Properties

Label 3120.2.g.q.961.1
Level $3120$
Weight $2$
Character 3120.961
Analytic conductor $24.913$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.9133254306\)
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 961.1
Root \(-2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 3120.961
Dual form 3120.2.g.q.961.4

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000i q^{5} -4.60555i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000i q^{5} -4.60555i q^{7} +1.00000 q^{9} +3.60555 q^{13} -1.00000i q^{15} +4.60555 q^{17} -4.60555i q^{19} -4.60555i q^{21} +1.39445 q^{23} -1.00000 q^{25} +1.00000 q^{27} +4.60555 q^{29} +6.00000i q^{31} -4.60555 q^{35} +9.21110i q^{37} +3.60555 q^{39} -3.21110i q^{41} -8.00000 q^{43} -1.00000i q^{45} -9.21110i q^{47} -14.2111 q^{49} +4.60555 q^{51} +6.00000 q^{53} -4.60555i q^{57} -9.21110i q^{59} -11.2111 q^{61} -4.60555i q^{63} -3.60555i q^{65} -3.21110i q^{67} +1.39445 q^{69} +9.21110i q^{71} +1.39445i q^{73} -1.00000 q^{75} +14.4222 q^{79} +1.00000 q^{81} +2.78890i q^{83} -4.60555i q^{85} +4.60555 q^{87} -15.2111i q^{89} -16.6056i q^{91} +6.00000i q^{93} -4.60555 q^{95} -1.39445i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 4q^{9} + 4q^{17} + 20q^{23} - 4q^{25} + 4q^{27} + 4q^{29} - 4q^{35} - 32q^{43} - 28q^{49} + 4q^{51} + 24q^{53} - 16q^{61} + 20q^{69} - 4q^{75} + 4q^{81} + 4q^{87} - 4q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times\).

\(n\) \(1951\) \(2081\) \(2341\) \(2497\) \(2641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) − 1.00000i − 0.447214i
\(6\) 0 0
\(7\) − 4.60555i − 1.74073i −0.492403 0.870367i \(-0.663881\pi\)
0.492403 0.870367i \(-0.336119\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 3.60555 1.00000
\(14\) 0 0
\(15\) − 1.00000i − 0.258199i
\(16\) 0 0
\(17\) 4.60555 1.11701 0.558505 0.829501i \(-0.311375\pi\)
0.558505 + 0.829501i \(0.311375\pi\)
\(18\) 0 0
\(19\) − 4.60555i − 1.05659i −0.849062 0.528293i \(-0.822832\pi\)
0.849062 0.528293i \(-0.177168\pi\)
\(20\) 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 \(0.453559\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 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 \(0.181128\pi\)
−0.842424 + 0.538816i \(0.818872\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.60555 −0.778480
\(36\) 0 0
\(37\) 9.21110i 1.51430i 0.653243 + 0.757148i \(0.273408\pi\)
−0.653243 + 0.757148i \(0.726592\pi\)
\(38\) 0 0
\(39\) 3.60555 0.577350
\(40\) 0 0
\(41\) − 3.21110i − 0.501490i −0.968053 0.250745i \(-0.919324\pi\)
0.968053 0.250745i \(-0.0806756\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) − 1.00000i − 0.149071i
\(46\) 0 0
\(47\) − 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
\(50\) 0 0
\(51\) 4.60555 0.644906
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.60555i − 0.610020i
\(58\) 0 0
\(59\) − 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 \(0.754813\pi\)
\(62\) 0 0
\(63\) − 4.60555i − 0.580245i
\(64\) 0 0
\(65\) − 3.60555i − 0.447214i
\(66\) 0 0
\(67\) − 3.21110i − 0.392299i −0.980574 0.196149i \(-0.937156\pi\)
0.980574 0.196149i \(-0.0628437\pi\)
\(68\) 0 0
\(69\) 1.39445 0.167872
\(70\) 0 0
\(71\) 9.21110i 1.09316i 0.837408 + 0.546578i \(0.184070\pi\)
−0.837408 + 0.546578i \(0.815930\pi\)
\(72\) 0 0
\(73\) 1.39445i 0.163208i 0.996665 + 0.0816039i \(0.0260043\pi\)
−0.996665 + 0.0816039i \(0.973996\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 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
