Properties

Label 1850.2.b.n.149.5
Level $1850$
Weight $2$
Character 1850.149
Analytic conductor $14.772$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1850.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(14.7723243739\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.5
Root \(-0.854638 - 0.854638i\) of defining polynomial
Character \(\chi\) \(=\) 1850.149
Dual form 1850.2.b.n.149.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.53919i q^{3} -1.00000 q^{4} -1.53919 q^{6} +2.87936i q^{7} -1.00000i q^{8} +0.630898 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.53919i q^{3} -1.00000 q^{4} -1.53919 q^{6} +2.87936i q^{7} -1.00000i q^{8} +0.630898 q^{9} -1.09171 q^{11} -1.53919i q^{12} +4.53919i q^{13} -2.87936 q^{14} +1.00000 q^{16} -2.80098i q^{17} +0.630898i q^{18} +5.04945 q^{19} -4.43188 q^{21} -1.09171i q^{22} +7.41855i q^{23} +1.53919 q^{24} -4.53919 q^{26} +5.58864i q^{27} -2.87936i q^{28} -6.68035 q^{29} +3.51026 q^{31} +1.00000i q^{32} -1.68035i q^{33} +2.80098 q^{34} -0.630898 q^{36} -1.00000i q^{37} +5.04945i q^{38} -6.98667 q^{39} -8.07838 q^{41} -4.43188i q^{42} +10.2329i q^{43} +1.09171 q^{44} -7.41855 q^{46} -8.68035i q^{47} +1.53919i q^{48} -1.29072 q^{49} +4.31124 q^{51} -4.53919i q^{52} -10.0989i q^{53} -5.58864 q^{54} +2.87936 q^{56} +7.77205i q^{57} -6.68035i q^{58} -10.2329 q^{59} +6.29791 q^{61} +3.51026i q^{62} +1.81658i q^{63} -1.00000 q^{64} +1.68035 q^{66} -13.2979i q^{67} +2.80098i q^{68} -11.4186 q^{69} +6.29791 q^{71} -0.630898i q^{72} +12.7093i q^{73} +1.00000 q^{74} -5.04945 q^{76} -3.14342i q^{77} -6.98667i q^{78} -2.58145 q^{79} -6.70928 q^{81} -8.07838i q^{82} +8.48360i q^{83} +4.43188 q^{84} -10.2329 q^{86} -10.2823i q^{87} +1.09171i q^{88} +6.51026 q^{89} -13.0700 q^{91} -7.41855i q^{92} +5.40295i q^{93} +8.68035 q^{94} -1.53919 q^{96} -3.07838i q^{97} -1.29072i q^{98} -0.688756 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} - 6 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{4} - 6 q^{6} - 4 q^{9} - 2 q^{11} + 8 q^{14} + 6 q^{16} - 6 q^{19} + 6 q^{24} - 24 q^{26} + 4 q^{29} - 12 q^{31} - 2 q^{34} + 4 q^{36} - 40 q^{39} - 42 q^{41} + 2 q^{44} - 16 q^{46} - 22 q^{49} - 26 q^{51} + 6 q^{54} - 8 q^{56} - 16 q^{59} - 16 q^{61} - 6 q^{64} - 34 q^{66} - 40 q^{69} - 16 q^{71} + 6 q^{74} + 6 q^{76} - 44 q^{79} - 26 q^{81} - 16 q^{86} + 6 q^{89} + 24 q^{91} + 8 q^{94} - 6 q^{96} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) 1.53919i 0.888651i 0.895865 + 0.444326i \(0.146557\pi\)
−0.895865 + 0.444326i \(0.853443\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.53919 −0.628371
\(7\) 2.87936i 1.08830i 0.838989 + 0.544148i \(0.183147\pi\)
−0.838989 + 0.544148i \(0.816853\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0.630898 0.210299
\(10\) 0 0
\(11\) −1.09171 −0.329163 −0.164581 0.986364i \(-0.552627\pi\)
−0.164581 + 0.986364i \(0.552627\pi\)
\(12\) − 1.53919i − 0.444326i
\(13\) 4.53919i 1.25894i 0.777023 + 0.629472i \(0.216729\pi\)
−0.777023 + 0.629472i \(0.783271\pi\)
\(14\) −2.87936 −0.769542
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 2.80098i − 0.679338i −0.940545 0.339669i \(-0.889685\pi\)
0.940545 0.339669i \(-0.110315\pi\)
\(18\) 0.630898i 0.148704i
\(19\) 5.04945 1.15842 0.579211 0.815177i \(-0.303361\pi\)
0.579211 + 0.815177i \(0.303361\pi\)
\(20\) 0 0
\(21\) −4.43188 −0.967116
\(22\) − 1.09171i − 0.232753i
\(23\) 7.41855i 1.54687i 0.633873 + 0.773437i \(0.281464\pi\)
−0.633873 + 0.773437i \(0.718536\pi\)
\(24\) 1.53919 0.314186
\(25\) 0 0
\(26\) −4.53919 −0.890208
\(27\) 5.58864i 1.07553i
\(28\) − 2.87936i − 0.544148i
\(29\) −6.68035 −1.24051 −0.620255 0.784401i \(-0.712971\pi\)
−0.620255 + 0.784401i \(0.712971\pi\)
\(30\) 0 0
\(31\) 3.51026 0.630461 0.315231 0.949015i \(-0.397918\pi\)
0.315231 + 0.949015i \(0.397918\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 1.68035i − 0.292511i
\(34\) 2.80098 0.480365
\(35\) 0 0
\(36\) −0.630898 −0.105150
\(37\) − 1.00000i − 0.164399i
\(38\) 5.04945i 0.819129i
\(39\) −6.98667 −1.11876
\(40\) 0 0
\(41\) −8.07838 −1.26163 −0.630815 0.775933i \(-0.717279\pi\)
−0.630815 + 0.775933i \(0.717279\pi\)
\(42\) − 4.43188i − 0.683854i
\(43\) 10.2329i 1.56050i 0.625469 + 0.780249i \(0.284908\pi\)
−0.625469 + 0.780249i \(0.715092\pi\)
\(44\) 1.09171 0.164581
\(45\) 0 0
\(46\) −7.41855 −1.09381
\(47\) − 8.68035i − 1.26616i −0.774087 0.633079i \(-0.781791\pi\)
0.774087 0.633079i \(-0.218209\pi\)
\(48\) 1.53919i 0.222163i
\(49\) −1.29072 −0.184389
\(50\) 0 0
\(51\) 4.31124 0.603695
\(52\) − 4.53919i − 0.629472i
\(53\) − 10.0989i − 1.38719i −0.720365 0.693595i \(-0.756026\pi\)
0.720365 0.693595i \(-0.243974\pi\)
\(54\) −5.58864 −0.760517
\(55\) 0 0
\(56\) 2.87936 0.384771
\(57\) 7.77205i 1.02943i
\(58\) − 6.68035i − 0.877172i
\(59\) −10.2329 −1.33221 −0.666103 0.745860i \(-0.732039\pi\)
−0.666103 + 0.745860i \(0.732039\pi\)
\(60\) 0 0
\(61\) 6.29791 0.806365 0.403183 0.915120i \(-0.367904\pi\)
