Properties

Label 300.3.b.d.149.2
Level $300$
Weight $3$
Character 300.149
Analytic conductor $8.174$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{35})\)
Defining polynomial: \(x^{4} - 17 x^{2} + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.2
Root \(-2.95804 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 300.149
Dual form 300.3.b.d.149.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-2.95804 + 0.500000i) q^{3} -8.00000i q^{7} +(8.50000 - 2.95804i) q^{9} +O(q^{10})\) \(q+(-2.95804 + 0.500000i) q^{3} -8.00000i q^{7} +(8.50000 - 2.95804i) q^{9} +17.7482i q^{11} +2.00000i q^{13} -17.7482 q^{17} -11.0000 q^{19} +(4.00000 + 23.6643i) q^{21} -35.4965 q^{23} +(-23.6643 + 13.0000i) q^{27} -35.4965i q^{29} -46.0000 q^{31} +(-8.87412 - 52.5000i) q^{33} +16.0000i q^{37} +(-1.00000 - 5.91608i) q^{39} +53.2447i q^{41} +62.0000i q^{43} -35.4965 q^{47} -15.0000 q^{49} +(52.5000 - 8.87412i) q^{51} +35.4965 q^{53} +(32.5384 - 5.50000i) q^{57} +70.9930i q^{59} -16.0000 q^{61} +(-23.6643 - 68.0000i) q^{63} -113.000i q^{67} +(105.000 - 17.7482i) q^{69} -106.489i q^{71} +101.000i q^{73} +141.986 q^{77} -68.0000 q^{79} +(63.5000 - 50.2867i) q^{81} +17.7482 q^{83} +(17.7482 + 105.000i) q^{87} -53.2447i q^{89} +16.0000 q^{91} +(136.070 - 23.0000i) q^{93} +22.0000i q^{97} +(52.5000 + 150.860i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 34q^{9} + O(q^{10}) \) \( 4q + 34q^{9} - 44q^{19} + 16q^{21} - 184q^{31} - 4q^{39} - 60q^{49} + 210q^{51} - 64q^{61} + 420q^{69} - 272q^{79} + 254q^{81} + 64q^{91} + 210q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) −2.95804 + 0.500000i −0.986013 + 0.166667i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 8.00000i 1.14286i −0.820652 0.571429i \(-0.806389\pi\)
0.820652 0.571429i \(-0.193611\pi\)
\(8\) 0 0
\(9\) 8.50000 2.95804i 0.944444 0.328671i
\(10\) 0 0
\(11\) 17.7482i 1.61348i 0.590909 + 0.806738i \(0.298769\pi\)
−0.590909 + 0.806738i \(0.701231\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.153846i 0.997037 + 0.0769231i \(0.0245096\pi\)
−0.997037 + 0.0769231i \(0.975490\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −17.7482 −1.04401 −0.522007 0.852941i \(-0.674817\pi\)
−0.522007 + 0.852941i \(0.674817\pi\)
\(18\) 0 0
\(19\) −11.0000 −0.578947 −0.289474 0.957186i \(-0.593480\pi\)
−0.289474 + 0.957186i \(0.593480\pi\)
\(20\) 0 0
\(21\) 4.00000 + 23.6643i 0.190476 + 1.12687i
\(22\) 0 0
\(23\) −35.4965 −1.54333 −0.771663 0.636032i \(-0.780575\pi\)
−0.771663 + 0.636032i \(0.780575\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −23.6643 + 13.0000i −0.876456 + 0.481481i
\(28\) 0 0
\(29\) 35.4965i 1.22402i −0.790851 0.612008i \(-0.790362\pi\)
0.790851 0.612008i \(-0.209638\pi\)
\(30\) 0 0
\(31\) −46.0000 −1.48387 −0.741935 0.670471i \(-0.766092\pi\)
−0.741935 + 0.670471i \(0.766092\pi\)
\(32\) 0 0
\(33\) −8.87412 52.5000i −0.268913 1.59091i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 16.0000i 0.432432i 0.976346 + 0.216216i \(0.0693716\pi\)
−0.976346 + 0.216216i \(0.930628\pi\)
\(38\) 0 0
\(39\) −1.00000 5.91608i −0.0256410 0.151694i
\(40\) 0 0
\(41\) 53.2447i 1.29865i 0.760510 + 0.649326i \(0.224949\pi\)
−0.760510 + 0.649326i \(0.775051\pi\)
\(42\) 0 0
\(43\) 62.0000i 1.44186i 0.693008 + 0.720930i \(0.256285\pi\)
−0.693008 + 0.720930i \(0.743715\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −35.4965 −0.755244 −0.377622 0.925960i \(-0.623258\pi\)
−0.377622 + 0.925960i \(0.623258\pi\)
\(48\) 0 0
\(49\) −15.0000 −0.306122
\(50\) 0 0
\(51\) 52.5000 8.87412i 1.02941 0.174002i
\(52\) 0 0
