Properties

Label 3381.2.a.bc.1.4
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.7997584.1
Defining polynomial: \(x^{6} - x^{5} - 7 x^{4} + 5 x^{3} + 12 x^{2} - 4 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.289957\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.289957 q^{2} -1.00000 q^{3} -1.91593 q^{4} +2.32621 q^{5} -0.289957 q^{6} -1.13545 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.289957 q^{2} -1.00000 q^{3} -1.91593 q^{4} +2.32621 q^{5} -0.289957 q^{6} -1.13545 q^{8} +1.00000 q^{9} +0.674501 q^{10} +0.730971 q^{11} +1.91593 q^{12} -0.115233 q^{13} -2.32621 q^{15} +3.50262 q^{16} +0.789022 q^{17} +0.289957 q^{18} -6.92342 q^{19} -4.45685 q^{20} +0.211950 q^{22} -1.00000 q^{23} +1.13545 q^{24} +0.411258 q^{25} -0.0334127 q^{26} -1.00000 q^{27} -6.26700 q^{29} -0.674501 q^{30} +1.49584 q^{31} +3.28651 q^{32} -0.730971 q^{33} +0.228782 q^{34} -1.91593 q^{36} +4.14294 q^{37} -2.00749 q^{38} +0.115233 q^{39} -2.64129 q^{40} +8.20805 q^{41} +1.50333 q^{43} -1.40049 q^{44} +2.32621 q^{45} -0.289957 q^{46} -3.78787 q^{47} -3.50262 q^{48} +0.119247 q^{50} -0.789022 q^{51} +0.220778 q^{52} -6.81735 q^{53} -0.289957 q^{54} +1.70039 q^{55} +6.92342 q^{57} -1.81716 q^{58} -6.18845 q^{59} +4.45685 q^{60} -2.39074 q^{61} +0.433729 q^{62} -6.05229 q^{64} -0.268057 q^{65} -0.211950 q^{66} -8.37206 q^{67} -1.51171 q^{68} +1.00000 q^{69} -1.53737 q^{71} -1.13545 q^{72} +9.90095 q^{73} +1.20127 q^{74} -0.411258 q^{75} +13.2647 q^{76} +0.0334127 q^{78} +0.483967 q^{79} +8.14783 q^{80} +1.00000 q^{81} +2.37998 q^{82} -1.84208 q^{83} +1.83543 q^{85} +0.435901 q^{86} +6.26700 q^{87} -0.829981 q^{88} +3.06457 q^{89} +0.674501 q^{90} +1.91593 q^{92} -1.49584 q^{93} -1.09832 q^{94} -16.1053 q^{95} -3.28651 q^{96} +14.8351 q^{97} +0.730971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{2} - 6q^{3} + 3q^{4} + 3q^{5} + q^{6} - 3q^{8} + 6q^{9} + O(q^{10}) \) \( 6q - q^{2} - 6q^{3} + 3q^{4} + 3q^{5} + q^{6} - 3q^{8} + 6q^{9} - 3q^{10} - 14q^{11} - 3q^{12} - 3q^{15} - 7q^{16} + 15q^{17} - q^{18} - q^{19} + 17q^{20} - 6q^{22} - 6q^{23} + 3q^{24} + 9q^{25} - 15q^{26} - 6q^{27} - 6q^{29} + 3q^{30} - 11q^{31} + 3q^{32} + 14q^{33} + 15q^{34} + 3q^{36} - 5q^{37} + 14q^{38} - 17q^{40} + 18q^{41} - 37q^{43} - 10q^{44} + 3q^{45} + q^{46} + 3q^{47} + 7q^{48} - 30q^{50} - 15q^{51} - 7q^{52} - 15q^{53} + q^{54} - 2q^{55} + q^{57} + 4q^{58} - 2q^{59} - 17q^{60} - 12q^{61} + 36q^{62} - 23q^{64} - 17q^{65} + 6q^{66} - 10q^{67} - q^{68} + 6q^{69} - 21q^{71} - 3q^{72} - 8q^{73} - 16q^{74} - 9q^{75} + 18q^{76} + 15q^{78} - 17q^{79} + 3q^{80} + 6q^{81} - 48q^{82} + 12q^{83} - 13q^{85} + 22q^{86} + 6q^{87} - 2q^{88} + 18q^{89} - 3q^{90} - 3q^{92} + 11q^{93} + 3q^{94} - 16q^{95} - 3q^{96} - 2q^{97} - 14q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.289957 0.205030 0.102515 0.994731i \(-0.467311\pi\)
0.102515 + 0.994731i \(0.467311\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.91593 −0.957963
\(5\) 2.32621 1.04031 0.520157 0.854071i \(-0.325874\pi\)
0.520157 + 0.854071i \(0.325874\pi\)
\(6\) −0.289957 −0.118374
\(7\) 0 0
\(8\) −1.13545 −0.401442
\(9\) 1.00000 0.333333
\(10\) 0.674501 0.213296
\(11\) 0.730971 0.220396 0.110198 0.993910i \(-0.464851\pi\)
0.110198 + 0.993910i \(0.464851\pi\)
\(12\) 1.91593 0.553080
\(13\) −0.115233 −0.0319599 −0.0159800 0.999872i \(-0.505087\pi\)
−0.0159800 + 0.999872i \(0.505087\pi\)
\(14\) 0 0
\(15\) −2.32621 −0.600625
\(16\) 3.50262 0.875655
\(17\) 0.789022 0.191366 0.0956830 0.995412i \(-0.469496\pi\)
0.0956830 + 0.995412i \(0.469496\pi\)
\(18\) 0.289957 0.0683435
\(19\) −6.92342 −1.58834 −0.794170 0.607695i \(-0.792094\pi\)
−0.794170 + 0.607695i \(0.792094\pi\)
\(20\) −4.45685 −0.996581
\(21\) 0 0
\(22\) 0.211950 0.0451879
\(23\) −1.00000 −0.208514
\(24\) 1.13545 0.231773
\(25\) 0.411258 0.0822516
\(26\) −0.0334127 −0.00655276
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.26700 −1.16375 −0.581877 0.813277i \(-0.697681\pi\)
−0.581877 + 0.813277i \(0.697681\pi\)
\(30\) −0.674501 −0.123146
\(31\) 1.49584 0.268661 0.134330 0.990937i \(-0.457112\pi\)
0.134330 + 0.990937i \(0.457112\pi\)
\(32\) 3.28651 0.580978
\(33\) −0.730971 −0.127246
\(34\) 0.228782 0.0392358
\(35\) 0 0
\(36\) −1.91593 −0.319321
\(37\) 4.14294 0.681095 0.340548 0.940227i \(-0.389388\pi\)
0.340548 + 0.940227i \(0.389388\pi\)
\(38\) −2.00749 −0.325658
\(39\) 0.115233 0.0184521
\(40\) −2.64129 −0.417625
\(41\) 8.20805 1.28188 0.640941 0.767590i \(-0.278544\pi\)
0.640941 + 0.767590i \(0.278544\pi\)
\(42\) 0 0
\(43\) 1.50333 0.229256 0.114628 0.993408i \(-0.463432\pi\)
0.114628 + 0.993408i \(0.463432\pi\)
\(44\) −1.40049 −0.211131
\(45\) 2.32621 0.346771
\(46\) −0.289957 −0.0427518
\(47\) −3.78787 −0.552518 −0.276259 0.961083i \(-0.589095\pi\)
−0.276259 + 0.961083i \(0.589095\pi\)
\(48\) −3.50262 −0.505559
\(49\) 0 0
\(50\) 0.119247 0.0168641
\(51\) −0.789022 −0.110485
\(52\) 0.220778 0.0306164
\(53\) −6.81735 −0.936434 −0.468217 0.883613i \(-0.655104\pi\)
