Properties

Label 1950.2.b.k.1351.3
Level $1950$
Weight $2$
Character 1950.1351
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.b (of order \(2\), degree \(1\), 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 1351.3
Root \(-2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 1950.1351
Dual form 1950.2.b.k.1351.2

$q$-expansion

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000i 0.408248i
\(7\) − 4.60555i − 1.74073i −0.492403 0.870367i \(-0.663881\pi\)
0.492403 0.870367i \(-0.336119\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −1.00000 −0.288675
\(13\) −3.60555 −1.00000
\(14\) 4.60555 1.23089
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.60555 −1.11701 −0.558505 0.829501i \(-0.688625\pi\)
−0.558505 + 0.829501i \(0.688625\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 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
\(23\) 1.39445 0.290763 0.145381 0.989376i \(-0.453559\pi\)
0.145381 + 0.989376i \(0.453559\pi\)
\(24\) − 1.00000i − 0.204124i
\(25\) 0 0
\(26\) − 3.60555i − 0.707107i
\(27\) 1.00000 0.192450
\(28\) 4.60555i 0.870367i
\(29\) 4.60555 0.855229 0.427615 0.903961i \(-0.359354\pi\)
0.427615 + 0.903961i \(0.359354\pi\)
\(30\) 0 0
\(31\) − 6.00000i − 1.07763i −0.842424 0.538816i \(-0.818872\pi\)
0.842424 0.538816i \(-0.181128\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) − 4.60555i − 0.789846i
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) − 9.21110i − 1.51430i −0.653243 0.757148i \(-0.726592\pi\)
0.653243 0.757148i \(-0.273408\pi\)
\(38\) −4.60555 −0.747119
\(39\) −3.60555 −0.577350
\(40\) 0 0
\(41\) − 3.21110i − 0.501490i −0.968053 0.250745i \(-0.919324\pi\)
0.968053 0.250745i \(-0.0806756\pi\)
\(42\) 4.60555 0.710652
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.39445i 0.205600i
\(47\) − 9.21110i − 1.34358i −0.740743 0.671789i \(-0.765526\pi\)
0.740743 0.671789i \(-0.234474\pi\)
\(48\) 1.00000 0.144338
\(49\) −14.2111 −2.03016
\(50\) 0 0
\(51\) −4.60555 −0.644906
\(52\) 3.60555 0.500000
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 0 0
\(56\) −4.60555 −0.615443
\(57\) 4.60555i 0.610020i
\(58\) 4.60555i 0.604739i
\(59\) 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 \(0.754813\pi\)
\(62\) 6.00000 0.762001
\(63\) − 4.60555i − 0.580245i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.21110i − 0.392299i −0.980574 0.196149i \(-0.937156\pi\)
0.980574 0.196149i \(-0.0628437\pi\)
\(68\) 4.60555 0.558505
\(69\) 1.39445 0.167872
\(70\) 0 0
\(71\) − 9.21110i − 1.09316i −0.837408 0.546578i \(-0.815930\pi\)
0.837408 0.546578i \(-0.184070\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 1.39445i − 0.163208i −0.996665 0.0816039i \(-0.973996\pi\)
0.996665 0.0816039i \(-0.0260043\pi\)
\(74\) 9.21110 1.07077
\(75\) 0 0
\(76\) − 4.60555i − 0.528293i
\(77\) 0 0
\(78\) − 3.60555i − 0.408248i
\(79\) −14.4222 −1.62262 −0.811312 0.584613i \(-0.801246\pi\)
−0.811312 + 0.584613i \(0.801246\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 3.21110 0.354607
\(83\) 2.78890i 0.306121i 0.988217 + 0.153061i \(0.0489130\pi\)
−0.988217 + 0.153061i \(0.951087\pi\)
\(84\) 4.60555i 0.502507i
\(85\) 0 0
\(86\) − 8.00000i − 0.862662i
\(87\) 4.60555 0.493767
\(88\) 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
\(92\) −1.39445 −0.145381
\(93\) − 6.00000i − 0.622171i
\(94\) 9.21110 0.950053
\(95\) 0 0
\(96\) 1.00000i 0.102062i
\(97\) 1.39445i 0.141585i 0.997491 + 0.0707924i \(0.0225528\pi\)
−0.997491 + 0.0707924i \(0.977447\pi\)
\(98\) − 14.2111i − 1.43554i
\(99\) 0 0
\(100\) 0 0
\(101\) −7.39445 −0.735775 −0.367888 0.929870i \(-0.619919\pi\)
−0.367888 + 0.929870i \(0.619919\pi\)
\(102\) − 4.60555i − 0.456018i
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 3.60555i 0.353553i
\(105\) 0 0
\(106\) − 6.00000i − 0.582772i
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1.00000 −0.0962250
\(109\) − 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
\(112\) − 4.60555i − 0.435184i
\(113\) 13.8167 1.29976 0.649881 0.760036i \(-0.274819\pi\)
0.649881 + 0.760036i \(0.274819\pi\)
\(114\) −4.60555 −0.431349
\(115\) 0 0
\(116\) −4.60555 −0.427615
\(117\) −3.60555 −0.333333
\(118\) −9.21110 −0.847951
\(119\) 21.2111i 1.94442i
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) − 11.2111i − 1.01501i
