Properties

Label 950.2.b.h.799.4
Level $950$
Weight $2$
Character 950.799
Analytic conductor $7.586$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.63107136.1
Defining polynomial: \(x^{6} + 13 x^{4} + 42 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 799.4
Root \(-2.25342i\) of defining polynomial
Character \(\chi\) \(=\) 950.799
Dual form 950.2.b.h.799.3

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -3.25342i q^{3} -1.00000 q^{4} +3.25342 q^{6} -0.0778929i q^{7} -1.00000i q^{8} -7.58473 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -3.25342i q^{3} -1.00000 q^{4} +3.25342 q^{6} -0.0778929i q^{7} -1.00000i q^{8} -7.58473 q^{9} -4.50684 q^{11} +3.25342i q^{12} +5.33131i q^{13} +0.0778929 q^{14} +1.00000 q^{16} +7.33131i q^{17} -7.58473i q^{18} -1.00000 q^{19} -0.253418 q^{21} -4.50684i q^{22} -3.40920i q^{23} -3.25342 q^{24} -5.33131 q^{26} +14.9160i q^{27} +0.0778929i q^{28} +1.33131 q^{29} -2.50684 q^{31} +1.00000i q^{32} +14.6626i q^{33} -7.33131 q^{34} +7.58473 q^{36} -5.50684i q^{37} -1.00000i q^{38} +17.3450 q^{39} -0.253418i q^{42} -0.506836i q^{43} +4.50684 q^{44} +3.40920 q^{46} -5.66262i q^{47} -3.25342i q^{48} +6.99393 q^{49} +23.8518 q^{51} -5.33131i q^{52} +12.9358i q^{53} -14.9160 q^{54} -0.0778929 q^{56} +3.25342i q^{57} +1.33131i q^{58} -7.56499 q^{59} -2.15579 q^{61} -2.50684i q^{62} +0.590796i q^{63} -1.00000 q^{64} -14.6626 q^{66} -4.58473i q^{67} -7.33131i q^{68} -11.0916 q^{69} -10.8579 q^{71} +7.58473i q^{72} +5.09763i q^{73} +5.50684 q^{74} +1.00000 q^{76} +0.351050i q^{77} +17.3450i q^{78} -17.0137 q^{79} +25.7739 q^{81} +13.1695i q^{83} +0.253418 q^{84} +0.506836 q^{86} -4.33131i q^{87} +4.50684i q^{88} -15.0137 q^{89} +0.415271 q^{91} +3.40920i q^{92} +8.15579i q^{93} +5.66262 q^{94} +3.25342 q^{96} +7.67629i q^{97} +6.99393i q^{98} +34.1831 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{4} + 4q^{6} - 10q^{9} + O(q^{10}) \) \( 6q - 6q^{4} + 4q^{6} - 10q^{9} + 4q^{11} - 4q^{14} + 6q^{16} - 6q^{19} + 14q^{21} - 4q^{24} - 12q^{26} - 12q^{29} + 16q^{31} - 24q^{34} + 10q^{36} + 22q^{39} - 4q^{44} - 4q^{46} - 18q^{49} + 30q^{51} - 34q^{54} + 4q^{56} - 12q^{59} - 4q^{61} - 6q^{64} - 48q^{66} - 12q^{71} + 2q^{74} + 6q^{76} - 40q^{79} + 46q^{81} - 14q^{84} - 28q^{86} - 28q^{89} + 38q^{91} - 6q^{94} + 4q^{96} + 72q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) − 3.25342i − 1.87836i −0.343423 0.939181i \(-0.611586\pi\)
0.343423 0.939181i \(-0.388414\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 3.25342 1.32820
\(7\) − 0.0778929i − 0.0294407i −0.999892 0.0147204i \(-0.995314\pi\)
0.999892 0.0147204i \(-0.00468581\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −7.58473 −2.52824
\(10\) 0 0
\(11\) −4.50684 −1.35886 −0.679431 0.733739i \(-0.737773\pi\)
−0.679431 + 0.733739i \(0.737773\pi\)
\(12\) 3.25342i 0.939181i
\(13\) 5.33131i 1.47864i 0.673354 + 0.739320i \(0.264853\pi\)
−0.673354 + 0.739320i \(0.735147\pi\)
\(14\) 0.0778929 0.0208177
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.33131i 1.77810i 0.457806 + 0.889052i \(0.348635\pi\)
−0.457806 + 0.889052i \(0.651365\pi\)
\(18\) − 7.58473i − 1.78774i
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −0.253418 −0.0553003
\(22\) − 4.50684i − 0.960861i
\(23\) − 3.40920i − 0.710868i −0.934701 0.355434i \(-0.884333\pi\)
0.934701 0.355434i \(-0.115667\pi\)
\(24\) −3.25342 −0.664101
\(25\) 0 0
\(26\) −5.33131 −1.04556
\(27\) 14.9160i 2.87059i
\(28\) 0.0778929i 0.0147204i
\(29\) 1.33131 0.247218 0.123609 0.992331i \(-0.460553\pi\)
0.123609 + 0.992331i \(0.460553\pi\)
\(30\) 0 0
\(31\) −2.50684 −0.450241 −0.225121 0.974331i \(-0.572278\pi\)
−0.225121 + 0.974331i \(0.572278\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 14.6626i 2.55243i
\(34\) −7.33131 −1.25731
\(35\) 0 0
\(36\) 7.58473 1.26412
\(37\) − 5.50684i − 0.905318i −0.891684 0.452659i \(-0.850475\pi\)
0.891684 0.452659i \(-0.149525\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) 17.3450 2.77742
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) − 0.253418i − 0.0391033i
\(43\) − 0.506836i − 0.0772918i −0.999253 0.0386459i \(-0.987696\pi\)
0.999253 0.0386459i \(-0.0123044\pi\)
\(44\) 4.50684 0.679431
\(45\) 0 0
\(46\) 3.40920 0.502660
\(47\) − 5.66262i − 0.825978i −0.910736 0.412989i \(-0.864485\pi\)
0.910736 0.412989i \(-0.135515\pi\)
\(48\) − 3.25342i − 0.469590i
\(49\) 6.99393 0.999133
\(50\) 0 0
\(51\) 23.8518 3.33992
\(52\) − 5.33131i − 0.739320i
\(53\) 12.9358i 1.77687i 0.459006 + 0.888433i \(0.348206\pi\)
−0.459006 + 0.888433i \(0.651794\pi\)
\(54\) −14.9160 −2.02982
\(55\) 0 0
\(56\) −0.0778929 −0.0104089
\(57\) 3.25342i 0.430926i
\(58\) 1.33131i 0.174810i
\(59\) −7.56499 −0.984878 −0.492439 0.870347i \(-0.663894\pi\)
−0.492439 + 0.870347i \(0.663894\pi\)
\(60\) 0 0
\(61\) −2.15579 −0.276020 −0.138010 0.990431i \(-0.544071\pi\)
−0.138010 + 0.990431i \(0.544071\pi\)
\(62\) − 2.50684i − 0.318369i
