Properties

Label 950.2.a.n.1.1
Level $950$
Weight $2$
Character 950.1
Self dual yes
Analytic conductor $7.586$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.12489\) of defining polynomial
Character \(\chi\) \(=\) 950.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.12489 q^{3} +1.00000 q^{4} -2.12489 q^{6} +4.12489 q^{7} +1.00000 q^{8} +1.51514 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.12489 q^{3} +1.00000 q^{4} -2.12489 q^{6} +4.12489 q^{7} +1.00000 q^{8} +1.51514 q^{9} -2.64002 q^{11} -2.12489 q^{12} +2.51514 q^{13} +4.12489 q^{14} +1.00000 q^{16} -0.515138 q^{17} +1.51514 q^{18} +1.00000 q^{19} -8.76491 q^{21} -2.64002 q^{22} +3.09461 q^{23} -2.12489 q^{24} +2.51514 q^{26} +3.15516 q^{27} +4.12489 q^{28} -7.79518 q^{29} +3.67030 q^{31} +1.00000 q^{32} +5.60975 q^{33} -0.515138 q^{34} +1.51514 q^{36} +10.2498 q^{37} +1.00000 q^{38} -5.34438 q^{39} +8.88979 q^{41} -8.76491 q^{42} +8.64002 q^{43} -2.64002 q^{44} +3.09461 q^{46} -4.96972 q^{47} -2.12489 q^{48} +10.0147 q^{49} +1.09461 q^{51} +2.51514 q^{52} +5.48486 q^{53} +3.15516 q^{54} +4.12489 q^{56} -2.12489 q^{57} -7.79518 q^{58} +3.15516 q^{59} -12.6400 q^{61} +3.67030 q^{62} +6.24977 q^{63} +1.00000 q^{64} +5.60975 q^{66} -7.40493 q^{67} -0.515138 q^{68} -6.57569 q^{69} +11.1396 q^{71} +1.51514 q^{72} -2.70436 q^{73} +10.2498 q^{74} +1.00000 q^{76} -10.8898 q^{77} -5.34438 q^{78} +16.7493 q^{79} -11.2498 q^{81} +8.88979 q^{82} +3.28005 q^{83} -8.76491 q^{84} +8.64002 q^{86} +16.5639 q^{87} -2.64002 q^{88} -7.60975 q^{89} +10.3747 q^{91} +3.09461 q^{92} -7.79897 q^{93} -4.96972 q^{94} -2.12489 q^{96} +3.93945 q^{97} +10.0147 q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + 4q^{7} + 3q^{8} + 5q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + 2q^{3} + 3q^{4} + 2q^{6} + 4q^{7} + 3q^{8} + 5q^{9} + 2q^{12} + 8q^{13} + 4q^{14} + 3q^{16} - 2q^{17} + 5q^{18} + 3q^{19} - 10q^{21} + 2q^{24} + 8q^{26} + 2q^{27} + 4q^{28} - 8q^{29} + 4q^{31} + 3q^{32} + 8q^{33} - 2q^{34} + 5q^{36} + 14q^{37} + 3q^{38} + 10q^{39} + 2q^{41} - 10q^{42} + 18q^{43} - 14q^{47} + 2q^{48} - 3q^{49} - 6q^{51} + 8q^{52} + 16q^{53} + 2q^{54} + 4q^{56} + 2q^{57} - 8q^{58} + 2q^{59} - 30q^{61} + 4q^{62} + 2q^{63} + 3q^{64} + 8q^{66} + 2q^{67} - 2q^{68} - 22q^{69} - 8q^{71} + 5q^{72} + 10q^{73} + 14q^{74} + 3q^{76} - 8q^{77} + 10q^{78} - 17q^{81} + 2q^{82} - 6q^{83} - 10q^{84} + 18q^{86} + 6q^{87} - 14q^{89} + 6q^{91} + 4q^{93} - 14q^{94} + 2q^{96} + 10q^{97} - 3q^{98} - 12q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −2.12489 −1.22680 −0.613402 0.789771i \(-0.710199\pi\)
−0.613402 + 0.789771i \(0.710199\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −2.12489 −0.867481
\(7\) 4.12489 1.55906 0.779530 0.626365i \(-0.215458\pi\)
0.779530 + 0.626365i \(0.215458\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.51514 0.505046
\(10\) 0 0
\(11\) −2.64002 −0.795997 −0.397999 0.917386i \(-0.630295\pi\)
−0.397999 + 0.917386i \(0.630295\pi\)
\(12\) −2.12489 −0.613402
\(13\) 2.51514 0.697574 0.348787 0.937202i \(-0.386594\pi\)
0.348787 + 0.937202i \(0.386594\pi\)
\(14\) 4.12489 1.10242
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.515138 −0.124939 −0.0624697 0.998047i \(-0.519898\pi\)
−0.0624697 + 0.998047i \(0.519898\pi\)
\(18\) 1.51514 0.357121
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −8.76491 −1.91266
\(22\) −2.64002 −0.562855
\(23\) 3.09461 0.645271 0.322635 0.946523i \(-0.395431\pi\)
0.322635 + 0.946523i \(0.395431\pi\)
\(24\) −2.12489 −0.433740
\(25\) 0 0
\(26\) 2.51514 0.493259
\(27\) 3.15516 0.607211
\(28\) 4.12489 0.779530
\(29\) −7.79518 −1.44753 −0.723765 0.690047i \(-0.757590\pi\)
−0.723765 + 0.690047i \(0.757590\pi\)
\(30\) 0 0
\(31\) 3.67030 0.659205 0.329603 0.944120i \(-0.393085\pi\)
0.329603 + 0.944120i \(0.393085\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.60975 0.976532
\(34\) −0.515138 −0.0883454
\(35\) 0 0
\(36\) 1.51514 0.252523
\(37\) 10.2498 1.68505 0.842526 0.538656i \(-0.181068\pi\)
0.842526 + 0.538656i \(0.181068\pi\)
\(38\) 1.00000 0.162221
\(39\) −5.34438 −0.855786
\(40\) 0 0
\(41\) 8.88979 1.38835 0.694176 0.719805i \(-0.255769\pi\)
0.694176 + 0.719805i \(0.255769\pi\)
\(42\) −8.76491 −1.35245
\(43\) 8.64002 1.31759 0.658796 0.752322i \(-0.271066\pi\)
0.658796 + 0.752322i \(0.271066\pi\)
\(44\) −2.64002 −0.397999
\(45\) 0 0
\(46\) 3.09461 0.456275
\(47\) −4.96972 −0.724909 −0.362454 0.932002i \(-0.618061\pi\)
−0.362454 + 0.932002i \(0.618061\pi\)
\(48\) −2.12489 −0.306701
\(49\) 10.0147 1.43067
\(50\) 0 0
\(51\) 1.09461 0.153276
\(52\) 2.51514 0.348787
\(53\) 5.48486 0.753404 0.376702 0.926335i \(-0.377058\pi\)
0.376702 + 0.926335i \(0.377058\pi\)
\(54\) 3.15516 0.429363
\(55\) 0 0
\(56\) 4.12489 0.551211
\(57\) −2.12489 −0.281448
\(58\) −7.79518 −1.02356
\(59\) 3.15516 0.410767 0.205384 0.978682i \(-0.434156\pi\)