\(123\) − 3.21110i − 0.289535i
\(124\) 6.00000i 0.538816i
\(125\) 0 0
\(126\) 4.60555 0.410295
\(127\) −1.21110 −0.107468 −0.0537340 0.998555i \(-0.517112\pi\)
−0.0537340 + 0.998555i \(0.517112\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −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
\(133\) 21.2111 1.83924
\(134\) 3.21110 0.277397
\(135\) 0 0
\(136\) 4.60555i 0.394923i
\(137\) 3.21110i 0.274343i 0.990547 + 0.137172i \(0.0438011\pi\)
−0.990547 + 0.137172i \(0.956199\pi\)
\(138\) 1.39445i 0.118703i
\(139\) 17.2111 1.45983 0.729913 0.683540i \(-0.239560\pi\)
0.729913 + 0.683540i \(0.239560\pi\)
\(140\) 0 0
\(141\) − 9.21110i − 0.775715i
\(142\) 9.21110 0.772979
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 1.39445 0.115405
\(147\) −14.2111 −1.17211
\(148\) 9.21110i 0.757148i
\(149\) − 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 \(-0.921495\pi\)
0.969741 0.244137i \(-0.0785045\pi\)
\(152\) 4.60555 0.373560
\(153\) −4.60555 −0.372337
\(154\) 0 0
\(155\) 0 0
\(156\) 3.60555 0.288675
\(157\) −20.4222 −1.62987 −0.814935 0.579553i \(-0.803227\pi\)
−0.814935 + 0.579553i \(0.803227\pi\)
\(158\) − 14.4222i − 1.14737i
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) − 6.42221i − 0.506141i
\(162\) 1.00000i 0.0785674i
\(163\) 24.4222i 1.91289i 0.291905 + 0.956447i \(0.405711\pi\)
−0.291905 + 0.956447i \(0.594289\pi\)
\(164\) 3.21110i 0.250745i
\(165\) 0 0
\(166\) −2.78890 −0.216460
\(167\) 9.21110i 0.712777i 0.934338 + 0.356388i \(0.115992\pi\)
−0.934338 + 0.356388i \(0.884008\pi\)
\(168\) −4.60555 −0.355326
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 4.60555i 0.352195i
\(172\) 8.00000 0.609994
\(173\) 12.4222 0.944443 0.472221 0.881480i \(-0.343452\pi\)
0.472221 + 0.881480i \(0.343452\pi\)
\(174\) 4.60555i 0.349146i
\(175\) 0 0
\(176\) 0 0
\(177\) 9.21110i 0.692349i
\(178\) 15.2111 1.14012
\(179\) −19.8167 −1.48117 −0.740583 0.671965i \(-0.765451\pi\)
−0.740583 + 0.671965i \(0.765451\pi\)
\(180\) 0 0
\(181\) 8.42221 0.626018 0.313009 0.949750i \(-0.398663\pi\)
0.313009 + 0.949750i \(0.398663\pi\)
\(182\) −16.6056 −1.23089
\(183\) −11.2111 −0.828749
\(184\) − 1.39445i − 0.102800i
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) 0 0
\(188\) 9.21110i 0.671789i
\(189\) − 4.60555i − 0.335005i
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 7.81665i 0.562655i 0.959612 + 0.281328i \(0.0907747\pi\)
−0.959612 + 0.281328i \(0.909225\pi\)
\(194\) −1.39445 −0.100116
\(195\) 0 0
\(196\) 14.2111 1.01508
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 0 0
\(199\) 22.4222 1.58947 0.794734 0.606958i \(-0.207610\pi\)
0.794734 + 0.606958i \(0.207610\pi\)
\(200\) 0 0
\(201\) − 3.21110i − 0.226494i
\(202\) − 7.39445i − 0.520272i
\(203\) − 21.2111i − 1.48873i
\(204\) 4.60555 0.322453
\(205\) 0 0
\(206\) 4.00000i 0.278693i
\(207\) 1.39445 0.0969209
\(208\) −3.60555 −0.250000
\(209\) 0 0
\(210\) 0 0
\(211\) −17.2111 −1.18486 −0.592431 0.805622i \(-0.701832\pi\)
−0.592431 + 0.805622i \(0.701832\pi\)
\(212\) 6.00000 0.412082
\(213\) − 9.21110i − 0.631134i
\(214\) 0 0
\(215\) 0 0
\(216\) − 1.00000i − 0.0680414i
\(217\) −27.6333 −1.87587
\(218\) 1.39445 0.0944440
\(219\) − 1.39445i − 0.0942281i
\(220\) 0 0
\(221\) 16.6056 1.11701
\(222\) 9.21110 0.618209
\(223\) 1.81665i 0.121652i 0.998148 + 0.0608261i \(0.0193735\pi\)
−0.998148 + 0.0608261i \(0.980627\pi\)
\(224\) 4.60555 0.307721
\(225\) 0 0
\(226\) 13.8167i 0.919070i
\(227\) 24.0000i 1.59294i 0.604681 + 0.796468i \(0.293301\pi\)
−0.604681 + 0.796468i \(0.706699\pi\)
\(228\) − 4.60555i − 0.305010i
\(229\) 19.8167i 1.30952i 0.755836 + 0.654761i \(0.227231\pi\)
−0.755836 + 0.654761i \(0.772769\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 4.60555i − 0.302369i
\(233\) 1.81665 0.119013 0.0595065 0.998228i \(-0.481047\pi\)
0.0595065 + 0.998228i \(0.481047\pi\)
\(234\) − 3.60555i − 0.235702i
\(235\) 0 0
\(236\) − 9.21110i − 0.599592i
\(237\) −14.4222 −0.936823
\(238\) −21.2111 −1.37491
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 6.42221i 0.413691i 0.978374 + 0.206845i \(0.0663197\pi\)
−0.978374 + 0.206845i \(0.933680\pi\)
\(242\) 11.0000i 0.707107i
\(243\) 1.00000 0.0641500
\(244\) 11.2111 0.717717
\(245\) 0 0
\(246\) 3.21110 0.204732
