Properties

Label 3120.2.l.n.1249.8
Level $3120$
Weight $2$
Character 3120.1249
Analytic conductor $24.913$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3120.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.9133254306\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.57815240704.2
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 89x^{4} - 170x^{3} + 162x^{2} - 72x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.8
Root \(-2.15569 - 2.15569i\) of defining polynomial
Character \(\chi\) \(=\) 3120.1249
Dual form 3120.2.l.n.1249.4

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +(2.15569 - 0.594137i) q^{5} -0.633776i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +(2.15569 - 0.594137i) q^{5} -0.633776i q^{7} -1.00000 q^{9} +0.177949 q^{11} +1.00000i q^{13} +(0.594137 + 2.15569i) q^{15} -4.48933i q^{17} +0.633776 q^{21} +6.48933i q^{23} +(4.29400 - 2.56155i) q^{25} -1.00000i q^{27} +3.49966 q^{29} +0.177949i q^{33} +(-0.376550 - 1.36622i) q^{35} -1.75688i q^{37} -1.00000 q^{39} +7.67760 q^{41} +7.49966i q^{43} +(-2.15569 + 0.594137i) q^{45} -3.18827i q^{47} +6.59833 q^{49} +4.48933 q^{51} -1.75688i q^{53} +(0.383604 - 0.105726i) q^{55} +3.93483 q^{59} +3.01033 q^{61} +0.633776i q^{63} +(0.594137 + 2.15569i) q^{65} -1.62345i q^{67} -6.48933 q^{69} +4.41005 q^{71} -3.85555i q^{73} +(2.56155 + 4.29400i) q^{75} -0.112780i q^{77} +9.63309 q^{79} +1.00000 q^{81} +1.72338i q^{83} +(-2.66728 - 9.67760i) q^{85} +3.49966i q^{87} +0.589258 q^{89} +0.633776 q^{91} +4.45457i q^{97} -0.177949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{5} - 8 q^{9} - 2 q^{11} + 2 q^{15} + 14 q^{21} - 16 q^{29} + 8 q^{35} - 8 q^{39} + 14 q^{41} + 2 q^{45} - 18 q^{49} - 6 q^{51} - 10 q^{55} + 4 q^{59} + 22 q^{61} + 2 q^{65} - 10 q^{69} - 30 q^{71} + 4 q^{75} - 2 q^{79} + 8 q^{81} + 24 q^{85} - 18 q^{89} + 14 q^{91} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times\).

\(n\) \(1951\) \(2081\) \(2341\) \(2497\) \(2641\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 2.15569 0.594137i 0.964054 0.265706i
\(6\) 0 0
\(7\) 0.633776i 0.239545i −0.992801 0.119772i \(-0.961784\pi\)
0.992801 0.119772i \(-0.0382165\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0.177949 0.0536537 0.0268269 0.999640i \(-0.491460\pi\)
0.0268269 + 0.999640i \(0.491460\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 0 0
\(15\) 0.594137 + 2.15569i 0.153406 + 0.556597i
\(16\) 0 0
\(17\) 4.48933i 1.08882i −0.838818 0.544411i \(-0.816753\pi\)
0.838818 0.544411i \(-0.183247\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0.633776 0.138301
