Properties

Label 1050.2.g.b.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.b.799.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -2.00000 q^{11} +1.00000i q^{12} +7.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.00000i q^{17} +1.00000i q^{18} -8.00000 q^{19} -1.00000 q^{21} +2.00000i q^{22} -5.00000i q^{23} +1.00000 q^{24} +7.00000 q^{26} +1.00000i q^{27} +1.00000i q^{28} -9.00000 q^{29} +1.00000 q^{31} -1.00000i q^{32} +2.00000i q^{33} +7.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +8.00000i q^{38} +7.00000 q^{39} +11.0000 q^{41} +1.00000i q^{42} +3.00000i q^{43} +2.00000 q^{44} -5.00000 q^{46} -4.00000i q^{47} -1.00000i q^{48} -1.00000 q^{49} +7.00000 q^{51} -7.00000i q^{52} +3.00000i q^{53} +1.00000 q^{54} +1.00000 q^{56} +8.00000i q^{57} +9.00000i q^{58} -7.00000 q^{59} -5.00000 q^{61} -1.00000i q^{62} +1.00000i q^{63} -1.00000 q^{64} +2.00000 q^{66} +12.0000i q^{67} -7.00000i q^{68} -5.00000 q^{69} -4.00000 q^{71} -1.00000i q^{72} +10.0000i q^{73} +2.00000 q^{74} +8.00000 q^{76} +2.00000i q^{77} -7.00000i q^{78} +6.00000 q^{79} +1.00000 q^{81} -11.0000i q^{82} -9.00000i q^{83} +1.00000 q^{84} +3.00000 q^{86} +9.00000i q^{87} -2.00000i q^{88} +10.0000 q^{89} +7.00000 q^{91} +5.00000i q^{92} -1.00000i q^{93} -4.00000 q^{94} -1.00000 q^{96} -10.0000i q^{97} +1.00000i q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} - 4q^{11} - 2q^{14} + 2q^{16} - 16q^{19} - 2q^{21} + 2q^{24} + 14q^{26} - 18q^{29} + 2q^{31} + 14q^{34} + 2q^{36} + 14q^{39} + 22q^{41} + 4q^{44} - 10q^{46} - 2q^{49} + 14q^{51} + 2q^{54} + 2q^{56} - 14q^{59} - 10q^{61} - 2q^{64} + 4q^{66} - 10q^{69} - 8q^{71} + 4q^{74} + 16q^{76} + 12q^{79} + 2q^{81} + 2q^{84} + 6q^{86} + 20q^{89} + 14q^{91} - 8q^{94} - 2q^{96} + 4q^{99} + O(q^{100}) \)

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
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 7.00000i 1.94145i 0.240192 + 0.970725i \(0.422790\pi\)
−0.240192 + 0.970725i \(0.577210\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.00000i 1.69775i 0.528594 + 0.848875i \(0.322719\pi\)
−0.528594 + 0.848875i \(0.677281\pi\)
\(18\) 1.00000i 0.235702i
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 2.00000i 0.426401i
\(23\) − 5.00000i − 1.04257i −0.853382 0.521286i \(-0.825452\pi\)
0.853382 0.521286i \(-0.174548\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 7.00000 1.37281
\(27\) 1.00000i 0.192450i
\(28\) 1.00000i 0.188982i
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 2.00000i 0.348155i
\(34\) 7.00000 1.20049
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 8.00000i 1.29777i
\(39\) 7.00000 1.12090
\(40\) 0 0
\(41\) 11.0000 1.71791 0.858956 0.512050i \(-0.171114\pi\)
0.858956 + 0.512050i \(0.171114\pi\)
\(42\) 1.00000i 0.154303i
\(43\) 3.00000i 0.457496i 0.973486 + 0.228748i \(0.0734631\pi\)
−0.973486 + 0.228748i \(0.926537\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) −5.00000 −0.737210
\(47\) − 4.00000i − 0.583460i −0.956501 0.291730i \(-0.905769\pi\)
0.956501 0.291730i \(-0.0942309\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 7.00000 0.980196
\(52\) − 7.00000i − 0.970725i
\(53\) 3.00000i 0.412082i 0.978543 + 0.206041i \(0.0660580\pi\)
−0.978543 + 0.206041i \(0.933942\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 8.00000i 1.05963i
\(58\) 9.00000i 1.18176i
\(59\) −7.00000 −0.911322 −0.455661 0.890153i \(-0.650597\pi\)
−0.455661 + 0.890153i \(0.650597\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) − 1.00000i − 0.127000i
\(63\) 1.00000i 0.125988i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) − 7.00000i − 0.848875i
\(69\) −5.00000 −0.601929
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) 2.00000i 0.227921i
\(78\) − 7.00000i − 0.792594i
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 11.0000i − 1.21475i
\(83\) − 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
\(86\) 3.00000 0.323498
\(87\) 9.00000i 0.964901i
\(88\) − 2.00000i − 0.213201i
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 7.00000 0.733799
\(92\) 5.00000i 0.521286i
\(93\) − 1.00000i − 0.103695i
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) − 7.00000i − 0.693103i
\(103\) 5.00000i 0.492665i 0.969185 + 0.246332i \(0.0792255\pi\)
−0.969185 + 0.246332i \(0.920775\pi\)
\(104\) −7.00000 −0.686406