\(53\) 35.4965 0.669745 0.334872 0.942263i \(-0.391307\pi\)
0.334872 + 0.942263i \(0.391307\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 32.5384 5.50000i 0.570850 0.0964912i
\(58\) 0 0
\(59\) 70.9930i 1.20327i 0.798771 + 0.601635i \(0.205484\pi\)
−0.798771 + 0.601635i \(0.794516\pi\)
\(60\) 0 0
\(61\) −16.0000 −0.262295 −0.131148 0.991363i \(-0.541866\pi\)
−0.131148 + 0.991363i \(0.541866\pi\)
\(62\) 0 0
\(63\) −23.6643 68.0000i −0.375624 1.07937i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 113.000i 1.68657i −0.537469 0.843284i \(-0.680619\pi\)
0.537469 0.843284i \(-0.319381\pi\)
\(68\) 0 0
\(69\) 105.000 17.7482i 1.52174 0.257221i
\(70\) 0 0
\(71\) 106.489i 1.49985i −0.661522 0.749926i \(-0.730089\pi\)
0.661522 0.749926i \(-0.269911\pi\)
\(72\) 0 0
\(73\) 101.000i 1.38356i 0.722108 + 0.691781i \(0.243174\pi\)
−0.722108 + 0.691781i \(0.756826\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 141.986 1.84397
\(78\) 0 0
\(79\) −68.0000 −0.860759 −0.430380 0.902648i \(-0.641620\pi\)
−0.430380 + 0.902648i \(0.641620\pi\)
\(80\) 0 0
\(81\) 63.5000 50.2867i 0.783951 0.620823i
\(82\) 0 0
\(83\) 17.7482 0.213834 0.106917 0.994268i \(-0.465902\pi\)
0.106917 + 0.994268i \(0.465902\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 17.7482 + 105.000i 0.204003 + 1.20690i
\(88\) 0 0
\(89\) 53.2447i 0.598255i −0.954213 0.299128i \(-0.903304\pi\)
0.954213 0.299128i \(-0.0966956\pi\)
\(90\) 0 0
\(91\) 16.0000 0.175824
\(92\) 0 0
\(93\) 136.070 23.0000i 1.46312 0.247312i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 22.0000i 0.226804i 0.993549 + 0.113402i \(0.0361748\pi\)
−0.993549 + 0.113402i \(0.963825\pi\)
\(98\) 0 0
\(99\) 52.5000 + 150.860i 0.530303 + 1.52384i
\(100\) 0 0
\(101\) 141.986i 1.40580i −0.711288 0.702901i \(-0.751888\pi\)
0.711288 0.702901i \(-0.248112\pi\)
\(102\) 0 0
\(103\) 26.0000i 0.252427i 0.992003 + 0.126214i \(0.0402825\pi\)
−0.992003 + 0.126214i \(0.959718\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −17.7482 −0.165871 −0.0829357 0.996555i \(-0.526430\pi\)
−0.0829357 + 0.996555i \(0.526430\pi\)
\(108\) 0 0
\(109\) −176.000 −1.61468 −0.807339 0.590087i \(-0.799093\pi\)
−0.807339 + 0.590087i \(0.799093\pi\)
\(110\) 0 0
\(111\) −8.00000 47.3286i −0.0720721 0.426384i
\(112\) 0 0
\(113\) 124.238 1.09945 0.549724 0.835346i \(-0.314733\pi\)
0.549724 + 0.835346i \(0.314733\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 5.91608 + 17.0000i 0.0505648 + 0.145299i
\(118\) 0 0
\(119\) 141.986i 1.19316i
\(120\) 0 0
\(121\) −194.000 −1.60331
\(122\) 0 0
\(123\) −26.6224 157.500i −0.216442 1.28049i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 106.000i 0.834646i 0.908758 + 0.417323i \(0.137032\pi\)
−0.908758 + 0.417323i \(0.862968\pi\)
\(128\) 0 0
\(129\) −31.0000 183.398i −0.240310 1.42169i
\(130\) 0 0
\(131\) 70.9930i 0.541931i −0.962589 0.270965i \(-0.912657\pi\)
0.962589 0.270965i \(-0.0873429\pi\)
\(132\) 0 0
\(133\) 88.0000i 0.661654i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 53.2447 0.388648 0.194324 0.980937i \(-0.437749\pi\)
0.194324 + 0.980937i \(0.437749\pi\)
\(138\) 0 0
\(139\) 127.000 0.913669 0.456835 0.889552i \(-0.348983\pi\)
0.456835 + 0.889552i \(0.348983\pi\)
\(140\) 0 0
\(141\) 105.000 17.7482i 0.744681 0.125874i
\(142\) 0 0
\(143\) −35.4965 −0.248227
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 44.3706 7.50000i 0.301841 0.0510204i
\(148\) 0 0
\(149\) 106.489i 0.714694i −0.933972 0.357347i \(-0.883681\pi\)
0.933972 0.357347i \(-0.116319\pi\)
\(150\) 0 0
\(151\) 164.000 1.08609 0.543046 0.839703i \(-0.317271\pi\)
