Properties

Label 920.2.a.h.1.1
Level $920$
Weight $2$
Character 920.1
Self dual yes
Analytic conductor $7.346$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.34623698596\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.2597.1
Defining polynomial: \( x^{3} - x^{2} - 9x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.95759\) of defining polynomial
Character \(\chi\) \(=\) 920.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.95759 q^{3} +1.00000 q^{5} +3.95759 q^{7} +5.74732 q^{9} +O(q^{10})\) \(q-2.95759 q^{3} +1.00000 q^{5} +3.95759 q^{7} +5.74732 q^{9} -0.957587 q^{11} -2.74732 q^{13} -2.95759 q^{15} +5.74732 q^{17} -6.74732 q^{19} -11.7049 q^{21} +1.00000 q^{23} +1.00000 q^{25} -8.12544 q^{27} +5.21027 q^{29} -5.95759 q^{31} +2.83215 q^{33} +3.95759 q^{35} +9.12544 q^{37} +8.12544 q^{39} +0.252679 q^{41} +8.00000 q^{43} +5.74732 q^{45} +5.49464 q^{47} +8.66249 q^{49} -16.9982 q^{51} -7.12544 q^{53} -0.957587 q^{55} +19.9558 q^{57} -4.78973 q^{59} +12.4522 q^{61} +22.7455 q^{63} -2.74732 q^{65} +9.12544 q^{67} -2.95759 q^{69} +1.66249 q^{71} +12.3357 q^{73} -2.95759 q^{75} -3.78973 q^{77} +11.8303 q^{79} +6.78973 q^{81} +0.704908 q^{83} +5.74732 q^{85} -15.4098 q^{87} -15.8303 q^{89} -10.8728 q^{91} +17.6201 q^{93} -6.74732 q^{95} -10.8728 q^{97} -5.50356 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} + 3 q^{5} + 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{3} + 3 q^{5} + 2 q^{7} + 10 q^{9} + 7 q^{11} - q^{13} + q^{15} + 10 q^{17} - 13 q^{19} - 18 q^{21} + 3 q^{23} + 3 q^{25} - 2 q^{27} + 13 q^{29} - 8 q^{31} + 21 q^{33} + 2 q^{35} + 5 q^{37} + 2 q^{39} + 8 q^{41} + 24 q^{43} + 10 q^{45} + 2 q^{47} - q^{49} + q^{51} + q^{53} + 7 q^{55} - 2 q^{57} - 17 q^{59} + 13 q^{61} + 9 q^{63} - q^{65} + 5 q^{67} + q^{69} - 22 q^{71} + 12 q^{73} + q^{75} - 14 q^{77} - 4 q^{79} + 23 q^{81} - 15 q^{83} + 10 q^{85} - 12 q^{87} - 8 q^{89} - 3 q^{91} + 16 q^{93} - 13 q^{95} - 3 q^{97} + 21 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −2.95759 −1.70756 −0.853782 0.520631i \(-0.825697\pi\)
−0.853782 + 0.520631i \(0.825697\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 3.95759 1.49583 0.747914 0.663796i \(-0.231056\pi\)
0.747914 + 0.663796i \(0.231056\pi\)
\(8\) 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
\(13\) −2.74732 −0.761970 −0.380985 0.924581i \(-0.624415\pi\)
−0.380985 + 0.924581i \(0.624415\pi\)
\(14\) 0 0
\(15\) −2.95759 −0.763646
\(16\) 0 0
\(17\) 5.74732 1.39393 0.696965 0.717105i \(-0.254533\pi\)
0.696965 + 0.717105i \(0.254533\pi\)
\(18\) 0 0
\(19\) −6.74732 −1.54794 −0.773971 0.633221i \(-0.781732\pi\)
−0.773971 + 0.633221i \(0.781732\pi\)
\(20\) 0 0
\(21\) −11.7049 −2.55422
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −8.12544 −1.56374
\(28\) 0 0
\(29\) 5.21027 0.967522 0.483761 0.875200i \(-0.339270\pi\)
0.483761 + 0.875200i \(0.339270\pi\)
\(30\) 0 0
\(31\) −5.95759 −1.07001 −0.535007 0.844848i \(-0.679691\pi\)
−0.535007 + 0.844848i \(0.679691\pi\)
\(32\) 0 0
\(33\) 2.83215 0.493013
\(34\) 0 0
\(35\) 3.95759 0.668954
\(36\) 0 0
\(37\) 9.12544 1.50021 0.750107 0.661317i \(-0.230002\pi\)
0.750107 + 0.661317i \(0.230002\pi\)
\(38\) 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
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 5.74732 0.856760
\(46\) 0 0
\(47\) 5.49464 0.801476 0.400738 0.916193i \(-0.368754\pi\)
0.400738 + 0.916193i \(0.368754\pi\)
\(48\) 0 0
\(49\) 8.66249 1.23750
\(50\) 0 0
\(51\) −16.9982 −2.38022
\(52\) 0 0
\(53\) −7.12544 −0.978754 −0.489377 0.872072i \(-0.662776\pi\)
−0.489377 + 0.872072i \(0.662776\pi\)
\(54\) 0 0
\(55\) −0.957587 −0.129121
\(56\) 0 0
\(57\) 19.9558 2.64321
\(58\) 0 0
\(59\) −4.78973 −0.623570 −0.311785 0.950153i \(-0.600927\pi\)
−0.311785 + 0.950153i \(0.600927\pi\)
\(60\) 0 0