\(22\) 0 0
\(23\) 6.48933i 1.35312i 0.736388 + 0.676559i \(0.236530\pi\)
−0.736388 + 0.676559i \(0.763470\pi\)
\(24\) 0 0
\(25\) 4.29400 2.56155i 0.858800 0.512311i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 3.49966 0.649870 0.324935 0.945736i \(-0.394658\pi\)
0.324935 + 0.945736i \(0.394658\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0.177949i 0.0309770i
\(34\) 0 0
\(35\) −0.376550 1.36622i −0.0636486 0.230934i
\(36\) 0 0
\(37\) 1.75688i 0.288830i −0.989517 0.144415i \(-0.953870\pi\)
0.989517 0.144415i \(-0.0461299\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 7.67760 1.19904 0.599520 0.800360i \(-0.295358\pi\)
0.599520 + 0.800360i \(0.295358\pi\)
\(42\) 0 0
\(43\) 7.49966i 1.14369i 0.820363 + 0.571843i \(0.193772\pi\)
−0.820363 + 0.571843i \(0.806228\pi\)
\(44\) 0 0
\(45\) −2.15569 + 0.594137i −0.321351 + 0.0885688i
\(46\) 0 0
\(47\) 3.18827i 0.465058i −0.972589 0.232529i \(-0.925300\pi\)
0.972589 0.232529i \(-0.0747000\pi\)
\(48\) 0 0
\(49\) 6.59833 0.942618
\(50\) 0 0
\(51\) 4.48933 0.628632
\(52\) 0 0
\(53\) 1.75688i 0.241326i −0.992694 0.120663i \(-0.961498\pi\)
0.992694 0.120663i \(-0.0385021\pi\)
\(54\) 0 0
\(55\) 0.383604 0.105726i 0.0517251 0.0142561i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.93483 0.512271 0.256136 0.966641i \(-0.417551\pi\)
0.256136 + 0.966641i \(0.417551\pi\)
\(60\) 0 0
\(61\) 3.01033 0.385433 0.192716 0.981255i \(-0.438270\pi\)
0.192716 + 0.981255i \(0.438270\pi\)
\(62\) 0 0
\(63\) 0.633776i 0.0798482i
\(64\) 0 0
\(65\) 0.594137 + 2.15569i 0.0736937 + 0.267380i
\(66\) 0 0
\(67\) 1.62345i 0.198336i −0.995071 0.0991680i \(-0.968382\pi\)
0.995071 0.0991680i \(-0.0316181\pi\)
\(68\) 0 0
\(69\) −6.48933 −0.781224
\(70\) 0 0
\(71\) 4.41005 0.523377 0.261689 0.965152i \(-0.415721\pi\)
0.261689 + 0.965152i \(0.415721\pi\)
\(72\) 0 0
\(73\) 3.85555i 0.451258i −0.974213 0.225629i \(-0.927556\pi\)
0.974213 0.225629i \(-0.0724438\pi\)
\(74\) 0 0
\(75\) 2.56155 + 4.29400i 0.295783 + 0.495829i
\(76\) 0 0
\(77\) 0.112780i 0.0128525i
\(78\) 0 0
\(79\) 9.63309 1.08381 0.541903 0.840441i \(-0.317704\pi\)
0.541903 + 0.840441i \(0.317704\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 1.72338i 0.189165i 0.995517 + 0.0945827i \(0.0301517\pi\)
−0.995517 + 0.0945827i \(0.969848\pi\)
\(84\) 0 0
\(85\) −2.66728 9.67760i −0.289307 1.04968i
\(86\) 0 0
\(87\) 3.49966i 0.375202i
\(88\) 0 0
\(89\) 0.589258 0.0624612 0.0312306 0.999512i \(-0.490057\pi\)
0.0312306 + 0.999512i \(0.490057\pi\)
\(90\) 0 0
