Properties

Label 4005.2.a.o.1.1
Level 4005
Weight 2
Character 4005.1
Self dual yes
Analytic conductor 31.980
Analytic rank 0
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4005.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9800860095\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 445)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.26266\)
Character \(\chi\) = 4005.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.11962 q^{2} +2.49279 q^{4} -1.00000 q^{5} -4.83304 q^{7} -1.04452 q^{8} +O(q^{10})\) \(q-2.11962 q^{2} +2.49279 q^{4} -1.00000 q^{5} -4.83304 q^{7} -1.04452 q^{8} +2.11962 q^{10} +2.21586 q^{11} -2.26162 q^{13} +10.2442 q^{14} -2.77159 q^{16} +3.10920 q^{17} -5.93952 q^{19} -2.49279 q^{20} -4.69678 q^{22} -5.79416 q^{23} +1.00000 q^{25} +4.79378 q^{26} -12.0477 q^{28} -2.32757 q^{29} +4.47624 q^{31} +7.96375 q^{32} -6.59033 q^{34} +4.83304 q^{35} -8.23124 q^{37} +12.5895 q^{38} +1.04452 q^{40} +0.278075 q^{41} -0.176109 q^{43} +5.52366 q^{44} +12.2814 q^{46} -1.91855 q^{47} +16.3583 q^{49} -2.11962 q^{50} -5.63775 q^{52} -8.65904 q^{53} -2.21586 q^{55} +5.04821 q^{56} +4.93357 q^{58} -5.39666 q^{59} -9.69150 q^{61} -9.48792 q^{62} -11.3370 q^{64} +2.26162 q^{65} +8.06916 q^{67} +7.75058 q^{68} -10.2442 q^{70} +5.00576 q^{71} -9.18909 q^{73} +17.4471 q^{74} -14.8060 q^{76} -10.7093 q^{77} -1.78949 q^{79} +2.77159 q^{80} -0.589414 q^{82} -13.8879 q^{83} -3.10920 q^{85} +0.373284 q^{86} -2.31451 q^{88} -1.00000 q^{89} +10.9305 q^{91} -14.4436 q^{92} +4.06660 q^{94} +5.93952 q^{95} -5.52008 q^{97} -34.6733 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} - 4q^{10} + 10q^{11} - 7q^{13} - 3q^{14} + 10q^{16} + 13q^{17} - 7q^{19} - 8q^{20} + 2q^{22} + 13q^{23} + 7q^{25} - q^{26} - 21q^{28} + 4q^{29} + q^{31} + 13q^{32} + 10q^{34} + 16q^{35} - 5q^{37} + 40q^{38} - 12q^{40} - 5q^{41} - 31q^{43} + 21q^{44} + 16q^{46} + 14q^{47} + 19q^{49} + 4q^{50} + 13q^{53} - 10q^{55} + q^{56} + 17q^{58} + 14q^{59} + 3q^{61} - 26q^{62} + 14q^{64} + 7q^{65} + q^{67} + 35q^{68} + 3q^{70} + 8q^{71} + 9q^{73} + 35q^{74} + 40q^{76} - 42q^{77} + 9q^{79} - 10q^{80} + 29q^{82} + 42q^{83} - 13q^{85} - 35q^{86} + 30q^{88} - 7q^{89} + 31q^{91} - 19q^{92} + 37q^{94} + 7q^{95} - 7q^{97} - 9q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.11962 −1.49880 −0.749399 0.662119i \(-0.769657\pi\)
−0.749399 + 0.662119i \(0.769657\pi\)
\(3\) 0 0
\(4\) 2.49279 1.24639
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.83304 −1.82672 −0.913358 0.407157i \(-0.866520\pi\)
−0.913358 + 0.407157i \(0.866520\pi\)
\(8\) −1.04452 −0.369294
\(9\) 0 0
\(10\) 2.11962 0.670283
\(11\) 2.21586 0.668106 0.334053 0.942554i \(-0.391583\pi\)
0.334053 + 0.942554i \(0.391583\pi\)
\(12\) 0 0
\(13\) −2.26162 −0.627261 −0.313631 0.949545i \(-0.601545\pi\)
−0.313631 + 0.949545i \(0.601545\pi\)
\(14\) 10.2442 2.73788
\(15\) 0 0
\(16\) −2.77159 −0.692896
\(17\) 3.10920 0.754092 0.377046 0.926194i \(-0.376940\pi\)
0.377046 + 0.926194i \(0.376940\pi\)
\(18\) 0 0
\(19\) −5.93952 −1.36262 −0.681309 0.731996i \(-0.738589\pi\)
−0.681309 + 0.731996i \(0.738589\pi\)
\(20\) −2.49279 −0.557404
\(21\) 0 0
\(22\) −4.69678 −1.00136
\(23\) −5.79416 −1.20817 −0.604083 0.796921i \(-0.706460\pi\)
−0.604083 + 0.796921i \(0.706460\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.79378 0.940138
\(27\) 0 0
\(28\) −12.0477 −2.27681
\(29\) −2.32757 −0.432220 −0.216110 0.976369i \(-0.569337\pi\)
−0.216110 + 0.976369i \(0.569337\pi\)
\(30\) 0 0
\(31\) 4.47624 0.803956 0.401978 0.915649i \(-0.368323\pi\)
0.401978 + 0.915649i \(0.368323\pi\)
\(32\) 7.96375 1.40781
\(33\) 0 0
\(34\) −6.59033 −1.13023
\(35\) 4.83304 0.816932
\(36\) 0 0
\(37\) −8.23124 −1.35321 −0.676604 0.736347i \(-0.736549\pi\)
−0.676604 + 0.736347i \(0.736549\pi\)
\(38\) 12.5895 2.04229
\(39\) 0 0
\(40\) 1.04452 0.165153
\(41\) 0.278075 0.0434281 0.0217140 0.999764i \(-0.493088\pi\)
0.0217140 + 0.999764i \(0.493088\pi\)
\(42\) 0 0
\(43\) −0.176109 −0.0268564 −0.0134282 0.999910i \(-0.504274\pi\)
−0.0134282 + 0.999910i \(0.504274\pi\)
\(44\) 5.52366 0.832724
\(45\) 0 0
\(46\) 12.2814 1.81080
\(47\) −1.91855 −0.279850 −0.139925 0.990162i \(-0.544686\pi\)
−0.139925 + 0.990162i \(0.544686\pi\)
\(48\) 0 0
\(49\) 16.3583 2.33689
\(50\) −2.11962 −0.299759
\(51\) 0 0
\(52\) −5.63775 −0.781815
\(53\) −8.65904 −1.18941 −0.594706 0.803944i \(-0.702731\pi\)
−0.594706 + 0.803944i \(0.702731\pi\)
\(54\) 0 0
\(55\) −2.21586 −0.298786
\(56\) 5.04821 0.674596
\(57\) 0 0
\(58\) 4.93357 0.647810
