Properties

Label 4225.2.a.bq.1.5
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 1.5
Root \(1.18733\) of defining polynomial
Character \(\chi\) \(=\) 4225.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.18733 q^{2} -0.345110 q^{3} -0.590239 q^{4} -0.409761 q^{6} +2.02956 q^{7} -3.07548 q^{8} -2.88090 q^{9} +O(q^{10})\) \(q+1.18733 q^{2} -0.345110 q^{3} -0.590239 q^{4} -0.409761 q^{6} +2.02956 q^{7} -3.07548 q^{8} -2.88090 q^{9} -3.88090 q^{11} +0.203698 q^{12} +2.40976 q^{14} -2.47114 q^{16} +5.45014 q^{17} -3.42059 q^{18} -5.88090 q^{19} -0.700420 q^{21} -4.60792 q^{22} +0.345110 q^{23} +1.06138 q^{24} +2.02956 q^{27} -1.19792 q^{28} +3.00000 q^{29} -1.18048 q^{31} +3.21689 q^{32} +1.33934 q^{33} +6.47114 q^{34} +1.70042 q^{36} +5.45014 q^{37} -6.98259 q^{38} -0.180479 q^{41} -0.831632 q^{42} +1.33934 q^{43} +2.29066 q^{44} +0.409761 q^{46} +12.2807 q^{47} +0.852815 q^{48} -2.88090 q^{49} -1.88090 q^{51} -2.42636 q^{53} +2.40976 q^{54} -6.24186 q^{56} +2.02956 q^{57} +3.56200 q^{58} +7.06138 q^{59} +6.76180 q^{61} -1.40162 q^{62} -5.84695 q^{63} +8.76180 q^{64} +1.59024 q^{66} -4.40422 q^{67} -3.21689 q^{68} -0.119101 q^{69} +1.88090 q^{71} +8.86014 q^{72} +8.86014 q^{73} +6.47114 q^{74} +3.47114 q^{76} -7.87651 q^{77} +11.1805 q^{79} +7.94228 q^{81} -0.214289 q^{82} +7.83540 q^{83} +0.413416 q^{84} +1.59024 q^{86} -1.03533 q^{87} +11.9356 q^{88} -12.2419 q^{89} -0.203698 q^{92} +0.407395 q^{93} +14.5813 q^{94} -1.11018 q^{96} -5.80585 q^{97} -3.42059 q^{98} +11.1805 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{4} - 10 q^{6} + 6 q^{9} + O(q^{10}) \) \( 6 q + 4 q^{4} - 10 q^{6} + 6 q^{9} + 22 q^{14} + 16 q^{16} - 12 q^{19} + 4 q^{21} - 32 q^{24} + 18 q^{29} + 8 q^{31} + 8 q^{34} + 2 q^{36} + 14 q^{41} - 2 q^{44} + 10 q^{46} + 6 q^{49} + 12 q^{51} + 22 q^{54} + 16 q^{56} + 4 q^{59} - 6 q^{61} + 6 q^{64} + 2 q^{66} - 24 q^{69} - 12 q^{71} + 8 q^{74} - 10 q^{76} + 52 q^{79} - 14 q^{81} - 90 q^{84} + 2 q^{86} - 20 q^{89} + 56 q^{94} - 6 q^{96} + 52 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.18733 0.839571 0.419786 0.907623i \(-0.362105\pi\)
0.419786 + 0.907623i \(0.362105\pi\)
\(3\) −0.345110 −0.199249 −0.0996247 0.995025i \(-0.531764\pi\)
−0.0996247 + 0.995025i \(0.531764\pi\)
\(4\) −0.590239 −0.295120
\(5\) 0 0
\(6\) −0.409761 −0.167284
\(7\) 2.02956 0.767100 0.383550 0.923520i \(-0.374701\pi\)
0.383550 + 0.923520i \(0.374701\pi\)
\(8\) −3.07548 −1.08735
\(9\) −2.88090 −0.960300
\(10\) 0 0
\(11\) −3.88090 −1.17014 −0.585068 0.810985i \(-0.698932\pi\)
−0.585068 + 0.810985i \(0.698932\pi\)
\(12\) 0.203698 0.0588024
\(13\) 0 0
\(14\) 2.40976 0.644036
\(15\) 0 0
\(16\) −2.47114 −0.617785
\(17\) 5.45014 1.32185 0.660927 0.750450i \(-0.270163\pi\)
0.660927 + 0.750450i \(0.270163\pi\)
\(18\) −3.42059 −0.806240
\(19\) −5.88090 −1.34917 −0.674585 0.738197i \(-0.735678\pi\)
−0.674585 + 0.738197i \(0.735678\pi\)
\(20\) 0 0
\(21\) −0.700420 −0.152844
\(22\) −4.60792 −0.982412
\(23\) 0.345110 0.0719604 0.0359802 0.999353i \(-0.488545\pi\)
0.0359802 + 0.999353i \(0.488545\pi\)
\(24\) 1.06138 0.216653
\(25\) 0 0
\(26\) 0 0
\(27\) 2.02956 0.390588
\(28\) −1.19792 −0.226386
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −1.18048 −0.212020 −0.106010 0.994365i \(-0.533808\pi\)
−0.106010 + 0.994365i \(0.533808\pi\)
\(32\) 3.21689 0.568671
\(33\) 1.33934 0.233149
\(34\) 6.47114 1.10979
\(35\) 0 0
\(36\) 1.70042 0.283403
\(37\) 5.45014 0.895998 0.447999 0.894034i \(-0.352137\pi\)
0.447999 + 0.894034i \(0.352137\pi\)
\(38\) −6.98259 −1.13273
\(39\) 0 0
\(40\) 0 0
\(41\) −0.180479 −0.0281861 −0.0140930 0.999901i \(-0.504486\pi\)
−0.0140930 + 0.999901i \(0.504486\pi\)
\(42\) −0.831632 −0.128324
\(43\) 1.33934 0.204247 0.102123 0.994772i \(-0.467436\pi\)
0.102123 + 0.994772i \(0.467436\pi\)
\(44\) 2.29066 0.345330
\(45\) 0 0
\(46\) 0.409761 0.0604159
\(47\) 12.2807 1.79133 0.895664 0.444731i \(-0.146701\pi\)
0.895664 + 0.444731i \(0.146701\pi\)
\(48\) 0.852815 0.123093
\(49\) −2.88090 −0.411557
\(50\) 0 0
\(51\) −1.88090 −0.263379
\(52\) 0 0
\(53\) −2.42636 −0.333286 −0.166643 0.986017i \(-0.553293\pi\)
−0.166643 + 0.986017i \(0.553293\pi\)
\(54\) 2.40976 0.327927
\(55\) 0 0
\(56\) −6.24186 −0.834103
\(57\) 2.02956 0.268821
\(58\) 3.56200 0.467714
\(59\) 7.06138 0.919313 0.459657 0.888097i \(-0.347973\pi\)
0.459657 + 0.888097i \(0.347973\pi\)
\(60\) 0 0
\(61\) 6.76180 0.865760 0.432880 0.901452i \(-0.357497\pi\)
0.432880 + 0.901452i \(0.357497\pi\)
\(62\) −1.40162 −0.178006
\(63\) −5.84695 −0.736646
\(64\) 8.76180 1.09522
\(65\) 0 0
\(66\) 1.59024 0.195745
\(67\) −4.40422 −0.538062 −0.269031 0.963132i \(-0.586703\pi\)
−0.269031 + 0.963132i \(0.586703\pi\)
\(68\) −3.21689 −0.390105
\(69\) −0.119101 −0.0143381
\(70\) 0 0
\(71\) 1.88090 0.223222 0.111611 0.993752i \(-0.464399\pi\)
0.111611 + 0.993752i \(0.464399\pi\)
\(72\) 8.86014 1.04418
\(73\) 8.86014 1.03700 0.518501 0.855077i \(-0.326490\pi\)
