Properties

Label 1170.2.b.d.181.4
Level $1170$
Weight $2$
Character 1170.181
Analytic conductor $9.342$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.34249703649\)
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 181.4
Root \(2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 1170.181
Dual form 1170.2.b.d.181.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} +4.60555i q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} +4.60555i q^{7} -1.00000i q^{8} -1.00000 q^{10} +3.60555 q^{13} -4.60555 q^{14} +1.00000 q^{16} -4.60555 q^{17} +4.60555i q^{19} -1.00000i q^{20} +1.39445 q^{23} -1.00000 q^{25} +3.60555i q^{26} -4.60555i q^{28} -4.60555 q^{29} -6.00000i q^{31} +1.00000i q^{32} -4.60555i q^{34} -4.60555 q^{35} +9.21110i q^{37} -4.60555 q^{38} +1.00000 q^{40} +3.21110i q^{41} +8.00000 q^{43} +1.39445i q^{46} -9.21110i q^{47} -14.2111 q^{49} -1.00000i q^{50} -3.60555 q^{52} -6.00000 q^{53} +4.60555 q^{56} -4.60555i q^{58} -9.21110i q^{59} -11.2111 q^{61} +6.00000 q^{62} -1.00000 q^{64} +3.60555i q^{65} +3.21110i q^{67} +4.60555 q^{68} -4.60555i q^{70} +9.21110i q^{71} +1.39445i q^{73} -9.21110 q^{74} -4.60555i q^{76} -14.4222 q^{79} +1.00000i q^{80} -3.21110 q^{82} +2.78890i q^{83} -4.60555i q^{85} +8.00000i q^{86} +15.2111i q^{89} +16.6056i q^{91} -1.39445 q^{92} +9.21110 q^{94} -4.60555 q^{95} -1.39445i q^{97} -14.2111i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + O(q^{10}) \) \( 4q - 4q^{4} - 4q^{10} - 4q^{14} + 4q^{16} - 4q^{17} + 20q^{23} - 4q^{25} - 4q^{29} - 4q^{35} - 4q^{38} + 4q^{40} + 32q^{43} - 28q^{49} - 24q^{53} + 4q^{56} - 16q^{61} + 24q^{62} - 4q^{64} + 4q^{68} - 8q^{74} + 16q^{82} - 20q^{92} + 8q^{94} - 4q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times\).

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 4.60555i 1.74073i 0.492403 + 0.870367i \(0.336119\pi\)
−0.492403 + 0.870367i \(0.663881\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 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 \(0.688625\pi\)
\(18\) 0 0
\(19\) 4.60555i 1.05659i 0.849062 + 0.528293i \(0.177168\pi\)
−0.849062 + 0.528293i \(0.822832\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) 0 0
\(22\) 0 0
\(23\) 1.39445 0.290763 0.145381 0.989376i \(-0.453559\pi\)
0.145381 + 0.989376i \(0.453559\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 3.60555i 0.707107i
\(27\) 0 0
\(28\) − 4.60555i − 0.870367i
\(29\) −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 \(-0.818872\pi\)
0.842424 0.538816i \(-0.181128\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) − 4.60555i − 0.789846i
\(35\) −4.60555 −0.778480
\(36\) 0 0
\(37\) 9.21110i 1.51430i 0.653243 + 0.757148i \(0.273408\pi\)
−0.653243 + 0.757148i \(0.726592\pi\)
\(38\) −4.60555 −0.747119
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 3.21110i 0.501490i 0.968053 + 0.250745i \(0.0806756\pi\)
−0.968053 + 0.250745i \(0.919324\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 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 \(-0.234474\pi\)
\(48\) 0 0
\(49\) −14.2111 −2.03016
\(50\) − 1.00000i − 0.141421i
\(51\) 0 0
\(52\) −3.60555 −0.500000
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.60555 0.615443
\(57\) 0 0
\(58\) − 4.60555i − 0.604739i
\(59\) − 9.21110i − 1.19918i −0.800306 0.599592i \(-0.795330\pi\)
0.800306 0.599592i \(-0.204670\pi\)
\(60\) 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
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 3.60555i 0.447214i
\(66\) 0 0
\(67\) 3.21110i 0.392299i 0.980574 + 0.196149i \(0.0628437\pi\)
−0.980574 + 0.196149i \(0.937156\pi\)
\(68\) 4.60555 0.558505
\(69\) 0 0
\(70\) − 4.60555i − 0.550469i
\(71\) 9.21110i 1.09316i 0.837408 + 0.546578i \(0.184070\pi\)
−0.837408 + 0.546578i \(0.815930\pi\)
\(72\) 0 0
\(73\) 1.39445i 0.163208i 0.996665 + 0.0816039i \(0.0260043\pi\)
