Properties

Label 4225.2.a.bq.1.6
Level $4225$
Weight $2$
Character 4225.1
Self dual yes
Analytic conductor $33.737$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4225 = 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(33.7367948540\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.199374400.1
Defining polynomial: \(x^{6} - 8 x^{4} + 10 x^{2} - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(2.54574\) of defining polynomial
Character \(\chi\) \(=\) 4225.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.54574 q^{2} -2.15293 q^{3} +4.48079 q^{4} -5.48079 q^{6} +2.93855 q^{7} +6.31544 q^{8} +1.63509 q^{9} +O(q^{10})\) \(q+2.54574 q^{2} -2.15293 q^{3} +4.48079 q^{4} -5.48079 q^{6} +2.93855 q^{7} +6.31544 q^{8} +1.63509 q^{9} +0.635089 q^{11} -9.64680 q^{12} +7.48079 q^{14} +7.11588 q^{16} -1.22396 q^{17} +4.16251 q^{18} -1.36491 q^{19} -6.32648 q^{21} +1.61677 q^{22} +2.15293 q^{23} -13.5967 q^{24} +2.93855 q^{27} +13.1670 q^{28} +3.00000 q^{29} +8.96157 q^{31} +5.48429 q^{32} -1.36730 q^{33} -3.11588 q^{34} +7.32648 q^{36} -1.22396 q^{37} -3.47471 q^{38} +9.96157 q^{41} -16.1056 q^{42} -1.36730 q^{43} +2.84570 q^{44} +5.48079 q^{46} +6.16379 q^{47} -15.3200 q^{48} +1.63509 q^{49} +2.63509 q^{51} +0.642285 q^{53} +7.48079 q^{54} +18.5582 q^{56} +2.93855 q^{57} +7.63722 q^{58} -7.59666 q^{59} -2.27018 q^{61} +22.8138 q^{62} +4.80479 q^{63} -0.270178 q^{64} -3.48079 q^{66} -8.03003 q^{67} -5.48429 q^{68} -4.63509 q^{69} -2.63509 q^{71} +10.3263 q^{72} +10.3263 q^{73} -3.11588 q^{74} -6.11588 q^{76} +1.86624 q^{77} +1.03843 q^{79} -11.2318 q^{81} +25.3596 q^{82} -11.8452 q^{83} -28.3476 q^{84} -3.48079 q^{86} -6.45878 q^{87} +4.01086 q^{88} +12.5582 q^{89} +9.64680 q^{92} -19.2936 q^{93} +15.6914 q^{94} -11.8073 q^{96} +14.7838 q^{97} +4.16251 q^{98} +1.03843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 4q^{4} - 10q^{6} + 6q^{9} + O(q^{10}) \) \( 6q + 4q^{4} - 10q^{6} + 6q^{9} + 22q^{14} + 16q^{16} - 12q^{19} + 4q^{21} - 32q^{24} + 18q^{29} + 8q^{31} + 8q^{34} + 2q^{36} + 14q^{41} - 2q^{44} + 10q^{46} + 6q^{49} + 12q^{51} + 22q^{54} + 16q^{56} + 4q^{59} - 6q^{61} + 6q^{64} + 2q^{66} - 24q^{69} - 12q^{71} + 8q^{74} - 10q^{76} + 52q^{79} - 14q^{81} - 90q^{84} + 2q^{86} - 20q^{89} + 56q^{94} - 6q^{96} + 52q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.54574 1.80011 0.900055 0.435777i \(-0.143526\pi\)
0.900055 + 0.435777i \(0.143526\pi\)
\(3\) −2.15293 −1.24299 −0.621496 0.783417i \(-0.713475\pi\)
−0.621496 + 0.783417i \(0.713475\pi\)
\(4\) 4.48079 2.24039
\(5\) 0 0
\(6\) −5.48079 −2.23752
\(7\) 2.93855 1.11067 0.555334 0.831627i \(-0.312590\pi\)
0.555334 + 0.831627i \(0.312590\pi\)
\(8\) 6.31544 2.23284
\(9\) 1.63509 0.545030
\(10\) 0 0
\(11\) 0.635089 0.191487 0.0957433 0.995406i \(-0.469477\pi\)
0.0957433 + 0.995406i \(0.469477\pi\)
\(12\) −9.64680 −2.78479
\(13\) 0 0
\(14\) 7.48079 1.99932
\(15\) 0 0
\(16\) 7.11588 1.77897
\(17\) −1.22396 −0.296853 −0.148427 0.988923i \(-0.547421\pi\)
−0.148427 + 0.988923i \(0.547421\pi\)
\(18\) 4.16251 0.981113
\(19\) −1.36491 −0.313132 −0.156566 0.987667i \(-0.550042\pi\)
−0.156566 + 0.987667i \(0.550042\pi\)
\(20\) 0 0
\(21\) −6.32648 −1.38055
\(22\) 1.61677 0.344697
\(23\) 2.15293 0.448916 0.224458 0.974484i \(-0.427939\pi\)
0.224458 + 0.974484i \(0.427939\pi\)
\(24\) −13.5967 −2.77541
\(25\) 0 0
\(26\) 0 0
\(27\) 2.93855 0.565525
\(28\) 13.1670 2.48833
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) 8.96157 1.60955 0.804773 0.593583i \(-0.202287\pi\)
0.804773 + 0.593583i \(0.202287\pi\)
\(32\) 5.48429 0.969495
\(33\) −1.36730 −0.238016
\(34\) −3.11588 −0.534368
\(35\) 0 0
\(36\) 7.32648 1.22108
\(37\) −1.22396 −0.201217 −0.100609 0.994926i \(-0.532079\pi\)
−0.100609 + 0.994926i \(0.532079\pi\)
\(38\) −3.47471 −0.563672
\(39\) 0 0
\(40\) 0 0
\(41\) 9.96157 1.55574 0.777868 0.628427i \(-0.216301\pi\)
0.777868 + 0.628427i \(0.216301\pi\)
\(42\) −16.1056 −2.48514
\(43\) −1.36730 −0.208511 −0.104256 0.994551i \(-0.533246\pi\)
−0.104256 + 0.994551i \(0.533246\pi\)
\(44\) 2.84570 0.429005
\(45\) 0 0
\(46\) 5.48079 0.808098
\(47\) 6.16379 0.899081 0.449540 0.893260i \(-0.351588\pi\)
0.449540 + 0.893260i \(0.351588\pi\)
\(48\) −15.3200 −2.21124
\(49\) 1.63509 0.233584
\(50\) 0 0
\(51\) 2.63509 0.368986
\(52\) 0 0
\(53\) 0.642285 0.0882246 0.0441123 0.999027i \(-0.485954\pi\)
0.0441123 + 0.999027i \(0.485954\pi\)
\(54\) 7.48079 1.01801
\(55\) 0 0
\(56\) 18.5582 2.47995
\(57\) 2.93855 0.389221
\(58\) 7.63722 1.00282
\(59\) −7.59666 −0.989001 −0.494501 0.869177i \(-0.664649\pi\)
−0.494501 + 0.869177i \(0.664649\pi\)
\(60\) 0 0
\(61\) −2.27018 −0.290666 −0.145333 0.989383i \(-0.546425\pi\)
−0.145333 + 0.989383i \(0.546425\pi\)
\(62\) 22.8138 2.89736
\(63\) 4.80479 0.605347
\(64\) −0.270178 −0.0337722
\(65\) 0 0
\(66\) −3.48079 −0.428455
\(67\) −8.03003 −0.981024 −0.490512 0.871434i \(-0.663190\pi\)
−0.490512 + 0.871434i \(0.663190\pi\)
\(68\) −5.48429 −0.665068
