Properties

Label 2850.2.d.a.799.1
Level $2850$
Weight $2$
Character 2850.799
Analytic conductor $22.757$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.7573645761\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 570)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2850.799
Dual form 2850.2.d.a.799.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +6.00000i q^{13} -4.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} -4.00000 q^{21} +4.00000i q^{22} -4.00000i q^{23} +1.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} +4.00000i q^{28} -6.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} +4.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} -1.00000i q^{38} +6.00000 q^{39} +10.0000 q^{41} +4.00000i q^{42} +8.00000i q^{43} +4.00000 q^{44} -4.00000 q^{46} +12.0000i q^{47} -1.00000i q^{48} -9.00000 q^{49} -6.00000 q^{51} -6.00000i q^{52} -2.00000i q^{53} +1.00000 q^{54} +4.00000 q^{56} -1.00000i q^{57} +6.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} +8.00000i q^{62} +4.00000i q^{63} -1.00000 q^{64} +4.00000 q^{66} -12.0000i q^{67} +6.00000i q^{68} -4.00000 q^{69} -16.0000 q^{71} -1.00000i q^{72} +14.0000i q^{73} +2.00000 q^{74} -1.00000 q^{76} +16.0000i q^{77} -6.00000i q^{78} -8.00000 q^{79} +1.00000 q^{81} -10.0000i q^{82} +4.00000 q^{84} +8.00000 q^{86} +6.00000i q^{87} -4.00000i q^{88} +6.00000 q^{89} +24.0000 q^{91} +4.00000i q^{92} +8.00000i q^{93} +12.0000 q^{94} -1.00000 q^{96} +14.0000i q^{97} +9.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 2q^{6} - 2q^{9} - 8q^{11} - 8q^{14} + 2q^{16} + 2q^{19} - 8q^{21} + 2q^{24} + 12q^{26} - 12q^{29} - 16q^{31} - 12q^{34} + 2q^{36} + 12q^{39} + 20q^{41} + 8q^{44} - 8q^{46} - 18q^{49} - 12q^{51} + 2q^{54} + 8q^{56} + 8q^{59} - 4q^{61} - 2q^{64} + 8q^{66} - 8q^{69} - 32q^{71} + 4q^{74} - 2q^{76} - 16q^{79} + 2q^{81} + 8q^{84} + 16q^{86} + 12q^{89} + 48q^{91} + 24q^{94} - 2q^{96} + 8q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2850\mathbb{Z}\right)^\times\).

\(n\) \(1027\) \(1351\) \(1901\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 4.00000i 0.852803i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) 1.00000i 0.192450i
\(28\) 4.00000i 0.755929i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 4.00000i 0.696311i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 4.00000i 0.617213i
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) − 6.00000i − 0.832050i
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) − 1.00000i − 0.132453i
\(58\) 6.00000i 0.787839i
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 8.00000i 1.01600i
\(63\) 4.00000i 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) − 12.0000i − 1.46603i −0.680211 0.733017i \(-0.738112\pi\)
0.680211 0.733017i \(-0.261888\pi\)
\(68\) 6.00000i 0.727607i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 14.0000i 1.63858i 0.573382 + 0.819288i \(0.305631\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 16.0000i 1.82337i
\(78\) − 6.00000i − 0.679366i
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 10.0000i − 1.10432i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 6.00000i 0.643268i
\(88\) − 4.00000i − 0.426401i
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 24.0000 2.51588
\(92\) 4.00000i 0.417029i
\(93\) 8.00000i 0.829561i
\(94\) 12.0000 1.23771
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) 9.00000i 0.909137i
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 6.00000i 0.594089i
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) − 4.00000i − 0.377964i
\(113\) 10.0000i 0.940721i 0.882474 + 0.470360i \(0.155876\pi\)
