Properties

Label 160.4.c.d.129.3
Level $160$
Weight $4$
Character 160.129
Analytic conductor $9.440$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 160.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.44030560092\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.359712057600.22
Defining polynomial: \( x^{8} + 17x^{6} + 82x^{4} + 96x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 129.3
Root \(-0.320221i\) of defining polynomial
Character \(\chi\) \(=\) 160.129
Dual form 160.4.c.d.129.6

$q$-expansion

\(f(q)\) \(=\) \(q-4.30169i q^{3} +(8.58258 - 7.16515i) q^{5} +28.3162i q^{7} +8.49545 q^{9} +O(q^{10})\) \(q-4.30169i q^{3} +(8.58258 - 7.16515i) q^{5} +28.3162i q^{7} +8.49545 q^{9} +65.2358 q^{11} -33.6697i q^{13} +(-30.8223 - 36.9196i) q^{15} -73.3212i q^{17} -134.063 q^{19} +121.808 q^{21} +14.7007i q^{23} +(22.3212 - 122.991i) q^{25} -152.690i q^{27} +224.642 q^{29} +68.8271 q^{31} -280.624i q^{33} +(202.890 + 243.026i) q^{35} +196.312i q^{37} -144.837 q^{39} -143.147 q^{41} -15.0755i q^{43} +(72.9129 - 60.8712i) q^{45} +134.399i q^{47} -458.808 q^{49} -315.405 q^{51} +262.955i q^{53} +(559.891 - 467.424i) q^{55} +576.697i q^{57} +119.698 q^{59} +16.5409 q^{61} +240.559i q^{63} +(-241.248 - 288.973i) q^{65} +545.565i q^{67} +63.2379 q^{69} -199.299 q^{71} +43.2667i q^{73} +(-529.069 - 96.0190i) q^{75} +1847.23i q^{77} -438.694 q^{79} -427.450 q^{81} -1220.89i q^{83} +(-525.358 - 629.285i) q^{85} -966.342i q^{87} -723.212 q^{89} +953.398 q^{91} -296.073i q^{93} +(-1150.60 + 960.581i) q^{95} +1136.00i q^{97} +554.208 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{5} - 152 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 32 q^{5} - 152 q^{9} - 272 q^{21} - 408 q^{25} + 624 q^{29} - 192 q^{41} + 400 q^{45} - 2424 q^{49} + 2112 q^{61} + 2176 q^{65} + 3952 q^{69} - 1000 q^{81} - 5376 q^{85} + 80 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 4.30169i 0.827861i −0.910309 0.413930i \(-0.864156\pi\)
0.910309 0.413930i \(-0.135844\pi\)
\(4\) 0 0
\(5\) 8.58258 7.16515i 0.767649 0.640871i
\(6\) 0 0
\(7\) 28.3162i 1.52893i 0.644664 + 0.764466i \(0.276997\pi\)
−0.644664 + 0.764466i \(0.723003\pi\)
\(8\) 0 0
\(9\) 8.49545 0.314646
\(10\) 0 0
\(11\) 65.2358 1.78812 0.894061 0.447946i \(-0.147844\pi\)
0.894061 + 0.447946i \(0.147844\pi\)
\(12\) 0 0
\(13\) 33.6697i 0.718330i −0.933274 0.359165i \(-0.883061\pi\)
0.933274 0.359165i \(-0.116939\pi\)
\(14\) 0 0
\(15\) −30.8223 36.9196i −0.530552 0.635506i
\(16\) 0 0
\(17\) 73.3212i 1.04606i −0.852315 0.523030i \(-0.824802\pi\)
0.852315 0.523030i \(-0.175198\pi\)
\(18\) 0 0
\(19\) −134.063 −1.61874 −0.809372 0.587297i \(-0.800192\pi\)
−0.809372 + 0.587297i \(0.800192\pi\)
\(20\) 0 0
\(21\) 121.808 1.26574
\(22\) 0 0
\(23\) 14.7007i 0.133274i 0.997777 + 0.0666371i \(0.0212270\pi\)
−0.997777 + 0.0666371i \(0.978773\pi\)
\(24\) 0 0
\(25\) 22.3212 122.991i 0.178570 0.983927i
\(26\) 0 0
\(27\) 152.690i 1.08834i
\(28\) 0 0
\(29\) 224.642 1.43845 0.719225 0.694777i \(-0.244497\pi\)
0.719225 + 0.694777i \(0.244497\pi\)
\(30\) 0 0
\(31\) 68.8271 0.398765 0.199382 0.979922i \(-0.436106\pi\)
0.199382 + 0.979922i \(0.436106\pi\)
\(32\) 0 0
\(33\) 280.624i 1.48032i
\(34\) 0 0
\(35\) 202.890 + 243.026i 0.979847 + 1.17368i
\(36\) 0 0
\(37\) 196.312i 0.872257i 0.899884 + 0.436129i \(0.143651\pi\)
−0.899884 + 0.436129i \(0.856349\pi\)
\(38\) 0 0
\(39\) −144.837 −0.594678
\(40\) 0 0
\(41\) −143.147 −0.545263 −0.272632 0.962118i \(-0.587894\pi\)
−0.272632 + 0.962118i \(0.587894\pi\)
\(42\) 0 0
\(43\) 15.0755i 0.0534648i −0.999643 0.0267324i \(-0.991490\pi\)
0.999643 0.0267324i \(-0.00851021\pi\)
