Properties

Label 4600.2.e.u.4049.3
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 28 x^{8} + 260 x^{6} + 897 x^{4} + 1056 x^{2} + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.3
Root \(-1.93283i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.u.4049.8

$q$-expansion

\(f(q)\) \(=\) \(q-1.93283i q^{3} +2.38236i q^{7} -0.735829 q^{9} +O(q^{10})\) \(q-1.93283i q^{3} +2.38236i q^{7} -0.735829 q^{9} +5.33368 q^{11} +4.53752i q^{13} +1.81464i q^{17} -7.00233 q^{19} +4.60469 q^{21} +1.00000i q^{23} -4.37626i q^{27} +0.118188 q^{29} -0.884147 q^{31} -10.3091i q^{33} +7.51903i q^{37} +8.77026 q^{39} -1.45186 q^{41} +10.4389i q^{47} +1.32437 q^{49} +3.50739 q^{51} +9.42167i q^{53} +13.5343i q^{57} -7.79239 q^{59} -2.80533 q^{61} -1.75301i q^{63} -3.11134i q^{67} +1.93283 q^{69} -13.5909 q^{71} -12.4389i q^{73} +12.7067i q^{77} -6.80169 q^{79} -10.6660 q^{81} +13.5190i q^{83} -0.228437i q^{87} -2.89906 q^{89} -10.8100 q^{91} +1.70890i q^{93} -1.97774i q^{97} -3.92468 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 26 q^{9} + O(q^{10}) \) \( 10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91} + 130 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 1.93283i − 1.11592i −0.829868 0.557960i \(-0.811584\pi\)
0.829868 0.557960i \(-0.188416\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.38236i 0.900447i 0.892916 + 0.450223i \(0.148656\pi\)
−0.892916 + 0.450223i \(0.851344\pi\)
\(8\) 0 0
\(9\) −0.735829 −0.245276
\(10\) 0 0
\(11\) 5.33368 1.60816 0.804082 0.594519i \(-0.202657\pi\)
0.804082 + 0.594519i \(0.202657\pi\)
\(12\) 0 0
\(13\) 4.53752i 1.25848i 0.777210 + 0.629241i \(0.216634\pi\)
−0.777210 + 0.629241i \(0.783366\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.81464i 0.440115i 0.975487 + 0.220058i \(0.0706245\pi\)
−0.975487 + 0.220058i \(0.929375\pi\)
\(18\) 0 0
\(19\) −7.00233 −1.60645 −0.803223 0.595679i \(-0.796883\pi\)
−0.803223 + 0.595679i \(0.796883\pi\)
\(20\) 0 0
\(21\) 4.60469 1.00483
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 4.37626i − 0.842211i
\(28\) 0 0
\(29\) 0.118188 0.0219469 0.0109735 0.999940i \(-0.496507\pi\)
0.0109735 + 0.999940i \(0.496507\pi\)
\(30\) 0 0
\(31\) −0.884147 −0.158797 −0.0793987 0.996843i \(-0.525300\pi\)
−0.0793987 + 0.996843i \(0.525300\pi\)
\(32\) 0 0
\(33\) − 10.3091i − 1.79458i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.51903i 1.23612i 0.786130 + 0.618061i \(0.212081\pi\)
−0.786130 + 0.618061i \(0.787919\pi\)
\(38\) 0 0
\(39\) 8.77026 1.40436
\(40\) 0 0
\(41\) −1.45186 −0.226743 −0.113371 0.993553i \(-0.536165\pi\)
−0.113371 + 0.993553i \(0.536165\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.4389i 1.52267i 0.648357 + 0.761336i \(0.275456\pi\)
−0.648357 + 0.761336i \(0.724544\pi\)
\(48\) 0 0
\(49\) 1.32437 0.189195
\(50\) 0 0
\(51\) 3.50739 0.491133
\(52\) 0 0
\(53\) 9.42167i 1.29417i 0.762420 + 0.647083i \(0.224011\pi\)
−0.762420 + 0.647083i \(0.775989\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 13.5343i 1.79266i
\(58\) 0 0
\(59\) −7.79239 −1.01448 −0.507241 0.861804i \(-0.669335\pi\)
−0.507241 + 0.861804i \(0.669335\pi\)
\(60\) 0 0
\(61\) −2.80533 −0.359186 −0.179593 0.983741i \(-0.557478\pi\)
−0.179593 + 0.983741i \(0.557478\pi\)
\(62\) 0 0
\(63\) − 1.75301i − 0.220858i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.11134i − 0.380111i −0.981773 0.190055i \(-0.939133\pi\)
0.981773 0.190055i \(-0.0608668\pi\)
\(68\) 0 0
\(69\) 1.93283 0.232685
\(70\) 0 0
\(71\) −13.5909 −1.61294 −0.806470 0.591275i \(-0.798625\pi\)
−0.806470 + 0.591275i \(0.798625\pi\)
\(72\) 0 0
\(73\) − 12.4389i − 1.45586i −0.685649 0.727932i \(-0.740481\pi\)
0.685649 0.727932i \(-0.259519\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.7067i 1.44807i
\(78\) 0 0
\(79\) −6.80169 −0.765250 −0.382625 0.923904i \(-0.624980\pi\)
−0.382625 + 0.923904i \(0.624980\pi\)
\(80\) 0 0
\(81\) −10.6660 −1.18512
\(82\) 0 0
\(83\) 13.5190i 1.48391i 0.670451 + 0.741953i \(0.266100\pi\)
−0.670451 + 0.741953i \(0.733900\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 0.228437i − 0.0244910i
