Properties

Label 4004.2.m.c.2157.13
Level 4004
Weight 2
Character 4004.2157
Analytic conductor 31.972
Analytic rank 0
Dimension 36
CM No

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Newspace parameters

Level: \( N \) = \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4004.m (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(36\)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2157.13
Character \(\chi\) = 4004.2157
Dual form 4004.2.m.c.2157.14

$q$-expansion

\(f(q)\) \(=\) \(q\)\(-1.07570 q^{3}\) \(-3.26058i q^{5}\) \(+1.00000i q^{7}\) \(-1.84286 q^{9}\) \(+O(q^{10})\) \(q\)\(-1.07570 q^{3}\) \(-3.26058i q^{5}\) \(+1.00000i q^{7}\) \(-1.84286 q^{9}\) \(-1.00000i q^{11}\) \(+(1.30427 - 3.36138i) q^{13}\) \(+3.50742i q^{15}\) \(+3.05222 q^{17}\) \(+3.15929i q^{19}\) \(-1.07570i q^{21}\) \(+5.49538 q^{23}\) \(-5.63139 q^{25}\) \(+5.20948 q^{27}\) \(+4.12550 q^{29}\) \(-6.96290i q^{31}\) \(+1.07570i q^{33}\) \(+3.26058 q^{35}\) \(+10.1344i q^{37}\) \(+(-1.40301 + 3.61585i) q^{39}\) \(+0.829912i q^{41}\) \(+10.0236 q^{43}\) \(+6.00880i q^{45}\) \(-9.75979i q^{47}\) \(-1.00000 q^{49}\) \(-3.28328 q^{51}\) \(+7.58699 q^{53}\) \(-3.26058 q^{55}\) \(-3.39846i q^{57}\) \(-15.0400i q^{59}\) \(-14.7376 q^{61}\) \(-1.84286i q^{63}\) \(+(-10.9601 - 4.25268i) q^{65}\) \(+9.94155i q^{67}\) \(-5.91140 q^{69}\) \(-14.2258i q^{71}\) \(+7.06972i q^{73}\) \(+6.05771 q^{75}\) \(+1.00000 q^{77}\) \(+11.7487 q^{79}\) \(-0.0752765 q^{81}\) \(-0.295776i q^{83}\) \(-9.95200i q^{85}\) \(-4.43781 q^{87}\) \(+8.17825i q^{89}\) \(+(3.36138 + 1.30427i) q^{91}\) \(+7.49002i q^{93}\) \(+10.3011 q^{95}\) \(-9.48378i q^{97}\) \(+1.84286i q^{99}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(36q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 40q^{9} \) \(\mathstrut -\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut -\mathstrut 80q^{25} \) \(\mathstrut +\mathstrut 8q^{27} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut -\mathstrut 24q^{39} \) \(\mathstrut +\mathstrut 32q^{43} \) \(\mathstrut -\mathstrut 36q^{49} \) \(\mathstrut -\mathstrut 20q^{51} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut +\mathstrut 32q^{61} \) \(\mathstrut -\mathstrut 24q^{65} \) \(\mathstrut +\mathstrut 80q^{69} \) \(\mathstrut -\mathstrut 36q^{75} \) \(\mathstrut +\mathstrut 36q^{77} \) \(\mathstrut +\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 132q^{81} \) \(\mathstrut +\mathstrut 8q^{91} \) \(\mathstrut +\mathstrut 56q^{95} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times\).

\(n\) \(365\) \(925\) \(2003\) \(3433\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) −1.07570 −0.621058 −0.310529 0.950564i \(-0.600506\pi\)
−0.310529 + 0.950564i \(0.600506\pi\)
\(4\) 0 0
\(5\) 3.26058i 1.45818i −0.684420 0.729088i \(-0.739944\pi\)
0.684420 0.729088i \(-0.260056\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −1.84286 −0.614287
\(10\) 0 0
\(11\) 1.00000i 0.301511i
\(12\) 0 0
\(13\) 1.30427 3.36138i 0.361739 0.932279i
\(14\) 0 0
\(15\) 3.50742i 0.905612i
\(16\) 0 0
\(17\) 3.05222 0.740271 0.370136 0.928978i \(-0.379311\pi\)
0.370136 + 0.928978i \(0.379311\pi\)
\(18\) 0 0
\(19\) 3.15929i 0.724791i 0.932024 + 0.362395i \(0.118041\pi\)
−0.932024 + 0.362395i \(0.881959\pi\)
\(20\) 0 0
\(21\) 1.07570i 0.234738i
\(22\) 0 0
\(23\) 5.49538 1.14587 0.572933 0.819602i \(-0.305806\pi\)
0.572933 + 0.819602i \(0.305806\pi\)
\(24\) 0 0
\(25\) −5.63139 −1.12628
\(26\) 0 0
\(27\) 5.20948 1.00257
\(28\) 0 0
\(29\) 4.12550 0.766085 0.383043 0.923731i \(-0.374876\pi\)