−0.996665 + 0.0816039i \(0.973996\pi\)
\(74\) −9.21110 −1.07077
\(75\) 0 0
\(76\) − 4.60555i − 0.528293i
\(77\) 0 0
\(78\) 0 0
\(79\) −14.4222 −1.62262 −0.811312 0.584613i \(-0.801246\pi\)
−0.811312 + 0.584613i \(0.801246\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 0 0
\(82\) −3.21110 −0.354607
\(83\) 2.78890i 0.306121i 0.988217 + 0.153061i \(0.0489130\pi\)
−0.988217 + 0.153061i \(0.951087\pi\)
\(84\) 0 0
\(85\) − 4.60555i − 0.499542i
\(86\) 8.00000i 0.862662i
\(87\) 0 0
\(88\) 0 0
\(89\) 15.2111i 1.61237i 0.591661 + 0.806187i \(0.298472\pi\)
−0.591661 + 0.806187i \(0.701528\pi\)
\(90\) 0 0
\(91\) 16.6056i 1.74073i
\(92\) −1.39445 −0.145381
\(93\) 0 0
\(94\) 9.21110 0.950053
\(95\) −4.60555 −0.472520
\(96\) 0 0
\(97\) − 1.39445i − 0.141585i −0.997491 0.0707924i \(-0.977447\pi\)
0.997491 0.0707924i \(-0.0225528\pi\)
\(98\) − 14.2111i − 1.43554i
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 7.39445 0.735775 0.367888 0.929870i \(-0.380081\pi\)
0.367888 + 0.929870i \(0.380081\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) − 3.60555i − 0.353553i
\(105\) 0 0
\(106\) − 6.00000i − 0.582772i
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) − 1.39445i − 0.133564i −0.997768 0.0667820i \(-0.978727\pi\)
0.997768 0.0667820i \(-0.0212732\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.60555i 0.435184i
\(113\) 13.8167 1.29976 0.649881 0.760036i \(-0.274819\pi\)
0.649881 + 0.760036i \(0.274819\pi\)
\(114\) 0 0
\(115\) 1.39445i 0.130033i
\(116\) 4.60555 0.427615
\(117\) 0 0
\(118\) 9.21110 0.847951
\(119\) − 21.2111i − 1.94442i
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) − 11.2111i − 1.01501i
\(123\) 0 0
\(124\) 6.00000i 0.538816i
\(125\) − 1.00000i − 0.0894427i
\(126\) 0 0
\(127\) 1.21110 0.107468 0.0537340 0.998555i \(-0.482888\pi\)
0.0537340 + 0.998555i \(0.482888\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 0 0
\(130\) −3.60555 −0.316228
\(131\) −22.6056 −1.97506 −0.987528 0.157443i \(-0.949675\pi\)
−0.987528 + 0.157443i \(0.949675\pi\)
\(132\) 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 \(0.956199\pi\)
\(138\) 0 0
\(139\) 17.2111 1.45983 0.729913 0.683540i \(-0.239560\pi\)
0.729913 + 0.683540i \(0.239560\pi\)
\(140\) 4.60555 0.389240
\(141\) 0 0
\(142\) −9.21110 −0.772979
\(143\) 0 0
\(144\) 0 0
\(145\) − 4.60555i − 0.382470i
\(146\) −1.39445 −0.115405
\(147\) 0 0
\(148\) − 9.21110i − 0.757148i
\(149\) 15.2111i 1.24614i 0.782165 + 0.623071i \(0.214115\pi\)
−0.782165 + 0.623071i \(0.785885\pi\)
\(150\) 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
\(153\) 0 0
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) 20.4222 1.62987 0.814935 0.579553i \(-0.196773\pi\)
0.814935 + 0.579553i \(0.196773\pi\)
\(158\) − 14.4222i − 1.14737i
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 6.42221i 0.506141i
\(162\) 0 0
\(163\) − 24.4222i − 1.91289i −0.291905 0.956447i \(-0.594289\pi\)
0.291905 0.956447i \(-0.405711\pi\)
\(164\) − 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 \(0.884008\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 4.60555 0.353230
\(171\) 0 0
\(172\) −8.00000 −0.609994
\(173\) 12.4222 0.944443 0.472221 0.881480i \(-0.343452\pi\)
0.472221 + 0.881480i \(0.343452\pi\)
\(174\) 0 0
\(175\) − 4.60555i − 0.348147i
\(176\) 0 0
\(177\) 0 0
\(178\) −15.2111 −1.14012
\(179\) 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 \(0.398663\pi\)
\(182\) −16.6056 −1.23089
\(183\) 0 0
\(184\) − 1.39445i − 0.102800i
\(185\) −9.21110 −0.677214
\(186\) 0 0
\(187\) 0 0
\(188\) 9.21110i 0.671789i
\(189\) 0 0
\(190\) − 4.60555i − 0.334122i
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) − 7.81665i − 0.562655i −0.959612 0.281328i \(-0.909225\pi\)
0.959612 0.281328i \(-0.0907747\pi\)
\(194\) 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 \(0.207610\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 0 0