\(59\) −5.39666 −0.702585 −0.351292 0.936266i \(-0.614258\pi\)
−0.351292 + 0.936266i \(0.614258\pi\)
\(60\) 0 0
\(61\) −9.69150 −1.24087 −0.620434 0.784258i \(-0.713044\pi\)
−0.620434 + 0.784258i \(0.713044\pi\)
\(62\) −9.48792 −1.20497
\(63\) 0 0
\(64\) −11.3370 −1.41712
\(65\) 2.26162 0.280520
\(66\) 0 0
\(67\) 8.06916 0.985805 0.492903 0.870084i \(-0.335936\pi\)
0.492903 + 0.870084i \(0.335936\pi\)
\(68\) 7.75058 0.939896
\(69\) 0 0
\(70\) −10.2442 −1.22442
\(71\) 5.00576 0.594075 0.297037 0.954866i \(-0.404001\pi\)
0.297037 + 0.954866i \(0.404001\pi\)
\(72\) 0 0
\(73\) −9.18909 −1.07550 −0.537751 0.843104i \(-0.680726\pi\)
−0.537751 + 0.843104i \(0.680726\pi\)
\(74\) 17.4471 2.02818
\(75\) 0 0
\(76\) −14.8060 −1.69836
\(77\) −10.7093 −1.22044
\(78\) 0 0
\(79\) −1.78949 −0.201333 −0.100666 0.994920i \(-0.532097\pi\)
−0.100666 + 0.994920i \(0.532097\pi\)
\(80\) 2.77159 0.309873
\(81\) 0 0
\(82\) −0.589414 −0.0650899
\(83\) −13.8879 −1.52439 −0.762197 0.647345i \(-0.775879\pi\)
−0.762197 + 0.647345i \(0.775879\pi\)
\(84\) 0 0
\(85\) −3.10920 −0.337240
\(86\) 0.373284 0.0402523
\(87\) 0 0
\(88\) −2.31451 −0.246728
\(89\) −1.00000 −0.106000
\(90\) 0 0
\(91\) 10.9305 1.14583
\(92\) −14.4436 −1.50585
\(93\) 0 0
\(94\) 4.06660 0.419438
\(95\) 5.93952 0.609382
\(96\) 0 0
\(97\) −5.52008 −0.560479 −0.280240 0.959930i \(-0.590414\pi\)
−0.280240 + 0.959930i \(0.590414\pi\)
\(98\) −34.6733 −3.50253
\(99\) 0 0
\(100\) 2.49279 0.249279
\(101\) 12.8409 1.27772 0.638860 0.769323i \(-0.279406\pi\)
0.638860 + 0.769323i \(0.279406\pi\)
\(102\) 0 0
\(103\) −11.7697 −1.15970 −0.579852 0.814722i \(-0.696890\pi\)
−0.579852 + 0.814722i \(0.696890\pi\)
\(104\) 2.36231 0.231644
\(105\) 0 0
\(106\) 18.3539 1.78269
\(107\) 15.4620 1.49477 0.747383 0.664393i \(-0.231310\pi\)
0.747383 + 0.664393i \(0.231310\pi\)
\(108\) 0 0
\(109\) −7.88060 −0.754824 −0.377412 0.926045i \(-0.623186\pi\)
−0.377412 + 0.926045i \(0.623186\pi\)
\(110\) 4.69678 0.447820
\(111\) 0 0
\(112\) 13.3952 1.26573
\(113\) 2.01004 0.189088 0.0945442 0.995521i \(-0.469861\pi\)
0.0945442 + 0.995521i \(0.469861\pi\)
\(114\) 0 0
\(115\) 5.79416 0.540308
\(116\) −5.80215 −0.538716
\(117\) 0 0
\(118\) 11.4389 1.05303
\(119\) −15.0269 −1.37751
\(120\) 0 0
\(121\) −6.08997 −0.553634
\(122\) 20.5423 1.85981
\(123\) 0 0
\(124\) 11.1583 1.00205
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −12.0158 −1.06623 −0.533113 0.846044i \(-0.678978\pi\)
−0.533113 + 0.846044i \(0.678978\pi\)
\(128\) 8.10252 0.716169
\(129\) 0 0
\(130\) −4.79378 −0.420442
\(131\) 20.6066 1.80040 0.900202 0.435472i \(-0.143419\pi\)
0.900202 + 0.435472i \(0.143419\pi\)
\(132\) 0 0
\(133\) 28.7059 2.48912
\(134\) −17.1036 −1.47752
\(135\) 0 0
\(136\) −3.24763 −0.278482
\(137\) −10.9369 −0.934400 −0.467200 0.884152i \(-0.654737\pi\)
−0.467200 + 0.884152i \(0.654737\pi\)
\(138\) 0 0
\(139\) −0.432073 −0.0366479 −0.0183240 0.999832i \(-0.505833\pi\)
−0.0183240 + 0.999832i \(0.505833\pi\)
\(140\) 12.0477 1.01822
\(141\) 0 0
\(142\) −10.6103 −0.890397
\(143\) −5.01144 −0.419077
\(144\) 0 0
\(145\) 2.32757 0.193294
\(146\) 19.4774 1.61196
\(147\) 0 0
\(148\) −20.5187 −1.68663
\(149\) 13.6417 1.11757 0.558785 0.829313i \(-0.311268\pi\)
0.558785 + 0.829313i \(0.311268\pi\)
\(150\) 0 0
\(151\) 22.3825 1.82146 0.910730 0.413002i \(-0.135520\pi\)
0.910730 + 0.413002i \(0.135520\pi\)
\(152\) 6.20396 0.503207
\(153\) 0 0
\(154\) 22.6997 1.82919
\(155\) −4.47624 −0.359540
\(156\) 0 0
\(157\) 11.7366 0.936683 0.468342 0.883547i \(-0.344852\pi\)
0.468342 + 0.883547i \(0.344852\pi\)
\(158\) 3.79303 0.301757
\(159\) 0 0
\(160\) −7.96375 −0.629590
\(161\) 28.0034 2.20698
\(162\) 0 0
\(163\) 1.49678 0.117237 0.0586184 0.998280i \(-0.481330\pi\)
0.0586184 + 0.998280i \(0.481330\pi\)
\(164\) 0.693183 0.0541285
\(165\) 0 0
\(166\) 29.4371 2.28476
\(167\) 0.555849 0.0430129 0.0215064 0.999769i \(-0.493154\pi\)
0.0215064 + 0.999769i \(0.493154\pi\)
\(168\) 0 0
\(169\) −7.88506 −0.606543
\(170\) 6.59033 0.505455
\(171\) 0 0
\(172\) −0.439002 −0.0334736
\(173\) −14.0371 −1.06722 −0.533611 0.845730i \(-0.679165\pi\)
−0.533611 + 0.845730i \(0.679165\pi\)
\(174\) 0 0
\(175\) −4.83304 −0.365343
\(176\) −6.14144 −0.462929
\(177\) 0 0
\(178\) 2.11962 0.158872
\(179\) −9.22570 −0.689561 −0.344780 0.938683i \(-0.612047\pi\)
−0.344780 + 0.938683i \(0.612047\pi\)
\(180\) 0 0
\(181\) −10.4704 −0.778259 −0.389129 0.921183i \(-0.627224\pi\)
−0.389129 + 0.921183i \(0.627224\pi\)
\(182\) −23.1685 −1.71737
\(183\) 0 0
\(184\) 6.05213 0.446169
\(185\) 8.23124 0.605173