−0.882474 + 0.470360i \(0.844124\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) − 6.00000i − 0.554700i
\(118\) − 4.00000i − 0.368230i
\(119\) −24.0000 −2.20008
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 2.00000i 0.181071i
\(123\) − 10.0000i − 0.901670i
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) − 4.00000i − 0.348155i
\(133\) − 4.00000i − 0.346844i
\(134\) −12.0000 −1.03664
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) 4.00000i 0.340503i
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 16.0000i 1.34269i
\(143\) − 24.0000i − 2.00698i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 14.0000 1.15865
\(147\) 9.00000i 0.742307i
\(148\) − 2.00000i − 0.164399i
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) 6.00000i 0.485071i
\(154\) 16.0000 1.28932
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) 22.0000i 1.75579i 0.478852 + 0.877896i \(0.341053\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) 8.00000i 0.636446i
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) − 1.00000i − 0.0785674i
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) − 4.00000i − 0.308607i
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) − 8.00000i − 0.609994i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) − 4.00000i − 0.300658i
\(178\) − 6.00000i − 0.449719i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) − 24.0000i − 1.77900i
\(183\) 2.00000i 0.147844i
\(184\) 4.00000 0.294884
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 24.0000i 1.75505i
\(188\) − 12.0000i − 0.875190i
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 2.00000i 0.143963i 0.997406 + 0.0719816i \(0.0229323\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) − 4.00000i − 0.284268i
\(199\) 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
\(202\) − 6.00000i − 0.422159i
\(203\) 24.0000i 1.68447i
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 4.00000i 0.278019i
\(208\) 6.00000i 0.416025i
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 2.00000i 0.137361i
\(213\) 16.0000i 1.09630i
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 32.0000i 2.17230i
\(218\) 6.00000i 0.406371i
\(219\) 14.0000 0.946032
\(220\) 0 0
\(221\) 36.0000 2.42162
\(222\) − 2.00000i − 0.134231i
\(223\) 16.0000i 1.07144i 0.844396 + 0.535720i \(0.179960\pi\)
−0.844396 + 0.535720i \(0.820040\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) 1.00000i 0.0662266i
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) − 6.00000i − 0.393919i
\(233\) − 10.0000i − 0.655122i −0.944830 0.327561i \(-0.893773\pi\)
0.944830 0.327561i \(-0.106227\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) 8.00000i 0.519656i
\(238\) 24.0000i 1.55569i
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) − 1.00000i − 0.0641500i
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −10.0000 −0.637577
\(247\) 6.00000i 0.381771i
\(248\) − 8.00000i − 0.508001i
\(249\) 0 0
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) − 4.00000i − 0.251976i
\(253\) 16.0000i 1.00591i
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) − 8.00000i − 0.498058i
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) − 12.0000i − 0.741362i
\(263\) − 20.0000i − 1.23325i −0.787256 0.616626i \(-0.788499\pi\)
0.787256 0.616626i \(-0.211501\pi\)
\(264\) −4.00000 −0.246183
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) − 6.00000i − 0.367194i
\(268\) 12.0000i 0.733017i
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) − 24.0000i − 1.45255i
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) 12.0000i 0.719712i