0.205384 + 0.978682i \(0.434156\pi\)
\(60\) 0 0
\(61\) −12.6400 −1.61839 −0.809195 0.587541i \(-0.800096\pi\)
−0.809195 + 0.587541i \(0.800096\pi\)
\(62\) 3.67030 0.466129
\(63\) 6.24977 0.787397
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 5.60975 0.690512
\(67\) −7.40493 −0.904656 −0.452328 0.891852i \(-0.649406\pi\)
−0.452328 + 0.891852i \(0.649406\pi\)
\(68\) −0.515138 −0.0624697
\(69\) −6.57569 −0.791620
\(70\) 0 0
\(71\) 11.1396 1.32202 0.661012 0.750376i \(-0.270127\pi\)
0.661012 + 0.750376i \(0.270127\pi\)
\(72\) 1.51514 0.178561
\(73\) −2.70436 −0.316521 −0.158261 0.987397i \(-0.550589\pi\)
−0.158261 + 0.987397i \(0.550589\pi\)
\(74\) 10.2498 1.19151
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −10.8898 −1.24101
\(78\) −5.34438 −0.605132
\(79\) 16.7493 1.88444 0.942222 0.334988i \(-0.108732\pi\)
0.942222 + 0.334988i \(0.108732\pi\)
\(80\) 0 0
\(81\) −11.2498 −1.24997
\(82\) 8.88979 0.981714
\(83\) 3.28005 0.360032 0.180016 0.983664i \(-0.442385\pi\)
0.180016 + 0.983664i \(0.442385\pi\)
\(84\) −8.76491 −0.956330
\(85\) 0 0
\(86\) 8.64002 0.931678
\(87\) 16.5639 1.77583
\(88\) −2.64002 −0.281427
\(89\) −7.60975 −0.806632 −0.403316 0.915061i \(-0.632142\pi\)
−0.403316 + 0.915061i \(0.632142\pi\)
\(90\) 0 0
\(91\) 10.3747 1.08756
\(92\) 3.09461 0.322635
\(93\) −7.79897 −0.808715
\(94\) −4.96972 −0.512588
\(95\) 0 0
\(96\) −2.12489 −0.216870
\(97\) 3.93945 0.399990 0.199995 0.979797i \(-0.435907\pi\)
0.199995 + 0.979797i \(0.435907\pi\)
\(98\) 10.0147 1.01164
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) −13.7990 −1.37305 −0.686524 0.727107i \(-0.740864\pi\)
−0.686524 + 0.727107i \(0.740864\pi\)
\(102\) 1.09461 0.108382
\(103\) 4.57947 0.451229 0.225614 0.974217i \(-0.427561\pi\)
0.225614 + 0.974217i \(0.427561\pi\)
\(104\) 2.51514 0.246630
\(105\) 0 0
\(106\) 5.48486 0.532737
\(107\) −10.3747 −1.00296 −0.501478 0.865170i \(-0.667210\pi\)
−0.501478 + 0.865170i \(0.667210\pi\)
\(108\) 3.15516 0.303606
\(109\) 3.01468 0.288754 0.144377 0.989523i \(-0.453882\pi\)
0.144377 + 0.989523i \(0.453882\pi\)
\(110\) 0 0
\(111\) −21.7796 −2.06723
\(112\) 4.12489 0.389765
\(113\) −19.2001 −1.80620 −0.903098 0.429435i \(-0.858713\pi\)
−0.903098 + 0.429435i \(0.858713\pi\)
\(114\) −2.12489 −0.199014
\(115\) 0 0
\(116\) −7.79518 −0.723765
\(117\) 3.81078 0.352307
\(118\) 3.15516 0.290456
\(119\) −2.12489 −0.194788
\(120\) 0 0
\(121\) −4.03028 −0.366389
\(122\) −12.6400 −1.14437
\(123\) −18.8898 −1.70324
\(124\) 3.67030 0.329603
\(125\) 0 0
\(126\) 6.24977 0.556774
\(127\) 14.3103 1.26984 0.634918 0.772580i \(-0.281034\pi\)
0.634918 + 0.772580i \(0.281034\pi\)
\(128\) 1.00000 0.0883883
\(129\) −18.3591 −1.61643
\(130\) 0 0
\(131\) −6.56009 −0.573158 −0.286579 0.958057i \(-0.592518\pi\)
−0.286579 + 0.958057i \(0.592518\pi\)
\(132\) 5.60975 0.488266
\(133\) 4.12489 0.357673
\(134\) −7.40493 −0.639689
\(135\) 0 0
\(136\) −0.515138 −0.0441727
\(137\) −6.45459 −0.551452 −0.275726 0.961236i \(-0.588918\pi\)
−0.275726 + 0.961236i \(0.588918\pi\)
\(138\) −6.57569 −0.559760
\(139\) −23.0596 −1.95589 −0.977946 0.208856i \(-0.933026\pi\)
−0.977946 + 0.208856i \(0.933026\pi\)
\(140\) 0 0
\(141\) 10.5601 0.889320
\(142\) 11.1396 0.934812
\(143\) −6.64002 −0.555267
\(144\) 1.51514 0.126262
\(145\) 0 0
\(146\) −2.70436 −0.223814
\(147\) −21.2800 −1.75515
\(148\) 10.2498 0.842526
\(149\) 16.0294 1.31318 0.656588 0.754249i \(-0.271999\pi\)
0.656588 + 0.754249i \(0.271999\pi\)
\(150\) 0 0
\(151\) −14.3103 −1.16456 −0.582279 0.812989i \(-0.697839\pi\)
−0.582279 + 0.812989i \(0.697839\pi\)
\(152\) 1.00000 0.0811107
\(153\) −0.780505 −0.0631001
\(154\) −10.8898 −0.877525
\(155\) 0 0
\(156\) −5.34438 −0.427893
\(157\) −18.0294 −1.43890 −0.719450 0.694544i \(-0.755606\pi\)
−0.719450 + 0.694544i \(0.755606\pi\)
\(158\) 16.7493 1.33250
\(159\) −11.6547 −0.924278
\(160\) 0 0
\(161\) 12.7649 1.00602
\(162\) −11.2498 −0.883865
\(163\) 2.70058 0.211525 0.105763 0.994391i \(-0.466272\pi\)
0.105763 + 0.994391i \(0.466272\pi\)
\(164\) 8.88979 0.694176
\(165\) 0 0
\(166\) 3.28005 0.254581
\(167\) 8.95035 0.692599 0.346299 0.938124i \(-0.387438\pi\)
0.346299 + 0.938124i \(0.387438\pi\)
\(168\) −8.76491 −0.676227
\(169\) −6.67408 −0.513391
\(170\) 0 0
\(171\) 1.51514 0.115866
\(172\) 8.64002 0.658796
\(173\) 0.310323 0.0235934 0.0117967 0.999930i \(-0.496245\pi\)
0.0117967 + 0.999930i \(0.496245\pi\)
\(174\) 16.5639 1.25570
\(175\) 0 0
\(176\) −2.64002 −0.198999
\(177\) −6.70436 −0.503930
\(178\) −7.60975 −0.570375
\(179\) −1.52982 −0.114344 −0.0571720 0.998364i \(-0.518208\pi\)
−0.0571720 + 0.998364i \(0.518208\pi\)
\(180\) 0 0
\(181\) −14.7493 −1.09631 −0.548154 0.836377i \(-0.684669\pi\)
−0.548154 + 0.836377i \(0.684669\pi\)
\(182\) 10.3747 0.769021
\(183\) 26.8586 1.98544
\(184\) 3.09461 0.228138
\(185\) 0 0
\(186\) −7.79897 −0.571848
\(187\) 1.35998 0.0994513