\(186\) 0 0
\(187\) 6.88955 0.503814
\(188\) −4.78255 −0.348803
\(189\) 0 0
\(190\) −12.5895 −0.913340
\(191\) 4.73905 0.342906 0.171453 0.985192i \(-0.445154\pi\)
0.171453 + 0.985192i \(0.445154\pi\)
\(192\) 0 0
\(193\) 1.48587 0.106955 0.0534775 0.998569i \(-0.482969\pi\)
0.0534775 + 0.998569i \(0.482969\pi\)
\(194\) 11.7005 0.840045
\(195\) 0 0
\(196\) 40.7777 2.91269
\(197\) 4.51312 0.321547 0.160773 0.986991i \(-0.448601\pi\)
0.160773 + 0.986991i \(0.448601\pi\)
\(198\) 0 0
\(199\) 7.28028 0.516085 0.258043 0.966134i \(-0.416923\pi\)
0.258043 + 0.966134i \(0.416923\pi\)
\(200\) −1.04452 −0.0738588
\(201\) 0 0
\(202\) −27.2179 −1.91504
\(203\) 11.2493 0.789543
\(204\) 0 0
\(205\) −0.278075 −0.0194216
\(206\) 24.9473 1.73816
\(207\) 0 0
\(208\) 6.26828 0.434627
\(209\) −13.1611 −0.910374
\(210\) 0 0
\(211\) 14.0051 0.964154 0.482077 0.876129i \(-0.339883\pi\)
0.482077 + 0.876129i \(0.339883\pi\)
\(212\) −21.5852 −1.48247
\(213\) 0 0
\(214\) −32.7735 −2.24035
\(215\) 0.176109 0.0120105
\(216\) 0 0
\(217\) −21.6338 −1.46860
\(218\) 16.7039 1.13133
\(219\) 0 0
\(220\) −5.52366 −0.372405
\(221\) −7.03184 −0.473013
\(222\) 0 0
\(223\) −8.76611 −0.587022 −0.293511 0.955956i \(-0.594824\pi\)
−0.293511 + 0.955956i \(0.594824\pi\)
\(224\) −38.4891 −2.57166
\(225\) 0 0
\(226\) −4.26051 −0.283405
\(227\) 21.8527 1.45042 0.725209 0.688529i \(-0.241743\pi\)
0.725209 + 0.688529i \(0.241743\pi\)
\(228\) 0 0
\(229\) −20.0211 −1.32303 −0.661514 0.749932i \(-0.730086\pi\)
−0.661514 + 0.749932i \(0.730086\pi\)
\(230\) −12.2814 −0.809812
\(231\) 0 0
\(232\) 2.43120 0.159616
\(233\) −15.3296 −1.00428 −0.502139 0.864787i \(-0.667453\pi\)
−0.502139 + 0.864787i \(0.667453\pi\)
\(234\) 0 0
\(235\) 1.91855 0.125153
\(236\) −13.4527 −0.875697
\(237\) 0 0
\(238\) 31.8513 2.06461
\(239\) −7.27188 −0.470379 −0.235190 0.971950i \(-0.575571\pi\)
−0.235190 + 0.971950i \(0.575571\pi\)
\(240\) 0 0
\(241\) 4.02887 0.259522 0.129761 0.991545i \(-0.458579\pi\)
0.129761 + 0.991545i \(0.458579\pi\)
\(242\) 12.9084 0.829785
\(243\) 0 0
\(244\) −24.1588 −1.54661
\(245\) −16.3583 −1.04509
\(246\) 0 0
\(247\) 13.4329 0.854718
\(248\) −4.67553 −0.296896
\(249\) 0 0
\(250\) 2.11962 0.134057
\(251\) 6.99410 0.441464 0.220732 0.975335i \(-0.429155\pi\)
0.220732 + 0.975335i \(0.429155\pi\)
\(252\) 0 0
\(253\) −12.8390 −0.807183
\(254\) 25.4688 1.59806
\(255\) 0 0
\(256\) 5.49964 0.343727
\(257\) 19.3582 1.20753 0.603767 0.797161i \(-0.293666\pi\)
0.603767 + 0.797161i \(0.293666\pi\)
\(258\) 0 0
\(259\) 39.7819 2.47193
\(260\) 5.63775 0.349638
\(261\) 0 0
\(262\) −43.6781 −2.69844
\(263\) 5.73446 0.353602 0.176801 0.984247i \(-0.443425\pi\)
0.176801 + 0.984247i \(0.443425\pi\)
\(264\) 0 0
\(265\) 8.65904 0.531921
\(266\) −60.8456 −3.73068
\(267\) 0 0
\(268\) 20.1147 1.22870
\(269\) −5.07725 −0.309565 −0.154783 0.987949i \(-0.549468\pi\)
−0.154783 + 0.987949i \(0.549468\pi\)
\(270\) 0 0
\(271\) 14.8909 0.904557 0.452278 0.891877i \(-0.350611\pi\)
0.452278 + 0.891877i \(0.350611\pi\)
\(272\) −8.61742 −0.522508
\(273\) 0 0
\(274\) 23.1820 1.40048
\(275\) 2.21586 0.133621
\(276\) 0 0
\(277\) −30.6930 −1.84417 −0.922083 0.386993i \(-0.873514\pi\)
−0.922083 + 0.386993i \(0.873514\pi\)
\(278\) 0.915829 0.0549278
\(279\) 0 0
\(280\) −5.04821 −0.301688
\(281\) −19.5982 −1.16913 −0.584564 0.811348i \(-0.698734\pi\)
−0.584564 + 0.811348i \(0.698734\pi\)
\(282\) 0 0
\(283\) 32.1638 1.91194 0.955968 0.293469i \(-0.0948099\pi\)
0.955968 + 0.293469i \(0.0948099\pi\)
\(284\) 12.4783 0.740451
\(285\) 0 0
\(286\) 10.6223 0.628112
\(287\) −1.34395 −0.0793308
\(288\) 0 0
\(289\) −7.33286 −0.431345
\(290\) −4.93357 −0.289709
\(291\) 0 0
\(292\) −22.9064 −1.34050
\(293\) 22.7569 1.32947 0.664737 0.747077i \(-0.268543\pi\)
0.664737 + 0.747077i \(0.268543\pi\)
\(294\) 0 0
\(295\) 5.39666 0.314205
\(296\) 8.59771 0.499732
\(297\) 0 0
\(298\) −28.9152 −1.67501
\(299\) 13.1042 0.757836
\(300\) 0 0
\(301\) 0.851142 0.0490590
\(302\) −47.4423 −2.73000
\(303\) 0 0
\(304\) 16.4619 0.944154
\(305\) 9.69150 0.554933
\(306\) 0 0
\(307\) 0.883192 0.0504064 0.0252032 0.999682i \(-0.491977\pi\)
0.0252032 + 0.999682i \(0.491977\pi\)
\(308\) −26.6961 −1.52115
\(309\) 0 0
\(310\) 9.48792 0.538878
\(311\) 2.98323 0.169163 0.0845816 0.996417i \(-0.473045\pi\)
0.0845816 + 0.996417i \(0.473045\pi\)
\(312\) 0 0
\(313\) 6.07859 0.343582 0.171791 0.985133i \(-0.445045\pi\)
0.171791 + 0.985133i \(0.445045\pi\)
\(314\) −24.8771 −1.40390
\(315\) 0 0
\(316\) −4.46081 −0.250940
