0.100251\pi\)
\(284\) 9.21110i 0.546578i
\(285\) 0 0
\(286\) 0 0
\(287\) 14.7889i 0.872961i
\(288\) 1.00000 0.0589256
\(289\) −4.21110 −0.247712
\(290\) 0 0
\(291\) − 1.39445i − 0.0817440i
\(292\) −1.39445 −0.0816039
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) − 14.2111i − 0.828808i
\(295\) 0 0
\(296\) −9.21110 −0.535384
\(297\) 0 0
\(298\) 15.2111i 0.881156i
\(299\) 5.02776 0.290763
\(300\) 0 0
\(301\) − 36.8444i − 2.12368i
\(302\) − 6.00000i − 0.345261i
\(303\) − 7.39445i − 0.424800i
\(304\) 4.60555i 0.264146i
\(305\) 0 0
\(306\) 4.60555i 0.263282i
\(307\) 8.78890 0.501609 0.250804 0.968038i \(-0.419305\pi\)
0.250804 + 0.968038i \(0.419305\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) −3.60555 −0.204124
\(313\) − 3.57779i − 0.202229i −0.994875 0.101114i \(-0.967759\pi\)
0.994875 0.101114i \(-0.0322408\pi\)
\(314\) − 20.4222i − 1.15249i
\(315\) 0 0
\(316\) 14.4222 0.811312
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −6.00000 −0.336463
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) − 6.42221i − 0.357895i
\(323\) −21.2111 −1.18022
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −24.4222 −1.35262
\(327\) 1.39445 0.0771132
\(328\) − 3.21110i − 0.177303i
\(329\) 42.4222 2.33881
\(330\) 0 0
\(331\) − 16.6056i − 0.912724i −0.889794 0.456362i \(-0.849152\pi\)
0.889794 0.456362i \(-0.150848\pi\)
\(332\) 2.78890 0.153061
\(333\) −9.21110 −0.504765
\(334\) 9.21110 0.504009
\(335\) 0 0
\(336\) 4.60555i 0.251253i
\(337\) 13.6333i 0.742654i 0.928502 + 0.371327i \(0.121097\pi\)
−0.928502 + 0.371327i \(0.878903\pi\)
\(338\) 13.0000 0.707107
\(339\) −13.8167 −0.750418
\(340\) 0 0
\(341\) 0 0
\(342\) 4.60555i 0.249040i
\(343\) 33.2111 1.79323
\(344\) 8.00000i 0.431331i
\(345\) 0 0
\(346\) − 12.4222i − 0.667822i
\(347\) − 27.6333i − 1.48343i −0.670713 0.741717i \(-0.734012\pi\)
0.670713 0.741717i \(-0.265988\pi\)
\(348\) − 4.60555i − 0.246883i
\(349\) 7.81665i 0.418416i 0.977871 + 0.209208i \(0.0670886\pi\)
−0.977871 + 0.209208i \(0.932911\pi\)
\(350\) 0 0
\(351\) −3.60555 −0.192450
\(352\) 0 0
\(353\) −8.78890 −0.467786 −0.233893 0.972262i \(-0.575147\pi\)
−0.233893 + 0.972262i \(0.575147\pi\)
\(354\) 9.21110 0.489565
\(355\) 0 0
\(356\) − 15.2111i − 0.806187i
\(357\) −21.2111 −1.12261
\(358\) −19.8167 −1.04734
\(359\) − 15.6333i − 0.825094i −0.910936 0.412547i \(-0.864639\pi\)
0.910936 0.412547i \(-0.135361\pi\)
\(360\) 0 0
\(361\) −2.21110 −0.116374
\(362\) −8.42221 −0.442661
\(363\) 11.0000i 0.577350i
\(364\) − 16.6056i − 0.870367i
\(365\) 0 0
\(366\) 11.2111i 0.586014i
\(367\) − 19.6333i − 1.02485i −0.858732 0.512425i \(-0.828747\pi\)
0.858732 0.512425i \(-0.171253\pi\)
\(368\) 1.39445i 0.0726907i
\(369\) − 3.21110i − 0.167163i
\(370\) 0 0
\(371\) − 27.6333i − 1.43465i
\(372\) −6.00000 −0.311086
\(373\) 20.4222i 1.05742i 0.848802 + 0.528711i \(0.177324\pi\)
−0.848802 + 0.528711i \(0.822676\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −9.21110 −0.475026
\(377\) 16.6056i 0.855229i
\(378\) 4.60555i 0.236884i
\(379\) − 35.0278i − 1.79925i −0.436658 0.899627i \(-0.643838\pi\)
0.436658 0.899627i \(-0.356162\pi\)
\(380\) 0 0
\(381\) −1.21110 −0.0620467