0.518501 + 0.855077i \(0.326490\pi\)
\(74\) 6.47114 0.752255
\(75\) 0 0
\(76\) 3.47114 0.398167
\(77\) −7.87651 −0.897611
\(78\) 0 0
\(79\) 11.1805 1.25790 0.628951 0.777445i \(-0.283485\pi\)
0.628951 + 0.777445i \(0.283485\pi\)
\(80\) 0 0
\(81\) 7.94228 0.882475
\(82\) −0.214289 −0.0236642
\(83\) 7.83540 0.860047 0.430024 0.902818i \(-0.358505\pi\)
0.430024 + 0.902818i \(0.358505\pi\)
\(84\) 0.413416 0.0451073
\(85\) 0 0
\(86\) 1.59024 0.171480
\(87\) −1.03533 −0.110999
\(88\) 11.9356 1.27234
\(89\) −12.2419 −1.29763 −0.648817 0.760944i \(-0.724736\pi\)
−0.648817 + 0.760944i \(0.724736\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −0.203698 −0.0212369
\(93\) 0.407395 0.0422449
\(94\) 14.5813 1.50395
\(95\) 0 0
\(96\) −1.11018 −0.113307
\(97\) −5.80585 −0.589494 −0.294747 0.955575i \(-0.595235\pi\)
−0.294747 + 0.955575i \(0.595235\pi\)
\(98\) −3.42059 −0.345532
\(99\) 11.1805 1.12368
\(100\) 0 0
\(101\) −5.94228 −0.591279 −0.295639 0.955300i \(-0.595533\pi\)
−0.295639 + 0.955300i \(0.595533\pi\)
\(102\) −2.23325 −0.221125
\(103\) 6.43378 0.633939 0.316970 0.948436i \(-0.397335\pi\)
0.316970 + 0.948436i \(0.397335\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2.88090 −0.279818
\(107\) −17.6792 −1.70911 −0.854555 0.519360i \(-0.826170\pi\)
−0.854555 + 0.519360i \(0.826170\pi\)
\(108\) −1.19792 −0.115270
\(109\) −5.76180 −0.551880 −0.275940 0.961175i \(-0.588989\pi\)
−0.275940 + 0.961175i \(0.588989\pi\)
\(110\) 0 0
\(111\) −1.88090 −0.178527
\(112\) −5.01532 −0.473903
\(113\) 4.75992 0.447776 0.223888 0.974615i \(-0.428125\pi\)
0.223888 + 0.974615i \(0.428125\pi\)
\(114\) 2.40976 0.225695
\(115\) 0 0
\(116\) −1.77072 −0.164407
\(117\) 0 0
\(118\) 8.38421 0.771829
\(119\) 11.0614 1.01399
\(120\) 0 0
\(121\) 4.06138 0.369216
\(122\) 8.02851 0.726867
\(123\) 0.0622851 0.00561605
\(124\) 0.696765 0.0625714
\(125\) 0 0
\(126\) −6.94228 −0.618467
\(127\) 16.7061 1.48243 0.741215 0.671268i \(-0.234250\pi\)
0.741215 + 0.671268i \(0.234250\pi\)
\(128\) 3.96940 0.350848
\(129\) −0.462218 −0.0406961
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) −0.790529 −0.0688068
\(133\) −11.9356 −1.03495
\(134\) −5.22928 −0.451741
\(135\) 0 0
\(136\) −16.7618 −1.43731
\(137\) 1.97786 0.168980 0.0844901 0.996424i \(-0.473074\pi\)
0.0844901 + 0.996424i \(0.473074\pi\)
\(138\) −0.141412 −0.0120378
\(139\) 8.70042 0.737960 0.368980 0.929437i \(-0.379707\pi\)
0.368980 + 0.929437i \(0.379707\pi\)
\(140\) 0 0
\(141\) −4.23820 −0.356921
\(142\) 2.23325 0.187411
\(143\) 0 0
\(144\) 7.11910 0.593258
\(145\) 0 0
\(146\) 10.5199 0.870637
\(147\) 0.994227 0.0820025
\(148\) −3.21689 −0.264427
\(149\) −22.3032 −1.82715 −0.913576 0.406668i \(-0.866691\pi\)
−0.913576 + 0.406668i \(0.866691\pi\)
\(150\) 0 0
\(151\) 19.1626 1.55943 0.779717 0.626132i \(-0.215363\pi\)
0.779717 + 0.626132i \(0.215363\pi\)
\(152\) 18.0866 1.46701
\(153\) −15.7013 −1.26938
\(154\) −9.35204 −0.753609
\(155\) 0 0
\(156\) 0 0
\(157\) 6.20265 0.495025 0.247513 0.968885i \(-0.420387\pi\)
0.247513 + 0.968885i \(0.420387\pi\)
\(158\) 13.2750 1.05610
\(159\) 0.837361 0.0664071
\(160\) 0 0
\(161\) 0.700420 0.0552008
\(162\) 9.43013 0.740901
\(163\) 11.9356 0.934870 0.467435 0.884027i \(-0.345178\pi\)
0.467435 + 0.884027i \(0.345178\pi\)
\(164\) 0.106526 0.00831826
\(165\) 0 0
\(166\) 9.30323 0.722071
\(167\) −2.02956 −0.157052 −0.0785259 0.996912i \(-0.525021\pi\)
−0.0785259 + 0.996912i \(0.525021\pi\)
\(168\) 2.15413 0.166195
\(169\) 0 0
\(170\) 0 0
\(171\) 16.9423 1.29561
\(172\) −0.790529 −0.0602773
\(173\) −1.33934 −0.101828 −0.0509139 0.998703i \(-0.516213\pi\)
−0.0509139 + 0.998703i \(0.516213\pi\)
\(174\) −1.22928 −0.0931916
\(175\) 0 0
\(176\) 9.59024 0.722891
\(177\) −2.43695 −0.183173
\(178\) −14.5352 −1.08946
\(179\) 20.2240 1.51161 0.755807 0.654794i \(-0.227245\pi\)
0.755807 + 0.654794i \(0.227245\pi\)
\(180\) 0 0
\(181\) 19.8232 1.47345 0.736723 0.676195i \(-0.236372\pi\)
0.736723 + 0.676195i \(0.236372\pi\)
\(182\) 0 0
\(183\) −2.33356 −0.172502
\(184\) −1.06138 −0.0782458
\(185\) 0 0
\(186\) 0.483714 0.0354676
\(187\) −21.1515 −1.54675
\(188\) −7.24857 −0.528656
\(189\) 4.11910 0.299621
\(190\) 0 0
\(191\) 1.53778 0.111270 0.0556350 0.998451i \(-0.482282\pi\)
0.0556350 + 0.998451i \(0.482282\pi\)
\(192\) −3.02378 −0.218223
\(193\) 21.2032 1.52624 0.763118 0.646259i \(-0.223667\pi\)
0.763118 + 0.646259i \(0.223667\pi\)
\(194\) −6.89347 −0.494923
\(195\) 0 0
\(196\) 1.70042 0.121459
\(197\) −9.25695 −0.659530 −0.329765 0.944063i \(-0.606970\pi\)
−0.329765 + 0.944063i \(0.606970\pi\)
\(198\) 13.2750 0.943410
\(199\) −17.4045 −1.23377 −0.616886 0.787053i \(-0.711606\pi\)
−0.616886 + 0.787053i \(0.711606\pi\)
\(200\) 0 0
\(201\) 1.51994 0.107208
\(202\) −7.05546 −0.496421
\(203\) 6.08867 0.427341
\(204\) 1.11018 0.0777282
\(205\) 0 0
\(206\) 7.63904 0.532237
\(207\) −0.994227 −0.0691036
\(208\) 0 0
\(209\) 22.8232 1.57871
\(210\) 0 0
\(211\) −7.28174 −0.501296 −0.250648 0.968078i \(-0.580644\pi\)
