Properties

Label 2475.2.f.d.2276.2
Level $2475$
Weight $2$
Character 2475.2276
Analytic conductor $19.763$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 495)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2276.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 2475.2276
Dual form 2475.2.f.d.2276.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} +1.41421i q^{7} -3.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} +1.41421i q^{7} -3.00000 q^{8} +(3.00000 + 1.41421i) q^{11} -2.82843i q^{13} +1.41421i q^{14} -1.00000 q^{16} +2.00000 q^{17} +4.24264i q^{19} +(3.00000 + 1.41421i) q^{22} -2.82843i q^{26} -1.41421i q^{28} -6.00000 q^{29} +5.00000 q^{32} +2.00000 q^{34} -6.00000 q^{37} +4.24264i q^{38} -6.00000 q^{41} +1.41421i q^{43} +(-3.00000 - 1.41421i) q^{44} +8.48528i q^{47} +5.00000 q^{49} +2.82843i q^{52} +4.24264i q^{53} -4.24264i q^{56} -6.00000 q^{58} +8.48528i q^{59} +8.48528i q^{61} +7.00000 q^{64} -2.00000 q^{68} +8.48528i q^{71} +11.3137i q^{73} -6.00000 q^{74} -4.24264i q^{76} +(-2.00000 + 4.24264i) q^{77} +12.7279i q^{79} -6.00000 q^{82} +14.0000 q^{83} +1.41421i q^{86} +(-9.00000 - 4.24264i) q^{88} -7.07107i q^{89} +4.00000 q^{91} +8.48528i q^{94} -12.0000 q^{97} +5.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} - 2 q^{4} - 6 q^{8} + 6 q^{11} - 2 q^{16} + 4 q^{17} + 6 q^{22} - 12 q^{29} + 10 q^{32} + 4 q^{34} - 12 q^{37} - 12 q^{41} - 6 q^{44} + 10 q^{49} - 12 q^{58} + 14 q^{64} - 4 q^{68} - 12 q^{74} - 4 q^{77} - 12 q^{82} + 28 q^{83} - 18 q^{88} + 8 q^{91} - 24 q^{97} + 10 q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.41421i 0.534522i 0.963624 + 0.267261i \(0.0861187\pi\)
−0.963624 + 0.267261i \(0.913881\pi\)
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000 + 1.41421i 0.904534 + 0.426401i
\(12\) 0 0
\(13\) 2.82843i 0.784465i −0.919866 0.392232i \(-0.871703\pi\)
0.919866 0.392232i \(-0.128297\pi\)
\(14\) 1.41421i 0.377964i
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) 4.24264i 0.973329i 0.873589 + 0.486664i \(0.161786\pi\)
−0.873589 + 0.486664i \(0.838214\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.00000 + 1.41421i 0.639602 + 0.301511i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.82843i 0.554700i
\(27\) 0 0
\(28\) 1.41421i 0.267261i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 4.24264i 0.688247i
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 1.41421i 0.215666i 0.994169 + 0.107833i \(0.0343911\pi\)
−0.994169 + 0.107833i \(0.965609\pi\)
\(44\) −3.00000 1.41421i −0.452267 0.213201i
\(45\) 0 0
\(46\) 0 0
\(47\) 8.48528i 1.23771i 0.785507 + 0.618853i \(0.212402\pi\)
−0.785507 + 0.618853i \(0.787598\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 2.82843i 0.392232i
\(53\) 4.24264i 0.582772i 0.956606 + 0.291386i \(0.0941163\pi\)
−0.956606 + 0.291386i \(0.905884\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.24264i 0.566947i
\(57\) 0 0
\(58\) −6.00000 −0.787839
\(59\) 8.48528i 1.10469i 0.833616 + 0.552345i \(0.186267\pi\)
−0.833616 + 0.552345i \(0.813733\pi\)
\(60\) 0 0
\(61\) 8.48528i 1.08643i 0.839594 + 0.543214i \(0.182793\pi\)
−0.839594 + 0.543214i \(0.817207\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) 8.48528i 1.00702i 0.863990 + 0.503509i \(0.167958\pi\)