\(317\) −2.02998 −0.114015 −0.0570075 0.998374i \(-0.518156\pi\)
−0.0570075 + 0.998374i \(0.518156\pi\)
\(318\) 0 0
\(319\) −5.15757 −0.288769
\(320\) 11.3370 0.633755
\(321\) 0 0
\(322\) −59.3565 −3.30781
\(323\) −18.4672 −1.02754
\(324\) 0 0
\(325\) −2.26162 −0.125452
\(326\) −3.17260 −0.175714
\(327\) 0 0
\(328\) −0.290456 −0.0160377
\(329\) 9.27244 0.511206
\(330\) 0 0
\(331\) −12.3999 −0.681559 −0.340779 0.940143i \(-0.610691\pi\)
−0.340779 + 0.940143i \(0.610691\pi\)
\(332\) −34.6196 −1.90000
\(333\) 0 0
\(334\) −1.17819 −0.0644676
\(335\) −8.06916 −0.440866
\(336\) 0 0
\(337\) 27.1425 1.47855 0.739274 0.673405i \(-0.235169\pi\)
0.739274 + 0.673405i \(0.235169\pi\)
\(338\) 16.7133 0.909085
\(339\) 0 0
\(340\) −7.75058 −0.420334
\(341\) 9.91871 0.537128
\(342\) 0 0
\(343\) −45.2288 −2.44213
\(344\) 0.183950 0.00991790
\(345\) 0 0
\(346\) 29.7533 1.59955
\(347\) 17.0635 0.916019 0.458009 0.888947i \(-0.348563\pi\)
0.458009 + 0.888947i \(0.348563\pi\)
\(348\) 0 0
\(349\) 25.3297 1.35587 0.677933 0.735124i \(-0.262876\pi\)
0.677933 + 0.735124i \(0.262876\pi\)
\(350\) 10.2442 0.547576
\(351\) 0 0
\(352\) 17.6465 0.940564
\(353\) −15.3713 −0.818132 −0.409066 0.912505i \(-0.634146\pi\)
−0.409066 + 0.912505i \(0.634146\pi\)
\(354\) 0 0
\(355\) −5.00576 −0.265678
\(356\) −2.49279 −0.132117
\(357\) 0 0
\(358\) 19.5550 1.03351
\(359\) 4.81197 0.253966 0.126983 0.991905i \(-0.459471\pi\)
0.126983 + 0.991905i \(0.459471\pi\)
\(360\) 0 0
\(361\) 16.2779 0.856730
\(362\) 22.1933 1.16645
\(363\) 0 0
\(364\) 27.2474 1.42815
\(365\) 9.18909 0.480979
\(366\) 0 0
\(367\) 23.4281 1.22294 0.611468 0.791269i \(-0.290579\pi\)
0.611468 + 0.791269i \(0.290579\pi\)
\(368\) 16.0590 0.837134
\(369\) 0 0
\(370\) −17.4471 −0.907031
\(371\) 41.8495 2.17272
\(372\) 0 0
\(373\) 10.4886 0.543078 0.271539 0.962427i \(-0.412467\pi\)
0.271539 + 0.962427i \(0.412467\pi\)
\(374\) −14.6032 −0.755115
\(375\) 0 0
\(376\) 2.00397 0.103347
\(377\) 5.26409 0.271115
\(378\) 0 0
\(379\) −20.6495 −1.06069 −0.530347 0.847781i \(-0.677938\pi\)
−0.530347 + 0.847781i \(0.677938\pi\)
\(380\) 14.8060 0.759529
\(381\) 0 0
\(382\) −10.0450 −0.513946
\(383\) −11.8529 −0.605654 −0.302827 0.953046i \(-0.597930\pi\)
−0.302827 + 0.953046i \(0.597930\pi\)
\(384\) 0 0
\(385\) 10.7093 0.545798
\(386\) −3.14947 −0.160304
\(387\) 0 0
\(388\) −13.7604 −0.698578
\(389\) 11.8172 0.599157 0.299578 0.954072i \(-0.403154\pi\)
0.299578 + 0.954072i \(0.403154\pi\)
\(390\) 0 0
\(391\) −18.0152 −0.911068
\(392\) −17.0866 −0.863001
\(393\) 0 0
\(394\) −9.56610 −0.481933
\(395\) 1.78949 0.0900388
\(396\) 0 0
\(397\) 36.5070 1.83223 0.916116 0.400912i \(-0.131307\pi\)
0.916116 + 0.400912i \(0.131307\pi\)
\(398\) −15.4314 −0.773507
\(399\) 0 0
\(400\) −2.77159 −0.138579
\(401\) −1.45221 −0.0725197 −0.0362599 0.999342i \(-0.511544\pi\)
−0.0362599 + 0.999342i \(0.511544\pi\)
\(402\) 0 0
\(403\) −10.1236 −0.504291
\(404\) 32.0097 1.59254
\(405\) 0 0
\(406\) −23.8441 −1.18336
\(407\) −18.2393 −0.904086
\(408\) 0 0
\(409\) 32.3981 1.60198 0.800992 0.598674i \(-0.204306\pi\)
0.800992 + 0.598674i \(0.204306\pi\)
\(410\) 0.589414 0.0291091
\(411\) 0 0
\(412\) −29.3394 −1.44545
\(413\) 26.0822 1.28342
\(414\) 0 0
\(415\) 13.8879 0.681730
\(416\) −18.0110 −0.883062
\(417\) 0 0
\(418\) 27.8966 1.36447
\(419\) 10.4032 0.508227 0.254114 0.967174i \(-0.418216\pi\)
0.254114 + 0.967174i \(0.418216\pi\)
\(420\) 0 0
\(421\) 1.28300 0.0625297 0.0312648 0.999511i \(-0.490046\pi\)
0.0312648 + 0.999511i \(0.490046\pi\)
\(422\) −29.6856 −1.44507
\(423\) 0 0
\(424\) 9.04456 0.439243
\(425\) 3.10920 0.150818
\(426\) 0 0
\(427\) 46.8394 2.26672
\(428\) 38.5434 1.86307
\(429\) 0 0
\(430\) −0.373284 −0.0180014
\(431\) 2.37582 0.114439 0.0572195 0.998362i \(-0.481777\pi\)
0.0572195 + 0.998362i \(0.481777\pi\)
\(432\) 0 0
\(433\) 31.3903 1.50852 0.754262 0.656574i \(-0.227995\pi\)
0.754262 + 0.656574i \(0.227995\pi\)
\(434\) 45.8555 2.20113
\(435\) 0 0
\(436\) −19.6447 −0.940808
\(437\) 34.4145 1.64627
\(438\) 0 0
\(439\) −8.32745 −0.397448 −0.198724 0.980056i \(-0.563680\pi\)
−0.198724 + 0.980056i \(0.563680\pi\)
\(440\) 2.31451 0.110340
\(441\) 0 0
\(442\) 14.9048 0.708950
\(443\) 27.1908 1.29188 0.645938 0.763390i \(-0.276466\pi\)
0.645938 + 0.763390i \(0.276466\pi\)
\(444\) 0 0
\(445\) 1.00000 0.0474045
\(446\) 18.5808 0.879827
\(447\) 0 0
\(448\) 54.7919 2.58867
\(449\) 20.2755 0.956861 0.478430 0.878126i \(-0.341206\pi\)
0.478430 + 0.878126i \(0.341206\pi\)
\(450\) 0 0
\(451\) 0.616176 0.0290146
\(452\) 5.01059 0.235679
\(453\) 0 0
