Properties

Label 4012.2.b.b.237.15
Level 4012
Weight 2
Character 4012.237
Analytic conductor 32.036
Analytic rank 0
Dimension 46
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4012.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(46\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 237.15
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.32

$q$-expansion

\(f(q)\) \(=\) \(q-1.45665i q^{3} -3.69601i q^{5} +3.33013i q^{7} +0.878183 q^{9} +O(q^{10})\) \(q-1.45665i q^{3} -3.69601i q^{5} +3.33013i q^{7} +0.878183 q^{9} +4.75976i q^{11} +2.51524 q^{13} -5.38377 q^{15} +(-3.76296 - 1.68527i) q^{17} -0.982947 q^{19} +4.85082 q^{21} +4.02346i q^{23} -8.66046 q^{25} -5.64914i q^{27} +0.375643i q^{29} +3.17061i q^{31} +6.93328 q^{33} +12.3082 q^{35} -1.30716i q^{37} -3.66382i q^{39} +7.88667i q^{41} +9.82393 q^{43} -3.24577i q^{45} +13.2756 q^{47} -4.08976 q^{49} +(-2.45484 + 5.48130i) q^{51} +1.89912 q^{53} +17.5921 q^{55} +1.43181i q^{57} +1.00000 q^{59} +0.836969i q^{61} +2.92446i q^{63} -9.29635i q^{65} +1.30621 q^{67} +5.86076 q^{69} +10.8431i q^{71} -0.00541582i q^{73} +12.6152i q^{75} -15.8506 q^{77} -10.6850i q^{79} -5.59424 q^{81} +7.42159 q^{83} +(-6.22878 + 13.9079i) q^{85} +0.547179 q^{87} -4.58338 q^{89} +8.37608i q^{91} +4.61846 q^{93} +3.63298i q^{95} +10.4572i q^{97} +4.17994i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times\).

\(n\) \(2007\) \(3129\) \(3777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.45665i 0.840995i −0.907294 0.420497i \(-0.861856\pi\)
0.907294 0.420497i \(-0.138144\pi\)
\(4\) 0 0
\(5\) 3.69601i 1.65290i −0.563007 0.826452i \(-0.690356\pi\)
0.563007 0.826452i \(-0.309644\pi\)
\(6\) 0 0
\(7\) 3.33013i 1.25867i 0.777134 + 0.629335i \(0.216673\pi\)
−0.777134 + 0.629335i \(0.783327\pi\)
\(8\) 0 0
\(9\) 0.878183 0.292728
\(10\) 0 0
\(11\) 4.75976i 1.43512i 0.696496 + 0.717561i \(0.254741\pi\)
−0.696496 + 0.717561i \(0.745259\pi\)
\(12\) 0 0
\(13\) 2.51524 0.697603 0.348801 0.937197i \(-0.386589\pi\)
0.348801 + 0.937197i \(0.386589\pi\)
\(14\) 0 0
\(15\) −5.38377 −1.39008
\(16\) 0 0
\(17\) −3.76296 1.68527i −0.912651 0.408739i
\(18\) 0 0
\(19\) −0.982947 −0.225504 −0.112752 0.993623i \(-0.535966\pi\)
−0.112752 + 0.993623i \(0.535966\pi\)
\(20\) 0 0
\(21\) 4.85082 1.05854
\(22\) 0 0
\(23\) 4.02346i 0.838950i 0.907767 + 0.419475i \(0.137786\pi\)
−0.907767 + 0.419475i \(0.862214\pi\)
\(24\) 0 0
\(25\) −8.66046 −1.73209
\(26\) 0 0
\(27\) 5.64914i 1.08718i
\(28\) 0 0
\(29\) 0.375643i 0.0697551i 0.999392 + 0.0348776i \(0.0111041\pi\)
−0.999392 + 0.0348776i \(0.988896\pi\)
\(30\) 0 0
\(31\) 3.17061i 0.569459i 0.958608 + 0.284729i \(0.0919037\pi\)
−0.958608 + 0.284729i \(0.908096\pi\)
\(32\) 0 0
\(33\) 6.93328 1.20693
\(34\) 0 0
\(35\) 12.3082 2.08046
\(36\) 0 0
\(37\) 1.30716i 0.214896i −0.994211 0.107448i \(-0.965732\pi\)
0.994211 0.107448i \(-0.0342679\pi\)
\(38\) 0 0
\(39\) 3.66382i 0.586680i
\(40\) 0 0
\(41\) 7.88667i 1.23169i 0.787867 + 0.615845i \(0.211185\pi\)
−0.787867 + 0.615845i \(0.788815\pi\)
\(42\) 0 0
\(43\) 9.82393 1.49814 0.749068 0.662494i \(-0.230502\pi\)
0.749068 + 0.662494i \(0.230502\pi\)
\(44\) 0 0
\(45\) 3.24577i 0.483851i
\(46\) 0 0
\(47\) 13.2756 1.93644 0.968221 0.250098i \(-0.0804628\pi\)
0.968221 + 0.250098i \(0.0804628\pi\)
\(48\) 0 0
\(49\) −4.08976 −0.584251
\(50\) 0 0
\(51\) −2.45484 + 5.48130i −0.343747 + 0.767535i
\(52\) 0 0