−0.250648 + 0.968078i \(0.580644\pi\)
\(212\) 1.43213 0.0983594
\(213\) −0.649117 −0.0444768
\(214\) −20.9911 −1.43492
\(215\) 0 0
\(216\) −6.24186 −0.424705
\(217\) −2.39585 −0.162641
\(218\) −6.84118 −0.463343
\(219\) −3.05772 −0.206622
\(220\) 0 0
\(221\) 0 0
\(222\) −2.23325 −0.149886
\(223\) −19.4670 −1.30361 −0.651804 0.758388i \(-0.725987\pi\)
−0.651804 + 0.758388i \(0.725987\pi\)
\(224\) 6.52886 0.436228
\(225\) 0 0
\(226\) 5.65162 0.375940
\(227\) 4.81162 0.319358 0.159679 0.987169i \(-0.448954\pi\)
0.159679 + 0.987169i \(0.448954\pi\)
\(228\) −1.19792 −0.0793345
\(229\) −1.52360 −0.100682 −0.0503410 0.998732i \(-0.516031\pi\)
−0.0503410 + 0.998732i \(0.516031\pi\)
\(230\) 0 0
\(231\) 2.71826 0.178848
\(232\) −9.22643 −0.605745
\(233\) 13.9652 0.914889 0.457445 0.889238i \(-0.348765\pi\)
0.457445 + 0.889238i \(0.348765\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −4.16790 −0.271307
\(237\) −3.85849 −0.250636
\(238\) 13.1335 0.851321
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 0 0
\(241\) 17.4659 1.12508 0.562538 0.826771i \(-0.309825\pi\)
0.562538 + 0.826771i \(0.309825\pi\)
\(242\) 4.82221 0.309983
\(243\) −8.82963 −0.566421
\(244\) −3.99108 −0.255503
\(245\) 0 0
\(246\) 0.0739531 0.00471508
\(247\) 0 0
\(248\) 3.63054 0.230539
\(249\) −2.70408 −0.171364
\(250\) 0 0
\(251\) 9.29958 0.586984 0.293492 0.955961i \(-0.405183\pi\)
0.293492 + 0.955961i \(0.405183\pi\)
\(252\) 3.45110 0.217399
\(253\) −1.33934 −0.0842034
\(254\) 19.8358 1.24461
\(255\) 0 0
\(256\) −12.8106 −0.800663
\(257\) −10.8897 −0.679281 −0.339640 0.940555i \(-0.610305\pi\)
−0.339640 + 0.940555i \(0.610305\pi\)
\(258\) −0.548807 −0.0341673
\(259\) 11.0614 0.687321
\(260\) 0 0
\(261\) −8.64270 −0.534970
\(262\) 11.8733 0.733537
\(263\) 13.4406 0.828785 0.414392 0.910098i \(-0.363994\pi\)
0.414392 + 0.910098i \(0.363994\pi\)
\(264\) −4.11910 −0.253513
\(265\) 0 0
\(266\) −14.1716 −0.868914
\(267\) 4.22479 0.258553
\(268\) 2.59955 0.158793
\(269\) 3.66054 0.223187 0.111593 0.993754i \(-0.464405\pi\)
0.111593 + 0.993754i \(0.464405\pi\)
\(270\) 0 0
\(271\) −22.0037 −1.33663 −0.668313 0.743880i \(-0.732983\pi\)
−0.668313 + 0.743880i \(0.732983\pi\)
\(272\) −13.4681 −0.816621
\(273\) 0 0
\(274\) 2.34838 0.141871
\(275\) 0 0
\(276\) 0.0702980 0.00423144
\(277\) −9.89547 −0.594561 −0.297281 0.954790i \(-0.596080\pi\)
−0.297281 + 0.954790i \(0.596080\pi\)
\(278\) 10.3303 0.619570
\(279\) 3.40084 0.203603
\(280\) 0 0
\(281\) −4.06138 −0.242281 −0.121141 0.992635i \(-0.538655\pi\)
−0.121141 + 0.992635i \(0.538655\pi\)
\(282\) −5.03216 −0.299661
\(283\) 6.08867 0.361934 0.180967 0.983489i \(-0.442077\pi\)
0.180967 + 0.983489i \(0.442077\pi\)
\(284\) −1.11018 −0.0658771
\(285\) 0 0
\(286\) 0 0
\(287\) −0.366292 −0.0216215
\(288\) −9.26754 −0.546095
\(289\) 12.7041 0.747299
\(290\) 0 0
\(291\) 2.00366 0.117456
\(292\) −5.22960 −0.306039
\(293\) −9.79208 −0.572060 −0.286030 0.958221i \(-0.592336\pi\)
−0.286030 + 0.958221i \(0.592336\pi\)
\(294\) 1.18048 0.0688469
\(295\) 0 0
\(296\) −16.7618 −0.974260
\(297\) −7.87651 −0.457041
\(298\) −26.4814 −1.53402
\(299\) 0 0
\(300\) 0 0
\(301\) 2.71826 0.156678
\(302\) 22.7524 1.30926
\(303\) 2.05074 0.117812
\(304\) 14.5325 0.833497
\(305\) 0 0
\(306\) −18.6427 −1.06573
\(307\) −22.1046 −1.26158 −0.630788 0.775955i \(-0.717268\pi\)
−0.630788 + 0.775955i \(0.717268\pi\)
\(308\) 4.64902 0.264903
\(309\) −2.22036 −0.126312
\(310\) 0 0
\(311\) 7.63904 0.433170 0.216585 0.976264i \(-0.430508\pi\)
0.216585 + 0.976264i \(0.430508\pi\)
\(312\) 0 0
\(313\) −26.1425 −1.47766 −0.738831 0.673891i \(-0.764622\pi\)
−0.738831 + 0.673891i \(0.764622\pi\)
\(314\) 7.36461 0.415609
\(315\) 0 0
\(316\) −6.59916 −0.371232
\(317\) 11.8428 0.665159 0.332580 0.943075i \(-0.392081\pi\)
0.332580 + 0.943075i \(0.392081\pi\)
\(318\) 0.994227 0.0557535
\(319\) −11.6427 −0.651866
\(320\) 0 0
\(321\) 6.10126 0.340539
\(322\) 0.831632 0.0463451
\(323\) −32.0518 −1.78341
\(324\) −4.68785 −0.260436
\(325\) 0 0
\(326\) 14.1716 0.784890
\(327\) 1.98845 0.109962
\(328\) 0.555059 0.0306480
\(329\) 24.9244 1.37413
\(330\) 0 0
\(331\) 12.7004 0.698078 0.349039 0.937108i \(-0.386508\pi\)
0.349039 + 0.937108i \(0.386508\pi\)
\(332\) −4.62476 −0.253817
\(333\) −15.7013 −0.860427
\(334\) −2.40976 −0.131856
\(335\) 0 0
\(336\) 1.73084 0.0944248
\(337\) 15.2939 0.833113 0.416556 0.909110i \(-0.363237\pi\)
0.416556 + 0.909110i \(0.363237\pi\)
\(338\) 0 0
\(339\) −1.64270 −0.0892191
\(340\) 0 0
\(341\) 4.58132 0.248092
\(342\) 20.1161 1.08776
\(343\) −20.0538 −1.08281
\(344\) −4.11910 −0.222087
\(345\) 0 0
\(346\) −1.59024 −0.0854918
\(347\) −12.2396 −0.657058 −0.328529 0.944494i \(-0.606553\pi\)
−0.328529 + 0.944494i \(0.606553\pi\)
\(348\) 0.611093 0.0327580
\(349\) −18.7004 −1.00101 −0.500505 0.865733i \(-0.666852\pi\)
−0.500505 + 0.865733i \(0.666852\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −12.4844 −0.665422