−0.468217 + 0.883613i \(0.655104\pi\)
\(54\) −0.289957 −0.0394581
\(55\) 1.70039 0.229281
\(56\) 0 0
\(57\) 6.92342 0.917029
\(58\) −1.81716 −0.238605
\(59\) −6.18845 −0.805668 −0.402834 0.915273i \(-0.631975\pi\)
−0.402834 + 0.915273i \(0.631975\pi\)
\(60\) 4.45685 0.575376
\(61\) −2.39074 −0.306103 −0.153051 0.988218i \(-0.548910\pi\)
−0.153051 + 0.988218i \(0.548910\pi\)
\(62\) 0.433729 0.0550836
\(63\) 0 0
\(64\) −6.05229 −0.756537
\(65\) −0.268057 −0.0332483
\(66\) −0.211950 −0.0260892
\(67\) −8.37206 −1.02281 −0.511405 0.859340i \(-0.670875\pi\)
−0.511405 + 0.859340i \(0.670875\pi\)
\(68\) −1.51171 −0.183321
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −1.53737 −0.182452 −0.0912260 0.995830i \(-0.529079\pi\)
−0.0912260 + 0.995830i \(0.529079\pi\)
\(72\) −1.13545 −0.133814
\(73\) 9.90095 1.15882 0.579409 0.815037i \(-0.303283\pi\)
0.579409 + 0.815037i \(0.303283\pi\)
\(74\) 1.20127 0.139645
\(75\) −0.411258 −0.0474880
\(76\) 13.2647 1.52157
\(77\) 0 0
\(78\) 0.0334127 0.00378324
\(79\) 0.483967 0.0544505 0.0272253 0.999629i \(-0.491333\pi\)
0.0272253 + 0.999629i \(0.491333\pi\)
\(80\) 8.14783 0.910955
\(81\) 1.00000 0.111111
\(82\) 2.37998 0.262825
\(83\) −1.84208 −0.202195 −0.101097 0.994877i \(-0.532235\pi\)
−0.101097 + 0.994877i \(0.532235\pi\)
\(84\) 0 0
\(85\) 1.83543 0.199081
\(86\) 0.435901 0.0470044
\(87\) 6.26700 0.671893
\(88\) −0.829981 −0.0884762
\(89\) 3.06457 0.324844 0.162422 0.986721i \(-0.448069\pi\)
0.162422 + 0.986721i \(0.448069\pi\)
\(90\) 0.674501 0.0710986
\(91\) 0 0
\(92\) 1.91593 0.199749
\(93\) −1.49584 −0.155111
\(94\) −1.09832 −0.113283
\(95\) −16.1053 −1.65237
\(96\) −3.28651 −0.335428
\(97\) 14.8351 1.50628 0.753140 0.657860i \(-0.228538\pi\)
0.753140 + 0.657860i \(0.228538\pi\)
\(98\) 0 0
\(99\) 0.730971 0.0734654
\(100\) −0.787940 −0.0787940
\(101\) 10.8469 1.07931 0.539654 0.841887i \(-0.318555\pi\)
0.539654 + 0.841887i \(0.318555\pi\)
\(102\) −0.228782 −0.0226528
\(103\) −9.39923 −0.926134 −0.463067 0.886323i \(-0.653251\pi\)
−0.463067 + 0.886323i \(0.653251\pi\)
\(104\) 0.130841 0.0128301
\(105\) 0 0
\(106\) −1.97674 −0.191998
\(107\) −17.7498 −1.71593 −0.857967 0.513704i \(-0.828273\pi\)
−0.857967 + 0.513704i \(0.828273\pi\)
\(108\) 1.91593 0.184360
\(109\) −9.77303 −0.936087 −0.468043 0.883705i \(-0.655041\pi\)
−0.468043 + 0.883705i \(0.655041\pi\)
\(110\) 0.493041 0.0470096
\(111\) −4.14294 −0.393231
\(112\) 0 0
\(113\) −3.68270 −0.346440 −0.173220 0.984883i \(-0.555417\pi\)
−0.173220 + 0.984883i \(0.555417\pi\)
\(114\) 2.00749 0.188019
\(115\) −2.32621 −0.216920
\(116\) 12.0071 1.11483
\(117\) −0.115233 −0.0106533
\(118\) −1.79438 −0.165186
\(119\) 0 0
\(120\) 2.64129 0.241116
\(121\) −10.4657 −0.951426
\(122\) −0.693211 −0.0627604
\(123\) −8.20805 −0.740095
\(124\) −2.86591 −0.257367
\(125\) −10.6744 −0.954746
\(126\) 0 0
\(127\) −3.53396 −0.313588 −0.156794 0.987631i \(-0.550116\pi\)
−0.156794 + 0.987631i \(0.550116\pi\)
\(128\) −8.32792 −0.736091
\(129\) −1.50333 −0.132361
\(130\) −0.0777249 −0.00681692
\(131\) −4.35221 −0.380254 −0.190127 0.981759i \(-0.560890\pi\)
−0.190127 + 0.981759i \(0.560890\pi\)
\(132\) 1.40049 0.121897
\(133\) 0 0
\(134\) −2.42754 −0.209707
\(135\) −2.32621 −0.200208
\(136\) −0.895894 −0.0768223
\(137\) −17.1154 −1.46227 −0.731133 0.682235i \(-0.761008\pi\)
−0.731133 + 0.682235i \(0.761008\pi\)
\(138\) 0.289957 0.0246828
\(139\) 5.41785 0.459536 0.229768 0.973245i \(-0.426203\pi\)
0.229768 + 0.973245i \(0.426203\pi\)
\(140\) 0 0
\(141\) 3.78787 0.318996
\(142\) −0.445770 −0.0374082
\(143\) −0.0842321 −0.00704385
\(144\) 3.50262 0.291885
\(145\) −14.5784 −1.21067
\(146\) 2.87085 0.237593
\(147\) 0 0
\(148\) −7.93756 −0.652464
\(149\) −15.2241 −1.24721 −0.623604 0.781740i \(-0.714332\pi\)
−0.623604 + 0.781740i \(0.714332\pi\)
\(150\) −0.119247 −0.00973649
\(151\) 14.3805 1.17027 0.585136 0.810935i \(-0.301041\pi\)
0.585136 + 0.810935i \(0.301041\pi\)
\(152\) 7.86119 0.637627
\(153\) 0.789022 0.0637887
\(154\) 0 0
\(155\) 3.47964 0.279491
\(156\) −0.220778 −0.0176764
\(157\) −4.46641 −0.356458 −0.178229 0.983989i \(-0.557037\pi\)
−0.178229 + 0.983989i \(0.557037\pi\)
\(158\) 0.140330 0.0111640
\(159\) 6.81735 0.540651
\(160\) 7.64511 0.604399
\(161\) 0 0
\(162\) 0.289957 0.0227812
\(163\) 0.785722 0.0615425 0.0307713 0.999526i \(-0.490204\pi\)
0.0307713 + 0.999526i \(0.490204\pi\)
\(164\) −15.7260 −1.22800
\(165\) −1.70039 −0.132375
\(166\) −0.534124 −0.0414560
\(167\) 5.52754 0.427734 0.213867 0.976863i \(-0.431394\pi\)
0.213867 + 0.976863i \(0.431394\pi\)
\(168\) 0 0
\(169\) −12.9867 −0.998979
\(170\) 0.532196 0.0408176
\(171\) −6.92342 −0.529447
\(172\) −2.88027 −0.219618
\(173\) −9.95116 −0.756573 −0.378286 0.925689i \(-0.623487\pi\)
−0.378286 + 0.925689i \(0.623487\pi\)
\(174\) 1.81716 0.137759
\(175\) 0 0
\(176\) 2.56031 0.192991
\(177\) 6.18845 0.465153
\(178\) 0.888593 0.0666029
\(179\) −17.4916 −1.30739 −0.653693 0.756760i \(-0.726781\pi\)
−0.653693 + 0.756760i \(0.726781\pi\)