0.543046 + 0.839703i \(0.317271\pi\)
\(152\) 0 0
\(153\) −150.860 + 52.5000i −0.986013 + 0.343137i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 38.0000i 0.242038i −0.992650 0.121019i \(-0.961384\pi\)
0.992650 0.121019i \(-0.0386162\pi\)
\(158\) 0 0
\(159\) −105.000 + 17.7482i −0.660377 + 0.111624i
\(160\) 0 0
\(161\) 283.972i 1.76380i
\(162\) 0 0
\(163\) 47.0000i 0.288344i 0.989553 + 0.144172i \(0.0460518\pi\)
−0.989553 + 0.144172i \(0.953948\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −177.482 −1.06277 −0.531384 0.847131i \(-0.678328\pi\)
−0.531384 + 0.847131i \(0.678328\pi\)
\(168\) 0 0
\(169\) 165.000 0.976331
\(170\) 0 0
\(171\) −93.5000 + 32.5384i −0.546784 + 0.190283i
\(172\) 0 0
\(173\) −70.9930 −0.410364 −0.205182 0.978724i \(-0.565779\pi\)
−0.205182 + 0.978724i \(0.565779\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −35.4965 210.000i −0.200545 1.18644i
\(178\) 0 0
\(179\) 17.7482i 0.0991522i 0.998770 + 0.0495761i \(0.0157870\pi\)
−0.998770 + 0.0495761i \(0.984213\pi\)
\(180\) 0 0
\(181\) −106.000 −0.585635 −0.292818 0.956168i \(-0.594593\pi\)
−0.292818 + 0.956168i \(0.594593\pi\)
\(182\) 0 0
\(183\) 47.3286 8.00000i 0.258626 0.0437158i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 315.000i 1.68449i
\(188\) 0 0
\(189\) 104.000 + 189.315i 0.550265 + 1.00166i
\(190\) 0 0
\(191\) 248.475i 1.30092i 0.759541 + 0.650459i \(0.225423\pi\)
−0.759541 + 0.650459i \(0.774577\pi\)
\(192\) 0 0
\(193\) 193.000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −35.4965 −0.180185 −0.0900926 0.995933i \(-0.528716\pi\)
−0.0900926 + 0.995933i \(0.528716\pi\)
\(198\) 0 0
\(199\) 4.00000 0.0201005 0.0100503 0.999949i \(-0.496801\pi\)
0.0100503 + 0.999949i \(0.496801\pi\)
\(200\) 0 0
\(201\) 56.5000 + 334.259i 0.281095 + 1.66298i
\(202\) 0 0
\(203\) −283.972 −1.39888
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −301.720 + 105.000i −1.45758 + 0.507246i
\(208\) 0 0
\(209\) 195.231i 0.934118i
\(210\) 0 0
\(211\) 209.000 0.990521 0.495261 0.868744i \(-0.335073\pi\)
0.495261 + 0.868744i \(0.335073\pi\)
\(212\) 0 0
\(213\) 53.2447 + 315.000i 0.249975 + 1.47887i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 368.000i 1.69585i
\(218\) 0 0
\(219\) −50.5000 298.762i −0.230594 1.36421i
\(220\) 0 0
\(221\) 35.4965i 0.160618i
\(222\) 0 0
\(223\) 148.000i 0.663677i −0.943336 0.331839i \(-0.892331\pi\)
0.943336 0.331839i \(-0.107669\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −116.000 −0.506550 −0.253275 0.967394i \(-0.581508\pi\)
−0.253275 + 0.967394i \(0.581508\pi\)
\(230\) 0 0
\(231\) −420.000 + 70.9930i −1.81818 + 0.307329i
\(232\) 0 0
\(233\) −212.979 −0.914072 −0.457036 0.889448i \(-0.651089\pi\)
−0.457036 + 0.889448i \(0.651089\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 201.147 34.0000i 0.848720 0.143460i
\(238\) 0 0
\(239\) 177.482i 0.742604i −0.928512 0.371302i \(-0.878911\pi\)
0.928512 0.371302i \(-0.121089\pi\)
\(240\) 0 0
\(241\) 59.0000 0.244813 0.122407 0.992480i \(-0.460939\pi\)
0.122407 + 0.992480i \(0.460939\pi\)
\(242\) 0 0
\(243\) −162.692 + 180.500i −0.669515 + 0.742798i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 22.0000i 0.0890688i
\(248\) 0 0
\(249\) −52.5000 + 8.87412i −0.210843 + 0.0356390i
\(250\) 0 0
\(251\) 124.238i 0.494971i 0.968892 + 0.247485i \(0.0796042\pi\)
−0.968892 + 0.247485i \(0.920396\pi\)
\(252\) 0 0
\(253\) 630.000i 2.49012i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −496.951 −1.93366 −0.966830 0.255420i \(-0.917786\pi\)
−0.966830 + 0.255420i \(0.917786\pi\)
\(258\) 0 0