\(454\) −46.3195 −2.17388
\(455\) −10.9305 −0.512430
\(456\) 0 0
\(457\) −27.8204 −1.30138 −0.650691 0.759343i \(-0.725521\pi\)
−0.650691 + 0.759343i \(0.725521\pi\)
\(458\) 42.4370 1.98295
\(459\) 0 0
\(460\) 14.4436 0.673437
\(461\) −3.25129 −0.151428 −0.0757138 0.997130i \(-0.524124\pi\)
−0.0757138 + 0.997130i \(0.524124\pi\)
\(462\) 0 0
\(463\) −13.8292 −0.642699 −0.321349 0.946961i \(-0.604136\pi\)
−0.321349 + 0.946961i \(0.604136\pi\)
\(464\) 6.45107 0.299483
\(465\) 0 0
\(466\) 32.4930 1.50521
\(467\) 19.5627 0.905253 0.452626 0.891700i \(-0.350487\pi\)
0.452626 + 0.891700i \(0.350487\pi\)
\(468\) 0 0
\(469\) −38.9986 −1.80079
\(470\) −4.06660 −0.187578
\(471\) 0 0
\(472\) 5.63692 0.259461
\(473\) −0.390233 −0.0179429
\(474\) 0 0
\(475\) −5.93952 −0.272524
\(476\) −37.4588 −1.71692
\(477\) 0 0
\(478\) 15.4136 0.705003
\(479\) −26.2070 −1.19743 −0.598714 0.800963i \(-0.704321\pi\)
−0.598714 + 0.800963i \(0.704321\pi\)
\(480\) 0 0
\(481\) 18.6160 0.848815
\(482\) −8.53968 −0.388972
\(483\) 0 0
\(484\) −15.1810 −0.690046
\(485\) 5.52008 0.250654
\(486\) 0 0
\(487\) 28.8600 1.30777 0.653887 0.756592i \(-0.273137\pi\)
0.653887 + 0.756592i \(0.273137\pi\)
\(488\) 10.1230 0.458246
\(489\) 0 0
\(490\) 34.6733 1.56638
\(491\) 27.5737 1.24438 0.622192 0.782865i \(-0.286242\pi\)
0.622192 + 0.782865i \(0.286242\pi\)
\(492\) 0 0
\(493\) −7.23690 −0.325933
\(494\) −28.4727 −1.28105
\(495\) 0 0
\(496\) −12.4063 −0.557058
\(497\) −24.1930 −1.08521
\(498\) 0 0
\(499\) 32.9455 1.47484 0.737422 0.675432i \(-0.236043\pi\)
0.737422 + 0.675432i \(0.236043\pi\)
\(500\) −2.49279 −0.111481
\(501\) 0 0
\(502\) −14.8248 −0.661665
\(503\) 25.6894 1.14543 0.572717 0.819753i \(-0.305890\pi\)
0.572717 + 0.819753i \(0.305890\pi\)
\(504\) 0 0
\(505\) −12.8409 −0.571414
\(506\) 27.2139 1.20980
\(507\) 0 0
\(508\) −29.9527 −1.32894
\(509\) −27.1811 −1.20478 −0.602390 0.798202i \(-0.705785\pi\)
−0.602390 + 0.798202i \(0.705785\pi\)
\(510\) 0 0
\(511\) 44.4112 1.96464
\(512\) −27.8622 −1.23135
\(513\) 0 0
\(514\) −41.0321 −1.80985
\(515\) 11.7697 0.518635
\(516\) 0 0
\(517\) −4.25124 −0.186969
\(518\) −84.3225 −3.70492
\(519\) 0 0
\(520\) −2.36231 −0.103594
\(521\) −0.901023 −0.0394745 −0.0197373 0.999805i \(-0.506283\pi\)
−0.0197373 + 0.999805i \(0.506283\pi\)
\(522\) 0 0
\(523\) 0.0126235 0.000551986 0 0.000275993 1.00000i \(-0.499912\pi\)
0.000275993 1.00000i \(0.499912\pi\)
\(524\) 51.3678 2.24401
\(525\) 0 0
\(526\) −12.1549 −0.529978
\(527\) 13.9175 0.606257
\(528\) 0 0
\(529\) 10.5723 0.459665
\(530\) −18.3539 −0.797242
\(531\) 0 0
\(532\) 71.5577 3.10242
\(533\) −0.628902 −0.0272408
\(534\) 0 0
\(535\) −15.4620 −0.668480
\(536\) −8.42842 −0.364052
\(537\) 0 0
\(538\) 10.7618 0.463976
\(539\) 36.2476 1.56129
\(540\) 0 0
\(541\) 9.28499 0.399193 0.199596 0.979878i \(-0.436037\pi\)
0.199596 + 0.979878i \(0.436037\pi\)
\(542\) −31.5630 −1.35575
\(543\) 0 0
\(544\) 24.7609 1.06162
\(545\) 7.88060 0.337568
\(546\) 0 0
\(547\) −12.6475 −0.540770 −0.270385 0.962752i \(-0.587151\pi\)
−0.270385 + 0.962752i \(0.587151\pi\)
\(548\) −27.2633 −1.16463
\(549\) 0 0
\(550\) −4.69678 −0.200271
\(551\) 13.8247 0.588951
\(552\) 0 0
\(553\) 8.64865 0.367778
\(554\) 65.0576 2.76403
\(555\) 0 0
\(556\) −1.07707 −0.0456777
\(557\) 1.89455 0.0802747 0.0401374 0.999194i \(-0.487220\pi\)
0.0401374 + 0.999194i \(0.487220\pi\)
\(558\) 0 0
\(559\) 0.398292 0.0168460
\(560\) −13.3952 −0.566050
\(561\) 0 0
\(562\) 41.5406 1.75229
\(563\) 39.9171 1.68231 0.841153 0.540797i \(-0.181877\pi\)
0.841153 + 0.540797i \(0.181877\pi\)
\(564\) 0 0
\(565\) −2.01004 −0.0845629
\(566\) −68.1749 −2.86561
\(567\) 0 0
\(568\) −5.22863 −0.219388
\(569\) 12.8214 0.537499 0.268750 0.963210i \(-0.413390\pi\)
0.268750 + 0.963210i \(0.413390\pi\)
\(570\) 0 0
\(571\) −24.0157 −1.00503 −0.502513 0.864570i \(-0.667591\pi\)
−0.502513 + 0.864570i \(0.667591\pi\)
\(572\) −12.4924 −0.522335
\(573\) 0 0
\(574\) 2.84866 0.118901
\(575\) −5.79416 −0.241633
\(576\) 0 0
\(577\) −28.5046 −1.18666 −0.593331 0.804959i \(-0.702187\pi\)
−0.593331 + 0.804959i \(0.702187\pi\)
\(578\) 15.5429 0.646499
\(579\) 0 0
\(580\) 5.80215 0.240921
\(581\) 67.1207 2.78464
\(582\) 0 0
\(583\) −19.1872 −0.794653
\(584\) 9.59820 0.397176
\(585\) 0 0
\(586\) −48.2361 −1.99261
\(587\) 2.40697 0.0993462 0.0496731 0.998766i \(-0.484182\pi\)
0.0496731 + 0.998766i \(0.484182\pi\)
\(588\) 0 0
\(589\) −26.5867 −1.09549
\(590\) −11.4389 −0.470930
\(591\) 0 0
\(592\) 22.8136 0.937633
\(593\) −10.4715 −0.430013 −0.215006 0.976613i \(-0.568977\pi\)