\(88\) 0 0
\(89\) −2.89906 −0.307300 −0.153650 0.988125i \(-0.549103\pi\)
−0.153650 + 0.988125i \(0.549103\pi\)
\(90\) 0 0
\(91\) −10.8100 −1.13320
\(92\) 0 0
\(93\) 1.70890i 0.177205i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 1.97774i − 0.200809i −0.994947 0.100405i \(-0.967986\pi\)
0.994947 0.100405i \(-0.0320138\pi\)
\(98\) 0 0
\(99\) −3.92468 −0.394445
\(100\) 0 0
\(101\) −12.2838 −1.22228 −0.611139 0.791523i \(-0.709289\pi\)
−0.611139 + 0.791523i \(0.709289\pi\)
\(102\) 0 0
\(103\) 12.5061i 1.23226i 0.787644 + 0.616131i \(0.211301\pi\)
−0.787644 + 0.616131i \(0.788699\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 17.3847i − 1.68064i −0.542089 0.840321i \(-0.682367\pi\)
0.542089 0.840321i \(-0.317633\pi\)
\(108\) 0 0
\(109\) −4.30908 −0.412735 −0.206368 0.978475i \(-0.566164\pi\)
−0.206368 + 0.978475i \(0.566164\pi\)
\(110\) 0 0
\(111\) 14.5330 1.37941
\(112\) 0 0
\(113\) 12.1494i 1.14292i 0.820630 + 0.571460i \(0.193623\pi\)
−0.820630 + 0.571460i \(0.806377\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 3.33884i − 0.308676i
\(118\) 0 0
\(119\) −4.32313 −0.396300
\(120\) 0 0
\(121\) 17.4481 1.58619
\(122\) 0 0
\(123\) 2.80620i 0.253027i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 1.22953i − 0.109103i −0.998511 0.0545515i \(-0.982627\pi\)
0.998511 0.0545515i \(-0.0173729\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.97345 0.172421 0.0862104 0.996277i \(-0.472524\pi\)
0.0862104 + 0.996277i \(0.472524\pi\)
\(132\) 0 0
\(133\) − 16.6821i − 1.44652i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.7671i 1.17620i 0.808789 + 0.588099i \(0.200124\pi\)
−0.808789 + 0.588099i \(0.799876\pi\)
\(138\) 0 0
\(139\) −5.30280 −0.449778 −0.224889 0.974384i \(-0.572202\pi\)
−0.224889 + 0.974384i \(0.572202\pi\)
\(140\) 0 0
\(141\) 20.1766 1.69918
\(142\) 0 0
\(143\) 24.2017i 2.02385i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 2.55978i − 0.211127i
\(148\) 0 0
\(149\) −7.73256 −0.633476 −0.316738 0.948513i \(-0.602588\pi\)
−0.316738 + 0.948513i \(0.602588\pi\)
\(150\) 0 0
\(151\) 21.4156 1.74277 0.871387 0.490596i \(-0.163221\pi\)
0.871387 + 0.490596i \(0.163221\pi\)
\(152\) 0 0
\(153\) − 1.33527i − 0.107950i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 4.04738i − 0.323016i −0.986871 0.161508i \(-0.948364\pi\)
0.986871 0.161508i \(-0.0516357\pi\)
\(158\) 0 0
\(159\) 18.2105 1.44418
\(160\) 0 0
\(161\) −2.38236 −0.187756
\(162\) 0 0
\(163\) − 9.04725i − 0.708635i −0.935125 0.354318i \(-0.884713\pi\)
0.935125 0.354318i \(-0.115287\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 12.5467i − 0.970893i −0.874266 0.485446i \(-0.838657\pi\)
0.874266 0.485446i \(-0.161343\pi\)
\(168\) 0 0
\(169\) −7.58910 −0.583777
\(170\) 0 0
\(171\) 5.15252 0.394023
\(172\) 0 0
\(173\) 14.1354i 1.07469i 0.843362 + 0.537346i \(0.180573\pi\)
−0.843362 + 0.537346i \(0.819427\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 15.0614i 1.13208i
\(178\) 0 0
\(179\) 26.2442 1.96158 0.980791 0.195061i \(-0.0624904\pi\)
0.980791 + 0.195061i \(0.0624904\pi\)
\(180\) 0 0
\(181\) −18.3461 −1.36365 −0.681826 0.731514i \(-0.738814\pi\)
−0.681826 + 0.731514i \(0.738814\pi\)
\(182\) 0 0
\(183\) 5.42223i 0.400823i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 9.67871i 0.707777i
\(188\) 0 0
\(189\) 10.4258 0.758366
\(190\) 0 0
\(191\) −3.86566 −0.279709 −0.139855 0.990172i \(-0.544664\pi\)
−0.139855 + 0.990172i \(0.544664\pi\)
\(192\) 0 0
\(193\) 23.8599i 1.71747i 0.512417 + 0.858737i \(0.328750\pi\)
−0.512417 + 0.858737i \(0.671250\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.7356i 0.836128i 0.908418 + 0.418064i \(0.137291\pi\)
−0.908418 + 0.418064i \(0.862709\pi\)
\(198\) 0 0
\(199\) 8.89906 0.630838 0.315419 0.948953i \(-0.397855\pi\)
0.315419 + 0.948953i \(0.397855\pi\)
\(200\) 0 0
\(201\) −6.01369 −0.424173
\(202\) 0 0
\(203\) 0.281566i 0.0197620i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 0.735829i − 0.0511437i
\(208\) 0 0
\(209\) −37.3482 −2.58343
\(210\) 0 0
\(211\) 23.1224 1.59181 0.795906 0.605420i \(-0.206995\pi\)