\(69\) −4.63509 −0.557999
\(70\) 0 0
\(71\) −2.63509 −0.312728 −0.156364 0.987700i \(-0.549977\pi\)
−0.156364 + 0.987700i \(0.549977\pi\)
\(72\) 10.3263 1.21697
\(73\) 10.3263 1.20860 0.604301 0.796756i \(-0.293453\pi\)
0.604301 + 0.796756i \(0.293453\pi\)
\(74\) −3.11588 −0.362213
\(75\) 0 0
\(76\) −6.11588 −0.701539
\(77\) 1.86624 0.212678
\(78\) 0 0
\(79\) 1.03843 0.116832 0.0584161 0.998292i \(-0.481395\pi\)
0.0584161 + 0.998292i \(0.481395\pi\)
\(80\) 0 0
\(81\) −11.2318 −1.24797
\(82\) 25.3596 2.80050
\(83\) −11.8452 −1.30018 −0.650092 0.759855i \(-0.725270\pi\)
−0.650092 + 0.759855i \(0.725270\pi\)
\(84\) −28.3476 −3.09298
\(85\) 0 0
\(86\) −3.48079 −0.375343
\(87\) −6.45878 −0.692454
\(88\) 4.01086 0.427559
\(89\) 12.5582 1.33117 0.665585 0.746322i \(-0.268182\pi\)
0.665585 + 0.746322i \(0.268182\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 9.64680 1.00575
\(93\) −19.2936 −2.00065
\(94\) 15.6914 1.61844
\(95\) 0 0
\(96\) −11.8073 −1.20507
\(97\) 14.7838 1.50107 0.750534 0.660832i \(-0.229797\pi\)
0.750534 + 0.660832i \(0.229797\pi\)
\(98\) 4.16251 0.420477
\(99\) 1.03843 0.104366
\(100\) 0 0
\(101\) 13.2318 1.31661 0.658304 0.752752i \(-0.271274\pi\)
0.658304 + 0.752752i \(0.271274\pi\)
\(102\) 6.70825 0.664216
\(103\) 10.9686 1.08077 0.540383 0.841419i \(-0.318279\pi\)
0.540383 + 0.841419i \(0.318279\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1.63509 0.158814
\(107\) −10.6736 −1.03186 −0.515928 0.856632i \(-0.672553\pi\)
−0.515928 + 0.856632i \(0.672553\pi\)
\(108\) 13.1670 1.26700
\(109\) 3.27018 0.313226 0.156613 0.987660i \(-0.449942\pi\)
0.156613 + 0.987660i \(0.449942\pi\)
\(110\) 0 0
\(111\) 2.63509 0.250112
\(112\) 20.9104 1.97584
\(113\) −5.52981 −0.520201 −0.260100 0.965582i \(-0.583756\pi\)
−0.260100 + 0.965582i \(0.583756\pi\)
\(114\) 7.48079 0.700640
\(115\) 0 0
\(116\) 13.4424 1.24809
\(117\) 0 0
\(118\) −19.3391 −1.78031
\(119\) −3.59666 −0.329705
\(120\) 0 0
\(121\) −10.5967 −0.963333
\(122\) −5.77928 −0.523231
\(123\) −21.4465 −1.93377
\(124\) 40.1549 3.60602
\(125\) 0 0
\(126\) 12.2318 1.08969
\(127\) −17.2317 −1.52907 −0.764534 0.644584i \(-0.777031\pi\)
−0.764534 + 0.644584i \(0.777031\pi\)
\(128\) −11.6564 −1.03029
\(129\) 2.94369 0.259178
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) −6.12658 −0.533250
\(133\) −4.01086 −0.347786
\(134\) −20.4424 −1.76595
\(135\) 0 0
\(136\) −7.72982 −0.662827
\(137\) 8.67231 0.740926 0.370463 0.928847i \(-0.379199\pi\)
0.370463 + 0.928847i \(0.379199\pi\)
\(138\) −11.7997 −1.00446
\(139\) 14.3265 1.21516 0.607578 0.794260i \(-0.292141\pi\)
0.607578 + 0.794260i \(0.292141\pi\)
\(140\) 0 0
\(141\) −13.2702 −1.11755
\(142\) −6.70825 −0.562944
\(143\) 0 0
\(144\) 11.6351 0.969591
\(145\) 0 0
\(146\) 26.2881 2.17562
\(147\) −3.52022 −0.290343
\(148\) −5.48429 −0.450806
\(149\) 17.1549 1.40538 0.702692 0.711494i \(-0.251981\pi\)
0.702692 + 0.711494i \(0.251981\pi\)
\(150\) 0 0
\(151\) 21.3828 1.74011 0.870053 0.492957i \(-0.164084\pi\)
0.870053 + 0.492957i \(0.164084\pi\)
\(152\) −8.62001 −0.699175
\(153\) −2.00128 −0.161794
\(154\) 4.75096 0.382844
\(155\) 0 0
\(156\) 0 0
\(157\) −18.3646 −1.46566 −0.732829 0.680413i \(-0.761800\pi\)
−0.732829 + 0.680413i \(0.761800\pi\)
\(158\) 2.64356 0.210311
\(159\) −1.38279 −0.109662
\(160\) 0 0
\(161\) 6.32648 0.498597
\(162\) −28.5931 −2.24649
\(163\) 4.01086 0.314155 0.157078 0.987586i \(-0.449793\pi\)
0.157078 + 0.987586i \(0.449793\pi\)
\(164\) 44.6357 3.48546
\(165\) 0 0
\(166\) −30.1549 −2.34047
\(167\) −2.93855 −0.227392 −0.113696 0.993516i \(-0.536269\pi\)
−0.113696 + 0.993516i \(0.536269\pi\)
\(168\) −39.9545 −3.08256
\(169\) 0 0
\(170\) 0 0
\(171\) −2.23175 −0.170666
\(172\) −6.12658 −0.467147
\(173\) 1.36730 0.103954 0.0519769 0.998648i \(-0.483448\pi\)
0.0519769 + 0.998648i \(0.483448\pi\)
\(174\) −16.4424 −1.24649
\(175\) 0 0
\(176\) 4.51921 0.340649
\(177\) 16.3550 1.22932
\(178\) 31.9700 2.39625
\(179\) 7.78613 0.581963 0.290981 0.956729i \(-0.406018\pi\)
0.290981 + 0.956729i \(0.406018\pi\)
\(180\) 0 0
\(181\) −3.86684 −0.287420 −0.143710 0.989620i \(-0.545903\pi\)
−0.143710 + 0.989620i \(0.545903\pi\)
\(182\) 0 0
\(183\) 4.88752 0.361296
\(184\) 13.5967 1.00236
\(185\) 0 0
\(186\) −49.1165 −3.60139
\(187\) −0.777322 −0.0568434
\(188\) 27.6186 2.01429
\(189\) 8.63509 0.628110
\(190\) 0 0
\(191\) 4.94369 0.357713 0.178857 0.983875i \(-0.442760\pi\)
0.178857 + 0.983875i \(0.442760\pi\)
\(192\) 0.581673 0.0419786
\(193\) −4.95644 −0.356772 −0.178386 0.983961i \(-0.557088\pi\)
−0.178386 + 0.983961i \(0.557088\pi\)
\(194\) 37.6357 2.70208
\(195\) 0 0
\(196\) 7.32648 0.523320
\(197\) −6.74546 −0.480594 −0.240297 0.970699i \(-0.577245\pi\)
−0.240297 + 0.970699i \(0.577245\pi\)
\(198\) 2.64356 0.187870
\(199\) 5.17544 0.366878 0.183439 0.983031i \(-0.441277\pi\)
0.183439 + 0.983031i \(0.441277\pi\)
\(200\) 0 0
\(201\) 17.2881 1.21941
\(202\) 33.6846 2.37004
\(203\) 8.81566 0.618738