\(247\) − 16.6056i − 1.05659i
\(248\) −6.00000 −0.381000
\(249\) 2.78890i 0.176739i
\(250\) 0 0
\(251\) −13.3944 −0.845450 −0.422725 0.906258i \(-0.638926\pi\)
−0.422725 + 0.906258i \(0.638926\pi\)
\(252\) 4.60555i 0.290122i
\(253\) 0 0
\(254\) − 1.21110i − 0.0759913i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 28.6056 1.78437 0.892183 0.451675i \(-0.149173\pi\)
0.892183 + 0.451675i \(0.149173\pi\)
\(258\) − 8.00000i − 0.498058i
\(259\) −42.4222 −2.63599
\(260\) 0 0
\(261\) 4.60555 0.285076
\(262\) 22.6056i 1.39658i
\(263\) 7.81665 0.481996 0.240998 0.970526i \(-0.422525\pi\)
0.240998 + 0.970526i \(0.422525\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 21.2111i 1.30054i
\(267\) − 15.2111i − 0.930904i
\(268\) 3.21110i 0.196149i
\(269\) 25.8167 1.57407 0.787035 0.616909i \(-0.211615\pi\)
0.787035 + 0.616909i \(0.211615\pi\)
\(270\) 0 0
\(271\) 0.422205i 0.0256471i 0.999918 + 0.0128236i \(0.00408198\pi\)
−0.999918 + 0.0128236i \(0.995918\pi\)
\(272\) −4.60555 −0.279253
\(273\) 16.6056i 1.00501i
\(274\) −3.21110 −0.193990
\(275\) 0 0
\(276\) −1.39445 −0.0839359
\(277\) 16.4222 0.986715 0.493357 0.869827i \(-0.335770\pi\)
0.493357 + 0.869827i \(0.335770\pi\)
\(278\) 17.2111i 1.03225i
\(279\) − 6.00000i − 0.359211i
\(280\) 0 0
\(281\) − 27.2111i − 1.62328i −0.584159 0.811639i \(-0.698576\pi\)
0.584159 0.811639i \(-0.301424\pi\)
\(282\) 9.21110 0.548513
\(283\) −10.4222 −0.619536 −0.309768 0.950812i \(-0.600251\pi\)
−0.309768 + 0.950812i \(0.600251\pi\)
\(284\) 9.21110i 0.546578i
\(285\) 0 0
\(286\) 0 0
\(287\) −14.7889 −0.872961
\(288\) 1.00000i 0.0589256i
\(289\) 4.21110 0.247712
\(290\) 0 0
\(291\) 1.39445i 0.0817440i
\(292\) 1.39445i 0.0816039i
\(293\) 18.0000i 1.05157i 0.850617 + 0.525786i \(0.176229\pi\)
−0.850617 + 0.525786i \(0.823771\pi\)
\(294\) − 14.2111i − 0.828808i
\(295\) 0 0
\(296\) −9.21110 −0.535384
\(297\) 0 0
\(298\) 15.2111 0.881156
\(299\) −5.02776 −0.290763
\(300\) 0 0
\(301\) 36.8444i 2.12368i
\(302\) 6.00000 0.345261
\(303\) −7.39445 −0.424800
\(304\) 4.60555i 0.264146i
\(305\) 0 0
\(306\) − 4.60555i − 0.263282i
\(307\) − 8.78890i − 0.501609i −0.968038 0.250804i \(-0.919305\pi\)
0.968038 0.250804i \(-0.0806951\pi\)
\(308\) 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 \(0.389495\pi\)
\(312\) 3.60555i 0.204124i
\(313\) −3.57779 −0.202229 −0.101114 0.994875i \(-0.532241\pi\)
−0.101114 + 0.994875i \(0.532241\pi\)
\(314\) − 20.4222i − 1.15249i
\(315\) 0 0
\(316\) 14.4222 0.811312
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 6.42221 0.357895
\(323\) − 21.2111i − 1.18022i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −24.4222 −1.35262
\(327\) − 1.39445i − 0.0771132i
\(328\) −3.21110 −0.177303
\(329\) −42.4222 −2.33881
\(330\) 0 0
\(331\) 16.6056i 0.912724i 0.889794 + 0.456362i \(0.150848\pi\)
−0.889794 + 0.456362i \(0.849152\pi\)
\(332\) − 2.78890i − 0.153061i
\(333\) − 9.21110i − 0.504765i
\(334\) −9.21110 −0.504009
\(335\) 0 0
\(336\) − 4.60555i − 0.251253i
\(337\) −13.6333 −0.742654 −0.371327 0.928502i \(-0.621097\pi\)
−0.371327 + 0.928502i \(0.621097\pi\)
\(338\) 13.0000i 0.707107i
\(339\) 13.8167 0.750418
\(340\) 0 0
\(341\) 0 0
\(342\) −4.60555 −0.249040
\(343\) 33.2111i 1.79323i
\(344\) 8.00000i 0.431331i
\(345\) 0 0
\(346\) 12.4222i 0.667822i
\(347\) 27.6333 1.48343 0.741717 0.670713i \(-0.234012\pi\)
0.741717 + 0.670713i \(0.234012\pi\)
\(348\) −4.60555 −0.246883
\(349\) 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
\(353\) − 8.78890i − 0.467786i −0.972262 0.233893i \(-0.924853\pi\)
0.972262 0.233893i \(-0.0751465\pi\)
\(354\) −9.21110 −0.489565
\(355\) 0 0
\(356\) 15.2111i 0.806187i
\(357\) 21.2111i 1.12261i
\(358\) − 19.8167i − 1.04734i
\(359\) − 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
\(362\) 8.42221i 0.442661i
\(363\) 11.0000 0.577350
\(364\) − 16.6056i − 0.870367i
\(365\) 0 0
\(366\) − 11.2111i − 0.586014i
\(367\) 19.6333 1.02485 0.512425 0.858732i \(-0.328747\pi\)
0.512425 + 0.858732i \(0.328747\pi\)
\(368\) 1.39445 0.0726907
\(369\) − 3.21110i − 0.167163i
\(370\) 0 0
\(371\) 27.6333i 1.43465i
\(372\) 6.00000i 0.311086i
\(373\) 20.4222 1.05742 0.528711 0.848802i \(-0.322676\pi\)
0.528711 + 0.848802i \(0.322676\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −9.21110 −0.475026