0.795906 + 0.605420i \(0.206995\pi\)
\(212\) 0 0
\(213\) 26.2688i 1.79991i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 2.10635i − 0.142989i
\(218\) 0 0
\(219\) −24.0423 −1.62463
\(220\) 0 0
\(221\) −8.23398 −0.553877
\(222\) 0 0
\(223\) − 1.50863i − 0.101026i −0.998723 0.0505128i \(-0.983914\pi\)
0.998723 0.0505128i \(-0.0160856\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 22.0047i − 1.46050i −0.683179 0.730251i \(-0.739403\pi\)
0.683179 0.730251i \(-0.260597\pi\)
\(228\) 0 0
\(229\) 11.2931 0.746266 0.373133 0.927778i \(-0.378283\pi\)
0.373133 + 0.927778i \(0.378283\pi\)
\(230\) 0 0
\(231\) 24.5599 1.61593
\(232\) 0 0
\(233\) 17.6789i 1.15818i 0.815263 + 0.579091i \(0.196592\pi\)
−0.815263 + 0.579091i \(0.803408\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 13.1465i 0.853958i
\(238\) 0 0
\(239\) 25.2583 1.63382 0.816911 0.576763i \(-0.195684\pi\)
0.816911 + 0.576763i \(0.195684\pi\)
\(240\) 0 0
\(241\) 0.790615 0.0509280 0.0254640 0.999676i \(-0.491894\pi\)
0.0254640 + 0.999676i \(0.491894\pi\)
\(242\) 0 0
\(243\) 7.48688i 0.480283i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 31.7732i − 2.02168i
\(248\) 0 0
\(249\) 26.1300 1.65592
\(250\) 0 0
\(251\) 12.3461 0.779276 0.389638 0.920968i \(-0.372600\pi\)
0.389638 + 0.920968i \(0.372600\pi\)
\(252\) 0 0
\(253\) 5.33368i 0.335325i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.40442i 0.399497i 0.979847 + 0.199748i \(0.0640125\pi\)
−0.979847 + 0.199748i \(0.935988\pi\)
\(258\) 0 0
\(259\) −17.9130 −1.11306
\(260\) 0 0
\(261\) −0.0869661 −0.00538307
\(262\) 0 0
\(263\) − 19.6914i − 1.21422i −0.794617 0.607111i \(-0.792328\pi\)
0.794617 0.607111i \(-0.207672\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 5.60338i 0.342922i
\(268\) 0 0
\(269\) 3.42494 0.208822 0.104411 0.994534i \(-0.466704\pi\)
0.104411 + 0.994534i \(0.466704\pi\)
\(270\) 0 0
\(271\) 13.7197 0.833411 0.416705 0.909042i \(-0.363185\pi\)
0.416705 + 0.909042i \(0.363185\pi\)
\(272\) 0 0
\(273\) 20.8939i 1.26456i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0.636129i 0.0382213i 0.999817 + 0.0191107i \(0.00608348\pi\)
−0.999817 + 0.0191107i \(0.993917\pi\)
\(278\) 0 0
\(279\) 0.650581 0.0389493
\(280\) 0 0
\(281\) 18.4124 1.09839 0.549195 0.835694i \(-0.314935\pi\)
0.549195 + 0.835694i \(0.314935\pi\)
\(282\) 0 0
\(283\) 7.95729i 0.473012i 0.971630 + 0.236506i \(0.0760023\pi\)
−0.971630 + 0.236506i \(0.923998\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 3.45886i − 0.204170i
\(288\) 0 0
\(289\) 13.7071 0.806299
\(290\) 0 0
\(291\) −3.82264 −0.224087
\(292\) 0 0
\(293\) 20.9166i 1.22196i 0.791645 + 0.610981i \(0.209225\pi\)
−0.791645 + 0.610981i \(0.790775\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 23.3415i − 1.35441i
\(298\) 0 0
\(299\) −4.53752 −0.262412
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 23.7424i 1.36396i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 5.78807i − 0.330342i −0.986265 0.165171i \(-0.947182\pi\)
0.986265 0.165171i \(-0.0528177\pi\)
\(308\) 0 0
\(309\) 24.1721 1.37510
\(310\) 0 0
\(311\) 18.3674 1.04152 0.520761 0.853702i \(-0.325648\pi\)
0.520761 + 0.853702i \(0.325648\pi\)
\(312\) 0 0
\(313\) − 9.85869i − 0.557246i −0.960401 0.278623i \(-0.910122\pi\)
0.960401 0.278623i \(-0.0898780\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 32.9488i − 1.85059i −0.379250 0.925294i \(-0.623818\pi\)
0.379250 0.925294i \(-0.376182\pi\)
\(318\) 0 0
\(319\) 0.630376 0.0352943
\(320\) 0 0
\(321\) −33.6016 −1.87546
\(322\) 0 0
\(323\) − 12.7067i − 0.707021i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 8.32873i 0.460580i
\(328\) 0 0
\(329\) −24.8692 −1.37109
\(330\) 0 0
\(331\) −6.85308 −0.376679 −0.188340 0.982104i \(-0.560311\pi\)
−0.188340 + 0.982104i \(0.560311\pi\)
\(332\) 0 0
\(333\) − 5.53273i − 0.303192i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 35.8432i 1.95250i 0.216641 + 0.976251i \(0.430490\pi\)
−0.216641 + 0.976251i \(0.569510\pi\)
\(338\) 0 0
\(339\) 23.4827 1.27541
\(340\) 0 0
\(341\) −4.71575 −0.255372
\(342\) 0 0