\(180\) −4.45685 −0.332194
\(181\) −19.8587 −1.47608 −0.738042 0.674754i \(-0.764250\pi\)
−0.738042 + 0.674754i \(0.764250\pi\)
\(182\) 0 0
\(183\) 2.39074 0.176729
\(184\) 1.13545 0.0837064
\(185\) 9.63736 0.708553
\(186\) −0.433729 −0.0318025
\(187\) 0.576752 0.0421763
\(188\) 7.25728 0.529291
\(189\) 0 0
\(190\) −4.66985 −0.338787
\(191\) 14.1099 1.02095 0.510477 0.859892i \(-0.329469\pi\)
0.510477 + 0.859892i \(0.329469\pi\)
\(192\) 6.05229 0.436787
\(193\) −12.7248 −0.915953 −0.457976 0.888964i \(-0.651426\pi\)
−0.457976 + 0.888964i \(0.651426\pi\)
\(194\) 4.30155 0.308833
\(195\) 0.268057 0.0191959
\(196\) 0 0
\(197\) −14.1582 −1.00873 −0.504364 0.863491i \(-0.668273\pi\)
−0.504364 + 0.863491i \(0.668273\pi\)
\(198\) 0.211950 0.0150626
\(199\) −4.23945 −0.300527 −0.150263 0.988646i \(-0.548012\pi\)
−0.150263 + 0.988646i \(0.548012\pi\)
\(200\) −0.466963 −0.0330193
\(201\) 8.37206 0.590520
\(202\) 3.14514 0.221291
\(203\) 0 0
\(204\) 1.51171 0.105841
\(205\) 19.0937 1.33356
\(206\) −2.72537 −0.189886
\(207\) −1.00000 −0.0695048
\(208\) −0.403618 −0.0279859
\(209\) −5.06082 −0.350064
\(210\) 0 0
\(211\) −26.3917 −1.81688 −0.908440 0.418015i \(-0.862726\pi\)
−0.908440 + 0.418015i \(0.862726\pi\)
\(212\) 13.0615 0.897069
\(213\) 1.53737 0.105339
\(214\) −5.14667 −0.351819
\(215\) 3.49706 0.238498
\(216\) 1.13545 0.0772575
\(217\) 0 0
\(218\) −2.83376 −0.191926
\(219\) −9.90095 −0.669044
\(220\) −3.25783 −0.219643
\(221\) −0.0909215 −0.00611604
\(222\) −1.20127 −0.0806242
\(223\) −9.57370 −0.641102 −0.320551 0.947231i \(-0.603868\pi\)
−0.320551 + 0.947231i \(0.603868\pi\)
\(224\) 0 0
\(225\) 0.411258 0.0274172
\(226\) −1.06783 −0.0710307
\(227\) −22.8593 −1.51723 −0.758614 0.651540i \(-0.774123\pi\)
−0.758614 + 0.651540i \(0.774123\pi\)
\(228\) −13.2647 −0.878479
\(229\) −2.08062 −0.137492 −0.0687458 0.997634i \(-0.521900\pi\)
−0.0687458 + 0.997634i \(0.521900\pi\)
\(230\) −0.674501 −0.0444753
\(231\) 0 0
\(232\) 7.11586 0.467179
\(233\) 5.01702 0.328676 0.164338 0.986404i \(-0.447451\pi\)
0.164338 + 0.986404i \(0.447451\pi\)
\(234\) −0.0334127 −0.00218425
\(235\) −8.81139 −0.574792
\(236\) 11.8566 0.771800
\(237\) −0.483967 −0.0314370
\(238\) 0 0
\(239\) 20.5640 1.33018 0.665089 0.746764i \(-0.268394\pi\)
0.665089 + 0.746764i \(0.268394\pi\)
\(240\) −8.14783 −0.525940
\(241\) 26.6259 1.71513 0.857563 0.514379i \(-0.171978\pi\)
0.857563 + 0.514379i \(0.171978\pi\)
\(242\) −3.03460 −0.195071
\(243\) −1.00000 −0.0641500
\(244\) 4.58048 0.293235
\(245\) 0 0
\(246\) −2.37998 −0.151742
\(247\) 0.797808 0.0507633
\(248\) −1.69845 −0.107852
\(249\) 1.84208 0.116737
\(250\) −3.09511 −0.195752
\(251\) −1.18435 −0.0747552 −0.0373776 0.999301i \(-0.511900\pi\)
−0.0373776 + 0.999301i \(0.511900\pi\)
\(252\) 0 0
\(253\) −0.730971 −0.0459558
\(254\) −1.02469 −0.0642951
\(255\) −1.83543 −0.114939
\(256\) 9.68985 0.605616
\(257\) −9.21806 −0.575006 −0.287503 0.957780i \(-0.592825\pi\)
−0.287503 + 0.957780i \(0.592825\pi\)
\(258\) −0.435901 −0.0271380
\(259\) 0 0
\(260\) 0.513577 0.0318507
\(261\) −6.26700 −0.387918
\(262\) −1.26195 −0.0779637
\(263\) −22.1016 −1.36284 −0.681422 0.731891i \(-0.738638\pi\)
−0.681422 + 0.731891i \(0.738638\pi\)
\(264\) 0.829981 0.0510818
\(265\) −15.8586 −0.974185
\(266\) 0 0
\(267\) −3.06457 −0.187549
\(268\) 16.0402 0.979813
\(269\) 29.3526 1.78966 0.894829 0.446408i \(-0.147297\pi\)
0.894829 + 0.446408i \(0.147297\pi\)
\(270\) −0.674501 −0.0410488
\(271\) 15.8453 0.962533 0.481266 0.876574i \(-0.340177\pi\)
0.481266 + 0.876574i \(0.340177\pi\)
\(272\) 2.76364 0.167571
\(273\) 0 0
\(274\) −4.96272 −0.299809
\(275\) 0.300618 0.0181279
\(276\) −1.91593 −0.115325
\(277\) 24.4377 1.46832 0.734160 0.678977i \(-0.237576\pi\)
0.734160 + 0.678977i \(0.237576\pi\)
\(278\) 1.57094 0.0942189
\(279\) 1.49584 0.0895535
\(280\) 0 0
\(281\) −9.53666 −0.568910 −0.284455 0.958689i \(-0.591813\pi\)
−0.284455 + 0.958689i \(0.591813\pi\)
\(282\) 1.09832 0.0654040
\(283\) 4.22846 0.251356 0.125678 0.992071i \(-0.459889\pi\)
0.125678 + 0.992071i \(0.459889\pi\)
\(284\) 2.94548 0.174782
\(285\) 16.1053 0.953997
\(286\) −0.0244237 −0.00144420
\(287\) 0 0
\(288\) 3.28651 0.193659
\(289\) −16.3774 −0.963379
\(290\) −4.22710 −0.248224
\(291\) −14.8351 −0.869652
\(292\) −18.9695 −1.11010
\(293\) 30.9919 1.81057 0.905283 0.424809i \(-0.139659\pi\)
0.905283 + 0.424809i \(0.139659\pi\)
\(294\) 0 0
\(295\) −14.3956 −0.838147
\(296\) −4.70410 −0.273420
\(297\) −0.730971 −0.0424152
\(298\) −4.41434 −0.255716
\(299\) 0.115233 0.00666411
\(300\) 0.787940 0.0454917
\(301\) 0 0
\(302\) 4.16974 0.239941
\(303\) −10.8469 −0.623139
\(304\) −24.2501 −1.39084
\(305\) −5.56136 −0.318443
\(306\) 0.228782 0.0130786
\(307\) −15.9601 −0.910888 −0.455444 0.890264i \(-0.650520\pi\)
−0.455444 + 0.890264i \(0.650520\pi\)
\(308\) 0 0
\(309\) 9.39923 0.534704
\(310\) 1.00894 0.0573042
\(311\) −15.3225 −0.868860 −0.434430 0.900706i \(-0.643050\pi\)
−0.434430 + 0.900706i \(0.643050\pi\)