\(53\) 1.89912 0.260864 0.130432 0.991457i \(-0.458364\pi\)
0.130432 + 0.991457i \(0.458364\pi\)
\(54\) 0 0
\(55\) 17.5921 2.37212
\(56\) 0 0
\(57\) 1.43181i 0.189647i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 0.836969i 0.107163i 0.998563 + 0.0535814i \(0.0170637\pi\)
−0.998563 + 0.0535814i \(0.982936\pi\)
\(62\) 0 0
\(63\) 2.92446i 0.368448i
\(64\) 0 0
\(65\) 9.29635i 1.15307i
\(66\) 0 0
\(67\) 1.30621 0.159579 0.0797894 0.996812i \(-0.474575\pi\)
0.0797894 + 0.996812i \(0.474575\pi\)
\(68\) 0 0
\(69\) 5.86076 0.705553
\(70\) 0 0
\(71\) 10.8431i 1.28683i 0.765516 + 0.643417i \(0.222484\pi\)
−0.765516 + 0.643417i \(0.777516\pi\)
\(72\) 0 0
\(73\) 0.00541582i 0.000633874i −1.00000 0.000316937i \(-0.999899\pi\)
1.00000 0.000316937i \(-0.000100884\pi\)
\(74\) 0 0
\(75\) 12.6152i 1.45668i
\(76\) 0 0
\(77\) −15.8506 −1.80634
\(78\) 0 0
\(79\) 10.6850i 1.20215i −0.799192 0.601076i \(-0.794739\pi\)
0.799192 0.601076i \(-0.205261\pi\)
\(80\) 0 0
\(81\) −5.59424 −0.621583
\(82\) 0 0
\(83\) 7.42159 0.814625 0.407313 0.913289i \(-0.366466\pi\)
0.407313 + 0.913289i \(0.366466\pi\)
\(84\) 0 0
\(85\) −6.22878 + 13.9079i −0.675606 + 1.50853i
\(86\) 0 0
\(87\) 0.547179 0.0586637
\(88\) 0 0
\(89\) −4.58338 −0.485838 −0.242919 0.970047i \(-0.578105\pi\)
−0.242919 + 0.970047i \(0.578105\pi\)
\(90\) 0 0
\(91\) 8.37608i 0.878052i
\(92\) 0 0
\(93\) 4.61846 0.478912
\(94\) 0 0
\(95\) 3.63298i 0.372736i
\(96\) 0 0
\(97\) 10.4572i 1.06177i 0.847445 + 0.530884i \(0.178140\pi\)
−0.847445 + 0.530884i \(0.821860\pi\)
\(98\) 0 0
\(99\) 4.17994i 0.420100i
\(100\) 0 0
\(101\) 7.08784 0.705266 0.352633 0.935762i \(-0.385286\pi\)
0.352633 + 0.935762i \(0.385286\pi\)
\(102\) 0 0
\(103\) 9.40257 0.926462 0.463231 0.886238i \(-0.346690\pi\)
0.463231 + 0.886238i \(0.346690\pi\)
\(104\) 0 0
\(105\) 17.9287i 1.74966i
\(106\) 0 0
\(107\) 8.68193i 0.839314i 0.907683 + 0.419657i \(0.137850\pi\)
−0.907683 + 0.419657i \(0.862150\pi\)
\(108\) 0 0
\(109\) 0.888172i 0.0850714i −0.999095 0.0425357i \(-0.986456\pi\)
0.999095 0.0425357i \(-0.0135436\pi\)
\(110\) 0 0
\(111\) −1.90407 −0.180726
\(112\) 0 0
\(113\) 2.57544i 0.242277i −0.992636 0.121138i \(-0.961345\pi\)
0.992636 0.121138i \(-0.0386545\pi\)
\(114\) 0 0
\(115\) 14.8707 1.38670
\(116\) 0 0
\(117\) 2.20884 0.204208
\(118\) 0 0
\(119\) 5.61217 12.5311i 0.514467 1.14873i
\(120\) 0 0
\(121\) −11.6553 −1.05957
\(122\) 0 0
\(123\) 11.4881 1.03585
\(124\) 0 0
\(125\) 13.5291i 1.21008i
\(126\) 0 0
\(127\) −15.5725 −1.38184 −0.690920 0.722931i \(-0.742794\pi\)
−0.690920 + 0.722931i \(0.742794\pi\)
\(128\) 0 0
\(129\) 14.3100i 1.25992i
\(130\) 0 0
\(131\) 1.44083i 0.125886i −0.998017 0.0629431i \(-0.979951\pi\)
0.998017 0.0629431i \(-0.0200487\pi\)
\(132\) 0 0
\(133\) 3.27334i 0.283835i
\(134\) 0 0
\(135\) −20.8793 −1.79700
\(136\) 0 0
\(137\) 14.2841 1.22037 0.610187 0.792257i \(-0.291094\pi\)
0.610187 + 0.792257i \(0.291094\pi\)
\(138\) 0 0
\(139\) 15.9480i 1.35269i −0.736586 0.676344i \(-0.763563\pi\)
0.736586 0.676344i \(-0.236437\pi\)
\(140\) 0 0
\(141\) 19.3378i 1.62854i
\(142\) 0 0
\(143\) 11.9720i 1.00115i
\(144\) 0 0
\(145\) 1.38838 0.115299
\(146\) 0 0
\(147\) 5.95732i 0.491352i
\(148\) 0 0
\(149\) −2.90208 −0.237748 −0.118874 0.992909i \(-0.537928\pi\)
−0.118874 + 0.992909i \(0.537928\pi\)
\(150\) 0 0
\(151\) 5.59425 0.455254 0.227627 0.973748i \(-0.426903\pi\)
0.227627 + 0.973748i \(0.426903\pi\)
\(152\) 0 0