\(82\) 0 0
\(83\) 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
\(86\) 0 0
\(87\) 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
\(92\) 0 0
\(93\) 6.00000i 0.622171i
\(94\) 0 0
\(95\) −4.60555 −0.472520
\(96\) 0 0
\(97\) − 1.39445i − 0.141585i −0.997491 0.0707924i \(-0.977447\pi\)
0.997491 0.0707924i \(-0.0225528\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.39445 −0.735775 −0.367888 0.929870i \(-0.619919\pi\)
−0.367888 + 0.929870i \(0.619919\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) −4.60555 −0.449456
\(106\) 0 0
\(107\) 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
\(111\) 9.21110i 0.874279i
\(112\) 0 0
\(113\) −13.8167 −1.29976 −0.649881 0.760036i \(-0.725181\pi\)
−0.649881 + 0.760036i \(0.725181\pi\)
\(114\) 0 0
\(115\) − 1.39445i − 0.130033i
\(116\) 0 0
\(117\) 3.60555 0.333333
\(118\) 0 0
\(119\) − 21.2111i − 1.94442i
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) − 3.21110i − 0.289535i
\(124\) 0 0
\(125\) 1.00000i 0.0894427i
\(126\) 0 0
\(127\) −1.21110 −0.107468 −0.0537340 0.998555i \(-0.517112\pi\)
−0.0537340 + 0.998555i \(0.517112\pi\)
\(128\) 0 0
\(129\) −8.00000 −0.704361
\(130\) 0 0
\(131\) −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
\(134\) 0 0
\(135\) − 1.00000i − 0.0860663i
\(136\) 0 0
\(137\) − 3.21110i − 0.274343i −0.990547 0.137172i \(-0.956199\pi\)
0.990547 0.137172i \(-0.0438011\pi\)
\(138\) 0 0
\(139\) −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
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) − 4.60555i − 0.382470i
\(146\) 0 0
\(147\) −14.2111 −1.17211
\(148\) 0 0
\(149\) − 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 \(0.921495\pi\)
\(152\) 0 0
\(153\) 4.60555 0.372337
\(154\) 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 \(0.196773\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) − 6.42221i − 0.506141i
\(162\) 0 0
\(163\) 24.4222i 1.91289i 0.291905 + 0.956447i \(0.405711\pi\)
−0.291905 + 0.956447i \(0.594289\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 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
\(170\) 0 0
\(171\) − 4.60555i − 0.352195i
\(172\) 0 0
\(173\) −12.4222 −0.944443 −0.472221 0.881480i \(-0.656548\pi\)
−0.472221 + 0.881480i \(0.656548\pi\)
\(174\) 0 0
\(175\) 4.60555i 0.348147i
\(176\) 0 0
\(177\) − 9.21110i − 0.692349i
\(178\) 0 0
\(179\) 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 \(0.398663\pi\)
\(182\) 0 0
\(183\) −11.2111 −0.828749
\(184\) 0 0
\(185\) 9.21110 0.677214
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) − 4.60555i − 0.335005i
\(190\) 0 0
\(191\) 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 \(-0.0907747\pi\)
\(194\) 0 0
\(195\) − 3.60555i − 0.258199i
\(196\) 0 0
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 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
\(202\) 0 0
\(203\) − 21.2111i − 1.48873i
\(204\) 0 0
\(205\) −3.21110 −0.224273
\(206\) 0 0
\(207\) 1.39445 0.0969209
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 17.2111 1.18486 0.592431 0.805622i \(-0.298168\pi\)
0.592431 + 0.805622i \(0.298168\pi\)
\(212\) 0 0
\(213\) 9.21110i 0.631134i
\(214\) 0 0
\(215\) 8.00000i 0.545595i
\(216\) 0 0
\(217\) 27.6333 1.87587
\(218\) 0 0
\(219\) 1.39445i 0.0942281i
\(220\) 0 0
\(221\) 16.6056 1.11701
\(222\) 0 0
\(223\) 1.81665i 0.121652i 0.998148 + 0.0608261i \(0.0193735\pi\)