−0.863990 + 0.503509i \(0.832042\pi\)
\(72\) 0 0
\(73\) 11.3137i 1.32417i 0.749429 + 0.662085i \(0.230328\pi\)
−0.749429 + 0.662085i \(0.769672\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 4.24264i 0.486664i
\(77\) −2.00000 + 4.24264i −0.227921 + 0.483494i
\(78\) 0 0
\(79\) 12.7279i 1.43200i 0.698099 + 0.716002i \(0.254030\pi\)
−0.698099 + 0.716002i \(0.745970\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 14.0000 1.53670 0.768350 0.640030i \(-0.221078\pi\)
0.768350 + 0.640030i \(0.221078\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.41421i 0.152499i
\(87\) 0 0
\(88\) −9.00000 4.24264i −0.959403 0.452267i
\(89\) 7.07107i 0.749532i −0.927119 0.374766i \(-0.877723\pi\)
0.927119 0.374766i \(-0.122277\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 8.48528i 0.875190i
\(95\) 0 0
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 5.00000 0.505076
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 8.48528i 0.832050i
\(105\) 0 0
\(106\) 4.24264i 0.412082i
\(107\) −14.0000 −1.35343 −0.676716 0.736245i \(-0.736597\pi\)
−0.676716 + 0.736245i \(0.736597\pi\)
\(108\) 0 0
\(109\) 16.9706i 1.62549i 0.582623 + 0.812743i \(0.302026\pi\)
−0.582623 + 0.812743i \(0.697974\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.41421i 0.133631i
\(113\) 12.7279i 1.19734i −0.800995 0.598671i \(-0.795696\pi\)
0.800995 0.598671i \(-0.204304\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) 8.48528i 0.781133i
\(119\) 2.82843i 0.259281i
\(120\) 0 0
\(121\) 7.00000 + 8.48528i 0.636364 + 0.771389i
\(122\) 8.48528i 0.768221i
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 18.3848i 1.63139i −0.578486 0.815693i \(-0.696356\pi\)
0.578486 0.815693i \(-0.303644\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) −6.00000 −0.520266
\(134\) 0 0
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 4.24264i 0.362473i −0.983440 0.181237i \(-0.941990\pi\)
0.983440 0.181237i \(-0.0580100\pi\)
\(138\) 0 0
\(139\) 4.24264i 0.359856i 0.983680 + 0.179928i \(0.0575865\pi\)
−0.983680 + 0.179928i \(0.942414\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.48528i 0.712069i
\(143\) 4.00000 8.48528i 0.334497 0.709575i
\(144\) 0 0
\(145\) 0 0
\(146\) 11.3137i 0.936329i
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 4.24264i 0.345261i 0.984987 + 0.172631i \(0.0552267\pi\)
−0.984987 + 0.172631i \(0.944773\pi\)
\(152\) 12.7279i 1.03237i
\(153\) 0 0
\(154\) −2.00000 + 4.24264i −0.161165 + 0.341882i
\(155\) 0 0
\(156\) 0 0
\(157\) 24.0000 1.91541 0.957704 0.287754i \(-0.0929087\pi\)
0.957704 + 0.287754i \(0.0929087\pi\)
\(158\) 12.7279i 1.01258i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 24.0000 1.87983 0.939913 0.341415i \(-0.110906\pi\)
0.939913 + 0.341415i \(0.110906\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 14.0000 1.08661
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 1.41421i 0.107833i
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.00000 1.41421i −0.226134 0.106600i
\(177\) 0 0
\(178\) 7.07107i 0.529999i
\(179\) 5.65685i 0.422813i −0.977398 0.211407i \(-0.932196\pi\)
0.977398 0.211407i \(-0.0678044\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.00000 + 2.82843i 0.438763 + 0.206835i
