Properties

Label 2475.2.a.o.1.2
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.30278 q^{2} -0.302776 q^{4} +0.697224 q^{7} -3.00000 q^{8} +O(q^{10})\) \(q+1.30278 q^{2} -0.302776 q^{4} +0.697224 q^{7} -3.00000 q^{8} +1.00000 q^{11} +5.00000 q^{13} +0.908327 q^{14} -3.30278 q^{16} -6.90833 q^{17} -1.00000 q^{19} +1.30278 q^{22} +7.30278 q^{23} +6.51388 q^{26} -0.211103 q^{28} -0.908327 q^{29} +10.2111 q^{31} +1.69722 q^{32} -9.00000 q^{34} +2.39445 q^{37} -1.30278 q^{38} +5.60555 q^{41} +7.21110 q^{43} -0.302776 q^{44} +9.51388 q^{46} +3.00000 q^{47} -6.51388 q^{49} -1.51388 q^{52} +1.30278 q^{53} -2.09167 q^{56} -1.18335 q^{58} +14.2111 q^{59} -7.90833 q^{61} +13.3028 q^{62} +8.81665 q^{64} -4.00000 q^{67} +2.09167 q^{68} +2.60555 q^{71} -7.90833 q^{73} +3.11943 q^{74} +0.302776 q^{76} +0.697224 q^{77} -10.9083 q^{79} +7.30278 q^{82} +3.51388 q^{83} +9.39445 q^{86} -3.00000 q^{88} -1.69722 q^{89} +3.48612 q^{91} -2.21110 q^{92} +3.90833 q^{94} +15.3028 q^{97} -8.48612 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + 3q^{4} + 5q^{7} - 6q^{8} + O(q^{10}) \) \( 2q - q^{2} + 3q^{4} + 5q^{7} - 6q^{8} + 2q^{11} + 10q^{13} - 9q^{14} - 3q^{16} - 3q^{17} - 2q^{19} - q^{22} + 11q^{23} - 5q^{26} + 14q^{28} + 9q^{29} + 6q^{31} + 7q^{32} - 18q^{34} + 12q^{37} + q^{38} + 4q^{41} + 3q^{44} + q^{46} + 6q^{47} + 5q^{49} + 15q^{52} - q^{53} - 15q^{56} - 24q^{58} + 14q^{59} - 5q^{61} + 23q^{62} - 4q^{64} - 8q^{67} + 15q^{68} - 2q^{71} - 5q^{73} - 19q^{74} - 3q^{76} + 5q^{77} - 11q^{79} + 11q^{82} - 11q^{83} + 26q^{86} - 6q^{88} - 7q^{89} + 25q^{91} + 10q^{92} - 3q^{94} + 27q^{97} - 35q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.30278 0.921201 0.460601 0.887607i \(-0.347634\pi\)
0.460601 + 0.887607i \(0.347634\pi\)
\(3\) 0 0
\(4\) −0.302776 −0.151388
\(5\) 0 0
\(6\) 0 0
\(7\) 0.697224 0.263526 0.131763 0.991281i \(-0.457936\pi\)
0.131763 + 0.991281i \(0.457936\pi\)
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 5.00000 1.38675 0.693375 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0.908327 0.242761
\(15\) 0 0
\(16\) −3.30278 −0.825694
\(17\) −6.90833 −1.67552 −0.837758 0.546042i \(-0.816134\pi\)
−0.837758 + 0.546042i \(0.816134\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.30278 0.277753
\(23\) 7.30278 1.52273 0.761367 0.648321i \(-0.224529\pi\)
0.761367 + 0.648321i \(0.224529\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 6.51388 1.27748
\(27\) 0 0
\(28\) −0.211103 −0.0398946
\(29\) −0.908327 −0.168672 −0.0843360 0.996437i \(-0.526877\pi\)
−0.0843360 + 0.996437i \(0.526877\pi\)
\(30\) 0 0
\(31\) 10.2111 1.83397 0.916984 0.398924i \(-0.130616\pi\)
0.916984 + 0.398924i \(0.130616\pi\)
\(32\) 1.69722 0.300030
\(33\) 0 0
\(34\) −9.00000 −1.54349
\(35\) 0 0
\(36\) 0 0
\(37\) 2.39445 0.393645 0.196822 0.980439i \(-0.436938\pi\)
0.196822 + 0.980439i \(0.436938\pi\)
\(38\) −1.30278 −0.211338
\(39\) 0 0
\(40\) 0 0
\(41\) 5.60555 0.875440 0.437720 0.899111i \(-0.355786\pi\)
0.437720 + 0.899111i \(0.355786\pi\)
\(42\) 0 0
\(43\) 7.21110 1.09968 0.549841 0.835269i \(-0.314688\pi\)
0.549841 + 0.835269i \(0.314688\pi\)
\(44\) −0.302776 −0.0456451
\(45\) 0 0
\(46\) 9.51388 1.40274
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) −6.51388 −0.930554
\(50\) 0 0
\(51\) 0 0
\(52\) −1.51388 −0.209937
\(53\) 1.30278 0.178950 0.0894750 0.995989i \(-0.471481\pi\)
0.0894750 + 0.995989i \(0.471481\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.09167 −0.279512
\(57\) 0 0
\(58\) −1.18335 −0.155381
\(59\) 14.2111 1.85013 0.925064 0.379811i \(-0.124011\pi\)
0.925064 + 0.379811i \(0.124011\pi\)
\(60\) 0 0
\(61\) −7.90833 −1.01256 −0.506279 0.862370i \(-0.668979\pi\)
−0.506279 + 0.862370i \(0.668979\pi\)
\(62\) 13.3028 1.68945
\(63\) 0 0
\(64\) 8.81665 1.10208
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 2.09167 0.253653
\(69\) 0 0
\(70\) 0 0
\(71\) 2.60555 0.309222 0.154611 0.987975i \(-0.450588\pi\)
0.154611 + 0.987975i \(0.450588\pi\)
\(72\) 0 0
\(73\) −7.90833 −0.925600 −0.462800 0.886463i \(-0.653155\pi\)
−0.462800 + 0.886463i \(0.653155\pi\)
\(74\) 3.11943 0.362626
\(75\) 0 0
\(76\) 0.302776 0.0347307
\(77\) 0.697224 0.0794561
\(78\) 0 0
\(79\) −10.9083 −1.22728 −0.613641 0.789585i \(-0.710296\pi\)
−0.613641 + 0.789585i \(0.710296\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 7.30278 0.806457