0.383043 + 0.923731i \(0.374876\pi\)
\(30\) 0 0
\(31\) 6.96290i 1.25057i −0.780395 0.625287i \(-0.784982\pi\)
0.780395 0.625287i \(-0.215018\pi\)
\(32\) 0 0
\(33\) 1.07570i 0.187256i
\(34\) 0 0
\(35\) 3.26058 0.551139
\(36\) 0 0
\(37\) 10.1344i 1.66608i 0.553211 + 0.833041i \(0.313402\pi\)
−0.553211 + 0.833041i \(0.686598\pi\)
\(38\) 0 0
\(39\) −1.40301 + 3.61585i −0.224661 + 0.578999i
\(40\) 0 0
\(41\) 0.829912i 0.129611i 0.997898 + 0.0648053i \(0.0206426\pi\)
−0.997898 + 0.0648053i \(0.979357\pi\)
\(42\) 0 0
\(43\) 10.0236 1.52858 0.764292 0.644870i \(-0.223089\pi\)
0.764292 + 0.644870i \(0.223089\pi\)
\(44\) 0 0
\(45\) 6.00880i 0.895739i
\(46\) 0 0
\(47\) 9.75979i 1.42361i −0.702377 0.711806i \(-0.747878\pi\)
0.702377 0.711806i \(-0.252122\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −3.28328 −0.459751
\(52\) 0 0
\(53\) 7.58699 1.04215 0.521077 0.853510i \(-0.325530\pi\)
0.521077 + 0.853510i \(0.325530\pi\)
\(54\) 0 0
\(55\) −3.26058 −0.439657
\(56\) 0 0
\(57\) 3.39846i 0.450137i
\(58\) 0 0
\(59\) 15.0400i 1.95804i −0.203773 0.979018i \(-0.565320\pi\)
0.203773 0.979018i \(-0.434680\pi\)
\(60\) 0 0
\(61\) −14.7376 −1.88696 −0.943478 0.331434i \(-0.892468\pi\)
−0.943478 + 0.331434i \(0.892468\pi\)
\(62\) 0 0
\(63\) 1.84286i 0.232179i
\(64\) 0 0
\(65\) −10.9601 4.25268i −1.35943 0.527479i
\(66\) 0 0
\(67\) 9.94155i 1.21455i 0.794490 + 0.607277i \(0.207738\pi\)
−0.794490 + 0.607277i \(0.792262\pi\)
\(68\) 0 0
\(69\) −5.91140 −0.711649
\(70\) 0 0
\(71\) 14.2258i 1.68829i −0.536111 0.844147i \(-0.680107\pi\)
0.536111 0.844147i \(-0.319893\pi\)
\(72\) 0 0
\(73\) 7.06972i 0.827448i 0.910402 + 0.413724i \(0.135772\pi\)
−0.910402 + 0.413724i \(0.864228\pi\)
\(74\) 0 0
\(75\) 6.05771 0.699484
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) 11.7487 1.32183 0.660916 0.750459i \(-0.270168\pi\)
0.660916 + 0.750459i \(0.270168\pi\)
\(80\) 0 0
\(81\) −0.0752765 −0.00836405
\(82\) 0 0
\(83\) 0.295776i 0.0324656i −0.999868 0.0162328i \(-0.994833\pi\)
0.999868 0.0162328i \(-0.00516728\pi\)
\(84\) 0 0
\(85\) 9.95200i 1.07945i
\(86\) 0 0
\(87\) −4.43781 −0.475783
\(88\) 0 0
\(89\) 8.17825i 0.866893i 0.901179 + 0.433446i \(0.142703\pi\)
−0.901179 + 0.433446i \(0.857297\pi\)
\(90\) 0 0
\(91\) 3.36138 + 1.30427i 0.352368 + 0.136725i
\(92\) 0 0
\(93\) 7.49002i 0.776679i
\(94\) 0 0
\(95\) 10.3011 1.05687
\(96\) 0 0
\(97\) 9.48378i 0.962932i −0.876465 0.481466i \(-0.840104\pi\)
0.876465 0.481466i \(-0.159896\pi\)
\(98\) 0 0
\(99\) 1.84286i 0.185215i
\(100\) 0 0
\(101\) −4.77226 −0.474858 −0.237429 0.971405i \(-0.576305\pi\)
−0.237429 + 0.971405i \(0.576305\pi\)
\(102\) 0 0
\(103\) 5.53648 0.545526 0.272763 0.962081i \(-0.412063\pi\)
0.272763 + 0.962081i \(0.412063\pi\)
\(104\) 0 0
\(105\) −3.50742 −0.342289
\(106\) 0 0
\(107\) −10.8337 −1.04733 −0.523665 0.851924i \(-0.675436\pi\)
−0.523665 + 0.851924i \(0.675436\pi\)
\(108\) 0 0
\(109\) 0.525322i 0.0503167i 0.999683 + 0.0251583i \(0.00800900\pi\)
−0.999683 + 0.0251583i \(0.991991\pi\)
\(110\) 0 0
\(111\) 10.9016i 1.03473i
\(112\) 0 0
\(113\) 0.688371 0.0647565 0.0323782 0.999476i \(-0.489692\pi\)
0.0323782 + 0.999476i \(0.489692\pi\)
\(114\) 0 0
\(115\) 17.9181i 1.67087i
\(116\) 0 0
\(117\) −2.40359 + 6.19456i −0.222212 + 0.572687i
\(118\) 0 0
\(119\) 3.05222i 0.279796i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 0.892740i 0.0804956i
\(124\) 0 0
\(125\) 2.05871i 0.184136i
\(126\) 0 0
\(127\) −11.4996 −1.02042 −0.510211 0.860050i \(-0.670433\pi\)
−0.510211 + 0.860050i \(0.670433\pi\)
\(128\) 0 0
\(129\) −10.7824 −0.949339
\(130\) 0 0