\(204\) 11.8073 0.826674
\(205\) 0 0
\(206\) 27.9231 1.94550
\(207\) 3.52022 0.244673
\(208\) 0 0
\(209\) −0.866840 −0.0599606
\(210\) 0 0
\(211\) −14.0179 −0.965031 −0.482515 0.875888i \(-0.660277\pi\)
−0.482515 + 0.875888i \(0.660277\pi\)
\(212\) 2.87794 0.197658
\(213\) 5.67315 0.388718
\(214\) −27.1722 −1.85745
\(215\) 0 0
\(216\) 18.5582 1.26273
\(217\) 26.3341 1.78767
\(218\) 8.32502 0.563841
\(219\) −22.2318 −1.50228
\(220\) 0 0
\(221\) 0 0
\(222\) 6.70825 0.450228
\(223\) 0.00830491 0.000556138 0 0.000278069 1.00000i \(-0.499911\pi\)
0.000278069 1.00000i \(0.499911\pi\)
\(224\) 16.1159 1.07679
\(225\) 0 0
\(226\) −14.0774 −0.936418
\(227\) −11.2636 −0.747589 −0.373795 0.927511i \(-0.621944\pi\)
−0.373795 + 0.927511i \(0.621944\pi\)
\(228\) 13.1670 0.872008
\(229\) 16.5404 1.09302 0.546509 0.837453i \(-0.315957\pi\)
0.546509 + 0.837453i \(0.315957\pi\)
\(230\) 0 0
\(231\) −4.01788 −0.264357
\(232\) 18.9463 1.24389
\(233\) 6.94941 0.455271 0.227636 0.973746i \(-0.426900\pi\)
0.227636 + 0.973746i \(0.426900\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −34.0390 −2.21575
\(237\) −2.23566 −0.145221
\(238\) −9.15616 −0.593506
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 0 0
\(241\) −19.7721 −1.27363 −0.636817 0.771015i \(-0.719749\pi\)
−0.636817 + 0.771015i \(0.719749\pi\)
\(242\) −26.9763 −1.73410
\(243\) 15.3655 0.985695
\(244\) −10.1722 −0.651207
\(245\) 0 0
\(246\) −54.5973 −3.48099
\(247\) 0 0
\(248\) 56.5962 3.59386
\(249\) 25.5019 1.61612
\(250\) 0 0
\(251\) 3.67352 0.231870 0.115935 0.993257i \(-0.463014\pi\)
0.115935 + 0.993257i \(0.463014\pi\)
\(252\) 21.5293 1.35622
\(253\) 1.36730 0.0859614
\(254\) −43.8674 −2.75249
\(255\) 0 0
\(256\) −29.1338 −1.82086
\(257\) −13.2648 −0.827439 −0.413719 0.910404i \(-0.635771\pi\)
−0.413719 + 0.910404i \(0.635771\pi\)
\(258\) 7.49387 0.466548
\(259\) −3.59666 −0.223486
\(260\) 0 0
\(261\) 4.90527 0.303628
\(262\) 25.4574 1.57276
\(263\) −30.2705 −1.86656 −0.933279 0.359152i \(-0.883066\pi\)
−0.933279 + 0.359152i \(0.883066\pi\)
\(264\) −8.63509 −0.531453
\(265\) 0 0
\(266\) −10.2106 −0.626053
\(267\) −27.0369 −1.65463
\(268\) −35.9809 −2.19788
\(269\) −22.2496 −1.35658 −0.678292 0.734792i \(-0.737280\pi\)
−0.678292 + 0.734792i \(0.737280\pi\)
\(270\) 0 0
\(271\) 11.8284 0.718525 0.359262 0.933237i \(-0.383028\pi\)
0.359262 + 0.933237i \(0.383028\pi\)
\(272\) −8.70953 −0.528093
\(273\) 0 0
\(274\) 22.0774 1.33375
\(275\) 0 0
\(276\) −20.7688 −1.25014
\(277\) −16.7851 −1.00852 −0.504259 0.863553i \(-0.668234\pi\)
−0.504259 + 0.863553i \(0.668234\pi\)
\(278\) 36.4715 2.18741
\(279\) 14.6530 0.877250
\(280\) 0 0
\(281\) 10.5967 0.632144 0.316072 0.948735i \(-0.397636\pi\)
0.316072 + 0.948735i \(0.397636\pi\)
\(282\) −33.7824 −2.01171
\(283\) 8.81566 0.524036 0.262018 0.965063i \(-0.415612\pi\)
0.262018 + 0.965063i \(0.415612\pi\)
\(284\) −11.8073 −0.700633
\(285\) 0 0
\(286\) 0 0
\(287\) 29.2726 1.72791
\(288\) 8.96730 0.528403
\(289\) −15.5019 −0.911878
\(290\) 0 0
\(291\) −31.8284 −1.86581
\(292\) 46.2699 2.70774
\(293\) −28.2526 −1.65053 −0.825267 0.564742i \(-0.808976\pi\)
−0.825267 + 0.564742i \(0.808976\pi\)
\(294\) −8.96157 −0.522650
\(295\) 0 0
\(296\) −7.72982 −0.449287
\(297\) 1.86624 0.108290
\(298\) 43.6719 2.52984
\(299\) 0 0
\(300\) 0 0
\(301\) −4.01788 −0.231587
\(302\) 54.4350 3.13238
\(303\) −28.4870 −1.63653
\(304\) −9.71254 −0.557052
\(305\) 0 0
\(306\) −5.09473 −0.291247
\(307\) 12.7219 0.726077 0.363039 0.931774i \(-0.381739\pi\)
0.363039 + 0.931774i \(0.381739\pi\)
\(308\) 8.36223 0.476482
\(309\) −23.6145 −1.34338
\(310\) 0 0
\(311\) 27.9231 1.58338 0.791688 0.610925i \(-0.209202\pi\)
0.791688 + 0.610925i \(0.209202\pi\)
\(312\) 0 0
\(313\) −24.5807 −1.38938 −0.694692 0.719307i \(-0.744460\pi\)
−0.694692 + 0.719307i \(0.744460\pi\)
\(314\) −46.7516 −2.63834
\(315\) 0 0
\(316\) 4.65297 0.261750
\(317\) −0.234377 −0.0131639 −0.00658196 0.999978i \(-0.502095\pi\)
−0.00658196 + 0.999978i \(0.502095\pi\)
\(318\) −3.52022 −0.197404
\(319\) 1.90527 0.106674
\(320\) 0 0
\(321\) 22.9795 1.28259
\(322\) 16.1056 0.897529
\(323\) 1.67059 0.0929543
\(324\) −50.3271 −2.79595
\(325\) 0 0
\(326\) 10.2106 0.565513
\(327\) −7.04045 −0.389338
\(328\) 62.9117 3.47372
\(329\) 18.1126 0.998581
\(330\) 0 0
\(331\) 18.3265 1.00731 0.503657 0.863904i \(-0.331987\pi\)
0.503657 + 0.863904i \(0.331987\pi\)
\(332\) −53.0760 −2.91292
\(333\) −2.00128 −0.109669
\(334\) −7.48079 −0.409330
\(335\) 0 0
\(336\) −45.0185 −2.45596
\(337\) 21.2949 1.16001 0.580003 0.814614i \(-0.303051\pi\)
0.580003 + 0.814614i \(0.303051\pi\)
\(338\) 0 0
\(339\) 11.9053 0.646605
\(340\) 0 0
\(341\) 5.69140 0.308206
\(342\) −5.68146 −0.307218
\(343\) −15.7651 −0.851234
\(344\) −8.63509 −0.465573
\(345\) 0 0
\(346\) 3.48079 0.187128
\(347\) 3.81521 0.204811 0.102406 0.994743i \(-0.467346\pi\)
0.102406 + 0.994743i \(0.467346\pi\)
\(348\) −28.9404 −1.55137
\(349\) −24.3265 −1.30217 −0.651083 0.759006i \(-0.725685\pi\)