\(353\) 1.30883 0.0696617 0.0348309 0.999393i \(-0.488911\pi\)
0.0348309 + 0.999393i \(0.488911\pi\)
\(354\) −2.89347 −0.153786
\(355\) 0 0
\(356\) 7.22563 0.382957
\(357\) −3.81739 −0.202038
\(358\) 24.0127 1.26911
\(359\) 29.4082 1.55210 0.776051 0.630670i \(-0.217220\pi\)
0.776051 + 0.630670i \(0.217220\pi\)
\(360\) 0 0
\(361\) 15.5850 0.820262
\(362\) 23.5367 1.23706
\(363\) −1.40162 −0.0735661
\(364\) 0 0
\(365\) 0 0
\(366\) −2.77072 −0.144828
\(367\) 33.4322 1.74515 0.872573 0.488484i \(-0.162450\pi\)
0.872573 + 0.488484i \(0.162450\pi\)
\(368\) −0.852815 −0.0444560
\(369\) 0.519941 0.0270671
\(370\) 0 0
\(371\) −4.92444 −0.255664
\(372\) −0.240461 −0.0124673
\(373\) 34.4781 1.78521 0.892604 0.450841i \(-0.148876\pi\)
0.892604 + 0.450841i \(0.148876\pi\)
\(374\) −25.1138 −1.29861
\(375\) 0 0
\(376\) −37.7691 −1.94779
\(377\) 0 0
\(378\) 4.89075 0.251553
\(379\) 17.4045 0.894009 0.447004 0.894532i \(-0.352491\pi\)
0.447004 + 0.894532i \(0.352491\pi\)
\(380\) 0 0
\(381\) −5.76545 −0.295373
\(382\) 1.82586 0.0934191
\(383\) 1.74673 0.0892538 0.0446269 0.999004i \(-0.485790\pi\)
0.0446269 + 0.999004i \(0.485790\pi\)
\(384\) −1.36988 −0.0699063
\(385\) 0 0
\(386\) 25.1752 1.28138
\(387\) −3.85849 −0.196138
\(388\) 3.42684 0.173971
\(389\) 22.0435 1.11765 0.558826 0.829285i \(-0.311252\pi\)
0.558826 + 0.829285i \(0.311252\pi\)
\(390\) 0 0
\(391\) 1.88090 0.0951212
\(392\) 8.86014 0.447505
\(393\) −3.45110 −0.174085
\(394\) −10.9911 −0.553723
\(395\) 0 0
\(396\) −6.59916 −0.331620
\(397\) −22.5319 −1.13084 −0.565422 0.824802i \(-0.691287\pi\)
−0.565422 + 0.824802i \(0.691287\pi\)
\(398\) −20.6649 −1.03584
\(399\) 4.11910 0.206213
\(400\) 0 0
\(401\) 3.70408 0.184973 0.0924863 0.995714i \(-0.470519\pi\)
0.0924863 + 0.995714i \(0.470519\pi\)
\(402\) 1.80468 0.0900091
\(403\) 0 0
\(404\) 3.50737 0.174498
\(405\) 0 0
\(406\) 7.22928 0.358783
\(407\) −21.1515 −1.04844
\(408\) 5.78466 0.286384
\(409\) 13.4837 0.666727 0.333363 0.942798i \(-0.391816\pi\)
0.333363 + 0.942798i \(0.391816\pi\)
\(410\) 0 0
\(411\) −0.682580 −0.0336692
\(412\) −3.79747 −0.187088
\(413\) 14.3315 0.705205
\(414\) −1.18048 −0.0580174
\(415\) 0 0
\(416\) 0 0
\(417\) −3.00260 −0.147038
\(418\) 27.0987 1.32544
\(419\) 16.8232 0.821866 0.410933 0.911666i \(-0.365203\pi\)
0.410933 + 0.911666i \(0.365203\pi\)
\(420\) 0 0
\(421\) −17.1013 −0.833464 −0.416732 0.909029i \(-0.636825\pi\)
−0.416732 + 0.909029i \(0.636825\pi\)
\(422\) −8.64585 −0.420874
\(423\) −35.3795 −1.72021
\(424\) 7.46222 0.362397
\(425\) 0 0
\(426\) −0.770718 −0.0373414
\(427\) 13.7235 0.664124
\(428\) 10.4349 0.504392
\(429\) 0 0
\(430\) 0 0
\(431\) −9.66054 −0.465332 −0.232666 0.972557i \(-0.574745\pi\)
−0.232666 + 0.972557i \(0.574745\pi\)
\(432\) −5.01532 −0.241300
\(433\) −24.7727 −1.19050 −0.595249 0.803541i \(-0.702947\pi\)
−0.595249 + 0.803541i \(0.702947\pi\)
\(434\) −2.84467 −0.136549
\(435\) 0 0
\(436\) 3.40084 0.162871
\(437\) −2.02956 −0.0970869
\(438\) −3.63054 −0.173474
\(439\) 7.06138 0.337021 0.168511 0.985700i \(-0.446104\pi\)
0.168511 + 0.985700i \(0.446104\pi\)
\(440\) 0 0
\(441\) 8.29958 0.395218
\(442\) 0 0
\(443\) 38.2438 1.81702 0.908509 0.417865i \(-0.137222\pi\)
0.908509 + 0.417865i \(0.137222\pi\)
\(444\) 1.11018 0.0526869
\(445\) 0 0
\(446\) −23.1138 −1.09447
\(447\) 7.69707 0.364059
\(448\) 17.7826 0.840147
\(449\) 12.4801 0.588970 0.294485 0.955656i \(-0.404852\pi\)
0.294485 + 0.955656i \(0.404852\pi\)
\(450\) 0 0
\(451\) 0.700420 0.0329815
\(452\) −2.80950 −0.132148
\(453\) −6.61322 −0.310716
\(454\) 5.71300 0.268124
\(455\) 0 0
\(456\) −6.24186 −0.292302
\(457\) 8.23221 0.385086 0.192543 0.981289i \(-0.438326\pi\)
0.192543 + 0.981289i \(0.438326\pi\)
\(458\) −1.80902 −0.0845298
\(459\) 11.0614 0.516301
\(460\) 0 0
\(461\) 4.54144 0.211516 0.105758 0.994392i \(-0.466273\pi\)
0.105758 + 0.994392i \(0.466273\pi\)
\(462\) 3.22748 0.150156
\(463\) −1.98845 −0.0924113 −0.0462056 0.998932i \(-0.514713\pi\)
−0.0462056 + 0.998932i \(0.514713\pi\)
\(464\) −7.41342 −0.344159
\(465\) 0 0
\(466\) 16.5813 0.768115
\(467\) 32.8043 1.51800 0.759000 0.651091i \(-0.225688\pi\)
0.759000 + 0.651091i \(0.225688\pi\)
\(468\) 0 0
\(469\) −8.93862 −0.412747
\(470\) 0 0
\(471\) −2.14060 −0.0986335
\(472\) −21.7171 −0.999611
\(473\) −5.19783 −0.238997
\(474\) −4.58132 −0.210427
\(475\) 0 0
\(476\) −6.52886 −0.299250
\(477\) 6.99010 0.320055
\(478\) −4.74933 −0.217229
\(479\) 30.8053 1.40753 0.703766 0.710432i \(-0.251500\pi\)
0.703766 + 0.710432i \(0.251500\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 20.7378 0.944582
\(483\) −0.241722 −0.0109987
\(484\) −2.39719 −0.108963
\(485\) 0 0
\(486\) −10.4837 −0.475551
\(487\) −22.3251 −1.01165 −0.505824 0.862637i \(-0.668811\pi\)
−0.505824 + 0.862637i \(0.668811\pi\)
\(488\) −20.7958 −0.941380
\(489\) −4.11910 −0.186272
\(490\) 0 0
\(491\) 10.6826 0.482098 0.241049 0.970513i \(-0.422509\pi\)
0.241049 + 0.970513i \(0.422509\pi\)