\(188\) 8.48528i 0.618853i
\(189\) 0 0
\(190\) 0 0
\(191\) 2.82843i 0.204658i 0.994751 + 0.102329i \(0.0326294\pi\)
−0.994751 + 0.102329i \(0.967371\pi\)
\(192\) 0 0
\(193\) 19.7990i 1.42516i −0.701590 0.712581i \(-0.747526\pi\)
0.701590 0.712581i \(-0.252474\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) −5.00000 −0.357143
\(197\) 14.0000 0.997459 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) 8.48528i 0.595550i
\(204\) 0 0
\(205\) 0 0
\(206\) 12.0000 0.836080
\(207\) 0 0
\(208\) 2.82843i 0.196116i
\(209\) −6.00000 + 12.7279i −0.415029 + 0.880409i
\(210\) 0 0
\(211\) 4.24264i 0.292075i 0.989279 + 0.146038i \(0.0466521\pi\)
−0.989279 + 0.146038i \(0.953348\pi\)
\(212\) 4.24264i 0.291386i
\(213\) 0 0
\(214\) −14.0000 −0.957020
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 16.9706i 1.14939i
\(219\) 0 0
\(220\) 0 0
\(221\) 5.65685i 0.380521i
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 7.07107i 0.472456i
\(225\) 0 0
\(226\) 12.7279i 0.846649i
\(227\) 2.00000 0.132745 0.0663723 0.997795i \(-0.478857\pi\)
0.0663723 + 0.997795i \(0.478857\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 18.0000 1.18176
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 8.48528i 0.552345i
\(237\) 0 0
\(238\) 2.82843i 0.183340i
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 8.48528i 0.546585i 0.961931 + 0.273293i \(0.0881127\pi\)
−0.961931 + 0.273293i \(0.911887\pi\)
\(242\) 7.00000 + 8.48528i 0.449977 + 0.545455i
\(243\) 0 0
\(244\) 8.48528i 0.543214i
\(245\) 0 0
\(246\) 0 0
\(247\) 12.0000 0.763542
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.9706i 1.07117i −0.844481 0.535586i \(-0.820091\pi\)
0.844481 0.535586i \(-0.179909\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 18.3848i 1.15356i
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 29.6985i 1.85254i −0.376860 0.926270i \(-0.622996\pi\)
0.376860 0.926270i \(-0.377004\pi\)
\(258\) 0 0
\(259\) 8.48528i 0.527250i
\(260\) 0 0
\(261\) 0 0
\(262\) −18.0000 −1.11204
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.00000 −0.367884
\(267\) 0 0
\(268\) 0 0
\(269\) 12.7279i 0.776035i 0.921652 + 0.388018i \(0.126840\pi\)
−0.921652 + 0.388018i \(0.873160\pi\)
\(270\) 0 0
\(271\) 12.7279i 0.773166i 0.922255 + 0.386583i \(0.126345\pi\)
−0.922255 + 0.386583i \(0.873655\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 4.24264i 0.256307i
\(275\) 0 0
\(276\) 0 0
\(277\) 31.1127i 1.86938i 0.355463 + 0.934690i \(0.384323\pi\)
−0.355463 + 0.934690i \(0.615677\pi\)
\(278\) 4.24264i 0.254457i
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 1.41421i 0.0840663i −0.999116 0.0420331i \(-0.986616\pi\)
0.999116 0.0420331i \(-0.0133835\pi\)
\(284\) 8.48528i 0.503509i
\(285\) 0 0
\(286\) 4.00000 8.48528i 0.236525 0.501745i
\(287\) 8.48528i 0.500870i
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 11.3137i 0.662085i
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 18.0000 1.04623
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 0 0
\(301\) −2.00000 −0.115278
\(302\) 4.24264i 0.244137i
\(303\) 0 0
\(304\) 4.24264i 0.243332i
\(305\) 0 0
\(306\) 0 0
\(307\) 24.0416i 1.37213i −0.727541 0.686064i \(-0.759337\pi\)
