Properties

Label 1050.2.g.e.799.1
Level 1050
Weight 2
Character 1050.799
Analytic conductor 8.384
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1050.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.38429221223\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(-1.00000i\)
Character \(\chi\) = 1050.799
Dual form 1050.2.g.e.799.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +6.00000 q^{11} +1.00000i q^{12} +1.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} +1.00000 q^{21} -6.00000i q^{22} +3.00000i q^{23} +1.00000 q^{24} +1.00000 q^{26} +1.00000i q^{27} -1.00000i q^{28} -3.00000 q^{29} +5.00000 q^{31} -1.00000i q^{32} -6.00000i q^{33} +3.00000 q^{34} +1.00000 q^{36} -10.0000i q^{37} -4.00000i q^{38} +1.00000 q^{39} +9.00000 q^{41} -1.00000i q^{42} +1.00000i q^{43} -6.00000 q^{44} +3.00000 q^{46} -1.00000i q^{48} -1.00000 q^{49} +3.00000 q^{51} -1.00000i q^{52} -9.00000i q^{53} +1.00000 q^{54} -1.00000 q^{56} -4.00000i q^{57} +3.00000i q^{58} -9.00000 q^{59} +11.0000 q^{61} -5.00000i q^{62} -1.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} -4.00000i q^{67} -3.00000i q^{68} +3.00000 q^{69} -12.0000 q^{71} -1.00000i q^{72} +10.0000i q^{73} -10.0000 q^{74} -4.00000 q^{76} +6.00000i q^{77} -1.00000i q^{78} +10.0000 q^{79} +1.00000 q^{81} -9.00000i q^{82} -9.00000i q^{83} -1.00000 q^{84} +1.00000 q^{86} +3.00000i q^{87} +6.00000i q^{88} +6.00000 q^{89} -1.00000 q^{91} -3.00000i q^{92} -5.00000i q^{93} -1.00000 q^{96} +14.0000i q^{97} +1.00000i q^{98} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + 12q^{11} + 2q^{14} + 2q^{16} + 8q^{19} + 2q^{21} + 2q^{24} + 2q^{26} - 6q^{29} + 10q^{31} + 6q^{34} + 2q^{36} + 2q^{39} + 18q^{41} - 12q^{44} + 6q^{46} - 2q^{49} + 6q^{51} + 2q^{54} - 2q^{56} - 18q^{59} + 22q^{61} - 2q^{64} - 12q^{66} + 6q^{69} - 24q^{71} - 20q^{74} - 8q^{76} + 20q^{79} + 2q^{81} - 2q^{84} + 2q^{86} + 12q^{89} - 2q^{91} - 2q^{96} - 12q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 1.00000i 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) − 6.00000i − 1.27920i
\(23\) 3.00000i 0.625543i 0.949828 + 0.312772i \(0.101257\pi\)
−0.949828 + 0.312772i \(0.898743\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) 1.00000i 0.192450i
\(28\) − 1.00000i − 0.188982i
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 6.00000i − 1.04447i
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) − 1.00000i − 0.154303i
\(43\) 1.00000i 0.152499i 0.997089 + 0.0762493i \(0.0242945\pi\)
−0.997089 + 0.0762493i \(0.975706\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) − 1.00000i − 0.138675i
\(53\) − 9.00000i − 1.23625i −0.786082 0.618123i \(-0.787894\pi\)
0.786082 0.618123i \(-0.212106\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) − 4.00000i − 0.529813i
\(58\) 3.00000i 0.393919i
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) 11.0000 1.40841 0.704203 0.709999i \(-0.251305\pi\)
0.704203 + 0.709999i \(0.251305\pi\)
\(62\) − 5.00000i − 0.635001i
\(63\) − 1.00000i − 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) − 3.00000i − 0.363803i
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) −10.0000 −1.16248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 6.00000i 0.683763i
\(78\) − 1.00000i − 0.113228i
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 9.00000i − 0.993884i
\(83\) − 9.00000i − 0.987878i −0.869496 0.493939i \(-0.835557\pi\)
0.869496 0.493939i \(-0.164443\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) 1.00000 0.107833
\(87\) 3.00000i 0.321634i
\(88\) 6.00000i 0.639602i
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) − 3.00000i − 0.312772i
\(93\) − 5.00000i − 0.518476i
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) 1.00000i 0.101015i
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) − 3.00000i − 0.297044i
\(103\) − 17.0000i − 1.67506i −0.546392 0.837530i \(-0.683999\pi\)
0.546392 0.837530i \(-0.316001\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) 18.0000i 1.74013i 0.492941 + 0.870063i \(0.335922\pi\)
−0.492941 + 0.870063i \(0.664078\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 0 0
\(111\) −10.0000 −0.949158