\(91\) 0.633776 0.0664378
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.45457i 0.452293i 0.974093 + 0.226147i \(0.0726128\pi\)
−0.974093 + 0.226147i \(0.927387\pi\)
\(98\) 0 0
\(99\) −0.177949 −0.0178846
\(100\) 0 0
\(101\) 16.4783 1.63965 0.819827 0.572612i \(-0.194070\pi\)
0.819827 + 0.572612i \(0.194070\pi\)
\(102\) 0 0
\(103\) 5.86966i 0.578355i −0.957276 0.289177i \(-0.906618\pi\)
0.957276 0.289177i \(-0.0933818\pi\)
\(104\) 0 0
\(105\) 1.36622 0.376550i 0.133330 0.0367475i
\(106\) 0 0
\(107\) 0.509981i 0.0493017i 0.999696 + 0.0246509i \(0.00784741\pi\)
−0.999696 + 0.0246509i \(0.992153\pi\)
\(108\) 0 0
\(109\) −6.35590 −0.608785 −0.304392 0.952547i \(-0.598453\pi\)
−0.304392 + 0.952547i \(0.598453\pi\)
\(110\) 0 0
\(111\) 1.75688 0.166756
\(112\) 0 0
\(113\) 9.71111i 0.913544i −0.889584 0.456772i \(-0.849005\pi\)
0.889584 0.456772i \(-0.150995\pi\)
\(114\) 0 0
\(115\) 3.85555 + 13.9890i 0.359532 + 1.30448i
\(116\) 0 0
\(117\) 1.00000i 0.0924500i
\(118\) 0 0
\(119\) −2.84523 −0.260822
\(120\) 0 0
\(121\) −10.9683 −0.997121
\(122\) 0 0
\(123\) 7.67760i 0.692266i
\(124\) 0 0
\(125\) 7.73462 8.07314i 0.691806 0.722084i
\(126\) 0 0
\(127\) 11.1231i 0.987016i −0.869741 0.493508i \(-0.835714\pi\)
0.869741 0.493508i \(-0.164286\pi\)
\(128\) 0 0
\(129\) −7.49966 −0.660308
\(130\) 0 0
\(131\) −12.9439 −1.13091 −0.565457 0.824778i \(-0.691300\pi\)
−0.565457 + 0.824778i \(0.691300\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −0.594137 2.15569i −0.0511352 0.185532i
\(136\) 0 0
\(137\) 11.7904i 1.00732i 0.863902 + 0.503660i \(0.168014\pi\)
−0.863902 + 0.503660i \(0.831986\pi\)
\(138\) 0 0
\(139\) 0.112780 0.00956587 0.00478293 0.999989i \(-0.498478\pi\)
0.00478293 + 0.999989i \(0.498478\pi\)
\(140\) 0 0
\(141\) 3.18827 0.268501
\(142\) 0 0
\(143\) 0.177949i 0.0148809i
\(144\) 0 0
\(145\) 7.54417 2.07928i 0.626510 0.172675i
\(146\) 0 0
\(147\) 6.59833i 0.544221i
\(148\) 0 0
\(149\) −2.30829 −0.189102 −0.0945512 0.995520i \(-0.530142\pi\)
−0.0945512 + 0.995520i \(0.530142\pi\)
\(150\) 0 0
\(151\) 17.7318 1.44299 0.721495 0.692420i \(-0.243455\pi\)
0.721495 + 0.692420i \(0.243455\pi\)
\(152\) 0 0
\(153\) 4.48933i 0.362941i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 12.8549i 1.02593i −0.858410 0.512965i \(-0.828547\pi\)
0.858410 0.512965i \(-0.171453\pi\)
\(158\) 0 0
\(159\) 1.75688 0.139330
\(160\) 0 0
\(161\) 4.11278 0.324132
\(162\) 0 0
\(163\) 14.3308i 1.12247i 0.827655 + 0.561237i \(0.189674\pi\)
−0.827655 + 0.561237i \(0.810326\pi\)
\(164\) 0 0
\(165\) 0.105726 + 0.383604i 0.00823078 + 0.0298635i