−0.998148 + 0.0608261i \(0.980627\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) 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 \(0.772769\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.81665 −0.119013 −0.0595065 0.998228i \(-0.518953\pi\)
−0.0595065 + 0.998228i \(0.518953\pi\)
\(234\) 0 0
\(235\) −9.21110 −0.600866
\(236\) 0 0
\(237\) 14.4222 0.936823
\(238\) 0 0
\(239\) 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 \(0.933680\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 14.2111i 0.907914i
\(246\) 0 0
\(247\) − 16.6056i − 1.05659i
\(248\) 0 0
\(249\) 2.78890i 0.176739i
\(250\) 0 0
\(251\) 13.3944 0.845450 0.422725 0.906258i \(-0.361074\pi\)
0.422725 + 0.906258i \(0.361074\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) − 4.60555i − 0.288411i
\(256\) 0 0
\(257\) −28.6056 −1.78437 −0.892183 0.451675i \(-0.850827\pi\)
−0.892183 + 0.451675i \(0.850827\pi\)
\(258\) 0 0
\(259\) 42.4222 2.63599
\(260\) 0 0
\(261\) 4.60555 0.285076
\(262\) 0 0
\(263\) 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
\(266\) 0 0
\(267\) − 15.2111i − 0.930904i
\(268\) 0 0
\(269\) 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 \(-0.995918\pi\)
0.999918 0.0128236i \(-0.00408198\pi\)
\(272\) 0 0
\(273\) − 16.6056i − 1.00501i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −16.4222 −0.986715 −0.493357 0.869827i \(-0.664230\pi\)
−0.493357 + 0.869827i \(0.664230\pi\)
\(278\) 0 0
\(279\) 6.00000i 0.359211i
\(280\) 0 0
\(281\) − 27.2111i − 1.62328i −0.584159 0.811639i \(-0.698576\pi\)
0.584159 0.811639i \(-0.301424\pi\)
\(282\) 0 0
\(283\) −10.4222 −0.619536 −0.309768 0.950812i \(-0.600251\pi\)
−0.309768 + 0.950812i \(0.600251\pi\)
\(284\) 0 0
\(285\) −4.60555 −0.272809
\(286\) 0 0
\(287\) −14.7889 −0.872961
\(288\) 0 0
\(289\) 4.21110 0.247712
\(290\) 0 0
\(291\) − 1.39445i − 0.0817440i
\(292\) 0 0
\(293\) − 18.0000i − 1.05157i −0.850617 0.525786i \(-0.823771\pi\)
0.850617 0.525786i \(-0.176229\pi\)
\(294\) 0 0
\(295\) −9.21110 −0.536291
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.02776 0.290763
\(300\) 0 0
\(301\) 36.8444i 2.12368i
\(302\) 0 0
\(303\) −7.39445 −0.424800
\(304\) 0 0
\(305\) 11.2111i 0.641946i
\(306\) 0 0
\(307\) − 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 \(-0.610505\pi\)
−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 \(0.467759\pi\)
\(314\) 0 0
\(315\) −4.60555 −0.259493
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 21.2111i − 1.18022i
\(324\) 0 0
\(325\) −3.60555 −0.200000
\(326\) 0 0
\(327\) − 1.39445i − 0.0771132i
\(328\) 0 0
\(329\) −42.4222 −2.33881
\(330\) 0 0
\(331\) − 16.6056i − 0.912724i −0.889794 0.456362i \(-0.849152\pi\)
0.889794 0.456362i \(-0.150848\pi\)
\(332\) 0 0
\(333\) 9.21110i 0.504765i
\(334\) 0 0
\(335\) −3.21110 −0.175441
\(336\) 0 0
\(337\) 13.6333 0.742654 0.371327 0.928502i \(-0.378903\pi\)
0.371327 + 0.928502i \(0.378903\pi\)
\(338\) 0 0
\(339\) −13.8167 −0.750418
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 33.2111i 1.79323i
\(344\) 0 0
\(345\) − 1.39445i − 0.0750746i
\(346\) 0 0
\(347\) 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 \(0.932911\pi\)
\(350\) 0 0
\(351\) 3.60555 0.192450
\(352\) 0 0
\(353\) 8.78890i 0.467786i 0.972262 + 0.233893i \(0.0751465\pi\)
−0.972262 + 0.233893i \(0.924853\pi\)
\(354\) 0 0
\(355\) 9.21110 0.488875
\(356\) 0 0
\(357\) − 21.2111i − 1.12261i
\(358\) 0 0
\(359\) 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