\(153\) −3.30457 1.47998i −0.267158 0.119649i
\(154\) 0 0
\(155\) 11.7186 0.941261
\(156\) 0 0
\(157\) 22.6290 1.80599 0.902997 0.429646i \(-0.141362\pi\)
0.902997 + 0.429646i \(0.141362\pi\)
\(158\) 0 0
\(159\) 2.76635i 0.219386i
\(160\) 0 0
\(161\) −13.3987 −1.05596
\(162\) 0 0
\(163\) 18.2245i 1.42745i 0.700425 + 0.713726i \(0.252994\pi\)
−0.700425 + 0.713726i \(0.747006\pi\)
\(164\) 0 0
\(165\) 25.6255i 1.99494i
\(166\) 0 0
\(167\) 2.07461i 0.160538i 0.996773 + 0.0802690i \(0.0255779\pi\)
−0.996773 + 0.0802690i \(0.974422\pi\)
\(168\) 0 0
\(169\) −6.67355 −0.513350
\(170\) 0 0
\(171\) −0.863208 −0.0660112
\(172\) 0 0
\(173\) 10.9810i 0.834872i 0.908706 + 0.417436i \(0.137071\pi\)
−0.908706 + 0.417436i \(0.862929\pi\)
\(174\) 0 0
\(175\) 28.8405i 2.18013i
\(176\) 0 0
\(177\) 1.45665i 0.109488i
\(178\) 0 0
\(179\) 9.18593 0.686588 0.343294 0.939228i \(-0.388457\pi\)
0.343294 + 0.939228i \(0.388457\pi\)
\(180\) 0 0
\(181\) 15.4332i 1.14714i −0.819156 0.573570i \(-0.805558\pi\)
0.819156 0.573570i \(-0.194442\pi\)
\(182\) 0 0
\(183\) 1.21917 0.0901234
\(184\) 0 0
\(185\) −4.83127 −0.355202
\(186\) 0 0
\(187\) 8.02149 17.9108i 0.586590 1.30977i
\(188\) 0 0
\(189\) 18.8124 1.36840
\(190\) 0 0
\(191\) 4.90307 0.354774 0.177387 0.984141i \(-0.443236\pi\)
0.177387 + 0.984141i \(0.443236\pi\)
\(192\) 0 0
\(193\) 9.14153i 0.658022i −0.944326 0.329011i \(-0.893285\pi\)
0.944326 0.329011i \(-0.106715\pi\)
\(194\) 0 0
\(195\) −13.5415 −0.969726
\(196\) 0 0
\(197\) 8.68513i 0.618790i 0.950934 + 0.309395i \(0.100127\pi\)
−0.950934 + 0.309395i \(0.899873\pi\)
\(198\) 0 0
\(199\) 18.1632i 1.28756i −0.765213 0.643778i \(-0.777366\pi\)
0.765213 0.643778i \(-0.222634\pi\)
\(200\) 0 0
\(201\) 1.90268i 0.134205i
\(202\) 0 0
\(203\) −1.25094 −0.0877987
\(204\) 0 0
\(205\) 29.1492 2.03587
\(206\) 0 0
\(207\) 3.53334i 0.245584i
\(208\) 0 0
\(209\) 4.67859i 0.323625i
\(210\) 0 0
\(211\) 4.81118i 0.331215i −0.986192 0.165608i \(-0.947041\pi\)
0.986192 0.165608i \(-0.0529586\pi\)
\(212\) 0 0
\(213\) 15.7945 1.08222
\(214\) 0 0
\(215\) 36.3093i 2.47627i
\(216\) 0 0
\(217\) −10.5585 −0.716761
\(218\) 0 0
\(219\) −0.00788893 −0.000533085
\(220\) 0 0
\(221\) −9.46475 4.23887i −0.636668 0.285137i
\(222\) 0 0
\(223\) −28.1279 −1.88358 −0.941792 0.336197i \(-0.890859\pi\)
−0.941792 + 0.336197i \(0.890859\pi\)
\(224\) 0 0
\(225\) −7.60547 −0.507032
\(226\) 0 0
\(227\) 12.9136i 0.857103i 0.903517 + 0.428551i \(0.140976\pi\)
−0.903517 + 0.428551i \(0.859024\pi\)
\(228\) 0 0
\(229\) 18.8142 1.24327 0.621637 0.783305i \(-0.286468\pi\)
0.621637 + 0.783305i \(0.286468\pi\)
\(230\) 0 0
\(231\) 23.0887i 1.51913i
\(232\) 0 0
\(233\) 7.70533i 0.504793i 0.967624 + 0.252396i \(0.0812187\pi\)
−0.967624 + 0.252396i \(0.918781\pi\)
\(234\) 0 0
\(235\) 49.0666i 3.20075i
\(236\) 0 0
\(237\) −15.5642 −1.01100
\(238\) 0 0
\(239\) −21.3317 −1.37983 −0.689916 0.723889i \(-0.742353\pi\)
−0.689916 + 0.723889i \(0.742353\pi\)
\(240\) 0 0
\(241\) 10.7602i 0.693126i −0.938027 0.346563i \(-0.887349\pi\)
0.938027 0.346563i \(-0.112651\pi\)
\(242\) 0 0
\(243\) 8.79859i 0.564430i
\(244\) 0 0
\(245\) 15.1158i 0.965711i
\(246\) 0 0
\(247\) −2.47235 −0.157312
\(248\) 0 0
\(249\) 10.8106i 0.685095i
\(250\) 0 0
\(251\) 2.59142 0.163569 0.0817846 0.996650i \(-0.473938\pi\)
0.0817846 + 0.996650i \(0.473938\pi\)
\(252\) 0 0
\(253\) −19.1507 −1.20400
\(254\) 0 0
\(255\) 20.2589 + 9.07312i 1.26866 + 0.568181i