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) − 12.0000i − 0.714590i
\(283\) 16.0000i 0.951101i 0.879688 + 0.475551i \(0.157751\pi\)
−0.879688 + 0.475551i \(0.842249\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) − 40.0000i − 2.36113i
\(288\) 1.00000i 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) − 14.0000i − 0.819288i
\(293\) − 2.00000i − 0.116841i −0.998292 0.0584206i \(-0.981394\pi\)
0.998292 0.0584206i \(-0.0186065\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) − 4.00000i − 0.232104i
\(298\) − 10.0000i − 0.579284i
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) 32.0000 1.84445
\(302\) − 8.00000i − 0.460348i
\(303\) − 6.00000i − 0.344691i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) − 20.0000i − 1.14146i −0.821138 0.570730i \(-0.806660\pi\)
0.821138 0.570730i \(-0.193340\pi\)
\(308\) − 16.0000i − 0.911685i
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 6.00000i 0.339683i
\(313\) 6.00000i 0.339140i 0.985518 + 0.169570i \(0.0542379\pi\)
−0.985518 + 0.169570i \(0.945762\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) − 14.0000i − 0.786318i −0.919470 0.393159i \(-0.871382\pi\)
0.919470 0.393159i \(-0.128618\pi\)
\(318\) 2.00000i 0.112154i
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 16.0000i 0.891645i
\(323\) − 6.00000i − 0.333849i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 16.0000 0.886158
\(327\) 6.00000i 0.331801i
\(328\) 10.0000i 0.552158i
\(329\) 48.0000 2.64633
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) − 2.00000i − 0.109599i
\(334\) 0 0
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) 23.0000i 1.25104i
\(339\) 10.0000 0.543125
\(340\) 0 0
\(341\) 32.0000 1.73290
\(342\) 1.00000i 0.0540738i
\(343\) 8.00000i 0.431959i
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 32.0000i 1.71785i 0.512101 + 0.858925i \(0.328867\pi\)
−0.512101 + 0.858925i \(0.671133\pi\)
\(348\) − 6.00000i − 0.321634i
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 4.00000i 0.213201i
\(353\) − 26.0000i − 1.38384i −0.721974 0.691920i \(-0.756765\pi\)
0.721974 0.691920i \(-0.243235\pi\)
\(354\) −4.00000 −0.212598
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 24.0000i 1.27021i
\(358\) − 12.0000i − 0.634220i
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 2.00000i 0.105118i
\(363\) − 5.00000i − 0.262432i
\(364\) −24.0000 −1.25794
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 4.00000i 0.208798i 0.994535 + 0.104399i \(0.0332919\pi\)
−0.994535 + 0.104399i \(0.966708\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) −8.00000 −0.415339
\(372\) − 8.00000i − 0.414781i
\(373\) − 10.0000i − 0.517780i −0.965907 0.258890i \(-0.916643\pi\)
0.965907 0.258890i \(-0.0833568\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) − 36.0000i − 1.85409i
\(378\) − 4.00000i − 0.205738i
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 8.00000i 0.409316i
\(383\) − 24.0000i − 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) − 8.00000i − 0.406663i
\(388\) − 14.0000i − 0.710742i
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) − 9.00000i − 0.454569i
\(393\) − 12.0000i − 0.605320i
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 22.0000i 1.10415i 0.833795 + 0.552074i \(0.186163\pi\)
−0.833795 + 0.552074i \(0.813837\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −22.0000 −1.09863 −0.549314 0.835616i \(-0.685111\pi\)
−0.549314 + 0.835616i \(0.685111\pi\)
\(402\) 12.0000i 0.598506i
\(403\) − 48.0000i − 2.39105i
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) − 8.00000i − 0.396545i
\(408\) − 6.00000i − 0.297044i