\(377\) −16.6056 −0.855229
\(378\) 4.60555 0.236884
\(379\) − 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
\(382\) − 12.0000i − 0.613973i
\(383\) − 27.6333i − 1.41200i −0.708214 0.705998i \(-0.750499\pi\)
0.708214 0.705998i \(-0.249501\pi\)
\(384\) − 1.00000i − 0.0510310i
\(385\) 0 0
\(386\) −7.81665 −0.397857
\(387\) −8.00000 −0.406663
\(388\) − 1.39445i − 0.0707924i
\(389\) 4.60555 0.233511 0.116755 0.993161i \(-0.462751\pi\)
0.116755 + 0.993161i \(0.462751\pi\)
\(390\) 0 0
\(391\) −6.42221 −0.324785
\(392\) 14.2111i 0.717769i
\(393\) 22.6056 1.14030
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) − 3.63331i − 0.182350i −0.995835 0.0911752i \(-0.970938\pi\)
0.995835 0.0911752i \(-0.0290623\pi\)
\(398\) 22.4222i 1.12392i
\(399\) 21.2111 1.06188
\(400\) 0 0
\(401\) − 8.78890i − 0.438897i −0.975624 0.219448i \(-0.929574\pi\)
0.975624 0.219448i \(-0.0704257\pi\)
\(402\) 3.21110 0.160155
\(403\) 21.6333i 1.07763i
\(404\) 7.39445 0.367888
\(405\) 0 0
\(406\) 21.2111 1.05269
\(407\) 0 0
\(408\) 4.60555i 0.228009i
\(409\) 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
\(412\) −4.00000 −0.197066
\(413\) 42.4222 2.08746
\(414\) 1.39445i 0.0685334i
\(415\) 0 0
\(416\) − 3.60555i − 0.176777i
\(417\) 17.2111 0.842831
\(418\) 0 0
\(419\) 4.18335 0.204370 0.102185 0.994765i \(-0.467417\pi\)
0.102185 + 0.994765i \(0.467417\pi\)
\(420\) 0 0
\(421\) 19.8167i 0.965805i 0.875674 + 0.482902i \(0.160417\pi\)
−0.875674 + 0.482902i \(0.839583\pi\)
\(422\) − 17.2111i − 0.837823i
\(423\) − 9.21110i − 0.447859i
\(424\) 6.00000i 0.291386i
\(425\) 0 0
\(426\) 9.21110 0.446279
\(427\) 51.6333i 2.49871i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 12.0000i − 0.578020i −0.957326 0.289010i \(-0.906674\pi\)
0.957326 0.289010i \(-0.0933260\pi\)
\(432\) 1.00000 0.0481125
\(433\) −19.2111 −0.923227 −0.461613 0.887081i \(-0.652729\pi\)
−0.461613 + 0.887081i \(0.652729\pi\)
\(434\) − 27.6333i − 1.32644i
\(435\) 0 0
\(436\) 1.39445i 0.0667820i
\(437\) 6.42221i 0.307216i
\(438\) 1.39445 0.0666293
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −14.2111 −0.676719
\(442\) 16.6056i 0.789846i
\(443\) −15.6333 −0.742761 −0.371380 0.928481i \(-0.621115\pi\)
−0.371380 + 0.928481i \(0.621115\pi\)
\(444\) 9.21110i 0.437140i
\(445\) 0 0
\(446\) −1.81665 −0.0860211
\(447\) − 15.2111i − 0.719460i
\(448\) 4.60555i 0.217592i
\(449\) − 33.6333i − 1.58725i −0.608405 0.793627i \(-0.708190\pi\)
0.608405 0.793627i \(-0.291810\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −13.8167 −0.649881
\(453\) − 6.00000i − 0.281905i
\(454\) −24.0000 −1.12638
\(455\) 0 0
\(456\) 4.60555 0.215675
\(457\) 38.2389i 1.78874i 0.447330 + 0.894369i \(0.352375\pi\)
−0.447330 + 0.894369i \(0.647625\pi\)
\(458\) −19.8167 −0.925971
\(459\) −4.60555 −0.214969
\(460\) 0 0
\(461\) 33.6333i 1.56646i 0.621733 + 0.783230i \(0.286429\pi\)
−0.621733 + 0.783230i \(0.713571\pi\)
\(462\) 0 0
\(463\) − 31.3944i − 1.45902i −0.683968 0.729512i \(-0.739747\pi\)
0.683968 0.729512i \(-0.260253\pi\)
\(464\) 4.60555 0.213807
\(465\) 0 0
\(466\) 1.81665i 0.0841549i
\(467\) 30.4222 1.40777 0.703886 0.710313i \(-0.251447\pi\)
0.703886 + 0.710313i \(0.251447\pi\)
\(468\) 3.60555 0.166667
\(469\) −14.7889 −0.682888
\(470\) 0 0
\(471\) −20.4222 −0.941006
\(472\) 9.21110 0.423975
\(473\) 0 0
\(474\) − 14.4222i − 0.662434i
\(475\) 0 0
\(476\) − 21.2111i − 0.972209i
\(477\) −6.00000 −0.274721
\(478\) 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
\(482\) −6.42221 −0.292523
\(483\) − 6.42221i − 0.292220i
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) 1.00000i 0.0453609i
\(487\) − 0.972244i − 0.0440566i −0.999757 0.0220283i \(-0.992988\pi\)
0.999757 0.0220283i \(-0.00701239\pi\)
\(488\) 11.2111i 0.507503i
\(489\) 24.4222i 1.10441i
\(490\) 0 0
\(491\) 7.81665 0.352761 0.176380 0.984322i \(-0.443561\pi\)
0.176380 + 0.984322i \(0.443561\pi\)
\(492\) 3.21110i 0.144768i
\(493\) −21.2111 −0.955300
\(494\) 16.6056 0.747119
\(495\) 0 0
\(496\) − 6.00000i − 0.269408i
\(497\) −42.4222 −1.90290
\(498\) −2.78890 −0.124973
\(499\) − 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
\(502\) − 13.3944i − 0.597824i
\(503\) −23.4500 −1.04558 −0.522791 0.852461i \(-0.675109\pi\)
−0.522791 + 0.852461i \(0.675109\pi\)