−0.215006 + 0.976613i \(0.568977\pi\)
\(594\) 0 0
\(595\) 15.0269 0.616042
\(596\) 34.0058 1.39293
\(597\) 0 0
\(598\) −27.7759 −1.13584
\(599\) −9.28858 −0.379521 −0.189760 0.981830i \(-0.560771\pi\)
−0.189760 + 0.981830i \(0.560771\pi\)
\(600\) 0 0
\(601\) −18.0877 −0.737814 −0.368907 0.929466i \(-0.620268\pi\)
−0.368907 + 0.929466i \(0.620268\pi\)
\(602\) −1.80410 −0.0735295
\(603\) 0 0
\(604\) 55.7948 2.27026
\(605\) 6.08997 0.247593
\(606\) 0 0
\(607\) −4.08445 −0.165783 −0.0828914 0.996559i \(-0.526415\pi\)
−0.0828914 + 0.996559i \(0.526415\pi\)
\(608\) −47.3008 −1.91830
\(609\) 0 0
\(610\) −20.5423 −0.831733
\(611\) 4.33904 0.175539
\(612\) 0 0
\(613\) 20.2800 0.819101 0.409551 0.912287i \(-0.365685\pi\)
0.409551 + 0.912287i \(0.365685\pi\)
\(614\) −1.87203 −0.0755490
\(615\) 0 0
\(616\) 11.1861 0.450702
\(617\) −3.23902 −0.130398 −0.0651990 0.997872i \(-0.520768\pi\)
−0.0651990 + 0.997872i \(0.520768\pi\)
\(618\) 0 0
\(619\) 2.62101 0.105347 0.0526737 0.998612i \(-0.483226\pi\)
0.0526737 + 0.998612i \(0.483226\pi\)
\(620\) −11.1583 −0.448129
\(621\) 0 0
\(622\) −6.32330 −0.253541
\(623\) 4.83304 0.193632
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −12.8843 −0.514960
\(627\) 0 0
\(628\) 29.2569 1.16748
\(629\) −25.5926 −1.02044
\(630\) 0 0
\(631\) −4.84085 −0.192711 −0.0963557 0.995347i \(-0.530719\pi\)
−0.0963557 + 0.995347i \(0.530719\pi\)
\(632\) 1.86916 0.0743511
\(633\) 0 0
\(634\) 4.30278 0.170885
\(635\) 12.0158 0.476831
\(636\) 0 0
\(637\) −36.9962 −1.46584
\(638\) 10.9321 0.432806
\(639\) 0 0
\(640\) −8.10252 −0.320280
\(641\) 41.4377 1.63669 0.818344 0.574728i \(-0.194892\pi\)
0.818344 + 0.574728i \(0.194892\pi\)
\(642\) 0 0
\(643\) −26.2789 −1.03634 −0.518168 0.855279i \(-0.673386\pi\)
−0.518168 + 0.855279i \(0.673386\pi\)
\(644\) 69.8065 2.75076
\(645\) 0 0
\(646\) 39.1434 1.54007
\(647\) −32.8680 −1.29218 −0.646088 0.763263i \(-0.723596\pi\)
−0.646088 + 0.763263i \(0.723596\pi\)
\(648\) 0 0
\(649\) −11.9582 −0.469401
\(650\) 4.79378 0.188028
\(651\) 0 0
\(652\) 3.73115 0.146123
\(653\) −34.0840 −1.33381 −0.666904 0.745143i \(-0.732381\pi\)
−0.666904 + 0.745143i \(0.732381\pi\)
\(654\) 0 0
\(655\) −20.6066 −0.805165
\(656\) −0.770710 −0.0300912
\(657\) 0 0
\(658\) −19.6540 −0.766195
\(659\) 22.3303 0.869864 0.434932 0.900463i \(-0.356772\pi\)
0.434932 + 0.900463i \(0.356772\pi\)
\(660\) 0 0
\(661\) −4.90059 −0.190611 −0.0953055 0.995448i \(-0.530383\pi\)
−0.0953055 + 0.995448i \(0.530383\pi\)
\(662\) 26.2830 1.02152
\(663\) 0 0
\(664\) 14.5062 0.562950
\(665\) −28.7059 −1.11317
\(666\) 0 0
\(667\) 13.4863 0.522193
\(668\) 1.38561 0.0536110
\(669\) 0 0
\(670\) 17.1036 0.660768
\(671\) −21.4750 −0.829032
\(672\) 0 0
\(673\) 17.8142 0.686689 0.343344 0.939210i \(-0.388440\pi\)
0.343344 + 0.939210i \(0.388440\pi\)
\(674\) −57.5318 −2.21604
\(675\) 0 0
\(676\) −19.6558 −0.755992
\(677\) 2.59037 0.0995561 0.0497780 0.998760i \(-0.484149\pi\)
0.0497780 + 0.998760i \(0.484149\pi\)
\(678\) 0 0
\(679\) 26.6788 1.02384
\(680\) 3.24763 0.124541
\(681\) 0 0
\(682\) −21.0239 −0.805046
\(683\) −24.9388 −0.954257 −0.477128 0.878834i \(-0.658322\pi\)
−0.477128 + 0.878834i \(0.658322\pi\)
\(684\) 0 0
\(685\) 10.9369 0.417876
\(686\) 95.8678 3.66025
\(687\) 0 0
\(688\) 0.488101 0.0186087
\(689\) 19.5835 0.746072
\(690\) 0 0
\(691\) −29.1377 −1.10845 −0.554226 0.832366i \(-0.686986\pi\)
−0.554226 + 0.832366i \(0.686986\pi\)
\(692\) −34.9915 −1.33018
\(693\) 0 0
\(694\) −36.1682 −1.37293
\(695\) 0.432073 0.0163894
\(696\) 0 0
\(697\) 0.864593 0.0327488
\(698\) −53.6893 −2.03217
\(699\) 0 0
\(700\) −12.0477 −0.455362
\(701\) 37.5559 1.41847 0.709233 0.704974i \(-0.249042\pi\)
0.709233 + 0.704974i \(0.249042\pi\)
\(702\) 0 0
\(703\) 48.8896 1.84391
\(704\) −25.1211 −0.946786
\(705\) 0 0
\(706\) 32.5813 1.22621
\(707\) −62.0607 −2.33403
\(708\) 0 0
\(709\) −16.0916 −0.604334 −0.302167 0.953255i \(-0.597710\pi\)
−0.302167 + 0.953255i \(0.597710\pi\)
\(710\) 10.6103 0.398198
\(711\) 0 0
\(712\) 1.04452 0.0391451
\(713\) −25.9360 −0.971312
\(714\) 0 0
\(715\) 5.01144 0.187417
\(716\) −22.9977 −0.859464
\(717\) 0 0
\(718\) −10.1995 −0.380644
\(719\) 3.86290 0.144062 0.0720310 0.997402i \(-0.477052\pi\)
0.0720310 + 0.997402i \(0.477052\pi\)
\(720\) 0 0
\(721\) 56.8834 2.11845
\(722\) −34.5029 −1.28406
\(723\) 0 0
\(724\) −26.1005 −0.970017
\(725\) −2.32757 −0.0864439
\(726\) 0 0
\(727\) −10.4042 −0.385871 −0.192935 0.981211i \(-0.561801\pi\)
−0.192935 + 0.981211i \(0.561801\pi\)
\(728\) −11.4172 −0.423148