\(83\) 3.51388 0.385698 0.192849 0.981228i \(-0.438227\pi\)
0.192849 + 0.981228i \(0.438227\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 9.39445 1.01303
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) −1.69722 −0.179905 −0.0899527 0.995946i \(-0.528672\pi\)
−0.0899527 + 0.995946i \(0.528672\pi\)
\(90\) 0 0
\(91\) 3.48612 0.365445
\(92\) −2.21110 −0.230523
\(93\) 0 0
\(94\) 3.90833 0.403113
\(95\) 0 0
\(96\) 0 0
\(97\) 15.3028 1.55376 0.776881 0.629648i \(-0.216801\pi\)
0.776881 + 0.629648i \(0.216801\pi\)
\(98\) −8.48612 −0.857228
\(99\) 0 0
\(100\) 0 0
\(101\) 0.513878 0.0511328 0.0255664 0.999673i \(-0.491861\pi\)
0.0255664 + 0.999673i \(0.491861\pi\)
\(102\) 0 0
\(103\) 2.90833 0.286566 0.143283 0.989682i \(-0.454234\pi\)
0.143283 + 0.989682i \(0.454234\pi\)
\(104\) −15.0000 −1.47087
\(105\) 0 0
\(106\) 1.69722 0.164849
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 0 0
\(109\) 11.5139 1.10283 0.551415 0.834231i \(-0.314088\pi\)
0.551415 + 0.834231i \(0.314088\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.30278 −0.217592
\(113\) 10.8167 1.01755 0.508773 0.860901i \(-0.330099\pi\)
0.508773 + 0.860901i \(0.330099\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.275019 0.0255349
\(117\) 0 0
\(118\) 18.5139 1.70434
\(119\) −4.81665 −0.441542
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −10.3028 −0.932769
\(123\) 0 0
\(124\) −3.09167 −0.277640
\(125\) 0 0
\(126\) 0 0
\(127\) 8.11943 0.720483 0.360241 0.932859i \(-0.382694\pi\)
0.360241 + 0.932859i \(0.382694\pi\)
\(128\) 8.09167 0.715210
\(129\) 0 0
\(130\) 0 0
\(131\) 9.90833 0.865695 0.432847 0.901467i \(-0.357509\pi\)
0.432847 + 0.901467i \(0.357509\pi\)
\(132\) 0 0
\(133\) −0.697224 −0.0604570
\(134\) −5.21110 −0.450171
\(135\) 0 0
\(136\) 20.7250 1.77715
\(137\) 12.9083 1.10283 0.551416 0.834230i \(-0.314088\pi\)
0.551416 + 0.834230i \(0.314088\pi\)
\(138\) 0 0
\(139\) −6.21110 −0.526819 −0.263409 0.964684i \(-0.584847\pi\)
−0.263409 + 0.964684i \(0.584847\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 3.39445 0.284856
\(143\) 5.00000 0.418121
\(144\) 0 0
\(145\) 0 0
\(146\) −10.3028 −0.852664
\(147\) 0 0
\(148\) −0.724981 −0.0595930
\(149\) −17.2111 −1.40999 −0.704994 0.709213i \(-0.749050\pi\)
−0.704994 + 0.709213i \(0.749050\pi\)
\(150\) 0 0
\(151\) 0.816654 0.0664583 0.0332292 0.999448i \(-0.489421\pi\)
0.0332292 + 0.999448i \(0.489421\pi\)
\(152\) 3.00000 0.243332
\(153\) 0 0
\(154\) 0.908327 0.0731951
\(155\) 0 0
\(156\) 0 0
\(157\) 19.2111 1.53321 0.766606 0.642117i \(-0.221944\pi\)
0.766606 + 0.642117i \(0.221944\pi\)
\(158\) −14.2111 −1.13057
\(159\) 0 0
\(160\) 0 0
\(161\) 5.09167 0.401280
\(162\) 0 0
\(163\) 9.30278 0.728650 0.364325 0.931272i \(-0.381300\pi\)
0.364325 + 0.931272i \(0.381300\pi\)
\(164\) −1.69722 −0.132531
\(165\) 0 0
\(166\) 4.57779 0.355306
\(167\) −13.4222 −1.03864 −0.519321 0.854579i \(-0.673815\pi\)
−0.519321 + 0.854579i \(0.673815\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) −2.18335 −0.166479
\(173\) 4.81665 0.366203 0.183102 0.983094i \(-0.441386\pi\)
0.183102 + 0.983094i \(0.441386\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.30278 −0.248956
\(177\) 0 0
\(178\) −2.21110 −0.165729
\(179\) −12.5139 −0.935331 −0.467666 0.883905i \(-0.654905\pi\)
−0.467666 + 0.883905i \(0.654905\pi\)
\(180\) 0 0
\(181\) −19.9083 −1.47977 −0.739887 0.672731i \(-0.765121\pi\)
−0.739887 + 0.672731i \(0.765121\pi\)
\(182\) 4.54163 0.336648
\(183\) 0 0
\(184\) −21.9083 −1.61510
\(185\) 0 0
\(186\) 0 0
\(187\) −6.90833 −0.505187
\(188\) −0.908327 −0.0662465
\(189\) 0 0
\(190\) 0 0
\(191\) −10.3028 −0.745483 −0.372741 0.927935i \(-0.621582\pi\)
−0.372741 + 0.927935i \(0.621582\pi\)
\(192\) 0 0
\(193\) 13.2111 0.950956 0.475478 0.879728i \(-0.342275\pi\)
0.475478 + 0.879728i \(0.342275\pi\)
\(194\) 19.9361 1.43133
\(195\) 0 0
\(196\) 1.97224 0.140875
\(197\) −13.3028 −0.947784 −0.473892 0.880583i \(-0.657151\pi\)
−0.473892 + 0.880583i \(0.657151\pi\)
\(198\) 0 0
\(199\) −6.48612 −0.459789 −0.229894 0.973216i \(-0.573838\pi\)
−0.229894 + 0.973216i \(0.573838\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0.669468 0.0471036
\(203\) −0.633308 −0.0444495
\(204\) 0 0
\(205\) 0 0
\(206\) 3.78890 0.263985
\(207\) 0 0
\(208\) −16.5139 −1.14503