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) 2.00000 0.0986527
\(412\) 8.00000i 0.394132i
\(413\) − 16.0000i − 0.787309i
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 6.00000 0.294174
\(417\) 12.0000i 0.587643i
\(418\) 4.00000i 0.195646i
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 20.0000i 0.973585i
\(423\) − 12.0000i − 0.583460i
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) 8.00000i 0.387147i
\(428\) 12.0000i 0.580042i
\(429\) −24.0000 −1.15873
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 14.0000i − 0.672797i −0.941720 0.336399i \(-0.890791\pi\)
0.941720 0.336399i \(-0.109209\pi\)
\(434\) 32.0000 1.53605
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) − 4.00000i − 0.191346i
\(438\) − 14.0000i − 0.668946i
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) − 36.0000i − 1.71235i
\(443\) − 8.00000i − 0.380091i −0.981775 0.190046i \(-0.939136\pi\)
0.981775 0.190046i \(-0.0608636\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 16.0000 0.757622
\(447\) − 10.0000i − 0.472984i
\(448\) 4.00000i 0.188982i
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) −40.0000 −1.88353
\(452\) − 10.0000i − 0.470360i
\(453\) − 8.00000i − 0.375873i
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 10.0000i 0.467780i 0.972263 + 0.233890i \(0.0751456\pi\)
−0.972263 + 0.233890i \(0.924854\pi\)
\(458\) 6.00000i 0.280362i
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) − 16.0000i − 0.744387i
\(463\) − 20.0000i − 0.929479i −0.885448 0.464739i \(-0.846148\pi\)
0.885448 0.464739i \(-0.153852\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) − 8.00000i − 0.370196i −0.982720 0.185098i \(-0.940740\pi\)
0.982720 0.185098i \(-0.0592602\pi\)
\(468\) 6.00000i 0.277350i
\(469\) −48.0000 −2.21643
\(470\) 0 0
\(471\) 22.0000 1.01371
\(472\) 4.00000i 0.184115i
\(473\) − 32.0000i − 1.47136i
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) 24.0000 1.10004
\(477\) 2.00000i 0.0915737i
\(478\) 16.0000i 0.731823i
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 22.0000i 1.00207i
\(483\) 16.0000i 0.728025i
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) − 16.0000i − 0.725029i −0.931978 0.362515i \(-0.881918\pi\)
0.931978 0.362515i \(-0.118082\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 10.0000i 0.450835i
\(493\) 36.0000i 1.62136i
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 64.0000i 2.87079i
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 4.00000i − 0.178529i
\(503\) 12.0000i 0.535054i 0.963550 + 0.267527i \(0.0862064\pi\)
−0.963550 + 0.267527i \(0.913794\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) 16.0000 0.711287
\(507\) 23.0000i 1.02147i
\(508\) 16.0000i 0.709885i
\(509\) −6.00000 −0.265945 −0.132973 0.991120i \(-0.542452\pi\)
−0.132973 + 0.991120i \(0.542452\pi\)
\(510\) 0 0
\(511\) 56.0000 2.47729
\(512\) − 1.00000i − 0.0441942i
\(513\) 1.00000i 0.0441511i
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) −8.00000 −0.352180
\(517\) − 48.0000i − 2.11104i
\(518\) − 8.00000i − 0.351500i
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) − 6.00000i − 0.262613i
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −20.0000 −0.872041
\(527\) 48.0000i 2.09091i
\(528\) 4.00000i 0.174078i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 4.00000i 0.173422i
\(533\) 60.0000i 2.59889i
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) − 12.0000i − 0.517838i
\(538\) 14.0000i 0.603583i
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 8.00000i 0.343629i
\(543\) 2.00000i 0.0858282i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) −24.0000 −1.02711
