Properties

Label 3150.2.b.e.251.4
Level 3150
Weight 2
Character 3150.251
Analytic conductor 25.153
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.7442857984.4
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 251.4
Root \(0.916813i\)
Character \(\chi\) = 3150.251
Dual form 3150.2.b.e.251.8

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.00000 q^{4} +(2.56510 + 0.648285i) q^{7} +1.00000i q^{8} +O(q^{10})\) \(q-1.00000i q^{2} -1.00000 q^{4} +(2.56510 + 0.648285i) q^{7} +1.00000i q^{8} +1.29657i q^{11} +3.13020i q^{13} +(0.648285 - 2.56510i) q^{14} +1.00000 q^{16} -5.53921 q^{17} -7.37284i q^{19} +1.29657 q^{22} +1.83363i q^{23} +3.13020 q^{26} +(-2.56510 - 0.648285i) q^{28} -1.83363i q^{29} -10.4268i q^{31} -1.00000i q^{32} +5.53921i q^{34} +10.6694 q^{37} -7.37284 q^{38} +3.13020 q^{41} -3.53921 q^{43} -1.29657i q^{44} +1.83363 q^{46} +10.7797 q^{47} +(6.15945 + 3.32583i) q^{49} -3.13020i q^{52} +4.42677i q^{53} +(-0.648285 + 2.56510i) q^{56} -1.83363 q^{58} +7.18871 q^{59} +4.88755i q^{61} -10.4268 q^{62} -1.00000 q^{64} +9.79960 q^{67} +5.53921 q^{68} +7.37284i q^{71} +3.40686i q^{73} -10.6694i q^{74} +7.37284i q^{76} +(-0.840546 + 3.32583i) q^{77} +9.01990 q^{79} -3.13020i q^{82} +6.26039 q^{83} +3.53921i q^{86} -1.29657 q^{88} -7.94822 q^{89} +(-2.02926 + 8.02926i) q^{91} -1.83363i q^{92} -10.7797i q^{94} -8.09402i q^{97} +(3.32583 - 6.15945i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + 4q^{7} + O(q^{10}) \) \( 8q - 8q^{4} + 4q^{7} + 8q^{16} - 8q^{26} - 4q^{28} + 8q^{37} + 8q^{38} - 8q^{41} + 16q^{43} - 8q^{46} + 40q^{47} + 4q^{49} + 8q^{58} - 40q^{62} - 8q^{64} - 32q^{67} - 52q^{77} + 8q^{79} - 16q^{83} - 8q^{89} - 4q^{91} + 4q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 2.56510 + 0.648285i 0.969516 + 0.245029i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.29657i 0.390930i 0.980711 + 0.195465i \(0.0626217\pi\)
−0.980711 + 0.195465i \(0.937378\pi\)
\(12\) 0 0
\(13\) 3.13020i 0.868160i 0.900874 + 0.434080i \(0.142926\pi\)
−0.900874 + 0.434080i \(0.857074\pi\)
\(14\) 0.648285 2.56510i 0.173261 0.685551i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.53921 −1.34346 −0.671728 0.740798i \(-0.734448\pi\)
−0.671728 + 0.740798i \(0.734448\pi\)
\(18\) 0 0
\(19\) 7.37284i 1.69144i −0.533623 0.845722i \(-0.679170\pi\)
0.533623 0.845722i \(-0.320830\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.29657 0.276430
\(23\) 1.83363i 0.382337i 0.981557 + 0.191169i \(0.0612278\pi\)
−0.981557 + 0.191169i \(0.938772\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3.13020 0.613882
\(27\) 0 0
\(28\) −2.56510 0.648285i −0.484758 0.122514i
\(29\) 1.83363i 0.340496i −0.985401 0.170248i \(-0.945543\pi\)
0.985401 0.170248i \(-0.0544569\pi\)
\(30\) 0 0
\(31\) 10.4268i 1.87270i −0.351065 0.936351i \(-0.614180\pi\)
0.351065 0.936351i \(-0.385820\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 5.53921i 0.949967i
\(35\) 0 0
\(36\) 0 0
\(37\) 10.6694 1.75404 0.877020 0.480454i \(-0.159528\pi\)
0.877020 + 0.480454i \(0.159528\pi\)
\(38\) −7.37284 −1.19603
\(39\) 0 0
\(40\) 0 0
\(41\) 3.13020 0.488854 0.244427 0.969668i \(-0.421400\pi\)
0.244427 + 0.969668i \(0.421400\pi\)
\(42\) 0 0
\(43\) −3.53921 −0.539724 −0.269862 0.962899i \(-0.586978\pi\)
−0.269862 + 0.962899i \(0.586978\pi\)
\(44\) 1.29657i 0.195465i
\(45\) 0 0
\(46\) 1.83363 0.270353
\(47\) 10.7797 1.57238 0.786190 0.617985i \(-0.212051\pi\)
0.786190 + 0.617985i \(0.212051\pi\)
\(48\) 0 0
\(49\) 6.15945 + 3.32583i 0.879922 + 0.475118i
\(50\) 0 0
\(51\) 0 0
\(52\) 3.13020i 0.434080i
\(53\) 4.42677i 0.608063i 0.952662 + 0.304031i \(0.0983328\pi\)