0.403183 + 0.915120i \(0.367904\pi\)
\(62\) 3.51026i 0.445803i
\(63\) 1.81658i 0.228868i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.68035 0.206836
\(67\) − 13.2979i − 1.62460i −0.583241 0.812299i \(-0.698216\pi\)
0.583241 0.812299i \(-0.301784\pi\)
\(68\) 2.80098i 0.339669i
\(69\) −11.4186 −1.37463
\(70\) 0 0
\(71\) 6.29791 0.747425 0.373712 0.927545i \(-0.378085\pi\)
0.373712 + 0.927545i \(0.378085\pi\)
\(72\) − 0.630898i − 0.0743520i
\(73\) 12.7093i 1.48751i 0.668453 + 0.743754i \(0.266957\pi\)
−0.668453 + 0.743754i \(0.733043\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −5.04945 −0.579211
\(77\) − 3.14342i − 0.358226i
\(78\) − 6.98667i − 0.791084i
\(79\) −2.58145 −0.290436 −0.145218 0.989400i \(-0.546388\pi\)
−0.145218 + 0.989400i \(0.546388\pi\)
\(80\) 0 0
\(81\) −6.70928 −0.745475
\(82\) − 8.07838i − 0.892108i
\(83\) 8.48360i 0.931196i 0.884996 + 0.465598i \(0.154161\pi\)
−0.884996 + 0.465598i \(0.845839\pi\)
\(84\) 4.43188 0.483558
\(85\) 0 0
\(86\) −10.2329 −1.10344
\(87\) − 10.2823i − 1.10238i
\(88\) 1.09171i 0.116377i
\(89\) 6.51026 0.690086 0.345043 0.938587i \(-0.387864\pi\)
0.345043 + 0.938587i \(0.387864\pi\)
\(90\) 0 0
\(91\) −13.0700 −1.37010
\(92\) − 7.41855i − 0.773437i
\(93\) 5.40295i 0.560260i
\(94\) 8.68035 0.895309
\(95\) 0 0
\(96\) −1.53919 −0.157093
\(97\) − 3.07838i − 0.312562i −0.987713 0.156281i \(-0.950049\pi\)
0.987713 0.156281i \(-0.0499505\pi\)
\(98\) − 1.29072i − 0.130383i
\(99\) −0.688756 −0.0692226
\(100\) 0 0
\(101\) −15.9155 −1.58365 −0.791825 0.610748i \(-0.790869\pi\)
−0.791825 + 0.610748i \(0.790869\pi\)
\(102\) 4.31124i 0.426877i
\(103\) − 10.5886i − 1.04333i −0.853151 0.521665i \(-0.825311\pi\)
0.853151 0.521665i \(-0.174689\pi\)
\(104\) 4.53919 0.445104
\(105\) 0 0
\(106\) 10.0989 0.980892
\(107\) 4.03612i 0.390186i 0.980785 + 0.195093i \(0.0625009\pi\)
−0.980785 + 0.195093i \(0.937499\pi\)
\(108\) − 5.58864i − 0.537767i
\(109\) 4.49693 0.430728 0.215364 0.976534i \(-0.430906\pi\)
0.215364 + 0.976534i \(0.430906\pi\)
\(110\) 0 0
\(111\) 1.53919 0.146093
\(112\) 2.87936i 0.272074i
\(113\) 3.58864i 0.337591i 0.985651 + 0.168795i \(0.0539877\pi\)
−0.985651 + 0.168795i \(0.946012\pi\)
\(114\) −7.77205 −0.727920
\(115\) 0 0
\(116\) 6.68035 0.620255
\(117\) 2.86376i 0.264755i
\(118\) − 10.2329i − 0.942012i
\(119\) 8.06505 0.739322
\(120\) 0 0
\(121\) −9.80817 −0.891652
\(122\) 6.29791i 0.570186i
\(123\) − 12.4341i − 1.12115i
\(124\) −3.51026 −0.315231
\(125\) 0 0
\(126\) −1.81658 −0.161834
\(127\) − 8.14116i − 0.722411i −0.932486 0.361205i \(-0.882365\pi\)
0.932486 0.361205i \(-0.117635\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −15.7503 −1.38674
\(130\) 0 0
\(131\) −1.46800 −0.128260 −0.0641298 0.997942i \(-0.520427\pi\)
−0.0641298 + 0.997942i \(0.520427\pi\)
\(132\) 1.68035i 0.146255i
\(133\) 14.5392i 1.26071i
\(134\) 13.2979 1.14876
\(135\) 0 0
\(136\) −2.80098 −0.240182
\(137\) 6.17727i 0.527760i 0.964555 + 0.263880i \(0.0850023\pi\)
−0.964555 + 0.263880i \(0.914998\pi\)
\(138\) − 11.4186i − 0.972012i
\(139\) −13.0338 −1.10552 −0.552758 0.833342i \(-0.686425\pi\)
−0.552758 + 0.833342i \(0.686425\pi\)
\(140\) 0 0
\(141\) 13.3607 1.12517
\(142\) 6.29791i 0.528509i
\(143\) − 4.95547i − 0.414397i
\(144\) 0.630898 0.0525748
\(145\) 0 0
\(146\) −12.7093 −1.05183
\(147\) − 1.98667i − 0.163858i
\(148\) 1.00000i 0.0821995i
\(149\) 4.40522 0.360890 0.180445 0.983585i \(-0.442246\pi\)
0.180445 + 0.983585i \(0.442246\pi\)
\(150\) 0 0
\(151\) 2.92162 0.237758 0.118879 0.992909i \(-0.462070\pi\)
0.118879 + 0.992909i \(0.462070\pi\)
\(152\) − 5.04945i − 0.409564i
\(153\) − 1.76713i − 0.142864i
\(154\) 3.14342 0.253304
\(155\) 0 0
\(156\) 6.98667 0.559381
\(157\) − 22.1906i − 1.77100i −0.464636 0.885502i \(-0.653815\pi\)
0.464636 0.885502i \(-0.346185\pi\)
\(158\) − 2.58145i − 0.205369i
\(159\) 15.5441 1.23273
\(160\) 0 0
\(161\) −21.3607 −1.68346
\(162\) − 6.70928i − 0.527130i
\(163\) − 14.6248i − 1.14550i −0.819730 0.572750i \(-0.805877\pi\)
0.819730 0.572750i \(-0.194123\pi\)
\(164\) 8.07838 0.630815
\(165\) 0 0
\(166\) −8.48360 −0.658455
\(167\) 6.34736i 0.491174i 0.969375 + 0.245587i \(0.0789806\pi\)
−0.969375 + 0.245587i \(0.921019\pi\)
\(168\) 4.43188i 0.341927i
\(169\) −7.60424 −0.584941
\(170\) 0 0
\(171\) 3.18568 0.243615
\(172\) − 10.2329i − 0.780249i
\(173\) − 1.23513i − 0.0939054i −0.998897 0.0469527i \(-0.985049\pi\)
0.998897 0.0469527i \(-0.0149510\pi\)
\(174\) 10.2823 0.779500
\(175\) 0 0
\(176\) −1.09171 −0.0822906
\(177\) − 15.7503i − 1.18387i
\(178\) 6.51026i 0.487965i
\(179\) −1.76487 −0.131912 −0.0659562 0.997823i \(-0.521010\pi\)
−0.0659562 + 0.997823i \(0.521010\pi\)
\(180\) 0 0
\(181\) 6.49693 0.482913 0.241456 0.970412i \(-0.422375\pi\)
0.241456 + 0.970412i \(0.422375\pi\)
\(182\) − 13.0700i − 0.968810i
\(183\) 9.69368i 0.716577i
\(184\) 7.41855 0.546903
\(185\) 0 0
\(186\) −5.40295 −0.396164
\(187\) 3.05786i 0.223613i
\(188\) 8.68035i 0.633079i