\(166\) 0 0
\(167\) 1.43449i 0.111004i 0.998459 + 0.0555019i \(0.0176759\pi\)
−0.998459 + 0.0555019i \(0.982324\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.24690i 0.0948001i −0.998876 0.0474000i \(-0.984906\pi\)
0.998876 0.0474000i \(-0.0150936\pi\)
\(174\) 0 0
\(175\) −1.62345 2.72143i −0.122721 0.205721i
\(176\) 0 0
\(177\) 3.93483i 0.295760i
\(178\) 0 0
\(179\) −7.87621 −0.588695 −0.294348 0.955698i \(-0.595102\pi\)
−0.294348 + 0.955698i \(0.595102\pi\)
\(180\) 0 0
\(181\) 10.3796 0.771513 0.385756 0.922601i \(-0.373941\pi\)
0.385756 + 0.922601i \(0.373941\pi\)
\(182\) 0 0
\(183\) 3.01033i 0.222530i
\(184\) 0 0
\(185\) −1.04383 3.78729i −0.0767438 0.278447i
\(186\) 0 0
\(187\) 0.798873i 0.0584194i
\(188\) 0 0
\(189\) −0.633776 −0.0461004
\(190\) 0 0
\(191\) −13.7318 −0.993595 −0.496798 0.867866i \(-0.665491\pi\)
−0.496798 + 0.867866i \(0.665491\pi\)
\(192\) 0 0
\(193\) 6.34488i 0.456715i −0.973577 0.228357i \(-0.926665\pi\)
0.973577 0.228357i \(-0.0733355\pi\)
\(194\) 0 0
\(195\) −2.15569 + 0.594137i −0.154372 + 0.0425471i
\(196\) 0 0
\(197\) 21.2966i 1.51732i 0.651487 + 0.758659i \(0.274145\pi\)
−0.651487 + 0.758659i \(0.725855\pi\)
\(198\) 0 0
\(199\) 11.2108 0.794710 0.397355 0.917665i \(-0.369928\pi\)
0.397355 + 0.917665i \(0.369928\pi\)
\(200\) 0 0
\(201\) 1.62345 0.114509
\(202\) 0 0
\(203\) 2.21800i 0.155673i
\(204\) 0 0
\(205\) 16.5505 4.56155i 1.15594 0.318593i
\(206\) 0 0
\(207\) 6.48933i 0.451040i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 20.2810 1.39620 0.698100 0.716000i \(-0.254029\pi\)
0.698100 + 0.716000i \(0.254029\pi\)
\(212\) 0 0
\(213\) 4.41005i 0.302172i
\(214\) 0 0
\(215\) 4.45583 + 16.1669i 0.303885 + 1.10258i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 3.85555 0.260534
\(220\) 0 0
\(221\) 4.48933 0.301985
\(222\) 0 0
\(223\) 11.3900i 0.762729i −0.924425 0.381364i \(-0.875454\pi\)
0.924425 0.381364i \(-0.124546\pi\)
\(224\) 0 0
\(225\) −4.29400 + 2.56155i −0.286267 + 0.170770i
\(226\) 0 0
\(227\) 14.7020i 0.975809i −0.872897 0.487904i \(-0.837762\pi\)
0.872897 0.487904i \(-0.162238\pi\)
\(228\) 0 0
\(229\) −9.84145 −0.650341 −0.325171 0.945655i \(-0.605422\pi\)
−0.325171 + 0.945655i \(0.605422\pi\)
\(230\) 0 0
\(231\) 0.112780 0.00742038
\(232\) 0 0
\(233\) 22.8439i 1.49655i 0.663388 + 0.748275i \(0.269118\pi\)
−0.663388 + 0.748275i \(0.730882\pi\)
\(234\) 0 0
\(235\) −1.89427 6.87293i −0.123569 0.448341i
\(236\) 0 0
\(237\) 9.63309i 0.625736i
\(238\) 0 0
\(239\) −24.1907 −1.56476 −0.782382 0.622798i \(-0.785996\pi\)
