Properties

Label 4600.2.e.p.4049.5
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.431642176.2
Defining polynomial: \( x^{6} + 19x^{4} + 97x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.5
Root \(2.95759i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.p.4049.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.95759i q^{3} +3.95759i q^{7} -5.74732 q^{9} +O(q^{10})\) \(q+2.95759i q^{3} +3.95759i q^{7} -5.74732 q^{9} -0.957587 q^{11} +2.74732i q^{13} +5.74732i q^{17} +6.74732 q^{19} -11.7049 q^{21} -1.00000i q^{23} -8.12544i q^{27} -5.21027 q^{29} -5.95759 q^{31} -2.83215i q^{33} +9.12544i q^{37} -8.12544 q^{39} +0.252679 q^{41} -8.00000i q^{43} +5.49464i q^{47} -8.66249 q^{49} -16.9982 q^{51} +7.12544i q^{53} +19.9558i q^{57} +4.78973 q^{59} +12.4522 q^{61} -22.7455i q^{63} +9.12544i q^{67} +2.95759 q^{69} +1.66249 q^{71} -12.3357i q^{73} -3.78973i q^{77} -11.8303 q^{79} +6.78973 q^{81} -0.704908i q^{83} -15.4098i q^{87} +15.8303 q^{89} -10.8728 q^{91} -17.6201i q^{93} -10.8728i q^{97} +5.50356 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 20 q^{9} + 14 q^{11} + 26 q^{19} - 36 q^{21} - 26 q^{29} - 16 q^{31} - 4 q^{39} + 16 q^{41} + 2 q^{49} + 2 q^{51} + 34 q^{59} + 26 q^{61} - 2 q^{69} - 44 q^{71} + 8 q^{79} + 46 q^{81} + 16 q^{89} - 6 q^{91} - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.95759i 1.70756i 0.520631 + 0.853782i \(0.325697\pi\)
−0.520631 + 0.853782i \(0.674303\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 3.95759i 1.49583i 0.663796 + 0.747914i \(0.268944\pi\)
−0.663796 + 0.747914i \(0.731056\pi\)
\(8\) 0 0
\(9\) −5.74732 −1.91577
\(10\) 0 0
\(11\) −0.957587 −0.288723 −0.144362 0.989525i \(-0.546113\pi\)
−0.144362 + 0.989525i \(0.546113\pi\)
\(12\) 0 0
\(13\) 2.74732i 0.761970i 0.924581 + 0.380985i \(0.124415\pi\)
−0.924581 + 0.380985i \(0.875585\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.74732i 1.39393i 0.717105 + 0.696965i \(0.245467\pi\)
−0.717105 + 0.696965i \(0.754533\pi\)
\(18\) 0 0
\(19\) 6.74732 1.54794 0.773971 0.633221i \(-0.218268\pi\)
0.773971 + 0.633221i \(0.218268\pi\)
\(20\) 0 0
\(21\) −11.7049 −2.55422
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 8.12544i − 1.56374i
\(28\) 0 0
\(29\) −5.21027 −0.967522 −0.483761 0.875200i \(-0.660730\pi\)
−0.483761 + 0.875200i \(0.660730\pi\)
\(30\) 0 0
\(31\) −5.95759 −1.07001 −0.535007 0.844848i \(-0.679691\pi\)
−0.535007 + 0.844848i \(0.679691\pi\)
\(32\) 0 0
\(33\) − 2.83215i − 0.493013i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 9.12544i 1.50021i 0.661317 + 0.750107i \(0.269998\pi\)
−0.661317 + 0.750107i \(0.730002\pi\)
\(38\) 0 0
\(39\) −8.12544 −1.30111
\(40\) 0 0
\(41\) 0.252679 0.0394619 0.0197309 0.999805i \(-0.493719\pi\)
0.0197309 + 0.999805i \(0.493719\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.49464i 0.801476i 0.916193 + 0.400738i \(0.131246\pi\)
−0.916193 + 0.400738i \(0.868754\pi\)
\(48\) 0 0
\(49\) −8.66249 −1.23750
\(50\) 0 0
\(51\) −16.9982 −2.38022
\(52\) 0 0
\(53\) 7.12544i 0.978754i 0.872072 + 0.489377i \(0.162776\pi\)
−0.872072 + 0.489377i \(0.837224\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 19.9558i 2.64321i
\(58\) 0 0
\(59\) 4.78973 0.623570 0.311785 0.950153i \(-0.399073\pi\)
0.311785 + 0.950153i \(0.399073\pi\)
\(60\) 0 0
\(61\) 12.4522 1.59434 0.797172 0.603752i \(-0.206328\pi\)
0.797172 + 0.603752i \(0.206328\pi\)
\(62\) 0 0
\(63\) − 22.7455i − 2.86567i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.12544i 1.11485i 0.830227 + 0.557425i \(0.188211\pi\)
−0.830227 + 0.557425i \(0.811789\pi\)
\(68\) 0 0
\(69\) 2.95759 0.356052
\(70\) 0 0
\(71\) 1.66249 0.197302 0.0986509 0.995122i \(-0.468547\pi\)
0.0986509 + 0.995122i \(0.468547\pi\)
\(72\) 0 0
\(73\) − 12.3357i − 1.44379i −0.692005 0.721893i \(-0.743272\pi\)
0.692005 0.721893i \(-0.256728\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.78973i − 0.431880i
\(78\) 0 0
\(79\) −11.8303 −1.33102 −0.665509 0.746390i \(-0.731786\pi\)
−0.665509 + 0.746390i \(0.731786\pi\)
\(80\) 0 0
\(81\) 6.78973 0.754415
\(82\) 0 0