\(63\) 0.590796i 0.0744333i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −14.6626 −1.80484
\(67\) − 4.58473i − 0.560114i −0.959983 0.280057i \(-0.909647\pi\)
0.959983 0.280057i \(-0.0903533\pi\)
\(68\) − 7.33131i − 0.889052i
\(69\) −11.0916 −1.33527
\(70\) 0 0
\(71\) −10.8579 −1.28859 −0.644297 0.764775i \(-0.722850\pi\)
−0.644297 + 0.764775i \(0.722850\pi\)
\(72\) 7.58473i 0.893869i
\(73\) 5.09763i 0.596633i 0.954467 + 0.298316i \(0.0964250\pi\)
−0.954467 + 0.298316i \(0.903575\pi\)
\(74\) 5.50684 0.640157
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0.351050i 0.0400059i
\(78\) 17.3450i 1.96393i
\(79\) −17.0137 −1.91419 −0.957094 0.289778i \(-0.906418\pi\)
−0.957094 + 0.289778i \(0.906418\pi\)
\(80\) 0 0
\(81\) 25.7739 2.86377
\(82\) 0 0
\(83\) 13.1695i 1.44554i 0.691091 + 0.722768i \(0.257130\pi\)
−0.691091 + 0.722768i \(0.742870\pi\)
\(84\) 0.253418 0.0276502
\(85\) 0 0
\(86\) 0.506836 0.0546535
\(87\) − 4.33131i − 0.464365i
\(88\) 4.50684i 0.480430i
\(89\) −15.0137 −1.59145 −0.795723 0.605661i \(-0.792909\pi\)
−0.795723 + 0.605661i \(0.792909\pi\)
\(90\) 0 0
\(91\) 0.415271 0.0435322
\(92\) 3.40920i 0.355434i
\(93\) 8.15579i 0.845716i
\(94\) 5.66262 0.584055
\(95\) 0 0
\(96\) 3.25342 0.332051
\(97\) 7.67629i 0.779410i 0.920940 + 0.389705i \(0.127423\pi\)
−0.920940 + 0.389705i \(0.872577\pi\)
\(98\) 6.99393i 0.706494i
\(99\) 34.1831 3.43553
\(100\) 0 0
\(101\) −4.15579 −0.413516 −0.206758 0.978392i \(-0.566291\pi\)
−0.206758 + 0.978392i \(0.566291\pi\)
\(102\) 23.8518i 2.36168i
\(103\) − 2.35105i − 0.231656i −0.993269 0.115828i \(-0.963048\pi\)
0.993269 0.115828i \(-0.0369521\pi\)
\(104\) 5.33131 0.522778
\(105\) 0 0
\(106\) −12.9358 −1.25643
\(107\) − 14.0334i − 1.35666i −0.734757 0.678331i \(-0.762704\pi\)
0.734757 0.678331i \(-0.237296\pi\)
\(108\) − 14.9160i − 1.43530i
\(109\) 0.0778929 0.00746078 0.00373039 0.999993i \(-0.498813\pi\)
0.00373039 + 0.999993i \(0.498813\pi\)
\(110\) 0 0
\(111\) −17.9160 −1.70052
\(112\) − 0.0778929i − 0.00736018i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) −3.25342 −0.304711
\(115\) 0 0
\(116\) −1.33131 −0.123609
\(117\) − 40.4365i − 3.73836i
\(118\) − 7.56499i − 0.696414i
\(119\) 0.571057 0.0523487
\(120\) 0 0
\(121\) 9.31157 0.846506
\(122\) − 2.15579i − 0.195176i
\(123\) 0 0
\(124\) 2.50684 0.225121
\(125\) 0 0
\(126\) −0.590796 −0.0526323
\(127\) 17.8321i 1.58234i 0.611596 + 0.791171i \(0.290528\pi\)
−0.611596 + 0.791171i \(0.709472\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −1.64895 −0.145182
\(130\) 0 0
\(131\) −1.49316 −0.130458 −0.0652292 0.997870i \(-0.520778\pi\)
−0.0652292 + 0.997870i \(0.520778\pi\)
\(132\) − 14.6626i − 1.27622i
\(133\) 0.0778929i 0.00675417i
\(134\) 4.58473 0.396060
\(135\) 0 0
\(136\) 7.33131 0.628655
\(137\) − 8.42894i − 0.720133i −0.932927 0.360067i \(-0.882754\pi\)
0.932927 0.360067i \(-0.117246\pi\)
\(138\) − 11.0916i − 0.944177i
\(139\) −8.81841 −0.747968 −0.373984 0.927435i \(-0.622008\pi\)
−0.373984 + 0.927435i \(0.622008\pi\)
\(140\) 0 0
\(141\) −18.4229 −1.55149
\(142\) − 10.8579i − 0.911174i
\(143\) − 24.0273i − 2.00927i
\(144\) −7.58473 −0.632061
\(145\) 0 0
\(146\) −5.09763 −0.421883
\(147\) − 22.7542i − 1.87673i
\(148\) 5.50684i 0.452659i
\(149\) −17.6763 −1.44810 −0.724049 0.689748i \(-0.757721\pi\)
−0.724049 + 0.689748i \(0.757721\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) − 55.6060i − 4.49548i
\(154\) −0.351050 −0.0282884
\(155\) 0 0
\(156\) −17.3450 −1.38871
\(157\) 0.506836i 0.0404499i 0.999795 + 0.0202250i \(0.00643824\pi\)
−0.999795 + 0.0202250i \(0.993562\pi\)
\(158\) − 17.0137i − 1.35354i
\(159\) 42.0855 3.33760
\(160\) 0 0
\(161\) −0.265553 −0.0209285
\(162\) 25.7739i 2.02499i
\(163\) − 0.830542i − 0.0650531i −0.999471 0.0325265i \(-0.989645\pi\)
0.999471 0.0325265i \(-0.0103553\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −13.1695 −1.02215
\(167\) 16.5068i 1.27734i 0.769482 + 0.638669i \(0.220515\pi\)
−0.769482 + 0.638669i \(0.779485\pi\)
\(168\) 0.253418i 0.0195516i
\(169\) −15.4229 −1.18638
\(170\) 0 0
\(171\) 7.58473 0.580019
\(172\) 0.506836i 0.0386459i
\(173\) − 18.8321i − 1.43178i −0.698215 0.715888i \(-0.746022\pi\)
0.698215 0.715888i \(-0.253978\pi\)
\(174\) 4.33131 0.328356
\(175\) 0 0
\(176\) −4.50684 −0.339716
\(177\) 24.6121i 1.84996i
\(178\) − 15.0137i − 1.12532i
\(179\) 7.15579 0.534849 0.267424 0.963579i \(-0.413827\pi\)
0.267424 + 0.963579i \(0.413827\pi\)
\(180\) 0 0
\(181\) 12.5205 0.930642 0.465321 0.885142i \(-0.345939\pi\)
0.465321 + 0.885142i \(0.345939\pi\)
\(182\) 0.415271i 0.0307819i
\(183\) 7.01367i 0.518466i
\(184\) −3.40920 −0.251330
\(185\) 0 0
\(186\) −8.15579 −0.598011
\(187\) − 33.0410i − 2.41620i
\(188\) 5.66262i 0.412989i
\(189\) 1.16185 0.0845124
\(190\) 0 0
\(191\) −4.90237 −0.354723 −0.177361 0.984146i \(-0.556756\pi\)
−0.177361 + 0.984146i \(0.556756\pi\)
\(192\) 3.25342i 0.234795i
\(193\) − 18.1558i − 1.30688i −0.756977 0.653441i \(-0.773325\pi\)