−0.782382 + 0.622798i \(0.785996\pi\)
\(240\) 0 0
\(241\) −22.8896 −1.47445 −0.737225 0.675647i \(-0.763864\pi\)
−0.737225 + 0.675647i \(0.763864\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 14.2240 3.92031i 0.908735 0.250460i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −1.72338 −0.109215
\(250\) 0 0
\(251\) 11.6763 0.737005 0.368502 0.929627i \(-0.379871\pi\)
0.368502 + 0.929627i \(0.379871\pi\)
\(252\) 0 0
\(253\) 1.15477i 0.0725999i
\(254\) 0 0
\(255\) 9.67760 2.66728i 0.606035 0.167031i
\(256\) 0 0
\(257\) 24.9993i 1.55941i 0.626144 + 0.779707i \(0.284632\pi\)
−0.626144 + 0.779707i \(0.715368\pi\)
\(258\) 0 0
\(259\) −1.11347 −0.0691876
\(260\) 0 0
\(261\) −3.49966 −0.216623
\(262\) 0 0
\(263\) 7.12965i 0.439633i 0.975541 + 0.219817i \(0.0705459\pi\)
−0.975541 + 0.219817i \(0.929454\pi\)
\(264\) 0 0
\(265\) −1.04383 3.78729i −0.0641219 0.232652i
\(266\) 0 0
\(267\) 0.589258i 0.0360620i
\(268\) 0 0
\(269\) 8.47832 0.516932 0.258466 0.966020i \(-0.416783\pi\)
0.258466 + 0.966020i \(0.416783\pi\)
\(270\) 0 0
\(271\) −5.60142 −0.340262 −0.170131 0.985421i \(-0.554419\pi\)
−0.170131 + 0.985421i \(0.554419\pi\)
\(272\) 0 0
\(273\) 0.633776i 0.0383579i
\(274\) 0 0
\(275\) 0.764114 0.455826i 0.0460778 0.0274874i
\(276\) 0 0
\(277\) 24.2991i 1.45999i 0.683451 + 0.729996i \(0.260478\pi\)
−0.683451 + 0.729996i \(0.739522\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 11.6318 0.693897 0.346948 0.937884i \(-0.387218\pi\)
0.346948 + 0.937884i \(0.387218\pi\)
\(282\) 0 0
\(283\) 0.197345i 0.0117310i −0.999983 0.00586549i \(-0.998133\pi\)
0.999983 0.00586549i \(-0.00186705\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.86588i 0.287224i
\(288\) 0 0
\(289\) −3.15408 −0.185534
\(290\) 0 0
\(291\) −4.45457 −0.261132
\(292\) 0 0
\(293\) 29.3918i 1.71709i 0.512740 + 0.858544i \(0.328630\pi\)
−0.512740 + 0.858544i \(0.671370\pi\)
\(294\) 0 0
\(295\) 8.48228 2.33783i 0.493857 0.136114i
\(296\) 0 0
\(297\) 0.177949i 0.0103257i
\(298\) 0 0
\(299\) −6.48933 −0.375288
\(300\) 0 0
\(301\) 4.75310 0.273964
\(302\) 0 0
\(303\) 16.4783i 0.946654i
\(304\) 0 0
\(305\) 6.48933 1.78855i 0.371578 0.102412i
\(306\) 0 0
\(307\) 29.1997i 1.66652i −0.552883 0.833259i \(-0.686472\pi\)
0.552883 0.833259i \(-0.313528\pi\)
\(308\) 0 0
\(309\) 5.86966 0.333913
\(310\) 0 0
\(311\) 13.3346 0.756133 0.378067 0.925778i \(-0.376589\pi\)
0.378067 + 0.925778i \(0.376589\pi\)
\(312\) 0 0
\(313\) 5.16441i 0.291910i −0.989291 0.145955i \(-0.953375\pi\)
0.989291 0.145955i \(-0.0466254\pi\)
\(314\) 0 0