\(259\) 128.000 0.494208
\(260\) 0 0
\(261\) −105.000 301.720i −0.402299 1.15602i
\(262\) 0 0
\(263\) 248.475 0.944773 0.472387 0.881391i \(-0.343393\pi\)
0.472387 + 0.881391i \(0.343393\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 26.6224 + 157.500i 0.0997092 + 0.589888i
\(268\) 0 0
\(269\) 354.965i 1.31957i 0.751454 + 0.659786i \(0.229353\pi\)
−0.751454 + 0.659786i \(0.770647\pi\)
\(270\) 0 0
\(271\) −178.000 −0.656827 −0.328413 0.944534i \(-0.606514\pi\)
−0.328413 + 0.944534i \(0.606514\pi\)
\(272\) 0 0
\(273\) −47.3286 + 8.00000i −0.173365 + 0.0293040i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 202.000i 0.729242i 0.931156 + 0.364621i \(0.118801\pi\)
−0.931156 + 0.364621i \(0.881199\pi\)
\(278\) 0 0
\(279\) −391.000 + 136.070i −1.40143 + 0.487706i
\(280\) 0 0
\(281\) 70.9930i 0.252644i 0.991989 + 0.126322i \(0.0403173\pi\)
−0.991989 + 0.126322i \(0.959683\pi\)
\(282\) 0 0
\(283\) 197.000i 0.696113i 0.937474 + 0.348057i \(0.113158\pi\)
−0.937474 + 0.348057i \(0.886842\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 425.958 1.48417
\(288\) 0 0
\(289\) 26.0000 0.0899654
\(290\) 0 0
\(291\) −11.0000 65.0769i −0.0378007 0.223632i
\(292\) 0 0
\(293\) 425.958 1.45378 0.726890 0.686754i \(-0.240965\pi\)
0.726890 + 0.686754i \(0.240965\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −230.727 420.000i −0.776859 1.41414i
\(298\) 0 0
\(299\) 70.9930i 0.237435i
\(300\) 0 0
\(301\) 496.000 1.64784
\(302\) 0 0
\(303\) 70.9930 + 420.000i 0.234300 + 1.38614i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 127.000i 0.413681i 0.978375 + 0.206840i \(0.0663181\pi\)
−0.978375 + 0.206840i \(0.933682\pi\)
\(308\) 0 0
\(309\) −13.0000 76.9090i −0.0420712 0.248897i
\(310\) 0 0
\(311\) 283.972i 0.913093i −0.889700 0.456546i \(-0.849086\pi\)
0.889700 0.456546i \(-0.150914\pi\)
\(312\) 0 0
\(313\) 58.0000i 0.185304i −0.995699 0.0926518i \(-0.970466\pi\)
0.995699 0.0926518i \(-0.0295343\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 283.972 0.895810 0.447905 0.894081i \(-0.352170\pi\)
0.447905 + 0.894081i \(0.352170\pi\)
\(318\) 0 0
\(319\) 630.000 1.97492
\(320\) 0 0
\(321\) 52.5000 8.87412i 0.163551 0.0276452i
\(322\) 0 0
\(323\) 195.231 0.604429
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 520.615 88.0000i 1.59209 0.269113i
\(328\) 0 0
\(329\) 283.972i 0.863136i
\(330\) 0 0
\(331\) 257.000 0.776435 0.388218 0.921568i \(-0.373091\pi\)
0.388218 + 0.921568i \(0.373091\pi\)
\(332\) 0 0
\(333\) 47.3286 + 136.000i 0.142128 + 0.408408i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 127.000i 0.376855i 0.982087 + 0.188427i \(0.0603390\pi\)
−0.982087 + 0.188427i \(0.939661\pi\)
\(338\) 0 0
\(339\) −367.500 + 62.1188i −1.08407 + 0.183241i
\(340\) 0 0
\(341\) 816.419i 2.39419i
\(342\) 0 0
\(343\) 272.000i 0.793003i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 372.713 1.07410 0.537050 0.843550i \(-0.319538\pi\)
0.537050 + 0.843550i \(0.319538\pi\)
\(348\) 0 0
\(349\) 292.000 0.836676 0.418338 0.908291i \(-0.362613\pi\)
0.418338 + 0.908291i \(0.362613\pi\)
\(350\) 0 0
\(351\) −26.0000 47.3286i −0.0740741 0.134839i
\(352\) 0 0
\(353\) −638.937 −1.81002 −0.905009 0.425392i \(-0.860136\pi\)
−0.905009 + 0.425392i \(0.860136\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −70.9930 420.000i −0.198860 1.17647i
\(358\) 0 0
\(359\) 283.972i 0.791008i −0.918464 0.395504i \(-0.870570\pi\)
0.918464 0.395504i \(-0.129430\pi\)
\(360\) 0 0
\(361\) −240.000 −0.664820
\(362\) 0 0
\(363\) 573.860 97.0000i 1.58088 0.267218i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 668.000i 1.82016i −0.414429 0.910082i \(-0.636019\pi\)