\(44\) 0 0
\(45\) 72.9129 60.8712i 0.241538 0.201648i
\(46\) 0 0
\(47\) 134.399i 0.417107i 0.978011 + 0.208554i \(0.0668756\pi\)
−0.978011 + 0.208554i \(0.933124\pi\)
\(48\) 0 0
\(49\) −458.808 −1.33763
\(50\) 0 0
\(51\) −315.405 −0.865991
\(52\) 0 0
\(53\) 262.955i 0.681502i 0.940154 + 0.340751i \(0.110681\pi\)
−0.940154 + 0.340751i \(0.889319\pi\)
\(54\) 0 0
\(55\) 559.891 467.424i 1.37265 1.14595i
\(56\) 0 0
\(57\) 576.697i 1.34009i
\(58\) 0 0
\(59\) 119.698 0.264124 0.132062 0.991241i \(-0.457840\pi\)
0.132062 + 0.991241i \(0.457840\pi\)
\(60\) 0 0
\(61\) 16.5409 0.0347188 0.0173594 0.999849i \(-0.494474\pi\)
0.0173594 + 0.999849i \(0.494474\pi\)
\(62\) 0 0
\(63\) 240.559i 0.481073i
\(64\) 0 0
\(65\) −241.248 288.973i −0.460357 0.551425i
\(66\) 0 0
\(67\) 545.565i 0.994797i 0.867522 + 0.497399i \(0.165711\pi\)
−0.867522 + 0.497399i \(0.834289\pi\)
\(68\) 0 0
\(69\) 63.2379 0.110333
\(70\) 0 0
\(71\) −199.299 −0.333132 −0.166566 0.986030i \(-0.553268\pi\)
−0.166566 + 0.986030i \(0.553268\pi\)
\(72\) 0 0
\(73\) 43.2667i 0.0693696i 0.999398 + 0.0346848i \(0.0110427\pi\)
−0.999398 + 0.0346848i \(0.988957\pi\)
\(74\) 0 0
\(75\) −529.069 96.0190i −0.814555 0.147831i
\(76\) 0 0
\(77\) 1847.23i 2.73391i
\(78\) 0 0
\(79\) −438.694 −0.624772 −0.312386 0.949955i \(-0.601128\pi\)
−0.312386 + 0.949955i \(0.601128\pi\)
\(80\) 0 0
\(81\) −427.450 −0.586351
\(82\) 0 0
\(83\) 1220.89i 1.61458i −0.590154 0.807291i \(-0.700933\pi\)
0.590154 0.807291i \(-0.299067\pi\)
\(84\) 0 0
\(85\) −525.358 629.285i −0.670389 0.803006i
\(86\) 0 0
\(87\) 966.342i 1.19084i
\(88\) 0 0
\(89\) −723.212 −0.861352 −0.430676 0.902507i \(-0.641725\pi\)
−0.430676 + 0.902507i \(0.641725\pi\)
\(90\) 0 0
\(91\) 953.398 1.09828
\(92\) 0 0
\(93\) 296.073i 0.330122i
\(94\) 0 0
\(95\) −1150.60 + 960.581i −1.24263 + 1.03741i
\(96\) 0 0
\(97\) 1136.00i 1.18911i 0.804056 + 0.594553i \(0.202671\pi\)
−0.804056 + 0.594553i \(0.797329\pi\)
\(98\) 0 0
\(99\) 554.208 0.562626
\(100\) 0 0
\(101\) −1175.21 −1.15780 −0.578901 0.815398i \(-0.696518\pi\)
−0.578901 + 0.815398i \(0.696518\pi\)
\(102\) 0 0
\(103\) 1752.43i 1.67643i 0.545344 + 0.838213i \(0.316399\pi\)
−0.545344 + 0.838213i \(0.683601\pi\)
\(104\) 0 0
\(105\) 1045.42 872.770i 0.971646 0.811177i
\(106\) 0 0
\(107\) 94.5981i 0.0854686i 0.999086 + 0.0427343i \(0.0136069\pi\)
−0.999086 + 0.0427343i \(0.986393\pi\)
\(108\) 0 0
\(109\) 1349.68 1.18602 0.593009 0.805196i \(-0.297940\pi\)
0.593009 + 0.805196i \(0.297940\pi\)
\(110\) 0 0
\(111\) 844.474 0.722108
\(112\) 0 0
\(113\) 529.248i 0.440597i −0.975432 0.220299i \(-0.929297\pi\)
0.975432 0.220299i \(-0.0707032\pi\)
\(114\) 0 0
\(115\) 105.333 + 126.170i 0.0854116 + 0.102308i
\(116\) 0 0
\(117\) 286.039i 0.226020i
\(118\) 0 0
\(119\) 2076.18 1.59935
\(120\) 0 0
\(121\) 2924.71 2.19738
\(122\) 0 0
\(123\) 615.774i 0.451402i
\(124\) 0 0
\(125\) −689.675 1215.51i −0.493491 0.869751i
\(126\) 0 0
\(127\) 814.066i 0.568793i −0.958707 0.284396i \(-0.908207\pi\)
0.958707 0.284396i \(-0.0917931\pi\)
\(128\) 0 0
\(129\) −64.8500 −0.0442614
\(130\) 0 0
\(131\) −1329.85 −0.886946 −0.443473 0.896288i \(-0.646254\pi\)
−0.443473 + 0.896288i \(0.646254\pi\)
\(132\) 0 0
\(133\) 3796.15i 2.47495i
\(134\) 0 0
\(135\) −1094.05 1310.48i −0.697488 0.835466i
\(136\) 0 0
\(137\) 2906.98i 1.81284i 0.422373 + 0.906422i \(0.361197\pi\)
−0.422373 + 0.906422i \(0.638803\pi\)
\(138\) 0 0
\(139\) −921.077 −0.562048 −0.281024 0.959701i \(-0.590674\pi\)
−0.281024 + 0.959701i \(0.590674\pi\)
\(140\) 0 0
\(141\) 578.141 0.345307
\(142\) 0 0
\(143\) 2196.47i 1.28446i
\(144\) 0 0
\(145\) 1928.01 1609.60i 1.10422 0.921860i