\(312\) −0.130841 −0.00740744
\(313\) −16.7734 −0.948086 −0.474043 0.880502i \(-0.657206\pi\)
−0.474043 + 0.880502i \(0.657206\pi\)
\(314\) −1.29507 −0.0730848
\(315\) 0 0
\(316\) −0.927244 −0.0521616
\(317\) 32.4011 1.81982 0.909912 0.414801i \(-0.136149\pi\)
0.909912 + 0.414801i \(0.136149\pi\)
\(318\) 1.97674 0.110850
\(319\) −4.58100 −0.256487
\(320\) −14.0789 −0.787035
\(321\) 17.7498 0.990695
\(322\) 0 0
\(323\) −5.46273 −0.303954
\(324\) −1.91593 −0.106440
\(325\) −0.0473906 −0.00262876
\(326\) 0.227825 0.0126181
\(327\) 9.77303 0.540450
\(328\) −9.31983 −0.514601
\(329\) 0 0
\(330\) −0.493041 −0.0271410
\(331\) 25.9585 1.42681 0.713404 0.700753i \(-0.247152\pi\)
0.713404 + 0.700753i \(0.247152\pi\)
\(332\) 3.52929 0.193695
\(333\) 4.14294 0.227032
\(334\) 1.60275 0.0876985
\(335\) −19.4752 −1.06404
\(336\) 0 0
\(337\) 19.2372 1.04791 0.523957 0.851744i \(-0.324455\pi\)
0.523957 + 0.851744i \(0.324455\pi\)
\(338\) −3.76559 −0.204821
\(339\) 3.68270 0.200017
\(340\) −3.51655 −0.190712
\(341\) 1.09341 0.0592117
\(342\) −2.00749 −0.108553
\(343\) 0 0
\(344\) −1.70696 −0.0920329
\(345\) 2.32621 0.125239
\(346\) −2.88541 −0.155120
\(347\) 21.7166 1.16581 0.582905 0.812541i \(-0.301916\pi\)
0.582905 + 0.812541i \(0.301916\pi\)
\(348\) −12.0071 −0.643649
\(349\) −33.3948 −1.78758 −0.893792 0.448482i \(-0.851965\pi\)
−0.893792 + 0.448482i \(0.851965\pi\)
\(350\) 0 0
\(351\) 0.115233 0.00615069
\(352\) 2.40234 0.128045
\(353\) 27.5424 1.46593 0.732967 0.680264i \(-0.238135\pi\)
0.732967 + 0.680264i \(0.238135\pi\)
\(354\) 1.79438 0.0953705
\(355\) −3.57624 −0.189807
\(356\) −5.87149 −0.311188
\(357\) 0 0
\(358\) −5.07182 −0.268054
\(359\) −13.1409 −0.693550 −0.346775 0.937948i \(-0.612723\pi\)
−0.346775 + 0.937948i \(0.612723\pi\)
\(360\) −2.64129 −0.139208
\(361\) 28.9337 1.52283
\(362\) −5.75816 −0.302642
\(363\) 10.4657 0.549306
\(364\) 0 0
\(365\) 23.0317 1.20553
\(366\) 0.693211 0.0362347
\(367\) −29.9839 −1.56514 −0.782572 0.622560i \(-0.786093\pi\)
−0.782572 + 0.622560i \(0.786093\pi\)
\(368\) −3.50262 −0.182587
\(369\) 8.20805 0.427294
\(370\) 2.79442 0.145275
\(371\) 0 0
\(372\) 2.86591 0.148591
\(373\) 15.4492 0.799930 0.399965 0.916530i \(-0.369022\pi\)
0.399965 + 0.916530i \(0.369022\pi\)
\(374\) 0.167233 0.00864743
\(375\) 10.6744 0.551223
\(376\) 4.30094 0.221804
\(377\) 0.722167 0.0371935
\(378\) 0 0
\(379\) −28.0665 −1.44168 −0.720840 0.693101i \(-0.756244\pi\)
−0.720840 + 0.693101i \(0.756244\pi\)
\(380\) 30.8566 1.58291
\(381\) 3.53396 0.181050
\(382\) 4.09125 0.209327
\(383\) −20.8640 −1.06610 −0.533051 0.846083i \(-0.678954\pi\)
−0.533051 + 0.846083i \(0.678954\pi\)
\(384\) 8.32792 0.424982
\(385\) 0 0
\(386\) −3.68965 −0.187798
\(387\) 1.50333 0.0764186
\(388\) −28.4230 −1.44296
\(389\) 3.07301 0.155808 0.0779039 0.996961i \(-0.475177\pi\)
0.0779039 + 0.996961i \(0.475177\pi\)
\(390\) 0.0777249 0.00393575
\(391\) −0.789022 −0.0399026
\(392\) 0 0
\(393\) 4.35221 0.219540
\(394\) −4.10526 −0.206820
\(395\) 1.12581 0.0566456
\(396\) −1.40049 −0.0703771
\(397\) −34.4854 −1.73077 −0.865385 0.501107i \(-0.832926\pi\)
−0.865385 + 0.501107i \(0.832926\pi\)
\(398\) −1.22926 −0.0616171
\(399\) 0 0
\(400\) 1.44048 0.0720240
\(401\) −33.9427 −1.69502 −0.847508 0.530783i \(-0.821898\pi\)
−0.847508 + 0.530783i \(0.821898\pi\)
\(402\) 2.42754 0.121074
\(403\) −0.172370 −0.00858638
\(404\) −20.7819 −1.03394
\(405\) 2.32621 0.115590
\(406\) 0 0
\(407\) 3.02837 0.150111
\(408\) 0.895894 0.0443534
\(409\) 14.3206 0.708109 0.354055 0.935225i \(-0.384803\pi\)
0.354055 + 0.935225i \(0.384803\pi\)
\(410\) 5.53634 0.273420
\(411\) 17.1154 0.844239
\(412\) 18.0082 0.887202
\(413\) 0 0
\(414\) −0.289957 −0.0142506
\(415\) −4.28507 −0.210346
\(416\) −0.378715 −0.0185680
\(417\) −5.41785 −0.265313
\(418\) −1.46742 −0.0717738
\(419\) 14.0322 0.685520 0.342760 0.939423i \(-0.388638\pi\)
0.342760 + 0.939423i \(0.388638\pi\)
\(420\) 0 0
\(421\) −33.9644 −1.65532 −0.827662 0.561227i \(-0.810329\pi\)
−0.827662 + 0.561227i \(0.810329\pi\)
\(422\) −7.65246 −0.372516
\(423\) −3.78787 −0.184173
\(424\) 7.74075 0.375924
\(425\) 0.324492 0.0157402
\(426\) 0.445770 0.0215976
\(427\) 0 0
\(428\) 34.0072 1.64380
\(429\) 0.0842321 0.00406677
\(430\) 1.01400 0.0488993
\(431\) 22.9625 1.10607 0.553033 0.833159i \(-0.313470\pi\)
0.553033 + 0.833159i \(0.313470\pi\)
\(432\) −3.50262 −0.168520
\(433\) 19.1710 0.921299 0.460650 0.887582i \(-0.347616\pi\)
0.460650 + 0.887582i \(0.347616\pi\)
\(434\) 0 0
\(435\) 14.5784 0.698979
\(436\) 18.7244 0.896736
\(437\) 6.92342 0.331192
\(438\) −2.87085 −0.137174
\(439\) −29.5226 −1.40904 −0.704518 0.709686i \(-0.748837\pi\)
−0.704518 + 0.709686i \(0.748837\pi\)
\(440\) −1.93071 −0.0920430
\(441\) 0 0
\(442\) −0.0263633 −0.00125398
\(443\) −12.2761 −0.583253 −0.291627 0.956532i \(-0.594196\pi\)
−0.291627 + 0.956532i \(0.594196\pi\)
\(444\) 7.93756 0.376700
\(445\) 7.12884 0.337939
\(446\) −2.77596 −0.131445
\(447\) 15.2241 0.720076
\(448\) 0 0