−0.952662 + 0.304031i \(0.901667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −0.648285 + 2.56510i −0.0866307 + 0.342776i
\(57\) 0 0
\(58\) −1.83363 −0.240767
\(59\) 7.18871 0.935891 0.467945 0.883757i \(-0.344994\pi\)
0.467945 + 0.883757i \(0.344994\pi\)
\(60\) 0 0
\(61\) 4.88755i 0.625787i 0.949788 + 0.312894i \(0.101298\pi\)
−0.949788 + 0.312894i \(0.898702\pi\)
\(62\) −10.4268 −1.32420
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 9.79960 1.19721 0.598606 0.801044i \(-0.295722\pi\)
0.598606 + 0.801044i \(0.295722\pi\)
\(68\) 5.53921 0.671728
\(69\) 0 0
\(70\) 0 0
\(71\) 7.37284i 0.874995i 0.899220 + 0.437497i \(0.144135\pi\)
−0.899220 + 0.437497i \(0.855865\pi\)
\(72\) 0 0
\(73\) 3.40686i 0.398743i 0.979924 + 0.199371i \(0.0638900\pi\)
−0.979924 + 0.199371i \(0.936110\pi\)
\(74\) 10.6694i 1.24029i
\(75\) 0 0
\(76\) 7.37284i 0.845722i
\(77\) −0.840546 + 3.32583i −0.0957891 + 0.379013i
\(78\) 0 0
\(79\) 9.01990 1.01482 0.507409 0.861705i \(-0.330603\pi\)
0.507409 + 0.861705i \(0.330603\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.13020i 0.345672i
\(83\) 6.26039 0.687167 0.343584 0.939122i \(-0.388359\pi\)
0.343584 + 0.939122i \(0.388359\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.53921i 0.381643i
\(87\) 0 0
\(88\) −1.29657 −0.138215
\(89\) −7.94822 −0.842510 −0.421255 0.906942i \(-0.638410\pi\)
−0.421255 + 0.906942i \(0.638410\pi\)
\(90\) 0 0
\(91\) −2.02926 + 8.02926i −0.212724 + 0.841695i
\(92\) 1.83363i 0.191169i
\(93\) 0 0
\(94\) 10.7797i 1.11184i
\(95\) 0 0
\(96\) 0 0
\(97\) 8.09402i 0.821823i −0.911675 0.410911i \(-0.865211\pi\)
0.911675 0.410911i \(-0.134789\pi\)
\(98\) 3.32583 6.15945i 0.335959 0.622199i
\(99\) 0 0
\(100\) 0 0
\(101\) 4.05852 0.403837 0.201919 0.979402i \(-0.435282\pi\)
0.201919 + 0.979402i \(0.435282\pi\)
\(102\) 0 0
\(103\) 5.61548i 0.553309i 0.960969 + 0.276655i \(0.0892258\pi\)
−0.960969 + 0.276655i \(0.910774\pi\)
\(104\) −3.13020 −0.306941
\(105\) 0 0
\(106\) 4.42677 0.429965
\(107\) 0.651655i 0.0629979i 0.999504 + 0.0314989i \(0.0100281\pi\)
−0.999504 + 0.0314989i \(0.989972\pi\)
\(108\) 0 0
\(109\) 15.3388 1.46919 0.734596 0.678505i \(-0.237372\pi\)
0.734596 + 0.678505i \(0.237372\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.56510 + 0.648285i 0.242379 + 0.0612572i
\(113\) 17.0784i 1.60660i 0.595573 + 0.803301i \(0.296925\pi\)
−0.595573 + 0.803301i \(0.703075\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.83363i 0.170248i
\(117\) 0 0
\(118\) 7.18871i 0.661775i
\(119\) −14.2086 3.59099i −1.30250 0.329185i
\(120\) 0 0
\(121\) 9.31891 0.847173
\(122\) 4.88755 0.442498
\(123\) 0 0
\(124\) 10.4268i 0.936351i
\(125\) 0 0
\(126\) 0 0
\(127\) 7.39059 0.655809 0.327904 0.944711i \(-0.393658\pi\)
0.327904 + 0.944711i \(0.393658\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −20.2086 −1.76563 −0.882817 0.469716i \(-0.844356\pi\)
−0.882817 + 0.469716i \(0.844356\pi\)
\(132\) 0 0
\(133\) 4.77970 18.9120i 0.414452 1.63988i
\(134\) 9.79960i 0.846557i
\(135\) 0 0
\(136\) 5.53921i 0.474983i
\(137\) 20.7457i 1.77242i −0.463281 0.886211i \(-0.653328\pi\)
0.463281 0.886211i \(-0.346672\pi\)
\(138\) 0 0
\(139\) 9.20646i 0.780882i −0.920628 0.390441i \(-0.872323\pi\)
0.920628 0.390441i \(-0.127677\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.37284 0.618715
\(143\) −4.05852 −0.339390
\(144\) 0 0
\(145\) 0 0
\(146\) 3.40686 0.281954
\(147\) 0 0
\(148\) −10.6694 −0.877020
\(149\) 10.3189i 0.845358i 0.906279 + 0.422679i \(0.138910\pi\)
−0.906279 + 0.422679i \(0.861090\pi\)
\(150\) 0 0
\(151\) −1.24049 −0.100949 −0.0504747 0.998725i \(-0.516073\pi\)
−0.0504747 + 0.998725i \(0.516073\pi\)
\(152\) 7.37284 0.598016
\(153\) 0 0
\(154\) 3.32583 + 0.840546i 0.268003 + 0.0677331i