\(188\) −4.96972 −0.362454
\(189\) 13.0147 0.946679
\(190\) 0 0
\(191\) 18.1249 1.31147 0.655735 0.754991i \(-0.272359\pi\)
0.655735 + 0.754991i \(0.272359\pi\)
\(192\) −2.12489 −0.153350
\(193\) −4.56009 −0.328243 −0.164121 0.986440i \(-0.552479\pi\)
−0.164121 + 0.986440i \(0.552479\pi\)
\(194\) 3.93945 0.282836
\(195\) 0 0
\(196\) 10.0147 0.715334
\(197\) −2.14048 −0.152503 −0.0762515 0.997089i \(-0.524295\pi\)
−0.0762515 + 0.997089i \(0.524295\pi\)
\(198\) −4.00000 −0.284268
\(199\) 16.5639 1.17418 0.587091 0.809521i \(-0.300273\pi\)
0.587091 + 0.809521i \(0.300273\pi\)
\(200\) 0 0
\(201\) 15.7346 1.10984
\(202\) −13.7990 −0.970892
\(203\) −32.1542 −2.25679
\(204\) 1.09461 0.0766380
\(205\) 0 0
\(206\) 4.57947 0.319067
\(207\) 4.68876 0.325891
\(208\) 2.51514 0.174393
\(209\) −2.64002 −0.182614
\(210\) 0 0
\(211\) −15.2838 −1.05218 −0.526091 0.850428i \(-0.676343\pi\)
−0.526091 + 0.850428i \(0.676343\pi\)
\(212\) 5.48486 0.376702
\(213\) −23.6703 −1.62186
\(214\) −10.3747 −0.709197
\(215\) 0 0
\(216\) 3.15516 0.214682
\(217\) 15.1396 1.02774
\(218\) 3.01468 0.204180
\(219\) 5.74645 0.388309
\(220\) 0 0
\(221\) −1.29564 −0.0871544
\(222\) −21.7796 −1.46175
\(223\) −3.75023 −0.251134 −0.125567 0.992085i \(-0.540075\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(224\) 4.12489 0.275606
\(225\) 0 0
\(226\) −19.2001 −1.27717
\(227\) −24.1542 −1.60317 −0.801587 0.597878i \(-0.796011\pi\)
−0.801587 + 0.597878i \(0.796011\pi\)
\(228\) −2.12489 −0.140724
\(229\) −13.0596 −0.863005 −0.431503 0.902112i \(-0.642016\pi\)
−0.431503 + 0.902112i \(0.642016\pi\)
\(230\) 0 0
\(231\) 23.1396 1.52247
\(232\) −7.79518 −0.511779
\(233\) −6.43899 −0.421832 −0.210916 0.977504i \(-0.567645\pi\)
−0.210916 + 0.977504i \(0.567645\pi\)
\(234\) 3.81078 0.249119
\(235\) 0 0
\(236\) 3.15516 0.205384
\(237\) −35.5904 −2.31184
\(238\) −2.12489 −0.137736
\(239\) 22.8742 1.47961 0.739804 0.672822i \(-0.234918\pi\)
0.739804 + 0.672822i \(0.234918\pi\)
\(240\) 0 0
\(241\) 4.96972 0.320128 0.160064 0.987107i \(-0.448830\pi\)
0.160064 + 0.987107i \(0.448830\pi\)
\(242\) −4.03028 −0.259076
\(243\) 14.4390 0.926262
\(244\) −12.6400 −0.809195
\(245\) 0 0
\(246\) −18.8898 −1.20437
\(247\) 2.51514 0.160034
\(248\) 3.67030 0.233064
\(249\) −6.96972 −0.441688
\(250\) 0 0
\(251\) 15.9201 1.00487 0.502433 0.864616i \(-0.332438\pi\)
0.502433 + 0.864616i \(0.332438\pi\)
\(252\) 6.24977 0.393699
\(253\) −8.16984 −0.513634
\(254\) 14.3103 0.897910
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −1.04965 −0.0654756 −0.0327378 0.999464i \(-0.510423\pi\)
−0.0327378 + 0.999464i \(0.510423\pi\)
\(258\) −18.3591 −1.14299
\(259\) 42.2791 2.62710
\(260\) 0 0
\(261\) −11.8108 −0.731069
\(262\) −6.56009 −0.405284
\(263\) −11.9394 −0.736218 −0.368109 0.929783i \(-0.619995\pi\)
−0.368109 + 0.929783i \(0.619995\pi\)
\(264\) 5.60975 0.345256
\(265\) 0 0
\(266\) 4.12489 0.252913
\(267\) 16.1698 0.989578
\(268\) −7.40493 −0.452328
\(269\) −15.9394 −0.971845 −0.485923 0.874002i \(-0.661516\pi\)
−0.485923 + 0.874002i \(0.661516\pi\)
\(270\) 0 0
\(271\) 1.46548 0.0890218 0.0445109 0.999009i \(-0.485827\pi\)
0.0445109 + 0.999009i \(0.485827\pi\)
\(272\) −0.515138 −0.0312348
\(273\) −22.0450 −1.33422
\(274\) −6.45459 −0.389936
\(275\) 0 0
\(276\) −6.57569 −0.395810
\(277\) 32.4995 1.95271 0.976354 0.216177i \(-0.0693590\pi\)
0.976354 + 0.216177i \(0.0693590\pi\)
\(278\) −23.0596 −1.38303
\(279\) 5.56101 0.332929
\(280\) 0 0
\(281\) 1.04965 0.0626171 0.0313085 0.999510i \(-0.490033\pi\)
0.0313085 + 0.999510i \(0.490033\pi\)
\(282\) 10.5601 0.628844
\(283\) −9.34060 −0.555241 −0.277620 0.960691i \(-0.589546\pi\)
−0.277620 + 0.960691i \(0.589546\pi\)
\(284\) 11.1396 0.661012
\(285\) 0 0
\(286\) −6.64002 −0.392633
\(287\) 36.6694 2.16453
\(288\) 1.51514 0.0892804
\(289\) −16.7346 −0.984390
\(290\) 0 0
\(291\) −8.37088 −0.490709
\(292\) −2.70436 −0.158261
\(293\) 25.5748 1.49409 0.747047 0.664771i \(-0.231471\pi\)
0.747047 + 0.664771i \(0.231471\pi\)
\(294\) −21.2800 −1.24108
\(295\) 0 0
\(296\) 10.2498 0.595756
\(297\) −8.32970 −0.483338
\(298\) 16.0294 0.928556
\(299\) 7.78337 0.450124
\(300\) 0 0
\(301\) 35.6391 2.05420
\(302\) −14.3103 −0.823467
\(303\) 29.3212 1.68446
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −0.780505 −0.0446185
\(307\) 9.43991 0.538764 0.269382 0.963033i \(-0.413181\pi\)
0.269382 + 0.963033i \(0.413181\pi\)
\(308\) −10.8898 −0.620504
\(309\) −9.73085 −0.553569
\(310\) 0 0
\(311\) −17.4655 −0.990377 −0.495188 0.868786i \(-0.664901\pi\)
−0.495188 + 0.868786i \(0.664901\pi\)
\(312\) −5.34438 −0.302566
\(313\) −10.4546 −0.590928 −0.295464 0.955354i \(-0.595474\pi\)
−0.295464 + 0.955354i \(0.595474\pi\)
\(314\) −18.0294 −1.01746
\(315\) 0 0
\(316\) 16.7493 0.942222
\(317\) 4.33348 0.243393 0.121696 0.992567i \(-0.461167\pi\)
0.121696 + 0.992567i \(0.461167\pi\)
\(318\) −11.6547 −0.653563