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −25.2389 −1.73751 −0.868757 0.495238i \(-0.835081\pi\)
−0.868757 + 0.495238i \(0.835081\pi\)
\(212\) −0.394449 −0.0270908
\(213\) 0 0
\(214\) 3.90833 0.267168
\(215\) 0 0
\(216\) 0 0
\(217\) 7.11943 0.483298
\(218\) 15.0000 1.01593
\(219\) 0 0
\(220\) 0 0
\(221\) −34.5416 −2.32352
\(222\) 0 0
\(223\) −22.6333 −1.51564 −0.757819 0.652465i \(-0.773735\pi\)
−0.757819 + 0.652465i \(0.773735\pi\)
\(224\) 1.18335 0.0790656
\(225\) 0 0
\(226\) 14.0917 0.937364
\(227\) −1.69722 −0.112649 −0.0563244 0.998413i \(-0.517938\pi\)
−0.0563244 + 0.998413i \(0.517938\pi\)
\(228\) 0 0
\(229\) −18.7250 −1.23738 −0.618691 0.785635i \(-0.712337\pi\)
−0.618691 + 0.785635i \(0.712337\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.72498 0.178904
\(233\) −15.9083 −1.04219 −0.521095 0.853499i \(-0.674476\pi\)
−0.521095 + 0.853499i \(0.674476\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −4.30278 −0.280087
\(237\) 0 0
\(238\) −6.27502 −0.406749
\(239\) −21.1194 −1.36610 −0.683051 0.730371i \(-0.739347\pi\)
−0.683051 + 0.730371i \(0.739347\pi\)
\(240\) 0 0
\(241\) 21.9361 1.41303 0.706514 0.707699i \(-0.250267\pi\)
0.706514 + 0.707699i \(0.250267\pi\)
\(242\) 1.30278 0.0837456
\(243\) 0 0
\(244\) 2.39445 0.153289
\(245\) 0 0
\(246\) 0 0
\(247\) −5.00000 −0.318142
\(248\) −30.6333 −1.94522
\(249\) 0 0
\(250\) 0 0
\(251\) −6.90833 −0.436050 −0.218025 0.975943i \(-0.569961\pi\)
−0.218025 + 0.975943i \(0.569961\pi\)
\(252\) 0 0
\(253\) 7.30278 0.459122
\(254\) 10.5778 0.663710
\(255\) 0 0
\(256\) −7.09167 −0.443230
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 1.66947 0.103736
\(260\) 0 0
\(261\) 0 0
\(262\) 12.9083 0.797479
\(263\) 22.8167 1.40694 0.703468 0.710727i \(-0.251634\pi\)
0.703468 + 0.710727i \(0.251634\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.908327 −0.0556931
\(267\) 0 0
\(268\) 1.21110 0.0739799
\(269\) −8.72498 −0.531971 −0.265986 0.963977i \(-0.585697\pi\)
−0.265986 + 0.963977i \(0.585697\pi\)
\(270\) 0 0
\(271\) −0.211103 −0.0128236 −0.00641178 0.999979i \(-0.502041\pi\)
−0.00641178 + 0.999979i \(0.502041\pi\)
\(272\) 22.8167 1.38346
\(273\) 0 0
\(274\) 16.8167 1.01593
\(275\) 0 0
\(276\) 0 0
\(277\) 14.3944 0.864879 0.432439 0.901663i \(-0.357653\pi\)
0.432439 + 0.901663i \(0.357653\pi\)
\(278\) −8.09167 −0.485306
\(279\) 0 0
\(280\) 0 0
\(281\) 1.18335 0.0705925 0.0352963 0.999377i \(-0.488763\pi\)
0.0352963 + 0.999377i \(0.488763\pi\)
\(282\) 0 0
\(283\) 6.30278 0.374661 0.187331 0.982297i \(-0.440016\pi\)
0.187331 + 0.982297i \(0.440016\pi\)
\(284\) −0.788897 −0.0468125
\(285\) 0 0
\(286\) 6.51388 0.385174
\(287\) 3.90833 0.230701
\(288\) 0 0
\(289\) 30.7250 1.80735
\(290\) 0 0
\(291\) 0 0
\(292\) 2.39445 0.140125
\(293\) −0.788897 −0.0460879 −0.0230439 0.999734i \(-0.507336\pi\)
−0.0230439 + 0.999734i \(0.507336\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −7.18335 −0.417524
\(297\) 0 0
\(298\) −22.4222 −1.29888
\(299\) 36.5139 2.11165
\(300\) 0 0
\(301\) 5.02776 0.289795
\(302\) 1.06392 0.0612215
\(303\) 0 0
\(304\) 3.30278 0.189427
\(305\) 0 0
\(306\) 0 0
\(307\) −16.9083 −0.965009 −0.482505 0.875893i \(-0.660273\pi\)
−0.482505 + 0.875893i \(0.660273\pi\)
\(308\) −0.211103 −0.0120287
\(309\) 0 0
\(310\) 0 0
\(311\) −4.81665 −0.273127 −0.136564 0.990631i \(-0.543606\pi\)
−0.136564 + 0.990631i \(0.543606\pi\)
\(312\) 0 0
\(313\) 0.183346 0.0103633 0.00518167 0.999987i \(-0.498351\pi\)
0.00518167 + 0.999987i \(0.498351\pi\)
\(314\) 25.0278 1.41240
\(315\) 0 0
\(316\) 3.30278 0.185796
\(317\) 0.908327 0.0510167 0.0255084 0.999675i \(-0.491880\pi\)
0.0255084 + 0.999675i \(0.491880\pi\)
\(318\) 0 0
\(319\) −0.908327 −0.0508565
\(320\) 0 0
\(321\) 0 0
\(322\) 6.63331 0.369660
\(323\) 6.90833 0.384390
\(324\) 0 0
\(325\) 0 0
\(326\) 12.1194 0.671233
\(327\) 0 0
\(328\) −16.8167 −0.928544
\(329\) 2.09167 0.115318
\(330\) 0 0
\(331\) −21.6056 −1.18755 −0.593774 0.804632i \(-0.702363\pi\)
−0.593774 + 0.804632i \(0.702363\pi\)
\(332\) −1.06392 −0.0583900
\(333\) 0 0
\(334\) −17.4861 −0.956798
\(335\) 0 0
\(336\) 0 0
\(337\) −30.8444 −1.68020 −0.840101 0.542430i \(-0.817504\pi\)
−0.840101 + 0.542430i \(0.817504\pi\)
\(338\) 15.6333 0.850340
\(339\) 0 0
\(340\) 0 0
\(341\) 10.2111 0.552962
\(342\) 0 0
\(343\) −9.42221 −0.508751