\(315\) 0.376550 + 1.36622i 0.0212162 + 0.0769780i
\(316\) 0 0
\(317\) 28.3461i 1.59208i 0.605245 + 0.796039i \(0.293075\pi\)
−0.605245 + 0.796039i \(0.706925\pi\)
\(318\) 0 0
\(319\) 0.622761 0.0348679
\(320\) 0 0
\(321\) −0.509981 −0.0284644
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 2.56155 + 4.29400i 0.142089 + 0.238188i
\(326\) 0 0
\(327\) 6.35590i 0.351482i
\(328\) 0 0
\(329\) −2.02065 −0.111402
\(330\) 0 0
\(331\) 11.1296 0.611741 0.305870 0.952073i \(-0.401053\pi\)
0.305870 + 0.952073i \(0.401053\pi\)
\(332\) 0 0
\(333\) 1.75688i 0.0962765i
\(334\) 0 0
\(335\) −0.964553 3.49966i −0.0526991 0.191207i
\(336\) 0 0
\(337\) 12.0529i 0.656563i −0.944580 0.328282i \(-0.893530\pi\)
0.944580 0.328282i \(-0.106470\pi\)
\(338\) 0 0
\(339\) 9.71111 0.527435
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 8.61829i 0.465344i
\(344\) 0 0
\(345\) −13.9890 + 3.85555i −0.753142 + 0.207576i
\(346\) 0 0
\(347\) 7.62654i 0.409414i 0.978823 + 0.204707i \(0.0656242\pi\)
−0.978823 + 0.204707i \(0.934376\pi\)
\(348\) 0 0
\(349\) 15.2455 0.816074 0.408037 0.912965i \(-0.366213\pi\)
0.408037 + 0.912965i \(0.366213\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 4.83238i 0.257201i −0.991696 0.128601i \(-0.958951\pi\)
0.991696 0.128601i \(-0.0410486\pi\)
\(354\) 0 0
\(355\) 9.50671 2.62018i 0.504564 0.139065i
\(356\) 0 0
\(357\) 2.84523i 0.150585i
\(358\) 0 0
\(359\) 7.25528 0.382919 0.191460 0.981501i \(-0.438678\pi\)
0.191460 + 0.981501i \(0.438678\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 10.9683i 0.575688i
\(364\) 0 0
\(365\) −2.29073 8.31138i −0.119902 0.435038i
\(366\) 0 0
\(367\) 33.5808i 1.75290i −0.481491 0.876451i \(-0.659905\pi\)
0.481491 0.876451i \(-0.340095\pi\)
\(368\) 0 0
\(369\) −7.67760 −0.399680
\(370\) 0 0
\(371\) −1.11347 −0.0578084
\(372\) 0 0
\(373\) 2.78131i 0.144011i 0.997404 + 0.0720055i \(0.0229399\pi\)
−0.997404 + 0.0720055i \(0.977060\pi\)
\(374\) 0 0
\(375\) 8.07314 + 7.73462i 0.416895 + 0.399414i
\(376\) 0 0
\(377\) 3.49966i 0.180241i
\(378\) 0 0
\(379\) −34.0268 −1.74784 −0.873921 0.486069i \(-0.838430\pi\)
−0.873921 + 0.486069i \(0.838430\pi\)
\(380\) 0 0
\(381\) 11.1231 0.569854
\(382\) 0 0
\(383\) 8.45332i 0.431944i 0.976400 + 0.215972i \(0.0692920\pi\)
−0.976400 + 0.215972i \(0.930708\pi\)
\(384\) 0 0
\(385\) −0.0670068 0.243119i −0.00341498 0.0123905i
\(386\) 0 0
\(387\) 7.49966i 0.381229i
\(388\) 0 0
\(389\) 14.2991 0.724994 0.362497 0.931985i \(-0.381924\pi\)
0.362497 + 0.931985i \(0.381924\pi\)
\(390\) 0 0
\(391\) 29.1327 1.47331
\(392\) 0 0