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) 2.00000i 0.193347i 0.995316 + 0.0966736i \(0.0308203\pi\)
−0.995316 + 0.0966736i \(0.969180\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) − 1.00000i − 0.0944911i
\(113\) − 14.0000i − 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) 9.00000 0.835629
\(117\) − 7.00000i − 0.647150i
\(118\) 7.00000i 0.644402i
\(119\) 7.00000 0.641689
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 5.00000i 0.452679i
\(123\) − 11.0000i − 0.991837i
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 18.0000i 1.59724i 0.601834 + 0.798621i \(0.294437\pi\)
−0.601834 + 0.798621i \(0.705563\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 3.00000 0.264135
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) − 2.00000i − 0.174078i
\(133\) 8.00000i 0.693688i
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) −7.00000 −0.600245
\(137\) − 16.0000i − 1.36697i −0.729964 0.683486i \(-0.760463\pi\)
0.729964 0.683486i \(-0.239537\pi\)
\(138\) 5.00000i 0.425628i
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) −4.00000 −0.336861
\(142\) 4.00000i 0.335673i
\(143\) − 14.0000i − 1.17074i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) 1.00000i 0.0824786i
\(148\) − 2.00000i − 0.164399i
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) −22.0000 −1.79033 −0.895167 0.445730i \(-0.852944\pi\)
−0.895167 + 0.445730i \(0.852944\pi\)
\(152\) − 8.00000i − 0.648886i
\(153\) − 7.00000i − 0.565916i
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) −7.00000 −0.560449
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) − 6.00000i − 0.477334i
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) −5.00000 −0.394055
\(162\) − 1.00000i − 0.0785674i
\(163\) − 15.0000i − 1.17489i −0.809264 0.587445i \(-0.800134\pi\)
0.809264 0.587445i \(-0.199866\pi\)
\(164\) −11.0000 −0.858956
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) − 2.00000i − 0.154765i −0.997001 0.0773823i \(-0.975344\pi\)
0.997001 0.0773823i \(-0.0246562\pi\)
\(168\) − 1.00000i − 0.0771517i
\(169\) −36.0000 −2.76923
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) − 3.00000i − 0.228748i
\(173\) − 16.0000i − 1.21646i −0.793762 0.608229i \(-0.791880\pi\)
0.793762 0.608229i \(-0.208120\pi\)
\(174\) 9.00000 0.682288
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 7.00000i 0.526152i
\(178\) − 10.0000i − 0.749532i
\(179\) 2.00000 0.149487 0.0747435 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) − 7.00000i − 0.518875i
\(183\) 5.00000i 0.369611i
\(184\) 5.00000 0.368605
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) − 14.0000i − 1.02378i
\(188\) 4.00000i 0.291730i
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −1.00000 −0.0723575 −0.0361787 0.999345i \(-0.511519\pi\)
−0.0361787 + 0.999345i \(0.511519\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 10.0000i 0.719816i 0.932988 + 0.359908i \(0.117192\pi\)
−0.932988 + 0.359908i \(0.882808\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 21.0000i 1.49619i 0.663593 + 0.748094i \(0.269031\pi\)
−0.663593 + 0.748094i \(0.730969\pi\)
\(198\) − 2.00000i − 0.142134i
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 0 0
\(203\) 9.00000i 0.631676i
\(204\) −7.00000 −0.490098
\(205\) 0 0
\(206\) 5.00000 0.348367
\(207\) 5.00000i 0.347524i
\(208\) 7.00000i 0.485363i
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) − 3.00000i − 0.206041i
\(213\) 4.00000i 0.274075i
\(214\) 2.00000 0.136717
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) − 1.00000i − 0.0678844i
\(218\) 4.00000i 0.270914i
\(219\) 10.0000 0.675737
\(220\) 0 0
\(221\) −49.0000 −3.29610
\(222\) − 2.00000i − 0.134231i
\(223\) 9.00000i 0.602685i 0.953516 + 0.301342i \(0.0974347\pi\)
−0.953516 + 0.301342i \(0.902565\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −14.0000 −0.931266
\(227\) 3.00000i 0.199117i 0.995032 + 0.0995585i \(0.0317430\pi\)
−0.995032 + 0.0995585i \(0.968257\pi\)
\(228\) − 8.00000i − 0.529813i
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) − 9.00000i − 0.590879i
\(233\) 16.0000i 1.04819i 0.851658 + 0.524097i \(0.175597\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) −7.00000 −0.457604
\(235\) 0 0
\(236\) 7.00000 0.455661
\(237\) − 6.00000i − 0.389742i
\(238\) − 7.00000i − 0.453743i
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) −30.0000 −1.93247 −0.966235 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) 7.00000i 0.449977i