\(83\) − 0.704908i − 0.0773737i −0.999251 0.0386868i \(-0.987683\pi\)
0.999251 0.0386868i \(-0.0123175\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 15.4098i − 1.65211i
\(88\) 0 0
\(89\) 15.8303 1.67801 0.839007 0.544121i \(-0.183137\pi\)
0.839007 + 0.544121i \(0.183137\pi\)
\(90\) 0 0
\(91\) −10.8728 −1.13978
\(92\) 0 0
\(93\) − 17.6201i − 1.82712i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.8728i − 1.10396i −0.833857 0.551981i \(-0.813872\pi\)
0.833857 0.551981i \(-0.186128\pi\)
\(98\) 0 0
\(99\) 5.50356 0.553129
\(100\) 0 0
\(101\) 9.21027 0.916456 0.458228 0.888835i \(-0.348484\pi\)
0.458228 + 0.888835i \(0.348484\pi\)
\(102\) 0 0
\(103\) − 12.4522i − 1.22695i −0.789712 0.613477i \(-0.789770\pi\)
0.789712 0.613477i \(-0.210230\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.63080i 0.351003i 0.984479 + 0.175501i \(0.0561546\pi\)
−0.984479 + 0.175501i \(0.943845\pi\)
\(108\) 0 0
\(109\) −16.6625 −1.59598 −0.797989 0.602672i \(-0.794103\pi\)
−0.797989 + 0.602672i \(0.794103\pi\)
\(110\) 0 0
\(111\) −26.9893 −2.56171
\(112\) 0 0
\(113\) 18.1147i 1.70409i 0.523469 + 0.852045i \(0.324638\pi\)
−0.523469 + 0.852045i \(0.675362\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 15.7897i − 1.45976i
\(118\) 0 0
\(119\) −22.7455 −2.08508
\(120\) 0 0
\(121\) −10.0830 −0.916639
\(122\) 0 0
\(123\) 0.747321i 0.0673836i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 5.91517i − 0.524887i −0.964947 0.262443i \(-0.915472\pi\)
0.964947 0.262443i \(-0.0845283\pi\)
\(128\) 0 0
\(129\) 23.6607 2.08321
\(130\) 0 0
\(131\) 3.40982 0.297917 0.148958 0.988843i \(-0.452408\pi\)
0.148958 + 0.988843i \(0.452408\pi\)
\(132\) 0 0
\(133\) 26.7031i 2.31545i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 1.67321i − 0.142952i −0.997442 0.0714761i \(-0.977229\pi\)
0.997442 0.0714761i \(-0.0227710\pi\)
\(138\) 0 0
\(139\) −8.62008 −0.731146 −0.365573 0.930783i \(-0.619127\pi\)
−0.365573 + 0.930783i \(0.619127\pi\)
\(140\) 0 0
\(141\) −16.2509 −1.36857
\(142\) 0 0
\(143\) − 2.63080i − 0.219998i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 25.6201i − 2.11311i
\(148\) 0 0
\(149\) 23.0830 1.89104 0.945518 0.325571i \(-0.105556\pi\)
0.945518 + 0.325571i \(0.105556\pi\)
\(150\) 0 0
\(151\) 10.7879 0.877910 0.438955 0.898509i \(-0.355349\pi\)
0.438955 + 0.898509i \(0.355349\pi\)
\(152\) 0 0
\(153\) − 33.0317i − 2.67045i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.19955i 0.654395i 0.944956 + 0.327198i \(0.106104\pi\)
−0.944956 + 0.327198i \(0.893896\pi\)
\(158\) 0 0
\(159\) −21.0741 −1.67129
\(160\) 0 0
\(161\) 3.95759 0.311902
\(162\) 0 0
\(163\) − 16.6625i − 1.30511i −0.757743 0.652554i \(-0.773698\pi\)
0.757743 0.652554i \(-0.226302\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 8.00000i − 0.619059i −0.950890 0.309529i \(-0.899829\pi\)
0.950890 0.309529i \(-0.100171\pi\)
\(168\) 0 0
\(169\) 5.45223 0.419402
\(170\) 0 0
\(171\) −38.7790 −2.96551
\(172\) 0 0
\(173\) − 8.87276i − 0.674584i −0.941400 0.337292i \(-0.890489\pi\)
0.941400 0.337292i \(-0.109511\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 14.1661i 1.06479i
\(178\) 0 0
\(179\) −1.49464 −0.111715 −0.0558574 0.998439i \(-0.517789\pi\)
−0.0558574 + 0.998439i \(0.517789\pi\)
\(180\) 0 0
\(181\) −12.5777 −0.934891 −0.467445 0.884022i \(-0.654826\pi\)
−0.467445 + 0.884022i \(0.654826\pi\)
\(182\) 0 0
\(183\) 36.8285i 2.72244i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 5.50356i − 0.402460i
\(188\) 0 0
\(189\) 32.1571 2.33909
\(190\) 0 0
\(191\) 16.9045 1.22316 0.611582 0.791181i \(-0.290534\pi\)
0.611582 + 0.791181i \(0.290534\pi\)
\(192\) 0 0
\(193\) − 9.91517i − 0.713710i −0.934160 0.356855i \(-0.883849\pi\)
0.934160 0.356855i \(-0.116151\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.53705i 0.323252i 0.986852 + 0.161626i \(0.0516738\pi\)
−0.986852 + 0.161626i \(0.948326\pi\)
\(198\) 0 0
\(199\) 14.0000 0.992434 0.496217 0.868199i \(-0.334722\pi\)
0.496217 + 0.868199i \(0.334722\pi\)
\(200\) 0 0
\(201\) −26.9893 −1.90368
\(202\) 0 0
\(203\) − 20.6201i − 1.44725i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 5.74732i 0.399466i
