Properties

Label 3150.2.d.a.3149.4
Level 3150
Weight 2
Character 3150.3149
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.d (of order \(2\), degree \(1\), not 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: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3149.4
Root \(-0.916813i\)
Character \(\chi\) = 3150.3149
Dual form 3150.2.d.a.3149.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +(-0.648285 + 2.56510i) q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +(-0.648285 + 2.56510i) q^{7} -1.00000 q^{8} -1.29657i q^{11} +3.13020 q^{13} +(0.648285 - 2.56510i) q^{14} +1.00000 q^{16} +5.53921i q^{17} +7.37284i q^{19} +1.29657i q^{22} -1.83363 q^{23} -3.13020 q^{26} +(-0.648285 + 2.56510i) q^{28} -1.83363i q^{29} -10.4268i q^{31} -1.00000 q^{32} -5.53921i q^{34} +10.6694i q^{37} -7.37284i q^{38} -3.13020 q^{41} +3.53921i q^{43} -1.29657i q^{44} +1.83363 q^{46} -10.7797i q^{47} +(-6.15945 - 3.32583i) q^{49} +3.13020 q^{52} -4.42677 q^{53} +(0.648285 - 2.56510i) q^{56} +1.83363i q^{58} +7.18871 q^{59} +4.88755i q^{61} +10.4268i q^{62} +1.00000 q^{64} +9.79960i q^{67} +5.53921i q^{68} -7.37284i q^{71} +3.40686 q^{73} -10.6694i q^{74} +7.37284i q^{76} +(3.32583 + 0.840546i) q^{77} -9.01990 q^{79} +3.13020 q^{82} +6.26039i q^{83} -3.53921i q^{86} +1.29657i q^{88} -7.94822 q^{89} +(-2.02926 + 8.02926i) q^{91} -1.83363 q^{92} +10.7797i q^{94} +8.09402 q^{97} +(6.15945 + 3.32583i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{2} + 8q^{4} - 8q^{8} + O(q^{10}) \) \( 8q - 8q^{2} + 8q^{4} - 8q^{8} - 8q^{13} + 8q^{16} + 8q^{23} + 8q^{26} - 8q^{32} + 8q^{41} - 8q^{46} - 4q^{49} - 8q^{52} + 8q^{53} + 8q^{64} + 48q^{73} + 4q^{77} - 8q^{79} - 8q^{82} - 8q^{89} - 4q^{91} + 8q^{92} - 24q^{97} + 4q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −0.648285 + 2.56510i −0.245029 + 0.969516i
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 1.29657i 0.390930i −0.980711 0.195465i \(-0.937378\pi\)
0.980711 0.195465i \(-0.0626217\pi\)
\(12\) 0 0
\(13\) 3.13020 0.868160 0.434080 0.900874i \(-0.357074\pi\)
0.434080 + 0.900874i \(0.357074\pi\)
\(14\) 0.648285 2.56510i 0.173261 0.685551i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.53921i 1.34346i 0.740798 + 0.671728i \(0.234448\pi\)
−0.740798 + 0.671728i \(0.765552\pi\)
\(18\) 0 0
\(19\) 7.37284i 1.69144i 0.533623 + 0.845722i \(0.320830\pi\)
−0.533623 + 0.845722i \(0.679170\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.29657i 0.276430i
\(23\) −1.83363 −0.382337 −0.191169 0.981557i \(-0.561228\pi\)
−0.191169 + 0.981557i \(0.561228\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −3.13020 −0.613882
\(27\) 0 0
\(28\) −0.648285 + 2.56510i −0.122514 + 0.484758i
\(29\) 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 \(-0.385820\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 5.53921i 0.949967i
\(35\) 0 0
\(36\) 0 0
\(37\) 10.6694i 1.75404i 0.480454 + 0.877020i \(0.340472\pi\)
−0.480454 + 0.877020i \(0.659528\pi\)
\(38\) 7.37284i 1.19603i
\(39\) 0 0
\(40\) 0 0
\(41\) −3.13020 −0.488854 −0.244427 0.969668i \(-0.578600\pi\)
−0.244427 + 0.969668i \(0.578600\pi\)
\(42\) 0 0
\(43\) 3.53921i 0.539724i 0.962899 + 0.269862i \(0.0869782\pi\)
−0.962899 + 0.269862i \(0.913022\pi\)
\(44\) 1.29657i 0.195465i
\(45\) 0 0
\(46\) 1.83363 0.270353
\(47\) 10.7797i 1.57238i −0.617985 0.786190i \(-0.712051\pi\)
0.617985 0.786190i \(-0.287949\pi\)