\(449\) 33.5155 1.58169 0.790847 0.612014i \(-0.209640\pi\)
0.790847 + 0.612014i \(0.209640\pi\)
\(450\) 0.119247 0.00562136
\(451\) 5.99985 0.282522
\(452\) 7.05579 0.331876
\(453\) −14.3805 −0.675657
\(454\) −6.62822 −0.311078
\(455\) 0 0
\(456\) −7.86119 −0.368134
\(457\) −26.8780 −1.25730 −0.628650 0.777688i \(-0.716392\pi\)
−0.628650 + 0.777688i \(0.716392\pi\)
\(458\) −0.603291 −0.0281900
\(459\) −0.789022 −0.0368284
\(460\) 4.45685 0.207802
\(461\) −33.3536 −1.55343 −0.776715 0.629852i \(-0.783115\pi\)
−0.776715 + 0.629852i \(0.783115\pi\)
\(462\) 0 0
\(463\) −15.9403 −0.740806 −0.370403 0.928871i \(-0.620780\pi\)
−0.370403 + 0.928871i \(0.620780\pi\)
\(464\) −21.9509 −1.01905
\(465\) −3.47964 −0.161364
\(466\) 1.45472 0.0673885
\(467\) 3.72521 0.172382 0.0861912 0.996279i \(-0.472530\pi\)
0.0861912 + 0.996279i \(0.472530\pi\)
\(468\) 0.220778 0.0102055
\(469\) 0 0
\(470\) −2.55492 −0.117850
\(471\) 4.46641 0.205801
\(472\) 7.02667 0.323429
\(473\) 1.09889 0.0505271
\(474\) −0.140330 −0.00644555
\(475\) −2.84731 −0.130644
\(476\) 0 0
\(477\) −6.81735 −0.312145
\(478\) 5.96269 0.272727
\(479\) 1.02488 0.0468278 0.0234139 0.999726i \(-0.492546\pi\)
0.0234139 + 0.999726i \(0.492546\pi\)
\(480\) −7.64511 −0.348950
\(481\) −0.477404 −0.0217678
\(482\) 7.72036 0.351653
\(483\) 0 0
\(484\) 20.0515 0.911430
\(485\) 34.5097 1.56700
\(486\) −0.289957 −0.0131527
\(487\) 8.12755 0.368294 0.184147 0.982899i \(-0.441048\pi\)
0.184147 + 0.982899i \(0.441048\pi\)
\(488\) 2.71456 0.122882
\(489\) −0.785722 −0.0355316
\(490\) 0 0
\(491\) −3.42514 −0.154574 −0.0772871 0.997009i \(-0.524626\pi\)
−0.0772871 + 0.997009i \(0.524626\pi\)
\(492\) 15.7260 0.708984
\(493\) −4.94480 −0.222703
\(494\) 0.231330 0.0104080
\(495\) 1.70039 0.0764270
\(496\) 5.23935 0.235254
\(497\) 0 0
\(498\) 0.534124 0.0239347
\(499\) −5.97802 −0.267613 −0.133806 0.991007i \(-0.542720\pi\)
−0.133806 + 0.991007i \(0.542720\pi\)
\(500\) 20.4513 0.914611
\(501\) −5.52754 −0.246952
\(502\) −0.343409 −0.0153271
\(503\) 30.6662 1.36734 0.683669 0.729793i \(-0.260383\pi\)
0.683669 + 0.729793i \(0.260383\pi\)
\(504\) 0 0
\(505\) 25.2322 1.12282
\(506\) −0.211950 −0.00942233
\(507\) 12.9867 0.576761
\(508\) 6.77079 0.300405
\(509\) 43.2957 1.91905 0.959523 0.281629i \(-0.0908747\pi\)
0.959523 + 0.281629i \(0.0908747\pi\)
\(510\) −0.532196 −0.0235660
\(511\) 0 0
\(512\) 19.4655 0.860260
\(513\) 6.92342 0.305676
\(514\) −2.67284 −0.117894
\(515\) −21.8646 −0.963469
\(516\) 2.88027 0.126797
\(517\) −2.76882 −0.121773
\(518\) 0 0
\(519\) 9.95116 0.436808
\(520\) 0.304365 0.0133473
\(521\) 39.8977 1.74795 0.873974 0.485972i \(-0.161534\pi\)
0.873974 + 0.485972i \(0.161534\pi\)
\(522\) −1.81716 −0.0795349
\(523\) 18.5692 0.811974 0.405987 0.913879i \(-0.366928\pi\)
0.405987 + 0.913879i \(0.366928\pi\)
\(524\) 8.33850 0.364269
\(525\) 0 0
\(526\) −6.40851 −0.279424
\(527\) 1.18025 0.0514125
\(528\) −2.56031 −0.111423
\(529\) 1.00000 0.0434783
\(530\) −4.59831 −0.199738
\(531\) −6.18845 −0.268556
\(532\) 0 0
\(533\) −0.945840 −0.0409689
\(534\) −0.888593 −0.0384532
\(535\) −41.2897 −1.78511
\(536\) 9.50605 0.410599
\(537\) 17.4916 0.754820
\(538\) 8.51098 0.366934
\(539\) 0 0
\(540\) 4.45685 0.191792
\(541\) −41.0597 −1.76529 −0.882646 0.470037i \(-0.844240\pi\)
−0.882646 + 0.470037i \(0.844240\pi\)
\(542\) 4.59445 0.197349
\(543\) 19.8587 0.852218
\(544\) 2.59313 0.111179
\(545\) −22.7341 −0.973824
\(546\) 0 0
\(547\) 13.7117 0.586268 0.293134 0.956071i \(-0.405302\pi\)
0.293134 + 0.956071i \(0.405302\pi\)
\(548\) 32.7918 1.40080
\(549\) −2.39074 −0.102034
\(550\) 0.0871662 0.00371678
\(551\) 43.3891 1.84844
\(552\) −1.13545 −0.0483279
\(553\) 0 0
\(554\) 7.08588 0.301050
\(555\) −9.63736 −0.409083
\(556\) −10.3802 −0.440219
\(557\) −0.0927102 −0.00392826 −0.00196413 0.999998i \(-0.500625\pi\)
−0.00196413 + 0.999998i \(0.500625\pi\)
\(558\) 0.433729 0.0183612
\(559\) −0.173234 −0.00732700
\(560\) 0 0
\(561\) −0.576752 −0.0243505
\(562\) −2.76522 −0.116644
\(563\) 41.4979 1.74893 0.874464 0.485091i \(-0.161214\pi\)
0.874464 + 0.485091i \(0.161214\pi\)
\(564\) −7.25728 −0.305587
\(565\) −8.56675 −0.360406
\(566\) 1.22607 0.0515355
\(567\) 0 0
\(568\) 1.74560 0.0732439
\(569\) 14.9403 0.626328 0.313164 0.949699i \(-0.398611\pi\)
0.313164 + 0.949699i \(0.398611\pi\)
\(570\) 4.66985 0.195599
\(571\) 24.4660 1.02387 0.511935 0.859024i \(-0.328929\pi\)
0.511935 + 0.859024i \(0.328929\pi\)
\(572\) 0.161382 0.00674774
\(573\) −14.1099 −0.589448
\(574\) 0 0
\(575\) −0.411258 −0.0171507
\(576\) −6.05229 −0.252179
\(577\) −13.9226 −0.579605 −0.289802 0.957086i \(-0.593590\pi\)
−0.289802 + 0.957086i \(0.593590\pi\)
\(578\) −4.74875 −0.197522
\(579\) 12.7248 0.528826
\(580\) 27.9311 1.15977
\(581\) 0 0
\(582\) −4.30155 −0.178305
\(583\) −4.98328 −0.206386
\(584\) −11.2420 −0.465198
\(585\) −0.268057 −0.0110828
\(586\) 8.98631 0.371221
\(587\) −16.3231 −0.673728 −0.336864 0.941553i \(-0.609366\pi\)
−0.336864 + 0.941553i \(0.609366\pi\)
\(588\) 0 0