\(131\) −10.4533 −0.913312 −0.456656 0.889643i \(-0.650953\pi\)
−0.456656 + 0.889643i \(0.650953\pi\)
\(132\) 0 0
\(133\) −3.15929 −0.273945
\(134\) 0 0
\(135\) 16.9859i 1.46192i
\(136\) 0 0
\(137\) 2.34884i 0.200675i 0.994953 + 0.100338i \(0.0319923\pi\)
−0.994953 + 0.100338i \(0.968008\pi\)
\(138\) 0 0
\(139\) 1.95608 0.165912 0.0829561 0.996553i \(-0.473564\pi\)
0.0829561 + 0.996553i \(0.473564\pi\)
\(140\) 0 0
\(141\) 10.4986i 0.884145i
\(142\) 0 0
\(143\) −3.36138 1.30427i −0.281093 0.109068i
\(144\) 0 0
\(145\) 13.4515i 1.11709i
\(146\) 0 0
\(147\) 1.07570 0.0887225
\(148\) 0 0
\(149\) 15.3388i 1.25661i −0.777969 0.628303i \(-0.783750\pi\)
0.777969 0.628303i \(-0.216250\pi\)
\(150\) 0 0
\(151\) 0.941516i 0.0766195i −0.999266 0.0383097i \(-0.987803\pi\)
0.999266 0.0383097i \(-0.0121974\pi\)
\(152\) 0 0
\(153\) −5.62481 −0.454739
\(154\) 0 0
\(155\) −22.7031 −1.82356
\(156\) 0 0
\(157\) −7.14676 −0.570373 −0.285187 0.958472i \(-0.592056\pi\)
−0.285187 + 0.958472i \(0.592056\pi\)
\(158\) 0 0
\(159\) −8.16136 −0.647238
\(160\) 0 0
\(161\) 5.49538i 0.433097i
\(162\) 0 0
\(163\) 7.15590i 0.560494i −0.959928 0.280247i \(-0.909584\pi\)
0.959928 0.280247i \(-0.0904163\pi\)
\(164\) 0 0
\(165\) 3.50742 0.273052
\(166\) 0 0
\(167\) 14.7620i 1.14232i 0.820838 + 0.571160i \(0.193507\pi\)
−0.820838 + 0.571160i \(0.806493\pi\)
\(168\) 0 0
\(169\) −9.59776 8.76829i −0.738290 0.674484i
\(170\) 0 0
\(171\) 5.82213i 0.445230i
\(172\) 0 0
\(173\) −11.0494 −0.840073 −0.420037 0.907507i \(-0.637983\pi\)
−0.420037 + 0.907507i \(0.637983\pi\)
\(174\) 0 0
\(175\) 5.63139i 0.425693i
\(176\) 0 0
\(177\) 16.1785i 1.21605i
\(178\) 0 0
\(179\) −9.24627 −0.691099 −0.345549 0.938401i \(-0.612307\pi\)
−0.345549 + 0.938401i \(0.612307\pi\)
\(180\) 0 0
\(181\) 17.8299 1.32528 0.662642 0.748937i \(-0.269435\pi\)
0.662642 + 0.748937i \(0.269435\pi\)
\(182\) 0 0
\(183\) 15.8533 1.17191
\(184\) 0 0
\(185\) 33.0440 2.42944
\(186\) 0 0
\(187\) 3.05222i 0.223200i
\(188\) 0 0
\(189\) 5.20948i 0.378934i
\(190\) 0 0
\(191\) 2.87147 0.207772 0.103886 0.994589i \(-0.466872\pi\)
0.103886 + 0.994589i \(0.466872\pi\)
\(192\) 0 0
\(193\) 14.4001i 1.03654i 0.855216 + 0.518272i \(0.173425\pi\)
−0.855216 + 0.518272i \(0.826575\pi\)
\(194\) 0 0
\(195\) 11.7898 + 4.57462i 0.844283 + 0.327595i
\(196\) 0 0
\(197\) 7.69489i 0.548238i −0.961696 0.274119i \(-0.911614\pi\)
0.961696 0.274119i \(-0.0883862\pi\)
\(198\) 0 0
\(199\) −6.11257 −0.433309 −0.216654 0.976248i \(-0.569514\pi\)
−0.216654 + 0.976248i \(0.569514\pi\)
\(200\) 0 0
\(201\) 10.6942i 0.754308i
\(202\) 0 0
\(203\) 4.12550i 0.289553i
\(204\) 0 0
\(205\) 2.70600 0.188995
\(206\) 0 0
\(207\) −10.1272 −0.703891
\(208\) 0 0
\(209\) 3.15929 0.218533
\(210\) 0 0
\(211\) −16.4170 −1.13019 −0.565095 0.825026i \(-0.691161\pi\)
−0.565095 + 0.825026i \(0.691161\pi\)
\(212\) 0 0
\(213\) 15.3028i 1.04853i
\(214\) 0 0
\(215\) 32.6827i 2.22894i
\(216\) 0 0
\(217\) 6.96290 0.472673
\(218\) 0 0
\(219\) 7.60493i 0.513893i
\(220\) 0 0
\(221\) 3.98091 10.2597i 0.267785 0.690140i
\(222\) 0 0
\(223\) 25.7760i 1.72609i −0.505126 0.863046i \(-0.668554\pi\)
0.505126 0.863046i \(-0.331446\pi\)
\(224\) 0 0
\(225\) 10.3779 0.691858
\(226\) 0 0
\(227\) 19.5995i 1.30087i 0.759563 + 0.650433i \(0.225413\pi\)
−0.759563 + 0.650433i \(0.774587\pi\)
\(228\) 0 0
\(229\) 14.9163i 0.985699i −0.870114 0.492850i \(-0.835955\pi\)
0.870114 0.492850i \(-0.164045\pi\)
\(230\) 0 0
\(231\) −1.07570 −0.0707761
\(232\) 0 0
\(233\) 10.8515 0.710903 0.355452 0.934695i \(-0.384327\pi\)
0.355452 + 0.934695i \(0.384327\pi\)