\(155\) 0 0
\(156\) 0 0
\(157\) 11.4629i 0.914842i 0.889250 + 0.457421i \(0.151227\pi\)
−0.889250 + 0.457421i \(0.848773\pi\)
\(158\) 9.01990i 0.717585i
\(159\) 0 0
\(160\) 0 0
\(161\) −1.18871 + 4.70343i −0.0936836 + 0.370682i
\(162\) 0 0
\(163\) −1.27882 −0.100165 −0.0500824 0.998745i \(-0.515948\pi\)
−0.0500824 + 0.998745i \(0.515948\pi\)
\(164\) −3.13020 −0.244427
\(165\) 0 0
\(166\) 6.26039i 0.485901i
\(167\) 6.72118 0.520101 0.260050 0.965595i \(-0.416261\pi\)
0.260050 + 0.965595i \(0.416261\pi\)
\(168\) 0 0
\(169\) 3.20188 0.246298
\(170\) 0 0
\(171\) 0 0
\(172\) 3.53921 0.269862
\(173\) 16.3189 1.24070 0.620352 0.784324i \(-0.286990\pi\)
0.620352 + 0.784324i \(0.286990\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.29657i 0.0977326i
\(177\) 0 0
\(178\) 7.94822i 0.595745i
\(179\) 19.1887i 1.43423i −0.696954 0.717116i \(-0.745462\pi\)
0.696954 0.717116i \(-0.254538\pi\)
\(180\) 0 0
\(181\) 10.7797i 0.801249i −0.916242 0.400624i \(-0.868793\pi\)
0.916242 0.400624i \(-0.131207\pi\)
\(182\) 8.02926 + 2.02926i 0.595168 + 0.150419i
\(183\) 0 0
\(184\) −1.83363 −0.135177
\(185\) 0 0
\(186\) 0 0
\(187\) 7.18197i 0.525198i
\(188\) −10.7797 −0.786190
\(189\) 0 0
\(190\) 0 0
\(191\) 19.3728i 1.40177i −0.713275 0.700885i \(-0.752789\pi\)
0.713275 0.700885i \(-0.247211\pi\)
\(192\) 0 0
\(193\) 6.81803 0.490772 0.245386 0.969425i \(-0.421085\pi\)
0.245386 + 0.969425i \(0.421085\pi\)
\(194\) −8.09402 −0.581117
\(195\) 0 0
\(196\) −6.15945 3.32583i −0.439961 0.237559i
\(197\) 14.7457i 1.05059i −0.850922 0.525293i \(-0.823956\pi\)
0.850922 0.525293i \(-0.176044\pi\)
\(198\) 0 0
\(199\) 1.94148i 0.137628i −0.997630 0.0688141i \(-0.978078\pi\)
0.997630 0.0688141i \(-0.0219215\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 4.05852i 0.285556i
\(203\) 1.18871 4.70343i 0.0834312 0.330116i
\(204\) 0 0
\(205\) 0 0
\(206\) 5.61548 0.391249
\(207\) 0 0
\(208\) 3.13020i 0.217040i
\(209\) 9.55939 0.661237
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 4.42677i 0.304031i
\(213\) 0 0
\(214\) 0.651655 0.0445462
\(215\) 0 0
\(216\) 0 0
\(217\) 6.75951 26.7457i 0.458866 1.81561i
\(218\) 15.3388i 1.03888i
\(219\) 0 0
\(220\) 0 0
\(221\) 17.3388i 1.16633i
\(222\) 0 0
\(223\) 8.76195i 0.586743i 0.955998 + 0.293372i \(0.0947773\pi\)
−0.955998 + 0.293372i \(0.905223\pi\)
\(224\) 0.648285 2.56510i 0.0433153 0.171388i
\(225\) 0 0
\(226\) 17.0784 1.13604
\(227\) −23.0784 −1.53177 −0.765884 0.642978i \(-0.777698\pi\)
−0.765884 + 0.642978i \(0.777698\pi\)
\(228\) 0 0
\(229\) 9.70558i 0.641363i 0.947187 + 0.320682i \(0.103912\pi\)
−0.947187 + 0.320682i \(0.896088\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.83363 0.120383
\(233\) 6.36825i 0.417198i 0.978001 + 0.208599i \(0.0668903\pi\)
−0.978001 + 0.208599i \(0.933110\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −7.18871 −0.467945
\(237\) 0 0
\(238\) −3.59099 + 14.2086i −0.232769 + 0.921008i
\(239\) 19.3728i 1.25312i 0.779371 + 0.626562i \(0.215539\pi\)
−0.779371 + 0.626562i \(0.784461\pi\)
\(240\) 0 0
\(241\) 24.3587i 1.56908i 0.620076 + 0.784541i \(0.287102\pi\)
−0.620076 + 0.784541i \(0.712898\pi\)
\(242\) 9.31891i 0.599042i
\(243\) 0 0
\(244\) 4.88755i 0.312894i
\(245\) 0 0
\(246\) 0 0
\(247\) 23.0784 1.46844
\(248\) 10.4268 0.662100
\(249\) 0 0
\(250\) 0 0
\(251\) 1.94822 0.122971 0.0614854 0.998108i \(-0.480416\pi\)
0.0614854 + 0.998108i \(0.480416\pi\)
\(252\) 0 0
\(253\) −2.37742 −0.149467
\(254\) 7.39059i 0.463727i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −25.2420 −1.57455 −0.787275 0.616602i \(-0.788509\pi\)
−0.787275 + 0.616602i \(0.788509\pi\)
\(258\) 0 0
\(259\) 27.3681 + 6.91681i 1.70057 + 0.429790i