\(256\) 0 0
\(257\) 15.9311 0.993754 0.496877 0.867821i \(-0.334480\pi\)
0.496877 + 0.867821i \(0.334480\pi\)
\(258\) 0 0
\(259\) 4.35301 0.270483
\(260\) 0 0
\(261\) 0.329883i 0.0204193i
\(262\) 0 0
\(263\) −18.3184 −1.12956 −0.564781 0.825241i \(-0.691039\pi\)
−0.564781 + 0.825241i \(0.691039\pi\)
\(264\) 0 0
\(265\) 7.01916i 0.431184i
\(266\) 0 0
\(267\) 6.67636i 0.408587i
\(268\) 0 0
\(269\) 6.62647i 0.404023i 0.979383 + 0.202011i \(0.0647478\pi\)
−0.979383 + 0.202011i \(0.935252\pi\)
\(270\) 0 0
\(271\) 29.1979 1.77364 0.886822 0.462111i \(-0.152908\pi\)
0.886822 + 0.462111i \(0.152908\pi\)
\(272\) 0 0
\(273\) 12.2010 0.738437
\(274\) 0 0
\(275\) 41.2217i 2.48576i
\(276\) 0 0
\(277\) 20.9842i 1.26082i −0.776264 0.630408i \(-0.782888\pi\)
0.776264 0.630408i \(-0.217112\pi\)
\(278\) 0 0
\(279\) 2.78438i 0.166696i
\(280\) 0 0
\(281\) −18.3190 −1.09282 −0.546409 0.837519i \(-0.684005\pi\)
−0.546409 + 0.837519i \(0.684005\pi\)
\(282\) 0 0
\(283\) 9.67937i 0.575379i −0.957724 0.287689i \(-0.907113\pi\)
0.957724 0.287689i \(-0.0928871\pi\)
\(284\) 0 0
\(285\) 5.29196 0.313469
\(286\) 0 0
\(287\) −26.2636 −1.55029
\(288\) 0 0
\(289\) 11.3197 + 12.6832i 0.665865 + 0.746072i
\(290\) 0 0
\(291\) 15.2324 0.892941
\(292\) 0 0
\(293\) −3.37269 −0.197035 −0.0985173 0.995135i \(-0.531410\pi\)
−0.0985173 + 0.995135i \(0.531410\pi\)
\(294\) 0 0
\(295\) 3.69601i 0.215190i
\(296\) 0 0
\(297\) 26.8885 1.56023
\(298\) 0 0
\(299\) 10.1200i 0.585254i
\(300\) 0 0
\(301\) 32.7149i 1.88566i
\(302\) 0 0
\(303\) 10.3245i 0.593125i
\(304\) 0 0
\(305\) 3.09344 0.177130
\(306\) 0 0
\(307\) −13.9557 −0.796492 −0.398246 0.917279i \(-0.630381\pi\)
−0.398246 + 0.917279i \(0.630381\pi\)
\(308\) 0 0
\(309\) 13.6962i 0.779150i
\(310\) 0 0
\(311\) 1.75361i 0.0994380i 0.998763 + 0.0497190i \(0.0158326\pi\)
−0.998763 + 0.0497190i \(0.984167\pi\)
\(312\) 0 0
\(313\) 9.60421i 0.542862i −0.962458 0.271431i \(-0.912503\pi\)
0.962458 0.271431i \(-0.0874969\pi\)
\(314\) 0 0
\(315\) 10.8088 0.609009
\(316\) 0 0
\(317\) 6.48874i 0.364444i 0.983257 + 0.182222i \(0.0583290\pi\)
−0.983257 + 0.182222i \(0.941671\pi\)
\(318\) 0 0
\(319\) −1.78797 −0.100107
\(320\) 0 0
\(321\) 12.6465 0.705859
\(322\) 0 0
\(323\) 3.69879 + 1.65653i 0.205806 + 0.0921720i
\(324\) 0 0
\(325\) −21.7832 −1.20831
\(326\) 0 0
\(327\) −1.29375 −0.0715446
\(328\) 0 0
\(329\) 44.2094i 2.43734i
\(330\) 0 0
\(331\) 10.6209 0.583776 0.291888 0.956452i \(-0.405716\pi\)
0.291888 + 0.956452i \(0.405716\pi\)
\(332\) 0 0
\(333\) 1.14793i 0.0629059i
\(334\) 0 0
\(335\) 4.82775i 0.263768i
\(336\) 0 0
\(337\) 18.8193i 1.02515i −0.858641 0.512577i \(-0.828691\pi\)
0.858641 0.512577i \(-0.171309\pi\)
\(338\) 0 0
\(339\) −3.75150 −0.203754
\(340\) 0 0
\(341\) −15.0914 −0.817242
\(342\) 0 0
\(343\) 9.69149i 0.523291i
\(344\) 0 0
\(345\) 21.6614i 1.16621i
\(346\) 0 0
\(347\) 19.1271i 1.02680i 0.858150 + 0.513400i \(0.171614\pi\)
−0.858150 + 0.513400i \(0.828386\pi\)
\(348\) 0 0
\(349\) −5.15560 −0.275973 −0.137987 0.990434i \(-0.544063\pi\)
−0.137987 + 0.990434i \(0.544063\pi\)
\(350\) 0 0
\(351\) 14.2090i 0.758418i
\(352\) 0 0
\(353\) 19.9898 1.06395 0.531975 0.846760i \(-0.321450\pi\)
0.531975 + 0.846760i \(0.321450\pi\)
\(354\) 0 0
\(355\) 40.0760 2.12701
\(356\) 0 0
\(357\) −18.2534 8.17495i −0.966074 0.432664i
\(358\) 0 0
\(359\) 21.6478 1.14253 0.571263 0.820767i \(-0.306454\pi\)