0.727541 0.686064i \(-0.240663\pi\)
\(308\) 2.00000 4.24264i 0.113961 0.241747i
\(309\) 0 0
\(310\) 0 0
\(311\) 16.9706i 0.962312i 0.876635 + 0.481156i \(0.159783\pi\)
−0.876635 + 0.481156i \(0.840217\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 24.0000 1.35440
\(315\) 0 0
\(316\) 12.7279i 0.716002i
\(317\) 21.2132i 1.19145i −0.803188 0.595726i \(-0.796864\pi\)
0.803188 0.595726i \(-0.203136\pi\)
\(318\) 0 0
\(319\) −18.0000 8.48528i −1.00781 0.475085i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 8.48528i 0.472134i
\(324\) 0 0
\(325\) 0 0
\(326\) 24.0000 1.32924
\(327\) 0 0
\(328\) 18.0000 0.993884
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −14.0000 −0.768350
\(333\) 0 0
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) 0 0
\(337\) 11.3137i 0.616297i 0.951338 + 0.308148i \(0.0997094\pi\)
−0.951338 + 0.308148i \(0.900291\pi\)
\(338\) 5.00000 0.271964
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 4.24264i 0.228748i
\(345\) 0 0
\(346\) −14.0000 −0.752645
\(347\) 14.0000 0.751559 0.375780 0.926709i \(-0.377375\pi\)
0.375780 + 0.926709i \(0.377375\pi\)
\(348\) 0 0
\(349\) 8.48528i 0.454207i 0.973871 + 0.227103i \(0.0729255\pi\)
−0.973871 + 0.227103i \(0.927074\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 15.0000 + 7.07107i 0.799503 + 0.376889i
\(353\) 4.24264i 0.225813i −0.993606 0.112906i \(-0.963984\pi\)
0.993606 0.112906i \(-0.0360161\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 7.07107i 0.374766i
\(357\) 0 0
\(358\) 5.65685i 0.298974i
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 0 0
\(367\) −12.0000 −0.626395 −0.313197 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) 22.6274i 1.17160i −0.810454 0.585802i \(-0.800780\pi\)
0.810454 0.585802i \(-0.199220\pi\)
\(374\) 6.00000 + 2.82843i 0.310253 + 0.146254i
\(375\) 0 0
\(376\) 25.4558i 1.31278i
\(377\) 16.9706i 0.874028i
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 2.82843i 0.144715i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 19.7990i 1.00774i
\(387\) 0 0
\(388\) 12.0000 0.609208
\(389\) 21.2132i 1.07555i 0.843088 + 0.537776i \(0.180735\pi\)
−0.843088 + 0.537776i \(0.819265\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −15.0000 −0.757614
\(393\) 0 0
\(394\) 14.0000 0.705310
\(395\) 0 0
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) 0 0
\(401\) 26.8701i 1.34183i 0.741536 + 0.670913i \(0.234098\pi\)
−0.741536 + 0.670913i \(0.765902\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 8.48528i 0.421117i
\(407\) −18.0000 8.48528i −0.892227 0.420600i
\(408\) 0 0
\(409\) 8.48528i 0.419570i −0.977748 0.209785i \(-0.932724\pi\)
0.977748 0.209785i \(-0.0672764\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −12.0000 −0.591198
\(413\) −12.0000 −0.590481
\(414\) 0 0
\(415\) 0 0
\(416\) 14.1421i 0.693375i
\(417\) 0 0
\(418\) −6.00000 + 12.7279i −0.293470 + 0.622543i
\(419\) 39.5980i 1.93449i −0.253849 0.967244i \(-0.581697\pi\)
0.253849 0.967244i \(-0.418303\pi\)
\(420\) 0 0
\(421\) 24.0000 1.16969 0.584844 0.811146i \(-0.301156\pi\)
0.584844 + 0.811146i \(0.301156\pi\)
\(422\) 4.24264i 0.206529i
\(423\) 0 0
\(424\) 12.7279i 0.618123i
\(425\) 0 0
\(426\) 0 0
\(427\) −12.0000 −0.580721