\(319\) 20.5795 1.15223
\(320\) 0 0
\(321\) 22.0450 1.23043
\(322\) 12.7649 0.711361
\(323\) −0.515138 −0.0286630
\(324\) −11.2498 −0.624987
\(325\) 0 0
\(326\) 2.70058 0.149571
\(327\) −6.40585 −0.354244
\(328\) 8.88979 0.490857
\(329\) −20.4995 −1.13018
\(330\) 0 0
\(331\) −4.56387 −0.250853 −0.125427 0.992103i \(-0.540030\pi\)
−0.125427 + 0.992103i \(0.540030\pi\)
\(332\) 3.28005 0.180016
\(333\) 15.5298 0.851029
\(334\) 8.95035 0.489741
\(335\) 0 0
\(336\) −8.76491 −0.478165
\(337\) 13.0109 0.708749 0.354374 0.935104i \(-0.384694\pi\)
0.354374 + 0.935104i \(0.384694\pi\)
\(338\) −6.67408 −0.363022
\(339\) 40.7980 2.21585
\(340\) 0 0
\(341\) −9.68968 −0.524725
\(342\) 1.51514 0.0819293
\(343\) 12.4352 0.671438
\(344\) 8.64002 0.465839
\(345\) 0 0
\(346\) 0.310323 0.0166831
\(347\) 13.2195 0.709660 0.354830 0.934931i \(-0.384539\pi\)
0.354830 + 0.934931i \(0.384539\pi\)
\(348\) 16.5639 0.887917
\(349\) −20.4390 −1.09407 −0.547037 0.837108i \(-0.684244\pi\)
−0.547037 + 0.837108i \(0.684244\pi\)
\(350\) 0 0
\(351\) 7.93567 0.423575
\(352\) −2.64002 −0.140714
\(353\) 10.7044 0.569735 0.284868 0.958567i \(-0.408050\pi\)
0.284868 + 0.958567i \(0.408050\pi\)
\(354\) −6.70436 −0.356333
\(355\) 0 0
\(356\) −7.60975 −0.403316
\(357\) 4.51514 0.238966
\(358\) −1.52982 −0.0808534
\(359\) −4.31410 −0.227690 −0.113845 0.993499i \(-0.536317\pi\)
−0.113845 + 0.993499i \(0.536317\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −14.7493 −0.775207
\(363\) 8.56387 0.449487
\(364\) 10.3747 0.543780
\(365\) 0 0
\(366\) 26.8586 1.40392
\(367\) −12.8099 −0.668669 −0.334335 0.942454i \(-0.608512\pi\)
−0.334335 + 0.942454i \(0.608512\pi\)
\(368\) 3.09461 0.161318
\(369\) 13.4693 0.701182
\(370\) 0 0
\(371\) 22.6244 1.17460
\(372\) −7.79897 −0.404358
\(373\) 0.704357 0.0364702 0.0182351 0.999834i \(-0.494195\pi\)
0.0182351 + 0.999834i \(0.494195\pi\)
\(374\) 1.35998 0.0703227
\(375\) 0 0
\(376\) −4.96972 −0.256294
\(377\) −19.6060 −1.00976
\(378\) 13.0147 0.669403
\(379\) 6.12489 0.314614 0.157307 0.987550i \(-0.449719\pi\)
0.157307 + 0.987550i \(0.449719\pi\)
\(380\) 0 0
\(381\) −30.4078 −1.55784
\(382\) 18.1249 0.927350
\(383\) 18.6400 0.952461 0.476230 0.879321i \(-0.342003\pi\)
0.476230 + 0.879321i \(0.342003\pi\)
\(384\) −2.12489 −0.108435
\(385\) 0 0
\(386\) −4.56009 −0.232103
\(387\) 13.0908 0.665444
\(388\) 3.93945 0.199995
\(389\) 13.9201 0.705776 0.352888 0.935666i \(-0.385200\pi\)
0.352888 + 0.935666i \(0.385200\pi\)
\(390\) 0 0
\(391\) −1.59415 −0.0806197
\(392\) 10.0147 0.505818
\(393\) 13.9394 0.703152
\(394\) −2.14048 −0.107836
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) −28.3784 −1.42427 −0.712136 0.702041i \(-0.752272\pi\)
−0.712136 + 0.702041i \(0.752272\pi\)
\(398\) 16.5639 0.830272
\(399\) −8.76491 −0.438794
\(400\) 0 0
\(401\) 5.54920 0.277114 0.138557 0.990354i \(-0.455754\pi\)
0.138557 + 0.990354i \(0.455754\pi\)
\(402\) 15.7346 0.784772
\(403\) 9.23131 0.459844
\(404\) −13.7990 −0.686524
\(405\) 0 0
\(406\) −32.1542 −1.59579
\(407\) −27.0596 −1.34130
\(408\) 1.09461 0.0541912
\(409\) −5.01090 −0.247773 −0.123886 0.992296i \(-0.539536\pi\)
−0.123886 + 0.992296i \(0.539536\pi\)
\(410\) 0 0
\(411\) 13.7153 0.676524
\(412\) 4.57947 0.225614
\(413\) 13.0147 0.640411
\(414\) 4.68876 0.230440
\(415\) 0 0
\(416\) 2.51514 0.123315
\(417\) 48.9991 2.39950
\(418\) −2.64002 −0.129128
\(419\) −15.8889 −0.776222 −0.388111 0.921613i \(-0.626872\pi\)
−0.388111 + 0.921613i \(0.626872\pi\)
\(420\) 0 0
\(421\) −2.38647 −0.116310 −0.0581548 0.998308i \(-0.518522\pi\)
−0.0581548 + 0.998308i \(0.518522\pi\)
\(422\) −15.2838 −0.744005
\(423\) −7.52982 −0.366112
\(424\) 5.48486 0.266368
\(425\) 0 0
\(426\) −23.6703 −1.14683
\(427\) −52.1386 −2.52317
\(428\) −10.3747 −0.501478
\(429\) 14.1093 0.681203
\(430\) 0 0
\(431\) −2.35906 −0.113632 −0.0568160 0.998385i \(-0.518095\pi\)
−0.0568160 + 0.998385i \(0.518095\pi\)
\(432\) 3.15516 0.151803
\(433\) −0.640023 −0.0307576 −0.0153788 0.999882i \(-0.504895\pi\)
−0.0153788 + 0.999882i \(0.504895\pi\)
\(434\) 15.1396 0.726722
\(435\) 0 0
\(436\) 3.01468 0.144377
\(437\) 3.09461 0.148035
\(438\) 5.74645 0.274576
\(439\) −25.4499 −1.21466 −0.607328 0.794451i \(-0.707759\pi\)
−0.607328 + 0.794451i \(0.707759\pi\)
\(440\) 0 0
\(441\) 15.1736 0.722553
\(442\) −1.29564 −0.0616275
\(443\) −35.6685 −1.69466 −0.847330 0.531067i \(-0.821791\pi\)
−0.847330 + 0.531067i \(0.821791\pi\)
\(444\) −21.7796 −1.03361
\(445\) 0 0
\(446\) −3.75023 −0.177578
\(447\) −34.0606 −1.61101
\(448\) 4.12489 0.194883
\(449\) −11.4399 −0.539883 −0.269941 0.962877i \(-0.587004\pi\)
−0.269941 + 0.962877i \(0.587004\pi\)
\(450\) 0 0
\(451\) −23.4693 −1.10512
\(452\) −19.2001 −0.903098
\(453\) 30.4078 1.42868
\(454\) −24.1542 −1.13361
\(455\) 0 0
\(456\) −2.12489 −0.0995069
\(457\) 19.4849 0.911463 0.455732 0.890117i \(-0.349378\pi\)