\(362\) 0 0
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) 1.39445 0.0729888
\(366\) 0 0
\(367\) 19.6333 1.02485 0.512425 0.858732i \(-0.328747\pi\)
0.512425 + 0.858732i \(0.328747\pi\)
\(368\) 0 0
\(369\) − 3.21110i − 0.167163i
\(370\) 0 0
\(371\) − 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
\(375\) 1.00000i 0.0516398i
\(376\) 0 0
\(377\) 16.6056 0.855229
\(378\) 0 0
\(379\) 35.0278i 1.79925i 0.436658 + 0.899627i \(0.356162\pi\)
−0.436658 + 0.899627i \(0.643838\pi\)
\(380\) 0 0
\(381\) −1.21110 −0.0620467
\(382\) 0 0
\(383\) − 27.6333i − 1.41200i −0.708214 0.705998i \(-0.750499\pi\)
0.708214 0.705998i \(-0.249501\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.00000 −0.406663
\(388\) 0 0
\(389\) 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
\(392\) 0 0
\(393\) −22.6056 −1.14030
\(394\) 0 0
\(395\) − 14.4222i − 0.725660i
\(396\) 0 0
\(397\) 3.63331i 0.182350i 0.995835 + 0.0911752i \(0.0290623\pi\)
−0.995835 + 0.0911752i \(0.970938\pi\)
\(398\) 0 0
\(399\) −21.2111 −1.06188
\(400\) 0 0
\(401\) − 8.78890i − 0.438897i −0.975624 0.219448i \(-0.929574\pi\)
0.975624 0.219448i \(-0.0704257\pi\)
\(402\) 0 0
\(403\) 21.6333i 1.07763i
\(404\) 0 0
\(405\) − 1.00000i − 0.0496904i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 14.7889i 0.731264i 0.930760 + 0.365632i \(0.119147\pi\)
−0.930760 + 0.365632i \(0.880853\pi\)
\(410\) 0 0
\(411\) − 3.21110i − 0.158392i
\(412\) 0 0
\(413\) −42.4222 −2.08746
\(414\) 0 0
\(415\) 2.78890 0.136902
\(416\) 0 0
\(417\) −17.2111 −0.842831
\(418\) 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 \(0.839583\pi\)
\(422\) 0 0
\(423\) − 9.21110i − 0.447859i
\(424\) 0 0
\(425\) −4.60555 −0.223402
\(426\) 0 0
\(427\) 51.6333i 2.49871i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000i 0.578020i 0.957326 + 0.289010i \(0.0933260\pi\)
−0.957326 + 0.289010i \(0.906674\pi\)
\(432\) 0 0
\(433\) 19.2111 0.923227 0.461613 0.887081i \(-0.347271\pi\)
0.461613 + 0.887081i \(0.347271\pi\)
\(434\) 0 0
\(435\) − 4.60555i − 0.220819i
\(436\) 0 0
\(437\) − 6.42221i − 0.307216i
\(438\) 0 0
\(439\) −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
\(442\) 0 0
\(443\) −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
\(446\) 0 0
\(447\) − 15.2111i − 0.719460i
\(448\) 0 0
\(449\) − 33.6333i − 1.58725i −0.608405 0.793627i \(-0.708190\pi\)
0.608405 0.793627i \(-0.291810\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 6.00000i 0.281905i
\(454\) 0 0
\(455\) −16.6056 −0.778480
\(456\) 0 0
\(457\) − 38.2389i − 1.78874i −0.447330 0.894369i \(-0.647625\pi\)
0.447330 0.894369i \(-0.352375\pi\)
\(458\) 0 0
\(459\) 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 \(-0.260253\pi\)
\(464\) 0 0
\(465\) 6.00000 0.278243
\(466\) 0 0
\(467\) 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
\(470\) 0 0
\(471\) 20.4222 0.941006
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.60555i 0.211317i
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 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
\(482\) 0 0
\(483\) − 6.42221i − 0.292220i
\(484\) 0 0
\(485\) −1.39445 −0.0633187
\(486\) 0 0
\(487\) − 0.972244i − 0.0440566i −0.999757 0.0220283i \(-0.992988\pi\)
0.999757 0.0220283i \(-0.00701239\pi\)
\(488\) 0 0
\(489\) 24.4222i 1.10441i
\(490\) 0 0
\(491\) −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
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 42.4222 1.90290