\(243\) − 1.00000i − 0.0641500i
\(244\) 5.00000 0.320092
\(245\) 0 0
\(246\) −11.0000 −0.701334
\(247\) − 56.0000i − 3.56319i
\(248\) 1.00000i 0.0635001i
\(249\) −9.00000 −0.570352
\(250\) 0 0
\(251\) −5.00000 −0.315597 −0.157799 0.987471i \(-0.550440\pi\)
−0.157799 + 0.987471i \(0.550440\pi\)
\(252\) − 1.00000i − 0.0629941i
\(253\) 10.0000i 0.628695i
\(254\) 18.0000 1.12942
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 5.00000i − 0.311891i −0.987766 0.155946i \(-0.950158\pi\)
0.987766 0.155946i \(-0.0498425\pi\)
\(258\) − 3.00000i − 0.186772i
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) 0 0
\(263\) − 9.00000i − 0.554964i −0.960731 0.277482i \(-0.910500\pi\)
0.960731 0.277482i \(-0.0894999\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 8.00000 0.490511
\(267\) − 10.0000i − 0.611990i
\(268\) − 12.0000i − 0.733017i
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 7.00000i 0.424437i
\(273\) − 7.00000i − 0.423659i
\(274\) −16.0000 −0.966595
\(275\) 0 0
\(276\) 5.00000 0.300965
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 8.00000i 0.479808i
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −8.00000 −0.477240 −0.238620 0.971113i \(-0.576695\pi\)
−0.238620 + 0.971113i \(0.576695\pi\)
\(282\) 4.00000i 0.238197i
\(283\) − 16.0000i − 0.951101i −0.879688 0.475551i \(-0.842249\pi\)
0.879688 0.475551i \(-0.157751\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) −14.0000 −0.827837
\(287\) − 11.0000i − 0.649309i
\(288\) 1.00000i 0.0589256i
\(289\) −32.0000 −1.88235
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) − 10.0000i − 0.585206i
\(293\) 24.0000i 1.40209i 0.713115 + 0.701047i \(0.247284\pi\)
−0.713115 + 0.701047i \(0.752716\pi\)
\(294\) 1.00000 0.0583212
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) − 2.00000i − 0.116052i
\(298\) − 3.00000i − 0.173785i
\(299\) 35.0000 2.02410
\(300\) 0 0
\(301\) 3.00000 0.172917
\(302\) 22.0000i 1.26596i
\(303\) 0 0
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) −7.00000 −0.400163
\(307\) 22.0000i 1.25561i 0.778372 + 0.627803i \(0.216046\pi\)
−0.778372 + 0.627803i \(0.783954\pi\)
\(308\) − 2.00000i − 0.113961i
\(309\) 5.00000 0.284440
\(310\) 0 0
\(311\) 2.00000 0.113410 0.0567048 0.998391i \(-0.481941\pi\)
0.0567048 + 0.998391i \(0.481941\pi\)
\(312\) 7.00000i 0.396297i
\(313\) 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
\(316\) −6.00000 −0.337526
\(317\) − 23.0000i − 1.29181i −0.763418 0.645904i \(-0.776480\pi\)
0.763418 0.645904i \(-0.223520\pi\)
\(318\) − 3.00000i − 0.168232i
\(319\) 18.0000 1.00781
\(320\) 0 0
\(321\) 2.00000 0.111629
\(322\) 5.00000i 0.278639i
\(323\) − 56.0000i − 3.11592i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −15.0000 −0.830773
\(327\) 4.00000i 0.221201i
\(328\) 11.0000i 0.607373i
\(329\) −4.00000 −0.220527
\(330\) 0 0
\(331\) 25.0000 1.37412 0.687062 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) 9.00000i 0.493939i
\(333\) − 2.00000i − 0.109599i
\(334\) −2.00000 −0.109435
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) 13.0000i 0.708155i 0.935216 + 0.354078i \(0.115205\pi\)
−0.935216 + 0.354078i \(0.884795\pi\)
\(338\) 36.0000i 1.95814i
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) −2.00000 −0.108306
\(342\) − 8.00000i − 0.432590i
\(343\) 1.00000i 0.0539949i
\(344\) −3.00000 −0.161749
\(345\) 0 0
\(346\) −16.0000 −0.860165
\(347\) 6.00000i 0.322097i 0.986947 + 0.161048i \(0.0514875\pi\)
−0.986947 + 0.161048i \(0.948512\pi\)
\(348\) − 9.00000i − 0.482451i
\(349\) 7.00000 0.374701 0.187351 0.982293i \(-0.440010\pi\)
0.187351 + 0.982293i \(0.440010\pi\)
\(350\) 0 0
\(351\) −7.00000 −0.373632
\(352\) 2.00000i 0.106600i
\(353\) 30.0000i 1.59674i 0.602168 + 0.798369i \(0.294304\pi\)
−0.602168 + 0.798369i \(0.705696\pi\)
\(354\) 7.00000 0.372046
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) − 7.00000i − 0.370479i
\(358\) − 2.00000i − 0.105703i
\(359\) −19.0000 −1.00278 −0.501391 0.865221i \(-0.667178\pi\)
−0.501391 + 0.865221i \(0.667178\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) − 2.00000i − 0.105118i
\(363\) 7.00000i 0.367405i
\(364\) −7.00000 −0.366900
\(365\) 0 0
\(366\) 5.00000 0.261354
\(367\) 17.0000i 0.887393i 0.896177 + 0.443696i \(0.146333\pi\)
−0.896177 + 0.443696i \(0.853667\pi\)