\(61\) 12.4522 1.59434 0.797172 0.603752i \(-0.206328\pi\)
0.797172 + 0.603752i \(0.206328\pi\)
\(62\) 0 0
\(63\) 22.7455 2.86567
\(64\) 0 0
\(65\) −2.74732 −0.340763
\(66\) 0 0
\(67\) 9.12544 1.11485 0.557425 0.830227i \(-0.311789\pi\)
0.557425 + 0.830227i \(0.311789\pi\)
\(68\) 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
\(73\) 12.3357 1.44379 0.721893 0.692005i \(-0.243272\pi\)
0.721893 + 0.692005i \(0.243272\pi\)
\(74\) 0 0
\(75\) −2.95759 −0.341513
\(76\) 0 0
\(77\) −3.78973 −0.431880
\(78\) 0 0
\(79\) 11.8303 1.33102 0.665509 0.746390i \(-0.268214\pi\)
0.665509 + 0.746390i \(0.268214\pi\)
\(80\) 0 0
\(81\) 6.78973 0.754415
\(82\) 0 0
\(83\) 0.704908 0.0773737 0.0386868 0.999251i \(-0.487683\pi\)
0.0386868 + 0.999251i \(0.487683\pi\)
\(84\) 0 0
\(85\) 5.74732 0.623384
\(86\) 0 0
\(87\) −15.4098 −1.65211
\(88\) 0 0
\(89\) −15.8303 −1.67801 −0.839007 0.544121i \(-0.816863\pi\)
−0.839007 + 0.544121i \(0.816863\pi\)
\(90\) 0 0
\(91\) −10.8728 −1.13978
\(92\) 0 0
\(93\) 17.6201 1.82712
\(94\) 0 0
\(95\) −6.74732 −0.692261
\(96\) 0 0
\(97\) −10.8728 −1.10396 −0.551981 0.833857i \(-0.686128\pi\)
−0.551981 + 0.833857i \(0.686128\pi\)
\(98\) 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
\(103\) 12.4522 1.22695 0.613477 0.789712i \(-0.289770\pi\)
0.613477 + 0.789712i \(0.289770\pi\)
\(104\) 0 0
\(105\) −11.7049 −1.14228
\(106\) 0 0
\(107\) 3.63080 0.351003 0.175501 0.984479i \(-0.443845\pi\)
0.175501 + 0.984479i \(0.443845\pi\)
\(108\) 0 0
\(109\) 16.6625 1.59598 0.797989 0.602672i \(-0.205897\pi\)
0.797989 + 0.602672i \(0.205897\pi\)
\(110\) 0 0
\(111\) −26.9893 −2.56171
\(112\) 0 0
\(113\) −18.1147 −1.70409 −0.852045 0.523469i \(-0.824638\pi\)
−0.852045 + 0.523469i \(0.824638\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) −15.7897 −1.45976
\(118\) 0 0
\(119\) 22.7455 2.08508
\(120\) 0 0
\(121\) −10.0830 −0.916639
\(122\) 0 0
\(123\) −0.747321 −0.0673836
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −5.91517 −0.524887 −0.262443 0.964947i \(-0.584528\pi\)
−0.262443 + 0.964947i \(0.584528\pi\)
\(128\) 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
\(133\) −26.7031 −2.31545
\(134\) 0 0
\(135\) −8.12544 −0.699327
\(136\) 0 0
\(137\) −1.67321 −0.142952 −0.0714761 0.997442i \(-0.522771\pi\)
−0.0714761 + 0.997442i \(0.522771\pi\)
\(138\) 0 0
\(139\) 8.62008 0.731146 0.365573 0.930783i \(-0.380873\pi\)
0.365573 + 0.930783i \(0.380873\pi\)
\(140\) 0 0
\(141\) −16.2509 −1.36857
\(142\) 0 0
\(143\) 2.63080 0.219998
\(144\) 0 0
\(145\) 5.21027 0.432689
\(146\) 0 0
\(147\) −25.6201 −2.11311
\(148\) 0 0
\(149\) −23.0830 −1.89104 −0.945518 0.325571i \(-0.894444\pi\)
−0.945518 + 0.325571i \(0.894444\pi\)
\(150\) 0 0
\(151\) 10.7879 0.877910 0.438955 0.898509i \(-0.355349\pi\)
0.438955 + 0.898509i \(0.355349\pi\)
\(152\) 0 0
\(153\) 33.0317 2.67045
\(154\) 0 0
\(155\) −5.95759 −0.478525
\(156\) 0 0
\(157\) 8.19955 0.654395 0.327198 0.944956i \(-0.393896\pi\)
0.327198 + 0.944956i \(0.393896\pi\)
\(158\) 0 0
\(159\) 21.0741 1.67129
\(160\) 0 0
\(161\) 3.95759 0.311902
\(162\) 0 0
\(163\) 16.6625 1.30511 0.652554 0.757743i \(-0.273698\pi\)
0.652554 + 0.757743i \(0.273698\pi\)
\(164\) 0 0
\(165\) 2.83215 0.220482
\(166\) 0 0
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) −5.45223 −0.419402
\(170\) 0 0
\(171\) −38.7790 −2.96551
\(172\) 0 0
\(173\) 8.87276 0.674584 0.337292 0.941400i \(-0.390489\pi\)
0.337292 + 0.941400i \(0.390489\pi\)
\(174\) 0 0
\(175\) 3.95759 0.299165
\(176\) 0 0
\(177\) 14.1661 1.06479
\(178\) 0 0
\(179\) 1.49464 0.111715 0.0558574 0.998439i \(-0.482211\pi\)
0.0558574 + 0.998439i \(0.482211\pi\)
\(180\) 0 0
\(181\) −12.5777 −0.934891 −0.467445 0.884022i \(-0.654826\pi\)
−0.467445 + 0.884022i \(0.654826\pi\)
\(182\) 0 0
\(183\) −36.8285 −2.72244
\(184\) 0 0
\(185\) 9.12544 0.670916