\(492\) −0.0367631 −0.00165741
\(493\) 16.3504 0.736386
\(494\) 0 0
\(495\) 0 0
\(496\) 2.91713 0.130983
\(497\) 3.81739 0.171233
\(498\) −3.21064 −0.143872
\(499\) −18.8195 −0.842477 −0.421239 0.906950i \(-0.638405\pi\)
−0.421239 + 0.906950i \(0.638405\pi\)
\(500\) 0 0
\(501\) 0.700420 0.0312925
\(502\) 11.0417 0.492815
\(503\) 5.68128 0.253316 0.126658 0.991946i \(-0.459575\pi\)
0.126658 + 0.991946i \(0.459575\pi\)
\(504\) 17.9822 0.800989
\(505\) 0 0
\(506\) −1.59024 −0.0706948
\(507\) 0 0
\(508\) −9.86062 −0.437494
\(509\) −27.9244 −1.23773 −0.618864 0.785498i \(-0.712407\pi\)
−0.618864 + 0.785498i \(0.712407\pi\)
\(510\) 0 0
\(511\) 17.9822 0.795484
\(512\) −23.1492 −1.02306
\(513\) −11.9356 −0.526970
\(514\) −12.9297 −0.570305
\(515\) 0 0
\(516\) 0.272820 0.0120102
\(517\) −47.6603 −2.09610
\(518\) 13.1335 0.577055
\(519\) 0.462218 0.0202891
\(520\) 0 0
\(521\) 6.29958 0.275990 0.137995 0.990433i \(-0.455934\pi\)
0.137995 + 0.990433i \(0.455934\pi\)
\(522\) −10.2618 −0.449145
\(523\) −22.8571 −0.999471 −0.499735 0.866178i \(-0.666569\pi\)
−0.499735 + 0.866178i \(0.666569\pi\)
\(524\) −5.90239 −0.257847
\(525\) 0 0
\(526\) 15.9585 0.695824
\(527\) −6.43378 −0.280260
\(528\) −3.30969 −0.144036
\(529\) −22.8809 −0.994822
\(530\) 0 0
\(531\) −20.3431 −0.882816
\(532\) 7.04487 0.305434
\(533\) 0 0
\(534\) 5.01623 0.217074
\(535\) 0 0
\(536\) 13.5451 0.585059
\(537\) −6.97951 −0.301188
\(538\) 4.34628 0.187381
\(539\) 11.1805 0.481577
\(540\) 0 0
\(541\) 9.48006 0.407580 0.203790 0.979015i \(-0.434674\pi\)
0.203790 + 0.979015i \(0.434674\pi\)
\(542\) −26.1257 −1.12219
\(543\) −6.84118 −0.293583
\(544\) 17.5325 0.751700
\(545\) 0 0
\(546\) 0 0
\(547\) −33.3911 −1.42770 −0.713850 0.700299i \(-0.753050\pi\)
−0.713850 + 0.700299i \(0.753050\pi\)
\(548\) −1.16741 −0.0498694
\(549\) −19.4801 −0.831389
\(550\) 0 0
\(551\) −17.6427 −0.751604
\(552\) 0.366292 0.0155904
\(553\) 22.6914 0.964937
\(554\) −11.7492 −0.499177
\(555\) 0 0
\(556\) −5.13533 −0.217787
\(557\) −37.7648 −1.60015 −0.800073 0.599903i \(-0.795206\pi\)
−0.800073 + 0.599903i \(0.795206\pi\)
\(558\) 4.03793 0.170939
\(559\) 0 0
\(560\) 0 0
\(561\) 7.29958 0.308188
\(562\) −4.82221 −0.203413
\(563\) −25.9008 −1.09159 −0.545794 0.837919i \(-0.683772\pi\)
−0.545794 + 0.837919i \(0.683772\pi\)
\(564\) 2.50155 0.105334
\(565\) 0 0
\(566\) 7.22928 0.303869
\(567\) 16.1193 0.676947
\(568\) −5.78466 −0.242719
\(569\) −21.5451 −0.903217 −0.451609 0.892216i \(-0.649150\pi\)
−0.451609 + 0.892216i \(0.649150\pi\)
\(570\) 0 0
\(571\) −2.22036 −0.0929192 −0.0464596 0.998920i \(-0.514794\pi\)
−0.0464596 + 0.998920i \(0.514794\pi\)
\(572\) 0 0
\(573\) −0.530704 −0.0221705
\(574\) −0.434911 −0.0181528
\(575\) 0 0
\(576\) −25.2419 −1.05174
\(577\) 6.20265 0.258220 0.129110 0.991630i \(-0.458788\pi\)
0.129110 + 0.991630i \(0.458788\pi\)
\(578\) 15.0840 0.627411
\(579\) −7.31742 −0.304102
\(580\) 0 0
\(581\) 15.9024 0.659742
\(582\) 2.37901 0.0986130
\(583\) 9.41646 0.389990
\(584\) −27.2492 −1.12758
\(585\) 0 0
\(586\) −11.6265 −0.480285
\(587\) 1.82894 0.0754883 0.0377442 0.999287i \(-0.487983\pi\)
0.0377442 + 0.999287i \(0.487983\pi\)
\(588\) −0.586832 −0.0242005
\(589\) 6.94228 0.286052
\(590\) 0 0
\(591\) 3.19466 0.131411
\(592\) −13.4681 −0.553534
\(593\) 0.0728761 0.00299266 0.00149633 0.999999i \(-0.499524\pi\)
0.00149633 + 0.999999i \(0.499524\pi\)
\(594\) −9.35204 −0.383719
\(595\) 0 0
\(596\) 13.1642 0.539229
\(597\) 6.00646 0.245828
\(598\) 0 0
\(599\) −14.5813 −0.595777 −0.297888 0.954601i \(-0.596282\pi\)
−0.297888 + 0.954601i \(0.596282\pi\)
\(600\) 0 0
\(601\) −44.4082 −1.81145 −0.905723 0.423870i \(-0.860671\pi\)
−0.905723 + 0.423870i \(0.860671\pi\)
\(602\) 3.22748 0.131542
\(603\) 12.6881 0.516700
\(604\) −11.3105 −0.460220
\(605\) 0 0
\(606\) 2.43491 0.0989115
\(607\) 36.2354 1.47075 0.735375 0.677660i \(-0.237006\pi\)
0.735375 + 0.677660i \(0.237006\pi\)
\(608\) −18.9182 −0.767235
\(609\) −2.10126 −0.0851474
\(610\) 0 0
\(611\) 0 0
\(612\) 9.26754 0.374618
\(613\) −3.35830 −0.135641 −0.0678203 0.997698i \(-0.521604\pi\)
−0.0678203 + 0.997698i \(0.521604\pi\)
\(614\) −26.2455 −1.05918
\(615\) 0 0
\(616\) 24.2240 0.976013
\(617\) 21.1820 0.852754 0.426377 0.904546i \(-0.359790\pi\)
0.426377 + 0.904546i \(0.359790\pi\)
\(618\) −2.63631 −0.106048
\(619\) −25.4082 −1.02124 −0.510620 0.859807i \(-0.670584\pi\)
−0.510620 + 0.859807i \(0.670584\pi\)
\(620\) 0 0
\(621\) 0.700420 0.0281069
\(622\) 9.07009 0.363677
\(623\) −24.8455 −0.995416
\(624\) 0 0
\(625\) 0 0
\(626\) −31.0399 −1.24060
\(627\) −7.87651 −0.314557
\(628\) −3.66105 −0.146092
\(629\) 29.7041 1.18438
\(630\) 0 0
\(631\) −43.5451 −1.73350 −0.866751 0.498740i \(-0.833796\pi\)
−0.866751 + 0.498740i \(0.833796\pi\)
\(632\) −34.3853 −1.36777
\(633\) 2.51300 0.0998828
\(634\) 14.0614 0.558449
\(635\) 0 0
\(636\) −0.494244 −0.0195980
\(637\) 0 0
\(638\) −13.8238 −0.547288
\(639\) −5.41868 −0.214360
\(640\) 0 0