\(189\) −16.0917 −1.17050
\(190\) 0 0
\(191\) 21.2039 1.53426 0.767131 0.641490i \(-0.221683\pi\)
0.767131 + 0.641490i \(0.221683\pi\)
\(192\) − 1.53919i − 0.111081i
\(193\) 8.14342i 0.586177i 0.956085 + 0.293088i \(0.0946830\pi\)
−0.956085 + 0.293088i \(0.905317\pi\)
\(194\) 3.07838 0.221015
\(195\) 0 0
\(196\) 1.29072 0.0921946
\(197\) 24.8443i 1.77008i 0.465513 + 0.885041i \(0.345870\pi\)
−0.465513 + 0.885041i \(0.654130\pi\)
\(198\) − 0.688756i − 0.0489478i
\(199\) 8.47027 0.600441 0.300221 0.953870i \(-0.402940\pi\)
0.300221 + 0.953870i \(0.402940\pi\)
\(200\) 0 0
\(201\) 20.4680 1.44370
\(202\) − 15.9155i − 1.11981i
\(203\) − 19.2351i − 1.35004i
\(204\) −4.31124 −0.301847
\(205\) 0 0
\(206\) 10.5886 0.737745
\(207\) 4.68035i 0.325307i
\(208\) 4.53919i 0.314736i
\(209\) −5.51253 −0.381309
\(210\) 0 0
\(211\) 21.7370 1.49644 0.748218 0.663453i \(-0.230910\pi\)
0.748218 + 0.663453i \(0.230910\pi\)
\(212\) 10.0989i 0.693595i
\(213\) 9.69368i 0.664200i
\(214\) −4.03612 −0.275903
\(215\) 0 0
\(216\) 5.58864 0.380259
\(217\) 10.1073i 0.686129i
\(218\) 4.49693i 0.304570i
\(219\) −19.5620 −1.32188
\(220\) 0 0
\(221\) 12.7142 0.855249
\(222\) 1.53919i 0.103304i
\(223\) 16.6537i 1.11521i 0.830105 + 0.557607i \(0.188280\pi\)
−0.830105 + 0.557607i \(0.811720\pi\)
\(224\) −2.87936 −0.192385
\(225\) 0 0
\(226\) −3.58864 −0.238713
\(227\) 11.2123i 0.744190i 0.928195 + 0.372095i \(0.121360\pi\)
−0.928195 + 0.372095i \(0.878640\pi\)
\(228\) − 7.77205i − 0.514717i
\(229\) 7.62863 0.504114 0.252057 0.967712i \(-0.418893\pi\)
0.252057 + 0.967712i \(0.418893\pi\)
\(230\) 0 0
\(231\) 4.83832 0.318338
\(232\) 6.68035i 0.438586i
\(233\) − 0.0494483i − 0.00323947i −0.999999 0.00161973i \(-0.999484\pi\)
0.999999 0.00161973i \(-0.000515578\pi\)
\(234\) −2.86376 −0.187210
\(235\) 0 0
\(236\) 10.2329 0.666103
\(237\) − 3.97334i − 0.258096i
\(238\) 8.06505i 0.522779i
\(239\) 14.7070 0.951317 0.475659 0.879630i \(-0.342210\pi\)
0.475659 + 0.879630i \(0.342210\pi\)
\(240\) 0 0
\(241\) −14.8999 −0.959786 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(242\) − 9.80817i − 0.630493i
\(243\) 6.43907i 0.413067i
\(244\) −6.29791 −0.403183
\(245\) 0 0
\(246\) 12.4341 0.792772
\(247\) 22.9204i 1.45839i
\(248\) − 3.51026i − 0.222902i
\(249\) −13.0579 −0.827508
\(250\) 0 0
\(251\) 6.23513 0.393558 0.196779 0.980448i \(-0.436952\pi\)
0.196779 + 0.980448i \(0.436952\pi\)
\(252\) − 1.81658i − 0.114434i
\(253\) − 8.09890i − 0.509173i
\(254\) 8.14116 0.510822
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 16.1568i − 1.00783i −0.863753 0.503915i \(-0.831892\pi\)
0.863753 0.503915i \(-0.168108\pi\)
\(258\) − 15.7503i − 0.980572i
\(259\) 2.87936 0.178915
\(260\) 0 0
\(261\) −4.21461 −0.260878
\(262\) − 1.46800i − 0.0906933i
\(263\) − 18.6537i − 1.15024i −0.818071 0.575118i \(-0.804956\pi\)
0.818071 0.575118i \(-0.195044\pi\)
\(264\) −1.68035 −0.103418
\(265\) 0 0
\(266\) −14.5392 −0.891455
\(267\) 10.0205i 0.613246i
\(268\) 13.2979i 0.812299i
\(269\) −21.8310 −1.33106 −0.665529 0.746372i \(-0.731794\pi\)
−0.665529 + 0.746372i \(0.731794\pi\)
\(270\) 0 0
\(271\) 24.9783 1.51732 0.758661 0.651486i \(-0.225854\pi\)
0.758661 + 0.651486i \(0.225854\pi\)
\(272\) − 2.80098i − 0.169835i
\(273\) − 20.1171i − 1.21755i
\(274\) −6.17727 −0.373183
\(275\) 0 0
\(276\) 11.4186 0.687316
\(277\) 0.822726i 0.0494328i 0.999695 + 0.0247164i \(0.00786827\pi\)
−0.999695 + 0.0247164i \(0.992132\pi\)
\(278\) − 13.0338i − 0.781718i
\(279\) 2.21461 0.132585
\(280\) 0 0
\(281\) −1.65142 −0.0985153 −0.0492576 0.998786i \(-0.515686\pi\)
−0.0492576 + 0.998786i \(0.515686\pi\)
\(282\) 13.3607i 0.795618i
\(283\) 15.1340i 0.899621i 0.893124 + 0.449811i \(0.148508\pi\)
−0.893124 + 0.449811i \(0.851492\pi\)
\(284\) −6.29791 −0.373712
\(285\) 0 0
\(286\) 4.95547 0.293023
\(287\) − 23.2606i − 1.37303i
\(288\) 0.630898i 0.0371760i
\(289\) 9.15449 0.538499
\(290\) 0 0
\(291\) 4.73820 0.277758
\(292\) − 12.7093i − 0.743754i
\(293\) − 25.4524i − 1.48695i −0.668766 0.743473i \(-0.733177\pi\)
0.668766 0.743473i \(-0.266823\pi\)
\(294\) 1.98667 0.115865
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) − 6.10116i − 0.354025i
\(298\) 4.40522i 0.255188i
\(299\) −33.6742 −1.94743
\(300\) 0 0
\(301\) −29.4641 −1.69828
\(302\) 2.92162i 0.168120i
\(303\) − 24.4969i − 1.40731i
\(304\) 5.04945 0.289606
\(305\) 0 0
\(306\) 1.76713 0.101020
\(307\) 27.8176i 1.58764i 0.608155 + 0.793818i \(0.291910\pi\)
−0.608155 + 0.793818i \(0.708090\pi\)
\(308\) 3.14342i 0.179113i
\(309\) 16.2979 0.927156
\(310\) 0 0
\(311\) 0.0650468 0.00368846 0.00184423 0.999998i \(-0.499413\pi\)
0.00184423 + 0.999998i \(0.499413\pi\)
\(312\) 6.98667i 0.395542i
\(313\) 22.5464i 1.27440i 0.770700 + 0.637198i \(0.219907\pi\)
−0.770700 + 0.637198i \(0.780093\pi\)
\(314\) 22.1906 1.25229
\(315\) 0 0
\(316\) 2.58145 0.145218
\(317\) 24.4775i 1.37479i 0.726283 + 0.687395i \(0.241246\pi\)
−0.726283 + 0.687395i \(0.758754\pi\)