\(208\) 0 0
\(209\) −6.46115 −0.446927
\(210\) 0 0
\(211\) 8.70491 0.599271 0.299635 0.954054i \(-0.403135\pi\)
0.299635 + 0.954054i \(0.403135\pi\)
\(212\) 0 0
\(213\) 4.91697i 0.336905i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 23.5777i − 1.60056i
\(218\) 0 0
\(219\) 36.4839 2.46536
\(220\) 0 0
\(221\) −15.7897 −1.06213
\(222\) 0 0
\(223\) 5.40982i 0.362268i 0.983458 + 0.181134i \(0.0579768\pi\)
−0.983458 + 0.181134i \(0.942023\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 17.3250i − 1.14990i −0.818189 0.574950i \(-0.805022\pi\)
0.818189 0.574950i \(-0.194978\pi\)
\(228\) 0 0
\(229\) 11.4098 0.753982 0.376991 0.926217i \(-0.376959\pi\)
0.376991 + 0.926217i \(0.376959\pi\)
\(230\) 0 0
\(231\) 11.2085 0.737463
\(232\) 0 0
\(233\) − 2.08483i − 0.136581i −0.997665 0.0682907i \(-0.978245\pi\)
0.997665 0.0682907i \(-0.0217546\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 34.9893i − 2.27280i
\(238\) 0 0
\(239\) 18.5353 1.19895 0.599473 0.800395i \(-0.295377\pi\)
0.599473 + 0.800395i \(0.295377\pi\)
\(240\) 0 0
\(241\) 10.2509 0.660317 0.330159 0.943925i \(-0.392898\pi\)
0.330159 + 0.943925i \(0.392898\pi\)
\(242\) 0 0
\(243\) − 4.29509i − 0.275530i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 18.5371i 1.17948i
\(248\) 0 0
\(249\) 2.08483 0.132120
\(250\) 0 0
\(251\) −17.2527 −1.08898 −0.544490 0.838768i \(-0.683277\pi\)
−0.544490 + 0.838768i \(0.683277\pi\)
\(252\) 0 0
\(253\) 0.957587i 0.0602030i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 28.9045i 1.80301i 0.432768 + 0.901505i \(0.357537\pi\)
−0.432768 + 0.901505i \(0.642463\pi\)
\(258\) 0 0
\(259\) −36.1147 −2.24406
\(260\) 0 0
\(261\) 29.9451 1.85355
\(262\) 0 0
\(263\) − 5.24196i − 0.323233i −0.986854 0.161617i \(-0.948329\pi\)
0.986854 0.161617i \(-0.0516708\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 46.8196i 2.86531i
\(268\) 0 0
\(269\) 5.37992 0.328019 0.164010 0.986459i \(-0.447557\pi\)
0.164010 + 0.986459i \(0.447557\pi\)
\(270\) 0 0
\(271\) −11.5371 −0.700826 −0.350413 0.936595i \(-0.613959\pi\)
−0.350413 + 0.936595i \(0.613959\pi\)
\(272\) 0 0
\(273\) − 32.1571i − 1.94624i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.83035i 0.350312i 0.984541 + 0.175156i \(0.0560429\pi\)
−0.984541 + 0.175156i \(0.943957\pi\)
\(278\) 0 0
\(279\) 34.2402 2.04990
\(280\) 0 0
\(281\) 11.0741 0.660626 0.330313 0.943871i \(-0.392846\pi\)
0.330313 + 0.943871i \(0.392846\pi\)
\(282\) 0 0
\(283\) 3.86384i 0.229682i 0.993384 + 0.114841i \(0.0366358\pi\)
−0.993384 + 0.114841i \(0.963364\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.00000i 0.0590281i
\(288\) 0 0
\(289\) −16.0317 −0.943041
\(290\) 0 0
\(291\) 32.1571 1.88508
\(292\) 0 0
\(293\) 1.54597i 0.0903167i 0.998980 + 0.0451583i \(0.0143792\pi\)
−0.998980 + 0.0451583i \(0.985621\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 7.78082i 0.451489i
\(298\) 0 0
\(299\) 2.74732 0.158882
\(300\) 0 0
\(301\) 31.6607 1.82489
\(302\) 0 0
\(303\) 27.2402i 1.56491i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 6.45223i − 0.368248i −0.982903 0.184124i \(-0.941055\pi\)
0.982903 0.184124i \(-0.0589448\pi\)
\(308\) 0 0
\(309\) 36.8285 2.09510
\(310\) 0 0
\(311\) 22.8196 1.29398 0.646991 0.762497i \(-0.276027\pi\)
0.646991 + 0.762497i \(0.276027\pi\)
\(312\) 0 0
\(313\) 16.8620i 0.953099i 0.879148 + 0.476550i \(0.158113\pi\)
−0.879148 + 0.476550i \(0.841887\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 1.33751i − 0.0751218i −0.999294 0.0375609i \(-0.988041\pi\)
0.999294 0.0375609i \(-0.0119588\pi\)
\(318\) 0 0
\(319\) 4.98928 0.279346
\(320\) 0 0
\(321\) −10.7384 −0.599359
\(322\) 0 0
\(323\) 38.7790i 2.15772i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 49.2808i − 2.72523i
\(328\) 0 0
\(329\) −21.7455 −1.19887
\(330\) 0 0
\(331\) −32.2808 −1.77431 −0.887156 0.461470i \(-0.847322\pi\)
−0.887156 + 0.461470i \(0.847322\pi\)
\(332\) 0 0
\(333\) − 52.4468i − 2.87407i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 4.83215i − 0.263224i −0.991301 0.131612i \(-0.957985\pi\)
0.991301 0.131612i \(-0.0420153\pi\)