\(234\) 0 0
\(235\) −31.8226 −2.07588
\(236\) 0 0
\(237\) −12.6381 −0.820935
\(238\) 0 0
\(239\) 11.4325i 0.739505i −0.929130 0.369752i \(-0.879443\pi\)
0.929130 0.369752i \(-0.120557\pi\)
\(240\) 0 0
\(241\) 5.20504i 0.335286i 0.985848 + 0.167643i \(0.0536156\pi\)
−0.985848 + 0.167643i \(0.946384\pi\)
\(242\) 0 0
\(243\) −15.5475 −0.997371
\(244\) 0 0
\(245\) 3.26058i 0.208311i
\(246\) 0 0
\(247\) 10.6196 + 4.12056i 0.675707 + 0.262185i
\(248\) 0 0
\(249\) 0.318167i 0.0201630i
\(250\) 0 0
\(251\) 16.8773 1.06529 0.532643 0.846340i \(-0.321199\pi\)
0.532643 + 0.846340i \(0.321199\pi\)
\(252\) 0 0
\(253\) 5.49538i 0.345491i
\(254\) 0 0
\(255\) 10.7054i 0.670398i
\(256\) 0 0
\(257\) 15.3254 0.955970 0.477985 0.878368i \(-0.341367\pi\)
0.477985 + 0.878368i \(0.341367\pi\)
\(258\) 0 0
\(259\) −10.1344 −0.629720
\(260\) 0 0
\(261\) −7.60272 −0.470596
\(262\) 0 0
\(263\) 14.3305 0.883658 0.441829 0.897099i \(-0.354330\pi\)
0.441829 + 0.897099i \(0.354330\pi\)
\(264\) 0 0
\(265\) 24.7380i 1.51964i
\(266\) 0 0
\(267\) 8.79737i 0.538390i
\(268\) 0 0
\(269\) −17.3313 −1.05671 −0.528353 0.849025i \(-0.677190\pi\)
−0.528353 + 0.849025i \(0.677190\pi\)
\(270\) 0 0
\(271\) 0.614183i 0.0373090i 0.999826 + 0.0186545i \(0.00593825\pi\)
−0.999826 + 0.0186545i \(0.994062\pi\)
\(272\) 0 0
\(273\) −3.61585 1.40301i −0.218841 0.0849138i
\(274\) 0 0
\(275\) 5.63139i 0.339586i
\(276\) 0 0
\(277\) −20.1543 −1.21095 −0.605477 0.795863i \(-0.707018\pi\)
−0.605477 + 0.795863i \(0.707018\pi\)
\(278\) 0 0
\(279\) 12.8317i 0.768212i
\(280\) 0 0
\(281\) 4.01985i 0.239804i 0.992786 + 0.119902i \(0.0382580\pi\)
−0.992786 + 0.119902i \(0.961742\pi\)
\(282\) 0 0
\(283\) 11.0278 0.655537 0.327768 0.944758i \(-0.393703\pi\)
0.327768 + 0.944758i \(0.393703\pi\)
\(284\) 0 0
\(285\) −11.0810 −0.656379
\(286\) 0 0
\(287\) −0.829912 −0.0489882
\(288\) 0 0
\(289\) −7.68397 −0.451998
\(290\) 0 0
\(291\) 10.2017i 0.598037i
\(292\) 0 0
\(293\) 32.5830i 1.90352i −0.306849 0.951758i \(-0.599275\pi\)
0.306849 0.951758i \(-0.400725\pi\)
\(294\) 0 0
\(295\) −49.0390 −2.85516
\(296\) 0 0
\(297\) 5.20948i 0.302285i
\(298\) 0 0
\(299\) 7.16745 18.4721i 0.414504 1.06827i
\(300\) 0 0
\(301\) 10.0236i 0.577750i
\(302\) 0 0
\(303\) 5.13354 0.294914
\(304\) 0 0
\(305\) 48.0532i 2.75152i
\(306\) 0 0
\(307\) 7.18948i 0.410325i −0.978728 0.205163i \(-0.934228\pi\)
0.978728 0.205163i \(-0.0657723\pi\)
\(308\) 0 0
\(309\) −5.95561 −0.338803
\(310\) 0 0
\(311\) 17.7964 1.00914 0.504572 0.863370i \(-0.331650\pi\)
0.504572 + 0.863370i \(0.331650\pi\)
\(312\) 0 0
\(313\) −19.2818 −1.08987 −0.544937 0.838477i \(-0.683446\pi\)
−0.544937 + 0.838477i \(0.683446\pi\)
\(314\) 0 0
\(315\) −6.00880 −0.338558
\(316\) 0 0
\(317\) 2.78137i 0.156217i 0.996945 + 0.0781086i \(0.0248881\pi\)
−0.996945 + 0.0781086i \(0.975112\pi\)
\(318\) 0 0
\(319\) 4.12550i 0.230983i
\(320\) 0 0
\(321\) 11.6538 0.650453
\(322\) 0 0
\(323\) 9.64283i 0.536542i
\(324\) 0 0
\(325\) −7.34485 + 18.9293i −0.407419 + 1.05001i
\(326\) 0 0
\(327\) 0.565090i 0.0312496i
\(328\) 0 0
\(329\) 9.75979 0.538074
\(330\) 0 0
\(331\) 1.67645i 0.0921461i 0.998938 + 0.0460731i \(0.0146707\pi\)
−0.998938 + 0.0460731i \(0.985329\pi\)
\(332\) 0 0
\(333\) 18.6763i 1.02345i
\(334\) 0 0
\(335\) 32.4152 1.77103
\(336\) 0 0
\(337\) −1.32141 −0.0719818 −0.0359909 0.999352i \(-0.511459\pi\)
−0.0359909 + 0.999352i \(0.511459\pi\)
\(338\) 0 0
\(339\) −0.740483 −0.0402175
\(340\) 0 0
\(341\) −6.96290 −0.377062
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 19.2746i 1.03771i
\(346\) 0 0