0.414429 0.910082i \(-0.363981\pi\)
\(368\) 0 0
\(369\) 157.500 + 452.580i 0.426829 + 1.22650i
\(370\) 0 0
\(371\) 283.972i 0.765423i
\(372\) 0 0
\(373\) 242.000i 0.648794i 0.945921 + 0.324397i \(0.105161\pi\)
−0.945921 + 0.324397i \(0.894839\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 70.9930 0.188310
\(378\) 0 0
\(379\) −191.000 −0.503958 −0.251979 0.967733i \(-0.581081\pi\)
−0.251979 + 0.967733i \(0.581081\pi\)
\(380\) 0 0
\(381\) −53.0000 313.552i −0.139108 0.822972i
\(382\) 0 0
\(383\) 283.972 0.741441 0.370720 0.928745i \(-0.379111\pi\)
0.370720 + 0.928745i \(0.379111\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 183.398 + 527.000i 0.473898 + 1.36176i
\(388\) 0 0
\(389\) 603.440i 1.55126i 0.631188 + 0.775630i \(0.282568\pi\)
−0.631188 + 0.775630i \(0.717432\pi\)
\(390\) 0 0
\(391\) 630.000 1.61125
\(392\) 0 0
\(393\) 35.4965 + 210.000i 0.0903218 + 0.534351i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 772.000i 1.94458i 0.233769 + 0.972292i \(0.424894\pi\)
−0.233769 + 0.972292i \(0.575106\pi\)
\(398\) 0 0
\(399\) −44.0000 260.308i −0.110276 0.652400i
\(400\) 0 0
\(401\) 266.224i 0.663899i 0.943297 + 0.331950i \(0.107706\pi\)
−0.943297 + 0.331950i \(0.892294\pi\)
\(402\) 0 0
\(403\) 92.0000i 0.228288i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −283.972 −0.697719
\(408\) 0 0
\(409\) −701.000 −1.71394 −0.856968 0.515369i \(-0.827655\pi\)
−0.856968 + 0.515369i \(0.827655\pi\)
\(410\) 0 0
\(411\) −157.500 + 26.6224i −0.383212 + 0.0647746i
\(412\) 0 0
\(413\) 567.944 1.37517
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −375.671 + 63.5000i −0.900890 + 0.152278i
\(418\) 0 0
\(419\) 301.720i 0.720096i −0.932934 0.360048i \(-0.882760\pi\)
0.932934 0.360048i \(-0.117240\pi\)
\(420\) 0 0
\(421\) −148.000 −0.351544 −0.175772 0.984431i \(-0.556242\pi\)
−0.175772 + 0.984431i \(0.556242\pi\)
\(422\) 0 0
\(423\) −301.720 + 105.000i −0.713286 + 0.248227i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 128.000i 0.299766i
\(428\) 0 0
\(429\) 105.000 17.7482i 0.244755 0.0413712i
\(430\) 0 0
\(431\) 212.979i 0.494151i 0.968996 + 0.247075i \(0.0794695\pi\)
−0.968996 + 0.247075i \(0.920531\pi\)
\(432\) 0 0
\(433\) 463.000i 1.06928i −0.845079 0.534642i \(-0.820446\pi\)
0.845079 0.534642i \(-0.179554\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 390.461 0.893504
\(438\) 0 0
\(439\) 394.000 0.897494 0.448747 0.893659i \(-0.351870\pi\)
0.448747 + 0.893659i \(0.351870\pi\)
\(440\) 0 0
\(441\) −127.500 + 44.3706i −0.289116 + 0.100614i
\(442\) 0 0
\(443\) 656.685 1.48236 0.741179 0.671307i \(-0.234267\pi\)
0.741179 + 0.671307i \(0.234267\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 53.2447 + 315.000i 0.119116 + 0.704698i
\(448\) 0 0
\(449\) 124.238i 0.276699i −0.990383 0.138349i \(-0.955820\pi\)
0.990383 0.138349i \(-0.0441797\pi\)
\(450\) 0 0
\(451\) −945.000 −2.09534
\(452\) 0 0
\(453\) −485.119 + 82.0000i −1.07090 + 0.181015i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 509.000i 1.11379i −0.830584 0.556893i \(-0.811993\pi\)
0.830584 0.556893i \(-0.188007\pi\)
\(458\) 0 0
\(459\) 420.000 230.727i 0.915033 0.502673i
\(460\) 0 0
\(461\) 212.979i 0.461993i −0.972955 0.230997i \(-0.925801\pi\)
0.972955 0.230997i \(-0.0741986\pi\)
\(462\) 0 0
\(463\) 64.0000i 0.138229i −0.997609 0.0691145i \(-0.977983\pi\)
0.997609 0.0691145i \(-0.0220174\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −638.937 −1.36817 −0.684086 0.729401i \(-0.739799\pi\)
−0.684086 + 0.729401i \(0.739799\pi\)