\(343\) 19.8316i 1.07081i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 4.09198i − 0.219669i −0.993950 0.109835i \(-0.964968\pi\)
0.993950 0.109835i \(-0.0350321\pi\)
\(348\) 0 0
\(349\) 11.3229 0.606101 0.303051 0.952974i \(-0.401995\pi\)
0.303051 + 0.952974i \(0.401995\pi\)
\(350\) 0 0
\(351\) 19.8574 1.05991
\(352\) 0 0
\(353\) − 26.1443i − 1.39152i −0.718273 0.695761i \(-0.755067\pi\)
0.718273 0.695761i \(-0.244933\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 8.35587i 0.442239i
\(358\) 0 0
\(359\) 12.2058 0.644198 0.322099 0.946706i \(-0.395612\pi\)
0.322099 + 0.946706i \(0.395612\pi\)
\(360\) 0 0
\(361\) 30.0327 1.58067
\(362\) 0 0
\(363\) − 33.7242i − 1.77006i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 3.15299i − 0.164585i −0.996608 0.0822923i \(-0.973776\pi\)
0.996608 0.0822923i \(-0.0262241\pi\)
\(368\) 0 0
\(369\) 1.06832 0.0556147
\(370\) 0 0
\(371\) −22.4458 −1.16533
\(372\) 0 0
\(373\) − 9.59872i − 0.497003i −0.968632 0.248501i \(-0.920062\pi\)
0.968632 0.248501i \(-0.0799380\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.536280i 0.0276198i
\(378\) 0 0
\(379\) −32.7841 −1.68401 −0.842003 0.539473i \(-0.818624\pi\)
−0.842003 + 0.539473i \(0.818624\pi\)
\(380\) 0 0
\(381\) −2.37647 −0.121750
\(382\) 0 0
\(383\) − 34.7378i − 1.77502i −0.460792 0.887508i \(-0.652435\pi\)
0.460792 0.887508i \(-0.347565\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −13.8965 −0.704581 −0.352290 0.935891i \(-0.614597\pi\)
−0.352290 + 0.935891i \(0.614597\pi\)
\(390\) 0 0
\(391\) −1.81464 −0.0917704
\(392\) 0 0
\(393\) − 3.81434i − 0.192408i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 26.1135i − 1.31060i −0.755370 0.655299i \(-0.772543\pi\)
0.755370 0.655299i \(-0.227457\pi\)
\(398\) 0 0
\(399\) −32.2436 −1.61420
\(400\) 0 0
\(401\) 18.0715 0.902446 0.451223 0.892411i \(-0.350988\pi\)
0.451223 + 0.892411i \(0.350988\pi\)
\(402\) 0 0
\(403\) − 4.01183i − 0.199844i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 40.1041i 1.98789i
\(408\) 0 0
\(409\) 0.865532 0.0427978 0.0213989 0.999771i \(-0.493188\pi\)
0.0213989 + 0.999771i \(0.493188\pi\)
\(410\) 0 0
\(411\) 26.6094 1.31254
\(412\) 0 0
\(413\) − 18.5643i − 0.913487i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 10.2494i 0.501916i
\(418\) 0 0
\(419\) 20.5675 1.00479 0.502394 0.864639i \(-0.332453\pi\)
0.502394 + 0.864639i \(0.332453\pi\)
\(420\) 0 0
\(421\) 26.2423 1.27897 0.639485 0.768803i \(-0.279147\pi\)
0.639485 + 0.768803i \(0.279147\pi\)
\(422\) 0 0
\(423\) − 7.68126i − 0.373476i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 6.68331i − 0.323428i
\(428\) 0 0
\(429\) 46.7777 2.25845
\(430\) 0 0
\(431\) 23.2654 1.12066 0.560328 0.828271i \(-0.310675\pi\)
0.560328 + 0.828271i \(0.310675\pi\)
\(432\) 0 0
\(433\) 34.7365i 1.66933i 0.550758 + 0.834665i \(0.314339\pi\)
−0.550758 + 0.834665i \(0.685661\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 7.00233i − 0.334967i
\(438\) 0 0
\(439\) −28.5238 −1.36137 −0.680684 0.732577i \(-0.738317\pi\)
−0.680684 + 0.732577i \(0.738317\pi\)
\(440\) 0 0
\(441\) −0.974509 −0.0464052
\(442\) 0 0
\(443\) − 26.3428i − 1.25158i −0.779990 0.625792i \(-0.784776\pi\)
0.779990 0.625792i \(-0.215224\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 14.9457i 0.706908i
\(448\) 0 0
\(449\) −29.5411 −1.39413 −0.697065 0.717008i \(-0.745511\pi\)
−0.697065 + 0.717008i \(0.745511\pi\)
\(450\) 0 0
\(451\) −7.74377 −0.364640
\(452\) 0 0
\(453\) − 41.3926i − 1.94480i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.16668i 0.148131i 0.997253 + 0.0740655i \(0.0235974\pi\)
−0.997253 + 0.0740655i \(0.976403\pi\)
\(458\) 0 0
\(459\) 7.94133 0.370670
\(460\) 0 0
\(461\) 8.43891 0.393039 0.196520 0.980500i \(-0.437036\pi\)
0.196520 + 0.980500i \(0.437036\pi\)
\(462\) 0 0
\(463\) − 9.62461i − 0.447294i −0.974670 0.223647i \(-0.928204\pi\)
0.974670 0.223647i \(-0.0717962\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 23.1041i − 1.06913i −0.845128 0.534564i \(-0.820476\pi\)
0.845128 0.534564i \(-0.179524\pi\)
\(468\) 0 0
\(469\) 7.41233 0.342270
\(470\) 0 0