\(112\) 1.00000i 0.0944911i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 3.00000 0.278543
\(117\) − 1.00000i − 0.0924500i
\(118\) 9.00000i 0.828517i
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) − 11.0000i − 0.995893i
\(123\) − 9.00000i − 0.811503i
\(124\) −5.00000 −0.449013
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) − 10.0000i − 0.887357i −0.896186 0.443678i \(-0.853673\pi\)
0.896186 0.443678i \(-0.146327\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 6.00000i 0.522233i
\(133\) 4.00000i 0.346844i
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −3.00000 −0.257248
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) − 3.00000i − 0.255377i
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 12.0000i 1.00702i
\(143\) 6.00000i 0.501745i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) 1.00000i 0.0824786i
\(148\) 10.0000i 0.821995i
\(149\) 9.00000 0.737309 0.368654 0.929567i \(-0.379819\pi\)
0.368654 + 0.929567i \(0.379819\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 4.00000i 0.324443i
\(153\) − 3.00000i − 0.242536i
\(154\) 6.00000 0.483494
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) − 10.0000i − 0.798087i −0.916932 0.399043i \(-0.869342\pi\)
0.916932 0.399043i \(-0.130658\pi\)
\(158\) − 10.0000i − 0.795557i
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) − 1.00000i − 0.0785674i
\(163\) − 5.00000i − 0.391630i −0.980641 0.195815i \(-0.937265\pi\)
0.980641 0.195815i \(-0.0627352\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) 18.0000i 1.39288i 0.717614 + 0.696441i \(0.245234\pi\)
−0.717614 + 0.696441i \(0.754766\pi\)
\(168\) 1.00000i 0.0771517i
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) − 1.00000i − 0.0762493i
\(173\) 12.0000i 0.912343i 0.889892 + 0.456172i \(0.150780\pi\)
−0.889892 + 0.456172i \(0.849220\pi\)
\(174\) 3.00000 0.227429
\(175\) 0 0
\(176\) 6.00000 0.452267
\(177\) 9.00000i 0.676481i
\(178\) − 6.00000i − 0.449719i
\(179\) −18.0000 −1.34538 −0.672692 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 1.00000i 0.0741249i
\(183\) − 11.0000i − 0.813143i
\(184\) −3.00000 −0.221163
\(185\) 0 0
\(186\) −5.00000 −0.366618
\(187\) 18.0000i 1.31629i
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −15.0000 −1.08536 −0.542681 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 22.0000i 1.58359i 0.610784 + 0.791797i \(0.290854\pi\)
−0.610784 + 0.791797i \(0.709146\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) − 15.0000i − 1.06871i −0.845262 0.534353i \(-0.820555\pi\)
0.845262 0.534353i \(-0.179445\pi\)
\(198\) 6.00000i 0.426401i
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 12.0000i 0.844317i
\(203\) − 3.00000i − 0.210559i
\(204\) −3.00000 −0.210042
\(205\) 0 0
\(206\) −17.0000 −1.18445
\(207\) − 3.00000i − 0.208514i
\(208\) 1.00000i 0.0693375i
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) −25.0000 −1.72107 −0.860535 0.509390i \(-0.829871\pi\)
−0.860535 + 0.509390i \(0.829871\pi\)
\(212\) 9.00000i 0.618123i
\(213\) 12.0000i 0.822226i
\(214\) 18.0000 1.23045
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 5.00000i 0.339422i
\(218\) 8.00000i 0.541828i
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) −3.00000 −0.201802
\(222\) 10.0000i 0.671156i
\(223\) 19.0000i 1.27233i 0.771551 + 0.636167i \(0.219481\pi\)
−0.771551 + 0.636167i \(0.780519\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 6.00000 0.399114
\(227\) 3.00000i 0.199117i 0.995032 + 0.0995585i \(0.0317430\pi\)
−0.995032 + 0.0995585i \(0.968257\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 6.00000 0.394771
\(232\) − 3.00000i − 0.196960i
\(233\) − 12.0000i − 0.786146i −0.919507 0.393073i \(-0.871412\pi\)
0.919507 0.393073i \(-0.128588\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) 9.00000 0.585850
\(237\) − 10.0000i − 0.649570i
\(238\) 3.00000i 0.194461i
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) − 25.0000i − 1.60706i
\(243\) − 1.00000i − 0.0641500i
\(244\) −11.0000 −0.704203
\(245\) 0 0
\(246\) −9.00000 −0.573819
\(247\) 4.00000i 0.254514i
\(248\) 5.00000i 0.317500i
\(249\) −9.00000 −0.570352
\(250\) 0 0