\(498\) 0 0
\(499\) 23.0278i 1.03086i 0.856930 + 0.515432i \(0.172369\pi\)
−0.856930 + 0.515432i \(0.827631\pi\)
\(500\) 0 0
\(501\) 9.21110i 0.411522i
\(502\) 0 0
\(503\) −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
\(507\) 13.0000 0.577350
\(508\) 0 0
\(509\) 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
\(512\) 0 0
\(513\) − 4.60555i − 0.203340i
\(514\) 0 0
\(515\) − 4.00000i − 0.176261i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −12.4222 −0.545274
\(520\) 0 0
\(521\) 21.6333 0.947772 0.473886 0.880586i \(-0.342851\pi\)
0.473886 + 0.880586i \(0.342851\pi\)
\(522\) 0 0
\(523\) −32.8444 −1.43619 −0.718093 0.695947i \(-0.754985\pi\)
−0.718093 + 0.695947i \(0.754985\pi\)
\(524\) 0 0
\(525\) 4.60555i 0.201003i
\(526\) 0 0
\(527\) 27.6333i 1.20373i
\(528\) 0 0
\(529\) −21.0555 −0.915457
\(530\) 0 0
\(531\) − 9.21110i − 0.399728i
\(532\) 0 0
\(533\) − 11.5778i − 0.501490i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 19.8167 0.855152
\(538\) 0 0
\(539\) 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 0
\(543\) 8.42221 0.361431
\(544\) 0 0
\(545\) −1.39445 −0.0597316
\(546\) 0 0
\(547\) −14.4222 −0.616649 −0.308324 0.951281i \(-0.599768\pi\)
−0.308324 + 0.951281i \(0.599768\pi\)
\(548\) 0 0
\(549\) −11.2111 −0.478478
\(550\) 0 0
\(551\) − 21.2111i − 0.903623i
\(552\) 0 0
\(553\) − 66.4222i − 2.82456i
\(554\) 0 0
\(555\) 9.21110 0.390990
\(556\) 0 0
\(557\) − 11.5778i − 0.490567i −0.969451 0.245283i \(-0.921119\pi\)
0.969451 0.245283i \(-0.0788810\pi\)
\(558\) 0 0
\(559\) −28.8444 −1.21999
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −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
\(566\) 0 0
\(567\) − 4.60555i − 0.193415i
\(568\) 0 0
\(569\) 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 \(-0.283509\pi\)
0.628892 + 0.777493i \(0.283509\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) −1.39445 −0.0581525
\(576\) 0 0
\(577\) 37.3944i 1.55675i 0.627799 + 0.778376i \(0.283956\pi\)
−0.627799 + 0.778376i \(0.716044\pi\)
\(578\) 0 0
\(579\) − 7.81665i − 0.324849i
\(580\) 0 0
\(581\) 12.8444 0.532876
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) − 3.60555i − 0.149071i
\(586\) 0 0
\(587\) − 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
\(590\) 0 0
\(591\) 6.00000i 0.246807i
\(592\) 0 0
\(593\) 24.4222i 1.00290i 0.865187 + 0.501450i \(0.167200\pi\)
−0.865187 + 0.501450i \(0.832800\pi\)
\(594\) 0 0
\(595\) −21.2111 −0.869570
\(596\) 0 0
\(597\) −22.4222 −0.917680
\(598\) 0 0
\(599\) 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 \(0.489394\pi\)
\(602\) 0 0
\(603\) − 3.21110i − 0.130766i
\(604\) 0 0
\(605\) − 11.0000i − 0.447214i
\(606\) 0 0
\(607\) −17.2111 −0.698577 −0.349289 0.937015i \(-0.613577\pi\)
−0.349289 + 0.937015i \(0.613577\pi\)
\(608\) 0 0
\(609\) − 21.2111i − 0.859517i
\(610\) 0 0
\(611\) − 33.2111i − 1.34358i
\(612\) 0 0
\(613\) 33.2111i 1.34138i 0.741736 + 0.670692i \(0.234003\pi\)
−0.741736 + 0.670692i \(0.765997\pi\)
\(614\) 0 0
\(615\) −3.21110 −0.129484
\(616\) 0 0
\(617\) − 12.4222i − 0.500099i −0.968233 0.250050i \(-0.919553\pi\)
0.968233 0.250050i \(-0.0804469\pi\)
\(618\) 0 0
\(619\) 25.8167i 1.03766i 0.854878 + 0.518829i \(0.173632\pi\)
−0.854878 + 0.518829i \(0.826368\pi\)
\(620\) 0 0
\(621\) 1.39445 0.0559573
\(622\) 0 0
\(623\) −70.0555 −2.80671
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 42.4222i 1.69148i
\(630\) 0 0