\(641\) 48.2854 1.90716 0.953579 0.301142i \(-0.0973679\pi\)
0.953579 + 0.301142i \(0.0973679\pi\)
\(642\) 7.24423 0.285907
\(643\) −42.3440 −1.66989 −0.834943 0.550337i \(-0.814499\pi\)
−0.834943 + 0.550337i \(0.814499\pi\)
\(644\) −0.413416 −0.0162909
\(645\) 0 0
\(646\) −38.0561 −1.49730
\(647\) 34.4052 1.35261 0.676305 0.736622i \(-0.263580\pi\)
0.676305 + 0.736622i \(0.263580\pi\)
\(648\) −24.4263 −0.959556
\(649\) −27.4045 −1.07572
\(650\) 0 0
\(651\) 0.826831 0.0324061
\(652\) −7.04487 −0.275899
\(653\) 14.3315 0.560834 0.280417 0.959878i \(-0.409527\pi\)
0.280417 + 0.959878i \(0.409527\pi\)
\(654\) 2.36096 0.0923208
\(655\) 0 0
\(656\) 0.445988 0.0174129
\(657\) −25.5252 −0.995832
\(658\) 29.5936 1.15368
\(659\) 22.8232 0.889065 0.444532 0.895763i \(-0.353370\pi\)
0.444532 + 0.895763i \(0.353370\pi\)
\(660\) 0 0
\(661\) −14.4187 −0.560822 −0.280411 0.959880i \(-0.590471\pi\)
−0.280411 + 0.959880i \(0.590471\pi\)
\(662\) 15.0796 0.586087
\(663\) 0 0
\(664\) −24.0976 −0.935168
\(665\) 0 0
\(666\) −18.6427 −0.722390
\(667\) 1.03533 0.0400881
\(668\) 1.19792 0.0463491
\(669\) 6.71826 0.259743
\(670\) 0 0
\(671\) −26.2419 −1.01306
\(672\) −2.25318 −0.0869181
\(673\) 34.1741 1.31731 0.658657 0.752443i \(-0.271125\pi\)
0.658657 + 0.752443i \(0.271125\pi\)
\(674\) 18.1590 0.699458
\(675\) 0 0
\(676\) 0 0
\(677\) −5.84695 −0.224716 −0.112358 0.993668i \(-0.535840\pi\)
−0.112358 + 0.993668i \(0.535840\pi\)
\(678\) −1.95043 −0.0749058
\(679\) −11.7833 −0.452201
\(680\) 0 0
\(681\) −1.66054 −0.0636319
\(682\) 5.43955 0.208291
\(683\) −11.3488 −0.434249 −0.217125 0.976144i \(-0.569668\pi\)
−0.217125 + 0.976144i \(0.569668\pi\)
\(684\) −10.0000 −0.382360
\(685\) 0 0
\(686\) −23.8106 −0.909093
\(687\) 0.525808 0.0200608
\(688\) −3.30969 −0.126181
\(689\) 0 0
\(690\) 0 0
\(691\) 18.8232 0.716067 0.358034 0.933709i \(-0.383447\pi\)
0.358034 + 0.933709i \(0.383447\pi\)
\(692\) 0.790529 0.0300514
\(693\) 22.6914 0.861976
\(694\) −14.5325 −0.551647
\(695\) 0 0
\(696\) 3.18413 0.120694
\(697\) −0.983636 −0.0372579
\(698\) −22.2036 −0.840420
\(699\) −4.81952 −0.182291
\(700\) 0 0
\(701\) 19.1626 0.723763 0.361881 0.932224i \(-0.382135\pi\)
0.361881 + 0.932224i \(0.382135\pi\)
\(702\) 0 0
\(703\) −32.0518 −1.20885
\(704\) −34.0037 −1.28156
\(705\) 0 0
\(706\) 1.55401 0.0584860
\(707\) −12.0602 −0.453570
\(708\) 1.43839 0.0540578
\(709\) −23.4837 −0.881949 −0.440975 0.897520i \(-0.645367\pi\)
−0.440975 + 0.897520i \(0.645367\pi\)
\(710\) 0 0
\(711\) −32.2098 −1.20796
\(712\) 37.6496 1.41098
\(713\) −0.407395 −0.0152571
\(714\) −4.53252 −0.169625
\(715\) 0 0
\(716\) −11.9370 −0.446107
\(717\) 1.38044 0.0515535
\(718\) 34.9173 1.30310
\(719\) −14.1086 −0.526161 −0.263080 0.964774i \(-0.584738\pi\)
−0.263080 + 0.964774i \(0.584738\pi\)
\(720\) 0 0
\(721\) 13.0577 0.486295
\(722\) 18.5046 0.688668
\(723\) −6.02765 −0.224171
\(724\) −11.7004 −0.434843
\(725\) 0 0
\(726\) −1.66419 −0.0617640
\(727\) −25.3762 −0.941153 −0.470576 0.882359i \(-0.655954\pi\)
−0.470576 + 0.882359i \(0.655954\pi\)
\(728\) 0 0
\(729\) −20.7796 −0.769616
\(730\) 0 0
\(731\) 7.29958 0.269985
\(732\) 1.37736 0.0509087
\(733\) −10.6692 −0.394074 −0.197037 0.980396i \(-0.563132\pi\)
−0.197037 + 0.980396i \(0.563132\pi\)
\(734\) 39.6952 1.46517
\(735\) 0 0
\(736\) 1.11018 0.0409218
\(737\) 17.0923 0.629605
\(738\) 0.617344 0.0227247
\(739\) 1.41503 0.0520526 0.0260263 0.999661i \(-0.491715\pi\)
0.0260263 + 0.999661i \(0.491715\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −5.84695 −0.214648
\(743\) −29.8777 −1.09611 −0.548053 0.836443i \(-0.684631\pi\)
−0.548053 + 0.836443i \(0.684631\pi\)
\(744\) −1.25293 −0.0459348
\(745\) 0 0
\(746\) 40.9370 1.49881
\(747\) −22.5730 −0.825903
\(748\) 12.4844 0.456476
\(749\) −35.8809 −1.31106
\(750\) 0 0
\(751\) −19.9858 −0.729293 −0.364646 0.931146i \(-0.618810\pi\)
−0.364646 + 0.931146i \(0.618810\pi\)
\(752\) −30.3474 −1.10666
\(753\) −3.20938 −0.116956
\(754\) 0 0
\(755\) 0 0
\(756\) −2.43126 −0.0884239
\(757\) 17.0923 0.621232 0.310616 0.950535i \(-0.399465\pi\)
0.310616 + 0.950535i \(0.399465\pi\)
\(758\) 20.6649 0.750584
\(759\) 0.462218 0.0167775
\(760\) 0 0
\(761\) 42.2240 1.53062 0.765310 0.643662i \(-0.222586\pi\)
0.765310 + 0.643662i \(0.222586\pi\)
\(762\) −6.84552 −0.247987
\(763\) −11.6939 −0.423347
\(764\) −0.907659 −0.0328380
\(765\) 0 0
\(766\) 2.07395 0.0749350
\(767\) 0 0
\(768\) 4.42107 0.159531
\(769\) 23.7655 0.857004 0.428502 0.903541i \(-0.359041\pi\)
0.428502 + 0.903541i \(0.359041\pi\)
\(770\) 0 0
\(771\) 3.75814 0.135346
\(772\) −12.5149 −0.450422
\(773\) −0.284086 −0.0102179 −0.00510894 0.999987i \(-0.501626\pi\)
−0.00510894 + 0.999987i \(0.501626\pi\)
\(774\) −4.58132 −0.164672
\(775\) 0 0
\(776\) 17.8557 0.640984
\(777\) −3.81739 −0.136948
\(778\) 26.1730 0.938349
\(779\) 1.06138 0.0380278
\(780\) 0 0
\(781\) −7.29958 −0.261199
\(782\) 2.23325 0.0798610
\(783\) 6.08867 0.217591
\(784\) 7.11910 0.254254