\(260\) 0 0
\(261\) 0 0
\(262\) 20.2086i 1.24849i
\(263\) 0.912047i 0.0562392i 0.999605 + 0.0281196i \(0.00895193\pi\)
−0.999605 + 0.0281196i \(0.991048\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −18.9120 4.77970i −1.15957 0.293062i
\(267\) 0 0
\(268\) −9.79960 −0.598606
\(269\) 16.5793 1.01086 0.505429 0.862868i \(-0.331334\pi\)
0.505429 + 0.862868i \(0.331334\pi\)
\(270\) 0 0
\(271\) 7.83363i 0.475859i −0.971282 0.237929i \(-0.923531\pi\)
0.971282 0.237929i \(-0.0764687\pi\)
\(272\) −5.53921 −0.335864
\(273\) 0 0
\(274\) −20.7457 −1.25329
\(275\) 0 0
\(276\) 0 0
\(277\) −11.5910 −0.696435 −0.348217 0.937414i \(-0.613213\pi\)
−0.348217 + 0.937414i \(0.613213\pi\)
\(278\) −9.20646 −0.552167
\(279\) 0 0
\(280\) 0 0
\(281\) 19.3566i 1.15472i −0.816491 0.577358i \(-0.804084\pi\)
0.816491 0.577358i \(-0.195916\pi\)
\(282\) 0 0
\(283\) 1.28983i 0.0766724i 0.999265 + 0.0383362i \(0.0122058\pi\)
−0.999265 + 0.0383362i \(0.987794\pi\)
\(284\) 7.37284i 0.437497i
\(285\) 0 0
\(286\) 4.05852i 0.239985i
\(287\) 8.02926 + 2.02926i 0.473952 + 0.119783i
\(288\) 0 0
\(289\) 13.6828 0.804873
\(290\) 0 0
\(291\) 0 0
\(292\) 3.40686i 0.199371i
\(293\) −17.5992 −1.02816 −0.514078 0.857743i \(-0.671866\pi\)
−0.514078 + 0.857743i \(0.671866\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 10.6694i 0.620147i
\(297\) 0 0
\(298\) 10.3189 0.597758
\(299\) −5.73961 −0.331930
\(300\) 0 0
\(301\) −9.07842 2.29442i −0.523271 0.132248i
\(302\) 1.24049i 0.0713820i
\(303\) 0 0
\(304\) 7.37284i 0.422861i
\(305\) 0 0
\(306\) 0 0
\(307\) 26.7457i 1.52646i 0.646129 + 0.763228i \(0.276387\pi\)
−0.646129 + 0.763228i \(0.723613\pi\)
\(308\) 0.840546 3.32583i 0.0478946 0.189507i
\(309\) 0 0
\(310\) 0 0
\(311\) 29.3388 1.66365 0.831826 0.555037i \(-0.187296\pi\)
0.831826 + 0.555037i \(0.187296\pi\)
\(312\) 0 0
\(313\) 34.4172i 1.94538i −0.232114 0.972688i \(-0.574564\pi\)
0.232114 0.972688i \(-0.425436\pi\)
\(314\) 11.4629 0.646891
\(315\) 0 0
\(316\) −9.01990 −0.507409
\(317\) 4.53462i 0.254690i −0.991858 0.127345i \(-0.959354\pi\)
0.991858 0.127345i \(-0.0406455\pi\)
\(318\) 0 0
\(319\) 2.37742 0.133110
\(320\) 0 0
\(321\) 0 0
\(322\) 4.70343 + 1.18871i 0.262112 + 0.0662443i
\(323\) 40.8397i 2.27238i
\(324\) 0 0
\(325\) 0 0
\(326\) 1.27882i 0.0708272i
\(327\) 0 0
\(328\) 3.13020i 0.172836i
\(329\) 27.6510 + 6.98831i 1.52445 + 0.385278i
\(330\) 0 0
\(331\) −3.94148 −0.216644 −0.108322 0.994116i \(-0.534548\pi\)
−0.108322 + 0.994116i \(0.534548\pi\)
\(332\) −6.26039 −0.343584
\(333\) 0 0
\(334\) 6.72118i 0.367767i
\(335\) 0 0
\(336\) 0 0
\(337\) −25.8198 −1.40649 −0.703247 0.710946i \(-0.748267\pi\)
−0.703247 + 0.710946i \(0.748267\pi\)
\(338\) 3.20188i 0.174159i
\(339\) 0 0
\(340\) 0 0
\(341\) 13.5190 0.732096
\(342\) 0 0
\(343\) 13.6435 + 12.5242i 0.736681 + 0.676241i
\(344\) 3.53921i 0.190821i
\(345\) 0 0
\(346\) 16.3189i 0.877310i
\(347\) 10.7102i 0.574952i −0.957788 0.287476i \(-0.907184\pi\)
0.957788 0.287476i \(-0.0928162\pi\)
\(348\) 0 0
\(349\) 14.5236i 0.777431i −0.921358 0.388716i \(-0.872919\pi\)
0.921358 0.388716i \(-0.127081\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.29657 0.0691074
\(353\) −28.6176 −1.52316 −0.761581 0.648069i \(-0.775577\pi\)
−0.761581 + 0.648069i \(0.775577\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 7.94822 0.421255
\(357\) 0 0
\(358\) −19.1887 −1.01415
\(359\) 29.3771i 1.55047i 0.631675 + 0.775233i \(0.282368\pi\)
−0.631675 + 0.775233i \(0.717632\pi\)
\(360\) 0 0
\(361\) −35.3587 −1.86099
\(362\) −10.7797 −0.566568
\(363\) 0 0
\(364\) 2.02926 8.02926i 0.106362 0.420847i
\(365\) 0 0
\(366\) 0 0
\(367\) 18.2671i 0.953537i −0.879029 0.476768i \(-0.841808\pi\)