\(146\) 0 0
\(147\) 1973.65i 1.10737i
\(148\) 0 0
\(149\) −265.259 −0.145845 −0.0729224 0.997338i \(-0.523233\pi\)
−0.0729224 + 0.997338i \(0.523233\pi\)
\(150\) 0 0
\(151\) −2507.69 −1.35148 −0.675738 0.737142i \(-0.736175\pi\)
−0.675738 + 0.737142i \(0.736175\pi\)
\(152\) 0 0
\(153\) 622.897i 0.329139i
\(154\) 0 0
\(155\) 590.713 493.156i 0.306111 0.255557i
\(156\) 0 0
\(157\) 14.8818i 0.00756495i −0.999993 0.00378248i \(-0.998796\pi\)
0.999993 0.00378248i \(-0.00120400\pi\)
\(158\) 0 0
\(159\) 1131.15 0.564188
\(160\) 0 0
\(161\) −416.268 −0.203767
\(162\) 0 0
\(163\) 1184.23i 0.569055i −0.958668 0.284528i \(-0.908163\pi\)
0.958668 0.284528i \(-0.0918368\pi\)
\(164\) 0 0
\(165\) −2010.72 2408.48i −0.948691 1.13636i
\(166\) 0 0
\(167\) 2688.78i 1.24589i 0.782266 + 0.622945i \(0.214064\pi\)
−0.782266 + 0.622945i \(0.785936\pi\)
\(168\) 0 0
\(169\) 1063.35 0.484002
\(170\) 0 0
\(171\) −1138.92 −0.509332
\(172\) 0 0
\(173\) 1940.72i 0.852891i 0.904513 + 0.426445i \(0.140234\pi\)
−0.904513 + 0.426445i \(0.859766\pi\)
\(174\) 0 0
\(175\) 3482.64 + 632.052i 1.50436 + 0.273021i
\(176\) 0 0
\(177\) 514.903i 0.218658i
\(178\) 0 0
\(179\) 1580.62 0.660005 0.330002 0.943980i \(-0.392950\pi\)
0.330002 + 0.943980i \(0.392950\pi\)
\(180\) 0 0
\(181\) −2561.93 −1.05208 −0.526040 0.850460i \(-0.676324\pi\)
−0.526040 + 0.850460i \(0.676324\pi\)
\(182\) 0 0
\(183\) 71.1539i 0.0287423i
\(184\) 0 0
\(185\) 1406.61 + 1684.86i 0.559004 + 0.669587i
\(186\) 0 0
\(187\) 4783.17i 1.87048i
\(188\) 0 0
\(189\) 4323.62 1.66400
\(190\) 0 0
\(191\) −137.654 −0.0521482 −0.0260741 0.999660i \(-0.508301\pi\)
−0.0260741 + 0.999660i \(0.508301\pi\)
\(192\) 0 0
\(193\) 845.503i 0.315340i −0.987492 0.157670i \(-0.949602\pi\)
0.987492 0.157670i \(-0.0503982\pi\)
\(194\) 0 0
\(195\) −1243.07 + 1037.78i −0.456504 + 0.381111i
\(196\) 0 0
\(197\) 2733.63i 0.988646i 0.869278 + 0.494323i \(0.164584\pi\)
−0.869278 + 0.494323i \(0.835416\pi\)
\(198\) 0 0
\(199\) −4649.70 −1.65632 −0.828162 0.560489i \(-0.810613\pi\)
−0.828162 + 0.560489i \(0.810613\pi\)
\(200\) 0 0
\(201\) 2346.85 0.823553
\(202\) 0 0
\(203\) 6361.02i 2.19929i
\(204\) 0 0
\(205\) −1228.57 + 1025.67i −0.418571 + 0.349443i
\(206\) 0 0
\(207\) 124.889i 0.0419343i
\(208\) 0 0
\(209\) −8745.70 −2.89451
\(210\) 0 0
\(211\) 3845.91 1.25480 0.627402 0.778696i \(-0.284118\pi\)
0.627402 + 0.778696i \(0.284118\pi\)
\(212\) 0 0
\(213\) 857.321i 0.275787i
\(214\) 0 0
\(215\) −108.018 129.386i −0.0342640 0.0410422i
\(216\) 0 0
\(217\) 1948.92i 0.609684i
\(218\) 0 0
\(219\) 186.120 0.0574284
\(220\) 0 0
\(221\) −2468.70 −0.751416
\(222\) 0 0
\(223\) 1460.74i 0.438648i −0.975652 0.219324i \(-0.929615\pi\)
0.975652 0.219324i \(-0.0703851\pi\)
\(224\) 0 0
\(225\) 189.629 1044.86i 0.0561863 0.309589i
\(226\) 0 0
\(227\) 3435.16i 1.00440i −0.864750 0.502202i \(-0.832523\pi\)
0.864750 0.502202i \(-0.167477\pi\)
\(228\) 0 0
\(229\) 595.358 0.171801 0.0859003 0.996304i \(-0.472623\pi\)
0.0859003 + 0.996304i \(0.472623\pi\)
\(230\) 0 0
\(231\) 7946.21 2.26330
\(232\) 0 0
\(233\) 5432.62i 1.52748i 0.645523 + 0.763740i \(0.276639\pi\)
−0.645523 + 0.763740i \(0.723361\pi\)
\(234\) 0 0
\(235\) 962.986 + 1153.49i 0.267312 + 0.320192i
\(236\) 0 0
\(237\) 1887.13i 0.517224i
\(238\) 0 0
\(239\) −3931.51 −1.06405 −0.532026 0.846728i \(-0.678569\pi\)
−0.532026 + 0.846728i \(0.678569\pi\)
\(240\) 0 0
\(241\) −2587.88 −0.691701 −0.345851 0.938290i \(-0.612410\pi\)
−0.345851 + 0.938290i \(0.612410\pi\)
\(242\) 0 0
\(243\) 2283.89i 0.602927i
\(244\) 0 0
\(245\) −3937.75 + 3287.43i −1.02683 + 0.857249i
\(246\) 0 0
\(247\) 4513.86i 1.16279i