\(589\) −10.3563 −0.426725
\(590\) −4.17412 −0.171846
\(591\) 14.1582 0.582389
\(592\) 14.5111 0.596404
\(593\) 43.6987 1.79449 0.897244 0.441534i \(-0.145566\pi\)
0.897244 + 0.441534i \(0.145566\pi\)
\(594\) −0.211950 −0.00869642
\(595\) 0 0
\(596\) 29.1683 1.19478
\(597\) 4.23945 0.173509
\(598\) 0.0334127 0.00136635
\(599\) 11.0256 0.450495 0.225247 0.974302i \(-0.427681\pi\)
0.225247 + 0.974302i \(0.427681\pi\)
\(600\) 0.466963 0.0190637
\(601\) −6.86484 −0.280023 −0.140011 0.990150i \(-0.544714\pi\)
−0.140011 + 0.990150i \(0.544714\pi\)
\(602\) 0 0
\(603\) −8.37206 −0.340937
\(604\) −27.5520 −1.12108
\(605\) −24.3454 −0.989781
\(606\) −3.14514 −0.127762
\(607\) 18.6522 0.757069 0.378534 0.925587i \(-0.376428\pi\)
0.378534 + 0.925587i \(0.376428\pi\)
\(608\) −22.7539 −0.922791
\(609\) 0 0
\(610\) −1.61256 −0.0652905
\(611\) 0.436489 0.0176584
\(612\) −1.51171 −0.0611071
\(613\) 16.0601 0.648663 0.324331 0.945944i \(-0.394861\pi\)
0.324331 + 0.945944i \(0.394861\pi\)
\(614\) −4.62773 −0.186760
\(615\) −19.0937 −0.769931
\(616\) 0 0
\(617\) 11.5349 0.464376 0.232188 0.972671i \(-0.425412\pi\)
0.232188 + 0.972671i \(0.425412\pi\)
\(618\) 2.72537 0.109631
\(619\) −20.2528 −0.814030 −0.407015 0.913421i \(-0.633430\pi\)
−0.407015 + 0.913421i \(0.633430\pi\)
\(620\) −6.66672 −0.267742
\(621\) 1.00000 0.0401286
\(622\) −4.44287 −0.178143
\(623\) 0 0
\(624\) 0.403618 0.0161577
\(625\) −26.8872 −1.07549
\(626\) −4.86355 −0.194387
\(627\) 5.06082 0.202110
\(628\) 8.55730 0.341473
\(629\) 3.26887 0.130338
\(630\) 0 0
\(631\) 9.38947 0.373789 0.186894 0.982380i \(-0.440158\pi\)
0.186894 + 0.982380i \(0.440158\pi\)
\(632\) −0.549520 −0.0218587
\(633\) 26.3917 1.04898
\(634\) 9.39491 0.373119
\(635\) −8.22073 −0.326230
\(636\) −13.0615 −0.517923
\(637\) 0 0
\(638\) −1.32829 −0.0525876
\(639\) −1.53737 −0.0608173
\(640\) −19.3725 −0.765765
\(641\) 2.03391 0.0803348 0.0401674 0.999193i \(-0.487211\pi\)
0.0401674 + 0.999193i \(0.487211\pi\)
\(642\) 5.14667 0.203123
\(643\) −11.7666 −0.464029 −0.232015 0.972712i \(-0.574532\pi\)
−0.232015 + 0.972712i \(0.574532\pi\)
\(644\) 0 0
\(645\) −3.49706 −0.137697
\(646\) −1.58396 −0.0623199
\(647\) −12.1599 −0.478053 −0.239027 0.971013i \(-0.576828\pi\)
−0.239027 + 0.971013i \(0.576828\pi\)
\(648\) −1.13545 −0.0446047
\(649\) −4.52358 −0.177566
\(650\) −0.0137412 −0.000538975 0
\(651\) 0 0
\(652\) −1.50538 −0.0589554
\(653\) −33.8366 −1.32413 −0.662065 0.749447i \(-0.730320\pi\)
−0.662065 + 0.749447i \(0.730320\pi\)
\(654\) 2.83376 0.110809
\(655\) −10.1242 −0.395583
\(656\) 28.7497 1.12249
\(657\) 9.90095 0.386273
\(658\) 0 0
\(659\) 15.7293 0.612727 0.306363 0.951915i \(-0.400888\pi\)
0.306363 + 0.951915i \(0.400888\pi\)
\(660\) 3.25783 0.126811
\(661\) 31.1757 1.21259 0.606297 0.795238i \(-0.292654\pi\)
0.606297 + 0.795238i \(0.292654\pi\)
\(662\) 7.52685 0.292539
\(663\) 0.0909215 0.00353110
\(664\) 2.09159 0.0811694
\(665\) 0 0
\(666\) 1.20127 0.0465484
\(667\) 6.26700 0.242659
\(668\) −10.5904 −0.409753
\(669\) 9.57370 0.370141
\(670\) −5.64696 −0.218161
\(671\) −1.74756 −0.0674639
\(672\) 0 0
\(673\) 9.23484 0.355977 0.177988 0.984033i \(-0.443041\pi\)
0.177988 + 0.984033i \(0.443041\pi\)
\(674\) 5.57795 0.214854
\(675\) −0.411258 −0.0158293
\(676\) 24.8816 0.956984
\(677\) 45.5403 1.75026 0.875128 0.483891i \(-0.160777\pi\)
0.875128 + 0.483891i \(0.160777\pi\)
\(678\) 1.06783 0.0410096
\(679\) 0 0
\(680\) −2.08404 −0.0799193
\(681\) 22.8593 0.875972
\(682\) 0.317043 0.0121402
\(683\) 12.2292 0.467938 0.233969 0.972244i \(-0.424829\pi\)
0.233969 + 0.972244i \(0.424829\pi\)
\(684\) 13.2647 0.507190
\(685\) −39.8140 −1.52121
\(686\) 0 0
\(687\) 2.08062 0.0793808
\(688\) 5.26559 0.200749
\(689\) 0.785585 0.0299284
\(690\) 0.674501 0.0256778
\(691\) −22.4785 −0.855123 −0.427562 0.903986i \(-0.640627\pi\)
−0.427562 + 0.903986i \(0.640627\pi\)
\(692\) 19.0657 0.724769
\(693\) 0 0
\(694\) 6.29688 0.239026
\(695\) 12.6031 0.478062
\(696\) −7.11586 −0.269726
\(697\) 6.47634 0.245309
\(698\) −9.68306 −0.366509
\(699\) −5.01702 −0.189761
\(700\) 0 0
\(701\) −27.0133 −1.02028 −0.510139 0.860092i \(-0.670406\pi\)
−0.510139 + 0.860092i \(0.670406\pi\)
\(702\) 0.0334127 0.00126108
\(703\) −28.6833 −1.08181
\(704\) −4.42405 −0.166738
\(705\) 8.81139 0.331856
\(706\) 7.98611 0.300561
\(707\) 0 0
\(708\) −11.8566 −0.445599
\(709\) 26.6975 1.00264 0.501322 0.865261i \(-0.332847\pi\)
0.501322 + 0.865261i \(0.332847\pi\)
\(710\) −1.03696 −0.0389163
\(711\) 0.483967 0.0181502
\(712\) −3.47966 −0.130406
\(713\) −1.49584 −0.0560196
\(714\) 0 0
\(715\) −0.195942 −0.00732781
\(716\) 33.5127 1.25243
\(717\) −20.5640 −0.767978
\(718\) −3.81029 −0.142199
\(719\) 34.7361 1.29544 0.647720 0.761879i \(-0.275723\pi\)
0.647720 + 0.761879i \(0.275723\pi\)
\(720\) 8.14783 0.303652
\(721\) 0 0
\(722\) 8.38953 0.312226
\(723\) −26.6259 −0.990228
\(724\) 38.0478 1.41403
\(725\) −2.57736 −0.0957206
\(726\) 3.03460 0.112624
\(727\) 6.54111 0.242596 0.121298 0.992616i \(-0.461294\pi\)