\(504\) −4.60555 −0.205148
\(505\) 0 0
\(506\) 0 0
\(507\) 13.0000 0.577350
\(508\) 1.21110 0.0537340
\(509\) 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
\(512\) 1.00000i 0.0441942i
\(513\) 4.60555i 0.203340i
\(514\) 28.6056i 1.26174i
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 0 0
\(518\) − 42.4222i − 1.86392i
\(519\) 12.4222 0.545274
\(520\) 0 0
\(521\) 21.6333 0.947772 0.473886 0.880586i \(-0.342851\pi\)
0.473886 + 0.880586i \(0.342851\pi\)
\(522\) 4.60555i 0.201580i
\(523\) −32.8444 −1.43619 −0.718093 0.695947i \(-0.754985\pi\)
−0.718093 + 0.695947i \(0.754985\pi\)
\(524\) −22.6056 −0.987528
\(525\) 0 0
\(526\) 7.81665i 0.340822i
\(527\) 27.6333i 1.20373i
\(528\) 0 0
\(529\) −21.0555 −0.915457
\(530\) 0 0
\(531\) 9.21110i 0.399728i
\(532\) −21.2111 −0.919618
\(533\) 11.5778i 0.501490i
\(534\) 15.2111 0.658249
\(535\) 0 0
\(536\) −3.21110 −0.138699
\(537\) −19.8167 −0.855152
\(538\) 25.8167i 1.11303i
\(539\) 0 0
\(540\) 0 0
\(541\) − 6.97224i − 0.299760i −0.988704 0.149880i \(-0.952111\pi\)
0.988704 0.149880i \(-0.0478888\pi\)
\(542\) −0.422205 −0.0181353
\(543\) 8.42221 0.361431
\(544\) − 4.60555i − 0.197461i
\(545\) 0 0
\(546\) −16.6056 −0.710652
\(547\) −14.4222 −0.616649 −0.308324 0.951281i \(-0.599768\pi\)
−0.308324 + 0.951281i \(0.599768\pi\)
\(548\) − 3.21110i − 0.137172i
\(549\) −11.2111 −0.478478
\(550\) 0 0
\(551\) 21.2111i 0.903623i
\(552\) − 1.39445i − 0.0593517i
\(553\) 66.4222i 2.82456i
\(554\) 16.4222i 0.697713i
\(555\) 0 0
\(556\) −17.2111 −0.729913
\(557\) 11.5778i 0.490567i 0.969451 + 0.245283i \(0.0788810\pi\)
−0.969451 + 0.245283i \(0.921119\pi\)
\(558\) 6.00000 0.254000
\(559\) 28.8444 1.21999
\(560\) 0 0
\(561\) 0 0
\(562\) 27.2111 1.14783
\(563\) −34.0555 −1.43527 −0.717634 0.696420i \(-0.754775\pi\)
−0.717634 + 0.696420i \(0.754775\pi\)
\(564\) 9.21110i 0.387857i
\(565\) 0 0
\(566\) − 10.4222i − 0.438078i
\(567\) − 4.60555i − 0.193415i
\(568\) −9.21110 −0.386489
\(569\) 33.6333 1.40998 0.704991 0.709216i \(-0.250951\pi\)
0.704991 + 0.709216i \(0.250951\pi\)
\(570\) 0 0
\(571\) −30.0555 −1.25778 −0.628892 0.777493i \(-0.716491\pi\)
−0.628892 + 0.777493i \(0.716491\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) − 14.7889i − 0.617277i
\(575\) 0 0
\(576\) −1.00000 −0.0416667
\(577\) − 37.3944i − 1.55675i −0.627799 0.778376i \(-0.716044\pi\)
0.627799 0.778376i \(-0.283956\pi\)
\(578\) 4.21110i 0.175159i
\(579\) 7.81665i 0.324849i
\(580\) 0 0
\(581\) 12.8444 0.532876
\(582\) −1.39445 −0.0578018
\(583\) 0 0
\(584\) −1.39445 −0.0577027
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) − 6.42221i − 0.265073i −0.991178 0.132536i \(-0.957688\pi\)
0.991178 0.132536i \(-0.0423121\pi\)
\(588\) 14.2111 0.586056
\(589\) 27.6333 1.13861
\(590\) 0 0
\(591\) − 6.00000i − 0.246807i
\(592\) − 9.21110i − 0.378574i
\(593\) − 24.4222i − 1.00290i −0.865187 0.501450i \(-0.832800\pi\)
0.865187 0.501450i \(-0.167200\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.2111i 0.623071i
\(597\) 22.4222 0.917680
\(598\) − 5.02776i − 0.205600i
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 1.63331 0.0666240 0.0333120 0.999445i \(-0.489394\pi\)
0.0333120 + 0.999445i \(0.489394\pi\)
\(602\) −36.8444 −1.50167
\(603\) − 3.21110i − 0.130766i
\(604\) 6.00000i 0.244137i
\(605\) 0 0
\(606\) − 7.39445i − 0.300379i
\(607\) −17.2111 −0.698577 −0.349289 0.937015i \(-0.613577\pi\)
−0.349289 + 0.937015i \(0.613577\pi\)
\(608\) −4.60555 −0.186780
\(609\) − 21.2111i − 0.859517i
\(610\) 0 0
\(611\) 33.2111i 1.34358i
\(612\) 4.60555 0.186168
\(613\) − 33.2111i − 1.34138i −0.741736 0.670692i \(-0.765997\pi\)
0.741736 0.670692i \(-0.234003\pi\)
\(614\) 8.78890 0.354691
\(615\) 0 0
\(616\) 0 0
\(617\) 12.4222i 0.500099i 0.968233 + 0.250050i \(0.0804469\pi\)
−0.968233 + 0.250050i \(0.919553\pi\)
\(618\) 4.00000i 0.160904i
\(619\) − 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
\(622\) 12.0000i 0.481156i
\(623\) −70.0555 −2.80671
\(624\) −3.60555 −0.144338
\(625\) 0 0
\(626\) − 3.57779i − 0.142997i
\(627\) 0 0
\(628\) 20.4222 0.814935
\(629\) 42.4222i 1.69148i
\(630\) 0 0
\(631\) − 3.21110i − 0.127832i −0.997955 0.0639160i \(-0.979641\pi\)
0.997955 0.0639160i \(-0.0203590\pi\)
\(632\) 14.4222i 0.573685i
\(633\) −17.2111 −0.684080