\(471\) −7.82289 −0.360460
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 6.93274i − 0.317428i
\(478\) 0 0
\(479\) −20.5981 −0.941150 −0.470575 0.882360i \(-0.655953\pi\)
−0.470575 + 0.882360i \(0.655953\pi\)
\(480\) 0 0
\(481\) −34.1178 −1.55564
\(482\) 0 0
\(483\) 4.60469i 0.209521i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 40.1145i 1.81776i 0.417056 + 0.908881i \(0.363062\pi\)
−0.417056 + 0.908881i \(0.636938\pi\)
\(488\) 0 0
\(489\) −17.4868 −0.790780
\(490\) 0 0
\(491\) −18.4799 −0.833985 −0.416993 0.908910i \(-0.636916\pi\)
−0.416993 + 0.908910i \(0.636916\pi\)
\(492\) 0 0
\(493\) 0.214469i 0.00965918i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 32.3783i − 1.45237i
\(498\) 0 0
\(499\) −0.0506452 −0.00226719 −0.00113360 0.999999i \(-0.500361\pi\)
−0.00113360 + 0.999999i \(0.500361\pi\)
\(500\) 0 0
\(501\) −24.2506 −1.08344
\(502\) 0 0
\(503\) 31.1887i 1.39064i 0.718702 + 0.695318i \(0.244737\pi\)
−0.718702 + 0.695318i \(0.755263\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 14.6684i 0.651448i
\(508\) 0 0
\(509\) −3.42426 −0.151778 −0.0758888 0.997116i \(-0.524179\pi\)
−0.0758888 + 0.997116i \(0.524179\pi\)
\(510\) 0 0
\(511\) 29.6340 1.31093
\(512\) 0 0
\(513\) 30.6440i 1.35297i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 55.6778i 2.44871i
\(518\) 0 0
\(519\) 27.3213 1.19927
\(520\) 0 0
\(521\) −9.78442 −0.428663 −0.214332 0.976761i \(-0.568757\pi\)
−0.214332 + 0.976761i \(0.568757\pi\)
\(522\) 0 0
\(523\) − 6.22884i − 0.272368i −0.990684 0.136184i \(-0.956516\pi\)
0.990684 0.136184i \(-0.0434838\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 1.60441i − 0.0698892i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 5.73387 0.248829
\(532\) 0 0
\(533\) − 6.58786i − 0.285352i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 50.7255i − 2.18897i
\(538\) 0 0
\(539\) 7.06375 0.304257
\(540\) 0 0
\(541\) 19.8477 0.853319 0.426660 0.904412i \(-0.359690\pi\)
0.426660 + 0.904412i \(0.359690\pi\)
\(542\) 0 0
\(543\) 35.4598i 1.52173i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 7.68024i − 0.328383i −0.986428 0.164192i \(-0.947498\pi\)
0.986428 0.164192i \(-0.0525016\pi\)
\(548\) 0 0
\(549\) 2.06425 0.0880999
\(550\) 0 0
\(551\) −0.827591 −0.0352566
\(552\) 0 0
\(553\) − 16.2041i − 0.689067i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.1260i 0.768023i 0.923328 + 0.384012i \(0.125458\pi\)
−0.923328 + 0.384012i \(0.874542\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 18.7073 0.789823
\(562\) 0 0
\(563\) 16.8772i 0.711287i 0.934622 + 0.355644i \(0.115738\pi\)
−0.934622 + 0.355644i \(0.884262\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 25.4103i − 1.06713i
\(568\) 0 0
\(569\) −36.0022 −1.50929 −0.754645 0.656133i \(-0.772191\pi\)
−0.754645 + 0.656133i \(0.772191\pi\)
\(570\) 0 0
\(571\) 26.6267 1.11429 0.557147 0.830414i \(-0.311896\pi\)
0.557147 + 0.830414i \(0.311896\pi\)
\(572\) 0 0
\(573\) 7.47166i 0.312133i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 30.0190i 1.24971i 0.780742 + 0.624854i \(0.214842\pi\)
−0.780742 + 0.624854i \(0.785158\pi\)
\(578\) 0 0
\(579\) 46.1171 1.91656
\(580\) 0 0
\(581\) −32.2072 −1.33618
\(582\) 0 0
\(583\) 50.2521i 2.08123i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 31.1814i − 1.28699i −0.765449 0.643497i \(-0.777483\pi\)
0.765449 0.643497i \(-0.222517\pi\)
\(588\) 0 0
\(589\) 6.19109 0.255099
\(590\) 0 0
\(591\) 22.6829 0.933051
\(592\) 0 0
\(593\) − 15.0334i − 0.617348i −0.951168 0.308674i \(-0.900115\pi\)
0.951168 0.308674i \(-0.0998852\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 17.2004i − 0.703964i
\(598\) 0 0
\(599\) −20.3534 −0.831617 −0.415808 0.909452i \(-0.636501\pi\)
−0.415808 + 0.909452i \(0.636501\pi\)
\(600\) 0 0
\(601\) 46.5338 1.89816 0.949078 0.315043i \(-0.102019\pi\)
0.949078 + 0.315043i \(0.102019\pi\)
\(602\) 0 0
\(603\) 2.28942i 0.0932323i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 7.52725i − 0.305522i −0.988263 0.152761i \(-0.951184\pi\)
0.988263 0.152761i \(-0.0488164\pi\)
\(608\) 0 0
\(609\) 0.544219 0.0220529