0.121298 + 0.992616i \(0.461294\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 6.67820 0.247171
\(731\) 1.18616 0.0438718
\(732\) −4.58048 −0.169299
\(733\) 51.7999 1.91327 0.956637 0.291283i \(-0.0940822\pi\)
0.956637 + 0.291283i \(0.0940822\pi\)
\(734\) −8.69402 −0.320902
\(735\) 0 0
\(736\) −3.28651 −0.121142
\(737\) −6.11973 −0.225423
\(738\) 2.37998 0.0876083
\(739\) −40.4751 −1.48890 −0.744451 0.667677i \(-0.767289\pi\)
−0.744451 + 0.667677i \(0.767289\pi\)
\(740\) −18.4645 −0.678767
\(741\) −0.797808 −0.0293082
\(742\) 0 0
\(743\) −41.4092 −1.51916 −0.759579 0.650415i \(-0.774595\pi\)
−0.759579 + 0.650415i \(0.774595\pi\)
\(744\) 1.69845 0.0622682
\(745\) −35.4145 −1.29749
\(746\) 4.47960 0.164010
\(747\) −1.84208 −0.0673982
\(748\) −1.10501 −0.0404033
\(749\) 0 0
\(750\) 3.09511 0.113017
\(751\) 25.5101 0.930876 0.465438 0.885081i \(-0.345897\pi\)
0.465438 + 0.885081i \(0.345897\pi\)
\(752\) −13.2675 −0.483815
\(753\) 1.18435 0.0431600
\(754\) 0.209397 0.00762580
\(755\) 33.4522 1.21745
\(756\) 0 0
\(757\) 45.9268 1.66924 0.834619 0.550827i \(-0.185688\pi\)
0.834619 + 0.550827i \(0.185688\pi\)
\(758\) −8.13808 −0.295588
\(759\) 0.730971 0.0265326
\(760\) 18.2868 0.663331
\(761\) 8.18381 0.296663 0.148331 0.988938i \(-0.452610\pi\)
0.148331 + 0.988938i \(0.452610\pi\)
\(762\) 1.02469 0.0371208
\(763\) 0 0
\(764\) −27.0334 −0.978035
\(765\) 1.83543 0.0663602
\(766\) −6.04966 −0.218583
\(767\) 0.713115 0.0257491
\(768\) −9.68985 −0.349652
\(769\) 15.3794 0.554596 0.277298 0.960784i \(-0.410561\pi\)
0.277298 + 0.960784i \(0.410561\pi\)
\(770\) 0 0
\(771\) 9.21806 0.331980
\(772\) 24.3798 0.877448
\(773\) −0.201716 −0.00725521 −0.00362760 0.999993i \(-0.501155\pi\)
−0.00362760 + 0.999993i \(0.501155\pi\)
\(774\) 0.435901 0.0156681
\(775\) 0.615176 0.0220978
\(776\) −16.8446 −0.604684
\(777\) 0 0
\(778\) 0.891041 0.0319454
\(779\) −56.8278 −2.03607
\(780\) −0.513577 −0.0183890
\(781\) −1.12377 −0.0402117
\(782\) −0.228782 −0.00818124
\(783\) 6.26700 0.223964
\(784\) 0 0
\(785\) −10.3898 −0.370828
\(786\) 1.26195 0.0450123
\(787\) −36.2719 −1.29296 −0.646478 0.762933i \(-0.723759\pi\)
−0.646478 + 0.762933i \(0.723759\pi\)
\(788\) 27.1260 0.966323
\(789\) 22.1016 0.786838
\(790\) 0.326436 0.0116141
\(791\) 0 0
\(792\) −0.829981 −0.0294921
\(793\) 0.275493 0.00978303
\(794\) −9.99926 −0.354861
\(795\) 15.8586 0.562446
\(796\) 8.12247 0.287893
\(797\) 25.4967 0.903138 0.451569 0.892236i \(-0.350864\pi\)
0.451569 + 0.892236i \(0.350864\pi\)
\(798\) 0 0
\(799\) −2.98871 −0.105733
\(800\) 1.35160 0.0477864
\(801\) 3.06457 0.108281
\(802\) −9.84190 −0.347530
\(803\) 7.23731 0.255399
\(804\) −16.0402 −0.565696
\(805\) 0 0
\(806\) −0.0499799 −0.00176047
\(807\) −29.3526 −1.03326
\(808\) −12.3161 −0.433280
\(809\) 16.5243 0.580963 0.290482 0.956881i \(-0.406184\pi\)
0.290482 + 0.956881i \(0.406184\pi\)
\(810\) 0.674501 0.0236995
\(811\) −35.0068 −1.22925 −0.614627 0.788818i \(-0.710694\pi\)
−0.614627 + 0.788818i \(0.710694\pi\)
\(812\) 0 0
\(813\) −15.8453 −0.555719
\(814\) 0.878097 0.0307773
\(815\) 1.82776 0.0640235
\(816\) −2.76364 −0.0967469
\(817\) −10.4082 −0.364136
\(818\) 4.15236 0.145184
\(819\) 0 0
\(820\) −36.5820 −1.27750
\(821\) 34.9634 1.22023 0.610116 0.792312i \(-0.291123\pi\)
0.610116 + 0.792312i \(0.291123\pi\)
\(822\) 4.96272 0.173095
\(823\) −29.7247 −1.03614 −0.518069 0.855339i \(-0.673349\pi\)
−0.518069 + 0.855339i \(0.673349\pi\)
\(824\) 10.6724 0.371789
\(825\) −0.300618 −0.0104662
\(826\) 0 0
\(827\) 8.29301 0.288376 0.144188 0.989550i \(-0.453943\pi\)
0.144188 + 0.989550i \(0.453943\pi\)
\(828\) 1.91593 0.0665830
\(829\) 1.76447 0.0612826 0.0306413 0.999530i \(-0.490245\pi\)
0.0306413 + 0.999530i \(0.490245\pi\)
\(830\) −1.24248 −0.0431273
\(831\) −24.4377 −0.847735
\(832\) 0.697425 0.0241789
\(833\) 0 0
\(834\) −1.57094 −0.0543973
\(835\) 12.8582 0.444977
\(836\) 9.69615 0.335348
\(837\) −1.49584 −0.0517037
\(838\) 4.06874 0.140552
\(839\) −25.9251 −0.895035 −0.447517 0.894275i \(-0.647692\pi\)
−0.447517 + 0.894275i \(0.647692\pi\)
\(840\) 0 0
\(841\) 10.2753 0.354321
\(842\) −9.84821 −0.339392
\(843\) 9.53666 0.328460
\(844\) 50.5645 1.74050
\(845\) −30.2099 −1.03925
\(846\) −1.09832 −0.0377610
\(847\) 0 0
\(848\) −23.8786 −0.819993
\(849\) −4.22846 −0.145120
\(850\) 0.0940886 0.00322721
\(851\) −4.14294 −0.142018
\(852\) −2.94548 −0.100911
\(853\) 23.2110 0.794729 0.397365 0.917661i \(-0.369925\pi\)
0.397365 + 0.917661i \(0.369925\pi\)
\(854\) 0 0
\(855\) −16.1053 −0.550791
\(856\) 20.1540 0.688848
\(857\) −12.4312 −0.424643 −0.212322 0.977200i \(-0.568102\pi\)
−0.212322 + 0.977200i \(0.568102\pi\)
\(858\) 0.0244237 0.000833811 0
\(859\) 5.94580 0.202868 0.101434 0.994842i \(-0.467657\pi\)
0.101434 + 0.994842i \(0.467657\pi\)
\(860\) −6.70011 −0.228472
\(861\) 0 0
\(862\) 6.65815 0.226777
\(863\) 46.2512 1.57441 0.787204 0.616692i \(-0.211528\pi\)
0.787204 + 0.616692i \(0.211528\pi\)
\(864\) −3.28651 −0.111809