\(368\) − 5.00000i − 0.260643i
\(369\) −11.0000 −0.572637
\(370\) 0 0
\(371\) 3.00000 0.155752
\(372\) 1.00000i 0.0518476i
\(373\) − 4.00000i − 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) −14.0000 −0.723923
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) − 63.0000i − 3.24467i
\(378\) − 1.00000i − 0.0514344i
\(379\) −7.00000 −0.359566 −0.179783 0.983706i \(-0.557540\pi\)
−0.179783 + 0.983706i \(0.557540\pi\)
\(380\) 0 0
\(381\) 18.0000 0.922168
\(382\) 1.00000i 0.0511645i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) − 3.00000i − 0.152499i
\(388\) 10.0000i 0.507673i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 35.0000 1.77003
\(392\) − 1.00000i − 0.0505076i
\(393\) 0 0
\(394\) 21.0000 1.05796
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) − 37.0000i − 1.85698i −0.371361 0.928488i \(-0.621109\pi\)
0.371361 0.928488i \(-0.378891\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) − 12.0000i − 0.598506i
\(403\) 7.00000i 0.348695i
\(404\) 0 0
\(405\) 0 0
\(406\) 9.00000 0.446663
\(407\) − 4.00000i − 0.198273i
\(408\) 7.00000i 0.346552i
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) −16.0000 −0.789222
\(412\) − 5.00000i − 0.246332i
\(413\) 7.00000i 0.344447i
\(414\) 5.00000 0.245737
\(415\) 0 0
\(416\) 7.00000 0.343203
\(417\) 8.00000i 0.391762i
\(418\) − 16.0000i − 0.782586i
\(419\) 3.00000 0.146560 0.0732798 0.997311i \(-0.476653\pi\)
0.0732798 + 0.997311i \(0.476653\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 13.0000i 0.632830i
\(423\) 4.00000i 0.194487i
\(424\) −3.00000 −0.145693
\(425\) 0 0
\(426\) 4.00000 0.193801
\(427\) 5.00000i 0.241967i
\(428\) − 2.00000i − 0.0966736i
\(429\) −14.0000 −0.675926
\(430\) 0 0
\(431\) 39.0000 1.87856 0.939282 0.343146i \(-0.111493\pi\)
0.939282 + 0.343146i \(0.111493\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 6.00000i 0.288342i 0.989553 + 0.144171i \(0.0460515\pi\)
−0.989553 + 0.144171i \(0.953949\pi\)
\(434\) −1.00000 −0.0480015
\(435\) 0 0
\(436\) 4.00000 0.191565
\(437\) 40.0000i 1.91346i
\(438\) − 10.0000i − 0.477818i
\(439\) 19.0000 0.906821 0.453410 0.891302i \(-0.350207\pi\)
0.453410 + 0.891302i \(0.350207\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 49.0000i 2.33069i
\(443\) − 4.00000i − 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 9.00000 0.426162
\(447\) − 3.00000i − 0.141895i
\(448\) 1.00000i 0.0472456i
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) −22.0000 −1.03594
\(452\) 14.0000i 0.658505i
\(453\) 22.0000i 1.03365i
\(454\) 3.00000 0.140797
\(455\) 0 0
\(456\) −8.00000 −0.374634
\(457\) − 7.00000i − 0.327446i −0.986506 0.163723i \(-0.947650\pi\)
0.986506 0.163723i \(-0.0523504\pi\)
\(458\) − 14.0000i − 0.654177i
\(459\) −7.00000 −0.326732
\(460\) 0 0
\(461\) 8.00000 0.372597 0.186299 0.982493i \(-0.440351\pi\)
0.186299 + 0.982493i \(0.440351\pi\)
\(462\) − 2.00000i − 0.0930484i
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) −9.00000 −0.417815
\(465\) 0 0
\(466\) 16.0000 0.741186
\(467\) 29.0000i 1.34196i 0.741475 + 0.670980i \(0.234126\pi\)
−0.741475 + 0.670980i \(0.765874\pi\)
\(468\) 7.00000i 0.323575i
\(469\) 12.0000 0.554109
\(470\) 0 0
\(471\) 10.0000 0.460776
\(472\) − 7.00000i − 0.322201i
\(473\) − 6.00000i − 0.275880i
\(474\) −6.00000 −0.275589
\(475\) 0 0
\(476\) −7.00000 −0.320844
\(477\) − 3.00000i − 0.137361i
\(478\) 8.00000i 0.365911i
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) −14.0000 −0.638345
\(482\) 30.0000i 1.36646i
\(483\) 5.00000i 0.227508i
\(484\) 7.00000 0.318182
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) − 6.00000i − 0.271886i −0.990717 0.135943i \(-0.956594\pi\)
0.990717 0.135943i \(-0.0434064\pi\)
\(488\) − 5.00000i − 0.226339i
\(489\) −15.0000 −0.678323
\(490\) 0 0
\(491\) −42.0000 −1.89543 −0.947717 0.319113i \(-0.896615\pi\)
−0.947717 + 0.319113i \(0.896615\pi\)
\(492\) 11.0000i 0.495918i
\(493\) − 63.0000i − 2.83738i
\(494\) −56.0000 −2.51956
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 4.00000i 0.179425i
\(498\) 9.00000i 0.403300i
\(499\) 23.0000 1.02962 0.514811 0.857304i \(-0.327862\pi\)
0.514811 + 0.857304i \(0.327862\pi\)
\(500\) 0 0
\(501\) −2.00000 −0.0893534
\(502\) 5.00000i 0.223161i
\(503\) − 26.0000i − 1.15928i −0.814872 0.579641i \(-0.803193\pi\)
0.814872 0.579641i \(-0.196807\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 10.0000 0.444554