\(48\) 0 0
\(49\) −6.15945 3.32583i −0.879922 0.475118i
\(50\) 0 0
\(51\) 0 0
\(52\) 3.13020 0.434080
\(53\) −4.42677 −0.608063 −0.304031 0.952662i \(-0.598333\pi\)
−0.304031 + 0.952662i \(0.598333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.648285 2.56510i 0.0866307 0.342776i
\(57\) 0 0
\(58\) 1.83363i 0.240767i
\(59\) 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 \(0.898702\pi\)
\(62\) 10.4268i 1.32420i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 9.79960i 1.19721i 0.801044 + 0.598606i \(0.204278\pi\)
−0.801044 + 0.598606i \(0.795722\pi\)
\(68\) 5.53921i 0.671728i
\(69\) 0 0
\(70\) 0 0
\(71\) 7.37284i 0.874995i −0.899220 0.437497i \(-0.855865\pi\)
0.899220 0.437497i \(-0.144135\pi\)
\(72\) 0 0
\(73\) 3.40686 0.398743 0.199371 0.979924i \(-0.436110\pi\)
0.199371 + 0.979924i \(0.436110\pi\)
\(74\) 10.6694i 1.24029i
\(75\) 0 0
\(76\) 7.37284i 0.845722i
\(77\) 3.32583 + 0.840546i 0.379013 + 0.0957891i
\(78\) 0 0
\(79\) −9.01990 −1.01482 −0.507409 0.861705i \(-0.669397\pi\)
−0.507409 + 0.861705i \(0.669397\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.13020 0.345672
\(83\) 6.26039i 0.687167i 0.939122 + 0.343584i \(0.111641\pi\)
−0.939122 + 0.343584i \(0.888359\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.53921i 0.381643i
\(87\) 0 0
\(88\) 1.29657i 0.138215i
\(89\) −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
\(92\) −1.83363 −0.191169
\(93\) 0 0
\(94\) 10.7797i 1.11184i
\(95\) 0 0
\(96\) 0 0
\(97\) 8.09402 0.821823 0.410911 0.911675i \(-0.365211\pi\)
0.410911 + 0.911675i \(0.365211\pi\)
\(98\) 6.15945 + 3.32583i 0.622199 + 0.335959i
\(99\) 0 0
\(100\) 0 0
\(101\) −4.05852 −0.403837 −0.201919 0.979402i \(-0.564718\pi\)
−0.201919 + 0.979402i \(0.564718\pi\)
\(102\) 0 0
\(103\) 5.61548 0.553309 0.276655 0.960969i \(-0.410774\pi\)
0.276655 + 0.960969i \(0.410774\pi\)
\(104\) −3.13020 −0.306941
\(105\) 0 0
\(106\) 4.42677 0.429965
\(107\) 0.651655 0.0629979 0.0314989 0.999504i \(-0.489972\pi\)
0.0314989 + 0.999504i \(0.489972\pi\)
\(108\) 0 0
\(109\) −15.3388 −1.46919 −0.734596 0.678505i \(-0.762628\pi\)
−0.734596 + 0.678505i \(0.762628\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −0.648285 + 2.56510i −0.0612572 + 0.242379i
\(113\) −17.0784 −1.60660 −0.803301 0.595573i \(-0.796925\pi\)
−0.803301 + 0.595573i \(0.796925\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.83363i 0.170248i
\(117\) 0 0
\(118\) −7.18871 −0.661775
\(119\) −14.2086 3.59099i −1.30250 0.329185i
\(120\) 0 0
\(121\) 9.31891 0.847173
\(122\) 4.88755i 0.442498i
\(123\) 0 0
\(124\) 10.4268i 0.936351i
\(125\) 0 0
\(126\) 0 0
\(127\) 7.39059i 0.655809i 0.944711 + 0.327904i \(0.106342\pi\)
−0.944711 + 0.327904i \(0.893658\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) 20.2086 1.76563 0.882817 0.469716i \(-0.155644\pi\)
0.882817 + 0.469716i \(0.155644\pi\)
\(132\) 0 0
\(133\) −18.9120 4.77970i −1.63988 0.414452i
\(134\) 9.79960i 0.846557i
\(135\) 0 0
\(136\) 5.53921i 0.474983i
\(137\) −20.7457 −1.77242 −0.886211 0.463281i \(-0.846672\pi\)
−0.886211 + 0.463281i \(0.846672\pi\)
\(138\) 0 0
\(139\) 9.20646i 0.780882i 0.920628 + 0.390441i \(0.127677\pi\)
−0.920628 + 0.390441i \(0.872323\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.37284i 0.618715i
\(143\) 4.05852i 0.339390i
\(144\) 0 0
\(145\) 0 0
\(146\) −3.40686 −0.281954
\(147\) 0 0
\(148\) 10.6694i 0.877020i