0.756977 0.653441i \(-0.226675\pi\)
\(194\) −7.67629 −0.551126
\(195\) 0 0
\(196\) −6.99393 −0.499567
\(197\) − 2.98633i − 0.212767i −0.994325 0.106384i \(-0.966073\pi\)
0.994325 0.106384i \(-0.0339271\pi\)
\(198\) 34.1831i 2.42929i
\(199\) 3.06422 0.217217 0.108608 0.994085i \(-0.465361\pi\)
0.108608 + 0.994085i \(0.465361\pi\)
\(200\) 0 0
\(201\) −14.9160 −1.05210
\(202\) − 4.15579i − 0.292400i
\(203\) − 0.103700i − 0.00727829i
\(204\) −23.8518 −1.66996
\(205\) 0 0
\(206\) 2.35105 0.163805
\(207\) 25.8579i 1.79725i
\(208\) 5.33131i 0.369660i
\(209\) 4.50684 0.311744
\(210\) 0 0
\(211\) 19.2534 1.32546 0.662730 0.748858i \(-0.269398\pi\)
0.662730 + 0.748858i \(0.269398\pi\)
\(212\) − 12.9358i − 0.888433i
\(213\) 35.3252i 2.42045i
\(214\) 14.0334 0.959304
\(215\) 0 0
\(216\) 14.9160 1.01491
\(217\) 0.195265i 0.0132554i
\(218\) 0.0778929i 0.00527557i
\(219\) 16.5847 1.12069
\(220\) 0 0
\(221\) −39.0855 −2.62918
\(222\) − 17.9160i − 1.20245i
\(223\) 14.5068i 0.971450i 0.874112 + 0.485725i \(0.161444\pi\)
−0.874112 + 0.485725i \(0.838556\pi\)
\(224\) 0.0778929 0.00520444
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) − 21.5984i − 1.43354i −0.697312 0.716768i \(-0.745621\pi\)
0.697312 0.716768i \(-0.254379\pi\)
\(228\) − 3.25342i − 0.215463i
\(229\) 19.0137 1.25646 0.628229 0.778028i \(-0.283780\pi\)
0.628229 + 0.778028i \(0.283780\pi\)
\(230\) 0 0
\(231\) 1.14211 0.0751456
\(232\) − 1.33131i − 0.0874048i
\(233\) 6.01367i 0.393969i 0.980407 + 0.196984i \(0.0631148\pi\)
−0.980407 + 0.196984i \(0.936885\pi\)
\(234\) 40.4365 2.64342
\(235\) 0 0
\(236\) 7.56499 0.492439
\(237\) 55.3526i 3.59554i
\(238\) 0.571057i 0.0370161i
\(239\) −15.4092 −0.996739 −0.498369 0.866965i \(-0.666068\pi\)
−0.498369 + 0.866965i \(0.666068\pi\)
\(240\) 0 0
\(241\) −4.81841 −0.310381 −0.155190 0.987885i \(-0.549599\pi\)
−0.155190 + 0.987885i \(0.549599\pi\)
\(242\) 9.31157i 0.598570i
\(243\) − 39.1052i − 2.50860i
\(244\) 2.15579 0.138010
\(245\) 0 0
\(246\) 0 0
\(247\) − 5.33131i − 0.339223i
\(248\) 2.50684i 0.159184i
\(249\) 42.8458 2.71524
\(250\) 0 0
\(251\) 1.52051 0.0959736 0.0479868 0.998848i \(-0.484719\pi\)
0.0479868 + 0.998848i \(0.484719\pi\)
\(252\) − 0.590796i − 0.0372167i
\(253\) 15.3647i 0.965972i
\(254\) −17.8321 −1.11888
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 16.5068i 1.02967i 0.857290 + 0.514834i \(0.172146\pi\)
−0.857290 + 0.514834i \(0.827854\pi\)
\(258\) − 1.64895i − 0.102659i
\(259\) −0.428943 −0.0266532
\(260\) 0 0
\(261\) −10.0976 −0.625028
\(262\) − 1.49316i − 0.0922480i
\(263\) 18.0273i 1.11161i 0.831311 + 0.555807i \(0.187591\pi\)
−0.831311 + 0.555807i \(0.812409\pi\)
\(264\) 14.6626 0.902422
\(265\) 0 0
\(266\) −0.0778929 −0.00477592
\(267\) 48.8458i 2.98931i
\(268\) 4.58473i 0.280057i
\(269\) −20.2089 −1.23216 −0.616080 0.787683i \(-0.711280\pi\)
−0.616080 + 0.787683i \(0.711280\pi\)
\(270\) 0 0
\(271\) 6.08396 0.369574 0.184787 0.982779i \(-0.440840\pi\)
0.184787 + 0.982779i \(0.440840\pi\)
\(272\) 7.33131i 0.444526i
\(273\) − 1.35105i − 0.0817693i
\(274\) 8.42894 0.509211
\(275\) 0 0
\(276\) 11.0916 0.667634
\(277\) − 31.3647i − 1.88452i −0.334877 0.942262i \(-0.608695\pi\)
0.334877 0.942262i \(-0.391305\pi\)
\(278\) − 8.81841i − 0.528893i
\(279\) 19.0137 1.13832
\(280\) 0 0
\(281\) 11.3252 0.675607 0.337804 0.941217i \(-0.390316\pi\)
0.337804 + 0.941217i \(0.390316\pi\)
\(282\) − 18.4229i − 1.09707i
\(283\) − 26.1437i − 1.55408i −0.629452 0.777039i \(-0.716721\pi\)
0.629452 0.777039i \(-0.283279\pi\)
\(284\) 10.8579 0.644297
\(285\) 0 0
\(286\) 24.0273 1.42077
\(287\) 0 0
\(288\) − 7.58473i − 0.446934i
\(289\) −36.7481 −2.16165
\(290\) 0 0
\(291\) 24.9742 1.46401
\(292\) − 5.09763i − 0.298316i
\(293\) − 1.33131i − 0.0777760i −0.999244 0.0388880i \(-0.987618\pi\)
0.999244 0.0388880i \(-0.0123816\pi\)
\(294\) 22.7542 1.32705
\(295\) 0 0
\(296\) −5.50684 −0.320078
\(297\) − 67.2241i − 3.90074i
\(298\) − 17.6763i − 1.02396i
\(299\) 18.1755 1.05112
\(300\) 0 0
\(301\) −0.0394789 −0.00227553
\(302\) 20.0000i 1.15087i
\(303\) 13.5205i 0.776733i
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 55.6060 3.17878
\(307\) − 2.84421i − 0.162328i −0.996701 0.0811639i \(-0.974136\pi\)
0.996701 0.0811639i \(-0.0258637\pi\)
\(308\) − 0.351050i − 0.0200030i
\(309\) −7.64895 −0.435134
\(310\) 0 0
\(311\) −16.3895 −0.929361 −0.464681 0.885478i \(-0.653831\pi\)
−0.464681 + 0.885478i \(0.653831\pi\)
\(312\) − 17.3450i − 0.981966i
\(313\) 21.0471i 1.18965i 0.803855 + 0.594826i \(0.202779\pi\)
−0.803855 + 0.594826i \(0.797221\pi\)
\(314\) −0.506836 −0.0286024
\(315\) 0 0
\(316\) 17.0137 0.957094
\(317\) − 30.8902i − 1.73497i −0.497465 0.867484i \(-0.665736\pi\)
0.497465 0.867484i \(-0.334264\pi\)
\(318\) 42.0855i 2.36004i
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) −45.6566 −2.54830
\(322\) − 0.265553i − 0.0147987i
\(323\) − 7.33131i − 0.407925i
\(324\) −25.7739 −1.43188
\(325\) 0 0