\(382\) 12.0000 0.613973
\(383\) −27.6333 −1.41200 −0.705998 0.708214i \(-0.749501\pi\)
−0.705998 + 0.708214i \(0.749501\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) 0 0
\(386\) −7.81665 −0.397857
\(387\) 8.00000i 0.406663i
\(388\) −1.39445 −0.0707924
\(389\) −4.60555 −0.233511 −0.116755 0.993161i \(-0.537249\pi\)
−0.116755 + 0.993161i \(0.537249\pi\)
\(390\) 0 0
\(391\) −6.42221 −0.324785
\(392\) −14.2111 −0.717769
\(393\) 22.6056i 1.14030i
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) 3.63331 0.182350 0.0911752 0.995835i \(-0.470938\pi\)
0.0911752 + 0.995835i \(0.470938\pi\)
\(398\) 22.4222 1.12392
\(399\) −21.2111 −1.06188
\(400\) 0 0
\(401\) 8.78890i 0.438897i 0.975624 + 0.219448i \(0.0704257\pi\)
−0.975624 + 0.219448i \(0.929574\pi\)
\(402\) − 3.21110i − 0.160155i
\(403\) 21.6333 1.07763
\(404\) −7.39445 −0.367888
\(405\) 0 0
\(406\) 21.2111 1.05269
\(407\) 0 0
\(408\) 4.60555 0.228009
\(409\) 14.7889i 0.731264i 0.930760 + 0.365632i \(0.119147\pi\)
−0.930760 + 0.365632i \(0.880853\pi\)
\(410\) 0 0
\(411\) − 3.21110i − 0.158392i
\(412\) 4.00000i 0.197066i
\(413\) 42.4222i 2.08746i
\(414\) 1.39445i 0.0685334i
\(415\) 0 0
\(416\) 3.60555i 0.176777i
\(417\) − 17.2111i − 0.842831i
\(418\) 0 0
\(419\) −4.18335 −0.204370 −0.102185 0.994765i \(-0.532583\pi\)
−0.102185 + 0.994765i \(0.532583\pi\)
\(420\) 0 0
\(421\) − 19.8167i − 0.965805i −0.875674 0.482902i \(-0.839583\pi\)
0.875674 0.482902i \(-0.160417\pi\)
\(422\) 17.2111 0.837823
\(423\) −9.21110 −0.447859
\(424\) 6.00000i 0.291386i
\(425\) 0 0
\(426\) 9.21110 0.446279
\(427\) −51.6333 −2.49871
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000i 0.578020i 0.957326 + 0.289010i \(0.0933260\pi\)
−0.957326 + 0.289010i \(0.906674\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 19.2111i − 0.923227i −0.887081 0.461613i \(-0.847271\pi\)
0.887081 0.461613i \(-0.152729\pi\)
\(434\) − 27.6333i − 1.32644i
\(435\) 0 0
\(436\) − 1.39445i − 0.0667820i
\(437\) −6.42221 −0.307216
\(438\) 1.39445i 0.0666293i
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −14.2111 −0.676719
\(442\) −16.6056 −0.789846
\(443\) − 15.6333i − 0.742761i −0.928481 0.371380i \(-0.878885\pi\)
0.928481 0.371380i \(-0.121115\pi\)
\(444\) 9.21110i 0.437140i
\(445\) 0 0
\(446\) −1.81665 −0.0860211
\(447\) 15.2111 0.719460
\(448\) 4.60555 0.217592
\(449\) − 33.6333i − 1.58725i −0.608405 0.793627i \(-0.708190\pi\)
0.608405 0.793627i \(-0.291810\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 13.8167i 0.649881i
\(453\) −6.00000 −0.281905
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) 4.60555 0.215675
\(457\) −38.2389 −1.78874 −0.894369 0.447330i \(-0.852375\pi\)
−0.894369 + 0.447330i \(0.852375\pi\)
\(458\) − 19.8167i − 0.925971i
\(459\) 4.60555 0.214969
\(460\) 0 0
\(461\) − 33.6333i − 1.56646i −0.621733 0.783230i \(-0.713571\pi\)
0.621733 0.783230i \(-0.286429\pi\)
\(462\) 0 0
\(463\) −31.3944 −1.45902 −0.729512 0.683968i \(-0.760253\pi\)
−0.729512 + 0.683968i \(0.760253\pi\)
\(464\) −4.60555 −0.213807
\(465\) 0 0
\(466\) − 1.81665i − 0.0841549i
\(467\) − 30.4222i − 1.40777i −0.710313 0.703886i \(-0.751447\pi\)
0.710313 0.703886i \(-0.248553\pi\)