0.571263 + 0.820767i \(0.306454\pi\)
\(360\) 0 0
\(361\) −18.0338 −0.949148
\(362\) 0 0
\(363\) 16.9777i 0.891096i
\(364\) 0 0
\(365\) −0.0200169 −0.00104773
\(366\) 0 0
\(367\) 25.4012i 1.32593i −0.748649 0.662967i \(-0.769297\pi\)
0.748649 0.662967i \(-0.230703\pi\)
\(368\) 0 0
\(369\) 6.92594i 0.360550i
\(370\) 0 0
\(371\) 6.32432i 0.328342i
\(372\) 0 0
\(373\) −2.83555 −0.146819 −0.0734096 0.997302i \(-0.523388\pi\)
−0.0734096 + 0.997302i \(0.523388\pi\)
\(374\) 0 0
\(375\) 19.7071 1.01767
\(376\) 0 0
\(377\) 0.944833i 0.0486614i
\(378\) 0 0
\(379\) 10.8572i 0.557696i 0.960335 + 0.278848i \(0.0899526\pi\)
−0.960335 + 0.278848i \(0.910047\pi\)
\(380\) 0 0
\(381\) 22.6837i 1.16212i
\(382\) 0 0
\(383\) −16.3714 −0.836539 −0.418269 0.908323i \(-0.637363\pi\)
−0.418269 + 0.908323i \(0.637363\pi\)
\(384\) 0 0
\(385\) 58.5840i 2.98572i
\(386\) 0 0
\(387\) 8.62721 0.438546
\(388\) 0 0
\(389\) 4.97521 0.252253 0.126127 0.992014i \(-0.459745\pi\)
0.126127 + 0.992014i \(0.459745\pi\)
\(390\) 0 0
\(391\) 6.78063 15.1401i 0.342911 0.765669i
\(392\) 0 0
\(393\) −2.09878 −0.105870
\(394\) 0 0
\(395\) −39.4917 −1.98704
\(396\) 0 0
\(397\) 19.2233i 0.964791i 0.875954 + 0.482395i \(0.160233\pi\)
−0.875954 + 0.482395i \(0.839767\pi\)
\(398\) 0 0
\(399\) −4.76810 −0.238703
\(400\) 0 0
\(401\) 14.9706i 0.747595i 0.927510 + 0.373798i \(0.121945\pi\)
−0.927510 + 0.373798i \(0.878055\pi\)
\(402\) 0 0
\(403\) 7.97486i 0.397256i
\(404\) 0 0
\(405\) 20.6764i 1.02742i
\(406\) 0 0
\(407\) 6.22176 0.308401
\(408\) 0 0
\(409\) −18.3553 −0.907610 −0.453805 0.891101i \(-0.649934\pi\)
−0.453805 + 0.891101i \(0.649934\pi\)
\(410\) 0 0
\(411\) 20.8069i 1.02633i
\(412\) 0 0
\(413\) 3.33013i 0.163865i
\(414\) 0 0
\(415\) 27.4302i 1.34650i
\(416\) 0 0
\(417\) −23.2305 −1.13760
\(418\) 0 0
\(419\) 5.34290i 0.261018i 0.991447 + 0.130509i \(0.0416611\pi\)
−0.991447 + 0.130509i \(0.958339\pi\)
\(420\) 0 0
\(421\) −25.8543 −1.26006 −0.630031 0.776570i \(-0.716958\pi\)
−0.630031 + 0.776570i \(0.716958\pi\)
\(422\) 0 0
\(423\) 11.6584 0.566850
\(424\) 0 0
\(425\) 32.5890 + 14.5952i 1.58080 + 0.707973i
\(426\) 0 0
\(427\) −2.78721 −0.134883
\(428\) 0 0
\(429\) 17.4389 0.841958
\(430\) 0 0
\(431\) 36.7833i 1.77179i 0.463886 + 0.885895i \(0.346455\pi\)
−0.463886 + 0.885895i \(0.653545\pi\)
\(432\) 0 0
\(433\) 12.3232 0.592214 0.296107 0.955155i \(-0.404312\pi\)
0.296107 + 0.955155i \(0.404312\pi\)
\(434\) 0 0
\(435\) 2.02238i 0.0969655i
\(436\) 0 0
\(437\) 3.95485i 0.189186i
\(438\) 0 0
\(439\) 16.6823i 0.796204i 0.917341 + 0.398102i \(0.130331\pi\)
−0.917341 + 0.398102i \(0.869669\pi\)
\(440\) 0 0
\(441\) −3.59155 −0.171026
\(442\) 0 0
\(443\) −13.5919 −0.645773 −0.322886 0.946438i \(-0.604653\pi\)
−0.322886 + 0.946438i \(0.604653\pi\)
\(444\) 0 0
\(445\) 16.9402i 0.803043i
\(446\) 0 0
\(447\) 4.22730i 0.199945i
\(448\) 0 0
\(449\) 34.3123i 1.61930i −0.586916 0.809648i \(-0.699658\pi\)
0.586916 0.809648i \(-0.300342\pi\)
\(450\) 0 0
\(451\) −37.5387 −1.76763
\(452\) 0 0
\(453\) 8.14884i 0.382866i
\(454\) 0 0
\(455\) 30.9581 1.45134
\(456\) 0 0
\(457\) −23.5737 −1.10273 −0.551367 0.834263i \(-0.685893\pi\)
−0.551367 + 0.834263i \(0.685893\pi\)
\(458\) 0 0
\(459\) −9.52034 + 21.2575i −0.444371 + 0.992214i
\(460\) 0 0
\(461\) −2.50142 −0.116503 −0.0582514 0.998302i \(-0.518552\pi\)
−0.0582514 + 0.998302i \(0.518552\pi\)
\(462\) 0 0
\(463\) −14.2151 −0.660632 −0.330316 0.943870i \(-0.607155\pi\)