\(610\) 0 0
\(611\) −47.3668 −1.91626
\(612\) 0 0
\(613\) − 32.2643i − 1.30314i −0.758587 0.651572i \(-0.774110\pi\)
0.758587 0.651572i \(-0.225890\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.52158i 0.182032i 0.995849 + 0.0910161i \(0.0290115\pi\)
−0.995849 + 0.0910161i \(0.970989\pi\)
\(618\) 0 0
\(619\) −17.7807 −0.714668 −0.357334 0.933977i \(-0.616314\pi\)
−0.357334 + 0.933977i \(0.616314\pi\)
\(620\) 0 0
\(621\) 4.37626 0.175613
\(622\) 0 0
\(623\) − 6.90660i − 0.276707i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 72.1877i 2.88290i
\(628\) 0 0
\(629\) −13.6444 −0.544036
\(630\) 0 0
\(631\) 27.4554 1.09298 0.546490 0.837466i \(-0.315964\pi\)
0.546490 + 0.837466i \(0.315964\pi\)
\(632\) 0 0
\(633\) − 44.6917i − 1.77634i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.00935i 0.238099i
\(638\) 0 0
\(639\) 10.0006 0.395616
\(640\) 0 0
\(641\) −48.9270 −1.93250 −0.966250 0.257605i \(-0.917067\pi\)
−0.966250 + 0.257605i \(0.917067\pi\)
\(642\) 0 0
\(643\) − 2.89333i − 0.114102i −0.998371 0.0570508i \(-0.981830\pi\)
0.998371 0.0570508i \(-0.0181697\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 15.0877i 0.593157i 0.955009 + 0.296578i \(0.0958456\pi\)
−0.955009 + 0.296578i \(0.904154\pi\)
\(648\) 0 0
\(649\) −41.5621 −1.63145
\(650\) 0 0
\(651\) −4.07122 −0.159564
\(652\) 0 0
\(653\) 13.3255i 0.521468i 0.965411 + 0.260734i \(0.0839645\pi\)
−0.965411 + 0.260734i \(0.916035\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 9.15292i 0.357089i
\(658\) 0 0
\(659\) 39.9000 1.55428 0.777142 0.629325i \(-0.216669\pi\)
0.777142 + 0.629325i \(0.216669\pi\)
\(660\) 0 0
\(661\) −37.3060 −1.45104 −0.725518 0.688203i \(-0.758400\pi\)
−0.725518 + 0.688203i \(0.758400\pi\)
\(662\) 0 0
\(663\) 15.9149i 0.618082i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0.118188i 0.00457625i
\(668\) 0 0
\(669\) −2.91593 −0.112736
\(670\) 0 0
\(671\) −14.9627 −0.577630
\(672\) 0 0
\(673\) 18.6641i 0.719447i 0.933059 + 0.359724i \(0.117129\pi\)
−0.933059 + 0.359724i \(0.882871\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19.4978i 0.749361i 0.927154 + 0.374681i \(0.122248\pi\)
−0.927154 + 0.374681i \(0.877752\pi\)
\(678\) 0 0
\(679\) 4.71169 0.180818
\(680\) 0 0
\(681\) −42.5313 −1.62980
\(682\) 0 0
\(683\) 41.6331i 1.59305i 0.604608 + 0.796523i \(0.293330\pi\)
−0.604608 + 0.796523i \(0.706670\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 21.8276i − 0.832773i
\(688\) 0 0
\(689\) −42.7510 −1.62868
\(690\) 0 0
\(691\) −9.53192 −0.362611 −0.181306 0.983427i \(-0.558032\pi\)
−0.181306 + 0.983427i \(0.558032\pi\)
\(692\) 0 0
\(693\) − 9.34998i − 0.355177i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 2.63461i − 0.0997930i
\(698\) 0 0
\(699\) 34.1702 1.29244
\(700\) 0 0
\(701\) 27.9897 1.05716 0.528579 0.848884i \(-0.322725\pi\)
0.528579 + 0.848884i \(0.322725\pi\)
\(702\) 0 0
\(703\) − 52.6508i − 1.98576i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 29.2643i − 1.10060i
\(708\) 0 0
\(709\) −27.6832 −1.03966 −0.519831 0.854269i \(-0.674005\pi\)
−0.519831 + 0.854269i \(0.674005\pi\)
\(710\) 0 0
\(711\) 5.00489 0.187698
\(712\) 0 0
\(713\) − 0.884147i − 0.0331115i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 48.8200i − 1.82322i
\(718\) 0 0
\(719\) −50.8364 −1.89588 −0.947938 0.318454i \(-0.896836\pi\)
−0.947938 + 0.318454i \(0.896836\pi\)
\(720\) 0 0
\(721\) −29.7940 −1.10959
\(722\) 0 0
\(723\) − 1.52812i − 0.0568316i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 37.5061i 1.39102i 0.718514 + 0.695512i \(0.244823\pi\)
−0.718514 + 0.695512i \(0.755177\pi\)
\(728\) 0 0
\(729\) −17.5273 −0.649158
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 36.9374i 1.36431i 0.731206 + 0.682157i \(0.238958\pi\)
−0.731206 + 0.682157i \(0.761042\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 16.5949i − 0.611281i
\(738\) 0 0
\(739\) 12.3287 0.453517 0.226759 0.973951i \(-0.427187\pi\)
0.226759 + 0.973951i \(0.427187\pi\)
\(740\) 0 0
\(741\) −61.4123 −2.25604
\(742\) 0 0
\(743\) 29.7534i 1.09154i 0.837933 + 0.545772i \(0.183764\pi\)