\(338\) 0 0
\(339\) −53.5759 −2.90984
\(340\) 0 0
\(341\) 5.70491 0.308938
\(342\) 0 0
\(343\) − 6.57947i − 0.355258i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 34.6183i − 1.85841i −0.369569 0.929203i \(-0.620495\pi\)
0.369569 0.929203i \(-0.379505\pi\)
\(348\) 0 0
\(349\) −26.9558 −1.44291 −0.721455 0.692461i \(-0.756526\pi\)
−0.721455 + 0.692461i \(0.756526\pi\)
\(350\) 0 0
\(351\) 22.3232 1.19152
\(352\) 0 0
\(353\) 8.00000i 0.425797i 0.977074 + 0.212899i \(0.0682904\pi\)
−0.977074 + 0.212899i \(0.931710\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 67.2719i − 3.56040i
\(358\) 0 0
\(359\) −0.420532 −0.0221949 −0.0110974 0.999938i \(-0.503532\pi\)
−0.0110974 + 0.999938i \(0.503532\pi\)
\(360\) 0 0
\(361\) 26.5263 1.39612
\(362\) 0 0
\(363\) − 29.8214i − 1.56522i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 11.7156i 0.611551i 0.952104 + 0.305775i \(0.0989156\pi\)
−0.952104 + 0.305775i \(0.901084\pi\)
\(368\) 0 0
\(369\) −1.45223 −0.0756000
\(370\) 0 0
\(371\) −28.1995 −1.46405
\(372\) 0 0
\(373\) 28.1661i 1.45838i 0.684310 + 0.729192i \(0.260104\pi\)
−0.684310 + 0.729192i \(0.739896\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 14.3143i − 0.737223i
\(378\) 0 0
\(379\) −27.3781 −1.40632 −0.703160 0.711032i \(-0.748228\pi\)
−0.703160 + 0.711032i \(0.748228\pi\)
\(380\) 0 0
\(381\) 17.4946 0.896278
\(382\) 0 0
\(383\) 31.7754i 1.62365i 0.583902 + 0.811824i \(0.301525\pi\)
−0.583902 + 0.811824i \(0.698475\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 45.9786i 2.33722i
\(388\) 0 0
\(389\) 18.7031 0.948285 0.474143 0.880448i \(-0.342758\pi\)
0.474143 + 0.880448i \(0.342758\pi\)
\(390\) 0 0
\(391\) 5.74732 0.290655
\(392\) 0 0
\(393\) 10.0848i 0.508712i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 39.6924i − 1.99210i −0.0887714 0.996052i \(-0.528294\pi\)
0.0887714 0.996052i \(-0.471706\pi\)
\(398\) 0 0
\(399\) −78.9768 −3.95378
\(400\) 0 0
\(401\) −11.3250 −0.565543 −0.282771 0.959187i \(-0.591254\pi\)
−0.282771 + 0.959187i \(0.591254\pi\)
\(402\) 0 0
\(403\) − 16.3674i − 0.815318i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 8.73840i − 0.433147i
\(408\) 0 0
\(409\) −9.45223 −0.467383 −0.233691 0.972311i \(-0.575081\pi\)
−0.233691 + 0.972311i \(0.575081\pi\)
\(410\) 0 0
\(411\) 4.94867 0.244100
\(412\) 0 0
\(413\) 18.9558i 0.932753i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 25.4946i − 1.24848i
\(418\) 0 0
\(419\) 33.2402 1.62389 0.811944 0.583735i \(-0.198409\pi\)
0.811944 + 0.583735i \(0.198409\pi\)
\(420\) 0 0
\(421\) −38.8285 −1.89239 −0.946194 0.323600i \(-0.895107\pi\)
−0.946194 + 0.323600i \(0.895107\pi\)
\(422\) 0 0
\(423\) − 31.5795i − 1.53545i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 49.2808i 2.38486i
\(428\) 0 0
\(429\) 7.78082 0.375661
\(430\) 0 0
\(431\) −12.4839 −0.601329 −0.300665 0.953730i \(-0.597209\pi\)
−0.300665 + 0.953730i \(0.597209\pi\)
\(432\) 0 0
\(433\) 5.78793i 0.278150i 0.990282 + 0.139075i \(0.0444130\pi\)
−0.990282 + 0.139075i \(0.955587\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 6.74732i − 0.322768i
\(438\) 0 0
\(439\) 11.9241 0.569106 0.284553 0.958660i \(-0.408155\pi\)
0.284553 + 0.958660i \(0.408155\pi\)
\(440\) 0 0
\(441\) 49.7861 2.37077
\(442\) 0 0
\(443\) − 39.3973i − 1.87182i −0.352236 0.935911i \(-0.614579\pi\)
0.352236 0.935911i \(-0.385421\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 68.2701i 3.22906i
\(448\) 0 0
\(449\) −21.3674 −1.00839 −0.504195 0.863590i \(-0.668211\pi\)
−0.504195 + 0.863590i \(0.668211\pi\)
\(450\) 0 0
\(451\) −0.241962 −0.0113936
\(452\) 0 0
\(453\) 31.9063i 1.49909i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 20.3656i − 0.952663i −0.879266 0.476331i \(-0.841966\pi\)
0.879266 0.476331i \(-0.158034\pi\)
\(458\) 0 0
\(459\) 46.6995 2.17975
\(460\) 0 0
\(461\) −27.4946 −1.28055 −0.640277 0.768144i \(-0.721180\pi\)
−0.640277 + 0.768144i \(0.721180\pi\)
\(462\) 0 0
\(463\) 16.8196i 0.781675i 0.920460 + 0.390837i \(0.127814\pi\)
−0.920460 + 0.390837i \(0.872186\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 24.4504i − 1.13143i −0.824601 0.565715i \(-0.808600\pi\)