\(186\) 0 0
\(187\) −5.50356 −0.402460
\(188\) 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
\(193\) 9.91517 0.713710 0.356855 0.934160i \(-0.383849\pi\)
0.356855 + 0.934160i \(0.383849\pi\)
\(194\) 0 0
\(195\) 8.12544 0.581875
\(196\) 0 0
\(197\) 4.53705 0.323252 0.161626 0.986852i \(-0.448326\pi\)
0.161626 + 0.986852i \(0.448326\pi\)
\(198\) 0 0
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) −26.9893 −1.90368
\(202\) 0 0
\(203\) 20.6201 1.44725
\(204\) 0 0
\(205\) 0.252679 0.0176479
\(206\) 0 0
\(207\) 5.74732 0.399466
\(208\) 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
\(213\) −4.91697 −0.336905
\(214\) 0 0
\(215\) 8.00000 0.545595
\(216\) 0 0
\(217\) −23.5777 −1.60056
\(218\) 0 0
\(219\) −36.4839 −2.46536
\(220\) 0 0
\(221\) −15.7897 −1.06213
\(222\) 0 0
\(223\) −5.40982 −0.362268 −0.181134 0.983458i \(-0.557977\pi\)
−0.181134 + 0.983458i \(0.557977\pi\)
\(224\) 0 0
\(225\) 5.74732 0.383155
\(226\) 0 0
\(227\) −17.3250 −1.14990 −0.574950 0.818189i \(-0.694978\pi\)
−0.574950 + 0.818189i \(0.694978\pi\)
\(228\) 0 0
\(229\) −11.4098 −0.753982 −0.376991 0.926217i \(-0.623041\pi\)
−0.376991 + 0.926217i \(0.623041\pi\)
\(230\) 0 0
\(231\) 11.2085 0.737463
\(232\) 0 0
\(233\) 2.08483 0.136581 0.0682907 0.997665i \(-0.478245\pi\)
0.0682907 + 0.997665i \(0.478245\pi\)
\(234\) 0 0
\(235\) 5.49464 0.358431
\(236\) 0 0
\(237\) −34.9893 −2.27280
\(238\) 0 0
\(239\) −18.5353 −1.19895 −0.599473 0.800395i \(-0.704623\pi\)
−0.599473 + 0.800395i \(0.704623\pi\)
\(240\) 0 0
\(241\) 10.2509 0.660317 0.330159 0.943925i \(-0.392898\pi\)
0.330159 + 0.943925i \(0.392898\pi\)
\(242\) 0 0
\(243\) 4.29509 0.275530
\(244\) 0 0
\(245\) 8.66249 0.553426
\(246\) 0 0
\(247\) 18.5371 1.17948
\(248\) 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
\(253\) −0.957587 −0.0602030
\(254\) 0 0
\(255\) −16.9982 −1.06447
\(256\) 0 0
\(257\) 28.9045 1.80301 0.901505 0.432768i \(-0.142463\pi\)
0.901505 + 0.432768i \(0.142463\pi\)
\(258\) 0 0
\(259\) 36.1147 2.24406
\(260\) 0 0
\(261\) 29.9451 1.85355
\(262\) 0 0
\(263\) 5.24196 0.323233 0.161617 0.986854i \(-0.448329\pi\)
0.161617 + 0.986854i \(0.448329\pi\)
\(264\) 0 0
\(265\) −7.12544 −0.437712
\(266\) 0 0
\(267\) 46.8196 2.86531
\(268\) 0 0
\(269\) −5.37992 −0.328019 −0.164010 0.986459i \(-0.552443\pi\)
−0.164010 + 0.986459i \(0.552443\pi\)
\(270\) 0 0
\(271\) −11.5371 −0.700826 −0.350413 0.936595i \(-0.613959\pi\)
−0.350413 + 0.936595i \(0.613959\pi\)
\(272\) 0 0
\(273\) 32.1571 1.94624
\(274\) 0 0
\(275\) −0.957587 −0.0577447
\(276\) 0 0
\(277\) 5.83035 0.350312 0.175156 0.984541i \(-0.443957\pi\)
0.175156 + 0.984541i \(0.443957\pi\)
\(278\) 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
\(283\) −3.86384 −0.229682 −0.114841 0.993384i \(-0.536636\pi\)
−0.114841 + 0.993384i \(0.536636\pi\)
\(284\) 0 0
\(285\) 19.9558 1.18208
\(286\) 0 0
\(287\) 1.00000 0.0590281
\(288\) 0 0
\(289\) 16.0317 0.943041
\(290\) 0 0
\(291\) 32.1571 1.88508
\(292\) 0 0
\(293\) −1.54597 −0.0903167 −0.0451583 0.998980i \(-0.514379\pi\)
−0.0451583 + 0.998980i \(0.514379\pi\)
\(294\) 0 0
\(295\) −4.78973 −0.278869
\(296\) 0 0
\(297\) 7.78082 0.451489
\(298\) 0 0
\(299\) −2.74732 −0.158882
\(300\) 0 0
\(301\) 31.6607 1.82489
\(302\) 0 0
\(303\) −27.2402 −1.56491
\(304\) 0 0
\(305\) 12.4522 0.713013
\(306\) 0 0
\(307\) −6.45223 −0.368248 −0.184124 0.982903i \(-0.558945\pi\)
−0.184124 + 0.982903i \(0.558945\pi\)
\(308\) 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
\(313\) −16.8620 −0.953099 −0.476550 0.879148i \(-0.658113\pi\)
−0.476550 + 0.879148i \(0.658113\pi\)
\(314\) 0 0
\(315\) 22.7455 1.28156
\(316\) 0 0
\(317\) −1.33751 −0.0751218 −0.0375609 0.999294i \(-0.511959\pi\)
−0.0375609 + 0.999294i \(0.511959\pi\)
\(318\) 0 0
\(319\) −4.98928 −0.279346
\(320\) 0 0
\(321\) −10.7384 −0.599359