\(344\) −21.6333 −1.16639
\(345\) 0 0
\(346\) 6.27502 0.337347
\(347\) 12.5139 0.671780 0.335890 0.941901i \(-0.390963\pi\)
0.335890 + 0.941901i \(0.390963\pi\)
\(348\) 0 0
\(349\) −5.18335 −0.277458 −0.138729 0.990330i \(-0.544302\pi\)
−0.138729 + 0.990330i \(0.544302\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.69722 0.0904624
\(353\) −18.6333 −0.991751 −0.495875 0.868394i \(-0.665153\pi\)
−0.495875 + 0.868394i \(0.665153\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0.513878 0.0272355
\(357\) 0 0
\(358\) −16.3028 −0.861628
\(359\) −0.788897 −0.0416364 −0.0208182 0.999783i \(-0.506627\pi\)
−0.0208182 + 0.999783i \(0.506627\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −25.9361 −1.36317
\(363\) 0 0
\(364\) −1.05551 −0.0553239
\(365\) 0 0
\(366\) 0 0
\(367\) −20.6972 −1.08039 −0.540193 0.841541i \(-0.681649\pi\)
−0.540193 + 0.841541i \(0.681649\pi\)
\(368\) −24.1194 −1.25731
\(369\) 0 0
\(370\) 0 0
\(371\) 0.908327 0.0471580
\(372\) 0 0
\(373\) 27.4222 1.41987 0.709934 0.704268i \(-0.248725\pi\)
0.709934 + 0.704268i \(0.248725\pi\)
\(374\) −9.00000 −0.465379
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) −4.54163 −0.233906
\(378\) 0 0
\(379\) 3.18335 0.163518 0.0817588 0.996652i \(-0.473946\pi\)
0.0817588 + 0.996652i \(0.473946\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −13.4222 −0.686740
\(383\) −21.6333 −1.10541 −0.552705 0.833377i \(-0.686404\pi\)
−0.552705 + 0.833377i \(0.686404\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 17.2111 0.876022
\(387\) 0 0
\(388\) −4.63331 −0.235221
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) −50.4500 −2.55136
\(392\) 19.5416 0.987002
\(393\) 0 0
\(394\) −17.3305 −0.873100
\(395\) 0 0
\(396\) 0 0
\(397\) 21.6972 1.08895 0.544476 0.838776i \(-0.316728\pi\)
0.544476 + 0.838776i \(0.316728\pi\)
\(398\) −8.44996 −0.423558
\(399\) 0 0
\(400\) 0 0
\(401\) 12.7889 0.638647 0.319324 0.947646i \(-0.396544\pi\)
0.319324 + 0.947646i \(0.396544\pi\)
\(402\) 0 0
\(403\) 51.0555 2.54326
\(404\) −0.155590 −0.00774088
\(405\) 0 0
\(406\) −0.825058 −0.0409469
\(407\) 2.39445 0.118688
\(408\) 0 0
\(409\) −6.21110 −0.307119 −0.153560 0.988139i \(-0.549074\pi\)
−0.153560 + 0.988139i \(0.549074\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −0.880571 −0.0433826
\(413\) 9.90833 0.487557
\(414\) 0 0
\(415\) 0 0
\(416\) 8.48612 0.416066
\(417\) 0 0
\(418\) −1.30278 −0.0637208
\(419\) −6.39445 −0.312389 −0.156195 0.987726i \(-0.549923\pi\)
−0.156195 + 0.987726i \(0.549923\pi\)
\(420\) 0 0
\(421\) 0.697224 0.0339806 0.0169903 0.999856i \(-0.494592\pi\)
0.0169903 + 0.999856i \(0.494592\pi\)
\(422\) −32.8806 −1.60060
\(423\) 0 0
\(424\) −3.90833 −0.189805
\(425\) 0 0
\(426\) 0 0
\(427\) −5.51388 −0.266835
\(428\) −0.908327 −0.0439056
\(429\) 0 0
\(430\) 0 0
\(431\) −33.0000 −1.58955 −0.794777 0.606902i \(-0.792412\pi\)
−0.794777 + 0.606902i \(0.792412\pi\)
\(432\) 0 0
\(433\) 5.00000 0.240285 0.120142 0.992757i \(-0.461665\pi\)
0.120142 + 0.992757i \(0.461665\pi\)
\(434\) 9.27502 0.445215
\(435\) 0 0
\(436\) −3.48612 −0.166955
\(437\) −7.30278 −0.349339
\(438\) 0 0
\(439\) 24.3028 1.15991 0.579954 0.814649i \(-0.303070\pi\)
0.579954 + 0.814649i \(0.303070\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −45.0000 −2.14043
\(443\) 8.60555 0.408862 0.204431 0.978881i \(-0.434466\pi\)
0.204431 + 0.978881i \(0.434466\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −29.4861 −1.39621
\(447\) 0 0
\(448\) 6.14719 0.290427
\(449\) −23.4861 −1.10838 −0.554189 0.832391i \(-0.686972\pi\)
−0.554189 + 0.832391i \(0.686972\pi\)
\(450\) 0 0
\(451\) 5.60555 0.263955
\(452\) −3.27502 −0.154044
\(453\) 0 0
\(454\) −2.21110 −0.103772
\(455\) 0 0
\(456\) 0 0
\(457\) −20.6972 −0.968175 −0.484088 0.875020i \(-0.660848\pi\)
−0.484088 + 0.875020i \(0.660848\pi\)
\(458\) −24.3944 −1.13988
\(459\) 0 0
\(460\) 0 0
\(461\) −32.2111 −1.50022 −0.750110 0.661313i \(-0.770000\pi\)
−0.750110 + 0.661313i \(0.770000\pi\)
\(462\) 0 0
\(463\) 11.7889 0.547877 0.273938 0.961747i \(-0.411674\pi\)
0.273938 + 0.961747i \(0.411674\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −20.7250 −0.960066
\(467\) 18.6333 0.862247 0.431123 0.902293i \(-0.358117\pi\)
0.431123 + 0.902293i \(0.358117\pi\)
\(468\) 0 0
\(469\) −2.78890 −0.128779
\(470\) 0 0
\(471\) 0 0