\(326\) 0.830542 0.0459995
\(327\) − 0.253418i − 0.0140140i
\(328\) 0 0
\(329\) −0.441078 −0.0243174
\(330\) 0 0
\(331\) 28.3845 1.56015 0.780076 0.625685i \(-0.215181\pi\)
0.780076 + 0.625685i \(0.215181\pi\)
\(332\) − 13.1695i − 0.722768i
\(333\) 41.7679i 2.28886i
\(334\) −16.5068 −0.903214
\(335\) 0 0
\(336\) −0.253418 −0.0138251
\(337\) − 17.8716i − 0.973526i −0.873534 0.486763i \(-0.838178\pi\)
0.873534 0.486763i \(-0.161822\pi\)
\(338\) − 15.4229i − 0.838894i
\(339\) 19.5205 1.06021
\(340\) 0 0
\(341\) 11.2979 0.611816
\(342\) 7.58473i 0.410135i
\(343\) − 1.09003i − 0.0588560i
\(344\) −0.506836 −0.0273268
\(345\) 0 0
\(346\) 18.8321 1.01242
\(347\) 33.0410i 1.77373i 0.462024 + 0.886867i \(0.347123\pi\)
−0.462024 + 0.886867i \(0.652877\pi\)
\(348\) 4.33131i 0.232183i
\(349\) 19.7158 1.05536 0.527681 0.849443i \(-0.323062\pi\)
0.527681 + 0.849443i \(0.323062\pi\)
\(350\) 0 0
\(351\) −79.5220 −4.24457
\(352\) − 4.50684i − 0.240215i
\(353\) 5.90997i 0.314556i 0.987554 + 0.157278i \(0.0502719\pi\)
−0.987554 + 0.157278i \(0.949728\pi\)
\(354\) −24.6121 −1.30812
\(355\) 0 0
\(356\) 15.0137 0.795723
\(357\) − 1.85789i − 0.0983298i
\(358\) 7.15579i 0.378195i
\(359\) 7.00760 0.369847 0.184924 0.982753i \(-0.440796\pi\)
0.184924 + 0.982753i \(0.440796\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 12.5205i 0.658063i
\(363\) − 30.2944i − 1.59005i
\(364\) −0.415271 −0.0217661
\(365\) 0 0
\(366\) −7.01367 −0.366611
\(367\) 17.7158i 0.924756i 0.886683 + 0.462378i \(0.153004\pi\)
−0.886683 + 0.462378i \(0.846996\pi\)
\(368\) − 3.40920i − 0.177717i
\(369\) 0 0
\(370\) 0 0
\(371\) 1.00760 0.0523122
\(372\) − 8.15579i − 0.422858i
\(373\) − 15.4487i − 0.799902i −0.916536 0.399951i \(-0.869027\pi\)
0.916536 0.399951i \(-0.130973\pi\)
\(374\) 33.0410 1.70851
\(375\) 0 0
\(376\) −5.66262 −0.292027
\(377\) 7.09763i 0.365547i
\(378\) 1.16185i 0.0597593i
\(379\) −34.4563 −1.76990 −0.884950 0.465685i \(-0.845808\pi\)
−0.884950 + 0.465685i \(0.845808\pi\)
\(380\) 0 0
\(381\) 58.0152 2.97221
\(382\) − 4.90237i − 0.250827i
\(383\) 12.0273i 0.614569i 0.951618 + 0.307284i \(0.0994203\pi\)
−0.951618 + 0.307284i \(0.900580\pi\)
\(384\) −3.25342 −0.166025
\(385\) 0 0
\(386\) 18.1558 0.924105
\(387\) 3.84421i 0.195412i
\(388\) − 7.67629i − 0.389705i
\(389\) 15.3647 0.779022 0.389511 0.921022i \(-0.372644\pi\)
0.389511 + 0.921022i \(0.372644\pi\)
\(390\) 0 0
\(391\) 24.9939 1.26400
\(392\) − 6.99393i − 0.353247i
\(393\) 4.85789i 0.245048i
\(394\) 2.98633 0.150449
\(395\) 0 0
\(396\) −34.1831 −1.71777
\(397\) 1.32524i 0.0665121i 0.999447 + 0.0332560i \(0.0105877\pi\)
−0.999447 + 0.0332560i \(0.989412\pi\)
\(398\) 3.06422i 0.153596i
\(399\) 0.253418 0.0126868
\(400\) 0 0
\(401\) −6.46736 −0.322964 −0.161482 0.986876i \(-0.551627\pi\)
−0.161482 + 0.986876i \(0.551627\pi\)
\(402\) − 14.9160i − 0.743944i
\(403\) − 13.3647i − 0.665744i
\(404\) 4.15579 0.206758
\(405\) 0 0
\(406\) 0.103700 0.00514653
\(407\) 24.8184i 1.23020i
\(408\) − 23.8518i − 1.18084i
\(409\) 1.36472 0.0674812 0.0337406 0.999431i \(-0.489258\pi\)
0.0337406 + 0.999431i \(0.489258\pi\)
\(410\) 0 0
\(411\) −27.4229 −1.35267
\(412\) 2.35105i 0.115828i
\(413\) 0.589259i 0.0289955i
\(414\) −25.8579 −1.27085
\(415\) 0 0
\(416\) −5.33131 −0.261389
\(417\) 28.6900i 1.40495i
\(418\) 4.50684i 0.220437i
\(419\) −16.6231 −0.812094 −0.406047 0.913852i \(-0.633093\pi\)
−0.406047 + 0.913852i \(0.633093\pi\)
\(420\) 0 0
\(421\) −31.1128 −1.51635 −0.758174 0.652053i \(-0.773908\pi\)
−0.758174 + 0.652053i \(0.773908\pi\)
\(422\) 19.2534i 0.937242i
\(423\) 42.9495i 2.08827i
\(424\) 12.9358 0.628217
\(425\) 0 0
\(426\) −35.3252 −1.71151
\(427\) 0.167920i 0.00812623i
\(428\) 14.0334i 0.678331i
\(429\) −78.1710 −3.77413
\(430\) 0 0
\(431\) −10.1831 −0.490504 −0.245252 0.969459i \(-0.578871\pi\)
−0.245252 + 0.969459i \(0.578871\pi\)
\(432\) 14.9160i 0.717648i
\(433\) − 22.1953i − 1.06664i −0.845915 0.533318i \(-0.820945\pi\)
0.845915 0.533318i \(-0.179055\pi\)
\(434\) −0.195265 −0.00937300
\(435\) 0 0
\(436\) −0.0778929 −0.00373039
\(437\) 3.40920i 0.163084i
\(438\) 16.5847i 0.792449i
\(439\) 9.32524 0.445070 0.222535 0.974925i \(-0.428567\pi\)
0.222535 + 0.974925i \(0.428567\pi\)
\(440\) 0 0
\(441\) −53.0471 −2.52605
\(442\) − 39.0855i − 1.85911i
\(443\) 13.9879i 0.664584i 0.943177 + 0.332292i \(0.107822\pi\)
−0.943177 + 0.332292i \(0.892178\pi\)
\(444\) 17.9160 0.850258
\(445\) 0 0
\(446\) −14.5068 −0.686919
\(447\) 57.5084i 2.72005i
\(448\) 0.0778929i 0.00368009i
\(449\) −13.4932 −0.636782 −0.318391 0.947960i \(-0.603142\pi\)
−0.318391 + 0.947960i \(0.603142\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 6.00000i − 0.282216i
\(453\) − 65.0684i − 3.05718i
\(454\) 21.5984 1.01366
\(455\) 0 0
\(456\) 3.25342 0.152355
\(457\) 9.68236i 0.452922i 0.974020 + 0.226461i \(0.0727155\pi\)
−0.974020 + 0.226461i \(0.927284\pi\)
\(458\) 19.0137i 0.888451i
\(459\) −109.354 −5.10421
\(460\) 0 0