\(318\) 15.5441i 0.871670i
\(319\) 7.29299 0.408329
\(320\) 0 0
\(321\) −6.21235 −0.346739
\(322\) − 21.3607i − 1.19038i
\(323\) − 14.1434i − 0.786961i
\(324\) 6.70928 0.372738
\(325\) 0 0
\(326\) 14.6248 0.809990
\(327\) 6.92162i 0.382767i
\(328\) 8.07838i 0.446054i
\(329\) 24.9939 1.37796
\(330\) 0 0
\(331\) −1.91321 −0.105160 −0.0525798 0.998617i \(-0.516744\pi\)
−0.0525798 + 0.998617i \(0.516744\pi\)
\(332\) − 8.48360i − 0.465598i
\(333\) − 0.630898i − 0.0345730i
\(334\) −6.34736 −0.347312
\(335\) 0 0
\(336\) −4.43188 −0.241779
\(337\) 12.6286i 0.687925i 0.938983 + 0.343963i \(0.111769\pi\)
−0.938983 + 0.343963i \(0.888231\pi\)
\(338\) − 7.60424i − 0.413616i
\(339\) −5.52359 −0.300000
\(340\) 0 0
\(341\) −3.83218 −0.207524
\(342\) 3.18568i 0.172262i
\(343\) 16.4391i 0.887626i
\(344\) 10.2329 0.551719
\(345\) 0 0
\(346\) 1.23513 0.0664012
\(347\) 8.33403i 0.447394i 0.974659 + 0.223697i \(0.0718126\pi\)
−0.974659 + 0.223697i \(0.928187\pi\)
\(348\) 10.2823i 0.551190i
\(349\) −2.11837 −0.113394 −0.0566969 0.998391i \(-0.518057\pi\)
−0.0566969 + 0.998391i \(0.518057\pi\)
\(350\) 0 0
\(351\) −25.3679 −1.35404
\(352\) − 1.09171i − 0.0581883i
\(353\) 28.4534i 1.51442i 0.653169 + 0.757212i \(0.273439\pi\)
−0.653169 + 0.757212i \(0.726561\pi\)
\(354\) 15.7503 0.837120
\(355\) 0 0
\(356\) −6.51026 −0.345043
\(357\) 12.4136i 0.656999i
\(358\) − 1.76487i − 0.0932761i
\(359\) −2.65368 −0.140056 −0.0700280 0.997545i \(-0.522309\pi\)
−0.0700280 + 0.997545i \(0.522309\pi\)
\(360\) 0 0
\(361\) 6.49693 0.341944
\(362\) 6.49693i 0.341471i
\(363\) − 15.0966i − 0.792368i
\(364\) 13.0700 0.685052
\(365\) 0 0
\(366\) −9.69368 −0.506697
\(367\) 31.7575i 1.65773i 0.559450 + 0.828864i \(0.311012\pi\)
−0.559450 + 0.828864i \(0.688988\pi\)
\(368\) 7.41855i 0.386719i
\(369\) −5.09663 −0.265320
\(370\) 0 0
\(371\) 29.0784 1.50967
\(372\) − 5.40295i − 0.280130i
\(373\) 25.1122i 1.30026i 0.759822 + 0.650131i \(0.225286\pi\)
−0.759822 + 0.650131i \(0.774714\pi\)
\(374\) −3.05786 −0.158118
\(375\) 0 0
\(376\) −8.68035 −0.447655
\(377\) − 30.3234i − 1.56173i
\(378\) − 16.0917i − 0.827668i
\(379\) −22.5330 −1.15744 −0.578722 0.815525i \(-0.696449\pi\)
−0.578722 + 0.815525i \(0.696449\pi\)
\(380\) 0 0
\(381\) 12.5308 0.641971
\(382\) 21.2039i 1.08489i
\(383\) − 12.5886i − 0.643249i −0.946867 0.321625i \(-0.895771\pi\)
0.946867 0.321625i \(-0.104229\pi\)
\(384\) 1.53919 0.0785464
\(385\) 0 0
\(386\) −8.14342 −0.414489
\(387\) 6.45589i 0.328171i
\(388\) 3.07838i 0.156281i
\(389\) 0.879362 0.0445854 0.0222927 0.999751i \(-0.492903\pi\)
0.0222927 + 0.999751i \(0.492903\pi\)
\(390\) 0 0
\(391\) 20.7792 1.05085
\(392\) 1.29072i 0.0651914i
\(393\) − 2.25953i − 0.113978i
\(394\) −24.8443 −1.25164
\(395\) 0 0
\(396\) 0.688756 0.0346113
\(397\) 29.6814i 1.48967i 0.667251 + 0.744833i \(0.267471\pi\)
−0.667251 + 0.744833i \(0.732529\pi\)
\(398\) 8.47027i 0.424576i
\(399\) −22.3786 −1.12033
\(400\) 0 0
\(401\) −13.9867 −0.698461 −0.349230 0.937037i \(-0.613557\pi\)
−0.349230 + 0.937037i \(0.613557\pi\)
\(402\) 20.4680i 1.02085i
\(403\) 15.9337i 0.793716i
\(404\) 15.9155 0.791825
\(405\) 0 0
\(406\) 19.2351 0.954624
\(407\) 1.09171i 0.0541140i
\(408\) − 4.31124i − 0.213438i
\(409\) 21.9060 1.08318 0.541592 0.840642i \(-0.317822\pi\)
0.541592 + 0.840642i \(0.317822\pi\)
\(410\) 0 0
\(411\) −9.50799 −0.468995
\(412\) 10.5886i 0.521665i
\(413\) − 29.4641i − 1.44983i
\(414\) −4.68035 −0.230026
\(415\) 0 0
\(416\) −4.53919 −0.222552
\(417\) − 20.0616i − 0.982419i
\(418\) − 5.51253i − 0.269627i
\(419\) 34.8648 1.70326 0.851629 0.524146i \(-0.175615\pi\)
0.851629 + 0.524146i \(0.175615\pi\)
\(420\) 0 0
\(421\) −4.76487 −0.232225 −0.116113 0.993236i \(-0.537043\pi\)
−0.116113 + 0.993236i \(0.537043\pi\)
\(422\) 21.7370i 1.05814i
\(423\) − 5.47641i − 0.266272i
\(424\) −10.0989 −0.490446
\(425\) 0 0
\(426\) −9.69368 −0.469660
\(427\) 18.1340i 0.877564i
\(428\) − 4.03612i − 0.195093i
\(429\) 7.62741 0.368255
\(430\) 0 0
\(431\) 2.66597 0.128415 0.0642076 0.997937i \(-0.479548\pi\)
0.0642076 + 0.997937i \(0.479548\pi\)
\(432\) 5.58864i 0.268883i
\(433\) 32.3074i 1.55259i 0.630368 + 0.776297i \(0.282904\pi\)
−0.630368 + 0.776297i \(0.717096\pi\)
\(434\) −10.1073 −0.485166
\(435\) 0 0
\(436\) −4.49693 −0.215364
\(437\) 37.4596i 1.79194i
\(438\) − 19.5620i − 0.934707i
\(439\) 22.1568 1.05748 0.528742 0.848783i \(-0.322664\pi\)
0.528742 + 0.848783i \(0.322664\pi\)
\(440\) 0 0
\(441\) −0.814315 −0.0387769
\(442\) 12.7142i 0.604753i
\(443\) 39.1061i 1.85799i 0.370097 + 0.928993i \(0.379324\pi\)
−0.370097 + 0.928993i \(0.620676\pi\)
\(444\) −1.53919 −0.0730467
\(445\) 0 0
\(446\) −16.6537 −0.788575
\(447\) 6.78047i 0.320705i
\(448\) − 2.87936i − 0.136037i
\(449\) −30.8915 −1.45786 −0.728929 0.684589i \(-0.759982\pi\)
−0.728929 + 0.684589i \(0.759982\pi\)
\(450\) 0 0
\(451\) 8.81924 0.415282
\(452\) − 3.58864i − 0.168795i
\(453\) 4.49693i 0.211284i
\(454\) −11.2123 −0.526222