−0.330316 + 0.943870i \(0.607155\pi\)
\(464\) 0 0
\(465\) 17.0698i 0.791595i
\(466\) 0 0
\(467\) −21.2130 −0.981623 −0.490811 0.871266i \(-0.663300\pi\)
−0.490811 + 0.871266i \(0.663300\pi\)
\(468\) 0 0
\(469\) 4.34984i 0.200857i
\(470\) 0 0
\(471\) 32.9625i 1.51883i
\(472\) 0 0
\(473\) 46.7596i 2.15001i
\(474\) 0 0
\(475\) 8.51278 0.390593
\(476\) 0 0
\(477\) 1.66778 0.0763622
\(478\) 0 0
\(479\) 8.51651i 0.389129i 0.980890 + 0.194565i \(0.0623294\pi\)
−0.980890 + 0.194565i \(0.937671\pi\)
\(480\) 0 0
\(481\) 3.28782i 0.149912i
\(482\) 0 0
\(483\) 19.5171i 0.888058i
\(484\) 0 0
\(485\) 38.6499 1.75500
\(486\) 0 0
\(487\) 8.90145i 0.403363i 0.979451 + 0.201682i \(0.0646406\pi\)
−0.979451 + 0.201682i \(0.935359\pi\)
\(488\) 0 0
\(489\) 26.5466 1.20048
\(490\) 0 0
\(491\) 19.9226 0.899093 0.449547 0.893257i \(-0.351586\pi\)
0.449547 + 0.893257i \(0.351586\pi\)
\(492\) 0 0
\(493\) 0.633061 1.41353i 0.0285116 0.0636621i
\(494\) 0 0
\(495\) 15.4491 0.694385
\(496\) 0 0
\(497\) −36.1088 −1.61970
\(498\) 0 0
\(499\) 16.1979i 0.725119i −0.931961 0.362559i \(-0.881903\pi\)
0.931961 0.362559i \(-0.118097\pi\)
\(500\) 0 0
\(501\) 3.02197 0.135012
\(502\) 0 0
\(503\) 39.8469i 1.77668i 0.459183 + 0.888342i \(0.348142\pi\)
−0.459183 + 0.888342i \(0.651858\pi\)
\(504\) 0 0
\(505\) 26.1967i 1.16574i
\(506\) 0 0
\(507\) 9.72100i 0.431725i
\(508\) 0 0
\(509\) 8.91248 0.395039 0.197519 0.980299i \(-0.436711\pi\)
0.197519 + 0.980299i \(0.436711\pi\)
\(510\) 0 0
\(511\) 0.0180354 0.000797838
\(512\) 0 0
\(513\) 5.55281i 0.245162i
\(514\) 0 0
\(515\) 34.7519i 1.53135i
\(516\) 0 0
\(517\) 63.1885i 2.77903i
\(518\) 0 0
\(519\) 15.9955 0.702123
\(520\) 0 0
\(521\) 1.16643i 0.0511022i −0.999674 0.0255511i \(-0.991866\pi\)
0.999674 0.0255511i \(-0.00813406\pi\)
\(522\) 0 0
\(523\) −28.3461 −1.23949 −0.619744 0.784804i \(-0.712764\pi\)
−0.619744 + 0.784804i \(0.712764\pi\)
\(524\) 0 0
\(525\) −42.0103 −1.83348
\(526\) 0 0
\(527\) 5.34334 11.9309i 0.232760 0.519717i
\(528\) 0 0
\(529\) 6.81174 0.296162
\(530\) 0 0
\(531\) 0.878183 0.0381099
\(532\) 0 0
\(533\) 19.8369i 0.859231i
\(534\) 0 0
\(535\) 32.0885 1.38731
\(536\) 0 0
\(537\) 13.3806i 0.577417i
\(538\) 0 0
\(539\) 19.4663i 0.838471i
\(540\) 0 0
\(541\) 16.8243i 0.723332i 0.932308 + 0.361666i \(0.117792\pi\)
−0.932308 + 0.361666i \(0.882208\pi\)
\(542\) 0 0
\(543\) −22.4807 −0.964739
\(544\) 0 0
\(545\) −3.28269 −0.140615
\(546\) 0 0
\(547\) 27.6741i 1.18326i −0.806210 0.591630i \(-0.798485\pi\)
0.806210 0.591630i \(-0.201515\pi\)
\(548\) 0 0
\(549\) 0.735012i 0.0313695i
\(550\) 0 0
\(551\) 0.369237i 0.0157300i
\(552\) 0 0
\(553\) 35.5823 1.51311
\(554\) 0 0
\(555\) 7.03745i 0.298723i
\(556\) 0 0
\(557\) −10.2906 −0.436028 −0.218014 0.975946i \(-0.569958\pi\)
−0.218014 + 0.975946i \(0.569958\pi\)
\(558\) 0 0
\(559\) 24.7096 1.04510
\(560\) 0 0
\(561\) −26.0897 11.6845i −1.10151 0.493319i
\(562\) 0 0
\(563\) 40.7751 1.71846 0.859232 0.511585i \(-0.170942\pi\)
0.859232 + 0.511585i \(0.170942\pi\)
\(564\) 0 0
\(565\) −9.51884 −0.400461
\(566\) 0 0
\(567\) 18.6296i 0.782368i
\(568\) 0 0
\(569\) 8.65394 0.362792 0.181396 0.983410i \(-0.441938\pi\)
0.181396 + 0.983410i \(0.441938\pi\)
\(570\) 0 0
\(571\) 44.4118i 1.85858i −0.369355 0.929288i \(-0.620421\pi\)
0.369355 0.929288i \(-0.379579\pi\)
\(572\) 0 0
\(573\) 7.14204i 0.298363i
\(574\) 0 0
\(575\) 34.8451i 1.45314i