\(461\) 8.66262 0.403459 0.201729 0.979441i \(-0.435344\pi\)
0.201729 + 0.979441i \(0.435344\pi\)
\(462\) 1.14211i 0.0531359i
\(463\) − 28.0015i − 1.30134i −0.759360 0.650671i \(-0.774488\pi\)
0.759360 0.650671i \(-0.225512\pi\)
\(464\) 1.33131 0.0618046
\(465\) 0 0
\(466\) −6.01367 −0.278578
\(467\) 16.9742i 0.785472i 0.919651 + 0.392736i \(0.128471\pi\)
−0.919651 + 0.392736i \(0.871529\pi\)
\(468\) 40.4365i 1.86918i
\(469\) −0.357118 −0.0164902
\(470\) 0 0
\(471\) 1.64895 0.0759796
\(472\) 7.56499i 0.348207i
\(473\) 2.28423i 0.105029i
\(474\) −55.3526 −2.54243
\(475\) 0 0
\(476\) −0.571057 −0.0261743
\(477\) − 98.1144i − 4.49235i
\(478\) − 15.4092i − 0.704801i
\(479\) −10.0532 −0.459340 −0.229670 0.973269i \(-0.573765\pi\)
−0.229670 + 0.973269i \(0.573765\pi\)
\(480\) 0 0
\(481\) 29.3587 1.33864
\(482\) − 4.81841i − 0.219472i
\(483\) 0.863954i 0.0393113i
\(484\) −9.31157 −0.423253
\(485\) 0 0
\(486\) 39.1052 1.77385
\(487\) 19.2089i 0.870440i 0.900324 + 0.435220i \(0.143329\pi\)
−0.900324 + 0.435220i \(0.856671\pi\)
\(488\) 2.15579i 0.0975878i
\(489\) −2.70210 −0.122193
\(490\) 0 0
\(491\) 5.32524 0.240325 0.120162 0.992754i \(-0.461658\pi\)
0.120162 + 0.992754i \(0.461658\pi\)
\(492\) 0 0
\(493\) 9.76025i 0.439580i
\(494\) 5.33131 0.239867
\(495\) 0 0
\(496\) −2.50684 −0.112560
\(497\) 0.845752i 0.0379372i
\(498\) 42.8458i 1.91996i
\(499\) 11.9605 0.535426 0.267713 0.963499i \(-0.413732\pi\)
0.267713 + 0.963499i \(0.413732\pi\)
\(500\) 0 0
\(501\) 53.7036 2.39930
\(502\) 1.52051i 0.0678636i
\(503\) 19.3313i 0.861941i 0.902366 + 0.430970i \(0.141829\pi\)
−0.902366 + 0.430970i \(0.858171\pi\)
\(504\) 0.590796 0.0263162
\(505\) 0 0
\(506\) −15.3647 −0.683045
\(507\) 50.1771i 2.22844i
\(508\) − 17.8321i − 0.791171i
\(509\) 36.1573 1.60265 0.801323 0.598232i \(-0.204130\pi\)
0.801323 + 0.598232i \(0.204130\pi\)
\(510\) 0 0
\(511\) 0.397069 0.0175653
\(512\) 1.00000i 0.0441942i
\(513\) − 14.9160i − 0.658559i
\(514\) −16.5068 −0.728085
\(515\) 0 0
\(516\) 1.64895 0.0725910
\(517\) 25.5205i 1.12239i
\(518\) − 0.428943i − 0.0188467i
\(519\) −61.2686 −2.68939
\(520\) 0 0
\(521\) −31.5205 −1.38094 −0.690469 0.723362i \(-0.742596\pi\)
−0.690469 + 0.723362i \(0.742596\pi\)
\(522\) − 10.0976i − 0.441961i
\(523\) − 5.59840i − 0.244801i −0.992481 0.122400i \(-0.960941\pi\)
0.992481 0.122400i \(-0.0390592\pi\)
\(524\) 1.49316 0.0652292
\(525\) 0 0
\(526\) −18.0273 −0.786030
\(527\) − 18.3784i − 0.800575i
\(528\) 14.6626i 0.638109i
\(529\) 11.3773 0.494667
\(530\) 0 0
\(531\) 57.3784 2.49001
\(532\) − 0.0778929i − 0.00337708i
\(533\) 0 0
\(534\) −48.8458 −2.11376
\(535\) 0 0
\(536\) −4.58473 −0.198030
\(537\) − 23.2808i − 1.00464i
\(538\) − 20.2089i − 0.871269i
\(539\) −31.5205 −1.35768
\(540\) 0 0
\(541\) 15.1968 0.653362 0.326681 0.945135i \(-0.394070\pi\)
0.326681 + 0.945135i \(0.394070\pi\)
\(542\) 6.08396i 0.261328i
\(543\) − 40.7344i − 1.74808i
\(544\) −7.33131 −0.314327
\(545\) 0 0
\(546\) 1.35105 0.0578196
\(547\) − 26.6505i − 1.13949i −0.821821 0.569746i \(-0.807041\pi\)
0.821821 0.569746i \(-0.192959\pi\)
\(548\) 8.42894i 0.360067i
\(549\) 16.3511 0.697846
\(550\) 0 0
\(551\) −1.33131 −0.0567158
\(552\) 11.0916i 0.472088i
\(553\) 1.32524i 0.0563551i
\(554\) 31.3647 1.33256
\(555\) 0 0
\(556\) 8.81841 0.373984
\(557\) 12.8458i 0.544292i 0.962256 + 0.272146i \(0.0877334\pi\)
−0.962256 + 0.272146i \(0.912267\pi\)
\(558\) 19.0137i 0.804913i
\(559\) 2.70210 0.114287
\(560\) 0 0
\(561\) −107.496 −4.53849
\(562\) 11.3252i 0.477727i
\(563\) − 10.1968i − 0.429744i −0.976642 0.214872i \(-0.931067\pi\)
0.976642 0.214872i \(-0.0689334\pi\)
\(564\) 18.4229 0.775743
\(565\) 0 0
\(566\) 26.1437 1.09890
\(567\) − 2.00760i − 0.0843115i
\(568\) 10.8579i 0.455587i
\(569\) 24.3784 1.02200 0.510998 0.859582i \(-0.329276\pi\)
0.510998 + 0.859582i \(0.329276\pi\)
\(570\) 0 0
\(571\) 8.32371 0.348336 0.174168 0.984716i \(-0.444276\pi\)
0.174168 + 0.984716i \(0.444276\pi\)
\(572\) 24.0273i 1.00463i
\(573\) 15.9495i 0.666298i
\(574\) 0 0
\(575\) 0 0
\(576\) 7.58473 0.316030
\(577\) − 7.29290i − 0.303607i −0.988411 0.151804i \(-0.951492\pi\)
0.988411 0.151804i \(-0.0485081\pi\)
\(578\) − 36.7481i − 1.52852i
\(579\) −59.0684 −2.45480
\(580\) 0 0
\(581\) 1.02581 0.0425576
\(582\) 24.9742i 1.03521i
\(583\) − 58.2994i − 2.41452i
\(584\) 5.09763 0.210942
\(585\) 0 0
\(586\) 1.33131 0.0549959
\(587\) 5.53264i 0.228357i 0.993460 + 0.114178i \(0.0364235\pi\)
−0.993460 + 0.114178i \(0.963576\pi\)
\(588\) 22.7542i 0.938367i
\(589\) 2.50684 0.103292
\(590\) 0 0
\(591\) −9.71577 −0.399653
\(592\) − 5.50684i − 0.226330i
\(593\) 26.3252i 1.08105i 0.841329 + 0.540524i \(0.181774\pi\)
−0.841329 + 0.540524i \(0.818226\pi\)
\(594\) 67.2241 2.75824
\(595\) 0 0
\(596\) 17.6763 0.724049
\(597\) − 9.96919i − 0.408012i
\(598\) 18.1755i 0.743252i
\(599\) −22.2994 −0.911130 −0.455565 0.890202i \(-0.650563\pi\)
−0.455565 + 0.890202i \(0.650563\pi\)
\(600\) 0 0