\(455\) 0 0
\(456\) 7.77205 0.363960
\(457\) 0.438025i 0.0204899i 0.999948 + 0.0102450i \(0.00326113\pi\)
−0.999948 + 0.0102450i \(0.996739\pi\)
\(458\) 7.62863i 0.356462i
\(459\) 15.6537 0.730651
\(460\) 0 0
\(461\) 0.523590 0.0243860 0.0121930 0.999926i \(-0.496119\pi\)
0.0121930 + 0.999926i \(0.496119\pi\)
\(462\) 4.83832i 0.225099i
\(463\) − 29.1845i − 1.35632i −0.734916 0.678158i \(-0.762778\pi\)
0.734916 0.678158i \(-0.237222\pi\)
\(464\) −6.68035 −0.310127
\(465\) 0 0
\(466\) 0.0494483 0.00229065
\(467\) − 9.49079i − 0.439181i −0.975592 0.219591i \(-0.929528\pi\)
0.975592 0.219591i \(-0.0704722\pi\)
\(468\) − 2.86376i − 0.132378i
\(469\) 38.2895 1.76804
\(470\) 0 0
\(471\) 34.1555 1.57380
\(472\) 10.2329i 0.471006i
\(473\) − 11.1713i − 0.513657i
\(474\) 3.97334 0.182501
\(475\) 0 0
\(476\) −8.06505 −0.369661
\(477\) − 6.37137i − 0.291725i
\(478\) 14.7070i 0.672683i
\(479\) 6.12291 0.279763 0.139881 0.990168i \(-0.455328\pi\)
0.139881 + 0.990168i \(0.455328\pi\)
\(480\) 0 0
\(481\) 4.53919 0.206969
\(482\) − 14.8999i − 0.678671i
\(483\) − 32.8781i − 1.49601i
\(484\) 9.80817 0.445826
\(485\) 0 0
\(486\) −6.43907 −0.292082
\(487\) − 10.3018i − 0.466819i −0.972379 0.233409i \(-0.925012\pi\)
0.972379 0.233409i \(-0.0749882\pi\)
\(488\) − 6.29791i − 0.285093i
\(489\) 22.5103 1.01795
\(490\) 0 0
\(491\) −10.5380 −0.475572 −0.237786 0.971318i \(-0.576422\pi\)
−0.237786 + 0.971318i \(0.576422\pi\)
\(492\) 12.4341i 0.560575i
\(493\) 18.7115i 0.842726i
\(494\) −22.9204 −1.03124
\(495\) 0 0
\(496\) 3.51026 0.157615
\(497\) 18.1340i 0.813420i
\(498\) − 13.0579i − 0.585137i
\(499\) 31.2123 1.39726 0.698628 0.715485i \(-0.253794\pi\)
0.698628 + 0.715485i \(0.253794\pi\)
\(500\) 0 0
\(501\) −9.76979 −0.436482
\(502\) 6.23513i 0.278288i
\(503\) − 36.1978i − 1.61398i −0.590565 0.806990i \(-0.701095\pi\)
0.590565 0.806990i \(-0.298905\pi\)
\(504\) 1.81658 0.0809170
\(505\) 0 0
\(506\) 8.09890 0.360040
\(507\) − 11.7044i − 0.519809i
\(508\) 8.14116i 0.361205i
\(509\) 33.6092 1.48970 0.744850 0.667232i \(-0.232521\pi\)
0.744850 + 0.667232i \(0.232521\pi\)
\(510\) 0 0
\(511\) −36.5946 −1.61885
\(512\) 1.00000i 0.0441942i
\(513\) 28.2195i 1.24592i
\(514\) 16.1568 0.712644
\(515\) 0 0
\(516\) 15.7503 0.693369
\(517\) 9.47641i 0.416772i
\(518\) 2.87936i 0.126512i
\(519\) 1.90110 0.0834492
\(520\) 0 0
\(521\) 5.42082 0.237490 0.118745 0.992925i \(-0.462113\pi\)
0.118745 + 0.992925i \(0.462113\pi\)
\(522\) − 4.21461i − 0.184469i
\(523\) − 31.3545i − 1.37104i −0.728054 0.685519i \(-0.759575\pi\)
0.728054 0.685519i \(-0.240425\pi\)
\(524\) 1.46800 0.0641298
\(525\) 0 0
\(526\) 18.6537 0.813339
\(527\) − 9.83218i − 0.428297i
\(528\) − 1.68035i − 0.0731277i
\(529\) −32.0349 −1.39282
\(530\) 0 0
\(531\) −6.45589 −0.280162
\(532\) − 14.5392i − 0.630354i
\(533\) − 36.6693i − 1.58832i
\(534\) −10.0205 −0.433630
\(535\) 0 0
\(536\) −13.2979 −0.574382
\(537\) − 2.71646i − 0.117224i
\(538\) − 21.8310i − 0.941199i
\(539\) 1.40910 0.0606940
\(540\) 0 0
\(541\) −9.98440 −0.429263 −0.214631 0.976695i \(-0.568855\pi\)
−0.214631 + 0.976695i \(0.568855\pi\)
\(542\) 24.9783i 1.07291i
\(543\) 10.0000i 0.429141i
\(544\) 2.80098 0.120091
\(545\) 0 0
\(546\) 20.1171 0.860934
\(547\) − 32.9649i − 1.40948i −0.709466 0.704739i \(-0.751064\pi\)
0.709466 0.704739i \(-0.248936\pi\)
\(548\) − 6.17727i − 0.263880i
\(549\) 3.97334 0.169578
\(550\) 0 0
\(551\) −33.7321 −1.43703
\(552\) 11.4186i 0.486006i
\(553\) − 7.43293i − 0.316080i
\(554\) −0.822726 −0.0349543
\(555\) 0 0
\(556\) 13.0338 0.552758
\(557\) 21.7009i 0.919495i 0.888050 + 0.459748i \(0.152060\pi\)
−0.888050 + 0.459748i \(0.847940\pi\)
\(558\) 2.21461i 0.0937521i
\(559\) −46.4489 −1.96458
\(560\) 0 0
\(561\) −4.70662 −0.198714
\(562\) − 1.65142i − 0.0696608i
\(563\) − 1.05559i − 0.0444879i −0.999753 0.0222439i \(-0.992919\pi\)
0.999753 0.0222439i \(-0.00708105\pi\)
\(564\) −13.3607 −0.562587
\(565\) 0 0
\(566\) −15.1340 −0.636128
\(567\) − 19.3184i − 0.811298i
\(568\) − 6.29791i − 0.264255i
\(569\) 41.4196 1.73640 0.868200 0.496215i \(-0.165277\pi\)
0.868200 + 0.496215i \(0.165277\pi\)
\(570\) 0 0
\(571\) 21.4452 0.897454 0.448727 0.893669i \(-0.351878\pi\)
0.448727 + 0.893669i \(0.351878\pi\)
\(572\) 4.95547i 0.207199i
\(573\) 32.6369i 1.36342i
\(574\) 23.2606 0.970878
\(575\) 0 0
\(576\) −0.630898 −0.0262874
\(577\) − 17.2667i − 0.718823i −0.933179 0.359411i \(-0.882977\pi\)
0.933179 0.359411i \(-0.117023\pi\)
\(578\) 9.15449i 0.380777i
\(579\) −12.5343 −0.520906
\(580\) 0 0
\(581\) −24.4273 −1.01342
\(582\) 4.73820i 0.196405i
\(583\) 11.0251i 0.456611i
\(584\) 12.7093 0.525914
\(585\) 0 0
\(586\) 25.4524 1.05143
\(587\) 9.63090i 0.397510i 0.980049 + 0.198755i \(0.0636898\pi\)
−0.980049 + 0.198755i \(0.936310\pi\)
\(588\) 1.98667i 0.0819288i
\(589\) 17.7249 0.730341
\(590\) 0 0
\(591\) −38.2401 −1.57299
\(592\) − 1.00000i − 0.0410997i
\(593\) 2.78992i 0.114568i 0.998358 + 0.0572842i \(0.0182441\pi\)