\(576\) 0 0
\(577\) 12.4836 0.519701 0.259850 0.965649i \(-0.416327\pi\)
0.259850 + 0.965649i \(0.416327\pi\)
\(578\) 0 0
\(579\) −13.3160 −0.553393
\(580\) 0 0
\(581\) 24.7148i 1.02534i
\(582\) 0 0
\(583\) 9.03936i 0.374372i
\(584\) 0 0
\(585\) 8.16390i 0.337536i
\(586\) 0 0
\(587\) −7.15235 −0.295209 −0.147605 0.989046i \(-0.547156\pi\)
−0.147605 + 0.989046i \(0.547156\pi\)
\(588\) 0 0
\(589\) 3.11654i 0.128415i
\(590\) 0 0
\(591\) 12.6512 0.520399
\(592\) 0 0
\(593\) 29.0737 1.19391 0.596957 0.802273i \(-0.296376\pi\)
0.596957 + 0.802273i \(0.296376\pi\)
\(594\) 0 0
\(595\) −46.3152 20.7426i −1.89874 0.850365i
\(596\) 0 0
\(597\) −26.4573 −1.08283
\(598\) 0 0
\(599\) 32.9059 1.34450 0.672250 0.740324i \(-0.265328\pi\)
0.672250 + 0.740324i \(0.265328\pi\)
\(600\) 0 0
\(601\) 34.5604i 1.40975i 0.709333 + 0.704873i \(0.248996\pi\)
−0.709333 + 0.704873i \(0.751004\pi\)
\(602\) 0 0
\(603\) 1.14709 0.0467131
\(604\) 0 0
\(605\) 43.0781i 1.75137i
\(606\) 0 0
\(607\) 32.5505i 1.32118i −0.750745 0.660592i \(-0.770305\pi\)
0.750745 0.660592i \(-0.229695\pi\)
\(608\) 0 0
\(609\) 1.82217i 0.0738382i
\(610\) 0 0
\(611\) 33.3913 1.35087
\(612\) 0 0
\(613\) 13.6816 0.552594 0.276297 0.961072i \(-0.410893\pi\)
0.276297 + 0.961072i \(0.410893\pi\)
\(614\) 0 0
\(615\) 42.4600i 1.71215i
\(616\) 0 0
\(617\) 48.7609i 1.96304i −0.191362 0.981520i \(-0.561290\pi\)
0.191362 0.981520i \(-0.438710\pi\)
\(618\) 0 0
\(619\) 9.51877i 0.382592i 0.981532 + 0.191296i \(0.0612690\pi\)
−0.981532 + 0.191296i \(0.938731\pi\)
\(620\) 0 0
\(621\) 22.7291 0.912088
\(622\) 0 0
\(623\) 15.2633i 0.611509i
\(624\) 0 0
\(625\) 6.70129 0.268052
\(626\) 0 0
\(627\) −6.81505 −0.272167
\(628\) 0 0
\(629\) −2.20292 + 4.91879i −0.0878362 + 0.196125i
\(630\) 0 0
\(631\) 21.5112 0.856348 0.428174 0.903696i \(-0.359157\pi\)
0.428174 + 0.903696i \(0.359157\pi\)
\(632\) 0 0
\(633\) −7.00819 −0.278550
\(634\) 0 0
\(635\) 57.5562i 2.28405i
\(636\) 0 0
\(637\) −10.2867 −0.407575
\(638\) 0 0
\(639\) 9.52219i 0.376692i
\(640\) 0 0
\(641\) 44.8593i 1.77184i −0.463843 0.885918i \(-0.653530\pi\)
0.463843 0.885918i \(-0.346470\pi\)
\(642\) 0 0
\(643\) 39.4001i 1.55379i 0.629632 + 0.776894i \(0.283206\pi\)
−0.629632 + 0.776894i \(0.716794\pi\)
\(644\) 0 0
\(645\) −52.8898 −2.08253
\(646\) 0 0
\(647\) −38.2810 −1.50498 −0.752491 0.658602i \(-0.771148\pi\)
−0.752491 + 0.658602i \(0.771148\pi\)
\(648\) 0 0
\(649\) 4.75976i 0.186837i
\(650\) 0 0
\(651\) 15.3801i 0.602792i
\(652\) 0 0
\(653\) 47.7182i 1.86736i 0.358111 + 0.933679i \(0.383421\pi\)
−0.358111 + 0.933679i \(0.616579\pi\)
\(654\) 0 0
\(655\) −5.32533 −0.208078
\(656\) 0 0
\(657\) 0.00475608i 0.000185553i
\(658\) 0 0
\(659\) 29.9368 1.16617 0.583087 0.812410i \(-0.301845\pi\)
0.583087 + 0.812410i \(0.301845\pi\)
\(660\) 0 0
\(661\) 7.71077 0.299914 0.149957 0.988693i \(-0.452086\pi\)
0.149957 + 0.988693i \(0.452086\pi\)
\(662\) 0 0
\(663\) −6.17453 + 13.7868i −0.239799 + 0.535435i
\(664\) 0 0
\(665\) −12.0983 −0.469151
\(666\) 0 0
\(667\) −1.51139 −0.0585211
\(668\) 0 0
\(669\) 40.9724i 1.58408i
\(670\) 0 0
\(671\) −3.98377 −0.153792
\(672\) 0 0
\(673\) 17.0441i 0.657001i 0.944504 + 0.328501i \(0.106543\pi\)
−0.944504 + 0.328501i \(0.893457\pi\)
\(674\) 0 0
\(675\) 48.9242i 1.88309i
\(676\) 0 0
\(677\) 34.9057i 1.34153i 0.741668 + 0.670767i \(0.234035\pi\)
−0.741668 + 0.670767i \(0.765965\pi\)
\(678\) 0 0
\(679\) −34.8238 −1.33642