0.455732 + 0.890117i \(0.349378\pi\)
\(458\) −13.0596 −0.610237
\(459\) −1.62534 −0.0758645
\(460\) 0 0
\(461\) −29.3893 −1.36880 −0.684399 0.729108i \(-0.739935\pi\)
−0.684399 + 0.729108i \(0.739935\pi\)
\(462\) 23.1396 1.07655
\(463\) 12.1892 0.566481 0.283241 0.959049i \(-0.408591\pi\)
0.283241 + 0.959049i \(0.408591\pi\)
\(464\) −7.79518 −0.361882
\(465\) 0 0
\(466\) −6.43899 −0.298280
\(467\) −17.8889 −0.827799 −0.413899 0.910323i \(-0.635833\pi\)
−0.413899 + 0.910323i \(0.635833\pi\)
\(468\) 3.81078 0.176153
\(469\) −30.5445 −1.41041
\(470\) 0 0
\(471\) 38.3103 1.76525
\(472\) 3.15516 0.145228
\(473\) −22.8099 −1.04880
\(474\) −35.5904 −1.63472
\(475\) 0 0
\(476\) −2.12489 −0.0973940
\(477\) 8.31032 0.380504
\(478\) 22.8742 1.04624
\(479\) −1.15894 −0.0529534 −0.0264767 0.999649i \(-0.508429\pi\)
−0.0264767 + 0.999649i \(0.508429\pi\)
\(480\) 0 0
\(481\) 25.7796 1.17545
\(482\) 4.96972 0.226365
\(483\) −27.1240 −1.23418
\(484\) −4.03028 −0.183194
\(485\) 0 0
\(486\) 14.4390 0.654966
\(487\) 30.6888 1.39064 0.695320 0.718700i \(-0.255263\pi\)
0.695320 + 0.718700i \(0.255263\pi\)
\(488\) −12.6400 −0.572187
\(489\) −5.73841 −0.259500
\(490\) 0 0
\(491\) 3.67030 0.165638 0.0828192 0.996565i \(-0.473608\pi\)
0.0828192 + 0.996565i \(0.473608\pi\)
\(492\) −18.8898 −0.851618
\(493\) 4.01560 0.180853
\(494\) 2.51514 0.113161
\(495\) 0 0
\(496\) 3.67030 0.164801
\(497\) 45.9494 2.06111
\(498\) −6.96972 −0.312321
\(499\) 31.3893 1.40518 0.702590 0.711595i \(-0.252027\pi\)
0.702590 + 0.711595i \(0.252027\pi\)
\(500\) 0 0
\(501\) −19.0185 −0.849682
\(502\) 15.9201 0.710548
\(503\) −18.1542 −0.809458 −0.404729 0.914437i \(-0.632634\pi\)
−0.404729 + 0.914437i \(0.632634\pi\)
\(504\) 6.24977 0.278387
\(505\) 0 0
\(506\) −8.16984 −0.363194
\(507\) 14.1817 0.629829
\(508\) 14.3103 0.634918
\(509\) 11.5298 0.511050 0.255525 0.966802i \(-0.417752\pi\)
0.255525 + 0.966802i \(0.417752\pi\)
\(510\) 0 0
\(511\) −11.1552 −0.493475
\(512\) 1.00000 0.0441942
\(513\) 3.15516 0.139304
\(514\) −1.04965 −0.0462982
\(515\) 0 0
\(516\) −18.3591 −0.808213
\(517\) 13.1202 0.577025
\(518\) 42.2791 1.85764
\(519\) −0.659401 −0.0289445
\(520\) 0 0
\(521\) −42.5895 −1.86588 −0.932939 0.360035i \(-0.882765\pi\)
−0.932939 + 0.360035i \(0.882765\pi\)
\(522\) −11.8108 −0.516944
\(523\) 1.21571 0.0531594 0.0265797 0.999647i \(-0.491538\pi\)
0.0265797 + 0.999647i \(0.491538\pi\)
\(524\) −6.56009 −0.286579
\(525\) 0 0
\(526\) −11.9394 −0.520585
\(527\) −1.89071 −0.0823607
\(528\) 5.60975 0.244133
\(529\) −13.4234 −0.583626
\(530\) 0 0
\(531\) 4.78051 0.207456
\(532\) 4.12489 0.178836
\(533\) 22.3591 0.968478
\(534\) 16.1698 0.699737
\(535\) 0 0
\(536\) −7.40493 −0.319844
\(537\) 3.25069 0.140278
\(538\) −15.9394 −0.687198
\(539\) −26.4390 −1.13881
\(540\) 0 0
\(541\) −1.92007 −0.0825503 −0.0412751 0.999148i \(-0.513142\pi\)
−0.0412751 + 0.999148i \(0.513142\pi\)
\(542\) 1.46548 0.0629480
\(543\) 31.3406 1.34495
\(544\) −0.515138 −0.0220864
\(545\) 0 0
\(546\) −22.0450 −0.943437
\(547\) 10.9697 0.469032 0.234516 0.972112i \(-0.424650\pi\)
0.234516 + 0.972112i \(0.424650\pi\)
\(548\) −6.45459 −0.275726
\(549\) −19.1514 −0.817361
\(550\) 0 0
\(551\) −7.79518 −0.332086
\(552\) −6.57569 −0.279880
\(553\) 69.0890 2.93796
\(554\) 32.4995 1.38077
\(555\) 0 0
\(556\) −23.0596 −0.977946
\(557\) 23.5005 0.995746 0.497873 0.867250i \(-0.334114\pi\)
0.497873 + 0.867250i \(0.334114\pi\)
\(558\) 5.56101 0.235416
\(559\) 21.7309 0.919117
\(560\) 0 0
\(561\) −2.88979 −0.122007
\(562\) 1.04965 0.0442770
\(563\) −7.87890 −0.332056 −0.166028 0.986121i \(-0.553094\pi\)
−0.166028 + 0.986121i \(0.553094\pi\)
\(564\) 10.5601 0.444660
\(565\) 0 0
\(566\) −9.34060 −0.392615
\(567\) −46.4040 −1.94879
\(568\) 11.1396 0.467406
\(569\) −1.09083 −0.0457299 −0.0228650 0.999739i \(-0.507279\pi\)
−0.0228650 + 0.999739i \(0.507279\pi\)
\(570\) 0 0
\(571\) 21.4886 0.899272 0.449636 0.893212i \(-0.351554\pi\)
0.449636 + 0.893212i \(0.351554\pi\)
\(572\) −6.64002 −0.277633
\(573\) −38.5133 −1.60892
\(574\) 36.6694 1.53055
\(575\) 0 0
\(576\) 1.51514 0.0631308
\(577\) −16.1055 −0.670481 −0.335241 0.942133i \(-0.608818\pi\)
−0.335241 + 0.942133i \(0.608818\pi\)
\(578\) −16.7346 −0.696069
\(579\) 9.68968 0.402689
\(580\) 0 0
\(581\) 13.5298 0.561311
\(582\) −8.37088 −0.346984
\(583\) −14.4802 −0.599707
\(584\) −2.70436 −0.111907
\(585\) 0 0
\(586\) 25.5748 1.05648
\(587\) 0.480164 0.0198185 0.00990925 0.999951i \(-0.496846\pi\)
0.00990925 + 0.999951i \(0.496846\pi\)
\(588\) −21.2800 −0.877574
\(589\) 3.67030 0.151232
\(590\) 0 0
\(591\) 4.54828 0.187091
\(592\) 10.2498 0.421263
\(593\) 40.4683 1.66184 0.830918 0.556395i \(-0.187816\pi\)
0.830918 + 0.556395i \(0.187816\pi\)
\(594\) −8.32970 −0.341772
\(595\) 0 0
\(596\) 16.0294 0.656588
\(597\) −35.1963 −1.44049
\(598\) 7.78337 0.318286