\(507\) 36.0000i 1.59882i
\(508\) − 18.0000i − 0.798621i
\(509\) 28.0000 1.24108 0.620539 0.784176i \(-0.286914\pi\)
0.620539 + 0.784176i \(0.286914\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) − 1.00000i − 0.0441942i
\(513\) − 8.00000i − 0.353209i
\(514\) −5.00000 −0.220541
\(515\) 0 0
\(516\) −3.00000 −0.132068
\(517\) 8.00000i 0.351840i
\(518\) − 2.00000i − 0.0878750i
\(519\) −16.0000 −0.702322
\(520\) 0 0
\(521\) 9.00000 0.394297 0.197149 0.980374i \(-0.436832\pi\)
0.197149 + 0.980374i \(0.436832\pi\)
\(522\) − 9.00000i − 0.393919i
\(523\) 8.00000i 0.349816i 0.984585 + 0.174908i \(0.0559627\pi\)
−0.984585 + 0.174908i \(0.944037\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −9.00000 −0.392419
\(527\) 7.00000i 0.304925i
\(528\) 2.00000i 0.0870388i
\(529\) −2.00000 −0.0869565
\(530\) 0 0
\(531\) 7.00000 0.303774
\(532\) − 8.00000i − 0.346844i
\(533\) 77.0000i 3.33524i
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) − 2.00000i − 0.0863064i
\(538\) − 6.00000i − 0.258678i
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) − 12.0000i − 0.515444i
\(543\) − 2.00000i − 0.0858282i
\(544\) 7.00000 0.300123
\(545\) 0 0
\(546\) −7.00000 −0.299572
\(547\) 35.0000i 1.49649i 0.663421 + 0.748246i \(0.269104\pi\)
−0.663421 + 0.748246i \(0.730896\pi\)
\(548\) 16.0000i 0.683486i
\(549\) 5.00000 0.213395
\(550\) 0 0
\(551\) 72.0000 3.06730
\(552\) − 5.00000i − 0.212814i
\(553\) − 6.00000i − 0.255146i
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 14.0000i 0.593199i 0.955002 + 0.296600i \(0.0958526\pi\)
−0.955002 + 0.296600i \(0.904147\pi\)
\(558\) 1.00000i 0.0423334i
\(559\) −21.0000 −0.888205
\(560\) 0 0
\(561\) −14.0000 −0.591080
\(562\) 8.00000i 0.337460i
\(563\) 27.0000i 1.13791i 0.822367 + 0.568957i \(0.192653\pi\)
−0.822367 + 0.568957i \(0.807347\pi\)
\(564\) 4.00000 0.168430
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) − 1.00000i − 0.0419961i
\(568\) − 4.00000i − 0.167836i
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 31.0000 1.29731 0.648655 0.761083i \(-0.275332\pi\)
0.648655 + 0.761083i \(0.275332\pi\)
\(572\) 14.0000i 0.585369i
\(573\) 1.00000i 0.0417756i
\(574\) −11.0000 −0.459131
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 38.0000i − 1.58196i −0.611842 0.790980i \(-0.709571\pi\)
0.611842 0.790980i \(-0.290429\pi\)
\(578\) 32.0000i 1.33102i
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) −9.00000 −0.373383
\(582\) 10.0000i 0.414513i
\(583\) − 6.00000i − 0.248495i
\(584\) −10.0000 −0.413803
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) − 39.0000i − 1.60970i −0.593477 0.804851i \(-0.702245\pi\)
0.593477 0.804851i \(-0.297755\pi\)
\(588\) − 1.00000i − 0.0412393i
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 21.0000 0.863825
\(592\) 2.00000i 0.0821995i
\(593\) 18.0000i 0.739171i 0.929197 + 0.369586i \(0.120500\pi\)
−0.929197 + 0.369586i \(0.879500\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) −3.00000 −0.122885
\(597\) − 16.0000i − 0.654836i
\(598\) − 35.0000i − 1.43126i
\(599\) 1.00000 0.0408589 0.0204294 0.999791i \(-0.493497\pi\)
0.0204294 + 0.999791i \(0.493497\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) − 3.00000i − 0.122271i
\(603\) − 12.0000i − 0.488678i
\(604\) 22.0000 0.895167
\(605\) 0 0
\(606\) 0 0
\(607\) 36.0000i 1.46119i 0.682808 + 0.730597i \(0.260758\pi\)
−0.682808 + 0.730597i \(0.739242\pi\)
\(608\) 8.00000i 0.324443i
\(609\) 9.00000 0.364698
\(610\) 0 0
\(611\) 28.0000 1.13276
\(612\) 7.00000i 0.282958i
\(613\) 2.00000i 0.0807792i 0.999184 + 0.0403896i \(0.0128599\pi\)
−0.999184 + 0.0403896i \(0.987140\pi\)
\(614\) 22.0000 0.887848
\(615\) 0 0
\(616\) −2.00000 −0.0805823
\(617\) 38.0000i 1.52982i 0.644136 + 0.764911i \(0.277217\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) − 5.00000i − 0.201129i
\(619\) 38.0000 1.52735 0.763674 0.645601i \(-0.223393\pi\)
0.763674 + 0.645601i \(0.223393\pi\)
\(620\) 0 0
\(621\) 5.00000 0.200643
\(622\) − 2.00000i − 0.0801927i
\(623\) − 10.0000i − 0.400642i
\(624\) 7.00000 0.280224
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) − 16.0000i − 0.638978i
\(628\) − 10.0000i − 0.399043i
\(629\) −14.0000 −0.558217
\(630\) 0 0
\(631\) 42.0000 1.67199 0.835997 0.548734i \(-0.184890\pi\)
0.835997 + 0.548734i \(0.184890\pi\)
\(632\) 6.00000i 0.238667i
\(633\) 13.0000i 0.516704i