\(248\) 0 0
\(249\) −5251.90 −1.33665
\(250\) 0 0
\(251\) −2231.79 −0.561232 −0.280616 0.959820i \(-0.590539\pi\)
−0.280616 + 0.959820i \(0.590539\pi\)
\(252\) 0 0
\(253\) 959.012i 0.238311i
\(254\) 0 0
\(255\) −2706.99 + 2259.93i −0.664777 + 0.554988i
\(256\) 0 0
\(257\) 2477.07i 0.601226i −0.953746 0.300613i \(-0.902809\pi\)
0.953746 0.300613i \(-0.0971913\pi\)
\(258\) 0 0
\(259\) −5558.81 −1.33362
\(260\) 0 0
\(261\) 1908.44 0.452603
\(262\) 0 0
\(263\) 4865.27i 1.14071i −0.821400 0.570353i \(-0.806806\pi\)
0.821400 0.570353i \(-0.193194\pi\)
\(264\) 0 0
\(265\) 1884.11 + 2256.83i 0.436754 + 0.523154i
\(266\) 0 0
\(267\) 3111.04i 0.713080i
\(268\) 0 0
\(269\) 6753.93 1.53084 0.765418 0.643534i \(-0.222532\pi\)
0.765418 + 0.643534i \(0.222532\pi\)
\(270\) 0 0
\(271\) −4315.74 −0.967390 −0.483695 0.875237i \(-0.660706\pi\)
−0.483695 + 0.875237i \(0.660706\pi\)
\(272\) 0 0
\(273\) 4101.22i 0.909221i
\(274\) 0 0
\(275\) 1456.14 8023.41i 0.319304 1.75938i
\(276\) 0 0
\(277\) 5799.07i 1.25788i −0.777454 0.628939i \(-0.783489\pi\)
0.777454 0.628939i \(-0.216511\pi\)
\(278\) 0 0
\(279\) 584.717 0.125470
\(280\) 0 0
\(281\) −538.674 −0.114358 −0.0571790 0.998364i \(-0.518211\pi\)
−0.0571790 + 0.998364i \(0.518211\pi\)
\(282\) 0 0
\(283\) 1648.50i 0.346265i −0.984899 0.173133i \(-0.944611\pi\)
0.984899 0.173133i \(-0.0553889\pi\)
\(284\) 0 0
\(285\) 4132.12 + 4949.55i 0.858827 + 1.02872i
\(286\) 0 0
\(287\) 4053.38i 0.833670i
\(288\) 0 0
\(289\) −463.000 −0.0942398
\(290\) 0 0
\(291\) 4886.72 0.984415
\(292\) 0 0
\(293\) 4963.73i 0.989707i 0.868976 + 0.494854i \(0.164778\pi\)
−0.868976 + 0.494854i \(0.835222\pi\)
\(294\) 0 0
\(295\) 1027.32 857.653i 0.202755 0.169269i
\(296\) 0 0
\(297\) 9960.88i 1.94609i
\(298\) 0 0
\(299\) 494.968 0.0957350
\(300\) 0 0
\(301\) 426.880 0.0817441
\(302\) 0 0
\(303\) 5055.40i 0.958499i
\(304\) 0 0
\(305\) 141.964 118.518i 0.0266518 0.0222503i
\(306\) 0 0
\(307\) 9906.61i 1.84169i 0.389925 + 0.920847i \(0.372501\pi\)
−0.389925 + 0.920847i \(0.627499\pi\)
\(308\) 0 0
\(309\) 7538.40 1.38785
\(310\) 0 0
\(311\) −3477.27 −0.634012 −0.317006 0.948424i \(-0.602677\pi\)
−0.317006 + 0.948424i \(0.602677\pi\)
\(312\) 0 0
\(313\) 3958.27i 0.714808i −0.933950 0.357404i \(-0.883662\pi\)
0.933950 0.357404i \(-0.116338\pi\)
\(314\) 0 0
\(315\) 1723.64 + 2064.62i 0.308305 + 0.369295i
\(316\) 0 0
\(317\) 3296.86i 0.584132i −0.956398 0.292066i \(-0.905657\pi\)
0.956398 0.292066i \(-0.0943428\pi\)
\(318\) 0 0
\(319\) 14654.7 2.57212
\(320\) 0 0
\(321\) 406.932 0.0707561
\(322\) 0 0
\(323\) 9829.65i 1.69330i
\(324\) 0 0
\(325\) −4141.07 751.548i −0.706785 0.128272i
\(326\) 0 0
\(327\) 5805.91i 0.981858i
\(328\) 0 0
\(329\) −3805.66 −0.637728
\(330\) 0 0
\(331\) −6048.38 −1.00438 −0.502189 0.864758i \(-0.667472\pi\)
−0.502189 + 0.864758i \(0.667472\pi\)
\(332\) 0 0
\(333\) 1667.76i 0.274453i
\(334\) 0 0
\(335\) 3909.06 + 4682.35i 0.637536 + 0.763655i
\(336\) 0 0
\(337\) 8539.53i 1.38035i −0.723643 0.690175i \(-0.757534\pi\)
0.723643 0.690175i \(-0.242466\pi\)
\(338\) 0 0
\(339\) −2276.66 −0.364753
\(340\) 0 0
\(341\) 4489.99 0.713040
\(342\) 0 0
\(343\) 3279.23i 0.516215i
\(344\) 0 0
\(345\) 542.744 453.109i 0.0846967 0.0707089i
\(346\) 0 0
\(347\) 6610.79i 1.02273i −0.859365 0.511363i \(-0.829141\pi\)
0.859365 0.511363i \(-0.170859\pi\)
\(348\) 0 0
\(349\) 2491.07 0.382074 0.191037 0.981583i \(-0.438815\pi\)
0.191037 + 0.981583i \(0.438815\pi\)
\(350\) 0 0
\(351\) −5141.04 −0.781791
\(352\) 0 0
\(353\) 3383.35i 0.510134i 0.966923 + 0.255067i \(0.0820975\pi\)
−0.966923 + 0.255067i \(0.917902\pi\)
\(354\) 0 0