0.879029 0.476768i \(-0.158192\pi\)
\(368\) 1.83363i 0.0955844i
\(369\) 0 0
\(370\) 0 0
\(371\) −2.86980 + 11.3551i −0.148993 + 0.589527i
\(372\) 0 0
\(373\) −22.6694 −1.17378 −0.586889 0.809668i \(-0.699647\pi\)
−0.586889 + 0.809668i \(0.699647\pi\)
\(374\) −7.18197 −0.371371
\(375\) 0 0
\(376\) 10.7797i 0.555920i
\(377\) 5.73961 0.295605
\(378\) 0 0
\(379\) −15.1955 −0.780538 −0.390269 0.920701i \(-0.627618\pi\)
−0.390269 + 0.920701i \(0.627618\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −19.3728 −0.991201
\(383\) −1.22030 −0.0623546 −0.0311773 0.999514i \(-0.509926\pi\)
−0.0311773 + 0.999514i \(0.509926\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 6.81803i 0.347029i
\(387\) 0 0
\(388\) 8.09402i 0.410911i
\(389\) 13.4654i 0.682722i −0.939932 0.341361i \(-0.889112\pi\)
0.939932 0.341361i \(-0.110888\pi\)
\(390\) 0 0
\(391\) 10.1568i 0.513654i
\(392\) −3.32583 + 6.15945i −0.167980 + 0.311099i
\(393\) 0 0
\(394\) −14.7457 −0.742876
\(395\) 0 0
\(396\) 0 0
\(397\) 12.9053i 0.647699i 0.946109 + 0.323849i \(0.104977\pi\)
−0.946109 + 0.323849i \(0.895023\pi\)
\(398\) −1.94148 −0.0973178
\(399\) 0 0
\(400\) 0 0
\(401\) 26.7677i 1.33672i −0.743839 0.668358i \(-0.766997\pi\)
0.743839 0.668358i \(-0.233003\pi\)
\(402\) 0 0
\(403\) 32.6378 1.62581
\(404\) −4.05852 −0.201919
\(405\) 0 0
\(406\) −4.70343 1.18871i −0.233427 0.0589948i
\(407\) 13.8336i 0.685707i
\(408\) 0 0
\(409\) 12.0000i 0.593362i −0.954977 0.296681i \(-0.904120\pi\)
0.954977 0.296681i \(-0.0958798\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 5.61548i 0.276655i
\(413\) 18.4397 + 4.66033i 0.907361 + 0.229320i
\(414\) 0 0
\(415\) 0 0
\(416\) 3.13020 0.153470
\(417\) 0 0
\(418\) 9.55939i 0.467565i
\(419\) −12.9283 −0.631590 −0.315795 0.948827i \(-0.602271\pi\)
−0.315795 + 0.948827i \(0.602271\pi\)
\(420\) 0 0
\(421\) −11.4424 −0.557667 −0.278833 0.960340i \(-0.589948\pi\)
−0.278833 + 0.960340i \(0.589948\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 0 0
\(424\) −4.42677 −0.214983
\(425\) 0 0
\(426\) 0 0
\(427\) −3.16853 + 12.5371i −0.153336 + 0.606711i
\(428\) 0.651655i 0.0314989i
\(429\) 0 0
\(430\) 0 0
\(431\) 19.7411i 0.950895i 0.879744 + 0.475447i \(0.157714\pi\)
−0.879744 + 0.475447i \(0.842286\pi\)
\(432\) 0 0
\(433\) 9.95066i 0.478198i 0.970995 + 0.239099i \(0.0768521\pi\)
−0.970995 + 0.239099i \(0.923148\pi\)
\(434\) −26.7457 6.75951i −1.28383 0.324467i
\(435\) 0 0
\(436\) −15.3388 −0.734596
\(437\) 13.5190 0.646703
\(438\) 0 0
\(439\) 2.64735i 0.126351i 0.998002 + 0.0631755i \(0.0201228\pi\)
−0.998002 + 0.0631755i \(0.979877\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −17.3388 −0.824723
\(443\) 26.8982i 1.27797i −0.769218 0.638986i \(-0.779354\pi\)
0.769218 0.638986i \(-0.220646\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 8.76195 0.414890
\(447\) 0 0
\(448\) −2.56510 0.648285i −0.121189 0.0306286i
\(449\) 41.1317i 1.94112i 0.240852 + 0.970562i \(0.422573\pi\)
−0.240852 + 0.970562i \(0.577427\pi\)
\(450\) 0 0
\(451\) 4.05852i 0.191108i
\(452\) 17.0784i 0.803301i
\(453\) 0 0
\(454\) 23.0784i 1.08312i
\(455\) 0 0
\(456\) 0 0
\(457\) −12.0398 −0.563198 −0.281599 0.959532i \(-0.590865\pi\)
−0.281599 + 0.959532i \(0.590865\pi\)
\(458\) 9.70558 0.453512
\(459\) 0 0
\(460\) 0 0
\(461\) −34.8397 −1.62265 −0.811323 0.584598i \(-0.801252\pi\)
−0.811323 + 0.584598i \(0.801252\pi\)
\(462\) 0 0
\(463\) −24.3256 −1.13051 −0.565254 0.824917i \(-0.691222\pi\)
−0.565254 + 0.824917i \(0.691222\pi\)
\(464\) 1.83363i 0.0851240i
\(465\) 0 0
\(466\) 6.36825 0.295003
\(467\) 18.7812 0.869089 0.434545 0.900650i \(-0.356909\pi\)
0.434545 + 0.900650i \(0.356909\pi\)