\(393\) 12.9439i 0.652933i
\(394\) 0 0
\(395\) 20.7660 5.72338i 1.04485 0.287974i
\(396\) 0 0
\(397\) 3.97488i 0.199493i −0.995013 0.0997467i \(-0.968197\pi\)
0.995013 0.0997467i \(-0.0318033\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10.2560 −0.512159 −0.256079 0.966656i \(-0.582431\pi\)
−0.256079 + 0.966656i \(0.582431\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 2.15569 0.594137i 0.107117 0.0295229i
\(406\) 0 0
\(407\) 0.312636i 0.0154968i
\(408\) 0 0
\(409\) −19.6296 −0.970623 −0.485311 0.874341i \(-0.661294\pi\)
−0.485311 + 0.874341i \(0.661294\pi\)
\(410\) 0 0
\(411\) −11.7904 −0.581577
\(412\) 0 0
\(413\) 2.49380i 0.122712i
\(414\) 0 0
\(415\) 1.02392 + 3.71507i 0.0502624 + 0.182366i
\(416\) 0 0
\(417\) 0.112780i 0.00552286i
\(418\) 0 0
\(419\) 23.7238 1.15899 0.579493 0.814977i \(-0.303251\pi\)
0.579493 + 0.814977i \(0.303251\pi\)
\(420\) 0 0
\(421\) 17.8697 0.870914 0.435457 0.900210i \(-0.356587\pi\)
0.435457 + 0.900210i \(0.356587\pi\)
\(422\) 0 0
\(423\) 3.18827i 0.155019i
\(424\) 0 0
\(425\) −11.4997 19.2772i −0.557815 0.935081i
\(426\) 0 0
\(427\) 1.90787i 0.0923284i
\(428\) 0 0
\(429\) −0.177949 −0.00859147
\(430\) 0 0
\(431\) −35.9476 −1.73153 −0.865767 0.500448i \(-0.833169\pi\)
−0.865767 + 0.500448i \(0.833169\pi\)
\(432\) 0 0
\(433\) 8.21396i 0.394738i −0.980329 0.197369i \(-0.936760\pi\)
0.980329 0.197369i \(-0.0632397\pi\)
\(434\) 0 0
\(435\) 2.07928 + 7.54417i 0.0996937 + 0.361715i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −25.0914 −1.19755 −0.598775 0.800918i \(-0.704345\pi\)
−0.598775 + 0.800918i \(0.704345\pi\)
\(440\) 0 0
\(441\) −6.59833 −0.314206
\(442\) 0 0
\(443\) 2.75619i 0.130951i −0.997854 0.0654753i \(-0.979144\pi\)
0.997854 0.0654753i \(-0.0208564\pi\)
\(444\) 0 0
\(445\) 1.27026 0.350100i 0.0602160 0.0165963i
\(446\) 0 0
\(447\) 2.30829i 0.109178i
\(448\) 0 0
\(449\) −39.7639 −1.87657 −0.938287 0.345858i \(-0.887588\pi\)
−0.938287 + 0.345858i \(0.887588\pi\)
\(450\) 0 0
\(451\) 1.36622 0.0643330
\(452\) 0 0
\(453\) 17.7318i 0.833111i
\(454\) 0 0
\(455\) 1.36622 0.376550i 0.0640496 0.0176529i
\(456\) 0 0
\(457\) 8.72143i 0.407971i −0.978974 0.203986i \(-0.934610\pi\)
0.978974 0.203986i \(-0.0653896\pi\)
\(458\) 0 0
\(459\) −4.48933 −0.209544
\(460\) 0 0
\(461\) 17.5332 0.816601 0.408300 0.912848i \(-0.366122\pi\)
0.408300 + 0.912848i \(0.366122\pi\)
\(462\) 0 0
\(463\) 18.1049i 0.841404i −0.907199 0.420702i \(-0.861784\pi\)
0.907199 0.420702i \(-0.138216\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11.0451i 0.511106i 0.966795 + 0.255553i \(0.0822575\pi\)