\(599\) −36.1698 −1.47786 −0.738930 0.673782i \(-0.764669\pi\)
−0.738930 + 0.673782i \(0.764669\pi\)
\(600\) 0 0
\(601\) 24.3903 0.994899 0.497450 0.867493i \(-0.334270\pi\)
0.497450 + 0.867493i \(0.334270\pi\)
\(602\) 35.6391 1.45254
\(603\) −11.2195 −0.456893
\(604\) −14.3103 −0.582279
\(605\) 0 0
\(606\) 29.3212 1.19109
\(607\) 21.6897 0.880357 0.440178 0.897910i \(-0.354915\pi\)
0.440178 + 0.897910i \(0.354915\pi\)
\(608\) 1.00000 0.0405554
\(609\) 68.3241 2.76863
\(610\) 0 0
\(611\) −12.4995 −0.505677
\(612\) −0.780505 −0.0315501
\(613\) −26.2304 −1.05944 −0.529718 0.848174i \(-0.677702\pi\)
−0.529718 + 0.848174i \(0.677702\pi\)
\(614\) 9.43991 0.380964
\(615\) 0 0
\(616\) −10.8898 −0.438762
\(617\) −22.7200 −0.914671 −0.457335 0.889294i \(-0.651196\pi\)
−0.457335 + 0.889294i \(0.651196\pi\)
\(618\) −9.73085 −0.391432
\(619\) −32.9192 −1.32313 −0.661566 0.749887i \(-0.730108\pi\)
−0.661566 + 0.749887i \(0.730108\pi\)
\(620\) 0 0
\(621\) 9.76399 0.391816
\(622\) −17.4655 −0.700302
\(623\) −31.3893 −1.25759
\(624\) −5.34438 −0.213946
\(625\) 0 0
\(626\) −10.4546 −0.417849
\(627\) 5.60975 0.224032
\(628\) −18.0294 −0.719450
\(629\) −5.28005 −0.210529
\(630\) 0 0
\(631\) 17.2876 0.688209 0.344104 0.938931i \(-0.388183\pi\)
0.344104 + 0.938931i \(0.388183\pi\)
\(632\) 16.7493 0.666252
\(633\) 32.4764 1.29082
\(634\) 4.33348 0.172105
\(635\) 0 0
\(636\) −11.6547 −0.462139
\(637\) 25.1883 0.997997
\(638\) 20.5795 0.814749
\(639\) 16.8780 0.667683
\(640\) 0 0
\(641\) −9.29942 −0.367305 −0.183653 0.982991i \(-0.558792\pi\)
−0.183653 + 0.982991i \(0.558792\pi\)
\(642\) 22.0450 0.870045
\(643\) −3.40115 −0.134128 −0.0670642 0.997749i \(-0.521363\pi\)
−0.0670642 + 0.997749i \(0.521363\pi\)
\(644\) 12.7649 0.503008
\(645\) 0 0
\(646\) −0.515138 −0.0202678
\(647\) 14.9348 0.587146 0.293573 0.955937i \(-0.405156\pi\)
0.293573 + 0.955937i \(0.405156\pi\)
\(648\) −11.2498 −0.441933
\(649\) −8.32970 −0.326969
\(650\) 0 0
\(651\) −32.1698 −1.26084
\(652\) 2.70058 0.105763
\(653\) 21.1202 0.826497 0.413248 0.910618i \(-0.364394\pi\)
0.413248 + 0.910618i \(0.364394\pi\)
\(654\) −6.40585 −0.250489
\(655\) 0 0
\(656\) 8.88979 0.347088
\(657\) −4.09747 −0.159858
\(658\) −20.4995 −0.799155
\(659\) 27.6547 1.07727 0.538637 0.842538i \(-0.318939\pi\)
0.538637 + 0.842538i \(0.318939\pi\)
\(660\) 0 0
\(661\) 10.2342 0.398063 0.199032 0.979993i \(-0.436220\pi\)
0.199032 + 0.979993i \(0.436220\pi\)
\(662\) −4.56387 −0.177380
\(663\) 2.75309 0.106921
\(664\) 3.28005 0.127291
\(665\) 0 0
\(666\) 15.5298 0.601768
\(667\) −24.1231 −0.934048
\(668\) 8.95035 0.346299
\(669\) 7.96881 0.308092
\(670\) 0 0
\(671\) 33.3700 1.28823
\(672\) −8.76491 −0.338114
\(673\) −13.4693 −0.519202 −0.259601 0.965716i \(-0.583591\pi\)
−0.259601 + 0.965716i \(0.583591\pi\)
\(674\) 13.0109 0.501161
\(675\) 0 0
\(676\) −6.67408 −0.256695
\(677\) 15.1433 0.582006 0.291003 0.956722i \(-0.406011\pi\)
0.291003 + 0.956722i \(0.406011\pi\)
\(678\) 40.7980 1.56684
\(679\) 16.2498 0.623609
\(680\) 0 0
\(681\) 51.3250 1.96678
\(682\) −9.68968 −0.371037
\(683\) 15.8789 0.607589 0.303795 0.952738i \(-0.401746\pi\)
0.303795 + 0.952738i \(0.401746\pi\)
\(684\) 1.51514 0.0579328
\(685\) 0 0
\(686\) 12.4352 0.474778
\(687\) 27.7502 1.05874
\(688\) 8.64002 0.329398
\(689\) 13.7952 0.525555
\(690\) 0 0
\(691\) 27.3094 1.03890 0.519449 0.854501i \(-0.326137\pi\)
0.519449 + 0.854501i \(0.326137\pi\)
\(692\) 0.310323 0.0117967
\(693\) −16.4995 −0.626766
\(694\) 13.2195 0.501805
\(695\) 0 0
\(696\) 16.5639 0.627852
\(697\) −4.57947 −0.173460
\(698\) −20.4390 −0.773627
\(699\) 13.6821 0.517505
\(700\) 0 0
\(701\) −20.9697 −0.792016 −0.396008 0.918247i \(-0.629605\pi\)
−0.396008 + 0.918247i \(0.629605\pi\)
\(702\) 7.93567 0.299512
\(703\) 10.2498 0.386577
\(704\) −2.64002 −0.0994996
\(705\) 0 0
\(706\) 10.7044 0.402864
\(707\) −56.9192 −2.14067
\(708\) −6.70436 −0.251965
\(709\) 37.5592 1.41056 0.705282 0.708927i \(-0.250820\pi\)
0.705282 + 0.708927i \(0.250820\pi\)
\(710\) 0 0
\(711\) 25.3775 0.951731
\(712\) −7.60975 −0.285187
\(713\) 11.3581 0.425366
\(714\) 4.51514 0.168975
\(715\) 0 0
\(716\) −1.52982 −0.0571720
\(717\) −48.6050 −1.81519
\(718\) −4.31410 −0.161001
\(719\) 3.94323 0.147058 0.0735288 0.997293i \(-0.476574\pi\)
0.0735288 + 0.997293i \(0.476574\pi\)
\(720\) 0 0
\(721\) 18.8898 0.703493
\(722\) 1.00000 0.0372161
\(723\) −10.5601 −0.392734
\(724\) −14.7493 −0.548154
\(725\) 0 0
\(726\) 8.56387 0.317835
\(727\) 33.2139 1.23183 0.615917 0.787811i \(-0.288786\pi\)
0.615917 + 0.787811i \(0.288786\pi\)
\(728\) 10.3747 0.384510
\(729\) 3.06811 0.113634
\(730\) 0 0
\(731\) −4.45080 −0.164619
\(732\) 26.8586 0.992722
\(733\) −8.62065 −0.318411 −0.159205 0.987245i \(-0.550893\pi\)
−0.159205 + 0.987245i \(0.550893\pi\)
\(734\) −12.8099 −0.472821
\(735\) 0 0
\(736\) 3.09461 0.114069
\(737\) 19.5492 0.720104
\(738\) 13.4693 0.495811