−0.651083 + 0.759006i \(0.725685\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 3.48301 0.185645
\(353\) −27.0591 −1.44021 −0.720104 0.693866i \(-0.755906\pi\)
−0.720104 + 0.693866i \(0.755906\pi\)
\(354\) 41.6357 2.21291
\(355\) 0 0
\(356\) 56.2708 2.98235
\(357\) 7.74335 0.409821
\(358\) 19.8215 1.04760
\(359\) −27.0039 −1.42521 −0.712605 0.701566i \(-0.752485\pi\)
−0.712605 + 0.701566i \(0.752485\pi\)
\(360\) 0 0
\(361\) −17.1370 −0.901948
\(362\) −9.84396 −0.517387
\(363\) 22.8138 1.19742
\(364\) 0 0
\(365\) 0 0
\(366\) 12.4424 0.650373
\(367\) 6.94111 0.362323 0.181161 0.983453i \(-0.442014\pi\)
0.181161 + 0.983453i \(0.442014\pi\)
\(368\) 15.3200 0.798608
\(369\) 16.2881 0.847922
\(370\) 0 0
\(371\) 1.88739 0.0979882
\(372\) −86.4505 −4.48225
\(373\) −2.31288 −0.119756 −0.0598781 0.998206i \(-0.519071\pi\)
−0.0598781 + 0.998206i \(0.519071\pi\)
\(374\) −1.97886 −0.102324
\(375\) 0 0
\(376\) 38.9270 2.00751
\(377\) 0 0
\(378\) 21.9827 1.13067
\(379\) −5.17544 −0.265845 −0.132922 0.991126i \(-0.542436\pi\)
−0.132922 + 0.991126i \(0.542436\pi\)
\(380\) 0 0
\(381\) 37.0986 1.90062
\(382\) 12.5854 0.643923
\(383\) −20.6609 −1.05572 −0.527861 0.849331i \(-0.677006\pi\)
−0.527861 + 0.849331i \(0.677006\pi\)
\(384\) 25.0953 1.28064
\(385\) 0 0
\(386\) −12.6178 −0.642229
\(387\) −2.23566 −0.113645
\(388\) 66.2430 3.36298
\(389\) 19.7477 1.00125 0.500624 0.865665i \(-0.333104\pi\)
0.500624 + 0.865665i \(0.333104\pi\)
\(390\) 0 0
\(391\) −2.63509 −0.133262
\(392\) 10.3263 0.521557
\(393\) −21.5293 −1.08601
\(394\) −17.1722 −0.865122
\(395\) 0 0
\(396\) 4.65297 0.233820
\(397\) −9.38902 −0.471222 −0.235611 0.971847i \(-0.575709\pi\)
−0.235611 + 0.971847i \(0.575709\pi\)
\(398\) 13.1753 0.660420
\(399\) 8.63509 0.432295
\(400\) 0 0
\(401\) −24.5019 −1.22357 −0.611784 0.791025i \(-0.709548\pi\)
−0.611784 + 0.791025i \(0.709548\pi\)
\(402\) 44.0109 2.19506
\(403\) 0 0
\(404\) 59.2887 2.94972
\(405\) 0 0
\(406\) 22.4424 1.11380
\(407\) −0.777322 −0.0385304
\(408\) 16.6417 0.823889
\(409\) −36.1165 −1.78584 −0.892922 0.450211i \(-0.851349\pi\)
−0.892922 + 0.450211i \(0.851349\pi\)
\(410\) 0 0
\(411\) −18.6708 −0.920965
\(412\) 49.1479 2.42134
\(413\) −22.3232 −1.09845
\(414\) 8.96157 0.440437
\(415\) 0 0
\(416\) 0 0
\(417\) −30.8439 −1.51043
\(418\) −2.20675 −0.107936
\(419\) −6.86684 −0.335467 −0.167734 0.985832i \(-0.553645\pi\)
−0.167734 + 0.985832i \(0.553645\pi\)
\(420\) 0 0
\(421\) −33.9795 −1.65606 −0.828029 0.560686i \(-0.810538\pi\)
−0.828029 + 0.560686i \(0.810538\pi\)
\(422\) −35.6859 −1.73716
\(423\) 10.0783 0.490026
\(424\) 4.05631 0.196992
\(425\) 0 0
\(426\) 14.4424 0.699735
\(427\) −6.67104 −0.322834
\(428\) −47.8261 −2.31176
\(429\) 0 0
\(430\) 0 0
\(431\) 16.2496 0.782717 0.391359 0.920238i \(-0.372005\pi\)
0.391359 + 0.920238i \(0.372005\pi\)
\(432\) 20.9104 1.00605
\(433\) −0.256261 −0.0123151 −0.00615756 0.999981i \(-0.501960\pi\)
−0.00615756 + 0.999981i \(0.501960\pi\)
\(434\) 67.0396 3.21800
\(435\) 0 0
\(436\) 14.6530 0.701750
\(437\) −2.93855 −0.140570
\(438\) −56.5962 −2.70427
\(439\) −7.59666 −0.362569 −0.181284 0.983431i \(-0.558025\pi\)
−0.181284 + 0.983431i \(0.558025\pi\)
\(440\) 0 0
\(441\) 2.67352 0.127310
\(442\) 0 0
\(443\) −4.32246 −0.205366 −0.102683 0.994714i \(-0.532743\pi\)
−0.102683 + 0.994714i \(0.532743\pi\)
\(444\) 11.8073 0.560348
\(445\) 0 0
\(446\) 0.0211421 0.00100111
\(447\) −36.9332 −1.74688
\(448\) −0.793931 −0.0375097
\(449\) −3.28806 −0.155173 −0.0775865 0.996986i \(-0.524721\pi\)
−0.0775865 + 0.996986i \(0.524721\pi\)
\(450\) 0 0
\(451\) 6.32648 0.297903
\(452\) −24.7779 −1.16545
\(453\) −46.0356 −2.16294
\(454\) −28.6741 −1.34574
\(455\) 0 0
\(456\) 18.5582 0.869069
\(457\) −15.4261 −0.721602 −0.360801 0.932643i \(-0.617497\pi\)
−0.360801 + 0.932643i \(0.617497\pi\)
\(458\) 42.1074 1.96755
\(459\) −3.59666 −0.167878
\(460\) 0 0
\(461\) −25.8847 −1.20557 −0.602786 0.797903i \(-0.705943\pi\)
−0.602786 + 0.797903i \(0.705943\pi\)
\(462\) −10.2285 −0.475872
\(463\) 7.04045 0.327197 0.163599 0.986527i \(-0.447690\pi\)
0.163599 + 0.986527i \(0.447690\pi\)
\(464\) 21.3476 0.991039
\(465\) 0 0
\(466\) 17.6914 0.819538
\(467\) −18.8113 −0.870482 −0.435241 0.900314i \(-0.643337\pi\)
−0.435241 + 0.900314i \(0.643337\pi\)
\(468\) 0 0
\(469\) −23.5967 −1.08959
\(470\) 0 0
\(471\) 39.5377 1.82180
\(472\) −47.9762 −2.20828
\(473\) −0.868356 −0.0399271
\(474\) −5.69140 −0.261414
\(475\) 0 0
\(476\) −16.1159 −0.738670
\(477\) 1.05019 0.0480850
\(478\) −10.1830 −0.465758
\(479\) 19.4775 0.889951 0.444975 0.895543i \(-0.353212\pi\)
0.444975 + 0.895543i \(0.353212\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −50.3346 −2.29268
\(483\) −13.6205 −0.619752
\(484\) −47.4814 −2.15824
\(485\) 0 0
\(486\) 39.1165 1.77436
\(487\) −32.3241 −1.46474 −0.732372 0.680905i \(-0.761587\pi\)
−0.732372 + 0.680905i \(0.761587\pi\)
\(488\) −14.3372 −0.649013