0.824601 0.565715i \(-0.191400\pi\)
\(468\) 0 0
\(469\) −36.1147 −1.66762
\(470\) 0 0
\(471\) −24.2509 −1.11742
\(472\) 0 0
\(473\) 7.66070i 0.352239i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 40.9522i − 1.87507i
\(478\) 0 0
\(479\) −10.5688 −0.482899 −0.241449 0.970413i \(-0.577623\pi\)
−0.241449 + 0.970413i \(0.577623\pi\)
\(480\) 0 0
\(481\) −25.0705 −1.14312
\(482\) 0 0
\(483\) 11.7049i 0.532592i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 13.0741i − 0.592444i −0.955119 0.296222i \(-0.904273\pi\)
0.955119 0.296222i \(-0.0957269\pi\)
\(488\) 0 0
\(489\) 49.2808 2.22855
\(490\) 0 0
\(491\) 2.22098 0.100232 0.0501158 0.998743i \(-0.484041\pi\)
0.0501158 + 0.998743i \(0.484041\pi\)
\(492\) 0 0
\(493\) − 29.9451i − 1.34866i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.57947i 0.295129i
\(498\) 0 0
\(499\) −2.87456 −0.128683 −0.0643415 0.997928i \(-0.520495\pi\)
−0.0643415 + 0.997928i \(0.520495\pi\)
\(500\) 0 0
\(501\) 23.6607 1.05708
\(502\) 0 0
\(503\) − 9.36740i − 0.417672i −0.977951 0.208836i \(-0.933033\pi\)
0.977951 0.208836i \(-0.0669675\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 16.1254i 0.716156i
\(508\) 0 0
\(509\) 40.5866 1.79897 0.899484 0.436953i \(-0.143942\pi\)
0.899484 + 0.436953i \(0.143942\pi\)
\(510\) 0 0
\(511\) 48.8196 2.15965
\(512\) 0 0
\(513\) − 54.8250i − 2.42058i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 5.26160i − 0.231405i
\(518\) 0 0
\(519\) 26.2420 1.15189
\(520\) 0 0
\(521\) 34.6714 1.51898 0.759491 0.650518i \(-0.225448\pi\)
0.759491 + 0.650518i \(0.225448\pi\)
\(522\) 0 0
\(523\) − 35.4098i − 1.54836i −0.632964 0.774182i \(-0.718162\pi\)
0.632964 0.774182i \(-0.281838\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 34.2402i − 1.49152i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −27.5281 −1.19462
\(532\) 0 0
\(533\) 0.694191i 0.0300687i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 4.42053i − 0.190760i
\(538\) 0 0
\(539\) 8.29509 0.357295
\(540\) 0 0
\(541\) 37.0705 1.59379 0.796893 0.604121i \(-0.206475\pi\)
0.796893 + 0.604121i \(0.206475\pi\)
\(542\) 0 0
\(543\) − 37.1995i − 1.59639i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 21.4187i 0.915799i 0.889004 + 0.457899i \(0.151398\pi\)
−0.889004 + 0.457899i \(0.848602\pi\)
\(548\) 0 0
\(549\) −71.5670 −3.05440
\(550\) 0 0
\(551\) −35.1553 −1.49767
\(552\) 0 0
\(553\) − 46.8196i − 1.99097i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20.9558i 0.887925i 0.896045 + 0.443963i \(0.146428\pi\)
−0.896045 + 0.443963i \(0.853572\pi\)
\(558\) 0 0
\(559\) 21.9786 0.929594
\(560\) 0 0
\(561\) 16.2773 0.687226
\(562\) 0 0
\(563\) − 6.70491i − 0.282578i −0.989968 0.141289i \(-0.954875\pi\)
0.989968 0.141289i \(-0.0451247\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 26.8710i 1.12847i
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 13.2933 0.556307 0.278154 0.960537i \(-0.410278\pi\)
0.278154 + 0.960537i \(0.410278\pi\)
\(572\) 0 0
\(573\) 49.9964i 2.08863i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 23.2402i 0.967501i 0.875206 + 0.483750i \(0.160726\pi\)
−0.875206 + 0.483750i \(0.839274\pi\)
\(578\) 0 0
\(579\) 29.3250 1.21870
\(580\) 0 0
\(581\) 2.78973 0.115738
\(582\) 0 0
\(583\) − 6.82323i − 0.282589i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17.6112i 0.726891i 0.931616 + 0.363445i \(0.118400\pi\)
−0.931616 + 0.363445i \(0.881600\pi\)
\(588\) 0 0
\(589\) −40.1978 −1.65632
\(590\) 0 0
\(591\) −13.4187 −0.551973
\(592\) 0 0
\(593\) 41.5759i 1.70732i 0.520834 + 0.853658i \(0.325621\pi\)
−0.520834 + 0.853658i \(0.674379\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 41.4062i 1.69464i
\(598\) 0 0
\(599\) −17.7138 −0.723767 −0.361884 0.932223i \(-0.617866\pi\)
−0.361884 + 0.932223i \(0.617866\pi\)
\(600\) 0 0
\(601\) −46.7772 −1.90808 −0.954041 0.299675i \(-0.903122\pi\)
−0.954041 + 0.299675i \(0.903122\pi\)
\(602\) 0 0
\(603\) − 52.4468i − 2.13580i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 42.9893i 1.74488i 0.488720 + 0.872441i \(0.337464\pi\)