\(202\) 7.39445i 0.520272i
\(203\) − 21.2111i − 1.48873i
\(204\) 0 0
\(205\) −3.21110 −0.224273
\(206\) − 4.00000i − 0.278693i
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) 0 0
\(215\) 8.00000i 0.545595i
\(216\) 0 0
\(217\) 27.6333 1.87587
\(218\) 1.39445 0.0944440
\(219\) 0 0
\(220\) 0 0
\(221\) −16.6056 −1.11701
\(222\) 0 0
\(223\) − 1.81665i − 0.121652i −0.998148 0.0608261i \(-0.980627\pi\)
0.998148 0.0608261i \(-0.0193735\pi\)
\(224\) −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 \(0.706699\pi\)
\(228\) 0 0
\(229\) 19.8167i 1.30952i 0.755836 + 0.654761i \(0.227231\pi\)
−0.755836 + 0.654761i \(0.772769\pi\)
\(230\) −1.39445 −0.0919472
\(231\) 0 0
\(232\) 4.60555i 0.302369i
\(233\) 1.81665 0.119013 0.0595065 0.998228i \(-0.481047\pi\)
0.0595065 + 0.998228i \(0.481047\pi\)
\(234\) 0 0
\(235\) 9.21110 0.600866
\(236\) 9.21110i 0.599592i
\(237\) 0 0
\(238\) 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
\(243\) 0 0
\(244\) 11.2111 0.717717
\(245\) − 14.2111i − 0.907914i
\(246\) 0 0
\(247\) 16.6056i 1.05659i
\(248\) −6.00000 −0.381000
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 13.3944 0.845450 0.422725 0.906258i \(-0.361074\pi\)
0.422725 + 0.906258i \(0.361074\pi\)
\(252\) 0 0
\(253\) 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 \(0.149173\pi\)
\(258\) 0 0
\(259\) −42.4222 −2.63599
\(260\) − 3.60555i − 0.223607i
\(261\) 0 0
\(262\) − 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
\(265\) − 6.00000i − 0.368577i
\(266\) − 21.2111i − 1.30054i
\(267\) 0 0
\(268\) − 3.21110i − 0.196149i
\(269\) −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 \(0.00408198\pi\)
−0.999918 + 0.0128236i \(0.995918\pi\)
\(272\) −4.60555 −0.279253
\(273\) 0 0
\(274\) −3.21110 −0.193990
\(275\) 0 0
\(276\) 0 0
\(277\) −16.4222 −0.986715 −0.493357 0.869827i \(-0.664230\pi\)
−0.493357 + 0.869827i \(0.664230\pi\)
\(278\) 17.2111i 1.03225i
\(279\) 0 0
\(280\) 4.60555i 0.275234i
\(281\) 27.2111i 1.62328i 0.584159 + 0.811639i \(0.301424\pi\)
−0.584159 + 0.811639i \(0.698576\pi\)
\(282\) 0 0
\(283\) 10.4222 0.619536 0.309768 0.950812i \(-0.399749\pi\)
0.309768 + 0.950812i \(0.399749\pi\)
\(284\) − 9.21110i − 0.546578i
\(285\) 0 0
\(286\) 0 0
\(287\) −14.7889 −0.872961
\(288\) 0 0
\(289\) 4.21110 0.247712
\(290\) 4.60555 0.270447
\(291\) 0 0
\(292\) − 1.39445i − 0.0816039i
\(293\) 18.0000i 1.05157i 0.850617 + 0.525786i \(0.176229\pi\)
−0.850617 + 0.525786i \(0.823771\pi\)
\(294\) 0 0
\(295\) 9.21110 0.536291
\(296\) 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
\(303\) 0 0
\(304\) 4.60555i 0.264146i
\(305\) − 11.2111i − 0.641946i
\(306\) 0 0
\(307\) 8.78890i 0.501609i 0.968038 + 0.250804i \(0.0806951\pi\)
−0.968038 + 0.250804i \(0.919305\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 6.00000i 0.340777i
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) 3.57779 0.202229 0.101114 0.994875i \(-0.467759\pi\)
0.101114 + 0.994875i \(0.467759\pi\)
\(314\) 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 \(-0.168688\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 1.00000i − 0.0559017i
\(321\) 0 0
\(322\) −6.42221 −0.357895
\(323\) − 21.2111i − 1.18022i
\(324\) 0 0
\(325\) −3.60555 −0.200000
\(326\) 24.4222 1.35262
\(327\) 0 0
\(328\) 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
\(333\) 0 0
\(334\) −9.21110 −0.504009
\(335\) −3.21110 −0.175441
\(336\) 0 0
\(337\) 13.6333 0.742654 0.371327 0.928502i \(-0.378903\pi\)
0.371327 + 0.928502i \(0.378903\pi\)
\(338\) 13.0000i 0.707107i
\(339\) 0 0
\(340\) 4.60555i 0.249771i
\(341\) 0 0
\(342\) 0 0
\(343\) − 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 \(0.234012\pi\)
\(348\) 0 0
\(349\) 7.81665i 0.418416i 0.977871 + 0.209208i \(0.0670886\pi\)
−0.977871 + 0.209208i \(0.932911\pi\)
\(350\) 4.60555 0.246177
\(351\) 0 0