\(149\) 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 \(0.516073\pi\)
\(152\) 7.37284i 0.598016i
\(153\) 0 0
\(154\) −3.32583 0.840546i −0.268003 0.0677331i
\(155\) 0 0
\(156\) 0 0
\(157\) −11.4629 −0.914842 −0.457421 0.889250i \(-0.651227\pi\)
−0.457421 + 0.889250i \(0.651227\pi\)
\(158\) 9.01990 0.717585
\(159\) 0 0
\(160\) 0 0
\(161\) 1.18871 4.70343i 0.0936836 0.370682i
\(162\) 0 0
\(163\) 1.27882i 0.100165i 0.998745 + 0.0500824i \(0.0159484\pi\)
−0.998745 + 0.0500824i \(0.984052\pi\)
\(164\) −3.13020 −0.244427
\(165\) 0 0
\(166\) 6.26039i 0.485901i
\(167\) 6.72118i 0.520101i −0.965595 0.260050i \(-0.916261\pi\)
0.965595 0.260050i \(-0.0837392\pi\)
\(168\) 0 0
\(169\) −3.20188 −0.246298
\(170\) 0 0
\(171\) 0 0
\(172\) 3.53921i 0.269862i
\(173\) 16.3189i 1.24070i 0.784324 + 0.620352i \(0.213010\pi\)
−0.784324 + 0.620352i \(0.786990\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.29657i 0.0977326i
\(177\) 0 0
\(178\) 7.94822 0.595745
\(179\) 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 \(-0.131207\pi\)
\(182\) 2.02926 8.02926i 0.150419 0.595168i
\(183\) 0 0
\(184\) 1.83363 0.135177
\(185\) 0 0
\(186\) 0 0
\(187\) 7.18197 0.525198
\(188\) 10.7797i 0.786190i
\(189\) 0 0
\(190\) 0 0
\(191\) 19.3728i 1.40177i 0.713275 + 0.700885i \(0.247211\pi\)
−0.713275 + 0.700885i \(0.752789\pi\)
\(192\) 0 0
\(193\) 6.81803i 0.490772i −0.969425 0.245386i \(-0.921085\pi\)
0.969425 0.245386i \(-0.0789148\pi\)
\(194\) −8.09402 −0.581117
\(195\) 0 0
\(196\) −6.15945 3.32583i −0.439961 0.237559i
\(197\) −14.7457 −1.05059 −0.525293 0.850922i \(-0.676044\pi\)
−0.525293 + 0.850922i \(0.676044\pi\)
\(198\) 0 0
\(199\) 1.94148i 0.137628i 0.997630 + 0.0688141i \(0.0219215\pi\)
−0.997630 + 0.0688141i \(0.978078\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 4.05852 0.285556
\(203\) 4.70343 + 1.18871i 0.330116 + 0.0834312i
\(204\) 0 0
\(205\) 0 0
\(206\) −5.61548 −0.391249
\(207\) 0 0
\(208\) 3.13020 0.217040
\(209\) 9.55939 0.661237
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −4.42677 −0.304031
\(213\) 0 0
\(214\) −0.651655 −0.0445462
\(215\) 0 0
\(216\) 0 0
\(217\) 26.7457 + 6.75951i 1.81561 + 0.458866i
\(218\) 15.3388 1.03888
\(219\) 0 0
\(220\) 0 0
\(221\) 17.3388i 1.16633i
\(222\) 0 0
\(223\) 8.76195 0.586743 0.293372 0.955998i \(-0.405223\pi\)
0.293372 + 0.955998i \(0.405223\pi\)
\(224\) 0.648285 2.56510i 0.0433153 0.171388i
\(225\) 0 0
\(226\) 17.0784 1.13604
\(227\) 23.0784i 1.53177i 0.642978 + 0.765884i \(0.277698\pi\)
−0.642978 + 0.765884i \(0.722302\pi\)
\(228\) 0 0
\(229\) 9.70558i 0.641363i −0.947187 0.320682i \(-0.896088\pi\)
0.947187 0.320682i \(-0.103912\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.83363i 0.120383i
\(233\) −6.36825 −0.417198 −0.208599 0.978001i \(-0.566890\pi\)
−0.208599 + 0.978001i \(0.566890\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 7.18871 0.467945
\(237\) 0 0
\(238\) 14.2086 + 3.59099i 0.921008 + 0.232769i
\(239\) 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 \(0.712898\pi\)
\(242\) −9.31891 −0.599042
\(243\) 0 0
\(244\) 4.88755i 0.312894i
\(245\) 0 0
\(246\) 0 0
\(247\) 23.0784i 1.46844i
\(248\) 10.4268i 0.662100i
\(249\) 0 0
\(250\) 0 0
\(251\) −1.94822 −0.122971 −0.0614854 0.998108i \(-0.519584\pi\)
−0.0614854 + 0.998108i \(0.519584\pi\)
\(252\) 0 0
\(253\) 2.37742i 0.149467i