\(251\) −27.0000 −1.70422 −0.852112 0.523359i \(-0.824679\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 1.00000i 0.0629941i
\(253\) 18.0000i 1.13165i
\(254\) −10.0000 −0.627456
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 15.0000i 0.935674i 0.883815 + 0.467837i \(0.154967\pi\)
−0.883815 + 0.467837i \(0.845033\pi\)
\(258\) − 1.00000i − 0.0622573i
\(259\) 10.0000 0.621370
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) − 9.00000i − 0.554964i −0.960731 0.277482i \(-0.910500\pi\)
0.960731 0.277482i \(-0.0894999\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) − 6.00000i − 0.367194i
\(268\) 4.00000i 0.244339i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 3.00000i 0.181902i
\(273\) 1.00000i 0.0605228i
\(274\) 0 0
\(275\) 0 0
\(276\) −3.00000 −0.180579
\(277\) − 10.0000i − 0.600842i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) − 16.0000i − 0.959616i
\(279\) −5.00000 −0.299342
\(280\) 0 0
\(281\) −24.0000 −1.43172 −0.715860 0.698244i \(-0.753965\pi\)
−0.715860 + 0.698244i \(0.753965\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) 9.00000i 0.531253i
\(288\) 1.00000i 0.0589256i
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) − 10.0000i − 0.585206i
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) 6.00000i 0.348155i
\(298\) − 9.00000i − 0.521356i
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) −1.00000 −0.0576390
\(302\) − 2.00000i − 0.115087i
\(303\) 12.0000i 0.689382i
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) −3.00000 −0.171499
\(307\) 2.00000i 0.114146i 0.998370 + 0.0570730i \(0.0181768\pi\)
−0.998370 + 0.0570730i \(0.981823\pi\)
\(308\) − 6.00000i − 0.341882i
\(309\) −17.0000 −0.967096
\(310\) 0 0
\(311\) −18.0000 −1.02069 −0.510343 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 1.00000i 0.0566139i
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) −10.0000 −0.564333
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) − 27.0000i − 1.51647i −0.651981 0.758236i \(-0.726062\pi\)
0.651981 0.758236i \(-0.273938\pi\)
\(318\) 9.00000i 0.504695i
\(319\) −18.0000 −1.00781
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 3.00000i 0.167183i
\(323\) 12.0000i 0.667698i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −5.00000 −0.276924
\(327\) 8.00000i 0.442401i
\(328\) 9.00000i 0.496942i
\(329\) 0 0
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) 9.00000i 0.493939i
\(333\) 10.0000i 0.547997i
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) − 13.0000i − 0.708155i −0.935216 0.354078i \(-0.884795\pi\)
0.935216 0.354078i \(-0.115205\pi\)
\(338\) − 12.0000i − 0.652714i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 30.0000 1.62459
\(342\) 4.00000i 0.216295i
\(343\) − 1.00000i − 0.0539949i
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) 12.0000 0.645124
\(347\) 18.0000i 0.966291i 0.875540 + 0.483145i \(0.160506\pi\)
−0.875540 + 0.483145i \(0.839494\pi\)
\(348\) − 3.00000i − 0.160817i
\(349\) −17.0000 −0.909989 −0.454995 0.890494i \(-0.650359\pi\)
−0.454995 + 0.890494i \(0.650359\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) − 6.00000i − 0.319801i
\(353\) 30.0000i 1.59674i 0.602168 + 0.798369i \(0.294304\pi\)
−0.602168 + 0.798369i \(0.705696\pi\)
\(354\) 9.00000 0.478345
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 3.00000i 0.158777i
\(358\) 18.0000i 0.951330i
\(359\) 3.00000 0.158334 0.0791670 0.996861i \(-0.474774\pi\)
0.0791670 + 0.996861i \(0.474774\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 2.00000i − 0.105118i
\(363\) − 25.0000i − 1.31216i
\(364\) 1.00000 0.0524142
\(365\) 0 0
\(366\) −11.0000 −0.574979
\(367\) − 37.0000i − 1.93138i −0.259690 0.965692i \(-0.583620\pi\)
0.259690 0.965692i \(-0.416380\pi\)
\(368\) 3.00000i 0.156386i
\(369\) −9.00000 −0.468521
\(370\) 0 0
\(371\) 9.00000 0.467257
\(372\) 5.00000i 0.259238i
\(373\) − 8.00000i − 0.414224i −0.978317 0.207112i \(-0.933593\pi\)
0.978317 0.207112i \(-0.0664065\pi\)
\(374\) 18.0000 0.930758
\(375\) 0 0
\(376\) 0 0
\(377\) − 3.00000i − 0.154508i
\(378\) 1.00000i 0.0514344i
\(379\) −35.0000 −1.79783 −0.898915 0.438124i \(-0.855643\pi\)
−0.898915 + 0.438124i \(0.855643\pi\)
\(380\) 0 0
\(381\) −10.0000 −0.512316
\(382\) 15.0000i 0.767467i