\(489\) −8.63509 −0.390492
\(490\) 0 0
\(491\) 28.6708 1.29390 0.646949 0.762534i \(-0.276045\pi\)
0.646949 + 0.762534i \(0.276045\pi\)
\(492\) −96.0973 −4.33240
\(493\) −3.67187 −0.165373
\(494\) 0 0
\(495\) 0 0
\(496\) 63.7694 2.86333
\(497\) −7.74335 −0.347337
\(498\) 64.9213 2.90919
\(499\) −28.9616 −1.29650 −0.648249 0.761428i \(-0.724498\pi\)
−0.648249 + 0.761428i \(0.724498\pi\)
\(500\) 0 0
\(501\) 6.32648 0.282646
\(502\) 9.35181 0.417392
\(503\) 28.1093 1.25333 0.626665 0.779289i \(-0.284419\pi\)
0.626665 + 0.779289i \(0.284419\pi\)
\(504\) 30.3444 1.35165
\(505\) 0 0
\(506\) 3.48079 0.154740
\(507\) 0 0
\(508\) −77.2116 −3.42571
\(509\) −21.1126 −0.935800 −0.467900 0.883781i \(-0.654989\pi\)
−0.467900 + 0.883781i \(0.654989\pi\)
\(510\) 0 0
\(511\) 30.3444 1.34236
\(512\) −50.8542 −2.24746
\(513\) −4.01086 −0.177084
\(514\) −33.7688 −1.48948
\(515\) 0 0
\(516\) 13.1901 0.580660
\(517\) 3.91455 0.172162
\(518\) −9.15616 −0.402299
\(519\) −2.94369 −0.129214
\(520\) 0 0
\(521\) 0.673516 0.0295073 0.0147536 0.999891i \(-0.495304\pi\)
0.0147536 + 0.999891i \(0.495304\pi\)
\(522\) 12.4875 0.546564
\(523\) 29.8626 1.30580 0.652900 0.757444i \(-0.273552\pi\)
0.652900 + 0.757444i \(0.273552\pi\)
\(524\) 44.8079 1.95744
\(525\) 0 0
\(526\) −77.0608 −3.36001
\(527\) −10.9686 −0.477799
\(528\) −9.72953 −0.423423
\(529\) −18.3649 −0.798474
\(530\) 0 0
\(531\) −12.4212 −0.539035
\(532\) −17.9718 −0.779177
\(533\) 0 0
\(534\) −68.8290 −2.97852
\(535\) 0 0
\(536\) −50.7131 −2.19047
\(537\) −16.7630 −0.723375
\(538\) −56.6418 −2.44200
\(539\) 1.03843 0.0447282
\(540\) 0 0
\(541\) −6.28806 −0.270345 −0.135172 0.990822i \(-0.543159\pi\)
−0.135172 + 0.990822i \(0.543159\pi\)
\(542\) 30.1121 1.29342
\(543\) 8.32502 0.357261
\(544\) −6.71254 −0.287798
\(545\) 0 0
\(546\) 0 0
\(547\) 3.03789 0.129891 0.0649454 0.997889i \(-0.479313\pi\)
0.0649454 + 0.997889i \(0.479313\pi\)
\(548\) 38.8588 1.65997
\(549\) −3.71194 −0.158422
\(550\) 0 0
\(551\) −4.09473 −0.174442
\(552\) −29.2726 −1.24592
\(553\) 3.05147 0.129762
\(554\) −42.7304 −1.81544
\(555\) 0 0
\(556\) 64.1939 2.72243
\(557\) 20.6996 0.877071 0.438536 0.898714i \(-0.355497\pi\)
0.438536 + 0.898714i \(0.355497\pi\)
\(558\) 37.3026 1.57915
\(559\) 0 0
\(560\) 0 0
\(561\) 1.67352 0.0706559
\(562\) 26.9763 1.13793
\(563\) −10.9603 −0.461921 −0.230960 0.972963i \(-0.574187\pi\)
−0.230960 + 0.972963i \(0.574187\pi\)
\(564\) −59.4608 −2.50375
\(565\) 0 0
\(566\) 22.4424 0.943323
\(567\) −33.0051 −1.38608
\(568\) −16.6417 −0.698272
\(569\) 42.7131 1.79063 0.895314 0.445436i \(-0.146951\pi\)
0.895314 + 0.445436i \(0.146951\pi\)
\(570\) 0 0
\(571\) −23.6145 −0.988238 −0.494119 0.869394i \(-0.664509\pi\)
−0.494119 + 0.869394i \(0.664509\pi\)
\(572\) 0 0
\(573\) −10.6434 −0.444635
\(574\) 74.5204 3.11042
\(575\) 0 0
\(576\) −0.441765 −0.0184069
\(577\) −18.3646 −0.764530 −0.382265 0.924053i \(-0.624856\pi\)
−0.382265 + 0.924053i \(0.624856\pi\)
\(578\) −39.4639 −1.64148
\(579\) 10.6708 0.443465
\(580\) 0 0
\(581\) −34.8079 −1.44407
\(582\) −81.0268 −3.35867
\(583\) 0.407908 0.0168938
\(584\) 65.2151 2.69862
\(585\) 0 0
\(586\) −71.9237 −2.97114
\(587\) −0.702897 −0.0290116 −0.0145058 0.999895i \(-0.504618\pi\)
−0.0145058 + 0.999895i \(0.504618\pi\)
\(588\) −15.7734 −0.650483
\(589\) −12.2318 −0.504001
\(590\) 0 0
\(591\) 14.5225 0.597375
\(592\) −8.70953 −0.357959
\(593\) −37.1593 −1.52595 −0.762975 0.646428i \(-0.776262\pi\)
−0.762975 + 0.646428i \(0.776262\pi\)
\(594\) 4.75096 0.194934
\(595\) 0 0
\(596\) 76.8674 3.14861
\(597\) −11.1423 −0.456026
\(598\) 0 0
\(599\) −15.6914 −0.641133 −0.320567 0.947226i \(-0.603873\pi\)
−0.320567 + 0.947226i \(0.603873\pi\)
\(600\) 0 0
\(601\) 12.0039 0.489648 0.244824 0.969568i \(-0.421270\pi\)
0.244824 + 0.969568i \(0.421270\pi\)
\(602\) −10.2285 −0.416881
\(603\) −13.1298 −0.534687
\(604\) 95.8117 3.89852
\(605\) 0 0
\(606\) −72.5204 −2.94594
\(607\) −38.6865 −1.57024 −0.785119 0.619345i \(-0.787398\pi\)
−0.785119 + 0.619345i \(0.787398\pi\)
\(608\) −7.48557 −0.303580
\(609\) −18.9795 −0.769086
\(610\) 0 0
\(611\) 0 0
\(612\) −8.96730 −0.362482
\(613\) −17.2840 −0.698095 −0.349047 0.937105i \(-0.613495\pi\)
−0.349047 + 0.937105i \(0.613495\pi\)
\(614\) 32.3866 1.30702
\(615\) 0 0
\(616\) 11.7861 0.474877
\(617\) 26.4691 1.06561 0.532803 0.846240i \(-0.321139\pi\)
0.532803 + 0.846240i \(0.321139\pi\)
\(618\) −60.1165 −2.41824
\(619\) 31.0039 1.24615 0.623075 0.782162i \(-0.285883\pi\)
0.623075 + 0.782162i \(0.285883\pi\)
\(620\) 0 0
\(621\) 6.32648 0.253873
\(622\) 71.0850 2.85025
\(623\) 36.9030 1.47849
\(624\) 0 0
\(625\) 0 0
\(626\) −62.5761 −2.50104
\(627\) 1.86624 0.0745305
\(628\) −82.2880 −3.28365
\(629\) 1.49807 0.0597320
\(630\) 0 0
\(631\) 20.7131 0.824577 0.412288 0.911053i \(-0.364730\pi\)
0.412288 + 0.911053i \(0.364730\pi\)
\(632\) 6.55812 0.260868
\(633\) 30.1795 1.19953
\(634\) −0.596662 −0.0236965
\(635\) 0 0
\(636\) −6.19599 −0.245687
\(637\) 0 0