\(634\) −23.0000 −0.913447
\(635\) 0 0
\(636\) −3.00000 −0.118958
\(637\) − 7.00000i − 0.277350i
\(638\) − 18.0000i − 0.712627i
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) − 2.00000i − 0.0789337i
\(643\) 10.0000i 0.394362i 0.980367 + 0.197181i \(0.0631786\pi\)
−0.980367 + 0.197181i \(0.936821\pi\)
\(644\) 5.00000 0.197028
\(645\) 0 0
\(646\) −56.0000 −2.20329
\(647\) − 46.0000i − 1.80845i −0.427060 0.904223i \(-0.640451\pi\)
0.427060 0.904223i \(-0.359549\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 14.0000 0.549548
\(650\) 0 0
\(651\) −1.00000 −0.0391931
\(652\) 15.0000i 0.587445i
\(653\) − 34.0000i − 1.33052i −0.746611 0.665261i \(-0.768320\pi\)
0.746611 0.665261i \(-0.231680\pi\)
\(654\) 4.00000 0.156412
\(655\) 0 0
\(656\) 11.0000 0.429478
\(657\) − 10.0000i − 0.390137i
\(658\) 4.00000i 0.155936i
\(659\) −34.0000 −1.32445 −0.662226 0.749304i \(-0.730388\pi\)
−0.662226 + 0.749304i \(0.730388\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) − 25.0000i − 0.971653i
\(663\) 49.0000i 1.90300i
\(664\) 9.00000 0.349268
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 45.0000i 1.74241i
\(668\) 2.00000i 0.0773823i
\(669\) 9.00000 0.347960
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) 1.00000i 0.0385758i
\(673\) − 13.0000i − 0.501113i −0.968102 0.250557i \(-0.919386\pi\)
0.968102 0.250557i \(-0.0806136\pi\)
\(674\) 13.0000 0.500741
\(675\) 0 0
\(676\) 36.0000 1.38462
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 14.0000i 0.537667i
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 3.00000 0.114960
\(682\) 2.00000i 0.0765840i
\(683\) 38.0000i 1.45403i 0.686622 + 0.727015i \(0.259093\pi\)
−0.686622 + 0.727015i \(0.740907\pi\)
\(684\) −8.00000 −0.305888
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) − 14.0000i − 0.534133i
\(688\) 3.00000i 0.114374i
\(689\) −21.0000 −0.800036
\(690\) 0 0
\(691\) −14.0000 −0.532585 −0.266293 0.963892i \(-0.585799\pi\)
−0.266293 + 0.963892i \(0.585799\pi\)
\(692\) 16.0000i 0.608229i
\(693\) − 2.00000i − 0.0759737i
\(694\) 6.00000 0.227757
\(695\) 0 0
\(696\) −9.00000 −0.341144
\(697\) 77.0000i 2.91658i
\(698\) − 7.00000i − 0.264954i
\(699\) 16.0000 0.605176
\(700\) 0 0
\(701\) −15.0000 −0.566542 −0.283271 0.959040i \(-0.591420\pi\)
−0.283271 + 0.959040i \(0.591420\pi\)
\(702\) 7.00000i 0.264198i
\(703\) − 16.0000i − 0.603451i
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 0 0
\(708\) − 7.00000i − 0.263076i
\(709\) 8.00000 0.300446 0.150223 0.988652i \(-0.452001\pi\)
0.150223 + 0.988652i \(0.452001\pi\)
\(710\) 0 0
\(711\) −6.00000 −0.225018
\(712\) 10.0000i 0.374766i
\(713\) − 5.00000i − 0.187251i
\(714\) −7.00000 −0.261968
\(715\) 0 0
\(716\) −2.00000 −0.0747435
\(717\) 8.00000i 0.298765i
\(718\) 19.0000i 0.709074i
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 5.00000 0.186210
\(722\) − 45.0000i − 1.67473i
\(723\) 30.0000i 1.11571i
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 7.00000 0.259794
\(727\) − 21.0000i − 0.778847i −0.921059 0.389423i \(-0.872674\pi\)
0.921059 0.389423i \(-0.127326\pi\)
\(728\) 7.00000i 0.259437i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −21.0000 −0.776713
\(732\) − 5.00000i − 0.184805i
\(733\) 45.0000i 1.66211i 0.556188 + 0.831056i \(0.312263\pi\)
−0.556188 + 0.831056i \(0.687737\pi\)
\(734\) 17.0000 0.627481
\(735\) 0 0
\(736\) −5.00000 −0.184302
\(737\) − 24.0000i − 0.884051i
\(738\) 11.0000i 0.404916i
\(739\) −15.0000 −0.551784 −0.275892 0.961189i \(-0.588973\pi\)
−0.275892 + 0.961189i \(0.588973\pi\)
\(740\) 0 0
\(741\) −56.0000 −2.05721
\(742\) − 3.00000i − 0.110133i
\(743\) − 39.0000i − 1.43077i −0.698730 0.715386i \(-0.746251\pi\)
0.698730 0.715386i \(-0.253749\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) −4.00000 −0.146450
\(747\) 9.00000i 0.329293i
\(748\) 14.0000i 0.511891i
\(749\) 2.00000 0.0730784
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) − 4.00000i − 0.145865i
\(753\) 5.00000i 0.182210i
\(754\) −63.0000 −2.29432
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) − 26.0000i − 0.944986i −0.881334 0.472493i \(-0.843354\pi\)
0.881334 0.472493i \(-0.156646\pi\)
\(758\) 7.00000i 0.254251i
\(759\) 10.0000 0.362977
\(760\) 0 0
\(761\) −34.0000 −1.23250 −0.616250 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(762\) − 18.0000i − 0.652071i
\(763\) 4.00000i 0.144810i