\(347\) −18.1529 −0.974500 −0.487250 0.873262i \(-0.662000\pi\)
−0.487250 + 0.873262i \(0.662000\pi\)
\(348\) 0 0
\(349\) 4.08323i 0.218570i −0.994010 0.109285i \(-0.965144\pi\)
0.994010 0.109285i \(-0.0348561\pi\)
\(350\) 0 0
\(351\) 6.79457 17.5111i 0.362667 0.934671i
\(352\) 0 0
\(353\) 28.2106i 1.50150i −0.660587 0.750750i \(-0.729693\pi\)
0.660587 0.750750i \(-0.270307\pi\)
\(354\) 0 0
\(355\) −46.3845 −2.46183
\(356\) 0 0
\(357\) 3.28328i 0.173770i
\(358\) 0 0
\(359\) 10.7635i 0.568077i 0.958813 + 0.284039i \(0.0916744\pi\)
−0.958813 + 0.284039i \(0.908326\pi\)
\(360\) 0 0
\(361\) 9.01889 0.474679
\(362\) 0 0
\(363\) 1.07570 0.0564598
\(364\) 0 0
\(365\) 23.0514 1.20657
\(366\) 0 0
\(367\) 27.9220 1.45752 0.728758 0.684771i \(-0.240098\pi\)
0.728758 + 0.684771i \(0.240098\pi\)
\(368\) 0 0
\(369\) 1.52941i 0.0796181i
\(370\) 0 0
\(371\) 7.58699i 0.393897i
\(372\) 0 0
\(373\) 8.95262 0.463549 0.231775 0.972770i \(-0.425547\pi\)
0.231775 + 0.972770i \(0.425547\pi\)
\(374\) 0 0
\(375\) 2.21456i 0.114359i
\(376\) 0 0
\(377\) 5.38076 13.8674i 0.277123 0.714206i
\(378\) 0 0
\(379\) 31.9416i 1.64073i −0.571841 0.820364i \(-0.693771\pi\)
0.571841 0.820364i \(-0.306229\pi\)
\(380\) 0 0
\(381\) 12.3701 0.633740
\(382\) 0 0
\(383\) 22.0246i 1.12540i −0.826660 0.562701i \(-0.809762\pi\)
0.826660 0.562701i \(-0.190238\pi\)
\(384\) 0 0
\(385\) 3.26058i 0.166175i
\(386\) 0 0
\(387\) −18.4721 −0.938990
\(388\) 0 0
\(389\) −4.17862 −0.211865 −0.105932 0.994373i \(-0.533783\pi\)
−0.105932 + 0.994373i \(0.533783\pi\)
\(390\) 0 0
\(391\) 16.7731 0.848251
\(392\) 0 0
\(393\) 11.2447 0.567220
\(394\) 0 0
\(395\) 38.3076i 1.92747i
\(396\) 0 0
\(397\) 12.0538i 0.604962i −0.953155 0.302481i \(-0.902185\pi\)
0.953155 0.302481i \(-0.0978149\pi\)
\(398\) 0 0
\(399\) 3.39846 0.170136
\(400\) 0 0
\(401\) 21.3073i 1.06403i 0.846734 + 0.532017i \(0.178566\pi\)
−0.846734 + 0.532017i \(0.821434\pi\)
\(402\) 0 0
\(403\) −23.4050 9.08150i −1.16588 0.452382i
\(404\) 0 0
\(405\) 0.245445i 0.0121963i
\(406\) 0 0
\(407\) 10.1344 0.502343
\(408\) 0 0
\(409\) 18.7180i 0.925544i −0.886477 0.462772i \(-0.846855\pi\)
0.886477 0.462772i \(-0.153145\pi\)
\(410\) 0 0
\(411\) 2.52666i 0.124631i
\(412\) 0 0
\(413\) 15.0400 0.740068
\(414\) 0 0
\(415\) −0.964400 −0.0473405
\(416\) 0 0
\(417\) −2.10416 −0.103041
\(418\) 0 0
\(419\) 29.6971 1.45080 0.725399 0.688329i \(-0.241655\pi\)
0.725399 + 0.688329i \(0.241655\pi\)
\(420\) 0 0
\(421\) 11.2123i 0.546456i −0.961949 0.273228i \(-0.911909\pi\)
0.961949 0.273228i \(-0.0880914\pi\)
\(422\) 0 0
\(423\) 17.9859i 0.874506i
\(424\) 0 0
\(425\) −17.1882 −0.833752
\(426\) 0 0
\(427\) 14.7376i 0.713203i
\(428\) 0 0
\(429\) 3.61585 + 1.40301i 0.174575 + 0.0677378i
\(430\) 0 0
\(431\) 0.176475i 0.00850050i −0.999991 0.00425025i \(-0.998647\pi\)
0.999991 0.00425025i \(-0.00135290\pi\)
\(432\) 0 0
\(433\) −10.1279 −0.486714 −0.243357 0.969937i \(-0.578249\pi\)
−0.243357 + 0.969937i \(0.578249\pi\)
\(434\) 0 0
\(435\) 14.4698i 0.693776i
\(436\) 0 0
\(437\) 17.3615i 0.830513i
\(438\) 0 0
\(439\) 12.1287 0.578869 0.289435 0.957198i \(-0.406533\pi\)
0.289435 + 0.957198i \(0.406533\pi\)
\(440\) 0 0
\(441\) 1.84286 0.0877553
\(442\) 0 0
\(443\) 0.685413 0.0325650 0.0162825 0.999867i \(-0.494817\pi\)
0.0162825 + 0.999867i \(0.494817\pi\)
\(444\) 0 0
\(445\) 26.6658 1.26408
\(446\) 0 0
\(447\) 16.5000i 0.780425i
\(448\) 0 0
\(449\) 35.4568i 1.67331i 0.547732 + 0.836654i \(0.315491\pi\)
−0.547732 + 0.836654i \(0.684509\pi\)
\(450\) 0 0
\(451\) 0.829912 0.0390790
\(452\) 0 0
\(453\) 1.01279i 0.0475851i