\(785\) 0 0
\(786\) −4.09761 −0.146157
\(787\) 24.1341 0.860289 0.430145 0.902760i \(-0.358463\pi\)
0.430145 + 0.902760i \(0.358463\pi\)
\(788\) 5.46381 0.194640
\(789\) −4.63849 −0.165135
\(790\) 0 0
\(791\) 9.66054 0.343489
\(792\) −34.3853 −1.22183
\(793\) 0 0
\(794\) −26.7529 −0.949424
\(795\) 0 0
\(796\) 10.2728 0.364110
\(797\) 34.3442 1.21653 0.608267 0.793732i \(-0.291865\pi\)
0.608267 + 0.793732i \(0.291865\pi\)
\(798\) 4.89075 0.173131
\(799\) 66.9317 2.36787
\(800\) 0 0
\(801\) 35.2676 1.24612
\(802\) 4.39797 0.155298
\(803\) −34.3853 −1.21343
\(804\) −0.897129 −0.0316393
\(805\) 0 0
\(806\) 0 0
\(807\) −1.26329 −0.0444698
\(808\) 18.2753 0.642924
\(809\) −47.6862 −1.67656 −0.838279 0.545241i \(-0.816438\pi\)
−0.838279 + 0.545241i \(0.816438\pi\)
\(810\) 0 0
\(811\) −24.5992 −0.863793 −0.431897 0.901923i \(-0.642155\pi\)
−0.431897 + 0.901923i \(0.642155\pi\)
\(812\) −3.59377 −0.126117
\(813\) 7.59368 0.266322
\(814\) −25.1138 −0.880239
\(815\) 0 0
\(816\) 4.64796 0.162711
\(817\) −7.87651 −0.275564
\(818\) 16.0097 0.559765
\(819\) 0 0
\(820\) 0 0
\(821\) −17.2996 −0.603759 −0.301880 0.953346i \(-0.597614\pi\)
−0.301880 + 0.953346i \(0.597614\pi\)
\(822\) −0.810450 −0.0282677
\(823\) 32.5625 1.13506 0.567529 0.823353i \(-0.307899\pi\)
0.567529 + 0.823353i \(0.307899\pi\)
\(824\) −19.7869 −0.689311
\(825\) 0 0
\(826\) 17.0162 0.592070
\(827\) −15.4702 −0.537951 −0.268976 0.963147i \(-0.586685\pi\)
−0.268976 + 0.963147i \(0.586685\pi\)
\(828\) 0.586832 0.0203938
\(829\) −14.5236 −0.504425 −0.252213 0.967672i \(-0.581158\pi\)
−0.252213 + 0.967672i \(0.581158\pi\)
\(830\) 0 0
\(831\) 3.41503 0.118466
\(832\) 0 0
\(833\) −15.7013 −0.544018
\(834\) −3.56509 −0.123449
\(835\) 0 0
\(836\) −13.4711 −0.465909
\(837\) −2.39585 −0.0828127
\(838\) 19.9747 0.690015
\(839\) −0.815866 −0.0281668 −0.0140834 0.999901i \(-0.504483\pi\)
−0.0140834 + 0.999901i \(0.504483\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) −20.3049 −0.699753
\(843\) 1.40162 0.0482744
\(844\) 4.29797 0.147942
\(845\) 0 0
\(846\) −42.0073 −1.44424
\(847\) 8.24280 0.283226
\(848\) 5.99587 0.205899
\(849\) −2.10126 −0.0721151
\(850\) 0 0
\(851\) 1.88090 0.0644764
\(852\) 0.383134 0.0131260
\(853\) 20.0856 0.687719 0.343859 0.939021i \(-0.388266\pi\)
0.343859 + 0.939021i \(0.388266\pi\)
\(854\) 16.2943 0.557580
\(855\) 0 0
\(856\) 54.3719 1.85839
\(857\) −40.7886 −1.39331 −0.696656 0.717406i \(-0.745329\pi\)
−0.696656 + 0.717406i \(0.745329\pi\)
\(858\) 0 0
\(859\) −40.1301 −1.36922 −0.684610 0.728909i \(-0.740028\pi\)
−0.684610 + 0.728909i \(0.740028\pi\)
\(860\) 0 0
\(861\) 0.126411 0.00430808
\(862\) −11.4703 −0.390679
\(863\) −20.8275 −0.708977 −0.354489 0.935060i \(-0.615345\pi\)
−0.354489 + 0.935060i \(0.615345\pi\)
\(864\) 6.52886 0.222116
\(865\) 0 0
\(866\) −29.4134 −0.999509
\(867\) −4.38430 −0.148899
\(868\) 1.41412 0.0479985
\(869\) −43.3903 −1.47192
\(870\) 0 0
\(871\) 0 0
\(872\) 17.7203 0.600084
\(873\) 16.7261 0.566091
\(874\) −2.40976 −0.0815114
\(875\) 0 0
\(876\) 1.80479 0.0609782
\(877\) 46.5944 1.57338 0.786691 0.617347i \(-0.211793\pi\)
0.786691 + 0.617347i \(0.211793\pi\)
\(878\) 8.38421 0.282953
\(879\) 3.37935 0.113982
\(880\) 0 0
\(881\) 23.8447 0.803347 0.401674 0.915783i \(-0.368429\pi\)
0.401674 + 0.915783i \(0.368429\pi\)
\(882\) 9.85437 0.331814
\(883\) 37.2496 1.25355 0.626774 0.779201i \(-0.284375\pi\)
0.626774 + 0.779201i \(0.284375\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 45.4082 1.52552
\(887\) −28.5795 −0.959605 −0.479802 0.877377i \(-0.659292\pi\)
−0.479802 + 0.877377i \(0.659292\pi\)
\(888\) 5.78466 0.194121
\(889\) 33.9060 1.13717
\(890\) 0 0
\(891\) −30.8232 −1.03262
\(892\) 11.4902 0.384720
\(893\) −72.2217 −2.41681
\(894\) 9.13899 0.305653
\(895\) 0 0
\(896\) 8.05611 0.269136
\(897\) 0 0
\(898\) 14.8180 0.494483
\(899\) −3.54144 −0.118114
\(900\) 0 0
\(901\) −13.2240 −0.440556
\(902\) 0.831632 0.0276903
\(903\) −0.938099 −0.0312180
\(904\) −14.6390 −0.486887
\(905\) 0 0
\(906\) −7.85209 −0.260868
\(907\) −6.41386 −0.212969 −0.106484 0.994314i \(-0.533959\pi\)
−0.106484 + 0.994314i \(0.533959\pi\)
\(908\) −2.84001 −0.0942489
\(909\) 17.1191 0.567805
\(910\) 0 0
\(911\) 22.2204 0.736193 0.368097 0.929788i \(-0.380010\pi\)
0.368097 + 0.929788i \(0.380010\pi\)
\(912\) −5.01532 −0.166074
\(913\) −30.4084 −1.00637
\(914\) 9.77437 0.323308
\(915\) 0 0
\(916\) 0.899287 0.0297133
\(917\) 20.2956 0.670219
\(918\) 13.1335 0.433472
\(919\) 26.1264 0.861831 0.430915 0.902392i \(-0.358191\pi\)
0.430915 + 0.902392i \(0.358191\pi\)
\(920\) 0 0
\(921\) 7.62851 0.251368
\(922\) 5.39220 0.177583
\(923\) 0 0
\(924\) −1.60442 −0.0527817
\(925\) 0 0
\(926\) −2.36096 −0.0775859
\(927\) −18.5351 −0.608772
\(928\) 9.65067 0.316799
\(929\) 23.9423 0.785521 0.392760 0.919641i \(-0.371520\pi\)
0.392760 + 0.919641i \(0.371520\pi\)
\(930\) 0 0
\(931\) 16.9423 0.555261
\(932\) −8.24280 −0.270002
\(933\) −2.63631 −0.0863089