\(355\) −1710.50 + 1428.00i −0.255729 + 0.213495i
\(356\) 0 0
\(357\) 8931.08i 1.32404i
\(358\) 0 0
\(359\) 1303.53 0.191637 0.0958185 0.995399i \(-0.469453\pi\)
0.0958185 + 0.995399i \(0.469453\pi\)
\(360\) 0 0
\(361\) 11113.8 1.62033
\(362\) 0 0
\(363\) 12581.2i 1.81912i
\(364\) 0 0
\(365\) 310.012 + 371.339i 0.0444569 + 0.0532515i
\(366\) 0 0
\(367\) 6672.42i 0.949040i −0.880245 0.474520i \(-0.842622\pi\)
0.880245 0.474520i \(-0.157378\pi\)
\(368\) 0 0
\(369\) −1216.10 −0.171565
\(370\) 0 0
\(371\) −7445.88 −1.04197
\(372\) 0 0
\(373\) 5945.42i 0.825314i −0.910886 0.412657i \(-0.864601\pi\)
0.910886 0.412657i \(-0.135399\pi\)
\(374\) 0 0
\(375\) −5228.76 + 2966.77i −0.720033 + 0.408542i
\(376\) 0 0
\(377\) 7563.64i 1.03328i
\(378\) 0 0
\(379\) 12553.4 1.70139 0.850693 0.525663i \(-0.176183\pi\)
0.850693 + 0.525663i \(0.176183\pi\)
\(380\) 0 0
\(381\) −3501.86 −0.470881
\(382\) 0 0
\(383\) 7602.57i 1.01429i 0.861861 + 0.507145i \(0.169299\pi\)
−0.861861 + 0.507145i \(0.830701\pi\)
\(384\) 0 0
\(385\) 13235.7 + 15854.0i 1.75209 + 2.09869i
\(386\) 0 0
\(387\) 128.073i 0.0168225i
\(388\) 0 0
\(389\) 484.765 0.0631840 0.0315920 0.999501i \(-0.489942\pi\)
0.0315920 + 0.999501i \(0.489942\pi\)
\(390\) 0 0
\(391\) 1077.87 0.139413
\(392\) 0 0
\(393\) 5720.62i 0.734268i
\(394\) 0 0
\(395\) −3765.13 + 3143.31i −0.479605 + 0.400398i
\(396\) 0 0
\(397\) 9367.85i 1.18428i −0.805835 0.592140i \(-0.798283\pi\)
0.805835 0.592140i \(-0.201717\pi\)
\(398\) 0 0
\(399\) −16329.9 −2.04891
\(400\) 0 0
\(401\) −6650.48 −0.828203 −0.414101 0.910231i \(-0.635904\pi\)
−0.414101 + 0.910231i \(0.635904\pi\)
\(402\) 0 0
\(403\) 2317.39i 0.286445i
\(404\) 0 0
\(405\) −3668.62 + 3062.74i −0.450112 + 0.375775i
\(406\) 0 0
\(407\) 12806.6i 1.55970i
\(408\) 0 0
\(409\) 1615.31 0.195286 0.0976432 0.995221i \(-0.468870\pi\)
0.0976432 + 0.995221i \(0.468870\pi\)
\(410\) 0 0
\(411\) 12504.9 1.50078
\(412\) 0 0
\(413\) 3389.39i 0.403828i
\(414\) 0 0
\(415\) −8747.87 10478.4i −1.03474 1.23943i
\(416\) 0 0
\(417\) 3962.19i 0.465298i
\(418\) 0 0
\(419\) 10181.0 1.18705 0.593526 0.804815i \(-0.297736\pi\)
0.593526 + 0.804815i \(0.297736\pi\)
\(420\) 0 0
\(421\) −4145.13 −0.479860 −0.239930 0.970790i \(-0.577124\pi\)
−0.239930 + 0.970790i \(0.577124\pi\)
\(422\) 0 0
\(423\) 1141.78i 0.131241i
\(424\) 0 0
\(425\) −9017.84 1636.62i −1.02925 0.186794i
\(426\) 0 0
\(427\) 468.376i 0.0530827i
\(428\) 0 0
\(429\) −9448.53 −1.06336
\(430\) 0 0
\(431\) 14283.0 1.59627 0.798133 0.602482i \(-0.205821\pi\)
0.798133 + 0.602482i \(0.205821\pi\)
\(432\) 0 0
\(433\) 7773.88i 0.862792i 0.902163 + 0.431396i \(0.141979\pi\)
−0.902163 + 0.431396i \(0.858021\pi\)
\(434\) 0 0
\(435\) −6923.99 8293.71i −0.763172 0.914144i
\(436\) 0 0
\(437\) 1970.82i 0.215737i
\(438\) 0 0
\(439\) 3841.14 0.417602 0.208801 0.977958i \(-0.433044\pi\)
0.208801 + 0.977958i \(0.433044\pi\)
\(440\) 0 0
\(441\) −3897.78 −0.420881
\(442\) 0 0
\(443\) 2955.34i 0.316958i −0.987362 0.158479i \(-0.949341\pi\)
0.987362 0.158479i \(-0.0506590\pi\)
\(444\) 0 0
\(445\) −6207.02 + 5181.92i −0.661216 + 0.552015i
\(446\) 0 0
\(447\) 1141.06i 0.120739i
\(448\) 0 0
\(449\) 10786.9 1.13377 0.566886 0.823796i \(-0.308148\pi\)
0.566886 + 0.823796i \(0.308148\pi\)
\(450\) 0 0
\(451\) −9338.31 −0.974997
\(452\) 0 0
\(453\) 10787.3i 1.11883i
\(454\) 0 0
\(455\) 8182.61 6831.24i 0.843092 0.703854i
\(456\) 0 0
\(457\) 15384.9i 1.57479i 0.616451 + 0.787393i \(0.288570\pi\)
−0.616451 + 0.787393i \(0.711430\pi\)
\(458\) 0 0
\(459\) −11195.5 −1.13847
\(460\) 0 0
\(461\) −14590.9 −1.47411 −0.737057 0.675830i \(-0.763785\pi\)
−0.737057 + 0.675830i \(0.763785\pi\)
\(462\) 0 0