\(428\) 14.0000 0.676716
\(429\) 0 0
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 16.9706i 0.812743i
\(437\) 0 0
\(438\) 0 0
\(439\) 38.1838i 1.82241i −0.411951 0.911206i \(-0.635153\pi\)
0.411951 0.911206i \(-0.364847\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 5.65685i 0.269069i
\(443\) 25.4558i 1.20944i −0.796437 0.604722i \(-0.793284\pi\)
0.796437 0.604722i \(-0.206716\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −24.0000 −1.13643
\(447\) 0 0
\(448\) 9.89949i 0.467707i
\(449\) 32.5269i 1.53504i −0.641025 0.767520i \(-0.721491\pi\)
0.641025 0.767520i \(-0.278509\pi\)
\(450\) 0 0
\(451\) −18.0000 8.48528i −0.847587 0.399556i
\(452\) 12.7279i 0.598671i
\(453\) 0 0
\(454\) 2.00000 0.0938647
\(455\) 0 0
\(456\) 0 0
\(457\) 28.2843i 1.32308i −0.749909 0.661541i \(-0.769903\pi\)
0.749909 0.661541i \(-0.230097\pi\)
\(458\) 14.0000 0.654177
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) 8.48528i 0.392652i 0.980539 + 0.196326i \(0.0629011\pi\)
−0.980539 + 0.196326i \(0.937099\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 25.4558i 1.17170i
\(473\) −2.00000 + 4.24264i −0.0919601 + 0.195077i
\(474\) 0 0
\(475\) 0 0
\(476\) 2.82843i 0.129641i
\(477\) 0 0
\(478\) 6.00000 0.274434
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) 16.9706i 0.773791i
\(482\) 8.48528i 0.386494i
\(483\) 0 0
\(484\) −7.00000 8.48528i −0.318182 0.385695i
\(485\) 0 0
\(486\) 0 0
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 25.4558i 1.15233i
\(489\) 0 0
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 12.0000 0.539906
\(495\) 0 0
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) −32.0000 −1.43252 −0.716258 0.697835i \(-0.754147\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 16.9706i 0.757433i
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 18.3848i 0.815693i
\(509\) 4.24264i 0.188052i 0.995570 + 0.0940259i \(0.0299736\pi\)
−0.995570 + 0.0940259i \(0.970026\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) 29.6985i 1.30994i
\(515\) 0 0
\(516\) 0 0
\(517\) −12.0000 + 25.4558i −0.527759 + 1.11955i
\(518\) 8.48528i 0.372822i
\(519\) 0 0
\(520\) 0 0
\(521\) 7.07107i 0.309789i −0.987931 0.154895i \(-0.950496\pi\)
0.987931 0.154895i \(-0.0495038\pi\)
\(522\) 0 0
\(523\) 7.07107i 0.309196i 0.987977 + 0.154598i \(0.0494083\pi\)
−0.987977 + 0.154598i \(0.950592\pi\)
\(524\) 18.0000 0.786334
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 6.00000 0.260133
\(533\) 16.9706i 0.735077i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 12.7279i 0.548740i
\(539\) 15.0000 + 7.07107i 0.646096 + 0.304572i
\(540\) 0 0
\(541\) 33.9411i 1.45924i 0.683851 + 0.729621i \(0.260304\pi\)
−0.683851 + 0.729621i \(0.739696\pi\)
\(542\) 12.7279i 0.546711i
\(543\) 0 0
\(544\) 10.0000 0.428746
\(545\) 0 0
\(546\) 0 0
\(547\) 9.89949i 0.423272i 0.977349 + 0.211636i \(0.0678791\pi\)
−0.977349 + 0.211636i \(0.932121\pi\)
\(548\) 4.24264i 0.181237i
\(549\) 0 0
\(550\) 0 0
\(551\) 25.4558i 1.08446i
\(552\) 0 0
\(553\) −18.0000 −0.765438
\(554\) 31.1127i 1.32185i
\(555\) 0 0
\(556\) 4.24264i 0.179928i
\(557\) 34.0000 1.44063 0.720313 0.693649i \(-0.243998\pi\)