\(934\) 38.9496 1.27447
\(935\) 0 0
\(936\) 0 0
\(937\) 18.5046 0.604518 0.302259 0.953226i \(-0.402259\pi\)
0.302259 + 0.953226i \(0.402259\pi\)
\(938\) −10.6131 −0.346531
\(939\) 9.02204 0.294423
\(940\) 0 0
\(941\) 14.3788 0.468735 0.234368 0.972148i \(-0.424698\pi\)
0.234368 + 0.972148i \(0.424698\pi\)
\(942\) −2.54160 −0.0828098
\(943\) −0.0622851 −0.00202828
\(944\) −17.4496 −0.567938
\(945\) 0 0
\(946\) −6.17156 −0.200655
\(947\) 58.3188 1.89511 0.947554 0.319596i \(-0.103547\pi\)
0.947554 + 0.319596i \(0.103547\pi\)
\(948\) 2.27744 0.0739677
\(949\) 0 0
\(950\) 0 0
\(951\) −4.08708 −0.132533
\(952\) −34.0190 −1.10256
\(953\) 13.7995 0.447010 0.223505 0.974703i \(-0.428250\pi\)
0.223505 + 0.974703i \(0.428250\pi\)
\(954\) 8.29958 0.268709
\(955\) 0 0
\(956\) 2.36096 0.0763588
\(957\) 4.01801 0.129884
\(958\) 36.5762 1.18172
\(959\) 4.01419 0.129625
\(960\) 0 0
\(961\) −29.6065 −0.955047
\(962\) 0 0
\(963\) 50.9319 1.64126
\(964\) −10.3090 −0.332032
\(965\) 0 0
\(966\) −0.287005 −0.00923422
\(967\) 30.3474 0.975906 0.487953 0.872870i \(-0.337744\pi\)
0.487953 + 0.872870i \(0.337744\pi\)
\(968\) −12.4907 −0.401466
\(969\) 11.0614 0.355343
\(970\) 0 0
\(971\) 44.1013 1.41528 0.707638 0.706575i \(-0.249761\pi\)
0.707638 + 0.706575i \(0.249761\pi\)
\(972\) 5.21160 0.167162
\(973\) 17.6580 0.566089
\(974\) −26.5074 −0.849351
\(975\) 0 0
\(976\) −16.7093 −0.534853
\(977\) −22.3220 −0.714143 −0.357071 0.934077i \(-0.616225\pi\)
−0.357071 + 0.934077i \(0.616225\pi\)
\(978\) −4.89075 −0.156389
\(979\) 47.5094 1.51841
\(980\) 0 0
\(981\) 16.5992 0.529970
\(982\) 12.6838 0.404756
\(983\) 4.03793 0.128790 0.0643950 0.997924i \(-0.479488\pi\)
0.0643950 + 0.997924i \(0.479488\pi\)
\(984\) −0.191556 −0.00610659
\(985\) 0 0
\(986\) 19.4134 0.618249
\(987\) −8.60167 −0.273794
\(988\) 0 0
\(989\) 0.462218 0.0146977
\(990\) 0 0
\(991\) 29.6500 0.941864 0.470932 0.882170i \(-0.343918\pi\)
0.470932 + 0.882170i \(0.343918\pi\)
\(992\) −3.79747 −0.120570
\(993\) −4.38304 −0.139092
\(994\) 4.53252 0.143763
\(995\) 0 0
\(996\) 1.59605 0.0505728
\(997\) −22.1762 −0.702327 −0.351164 0.936314i \(-0.614214\pi\)
−0.351164 + 0.936314i \(0.614214\pi\)
\(998\) −22.3450 −0.707320
\(999\) 11.0614 0.349967
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.1.5 6
5.2 odd 4 845.2.b.e.339.5 6
5.3 odd 4 845.2.b.e.339.2 6
5.4 even 2 inner 4225.2.a.bq.1.2 6
13.4 even 6 325.2.e.e.276.5 12
13.10 even 6 325.2.e.e.126.5 12
13.12 even 2 4225.2.a.br.1.2 6
65.2 even 12 845.2.l.f.654.3 24
65.3 odd 12 845.2.n.e.529.5 12
65.4 even 6 325.2.e.e.276.2 12
65.7 even 12 845.2.l.f.699.10 24
65.8 even 4 845.2.d.d.844.9 12
65.12 odd 4 845.2.b.d.339.2 6
65.17 odd 12 65.2.n.a.29.2 yes 12
65.18 even 4 845.2.d.d.844.3 12
65.22 odd 12 845.2.n.e.484.5 12
65.23 odd 12 65.2.n.a.9.2 12
65.28 even 12 845.2.l.f.654.10 24
65.32 even 12 845.2.l.f.699.4 24
65.33 even 12 845.2.l.f.699.3 24
65.37 even 12 845.2.l.f.654.9 24
65.38 odd 4 845.2.b.d.339.5 6
65.42 odd 12 845.2.n.e.529.2 12
65.43 odd 12 65.2.n.a.29.5 yes 12
65.47 even 4 845.2.d.d.844.4 12
65.48 odd 12 845.2.n.e.484.2 12
65.49 even 6 325.2.e.e.126.2 12
65.57 even 4 845.2.d.d.844.10 12
65.58 even 12 845.2.l.f.699.9 24
65.62 odd 12 65.2.n.a.9.5 yes 12
65.63 even 12 845.2.l.f.654.4 24
65.64 even 2 4225.2.a.br.1.5 6
195.17 even 12 585.2.bs.a.289.5 12
195.23 even 12 585.2.bs.a.334.5 12
195.62 even 12 585.2.bs.a.334.2 12
195.173 even 12 585.2.bs.a.289.2 12
260.23 even 12 1040.2.dh.a.529.3 12
260.43 even 12 1040.2.dh.a.289.4 12
260.127 even 12 1040.2.dh.a.529.4 12
260.147 even 12 1040.2.dh.a.289.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.n.a.9.2 12 65.23 odd 12
65.2.n.a.9.5 yes 12 65.62 odd 12
65.2.n.a.29.2 yes 12 65.17 odd 12
65.2.n.a.29.5 yes 12 65.43 odd 12
325.2.e.e.126.2 12 65.49 even 6
325.2.e.e.126.5 12 13.10 even 6
325.2.e.e.276.2 12 65.4 even 6
325.2.e.e.276.5 12 13.4 even 6
585.2.bs.a.289.2 12 195.173 even 12
585.2.bs.a.289.5 12 195.17 even 12
585.2.bs.a.334.2 12 195.62 even 12
585.2.bs.a.334.5 12 195.23 even 12
845.2.b.d.339.2 6 65.12 odd 4
845.2.b.d.339.5 6 65.38 odd 4
845.2.b.e.339.2 6 5.3 odd 4
845.2.b.e.339.5 6 5.2 odd 4
845.2.d.d.844.3 12 65.18 even 4
845.2.d.d.844.4 12 65.47 even 4
845.2.d.d.844.9 12 65.8 even 4
845.2.d.d.844.10 12 65.57 even 4
845.2.l.f.654.3 24 65.2 even 12
845.2.l.f.654.4 24 65.63 even 12
845.2.l.f.654.9 24 65.37 even 12
845.2.l.f.654.10 24 65.28 even 12
845.2.l.f.699.3 24 65.33 even 12
845.2.l.f.699.4 24 65.32 even 12
845.2.l.f.699.9 24 65.58 even 12
845.2.l.f.699.10 24 65.7 even 12
845.2.n.e.484.2 12 65.48 odd 12
845.2.n.e.484.5 12 65.22 odd 12
845.2.n.e.529.2 12 65.42 odd 12
845.2.n.e.529.5 12 65.3 odd 12
1040.2.dh.a.289.3 12 260.147 even 12
1040.2.dh.a.289.4 12 260.43 even 12
1040.2.dh.a.529.3 12 260.23 even 12
1040.2.dh.a.529.4 12 260.127 even 12
4225.2.a.bq.1.2 6 5.4 even 2 inner
4225.2.a.bq.1.5 6 1.1 even 1 trivial
4225.2.a.br.1.2 6 13.12 even 2
4225.2.a.br.1.5 6 65.64 even 2