\(454\) 0 0
\(455\) 4.25268 10.9601i 0.199369 0.513815i
\(456\) 0 0
\(457\) 25.7743i 1.20567i 0.797865 + 0.602836i \(0.205963\pi\)
−0.797865 + 0.602836i \(0.794037\pi\)
\(458\) 0 0
\(459\) 15.9005 0.742171
\(460\) 0 0
\(461\) 2.29626i 0.106947i 0.998569 + 0.0534737i \(0.0170293\pi\)
−0.998569 + 0.0534737i \(0.982971\pi\)
\(462\) 0 0
\(463\) 10.2474i 0.476239i −0.971236 0.238119i \(-0.923469\pi\)
0.971236 0.238119i \(-0.0765310\pi\)
\(464\) 0 0
\(465\) 24.4218 1.13253
\(466\) 0 0
\(467\) −35.7823 −1.65581 −0.827904 0.560870i \(-0.810467\pi\)
−0.827904 + 0.560870i \(0.810467\pi\)
\(468\) 0 0
\(469\) −9.94155 −0.459058
\(470\) 0 0
\(471\) 7.68779 0.354235
\(472\) 0 0
\(473\) 10.0236i 0.460885i
\(474\) 0 0
\(475\) 17.7912i 0.816316i
\(476\) 0 0
\(477\) −13.9818 −0.640182
\(478\) 0 0
\(479\) 11.4723i 0.524184i −0.965043 0.262092i \(-0.915588\pi\)
0.965043 0.262092i \(-0.0844123\pi\)
\(480\) 0 0
\(481\) 34.0655 + 13.2180i 1.55325 + 0.602687i
\(482\) 0 0
\(483\) 5.91140i 0.268978i
\(484\) 0 0
\(485\) −30.9226 −1.40412
\(486\) 0 0
\(487\) 30.6514i 1.38895i −0.719519 0.694473i \(-0.755638\pi\)
0.719519 0.694473i \(-0.244362\pi\)
\(488\) 0 0
\(489\) 7.69763i 0.348099i
\(490\) 0 0
\(491\) 35.3578 1.59567 0.797837 0.602873i \(-0.205978\pi\)
0.797837 + 0.602873i \(0.205978\pi\)
\(492\) 0 0
\(493\) 12.5919 0.567111
\(494\) 0 0
\(495\) 6.00880 0.270075
\(496\) 0 0
\(497\) 14.2258 0.638115
\(498\) 0 0
\(499\) 8.19506i 0.366861i −0.983033 0.183431i \(-0.941280\pi\)
0.983033 0.183431i \(-0.0587202\pi\)
\(500\) 0 0
\(501\) 15.8796i 0.709447i
\(502\) 0 0
\(503\) 12.2611 0.546698 0.273349 0.961915i \(-0.411869\pi\)
0.273349 + 0.961915i \(0.411869\pi\)
\(504\) 0 0
\(505\) 15.5603i 0.692426i
\(506\) 0 0
\(507\) 10.3244 + 9.43208i 0.458521 + 0.418893i
\(508\) 0 0
\(509\) 18.8793i 0.836810i 0.908261 + 0.418405i \(0.137411\pi\)
−0.908261 + 0.418405i \(0.862589\pi\)
\(510\) 0 0
\(511\) −7.06972 −0.312746
\(512\) 0 0
\(513\) 16.4583i 0.726650i
\(514\) 0 0
\(515\) 18.0521i 0.795473i
\(516\) 0 0
\(517\) −9.75979 −0.429235
\(518\) 0 0
\(519\) 11.8859 0.521734
\(520\) 0 0
\(521\) 15.1763 0.664885 0.332443 0.943123i \(-0.392127\pi\)
0.332443 + 0.943123i \(0.392127\pi\)
\(522\) 0 0
\(523\) −26.6123 −1.16368 −0.581838 0.813305i \(-0.697666\pi\)
−0.581838 + 0.813305i \(0.697666\pi\)
\(524\) 0 0
\(525\) 6.05771i 0.264380i
\(526\) 0 0
\(527\) 21.2523i 0.925764i
\(528\) 0 0
\(529\) 7.19919 0.313008
\(530\) 0 0
\(531\) 27.7166i 1.20280i
\(532\) 0 0
\(533\) 2.78965 + 1.08243i 0.120833 + 0.0468852i
\(534\) 0 0
\(535\) 35.3241i 1.52719i
\(536\) 0 0
\(537\) 9.94625 0.429212
\(538\) 0 0
\(539\) 1.00000i 0.0430730i
\(540\) 0 0
\(541\) 23.1326i 0.994546i 0.867594 + 0.497273i \(0.165665\pi\)
−0.867594 + 0.497273i \(0.834335\pi\)
\(542\) 0 0
\(543\) −19.1796 −0.823078
\(544\) 0 0
\(545\) 1.71285 0.0733706
\(546\) 0 0
\(547\) 6.88107 0.294213 0.147107 0.989121i \(-0.453004\pi\)
0.147107 + 0.989121i \(0.453004\pi\)
\(548\) 0 0
\(549\) 27.1594 1.15913
\(550\) 0 0
\(551\) 13.0336i 0.555251i
\(552\) 0 0
\(553\) 11.7487i 0.499606i
\(554\) 0 0
\(555\) −35.5455 −1.50882
\(556\) 0 0
\(557\) 29.3101i 1.24191i 0.783846 + 0.620955i \(0.213255\pi\)
−0.783846 + 0.620955i \(0.786745\pi\)
\(558\) 0 0
\(559\) 13.0735 33.6931i 0.552949 1.42507i
\(560\) 0 0
\(561\) 3.28328i 0.138620i
\(562\) 0 0
\(563\) 32.0739 1.35176 0.675878 0.737014i \(-0.263765\pi\)
0.675878 + 0.737014i \(0.263765\pi\)
\(564\) 0 0
\(565\) 2.24449i 0.0944263i
\(566\) 0 0
\(567\) 0.0752765i 0.00316131i
\(568\) 0 0
\(569\) −34.1986 −1.43368 −0.716841 0.697237i \(-0.754413\pi\)