−0.837933 + 0.545772i \(0.816236\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 9.94770i − 0.363967i
\(748\) 0 0
\(749\) 41.4166 1.51333
\(750\) 0 0
\(751\) −41.9834 −1.53200 −0.765999 0.642842i \(-0.777755\pi\)
−0.765999 + 0.642842i \(0.777755\pi\)
\(752\) 0 0
\(753\) − 23.8628i − 0.869610i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 11.4708i 0.416915i 0.978031 + 0.208458i \(0.0668443\pi\)
−0.978031 + 0.208458i \(0.933156\pi\)
\(758\) 0 0
\(759\) 10.3091 0.374196
\(760\) 0 0
\(761\) −13.4122 −0.486193 −0.243097 0.970002i \(-0.578163\pi\)
−0.243097 + 0.970002i \(0.578163\pi\)
\(762\) 0 0
\(763\) − 10.2658i − 0.371646i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 35.3581i − 1.27671i
\(768\) 0 0
\(769\) −2.18327 −0.0787307 −0.0393654 0.999225i \(-0.512534\pi\)
−0.0393654 + 0.999225i \(0.512534\pi\)
\(770\) 0 0
\(771\) 12.3787 0.445806
\(772\) 0 0
\(773\) − 5.73306i − 0.206204i −0.994671 0.103102i \(-0.967123\pi\)
0.994671 0.103102i \(-0.0328768\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 34.6228i 1.24209i
\(778\) 0 0
\(779\) 10.1664 0.364250
\(780\) 0 0
\(781\) −72.4893 −2.59387
\(782\) 0 0
\(783\) − 0.517220i − 0.0184839i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 8.13502i − 0.289982i −0.989433 0.144991i \(-0.953685\pi\)
0.989433 0.144991i \(-0.0463153\pi\)
\(788\) 0 0
\(789\) −38.0601 −1.35497
\(790\) 0 0
\(791\) −28.9442 −1.02914
\(792\) 0 0
\(793\) − 12.7293i − 0.452029i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 41.5139i − 1.47050i −0.677797 0.735250i \(-0.737065\pi\)
0.677797 0.735250i \(-0.262935\pi\)
\(798\) 0 0
\(799\) −18.9429 −0.670151
\(800\) 0 0
\(801\) 2.13321 0.0753733
\(802\) 0 0
\(803\) − 66.3451i − 2.34127i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 6.61982i − 0.233029i
\(808\) 0 0
\(809\) 22.8566 0.803596 0.401798 0.915728i \(-0.368385\pi\)
0.401798 + 0.915728i \(0.368385\pi\)
\(810\) 0 0
\(811\) −42.1917 −1.48155 −0.740776 0.671753i \(-0.765542\pi\)
−0.740776 + 0.671753i \(0.765542\pi\)
\(812\) 0 0
\(813\) − 26.5178i − 0.930020i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 7.95432 0.277946
\(820\) 0 0
\(821\) 27.0195 0.942985 0.471493 0.881870i \(-0.343715\pi\)
0.471493 + 0.881870i \(0.343715\pi\)
\(822\) 0 0
\(823\) − 27.2435i − 0.949650i −0.880080 0.474825i \(-0.842511\pi\)
0.880080 0.474825i \(-0.157489\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 10.5732i 0.367667i 0.982957 + 0.183834i \(0.0588508\pi\)
−0.982957 + 0.183834i \(0.941149\pi\)
\(828\) 0 0
\(829\) 1.16445 0.0404429 0.0202215 0.999796i \(-0.493563\pi\)
0.0202215 + 0.999796i \(0.493563\pi\)
\(830\) 0 0
\(831\) 1.22953 0.0426519
\(832\) 0 0
\(833\) 2.40325i 0.0832678i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.86925i 0.133741i
\(838\) 0 0
\(839\) 20.2920 0.700556 0.350278 0.936646i \(-0.386087\pi\)
0.350278 + 0.936646i \(0.386087\pi\)
\(840\) 0 0
\(841\) −28.9860 −0.999518
\(842\) 0 0
\(843\) − 35.5880i − 1.22571i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 41.5676i 1.42828i
\(848\) 0 0
\(849\) 15.3801 0.527843
\(850\) 0 0
\(851\) −7.51903 −0.257749
\(852\) 0 0
\(853\) − 26.6154i − 0.911295i −0.890160 0.455648i \(-0.849408\pi\)
0.890160 0.455648i \(-0.150592\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 10.4342i − 0.356427i −0.983992 0.178214i \(-0.942968\pi\)
0.983992 0.178214i \(-0.0570318\pi\)
\(858\) 0 0
\(859\) 24.6623 0.841466 0.420733 0.907185i \(-0.361773\pi\)
0.420733 + 0.907185i \(0.361773\pi\)
\(860\) 0 0
\(861\) −6.68538 −0.227837
\(862\) 0 0
\(863\) − 8.48317i − 0.288771i −0.989522 0.144385i \(-0.953880\pi\)
0.989522 0.144385i \(-0.0461205\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 26.4934i − 0.899764i
\(868\) 0 0
\(869\) −36.2780 −1.23065
\(870\) 0 0
\(871\) 14.1178 0.478363
\(872\) 0 0
\(873\) 1.45528i 0.0492538i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.08796i 0.104273i 0.998640 + 0.0521365i \(0.0166031\pi\)
−0.998640 + 0.0521365i \(0.983397\pi\)
\(878\) 0 0
\(879\) 40.4282 1.36361
\(880\) 0 0
\(881\) 42.5879 1.43482 0.717412 0.696649i \(-0.245327\pi\)