\(322\) 0 0
\(323\) −38.7790 −2.15772
\(324\) 0 0
\(325\) −2.74732 −0.152394
\(326\) 0 0
\(327\) −49.2808 −2.72523
\(328\) 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
\(333\) 52.4468 2.87407
\(334\) 0 0
\(335\) 9.12544 0.498576
\(336\) 0 0
\(337\) −4.83215 −0.263224 −0.131612 0.991301i \(-0.542015\pi\)
−0.131612 + 0.991301i \(0.542015\pi\)
\(338\) 0 0
\(339\) 53.5759 2.90984
\(340\) 0 0
\(341\) 5.70491 0.308938
\(342\) 0 0
\(343\) 6.57947 0.355258
\(344\) 0 0
\(345\) −2.95759 −0.159231
\(346\) 0 0
\(347\) −34.6183 −1.85841 −0.929203 0.369569i \(-0.879505\pi\)
−0.929203 + 0.369569i \(0.879505\pi\)
\(348\) 0 0
\(349\) 26.9558 1.44291 0.721455 0.692461i \(-0.243474\pi\)
0.721455 + 0.692461i \(0.243474\pi\)
\(350\) 0 0
\(351\) 22.3232 1.19152
\(352\) 0 0
\(353\) −8.00000 −0.425797 −0.212899 0.977074i \(-0.568290\pi\)
−0.212899 + 0.977074i \(0.568290\pi\)
\(354\) 0 0
\(355\) 1.66249 0.0882361
\(356\) 0 0
\(357\) −67.2719 −3.56040
\(358\) 0 0
\(359\) 0.420532 0.0221949 0.0110974 0.999938i \(-0.496468\pi\)
0.0110974 + 0.999938i \(0.496468\pi\)
\(360\) 0 0
\(361\) 26.5263 1.39612
\(362\) 0 0
\(363\) 29.8214 1.56522
\(364\) 0 0
\(365\) 12.3357 0.645680
\(366\) 0 0
\(367\) 11.7156 0.611551 0.305775 0.952104i \(-0.401084\pi\)
0.305775 + 0.952104i \(0.401084\pi\)
\(368\) 0 0
\(369\) 1.45223 0.0756000
\(370\) 0 0
\(371\) −28.1995 −1.46405
\(372\) 0 0
\(373\) −28.1661 −1.45838 −0.729192 0.684310i \(-0.760104\pi\)
−0.729192 + 0.684310i \(0.760104\pi\)
\(374\) 0 0
\(375\) −2.95759 −0.152729
\(376\) 0 0
\(377\) −14.3143 −0.737223
\(378\) 0 0
\(379\) 27.3781 1.40632 0.703160 0.711032i \(-0.251772\pi\)
0.703160 + 0.711032i \(0.251772\pi\)
\(380\) 0 0
\(381\) 17.4946 0.896278
\(382\) 0 0
\(383\) −31.7754 −1.62365 −0.811824 0.583902i \(-0.801525\pi\)
−0.811824 + 0.583902i \(0.801525\pi\)
\(384\) 0 0
\(385\) −3.78973 −0.193143
\(386\) 0 0
\(387\) 45.9786 2.33722
\(388\) 0 0
\(389\) −18.7031 −0.948285 −0.474143 0.880448i \(-0.657242\pi\)
−0.474143 + 0.880448i \(0.657242\pi\)
\(390\) 0 0
\(391\) 5.74732 0.290655
\(392\) 0 0
\(393\) −10.0848 −0.508712
\(394\) 0 0
\(395\) 11.8303 0.595249
\(396\) 0 0
\(397\) −39.6924 −1.99210 −0.996052 0.0887714i \(-0.971706\pi\)
−0.996052 + 0.0887714i \(0.971706\pi\)
\(398\) 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
\(403\) 16.3674 0.815318
\(404\) 0 0
\(405\) 6.78973 0.337385
\(406\) 0 0
\(407\) −8.73840 −0.433147
\(408\) 0 0
\(409\) 9.45223 0.467383 0.233691 0.972311i \(-0.424919\pi\)
0.233691 + 0.972311i \(0.424919\pi\)
\(410\) 0 0
\(411\) 4.94867 0.244100
\(412\) 0 0
\(413\) −18.9558 −0.932753
\(414\) 0 0
\(415\) 0.704908 0.0346026
\(416\) 0 0
\(417\) −25.4946 −1.24848
\(418\) 0 0
\(419\) −33.2402 −1.62389 −0.811944 0.583735i \(-0.801591\pi\)
−0.811944 + 0.583735i \(0.801591\pi\)
\(420\) 0 0
\(421\) −38.8285 −1.89239 −0.946194 0.323600i \(-0.895107\pi\)
−0.946194 + 0.323600i \(0.895107\pi\)
\(422\) 0 0
\(423\) 31.5795 1.53545
\(424\) 0 0
\(425\) 5.74732 0.278786
\(426\) 0 0
\(427\) 49.2808 2.38486
\(428\) 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
\(433\) −5.78793 −0.278150 −0.139075 0.990282i \(-0.544413\pi\)
−0.139075 + 0.990282i \(0.544413\pi\)
\(434\) 0 0
\(435\) −15.4098 −0.738844
\(436\) 0 0
\(437\) −6.74732 −0.322768
\(438\) 0 0
\(439\) −11.9241 −0.569106 −0.284553 0.958660i \(-0.591845\pi\)
−0.284553 + 0.958660i \(0.591845\pi\)
\(440\) 0 0
\(441\) 49.7861 2.37077
\(442\) 0 0
\(443\) 39.3973 1.87182 0.935911 0.352236i \(-0.114579\pi\)
0.935911 + 0.352236i \(0.114579\pi\)
\(444\) 0 0
\(445\) −15.8303 −0.750430
\(446\) 0 0
\(447\) 68.2701 3.22906
\(448\) 0 0
\(449\) 21.3674 1.00839 0.504195 0.863590i \(-0.331789\pi\)
0.504195 + 0.863590i \(0.331789\pi\)
\(450\) 0 0
\(451\) −0.241962 −0.0113936
\(452\) 0 0
\(453\) −31.9063 −1.49909
\(454\) 0 0