\(547\) − 36.0000i − 1.53925i −0.638497 0.769624i \(-0.720443\pi\)
0.638497 0.769624i \(-0.279557\pi\)
\(548\) − 2.00000i − 0.0854358i
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) − 4.00000i − 0.170251i
\(553\) 32.0000i 1.36078i
\(554\) −2.00000 −0.0849719
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) − 26.0000i − 1.10166i −0.834619 0.550828i \(-0.814312\pi\)
0.834619 0.550828i \(-0.185688\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) −48.0000 −2.03018
\(560\) 0 0
\(561\) 24.0000 1.01328
\(562\) 22.0000i 0.928014i
\(563\) 20.0000i 0.842900i 0.906852 + 0.421450i \(0.138479\pi\)
−0.906852 + 0.421450i \(0.861521\pi\)
\(564\) −12.0000 −0.505291
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) − 4.00000i − 0.167984i
\(568\) − 16.0000i − 0.671345i
\(569\) −34.0000 −1.42535 −0.712677 0.701492i \(-0.752517\pi\)
−0.712677 + 0.701492i \(0.752517\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 24.0000i 1.00349i
\(573\) 8.00000i 0.334205i
\(574\) −40.0000 −1.66957
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 18.0000i 0.749350i 0.927156 + 0.374675i \(0.122246\pi\)
−0.927156 + 0.374675i \(0.877754\pi\)
\(578\) 19.0000i 0.790296i
\(579\) 2.00000 0.0831172
\(580\) 0 0
\(581\) 0 0
\(582\) − 14.0000i − 0.580319i
\(583\) 8.00000i 0.331326i
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) −2.00000 −0.0826192
\(587\) 32.0000i 1.32078i 0.750922 + 0.660391i \(0.229609\pi\)
−0.750922 + 0.660391i \(0.770391\pi\)
\(588\) − 9.00000i − 0.371154i
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 2.00000i 0.0821995i
\(593\) − 34.0000i − 1.39621i −0.715994 0.698106i \(-0.754026\pi\)
0.715994 0.698106i \(-0.245974\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) − 16.0000i − 0.654836i
\(598\) − 24.0000i − 0.981433i
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) − 32.0000i − 1.30422i
\(603\) 12.0000i 0.488678i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) 24.0000i 0.974130i 0.873366 + 0.487065i \(0.161933\pi\)
−0.873366 + 0.487065i \(0.838067\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) 24.0000 0.972529
\(610\) 0 0
\(611\) −72.0000 −2.91281
\(612\) − 6.00000i − 0.242536i
\(613\) − 46.0000i − 1.85792i −0.370177 0.928961i \(-0.620703\pi\)
0.370177 0.928961i \(-0.379297\pi\)
\(614\) −20.0000 −0.807134
\(615\) 0 0
\(616\) −16.0000 −0.644658
\(617\) − 14.0000i − 0.563619i −0.959470 0.281809i \(-0.909065\pi\)
0.959470 0.281809i \(-0.0909346\pi\)
\(618\) 8.00000i 0.321807i
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 8.00000i 0.320771i
\(623\) − 24.0000i − 0.961540i
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) 6.00000 0.239808
\(627\) 4.00000i 0.159745i
\(628\) − 22.0000i − 0.877896i
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) −32.0000 −1.27390 −0.636950 0.770905i \(-0.719804\pi\)
−0.636950 + 0.770905i \(0.719804\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) 20.0000i 0.794929i
\(634\) −14.0000 −0.556011
\(635\) 0 0
\(636\) 2.00000 0.0793052
\(637\) − 54.0000i − 2.13956i
\(638\) − 24.0000i − 0.950169i
\(639\) 16.0000 0.632950
\(640\) 0 0
\(641\) −14.0000 −0.552967 −0.276483 0.961019i \(-0.589169\pi\)
−0.276483 + 0.961019i \(0.589169\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 40.0000i 1.57745i 0.614749 + 0.788723i \(0.289257\pi\)
−0.614749 + 0.788723i \(0.710743\pi\)
\(644\) 16.0000 0.630488
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) 12.0000i 0.471769i 0.971781 + 0.235884i \(0.0757987\pi\)
−0.971781 + 0.235884i \(0.924201\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 32.0000 1.25418