\(468\) 0 0
\(469\) −904.000 −1.92751
\(470\) 0 0
\(471\) 19.0000 + 112.406i 0.0403397 + 0.238653i
\(472\) 0 0
\(473\) −1100.39 −2.32641
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 301.720 105.000i 0.632537 0.220126i
\(478\) 0 0
\(479\) 248.475i 0.518738i −0.965778 0.259369i \(-0.916485\pi\)
0.965778 0.259369i \(-0.0835145\pi\)
\(480\) 0 0
\(481\) −32.0000 −0.0665281
\(482\) 0 0
\(483\) −141.986 840.000i −0.293967 1.73913i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 382.000i 0.784394i 0.919881 + 0.392197i \(0.128285\pi\)
−0.919881 + 0.392197i \(0.871715\pi\)
\(488\) 0 0
\(489\) −23.5000 139.028i −0.0480573 0.284311i
\(490\) 0 0
\(491\) 709.930i 1.44589i 0.690908 + 0.722943i \(0.257211\pi\)
−0.690908 + 0.722943i \(0.742789\pi\)
\(492\) 0 0
\(493\) 630.000i 1.27789i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −851.915 −1.71412
\(498\) 0 0
\(499\) 94.0000 0.188377 0.0941884 0.995554i \(-0.469974\pi\)
0.0941884 + 0.995554i \(0.469974\pi\)
\(500\) 0 0
\(501\) 525.000 88.7412i 1.04790 0.177128i
\(502\) 0 0
\(503\) −780.923 −1.55253 −0.776265 0.630407i \(-0.782888\pi\)
−0.776265 + 0.630407i \(0.782888\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −488.077 + 82.5000i −0.962676 + 0.162722i
\(508\) 0 0
\(509\) 319.468i 0.627639i 0.949483 + 0.313820i \(0.101609\pi\)
−0.949483 + 0.313820i \(0.898391\pi\)
\(510\) 0 0
\(511\) 808.000 1.58121
\(512\) 0 0
\(513\) 260.308 143.000i 0.507422 0.278752i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 630.000i 1.21857i
\(518\) 0 0
\(519\) 210.000 35.4965i 0.404624 0.0683940i
\(520\) 0 0
\(521\) 798.671i 1.53296i 0.642270 + 0.766479i \(0.277993\pi\)
−0.642270 + 0.766479i \(0.722007\pi\)
\(522\) 0 0
\(523\) 467.000i 0.892925i 0.894802 + 0.446463i \(0.147316\pi\)
−0.894802 + 0.446463i \(0.852684\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 816.419 1.54918
\(528\) 0 0
\(529\) 731.000 1.38185
\(530\) 0 0
\(531\) 210.000 + 603.440i 0.395480 + 1.13642i
\(532\) 0 0
\(533\) −106.489 −0.199793
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −8.87412 52.5000i −0.0165254 0.0977654i
\(538\) 0 0
\(539\) 266.224i 0.493921i
\(540\) 0 0
\(541\) −976.000 −1.80407 −0.902033 0.431667i \(-0.857926\pi\)
−0.902033 + 0.431667i \(0.857926\pi\)
\(542\) 0 0
\(543\) 313.552 53.0000i 0.577444 0.0976059i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 779.000i 1.42413i −0.702113 0.712066i \(-0.747760\pi\)
0.702113 0.712066i \(-0.252240\pi\)
\(548\) 0 0
\(549\) −136.000 + 47.3286i −0.247723 + 0.0862088i
\(550\) 0 0
\(551\) 390.461i 0.708641i
\(552\) 0 0
\(553\) 544.000i 0.983725i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −745.426 −1.33829 −0.669144 0.743133i \(-0.733339\pi\)
−0.669144 + 0.743133i \(0.733339\pi\)
\(558\) 0 0
\(559\) −124.000 −0.221825
\(560\) 0 0
\(561\) 157.500 + 931.783i 0.280749 + 1.66093i
\(562\) 0 0
\(563\) −70.9930 −0.126098 −0.0630488 0.998010i \(-0.520082\pi\)
−0.0630488 + 0.998010i \(0.520082\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −402.293 508.000i −0.709512 0.895944i
\(568\) 0 0
\(569\) 976.153i 1.71556i −0.514018 0.857780i \(-0.671843\pi\)
0.514018 0.857780i \(-0.328157\pi\)
\(570\) 0 0
\(571\) −286.000 −0.500876 −0.250438 0.968133i \(-0.580575\pi\)
−0.250438 + 0.968133i \(0.580575\pi\)
\(572\) 0 0
\(573\) −124.238 735.000i −0.216820 1.28272i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 487.000i 0.844021i 0.906591 + 0.422010i \(0.138675\pi\)
−0.906591 + 0.422010i \(0.861325\pi\)
\(578\) 0 0
\(579\) 96.5000 + 570.902i 0.166667 + 0.986013i
\(580\) 0 0
\(581\) 141.986i 0.244382i