\(468\) 3.60555i 0.166667i
\(469\) 14.7889 0.682888
\(470\) 0 0
\(471\) −20.4222 −0.941006
\(472\) − 9.21110i − 0.423975i
\(473\) 0 0
\(474\) − 14.4222i − 0.662434i
\(475\) 0 0
\(476\) 21.2111i 0.972209i
\(477\) 6.00000i 0.274721i
\(478\) 0 0
\(479\) − 5.57779i − 0.254856i −0.991848 0.127428i \(-0.959328\pi\)
0.991848 0.127428i \(-0.0406722\pi\)
\(480\) 0 0
\(481\) − 33.2111i − 1.51430i
\(482\) 6.42221i 0.292523i
\(483\) −6.42221 −0.292220
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) − 1.00000i − 0.0453609i
\(487\) 0.972244 0.0440566 0.0220283 0.999757i \(-0.492988\pi\)
0.0220283 + 0.999757i \(0.492988\pi\)
\(488\) 11.2111 0.507503
\(489\) 24.4222i 1.10441i
\(490\) 0 0
\(491\) 7.81665 0.352761 0.176380 0.984322i \(-0.443561\pi\)
0.176380 + 0.984322i \(0.443561\pi\)
\(492\) −3.21110 −0.144768
\(493\) − 21.2111i − 0.955300i
\(494\) −16.6056 −0.747119
\(495\) 0 0
\(496\) 6.00000i 0.269408i
\(497\) 42.4222i 1.90290i
\(498\) − 2.78890i − 0.124973i
\(499\) − 23.0278i − 1.03086i −0.856930 0.515432i \(-0.827631\pi\)
0.856930 0.515432i \(-0.172369\pi\)
\(500\) 0 0
\(501\) − 9.21110i − 0.411522i
\(502\) 13.3944 0.597824
\(503\) − 23.4500i − 1.04558i −0.852461 0.522791i \(-0.824891\pi\)
0.852461 0.522791i \(-0.175109\pi\)
\(504\) 4.60555 0.205148
\(505\) 0 0
\(506\) 0 0
\(507\) − 13.0000i − 0.577350i
\(508\) 1.21110i 0.0537340i
\(509\) 33.6333i 1.49077i 0.666634 + 0.745385i \(0.267734\pi\)
−0.666634 + 0.745385i \(0.732266\pi\)
\(510\) 0 0
\(511\) −6.42221 −0.284102
\(512\) −1.00000 −0.0441942
\(513\) 4.60555 0.203340
\(514\) 28.6056i 1.26174i
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) −42.4222 −1.86392
\(519\) −12.4222 −0.545274
\(520\) 0 0
\(521\) 21.6333 0.947772 0.473886 0.880586i \(-0.342851\pi\)
0.473886 + 0.880586i \(0.342851\pi\)
\(522\) −4.60555 −0.201580
\(523\) − 32.8444i − 1.43619i −0.695947 0.718093i \(-0.745015\pi\)
0.695947 0.718093i \(-0.254985\pi\)
\(524\) 22.6056 0.987528
\(525\) 0 0
\(526\) − 7.81665i − 0.340822i
\(527\) −27.6333 −1.20373
\(528\) 0 0
\(529\) 21.0555 0.915457
\(530\) 0 0
\(531\) − 9.21110i − 0.399728i
\(532\) 21.2111i 0.919618i
\(533\) 11.5778 0.501490
\(534\) −15.2111 −0.658249
\(535\) 0 0
\(536\) −3.21110 −0.138699
\(537\) 19.8167i 0.855152i
\(538\) 25.8167 1.11303
\(539\) 0 0
\(540\) 0 0
\(541\) 6.97224i 0.299760i 0.988704 + 0.149880i \(0.0478888\pi\)
−0.988704 + 0.149880i \(0.952111\pi\)
\(542\) 0.422205i 0.0181353i
\(543\) 8.42221i 0.361431i
\(544\) − 4.60555i − 0.197461i
\(545\) 0 0
\(546\) −16.6056 −0.710652
\(547\) 14.4222i 0.616649i 0.951281 + 0.308324i \(0.0997682\pi\)
−0.951281 + 0.308324i \(0.900232\pi\)
\(548\) −3.21110 −0.137172
\(549\) 11.2111 0.478478
\(550\) 0 0
\(551\) − 21.2111i − 0.903623i
\(552\) 1.39445 0.0593517
\(553\) 66.4222 2.82456
\(554\) 16.4222i 0.697713i
\(555\) 0 0
\(556\) −17.2111 −0.729913
\(557\) −11.5778 −0.490567 −0.245283 0.969451i \(-0.578881\pi\)
−0.245283 + 0.969451i \(0.578881\pi\)
\(558\) 6.00000i 0.254000i
\(559\) −28.8444 −1.21999
\(560\) 0 0
\(561\) 0 0
\(562\) − 27.2111i − 1.14783i
\(563\) − 34.0555i − 1.43527i −0.696420 0.717634i \(-0.745225\pi\)
0.696420 0.717634i \(-0.254775\pi\)
\(564\) 9.21110i 0.387857i