\(652\) − 16.0000i − 0.626608i
\(653\) − 6.00000i − 0.234798i −0.993085 0.117399i \(-0.962544\pi\)
0.993085 0.117399i \(-0.0374557\pi\)
\(654\) 6.00000 0.234619
\(655\) 0 0
\(656\) 10.0000 0.390434
\(657\) − 14.0000i − 0.546192i
\(658\) − 48.0000i − 1.87123i
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) − 4.00000i − 0.155464i
\(663\) − 36.0000i − 1.39812i
\(664\) 0 0
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 24.0000i 0.929284i
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) 8.00000 0.308837
\(672\) 4.00000i 0.154303i
\(673\) − 22.0000i − 0.848038i −0.905653 0.424019i \(-0.860619\pi\)
0.905653 0.424019i \(-0.139381\pi\)
\(674\) 14.0000 0.539260
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) − 30.0000i − 1.15299i −0.817099 0.576497i \(-0.804419\pi\)
0.817099 0.576497i \(-0.195581\pi\)
\(678\) − 10.0000i − 0.384048i
\(679\) 56.0000 2.14908
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) − 32.0000i − 1.22534i
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) 6.00000i 0.228914i
\(688\) 8.00000i 0.304997i
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) − 6.00000i − 0.228086i
\(693\) − 16.0000i − 0.607790i
\(694\) 32.0000 1.21470
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) − 60.0000i − 2.27266i
\(698\) − 2.00000i − 0.0757011i
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 6.00000i 0.226455i
\(703\) 2.00000i 0.0754314i
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) −26.0000 −0.978523
\(707\) − 24.0000i − 0.902613i
\(708\) 4.00000i 0.150329i
\(709\) −6.00000 −0.225335 −0.112667 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 6.00000i 0.224860i
\(713\) 32.0000i 1.19841i
\(714\) 24.0000 0.898177
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 16.0000i 0.597531i
\(718\) − 16.0000i − 0.597115i
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) − 1.00000i − 0.0372161i
\(723\) 22.0000i 0.818189i
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) 28.0000i 1.03846i 0.854634 + 0.519231i \(0.173782\pi\)
−0.854634 + 0.519231i \(0.826218\pi\)
\(728\) 24.0000i 0.889499i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) − 2.00000i − 0.0739221i
\(733\) 50.0000i 1.84679i 0.383849 + 0.923396i \(0.374598\pi\)
−0.383849 + 0.923396i \(0.625402\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 48.0000i 1.76810i
\(738\) 10.0000i 0.368105i
\(739\) −44.0000 −1.61857 −0.809283 0.587419i \(-0.800144\pi\)
−0.809283 + 0.587419i \(0.800144\pi\)
\(740\) 0 0
\(741\) 6.00000 0.220416
\(742\) 8.00000i 0.293689i
\(743\) − 16.0000i − 0.586983i −0.955962 0.293492i \(-0.905183\pi\)
0.955962 0.293492i \(-0.0948173\pi\)
\(744\) −8.00000 −0.293294
\(745\) 0 0
\(746\) −10.0000 −0.366126
\(747\) 0 0
\(748\) − 24.0000i − 0.877527i
\(749\) −48.0000 −1.75388
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 12.0000i 0.437595i
\(753\) − 4.00000i − 0.145768i
\(754\) −36.0000 −1.31104
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) − 10.0000i − 0.363456i −0.983349 0.181728i \(-0.941831\pi\)
0.983349 0.181728i \(-0.0581691\pi\)
\(758\) 4.00000i 0.145287i
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 16.0000i 0.579619i
\(763\) 24.0000i 0.868858i
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 24.0000i 0.866590i
\(768\) − 1.00000i − 0.0360844i
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) − 2.00000i − 0.0719816i
\(773\) − 10.0000i − 0.359675i −0.983696 0.179838i \(-0.942443\pi\)
0.983696 0.179838i \(-0.0575572\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) −14.0000 −0.502571
\(777\) − 8.00000i − 0.286998i