\(352\) 0 0
\(353\) − 8.78890i − 0.467786i −0.972262 0.233893i \(-0.924853\pi\)
0.972262 0.233893i \(-0.0751465\pi\)
\(354\) 0 0
\(355\) −9.21110 −0.488875
\(356\) − 15.2111i − 0.806187i
\(357\) 0 0
\(358\) 19.8167i 1.04734i
\(359\) 15.6333i 0.825094i 0.910936 + 0.412547i \(0.135361\pi\)
−0.910936 + 0.412547i \(0.864639\pi\)
\(360\) 0 0
\(361\) −2.21110 −0.116374
\(362\) 8.42221i 0.442661i
\(363\) 0 0
\(364\) − 16.6056i − 0.870367i
\(365\) −1.39445 −0.0729888
\(366\) 0 0
\(367\) −19.6333 −1.02485 −0.512425 0.858732i \(-0.671253\pi\)
−0.512425 + 0.858732i \(0.671253\pi\)
\(368\) 1.39445 0.0726907
\(369\) 0 0
\(370\) − 9.21110i − 0.478862i
\(371\) − 27.6333i − 1.43465i
\(372\) 0 0
\(373\) −20.4222 −1.05742 −0.528711 0.848802i \(-0.677324\pi\)
−0.528711 + 0.848802i \(0.677324\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −9.21110 −0.475026
\(377\) −16.6056 −0.855229
\(378\) 0 0
\(379\) − 35.0278i − 1.79925i −0.436658 0.899627i \(-0.643838\pi\)
0.436658 0.899627i \(-0.356162\pi\)
\(380\) 4.60555 0.236260
\(381\) 0 0
\(382\) 12.0000i 0.613973i
\(383\) − 27.6333i − 1.41200i −0.708214 0.705998i \(-0.750499\pi\)
0.708214 0.705998i \(-0.249501\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 7.81665 0.397857
\(387\) 0 0
\(388\) 1.39445i 0.0707924i
\(389\) −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
\(392\) 14.2111i 0.717769i
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) − 14.4222i − 0.725660i
\(396\) 0 0
\(397\) 3.63331i 0.182350i 0.995835 + 0.0911752i \(0.0290623\pi\)
−0.995835 + 0.0911752i \(0.970938\pi\)
\(398\) 22.4222i 1.12392i
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 8.78890i 0.438897i 0.975624 + 0.219448i \(0.0704257\pi\)
−0.975624 + 0.219448i \(0.929574\pi\)
\(402\) 0 0
\(403\) − 21.6333i − 1.07763i
\(404\) −7.39445 −0.367888
\(405\) 0 0
\(406\) 21.2111 1.05269
\(407\) 0 0
\(408\) 0 0
\(409\) 14.7889i 0.731264i 0.930760 + 0.365632i \(0.119147\pi\)
−0.930760 + 0.365632i \(0.880853\pi\)
\(410\) − 3.21110i − 0.158585i
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) 42.4222 2.08746
\(414\) 0 0
\(415\) −2.78890 −0.136902
\(416\) 3.60555i 0.176777i
\(417\) 0 0
\(418\) 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 \(0.160417\pi\)
−0.875674 + 0.482902i \(0.839583\pi\)
\(422\) − 17.2111i − 0.837823i
\(423\) 0 0
\(424\) 6.00000i 0.291386i
\(425\) 4.60555 0.223402
\(426\) 0 0
\(427\) − 51.6333i − 2.49871i
\(428\) 0 0
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 12.0000i 0.578020i 0.957326 + 0.289010i \(0.0933260\pi\)
−0.957326 + 0.289010i \(0.906674\pi\)
\(432\) 0 0
\(433\) 19.2111 0.923227 0.461613 0.887081i \(-0.347271\pi\)
0.461613 + 0.887081i \(0.347271\pi\)
\(434\) 27.6333i 1.32644i
\(435\) 0 0
\(436\) 1.39445i 0.0667820i
\(437\) 6.42221i 0.307216i
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 16.6056i − 0.789846i
\(443\) −15.6333 −0.742761 −0.371380 0.928481i \(-0.621115\pi\)
−0.371380 + 0.928481i \(0.621115\pi\)
\(444\) 0 0
\(445\) −15.2111 −0.721075
\(446\) 1.81665 0.0860211
\(447\) 0 0
\(448\) − 4.60555i − 0.217592i
\(449\) 33.6333i 1.58725i 0.608405 + 0.793627i \(0.291810\pi\)
−0.608405 + 0.793627i \(0.708190\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −13.8167 −0.649881
\(453\) 0 0
\(454\) −24.0000 −1.12638
\(455\) −16.6056 −0.778480
\(456\) 0 0
\(457\) − 38.2389i − 1.78874i −0.447330 0.894369i \(-0.647625\pi\)
0.447330 0.894369i \(-0.352375\pi\)
\(458\) −19.8167 −0.925971
\(459\) 0 0
\(460\) − 1.39445i − 0.0650165i
\(461\) − 33.6333i − 1.56646i −0.621733 0.783230i \(-0.713571\pi\)
0.621733 0.783230i \(-0.286429\pi\)
\(462\) 0 0
\(463\) 31.3944i 1.45902i 0.683968 + 0.729512i \(0.260253\pi\)
−0.683968 + 0.729512i \(0.739747\pi\)
\(464\) −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 \(0.251447\pi\)
\(468\) 0 0
\(469\) −14.7889 −0.682888
\(470\) 9.21110i 0.424876i