\(565\) 0 0
\(566\) 10.4222i 0.438078i
\(567\) 4.60555 0.193415
\(568\) − 9.21110i − 0.386489i
\(569\) −33.6333 −1.40998 −0.704991 0.709216i \(-0.749049\pi\)
−0.704991 + 0.709216i \(0.749049\pi\)
\(570\) 0 0
\(571\) −30.0555 −1.25778 −0.628892 0.777493i \(-0.716491\pi\)
−0.628892 + 0.777493i \(0.716491\pi\)
\(572\) 0 0
\(573\) − 12.0000i − 0.501307i
\(574\) − 14.7889i − 0.617277i
\(575\) 0 0
\(576\) −1.00000 −0.0416667
\(577\) 37.3944 1.55675 0.778376 0.627799i \(-0.216044\pi\)
0.778376 + 0.627799i \(0.216044\pi\)
\(578\) 4.21110 0.175159
\(579\) 7.81665i 0.324849i
\(580\) 0 0
\(581\) 12.8444 0.532876
\(582\) 1.39445i 0.0578018i
\(583\) 0 0
\(584\) 1.39445 0.0577027
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) 6.42221 0.265073 0.132536 0.991178i \(-0.457688\pi\)
0.132536 + 0.991178i \(0.457688\pi\)
\(588\) 14.2111i 0.586056i
\(589\) −27.6333 −1.13861
\(590\) 0 0
\(591\) 6.00000i 0.246807i
\(592\) 9.21110 0.378574
\(593\) −24.4222 −1.00290 −0.501450 0.865187i \(-0.667200\pi\)
−0.501450 + 0.865187i \(0.667200\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 15.2111i − 0.623071i
\(597\) − 22.4222i − 0.917680i
\(598\) −5.02776 −0.205600
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 1.63331 0.0666240 0.0333120 0.999445i \(-0.489394\pi\)
0.0333120 + 0.999445i \(0.489394\pi\)
\(602\) 36.8444i 1.50167i
\(603\) −3.21110 −0.130766
\(604\) 6.00000i 0.244137i
\(605\) 0 0
\(606\) 7.39445i 0.300379i
\(607\) 17.2111i 0.698577i 0.937015 + 0.349289i \(0.113577\pi\)
−0.937015 + 0.349289i \(0.886423\pi\)
\(608\) − 4.60555i − 0.186780i
\(609\) − 21.2111i − 0.859517i
\(610\) 0 0
\(611\) − 33.2111i − 1.34358i
\(612\) − 4.60555i − 0.186168i
\(613\) −33.2111 −1.34138 −0.670692 0.741736i \(-0.734003\pi\)
−0.670692 + 0.741736i \(0.734003\pi\)
\(614\) −8.78890 −0.354691
\(615\) 0 0
\(616\) 0 0
\(617\) −12.4222 −0.500099 −0.250050 0.968233i \(-0.580447\pi\)
−0.250050 + 0.968233i \(0.580447\pi\)
\(618\) 4.00000 0.160904
\(619\) − 25.8167i − 1.03766i −0.854878 0.518829i \(-0.826368\pi\)
0.854878 0.518829i \(-0.173632\pi\)
\(620\) 0 0
\(621\) 1.39445 0.0559573
\(622\) −12.0000 −0.481156
\(623\) − 70.0555i − 2.80671i
\(624\) 3.60555 0.144338
\(625\) 0 0
\(626\) 3.57779i 0.142997i
\(627\) 0 0
\(628\) 20.4222i 0.814935i
\(629\) 42.4222i 1.69148i
\(630\) 0 0
\(631\) 3.21110i 0.127832i 0.997955 + 0.0639160i \(0.0203590\pi\)
−0.997955 + 0.0639160i \(0.979641\pi\)
\(632\) −14.4222 −0.573685
\(633\) − 17.2111i − 0.684080i
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) − 51.2389i − 2.03016i
\(638\) 0 0
\(639\) − 9.21110i − 0.364386i
\(640\) 0 0
\(641\) 0.422205 0.0166761 0.00833805 0.999965i \(-0.497346\pi\)
0.00833805 + 0.999965i \(0.497346\pi\)
\(642\) 0 0
\(643\) 9.63331 0.379901 0.189950 0.981794i \(-0.439167\pi\)
0.189950 + 0.981794i \(0.439167\pi\)
\(644\) 6.42221i 0.253070i
\(645\) 0 0
\(646\) 21.2111 0.834540
\(647\) − 34.6056i − 1.36048i −0.732987 0.680242i \(-0.761875\pi\)
0.732987 0.680242i \(-0.238125\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) −27.6333 −1.08303
\(652\) 24.4222 0.956447
\(653\) − 39.2111i − 1.53445i −0.641379 0.767225i \(-0.721637\pi\)