\(463\) 8828.72i 0.886188i 0.896475 + 0.443094i \(0.146119\pi\)
−0.896475 + 0.443094i \(0.853881\pi\)
\(464\) 0 0
\(465\) −2121.41 2541.07i −0.211565 0.253418i
\(466\) 0 0
\(467\) 4110.64i 0.407319i 0.979042 + 0.203659i \(0.0652835\pi\)
−0.979042 + 0.203659i \(0.934716\pi\)
\(468\) 0 0
\(469\) −15448.3 −1.52098
\(470\) 0 0
\(471\) −64.0169 −0.00626273
\(472\) 0 0
\(473\) 983.461i 0.0956016i
\(474\) 0 0
\(475\) −2992.45 + 16488.5i −0.289059 + 1.59273i
\(476\) 0 0
\(477\) 2233.92i 0.214432i
\(478\) 0 0
\(479\) −1852.96 −0.176751 −0.0883757 0.996087i \(-0.528168\pi\)
−0.0883757 + 0.996087i \(0.528168\pi\)
\(480\) 0 0
\(481\) 6609.77 0.626569
\(482\) 0 0
\(483\) 1790.66i 0.168691i
\(484\) 0 0
\(485\) 8139.61 + 9749.81i 0.762063 + 0.912816i
\(486\) 0 0
\(487\) 5285.78i 0.491831i 0.969291 + 0.245915i \(0.0790885\pi\)
−0.969291 + 0.245915i \(0.920912\pi\)
\(488\) 0 0
\(489\) −5094.19 −0.471099
\(490\) 0 0
\(491\) 12149.4 1.11669 0.558347 0.829608i \(-0.311436\pi\)
0.558347 + 0.829608i \(0.311436\pi\)
\(492\) 0 0
\(493\) 16471.1i 1.50470i
\(494\) 0 0
\(495\) 4756.53 3970.98i 0.431899 0.360570i
\(496\) 0 0
\(497\) 5643.38i 0.509337i
\(498\) 0 0
\(499\) 3587.97 0.321883 0.160941 0.986964i \(-0.448547\pi\)
0.160941 + 0.986964i \(0.448547\pi\)
\(500\) 0 0
\(501\) 11566.3 1.03142
\(502\) 0 0
\(503\) 18351.9i 1.62678i −0.581720 0.813389i \(-0.697620\pi\)
0.581720 0.813389i \(-0.302380\pi\)
\(504\) 0 0
\(505\) −10086.3 + 8420.57i −0.888785 + 0.742001i
\(506\) 0 0
\(507\) 4574.21i 0.400686i
\(508\) 0 0
\(509\) −992.521 −0.0864297 −0.0432149 0.999066i \(-0.513760\pi\)
−0.0432149 + 0.999066i \(0.513760\pi\)
\(510\) 0 0
\(511\) −1225.15 −0.106061
\(512\) 0 0
\(513\) 20470.1i 1.76175i
\(514\) 0 0
\(515\) 12556.4 + 15040.3i 1.07437 + 1.28691i
\(516\) 0 0
\(517\) 8767.59i 0.745838i
\(518\) 0 0
\(519\) 8348.37 0.706075
\(520\) 0 0
\(521\) 11192.3 0.941155 0.470577 0.882359i \(-0.344046\pi\)
0.470577 + 0.882359i \(0.344046\pi\)
\(522\) 0 0
\(523\) 4959.46i 0.414650i 0.978272 + 0.207325i \(0.0664758\pi\)
−0.978272 + 0.207325i \(0.933524\pi\)
\(524\) 0 0
\(525\) 2718.89 14981.2i 0.226023 1.24540i
\(526\) 0 0
\(527\) 5046.48i 0.417131i
\(528\) 0 0
\(529\) 11950.9 0.982238
\(530\) 0 0
\(531\) 1016.89 0.0831057
\(532\) 0 0
\(533\) 4819.72i 0.391679i
\(534\) 0 0
\(535\) 677.810 + 811.895i 0.0547743 + 0.0656099i
\(536\) 0 0
\(537\) 6799.33i 0.546392i
\(538\) 0 0
\(539\) −29930.7 −2.39185
\(540\) 0 0
\(541\) −18555.9 −1.47464 −0.737319 0.675545i \(-0.763908\pi\)
−0.737319 + 0.675545i \(0.763908\pi\)
\(542\) 0 0
\(543\) 11020.6i 0.870976i
\(544\) 0 0
\(545\) 11583.7 9670.66i 0.910445 0.760084i
\(546\) 0 0
\(547\) 24604.8i 1.92326i −0.274345 0.961631i \(-0.588461\pi\)
0.274345 0.961631i \(-0.411539\pi\)
\(548\) 0 0
\(549\) 140.523 0.0109241
\(550\) 0 0
\(551\) −30116.2 −2.32848
\(552\) 0 0
\(553\) 12422.2i 0.955233i
\(554\) 0 0
\(555\) 7247.76 6050.78i 0.554325 0.462777i
\(556\) 0 0
\(557\) 13010.2i 0.989694i −0.868980 0.494847i \(-0.835224\pi\)
0.868980 0.494847i \(-0.164776\pi\)
\(558\) 0 0
\(559\) −507.587 −0.0384054
\(560\) 0 0
\(561\) −20575.7 −1.54850
\(562\) 0 0
\(563\) 9751.95i 0.730010i −0.931006 0.365005i \(-0.881067\pi\)
0.931006 0.365005i \(-0.118933\pi\)
\(564\) 0 0
\(565\) −3792.15 4542.32i −0.282366 0.338224i
\(566\) 0 0
\(567\) 12103.8i 0.896491i
\(568\) 0 0
\(569\) −5645.90 −0.415973 −0.207986 0.978132i \(-0.566691\pi\)
−0.207986 + 0.978132i \(0.566691\pi\)
\(570\) 0 0
\(571\) −17188.1 −1.25972 −0.629860 0.776709i \(-0.716888\pi\)
−0.629860 + 0.776709i \(0.716888\pi\)
\(572\) 0 0
\(573\) 592.145i 0.0431714i
\(574\) 0 0
\(575\) 1808.05 + 328.138i 0.131132 + 0.0237987i