\(455\) −10.8728 −0.509723
\(456\) 0 0
\(457\) −20.3656 −0.952663 −0.476331 0.879266i \(-0.658034\pi\)
−0.476331 + 0.879266i \(0.658034\pi\)
\(458\) 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
\(463\) −16.8196 −0.781675 −0.390837 0.920460i \(-0.627814\pi\)
−0.390837 + 0.920460i \(0.627814\pi\)
\(464\) 0 0
\(465\) 17.6201 0.817112
\(466\) 0 0
\(467\) −24.4504 −1.13143 −0.565715 0.824601i \(-0.691400\pi\)
−0.565715 + 0.824601i \(0.691400\pi\)
\(468\) 0 0
\(469\) 36.1147 1.66762
\(470\) 0 0
\(471\) −24.2509 −1.11742
\(472\) 0 0
\(473\) −7.66070 −0.352239
\(474\) 0 0
\(475\) −6.74732 −0.309588
\(476\) 0 0
\(477\) −40.9522 −1.87507
\(478\) 0 0
\(479\) 10.5688 0.482899 0.241449 0.970413i \(-0.422377\pi\)
0.241449 + 0.970413i \(0.422377\pi\)
\(480\) 0 0
\(481\) −25.0705 −1.14312
\(482\) 0 0
\(483\) −11.7049 −0.532592
\(484\) 0 0
\(485\) −10.8728 −0.493707
\(486\) 0 0
\(487\) −13.0741 −0.592444 −0.296222 0.955119i \(-0.595727\pi\)
−0.296222 + 0.955119i \(0.595727\pi\)
\(488\) 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
\(493\) 29.9451 1.34866
\(494\) 0 0
\(495\) −5.50356 −0.247367
\(496\) 0 0
\(497\) 6.57947 0.295129
\(498\) 0 0
\(499\) 2.87456 0.128683 0.0643415 0.997928i \(-0.479505\pi\)
0.0643415 + 0.997928i \(0.479505\pi\)
\(500\) 0 0
\(501\) 23.6607 1.05708
\(502\) 0 0
\(503\) 9.36740 0.417672 0.208836 0.977951i \(-0.433033\pi\)
0.208836 + 0.977951i \(0.433033\pi\)
\(504\) 0 0
\(505\) 9.21027 0.409851
\(506\) 0 0
\(507\) 16.1254 0.716156
\(508\) 0 0
\(509\) −40.5866 −1.79897 −0.899484 0.436953i \(-0.856058\pi\)
−0.899484 + 0.436953i \(0.856058\pi\)
\(510\) 0 0
\(511\) 48.8196 2.15965
\(512\) 0 0
\(513\) 54.8250 2.42058
\(514\) 0 0
\(515\) 12.4522 0.548711
\(516\) 0 0
\(517\) −5.26160 −0.231405
\(518\) 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
\(523\) 35.4098 1.54836 0.774182 0.632964i \(-0.218162\pi\)
0.774182 + 0.632964i \(0.218162\pi\)
\(524\) 0 0
\(525\) −11.7049 −0.510844
\(526\) 0 0
\(527\) −34.2402 −1.49152
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −27.5281 −1.19462
\(532\) 0 0
\(533\) −0.694191 −0.0300687
\(534\) 0 0
\(535\) 3.63080 0.156973
\(536\) 0 0
\(537\) −4.42053 −0.190760
\(538\) 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
\(543\) 37.1995 1.59639
\(544\) 0 0
\(545\) 16.6625 0.713743
\(546\) 0 0
\(547\) 21.4187 0.915799 0.457899 0.889004i \(-0.348602\pi\)
0.457899 + 0.889004i \(0.348602\pi\)
\(548\) 0 0
\(549\) 71.5670 3.05440
\(550\) 0 0
\(551\) −35.1553 −1.49767
\(552\) 0 0
\(553\) 46.8196 1.99097
\(554\) 0 0
\(555\) −26.9893 −1.14563
\(556\) 0 0
\(557\) 20.9558 0.887925 0.443963 0.896045i \(-0.353572\pi\)
0.443963 + 0.896045i \(0.353572\pi\)
\(558\) 0 0
\(559\) −21.9786 −0.929594
\(560\) 0 0
\(561\) 16.2773 0.687226
\(562\) 0 0
\(563\) 6.70491 0.282578 0.141289 0.989968i \(-0.454875\pi\)
0.141289 + 0.989968i \(0.454875\pi\)
\(564\) 0 0
\(565\) −18.1147 −0.762092
\(566\) 0 0
\(567\) 26.8710 1.12847
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 13.2933 0.556307 0.278154 0.960537i \(-0.410278\pi\)
0.278154 + 0.960537i \(0.410278\pi\)
\(572\) 0 0
\(573\) −49.9964 −2.08863
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 23.2402 0.967501 0.483750 0.875206i \(-0.339274\pi\)
0.483750 + 0.875206i \(0.339274\pi\)
\(578\) 0 0
\(579\) −29.3250 −1.21870
\(580\) 0 0
\(581\) 2.78973 0.115738
\(582\) 0 0
\(583\) 6.82323 0.282589
\(584\) 0 0
\(585\) −15.7897 −0.652825
\(586\) 0 0
\(587\) 17.6112 0.726891 0.363445 0.931616i \(-0.381600\pi\)
0.363445 + 0.931616i \(0.381600\pi\)
\(588\) 0 0
\(589\) 40.1978 1.65632
\(590\) 0 0
\(591\) −13.4187 −0.551973
\(592\) 0 0
\(593\) −41.5759 −1.70732 −0.853658 0.520834i \(-0.825621\pi\)
−0.853658 + 0.520834i \(0.825621\pi\)
\(594\) 0 0
\(595\) 22.7455 0.932475
\(596\) 0 0
\(597\) 41.4062 1.69464
\(598\) 0 0