0.641379 0.767225i \(-0.278363\pi\)
\(654\) −1.39445 −0.0545273
\(655\) 0 0
\(656\) 3.21110i 0.125372i
\(657\) 1.39445 0.0544026
\(658\) −42.4222 −1.65379
\(659\) 26.2389 1.02212 0.511060 0.859545i \(-0.329253\pi\)
0.511060 + 0.859545i \(0.329253\pi\)
\(660\) 0 0
\(661\) 50.2389i 1.95407i 0.213090 + 0.977033i \(0.431647\pi\)
−0.213090 + 0.977033i \(0.568353\pi\)
\(662\) 16.6056i 0.645393i
\(663\) 16.6056i 0.644906i
\(664\) −2.78890 −0.108230
\(665\) 0 0
\(666\) 9.21110 0.356923
\(667\) − 6.42221i − 0.248669i
\(668\) −9.21110 −0.356388
\(669\) 1.81665i 0.0702359i
\(670\) 0 0
\(671\) 0 0
\(672\) − 4.60555i − 0.177663i
\(673\) 37.6333i 1.45066i 0.688403 + 0.725329i \(0.258312\pi\)
−0.688403 + 0.725329i \(0.741688\pi\)
\(674\) − 13.6333i − 0.525135i
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) − 28.0555i − 1.07826i −0.842222 0.539130i \(-0.818753\pi\)
0.842222 0.539130i \(-0.181247\pi\)
\(678\) 13.8167 0.530625
\(679\) −6.42221 −0.246462
\(680\) 0 0
\(681\) − 24.0000i − 0.919682i
\(682\) 0 0
\(683\) 9.21110 0.352453 0.176227 0.984350i \(-0.443611\pi\)
0.176227 + 0.984350i \(0.443611\pi\)
\(684\) − 4.60555i − 0.176098i
\(685\) 0 0
\(686\) −33.2111 −1.26801
\(687\) −19.8167 −0.756053
\(688\) − 8.00000i − 0.304997i
\(689\) −21.6333 −0.824163
\(690\) 0 0
\(691\) 20.2389i 0.769922i 0.922933 + 0.384961i \(0.125785\pi\)
−0.922933 + 0.384961i \(0.874215\pi\)
\(692\) 12.4222i 0.472221i
\(693\) 0 0
\(694\) 27.6333i 1.04895i
\(695\) 0 0
\(696\) 4.60555i 0.174573i
\(697\) −14.7889 −0.560169
\(698\) − 7.81665i − 0.295865i
\(699\) −1.81665 −0.0687122
\(700\) 0 0
\(701\) 47.0278 1.77621 0.888107 0.459637i \(-0.152020\pi\)
0.888107 + 0.459637i \(0.152020\pi\)
\(702\) 3.60555 0.136083
\(703\) 42.4222i 1.59998i
\(704\) 0 0
\(705\) 0 0
\(706\) 8.78890 0.330775
\(707\) −34.0555 −1.28079
\(708\) −9.21110 −0.346174
\(709\) 1.39445i 0.0523696i 0.999657 + 0.0261848i \(0.00833584\pi\)
−0.999657 + 0.0261848i \(0.991664\pi\)
\(710\) 0 0
\(711\) −14.4222 −0.540875
\(712\) 15.2111i 0.570060i
\(713\) −8.36669 −0.313335
\(714\) 21.2111 0.793806
\(715\) 0 0
\(716\) 19.8167 0.740583
\(717\) 0 0
\(718\) 15.6333i 0.583430i
\(719\) 51.6333 1.92560 0.962799 0.270220i \(-0.0870963\pi\)
0.962799 + 0.270220i \(0.0870963\pi\)
\(720\) 0 0
\(721\) 18.4222i 0.686079i
\(722\) 2.21110 0.0822887
\(723\) 6.42221 0.238844
\(724\) 8.42221 0.313009
\(725\) 0 0
\(726\) − 11.0000i − 0.408248i
\(727\) − 14.4222i − 0.534890i −0.963573 0.267445i \(-0.913821\pi\)
0.963573 0.267445i \(-0.0861794\pi\)
\(728\) 16.6056i 0.615443i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 36.8444 1.36274
\(732\) − 11.2111i − 0.414374i
\(733\) −34.0555 −1.25787 −0.628935 0.777458i \(-0.716509\pi\)
−0.628935 + 0.777458i \(0.716509\pi\)
\(734\) 19.6333i 0.724679i
\(735\) 0 0
\(736\) − 1.39445i − 0.0514001i
\(737\) 0 0
\(738\) 3.21110i 0.118202i
\(739\) − 20.2389i − 0.744498i −0.928133 0.372249i \(-0.878587\pi\)
0.928133 0.372249i \(-0.121413\pi\)
\(740\) 0 0
\(741\) 16.6056i 0.610020i
\(742\) 27.6333i 1.01445i
\(743\) 36.8444 1.35169 0.675845 0.737044i \(-0.263779\pi\)
0.675845 + 0.737044i \(0.263779\pi\)
\(744\) 6.00000 0.219971
\(745\) 0 0
\(746\) − 20.4222i − 0.747710i