\(472\) −42.6333 −1.96236
\(473\) 7.21110 0.331567
\(474\) 0 0
\(475\) 0 0
\(476\) 1.45837 0.0668441
\(477\) 0 0
\(478\) −27.5139 −1.25846
\(479\) 34.8167 1.59081 0.795407 0.606076i \(-0.207257\pi\)
0.795407 + 0.606076i \(0.207257\pi\)
\(480\) 0 0
\(481\) 11.9722 0.545887
\(482\) 28.5778 1.30168
\(483\) 0 0
\(484\) −0.302776 −0.0137625
\(485\) 0 0
\(486\) 0 0
\(487\) 4.21110 0.190823 0.0954116 0.995438i \(-0.469583\pi\)
0.0954116 + 0.995438i \(0.469583\pi\)
\(488\) 23.7250 1.07398
\(489\) 0 0
\(490\) 0 0
\(491\) 9.78890 0.441767 0.220883 0.975300i \(-0.429106\pi\)
0.220883 + 0.975300i \(0.429106\pi\)
\(492\) 0 0
\(493\) 6.27502 0.282613
\(494\) −6.51388 −0.293073
\(495\) 0 0
\(496\) −33.7250 −1.51430
\(497\) 1.81665 0.0814881
\(498\) 0 0
\(499\) −3.48612 −0.156060 −0.0780301 0.996951i \(-0.524863\pi\)
−0.0780301 + 0.996951i \(0.524863\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −9.00000 −0.401690
\(503\) −9.39445 −0.418878 −0.209439 0.977822i \(-0.567164\pi\)
−0.209439 + 0.977822i \(0.567164\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 9.51388 0.422943
\(507\) 0 0
\(508\) −2.45837 −0.109072
\(509\) 22.6972 1.00604 0.503018 0.864276i \(-0.332223\pi\)
0.503018 + 0.864276i \(0.332223\pi\)
\(510\) 0 0
\(511\) −5.51388 −0.243920
\(512\) −25.4222 −1.12351
\(513\) 0 0
\(514\) 23.4500 1.03433
\(515\) 0 0
\(516\) 0 0
\(517\) 3.00000 0.131940
\(518\) 2.17494 0.0955615
\(519\) 0 0
\(520\) 0 0
\(521\) 41.4500 1.81596 0.907978 0.419018i \(-0.137626\pi\)
0.907978 + 0.419018i \(0.137626\pi\)
\(522\) 0 0
\(523\) −32.4222 −1.41772 −0.708862 0.705347i \(-0.750791\pi\)
−0.708862 + 0.705347i \(0.750791\pi\)
\(524\) −3.00000 −0.131056
\(525\) 0 0
\(526\) 29.7250 1.29607
\(527\) −70.5416 −3.07284
\(528\) 0 0
\(529\) 30.3305 1.31872
\(530\) 0 0
\(531\) 0 0
\(532\) 0.211103 0.00915246
\(533\) 28.0278 1.21402
\(534\) 0 0
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) −11.3667 −0.490053
\(539\) −6.51388 −0.280573
\(540\) 0 0
\(541\) 25.7250 1.10600 0.553002 0.833180i \(-0.313482\pi\)
0.553002 + 0.833180i \(0.313482\pi\)
\(542\) −0.275019 −0.0118131
\(543\) 0 0
\(544\) −11.7250 −0.502704
\(545\) 0 0
\(546\) 0 0
\(547\) −7.11943 −0.304405 −0.152202 0.988349i \(-0.548637\pi\)
−0.152202 + 0.988349i \(0.548637\pi\)
\(548\) −3.90833 −0.166955
\(549\) 0 0
\(550\) 0 0
\(551\) 0.908327 0.0386960
\(552\) 0 0
\(553\) −7.60555 −0.323421
\(554\) 18.7527 0.796727
\(555\) 0 0
\(556\) 1.88057 0.0797540
\(557\) −19.4222 −0.822945 −0.411473 0.911422i \(-0.634985\pi\)
−0.411473 + 0.911422i \(0.634985\pi\)
\(558\) 0 0
\(559\) 36.0555 1.52499
\(560\) 0 0
\(561\) 0 0
\(562\) 1.54163 0.0650299
\(563\) −8.09167 −0.341023 −0.170512 0.985356i \(-0.554542\pi\)
−0.170512 + 0.985356i \(0.554542\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 8.21110 0.345138
\(567\) 0 0
\(568\) −7.81665 −0.327980
\(569\) 46.1472 1.93459 0.967295 0.253653i \(-0.0816321\pi\)
0.967295 + 0.253653i \(0.0816321\pi\)
\(570\) 0 0
\(571\) 22.3305 0.934504 0.467252 0.884124i \(-0.345244\pi\)
0.467252 + 0.884124i \(0.345244\pi\)
\(572\) −1.51388 −0.0632984
\(573\) 0 0
\(574\) 5.09167 0.212522
\(575\) 0 0
\(576\) 0 0
\(577\) 44.3583 1.84666 0.923330 0.384008i \(-0.125456\pi\)
0.923330 + 0.384008i \(0.125456\pi\)
\(578\) 40.0278 1.66494
\(579\) 0 0
\(580\) 0 0
\(581\) 2.44996 0.101642
\(582\) 0 0
\(583\) 1.30278 0.0539555
\(584\) 23.7250 0.981747
\(585\) 0 0
\(586\) −1.02776 −0.0424562
\(587\) −16.5416 −0.682746 −0.341373 0.939928i \(-0.610892\pi\)
−0.341373 + 0.939928i \(0.610892\pi\)
\(588\) 0 0
\(589\) −10.2111 −0.420741
\(590\) 0 0
\(591\) 0 0
\(592\) −7.90833 −0.325030
\(593\) 6.39445 0.262589 0.131294 0.991343i \(-0.458087\pi\)
0.131294 + 0.991343i \(0.458087\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 5.21110 0.213455
\(597\) 0 0
\(598\) 47.5694 1.94526
\(599\) 24.9083 1.01773 0.508863 0.860847i \(-0.330066\pi\)
0.508863 + 0.860847i \(0.330066\pi\)
\(600\) 0 0
\(601\) −1.90833 −0.0778423 −0.0389211 0.999242i \(-0.512392\pi\)
−0.0389211 + 0.999242i \(0.512392\pi\)
\(602\) 6.55004 0.266960
\(603\) 0 0
\(604\) −0.247263 −0.0100610
\(605\) 0 0
\(606\) 0 0
\(607\) 7.21110 0.292690 0.146345 0.989234i \(-0.453249\pi\)
0.146345 + 0.989234i \(0.453249\pi\)
\(608\) −1.69722 −0.0688315
\(609\) 0 0
\(610\) 0 0
\(611\) 15.0000 0.606835
\(612\) 0 0