\(582\) 0 0
\(583\) 630.000i 1.08062i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 585.692 0.997772 0.498886 0.866668i \(-0.333743\pi\)
0.498886 + 0.866668i \(0.333743\pi\)
\(588\) 0 0
\(589\) 506.000 0.859083
\(590\) 0 0
\(591\) 105.000 17.7482i 0.177665 0.0300309i
\(592\) 0 0
\(593\) 88.7412 0.149648 0.0748239 0.997197i \(-0.476161\pi\)
0.0748239 + 0.997197i \(0.476161\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −11.8322 + 2.00000i −0.0198194 + 0.00335008i
\(598\) 0 0
\(599\) 638.937i 1.06667i 0.845903 + 0.533336i \(0.179062\pi\)
−0.845903 + 0.533336i \(0.820938\pi\)
\(600\) 0 0
\(601\) −271.000 −0.450915 −0.225458 0.974253i \(-0.572388\pi\)
−0.225458 + 0.974253i \(0.572388\pi\)
\(602\) 0 0
\(603\) −334.259 960.500i −0.554326 1.59287i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 404.000i 0.665568i −0.943003 0.332784i \(-0.892012\pi\)
0.943003 0.332784i \(-0.107988\pi\)
\(608\) 0 0
\(609\) 840.000 141.986i 1.37931 0.233146i
\(610\) 0 0
\(611\) 70.9930i 0.116191i
\(612\) 0 0
\(613\) 34.0000i 0.0554649i −0.999615 0.0277325i \(-0.991171\pi\)
0.999615 0.0277325i \(-0.00882865\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −425.958 −0.690369 −0.345185 0.938535i \(-0.612184\pi\)
−0.345185 + 0.938535i \(0.612184\pi\)
\(618\) 0 0
\(619\) −1046.00 −1.68982 −0.844911 0.534907i \(-0.820347\pi\)
−0.844911 + 0.534907i \(0.820347\pi\)
\(620\) 0 0
\(621\) 840.000 461.454i 1.35266 0.743082i
\(622\) 0 0
\(623\) −425.958 −0.683720
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 97.6153 + 577.500i 0.155686 + 0.921053i
\(628\) 0 0
\(629\) 283.972i 0.451466i
\(630\) 0 0
\(631\) −106.000 −0.167987 −0.0839937 0.996466i \(-0.526768\pi\)
−0.0839937 + 0.996466i \(0.526768\pi\)
\(632\) 0 0
\(633\) −618.230 + 104.500i −0.976667 + 0.165087i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 30.0000i 0.0470958i
\(638\) 0 0
\(639\) −315.000 905.160i −0.492958 1.41653i
\(640\) 0 0
\(641\) 212.979i 0.332260i 0.986104 + 0.166130i \(0.0531272\pi\)
−0.986104 + 0.166130i \(0.946873\pi\)
\(642\) 0 0
\(643\) 566.000i 0.880249i 0.897937 + 0.440124i \(0.145066\pi\)
−0.897937 + 0.440124i \(0.854934\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −496.951 −0.768085 −0.384042 0.923316i \(-0.625468\pi\)
−0.384042 + 0.923316i \(0.625468\pi\)
\(648\) 0 0
\(649\) −1260.00 −1.94145
\(650\) 0 0
\(651\) −184.000 1088.56i −0.282642 1.67213i
\(652\) 0 0
\(653\) 496.951 0.761027 0.380514 0.924775i \(-0.375747\pi\)
0.380514 + 0.924775i \(0.375747\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 298.762 + 858.500i 0.454737 + 1.30670i
\(658\) 0 0
\(659\) 550.195i 0.834894i 0.908701 + 0.417447i \(0.137075\pi\)
−0.908701 + 0.417447i \(0.862925\pi\)
\(660\) 0 0
\(661\) −298.000 −0.450832 −0.225416 0.974263i \(-0.572374\pi\)
−0.225416 + 0.974263i \(0.572374\pi\)
\(662\) 0 0
\(663\) 17.7482 + 105.000i 0.0267696 + 0.158371i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1260.00i 1.88906i
\(668\) 0 0
\(669\) 74.0000 + 437.790i 0.110613 + 0.654394i
\(670\) 0 0
\(671\) 283.972i 0.423207i
\(672\) 0 0
\(673\) 274.000i 0.407132i −0.979061 0.203566i \(-0.934747\pi\)
0.979061 0.203566i \(-0.0652532\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 993.901 1.46810 0.734048 0.679097i \(-0.237629\pi\)
0.734048 + 0.679097i \(0.237629\pi\)
\(678\) 0 0
\(679\) 176.000 0.259205
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 124.238 0.181900 0.0909500 0.995855i \(-0.471010\pi\)
0.0909500 + 0.995855i \(0.471010\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 343.133 58.0000i 0.499465 0.0844250i