\(254\) 7.39059i 0.463727i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 25.2420i 1.57455i 0.616602 + 0.787275i \(0.288509\pi\)
−0.616602 + 0.787275i \(0.711491\pi\)
\(258\) 0 0
\(259\) −27.3681 6.91681i −1.70057 0.429790i
\(260\) 0 0
\(261\) 0 0
\(262\) −20.2086 −1.24849
\(263\) −0.912047 −0.0562392 −0.0281196 0.999605i \(-0.508952\pi\)
−0.0281196 + 0.999605i \(0.508952\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 18.9120 + 4.77970i 1.15957 + 0.293062i
\(267\) 0 0
\(268\) 9.79960i 0.598606i
\(269\) 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 \(-0.0764687\pi\)
\(272\) 5.53921i 0.335864i
\(273\) 0 0
\(274\) 20.7457 1.25329
\(275\) 0 0
\(276\) 0 0
\(277\) 11.5910i 0.696435i −0.937414 0.348217i \(-0.886787\pi\)
0.937414 0.348217i \(-0.113213\pi\)
\(278\) 9.20646i 0.552167i
\(279\) 0 0
\(280\) 0 0
\(281\) 19.3566i 1.15472i 0.816491 + 0.577358i \(0.195916\pi\)
−0.816491 + 0.577358i \(0.804084\pi\)
\(282\) 0 0
\(283\) 1.28983 0.0766724 0.0383362 0.999265i \(-0.487794\pi\)
0.0383362 + 0.999265i \(0.487794\pi\)
\(284\) 7.37284i 0.437497i
\(285\) 0 0
\(286\) 4.05852i 0.239985i
\(287\) 2.02926 8.02926i 0.119783 0.473952i
\(288\) 0 0
\(289\) −13.6828 −0.804873
\(290\) 0 0
\(291\) 0 0
\(292\) 3.40686 0.199371
\(293\) 17.5992i 1.02816i −0.857743 0.514078i \(-0.828134\pi\)
0.857743 0.514078i \(-0.171866\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 10.6694i 0.620147i
\(297\) 0 0
\(298\) 10.3189i 0.597758i
\(299\) −5.73961 −0.331930
\(300\) 0 0
\(301\) −9.07842 2.29442i −0.523271 0.132248i
\(302\) 1.24049 0.0713820
\(303\) 0 0
\(304\) 7.37284i 0.422861i
\(305\) 0 0
\(306\) 0 0
\(307\) −26.7457 −1.52646 −0.763228 0.646129i \(-0.776387\pi\)
−0.763228 + 0.646129i \(0.776387\pi\)
\(308\) 3.32583 + 0.840546i 0.189507 + 0.0478946i
\(309\) 0 0
\(310\) 0 0
\(311\) −29.3388 −1.66365 −0.831826 0.555037i \(-0.812704\pi\)
−0.831826 + 0.555037i \(0.812704\pi\)
\(312\) 0 0
\(313\) −34.4172 −1.94538 −0.972688 0.232114i \(-0.925436\pi\)
−0.972688 + 0.232114i \(0.925436\pi\)
\(314\) 11.4629 0.646891
\(315\) 0 0
\(316\) −9.01990 −0.507409
\(317\) −4.53462 −0.254690 −0.127345 0.991858i \(-0.540646\pi\)
−0.127345 + 0.991858i \(0.540646\pi\)
\(318\) 0 0
\(319\) −2.37742 −0.133110
\(320\) 0 0
\(321\) 0 0
\(322\) −1.18871 + 4.70343i −0.0662443 + 0.262112i
\(323\) −40.8397 −2.27238
\(324\) 0 0
\(325\) 0 0
\(326\) 1.27882i 0.0708272i
\(327\) 0 0
\(328\) 3.13020 0.172836
\(329\) 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 \(0.534548\pi\)
\(332\) 6.26039i 0.343584i
\(333\) 0 0
\(334\) 6.72118i 0.367767i
\(335\) 0 0
\(336\) 0 0
\(337\) 25.8198i 1.40649i −0.710946 0.703247i \(-0.751733\pi\)
0.710946 0.703247i \(-0.248267\pi\)
\(338\) 3.20188 0.174159
\(339\) 0 0
\(340\) 0 0
\(341\) −13.5190 −0.732096
\(342\) 0 0
\(343\) 12.5242 13.6435i 0.676241 0.736681i
\(344\) 3.53921i 0.190821i
\(345\) 0 0
\(346\) 16.3189i 0.877310i
\(347\) −10.7102 −0.574952 −0.287476 0.957788i \(-0.592816\pi\)
−0.287476 + 0.957788i \(0.592816\pi\)
\(348\) 0 0
\(349\) 14.5236i 0.777431i 0.921358 + 0.388716i \(0.127081\pi\)
−0.921358 + 0.388716i \(0.872919\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.29657i 0.0691074i
\(353\) 28.6176i 1.52316i −0.648069 0.761581i \(-0.724423\pi\)
0.648069 0.761581i \(-0.275577\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −7.94822 −0.421255
\(357\) 0 0
\(358\) 19.1887i 1.01415i
\(359\) 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