\(631\) 3.21110i 0.127832i 0.997955 + 0.0639160i \(0.0203590\pi\)
−0.997955 + 0.0639160i \(0.979641\pi\)
\(632\) 0 0
\(633\) 17.2111 0.684080
\(634\) 0 0
\(635\) 1.21110i 0.0480611i
\(636\) 0 0
\(637\) −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 \(0.939167\pi\)
\(644\) 0 0
\(645\) 8.00000i 0.315000i
\(646\) 0 0
\(647\) 34.6056 1.36048 0.680242 0.732987i \(-0.261875\pi\)
0.680242 + 0.732987i \(0.261875\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 27.6333 1.08303
\(652\) 0 0
\(653\) 39.2111 1.53445 0.767225 0.641379i \(-0.221637\pi\)
0.767225 + 0.641379i \(0.221637\pi\)
\(654\) 0 0
\(655\) 22.6056i 0.883272i
\(656\) 0 0
\(657\) 1.39445i 0.0544026i
\(658\) 0 0
\(659\) 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 \(-0.431647\pi\)
\(662\) 0 0
\(663\) 16.6056 0.644906
\(664\) 0 0
\(665\) 21.2111i 0.822531i
\(666\) 0 0
\(667\) 6.42221 0.248669
\(668\) 0 0
\(669\) 1.81665i 0.0702359i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −37.6333 −1.45066 −0.725329 0.688403i \(-0.758312\pi\)
−0.725329 + 0.688403i \(0.758312\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −28.0555 −1.07826 −0.539130 0.842222i \(-0.681247\pi\)
−0.539130 + 0.842222i \(0.681247\pi\)
\(678\) 0 0
\(679\) −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 \(0.943611\pi\)
\(684\) 0 0
\(685\) −3.21110 −0.122690
\(686\) 0 0
\(687\) 19.8167i 0.756053i
\(688\) 0 0
\(689\) 21.6333 0.824163
\(690\) 0 0
\(691\) 20.2389i 0.769922i 0.922933 + 0.384961i \(0.125785\pi\)
−0.922933 + 0.384961i \(0.874215\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 17.2111i 0.652854i
\(696\) 0 0
\(697\) − 14.7889i − 0.560169i
\(698\) 0 0
\(699\) −1.81665 −0.0687122
\(700\) 0 0
\(701\) 47.0278 1.77621 0.888107 0.459637i \(-0.152020\pi\)
0.888107 + 0.459637i \(0.152020\pi\)
\(702\) 0 0
\(703\) 42.4222 1.59998
\(704\) 0 0
\(705\) −9.21110 −0.346910
\(706\) 0 0
\(707\) 34.0555i 1.28079i
\(708\) 0 0
\(709\) 1.39445i 0.0523696i 0.999657 + 0.0261848i \(0.00833584\pi\)
−0.999657 + 0.0261848i \(0.991664\pi\)
\(710\) 0 0
\(711\) 14.4222 0.540875
\(712\) 0 0
\(713\) 8.36669i 0.313335i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 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
\(722\) 0 0
\(723\) 6.42221i 0.238844i
\(724\) 0 0
\(725\) −4.60555 −0.171046
\(726\) 0 0
\(727\) 14.4222 0.534890 0.267445 0.963573i \(-0.413821\pi\)
0.267445 + 0.963573i \(0.413821\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −36.8444 −1.36274
\(732\) 0 0
\(733\) 34.0555i 1.25787i 0.777458 + 0.628935i \(0.216509\pi\)
−0.777458 + 0.628935i \(0.783491\pi\)
\(734\) 0 0
\(735\) 14.2111i 0.524184i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 20.2389i 0.744498i 0.928133 + 0.372249i \(0.121413\pi\)
−0.928133 + 0.372249i \(0.878587\pi\)
\(740\) 0 0
\(741\) − 16.6056i − 0.610020i
\(742\) 0 0
\(743\) 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
\(746\) 0 0
\(747\) 2.78890i 0.102040i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 10.4222 0.380312 0.190156 0.981754i \(-0.439101\pi\)
0.190156 + 0.981754i \(0.439101\pi\)
\(752\) 0 0
\(753\) 13.3944 0.488121
\(754\) 0 0
\(755\) 6.00000 0.218362
\(756\) 0 0
\(757\) 12.7889 0.464820 0.232410 0.972618i \(-0.425339\pi\)
0.232410 + 0.972618i \(0.425339\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 33.6333i − 1.21921i −0.792707 0.609603i \(-0.791329\pi\)
0.792707 0.609603i \(-0.208671\pi\)
\(762\) 0 0
\(763\) −6.42221 −0.232499
\(764\) 0 0