\(680\) 0 0
\(681\) 18.8105 0.720819
\(682\) 0 0
\(683\) 11.4915i 0.439709i 0.975533 + 0.219855i \(0.0705583\pi\)
−0.975533 + 0.219855i \(0.929442\pi\)
\(684\) 0 0
\(685\) 52.7942i 2.01716i
\(686\) 0 0
\(687\) 27.4056i 1.04559i
\(688\) 0 0
\(689\) 4.77675 0.181980
\(690\) 0 0
\(691\) 24.6427i 0.937451i −0.883344 0.468725i \(-0.844713\pi\)
0.883344 0.468725i \(-0.155287\pi\)
\(692\) 0 0
\(693\) −13.9197 −0.528767
\(694\) 0 0
\(695\) −58.9437 −2.23586
\(696\) 0 0
\(697\) 13.2912 29.6772i 0.503440 1.12410i
\(698\) 0 0
\(699\) 11.2239 0.424528
\(700\) 0 0
\(701\) −21.0000 −0.793157 −0.396579 0.918001i \(-0.629803\pi\)
−0.396579 + 0.918001i \(0.629803\pi\)
\(702\) 0 0
\(703\) 1.28487i 0.0484597i
\(704\) 0 0
\(705\) −71.4726 −2.69182
\(706\) 0 0
\(707\) 23.6034i 0.887698i
\(708\) 0 0
\(709\) 11.8482i 0.444967i 0.974936 + 0.222484i \(0.0714164\pi\)
−0.974936 + 0.222484i \(0.928584\pi\)
\(710\) 0 0
\(711\) 9.38335i 0.351903i
\(712\) 0 0
\(713\) −12.7568 −0.477747
\(714\) 0 0
\(715\) 44.2484 1.65480
\(716\) 0 0
\(717\) 31.0727i 1.16043i
\(718\) 0 0
\(719\) 37.9487i 1.41525i −0.706589 0.707625i \(-0.749767\pi\)
0.706589 0.707625i \(-0.250233\pi\)
\(720\) 0 0
\(721\) 31.3117i 1.16611i
\(722\) 0 0
\(723\) −15.6738 −0.582916
\(724\) 0 0
\(725\) 3.25324i 0.120822i
\(726\) 0 0
\(727\) −0.354944 −0.0131641 −0.00658207 0.999978i \(-0.502095\pi\)
−0.00658207 + 0.999978i \(0.502095\pi\)
\(728\) 0 0
\(729\) −29.5992 −1.09626
\(730\) 0 0
\(731\) −36.9670 16.5560i −1.36728 0.612346i
\(732\) 0 0
\(733\) −29.6022 −1.09338 −0.546692 0.837334i \(-0.684113\pi\)
−0.546692 + 0.837334i \(0.684113\pi\)
\(734\) 0 0
\(735\) 22.0183 0.812158
\(736\) 0 0
\(737\) 6.21724i 0.229015i
\(738\) 0 0
\(739\) 1.73448 0.0638040 0.0319020 0.999491i \(-0.489844\pi\)
0.0319020 + 0.999491i \(0.489844\pi\)
\(740\) 0 0
\(741\) 3.60134i 0.132299i
\(742\) 0 0
\(743\) 47.4202i 1.73968i 0.493335 + 0.869839i \(0.335778\pi\)
−0.493335 + 0.869839i \(0.664222\pi\)
\(744\) 0 0
\(745\) 10.7261i 0.392974i
\(746\) 0 0
\(747\) 6.51751 0.238463
\(748\) 0 0
\(749\) −28.9119 −1.05642
\(750\) 0 0
\(751\) 37.8765i 1.38213i 0.722792 + 0.691066i \(0.242859\pi\)
−0.722792 + 0.691066i \(0.757141\pi\)
\(752\) 0 0
\(753\) 3.77478i 0.137561i
\(754\) 0 0
\(755\) 20.6764i 0.752491i
\(756\) 0 0
\(757\) −26.6011 −0.966834 −0.483417 0.875390i \(-0.660604\pi\)
−0.483417 + 0.875390i \(0.660604\pi\)
\(758\) 0 0
\(759\) 27.8958i 1.01255i
\(760\) 0 0
\(761\) 9.39591 0.340602 0.170301 0.985392i \(-0.445526\pi\)
0.170301 + 0.985392i \(0.445526\pi\)
\(762\) 0 0
\(763\) 2.95773 0.107077
\(764\) 0 0
\(765\) −5.47001 + 12.2137i −0.197769 + 0.441587i
\(766\) 0 0
\(767\) 2.51524 0.0908202
\(768\) 0 0
\(769\) 2.29722 0.0828400 0.0414200 0.999142i \(-0.486812\pi\)
0.0414200 + 0.999142i \(0.486812\pi\)
\(770\) 0 0
\(771\) 23.2059i 0.835742i
\(772\) 0 0
\(773\) −22.3295 −0.803135 −0.401568 0.915829i \(-0.631535\pi\)
−0.401568 + 0.915829i \(0.631535\pi\)
\(774\) 0 0
\(775\) 27.4590i 0.986355i
\(776\) 0 0
\(777\) 6.34079i 0.227475i
\(778\) 0 0
\(779\) 7.75218i 0.277751i
\(780\) 0 0
\(781\) −51.6103 −1.84676
\(782\) 0 0
\(783\) 2.12206 0.0758362
\(784\) 0 0
\(785\) 83.6371i 2.98514i
\(786\) 0 0
\(787\) 28.5559i 1.01791i 0.860794 + 0.508953i \(0.169967\pi\)
−0.860794 + 0.508953i \(0.830033\pi\)
\(788\) 0 0
\(789\) 26.6834i 0.949955i
\(790\) 0 0
\(791\) 8.57654 0.304947
\(792\) 0 0
\(793\) 2.10518i 0.0747571i
\(794\)