\(362\) 10.7797i 0.566568i
\(363\) 0 0
\(364\) −2.02926 + 8.02926i −0.106362 + 0.420847i
\(365\) 0 0
\(366\) 0 0
\(367\) 18.2671 0.953537 0.476768 0.879029i \(-0.341808\pi\)
0.476768 + 0.879029i \(0.341808\pi\)
\(368\) −1.83363 −0.0955844
\(369\) 0 0
\(370\) 0 0
\(371\) 2.86980 11.3551i 0.148993 0.589527i
\(372\) 0 0
\(373\) 22.6694i 1.17378i 0.809668 + 0.586889i \(0.199647\pi\)
−0.809668 + 0.586889i \(0.800353\pi\)
\(374\) −7.18197 −0.371371
\(375\) 0 0
\(376\) 10.7797i 0.555920i
\(377\) 5.73961i 0.295605i
\(378\) 0 0
\(379\) 15.1955 0.780538 0.390269 0.920701i \(-0.372382\pi\)
0.390269 + 0.920701i \(0.372382\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 19.3728i 0.991201i
\(383\) 1.22030i 0.0623546i −0.999514 0.0311773i \(-0.990074\pi\)
0.999514 0.0311773i \(-0.00992564\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 6.81803i 0.347029i
\(387\) 0 0
\(388\) 8.09402 0.410911
\(389\) 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
\(392\) 6.15945 + 3.32583i 0.311099 + 0.167980i
\(393\) 0 0
\(394\) 14.7457 0.742876
\(395\) 0 0
\(396\) 0 0
\(397\) −12.9053 −0.647699 −0.323849 0.946109i \(-0.604977\pi\)
−0.323849 + 0.946109i \(0.604977\pi\)
\(398\) 1.94148i 0.0973178i
\(399\) 0 0
\(400\) 0 0
\(401\) 26.7677i 1.33672i 0.743839 + 0.668358i \(0.233003\pi\)
−0.743839 + 0.668358i \(0.766997\pi\)
\(402\) 0 0
\(403\) 32.6378i 1.62581i
\(404\) −4.05852 −0.201919
\(405\) 0 0
\(406\) −4.70343 1.18871i −0.233427 0.0589948i
\(407\) 13.8336 0.685707
\(408\) 0 0
\(409\) 12.0000i 0.593362i 0.954977 + 0.296681i \(0.0958798\pi\)
−0.954977 + 0.296681i \(0.904120\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 5.61548 0.276655
\(413\) −4.66033 + 18.4397i −0.229320 + 0.907361i
\(414\) 0 0
\(415\) 0 0
\(416\) −3.13020 −0.153470
\(417\) 0 0
\(418\) −9.55939 −0.467565
\(419\) −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 \(0.589948\pi\)
\(422\) −4.00000 −0.194717
\(423\) 0 0
\(424\) 4.42677 0.214983
\(425\) 0 0
\(426\) 0 0
\(427\) −12.5371 3.16853i −0.606711 0.153336i
\(428\) 0.651655 0.0314989
\(429\) 0 0
\(430\) 0 0
\(431\) 19.7411i 0.950895i −0.879744 0.475447i \(-0.842286\pi\)
0.879744 0.475447i \(-0.157714\pi\)
\(432\) 0 0
\(433\) 9.95066 0.478198 0.239099 0.970995i \(-0.423148\pi\)
0.239099 + 0.970995i \(0.423148\pi\)
\(434\) −26.7457 6.75951i −1.28383 0.324467i
\(435\) 0 0
\(436\) −15.3388 −0.734596
\(437\) 13.5190i 0.646703i
\(438\) 0 0
\(439\) 2.64735i 0.126351i −0.998002 0.0631755i \(-0.979877\pi\)
0.998002 0.0631755i \(-0.0201228\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 17.3388i 0.824723i
\(443\) 26.8982 1.27797 0.638986 0.769218i \(-0.279354\pi\)
0.638986 + 0.769218i \(0.279354\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −8.76195 −0.414890
\(447\) 0 0
\(448\) −0.648285 + 2.56510i −0.0306286 + 0.121189i
\(449\) 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
\(452\) −17.0784 −0.803301
\(453\) 0 0
\(454\) 23.0784i 1.08312i
\(455\) 0 0
\(456\) 0 0
\(457\) 12.0398i 0.563198i −0.959532 0.281599i \(-0.909135\pi\)
0.959532 0.281599i \(-0.0908649\pi\)
\(458\) 9.70558i 0.453512i
\(459\) 0 0
\(460\) 0 0
\(461\) 34.8397 1.62265 0.811323 0.584598i \(-0.198748\pi\)
0.811323 + 0.584598i \(0.198748\pi\)
\(462\) 0 0
\(463\) 24.3256i 1.13051i 0.824917 + 0.565254i \(0.191222\pi\)
−0.824917 + 0.565254i \(0.808778\pi\)
\(464\) 1.83363i 0.0851240i
\(465\) 0 0
\(466\) 6.36825 0.295003