\(601\) 32.4947 1.32549 0.662743 0.748847i \(-0.269392\pi\)
0.662743 + 0.748847i \(0.269392\pi\)
\(602\) − 0.0394789i − 0.00160904i
\(603\) 34.7739i 1.41610i
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) −13.5205 −0.549233
\(607\) 8.35105i 0.338959i 0.985534 + 0.169479i \(0.0542086\pi\)
−0.985534 + 0.169479i \(0.945791\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) −0.337378 −0.0136713
\(610\) 0 0
\(611\) 30.1892 1.22132
\(612\) 55.6060i 2.24774i
\(613\) 38.2994i 1.54690i 0.633858 + 0.773450i \(0.281471\pi\)
−0.633858 + 0.773450i \(0.718529\pi\)
\(614\) 2.84421 0.114783
\(615\) 0 0
\(616\) 0.351050 0.0141442
\(617\) − 14.3526i − 0.577813i −0.957357 0.288907i \(-0.906708\pi\)
0.957357 0.288907i \(-0.0932917\pi\)
\(618\) − 7.64895i − 0.307686i
\(619\) 8.62314 0.346593 0.173297 0.984870i \(-0.444558\pi\)
0.173297 + 0.984870i \(0.444558\pi\)
\(620\) 0 0
\(621\) 50.8518 2.04061
\(622\) − 16.3895i − 0.657158i
\(623\) 1.16946i 0.0468533i
\(624\) 17.3450 0.694355
\(625\) 0 0
\(626\) −21.0471 −0.841211
\(627\) − 14.6626i − 0.585569i
\(628\) − 0.506836i − 0.0202250i
\(629\) 40.3723 1.60975
\(630\) 0 0
\(631\) −10.3374 −0.411525 −0.205762 0.978602i \(-0.565967\pi\)
−0.205762 + 0.978602i \(0.565967\pi\)
\(632\) 17.0137i 0.676768i
\(633\) − 62.6394i − 2.48969i
\(634\) 30.8902 1.22681
\(635\) 0 0
\(636\) −42.0855 −1.66880
\(637\) 37.2868i 1.47736i
\(638\) − 6.00000i − 0.237542i
\(639\) 82.3541 3.25788
\(640\) 0 0
\(641\) 5.88369 0.232392 0.116196 0.993226i \(-0.462930\pi\)
0.116196 + 0.993226i \(0.462930\pi\)
\(642\) − 45.6566i − 1.80192i
\(643\) 25.5084i 1.00595i 0.864300 + 0.502976i \(0.167762\pi\)
−0.864300 + 0.502976i \(0.832238\pi\)
\(644\) 0.265553 0.0104642
\(645\) 0 0
\(646\) 7.33131 0.288447
\(647\) − 5.09157i − 0.200170i −0.994979 0.100085i \(-0.968089\pi\)
0.994979 0.100085i \(-0.0319115\pi\)
\(648\) − 25.7739i − 1.01250i
\(649\) 34.0942 1.33831
\(650\) 0 0
\(651\) 0.635277 0.0248985
\(652\) 0.830542i 0.0325265i
\(653\) 11.1816i 0.437570i 0.975773 + 0.218785i \(0.0702093\pi\)
−0.975773 + 0.218785i \(0.929791\pi\)
\(654\) 0.253418 0.00990943
\(655\) 0 0
\(656\) 0 0
\(657\) − 38.6642i − 1.50843i
\(658\) − 0.441078i − 0.0171950i
\(659\) 13.7542 0.535787 0.267894 0.963449i \(-0.413672\pi\)
0.267894 + 0.963449i \(0.413672\pi\)
\(660\) 0 0
\(661\) −12.6171 −0.490747 −0.245374 0.969429i \(-0.578911\pi\)
−0.245374 + 0.969429i \(0.578911\pi\)
\(662\) 28.3845i 1.10319i
\(663\) 127.161i 4.93854i
\(664\) 13.1695 0.511074
\(665\) 0 0
\(666\) −41.7679 −1.61847
\(667\) − 4.53871i − 0.175740i
\(668\) − 16.5068i − 0.638669i
\(669\) 47.1968 1.82473
\(670\) 0 0
\(671\) 9.71577 0.375073
\(672\) − 0.253418i − 0.00977581i
\(673\) − 2.35105i − 0.0906263i −0.998973 0.0453132i \(-0.985571\pi\)
0.998973 0.0453132i \(-0.0144286\pi\)
\(674\) 17.8716 0.688387
\(675\) 0 0
\(676\) 15.4229 0.593188
\(677\) − 23.7663i − 0.913414i −0.889617 0.456707i \(-0.849029\pi\)
0.889617 0.456707i \(-0.150971\pi\)
\(678\) 19.5205i 0.749681i
\(679\) 0.597928 0.0229464
\(680\) 0 0
\(681\) −70.2686 −2.69270
\(682\) 11.2979i 0.432619i
\(683\) − 28.1695i − 1.07787i −0.842346 0.538937i \(-0.818826\pi\)
0.842346 0.538937i \(-0.181174\pi\)
\(684\) −7.58473 −0.290009
\(685\) 0 0
\(686\) 1.09003 0.0416174
\(687\) − 61.8594i − 2.36008i
\(688\) − 0.506836i − 0.0193229i
\(689\) −68.9647 −2.62734
\(690\) 0 0
\(691\) 2.32371 0.0883979 0.0441990 0.999023i \(-0.485926\pi\)
0.0441990 + 0.999023i \(0.485926\pi\)
\(692\) 18.8321i 0.715888i
\(693\) − 2.66262i − 0.101145i
\(694\) −33.0410 −1.25422
\(695\) 0 0
\(696\) −4.33131 −0.164178
\(697\) 0 0
\(698\) 19.7158i 0.746253i
\(699\) 19.5650 0.740016
\(700\) 0 0
\(701\) 23.7036 0.895274 0.447637 0.894215i \(-0.352266\pi\)
0.447637 + 0.894215i \(0.352266\pi\)
\(702\) − 79.5220i − 3.00137i
\(703\) 5.50684i 0.207694i
\(704\) 4.50684 0.169858
\(705\) 0 0
\(706\) −5.90997 −0.222425
\(707\) 0.323706i 0.0121742i
\(708\) − 24.6121i − 0.924978i
\(709\) −9.05315 −0.339998 −0.169999 0.985444i \(-0.554376\pi\)
−0.169999 + 0.985444i \(0.554376\pi\)
\(710\) 0 0
\(711\) 129.044 4.83953
\(712\) 15.0137i 0.562661i
\(713\) 8.54631i 0.320062i
\(714\) 1.85789 0.0695297
\(715\) 0 0
\(716\) −7.15579 −0.267424
\(717\) 50.1326i 1.87224i
\(718\) 7.00760i 0.261521i
\(719\) −35.0734 −1.30802 −0.654008 0.756488i \(-0.726914\pi\)
−0.654008 + 0.756488i \(0.726914\pi\)
\(720\) 0 0
\(721\) −0.183130 −0.00682012
\(722\) 1.00000i 0.0372161i
\(723\) 15.6763i 0.583008i
\(724\) −12.5205 −0.465321
\(725\) 0 0
\(726\) 30.2944 1.12433
\(727\) 30.1386i 1.11778i 0.829242 + 0.558890i \(0.188773\pi\)
−0.829242 + 0.558890i \(0.811227\pi\)
\(728\) − 0.415271i − 0.0153910i
\(729\) −49.9039 −1.84829
\(730\) 0 0
\(731\) 3.71577 0.137433
\(732\) − 7.01367i − 0.259233i
\(733\) − 47.8321i − 1.76672i −0.468697 0.883359i \(-0.655276\pi\)
0.468697 0.883359i \(-0.344724\pi\)
\(734\) −17.7158 −0.653901
\(735\) 0 0
\(736\) 3.40920 0.125665
\(737\) 20.6626i 0.761117i