\(739\) −45.1689 −1.66157 −0.830783 0.556597i \(-0.812107\pi\)
−0.830783 + 0.556597i \(0.812107\pi\)
\(740\) 0 0
\(741\) −5.34438 −0.196331
\(742\) 22.6244 0.830569
\(743\) −24.7905 −0.909475 −0.454737 0.890626i \(-0.650267\pi\)
−0.454737 + 0.890626i \(0.650267\pi\)
\(744\) −7.79897 −0.285924
\(745\) 0 0
\(746\) 0.704357 0.0257883
\(747\) 4.96972 0.181833
\(748\) 1.35998 0.0497257
\(749\) −42.7943 −1.56367
\(750\) 0 0
\(751\) −6.76869 −0.246993 −0.123497 0.992345i \(-0.539411\pi\)
−0.123497 + 0.992345i \(0.539411\pi\)
\(752\) −4.96972 −0.181227
\(753\) −33.8283 −1.23277
\(754\) −19.6060 −0.714007
\(755\) 0 0
\(756\) 13.0147 0.473339
\(757\) 45.2101 1.64319 0.821595 0.570072i \(-0.193085\pi\)
0.821595 + 0.570072i \(0.193085\pi\)
\(758\) 6.12489 0.222466
\(759\) 17.3600 0.630127
\(760\) 0 0
\(761\) 19.8851 0.720834 0.360417 0.932791i \(-0.382634\pi\)
0.360417 + 0.932791i \(0.382634\pi\)
\(762\) −30.4078 −1.10156
\(763\) 12.4352 0.450185
\(764\) 18.1249 0.655735
\(765\) 0 0
\(766\) 18.6400 0.673491
\(767\) 7.93567 0.286540
\(768\) −2.12489 −0.0766752
\(769\) 8.07615 0.291233 0.145617 0.989341i \(-0.453483\pi\)
0.145617 + 0.989341i \(0.453483\pi\)
\(770\) 0 0
\(771\) 2.23039 0.0803257
\(772\) −4.56009 −0.164121
\(773\) 45.9456 1.65255 0.826275 0.563267i \(-0.190456\pi\)
0.826275 + 0.563267i \(0.190456\pi\)
\(774\) 13.0908 0.470540
\(775\) 0 0
\(776\) 3.93945 0.141418
\(777\) −89.8383 −3.22293
\(778\) 13.9201 0.499059
\(779\) 8.88979 0.318510
\(780\) 0 0
\(781\) −29.4087 −1.05233
\(782\) −1.59415 −0.0570067
\(783\) −24.5951 −0.878956
\(784\) 10.0147 0.357667
\(785\) 0 0
\(786\) 13.9394 0.497204
\(787\) −17.8439 −0.636067 −0.318034 0.948079i \(-0.603022\pi\)
−0.318034 + 0.948079i \(0.603022\pi\)
\(788\) −2.14048 −0.0762515
\(789\) 25.3700 0.903194
\(790\) 0 0
\(791\) −79.1983 −2.81597
\(792\) −4.00000 −0.142134
\(793\) −31.7914 −1.12895
\(794\) −28.3784 −1.00711
\(795\) 0 0
\(796\) 16.5639 0.587091
\(797\) −37.3326 −1.32239 −0.661194 0.750215i \(-0.729950\pi\)
−0.661194 + 0.750215i \(0.729950\pi\)
\(798\) −8.76491 −0.310274
\(799\) 2.56009 0.0905696
\(800\) 0 0
\(801\) −11.5298 −0.407386
\(802\) 5.54920 0.195949
\(803\) 7.13957 0.251950
\(804\) 15.7346 0.554918
\(805\) 0 0
\(806\) 9.23131 0.325159
\(807\) 33.8695 1.19226
\(808\) −13.7990 −0.485446
\(809\) −38.6950 −1.36044 −0.680221 0.733007i \(-0.738116\pi\)
−0.680221 + 0.733007i \(0.738116\pi\)
\(810\) 0 0
\(811\) 13.3444 0.468585 0.234292 0.972166i \(-0.424723\pi\)
0.234292 + 0.972166i \(0.424723\pi\)
\(812\) −32.1542 −1.12839
\(813\) −3.11399 −0.109212
\(814\) −27.0596 −0.948440
\(815\) 0 0
\(816\) 1.09461 0.0383190
\(817\) 8.64002 0.302276
\(818\) −5.01090 −0.175202
\(819\) 15.7190 0.549268
\(820\) 0 0
\(821\) −37.3482 −1.30346 −0.651730 0.758451i \(-0.725956\pi\)
−0.651730 + 0.758451i \(0.725956\pi\)
\(822\) 13.7153 0.478374
\(823\) −51.5630 −1.79737 −0.898686 0.438593i \(-0.855477\pi\)
−0.898686 + 0.438593i \(0.855477\pi\)
\(824\) 4.57947 0.159533
\(825\) 0 0
\(826\) 13.0147 0.452839
\(827\) −48.5639 −1.68873 −0.844366 0.535767i \(-0.820022\pi\)
−0.844366 + 0.535767i \(0.820022\pi\)
\(828\) 4.68876 0.162946
\(829\) 0.325919 0.0113196 0.00565982 0.999984i \(-0.498198\pi\)
0.00565982 + 0.999984i \(0.498198\pi\)
\(830\) 0 0
\(831\) −69.0578 −2.39559
\(832\) 2.51514 0.0871967
\(833\) −5.15894 −0.178747
\(834\) 48.9991 1.69670
\(835\) 0 0
\(836\) −2.64002 −0.0913071
\(837\) 11.5804 0.400277
\(838\) −15.8889 −0.548872
\(839\) −17.1202 −0.591055 −0.295527 0.955334i \(-0.595495\pi\)
−0.295527 + 0.955334i \(0.595495\pi\)
\(840\) 0 0
\(841\) 31.7649 1.09534
\(842\) −2.38647 −0.0822432
\(843\) −2.23039 −0.0768188
\(844\) −15.2838 −0.526091
\(845\) 0 0
\(846\) −7.52982 −0.258880
\(847\) −16.6244 −0.571222
\(848\) 5.48486 0.188351
\(849\) 19.8477 0.681171
\(850\) 0 0
\(851\) 31.7190 1.08731
\(852\) −23.6703 −0.810931
\(853\) −24.9092 −0.852874 −0.426437 0.904517i \(-0.640231\pi\)
−0.426437 + 0.904517i \(0.640231\pi\)
\(854\) −52.1386 −1.78415
\(855\) 0 0
\(856\) −10.3747 −0.354598
\(857\) −11.6509 −0.397988 −0.198994 0.980001i \(-0.563767\pi\)
−0.198994 + 0.980001i \(0.563767\pi\)
\(858\) 14.1093 0.481683
\(859\) −5.35998 −0.182880 −0.0914400 0.995811i \(-0.529147\pi\)
−0.0914400 + 0.995811i \(0.529147\pi\)
\(860\) 0 0
\(861\) −77.9182 −2.65545
\(862\) −2.35906 −0.0803499
\(863\) −41.0790 −1.39835 −0.699173 0.714953i \(-0.746448\pi\)
−0.699173 + 0.714953i \(0.746448\pi\)
\(864\) 3.15516 0.107341
\(865\) 0 0
\(866\) −0.640023 −0.0217489
\(867\) 35.5592 1.20765
\(868\) 15.1396 0.513870
\(869\) −44.2186 −1.50001
\(870\) 0 0
\(871\) −18.6244 −0.631065
\(872\) 3.01468 0.102090
\(873\) 5.96881 0.202014
\(874\) 3.09461 0.104677
\(875\) 0 0
\(876\) 5.74645 0.194154
\(877\) 45.9532 1.55173 0.775865 0.630899i \(-0.217314\pi\)
0.775865 + 0.630899i \(0.217314\pi\)
\(878\) −25.4499 −0.858892
\(879\) −54.3435 −1.83296