\(747\) −2.78890 −0.102040
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −10.4222 −0.380312 −0.190156 0.981754i \(-0.560899\pi\)
−0.190156 + 0.981754i \(0.560899\pi\)
\(752\) 9.21110 0.335894
\(753\) − 13.3944i − 0.488121i
\(754\) − 16.6056i − 0.604739i
\(755\) 0 0
\(756\) − 4.60555i − 0.167502i
\(757\) 12.7889i 0.464820i 0.972618 + 0.232410i \(0.0746612\pi\)
−0.972618 + 0.232410i \(0.925339\pi\)
\(758\) 35.0278i 1.27227i
\(759\) 0 0
\(760\) 0 0
\(761\) 33.6333i 1.21921i 0.792707 + 0.609603i \(0.208671\pi\)
−0.792707 + 0.609603i \(0.791329\pi\)
\(762\) 1.21110 0.0438736
\(763\) − 6.42221i − 0.232499i
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 27.6333 0.998432
\(767\) 33.2111 1.19918
\(768\) 1.00000i 0.0360844i
\(769\) 12.8444i 0.463181i 0.972813 + 0.231591i \(0.0743930\pi\)
−0.972813 + 0.231591i \(0.925607\pi\)
\(770\) 0 0
\(771\) 28.6056 1.03020
\(772\) 7.81665 0.281328
\(773\) 30.0000 1.07903 0.539513 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) − 8.00000i − 0.287554i
\(775\) 0 0
\(776\) 1.39445 0.0500578
\(777\) 42.4222i 1.52189i
\(778\) 4.60555 0.165117
\(779\) −14.7889 −0.529867
\(780\) 0 0
\(781\) 0 0
\(782\) 6.42221 0.229658
\(783\) 4.60555i 0.164589i
\(784\) 14.2111 0.507539
\(785\) 0 0
\(786\) − 22.6056i − 0.806313i
\(787\) −49.2666 −1.75617 −0.878083 0.478509i \(-0.841177\pi\)
−0.878083 + 0.478509i \(0.841177\pi\)
\(788\) 6.00000 0.213741
\(789\) −7.81665 −0.278280
\(790\) 0 0
\(791\) 63.6333i 2.26254i
\(792\) 0 0
\(793\) 40.4222i 1.43543i
\(794\) −3.63331 −0.128941
\(795\) 0 0
\(796\) −22.4222 −0.794734
\(797\) − 6.00000i − 0.212531i −0.994338 0.106265i \(-0.966111\pi\)
0.994338 0.106265i \(-0.0338893\pi\)
\(798\) 21.2111 0.750865
\(799\) 42.4222i 1.50079i
\(800\) 0 0
\(801\) 15.2111i 0.537458i
\(802\) − 8.78890i − 0.310347i
\(803\) 0 0
\(804\) 3.21110i 0.113247i
\(805\) 0 0
\(806\) −21.6333 −0.762001
\(807\) − 25.8167i − 0.908789i
\(808\) 7.39445 0.260136
\(809\) 6.84441 0.240637 0.120318 0.992735i \(-0.461608\pi\)
0.120318 + 0.992735i \(0.461608\pi\)
\(810\) 0 0
\(811\) 32.2389i 1.13206i 0.824385 + 0.566030i \(0.191521\pi\)
−0.824385 + 0.566030i \(0.808479\pi\)
\(812\) −21.2111 −0.744364
\(813\) 0.422205 0.0148074
\(814\) 0 0
\(815\) 0 0
\(816\) −4.60555 −0.161227
\(817\) 36.8444 1.28902
\(818\) − 14.7889i − 0.517082i
\(819\) 16.6056i 0.580245i
\(820\) 0 0
\(821\) − 3.21110i − 0.112068i −0.998429 0.0560341i \(-0.982154\pi\)
0.998429 0.0560341i \(-0.0178456\pi\)
\(822\) 3.21110i 0.112000i
\(823\) 4.00000i 0.139431i 0.997567 + 0.0697156i \(0.0222092\pi\)
−0.997567 + 0.0697156i \(0.977791\pi\)
\(824\) − 4.00000i − 0.139347i
\(825\) 0 0
\(826\) − 42.4222i − 1.47606i
\(827\) 27.6333 0.960904 0.480452 0.877021i \(-0.340473\pi\)
0.480452 + 0.877021i \(0.340473\pi\)
\(828\) − 1.39445i − 0.0484604i
\(829\) −46.8444 −1.62697 −0.813487 0.581583i \(-0.802433\pi\)
−0.813487 + 0.581583i \(0.802433\pi\)
\(830\) 0 0
\(831\) 16.4222 0.569680
\(832\) − 3.60555i − 0.125000i
\(833\) 65.4500i 2.26771i
\(834\) 17.2111i 0.595972i
\(835\) 0 0
\(836\) 0 0
\(837\) 6.00000 0.207390
\(838\) 4.18335 0.144511
\(839\) 18.4222i 0.636005i 0.948090 + 0.318003i \(0.103012\pi\)