\(778\) − 26.0000i − 0.932145i
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) 64.0000 2.29010
\(782\) 24.0000i 0.858238i
\(783\) − 6.00000i − 0.214423i
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) − 52.0000i − 1.85360i −0.375555 0.926800i \(-0.622548\pi\)
0.375555 0.926800i \(-0.377452\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) −20.0000 −0.712019
\(790\) 0 0
\(791\) 40.0000 1.42224
\(792\) 4.00000i 0.142134i
\(793\) − 12.0000i − 0.426132i
\(794\) 22.0000 0.780751
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) − 38.0000i − 1.34603i −0.739629 0.673015i \(-0.764999\pi\)
0.739629 0.673015i \(-0.235001\pi\)
\(798\) 4.00000i 0.141598i
\(799\) 72.0000 2.54718
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 22.0000i 0.776847i
\(803\) − 56.0000i − 1.97620i
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) −48.0000 −1.69073
\(807\) 14.0000i 0.492823i
\(808\) 6.00000i 0.211079i
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) 28.0000 0.983213 0.491606 0.870817i \(-0.336410\pi\)
0.491606 + 0.870817i \(0.336410\pi\)
\(812\) − 24.0000i − 0.842235i
\(813\) 8.00000i 0.280572i
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) 8.00000i 0.279885i
\(818\) − 6.00000i − 0.209785i
\(819\) −24.0000 −0.838628
\(820\) 0 0
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) − 2.00000i − 0.0697580i
\(823\) − 4.00000i − 0.139431i −0.997567 0.0697156i \(-0.977791\pi\)
0.997567 0.0697156i \(-0.0222092\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) − 4.00000i − 0.139010i
\(829\) 26.0000 0.903017 0.451509 0.892267i \(-0.350886\pi\)
0.451509 + 0.892267i \(0.350886\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) − 6.00000i − 0.208013i
\(833\) 54.0000i 1.87099i
\(834\) 12.0000 0.415526
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) − 8.00000i − 0.276520i
\(838\) 20.0000i 0.690889i
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 26.0000i 0.896019i
\(843\) 22.0000i 0.757720i
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) −12.0000 −0.412568
\(847\) − 20.0000i − 0.687208i
\(848\) − 2.00000i − 0.0686803i
\(849\) 16.0000 0.549119
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) − 16.0000i − 0.548151i
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 6.00000i 0.204956i 0.994735 + 0.102478i \(0.0326771\pi\)
−0.994735 + 0.102478i \(0.967323\pi\)
\(858\) 24.0000i 0.819346i
\(859\) 44.0000 1.50126 0.750630 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 0 0
\(861\) −40.0000 −1.36320
\(862\) 32.0000i 1.08992i
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) 19.0000i 0.645274i
\(868\) − 32.0000i − 1.08615i
\(869\) 32.0000 1.08553
\(870\) 0 0
\(871\) 72.0000 2.43963
\(872\) − 6.00000i − 0.203186i
\(873\) − 14.0000i − 0.473828i
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) 18.0000i 0.607817i 0.952701 + 0.303908i \(0.0982917\pi\)
−0.952701 + 0.303908i \(0.901708\pi\)
\(878\) 32.0000i 1.07995i
\(879\) −2.00000 −0.0674583
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) − 9.00000i − 0.303046i
\(883\) 16.0000i 0.538443i 0.963078 + 0.269221i \(0.0867663\pi\)
−0.963078 + 0.269221i \(0.913234\pi\)
\(884\) −36.0000 −1.21081
\(885\) 0 0
\(886\) −8.00000 −0.268765
\(887\) − 8.00000i − 0.268614i −0.990940 0.134307i \(-0.957119\pi\)
0.990940 0.134307i \(-0.0428808\pi\)
\(888\) 2.00000i 0.0671156i
\(889\) −64.0000 −2.14649
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) − 16.0000i − 0.535720i
\(893\) 12.0000i 0.401565i
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) − 24.0000i − 0.801337i
\(898\) 10.0000i 0.333704i