\(471\) 0 0
\(472\) −9.21110 −0.423975
\(473\) 0 0
\(474\) 0 0
\(475\) − 4.60555i − 0.211317i
\(476\) 21.2111i 0.972209i
\(477\) 0 0
\(478\) 0 0
\(479\) 5.57779i 0.254856i 0.991848 + 0.127428i \(0.0406722\pi\)
−0.991848 + 0.127428i \(0.959328\pi\)
\(480\) 0 0
\(481\) 33.2111i 1.51430i
\(482\) −6.42221 −0.292523
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) 1.39445 0.0633187
\(486\) 0 0
\(487\) 0.972244i 0.0440566i 0.999757 + 0.0220283i \(0.00701239\pi\)
−0.999757 + 0.0220283i \(0.992988\pi\)
\(488\) 11.2111i 0.507503i
\(489\) 0 0
\(490\) 14.2111 0.641992
\(491\) −7.81665 −0.352761 −0.176380 0.984322i \(-0.556439\pi\)
−0.176380 + 0.984322i \(0.556439\pi\)
\(492\) 0 0
\(493\) 21.2111 0.955300
\(494\) −16.6056 −0.747119
\(495\) 0 0
\(496\) − 6.00000i − 0.269408i
\(497\) −42.4222 −1.90290
\(498\) 0 0
\(499\) − 23.0278i − 1.03086i −0.856930 0.515432i \(-0.827631\pi\)
0.856930 0.515432i \(-0.172369\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) 0 0
\(502\) 13.3944i 0.597824i
\(503\) −23.4500 −1.04558 −0.522791 0.852461i \(-0.675109\pi\)
−0.522791 + 0.852461i \(0.675109\pi\)
\(504\) 0 0
\(505\) 7.39445i 0.329049i
\(506\) 0 0
\(507\) 0 0
\(508\) −1.21110 −0.0537340
\(509\) − 33.6333i − 1.49077i −0.666634 0.745385i \(-0.732266\pi\)
0.666634 0.745385i \(-0.267734\pi\)
\(510\) 0 0
\(511\) −6.42221 −0.284102
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 28.6056i 1.26174i
\(515\) − 4.00000i − 0.176261i
\(516\) 0 0
\(517\) 0 0
\(518\) − 42.4222i − 1.86392i
\(519\) 0 0
\(520\) 3.60555 0.158114
\(521\) −21.6333 −0.947772 −0.473886 0.880586i \(-0.657149\pi\)
−0.473886 + 0.880586i \(0.657149\pi\)
\(522\) 0 0
\(523\) 32.8444 1.43619 0.718093 0.695947i \(-0.245015\pi\)
0.718093 + 0.695947i \(0.245015\pi\)
\(524\) 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
\(530\) 6.00000 0.260623
\(531\) 0 0
\(532\) 21.2111 0.919618
\(533\) 11.5778i 0.501490i
\(534\) 0 0
\(535\) 0 0
\(536\) 3.21110 0.138699
\(537\) 0 0
\(538\) − 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
\(543\) 0 0
\(544\) − 4.60555i − 0.197461i
\(545\) 1.39445 0.0597316
\(546\) 0 0
\(547\) 14.4222 0.616649 0.308324 0.951281i \(-0.400232\pi\)
0.308324 + 0.951281i \(0.400232\pi\)
\(548\) − 3.21110i − 0.137172i
\(549\) 0 0
\(550\) 0 0
\(551\) − 21.2111i − 0.903623i
\(552\) 0 0
\(553\) − 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 \(0.921119\pi\)
\(558\) 0 0
\(559\) 28.8444 1.21999
\(560\) −4.60555 −0.194620
\(561\) 0 0
\(562\) −27.2111 −1.14783
\(563\) −34.0555 −1.43527 −0.717634 0.696420i \(-0.754775\pi\)
−0.717634 + 0.696420i \(0.754775\pi\)
\(564\) 0 0
\(565\) 13.8167i 0.581271i
\(566\) 10.4222i 0.438078i
\(567\) 0 0
\(568\) 9.21110 0.386489
\(569\) −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
\(573\) 0 0
\(574\) − 14.7889i − 0.617277i
\(575\) −1.39445 −0.0581525
\(576\) 0 0
\(577\) 37.3944i 1.55675i 0.627799 + 0.778376i \(0.283956\pi\)
−0.627799 + 0.778376i \(0.716044\pi\)
\(578\) 4.21110i 0.175159i
\(579\) 0 0
\(580\) 4.60555i 0.191235i
\(581\) −12.8444 −0.532876
\(582\) 0 0
\(583\) 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 \(-0.0423121\pi\)
\(588\) 0 0
\(589\) 27.6333 1.13861
\(590\) 9.21110i 0.379215i
\(591\) 0 0
\(592\) 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
\(595\) 21.2111 0.869570
\(596\) − 15.2111i − 0.623071i
\(597\) 0 0
\(598\) 5.02776i 0.205600i
\(599\) 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 \(0.489394\pi\)
\(602\) −36.8444 −1.50167
\(603\) 0 0
\(604\) 6.00000i 0.244137i
\(605\) 11.0000i 0.447214i
\(606\) 0 0
\(607\) 17.2111 0.698577 0.349289 0.937015i \(-0.386423\pi\)
0.349289 + 0.937015i \(0.386423\pi\)
\(608\) −4.60555 −0.186780
\(609\) 0 0
\(610\) 11.2111 0.453924
\(611\) − 33.2111i − 1.34358i
\(612\) 0 0
\(613\) 33.2111i 1.34138i 0.741736 + 0.670692i \(0.234003\pi\)
−0.741736 + 0.670692i \(0.765997\pi\)