−0.998358 + 0.0572842i \(0.981756\pi\)
\(594\) 6.10116 0.250334
\(595\) 0 0
\(596\) −4.40522 −0.180445
\(597\) 13.0373i 0.533583i
\(598\) − 33.6742i − 1.37704i
\(599\) −25.2606 −1.03212 −0.516060 0.856553i \(-0.672602\pi\)
−0.516060 + 0.856553i \(0.672602\pi\)
\(600\) 0 0
\(601\) 35.4040 1.44416 0.722080 0.691810i \(-0.243186\pi\)
0.722080 + 0.691810i \(0.243186\pi\)
\(602\) − 29.4641i − 1.20087i
\(603\) − 8.38962i − 0.341652i
\(604\) −2.92162 −0.118879
\(605\) 0 0
\(606\) 24.4969 0.995120
\(607\) − 38.2485i − 1.55246i −0.630452 0.776229i \(-0.717130\pi\)
0.630452 0.776229i \(-0.282870\pi\)
\(608\) 5.04945i 0.204782i
\(609\) 29.6065 1.19972
\(610\) 0 0
\(611\) 39.4017 1.59402
\(612\) 1.76713i 0.0714322i
\(613\) − 46.3279i − 1.87117i −0.353107 0.935583i \(-0.614875\pi\)
0.353107 0.935583i \(-0.385125\pi\)
\(614\) −27.8176 −1.12263
\(615\) 0 0
\(616\) −3.14342 −0.126652
\(617\) − 23.7503i − 0.956152i −0.878319 0.478076i \(-0.841334\pi\)
0.878319 0.478076i \(-0.158666\pi\)
\(618\) 16.2979i 0.655598i
\(619\) 6.53797 0.262783 0.131392 0.991331i \(-0.458055\pi\)
0.131392 + 0.991331i \(0.458055\pi\)
\(620\) 0 0
\(621\) −41.4596 −1.66372
\(622\) 0.0650468i 0.00260814i
\(623\) 18.7454i 0.751018i
\(624\) −6.98667 −0.279691
\(625\) 0 0
\(626\) −22.5464 −0.901134
\(627\) − 8.48482i − 0.338851i
\(628\) 22.1906i 0.885502i
\(629\) −2.80098 −0.111683
\(630\) 0 0
\(631\) 27.1506 1.08085 0.540424 0.841393i \(-0.318264\pi\)
0.540424 + 0.841393i \(0.318264\pi\)
\(632\) 2.58145i 0.102685i
\(633\) 33.4573i 1.32981i
\(634\) −24.4775 −0.972124
\(635\) 0 0
\(636\) −15.5441 −0.616364
\(637\) − 5.85884i − 0.232136i
\(638\) 7.29299i 0.288732i
\(639\) 3.97334 0.157183
\(640\) 0 0
\(641\) 31.9877 1.26344 0.631719 0.775197i \(-0.282350\pi\)
0.631719 + 0.775197i \(0.282350\pi\)
\(642\) − 6.21235i − 0.245182i
\(643\) 26.2784i 1.03632i 0.855284 + 0.518160i \(0.173383\pi\)
−0.855284 + 0.518160i \(0.826617\pi\)
\(644\) 21.3607 0.841729
\(645\) 0 0
\(646\) 14.1434 0.556466
\(647\) − 31.3679i − 1.23320i −0.787277 0.616599i \(-0.788510\pi\)
0.787277 0.616599i \(-0.211490\pi\)
\(648\) 6.70928i 0.263565i
\(649\) 11.1713 0.438512
\(650\) 0 0
\(651\) −15.5571 −0.609729
\(652\) 14.6248i 0.572750i
\(653\) − 25.3184i − 0.990787i −0.868669 0.495393i \(-0.835024\pi\)
0.868669 0.495393i \(-0.164976\pi\)
\(654\) −6.92162 −0.270657
\(655\) 0 0
\(656\) −8.07838 −0.315408
\(657\) 8.01825i 0.312822i
\(658\) 24.9939i 0.974362i
\(659\) 38.2072 1.48834 0.744172 0.667989i \(-0.232844\pi\)
0.744172 + 0.667989i \(0.232844\pi\)
\(660\) 0 0
\(661\) 13.7899 0.536366 0.268183 0.963368i \(-0.413577\pi\)
0.268183 + 0.963368i \(0.413577\pi\)
\(662\) − 1.91321i − 0.0743591i
\(663\) 19.5695i 0.760018i
\(664\) 8.48360 0.329227
\(665\) 0 0
\(666\) 0.630898 0.0244468
\(667\) − 49.5585i − 1.91891i
\(668\) − 6.34736i − 0.245587i
\(669\) −25.6332 −0.991035
\(670\) 0 0
\(671\) −6.87549 −0.265425
\(672\) − 4.43188i − 0.170964i
\(673\) 4.41628i 0.170235i 0.996371 + 0.0851176i \(0.0271266\pi\)
−0.996371 + 0.0851176i \(0.972873\pi\)
\(674\) −12.6286 −0.486437
\(675\) 0 0
\(676\) 7.60424 0.292471
\(677\) 38.1399i 1.46584i 0.680317 + 0.732918i \(0.261842\pi\)
−0.680317 + 0.732918i \(0.738158\pi\)
\(678\) − 5.52359i − 0.212132i
\(679\) 8.86376 0.340160
\(680\) 0 0
\(681\) −17.2579 −0.661325
\(682\) − 3.83218i − 0.146742i
\(683\) − 22.6514i − 0.866732i −0.901218 0.433366i \(-0.857326\pi\)
0.901218 0.433366i \(-0.142674\pi\)
\(684\) −3.18568 −0.121808
\(685\) 0 0
\(686\) −16.4391 −0.627647
\(687\) 11.7419i 0.447982i
\(688\) 10.2329i 0.390124i
\(689\) 45.8408 1.74640
\(690\) 0 0
\(691\) −15.8443 −0.602745 −0.301373 0.953506i \(-0.597445\pi\)
−0.301373 + 0.953506i \(0.597445\pi\)
\(692\) 1.23513i 0.0469527i
\(693\) − 1.98318i − 0.0753347i
\(694\) −8.33403 −0.316355
\(695\) 0 0
\(696\) −10.2823 −0.389750
\(697\) 22.6274i 0.857074i
\(698\) − 2.11837i − 0.0801815i
\(699\) 0.0761103 0.00287876
\(700\) 0 0
\(701\) −2.92284 −0.110394 −0.0551972 0.998475i \(-0.517579\pi\)
−0.0551972 + 0.998475i \(0.517579\pi\)
\(702\) − 25.3679i − 0.957449i
\(703\) − 5.04945i − 0.190444i
\(704\) 1.09171 0.0411453
\(705\) 0 0
\(706\) −28.4534 −1.07086
\(707\) − 45.8264i − 1.72348i
\(708\) 15.7503i 0.591933i
\(709\) 8.45136 0.317397 0.158699 0.987327i \(-0.449270\pi\)
0.158699 + 0.987327i \(0.449270\pi\)
\(710\) 0 0
\(711\) −1.62863 −0.0610784
\(712\) − 6.51026i − 0.243982i
\(713\) 26.0410i 0.975245i
\(714\) −12.4136 −0.464568
\(715\) 0 0
\(716\) 1.76487 0.0659562
\(717\) 22.6369i 0.845389i
\(718\) − 2.65368i − 0.0990346i
\(719\) 15.3763 0.573439 0.286719 0.958015i \(-0.407435\pi\)
0.286719 + 0.958015i \(0.407435\pi\)
\(720\) 0 0
\(721\) 30.4885 1.13545
\(722\) 6.49693i 0.241791i
\(723\) − 22.9337i − 0.852915i
\(724\) −6.49693 −0.241456
\(725\) 0 0
\(726\) 15.0966 0.560288
\(727\) 15.7275i 0.583302i 0.956525 + 0.291651i \(0.0942045\pi\)
−0.956525 + 0.291651i \(0.905795\pi\)
\(728\) 13.0700i 0.484405i
\(729\) −30.0388 −1.11255