\(576\) 0 0
\(577\) 23294.0i 1.68066i 0.542076 + 0.840330i \(0.317639\pi\)
−0.542076 + 0.840330i \(0.682361\pi\)
\(578\) 0 0
\(579\) −3637.09 −0.261058
\(580\) 0 0
\(581\) 34571.0 2.46858
\(582\) 0 0
\(583\) 17154.0i 1.21861i
\(584\) 0 0
\(585\) −2049.52 2454.95i −0.144850 0.173504i
\(586\) 0 0
\(587\) 15182.5i 1.06754i 0.845629 + 0.533771i \(0.179226\pi\)
−0.845629 + 0.533771i \(0.820774\pi\)
\(588\) 0 0
\(589\) −9227.15 −0.645498
\(590\) 0 0
\(591\) 11759.2 0.818461
\(592\) 0 0
\(593\) 26734.6i 1.85137i −0.378301 0.925683i \(-0.623492\pi\)
0.378301 0.925683i \(-0.376508\pi\)
\(594\) 0 0
\(595\) 17819.0 14876.1i 1.22774 1.02498i
\(596\) 0 0
\(597\) 20001.6i 1.37121i
\(598\) 0 0
\(599\) −2868.56 −0.195670 −0.0978350 0.995203i \(-0.531192\pi\)
−0.0978350 + 0.995203i \(0.531192\pi\)
\(600\) 0 0
\(601\) −4242.60 −0.287952 −0.143976 0.989581i \(-0.545989\pi\)
−0.143976 + 0.989581i \(0.545989\pi\)
\(602\) 0 0
\(603\) 4634.82i 0.313009i
\(604\) 0 0
\(605\) 25101.5 20956.0i 1.68681 1.40823i
\(606\) 0 0
\(607\) 2058.87i 0.137672i 0.997628 + 0.0688361i \(0.0219285\pi\)
−0.997628 + 0.0688361i \(0.978071\pi\)
\(608\) 0 0
\(609\) 27363.1 1.82071
\(610\) 0 0
\(611\) 4525.16 0.299621
\(612\) 0 0
\(613\) 16492.5i 1.08667i −0.839518 0.543333i \(-0.817162\pi\)
0.839518 0.543333i \(-0.182838\pi\)
\(614\) 0 0
\(615\) 4412.11 + 5284.93i 0.289290 + 0.346518i
\(616\) 0 0
\(617\) 12562.1i 0.819662i −0.912161 0.409831i \(-0.865588\pi\)
0.912161 0.409831i \(-0.134412\pi\)
\(618\) 0 0
\(619\) 18297.2 1.18809 0.594043 0.804434i \(-0.297531\pi\)
0.594043 + 0.804434i \(0.297531\pi\)
\(620\) 0 0
\(621\) 2244.66 0.145048
\(622\) 0 0
\(623\) 20478.6i 1.31695i
\(624\) 0 0
\(625\) −14628.5 5490.61i −0.936226 0.351399i
\(626\) 0 0
\(627\) 37621.3i 2.39625i
\(628\) 0 0
\(629\) 14393.8 0.912433
\(630\) 0 0
\(631\) 25681.4 1.62022 0.810112 0.586275i \(-0.199406\pi\)
0.810112 + 0.586275i \(0.199406\pi\)
\(632\) 0 0
\(633\) 16543.9i 1.03880i
\(634\) 0 0
\(635\) −5832.90 6986.78i −0.364522 0.436633i
\(636\) 0 0
\(637\) 15447.9i 0.960861i
\(638\) 0 0
\(639\) −1693.13 −0.104819
\(640\) 0 0
\(641\) 12994.7 0.800720 0.400360 0.916358i \(-0.368885\pi\)
0.400360 + 0.916358i \(0.368885\pi\)
\(642\) 0 0
\(643\) 4625.71i 0.283701i −0.989888 0.141851i \(-0.954695\pi\)
0.989888 0.141851i \(-0.0453053\pi\)
\(644\) 0 0
\(645\) −556.580 + 464.660i −0.0339773 + 0.0283659i
\(646\) 0 0
\(647\) 8771.72i 0.533002i 0.963835 + 0.266501i \(0.0858675\pi\)
−0.963835 + 0.266501i \(0.914132\pi\)
\(648\) 0 0
\(649\) 7808.58 0.472286
\(650\) 0 0
\(651\) 8383.66 0.504733
\(652\) 0 0
\(653\) 2239.49i 0.134209i 0.997746 + 0.0671043i \(0.0213760\pi\)
−0.997746 + 0.0671043i \(0.978624\pi\)
\(654\) 0 0
\(655\) −11413.6 + 9528.61i −0.680863 + 0.568418i
\(656\) 0 0
\(657\) 367.570i 0.0218269i
\(658\) 0 0
\(659\) −4539.62 −0.268344 −0.134172 0.990958i \(-0.542837\pi\)
−0.134172 + 0.990958i \(0.542837\pi\)
\(660\) 0 0
\(661\) 28228.4 1.66105 0.830526 0.556980i \(-0.188040\pi\)
0.830526 + 0.556980i \(0.188040\pi\)
\(662\) 0 0
\(663\) 10619.6i 0.622068i
\(664\) 0 0
\(665\) −27200.0 32580.8i −1.58612 1.89989i
\(666\) 0 0
\(667\) 3302.40i 0.191708i
\(668\) 0 0
\(669\) −6283.65 −0.363139
\(670\) 0 0
\(671\) 1079.06 0.0620814
\(672\) 0 0
\(673\) 7649.89i 0.438160i −0.975707 0.219080i \(-0.929694\pi\)
0.975707 0.219080i \(-0.0703056\pi\)
\(674\) 0 0
\(675\) −18779.5 3408.24i −1.07085 0.194345i
\(676\) 0 0
\(677\) 12087.9i 0.686228i 0.939294 + 0.343114i \(0.111482\pi\)
−0.939294 + 0.343114i \(0.888518\pi\)
\(678\) 0 0
\(679\) −32167.2 −1.81806
\(680\) 0 0
\(681\) −14777.0 −0.831506
\(682\) 0 0