\(614\) −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 \(0.919553\pi\)
\(618\) 0 0
\(619\) − 25.8167i − 1.03766i −0.854878 0.518829i \(-0.826368\pi\)
0.854878 0.518829i \(-0.173632\pi\)
\(620\) −6.00000 −0.240966
\(621\) 0 0
\(622\) − 12.0000i − 0.481156i
\(623\) −70.0555 −2.80671
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 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
\(633\) 0 0
\(634\) 18.0000 0.714871
\(635\) 1.21110i 0.0480611i
\(636\) 0 0
\(637\) −51.2389 −2.03016
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −0.422205 −0.0166761 −0.00833805 0.999965i \(-0.502654\pi\)
−0.00833805 + 0.999965i \(0.502654\pi\)
\(642\) 0 0
\(643\) − 9.63331i − 0.379901i −0.981794 0.189950i \(-0.939167\pi\)
0.981794 0.189950i \(-0.0608327\pi\)
\(644\) − 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 \(0.261875\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) − 3.60555i − 0.141421i
\(651\) 0 0
\(652\) 24.4222i 0.956447i
\(653\) −39.2111 −1.53445 −0.767225 0.641379i \(-0.778363\pi\)
−0.767225 + 0.641379i \(0.778363\pi\)
\(654\) 0 0
\(655\) − 22.6056i − 0.883272i
\(656\) 3.21110i 0.125372i
\(657\) 0 0
\(658\) 42.4222i 1.65379i
\(659\) 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 \(-0.568353\pi\)
0.213090 0.977033i \(-0.431647\pi\)
\(662\) −16.6056 −0.645393
\(663\) 0 0
\(664\) 2.78890 0.108230
\(665\) − 21.2111i − 0.822531i
\(666\) 0 0
\(667\) −6.42221 −0.248669
\(668\) − 9.21110i − 0.356388i
\(669\) 0 0
\(670\) − 3.21110i − 0.124056i
\(671\) 0 0
\(672\) 0 0
\(673\) −37.6333 −1.45066 −0.725329 0.688403i \(-0.758312\pi\)
−0.725329 + 0.688403i \(0.758312\pi\)
\(674\) 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 \(0.318753\pi\)
\(678\) 0 0
\(679\) 6.42221 0.246462
\(680\) −4.60555 −0.176615
\(681\) 0 0
\(682\) 0 0
\(683\) 9.21110i 0.352453i 0.984350 + 0.176227i \(0.0563891\pi\)
−0.984350 + 0.176227i \(0.943611\pi\)
\(684\) 0 0
\(685\) −3.21110 −0.122690
\(686\) 33.2111 1.26801
\(687\) 0 0
\(688\) 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
\(695\) 17.2111i 0.652854i
\(696\) 0 0
\(697\) − 14.7889i − 0.560169i
\(698\) −7.81665 −0.295865
\(699\) 0 0
\(700\) 4.60555i 0.174073i
\(701\) −47.0278 −1.77621 −0.888107 0.459637i \(-0.847980\pi\)
−0.888107 + 0.459637i \(0.847980\pi\)
\(702\) 0 0
\(703\) −42.4222 −1.59998
\(704\) 0 0
\(705\) 0 0
\(706\) 8.78890 0.330775
\(707\) 34.0555i 1.28079i
\(708\) 0 0
\(709\) 1.39445i 0.0523696i 0.999657 + 0.0261848i \(0.00833584\pi\)
−0.999657 + 0.0261848i \(0.991664\pi\)
\(710\) − 9.21110i − 0.345687i
\(711\) 0 0
\(712\) 15.2111 0.570060
\(713\) − 8.36669i − 0.313335i
\(714\) 0 0
\(715\) 0 0
\(716\) −19.8167 −0.740583
\(717\) 0 0
\(718\) −15.6333 −0.583430
\(719\) 51.6333 1.92560 0.962799 0.270220i \(-0.0870963\pi\)
0.962799 + 0.270220i \(0.0870963\pi\)
\(720\) 0 0
\(721\) − 18.4222i − 0.686079i
\(722\) − 2.21110i − 0.0822887i
\(723\) 0 0
\(724\) −8.42221 −0.313009
\(725\) 4.60555 0.171046
\(726\) 0 0
\(727\) −14.4222 −0.534890 −0.267445 0.963573i \(-0.586179\pi\)
−0.267445 + 0.963573i \(0.586179\pi\)
\(728\) 16.6056 0.615443
\(729\) 0 0
\(730\) − 1.39445i − 0.0516109i
\(731\) −36.8444 −1.36274
\(732\) 0 0
\(733\) 34.0555i 1.25787i 0.777458 + 0.628935i \(0.216509\pi\)
−0.777458 + 0.628935i \(0.783491\pi\)
\(734\) − 19.6333i − 0.724679i
\(735\) 0 0
\(736\) 1.39445i 0.0514001i
\(737\) 0 0
\(738\) 0 0
\(739\) − 20.2389i − 0.744498i −0.928133 0.372249i \(-0.878587\pi\)
0.928133 0.372249i \(-0.121413\pi\)
\(740\) 9.21110 0.338607
\(741\) 0 0
\(742\) 27.6333 1.01445
\(743\) 36.8444i 1.35169i 0.737044 + 0.675845i \(0.236221\pi\)
−0.737044 + 0.675845i \(0.763779\pi\)
\(744\) 0 0
\(745\) −15.2111 −0.557292
\(746\) − 20.4222i − 0.747710i
\(747\) 0 0
\(748\) 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
\(753\) 0 0
\(754\) − 16.6056i − 0.604739i