\(468\) 0 0
\(469\) 25.1369 + 6.35293i 1.16072 + 0.293351i
\(470\) 0 0
\(471\) 0 0
\(472\) 7.18871i 0.330887i
\(473\) 4.58883i 0.210995i
\(474\) 0 0
\(475\) 0 0
\(476\) 14.2086 + 3.59099i 0.651251 + 0.164593i
\(477\) 0 0
\(478\) 19.3728 0.886093
\(479\) 6.66119 0.304357 0.152179 0.988353i \(-0.451371\pi\)
0.152179 + 0.988353i \(0.451371\pi\)
\(480\) 0 0
\(481\) 33.3973i 1.52279i
\(482\) 24.3587 1.10951
\(483\) 0 0
\(484\) −9.31891 −0.423587
\(485\) 0 0
\(486\) 0 0
\(487\) −20.9096 −0.947505 −0.473752 0.880658i \(-0.657101\pi\)
−0.473752 + 0.880658i \(0.657101\pi\)
\(488\) −4.88755 −0.221249
\(489\) 0 0
\(490\) 0 0
\(491\) 0.374990i 0.0169231i 0.999964 + 0.00846153i \(0.00269342\pi\)
−0.999964 + 0.00846153i \(0.997307\pi\)
\(492\) 0 0
\(493\) 10.1568i 0.457441i
\(494\) 23.0784i 1.03835i
\(495\) 0 0
\(496\) 10.4268i 0.468176i
\(497\) −4.77970 + 18.9120i −0.214399 + 0.848321i
\(498\) 0 0
\(499\) −36.1785 −1.61957 −0.809786 0.586725i \(-0.800417\pi\)
−0.809786 + 0.586725i \(0.800417\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 1.94822i 0.0869535i
\(503\) 36.8195 1.64170 0.820850 0.571143i \(-0.193500\pi\)
0.820850 + 0.571143i \(0.193500\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.37742i 0.105689i
\(507\) 0 0
\(508\) −7.39059 −0.327904
\(509\) 11.7396 0.520349 0.260175 0.965562i \(-0.416220\pi\)
0.260175 + 0.965562i \(0.416220\pi\)
\(510\) 0 0
\(511\) −2.20862 + 8.73893i −0.0977034 + 0.386588i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 25.2420i 1.11338i
\(515\) 0 0
\(516\) 0 0
\(517\) 13.9766i 0.614691i
\(518\) 6.91681 27.3681i 0.303907 1.20248i
\(519\) 0 0
\(520\) 0 0
\(521\) −30.9500 −1.35594 −0.677972 0.735088i \(-0.737141\pi\)
−0.677972 + 0.735088i \(0.737141\pi\)
\(522\) 0 0
\(523\) 20.4853i 0.895759i 0.894094 + 0.447879i \(0.147821\pi\)
−0.894094 + 0.447879i \(0.852179\pi\)
\(524\) 20.2086 0.882817
\(525\) 0 0
\(526\) 0.912047 0.0397671
\(527\) 57.7560i 2.51589i
\(528\) 0 0
\(529\) 19.6378 0.853818
\(530\) 0 0
\(531\) 0 0
\(532\) −4.77970 + 18.9120i −0.207226 + 0.819941i
\(533\) 9.79812i 0.424404i
\(534\) 0 0
\(535\) 0 0
\(536\) 9.79960i 0.423278i
\(537\) 0 0
\(538\) 16.5793i 0.714784i
\(539\) −4.31217 + 7.98616i −0.185738 + 0.343988i
\(540\) 0 0
\(541\) −14.7414 −0.633781 −0.316890 0.948462i \(-0.602639\pi\)
−0.316890 + 0.948462i \(0.602639\pi\)
\(542\) −7.83363 −0.336483
\(543\) 0 0
\(544\) 5.53921i 0.237492i
\(545\) 0 0
\(546\) 0 0
\(547\) −6.61763 −0.282949 −0.141475 0.989942i \(-0.545184\pi\)
−0.141475 + 0.989942i \(0.545184\pi\)
\(548\) 20.7457i 0.886211i
\(549\) 0 0
\(550\) 0 0
\(551\) −13.5190 −0.575930
\(552\) 0 0
\(553\) 23.1369 + 5.84747i 0.983883 + 0.248660i
\(554\) 11.5910i 0.492454i
\(555\) 0 0
\(556\) 9.20646i 0.390441i
\(557\) 16.1568i 0.684587i −0.939593 0.342294i \(-0.888796\pi\)
0.939593 0.342294i \(-0.111204\pi\)
\(558\) 0 0
\(559\) 11.0784i 0.468567i
\(560\) 0 0
\(561\) 0 0
\(562\) −19.3566 −0.816507
\(563\) 9.55939 0.402880 0.201440 0.979501i \(-0.435438\pi\)
0.201440 + 0.979501i \(0.435438\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.28983 0.0542155
\(567\) 0 0
\(568\) −7.37284 −0.309357
\(569\) 30.9883i 1.29910i 0.760320 + 0.649549i \(0.225042\pi\)
−0.760320 + 0.649549i \(0.774958\pi\)
\(570\) 0 0
\(571\) 16.4623 0.688924 0.344462 0.938800i \(-0.388061\pi\)
0.344462 + 0.938800i \(0.388061\pi\)
\(572\) 4.05852 0.169695
\(573\) 0 0
\(574\) 2.02926 8.02926i 0.0846996 0.335135i
\(575\) 0 0
\(576\) 0 0
\(577\) 23.1863i 0.965257i −0.875825 0.482629i \(-0.839682\pi\)
0.875825 0.482629i \(-0.160318\pi\)
\(578\) 13.6828i 0.569132i
\(579\) 0 0
\(580\) 0 0
\(581\) 16.0585 + 4.05852i 0.666220 + 0.168376i
\(582\) 0 0
\(583\) −5.73961 −0.237710