\(383\) 24.0000i 1.22634i 0.789950 + 0.613171i \(0.210106\pi\)
−0.789950 + 0.613171i \(0.789894\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) − 1.00000i − 0.0508329i
\(388\) − 14.0000i − 0.710742i
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −9.00000 −0.455150
\(392\) − 1.00000i − 0.0505076i
\(393\) 0 0
\(394\) −15.0000 −0.755689
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) 29.0000i 1.45547i 0.685859 + 0.727734i \(0.259427\pi\)
−0.685859 + 0.727734i \(0.740573\pi\)
\(398\) 8.00000i 0.401004i
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 4.00000i 0.199502i
\(403\) 5.00000i 0.249068i
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) −3.00000 −0.148888
\(407\) − 60.0000i − 2.97409i
\(408\) 3.00000i 0.148522i
\(409\) −38.0000 −1.87898 −0.939490 0.342578i \(-0.888700\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 17.0000i 0.837530i
\(413\) − 9.00000i − 0.442861i
\(414\) −3.00000 −0.147442
\(415\) 0 0
\(416\) 1.00000 0.0490290
\(417\) − 16.0000i − 0.783523i
\(418\) − 24.0000i − 1.17388i
\(419\) 21.0000 1.02592 0.512959 0.858413i \(-0.328549\pi\)
0.512959 + 0.858413i \(0.328549\pi\)
\(420\) 0 0
\(421\) −34.0000 −1.65706 −0.828529 0.559946i \(-0.810822\pi\)
−0.828529 + 0.559946i \(0.810822\pi\)
\(422\) 25.0000i 1.21698i
\(423\) 0 0
\(424\) 9.00000 0.437079
\(425\) 0 0
\(426\) 12.0000 0.581402
\(427\) 11.0000i 0.532327i
\(428\) − 18.0000i − 0.870063i
\(429\) 6.00000 0.289683
\(430\) 0 0
\(431\) −15.0000 −0.722525 −0.361262 0.932464i \(-0.617654\pi\)
−0.361262 + 0.932464i \(0.617654\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 38.0000i − 1.82616i −0.407777 0.913082i \(-0.633696\pi\)
0.407777 0.913082i \(-0.366304\pi\)
\(434\) 5.00000 0.240008
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) 12.0000i 0.574038i
\(438\) − 10.0000i − 0.477818i
\(439\) −41.0000 −1.95682 −0.978412 0.206666i \(-0.933739\pi\)
−0.978412 + 0.206666i \(0.933739\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 3.00000i 0.142695i
\(443\) − 24.0000i − 1.14027i −0.821549 0.570137i \(-0.806890\pi\)
0.821549 0.570137i \(-0.193110\pi\)
\(444\) 10.0000 0.474579
\(445\) 0 0
\(446\) 19.0000 0.899676
\(447\) − 9.00000i − 0.425685i
\(448\) − 1.00000i − 0.0472456i
\(449\) −24.0000 −1.13263 −0.566315 0.824189i \(-0.691631\pi\)
−0.566315 + 0.824189i \(0.691631\pi\)
\(450\) 0 0
\(451\) 54.0000 2.54276
\(452\) − 6.00000i − 0.282216i
\(453\) − 2.00000i − 0.0939682i
\(454\) 3.00000 0.140797
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) − 25.0000i − 1.16945i −0.811231 0.584725i \(-0.801202\pi\)
0.811231 0.584725i \(-0.198798\pi\)
\(458\) − 22.0000i − 1.02799i
\(459\) −3.00000 −0.140028
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) − 6.00000i − 0.279145i
\(463\) − 32.0000i − 1.48717i −0.668644 0.743583i \(-0.733125\pi\)
0.668644 0.743583i \(-0.266875\pi\)
\(464\) −3.00000 −0.139272
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) − 3.00000i − 0.138823i −0.997588 0.0694117i \(-0.977888\pi\)
0.997588 0.0694117i \(-0.0221122\pi\)
\(468\) 1.00000i 0.0462250i
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) − 9.00000i − 0.414259i
\(473\) 6.00000i 0.275880i
\(474\) −10.0000 −0.459315
\(475\) 0 0
\(476\) 3.00000 0.137505
\(477\) 9.00000i 0.412082i
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 10.0000 0.455961
\(482\) − 2.00000i − 0.0910975i
\(483\) 3.00000i 0.136505i
\(484\) −25.0000 −1.13636
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) 11.0000i 0.497947i
\(489\) −5.00000 −0.226108
\(490\) 0 0
\(491\) 42.0000 1.89543 0.947717 0.319113i \(-0.103385\pi\)
0.947717 + 0.319113i \(0.103385\pi\)
\(492\) 9.00000i 0.405751i
\(493\) − 9.00000i − 0.405340i
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) 5.00000 0.224507
\(497\) − 12.0000i − 0.538274i
\(498\) 9.00000i 0.403300i
\(499\) 19.0000 0.850557 0.425278 0.905063i \(-0.360176\pi\)
0.425278 + 0.905063i \(0.360176\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 27.0000i 1.20507i
\(503\) − 6.00000i − 0.267527i −0.991013 0.133763i \(-0.957294\pi\)
0.991013 0.133763i \(-0.0427062\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 18.0000 0.800198