\(613\) −15.8806 −0.641410 −0.320705 0.947179i \(-0.603920\pi\)
−0.320705 + 0.947179i \(0.603920\pi\)
\(614\) −22.0278 −0.888968
\(615\) 0 0
\(616\) −2.09167 −0.0842759
\(617\) −3.39445 −0.136655 −0.0683277 0.997663i \(-0.521766\pi\)
−0.0683277 + 0.997663i \(0.521766\pi\)
\(618\) 0 0
\(619\) −11.4222 −0.459097 −0.229549 0.973297i \(-0.573725\pi\)
−0.229549 + 0.973297i \(0.573725\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −6.27502 −0.251605
\(623\) −1.18335 −0.0474098
\(624\) 0 0
\(625\) 0 0
\(626\) 0.238859 0.00954672
\(627\) 0 0
\(628\) −5.81665 −0.232110
\(629\) −16.5416 −0.659558
\(630\) 0 0
\(631\) 6.93608 0.276121 0.138061 0.990424i \(-0.455913\pi\)
0.138061 + 0.990424i \(0.455913\pi\)
\(632\) 32.7250 1.30173
\(633\) 0 0
\(634\) 1.18335 0.0469967
\(635\) 0 0
\(636\) 0 0
\(637\) −32.5694 −1.29045
\(638\) −1.18335 −0.0468491
\(639\) 0 0
\(640\) 0 0
\(641\) −27.7889 −1.09760 −0.548798 0.835955i \(-0.684914\pi\)
−0.548798 + 0.835955i \(0.684914\pi\)
\(642\) 0 0
\(643\) −22.0000 −0.867595 −0.433798 0.901010i \(-0.642827\pi\)
−0.433798 + 0.901010i \(0.642827\pi\)
\(644\) −1.54163 −0.0607489
\(645\) 0 0
\(646\) 9.00000 0.354100
\(647\) 33.2389 1.30675 0.653377 0.757033i \(-0.273352\pi\)
0.653377 + 0.757033i \(0.273352\pi\)
\(648\) 0 0
\(649\) 14.2111 0.557835
\(650\) 0 0
\(651\) 0 0
\(652\) −2.81665 −0.110309
\(653\) −6.11943 −0.239472 −0.119736 0.992806i \(-0.538205\pi\)
−0.119736 + 0.992806i \(0.538205\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −18.5139 −0.722846
\(657\) 0 0
\(658\) 2.72498 0.106231
\(659\) 30.9083 1.20402 0.602009 0.798489i \(-0.294367\pi\)
0.602009 + 0.798489i \(0.294367\pi\)
\(660\) 0 0
\(661\) −8.81665 −0.342928 −0.171464 0.985190i \(-0.554850\pi\)
−0.171464 + 0.985190i \(0.554850\pi\)
\(662\) −28.1472 −1.09397
\(663\) 0 0
\(664\) −10.5416 −0.409095
\(665\) 0 0
\(666\) 0 0
\(667\) −6.63331 −0.256843
\(668\) 4.06392 0.157238
\(669\) 0 0
\(670\) 0 0
\(671\) −7.90833 −0.305298
\(672\) 0 0
\(673\) 30.0278 1.15748 0.578742 0.815510i \(-0.303544\pi\)
0.578742 + 0.815510i \(0.303544\pi\)
\(674\) −40.1833 −1.54780
\(675\) 0 0
\(676\) −3.63331 −0.139743
\(677\) −24.2389 −0.931575 −0.465788 0.884897i \(-0.654229\pi\)
−0.465788 + 0.884897i \(0.654229\pi\)
\(678\) 0 0
\(679\) 10.6695 0.409457
\(680\) 0 0
\(681\) 0 0
\(682\) 13.3028 0.509390
\(683\) −47.8444 −1.83072 −0.915358 0.402642i \(-0.868092\pi\)
−0.915358 + 0.402642i \(0.868092\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −12.2750 −0.468662
\(687\) 0 0
\(688\) −23.8167 −0.908001
\(689\) 6.51388 0.248159
\(690\) 0 0
\(691\) 27.5416 1.04773 0.523867 0.851800i \(-0.324489\pi\)
0.523867 + 0.851800i \(0.324489\pi\)
\(692\) −1.45837 −0.0554387
\(693\) 0 0
\(694\) 16.3028 0.618845
\(695\) 0 0
\(696\) 0 0
\(697\) −38.7250 −1.46681
\(698\) −6.75274 −0.255595
\(699\) 0 0
\(700\) 0 0
\(701\) 41.2111 1.55652 0.778261 0.627941i \(-0.216102\pi\)
0.778261 + 0.627941i \(0.216102\pi\)
\(702\) 0 0
\(703\) −2.39445 −0.0903083
\(704\) 8.81665 0.332290
\(705\) 0 0
\(706\) −24.2750 −0.913602
\(707\) 0.358288 0.0134748
\(708\) 0 0
\(709\) −31.6333 −1.18801 −0.594007 0.804460i \(-0.702455\pi\)
−0.594007 + 0.804460i \(0.702455\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 5.09167 0.190819
\(713\) 74.5694 2.79265
\(714\) 0 0
\(715\) 0 0
\(716\) 3.78890 0.141598
\(717\) 0 0
\(718\) −1.02776 −0.0383555
\(719\) −7.18335 −0.267894 −0.133947 0.990989i \(-0.542765\pi\)
−0.133947 + 0.990989i \(0.542765\pi\)
\(720\) 0 0
\(721\) 2.02776 0.0755176
\(722\) −23.4500 −0.872717
\(723\) 0 0
\(724\) 6.02776 0.224020
\(725\) 0 0
\(726\) 0 0
\(727\) −39.3305 −1.45869 −0.729344 0.684147i \(-0.760175\pi\)
−0.729344 + 0.684147i \(0.760175\pi\)
\(728\) −10.4584 −0.387613
\(729\) 0 0
\(730\) 0 0
\(731\) −49.8167 −1.84254
\(732\) 0 0
\(733\) 19.6056 0.724148 0.362074 0.932149i \(-0.382069\pi\)
0.362074 + 0.932149i \(0.382069\pi\)
\(734\) −26.9638 −0.995253
\(735\) 0 0
\(736\) 12.3944 0.456865
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) 35.1194 1.29189 0.645945 0.763384i \(-0.276464\pi\)
0.645945 + 0.763384i \(0.276464\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.18335 0.0434420
\(743\) −40.6972 −1.49304 −0.746518 0.665365i \(-0.768276\pi\)
−0.746518 + 0.665365i \(0.768276\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 35.7250 1.30798