\(738\) 0 0
\(739\) 22.4674 0.826475 0.413238 0.910623i \(-0.364398\pi\)
0.413238 + 0.910623i \(0.364398\pi\)
\(740\) 0 0
\(741\) −17.3450 −0.637184
\(742\) 1.00760i 0.0369903i
\(743\) 11.7926i 0.432629i 0.976324 + 0.216314i \(0.0694036\pi\)
−0.976324 + 0.216314i \(0.930596\pi\)
\(744\) 8.15579 0.299006
\(745\) 0 0
\(746\) 15.4487 0.565616
\(747\) − 99.8868i − 3.65467i
\(748\) 33.0410i 1.20810i
\(749\) −1.09310 −0.0399411
\(750\) 0 0
\(751\) −2.03948 −0.0744216 −0.0372108 0.999307i \(-0.511847\pi\)
−0.0372108 + 0.999307i \(0.511847\pi\)
\(752\) − 5.66262i − 0.206495i
\(753\) − 4.94685i − 0.180273i
\(754\) −7.09763 −0.258481
\(755\) 0 0
\(756\) −1.16185 −0.0422562
\(757\) 43.3526i 1.57568i 0.615882 + 0.787838i \(0.288800\pi\)
−0.615882 + 0.787838i \(0.711200\pi\)
\(758\) − 34.4563i − 1.25151i
\(759\) 49.9879 1.81444
\(760\) 0 0
\(761\) 6.62921 0.240309 0.120154 0.992755i \(-0.461661\pi\)
0.120154 + 0.992755i \(0.461661\pi\)
\(762\) 58.0152i 2.10167i
\(763\) − 0.00606730i 0 0.000219651i
\(764\) 4.90237 0.177361
\(765\) 0 0
\(766\) −12.0273 −0.434566
\(767\) − 40.3313i − 1.45628i
\(768\) − 3.25342i − 0.117398i
\(769\) 19.6429 0.708340 0.354170 0.935181i \(-0.384763\pi\)
0.354170 + 0.935181i \(0.384763\pi\)
\(770\) 0 0
\(771\) 53.7036 1.93409
\(772\) 18.1558i 0.653441i
\(773\) − 14.4107i − 0.518318i −0.965835 0.259159i \(-0.916555\pi\)
0.965835 0.259159i \(-0.0834454\pi\)
\(774\) −3.84421 −0.138177
\(775\) 0 0
\(776\) 7.67629 0.275563
\(777\) 1.39553i 0.0500644i
\(778\) 15.3647i 0.550852i
\(779\) 0 0
\(780\) 0 0
\(781\) 48.9347 1.75102
\(782\) 24.9939i 0.893781i
\(783\) 19.8579i 0.709663i
\(784\) 6.99393 0.249783
\(785\) 0 0
\(786\) −4.85789 −0.173275
\(787\) 24.3176i 0.866830i 0.901194 + 0.433415i \(0.142692\pi\)
−0.901194 + 0.433415i \(0.857308\pi\)
\(788\) 2.98633i 0.106384i
\(789\) 58.6505 2.08801
\(790\) 0 0
\(791\) 0.467357 0.0166173
\(792\) − 34.1831i − 1.21464i
\(793\) − 11.4932i − 0.408134i
\(794\) −1.32524 −0.0470311
\(795\) 0 0
\(796\) −3.06422 −0.108608
\(797\) − 7.30397i − 0.258720i −0.991598 0.129360i \(-0.958708\pi\)
0.991598 0.129360i \(-0.0412922\pi\)
\(798\) 0.253418i 0.00897090i
\(799\) 41.5144 1.46868
\(800\) 0 0
\(801\) 113.875 4.02356
\(802\) − 6.46736i − 0.228370i
\(803\) − 22.9742i − 0.810742i
\(804\) 14.9160 0.526048
\(805\) 0 0
\(806\) 13.3647 0.470752
\(807\) 65.7481i 2.31444i
\(808\) 4.15579i 0.146200i
\(809\) 0.584729 0.0205580 0.0102790 0.999947i \(-0.496728\pi\)
0.0102790 + 0.999947i \(0.496728\pi\)
\(810\) 0 0
\(811\) 1.90997 0.0670682 0.0335341 0.999438i \(-0.489324\pi\)
0.0335341 + 0.999438i \(0.489324\pi\)
\(812\) 0.103700i 0.00363914i
\(813\) − 19.7937i − 0.694194i
\(814\) −24.8184 −0.869885
\(815\) 0 0
\(816\) 23.8518 0.834981
\(817\) 0.506836i 0.0177319i
\(818\) 1.36472i 0.0477164i
\(819\) −3.14972 −0.110060
\(820\) 0 0
\(821\) −38.2226 −1.33398 −0.666989 0.745067i \(-0.732417\pi\)
−0.666989 + 0.745067i \(0.732417\pi\)
\(822\) − 27.4229i − 0.956483i
\(823\) 45.7481i 1.59468i 0.603531 + 0.797340i \(0.293760\pi\)
−0.603531 + 0.797340i \(0.706240\pi\)
\(824\) −2.35105 −0.0819027
\(825\) 0 0
\(826\) −0.589259 −0.0205029
\(827\) − 22.6687i − 0.788268i −0.919053 0.394134i \(-0.871045\pi\)
0.919053 0.394134i \(-0.128955\pi\)
\(828\) − 25.8579i − 0.898624i
\(829\) 4.63028 0.160816 0.0804081 0.996762i \(-0.474378\pi\)
0.0804081 + 0.996762i \(0.474378\pi\)
\(830\) 0 0
\(831\) −102.043 −3.53982
\(832\) − 5.33131i − 0.184830i
\(833\) 51.2747i 1.77656i
\(834\) −28.6900 −0.993452
\(835\) 0 0
\(836\) −4.50684 −0.155872
\(837\) − 37.3921i − 1.29246i
\(838\) − 16.6231i − 0.574237i
\(839\) 50.4826 1.74285 0.871426 0.490527i \(-0.163196\pi\)
0.871426 + 0.490527i \(0.163196\pi\)
\(840\) 0 0
\(841\) −27.2276 −0.938883
\(842\) − 31.1128i − 1.07222i
\(843\) − 36.8458i − 1.26904i
\(844\) −19.2534 −0.662730
\(845\) 0 0
\(846\) −42.9495 −1.47663
\(847\) − 0.725305i − 0.0249218i
\(848\) 12.9358i 0.444216i
\(849\) −85.0562 −2.91912
\(850\) 0 0
\(851\) −18.7739 −0.643562
\(852\) − 35.3252i − 1.21022i
\(853\) − 36.2105i − 1.23982i −0.784672 0.619912i \(-0.787168\pi\)
0.784672 0.619912i \(-0.212832\pi\)
\(854\) −0.167920 −0.00574611
\(855\) 0 0
\(856\) −14.0334 −0.479652
\(857\) 25.1968i 0.860706i 0.902661 + 0.430353i \(0.141611\pi\)
−0.902661 + 0.430353i \(0.858389\pi\)
\(858\) − 78.1710i − 2.66871i
\(859\) −18.8579 −0.643423 −0.321711 0.946838i \(-0.604258\pi\)
−0.321711 + 0.946838i \(0.604258\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 10.1831i − 0.346839i
\(863\) − 15.0137i − 0.511071i −0.966800 0.255536i \(-0.917748\pi\)
0.966800 0.255536i \(-0.0822518\pi\)
\(864\) −14.9160 −0.507454
\(865\) 0 0
\(866\) 22.1953 0.754226
\(867\) 119.557i 4.06037i
\(868\) − 0.195265i − 0.00662771i
\(869\) 76.6778 2.60112
\(870\) 0 0
\(871\) 24.4426 0.828206
\(872\) − 0.0778929i − 0.00263779i
\(873\) − 58.2226i − 1.97054i
\(874\) −3.40920 −0.115318
\(875\) 0 0
\(876\) −16.5847 −0.560346