\(730\) 0 0
\(731\) 28.6621 1.06011
\(732\) − 9.69368i − 0.358289i
\(733\) 15.7275i 0.580909i 0.956889 + 0.290455i \(0.0938066\pi\)
−0.956889 + 0.290455i \(0.906193\pi\)
\(734\) −31.7575 −1.17219
\(735\) 0 0
\(736\) −7.41855 −0.273451
\(737\) 14.5174i 0.534757i
\(738\) − 5.09663i − 0.187610i
\(739\) −45.0472 −1.65709 −0.828544 0.559924i \(-0.810830\pi\)
−0.828544 + 0.559924i \(0.810830\pi\)
\(740\) 0 0
\(741\) −35.2788 −1.29600
\(742\) 29.0784i 1.06750i
\(743\) − 9.31965i − 0.341905i −0.985279 0.170952i \(-0.945316\pi\)
0.985279 0.170952i \(-0.0546845\pi\)
\(744\) 5.40295 0.198082
\(745\) 0 0
\(746\) −25.1122 −0.919424
\(747\) 5.35228i 0.195830i
\(748\) − 3.05786i − 0.111806i
\(749\) −11.6214 −0.424638
\(750\) 0 0
\(751\) 11.6020 0.423362 0.211681 0.977339i \(-0.432106\pi\)
0.211681 + 0.977339i \(0.432106\pi\)
\(752\) − 8.68035i − 0.316540i
\(753\) 9.59705i 0.349736i
\(754\) 30.3234 1.10431
\(755\) 0 0
\(756\) 16.0917 0.585250
\(757\) − 38.3545i − 1.39402i −0.717062 0.697010i \(-0.754513\pi\)
0.717062 0.697010i \(-0.245487\pi\)
\(758\) − 22.5330i − 0.818437i
\(759\) 12.4657 0.452477
\(760\) 0 0
\(761\) −45.3318 −1.64328 −0.821638 0.570010i \(-0.806939\pi\)
−0.821638 + 0.570010i \(0.806939\pi\)
\(762\) 12.5308i 0.453942i
\(763\) 12.9483i 0.468759i
\(764\) −21.2039 −0.767131
\(765\) 0 0
\(766\) 12.5886 0.454846
\(767\) − 46.4489i − 1.67717i
\(768\) 1.53919i 0.0555407i
\(769\) 31.7454 1.14477 0.572384 0.819986i \(-0.306019\pi\)
0.572384 + 0.819986i \(0.306019\pi\)
\(770\) 0 0
\(771\) 24.8683 0.895610
\(772\) − 8.14342i − 0.293088i
\(773\) − 38.0482i − 1.36850i −0.729248 0.684250i \(-0.760130\pi\)
0.729248 0.684250i \(-0.239870\pi\)
\(774\) −6.45589 −0.232052
\(775\) 0 0
\(776\) −3.07838 −0.110507
\(777\) 4.43188i 0.158993i
\(778\) 0.879362i 0.0315266i
\(779\) −40.7914 −1.46150
\(780\) 0 0
\(781\) −6.87549 −0.246024
\(782\) 20.7792i 0.743064i
\(783\) − 37.3340i − 1.33421i
\(784\) −1.29072 −0.0460973
\(785\) 0 0
\(786\) 2.25953 0.0805947
\(787\) 15.1506i 0.540061i 0.962852 + 0.270031i \(0.0870338\pi\)
−0.962852 + 0.270031i \(0.912966\pi\)
\(788\) − 24.8443i − 0.885041i
\(789\) 28.7115 1.02216
\(790\) 0 0
\(791\) −10.3330 −0.367399
\(792\) 0.688756i 0.0244739i
\(793\) 28.5874i 1.01517i
\(794\) −29.6814 −1.05335
\(795\) 0 0
\(796\) −8.47027 −0.300221
\(797\) − 43.1350i − 1.52792i −0.645263 0.763960i \(-0.723252\pi\)
0.645263 0.763960i \(-0.276748\pi\)
\(798\) − 22.3786i − 0.792192i
\(799\) −24.3135 −0.860150
\(800\) 0 0
\(801\) 4.10731 0.145125
\(802\) − 13.9867i − 0.493886i
\(803\) − 13.8748i − 0.489632i
\(804\) −20.4680 −0.721851
\(805\) 0 0
\(806\) −15.9337 −0.561242
\(807\) − 33.6020i − 1.18285i
\(808\) 15.9155i 0.559905i
\(809\) −33.3874 −1.17384 −0.586918 0.809646i \(-0.699659\pi\)
−0.586918 + 0.809646i \(0.699659\pi\)
\(810\) 0 0
\(811\) 10.9216 0.383510 0.191755 0.981443i \(-0.438582\pi\)
0.191755 + 0.981443i \(0.438582\pi\)
\(812\) 19.2351i 0.675021i
\(813\) 38.4463i 1.34837i
\(814\) −1.09171 −0.0382644
\(815\) 0 0
\(816\) 4.31124 0.150924
\(817\) 51.6703i 1.80772i
\(818\) 21.9060i 0.765926i
\(819\) −8.24581 −0.288132
\(820\) 0 0
\(821\) −32.7910 −1.14441 −0.572206 0.820110i \(-0.693912\pi\)
−0.572206 + 0.820110i \(0.693912\pi\)
\(822\) − 9.50799i − 0.331629i
\(823\) 9.83218i 0.342728i 0.985208 + 0.171364i \(0.0548175\pi\)
−0.985208 + 0.171364i \(0.945183\pi\)
\(824\) −10.5886 −0.368873
\(825\) 0 0
\(826\) 29.4641 1.02519
\(827\) − 17.0228i − 0.591940i −0.955197 0.295970i \(-0.904357\pi\)
0.955197 0.295970i \(-0.0956429\pi\)
\(828\) − 4.68035i − 0.162653i
\(829\) −12.8260 −0.445467 −0.222733 0.974879i \(-0.571498\pi\)
−0.222733 + 0.974879i \(0.571498\pi\)
\(830\) 0 0
\(831\) −1.26633 −0.0439285
\(832\) − 4.53919i − 0.157368i
\(833\) 3.61530i 0.125263i
\(834\) 20.0616 0.694675
\(835\) 0 0
\(836\) 5.51253 0.190655
\(837\) 19.6176i 0.678082i
\(838\) 34.8648i 1.20438i
\(839\) 11.5018 0.397088 0.198544 0.980092i \(-0.436379\pi\)
0.198544 + 0.980092i \(0.436379\pi\)
\(840\) 0 0
\(841\) 15.6270 0.538863
\(842\) − 4.76487i − 0.164208i
\(843\) − 2.54184i − 0.0875457i
\(844\) −21.7370 −0.748218
\(845\) 0 0
\(846\) 5.47641 0.188283
\(847\) − 28.2413i − 0.970382i
\(848\) − 10.0989i − 0.346798i
\(849\) −23.2940 −0.799449
\(850\) 0 0
\(851\) 7.41855 0.254305
\(852\) − 9.69368i − 0.332100i
\(853\) 40.5946i 1.38993i 0.719042 + 0.694966i \(0.244581\pi\)
−0.719042 + 0.694966i \(0.755419\pi\)
\(854\) −18.1340 −0.620532
\(855\) 0 0
\(856\) 4.03612 0.137952
\(857\) − 38.0361i − 1.29929i −0.760238 0.649645i \(-0.774918\pi\)
0.760238 0.649645i \(-0.225082\pi\)
\(858\) 7.62741i 0.260395i
\(859\) 40.9506 1.39721 0.698607 0.715505i \(-0.253803\pi\)
0.698607 + 0.715505i \(0.253803\pi\)
\(860\) 0 0
\(861\) 35.8024 1.22014
\(862\) 2.66597i 0.0908033i
\(863\) 33.0817i 1.12611i 0.826418 + 0.563057i \(0.190375\pi\)
−0.826418 + 0.563057i \(0.809625\pi\)
\(864\) −5.58864 −0.190129
\(865\) 0 0
\(866\) −32.3074 −1.09785
\(867\) 14.0905i 0.478538i