0.720313 + 0.693649i \(0.243998\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.41421i 0.0594438i
\(567\) 0 0
\(568\) 25.4558i 1.06810i
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 4.24264i 0.177549i 0.996052 + 0.0887745i \(0.0282950\pi\)
−0.996052 + 0.0887745i \(0.971705\pi\)
\(572\) −4.00000 + 8.48528i −0.167248 + 0.354787i
\(573\) 0 0
\(574\) 8.48528i 0.354169i
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −13.0000 −0.540729
\(579\) 0 0
\(580\) 0 0
\(581\) 19.7990i 0.821401i
\(582\) 0 0
\(583\) −6.00000 + 12.7279i −0.248495 + 0.527137i
\(584\) 33.9411i 1.40449i
\(585\) 0 0
\(586\) −10.0000 −0.413096
\(587\) 42.4264i 1.75113i 0.483105 + 0.875563i \(0.339509\pi\)
−0.483105 + 0.875563i \(0.660491\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 6.00000 0.246598
\(593\) 2.00000 0.0821302 0.0410651 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) 0 0
\(598\) 0 0
\(599\) 11.3137i 0.462266i 0.972922 + 0.231133i \(0.0742432\pi\)
−0.972922 + 0.231133i \(0.925757\pi\)
\(600\) 0 0
\(601\) 33.9411i 1.38449i −0.721664 0.692244i \(-0.756622\pi\)
0.721664 0.692244i \(-0.243378\pi\)
\(602\) −2.00000 −0.0815139
\(603\) 0 0
\(604\) 4.24264i 0.172631i
\(605\) 0 0
\(606\) 0 0
\(607\) 32.5269i 1.32023i 0.751166 + 0.660113i \(0.229492\pi\)
−0.751166 + 0.660113i \(0.770508\pi\)
\(608\) 21.2132i 0.860309i
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) 28.2843i 1.14239i −0.820814 0.571195i \(-0.806480\pi\)
0.820814 0.571195i \(-0.193520\pi\)
\(614\) 24.0416i 0.970241i
\(615\) 0 0
\(616\) 6.00000 12.7279i 0.241747 0.512823i
\(617\) 4.24264i 0.170802i 0.996347 + 0.0854011i \(0.0272172\pi\)
−0.996347 + 0.0854011i \(0.972783\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 16.9706i 0.680458i
\(623\) 10.0000 0.400642
\(624\) 0 0
\(625\) 0 0
\(626\) −6.00000 −0.239808
\(627\) 0 0
\(628\) −24.0000 −0.957704
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 38.1838i 1.51887i
\(633\) 0 0
\(634\) 21.2132i 0.842484i
\(635\) 0 0
\(636\) 0 0
\(637\) 14.1421i 0.560332i
\(638\) −18.0000 8.48528i −0.712627 0.335936i
\(639\) 0 0
\(640\) 0 0
\(641\) 15.5563i 0.614439i −0.951639 0.307219i \(-0.900601\pi\)
0.951639 0.307219i \(-0.0993986\pi\)
\(642\) 0 0
\(643\) −12.0000 −0.473234 −0.236617 0.971603i \(-0.576039\pi\)
−0.236617 + 0.971603i \(0.576039\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 8.48528i 0.333849i
\(647\) 16.9706i 0.667182i 0.942718 + 0.333591i \(0.108260\pi\)
−0.942718 + 0.333591i \(0.891740\pi\)
\(648\) 0 0
\(649\) −12.0000 + 25.4558i −0.471041 + 0.999229i
\(650\) 0 0
\(651\) 0 0
\(652\) −24.0000 −0.939913
\(653\) 29.6985i 1.16219i 0.813835 + 0.581096i \(0.197376\pi\)
−0.813835 + 0.581096i \(0.802624\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 0 0
\(658\) −12.0000 −0.467809
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 12.0000 0.466393
\(663\) 0 0
\(664\) −42.0000 −1.62992
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 8.00000 0.309529
\(669\) 0 0
\(670\) 0 0
\(671\) −12.0000 + 25.4558i −0.463255 + 0.982712i
\(672\) 0 0
\(673\) 36.7696i 1.41736i 0.705529 + 0.708681i \(0.250709\pi\)
−0.705529 + 0.708681i \(0.749291\pi\)
\(674\) 11.3137i 0.435788i
\(675\) 0 0
\(676\) −5.00000 −0.192308
\(677\) 46.0000 1.76792 0.883962 0.467559i \(-0.154866\pi\)