\(638\) 4.85031 0.192026
\(639\) −4.30860 −0.170446
\(640\) 0 0
\(641\) 21.1895 0.836934 0.418467 0.908232i \(-0.362568\pi\)
0.418467 + 0.908232i \(0.362568\pi\)
\(642\) 58.4997 2.30880
\(643\) −11.5336 −0.454843 −0.227421 0.973796i \(-0.573029\pi\)
−0.227421 + 0.973796i \(0.573029\pi\)
\(644\) 28.3476 1.11705
\(645\) 0 0
\(646\) 4.25289 0.167328
\(647\) 34.8464 1.36995 0.684977 0.728565i \(-0.259812\pi\)
0.684977 + 0.728565i \(0.259812\pi\)
\(648\) −70.9334 −2.78653
\(649\) −4.82456 −0.189380
\(650\) 0 0
\(651\) −56.6953 −2.22206
\(652\) 17.9718 0.703831
\(653\) −22.3232 −0.873574 −0.436787 0.899565i \(-0.643884\pi\)
−0.436787 + 0.899565i \(0.643884\pi\)
\(654\) −17.9231 −0.700850
\(655\) 0 0
\(656\) 70.8853 2.76761
\(657\) 16.8844 0.658724
\(658\) 46.1100 1.79755
\(659\) −0.866840 −0.0337673 −0.0168836 0.999857i \(-0.505374\pi\)
−0.0168836 + 0.999857i \(0.505374\pi\)
\(660\) 0 0
\(661\) −13.3086 −0.517645 −0.258822 0.965925i \(-0.583334\pi\)
−0.258822 + 0.965925i \(0.583334\pi\)
\(662\) 46.6544 1.81328
\(663\) 0 0
\(664\) −74.8079 −2.90311
\(665\) 0 0
\(666\) −5.09473 −0.197417
\(667\) 6.45878 0.250085
\(668\) −13.1670 −0.509448
\(669\) −0.0178799 −0.000691275 0
\(670\) 0 0
\(671\) −1.44176 −0.0556587
\(672\) −34.6963 −1.33844
\(673\) 5.51320 0.212518 0.106259 0.994338i \(-0.466113\pi\)
0.106259 + 0.994338i \(0.466113\pi\)
\(674\) 54.2112 2.08814
\(675\) 0 0
\(676\) 0 0
\(677\) 4.80479 0.184663 0.0923316 0.995728i \(-0.470568\pi\)
0.0923316 + 0.995728i \(0.470568\pi\)
\(678\) 30.3077 1.16396
\(679\) 43.4430 1.66719
\(680\) 0 0
\(681\) 24.2496 0.929248
\(682\) 14.4888 0.554805
\(683\) 11.7625 0.450080 0.225040 0.974350i \(-0.427749\pi\)
0.225040 + 0.974350i \(0.427749\pi\)
\(684\) −10.0000 −0.382360
\(685\) 0 0
\(686\) −40.1338 −1.53231
\(687\) −35.6102 −1.35861
\(688\) −9.72953 −0.370935
\(689\) 0 0
\(690\) 0 0
\(691\) −4.86684 −0.185143 −0.0925717 0.995706i \(-0.529509\pi\)
−0.0925717 + 0.995706i \(0.529509\pi\)
\(692\) 6.12658 0.232897
\(693\) 3.05147 0.115916
\(694\) 9.71254 0.368683
\(695\) 0 0
\(696\) −40.7900 −1.54614
\(697\) −12.1925 −0.461825
\(698\) −61.9289 −2.34404
\(699\) −14.9616 −0.565899
\(700\) 0 0
\(701\) 21.3828 0.807617 0.403808 0.914844i \(-0.367686\pi\)
0.403808 + 0.914844i \(0.367686\pi\)
\(702\) 0 0
\(703\) 1.67059 0.0630076
\(704\) −0.171587 −0.00646692
\(705\) 0 0
\(706\) −68.8853 −2.59253
\(707\) 38.8822 1.46232
\(708\) 73.2835 2.75416
\(709\) 26.1165 0.980825 0.490412 0.871491i \(-0.336846\pi\)
0.490412 + 0.871491i \(0.336846\pi\)
\(710\) 0 0
\(711\) 1.69792 0.0636770
\(712\) 79.3107 2.97229
\(713\) 19.2936 0.722551
\(714\) 19.7125 0.737723
\(715\) 0 0
\(716\) 34.8880 1.30383
\(717\) 8.61170 0.321610
\(718\) −68.7448 −2.56553
\(719\) 36.6774 1.36784 0.683918 0.729559i \(-0.260275\pi\)
0.683918 + 0.729559i \(0.260275\pi\)
\(720\) 0 0
\(721\) 32.2318 1.20037
\(722\) −43.6264 −1.62361
\(723\) 42.5679 1.58312
\(724\) −17.3265 −0.643934
\(725\) 0 0
\(726\) 58.0780 2.15548
\(727\) 26.2596 0.973916 0.486958 0.873425i \(-0.338107\pi\)
0.486958 + 0.873425i \(0.338107\pi\)
\(728\) 0 0
\(729\) 0.614542 0.0227608
\(730\) 0 0
\(731\) 1.67352 0.0618972
\(732\) 21.9000 0.809446
\(733\) 31.7811 1.17386 0.586931 0.809637i \(-0.300336\pi\)
0.586931 + 0.809637i \(0.300336\pi\)
\(734\) 17.6703 0.652221
\(735\) 0 0
\(736\) 11.8073 0.435222
\(737\) −5.09978 −0.187853
\(738\) 41.4651 1.52635
\(739\) 34.1370 1.25575 0.627875 0.778314i \(-0.283925\pi\)
0.627875 + 0.778314i \(0.283925\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4.80479 0.176390
\(743\) 3.12062 0.114485 0.0572423 0.998360i \(-0.481769\pi\)
0.0572423 + 0.998360i \(0.481769\pi\)
\(744\) −121.847 −4.46715
\(745\) 0 0
\(746\) −5.88798 −0.215574
\(747\) −19.3680 −0.708639
\(748\) −3.48301 −0.127352
\(749\) −31.3649 −1.14605
\(750\) 0 0
\(751\) 1.48405 0.0541537 0.0270769 0.999633i \(-0.491380\pi\)
0.0270769 + 0.999633i \(0.491380\pi\)
\(752\) 43.8607 1.59944
\(753\) −7.90881 −0.288213
\(754\) 0 0
\(755\) 0 0
\(756\) 38.6920 1.40721
\(757\) −5.09978 −0.185355 −0.0926774 0.995696i \(-0.529543\pi\)
−0.0926774 + 0.995696i \(0.529543\pi\)
\(758\) −13.1753 −0.478550
\(759\) −2.94369 −0.106849
\(760\) 0 0
\(761\) 29.7861 1.07975 0.539873 0.841746i \(-0.318472\pi\)
0.539873 + 0.841746i \(0.318472\pi\)
\(762\) 94.4433 3.42132
\(763\) 9.60959 0.347890
\(764\) 22.1516 0.801418
\(765\) 0 0
\(766\) −52.5973 −1.90042
\(767\) 0 0
\(768\) 62.7228 2.26331
\(769\) −19.0986 −0.688713 −0.344356 0.938839i \(-0.611903\pi\)
−0.344356 + 0.938839i \(0.611903\pi\)
\(770\) 0 0
\(771\) 28.5582 1.02850
\(772\) −22.2088 −0.799311
\(773\) 49.2306 1.77070 0.885351 0.464923i \(-0.153918\pi\)
0.885351 + 0.464923i \(0.153918\pi\)
\(774\) −5.69140 −0.204573
\(775\) 0 0
\(776\) 93.3661 3.35165
\(777\) 7.74335 0.277791
\(778\) 50.2725 1.80236
\(779\) −13.5967 −0.487151
\(780\) 0 0
\(781\) −1.67352 −0.0598831
\(782\) −6.70825 −0.239886
\(783\) 8.81566 0.315046
\(784\) 11.6351 0.415539