\(729\) 0 0
\(730\) −19.4774 −0.720890
\(731\) −0.547558 −0.0202522
\(732\) 0 0
\(733\) −16.9908 −0.627569 −0.313784 0.949494i \(-0.601597\pi\)
−0.313784 + 0.949494i \(0.601597\pi\)
\(734\) −49.6587 −1.83293
\(735\) 0 0
\(736\) −46.1432 −1.70086
\(737\) 17.8801 0.658623
\(738\) 0 0
\(739\) 10.1630 0.373854 0.186927 0.982374i \(-0.440147\pi\)
0.186927 + 0.982374i \(0.440147\pi\)
\(740\) 20.5187 0.754283
\(741\) 0 0
\(742\) −88.7050 −3.25646
\(743\) 25.6666 0.941617 0.470809 0.882235i \(-0.343962\pi\)
0.470809 + 0.882235i \(0.343962\pi\)
\(744\) 0 0
\(745\) −13.6417 −0.499793
\(746\) −22.2318 −0.813964
\(747\) 0 0
\(748\) 17.1742 0.627950
\(749\) −74.7284 −2.73051
\(750\) 0 0
\(751\) −1.88134 −0.0686510 −0.0343255 0.999411i \(-0.510928\pi\)
−0.0343255 + 0.999411i \(0.510928\pi\)
\(752\) 5.31744 0.193907
\(753\) 0 0
\(754\) −11.1579 −0.406346
\(755\) −22.3825 −0.814582
\(756\) 0 0
\(757\) 11.6427 0.423161 0.211580 0.977361i \(-0.432139\pi\)
0.211580 + 0.977361i \(0.432139\pi\)
\(758\) 43.7691 1.58977
\(759\) 0 0
\(760\) −6.20396 −0.225041
\(761\) −20.9113 −0.758035 −0.379017 0.925390i \(-0.623738\pi\)
−0.379017 + 0.925390i \(0.623738\pi\)
\(762\) 0 0
\(763\) 38.0872 1.37885
\(764\) 11.8134 0.427395
\(765\) 0 0
\(766\) 25.1236 0.907753
\(767\) 12.2052 0.440704
\(768\) 0 0
\(769\) −21.0699 −0.759800 −0.379900 0.925028i \(-0.624042\pi\)
−0.379900 + 0.925028i \(0.624042\pi\)
\(770\) −22.6997 −0.818040
\(771\) 0 0
\(772\) 3.70395 0.133308
\(773\) −4.26541 −0.153416 −0.0767080 0.997054i \(-0.524441\pi\)
−0.0767080 + 0.997054i \(0.524441\pi\)
\(774\) 0 0
\(775\) 4.47624 0.160791
\(776\) 5.76584 0.206982
\(777\) 0 0
\(778\) −25.0480 −0.898015
\(779\) −1.65163 −0.0591759
\(780\) 0 0
\(781\) 11.0921 0.396905
\(782\) 38.1854 1.36551
\(783\) 0 0
\(784\) −45.3383 −1.61923
\(785\) −11.7366 −0.418898
\(786\) 0 0
\(787\) 46.9571 1.67384 0.836920 0.547325i \(-0.184354\pi\)
0.836920 + 0.547325i \(0.184354\pi\)
\(788\) 11.2503 0.400774
\(789\) 0 0
\(790\) −3.79303 −0.134950
\(791\) −9.71458 −0.345411
\(792\) 0 0
\(793\) 21.9185 0.778349
\(794\) −77.3809 −2.74615
\(795\) 0 0
\(796\) 18.1482 0.643245
\(797\) 36.9401 1.30849 0.654243 0.756284i \(-0.272987\pi\)
0.654243 + 0.756284i \(0.272987\pi\)
\(798\) 0 0
\(799\) −5.96517 −0.211033
\(800\) 7.96375 0.281561
\(801\) 0 0
\(802\) 3.07813 0.108692
\(803\) −20.3617 −0.718549
\(804\) 0 0
\(805\) −28.0034 −0.986990
\(806\) 21.4581 0.755829
\(807\) 0 0
\(808\) −13.4126 −0.471855
\(809\) −17.3676 −0.610613 −0.305306 0.952254i \(-0.598759\pi\)
−0.305306 + 0.952254i \(0.598759\pi\)
\(810\) 0 0
\(811\) −44.5363 −1.56388 −0.781941 0.623352i \(-0.785770\pi\)
−0.781941 + 0.623352i \(0.785770\pi\)
\(812\) 28.0420 0.984081
\(813\) 0 0
\(814\) 38.6603 1.35504
\(815\) −1.49678 −0.0524299
\(816\) 0 0
\(817\) 1.04600 0.0365950
\(818\) −68.6717 −2.40105
\(819\) 0 0
\(820\) −0.693183 −0.0242070
\(821\) 17.7581 0.619761 0.309880 0.950776i \(-0.399711\pi\)
0.309880 + 0.950776i \(0.399711\pi\)
\(822\) 0 0
\(823\) −27.7787 −0.968304 −0.484152 0.874984i \(-0.660872\pi\)
−0.484152 + 0.874984i \(0.660872\pi\)
\(824\) 12.2937 0.428272
\(825\) 0 0
\(826\) −55.2844 −1.92359
\(827\) 28.8725 1.00400 0.501998 0.864869i \(-0.332599\pi\)
0.501998 + 0.864869i \(0.332599\pi\)
\(828\) 0 0
\(829\) 41.4440 1.43941 0.719704 0.694281i \(-0.244277\pi\)
0.719704 + 0.694281i \(0.244277\pi\)
\(830\) −29.4371 −1.02178
\(831\) 0 0
\(832\) 25.6399 0.888904
\(833\) 50.8611 1.76223
\(834\) 0 0
\(835\) −0.555849 −0.0192360
\(836\) −32.8079 −1.13468
\(837\) 0 0
\(838\) −22.0507 −0.761730
\(839\) 15.0790 0.520583 0.260292 0.965530i \(-0.416181\pi\)
0.260292 + 0.965530i \(0.416181\pi\)
\(840\) 0 0
\(841\) −23.5824 −0.813186
\(842\) −2.71948 −0.0937193
\(843\) 0 0
\(844\) 34.9118 1.20172
\(845\) 7.88506 0.271254
\(846\) 0 0
\(847\) 29.4331 1.01133
\(848\) 23.9993 0.824139
\(849\) 0 0
\(850\) −6.59033 −0.226046
\(851\) 47.6931 1.63490
\(852\) 0 0
\(853\) 36.1724 1.23852 0.619260 0.785186i \(-0.287433\pi\)
0.619260 + 0.785186i \(0.287433\pi\)
\(854\) −99.2816 −3.39735
\(855\) 0 0
\(856\) −16.1504 −0.552009
\(857\) −14.1138 −0.482117 −0.241058 0.970511i \(-0.577495\pi\)
−0.241058 + 0.970511i \(0.577495\pi\)
\(858\) 0 0
\(859\) −3.04904 −0.104032 −0.0520159 0.998646i \(-0.516565\pi\)
−0.0520159 + 0.998646i \(0.516565\pi\)
\(860\) 0.439002 0.0149699
\(861\) 0 0
\(862\) −5.03582 −0.171521
\(863\) 44.3700 1.51037 0.755187 0.655510i \(-0.227546\pi\)
0.755187 + 0.655510i \(0.227546\pi\)
\(864\) 0 0
\(865\) 14.0371 0.477276
\(866\) −66.5356 −2.26097