\(765\) − 4.60555i − 0.166514i
\(766\) 0 0
\(767\) − 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
\(771\) −28.6056 −1.03020
\(772\) 0 0
\(773\) − 30.0000i − 1.07903i −0.841978 0.539513i \(-0.818609\pi\)
0.841978 0.539513i \(-0.181391\pi\)
\(774\) 0 0
\(775\) − 6.00000i − 0.215526i
\(776\) 0 0
\(777\) 42.4222 1.52189
\(778\) 0 0
\(779\) −14.7889 −0.529867
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 4.60555 0.164589
\(784\) 0 0
\(785\) − 20.4222i − 0.728900i
\(786\) 0 0
\(787\) 49.2666i 1.75617i 0.478509 + 0.878083i \(0.341177\pi\)
−0.478509 + 0.878083i \(0.658823\pi\)
\(788\) 0 0
\(789\) 7.81665 0.278280
\(790\) 0 0
\(791\) 63.6333i 2.26254i
\(792\) 0 0
\(793\) −40.4222 −1.43543
\(794\) 0 0
\(795\) − 6.00000i − 0.212798i
\(796\) 0 0
\(797\) −6.00000 −0.212531 −0.106265 0.994338i \(-0.533889\pi\)
−0.106265 + 0.994338i \(0.533889\pi\)
\(798\) 0 0
\(799\) − 42.4222i − 1.50079i
\(800\) 0 0
\(801\) − 15.2111i − 0.537458i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −6.42221 −0.226353
\(806\) 0 0
\(807\) 25.8167 0.908789
\(808\) 0 0
\(809\) −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 \(0.191521\pi\)
−0.824385 + 0.566030i \(0.808479\pi\)
\(812\) 0 0
\(813\) − 0.422205i − 0.0148074i
\(814\) 0 0
\(815\) 24.4222 0.855473
\(816\) 0 0
\(817\) 36.8444i 1.28902i
\(818\) 0 0
\(819\) − 16.6056i − 0.580245i
\(820\) 0 0
\(821\) 3.21110i 0.112068i 0.998429 + 0.0560341i \(0.0178456\pi\)
−0.998429 + 0.0560341i \(0.982154\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 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 \(0.197567\pi\)
\(830\) 0 0
\(831\) −16.4222 −0.569680
\(832\) 0 0
\(833\) −65.4500 −2.26771
\(834\) 0 0
\(835\) 9.21110 0.318763
\(836\) 0 0
\(837\) 6.00000i 0.207390i
\(838\) 0 0
\(839\) − 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
\(842\) 0 0
\(843\) − 27.2111i − 0.937200i
\(844\) 0 0
\(845\) − 13.0000i − 0.447214i
\(846\) 0 0
\(847\) − 50.6611i − 1.74073i
\(848\) 0 0
\(849\) −10.4222 −0.357689
\(850\) 0 0
\(851\) 12.8444i 0.440301i
\(852\) 0 0
\(853\) 14.7889i 0.506362i 0.967419 + 0.253181i \(0.0814769\pi\)
−0.967419 + 0.253181i \(0.918523\pi\)
\(854\) 0 0
\(855\) −4.60555 −0.157507
\(856\) 0 0
\(857\) −23.0278 −0.786613 −0.393307 0.919407i \(-0.628669\pi\)
−0.393307 + 0.919407i \(0.628669\pi\)
\(858\) 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
\(862\) 0 0
\(863\) 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
\(866\) 0 0
\(867\) 4.21110 0.143017
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) − 11.5778i − 0.392299i
\(872\) 0 0
\(873\) − 1.39445i − 0.0471949i
\(874\) 0 0
\(875\) 4.60555 0.155696
\(876\) 0 0
\(877\) 24.8444i 0.838936i 0.907770 + 0.419468i \(0.137783\pi\)
−0.907770 + 0.419468i \(0.862217\pi\)
\(878\) 0 0
\(879\) − 18.0000i − 0.607125i
\(880\) 0 0
\(881\) 39.2111 1.32106 0.660528 0.750802i \(-0.270333\pi\)
0.660528 + 0.750802i \(0.270333\pi\)
\(882\) 0 0
\(883\) −9.57779 −0.322318 −0.161159 0.986928i \(-0.551523\pi\)
−0.161159 + 0.986928i \(0.551523\pi\)
\(884\) 0 0
\(885\) −9.21110 −0.309628
\(886\) 0 0
\(887\) −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
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −42.4222 −1.41960
\(894\) 0 0
\(895\) − 19.8167i − 0.662398i
\(896\) 0 0
\(897\) 5.02776 0.167872
\(898\) 0 0
\(899\) 27.6333i 0.921622i
\(900\) 0 0
\(901\) 27.6333 0.920599
\(902\) 0 0
\(903\) 36.8444i 1.22611i
\(904\) 0 0
\(905\) − 8.42221i − 0.279964i