\(880\) 0 0
\(881\) −6.90917 −0.232776 −0.116388 0.993204i \(-0.537132\pi\)
−0.116388 + 0.993204i \(0.537132\pi\)
\(882\) 15.1736 0.510922
\(883\) 48.5213 1.63287 0.816437 0.577435i \(-0.195946\pi\)
0.816437 + 0.577435i \(0.195946\pi\)
\(884\) −1.29564 −0.0435772
\(885\) 0 0
\(886\) −35.6685 −1.19831
\(887\) 23.7990 0.799091 0.399546 0.916713i \(-0.369168\pi\)
0.399546 + 0.916713i \(0.369168\pi\)
\(888\) −21.7796 −0.730875
\(889\) 59.0284 1.97975
\(890\) 0 0
\(891\) 29.6997 0.994976
\(892\) −3.75023 −0.125567
\(893\) −4.96972 −0.166305
\(894\) −34.0606 −1.13916
\(895\) 0 0
\(896\) 4.12489 0.137803
\(897\) −16.5388 −0.552213
\(898\) −11.4399 −0.381755
\(899\) −28.6107 −0.954219
\(900\) 0 0
\(901\) −2.82546 −0.0941298
\(902\) −23.4693 −0.781441
\(903\) −75.7290 −2.52010
\(904\) −19.2001 −0.638586
\(905\) 0 0
\(906\) 30.4078 1.01023
\(907\) 17.9726 0.596770 0.298385 0.954446i \(-0.403552\pi\)
0.298385 + 0.954446i \(0.403552\pi\)
\(908\) −24.1542 −0.801587
\(909\) −20.9073 −0.693453
\(910\) 0 0
\(911\) 19.5592 0.648024 0.324012 0.946053i \(-0.394968\pi\)
0.324012 + 0.946053i \(0.394968\pi\)
\(912\) −2.12489 −0.0703620
\(913\) −8.65940 −0.286584
\(914\) 19.4849 0.644502
\(915\) 0 0
\(916\) −13.0596 −0.431503
\(917\) −27.0596 −0.893588
\(918\) −1.62534 −0.0536443
\(919\) 35.4948 1.17087 0.585433 0.810720i \(-0.300924\pi\)
0.585433 + 0.810720i \(0.300924\pi\)
\(920\) 0 0
\(921\) −20.0587 −0.660957
\(922\) −29.3893 −0.967886
\(923\) 28.0175 0.922209
\(924\) 23.1396 0.761236
\(925\) 0 0
\(926\) 12.1892 0.400563
\(927\) 6.93853 0.227891
\(928\) −7.79518 −0.255889
\(929\) 17.9532 0.589026 0.294513 0.955648i \(-0.404843\pi\)
0.294513 + 0.955648i \(0.404843\pi\)
\(930\) 0 0
\(931\) 10.0147 0.328218
\(932\) −6.43899 −0.210916
\(933\) 37.1122 1.21500
\(934\) −17.8889 −0.585342
\(935\) 0 0
\(936\) 3.81078 0.124559
\(937\) −5.66652 −0.185117 −0.0925585 0.995707i \(-0.529505\pi\)
−0.0925585 + 0.995707i \(0.529505\pi\)
\(938\) −30.5445 −0.997313
\(939\) 22.2148 0.724953
\(940\) 0 0
\(941\) −24.7044 −0.805339 −0.402670 0.915345i \(-0.631918\pi\)
−0.402670 + 0.915345i \(0.631918\pi\)
\(942\) 38.3103 1.24822
\(943\) 27.5104 0.895863
\(944\) 3.15516 0.102692
\(945\) 0 0
\(946\) −22.8099 −0.741613
\(947\) 13.5904 0.441628 0.220814 0.975316i \(-0.429129\pi\)
0.220814 + 0.975316i \(0.429129\pi\)
\(948\) −35.5904 −1.15592
\(949\) −6.80183 −0.220797
\(950\) 0 0
\(951\) −9.20815 −0.298595
\(952\) −2.12489 −0.0688679
\(953\) 24.7375 0.801326 0.400663 0.916225i \(-0.368780\pi\)
0.400663 + 0.916225i \(0.368780\pi\)
\(954\) 8.31032 0.269057
\(955\) 0 0
\(956\) 22.8742 0.739804
\(957\) −43.7290 −1.41356
\(958\) −1.15894 −0.0374437
\(959\) −26.6244 −0.859748
\(960\) 0 0
\(961\) −17.5289 −0.565448
\(962\) 25.7796 0.831167
\(963\) −15.7190 −0.506539
\(964\) 4.96972 0.160064
\(965\) 0 0
\(966\) −27.1240 −0.872699
\(967\) −35.6897 −1.14770 −0.573851 0.818960i \(-0.694551\pi\)
−0.573851 + 0.818960i \(0.694551\pi\)
\(968\) −4.03028 −0.129538
\(969\) 1.09461 0.0351639
\(970\) 0 0
\(971\) 16.4995 0.529495 0.264748 0.964318i \(-0.414711\pi\)
0.264748 + 0.964318i \(0.414711\pi\)
\(972\) 14.4390 0.463131
\(973\) −95.1184 −3.04935
\(974\) 30.6888 0.983331
\(975\) 0 0
\(976\) −12.6400 −0.404597
\(977\) 49.8501 1.59485 0.797423 0.603420i \(-0.206196\pi\)
0.797423 + 0.603420i \(0.206196\pi\)
\(978\) −5.73841 −0.183494
\(979\) 20.0899 0.642076
\(980\) 0 0
\(981\) 4.56766 0.145834
\(982\) 3.67030 0.117124
\(983\) −5.19014 −0.165540 −0.0827698 0.996569i \(-0.526377\pi\)
−0.0827698 + 0.996569i \(0.526377\pi\)
\(984\) −18.8898 −0.602185
\(985\) 0 0
\(986\) 4.01560 0.127883
\(987\) 43.5592 1.38650
\(988\) 2.51514 0.0800172
\(989\) 26.7375 0.850203
\(990\) 0 0
\(991\) −32.4272 −1.03008 −0.515042 0.857165i \(-0.672224\pi\)
−0.515042 + 0.857165i \(0.672224\pi\)
\(992\) 3.67030 0.116532
\(993\) 9.69771 0.307748
\(994\) 45.9494 1.45743
\(995\) 0 0
\(996\) −6.96972 −0.220844
\(997\) 10.1992 0.323012 0.161506 0.986872i \(-0.448365\pi\)
0.161506 + 0.986872i \(0.448365\pi\)
\(998\) 31.3893 0.993612
\(999\) 32.3397 1.02318
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.a.n.1.1 3
3.2 odd 2 8550.2.a.ck.1.3 3
4.3 odd 2 7600.2.a.bi.1.3 3
5.2 odd 4 190.2.b.b.39.6 yes 6
5.3 odd 4 190.2.b.b.39.1 6
5.4 even 2 950.2.a.i.1.3 3
15.2 even 4 1710.2.d.d.1369.1 6
15.8 even 4 1710.2.d.d.1369.4 6
15.14 odd 2 8550.2.a.cl.1.1 3
20.3 even 4 1520.2.d.j.609.5 6
20.7 even 4 1520.2.d.j.609.2 6
20.19 odd 2 7600.2.a.cd.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
190.2.b.b.39.1 6 5.3 odd 4
190.2.b.b.39.6 yes 6 5.2 odd 4
950.2.a.i.1.3 3 5.4 even 2
950.2.a.n.1.1 3 1.1 even 1 trivial
1520.2.d.j.609.2 6 20.7 even 4
1520.2.d.j.609.5 6 20.3 even 4
1710.2.d.d.1369.1 6 15.2 even 4
1710.2.d.d.1369.4 6 15.8 even 4
7600.2.a.bi.1.3 3 4.3 odd 2
7600.2.a.cd.1.1 3 20.19 odd 2
8550.2.a.ck.1.3 3 3.2 odd 2
8550.2.a.cl.1.1 3 15.14 odd 2