\(868\) − 10.1073i − 0.343064i
\(869\) 2.81819 0.0956006
\(870\) 0 0
\(871\) 60.3617 2.04528
\(872\) − 4.49693i − 0.152285i
\(873\) − 1.94214i − 0.0657315i
\(874\) −37.4596 −1.26709
\(875\) 0 0
\(876\) 19.5620 0.660938
\(877\) − 25.5057i − 0.861267i −0.902527 0.430634i \(-0.858290\pi\)
0.902527 0.430634i \(-0.141710\pi\)
\(878\) 22.1568i 0.747754i
\(879\) 39.1761 1.32138
\(880\) 0 0
\(881\) 15.6514 0.527310 0.263655 0.964617i \(-0.415072\pi\)
0.263655 + 0.964617i \(0.415072\pi\)
\(882\) − 0.814315i − 0.0274194i
\(883\) 56.1071i 1.88816i 0.329723 + 0.944078i \(0.393045\pi\)
−0.329723 + 0.944078i \(0.606955\pi\)
\(884\) −12.7142 −0.427625
\(885\) 0 0
\(886\) −39.1061 −1.31379
\(887\) 47.5052i 1.59507i 0.603275 + 0.797534i \(0.293862\pi\)
−0.603275 + 0.797534i \(0.706138\pi\)
\(888\) − 1.53919i − 0.0516518i
\(889\) 23.4413 0.786197
\(890\) 0 0
\(891\) 7.32457 0.245382
\(892\) − 16.6537i − 0.557607i
\(893\) − 43.8310i − 1.46675i
\(894\) −6.78047 −0.226773
\(895\) 0 0
\(896\) 2.87936 0.0961927
\(897\) − 51.8310i − 1.73059i
\(898\) − 30.8915i − 1.03086i
\(899\) −23.4497 −0.782093
\(900\) 0 0
\(901\) −28.2868 −0.942372
\(902\) 8.81924i 0.293648i
\(903\) − 45.3509i − 1.50918i
\(904\) 3.58864 0.119356
\(905\) 0 0
\(906\) −4.49693 −0.149400
\(907\) − 15.8394i − 0.525938i −0.964804 0.262969i \(-0.915298\pi\)
0.964804 0.262969i \(-0.0847016\pi\)
\(908\) − 11.2123i − 0.372095i
\(909\) −10.0410 −0.333040
\(910\) 0 0
\(911\) 11.9539 0.396049 0.198025 0.980197i \(-0.436547\pi\)
0.198025 + 0.980197i \(0.436547\pi\)
\(912\) 7.77205i 0.257358i
\(913\) − 9.26162i − 0.306515i
\(914\) −0.438025 −0.0144886
\(915\) 0 0
\(916\) −7.62863 −0.252057
\(917\) − 4.22690i − 0.139585i
\(918\) 15.6537i 0.516649i
\(919\) 34.8781 1.15052 0.575262 0.817969i \(-0.304900\pi\)
0.575262 + 0.817969i \(0.304900\pi\)
\(920\) 0 0
\(921\) −42.8166 −1.41085
\(922\) 0.523590i 0.0172435i
\(923\) 28.5874i 0.940966i
\(924\) −4.83832 −0.159169
\(925\) 0 0
\(926\) 29.1845 0.959061
\(927\) − 6.68035i − 0.219411i
\(928\) − 6.68035i − 0.219293i
\(929\) 52.5439 1.72391 0.861955 0.506984i \(-0.169240\pi\)
0.861955 + 0.506984i \(0.169240\pi\)
\(930\) 0 0
\(931\) −6.51745 −0.213601
\(932\) 0.0494483i 0.00161973i
\(933\) 0.100119i 0.00327776i
\(934\) 9.49079 0.310548
\(935\) 0 0
\(936\) 2.86376 0.0936050
\(937\) 13.3318i 0.435530i 0.976001 + 0.217765i \(0.0698766\pi\)
−0.976001 + 0.217765i \(0.930123\pi\)
\(938\) 38.2895i 1.25020i
\(939\) −34.7031 −1.13249
\(940\) 0 0
\(941\) 14.5281 0.473603 0.236802 0.971558i \(-0.423901\pi\)
0.236802 + 0.971558i \(0.423901\pi\)
\(942\) 34.1555i 1.11285i
\(943\) − 59.9299i − 1.95158i
\(944\) −10.2329 −0.333051
\(945\) 0 0
\(946\) 11.1713 0.363211
\(947\) 24.6803i 0.802003i 0.916077 + 0.401002i \(0.131338\pi\)
−0.916077 + 0.401002i \(0.868662\pi\)
\(948\) 3.97334i 0.129048i
\(949\) −57.6898 −1.87269
\(950\) 0 0
\(951\) −37.6754 −1.22171
\(952\) − 8.06505i − 0.261390i
\(953\) − 40.5936i − 1.31495i −0.753474 0.657477i \(-0.771624\pi\)
0.753474 0.657477i \(-0.228376\pi\)
\(954\) 6.37137 0.206281
\(955\) 0 0
\(956\) −14.7070 −0.475659
\(957\) 11.2253i 0.362862i
\(958\) 6.12291i 0.197822i
\(959\) −17.7866 −0.574360
\(960\) 0 0
\(961\) −18.6781 −0.602519
\(962\) 4.53919i 0.146349i
\(963\) 2.54638i 0.0820558i
\(964\) 14.8999 0.479893
\(965\) 0 0
\(966\) 32.8781 1.05784
\(967\) 1.06400i 0.0342160i 0.999854 + 0.0171080i \(0.00544591\pi\)
−0.999854 + 0.0171080i \(0.994554\pi\)
\(968\) 9.80817i 0.315247i
\(969\) 21.7694 0.699334
\(970\) 0 0
\(971\) 11.9333 0.382959 0.191480 0.981497i \(-0.438671\pi\)
0.191480 + 0.981497i \(0.438671\pi\)
\(972\) − 6.43907i − 0.206533i
\(973\) − 37.5292i − 1.20313i
\(974\) 10.3018 0.330091
\(975\) 0 0
\(976\) 6.29791 0.201591
\(977\) 19.1867i 0.613838i 0.951736 + 0.306919i \(0.0992980\pi\)
−0.951736 + 0.306919i \(0.900702\pi\)
\(978\) 22.5103i 0.719799i
\(979\) −7.10731 −0.227151
\(980\) 0 0
\(981\) 2.83710 0.0905817
\(982\) − 10.5380i − 0.336280i
\(983\) − 22.9171i − 0.730942i −0.930823 0.365471i \(-0.880908\pi\)
0.930823 0.365471i \(-0.119092\pi\)
\(984\) −12.4341 −0.396386
\(985\) 0 0
\(986\) −18.7115 −0.595897
\(987\) 38.4703i 1.22452i
\(988\) − 22.9204i − 0.729195i
\(989\) −75.9130 −2.41389
\(990\) 0 0
\(991\) −52.2772 −1.66064 −0.830320 0.557287i \(-0.811842\pi\)
−0.830320 + 0.557287i \(0.811842\pi\)
\(992\) 3.51026i 0.111451i
\(993\) − 2.94479i − 0.0934502i
\(994\) −18.1340 −0.575175
\(995\) 0 0
\(996\) 13.0579 0.413754
\(997\) − 48.8892i − 1.54834i −0.632980 0.774168i \(-0.718168\pi\)
0.632980 0.774168i \(-0.281832\pi\)
\(998\) 31.2123i 0.988010i
\(999\) 5.58864 0.176817
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 1850.2.b.n.149.5 6
5.2 odd 4 1850.2.a.ba.1.2 3
5.3 odd 4 1850.2.a.bb.1.2 yes 3
5.4 even 2 inner 1850.2.b.n.149.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1850.2.a.ba.1.2 3 5.2 odd 4
1850.2.a.bb.1.2 yes 3 5.3 odd 4
1850.2.b.n.149.2 6 5.4 even 2 inner
1850.2.b.n.149.5 6 1.1 even 1 trivial