\(747\) 0 0
\(748\) 2.09167 0.0764791
\(749\) 2.09167 0.0764281
\(750\) 0 0
\(751\) −45.3305 −1.65413 −0.827067 0.562103i \(-0.809992\pi\)
−0.827067 + 0.562103i \(0.809992\pi\)
\(752\) −9.90833 −0.361320
\(753\) 0 0
\(754\) −5.91673 −0.215475
\(755\) 0 0
\(756\) 0 0
\(757\) 49.0555 1.78295 0.891476 0.453067i \(-0.149670\pi\)
0.891476 + 0.453067i \(0.149670\pi\)
\(758\) 4.14719 0.150633
\(759\) 0 0
\(760\) 0 0
\(761\) 13.5778 0.492195 0.246097 0.969245i \(-0.420852\pi\)
0.246097 + 0.969245i \(0.420852\pi\)
\(762\) 0 0
\(763\) 8.02776 0.290624
\(764\) 3.11943 0.112857
\(765\) 0 0
\(766\) −28.1833 −1.01831
\(767\) 71.0555 2.56567
\(768\) 0 0
\(769\) −5.18335 −0.186916 −0.0934581 0.995623i \(-0.529792\pi\)
−0.0934581 + 0.995623i \(0.529792\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.00000 −0.143963
\(773\) −3.11943 −0.112198 −0.0560990 0.998425i \(-0.517866\pi\)
−0.0560990 + 0.998425i \(0.517866\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −45.9083 −1.64801
\(777\) 0 0
\(778\) −15.6333 −0.560481
\(779\) −5.60555 −0.200840
\(780\) 0 0
\(781\) 2.60555 0.0932340
\(782\) −65.7250 −2.35032
\(783\) 0 0
\(784\) 21.5139 0.768353
\(785\) 0 0
\(786\) 0 0
\(787\) 10.2111 0.363986 0.181993 0.983300i \(-0.441745\pi\)
0.181993 + 0.983300i \(0.441745\pi\)
\(788\) 4.02776 0.143483
\(789\) 0 0
\(790\) 0 0
\(791\) 7.54163 0.268150
\(792\) 0 0
\(793\) −39.5416 −1.40416
\(794\) 28.2666 1.00314
\(795\) 0 0
\(796\) 1.96384 0.0696065
\(797\) −3.51388 −0.124468 −0.0622340 0.998062i \(-0.519822\pi\)
−0.0622340 + 0.998062i \(0.519822\pi\)
\(798\) 0 0
\(799\) −20.7250 −0.733197
\(800\) 0 0
\(801\) 0 0
\(802\) 16.6611 0.588323
\(803\) −7.90833 −0.279079
\(804\) 0 0
\(805\) 0 0
\(806\) 66.5139 2.34285
\(807\) 0 0
\(808\) −1.54163 −0.0542345
\(809\) −39.6333 −1.39343 −0.696716 0.717347i \(-0.745356\pi\)
−0.696716 + 0.717347i \(0.745356\pi\)
\(810\) 0 0
\(811\) 38.8722 1.36499 0.682493 0.730892i \(-0.260896\pi\)
0.682493 + 0.730892i \(0.260896\pi\)
\(812\) 0.191750 0.00672911
\(813\) 0 0
\(814\) 3.11943 0.109336
\(815\) 0 0
\(816\) 0 0
\(817\) −7.21110 −0.252285
\(818\) −8.09167 −0.282919
\(819\) 0 0
\(820\) 0 0
\(821\) −12.0000 −0.418803 −0.209401 0.977830i \(-0.567152\pi\)
−0.209401 + 0.977830i \(0.567152\pi\)
\(822\) 0 0
\(823\) 18.4222 0.642158 0.321079 0.947052i \(-0.395955\pi\)
0.321079 + 0.947052i \(0.395955\pi\)
\(824\) −8.72498 −0.303949
\(825\) 0 0
\(826\) 12.9083 0.449138
\(827\) −13.8167 −0.480452 −0.240226 0.970717i \(-0.577222\pi\)
−0.240226 + 0.970717i \(0.577222\pi\)
\(828\) 0 0
\(829\) 29.7527 1.03336 0.516678 0.856180i \(-0.327169\pi\)
0.516678 + 0.856180i \(0.327169\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 44.0833 1.52831
\(833\) 45.0000 1.55916
\(834\) 0 0
\(835\) 0 0
\(836\) 0.302776 0.0104717
\(837\) 0 0
\(838\) −8.33053 −0.287773
\(839\) 9.11943 0.314838 0.157419 0.987532i \(-0.449683\pi\)
0.157419 + 0.987532i \(0.449683\pi\)
\(840\) 0 0
\(841\) −28.1749 −0.971550
\(842\) 0.908327 0.0313030
\(843\) 0 0
\(844\) 7.64171 0.263039
\(845\) 0 0
\(846\) 0 0
\(847\) 0.697224 0.0239569
\(848\) −4.30278 −0.147758
\(849\) 0 0
\(850\) 0 0
\(851\) 17.4861 0.599417
\(852\) 0 0
\(853\) −12.7250 −0.435695 −0.217848 0.975983i \(-0.569904\pi\)
−0.217848 + 0.975983i \(0.569904\pi\)
\(854\) −7.18335 −0.245809
\(855\) 0 0
\(856\) −9.00000 −0.307614
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) 41.3944 1.41236 0.706180 0.708032i \(-0.250417\pi\)
0.706180 + 0.708032i \(0.250417\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −42.9916 −1.46430
\(863\) 12.3944 0.421912 0.210956 0.977496i \(-0.432342\pi\)
0.210956 + 0.977496i \(0.432342\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 6.51388 0.221351
\(867\) 0 0
\(868\) −2.15559 −0.0731655
\(869\) −10.9083 −0.370040
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) −34.5416 −1.16973
\(873\) 0 0
\(874\) −9.51388 −0.321812
\(875\) 0 0
\(876\) 0 0
\(877\) 20.0000 0.675352 0.337676 0.941262i \(-0.390359\pi\)
0.337676 + 0.941262i \(0.390359\pi\)
\(878\) 31.6611 1.06851
\(879\) 0 0
\(880\) 0 0
\(881\) −19.5416 −0.658374 −0.329187 0.944265i \(-0.606775\pi\)
−0.329187 + 0.944265i \(0.606775\pi\)
\(882\) 0 0
\(883\) 52.4500 1.76508 0.882541 0.470236i \(-0.155831\pi\)
0.882541 + 0.470236i \(0.155831\pi\)
\(884\) 10.4584 0.351753