\(599\) 17.7138 0.723767 0.361884 0.932223i \(-0.382134\pi\)
0.361884 + 0.932223i \(0.382134\pi\)
\(600\) 0 0
\(601\) −46.7772 −1.90808 −0.954041 0.299675i \(-0.903122\pi\)
−0.954041 + 0.299675i \(0.903122\pi\)
\(602\) 0 0
\(603\) 52.4468 2.13580
\(604\) 0 0
\(605\) −10.0830 −0.409933
\(606\) 0 0
\(607\) 42.9893 1.74488 0.872441 0.488720i \(-0.162536\pi\)
0.872441 + 0.488720i \(0.162536\pi\)
\(608\) 0 0
\(609\) −60.9857 −2.47126
\(610\) 0 0
\(611\) −15.0955 −0.610700
\(612\) 0 0
\(613\) −32.6500 −1.31872 −0.659360 0.751827i \(-0.729173\pi\)
−0.659360 + 0.751827i \(0.729173\pi\)
\(614\) 0 0
\(615\) −0.747321 −0.0301349
\(616\) 0 0
\(617\) −11.3232 −0.455854 −0.227927 0.973678i \(-0.573195\pi\)
−0.227927 + 0.973678i \(0.573195\pi\)
\(618\) 0 0
\(619\) −10.6625 −0.428562 −0.214281 0.976772i \(-0.568741\pi\)
−0.214281 + 0.976772i \(0.568741\pi\)
\(620\) 0 0
\(621\) −8.12544 −0.326063
\(622\) 0 0
\(623\) −62.6500 −2.51002
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −19.1094 −0.763156
\(628\) 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
\(633\) −25.7455 −1.02329
\(634\) 0 0
\(635\) −5.91517 −0.234737
\(636\) 0 0
\(637\) −23.7987 −0.942937
\(638\) 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
\(643\) −9.37992 −0.369908 −0.184954 0.982747i \(-0.559214\pi\)
−0.184954 + 0.982747i \(0.559214\pi\)
\(644\) 0 0
\(645\) −23.6607 −0.931639
\(646\) 0 0
\(647\) −18.6714 −0.734049 −0.367024 0.930211i \(-0.619623\pi\)
−0.367024 + 0.930211i \(0.619623\pi\)
\(648\) 0 0
\(649\) 4.58659 0.180039
\(650\) 0 0
\(651\) 69.7330 2.73305
\(652\) 0 0
\(653\) 12.0723 0.472426 0.236213 0.971701i \(-0.424094\pi\)
0.236213 + 0.971701i \(0.424094\pi\)
\(654\) 0 0
\(655\) 3.40982 0.133233
\(656\) 0 0
\(657\) 70.8973 2.76597
\(658\) 0 0
\(659\) 21.2402 0.827399 0.413700 0.910413i \(-0.364236\pi\)
0.413700 + 0.910413i \(0.364236\pi\)
\(660\) 0 0
\(661\) −26.9362 −1.04769 −0.523847 0.851812i \(-0.675504\pi\)
−0.523847 + 0.851812i \(0.675504\pi\)
\(662\) 0 0
\(663\) 46.6995 1.81366
\(664\) 0 0
\(665\) −26.7031 −1.03550
\(666\) 0 0
\(667\) 5.21027 0.201742
\(668\) 0 0
\(669\) 16.0000 0.618596
\(670\) 0 0
\(671\) −11.9241 −0.460324
\(672\) 0 0
\(673\) −11.4098 −0.439816 −0.219908 0.975521i \(-0.570576\pi\)
−0.219908 + 0.975521i \(0.570576\pi\)
\(674\) 0 0
\(675\) −8.12544 −0.312748
\(676\) 0 0
\(677\) −11.8638 −0.455965 −0.227982 0.973665i \(-0.573213\pi\)
−0.227982 + 0.973665i \(0.573213\pi\)
\(678\) 0 0
\(679\) −43.0299 −1.65134
\(680\) 0 0
\(681\) 51.2402 1.96353
\(682\) 0 0
\(683\) 10.0723 0.385406 0.192703 0.981257i \(-0.438275\pi\)
0.192703 + 0.981257i \(0.438275\pi\)
\(684\) 0 0
\(685\) −1.67321 −0.0639301
\(686\) 0 0
\(687\) 33.7455 1.28747
\(688\) 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
\(693\) −21.7808 −0.827385
\(694\) 0 0
\(695\) 8.62008 0.326978
\(696\) 0 0
\(697\) 1.45223 0.0550071
\(698\) 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
\(703\) −61.5723 −2.32224
\(704\) 0 0
\(705\) −16.2509 −0.612044
\(706\) 0 0
\(707\) 36.4504 1.37086
\(708\) 0 0
\(709\) 31.3781 1.17843 0.589215 0.807976i \(-0.299437\pi\)
0.589215 + 0.807976i \(0.299437\pi\)
\(710\) 0 0
\(711\) 67.9928 2.54993
\(712\) 0 0
\(713\) −5.95759 −0.223113
\(714\) 0 0
\(715\) 2.63080 0.0983863
\(716\) 0 0
\(717\) 54.8196 2.04728
\(718\) 0 0
\(719\) −20.4629 −0.763139 −0.381570 0.924340i \(-0.624616\pi\)
−0.381570 + 0.924340i \(0.624616\pi\)
\(720\) 0 0
\(721\) 49.2808 1.83531
\(722\) 0 0
\(723\) −30.3179 −1.12753
\(724\) 0 0
\(725\) 5.21027 0.193504
\(726\) 0 0
\(727\) 14.9875 0.555855 0.277928 0.960602i \(-0.410352\pi\)
0.277928 + 0.960602i \(0.410352\pi\)
\(728\) 0 0
\(729\) −33.0723 −1.22490
\(730\) 0 0
\(731\) 45.9786 1.70058
\(732\) 0 0
\(733\) −22.2844 −0.823092 −0.411546 0.911389i \(-0.635011\pi\)
−0.411546 + 0.911389i \(0.635011\pi\)