\(584\) −3.40686 −0.140977
\(585\) 0 0
\(586\) 17.5992i 0.727016i
\(587\) −7.18197 −0.296432 −0.148216 0.988955i \(-0.547353\pi\)
−0.148216 + 0.988955i \(0.547353\pi\)
\(588\) 0 0
\(589\) −76.8748 −3.16757
\(590\) 0 0
\(591\) 0 0
\(592\) 10.6694 0.438510
\(593\) −9.75979 −0.400787 −0.200393 0.979716i \(-0.564222\pi\)
−0.200393 + 0.979716i \(0.564222\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.3189i 0.422679i
\(597\) 0 0
\(598\) 5.73961i 0.234710i
\(599\) 14.4158i 0.589012i −0.955650 0.294506i \(-0.904845\pi\)
0.955650 0.294506i \(-0.0951551\pi\)
\(600\) 0 0
\(601\) 17.1326i 0.698855i −0.936963 0.349427i \(-0.886376\pi\)
0.936963 0.349427i \(-0.113624\pi\)
\(602\) −2.29442 + 9.07842i −0.0935134 + 0.370009i
\(603\) 0 0
\(604\) 1.24049 0.0504747
\(605\) 0 0
\(606\) 0 0
\(607\) 19.1345i 0.776646i −0.921523 0.388323i \(-0.873055\pi\)
0.921523 0.388323i \(-0.126945\pi\)
\(608\) −7.37284 −0.299008
\(609\) 0 0
\(610\) 0 0
\(611\) 33.7426i 1.36508i
\(612\) 0 0
\(613\) 33.5676 1.35578 0.677892 0.735162i \(-0.262894\pi\)
0.677892 + 0.735162i \(0.262894\pi\)
\(614\) 26.7457 1.07937
\(615\) 0 0
\(616\) −3.32583 0.840546i −0.134001 0.0338666i
\(617\) 0.476107i 0.0191673i 0.999954 + 0.00958366i \(0.00305062\pi\)
−0.999954 + 0.00958366i \(0.996949\pi\)
\(618\) 0 0
\(619\) 35.9521i 1.44504i 0.691351 + 0.722519i \(0.257016\pi\)
−0.691351 + 0.722519i \(0.742984\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 29.3388i 1.17638i
\(623\) −20.3880 5.15271i −0.816827 0.206439i
\(624\) 0 0
\(625\) 0 0
\(626\) −34.4172 −1.37559
\(627\) 0 0
\(628\) 11.4629i 0.457421i
\(629\) −59.1001 −2.35647
\(630\) 0 0
\(631\) 46.8982 1.86699 0.933494 0.358593i \(-0.116743\pi\)
0.933494 + 0.358593i \(0.116743\pi\)
\(632\) 9.01990i 0.358793i
\(633\) 0 0
\(634\) −4.53462 −0.180093
\(635\) 0 0
\(636\) 0 0
\(637\) −10.4105 + 19.2803i −0.412479 + 0.763913i
\(638\) 2.37742i 0.0941231i
\(639\) 0 0
\(640\) 0 0
\(641\) 15.6893i 0.619690i −0.950787 0.309845i \(-0.899723\pi\)
0.950787 0.309845i \(-0.100277\pi\)
\(642\) 0 0
\(643\) 39.8198i 1.57034i −0.619281 0.785170i \(-0.712576\pi\)
0.619281 0.785170i \(-0.287424\pi\)
\(644\) 1.18871 4.70343i 0.0468418 0.185341i
\(645\) 0 0
\(646\) 40.8397 1.60682
\(647\) 11.3005 0.444268 0.222134 0.975016i \(-0.428698\pi\)
0.222134 + 0.975016i \(0.428698\pi\)
\(648\) 0 0
\(649\) 9.32066i 0.365868i
\(650\) 0 0
\(651\) 0 0
\(652\) 1.27882 0.0500824
\(653\) 42.1430i 1.64918i −0.565729 0.824592i \(-0.691405\pi\)
0.565729 0.824592i \(-0.308595\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 3.13020 0.122214
\(657\) 0 0
\(658\) 6.98831 27.6510i 0.272433 1.07795i
\(659\) 17.5172i 0.682371i 0.939996 + 0.341186i \(0.110828\pi\)
−0.939996 + 0.341186i \(0.889172\pi\)
\(660\) 0 0
\(661\) 32.2307i 1.25363i −0.779169 0.626814i \(-0.784359\pi\)
0.779169 0.626814i \(-0.215641\pi\)
\(662\) 3.94148i 0.153190i
\(663\) 0 0
\(664\) 6.26039i 0.242950i
\(665\) 0 0
\(666\) 0 0
\(667\) 3.36218 0.130184
\(668\) −6.72118 −0.260050
\(669\) 0 0
\(670\) 0 0
\(671\) −6.33705 −0.244639
\(672\) 0 0
\(673\) −10.9351 −0.421516 −0.210758 0.977538i \(-0.567593\pi\)
−0.210758 + 0.977538i \(0.567593\pi\)
\(674\) 25.8198i 0.994541i
\(675\) 0 0
\(676\) −3.20188 −0.123149
\(677\) 44.5606 1.71260 0.856301 0.516477i \(-0.172757\pi\)
0.856301 + 0.516477i \(0.172757\pi\)
\(678\) 0 0
\(679\) 5.24723 20.7619i 0.201370 0.796770i
\(680\) 0 0
\(681\) 0 0
\(682\) 13.5190i 0.517670i
\(683\) 7.98616i 0.305582i 0.988259 + 0.152791i \(0.0488261\pi\)
−0.988259 + 0.152791i \(0.951174\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 12.5242 13.6435i 0.478174 0.520912i
\(687\) 0 0
\(688\) −3.53921 −0.134931
\(689\) −13.8566 −0.527896