−0.716841 + 0.697237i \(0.754413\pi\)
\(570\) 0 0
\(571\) −29.6809 −1.24211 −0.621054 0.783767i \(-0.713295\pi\)
−0.621054 + 0.783767i \(0.713295\pi\)
\(572\) 0 0
\(573\) −3.08885 −0.129039
\(574\) 0 0
\(575\) −30.9466 −1.29056
\(576\) 0 0
\(577\) 14.4269i 0.600601i −0.953845 0.300300i \(-0.902913\pi\)
0.953845 0.300300i \(-0.0970870\pi\)
\(578\) 0 0
\(579\) 15.4903i 0.643754i
\(580\) 0 0
\(581\) 0.295776 0.0122708
\(582\) 0 0
\(583\) 7.58699i 0.314221i
\(584\) 0 0
\(585\) 20.1979 + 7.83709i 0.835079 + 0.324024i
\(586\) 0 0
\(587\) 29.2819i 1.20859i 0.796759 + 0.604297i \(0.206546\pi\)
−0.796759 + 0.604297i \(0.793454\pi\)
\(588\) 0 0
\(589\) 21.9978 0.906404
\(590\) 0 0
\(591\) 8.27742i 0.340488i
\(592\) 0 0
\(593\) 45.9708i 1.88780i 0.330239 + 0.943898i \(0.392871\pi\)
−0.330239 + 0.943898i \(0.607129\pi\)
\(594\) 0 0
\(595\) 9.95200 0.407992
\(596\) 0 0
\(597\) 6.57531 0.269110
\(598\) 0 0
\(599\) −4.54448 −0.185682 −0.0928411 0.995681i \(-0.529595\pi\)
−0.0928411 + 0.995681i \(0.529595\pi\)
\(600\) 0 0
\(601\) −2.93195 −0.119597 −0.0597984 0.998210i \(-0.519046\pi\)
−0.0597984 + 0.998210i \(0.519046\pi\)
\(602\) 0 0
\(603\) 18.3209i 0.746085i
\(604\) 0 0
\(605\) 3.26058i 0.132561i
\(606\) 0 0
\(607\) 43.9586 1.78422 0.892112 0.451815i \(-0.149223\pi\)
0.892112 + 0.451815i \(0.149223\pi\)
\(608\) 0 0
\(609\) 4.43781i 0.179829i
\(610\) 0 0
\(611\) −32.8064 12.7294i −1.32720 0.514976i
\(612\) 0 0
\(613\) 21.5218i 0.869257i −0.900610 0.434628i \(-0.856880\pi\)
0.900610 0.434628i \(-0.143120\pi\)
\(614\) 0 0
\(615\) −2.91085 −0.117377
\(616\) 0 0
\(617\) 25.3070i 1.01882i −0.860523 0.509412i \(-0.829863\pi\)
0.860523 0.509412i \(-0.170137\pi\)
\(618\) 0 0
\(619\) 20.8075i 0.836324i 0.908373 + 0.418162i \(0.137326\pi\)
−0.908373 + 0.418162i \(0.862674\pi\)
\(620\) 0 0
\(621\) 28.6281 1.14881
\(622\) 0 0
\(623\) −8.17825 −0.327655
\(624\) 0 0
\(625\) −21.4444 −0.857775
\(626\) 0 0
\(627\) −3.39846 −0.135721
\(628\) 0 0
\(629\) 30.9323i 1.23335i
\(630\) 0 0
\(631\) 19.3787i 0.771455i 0.922613 + 0.385727i \(0.126049\pi\)
−0.922613 + 0.385727i \(0.873951\pi\)
\(632\) 0 0
\(633\) 17.6598 0.701914
\(634\) 0 0
\(635\) 37.4953i 1.48795i
\(636\) 0 0
\(637\) −1.30427 + 3.36138i −0.0516770 + 0.133183i
\(638\) 0 0
\(639\) 26.2162i 1.03710i
\(640\) 0 0
\(641\) 19.7551 0.780279 0.390140 0.920756i \(-0.372427\pi\)
0.390140 + 0.920756i \(0.372427\pi\)
\(642\) 0 0
\(643\) 45.6637i 1.80080i −0.435063 0.900400i \(-0.643274\pi\)
0.435063 0.900400i \(-0.356726\pi\)
\(644\) 0 0
\(645\) 35.1570i 1.38430i
\(646\) 0 0
\(647\) 13.6121 0.535145 0.267573 0.963538i \(-0.413778\pi\)
0.267573 + 0.963538i \(0.413778\pi\)
\(648\) 0 0
\(649\) −15.0400 −0.590370
\(650\) 0 0
\(651\) −7.49002 −0.293557
\(652\) 0 0
\(653\) 6.97739 0.273046 0.136523 0.990637i \(-0.456407\pi\)
0.136523 + 0.990637i \(0.456407\pi\)
\(654\) 0 0
\(655\) 34.0840i 1.33177i
\(656\) 0 0
\(657\) 13.0285i 0.508291i
\(658\) 0 0
\(659\) 0.333655 0.0129974 0.00649869 0.999979i \(-0.497931\pi\)
0.00649869 + 0.999979i \(0.497931\pi\)
\(660\) 0 0
\(661\) 39.0873i 1.52032i 0.649737 + 0.760159i \(0.274879\pi\)
−0.649737 + 0.760159i \(0.725121\pi\)
\(662\) 0 0
\(663\) −4.28228 + 11.0364i −0.166310 + 0.428617i
\(664\) 0 0
\(665\) 10.3011i 0.399460i
\(666\) 0 0
\(667\) 22.6712 0.877831
\(668\) 0 0
\(669\) 27.7274i 1.07200i
\(670\) 0 0
\(671\) 14.7376i 0.568939i
\(672\) 0 0
\(673\) 30.7921 1.18695 0.593474 0.804853i \(-0.297756\pi\)
0.593474 + 0.804853i \(0.297756\pi\)
\(674\) 0 0
\(675\) −29.3366 −1.12917
\(676\) 0 0
\(677\) −29.8179 −1.14599 −0.572997 0.819557i \(-0.694219\pi\)