\(867\) 0 0
\(868\) −53.9285 −1.83045
\(869\) −3.96525 −0.134512
\(870\) 0 0
\(871\) −18.2494 −0.618358
\(872\) 8.23146 0.278752
\(873\) 0 0
\(874\) −72.9457 −2.46742
\(875\) 4.83304 0.163386
\(876\) 0 0
\(877\) −39.0474 −1.31854 −0.659268 0.751908i \(-0.729134\pi\)
−0.659268 + 0.751908i \(0.729134\pi\)
\(878\) 17.6510 0.595693
\(879\) 0 0
\(880\) 6.14144 0.207028
\(881\) −8.96712 −0.302110 −0.151055 0.988525i \(-0.548267\pi\)
−0.151055 + 0.988525i \(0.548267\pi\)
\(882\) 0 0
\(883\) 6.30062 0.212033 0.106016 0.994364i \(-0.466190\pi\)
0.106016 + 0.994364i \(0.466190\pi\)
\(884\) −17.5289 −0.589560
\(885\) 0 0
\(886\) −57.6343 −1.93626
\(887\) −51.6559 −1.73443 −0.867217 0.497931i \(-0.834093\pi\)
−0.867217 + 0.497931i \(0.834093\pi\)
\(888\) 0 0
\(889\) 58.0726 1.94769
\(890\) −2.11962 −0.0710498
\(891\) 0 0
\(892\) −21.8520 −0.731660
\(893\) 11.3953 0.381329
\(894\) 0 0
\(895\) 9.22570 0.308381
\(896\) −39.1598 −1.30824
\(897\) 0 0
\(898\) −42.9764 −1.43414
\(899\) −10.4188 −0.347486
\(900\) 0 0
\(901\) −26.9227 −0.896926
\(902\) −1.30606 −0.0434870
\(903\) 0 0
\(904\) −2.09953 −0.0698292
\(905\) 10.4704 0.348048
\(906\) 0 0
\(907\) −30.5492 −1.01437 −0.507185 0.861837i \(-0.669314\pi\)
−0.507185 + 0.861837i \(0.669314\pi\)
\(908\) 54.4742 1.80779
\(909\) 0 0
\(910\) 23.1685 0.768029
\(911\) −13.9379 −0.461784 −0.230892 0.972979i \(-0.574164\pi\)
−0.230892 + 0.972979i \(0.574164\pi\)
\(912\) 0 0
\(913\) −30.7736 −1.01846
\(914\) 58.9686 1.95051
\(915\) 0 0
\(916\) −49.9082 −1.64901
\(917\) −99.5923 −3.28883
\(918\) 0 0
\(919\) 21.5909 0.712217 0.356108 0.934445i \(-0.384103\pi\)
0.356108 + 0.934445i \(0.384103\pi\)
\(920\) −6.05213 −0.199533
\(921\) 0 0
\(922\) 6.89150 0.226959
\(923\) −11.3211 −0.372640
\(924\) 0 0
\(925\) −8.23124 −0.270641
\(926\) 29.3127 0.963275
\(927\) 0 0
\(928\) −18.5362 −0.608481
\(929\) 34.8126 1.14216 0.571082 0.820893i \(-0.306524\pi\)
0.571082 + 0.820893i \(0.306524\pi\)
\(930\) 0 0
\(931\) −97.1601 −3.18429
\(932\) −38.2135 −1.25172
\(933\) 0 0
\(934\) −41.4654 −1.35679
\(935\) −6.88955 −0.225312
\(936\) 0 0
\(937\) 29.3529 0.958918 0.479459 0.877564i \(-0.340833\pi\)
0.479459 + 0.877564i \(0.340833\pi\)
\(938\) 82.6622 2.69901
\(939\) 0 0
\(940\) 4.78255 0.155989
\(941\) −39.9726 −1.30307 −0.651534 0.758619i \(-0.725874\pi\)
−0.651534 + 0.758619i \(0.725874\pi\)
\(942\) 0 0
\(943\) −1.61121 −0.0524683
\(944\) 14.9573 0.486819
\(945\) 0 0
\(946\) 0.827145 0.0268928
\(947\) −57.0642 −1.85434 −0.927168 0.374646i \(-0.877764\pi\)
−0.927168 + 0.374646i \(0.877764\pi\)
\(948\) 0 0
\(949\) 20.7822 0.674620
\(950\) 12.5895 0.408458
\(951\) 0 0
\(952\) 15.6959 0.508707
\(953\) −47.9488 −1.55321 −0.776607 0.629985i \(-0.783061\pi\)
−0.776607 + 0.629985i \(0.783061\pi\)
\(954\) 0 0
\(955\) −4.73905 −0.153352
\(956\) −18.1273 −0.586278
\(957\) 0 0
\(958\) 55.5488 1.79470
\(959\) 52.8583 1.70688
\(960\) 0 0
\(961\) −10.9633 −0.353655
\(962\) −39.4587 −1.27220
\(963\) 0 0
\(964\) 10.0431 0.323467
\(965\) −1.48587 −0.0478317
\(966\) 0 0
\(967\) −22.9873 −0.739222 −0.369611 0.929187i \(-0.620509\pi\)
−0.369611 + 0.929187i \(0.620509\pi\)
\(968\) 6.36111 0.204454
\(969\) 0 0
\(970\) −11.7005 −0.375679
\(971\) −4.86938 −0.156266 −0.0781330 0.996943i \(-0.524896\pi\)
−0.0781330 + 0.996943i \(0.524896\pi\)
\(972\) 0 0
\(973\) 2.08822 0.0669454
\(974\) −61.1723 −1.96009
\(975\) 0 0
\(976\) 26.8608 0.859794
\(977\) −5.28623 −0.169121 −0.0845607 0.996418i \(-0.526949\pi\)
−0.0845607 + 0.996418i \(0.526949\pi\)
\(978\) 0 0
\(979\) −2.21586 −0.0708191
\(980\) −40.7777 −1.30259
\(981\) 0 0
\(982\) −58.4458 −1.86508
\(983\) −21.4830 −0.685202 −0.342601 0.939481i \(-0.611308\pi\)
−0.342601 + 0.939481i \(0.611308\pi\)
\(984\) 0 0
\(985\) −4.51312 −0.143800
\(986\) 15.3395 0.488508
\(987\) 0 0
\(988\) 33.4855 1.06532
\(989\) 1.02040 0.0324470
\(990\) 0 0
\(991\) −39.9796 −1.26999 −0.634997 0.772514i \(-0.718999\pi\)
−0.634997 + 0.772514i \(0.718999\pi\)
\(992\) 35.6476 1.13181
\(993\) 0 0
\(994\) 51.2800 1.62650
\(995\) −7.28028 −0.230800
\(996\) 0 0
\(997\) −40.8157 −1.29265 −0.646323 0.763064i \(-0.723694\pi\)
−0.646323 + 0.763064i \(0.723694\pi\)
\(998\) −69.8320 −2.21049
\(999\) 0 0
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 4005.2.a.o.1.1 7
3.2 odd 2 445.2.a.f.1.7 7
12.11 even 2 7120.2.a.bj.1.7 7
15.14 odd 2 2225.2.a.k.1.1 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
445.2.a.f.1.7 7 3.2 odd 2
2225.2.a.k.1.1 7 15.14 odd 2
4005.2.a.o.1.1 7 1.1 even 1 trivial
7120.2.a.bj.1.7 7 12.11 even 2