−0.488720 + 0.872441i \(0.662536\pi\)
\(608\) 0 0
\(609\) 60.9857 2.47126
\(610\) 0 0
\(611\) −15.0955 −0.610700
\(612\) 0 0
\(613\) 32.6500i 1.31872i 0.751827 + 0.659360i \(0.229173\pi\)
−0.751827 + 0.659360i \(0.770827\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 11.3232i − 0.455854i −0.973678 0.227927i \(-0.926805\pi\)
0.973678 0.227927i \(-0.0731948\pi\)
\(618\) 0 0
\(619\) 10.6625 0.428562 0.214281 0.976772i \(-0.431259\pi\)
0.214281 + 0.976772i \(0.431259\pi\)
\(620\) 0 0
\(621\) −8.12544 −0.326063
\(622\) 0 0
\(623\) 62.6500i 2.51002i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 19.1094i − 0.763156i
\(628\) 0 0
\(629\) −52.4468 −2.09119
\(630\) 0 0
\(631\) 5.24376 0.208751 0.104375 0.994538i \(-0.466716\pi\)
0.104375 + 0.994538i \(0.466716\pi\)
\(632\) 0 0
\(633\) 25.7455i 1.02329i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 23.7987i − 0.942937i
\(638\) 0 0
\(639\) −9.55489 −0.377986
\(640\) 0 0
\(641\) 30.4205 1.20154 0.600769 0.799422i \(-0.294861\pi\)
0.600769 + 0.799422i \(0.294861\pi\)
\(642\) 0 0
\(643\) 9.37992i 0.369908i 0.982747 + 0.184954i \(0.0592136\pi\)
−0.982747 + 0.184954i \(0.940786\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 18.6714i − 0.734049i −0.930211 0.367024i \(-0.880377\pi\)
0.930211 0.367024i \(-0.119623\pi\)
\(648\) 0 0
\(649\) −4.58659 −0.180039
\(650\) 0 0
\(651\) 69.7330 2.73305
\(652\) 0 0
\(653\) − 12.0723i − 0.472426i −0.971701 0.236213i \(-0.924094\pi\)
0.971701 0.236213i \(-0.0759063\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 70.8973i 2.76597i
\(658\) 0 0
\(659\) −21.2402 −0.827399 −0.413700 0.910413i \(-0.635764\pi\)
−0.413700 + 0.910413i \(0.635764\pi\)
\(660\) 0 0
\(661\) −26.9362 −1.04769 −0.523847 0.851812i \(-0.675504\pi\)
−0.523847 + 0.851812i \(0.675504\pi\)
\(662\) 0 0
\(663\) − 46.6995i − 1.81366i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 5.21027i 0.201742i
\(668\) 0 0
\(669\) −16.0000 −0.618596
\(670\) 0 0
\(671\) −11.9241 −0.460324
\(672\) 0 0
\(673\) 11.4098i 0.439816i 0.975521 + 0.219908i \(0.0705757\pi\)
−0.975521 + 0.219908i \(0.929424\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 11.8638i − 0.455965i −0.973665 0.227982i \(-0.926787\pi\)
0.973665 0.227982i \(-0.0732128\pi\)
\(678\) 0 0
\(679\) 43.0299 1.65134
\(680\) 0 0
\(681\) 51.2402 1.96353
\(682\) 0 0
\(683\) − 10.0723i − 0.385406i −0.981257 0.192703i \(-0.938275\pi\)
0.981257 0.192703i \(-0.0617254\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 33.7455i 1.28747i
\(688\) 0 0
\(689\) −19.5759 −0.745781
\(690\) 0 0
\(691\) −45.9964 −1.74979 −0.874893 0.484317i \(-0.839068\pi\)
−0.874893 + 0.484317i \(0.839068\pi\)
\(692\) 0 0
\(693\) 21.7808i 0.827385i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.45223i 0.0550071i
\(698\) 0 0
\(699\) 6.16605 0.233222
\(700\) 0 0
\(701\) −21.2312 −0.801893 −0.400947 0.916101i \(-0.631319\pi\)
−0.400947 + 0.916101i \(0.631319\pi\)
\(702\) 0 0
\(703\) 61.5723i 2.32224i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 36.4504i 1.37086i
\(708\) 0 0
\(709\) −31.3781 −1.17843 −0.589215 0.807976i \(-0.700563\pi\)
−0.589215 + 0.807976i \(0.700563\pi\)
\(710\) 0 0
\(711\) 67.9928 2.54993
\(712\) 0 0
\(713\) 5.95759i 0.223113i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 54.8196i 2.04728i
\(718\) 0 0
\(719\) 20.4629 0.763139 0.381570 0.924340i \(-0.375384\pi\)
0.381570 + 0.924340i \(0.375384\pi\)
\(720\) 0 0
\(721\) 49.2808 1.83531
\(722\) 0 0
\(723\) 30.3179i 1.12753i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 14.9875i 0.555855i 0.960602 + 0.277928i \(0.0896475\pi\)
−0.960602 + 0.277928i \(0.910352\pi\)
\(728\) 0 0
\(729\) 33.0723 1.22490
\(730\) 0 0
\(731\) 45.9786 1.70058
\(732\) 0 0
\(733\) 22.2844i 0.823092i 0.911389 + 0.411546i \(0.135011\pi\)
−0.911389 + 0.411546i \(0.864989\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 8.73840i − 0.321883i
\(738\) 0 0
\(739\) −30.9344 −1.13794 −0.568969 0.822359i \(-0.692658\pi\)
−0.568969 + 0.822359i \(0.692658\pi\)
\(740\) 0 0
\(741\) −54.8250 −2.01404
\(742\) 0 0