\(507\) − 12.0000i − 0.532939i
\(508\) 10.0000i 0.443678i
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) − 1.00000i − 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) 15.0000 0.661622
\(515\) 0 0
\(516\) −1.00000 −0.0440225
\(517\) 0 0
\(518\) − 10.0000i − 0.439375i
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) −21.0000 −0.920027 −0.460013 0.887912i \(-0.652155\pi\)
−0.460013 + 0.887912i \(0.652155\pi\)
\(522\) − 3.00000i − 0.131306i
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −9.00000 −0.392419
\(527\) 15.0000i 0.653410i
\(528\) − 6.00000i − 0.261116i
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) 9.00000 0.390567
\(532\) − 4.00000i − 0.173422i
\(533\) 9.00000i 0.389833i
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 18.0000i 0.776757i
\(538\) − 18.0000i − 0.776035i
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) − 20.0000i − 0.859074i
\(543\) − 2.00000i − 0.0858282i
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) 1.00000 0.0427960
\(547\) 17.0000i 0.726868i 0.931620 + 0.363434i \(0.118396\pi\)
−0.931620 + 0.363434i \(0.881604\pi\)
\(548\) 0 0
\(549\) −11.0000 −0.469469
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 3.00000i 0.127688i
\(553\) 10.0000i 0.425243i
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) − 18.0000i − 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 5.00000i 0.211667i
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) 18.0000 0.759961
\(562\) 24.0000i 1.01238i
\(563\) − 21.0000i − 0.885044i −0.896758 0.442522i \(-0.854084\pi\)
0.896758 0.442522i \(-0.145916\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 4.00000 0.168133
\(567\) 1.00000i 0.0419961i
\(568\) − 12.0000i − 0.503509i
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −13.0000 −0.544033 −0.272017 0.962293i \(-0.587691\pi\)
−0.272017 + 0.962293i \(0.587691\pi\)
\(572\) − 6.00000i − 0.250873i
\(573\) 15.0000i 0.626634i
\(574\) 9.00000 0.375653
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 34.0000i − 1.41544i −0.706494 0.707719i \(-0.749724\pi\)
0.706494 0.707719i \(-0.250276\pi\)
\(578\) − 8.00000i − 0.332756i
\(579\) 22.0000 0.914289
\(580\) 0 0
\(581\) 9.00000 0.373383
\(582\) − 14.0000i − 0.580319i
\(583\) − 54.0000i − 2.23645i
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) 0 0
\(587\) 33.0000i 1.36206i 0.732257 + 0.681028i \(0.238467\pi\)
−0.732257 + 0.681028i \(0.761533\pi\)
\(588\) − 1.00000i − 0.0412393i
\(589\) 20.0000 0.824086
\(590\) 0 0
\(591\) −15.0000 −0.617018
\(592\) − 10.0000i − 0.410997i
\(593\) 18.0000i 0.739171i 0.929197 + 0.369586i \(0.120500\pi\)
−0.929197 + 0.369586i \(0.879500\pi\)
\(594\) 6.00000 0.246183
\(595\) 0 0
\(596\) −9.00000 −0.368654
\(597\) 8.00000i 0.327418i
\(598\) 3.00000i 0.122679i
\(599\) −9.00000 −0.367730 −0.183865 0.982952i \(-0.558861\pi\)
−0.183865 + 0.982952i \(0.558861\pi\)
\(600\) 0 0
\(601\) −4.00000 −0.163163 −0.0815817 0.996667i \(-0.525997\pi\)
−0.0815817 + 0.996667i \(0.525997\pi\)
\(602\) 1.00000i 0.0407570i
\(603\) 4.00000i 0.162893i
\(604\) −2.00000 −0.0813788
\(605\) 0 0
\(606\) 12.0000 0.487467
\(607\) − 28.0000i − 1.13648i −0.822861 0.568242i \(-0.807624\pi\)
0.822861 0.568242i \(-0.192376\pi\)
\(608\) − 4.00000i − 0.162221i
\(609\) −3.00000 −0.121566
\(610\) 0 0
\(611\) 0 0
\(612\) 3.00000i 0.121268i
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 2.00000 0.0807134
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 17.0000i 0.683840i
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) −3.00000 −0.120386
\(622\) 18.0000i 0.721734i
\(623\) 6.00000i 0.240385i
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) − 24.0000i − 0.958468i
\(628\) 10.0000i 0.399043i
\(629\) 30.0000 1.19618
\(630\) 0 0
\(631\) −22.0000 −0.875806 −0.437903 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(632\) 10.0000i 0.397779i
\(633\) 25.0000i 0.993661i
\(634\) −27.0000 −1.07231
\(635\) 0 0
\(636\) 9.00000 0.356873
\(637\) − 1.00000i − 0.0396214i
\(638\) 18.0000i 0.712627i
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) − 18.0000i − 0.710403i
\(643\) − 14.0000i − 0.552106i −0.961142 0.276053i \(-0.910973\pi\)