−0.948090 + 0.318003i \(0.896988\pi\)
\(840\) 0 0
\(841\) −7.78890 −0.268583
\(842\) 19.8167i 0.682927i
\(843\) −27.2111 −0.937200
\(844\) −17.2111 −0.592431
\(845\) 0 0
\(846\) 9.21110 0.316684
\(847\) 50.6611 1.74073
\(848\) − 6.00000i − 0.206041i
\(849\) 10.4222 0.357689
\(850\) 0 0
\(851\) 12.8444i 0.440301i
\(852\) −9.21110 −0.315567
\(853\) −14.7889 −0.506362 −0.253181 0.967419i \(-0.581477\pi\)
−0.253181 + 0.967419i \(0.581477\pi\)
\(854\) 51.6333 1.76686
\(855\) 0 0
\(856\) 0 0
\(857\) − 23.0278i − 0.786613i −0.919407 0.393307i \(-0.871331\pi\)
0.919407 0.393307i \(-0.128669\pi\)
\(858\) 0 0
\(859\) −25.2111 −0.860192 −0.430096 0.902783i \(-0.641520\pi\)
−0.430096 + 0.902783i \(0.641520\pi\)
\(860\) 0 0
\(861\) −14.7889 −0.504004
\(862\) − 12.0000i − 0.408722i
\(863\) 51.6333 1.75762 0.878809 0.477173i \(-0.158339\pi\)
0.878809 + 0.477173i \(0.158339\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 0 0
\(866\) 19.2111i 0.652820i
\(867\) − 4.21110i − 0.143017i
\(868\) 27.6333i 0.937936i
\(869\) 0 0
\(870\) 0 0
\(871\) − 11.5778i − 0.392299i
\(872\) 1.39445i 0.0472220i
\(873\) 1.39445 0.0471949
\(874\) 6.42221 0.217234
\(875\) 0 0
\(876\) − 1.39445i − 0.0471141i
\(877\) 24.8444 0.838936 0.419468 0.907770i \(-0.362217\pi\)
0.419468 + 0.907770i \(0.362217\pi\)
\(878\) 8.00000 0.269987
\(879\) 18.0000i 0.607125i
\(880\) 0 0
\(881\) 39.2111 1.32106 0.660528 0.750802i \(-0.270333\pi\)
0.660528 + 0.750802i \(0.270333\pi\)
\(882\) 14.2111 0.478513
\(883\) − 9.57779i − 0.322318i −0.986928 0.161159i \(-0.948477\pi\)
0.986928 0.161159i \(-0.0515233\pi\)
\(884\) 16.6056 0.558505
\(885\) 0 0
\(886\) 15.6333i 0.525211i
\(887\) 6.97224i 0.234105i 0.993126 + 0.117053i \(0.0373446\pi\)
−0.993126 + 0.117053i \(0.962655\pi\)
\(888\) − 9.21110i − 0.309104i
\(889\) 5.57779i 0.187073i
\(890\) 0 0
\(891\) 0 0
\(892\) 1.81665 0.0608261
\(893\) 42.4222i 1.41960i
\(894\) −15.2111 −0.508735
\(895\) 0 0
\(896\) −4.60555 −0.153861
\(897\) 5.02776i 0.167872i
\(898\) 33.6333i 1.12236i
\(899\) − 27.6333i − 0.921622i
\(900\) 0 0
\(901\) 27.6333 0.920599
\(902\) 0 0
\(903\) 36.8444 1.22611
\(904\) − 13.8167i − 0.459535i
\(905\) 0 0
\(906\) 6.00000 0.199337
\(907\) 21.5778i 0.716479i 0.933630 + 0.358239i \(0.116623\pi\)
−0.933630 + 0.358239i \(0.883377\pi\)
\(908\) −24.0000 −0.796468
\(909\) 7.39445 0.245258
\(910\) 0 0
\(911\) −27.6333 −0.915532 −0.457766 0.889073i \(-0.651350\pi\)
−0.457766 + 0.889073i \(0.651350\pi\)
\(912\) −4.60555 −0.152505
\(913\) 0 0
\(914\) 38.2389 1.26483
\(915\) 0 0
\(916\) 19.8167i 0.654761i
\(917\) 104.111 3.43805
\(918\) −4.60555 −0.152006
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 8.78890i 0.289604i
\(922\) 33.6333i 1.10765i
\(923\) 33.2111 1.09316
\(924\) 0 0
\(925\) 0 0
\(926\) 31.3944 1.03169
\(927\) − 4.00000i − 0.131377i
\(928\) 4.60555 0.151185
\(929\) − 39.2111i − 1.28647i −0.765667 0.643237i \(-0.777591\pi\)
0.765667 0.643237i \(-0.222409\pi\)
\(930\) 0 0
\(931\) 65.4500i 2.14504i
\(932\) 1.81665i 0.0595065i
\(933\) 12.0000i 0.392862i
\(934\) 30.4222i 0.995445i
\(935\) 0 0
\(936\) − 3.60555i − 0.117851i
\(937\) − 10.3667i − 0.338665i −0.985559 0.169333i \(-0.945839\pi\)