\(467\) 18.7812i 0.869089i −0.900650 0.434545i \(-0.856909\pi\)
0.900650 0.434545i \(-0.143091\pi\)
\(468\) 0 0
\(469\) −25.1369 6.35293i −1.16072 0.293351i
\(470\) 0 0
\(471\) 0 0
\(472\) −7.18871 −0.330887
\(473\) 4.58883 0.210995
\(474\) 0 0
\(475\) 0 0
\(476\) −14.2086 3.59099i −0.651251 0.164593i
\(477\) 0 0
\(478\) 19.3728i 0.886093i
\(479\) 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
\(482\) 24.3587i 1.10951i
\(483\) 0 0
\(484\) 9.31891 0.423587
\(485\) 0 0
\(486\) 0 0
\(487\) 20.9096i 0.947505i −0.880658 0.473752i \(-0.842899\pi\)
0.880658 0.473752i \(-0.157101\pi\)
\(488\) 4.88755i 0.221249i
\(489\) 0 0
\(490\) 0 0
\(491\) 0.374990i 0.0169231i −0.999964 0.00846153i \(-0.997307\pi\)
0.999964 0.00846153i \(-0.00269342\pi\)
\(492\) 0 0
\(493\) 10.1568 0.457441
\(494\) 23.0784i 1.03835i
\(495\) 0 0
\(496\) 10.4268i 0.468176i
\(497\) 18.9120 + 4.77970i 0.848321 + 0.214399i
\(498\) 0 0
\(499\) 36.1785 1.61957 0.809786 0.586725i \(-0.199583\pi\)
0.809786 + 0.586725i \(0.199583\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 1.94822 0.0869535
\(503\) 36.8195i 1.64170i 0.571143 + 0.820850i \(0.306500\pi\)
−0.571143 + 0.820850i \(0.693500\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.37742i 0.105689i
\(507\) 0 0
\(508\) 7.39059i 0.327904i
\(509\) 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
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 25.2420i 1.11338i
\(515\) 0 0
\(516\) 0 0
\(517\) −13.9766 −0.614691
\(518\) 27.3681 + 6.91681i 1.20248 + 0.303907i
\(519\) 0 0
\(520\) 0 0
\(521\) 30.9500 1.35594 0.677972 0.735088i \(-0.262859\pi\)
0.677972 + 0.735088i \(0.262859\pi\)
\(522\) 0 0
\(523\) 20.4853 0.895759 0.447879 0.894094i \(-0.352179\pi\)
0.447879 + 0.894094i \(0.352179\pi\)
\(524\) 20.2086 0.882817
\(525\) 0 0
\(526\) 0.912047 0.0397671
\(527\) 57.7560 2.51589
\(528\) 0 0
\(529\) −19.6378 −0.853818
\(530\) 0 0
\(531\) 0 0
\(532\) −18.9120 4.77970i −0.819941 0.207226i
\(533\) −9.79812 −0.424404
\(534\) 0 0
\(535\) 0 0
\(536\) 9.79960i 0.423278i
\(537\) 0 0
\(538\) −16.5793 −0.714784
\(539\) −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 \(0.602639\pi\)
\(542\) 7.83363i 0.336483i
\(543\) 0 0
\(544\) 5.53921i 0.237492i
\(545\) 0 0
\(546\) 0 0
\(547\) 6.61763i 0.282949i −0.989942 0.141475i \(-0.954816\pi\)
0.989942 0.141475i \(-0.0451844\pi\)
\(548\) −20.7457 −0.886211
\(549\) 0 0
\(550\) 0 0
\(551\) 13.5190 0.575930
\(552\) 0 0
\(553\) 5.84747 23.1369i 0.248660 0.983883i
\(554\) 11.5910i 0.492454i
\(555\) 0 0
\(556\) 9.20646i 0.390441i
\(557\) −16.1568 −0.684587 −0.342294 0.939593i \(-0.611204\pi\)
−0.342294 + 0.939593i \(0.611204\pi\)
\(558\) 0 0
\(559\) 11.0784i 0.468567i
\(560\) 0 0
\(561\) 0 0
\(562\) 19.3566i 0.816507i
\(563\) 9.55939i 0.402880i 0.979501 + 0.201440i \(0.0645622\pi\)
−0.979501 + 0.201440i \(0.935438\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −1.28983 −0.0542155
\(567\) 0 0
\(568\) 7.37284i 0.309357i
\(569\) 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 \(0.388061\pi\)
\(572\) 4.05852i 0.169695i
\(573\) 0 0
\(574\) −2.02926 + 8.02926i −0.0846996 + 0.335135i
\(575\) 0 0
\(576\) 0 0
\(577\) 23.1863 0.965257 0.482629 0.875825i \(-0.339682\pi\)
0.482629 + 0.875825i \(0.339682\pi\)
\(578\) 13.6828 0.569132
\(579\) 0 0
\(580\) 0 0
\(581\) −16.0585 4.05852i −0.666220 0.168376i
\(582\) 0 0
\(583\) 5.73961i 0.237710i
\(584\) −3.40686 −0.140977