\(755\) 6.00000 0.218362
\(756\) 0 0
\(757\) 12.7889 0.464820 0.232410 0.972618i \(-0.425339\pi\)
0.232410 + 0.972618i \(0.425339\pi\)
\(758\) 35.0278 1.27227
\(759\) 0 0
\(760\) 4.60555i 0.167061i
\(761\) 33.6333i 1.21921i 0.792707 + 0.609603i \(0.208671\pi\)
−0.792707 + 0.609603i \(0.791329\pi\)
\(762\) 0 0
\(763\) 6.42221 0.232499
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 27.6333 0.998432
\(767\) − 33.2111i − 1.19918i
\(768\) 0 0
\(769\) 12.8444i 0.463181i 0.972813 + 0.231591i \(0.0743930\pi\)
−0.972813 + 0.231591i \(0.925607\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7.81665i 0.281328i
\(773\) 30.0000i 1.07903i 0.841978 + 0.539513i \(0.181391\pi\)
−0.841978 + 0.539513i \(0.818609\pi\)
\(774\) 0 0
\(775\) 6.00000i 0.215526i
\(776\) −1.39445 −0.0500578
\(777\) 0 0
\(778\) − 4.60555i − 0.165117i
\(779\) −14.7889 −0.529867
\(780\) 0 0
\(781\) 0 0
\(782\) − 6.42221i − 0.229658i
\(783\) 0 0
\(784\) −14.2111 −0.507539
\(785\) 20.4222i 0.728900i
\(786\) 0 0
\(787\) − 49.2666i − 1.75617i −0.478509 0.878083i \(-0.658823\pi\)
0.478509 0.878083i \(-0.341177\pi\)
\(788\) 6.00000i 0.213741i
\(789\) 0 0
\(790\) 14.4222 0.513119
\(791\) 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 \(0.466111\pi\)
\(798\) 0 0
\(799\) 42.4222i 1.50079i
\(800\) − 1.00000i − 0.0353553i
\(801\) 0 0
\(802\) −8.78890 −0.310347
\(803\) 0 0
\(804\) 0 0
\(805\) −6.42221 −0.226353
\(806\) 21.6333 0.762001
\(807\) 0 0
\(808\) − 7.39445i − 0.260136i
\(809\) 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 \(-0.808479\pi\)
0.824385 0.566030i \(-0.191521\pi\)
\(812\) 21.2111i 0.744364i
\(813\) 0 0
\(814\) 0 0
\(815\) 24.4222 0.855473
\(816\) 0 0
\(817\) 36.8444i 1.28902i
\(818\) −14.7889 −0.517082
\(819\) 0 0
\(820\) 3.21110 0.112137
\(821\) − 3.21110i − 0.112068i −0.998429 0.0560341i \(-0.982154\pi\)
0.998429 0.0560341i \(-0.0178456\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 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 \(-0.159527\pi\)
\(828\) 0 0
\(829\) 46.8444 1.62697 0.813487 0.581583i \(-0.197567\pi\)
0.813487 + 0.581583i \(0.197567\pi\)
\(830\) − 2.78890i − 0.0968040i
\(831\) 0 0
\(832\) −3.60555 −0.125000
\(833\) 65.4500 2.26771
\(834\) 0 0
\(835\) −9.21110 −0.318763
\(836\) 0 0
\(837\) 0 0
\(838\) − 4.18335i − 0.144511i
\(839\) − 18.4222i − 0.636005i −0.948090 0.318003i \(-0.896988\pi\)
0.948090 0.318003i \(-0.103012\pi\)
\(840\) 0 0
\(841\) −7.78890 −0.268583
\(842\) −19.8167 −0.682927
\(843\) 0 0
\(844\) 17.2111 0.592431
\(845\) 13.0000i 0.447214i
\(846\) 0 0
\(847\) 50.6611i 1.74073i
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 4.60555i 0.157969i
\(851\) 12.8444i 0.440301i
\(852\) 0 0
\(853\) 14.7889i 0.506362i 0.967419 + 0.253181i \(0.0814769\pi\)
−0.967419 + 0.253181i \(0.918523\pi\)
\(854\) 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 \(0.358480\pi\)
\(860\) − 8.00000i − 0.272798i
\(861\) 0 0
\(862\) −12.0000 −0.408722
\(863\) 51.6333i 1.75762i 0.477173 + 0.878809i \(0.341661\pi\)
−0.477173 + 0.878809i \(0.658339\pi\)
\(864\) 0 0
\(865\) 12.4222i 0.422368i
\(866\) 19.2111i 0.652820i
\(867\) 0 0
\(868\) −27.6333 −0.937936
\(869\) 0 0
\(870\) 0 0
\(871\) 11.5778i 0.392299i
\(872\) −1.39445 −0.0472220
\(873\) 0 0
\(874\) −6.42221 −0.217234
\(875\) 4.60555 0.155696
\(876\) 0 0
\(877\) 24.8444i 0.838936i 0.907770 + 0.419468i \(0.137783\pi\)
−0.907770 + 0.419468i \(0.862217\pi\)
\(878\) 8.00000i 0.269987i
\(879\) 0 0
\(880\) 0 0
\(881\) −39.2111 −1.32106 −0.660528 0.750802i \(-0.729667\pi\)
−0.660528 + 0.750802i \(0.729667\pi\)
\(882\) 0 0
\(883\) 9.57779 0.322318 0.161159 0.986928i \(-0.448477\pi\)
0.161159 + 0.986928i \(0.448477\pi\)
\(884\) 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 \(0.537345\pi\)
\(888\) 0 0
\(889\) 5.57779i 0.187073i
\(890\) − 15.2111i − 0.509877i
\(891\) 0 0
\(892\) 1.81665i 0.0608261i