\(683\) 8827.56i 0.494549i −0.968945 0.247275i \(-0.920465\pi\)
0.968945 0.247275i \(-0.0795350\pi\)
\(684\) 0 0
\(685\) 20828.9 + 24949.3i 1.16180 + 1.39163i
\(686\) 0 0
\(687\) 2561.04i 0.142227i
\(688\) 0 0
\(689\) 8853.60 0.489543
\(690\) 0 0
\(691\) 12913.1 0.710908 0.355454 0.934694i \(-0.384326\pi\)
0.355454 + 0.934694i \(0.384326\pi\)
\(692\) 0 0
\(693\) 15693.1i 0.860217i
\(694\) 0 0
\(695\) −7905.21 + 6599.65i −0.431456 + 0.360200i
\(696\) 0 0
\(697\) 10495.7i 0.570378i
\(698\) 0 0
\(699\) 23369.5 1.26454
\(700\) 0 0
\(701\) −12297.2 −0.662567 −0.331283 0.943531i \(-0.607482\pi\)
−0.331283 + 0.943531i \(0.607482\pi\)
\(702\) 0 0
\(703\) 26318.2i 1.41196i
\(704\) 0 0
\(705\) 4961.94 4142.47i 0.265074 0.221297i
\(706\) 0 0
\(707\) 33277.5i 1.77020i
\(708\) 0 0
\(709\) −34308.2 −1.81731 −0.908653 0.417552i \(-0.862888\pi\)
−0.908653 + 0.417552i \(0.862888\pi\)
\(710\) 0 0
\(711\) −3726.91 −0.196582
\(712\) 0 0
\(713\) 1011.81i 0.0531451i
\(714\) 0 0
\(715\) −15738.0 18851.4i −0.823174 0.986016i
\(716\) 0 0
\(717\) 16912.1i 0.880887i
\(718\) 0 0
\(719\) 21631.4 1.12200 0.560999 0.827817i \(-0.310417\pi\)
0.560999 + 0.827817i \(0.310417\pi\)
\(720\) 0 0
\(721\) −49622.1 −2.56314
\(722\) 0 0
\(723\) 11132.3i 0.572632i
\(724\) 0 0
\(725\) 5014.29 27629.0i 0.256864 1.41533i
\(726\) 0 0
\(727\) 12049.8i 0.614723i −0.951593 0.307361i \(-0.900554\pi\)
0.951593 0.307361i \(-0.0994460\pi\)
\(728\) 0 0
\(729\) −21365.7 −1.08549
\(730\) 0 0
\(731\) −1105.35 −0.0559274
\(732\) 0 0
\(733\) 15221.6i 0.767017i −0.923537 0.383508i \(-0.874716\pi\)
0.923537 0.383508i \(-0.125284\pi\)
\(734\) 0 0
\(735\) 14141.5 + 16939.0i 0.709683 + 0.850073i
\(736\) 0 0
\(737\) 35590.4i 1.77882i
\(738\) 0 0
\(739\) −9845.80 −0.490099 −0.245050 0.969511i \(-0.578804\pi\)
−0.245050 + 0.969511i \(0.578804\pi\)
\(740\) 0 0
\(741\) 19417.2 0.962630
\(742\) 0 0
\(743\) 35081.5i 1.73219i 0.499881 + 0.866094i \(0.333377\pi\)
−0.499881 + 0.866094i \(0.666623\pi\)
\(744\) 0 0
\(745\) −2276.61 + 1900.62i −0.111958 + 0.0934676i
\(746\) 0 0
\(747\) 10372.0i 0.508022i
\(748\) 0 0
\(749\) −2678.66 −0.130676
\(750\) 0 0
\(751\) −20007.7 −0.972158 −0.486079 0.873915i \(-0.661573\pi\)
−0.486079 + 0.873915i \(0.661573\pi\)
\(752\) 0 0
\(753\) 9600.47i 0.464622i
\(754\) 0 0
\(755\) −21522.4 + 17968.0i −1.03746 + 0.866122i
\(756\) 0 0
\(757\) 33478.7i 1.60741i 0.595031 + 0.803703i \(0.297140\pi\)
−0.595031 + 0.803703i \(0.702860\pi\)
\(758\) 0 0
\(759\) 4125.37 0.197288
\(760\) 0 0
\(761\) 12728.1 0.606298 0.303149 0.952943i \(-0.401962\pi\)
0.303149 + 0.952943i \(0.401962\pi\)
\(762\) 0 0
\(763\) 38217.8i 1.81334i
\(764\) 0 0
\(765\) −4463.15 5346.06i −0.210935 0.252663i
\(766\) 0 0
\(767\) 4030.19i 0.189728i
\(768\) 0 0
\(769\) 20061.4 0.940747 0.470373 0.882468i \(-0.344119\pi\)
0.470373 + 0.882468i \(0.344119\pi\)
\(770\) 0 0
\(771\) −10655.6 −0.497732
\(772\) 0 0
\(773\) 672.445i 0.0312887i 0.999878 + 0.0156444i \(0.00497996\pi\)
−0.999878 + 0.0156444i \(0.995020\pi\)
\(774\) 0 0
\(775\) 1536.30 8465.10i 0.0712073 0.392355i
\(776\) 0 0
\(777\) 23912.3i 1.10405i
\(778\) 0 0
\(779\) 19190.7 0.882642
\(780\) 0 0
\(781\) −13001.4 −0.595681
\(782\) 0 0
\(783\) 34300.8i 1.56553i
\(784\) 0 0
\(785\) −106.630 127.724i −0.00484816 0.00580723i
\(786\) 0 0
\(787\) 7928.23i 0.359099i 0.983749 + 0.179550i \(0.0574640\pi\)
−0.983749 + 0.179550i \(0.942536\pi\)
\(788\) 0 0
\(789\) −20928.9 −0.944346
\(790\) 0 0
\(791\) 14986.3 0.673643
\(792\) 0 0
\(793\) 556.928i 0.0249396i
\(794\) 0 0
\(795\) 9708.17 8104.86i 0.433099 0.361572i
\(796\) 0 0
\(797\) 3196.35i 0.142059i 0.997474 + 0.0710293i \(0.0226284\pi\)