0.961142 0.276053i \(-0.0890266\pi\)
\(644\) 3.00000 0.118217
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 6.00000i 0.235884i 0.993020 + 0.117942i \(0.0376297\pi\)
−0.993020 + 0.117942i \(0.962370\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −54.0000 −2.11969
\(650\) 0 0
\(651\) 5.00000 0.195965
\(652\) 5.00000i 0.195815i
\(653\) − 18.0000i − 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) 8.00000 0.312825
\(655\) 0 0
\(656\) 9.00000 0.351391
\(657\) − 10.0000i − 0.390137i
\(658\) 0 0
\(659\) −42.0000 −1.63609 −0.818044 0.575156i \(-0.804941\pi\)
−0.818044 + 0.575156i \(0.804941\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 19.0000i 0.738456i
\(663\) 3.00000i 0.116510i
\(664\) 9.00000 0.349268
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) − 9.00000i − 0.348481i
\(668\) − 18.0000i − 0.696441i
\(669\) 19.0000 0.734582
\(670\) 0 0
\(671\) 66.0000 2.54790
\(672\) − 1.00000i − 0.0385758i
\(673\) 37.0000i 1.42625i 0.701039 + 0.713123i \(0.252720\pi\)
−0.701039 + 0.713123i \(0.747280\pi\)
\(674\) −13.0000 −0.500741
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) − 12.0000i − 0.461197i −0.973049 0.230599i \(-0.925932\pi\)
0.973049 0.230599i \(-0.0740685\pi\)
\(678\) − 6.00000i − 0.230429i
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) − 30.0000i − 1.14876i
\(683\) − 18.0000i − 0.688751i −0.938832 0.344375i \(-0.888091\pi\)
0.938832 0.344375i \(-0.111909\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) − 22.0000i − 0.839352i
\(688\) 1.00000i 0.0381246i
\(689\) 9.00000 0.342873
\(690\) 0 0
\(691\) −34.0000 −1.29342 −0.646710 0.762736i \(-0.723856\pi\)
−0.646710 + 0.762736i \(0.723856\pi\)
\(692\) − 12.0000i − 0.456172i
\(693\) − 6.00000i − 0.227921i
\(694\) 18.0000 0.683271
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) 27.0000i 1.02270i
\(698\) 17.0000i 0.643459i
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) 3.00000 0.113308 0.0566542 0.998394i \(-0.481957\pi\)
0.0566542 + 0.998394i \(0.481957\pi\)
\(702\) 1.00000i 0.0377426i
\(703\) − 40.0000i − 1.50863i
\(704\) −6.00000 −0.226134
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) − 12.0000i − 0.451306i
\(708\) − 9.00000i − 0.338241i
\(709\) 40.0000 1.50223 0.751116 0.660171i \(-0.229516\pi\)
0.751116 + 0.660171i \(0.229516\pi\)
\(710\) 0 0
\(711\) −10.0000 −0.375029
\(712\) 6.00000i 0.224860i
\(713\) 15.0000i 0.561754i
\(714\) 3.00000 0.112272
\(715\) 0 0
\(716\) 18.0000 0.672692
\(717\) 0 0
\(718\) − 3.00000i − 0.111959i
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 17.0000 0.633113
\(722\) 3.00000i 0.111648i
\(723\) − 2.00000i − 0.0743808i
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) −25.0000 −0.927837
\(727\) − 7.00000i − 0.259616i −0.991539 0.129808i \(-0.958564\pi\)
0.991539 0.129808i \(-0.0414360\pi\)
\(728\) − 1.00000i − 0.0370625i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −3.00000 −0.110959
\(732\) 11.0000i 0.406572i
\(733\) − 29.0000i − 1.07114i −0.844491 0.535570i \(-0.820097\pi\)
0.844491 0.535570i \(-0.179903\pi\)
\(734\) −37.0000 −1.36569
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) − 24.0000i − 0.884051i
\(738\) 9.00000i 0.331295i
\(739\) 37.0000 1.36107 0.680534 0.732717i \(-0.261748\pi\)
0.680534 + 0.732717i \(0.261748\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) − 9.00000i − 0.330400i
\(743\) 9.00000i 0.330178i 0.986279 + 0.165089i \(0.0527911\pi\)
−0.986279 + 0.165089i \(0.947209\pi\)
\(744\) 5.00000 0.183309
\(745\) 0 0
\(746\) −8.00000 −0.292901
\(747\) 9.00000i 0.329293i
\(748\) − 18.0000i − 0.658145i
\(749\) −18.0000 −0.657706
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 27.0000i 0.983935i
\(754\) −3.00000 −0.109254
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) 14.0000i 0.508839i 0.967094 + 0.254419i \(0.0818843\pi\)
−0.967094 + 0.254419i \(0.918116\pi\)
\(758\) 35.0000i 1.27126i
\(759\) 18.0000 0.653359
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 10.0000i 0.362262i
\(763\) − 8.00000i − 0.289619i
\(764\) 15.0000 0.542681
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) − 9.00000i − 0.324971i
\(768\) − 1.00000i − 0.0360844i