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 51.2389 2.03016
\(638\) 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
\(643\) 9.63331i 0.379901i 0.981794 + 0.189950i \(0.0608327\pi\)
−0.981794 + 0.189950i \(0.939167\pi\)
\(644\) 6.42221i 0.253070i
\(645\) 0 0
\(646\) 21.2111 0.834540
\(647\) 34.6056 1.36048 0.680242 0.732987i \(-0.261875\pi\)
0.680242 + 0.732987i \(0.261875\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) −27.6333 −1.08303
\(652\) − 24.4222i − 0.956447i
\(653\) −39.2111 −1.53445 −0.767225 0.641379i \(-0.778363\pi\)
−0.767225 + 0.641379i \(0.778363\pi\)
\(654\) 1.39445 0.0545273
\(655\) 0 0
\(656\) − 3.21110i − 0.125372i
\(657\) − 1.39445i − 0.0544026i
\(658\) − 42.4222i − 1.65379i
\(659\) −26.2389 −1.02212 −0.511060 0.859545i \(-0.670747\pi\)
−0.511060 + 0.859545i \(0.670747\pi\)
\(660\) 0 0
\(661\) − 50.2389i − 1.95407i −0.213090 0.977033i \(-0.568353\pi\)
0.213090 0.977033i \(-0.431647\pi\)
\(662\) −16.6056 −0.645393
\(663\) 16.6056 0.644906
\(664\) 2.78890 0.108230
\(665\) 0 0
\(666\) 9.21110 0.356923
\(667\) 6.42221 0.248669
\(668\) − 9.21110i − 0.356388i
\(669\) 1.81665i 0.0702359i
\(670\) 0 0
\(671\) 0 0
\(672\) 4.60555 0.177663
\(673\) 37.6333 1.45066 0.725329 0.688403i \(-0.241688\pi\)
0.725329 + 0.688403i \(0.241688\pi\)
\(674\) − 13.6333i − 0.525135i
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 28.0555 1.07826 0.539130 0.842222i \(-0.318753\pi\)
0.539130 + 0.842222i \(0.318753\pi\)
\(678\) 13.8167i 0.530625i
\(679\) 6.42221 0.246462
\(680\) 0 0
\(681\) 24.0000i 0.919682i
\(682\) 0 0
\(683\) 9.21110i 0.352453i 0.984350 + 0.176227i \(0.0563891\pi\)
−0.984350 + 0.176227i \(0.943611\pi\)
\(684\) − 4.60555i − 0.176098i
\(685\) 0 0
\(686\) −33.2111 −1.26801
\(687\) 19.8167i 0.756053i
\(688\) −8.00000 −0.304997
\(689\) 21.6333 0.824163
\(690\) 0 0
\(691\) − 20.2389i − 0.769922i −0.922933 0.384961i \(-0.874215\pi\)
0.922933 0.384961i \(-0.125785\pi\)
\(692\) −12.4222 −0.472221
\(693\) 0 0
\(694\) 27.6333i 1.04895i
\(695\) 0 0
\(696\) − 4.60555i − 0.174573i
\(697\) 14.7889i 0.560169i
\(698\) −7.81665 −0.295865
\(699\) 1.81665 0.0687122
\(700\) 0 0
\(701\) 47.0278 1.77621 0.888107 0.459637i \(-0.152020\pi\)
0.888107 + 0.459637i \(0.152020\pi\)
\(702\) − 3.60555i − 0.136083i
\(703\) 42.4222 1.59998
\(704\) 0 0
\(705\) 0 0
\(706\) 8.78890 0.330775
\(707\) 34.0555i 1.28079i
\(708\) − 9.21110i − 0.346174i
\(709\) 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
\(712\) −15.2111 −0.570060
\(713\) − 8.36669i − 0.313335i
\(714\) −21.2111 −0.793806
\(715\) 0 0
\(716\) 19.8167 0.740583
\(717\) 0 0
\(718\) 15.6333 0.583430
\(719\) −51.6333 −1.92560 −0.962799 0.270220i \(-0.912904\pi\)
−0.962799 + 0.270220i \(0.912904\pi\)
\(720\) 0 0
\(721\) − 18.4222i − 0.686079i
\(722\) − 2.21110i − 0.0822887i
\(723\) 6.42221i 0.238844i
\(724\) −8.42221 −0.313009
\(725\) 0 0
\(726\) 11.0000i 0.408248i
\(727\) 14.4222 0.534890 0.267445 0.963573i \(-0.413821\pi\)
0.267445 + 0.963573i \(0.413821\pi\)
\(728\) 16.6056 0.615443
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 36.8444 1.36274
\(732\) 11.2111 0.414374
\(733\) − 34.0555i − 1.25787i −0.777458 0.628935i \(-0.783491\pi\)
0.777458 0.628935i \(-0.216509\pi\)
\(734\) 19.6333i 0.724679i
\(735\) 0 0
\(736\) 1.39445i 0.0514001i
\(737\) 0 0
\(738\) 3.21110 0.118202
\(739\) − 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
\(742\) −27.6333 −1.01445
\(743\) 36.8444i 1.35169i 0.737044 + 0.675845i \(0.236221\pi\)
−0.737044 + 0.675845i \(0.763779\pi\)
\(744\) −6.00000 −0.219971
\(745\) 0 0
\(746\) 20.4222i 0.747710i
\(747\) 2.78890i 0.102040i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −10.4222 −0.380312 −0.190156 0.981754i \(-0.560899\pi\)
−0.190156 + 0.981754i \(0.560899\pi\)
\(752\) − 9.21110i − 0.335894i
\(753\) −13.3944 −0.488121
\(754\) − 16.6056i − 0.604739i
\(755\) 0 0
\(756\) 4.60555i 0.167502i
\(757\) −12.7889 −0.464820 −0.232410 0.972618i \(-0.574661\pi\)
−0.232410 + 0.972618i \(0.574661\pi\)
\(758\) 35.0278 1.27227
\(759\) 0 0
\(760\) 0 0
\(761\) − 33.6333i − 1.21921i −0.792707 0.609603i \(-0.791329\pi\)
0.792707 0.609603i \(-0.208671\pi\)
\(762\) − 1.21110i − 0.0438736i
\(763\) −6.42221 −0.232499
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) 27.6333 0.998432