\(764\) 1.00000 0.0361787
\(765\) 0 0
\(766\) 0 0
\(767\) − 49.0000i − 1.76929i
\(768\) − 1.00000i − 0.0360844i
\(769\) 32.0000 1.15395 0.576975 0.816762i \(-0.304233\pi\)
0.576975 + 0.816762i \(0.304233\pi\)
\(770\) 0 0
\(771\) −5.00000 −0.180071
\(772\) − 10.0000i − 0.359908i
\(773\) 28.0000i 1.00709i 0.863969 + 0.503545i \(0.167971\pi\)
−0.863969 + 0.503545i \(0.832029\pi\)
\(774\) −3.00000 −0.107833
\(775\) 0 0
\(776\) 10.0000 0.358979
\(777\) − 2.00000i − 0.0717496i
\(778\) 6.00000i 0.215110i
\(779\) −88.0000 −3.15293
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) − 35.0000i − 1.25160i
\(783\) − 9.00000i − 0.321634i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) − 26.0000i − 0.926800i −0.886149 0.463400i \(-0.846629\pi\)
0.886149 0.463400i \(-0.153371\pi\)
\(788\) − 21.0000i − 0.748094i
\(789\) −9.00000 −0.320408
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 2.00000i 0.0710669i
\(793\) − 35.0000i − 1.24289i
\(794\) −37.0000 −1.31308
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 30.0000i 1.06265i 0.847167 + 0.531327i \(0.178307\pi\)
−0.847167 + 0.531327i \(0.821693\pi\)
\(798\) − 8.00000i − 0.283197i
\(799\) 28.0000 0.990569
\(800\) 0 0
\(801\) −10.0000 −0.353333
\(802\) 12.0000i 0.423735i
\(803\) − 20.0000i − 0.705785i
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 7.00000 0.246564
\(807\) − 6.00000i − 0.211210i
\(808\) 0 0
\(809\) 44.0000 1.54696 0.773479 0.633822i \(-0.218515\pi\)
0.773479 + 0.633822i \(0.218515\pi\)
\(810\) 0 0
\(811\) −6.00000 −0.210688 −0.105344 0.994436i \(-0.533594\pi\)
−0.105344 + 0.994436i \(0.533594\pi\)
\(812\) − 9.00000i − 0.315838i
\(813\) − 12.0000i − 0.420858i
\(814\) −4.00000 −0.140200
\(815\) 0 0
\(816\) 7.00000 0.245049
\(817\) − 24.0000i − 0.839654i
\(818\) 10.0000i 0.349642i
\(819\) −7.00000 −0.244600
\(820\) 0 0
\(821\) 34.0000 1.18661 0.593304 0.804978i \(-0.297823\pi\)
0.593304 + 0.804978i \(0.297823\pi\)
\(822\) 16.0000i 0.558064i
\(823\) − 14.0000i − 0.488009i −0.969774 0.244005i \(-0.921539\pi\)
0.969774 0.244005i \(-0.0784612\pi\)
\(824\) −5.00000 −0.174183
\(825\) 0 0
\(826\) 7.00000 0.243561
\(827\) 6.00000i 0.208640i 0.994544 + 0.104320i \(0.0332667\pi\)
−0.994544 + 0.104320i \(0.966733\pi\)
\(828\) − 5.00000i − 0.173762i
\(829\) 19.0000 0.659897 0.329949 0.943999i \(-0.392969\pi\)
0.329949 + 0.943999i \(0.392969\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) − 7.00000i − 0.242681i
\(833\) − 7.00000i − 0.242536i
\(834\) 8.00000 0.277017
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) 1.00000i 0.0345651i
\(838\) − 3.00000i − 0.103633i
\(839\) 36.0000 1.24286 0.621429 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) − 10.0000i − 0.344623i
\(843\) 8.00000i 0.275535i
\(844\) 13.0000 0.447478
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) 7.00000i 0.240523i
\(848\) 3.00000i 0.103020i
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) 10.0000 0.342796
\(852\) − 4.00000i − 0.137038i
\(853\) − 35.0000i − 1.19838i −0.800608 0.599189i \(-0.795490\pi\)
0.800608 0.599189i \(-0.204510\pi\)
\(854\) 5.00000 0.171096
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) 10.0000i 0.341593i 0.985306 + 0.170797i \(0.0546341\pi\)
−0.985306 + 0.170797i \(0.945366\pi\)
\(858\) 14.0000i 0.477952i
\(859\) 16.0000 0.545913 0.272956 0.962026i \(-0.411998\pi\)
0.272956 + 0.962026i \(0.411998\pi\)
\(860\) 0 0
\(861\) −11.0000 −0.374879
\(862\) − 39.0000i − 1.32835i
\(863\) 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
\(866\) 6.00000 0.203888
\(867\) 32.0000i 1.08678i
\(868\) 1.00000i 0.0339422i
\(869\) −12.0000 −0.407072
\(870\) 0 0
\(871\) −84.0000 −2.84623
\(872\) − 4.00000i − 0.135457i
\(873\) 10.0000i 0.338449i
\(874\) 40.0000 1.35302
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) − 32.0000i − 1.08056i −0.841484 0.540282i \(-0.818318\pi\)
0.841484 0.540282i \(-0.181682\pi\)
\(878\) − 19.0000i − 0.641219i
\(879\) 24.0000 0.809500
\(880\) 0 0
\(881\) −21.0000 −0.707508 −0.353754 0.935339i \(-0.615095\pi\)
−0.353754 + 0.935339i \(0.615095\pi\)
\(882\) − 1.00000i − 0.0336718i
\(883\) − 23.0000i − 0.774012i −0.922077 0.387006i \(-0.873509\pi\)
0.922077 0.387006i \(-0.126491\pi\)
\(884\) 49.0000 1.64805
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 14.0000i 0.470074i 0.971986 + 0.235037i \(0.0755211\pi\)