\(877\) 35.1128i 1.18568i 0.805322 + 0.592838i \(0.201993\pi\)
−0.805322 + 0.592838i \(0.798007\pi\)
\(878\) 9.32524i 0.314712i
\(879\) −4.33131 −0.146091
\(880\) 0 0
\(881\) −41.3389 −1.39274 −0.696372 0.717681i \(-0.745203\pi\)
−0.696372 + 0.717681i \(0.745203\pi\)
\(882\) − 53.0471i − 1.78619i
\(883\) 12.6353i 0.425211i 0.977138 + 0.212605i \(0.0681949\pi\)
−0.977138 + 0.212605i \(0.931805\pi\)
\(884\) 39.0855 1.31459
\(885\) 0 0
\(886\) −13.9879 −0.469932
\(887\) 4.15579i 0.139538i 0.997563 + 0.0697688i \(0.0222262\pi\)
−0.997563 + 0.0697688i \(0.977774\pi\)
\(888\) 17.9160i 0.601223i
\(889\) 1.38899 0.0465853
\(890\) 0 0
\(891\) −116.159 −3.89147
\(892\) − 14.5068i − 0.485725i
\(893\) 5.66262i 0.189492i
\(894\) −57.5084 −1.92337
\(895\) 0 0
\(896\) −0.0778929 −0.00260222
\(897\) − 59.1326i − 1.97438i
\(898\) − 13.4932i − 0.450273i
\(899\) −3.33738 −0.111308
\(900\) 0 0
\(901\) −94.8362 −3.15945
\(902\) 0 0
\(903\) 0.128441i 0.00427426i
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 65.0684 2.16175
\(907\) − 41.7542i − 1.38643i −0.720733 0.693213i \(-0.756195\pi\)
0.720733 0.693213i \(-0.243805\pi\)
\(908\) 21.5984i 0.716768i
\(909\) 31.5205 1.04547
\(910\) 0 0
\(911\) −9.12998 −0.302490 −0.151245 0.988496i \(-0.548328\pi\)
−0.151245 + 0.988496i \(0.548328\pi\)
\(912\) 3.25342i 0.107731i
\(913\) − 59.3526i − 1.96428i
\(914\) −9.68236 −0.320264
\(915\) 0 0
\(916\) −19.0137 −0.628229
\(917\) 0.116307i 0.00384079i
\(918\) − 109.354i − 3.60922i
\(919\) −19.0197 −0.627403 −0.313702 0.949522i \(-0.601569\pi\)
−0.313702 + 0.949522i \(0.601569\pi\)
\(920\) 0 0
\(921\) −9.25342 −0.304910
\(922\) 8.66262i 0.285288i
\(923\) − 57.8868i − 1.90537i
\(924\) −1.14211 −0.0375728
\(925\) 0 0
\(926\) 28.0015 0.920188
\(927\) 17.8321i 0.585682i
\(928\) 1.33131i 0.0437024i
\(929\) 7.77239 0.255004 0.127502 0.991838i \(-0.459304\pi\)
0.127502 + 0.991838i \(0.459304\pi\)
\(930\) 0 0
\(931\) −6.99393 −0.229217
\(932\) − 6.01367i − 0.196984i
\(933\) 53.3218i 1.74568i
\(934\) −16.9742 −0.555413
\(935\) 0 0
\(936\) −40.4365 −1.32171
\(937\) − 45.9818i − 1.50216i −0.660211 0.751080i \(-0.729533\pi\)
0.660211 0.751080i \(-0.270467\pi\)
\(938\) − 0.357118i − 0.0116603i
\(939\) 68.4750 2.23460
\(940\) 0 0
\(941\) −40.6242 −1.32431 −0.662156 0.749366i \(-0.730358\pi\)
−0.662156 + 0.749366i \(0.730358\pi\)
\(942\) 1.64895i 0.0537257i
\(943\) 0 0
\(944\) −7.56499 −0.246219
\(945\) 0 0
\(946\) −2.28423 −0.0742666
\(947\) − 2.28423i − 0.0742274i −0.999311 0.0371137i \(-0.988184\pi\)
0.999311 0.0371137i \(-0.0118164\pi\)
\(948\) − 55.3526i − 1.79777i
\(949\) −27.1771 −0.882205
\(950\) 0 0
\(951\) −100.499 −3.25890
\(952\) − 0.571057i − 0.0185081i
\(953\) 57.5084i 1.86288i 0.363896 + 0.931439i \(0.381446\pi\)
−0.363896 + 0.931439i \(0.618554\pi\)
\(954\) 98.1144 3.17657
\(955\) 0 0
\(956\) 15.4092 0.498369
\(957\) 19.5205i 0.631008i
\(958\) − 10.0532i − 0.324803i
\(959\) −0.656555 −0.0212013
\(960\) 0 0
\(961\) −24.7158 −0.797283
\(962\) 29.3587i 0.946561i
\(963\) 106.440i 3.42997i
\(964\) 4.81841 0.155190
\(965\) 0 0
\(966\) −0.863954 −0.0277973
\(967\) − 4.70210i − 0.151209i −0.997138 0.0756047i \(-0.975911\pi\)
0.997138 0.0756047i \(-0.0240887\pi\)
\(968\) − 9.31157i − 0.299285i
\(969\) −23.8518 −0.766231
\(970\) 0 0
\(971\) 14.9863 0.480934 0.240467 0.970657i \(-0.422699\pi\)
0.240467 + 0.970657i \(0.422699\pi\)
\(972\) 39.1052i 1.25430i
\(973\) 0.686891i 0.0220207i
\(974\) −19.2089 −0.615494
\(975\) 0 0
\(976\) −2.15579 −0.0690050
\(977\) 8.89737i 0.284652i 0.989820 + 0.142326i \(0.0454581\pi\)
−0.989820 + 0.142326i \(0.954542\pi\)
\(978\) − 2.70210i − 0.0864037i
\(979\) 67.6642 2.16256
\(980\) 0 0
\(981\) −0.590796 −0.0188627
\(982\) 5.32524i 0.169935i
\(983\) 1.60947i 0.0513341i 0.999671 + 0.0256671i \(0.00817098\pi\)
−0.999671 + 0.0256671i \(0.991829\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −9.76025 −0.310830
\(987\) 1.43501i 0.0456769i
\(988\) 5.33131i 0.169612i
\(989\) −1.72791 −0.0549443
\(990\) 0 0
\(991\) 16.8974 0.536763 0.268381 0.963313i \(-0.413511\pi\)
0.268381 + 0.963313i \(0.413511\pi\)
\(992\) − 2.50684i − 0.0795921i
\(993\) − 92.3465i − 2.93053i
\(994\) −0.845752 −0.0268256
\(995\) 0 0
\(996\) −42.8458 −1.35762
\(997\) 24.1558i 0.765021i 0.923951 + 0.382511i \(0.124940\pi\)
−0.923951 + 0.382511i \(0.875060\pi\)
\(998\) 11.9605i 0.378604i
\(999\) 82.1402 2.59880
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 950.2.b.h.799.4 6
5.2 odd 4 950.2.a.j.1.1 3
5.3 odd 4 950.2.a.l.1.3 yes 3
5.4 even 2 inner 950.2.b.h.799.3 6
15.2 even 4 8550.2.a.cp.1.2 3
15.8 even 4 8550.2.a.ci.1.2 3
20.3 even 4 7600.2.a.bk.1.1 3
20.7 even 4 7600.2.a.bz.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
950.2.a.j.1.1 3 5.2 odd 4
950.2.a.l.1.3 yes 3 5.3 odd 4
950.2.b.h.799.3 6 5.4 even 2 inner
950.2.b.h.799.4 6 1.1 even 1 trivial
7600.2.a.bk.1.1 3 20.3 even 4
7600.2.a.bz.1.3 3 20.7 even 4
8550.2.a.ci.1.2 3 15.8 even 4
8550.2.a.cp.1.2 3 15.2 even 4