\(688\) 0 0
\(689\) 70.9930i 0.103038i
\(690\) 0 0
\(691\) −373.000 −0.539797 −0.269899 0.962889i \(-0.586990\pi\)
−0.269899 + 0.962889i \(0.586990\pi\)
\(692\) 0 0
\(693\) 1206.88 420.000i 1.74153 0.606061i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 945.000i 1.35581i
\(698\) 0 0
\(699\) 630.000 106.489i 0.901288 0.152345i
\(700\) 0 0
\(701\) 674.433i 0.962101i −0.876693 0.481051i \(-0.840255\pi\)
0.876693 0.481051i \(-0.159745\pi\)
\(702\) 0 0
\(703\) 176.000i 0.250356i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1135.89 −1.60663
\(708\) 0 0
\(709\) 184.000 0.259520 0.129760 0.991545i \(-0.458579\pi\)
0.129760 + 0.991545i \(0.458579\pi\)
\(710\) 0 0
\(711\) −578.000 + 201.147i −0.812940 + 0.282907i
\(712\) 0 0
\(713\) 1632.84 2.29010
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 88.7412 + 525.000i 0.123767 + 0.732218i
\(718\) 0 0
\(719\) 638.937i 0.888646i 0.895867 + 0.444323i \(0.146556\pi\)
−0.895867 + 0.444323i \(0.853444\pi\)
\(720\) 0 0
\(721\) 208.000 0.288488
\(722\) 0 0
\(723\) −174.524 + 29.5000i −0.241389 + 0.0408022i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 758.000i 1.04264i −0.853361 0.521320i \(-0.825440\pi\)
0.853361 0.521320i \(-0.174560\pi\)
\(728\) 0 0
\(729\) 391.000 615.272i 0.536351 0.843995i
\(730\) 0 0
\(731\) 1100.39i 1.50532i
\(732\) 0 0
\(733\) 26.0000i 0.0354707i 0.999843 + 0.0177353i \(0.00564563\pi\)
−0.999843 + 0.0177353i \(0.994354\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2005.55 2.72124
\(738\) 0 0
\(739\) −1298.00 −1.75643 −0.878214 0.478268i \(-0.841265\pi\)
−0.878214 + 0.478268i \(0.841265\pi\)
\(740\) 0 0
\(741\) 11.0000 + 65.0769i 0.0148448 + 0.0878230i
\(742\) 0 0
\(743\) 532.447 0.716618 0.358309 0.933603i \(-0.383353\pi\)
0.358309 + 0.933603i \(0.383353\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 150.860 52.5000i 0.201955 0.0702811i
\(748\) 0 0
\(749\) 141.986i 0.189567i
\(750\) 0 0
\(751\) −478.000 −0.636485 −0.318242 0.948009i \(-0.603093\pi\)
−0.318242 + 0.948009i \(0.603093\pi\)
\(752\) 0 0
\(753\) −62.1188 367.500i −0.0824951 0.488048i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1072.00i 1.41612i 0.706154 + 0.708058i \(0.250429\pi\)
−0.706154 + 0.708058i \(0.749571\pi\)
\(758\) 0 0
\(759\) 315.000 + 1863.57i 0.415020 + 2.45529i
\(760\) 0 0
\(761\) 550.195i 0.722990i −0.932374 0.361495i \(-0.882266\pi\)
0.932374 0.361495i \(-0.117734\pi\)
\(762\) 0 0
\(763\) 1408.00i 1.84535i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −141.986 −0.185119
\(768\) 0 0
\(769\) 169.000 0.219766 0.109883 0.993945i \(-0.464952\pi\)
0.109883 + 0.993945i \(0.464952\pi\)
\(770\) 0 0
\(771\) 1470.00 248.475i 1.90661 0.322277i
\(772\) 0 0
\(773\) −461.454 −0.596965 −0.298483 0.954415i \(-0.596481\pi\)
−0.298483 + 0.954415i \(0.596481\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −378.629 + 64.0000i −0.487296 + 0.0823681i
\(778\) 0 0
\(779\) 585.692i 0.751851i
\(780\) 0 0
\(781\) 1890.00 2.41997
\(782\) 0 0
\(783\) 461.454 + 840.000i 0.589341 + 1.07280i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 878.000i 1.11563i −0.829966 0.557814i \(-0.811640\pi\)
0.829966 0.557814i \(-0.188360\pi\)
\(788\) 0 0
\(789\) −735.000 + 124.238i −0.931559 + 0.157462i
\(790\) 0 0
\(791\) 993.901i 1.25651i
\(792\) 0 0
\(793\) 32.0000i 0.0403531i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 816.419 1.02437 0.512183 0.858877i \(-0.328837\pi\)
0.512183 + 0.858877i \(0.328837\pi\)
\(798\) 0 0
\(799\) 630.000 0.788486
\(800\) 0 0
\(801\) −157.500 452.580i −0.196629 0.565019i
\(802\) 0 0