−0.971986 + 0.235037i \(0.924479\pi\)
\(888\) 2.00000i 0.0671156i
\(889\) 18.0000 0.603701
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) − 9.00000i − 0.301342i
\(893\) 32.0000i 1.07084i
\(894\) −3.00000 −0.100335
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) − 35.0000i − 1.16862i
\(898\) − 20.0000i − 0.667409i
\(899\) −9.00000 −0.300167
\(900\) 0 0
\(901\) −21.0000 −0.699611
\(902\) 22.0000i 0.732520i
\(903\) − 3.00000i − 0.0998337i
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) 22.0000 0.730901
\(907\) 39.0000i 1.29497i 0.762077 + 0.647487i \(0.224180\pi\)
−0.762077 + 0.647487i \(0.775820\pi\)
\(908\) − 3.00000i − 0.0995585i
\(909\) 0 0
\(910\) 0 0
\(911\) −39.0000 −1.29213 −0.646064 0.763283i \(-0.723586\pi\)
−0.646064 + 0.763283i \(0.723586\pi\)
\(912\) 8.00000i 0.264906i
\(913\) 18.0000i 0.595713i
\(914\) −7.00000 −0.231539
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) 7.00000i 0.231034i
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 22.0000 0.724925
\(922\) − 8.00000i − 0.263466i
\(923\) − 28.0000i − 0.921631i
\(924\) −2.00000 −0.0657952
\(925\) 0 0
\(926\) 8.00000 0.262896
\(927\) − 5.00000i − 0.164222i
\(928\) 9.00000i 0.295439i
\(929\) −7.00000 −0.229663 −0.114831 0.993385i \(-0.536633\pi\)
−0.114831 + 0.993385i \(0.536633\pi\)
\(930\) 0 0
\(931\) 8.00000 0.262189
\(932\) − 16.0000i − 0.524097i
\(933\) − 2.00000i − 0.0654771i
\(934\) 29.0000 0.948909
\(935\) 0 0
\(936\) 7.00000 0.228802
\(937\) 6.00000i 0.196011i 0.995186 + 0.0980057i \(0.0312463\pi\)
−0.995186 + 0.0980057i \(0.968754\pi\)
\(938\) − 12.0000i − 0.391814i
\(939\) 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
\(943\) − 55.0000i − 1.79105i
\(944\) −7.00000 −0.227831
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) − 12.0000i − 0.389948i −0.980808 0.194974i \(-0.937538\pi\)
0.980808 0.194974i \(-0.0624622\pi\)
\(948\) 6.00000i 0.194871i
\(949\) −70.0000 −2.27230
\(950\) 0 0
\(951\) −23.0000 −0.745826
\(952\) 7.00000i 0.226871i
\(953\) − 48.0000i − 1.55487i −0.628962 0.777436i \(-0.716520\pi\)
0.628962 0.777436i \(-0.283480\pi\)
\(954\) −3.00000 −0.0971286
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) − 18.0000i − 0.581857i
\(958\) − 12.0000i − 0.387702i
\(959\) −16.0000 −0.516667
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 14.0000i 0.451378i
\(963\) − 2.00000i − 0.0644491i
\(964\) 30.0000 0.966235
\(965\) 0 0
\(966\) 5.00000 0.160872
\(967\) − 16.0000i − 0.514525i −0.966342 0.257263i \(-0.917179\pi\)
0.966342 0.257263i \(-0.0828206\pi\)
\(968\) − 7.00000i − 0.224989i
\(969\) −56.0000 −1.79898
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 8.00000i 0.256468i
\(974\) −6.00000 −0.192252
\(975\) 0 0
\(976\) −5.00000 −0.160046
\(977\) 18.0000i 0.575871i 0.957650 + 0.287936i \(0.0929689\pi\)
−0.957650 + 0.287936i \(0.907031\pi\)
\(978\) 15.0000i 0.479647i
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 42.0000i 1.34027i
\(983\) 32.0000i 1.02064i 0.859984 + 0.510321i \(0.170473\pi\)
−0.859984 + 0.510321i \(0.829527\pi\)
\(984\) 11.0000 0.350667
\(985\) 0 0
\(986\) −63.0000 −2.00633
\(987\) 4.00000i 0.127321i
\(988\) 56.0000i 1.78160i
\(989\) 15.0000 0.476972
\(990\) 0 0
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) − 1.00000i − 0.0317500i
\(993\) − 25.0000i − 0.793351i
\(994\) 4.00000 0.126872
\(995\) 0 0
\(996\) 9.00000 0.285176
\(997\) − 2.00000i − 0.0633406i −0.999498 0.0316703i \(-0.989917\pi\)
0.999498 0.0316703i \(-0.0100827\pi\)
\(998\) − 23.0000i − 0.728052i
\(999\) −2.00000 −0.0632772
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.b.799.1 2
3.2 odd 2 3150.2.g.p.2899.2 2
5.2 odd 4 1050.2.a.n.1.1 yes 1
5.3 odd 4 1050.2.a.f.1.1 1
5.4 even 2 inner 1050.2.g.b.799.2 2
15.2 even 4 3150.2.a.r.1.1 1
15.8 even 4 3150.2.a.bd.1.1 1
15.14 odd 2 3150.2.g.p.2899.1 2
20.3 even 4 8400.2.a.bb.1.1 1
20.7 even 4 8400.2.a.cb.1.1 1
35.13 even 4 7350.2.a.i.1.1 1
35.27 even 4 7350.2.a.cm.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.f.1.1 1 5.3 odd 4
1050.2.a.n.1.1 yes 1 5.2 odd 4
1050.2.g.b.799.1 2 1.1 even 1 trivial
1050.2.g.b.799.2 2 5.4 even 2 inner
3150.2.a.r.1.1 1 15.2 even 4
3150.2.a.bd.1.1 1 15.8 even 4
3150.2.g.p.2899.1 2 15.14 odd 2
3150.2.g.p.2899.2 2 3.2 odd 2
7350.2.a.i.1.1 1 35.13 even 4
7350.2.a.cm.1.1 1 35.27 even 4
8400.2.a.bb.1.1 1 20.3 even 4
8400.2.a.cb.1.1 1 20.7 even 4