\(585\) 0 0
\(586\) 17.5992i 0.727016i
\(587\) 7.18197i 0.296432i 0.988955 + 0.148216i \(0.0473530\pi\)
−0.988955 + 0.148216i \(0.952647\pi\)
\(588\) 0 0
\(589\) 76.8748 3.16757
\(590\) 0 0
\(591\) 0 0
\(592\) 10.6694i 0.438510i
\(593\) 9.75979i 0.400787i −0.979716 0.200393i \(-0.935778\pi\)
0.979716 0.200393i \(-0.0642220\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.3189i 0.422679i
\(597\) 0 0
\(598\) 5.73961 0.234710
\(599\) 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 \(-0.113624\pi\)
\(602\) 9.07842 + 2.29442i 0.370009 + 0.0935134i
\(603\) 0 0
\(604\) −1.24049 −0.0504747
\(605\) 0 0
\(606\) 0 0
\(607\) 19.1345 0.776646 0.388323 0.921523i \(-0.373055\pi\)
0.388323 + 0.921523i \(0.373055\pi\)
\(608\) 7.37284i 0.299008i
\(609\) 0 0
\(610\) 0 0
\(611\) 33.7426i 1.36508i
\(612\) 0 0
\(613\) 33.5676i 1.35578i −0.735162 0.677892i \(-0.762894\pi\)
0.735162 0.677892i \(-0.237106\pi\)
\(614\) 26.7457 1.07937
\(615\) 0 0
\(616\) −3.32583 0.840546i −0.134001 0.0338666i
\(617\) 0.476107 0.0191673 0.00958366 0.999954i \(-0.496949\pi\)
0.00958366 + 0.999954i \(0.496949\pi\)
\(618\) 0 0
\(619\) 35.9521i 1.44504i −0.691351 0.722519i \(-0.742984\pi\)
0.691351 0.722519i \(-0.257016\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 29.3388 1.17638
\(623\) 5.15271 20.3880i 0.206439 0.816827i
\(624\) 0 0
\(625\) 0 0
\(626\) 34.4172 1.37559
\(627\) 0 0
\(628\) −11.4629 −0.457421
\(629\) −59.1001 −2.35647
\(630\) 0 0
\(631\) 46.8982 1.86699 0.933494 0.358593i \(-0.116743\pi\)
0.933494 + 0.358593i \(0.116743\pi\)
\(632\) 9.01990 0.358793
\(633\) 0 0
\(634\) 4.53462 0.180093
\(635\) 0 0
\(636\) 0 0
\(637\) −19.2803 10.4105i −0.763913 0.412479i
\(638\) 2.37742 0.0941231
\(639\) 0 0
\(640\) 0 0
\(641\) 15.6893i 0.619690i 0.950787 + 0.309845i \(0.100277\pi\)
−0.950787 + 0.309845i \(0.899723\pi\)
\(642\) 0 0
\(643\) −39.8198 −1.57034 −0.785170 0.619281i \(-0.787424\pi\)
−0.785170 + 0.619281i \(0.787424\pi\)
\(644\) 1.18871 4.70343i 0.0468418 0.185341i
\(645\) 0 0
\(646\) 40.8397 1.60682
\(647\) 11.3005i 0.444268i −0.975016 0.222134i \(-0.928698\pi\)
0.975016 0.222134i \(-0.0713022\pi\)
\(648\) 0 0
\(649\) 9.32066i 0.365868i
\(650\) 0 0
\(651\) 0 0
\(652\) 1.27882i 0.0500824i
\(653\) 42.1430 1.64918 0.824592 0.565729i \(-0.191405\pi\)
0.824592 + 0.565729i \(0.191405\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3.13020 −0.122214
\(657\) 0 0
\(658\) −27.6510 6.98831i −1.07795 0.272433i
\(659\) 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 \(-0.215641\pi\)
\(662\) 3.94148 0.153190
\(663\) 0 0
\(664\) 6.26039i 0.242950i
\(665\) 0 0
\(666\) 0 0
\(667\) 3.36218i 0.130184i
\(668\) 6.72118i 0.260050i
\(669\) 0 0
\(670\) 0 0
\(671\) 6.33705 0.244639
\(672\) 0 0
\(673\) 10.9351i 0.421516i 0.977538 + 0.210758i \(0.0675931\pi\)
−0.977538 + 0.210758i \(0.932407\pi\)
\(674\) 25.8198i 0.994541i
\(675\) 0 0
\(676\) −3.20188 −0.123149
\(677\) 44.5606i 1.71260i −0.516477 0.856301i \(-0.672757\pi\)
0.516477 0.856301i \(-0.327243\pi\)
\(678\) 0 0
\(679\) −5.24723 + 20.7619i −0.201370 + 0.796770i
\(680\) 0 0
\(681\) 0 0
\(682\) 13.5190 0.517670
\(683\) −7.98616 −0.305582 −0.152791 0.988259i \(-0.548826\pi\)
−0.152791 + 0.988259i \(0.548826\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −12.5242 + 13.6435i −0.478174 + 0.520912i
\(687\) 0 0
\(688\) 3.53921i 0.134931i
\(689\) −13.8566 −0.527896
\(690\) 0 0