\(885\) 0 0
\(886\) 11.2111 0.376644
\(887\) 3.23886 0.108750 0.0543751 0.998521i \(-0.482683\pi\)
0.0543751 + 0.998521i \(0.482683\pi\)
\(888\) 0 0
\(889\) 5.66106 0.189866
\(890\) 0 0
\(891\) 0 0
\(892\) 6.85281 0.229449
\(893\) −3.00000 −0.100391
\(894\) 0 0
\(895\) 0 0
\(896\) 5.64171 0.188476
\(897\) 0 0
\(898\) −30.5971 −1.02104
\(899\) −9.27502 −0.309339
\(900\) 0 0
\(901\) −9.00000 −0.299833
\(902\) 7.30278 0.243156
\(903\) 0 0
\(904\) −32.4500 −1.07927
\(905\) 0 0
\(906\) 0 0
\(907\) −4.00000 −0.132818 −0.0664089 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) 0.513878 0.0170536
\(909\) 0 0
\(910\) 0 0
\(911\) 24.7889 0.821293 0.410646 0.911795i \(-0.365303\pi\)
0.410646 + 0.911795i \(0.365303\pi\)
\(912\) 0 0
\(913\) 3.51388 0.116292
\(914\) −26.9638 −0.891885
\(915\) 0 0
\(916\) 5.66947 0.187324
\(917\) 6.90833 0.228133
\(918\) 0 0
\(919\) 26.7889 0.883684 0.441842 0.897093i \(-0.354325\pi\)
0.441842 + 0.897093i \(0.354325\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −41.9638 −1.38201
\(923\) 13.0278 0.428814
\(924\) 0 0
\(925\) 0 0
\(926\) 15.3583 0.504705
\(927\) 0 0
\(928\) −1.54163 −0.0506066
\(929\) −53.6056 −1.75874 −0.879371 0.476138i \(-0.842036\pi\)
−0.879371 + 0.476138i \(0.842036\pi\)
\(930\) 0 0
\(931\) 6.51388 0.213484
\(932\) 4.81665 0.157775
\(933\) 0 0
\(934\) 24.2750 0.794303
\(935\) 0 0
\(936\) 0 0
\(937\) −9.21110 −0.300914 −0.150457 0.988617i \(-0.548074\pi\)
−0.150457 + 0.988617i \(0.548074\pi\)
\(938\) −3.63331 −0.118632
\(939\) 0 0
\(940\) 0 0
\(941\) −59.6056 −1.94309 −0.971543 0.236864i \(-0.923880\pi\)
−0.971543 + 0.236864i \(0.923880\pi\)
\(942\) 0 0
\(943\) 40.9361 1.33306
\(944\) −46.9361 −1.52764
\(945\) 0 0
\(946\) 9.39445 0.305440
\(947\) −6.63331 −0.215554 −0.107777 0.994175i \(-0.534373\pi\)
−0.107777 + 0.994175i \(0.534373\pi\)
\(948\) 0 0
\(949\) −39.5416 −1.28358
\(950\) 0 0
\(951\) 0 0
\(952\) 14.4500 0.468326
\(953\) −37.2666 −1.20718 −0.603592 0.797293i \(-0.706264\pi\)
−0.603592 + 0.797293i \(0.706264\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 6.39445 0.206811
\(957\) 0 0
\(958\) 45.3583 1.46546
\(959\) 9.00000 0.290625
\(960\) 0 0
\(961\) 73.2666 2.36344
\(962\) 15.5971 0.502872
\(963\) 0 0
\(964\) −6.64171 −0.213915
\(965\) 0 0
\(966\) 0 0
\(967\) 14.9083 0.479419 0.239710 0.970845i \(-0.422948\pi\)
0.239710 + 0.970845i \(0.422948\pi\)
\(968\) −3.00000 −0.0964237
\(969\) 0 0
\(970\) 0 0
\(971\) −45.3583 −1.45562 −0.727808 0.685781i \(-0.759461\pi\)
−0.727808 + 0.685781i \(0.759461\pi\)
\(972\) 0 0
\(973\) −4.33053 −0.138830
\(974\) 5.48612 0.175787
\(975\) 0 0
\(976\) 26.1194 0.836063
\(977\) 52.0278 1.66452 0.832258 0.554389i \(-0.187048\pi\)
0.832258 + 0.554389i \(0.187048\pi\)
\(978\) 0 0
\(979\) −1.69722 −0.0542435
\(980\) 0 0
\(981\) 0 0
\(982\) 12.7527 0.406956
\(983\) −8.84441 −0.282093 −0.141046 0.990003i \(-0.545047\pi\)
−0.141046 + 0.990003i \(0.545047\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 8.17494 0.260343
\(987\) 0 0
\(988\) 1.51388 0.0481629
\(989\) 52.6611 1.67452
\(990\) 0 0
\(991\) −16.9083 −0.537111 −0.268555 0.963264i \(-0.586546\pi\)
−0.268555 + 0.963264i \(0.586546\pi\)
\(992\) 17.3305 0.550245
\(993\) 0 0
\(994\) 2.36669 0.0750669
\(995\) 0 0
\(996\) 0 0
\(997\) 46.7250 1.47979 0.739897 0.672720i \(-0.234874\pi\)
0.739897 + 0.672720i \(0.234874\pi\)
\(998\) −4.54163 −0.143763
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2475.2.a.o.1.2 2
3.2 odd 2 275.2.a.f.1.1 yes 2
5.2 odd 4 2475.2.c.k.199.3 4
5.3 odd 4 2475.2.c.k.199.2 4
5.4 even 2 2475.2.a.t.1.1 2
12.11 even 2 4400.2.a.bh.1.1 2
15.2 even 4 275.2.b.c.199.2 4
15.8 even 4 275.2.b.c.199.3 4
15.14 odd 2 275.2.a.e.1.2 2
33.32 even 2 3025.2.a.h.1.2 2
60.23 odd 4 4400.2.b.y.4049.1 4
60.47 odd 4 4400.2.b.y.4049.4 4
60.59 even 2 4400.2.a.bs.1.2 2
165.164 even 2 3025.2.a.n.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.2 2 15.14 odd 2
275.2.a.f.1.1 yes 2 3.2 odd 2
275.2.b.c.199.2 4 15.2 even 4
275.2.b.c.199.3 4 15.8 even 4
2475.2.a.o.1.2 2 1.1 even 1 trivial
2475.2.a.t.1.1 2 5.4 even 2
2475.2.c.k.199.2 4 5.3 odd 4
2475.2.c.k.199.3 4 5.2 odd 4
3025.2.a.h.1.2 2 33.32 even 2
3025.2.a.n.1.1 2 165.164 even 2
4400.2.a.bh.1.1 2 12.11 even 2
4400.2.a.bs.1.2 2 60.59 even 2
4400.2.b.y.4049.1 4 60.23 odd 4
4400.2.b.y.4049.4 4 60.47 odd 4