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) −12.0000 −0.399778
\(902\) 40.0000i 1.33185i
\(903\) − 32.0000i − 1.06489i
\(904\) −10.0000 −0.332595
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) − 12.0000i − 0.398453i −0.979953 0.199227i \(-0.936157\pi\)
0.979953 0.199227i \(-0.0638430\pi\)
\(908\) 4.00000i 0.132745i
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 16.0000 0.530104 0.265052 0.964234i \(-0.414611\pi\)
0.265052 + 0.964234i \(0.414611\pi\)
\(912\) − 1.00000i − 0.0331133i
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) − 48.0000i − 1.58510i
\(918\) − 6.00000i − 0.198030i
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) −20.0000 −0.659022
\(922\) 18.0000i 0.592798i
\(923\) − 96.0000i − 3.15988i
\(924\) −16.0000 −0.526361
\(925\) 0 0
\(926\) −20.0000 −0.657241
\(927\) 8.00000i 0.262754i
\(928\) 6.00000i 0.196960i
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) −9.00000 −0.294963
\(932\) 10.0000i 0.327561i
\(933\) 8.00000i 0.261908i
\(934\) −8.00000 −0.261768
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) − 30.0000i − 0.980057i −0.871706 0.490029i \(-0.836986\pi\)
0.871706 0.490029i \(-0.163014\pi\)
\(938\) 48.0000i 1.56726i
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 46.0000 1.49956 0.749779 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) − 22.0000i − 0.716799i
\(943\) − 40.0000i − 1.30258i
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −32.0000 −1.04041
\(947\) − 24.0000i − 0.779895i −0.920837 0.389948i \(-0.872493\pi\)
0.920837 0.389948i \(-0.127507\pi\)
\(948\) − 8.00000i − 0.259828i
\(949\) −84.0000 −2.72676
\(950\) 0 0
\(951\) −14.0000 −0.453981
\(952\) − 24.0000i − 0.777844i
\(953\) 42.0000i 1.36051i 0.732974 + 0.680257i \(0.238132\pi\)
−0.732974 + 0.680257i \(0.761868\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) 16.0000 0.517477
\(957\) − 24.0000i − 0.775810i
\(958\) 24.0000i 0.775405i
\(959\) 8.00000 0.258333
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 12.0000i 0.386896i
\(963\) 12.0000i 0.386695i
\(964\) 22.0000 0.708572
\(965\) 0 0
\(966\) 16.0000 0.514792
\(967\) 52.0000i 1.67221i 0.548572 + 0.836104i \(0.315172\pi\)
−0.548572 + 0.836104i \(0.684828\pi\)
\(968\) 5.00000i 0.160706i
\(969\) −6.00000 −0.192748
\(970\) 0 0
\(971\) 28.0000 0.898563 0.449281 0.893390i \(-0.351680\pi\)
0.449281 + 0.893390i \(0.351680\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 48.0000i 1.53881i
\(974\) −16.0000 −0.512673
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) − 2.00000i − 0.0639857i −0.999488 0.0319928i \(-0.989815\pi\)
0.999488 0.0319928i \(-0.0101854\pi\)
\(978\) − 16.0000i − 0.511624i
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 12.0000i 0.382935i
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) − 48.0000i − 1.52786i
\(988\) − 6.00000i − 0.190885i
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 8.00000i 0.254000i
\(993\) − 4.00000i − 0.126936i
\(994\) 64.0000 2.02996
\(995\) 0 0
\(996\) 0 0
\(997\) 46.0000i 1.45683i 0.685134 + 0.728417i \(0.259744\pi\)
−0.685134 + 0.728417i \(0.740256\pi\)
\(998\) − 28.0000i − 0.886325i
\(999\) −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 2850.2.d.a.799.1 2
5.2 odd 4 2850.2.a.v.1.1 1
5.3 odd 4 570.2.a.e.1.1 1
5.4 even 2 inner 2850.2.d.a.799.2 2
15.2 even 4 8550.2.a.r.1.1 1
15.8 even 4 1710.2.a.l.1.1 1
20.3 even 4 4560.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.e.1.1 1 5.3 odd 4
1710.2.a.l.1.1 1 15.8 even 4
2850.2.a.v.1.1 1 5.2 odd 4
2850.2.d.a.799.1 2 1.1 even 1 trivial
2850.2.d.a.799.2 2 5.4 even 2 inner
4560.2.a.q.1.1 1 20.3 even 4
8550.2.a.r.1.1 1 15.2 even 4