\(743\) 44.8285i 1.64460i 0.569054 + 0.822300i \(0.307309\pi\)
−0.569054 + 0.822300i \(0.692691\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 4.05133i 0.148230i
\(748\) 0 0
\(749\) −14.3692 −0.525039
\(750\) 0 0
\(751\) 32.1661 1.17376 0.586878 0.809675i \(-0.300357\pi\)
0.586878 + 0.809675i \(0.300357\pi\)
\(752\) 0 0
\(753\) − 51.0263i − 1.85950i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 40.6835i 1.47867i 0.673340 + 0.739333i \(0.264859\pi\)
−0.673340 + 0.739333i \(0.735141\pi\)
\(758\) 0 0
\(759\) −2.83215 −0.102800
\(760\) 0 0
\(761\) 28.4415 1.03100 0.515502 0.856888i \(-0.327605\pi\)
0.515502 + 0.856888i \(0.327605\pi\)
\(762\) 0 0
\(763\) − 65.9433i − 2.38731i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 13.1589i 0.475142i
\(768\) 0 0
\(769\) 40.4205 1.45760 0.728801 0.684726i \(-0.240078\pi\)
0.728801 + 0.684726i \(0.240078\pi\)
\(770\) 0 0
\(771\) −85.4874 −3.07876
\(772\) 0 0
\(773\) − 24.5688i − 0.883677i −0.897095 0.441838i \(-0.854327\pi\)
0.897095 0.441838i \(-0.145673\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 106.812i − 3.83187i
\(778\) 0 0
\(779\) 1.70491 0.0610847
\(780\) 0 0
\(781\) −1.59198 −0.0569656
\(782\) 0 0
\(783\) 42.3357i 1.51295i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 11.7790i − 0.419877i −0.977715 0.209938i \(-0.932674\pi\)
0.977715 0.209938i \(-0.0673263\pi\)
\(788\) 0 0
\(789\) 15.5036 0.551941
\(790\) 0 0
\(791\) −71.6906 −2.54902
\(792\) 0 0
\(793\) 34.2103i 1.21484i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 13.8602i − 0.490955i −0.969402 0.245478i \(-0.921055\pi\)
0.969402 0.245478i \(-0.0789448\pi\)
\(798\) 0 0
\(799\) −31.5795 −1.11720
\(800\) 0 0
\(801\) −90.9821 −3.21469
\(802\) 0 0
\(803\) 11.8125i 0.416854i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 15.9116i 0.560114i
\(808\) 0 0
\(809\) 52.0830 1.83114 0.915571 0.402157i \(-0.131739\pi\)
0.915571 + 0.402157i \(0.131739\pi\)
\(810\) 0 0
\(811\) −40.7013 −1.42922 −0.714608 0.699525i \(-0.753395\pi\)
−0.714608 + 0.699525i \(0.753395\pi\)
\(812\) 0 0
\(813\) − 34.1218i − 1.19671i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 53.9786i − 1.88847i
\(818\) 0 0
\(819\) 62.4892 2.18355
\(820\) 0 0
\(821\) 10.0000 0.349002 0.174501 0.984657i \(-0.444169\pi\)
0.174501 + 0.984657i \(0.444169\pi\)
\(822\) 0 0
\(823\) 42.9009i 1.49543i 0.664020 + 0.747715i \(0.268849\pi\)
−0.664020 + 0.747715i \(0.731151\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 39.4397i 1.37145i 0.727859 + 0.685727i \(0.240515\pi\)
−0.727859 + 0.685727i \(0.759485\pi\)
\(828\) 0 0
\(829\) −29.5460 −1.02617 −0.513087 0.858337i \(-0.671498\pi\)
−0.513087 + 0.858337i \(0.671498\pi\)
\(830\) 0 0
\(831\) −17.2438 −0.598179
\(832\) 0 0
\(833\) − 49.7861i − 1.72499i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 48.4080i 1.67323i
\(838\) 0 0
\(839\) 1.09554 0.0378223 0.0189112 0.999821i \(-0.493980\pi\)
0.0189112 + 0.999821i \(0.493980\pi\)
\(840\) 0 0
\(841\) −1.85313 −0.0639009
\(842\) 0 0
\(843\) 32.7526i 1.12806i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 39.9045i − 1.37113i
\(848\) 0 0
\(849\) −11.4277 −0.392196
\(850\) 0 0
\(851\) 9.12544 0.312816
\(852\) 0 0
\(853\) 10.0937i 0.345603i 0.984957 + 0.172802i \(0.0552819\pi\)
−0.984957 + 0.172802i \(0.944718\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 41.8303i 1.42890i 0.699688 + 0.714449i \(0.253322\pi\)
−0.699688 + 0.714449i \(0.746678\pi\)
\(858\) 0 0
\(859\) 3.35848 0.114590 0.0572950 0.998357i \(-0.481752\pi\)
0.0572950 + 0.998357i \(0.481752\pi\)
\(860\) 0 0
\(861\) −2.95759 −0.100794
\(862\) 0 0
\(863\) 3.83395i 0.130509i 0.997869 + 0.0652545i \(0.0207859\pi\)
−0.997869 + 0.0652545i \(0.979214\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 47.4151i − 1.61030i
\(868\) 0 0
\(869\) 11.3286 0.384296
\(870\) 0 0
\(871\) −25.0705 −0.849482
\(872\) 0 0
\(873\) 62.4892i 2.11494i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 15.5263i 0.524287i 0.965029 + 0.262144i \(0.0844294\pi\)
−0.965029 + 0.262144i \(0.915571\pi\)
\(878\) 0 0
\(879\) −4.57235 −0.154221
\(880\) 0 0