0.883962 + 0.467559i \(0.154866\pi\)
\(678\) 0 0
\(679\) 16.9706i 0.651270i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 33.9411i 1.29872i 0.760481 + 0.649361i \(0.224963\pi\)
−0.760481 + 0.649361i \(0.775037\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 16.9706i 0.647939i
\(687\) 0 0
\(688\) 1.41421i 0.0539164i
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) 14.0000 0.531433
\(695\) 0 0
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 8.48528i 0.321173i
\(699\) 0 0
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 25.4558i 0.960085i
\(704\) 21.0000 + 9.89949i 0.791467 + 0.373101i
\(705\) 0 0
\(706\) 4.24264i 0.159674i
\(707\) 8.48528i 0.319122i
\(708\) 0 0
\(709\) −30.0000 −1.12667 −0.563337 0.826227i \(-0.690483\pi\)
−0.563337 + 0.826227i \(0.690483\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 21.2132i 0.794998i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 5.65685i 0.211407i
\(717\) 0 0
\(718\) 24.0000 0.895672
\(719\) 31.1127i 1.16031i 0.814507 + 0.580154i \(0.197008\pi\)
−0.814507 + 0.580154i \(0.802992\pi\)
\(720\) 0 0
\(721\) 16.9706i 0.632017i
\(722\) 1.00000 0.0372161
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −48.0000 −1.78022 −0.890111 0.455744i \(-0.849373\pi\)
−0.890111 + 0.455744i \(0.849373\pi\)
\(728\) −12.0000 −0.444750
\(729\) 0 0
\(730\) 0 0
\(731\) 2.82843i 0.104613i
\(732\) 0 0
\(733\) 31.1127i 1.14917i −0.818444 0.574587i \(-0.805163\pi\)
0.818444 0.574587i \(-0.194837\pi\)
\(734\) −12.0000 −0.442928
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 46.6690i 1.71675i −0.513024 0.858374i \(-0.671475\pi\)
0.513024 0.858374i \(-0.328525\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) 10.0000 0.366864 0.183432 0.983032i \(-0.441279\pi\)
0.183432 + 0.983032i \(0.441279\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 22.6274i 0.828449i
\(747\) 0 0
\(748\) −6.00000 2.82843i −0.219382 0.103418i
\(749\) 19.7990i 0.723439i
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 8.48528i 0.309426i
\(753\) 0 0
\(754\) 16.9706i 0.618031i
\(755\) 0 0
\(756\) 0 0
\(757\) 42.0000 1.52652 0.763258 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) −24.0000 −0.868858
\(764\) 2.82843i 0.102329i
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 19.7990i 0.712581i
\(773\) 4.24264i 0.152597i 0.997085 + 0.0762986i \(0.0243102\pi\)
−0.997085 + 0.0762986i \(0.975690\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 36.0000 1.29232
\(777\) 0 0
\(778\) 21.2132i 0.760530i
\(779\) 25.4558i 0.912050i
\(780\) 0 0
\(781\) −12.0000 + 25.4558i −0.429394 + 0.910882i
\(782\) 0 0
\(783\) 0 0
\(784\) −5.00000 −0.178571
\(785\) 0 0
\(786\) 0 0
\(787\) 9.89949i 0.352879i −0.984311 0.176439i \(-0.943542\pi\)
0.984311 0.176439i \(-0.0564580\pi\)
\(788\) −14.0000 −0.498729
\(789\) 0 0
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) 24.0000 0.852265
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) −20.0000 −0.708881
\(797\) 29.6985i 1.05197i 0.850493 + 0.525987i \(0.176304\pi\)
−0.850493 + 0.525987i \(0.823696\pi\)
\(798\) 0 0
\(799\) 16.9706i 0.600375i
\(800\) 0 0
\(801\) 0 0
\(802\) 26.8701i 0.948815i
\(803\) −16.0000 + 33.9411i −0.564628