\(785\) 0 0
\(786\) −54.8079 −1.95493
\(787\) −9.78335 −0.348739 −0.174369 0.984680i \(-0.555789\pi\)
−0.174369 + 0.984680i \(0.555789\pi\)
\(788\) −30.2250 −1.07672
\(789\) 65.1701 2.32012
\(790\) 0 0
\(791\) −16.2496 −0.577770
\(792\) 6.55812 0.233033
\(793\) 0 0
\(794\) −23.9020 −0.848250
\(795\) 0 0
\(796\) 23.1901 0.821950
\(797\) −16.5371 −0.585775 −0.292887 0.956147i \(-0.594616\pi\)
−0.292887 + 0.956147i \(0.594616\pi\)
\(798\) 21.9827 0.778178
\(799\) −7.54421 −0.266895
\(800\) 0 0
\(801\) 20.5338 0.725527
\(802\) −62.3755 −2.20256
\(803\) 6.55812 0.231431
\(804\) 77.4641 2.73195
\(805\) 0 0
\(806\) 0 0
\(807\) 47.9018 1.68622
\(808\) 83.5643 2.93978
\(809\) −31.8424 −1.11952 −0.559760 0.828655i \(-0.689107\pi\)
−0.559760 + 0.828655i \(0.689107\pi\)
\(810\) 0 0
\(811\) −13.3470 −0.468678 −0.234339 0.972155i \(-0.575293\pi\)
−0.234339 + 0.972155i \(0.575293\pi\)
\(812\) 39.5011 1.38622
\(813\) −25.4657 −0.893121
\(814\) −1.97886 −0.0693589
\(815\) 0 0
\(816\) 18.7510 0.656415
\(817\) 1.86624 0.0652915
\(818\) −91.9431 −3.21472
\(819\) 0 0
\(820\) 0 0
\(821\) −11.6735 −0.407409 −0.203704 0.979032i \(-0.565298\pi\)
−0.203704 + 0.979032i \(0.565298\pi\)
\(822\) −47.5311 −1.65784
\(823\) −32.4317 −1.13050 −0.565249 0.824920i \(-0.691220\pi\)
−0.565249 + 0.824920i \(0.691220\pi\)
\(824\) 69.2714 2.41318
\(825\) 0 0
\(826\) −56.8290 −1.97733
\(827\) 27.3319 0.950425 0.475212 0.879871i \(-0.342371\pi\)
0.475212 + 0.879871i \(0.342371\pi\)
\(828\) 15.7734 0.548163
\(829\) 3.54036 0.122962 0.0614808 0.998108i \(-0.480418\pi\)
0.0614808 + 0.998108i \(0.480418\pi\)
\(830\) 0 0
\(831\) 36.1370 1.25358
\(832\) 0 0
\(833\) −2.00128 −0.0693402
\(834\) −78.5204 −2.71894
\(835\) 0 0
\(836\) −3.88412 −0.134335
\(837\) 26.3341 0.910238
\(838\) −17.4812 −0.603877
\(839\) −44.7900 −1.54632 −0.773161 0.634210i \(-0.781326\pi\)
−0.773161 + 0.634210i \(0.781326\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −86.5028 −2.98108
\(843\) −22.8138 −0.785750
\(844\) −62.8111 −2.16205
\(845\) 0 0
\(846\) 25.6568 0.882100
\(847\) −31.1388 −1.06994
\(848\) 4.57042 0.156949
\(849\) −18.9795 −0.651373
\(850\) 0 0
\(851\) −2.63509 −0.0903297
\(852\) 25.4202 0.870881
\(853\) −31.3732 −1.07420 −0.537099 0.843519i \(-0.680480\pi\)
−0.537099 + 0.843519i \(0.680480\pi\)
\(854\) −16.9827 −0.581137
\(855\) 0 0
\(856\) −67.4084 −2.30397
\(857\) 21.2813 0.726955 0.363478 0.931603i \(-0.381589\pi\)
0.363478 + 0.931603i \(0.381589\pi\)
\(858\) 0 0
\(859\) 56.8502 1.93970 0.969851 0.243698i \(-0.0783607\pi\)
0.969851 + 0.243698i \(0.0783607\pi\)
\(860\) 0 0
\(861\) −63.0217 −2.14778
\(862\) 41.3673 1.40898
\(863\) 32.8011 1.11656 0.558282 0.829651i \(-0.311461\pi\)
0.558282 + 0.829651i \(0.311461\pi\)
\(864\) 16.1159 0.548273
\(865\) 0 0
\(866\) −0.652374 −0.0221686
\(867\) 33.3745 1.13346
\(868\) 117.997 4.00509
\(869\) 0.659493 0.0223718
\(870\) 0 0
\(871\) 0 0
\(872\) 20.6526 0.699385
\(873\) 24.1728 0.818126
\(874\) −7.48079 −0.253041
\(875\) 0 0
\(876\) −99.6157 −3.36570
\(877\) −36.0651 −1.21783 −0.608916 0.793235i \(-0.708395\pi\)
−0.608916 + 0.793235i \(0.708395\pi\)
\(878\) −19.3391 −0.652664
\(879\) 60.8257 2.05160
\(880\) 0 0
\(881\) −46.0396 −1.55111 −0.775557 0.631277i \(-0.782531\pi\)
−0.775557 + 0.631277i \(0.782531\pi\)
\(882\) 6.80607 0.229172
\(883\) −0.802236 −0.0269974 −0.0134987 0.999909i \(-0.504297\pi\)
−0.0134987 + 0.999909i \(0.504297\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −11.0039 −0.369682
\(887\) −8.22568 −0.276191 −0.138096 0.990419i \(-0.544098\pi\)
−0.138096 + 0.990419i \(0.544098\pi\)
\(888\) 16.6417 0.558460
\(889\) −50.6363 −1.69829
\(890\) 0 0
\(891\) −7.13316 −0.238970
\(892\) 0.0372125 0.00124597
\(893\) −8.41302 −0.281531
\(894\) −94.0223 −3.14458
\(895\) 0 0
\(896\) −34.2529 −1.14431
\(897\) 0 0
\(898\) −8.37054 −0.279328
\(899\) 26.8847 0.896656
\(900\) 0 0
\(901\) −0.786129 −0.0261897
\(902\) 16.1056 0.536257
\(903\) 8.65020 0.287861
\(904\) −34.9231 −1.16153
\(905\) 0 0
\(906\) −117.195 −3.89353
\(907\) 30.4359 1.01061 0.505305 0.862941i \(-0.331380\pi\)
0.505305 + 0.862941i \(0.331380\pi\)
\(908\) −50.4697 −1.67489
\(909\) 21.6351 0.717591
\(910\) 0 0
\(911\) 43.6145 1.44501 0.722507 0.691363i \(-0.242990\pi\)
0.722507 + 0.691363i \(0.242990\pi\)
\(912\) 20.9104 0.692412
\(913\) −7.52278 −0.248968
\(914\) −39.2708 −1.29896
\(915\) 0 0
\(916\) 74.1138 2.44879
\(917\) 29.3855 0.970395
\(918\) −9.15616 −0.302198
\(919\) −37.0217 −1.22123 −0.610617 0.791926i \(-0.709079\pi\)
−0.610617 + 0.791926i \(0.709079\pi\)
\(920\) 0 0
\(921\) −27.3893 −0.902509
\(922\) −65.8957 −2.17016
\(923\) 0 0
\(924\) −18.0033 −0.592264
\(925\) 0 0
\(926\) 17.9231 0.588991
\(927\) 17.9346 0.589050
\(928\) 16.4529 0.540092
\(929\) 4.76825 0.156441 0.0782206 0.996936i \(-0.475076\pi\)
0.0782206 + 0.996936i \(0.475076\pi\)
\(930\) 0 0
\(931\) −2.23175 −0.0731427
\(932\) 31.1388 1.01999
\(933\) −60.1165 −1.96812