0.717412 + 0.696649i \(0.245327\pi\)
\(882\) 0 0
\(883\) 48.3342i 1.62657i 0.581863 + 0.813287i \(0.302324\pi\)
−0.581863 + 0.813287i \(0.697676\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 2.12173i − 0.0712408i −0.999365 0.0356204i \(-0.988659\pi\)
0.999365 0.0356204i \(-0.0113407\pi\)
\(888\) 0 0
\(889\) 2.92918 0.0982415
\(890\) 0 0
\(891\) −56.8892 −1.90586
\(892\) 0 0
\(893\) − 73.0968i − 2.44609i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 8.77026i 0.292830i
\(898\) 0 0
\(899\) −0.104495 −0.00348512
\(900\) 0 0
\(901\) −17.0970 −0.569582
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 28.6914i 0.952684i 0.879260 + 0.476342i \(0.158038\pi\)
−0.879260 + 0.476342i \(0.841962\pi\)
\(908\) 0 0
\(909\) 9.03875 0.299796
\(910\) 0 0
\(911\) −1.14899 −0.0380679 −0.0190339 0.999819i \(-0.506059\pi\)
−0.0190339 + 0.999819i \(0.506059\pi\)
\(912\) 0 0
\(913\) 72.1061i 2.38636i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.70146i 0.155256i
\(918\) 0 0
\(919\) 55.4798 1.83011 0.915055 0.403329i \(-0.132147\pi\)
0.915055 + 0.403329i \(0.132147\pi\)
\(920\) 0 0
\(921\) −11.1873 −0.368636
\(922\) 0 0
\(923\) − 61.6689i − 2.02986i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 9.20235i − 0.302245i
\(928\) 0 0
\(929\) 4.05780 0.133132 0.0665660 0.997782i \(-0.478796\pi\)
0.0665660 + 0.997782i \(0.478796\pi\)
\(930\) 0 0
\(931\) −9.27367 −0.303932
\(932\) 0 0
\(933\) − 35.5011i − 1.16226i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 16.0588i 0.524617i 0.964984 + 0.262308i \(0.0844838\pi\)
−0.964984 + 0.262308i \(0.915516\pi\)
\(938\) 0 0
\(939\) −19.0552 −0.621842
\(940\) 0 0
\(941\) 4.73779 0.154448 0.0772238 0.997014i \(-0.475394\pi\)
0.0772238 + 0.997014i \(0.475394\pi\)
\(942\) 0 0
\(943\) − 1.45186i − 0.0472792i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 3.38634i − 0.110041i −0.998485 0.0550207i \(-0.982478\pi\)
0.998485 0.0550207i \(-0.0175225\pi\)
\(948\) 0 0
\(949\) 56.4418 1.83218
\(950\) 0 0
\(951\) −63.6844 −2.06511
\(952\) 0 0
\(953\) − 29.2967i − 0.949013i −0.880252 0.474507i \(-0.842627\pi\)
0.880252 0.474507i \(-0.157373\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 1.21841i − 0.0393856i
\(958\) 0 0
\(959\) −32.7981 −1.05910
\(960\) 0 0
\(961\) −30.2183 −0.974783
\(962\) 0 0
\(963\) 12.7922i 0.412222i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 15.5743i − 0.500837i −0.968138 0.250419i \(-0.919432\pi\)
0.968138 0.250419i \(-0.0805683\pi\)
\(968\) 0 0
\(969\) −24.5599 −0.788979
\(970\) 0 0
\(971\) 51.5951 1.65577 0.827883 0.560900i \(-0.189545\pi\)
0.827883 + 0.560900i \(0.189545\pi\)
\(972\) 0 0
\(973\) − 12.6332i − 0.405001i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.04984i 0.289530i 0.989466 + 0.144765i \(0.0462426\pi\)
−0.989466 + 0.144765i \(0.953757\pi\)
\(978\) 0 0
\(979\) −15.4626 −0.494188
\(980\) 0 0
\(981\) 3.17075 0.101234
\(982\) 0 0
\(983\) 39.3855i 1.25620i 0.778132 + 0.628101i \(0.216167\pi\)
−0.778132 + 0.628101i \(0.783833\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 48.0680i 1.53002i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −18.9152 −0.600860 −0.300430 0.953804i \(-0.597130\pi\)
−0.300430 + 0.953804i \(0.597130\pi\)
\(992\) 0 0
\(993\) 13.2458i 0.420344i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 12.2966i − 0.389438i −0.980859 0.194719i \(-0.937620\pi\)
0.980859 0.194719i \(-0.0623795\pi\)
\(998\) 0 0
\(999\) 32.9052 1.04107
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 4600.2.e.u.4049.3 10
5.2 odd 4 4600.2.a.be.1.2 5
5.3 odd 4 920.2.a.j.1.4 5
5.4 even 2 inner 4600.2.e.u.4049.8 10
15.8 even 4 8280.2.a.bs.1.4 5
20.3 even 4 1840.2.a.v.1.2 5
20.7 even 4 9200.2.a.cu.1.4 5
40.3 even 4 7360.2.a.cp.1.4 5
40.13 odd 4 7360.2.a.co.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.4 5 5.3 odd 4
1840.2.a.v.1.2 5 20.3 even 4
4600.2.a.be.1.2 5 5.2 odd 4
4600.2.e.u.4049.3 10 1.1 even 1 trivial
4600.2.e.u.4049.8 10 5.4 even 2 inner
7360.2.a.co.1.2 5 40.13 odd 4
7360.2.a.cp.1.4 5 40.3 even 4
8280.2.a.bs.1.4 5 15.8 even 4
9200.2.a.cu.1.4 5 20.7 even 4