\(893\) 42.4222 1.41960
\(894\) 0 0
\(895\) 19.8167i 0.662398i
\(896\) 4.60555 0.153861
\(897\) 0 0
\(898\) −33.6333 −1.12236
\(899\) 27.6333i 0.921622i
\(900\) 0 0
\(901\) 27.6333 0.920599
\(902\) 0 0
\(903\) 0 0
\(904\) − 13.8167i − 0.459535i
\(905\) 8.42221i 0.279964i
\(906\) 0 0
\(907\) 21.5778 0.716479 0.358239 0.933630i \(-0.383377\pi\)
0.358239 + 0.933630i \(0.383377\pi\)
\(908\) − 24.0000i − 0.796468i
\(909\) 0 0
\(910\) − 16.6056i − 0.550469i
\(911\) 27.6333 0.915532 0.457766 0.889073i \(-0.348650\pi\)
0.457766 + 0.889073i \(0.348650\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 38.2389 1.26483
\(915\) 0 0
\(916\) − 19.8167i − 0.654761i
\(917\) − 104.111i − 3.43805i
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 1.39445 0.0459736
\(921\) 0 0
\(922\) 33.6333 1.10765
\(923\) 33.2111i 1.09316i
\(924\) 0 0
\(925\) − 9.21110i − 0.302859i
\(926\) −31.3944 −1.03169
\(927\) 0 0
\(928\) − 4.60555i − 0.151185i
\(929\) 39.2111i 1.28647i 0.765667 + 0.643237i \(0.222409\pi\)
−0.765667 + 0.643237i \(0.777591\pi\)
\(930\) 0 0
\(931\) − 65.4500i − 2.14504i
\(932\) −1.81665 −0.0595065
\(933\) 0 0
\(934\) 30.4222i 0.995445i
\(935\) 0 0
\(936\) 0 0
\(937\) −10.3667 −0.338665 −0.169333 0.985559i \(-0.554161\pi\)
−0.169333 + 0.985559i \(0.554161\pi\)
\(938\) − 14.7889i − 0.482875i
\(939\) 0 0
\(940\) −9.21110 −0.300433
\(941\) − 54.0000i − 1.76035i −0.474650 0.880175i \(-0.657425\pi\)
0.474650 0.880175i \(-0.342575\pi\)
\(942\) 0 0
\(943\) 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 \(-0.0817487\pi\)
\(948\) 0 0
\(949\) 5.02776i 0.163208i
\(950\) 4.60555 0.149424
\(951\) 0 0
\(952\) −21.2111 −0.687456
\(953\) 20.2389 0.655601 0.327800 0.944747i \(-0.393693\pi\)
0.327800 + 0.944747i \(0.393693\pi\)
\(954\) 0 0
\(955\) 12.0000i 0.388311i
\(956\) 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
\(965\) 7.81665 0.251627
\(966\) 0 0
\(967\) 8.23886i 0.264944i 0.991187 + 0.132472i \(0.0422914\pi\)
−0.991187 + 0.132472i \(0.957709\pi\)
\(968\) − 11.0000i − 0.353553i
\(969\) 0 0
\(970\) 1.39445i 0.0447731i
\(971\) 53.0278 1.70174 0.850871 0.525375i \(-0.176075\pi\)
0.850871 + 0.525375i \(0.176075\pi\)
\(972\) 0 0
\(973\) 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 \(0.902532\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 14.2111i 0.453957i
\(981\) 0 0
\(982\) − 7.81665i − 0.249439i
\(983\) − 42.4222i − 1.35306i −0.736416 0.676529i \(-0.763483\pi\)
0.736416 0.676529i \(-0.236517\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 21.2111i 0.675499i
\(987\) 0 0
\(988\) − 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
\(993\) 0 0
\(994\) − 42.4222i − 1.34555i
\(995\) 22.4222i 0.710832i
\(996\) 0 0
\(997\) −16.4222 −0.520096 −0.260048 0.965596i \(-0.583738\pi\)
−0.260048 + 0.965596i \(0.583738\pi\)
\(998\) 23.0278 0.728931
\(999\) 0 0
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 1170.2.b.d.181.4 4
3.2 odd 2 390.2.b.c.181.2 4
12.11 even 2 3120.2.g.q.961.1 4
13.12 even 2 inner 1170.2.b.d.181.1 4
15.2 even 4 1950.2.f.n.649.3 4
15.8 even 4 1950.2.f.m.649.2 4
15.14 odd 2 1950.2.b.k.1351.3 4
39.5 even 4 5070.2.a.z.1.1 2
39.8 even 4 5070.2.a.bf.1.2 2
39.38 odd 2 390.2.b.c.181.3 yes 4
156.155 even 2 3120.2.g.q.961.4 4
195.38 even 4 1950.2.f.n.649.1 4
195.77 even 4 1950.2.f.m.649.4 4
195.194 odd 2 1950.2.b.k.1351.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
390.2.b.c.181.2 4 3.2 odd 2
390.2.b.c.181.3 yes 4 39.38 odd 2
1170.2.b.d.181.1 4 13.12 even 2 inner
1170.2.b.d.181.4 4 1.1 even 1 trivial
1950.2.b.k.1351.2 4 195.194 odd 2
1950.2.b.k.1351.3 4 15.14 odd 2
1950.2.f.m.649.2 4 15.8 even 4
1950.2.f.m.649.4 4 195.77 even 4
1950.2.f.n.649.1 4 195.38 even 4
1950.2.f.n.649.3 4 15.2 even 4
3120.2.g.q.961.1 4 12.11 even 2
3120.2.g.q.961.4 4 156.155 even 2
5070.2.a.z.1.1 2 39.5 even 4
5070.2.a.bf.1.2 2 39.8 even 4