\(734\) 0 0
\(735\) −25.6201 −0.945011
\(736\) 0 0
\(737\) −8.73840 −0.321883
\(738\) 0 0
\(739\) 30.9344 1.13794 0.568969 0.822359i \(-0.307342\pi\)
0.568969 + 0.822359i \(0.307342\pi\)
\(740\) 0 0
\(741\) −54.8250 −2.01404
\(742\) 0 0
\(743\) −44.8285 −1.64460 −0.822300 0.569054i \(-0.807309\pi\)
−0.822300 + 0.569054i \(0.807309\pi\)
\(744\) 0 0
\(745\) −23.0830 −0.845697
\(746\) 0 0
\(747\) 4.05133 0.148230
\(748\) 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
\(753\) 51.0263 1.85950
\(754\) 0 0
\(755\) 10.7879 0.392613
\(756\) 0 0
\(757\) 40.6835 1.47867 0.739333 0.673340i \(-0.235141\pi\)
0.739333 + 0.673340i \(0.235141\pi\)
\(758\) 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
\(763\) 65.9433 2.38731
\(764\) 0 0
\(765\) 33.0317 1.19426
\(766\) 0 0
\(767\) 13.1589 0.475142
\(768\) 0 0
\(769\) −40.4205 −1.45760 −0.728801 0.684726i \(-0.759922\pi\)
−0.728801 + 0.684726i \(0.759922\pi\)
\(770\) 0 0
\(771\) −85.4874 −3.07876
\(772\) 0 0
\(773\) 24.5688 0.883677 0.441838 0.897095i \(-0.354327\pi\)
0.441838 + 0.897095i \(0.354327\pi\)
\(774\) 0 0
\(775\) −5.95759 −0.214003
\(776\) 0 0
\(777\) −106.812 −3.83187
\(778\) 0 0
\(779\) −1.70491 −0.0610847
\(780\) 0 0
\(781\) −1.59198 −0.0569656
\(782\) 0 0
\(783\) −42.3357 −1.51295
\(784\) 0 0
\(785\) 8.19955 0.292654
\(786\) 0 0
\(787\) −11.7790 −0.419877 −0.209938 0.977715i \(-0.567326\pi\)
−0.209938 + 0.977715i \(0.567326\pi\)
\(788\) 0 0
\(789\) −15.5036 −0.551941
\(790\) 0 0
\(791\) −71.6906 −2.54902
\(792\) 0 0
\(793\) −34.2103 −1.21484
\(794\) 0 0
\(795\) 21.0741 0.747422
\(796\) 0 0
\(797\) −13.8602 −0.490955 −0.245478 0.969402i \(-0.578945\pi\)
−0.245478 + 0.969402i \(0.578945\pi\)
\(798\) 0 0
\(799\) 31.5795 1.11720
\(800\) 0 0
\(801\) −90.9821 −3.21469
\(802\) 0 0
\(803\) −11.8125 −0.416854
\(804\) 0 0
\(805\) 3.95759 0.139487
\(806\) 0 0
\(807\) 15.9116 0.560114
\(808\) 0 0
\(809\) −52.0830 −1.83114 −0.915571 0.402157i \(-0.868261\pi\)
−0.915571 + 0.402157i \(0.868261\pi\)
\(810\) 0 0
\(811\) −40.7013 −1.42922 −0.714608 0.699525i \(-0.753395\pi\)
−0.714608 + 0.699525i \(0.753395\pi\)
\(812\) 0 0
\(813\) 34.1218 1.19671
\(814\) 0 0
\(815\) 16.6625 0.583662
\(816\) 0 0
\(817\) −53.9786 −1.88847
\(818\) 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
\(823\) −42.9009 −1.49543 −0.747715 0.664020i \(-0.768849\pi\)
−0.747715 + 0.664020i \(0.768849\pi\)
\(824\) 0 0
\(825\) 2.83215 0.0986027
\(826\) 0 0
\(827\) 39.4397 1.37145 0.685727 0.727859i \(-0.259485\pi\)
0.685727 + 0.727859i \(0.259485\pi\)
\(828\) 0 0
\(829\) 29.5460 1.02617 0.513087 0.858337i \(-0.328502\pi\)
0.513087 + 0.858337i \(0.328502\pi\)
\(830\) 0 0
\(831\) −17.2438 −0.598179
\(832\) 0 0
\(833\) 49.7861 1.72499
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 0 0
\(837\) 48.4080 1.67323
\(838\) 0 0
\(839\) −1.09554 −0.0378223 −0.0189112 0.999821i \(-0.506020\pi\)
−0.0189112 + 0.999821i \(0.506020\pi\)
\(840\) 0 0
\(841\) −1.85313 −0.0639009
\(842\) 0 0
\(843\) −32.7526 −1.12806
\(844\) 0 0
\(845\) −5.45223 −0.187562
\(846\) 0 0
\(847\) −39.9045 −1.37113
\(848\) 0 0
\(849\) 11.4277 0.392196
\(850\) 0 0
\(851\) 9.12544 0.312816
\(852\) 0 0
\(853\) −10.0937 −0.345603 −0.172802 0.984957i \(-0.555282\pi\)
−0.172802 + 0.984957i \(0.555282\pi\)
\(854\) 0 0
\(855\) −38.7790 −1.32621
\(856\) 0 0
\(857\) 41.8303 1.42890 0.714449 0.699688i \(-0.246678\pi\)
0.714449 + 0.699688i \(0.246678\pi\)
\(858\) 0 0
\(859\) −3.35848 −0.114590 −0.0572950 0.998357i \(-0.518248\pi\)
−0.0572950 + 0.998357i \(0.518248\pi\)
\(860\) 0 0
\(861\) −2.95759 −0.100794
\(862\) 0 0
\(863\) −3.83395 −0.130509 −0.0652545 0.997869i \(-0.520786\pi\)
−0.0652545 + 0.997869i \(0.520786\pi\)
\(864\) 0 0
\(865\) 8.87276 0.301683
\(866\) 0 0
\(867\) −47.4151 −1.61030
\(868\) 0 0
\(869\) −11.3286 −0.384296
\(870\) 0 0