\(769\) 4.00000 0.144244 0.0721218 0.997396i \(-0.477023\pi\)
0.0721218 + 0.997396i \(0.477023\pi\)
\(770\) 0 0
\(771\) 15.0000 0.540212
\(772\) − 22.0000i − 0.791797i
\(773\) 48.0000i 1.72644i 0.504828 + 0.863220i \(0.331556\pi\)
−0.504828 + 0.863220i \(0.668444\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) − 10.0000i − 0.358748i
\(778\) − 6.00000i − 0.215110i
\(779\) 36.0000 1.28983
\(780\) 0 0
\(781\) −72.0000 −2.57636
\(782\) 9.00000i 0.321839i
\(783\) − 3.00000i − 0.107211i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) 26.0000i 0.926800i 0.886149 + 0.463400i \(0.153371\pi\)
−0.886149 + 0.463400i \(0.846629\pi\)
\(788\) 15.0000i 0.534353i
\(789\) −9.00000 −0.320408
\(790\) 0 0
\(791\) −6.00000 −0.213335
\(792\) − 6.00000i − 0.213201i
\(793\) 11.0000i 0.390621i
\(794\) 29.0000 1.02917
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) − 6.00000i − 0.212531i −0.994338 0.106265i \(-0.966111\pi\)
0.994338 0.106265i \(-0.0338893\pi\)
\(798\) − 4.00000i − 0.141598i
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 12.0000i 0.423735i
\(803\) 60.0000i 2.11735i
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) 5.00000 0.176117
\(807\) − 18.0000i − 0.633630i
\(808\) − 12.0000i − 0.422159i
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −34.0000 −1.19390 −0.596951 0.802278i \(-0.703621\pi\)
−0.596951 + 0.802278i \(0.703621\pi\)
\(812\) 3.00000i 0.105279i
\(813\) − 20.0000i − 0.701431i
\(814\) −60.0000 −2.10300
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) 4.00000i 0.139942i
\(818\) 38.0000i 1.32864i
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) 34.0000i 1.18517i 0.805510 + 0.592583i \(0.201892\pi\)
−0.805510 + 0.592583i \(0.798108\pi\)
\(824\) 17.0000 0.592223
\(825\) 0 0
\(826\) −9.00000 −0.313150
\(827\) 6.00000i 0.208640i 0.994544 + 0.104320i \(0.0332667\pi\)
−0.994544 + 0.104320i \(0.966733\pi\)
\(828\) 3.00000i 0.104257i
\(829\) −29.0000 −1.00721 −0.503606 0.863934i \(-0.667994\pi\)
−0.503606 + 0.863934i \(0.667994\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) − 1.00000i − 0.0346688i
\(833\) − 3.00000i − 0.103944i
\(834\) −16.0000 −0.554035
\(835\) 0 0
\(836\) −24.0000 −0.830057
\(837\) 5.00000i 0.172825i
\(838\) − 21.0000i − 0.725433i
\(839\) −36.0000 −1.24286 −0.621429 0.783470i \(-0.713448\pi\)
−0.621429 + 0.783470i \(0.713448\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 34.0000i 1.17172i
\(843\) 24.0000i 0.826604i
\(844\) 25.0000 0.860535
\(845\) 0 0
\(846\) 0 0
\(847\) 25.0000i 0.859010i
\(848\) − 9.00000i − 0.309061i
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 30.0000 1.02839
\(852\) − 12.0000i − 0.411113i
\(853\) 19.0000i 0.650548i 0.945620 + 0.325274i \(0.105456\pi\)
−0.945620 + 0.325274i \(0.894544\pi\)
\(854\) 11.0000 0.376412
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) − 6.00000i − 0.204837i
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 9.00000 0.306719
\(862\) 15.0000i 0.510902i
\(863\) 48.0000i 1.63394i 0.576681 + 0.816970i \(0.304348\pi\)
−0.576681 + 0.816970i \(0.695652\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −38.0000 −1.29129
\(867\) − 8.00000i − 0.271694i
\(868\) − 5.00000i − 0.169711i
\(869\) 60.0000 2.03536
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) − 8.00000i − 0.270914i
\(873\) − 14.0000i − 0.473828i
\(874\) 12.0000 0.405906
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) 32.0000i 1.08056i 0.841484 + 0.540282i \(0.181682\pi\)
−0.841484 + 0.540282i \(0.818318\pi\)
\(878\) 41.0000i 1.38368i
\(879\) 0 0
\(880\) 0 0
\(881\) 33.0000 1.11180 0.555899 0.831250i \(-0.312374\pi\)
0.555899 + 0.831250i \(0.312374\pi\)
\(882\) − 1.00000i − 0.0336718i
\(883\) − 5.00000i − 0.168263i −0.996455 0.0841317i \(-0.973188\pi\)
0.996455 0.0841317i \(-0.0268116\pi\)
\(884\) 3.00000 0.100901
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) 54.0000i 1.81314i 0.422053 + 0.906571i \(0.361310\pi\)
−0.422053 + 0.906571i \(0.638690\pi\)
\(888\) − 10.0000i − 0.335578i
\(889\) 10.0000 0.335389
\(890\) 0 0
\(891\) 6.00000 0.201008
\(892\) − 19.0000i − 0.636167i
\(893\) 0 0
\(894\) −9.00000 −0.301005
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 3.00000i 0.100167i