\(865\) −23.1485 −0.787073
\(866\) 5.55876 0.188894
\(867\) 16.3774 0.556207
\(868\) 0 0
\(869\) 0.353766 0.0120007
\(870\) 4.22710 0.143312
\(871\) 0.964739 0.0326889
\(872\) 11.0968 0.375784
\(873\) 14.8351 0.502094
\(874\) 2.00749 0.0679044
\(875\) 0 0
\(876\) 18.9695 0.640919
\(877\) −12.7311 −0.429899 −0.214950 0.976625i \(-0.568959\pi\)
−0.214950 + 0.976625i \(0.568959\pi\)
\(878\) −8.56028 −0.288895
\(879\) −30.9919 −1.04533
\(880\) 5.95583 0.200771
\(881\) 22.2712 0.750334 0.375167 0.926957i \(-0.377585\pi\)
0.375167 + 0.926957i \(0.377585\pi\)
\(882\) 0 0
\(883\) 39.1941 1.31899 0.659493 0.751711i \(-0.270771\pi\)
0.659493 + 0.751711i \(0.270771\pi\)
\(884\) 0.174199 0.00585894
\(885\) 14.3956 0.483904
\(886\) −3.55953 −0.119585
\(887\) −7.54606 −0.253372 −0.126686 0.991943i \(-0.540434\pi\)
−0.126686 + 0.991943i \(0.540434\pi\)
\(888\) 4.70410 0.157859
\(889\) 0 0
\(890\) 2.06706 0.0692879
\(891\) 0.730971 0.0244885
\(892\) 18.3425 0.614152
\(893\) 26.2250 0.877587
\(894\) 4.41434 0.147638
\(895\) −40.6892 −1.36009
\(896\) 0 0
\(897\) −0.115233 −0.00384752
\(898\) 9.71805 0.324295
\(899\) −9.37442 −0.312655
\(900\) −0.787940 −0.0262647
\(901\) −5.37904 −0.179202
\(902\) 1.73970 0.0579256
\(903\) 0 0
\(904\) 4.18152 0.139075
\(905\) −46.1955 −1.53559
\(906\) −4.16974 −0.138530
\(907\) 17.4927 0.580836 0.290418 0.956900i \(-0.406206\pi\)
0.290418 + 0.956900i \(0.406206\pi\)
\(908\) 43.7968 1.45345
\(909\) 10.8469 0.359770
\(910\) 0 0
\(911\) 1.76174 0.0583691 0.0291846 0.999574i \(-0.490709\pi\)
0.0291846 + 0.999574i \(0.490709\pi\)
\(912\) 24.2501 0.803001
\(913\) −1.34651 −0.0445629
\(914\) −7.79347 −0.257785
\(915\) 5.56136 0.183853
\(916\) 3.98632 0.131712
\(917\) 0 0
\(918\) −0.228782 −0.00755094
\(919\) −19.9587 −0.658377 −0.329189 0.944264i \(-0.606775\pi\)
−0.329189 + 0.944264i \(0.606775\pi\)
\(920\) 2.64129 0.0870809
\(921\) 15.9601 0.525902
\(922\) −9.67109 −0.318500
\(923\) 0.177156 0.00583115
\(924\) 0 0
\(925\) 1.70382 0.0560212
\(926\) −4.62199 −0.151888
\(927\) −9.39923 −0.308711
\(928\) −20.5965 −0.676115
\(929\) −16.9400 −0.555783 −0.277892 0.960612i \(-0.589636\pi\)
−0.277892 + 0.960612i \(0.589636\pi\)
\(930\) −1.00894 −0.0330846
\(931\) 0 0
\(932\) −9.61223 −0.314859
\(933\) 15.3225 0.501637
\(934\) 1.08015 0.0353436
\(935\) 1.34165 0.0438766
\(936\) 0.130841 0.00427669
\(937\) −2.54779 −0.0832327 −0.0416163 0.999134i \(-0.513251\pi\)
−0.0416163 + 0.999134i \(0.513251\pi\)
\(938\) 0 0
\(939\) 16.7734 0.547378
\(940\) 16.8820 0.550629
\(941\) −43.0016 −1.40181 −0.700906 0.713253i \(-0.747221\pi\)
−0.700906 + 0.713253i \(0.747221\pi\)
\(942\) 1.29507 0.0421955
\(943\) −8.20805 −0.267291
\(944\) −21.6758 −0.705487
\(945\) 0 0
\(946\) 0.318631 0.0103596
\(947\) −46.4295 −1.50875 −0.754377 0.656441i \(-0.772061\pi\)
−0.754377 + 0.656441i \(0.772061\pi\)
\(948\) 0.927244 0.0301155
\(949\) −1.14092 −0.0370358
\(950\) −0.825598 −0.0267859
\(951\) −32.4011 −1.05068
\(952\) 0 0
\(953\) −23.0858 −0.747824 −0.373912 0.927464i \(-0.621984\pi\)
−0.373912 + 0.927464i \(0.621984\pi\)
\(954\) −1.97674 −0.0639992
\(955\) 32.8225 1.06211
\(956\) −39.3992 −1.27426
\(957\) 4.58100 0.148083
\(958\) 0.297170 0.00960112
\(959\) 0 0
\(960\) 14.0789 0.454395
\(961\) −28.7625 −0.927822
\(962\) −0.138427 −0.00446305
\(963\) −17.7498 −0.571978
\(964\) −51.0132 −1.64303
\(965\) −29.6006 −0.952878
\(966\) 0 0
\(967\) −6.48746 −0.208623 −0.104311 0.994545i \(-0.533264\pi\)
−0.104311 + 0.994545i \(0.533264\pi\)
\(968\) 11.8832 0.381942
\(969\) 5.46273 0.175488
\(970\) 10.0063 0.321283
\(971\) 23.6355 0.758499 0.379250 0.925294i \(-0.376182\pi\)
0.379250 + 0.925294i \(0.376182\pi\)
\(972\) 1.91593 0.0614533
\(973\) 0 0
\(974\) 2.35664 0.0755116
\(975\) 0.0473906 0.00151771
\(976\) −8.37385 −0.268040
\(977\) 9.62430 0.307909 0.153954 0.988078i \(-0.450799\pi\)
0.153954 + 0.988078i \(0.450799\pi\)
\(978\) −0.227825 −0.00728506
\(979\) 2.24011 0.0715943
\(980\) 0 0
\(981\) −9.77303 −0.312029
\(982\) −0.993142 −0.0316924
\(983\) 31.7316 1.01208 0.506041 0.862509i \(-0.331108\pi\)
0.506041 + 0.862509i \(0.331108\pi\)
\(984\) 9.31983 0.297105
\(985\) −32.9349 −1.04939
\(986\) −1.43378 −0.0456608
\(987\) 0 0
\(988\) −1.52854 −0.0486293
\(989\) −1.50333 −0.0478031
\(990\) 0.493041 0.0156699
\(991\) 4.46149 0.141724 0.0708620 0.997486i \(-0.477425\pi\)
0.0708620 + 0.997486i \(0.477425\pi\)
\(992\) 4.91608 0.156086
\(993\) −25.9585 −0.823768
\(994\) 0 0
\(995\) −9.86186 −0.312642
\(996\) −3.52929 −0.111830
\(997\) −25.4916 −0.807328 −0.403664 0.914907i \(-0.632264\pi\)
−0.403664 + 0.914907i \(0.632264\pi\)
\(998\) −1.73337 −0.0548688
\(999\) −4.14294 −0.131077
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 3381.2.a.bc.1.4 6
7.2 even 3 483.2.i.f.277.3 12
7.4 even 3 483.2.i.f.415.3 yes 12
7.6 odd 2 3381.2.a.bd.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.f.277.3 12 7.2 even 3
483.2.i.f.415.3 yes 12 7.4 even 3
3381.2.a.bc.1.4 6 1.1 even 1 trivial
3381.2.a.bd.1.4 6 7.6 odd 2