\(767\) − 33.2111i − 1.19918i
\(768\) 1.00000 0.0360844
\(769\) 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
\(772\) − 7.81665i − 0.281328i
\(773\) 30.0000i 1.07903i 0.841978 + 0.539513i \(0.181391\pi\)
−0.841978 + 0.539513i \(0.818609\pi\)
\(774\) − 8.00000i − 0.287554i
\(775\) 0 0
\(776\) 1.39445 0.0500578
\(777\) −42.4222 −1.52189
\(778\) 4.60555i 0.165117i
\(779\) 14.7889 0.529867
\(780\) 0 0
\(781\) 0 0
\(782\) − 6.42221i − 0.229658i
\(783\) 4.60555 0.164589
\(784\) −14.2111 −0.507539
\(785\) 0 0
\(786\) 22.6056i 0.806313i
\(787\) 49.2666i 1.75617i 0.478509 + 0.878083i \(0.341177\pi\)
−0.478509 + 0.878083i \(0.658823\pi\)
\(788\) 6.00000i 0.213741i
\(789\) 7.81665 0.278280
\(790\) 0 0
\(791\) − 63.6333i − 2.26254i
\(792\) 0 0
\(793\) 40.4222 1.43543
\(794\) 3.63331 0.128941
\(795\) 0 0
\(796\) −22.4222 −0.794734
\(797\) 6.00000 0.212531 0.106265 0.994338i \(-0.466111\pi\)
0.106265 + 0.994338i \(0.466111\pi\)
\(798\) 21.2111i 0.750865i
\(799\) 42.4222i 1.50079i
\(800\) 0 0
\(801\) − 15.2111i − 0.537458i
\(802\) 8.78890 0.310347
\(803\) 0 0
\(804\) 3.21110i 0.113247i
\(805\) 0 0
\(806\) −21.6333 −0.762001
\(807\) 25.8167 0.908789
\(808\) 7.39445i 0.260136i
\(809\) −6.84441 −0.240637 −0.120318 0.992735i \(-0.538392\pi\)
−0.120318 + 0.992735i \(0.538392\pi\)
\(810\) 0 0
\(811\) − 32.2389i − 1.13206i −0.824385 0.566030i \(-0.808479\pi\)
0.824385 0.566030i \(-0.191521\pi\)
\(812\) 21.2111i 0.744364i
\(813\) 0.422205i 0.0148074i
\(814\) 0 0
\(815\) 0 0
\(816\) −4.60555 −0.161227
\(817\) − 36.8444i − 1.28902i
\(818\) −14.7889 −0.517082
\(819\) 16.6056i 0.580245i
\(820\) 0 0
\(821\) 3.21110i 0.112068i 0.998429 + 0.0560341i \(0.0178456\pi\)
−0.998429 + 0.0560341i \(0.982154\pi\)
\(822\) −3.21110 −0.112000
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) − 4.00000i − 0.139347i
\(825\) 0 0
\(826\) 42.4222i 1.47606i
\(827\) − 27.6333i − 0.960904i −0.877021 0.480452i \(-0.840473\pi\)
0.877021 0.480452i \(-0.159527\pi\)
\(828\) −1.39445 −0.0484604
\(829\) 46.8444 1.62697 0.813487 0.581583i \(-0.197567\pi\)
0.813487 + 0.581583i \(0.197567\pi\)
\(830\) 0 0
\(831\) 16.4222 0.569680
\(832\) 3.60555 0.125000
\(833\) 65.4500 2.26771
\(834\) 17.2111i 0.595972i
\(835\) 0 0
\(836\) 0 0
\(837\) − 6.00000i − 0.207390i
\(838\) 4.18335i 0.144511i
\(839\) 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
\(842\) −19.8167 −0.682927
\(843\) − 27.2111i − 0.937200i
\(844\) 17.2111 0.592431
\(845\) 0 0
\(846\) 9.21110 0.316684
\(847\) − 50.6611i − 1.74073i
\(848\) −6.00000 −0.206041
\(849\) −10.4222 −0.357689
\(850\) 0 0
\(851\) − 12.8444i − 0.440301i
\(852\) 9.21110i 0.315567i
\(853\) − 14.7889i − 0.506362i −0.967419 0.253181i \(-0.918523\pi\)
0.967419 0.253181i \(-0.0814769\pi\)
\(854\) −51.6333 −1.76686
\(855\) 0 0
\(856\) 0 0
\(857\) 23.0278 0.786613 0.393307 0.919407i \(-0.371331\pi\)
0.393307 + 0.919407i \(0.371331\pi\)
\(858\) 0 0
\(859\) 25.2111 0.860192 0.430096 0.902783i \(-0.358480\pi\)
0.430096 + 0.902783i \(0.358480\pi\)
\(860\) 0 0
\(861\) −14.7889 −0.504004
\(862\) 12.0000 0.408722
\(863\) 51.6333i 1.75762i 0.477173 + 0.878809i \(0.341661\pi\)
−0.477173 + 0.878809i \(0.658339\pi\)
\(864\) 1.00000i 0.0340207i
\(865\) 0 0
\(866\) − 19.2111i − 0.652820i
\(867\) 4.21110 0.143017
\(868\) 27.6333 0.937936
\(869\) 0 0
\(870\) 0 0
\(871\) 11.5778i 0.392299i
\(872\) −1.39445 −0.0472220
\(873\) 1.39445i 0.0471949i
\(874\) −6.42221 −0.217234
\(875\) 0 0
\(876\) 1.39445i 0.0471141i
\(877\) − 24.8444i − 0.838936i −0.907770 0.419468i \(-0.862217\pi\)
0.907770 0.419468i \(-0.137783\pi\)
\(878\) 8.00000i 0.269987i
\(879\) 18.0000i 0.607125i
\(880\) 0 0
\(881\) 39.2111 1.32106 0.660528 0.750802i \(-0.270333\pi\)
0.660528 + 0.750802i \(0.270333\pi\)
\(882\) − 14.2111i − 0.478513i
\(883\) −9.57779 −0.322318 −0.161159 0.986928i \(-0.551523\pi\)
−0.161159 + 0.986928i \(0.551523\pi\)
\(884\) −16.6056 −0.558505
\(885\) 0 0
\(886\) − 15.6333i − 0.525211i
\(887\) −6.97224 −0.234105 −0.117053 0.993126i \(-0.537345\pi\)
−0.117053 + 0.993126i \(0.537345\pi\)
\(888\) −9.21110 −0.309104
\(889\) 5.57779i 0.187073i
\(890\) 0 0
\(891\) 0 0
\(892\) − 1.81665i − 0.0608261i
\(893\) 42.4222 1.41960
\(894\) 15.2111 0.508735
\(895\) 0 0
\(896\) −4.60555 −0.153861
\(897\) −5.02776 −0.167872