\(690\) 0 0
\(691\) 45.1969i 1.71937i −0.510823 0.859686i \(-0.670659\pi\)
0.510823 0.859686i \(-0.329341\pi\)
\(692\) −16.3189 −0.620352
\(693\) 0 0
\(694\) −10.7102 −0.406553
\(695\) 0 0
\(696\) 0 0
\(697\) −17.3388 −0.656754
\(698\) −14.5236 −0.549727
\(699\) 0 0
\(700\) 0 0
\(701\) 1.31284i 0.0495854i −0.999693 0.0247927i \(-0.992107\pi\)
0.999693 0.0247927i \(-0.00789257\pi\)
\(702\) 0 0
\(703\) 78.6638i 2.96686i
\(704\) 1.29657i 0.0488663i
\(705\) 0 0
\(706\) 28.6176i 1.07704i
\(707\) 10.4105 + 2.63107i 0.391527 + 0.0989517i
\(708\) 0 0
\(709\) −13.0784 −0.491170 −0.245585 0.969375i \(-0.578980\pi\)
−0.245585 + 0.969375i \(0.578980\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 7.94822i 0.297872i
\(713\) 19.1188 0.716004
\(714\) 0 0
\(715\) 0 0
\(716\) 19.1887i 0.717116i
\(717\) 0 0
\(718\) 29.3771 1.09635
\(719\) 34.1568 1.27384 0.636918 0.770932i \(-0.280209\pi\)
0.636918 + 0.770932i \(0.280209\pi\)
\(720\) 0 0
\(721\) −3.64043 + 14.4042i −0.135577 + 0.536442i
\(722\) 35.3587i 1.31592i
\(723\) 0 0
\(724\) 10.7797i 0.400624i
\(725\) 0 0
\(726\) 0 0
\(727\) 16.0965i 0.596984i 0.954412 + 0.298492i \(0.0964837\pi\)
−0.954412 + 0.298492i \(0.903516\pi\)
\(728\) −8.02926 2.02926i −0.297584 0.0752093i
\(729\) 0 0
\(730\) 0 0
\(731\) 19.6044 0.725096
\(732\) 0 0
\(733\) 40.2484i 1.48661i 0.668953 + 0.743305i \(0.266743\pi\)
−0.668953 + 0.743305i \(0.733257\pi\)
\(734\) −18.2671 −0.674252
\(735\) 0 0
\(736\) 1.83363 0.0675884
\(737\) 12.7059i 0.468027i
\(738\) 0 0
\(739\) −13.3838 −0.492333 −0.246166 0.969228i \(-0.579171\pi\)
−0.246166 + 0.969228i \(0.579171\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 11.3551 + 2.86980i 0.416858 + 0.105354i
\(743\) 4.18803i 0.153644i −0.997045 0.0768221i \(-0.975523\pi\)
0.997045 0.0768221i \(-0.0244773\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 22.6694i 0.829986i
\(747\) 0 0
\(748\) 7.18197i 0.262599i
\(749\) −0.422458 + 1.67156i −0.0154363 + 0.0610774i
\(750\) 0 0
\(751\) 22.7812 0.831297 0.415648 0.909525i \(-0.363555\pi\)
0.415648 + 0.909525i \(0.363555\pi\)
\(752\) 10.7797 0.393095
\(753\) 0 0
\(754\) 5.73961i 0.209024i
\(755\) 0 0
\(756\) 0 0
\(757\) −44.7092 −1.62498 −0.812492 0.582972i \(-0.801890\pi\)
−0.812492 + 0.582972i \(0.801890\pi\)
\(758\) 15.1955i 0.551924i
\(759\) 0 0
\(760\) 0 0
\(761\) 43.5474 1.57859 0.789297 0.614012i \(-0.210445\pi\)
0.789297 + 0.614012i \(0.210445\pi\)
\(762\) 0 0
\(763\) 39.3456 + 9.94392i 1.42440 + 0.359994i
\(764\) 19.3728i 0.700885i
\(765\) 0 0
\(766\) 1.22030i 0.0440913i
\(767\) 22.5021i 0.812503i
\(768\) 0 0
\(769\) 37.2803i 1.34436i 0.740387 + 0.672181i \(0.234642\pi\)
−0.740387 + 0.672181i \(0.765358\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −6.81803 −0.245386
\(773\) 39.2353 1.41119 0.705597 0.708613i \(-0.250679\pi\)
0.705597 + 0.708613i \(0.250679\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 8.09402 0.290558
\(777\) 0 0
\(778\) −13.4654 −0.482757
\(779\) 23.0784i 0.826870i
\(780\) 0 0
\(781\) −9.55939 −0.342062
\(782\) −10.1568 −0.363208
\(783\) 0 0
\(784\) 6.15945 + 3.32583i 0.219980 + 0.118780i
\(785\) 0 0
\(786\) 0 0
\(787\) 1.15078i 0.0410208i −0.999790 0.0205104i \(-0.993471\pi\)
0.999790 0.0205104i \(-0.00652911\pi\)
\(788\) 14.7457i 0.525293i
\(789\) 0 0
\(790\) 0 0
\(791\) −11.0717 + 43.8078i −0.393664 + 1.55763i
\(792\) 0 0
\(793\) −15.2990 −0.543284
\(794\) 12.9053 0.457992
\(795\) 0 0
\(796\) 1.94148i 0.0688141i
\(797\) −16.3189 −0.578045 −0.289023 0.957322i \(-0.593330\pi\)
−0.289023 + 0.957322i \(0.593330\pi\)
\(798\) 0 0
\(799\) −59.7110 −2.11242
\(800\) 0 0
\(801\) 0 0
\(802\) −26.7677 −0.945201
\(803\) −4.41723 −0.155881