\(906\) 0 0
\(907\) −21.5778 −0.716479 −0.358239 0.933630i \(-0.616623\pi\)
−0.358239 + 0.933630i \(0.616623\pi\)
\(908\) 0 0
\(909\) −7.39445 −0.245258
\(910\) 0 0
\(911\) 27.6333 0.915532 0.457766 0.889073i \(-0.348650\pi\)
0.457766 + 0.889073i \(0.348650\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 11.2111i 0.370628i
\(916\) 0 0
\(917\) 104.111i 3.43805i
\(918\) 0 0
\(919\) −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
\(922\) 0 0
\(923\) 33.2111i 1.09316i
\(924\) 0 0
\(925\) − 9.21110i − 0.302859i
\(926\) 0 0
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) − 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
\(932\) 0 0
\(933\) −12.0000 −0.392862
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −10.3667 −0.338665 −0.169333 0.985559i \(-0.554161\pi\)
−0.169333 + 0.985559i \(0.554161\pi\)
\(938\) 0 0
\(939\) 3.57779 0.116757
\(940\) 0 0
\(941\) 54.0000i 1.76035i 0.474650 + 0.880175i \(0.342575\pi\)
−0.474650 + 0.880175i \(0.657425\pi\)
\(942\) 0 0
\(943\) − 4.47772i − 0.145815i
\(944\) 0 0
\(945\) −4.60555 −0.149819
\(946\) 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
\(950\) 0 0
\(951\) 18.0000i 0.583690i
\(952\) 0 0
\(953\) −20.2389 −0.655601 −0.327800 0.944747i \(-0.606307\pi\)
−0.327800 + 0.944747i \(0.606307\pi\)
\(954\) 0 0
\(955\) − 12.0000i − 0.388311i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −14.7889 −0.477558
\(960\) 0 0
\(961\) −5.00000 −0.161290
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −7.81665 −0.251627
\(966\) 0 0
\(967\) − 8.23886i − 0.264944i −0.991187 0.132472i \(-0.957709\pi\)
0.991187 0.132472i \(-0.0422914\pi\)
\(968\) 0 0
\(969\) − 21.2111i − 0.681399i
\(970\) 0 0
\(971\) 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 0
\(975\) −3.60555 −0.115470
\(976\) 0 0
\(977\) − 18.8444i − 0.602886i −0.953484 0.301443i \(-0.902532\pi\)
0.953484 0.301443i \(-0.0974683\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) − 1.39445i − 0.0445213i
\(982\) 0 0
\(983\) − 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
\(986\) 0 0
\(987\) −42.4222 −1.35031
\(988\) 0 0
\(989\) −11.1556 −0.354727
\(990\) 0 0
\(991\) 22.4222 0.712265 0.356132 0.934436i \(-0.384095\pi\)
0.356132 + 0.934436i \(0.384095\pi\)
\(992\) 0 0
\(993\) − 16.6056i − 0.526961i
\(994\) 0 0
\(995\) 22.4222i 0.710832i
\(996\) 0 0
\(997\) −16.4222 −0.520096 −0.260048 0.965596i \(-0.583738\pi\)
−0.260048 + 0.965596i \(0.583738\pi\)
\(998\) 0 0
\(999\) 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 3120.2.g.q.961.1 4
4.3 odd 2 390.2.b.c.181.2 4
12.11 even 2 1170.2.b.d.181.4 4
13.12 even 2 inner 3120.2.g.q.961.4 4
20.3 even 4 1950.2.f.m.649.2 4
20.7 even 4 1950.2.f.n.649.3 4
20.19 odd 2 1950.2.b.k.1351.3 4
52.31 even 4 5070.2.a.z.1.1 2
52.47 even 4 5070.2.a.bf.1.2 2
52.51 odd 2 390.2.b.c.181.3 yes 4
156.155 even 2 1170.2.b.d.181.1 4
260.103 even 4 1950.2.f.n.649.1 4
260.207 even 4 1950.2.f.m.649.4 4
260.259 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 4.3 odd 2
390.2.b.c.181.3 yes 4 52.51 odd 2
1170.2.b.d.181.1 4 156.155 even 2
1170.2.b.d.181.4 4 12.11 even 2
1950.2.b.k.1351.2 4 260.259 odd 2
1950.2.b.k.1351.3 4 20.19 odd 2
1950.2.f.m.649.2 4 20.3 even 4
1950.2.f.m.649.4 4 260.207 even 4
1950.2.f.n.649.1 4 260.103 even 4
1950.2.f.n.649.3 4 20.7 even 4
3120.2.g.q.961.1 4 1.1 even 1 trivial
3120.2.g.q.961.4 4 13.12 even 2 inner
5070.2.a.z.1.1 2 52.31 even 4
5070.2.a.bf.1.2 2 52.47 even 4