\(898\) 33.6333 1.12236
\(899\) − 27.6333i − 0.921622i
\(900\) 0 0
\(901\) 27.6333 0.920599
\(902\) 0 0
\(903\) 36.8444i 1.22611i
\(904\) − 13.8167i − 0.459535i
\(905\) 0 0
\(906\) 6.00000 0.199337
\(907\) −21.5778 −0.716479 −0.358239 0.933630i \(-0.616623\pi\)
−0.358239 + 0.933630i \(0.616623\pi\)
\(908\) − 24.0000i − 0.796468i
\(909\) −7.39445 −0.245258
\(910\) 0 0
\(911\) −27.6333 −0.915532 −0.457766 0.889073i \(-0.651350\pi\)
−0.457766 + 0.889073i \(0.651350\pi\)
\(912\) 4.60555i 0.152505i
\(913\) 0 0
\(914\) −38.2389 −1.26483
\(915\) 0 0
\(916\) − 19.8167i − 0.654761i
\(917\) − 104.111i − 3.43805i
\(918\) − 4.60555i − 0.152006i
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) − 8.78890i − 0.289604i
\(922\) −33.6333 −1.10765
\(923\) 33.2111i 1.09316i
\(924\) 0 0
\(925\) 0 0
\(926\) 31.3944 1.03169
\(927\) 4.00000 0.131377
\(928\) 4.60555i 0.151185i
\(929\) − 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
\(932\) −1.81665 −0.0595065
\(933\) 12.0000 0.392862
\(934\) 30.4222i 0.995445i
\(935\) 0 0
\(936\) 3.60555i 0.117851i
\(937\) 10.3667 0.338665 0.169333 0.985559i \(-0.445839\pi\)
0.169333 + 0.985559i \(0.445839\pi\)
\(938\) − 14.7889i − 0.482875i
\(939\) −3.57779 −0.116757
\(940\) 0 0
\(941\) 54.0000i 1.76035i 0.474650 + 0.880175i \(0.342575\pi\)
−0.474650 + 0.880175i \(0.657425\pi\)
\(942\) − 20.4222i − 0.665391i
\(943\) − 4.47772i − 0.145815i
\(944\) 9.21110i 0.299796i
\(945\) 0 0
\(946\) 0 0
\(947\) − 15.6333i − 0.508014i −0.967202 0.254007i \(-0.918251\pi\)
0.967202 0.254007i \(-0.0817487\pi\)
\(948\) 14.4222 0.468411
\(949\) 5.02776i 0.163208i
\(950\) 0 0
\(951\) − 18.0000i − 0.583690i
\(952\) 21.2111 0.687456
\(953\) 20.2389 0.655601 0.327800 0.944747i \(-0.393693\pi\)
0.327800 + 0.944747i \(0.393693\pi\)
\(954\) − 6.00000i − 0.194257i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 5.57779 0.180210
\(959\) 14.7889 0.477558
\(960\) 0 0
\(961\) −5.00000 −0.161290
\(962\) −33.2111 −1.07077
\(963\) 0 0
\(964\) − 6.42221i − 0.206845i
\(965\) 0 0
\(966\) 6.42221 0.206631
\(967\) − 8.23886i − 0.264944i −0.991187 0.132472i \(-0.957709\pi\)
0.991187 0.132472i \(-0.0422914\pi\)
\(968\) − 11.0000i − 0.353553i
\(969\) − 21.2111i − 0.681399i
\(970\) 0 0
\(971\) −53.0278 −1.70174 −0.850871 0.525375i \(-0.823925\pi\)
−0.850871 + 0.525375i \(0.823925\pi\)
\(972\) −1.00000 −0.0320750
\(973\) − 79.2666i − 2.54117i
\(974\) 0.972244 0.0311527
\(975\) 0 0
\(976\) −11.2111 −0.358859
\(977\) 18.8444i 0.602886i 0.953484 + 0.301443i \(0.0974683\pi\)
−0.953484 + 0.301443i \(0.902532\pi\)
\(978\) −24.4222 −0.780936
\(979\) 0 0
\(980\) 0 0
\(981\) − 1.39445i − 0.0445213i
\(982\) 7.81665i 0.249439i
\(983\) − 42.4222i − 1.35306i −0.736416 0.676529i \(-0.763483\pi\)
0.736416 0.676529i \(-0.236517\pi\)
\(984\) −3.21110 −0.102366
\(985\) 0 0
\(986\) − 21.2111i − 0.675499i
\(987\) −42.4222 −1.35031
\(988\) 16.6056i 0.528293i
\(989\) −11.1556 −0.354727
\(990\) 0 0
\(991\) −22.4222 −0.712265 −0.356132 0.934436i \(-0.615905\pi\)
−0.356132 + 0.934436i \(0.615905\pi\)
\(992\) 6.00000 0.190500
\(993\) 16.6056i 0.526961i
\(994\) − 42.4222i − 1.34555i
\(995\) 0 0
\(996\) − 2.78890i − 0.0883696i
\(997\) 16.4222 0.520096 0.260048 0.965596i \(-0.416262\pi\)
0.260048 + 0.965596i \(0.416262\pi\)
\(998\) 23.0278 0.728931
\(999\) − 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.b.k.1351.3 4
5.2 odd 4 1950.2.f.m.649.2 4
5.3 odd 4 1950.2.f.n.649.3 4
5.4 even 2 390.2.b.c.181.2 4
13.12 even 2 inner 1950.2.b.k.1351.2 4
15.14 odd 2 1170.2.b.d.181.4 4
20.19 odd 2 3120.2.g.q.961.1 4
65.12 odd 4 1950.2.f.n.649.1 4
65.34 odd 4 5070.2.a.bf.1.2 2
65.38 odd 4 1950.2.f.m.649.4 4
65.44 odd 4 5070.2.a.z.1.1 2
65.64 even 2 390.2.b.c.181.3 yes 4
195.194 odd 2 1170.2.b.d.181.1 4
260.259 odd 2 3120.2.g.q.961.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.c.181.2 4 5.4 even 2
390.2.b.c.181.3 yes 4 65.64 even 2
1170.2.b.d.181.1 4 195.194 odd 2
1170.2.b.d.181.4 4 15.14 odd 2
1950.2.b.k.1351.2 4 13.12 even 2 inner
1950.2.b.k.1351.3 4 1.1 even 1 trivial
1950.2.f.m.649.2 4 5.2 odd 4
1950.2.f.m.649.4 4 65.38 odd 4
1950.2.f.n.649.1 4 65.12 odd 4
1950.2.f.n.649.3 4 5.3 odd 4
3120.2.g.q.961.1 4 20.19 odd 2
3120.2.g.q.961.4 4 260.259 odd 2
5070.2.a.z.1.1 2 65.44 odd 4
5070.2.a.bf.1.2 2 65.34 odd 4