\(881\) 26.9893 0.909292 0.454646 0.890672i \(-0.349766\pi\)
0.454646 + 0.890672i \(0.349766\pi\)
\(882\) 0 0
\(883\) − 21.5884i − 0.726507i −0.931690 0.363254i \(-0.881666\pi\)
0.931690 0.363254i \(-0.118334\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 49.4098i 1.65902i 0.558492 + 0.829510i \(0.311380\pi\)
−0.558492 + 0.829510i \(0.688620\pi\)
\(888\) 0 0
\(889\) 23.4098 0.785140
\(890\) 0 0
\(891\) −6.50176 −0.217817
\(892\) 0 0
\(893\) 37.0741i 1.24064i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 8.12544i 0.271301i
\(898\) 0 0
\(899\) 31.0406 1.03526
\(900\) 0 0
\(901\) −40.9522 −1.36432
\(902\) 0 0
\(903\) 93.6393i 3.11612i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 9.54957i − 0.317088i −0.987352 0.158544i \(-0.949320\pi\)
0.987352 0.158544i \(-0.0506800\pi\)
\(908\) 0 0
\(909\) −52.9344 −1.75572
\(910\) 0 0
\(911\) −47.3036 −1.56724 −0.783618 0.621243i \(-0.786628\pi\)
−0.783618 + 0.621243i \(0.786628\pi\)
\(912\) 0 0
\(913\) 0.675011i 0.0223396i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 13.4946i 0.445632i
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 19.0830 0.628807
\(922\) 0 0
\(923\) 4.56741i 0.150338i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 71.5670i 2.35057i
\(928\) 0 0
\(929\) 19.6942 0.646145 0.323073 0.946374i \(-0.395284\pi\)
0.323073 + 0.946374i \(0.395284\pi\)
\(930\) 0 0
\(931\) −58.4486 −1.91558
\(932\) 0 0
\(933\) 67.4910i 2.20956i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 31.8214i − 1.03956i −0.854300 0.519780i \(-0.826014\pi\)
0.854300 0.519780i \(-0.173986\pi\)
\(938\) 0 0
\(939\) −49.8710 −1.62748
\(940\) 0 0
\(941\) −2.36740 −0.0771751 −0.0385876 0.999255i \(-0.512286\pi\)
−0.0385876 + 0.999255i \(0.512286\pi\)
\(942\) 0 0
\(943\) − 0.252679i − 0.00822837i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 9.90266i − 0.321793i −0.986971 0.160897i \(-0.948561\pi\)
0.986971 0.160897i \(-0.0514386\pi\)
\(948\) 0 0
\(949\) 33.8901 1.10012
\(950\) 0 0
\(951\) 3.95579 0.128275
\(952\) 0 0
\(953\) 31.5442i 1.02182i 0.859635 + 0.510908i \(0.170691\pi\)
−0.859635 + 0.510908i \(0.829309\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 14.7562i 0.477001i
\(958\) 0 0
\(959\) 6.62188 0.213832
\(960\) 0 0
\(961\) 4.49284 0.144930
\(962\) 0 0
\(963\) − 20.8674i − 0.672441i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 15.5973i − 0.501575i −0.968042 0.250788i \(-0.919310\pi\)
0.968042 0.250788i \(-0.0806896\pi\)
\(968\) 0 0
\(969\) −114.692 −3.68445
\(970\) 0 0
\(971\) −47.1022 −1.51158 −0.755791 0.654813i \(-0.772747\pi\)
−0.755791 + 0.654813i \(0.772747\pi\)
\(972\) 0 0
\(973\) − 34.1147i − 1.09367i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 28.9469i 0.926092i 0.886334 + 0.463046i \(0.153244\pi\)
−0.886334 + 0.463046i \(0.846756\pi\)
\(978\) 0 0
\(979\) −15.1589 −0.484482
\(980\) 0 0
\(981\) 95.7647 3.05753
\(982\) 0 0
\(983\) 36.2312i 1.15560i 0.816179 + 0.577799i \(0.196088\pi\)
−0.816179 + 0.577799i \(0.803912\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 64.3143i − 2.04715i
\(988\) 0 0
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) −35.1750 −1.11737 −0.558685 0.829380i \(-0.688694\pi\)
−0.558685 + 0.829380i \(0.688694\pi\)
\(992\) 0 0
\(993\) − 95.4732i − 3.02975i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 19.2402i 0.609342i 0.952458 + 0.304671i \(0.0985465\pi\)
−0.952458 + 0.304671i \(0.901453\pi\)
\(998\) 0 0
\(999\) 74.1482 2.34595
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 4600.2.e.p.4049.5 6
5.2 odd 4 4600.2.a.x.1.3 3
5.3 odd 4 920.2.a.h.1.1 3
5.4 even 2 inner 4600.2.e.p.4049.2 6
15.8 even 4 8280.2.a.bj.1.3 3
20.3 even 4 1840.2.a.s.1.3 3
20.7 even 4 9200.2.a.ce.1.1 3
40.3 even 4 7360.2.a.cc.1.1 3
40.13 odd 4 7360.2.a.by.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.h.1.1 3 5.3 odd 4
1840.2.a.s.1.3 3 20.3 even 4
4600.2.a.x.1.3 3 5.2 odd 4
4600.2.e.p.4049.2 6 5.4 even 2 inner
4600.2.e.p.4049.5 6 1.1 even 1 trivial
7360.2.a.by.1.3 3 40.13 odd 4
7360.2.a.cc.1.1 3 40.3 even 4
8280.2.a.bj.1.3 3 15.8 even 4
9200.2.a.ce.1.1 3 20.7 even 4