\(934\) −47.8886 −1.56696
\(935\) 0 0
\(936\) 0 0
\(937\) −43.6264 −1.42521 −0.712606 0.701565i \(-0.752485\pi\)
−0.712606 + 0.701565i \(0.752485\pi\)
\(938\) −60.0709 −1.96139
\(939\) 52.9205 1.72699
\(940\) 0 0
\(941\) −18.2675 −0.595504 −0.297752 0.954643i \(-0.596237\pi\)
−0.297752 + 0.954643i \(0.596237\pi\)
\(942\) 100.653 3.27944
\(943\) 21.4465 0.698395
\(944\) −54.0569 −1.75940
\(945\) 0 0
\(946\) −2.21061 −0.0718731
\(947\) −19.9829 −0.649358 −0.324679 0.945824i \(-0.605256\pi\)
−0.324679 + 0.945824i \(0.605256\pi\)
\(948\) −10.0175 −0.325353
\(949\) 0 0
\(950\) 0 0
\(951\) 0.504596 0.0163626
\(952\) −22.7145 −0.736181
\(953\) 39.8635 1.29130 0.645652 0.763632i \(-0.276586\pi\)
0.645652 + 0.763632i \(0.276586\pi\)
\(954\) 2.67352 0.0865583
\(955\) 0 0
\(956\) −17.9231 −0.579676
\(957\) −4.10190 −0.132596
\(958\) 49.5847 1.60201
\(959\) 25.4840 0.822923
\(960\) 0 0
\(961\) 49.3098 1.59064
\(962\) 0 0
\(963\) −17.4523 −0.562392
\(964\) −88.5946 −2.85344
\(965\) 0 0
\(966\) −34.6741 −1.11562
\(967\) −43.8607 −1.41047 −0.705233 0.708975i \(-0.749158\pi\)
−0.705233 + 0.708975i \(0.749158\pi\)
\(968\) −66.9225 −2.15097
\(969\) −3.59666 −0.115541
\(970\) 0 0
\(971\) 60.9795 1.95692 0.978462 0.206428i \(-0.0661838\pi\)
0.978462 + 0.206428i \(0.0661838\pi\)
\(972\) 68.8494 2.20835
\(973\) 42.0991 1.34964
\(974\) −82.2887 −2.63670
\(975\) 0 0
\(976\) −16.1543 −0.517087
\(977\) 51.3697 1.64346 0.821731 0.569875i \(-0.193008\pi\)
0.821731 + 0.569875i \(0.193008\pi\)
\(978\) −21.9827 −0.702929
\(979\) 7.97560 0.254901
\(980\) 0 0
\(981\) 5.34703 0.170718
\(982\) 72.9885 2.32916
\(983\) 37.3026 1.18977 0.594885 0.803811i \(-0.297198\pi\)
0.594885 + 0.803811i \(0.297198\pi\)
\(984\) −135.444 −4.31780
\(985\) 0 0
\(986\) −9.34763 −0.297689
\(987\) −38.9951 −1.24123
\(988\) 0 0
\(989\) −2.94369 −0.0936040
\(990\) 0 0
\(991\) −51.5621 −1.63792 −0.818962 0.573848i \(-0.805450\pi\)
−0.818962 + 0.573848i \(0.805450\pi\)
\(992\) 49.1479 1.56045
\(993\) −39.4556 −1.25208
\(994\) −19.7125 −0.625244
\(995\) 0 0
\(996\) 114.269 3.62074
\(997\) −22.9489 −0.726798 −0.363399 0.931634i \(-0.618384\pi\)
−0.363399 + 0.931634i \(0.618384\pi\)
\(998\) −73.7286 −2.33384
\(999\) −3.59666 −0.113793
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 4225.2.a.bq.1.6 6
5.2 odd 4 845.2.b.e.339.6 6
5.3 odd 4 845.2.b.e.339.1 6
5.4 even 2 inner 4225.2.a.bq.1.1 6
13.4 even 6 325.2.e.e.276.6 12
13.10 even 6 325.2.e.e.126.6 12
13.12 even 2 4225.2.a.br.1.1 6
65.2 even 12 845.2.l.f.654.1 24
65.3 odd 12 845.2.n.e.529.6 12
65.4 even 6 325.2.e.e.276.1 12
65.7 even 12 845.2.l.f.699.12 24
65.8 even 4 845.2.d.d.844.11 12
65.12 odd 4 845.2.b.d.339.1 6
65.17 odd 12 65.2.n.a.29.1 yes 12
65.18 even 4 845.2.d.d.844.1 12
65.22 odd 12 845.2.n.e.484.6 12
65.23 odd 12 65.2.n.a.9.1 12
65.28 even 12 845.2.l.f.654.12 24
65.32 even 12 845.2.l.f.699.2 24
65.33 even 12 845.2.l.f.699.1 24
65.37 even 12 845.2.l.f.654.11 24
65.38 odd 4 845.2.b.d.339.6 6
65.42 odd 12 845.2.n.e.529.1 12
65.43 odd 12 65.2.n.a.29.6 yes 12
65.47 even 4 845.2.d.d.844.2 12
65.48 odd 12 845.2.n.e.484.1 12
65.49 even 6 325.2.e.e.126.1 12
65.57 even 4 845.2.d.d.844.12 12
65.58 even 12 845.2.l.f.699.11 24
65.62 odd 12 65.2.n.a.9.6 yes 12
65.63 even 12 845.2.l.f.654.2 24
65.64 even 2 4225.2.a.br.1.6 6
195.17 even 12 585.2.bs.a.289.6 12
195.23 even 12 585.2.bs.a.334.6 12
195.62 even 12 585.2.bs.a.334.1 12
195.173 even 12 585.2.bs.a.289.1 12
260.23 even 12 1040.2.dh.a.529.2 12
260.43 even 12 1040.2.dh.a.289.5 12
260.127 even 12 1040.2.dh.a.529.5 12
260.147 even 12 1040.2.dh.a.289.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.n.a.9.1 12 65.23 odd 12
65.2.n.a.9.6 yes 12 65.62 odd 12
65.2.n.a.29.1 yes 12 65.17 odd 12
65.2.n.a.29.6 yes 12 65.43 odd 12
325.2.e.e.126.1 12 65.49 even 6
325.2.e.e.126.6 12 13.10 even 6
325.2.e.e.276.1 12 65.4 even 6
325.2.e.e.276.6 12 13.4 even 6
585.2.bs.a.289.1 12 195.173 even 12
585.2.bs.a.289.6 12 195.17 even 12
585.2.bs.a.334.1 12 195.62 even 12
585.2.bs.a.334.6 12 195.23 even 12
845.2.b.d.339.1 6 65.12 odd 4
845.2.b.d.339.6 6 65.38 odd 4
845.2.b.e.339.1 6 5.3 odd 4
845.2.b.e.339.6 6 5.2 odd 4
845.2.d.d.844.1 12 65.18 even 4
845.2.d.d.844.2 12 65.47 even 4
845.2.d.d.844.11 12 65.8 even 4
845.2.d.d.844.12 12 65.57 even 4
845.2.l.f.654.1 24 65.2 even 12
845.2.l.f.654.2 24 65.63 even 12
845.2.l.f.654.11 24 65.37 even 12
845.2.l.f.654.12 24 65.28 even 12
845.2.l.f.699.1 24 65.33 even 12
845.2.l.f.699.2 24 65.32 even 12
845.2.l.f.699.11 24 65.58 even 12
845.2.l.f.699.12 24 65.7 even 12
845.2.n.e.484.1 12 65.48 odd 12
845.2.n.e.484.6 12 65.22 odd 12
845.2.n.e.529.1 12 65.42 odd 12
845.2.n.e.529.6 12 65.3 odd 12
1040.2.dh.a.289.2 12 260.147 even 12
1040.2.dh.a.289.5 12 260.43 even 12
1040.2.dh.a.529.2 12 260.23 even 12
1040.2.dh.a.529.5 12 260.127 even 12
4225.2.a.bq.1.1 6 5.4 even 2 inner
4225.2.a.bq.1.6 6 1.1 even 1 trivial
4225.2.a.br.1.1 6 13.12 even 2
4225.2.a.br.1.6 6 65.64 even 2