\(691\) 45.1969i 1.71937i −0.510823 0.859686i \(-0.670659\pi\)
0.510823 0.859686i \(-0.329341\pi\)
\(692\) 16.3189i 0.620352i
\(693\) 0 0
\(694\) 10.7102 0.406553
\(695\) 0 0
\(696\) 0 0
\(697\) 17.3388i 0.656754i
\(698\) 14.5236i 0.549727i
\(699\) 0 0
\(700\) 0 0
\(701\) 1.31284i 0.0495854i 0.999693 + 0.0247927i \(0.00789257\pi\)
−0.999693 + 0.0247927i \(0.992107\pi\)
\(702\) 0 0
\(703\) −78.6638 −2.96686
\(704\) 1.29657i 0.0488663i
\(705\) 0 0
\(706\) 28.6176i 1.07704i
\(707\) 2.63107 10.4105i 0.0989517 0.391527i
\(708\) 0 0
\(709\) 13.0784 0.491170 0.245585 0.969375i \(-0.421020\pi\)
0.245585 + 0.969375i \(0.421020\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 7.94822 0.297872
\(713\) 19.1188i 0.716004i
\(714\) 0 0
\(715\) 0 0
\(716\) 19.1887i 0.717116i
\(717\) 0 0
\(718\) 29.3771i 1.09635i
\(719\) 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
\(722\) 35.3587 1.31592
\(723\) 0 0
\(724\) 10.7797i 0.400624i
\(725\) 0 0
\(726\) 0 0
\(727\) −16.0965 −0.596984 −0.298492 0.954412i \(-0.596484\pi\)
−0.298492 + 0.954412i \(0.596484\pi\)
\(728\) 2.02926 8.02926i 0.0752093 0.297584i
\(729\) 0 0
\(730\) 0 0
\(731\) −19.6044 −0.725096
\(732\) 0 0
\(733\) 40.2484 1.48661 0.743305 0.668953i \(-0.233257\pi\)
0.743305 + 0.668953i \(0.233257\pi\)
\(734\) −18.2671 −0.674252
\(735\) 0 0
\(736\) 1.83363 0.0675884
\(737\) 12.7059 0.468027
\(738\) 0 0
\(739\) 13.3838 0.492333 0.246166 0.969228i \(-0.420829\pi\)
0.246166 + 0.969228i \(0.420829\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.86980 + 11.3551i −0.105354 + 0.416858i
\(743\) 4.18803 0.153644 0.0768221 0.997045i \(-0.475523\pi\)
0.0768221 + 0.997045i \(0.475523\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 22.6694i 0.829986i
\(747\) 0 0
\(748\) 7.18197 0.262599
\(749\) −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 \(0.363555\pi\)
\(752\) 10.7797i 0.393095i
\(753\) 0 0
\(754\) 5.73961i 0.209024i
\(755\) 0 0
\(756\) 0 0
\(757\) 44.7092i 1.62498i −0.582972 0.812492i \(-0.698110\pi\)
0.582972 0.812492i \(-0.301890\pi\)
\(758\) −15.1955 −0.551924
\(759\) 0 0
\(760\) 0 0
\(761\) −43.5474 −1.57859 −0.789297 0.614012i \(-0.789555\pi\)
−0.789297 + 0.614012i \(0.789555\pi\)
\(762\) 0 0
\(763\) 9.94392 39.3456i 0.359994 1.42440i
\(764\) 19.3728i 0.700885i
\(765\) 0 0
\(766\) 1.22030i 0.0440913i
\(767\) 22.5021 0.812503
\(768\) 0 0
\(769\) 37.2803i 1.34436i −0.740387 0.672181i \(-0.765358\pi\)
0.740387 0.672181i \(-0.234642\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 6.81803i 0.245386i
\(773\) 39.2353i 1.41119i 0.708613 + 0.705597i \(0.249321\pi\)
−0.708613 + 0.705597i \(0.750679\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −8.09402 −0.290558
\(777\) 0 0
\(778\) 13.4654i 0.482757i
\(779\) 23.0784i 0.826870i
\(780\) 0 0
\(781\) −9.55939 −0.342062
\(782\) 10.1568i 0.363208i
\(783\) 0 0
\(784\) −6.15945 3.32583i −0.219980 0.118780i
\(785\) 0 0
\(786\) 0 0
\(787\) 1.15078 0.0410208 0.0205104 0.999790i \(-0.493471\pi\)
0.0205104 + 0.999790i \(0.493471\pi\)
\(788\) −14.7457 −0.525293
\(789\) 0 0
\(790\) 0 0
\(791\) 11.0717 43.8078i 0.393664 1.55763i
\(792\) 0 0
\(793\) 15.2990i 0.543284i
\(794\) 12.9053 0.457992
\(795\) 0 0
\(796\) 1.94148i 0.0688141i
\(797\) 16.3189i 0.578045i 0.957322 + 0.289023i \(0.0933303\pi\)
−0.957322 + 0.289023i \(0.906670\pi\)
\(798\) 0 0
\(799\) 59.7110 2.11242
\(800\) 0 0
\(801\) 0 0
\(802\) 26.7677i