−0.572997 + 0.819557i \(0.694219\pi\)
\(678\) 0 0
\(679\) 9.48378 0.363954
\(680\) 0 0
\(681\) 21.0833i 0.807913i
\(682\) 0 0
\(683\) 22.9745i 0.879093i 0.898220 + 0.439547i \(0.144861\pi\)
−0.898220 + 0.439547i \(0.855139\pi\)
\(684\) 0 0
\(685\) 7.65860 0.292620
\(686\) 0 0
\(687\) 16.0456i 0.612176i
\(688\) 0 0
\(689\) 9.89548 25.5028i 0.376988 0.971579i
\(690\) 0 0
\(691\) 9.75760i 0.371197i −0.982626 0.185598i \(-0.940578\pi\)
0.982626 0.185598i \(-0.0594223\pi\)
\(692\) 0 0
\(693\) −1.84286 −0.0700045
\(694\) 0 0
\(695\) 6.37794i 0.241929i
\(696\) 0 0
\(697\) 2.53307i 0.0959469i
\(698\) 0 0
\(699\) −11.6730 −0.441512
\(700\) 0 0
\(701\) 1.23964 0.0468207 0.0234104 0.999726i \(-0.492548\pi\)
0.0234104 + 0.999726i \(0.492548\pi\)
\(702\) 0 0
\(703\) −32.0174 −1.20756
\(704\) 0 0
\(705\) 34.2317 1.28924
\(706\) 0 0
\(707\) 4.77226i 0.179479i
\(708\) 0 0
\(709\) 11.9775i 0.449825i 0.974379 + 0.224912i \(0.0722096\pi\)
−0.974379 + 0.224912i \(0.927790\pi\)
\(710\) 0 0
\(711\) −21.6512 −0.811985
\(712\) 0 0
\(713\) 38.2638i 1.43299i
\(714\) 0 0
\(715\) −4.25268 + 10.9601i −0.159041 + 0.409883i
\(716\) 0 0
\(717\) 12.2979i 0.459275i
\(718\) 0 0
\(719\) −12.9151 −0.481653 −0.240826 0.970568i \(-0.577418\pi\)
−0.240826 + 0.970568i \(0.577418\pi\)
\(720\) 0 0
\(721\) 5.53648i 0.206189i
\(722\) 0 0
\(723\) 5.59908i 0.208232i
\(724\) 0 0
\(725\) −23.2323 −0.862825
\(726\) 0 0
\(727\) −47.8067 −1.77305 −0.886527 0.462678i \(-0.846889\pi\)
−0.886527 + 0.462678i \(0.846889\pi\)
\(728\) 0 0
\(729\) 16.9503 0.627789
\(730\) 0 0
\(731\) 30.5942 1.13157
\(732\) 0 0
\(733\) 2.43540i 0.0899534i 0.998988 + 0.0449767i \(0.0143214\pi\)
−0.998988 + 0.0449767i \(0.985679\pi\)
\(734\) 0 0
\(735\) 3.50742i 0.129373i
\(736\) 0 0
\(737\) 9.94155 0.366202
\(738\) 0 0
\(739\) 1.76085i 0.0647739i 0.999475 + 0.0323870i \(0.0103109\pi\)
−0.999475 + 0.0323870i \(0.989689\pi\)
\(740\) 0 0
\(741\) −11.4235 4.43250i −0.419653 0.162832i
\(742\) 0 0
\(743\) 38.7004i 1.41978i 0.704312 + 0.709890i \(0.251256\pi\)
−0.704312 + 0.709890i \(0.748744\pi\)
\(744\) 0 0
\(745\) −50.0135 −1.83235
\(746\) 0 0
\(747\) 0.545073i 0.0199432i
\(748\) 0 0
\(749\) 10.8337i 0.395854i
\(750\) 0 0
\(751\) −36.5856 −1.33503 −0.667514 0.744597i \(-0.732642\pi\)
−0.667514 + 0.744597i \(0.732642\pi\)
\(752\) 0 0
\(753\) −18.1550 −0.661604
\(754\) 0 0
\(755\) −3.06989 −0.111725
\(756\) 0 0
\(757\) 11.9777 0.435336 0.217668 0.976023i \(-0.430155\pi\)
0.217668 + 0.976023i \(0.430155\pi\)
\(758\) 0 0
\(759\) 5.91140i 0.214570i
\(760\) 0 0
\(761\) 43.4044i 1.57341i 0.617331 + 0.786703i \(0.288214\pi\)
−0.617331 + 0.786703i \(0.711786\pi\)
\(762\) 0 0
\(763\) −0.525322 −0.0190179
\(764\) 0 0
\(765\) 18.3402i 0.663090i
\(766\) 0 0
\(767\) −50.5550 19.6162i −1.82544 0.708298i
\(768\) 0 0
\(769\) 19.2992i 0.695946i 0.937504 + 0.347973i \(0.113130\pi\)
−0.937504 + 0.347973i \(0.886870\pi\)
\(770\) 0 0
\(771\) −16.4856 −0.593713
\(772\) 0 0
\(773\) 15.9785i 0.574706i −0.957825 0.287353i \(-0.907225\pi\)
0.957825 0.287353i \(-0.0927752\pi\)
\(774\) 0 0
\(775\) 39.2108i 1.40849i
\(776\) 0 0
\(777\) 10.9016 0.391092
\(778\) 0 0
\(779\) −2.62193 −0.0939405
\(780\) 0 0
\(781\) −14.2258 −0.509040
\(782\) 0 0
\(783\) 21.4917 0.768051
\(784\) 0 0
\(785\) 23.3026i 0.831705i
\(786\) 0 0
\(787\) 16.4732i 0.587206i −0.955927 0.293603i \(-0.905146\pi\)
0.955927 0.293603i \(-0.0948543\pi\)
\(788\) 0 0
\(789\) −15.4154 −0.548803
\(790\) 0 0
\(791\) 0.688371i 0.0244756i
\(792\) 0 0
\(793\) −19.2218 + 49.5387i −0.682586 + 1.75917i
\(794\) 0 0