\(898\) 24.0000i 0.800890i
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) 27.0000 0.899500
\(902\) − 54.0000i − 1.79800i
\(903\) 1.00000i 0.0332779i
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) −2.00000 −0.0664455
\(907\) 53.0000i 1.75984i 0.475125 + 0.879918i \(0.342403\pi\)
−0.475125 + 0.879918i \(0.657597\pi\)
\(908\) − 3.00000i − 0.0995585i
\(909\) 12.0000 0.398015
\(910\) 0 0
\(911\) 15.0000 0.496972 0.248486 0.968635i \(-0.420067\pi\)
0.248486 + 0.968635i \(0.420067\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) − 54.0000i − 1.78714i
\(914\) −25.0000 −0.826927
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) 0 0
\(918\) 3.00000i 0.0990148i
\(919\) −20.0000 −0.659739 −0.329870 0.944027i \(-0.607005\pi\)
−0.329870 + 0.944027i \(0.607005\pi\)
\(920\) 0 0
\(921\) 2.00000 0.0659022
\(922\) − 12.0000i − 0.395199i
\(923\) − 12.0000i − 0.394985i
\(924\) −6.00000 −0.197386
\(925\) 0 0
\(926\) −32.0000 −1.05159
\(927\) 17.0000i 0.558353i
\(928\) 3.00000i 0.0984798i
\(929\) 27.0000 0.885841 0.442921 0.896561i \(-0.353942\pi\)
0.442921 + 0.896561i \(0.353942\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 12.0000i 0.393073i
\(933\) 18.0000i 0.589294i
\(934\) −3.00000 −0.0981630
\(935\) 0 0
\(936\) 1.00000 0.0326860
\(937\) 2.00000i 0.0653372i 0.999466 + 0.0326686i \(0.0104006\pi\)
−0.999466 + 0.0326686i \(0.989599\pi\)
\(938\) − 4.00000i − 0.130605i
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 24.0000 0.782378 0.391189 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) 10.0000i 0.325818i
\(943\) 27.0000i 0.879241i
\(944\) −9.00000 −0.292925
\(945\) 0 0
\(946\) 6.00000 0.195077
\(947\) − 24.0000i − 0.779895i −0.920837 0.389948i \(-0.872493\pi\)
0.920837 0.389948i \(-0.127507\pi\)
\(948\) 10.0000i 0.324785i
\(949\) −10.0000 −0.324614
\(950\) 0 0
\(951\) −27.0000 −0.875535
\(952\) − 3.00000i − 0.0972306i
\(953\) − 24.0000i − 0.777436i −0.921357 0.388718i \(-0.872918\pi\)
0.921357 0.388718i \(-0.127082\pi\)
\(954\) 9.00000 0.291386
\(955\) 0 0
\(956\) 0 0
\(957\) 18.0000i 0.581857i
\(958\) 24.0000i 0.775405i
\(959\) 0 0
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) − 10.0000i − 0.322413i
\(963\) − 18.0000i − 0.580042i
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 3.00000 0.0965234
\(967\) − 28.0000i − 0.900419i −0.892923 0.450210i \(-0.851349\pi\)
0.892923 0.450210i \(-0.148651\pi\)
\(968\) 25.0000i 0.803530i
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 16.0000i 0.512936i
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) 11.0000 0.352101
\(977\) 42.0000i 1.34370i 0.740688 + 0.671850i \(0.234500\pi\)
−0.740688 + 0.671850i \(0.765500\pi\)
\(978\) 5.00000i 0.159882i
\(979\) 36.0000 1.15056
\(980\) 0 0
\(981\) 8.00000 0.255420
\(982\) − 42.0000i − 1.34027i
\(983\) − 24.0000i − 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 9.00000 0.286910
\(985\) 0 0
\(986\) −9.00000 −0.286618
\(987\) 0 0
\(988\) − 4.00000i − 0.127257i
\(989\) −3.00000 −0.0953945
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) − 5.00000i − 0.158750i
\(993\) 19.0000i 0.602947i
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) 9.00000 0.285176
\(997\) 26.0000i 0.823428i 0.911313 + 0.411714i \(0.135070\pi\)
−0.911313 + 0.411714i \(0.864930\pi\)
\(998\) − 19.0000i − 0.601434i
\(999\) 10.0000 0.316386
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 1050.2.g.e.799.1 2
3.2 odd 2 3150.2.g.a.2899.2 2
5.2 odd 4 1050.2.a.l.1.1 yes 1
5.3 odd 4 1050.2.a.j.1.1 1
5.4 even 2 inner 1050.2.g.e.799.2 2
15.2 even 4 3150.2.a.a.1.1 1
15.8 even 4 3150.2.a.bg.1.1 1
15.14 odd 2 3150.2.g.a.2899.1 2
20.3 even 4 8400.2.a.a.1.1 1
20.7 even 4 8400.2.a.ci.1.1 1
35.13 even 4 7350.2.a.r.1.1 1
35.27 even 4 7350.2.a.cz.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.j.1.1 1 5.3 odd 4
1050.2.a.l.1.1 yes 1 5.2 odd 4
1050.2.g.e.799.1 2 1.1 even 1 trivial
1050.2.g.e.799.2 2 5.4 even 2 inner
3150.2.a.a.1.1 1 15.2 even 4
3150.2.a.bg.1.1 1 15.8 even 4
3150.2.g.a.2899.1 2 15.14 odd 2
3150.2.g.a.2899.2 2 3.2 odd 2
7350.2.a.r.1.1 1 35.13 even 4
7350.2.a.cz.1.1 1 35.27 even 4
8400.2.a.a.1.1 1 20.3 even 4
8400.2.a.ci.1.1 1 20.7 even 4