0.985559 0.169333i \(-0.0541612\pi\)
\(938\) −14.7889 −0.482875
\(939\) 3.57779 0.116757
\(940\) 0 0
\(941\) − 54.0000i − 1.76035i −0.474650 0.880175i \(-0.657425\pi\)
0.474650 0.880175i \(-0.342575\pi\)
\(942\) 20.4222 0.665391
\(943\) −4.47772 −0.145815
\(944\) 9.21110i 0.299796i
\(945\) 0 0
\(946\) 0 0
\(947\) 15.6333 0.508014 0.254007 0.967202i \(-0.418251\pi\)
0.254007 + 0.967202i \(0.418251\pi\)
\(948\) 14.4222i 0.468411i
\(949\) 5.02776i 0.163208i
\(950\) 0 0
\(951\) 18.0000i 0.583690i
\(952\) − 21.2111i − 0.687456i
\(953\) 20.2389i 0.655601i 0.944747 + 0.327800i \(0.106307\pi\)
−0.944747 + 0.327800i \(0.893693\pi\)
\(954\) − 6.00000i − 0.194257i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 5.57779i 0.180210i
\(959\) −14.7889 −0.477558
\(960\) 0 0
\(961\) −5.00000 −0.161290
\(962\) 33.2111i 1.07077i
\(963\) 0 0
\(964\) − 6.42221i − 0.206845i
\(965\) 0 0
\(966\) 6.42221 0.206631
\(967\) 8.23886 0.264944 0.132472 0.991187i \(-0.457709\pi\)
0.132472 + 0.991187i \(0.457709\pi\)
\(968\) −11.0000 −0.353553
\(969\) − 21.2111i − 0.681399i
\(970\) 0 0
\(971\) −53.0278 −1.70174 −0.850871 0.525375i \(-0.823925\pi\)
−0.850871 + 0.525375i \(0.823925\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) −79.2666 −2.54117
\(974\) −0.972244 −0.0311527
\(975\) 0 0
\(976\) −11.2111 −0.358859
\(977\) −18.8444 −0.602886 −0.301443 0.953484i \(-0.597468\pi\)
−0.301443 + 0.953484i \(0.597468\pi\)
\(978\) − 24.4222i − 0.780936i
\(979\) 0 0
\(980\) 0 0
\(981\) 1.39445i 0.0445213i
\(982\) −7.81665 −0.249439
\(983\) −42.4222 −1.35306 −0.676529 0.736416i \(-0.736517\pi\)
−0.676529 + 0.736416i \(0.736517\pi\)
\(984\) 3.21110 0.102366
\(985\) 0 0
\(986\) 21.2111i 0.675499i
\(987\) 42.4222i 1.35031i
\(988\) 16.6056 0.528293
\(989\) 11.1556 0.354727
\(990\) 0 0
\(991\) −22.4222 −0.712265 −0.356132 0.934436i \(-0.615905\pi\)
−0.356132 + 0.934436i \(0.615905\pi\)
\(992\) − 6.00000i − 0.190500i
\(993\) 16.6056 0.526961
\(994\) − 42.4222i − 1.34555i
\(995\) 0 0
\(996\) 2.78890i 0.0883696i
\(997\) − 16.4222i − 0.520096i −0.965596 0.260048i \(-0.916262\pi\)
0.965596 0.260048i \(-0.0837385\pi\)
\(998\) 23.0278i 0.728931i
\(999\) − 9.21110i − 0.291426i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1950.2.f.m.649.4 4
5.2 odd 4 1950.2.b.k.1351.2 4
5.3 odd 4 390.2.b.c.181.3 yes 4
5.4 even 2 1950.2.f.n.649.1 4
13.12 even 2 1950.2.f.n.649.3 4
15.8 even 4 1170.2.b.d.181.1 4
20.3 even 4 3120.2.g.q.961.4 4
65.8 even 4 5070.2.a.z.1.1 2
65.12 odd 4 1950.2.b.k.1351.3 4
65.18 even 4 5070.2.a.bf.1.2 2
65.38 odd 4 390.2.b.c.181.2 4
65.64 even 2 inner 1950.2.f.m.649.2 4
195.38 even 4 1170.2.b.d.181.4 4
260.103 even 4 3120.2.g.q.961.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.c.181.2 4 65.38 odd 4
390.2.b.c.181.3 yes 4 5.3 odd 4
1170.2.b.d.181.1 4 15.8 even 4
1170.2.b.d.181.4 4 195.38 even 4
1950.2.b.k.1351.2 4 5.2 odd 4
1950.2.b.k.1351.3 4 65.12 odd 4
1950.2.f.m.649.2 4 65.64 even 2 inner
1950.2.f.m.649.4 4 1.1 even 1 trivial
1950.2.f.n.649.1 4 5.4 even 2
1950.2.f.n.649.3 4 13.12 even 2
3120.2.g.q.961.1 4 260.103 even 4
3120.2.g.q.961.4 4 20.3 even 4
5070.2.a.z.1.1 2 65.8 even 4
5070.2.a.bf.1.2 2 65.18 even 4