\(871\) −25.0705 −0.849482
\(872\) 0 0
\(873\) −62.4892 −2.11494
\(874\) 0 0
\(875\) 3.95759 0.133791
\(876\) 0 0
\(877\) 15.5263 0.524287 0.262144 0.965029i \(-0.415571\pi\)
0.262144 + 0.965029i \(0.415571\pi\)
\(878\) 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
\(883\) 21.5884 0.726507 0.363254 0.931690i \(-0.381666\pi\)
0.363254 + 0.931690i \(0.381666\pi\)
\(884\) 0 0
\(885\) 14.1661 0.476187
\(886\) 0 0
\(887\) 49.4098 1.65902 0.829510 0.558492i \(-0.188620\pi\)
0.829510 + 0.558492i \(0.188620\pi\)
\(888\) 0 0
\(889\) −23.4098 −0.785140
\(890\) 0 0
\(891\) −6.50176 −0.217817
\(892\) 0 0
\(893\) −37.0741 −1.24064
\(894\) 0 0
\(895\) 1.49464 0.0499604
\(896\) 0 0
\(897\) 8.12544 0.271301
\(898\) 0 0
\(899\) −31.0406 −1.03526
\(900\) 0 0
\(901\) −40.9522 −1.36432
\(902\) 0 0
\(903\) −93.6393 −3.11612
\(904\) 0 0
\(905\) −12.5777 −0.418096
\(906\) 0 0
\(907\) −9.54957 −0.317088 −0.158544 0.987352i \(-0.550680\pi\)
−0.158544 + 0.987352i \(0.550680\pi\)
\(908\) 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
\(913\) −0.675011 −0.0223396
\(914\) 0 0
\(915\) −36.8285 −1.21751
\(916\) 0 0
\(917\) 13.4946 0.445632
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 19.0830 0.628807
\(922\) 0 0
\(923\) −4.56741 −0.150338
\(924\) 0 0
\(925\) 9.12544 0.300043
\(926\) 0 0
\(927\) 71.5670 2.35057
\(928\) 0 0
\(929\) −19.6942 −0.646145 −0.323073 0.946374i \(-0.604716\pi\)
−0.323073 + 0.946374i \(0.604716\pi\)
\(930\) 0 0
\(931\) −58.4486 −1.91558
\(932\) 0 0
\(933\) −67.4910 −2.20956
\(934\) 0 0
\(935\) −5.50356 −0.179986
\(936\) 0 0
\(937\) −31.8214 −1.03956 −0.519780 0.854300i \(-0.673986\pi\)
−0.519780 + 0.854300i \(0.673986\pi\)
\(938\) 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
\(943\) 0.252679 0.00822837
\(944\) 0 0
\(945\) −32.1571 −1.04607
\(946\) 0 0
\(947\) −9.90266 −0.321793 −0.160897 0.986971i \(-0.551439\pi\)
−0.160897 + 0.986971i \(0.551439\pi\)
\(948\) 0 0
\(949\) −33.8901 −1.10012
\(950\) 0 0
\(951\) 3.95579 0.128275
\(952\) 0 0
\(953\) −31.5442 −1.02182 −0.510908 0.859635i \(-0.670691\pi\)
−0.510908 + 0.859635i \(0.670691\pi\)
\(954\) 0 0
\(955\) 16.9045 0.547015
\(956\) 0 0
\(957\) 14.7562 0.477001
\(958\) 0 0
\(959\) −6.62188 −0.213832
\(960\) 0 0
\(961\) 4.49284 0.144930
\(962\) 0 0
\(963\) 20.8674 0.672441
\(964\) 0 0
\(965\) 9.91517 0.319181
\(966\) 0 0
\(967\) −15.5973 −0.501575 −0.250788 0.968042i \(-0.580690\pi\)
−0.250788 + 0.968042i \(0.580690\pi\)
\(968\) 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
\(973\) 34.1147 1.09367
\(974\) 0 0
\(975\) 8.12544 0.260222
\(976\) 0 0
\(977\) 28.9469 0.926092 0.463046 0.886334i \(-0.346756\pi\)
0.463046 + 0.886334i \(0.346756\pi\)
\(978\) 0 0
\(979\) 15.1589 0.484482
\(980\) 0 0
\(981\) 95.7647 3.05753
\(982\) 0 0
\(983\) −36.2312 −1.15560 −0.577799 0.816179i \(-0.696088\pi\)
−0.577799 + 0.816179i \(0.696088\pi\)
\(984\) 0 0
\(985\) 4.53705 0.144563
\(986\) 0 0
\(987\) −64.3143 −2.04715
\(988\) 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
\(993\) 95.4732 3.02975
\(994\) 0 0
\(995\) −14.0000 −0.443830
\(996\) 0 0
\(997\) 19.2402 0.609342 0.304671 0.952458i \(-0.401453\pi\)
0.304671 + 0.952458i \(0.401453\pi\)
\(998\) 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 920.2.a.h.1.1 3
3.2 odd 2 8280.2.a.bj.1.3 3
4.3 odd 2 1840.2.a.s.1.3 3
5.2 odd 4 4600.2.e.p.4049.5 6
5.3 odd 4 4600.2.e.p.4049.2 6
5.4 even 2 4600.2.a.x.1.3 3
8.3 odd 2 7360.2.a.cc.1.1 3
8.5 even 2 7360.2.a.by.1.3 3
20.19 odd 2 9200.2.a.ce.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.h.1.1 3 1.1 even 1 trivial
1840.2.a.s.1.3 3 4.3 odd 2
4600.2.a.x.1.3 3 5.4 even 2
4600.2.e.p.4049.2 6 5.3 odd 4
4600.2.e.p.4049.5 6 5.2 odd 4
7360.2.a.by.1.3 3 8.5 even 2
7360.2.a.cc.1.1 3 8.3 odd 2
8280.2.a.bj.1.3 3 3.2 odd 2
9200.2.a.ce.1.1 3 20.19 odd 2