−0.966795 + 0.255553i \(0.917743\pi\)
\(468\) 0 0
\(469\) −1.02890 −0.0475103
\(470\) 0 0
\(471\) 12.8549 0.592321
\(472\) 0 0
\(473\) 1.33456i 0.0613631i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.75688i 0.0804421i
\(478\) 0 0
\(479\) −33.3887 −1.52557 −0.762785 0.646653i \(-0.776168\pi\)
−0.762785 + 0.646653i \(0.776168\pi\)
\(480\) 0 0
\(481\) 1.75688 0.0801069
\(482\) 0 0
\(483\) 4.11278i 0.187138i
\(484\) 0 0
\(485\) 2.64663 + 9.60268i 0.120177 + 0.436035i
\(486\) 0 0
\(487\) 35.7414i 1.61960i −0.586708 0.809799i \(-0.699576\pi\)
0.586708 0.809799i \(-0.300424\pi\)
\(488\) 0 0
\(489\) −14.3308 −0.648060
\(490\) 0 0
\(491\) −36.9914 −1.66940 −0.834699 0.550706i \(-0.814358\pi\)
−0.834699 + 0.550706i \(0.814358\pi\)
\(492\) 0 0
\(493\) 15.7111i 0.707593i
\(494\) 0 0
\(495\) −0.383604 + 0.105726i −0.0172417 + 0.00475205i
\(496\) 0 0
\(497\) 2.79498i 0.125372i
\(498\) 0 0
\(499\) −20.0670 −0.898323 −0.449161 0.893451i \(-0.648277\pi\)
−0.449161 + 0.893451i \(0.648277\pi\)
\(500\) 0 0
\(501\) −1.43449 −0.0640881
\(502\) 0 0
\(503\) 15.3332i 0.683673i 0.939759 + 0.341836i \(0.111049\pi\)
−0.939759 + 0.341836i \(0.888951\pi\)
\(504\) 0 0
\(505\) 35.5221 9.79038i 1.58071 0.435666i
\(506\) 0 0
\(507\) 1.00000i 0.0444116i
\(508\) 0 0
\(509\) −30.0476 −1.33184 −0.665918 0.746025i \(-0.731960\pi\)
−0.665918 + 0.746025i \(0.731960\pi\)
\(510\) 0 0
\(511\) −2.44356 −0.108097
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3.48739 12.6532i −0.153673 0.557565i
\(516\) 0 0
\(517\) 0.567351i 0.0249521i
\(518\) 0 0
\(519\) 1.24690 0.0547328
\(520\) 0 0
\(521\) −30.8263 −1.35052 −0.675262 0.737578i \(-0.735970\pi\)
−0.675262 + 0.737578i \(0.735970\pi\)
\(522\) 0 0
\(523\) 12.0141i 0.525340i 0.964886 + 0.262670i \(0.0846031\pi\)
−0.964886 + 0.262670i \(0.915397\pi\)
\(524\) 0 0
\(525\) 2.72143 1.62345i 0.118773 0.0708532i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −19.1114 −0.830931
\(530\) 0 0
\(531\) −3.93483 −0.170757
\(532\) 0 0
\(533\) 7.67760i 0.332554i
\(534\) 0 0
\(535\) 0.302999 + 1.09936i 0.0130998 + 0.0475295i
\(536\) 0 0
\(537\) 7.87621i 0.339883i
\(538\) 0 0
\(539\) 1.17417 0.0505750
\(540\) 0 0
\(541\) 16.3132 0.701360 0.350680 0.936495i \(-0.385950\pi\)
0.350680 + 0.936495i \(0.385950\pi\)
\(542\) 0 0
\(543\) 10.3796i 0.445433i
\(544\) 0 0
\(545\) −13.7013 + 3.77628i −0.586901 + 0.161758i
\(546\) 0 0
\(547\) 0.239667i 0.0102474i −0.999987 0.00512371i \(-0.998369\pi\)
0.999987 0.00512371i \(-0.00163094\pi\)
\(548\) 0 0
\(549\) −3.01033 −0.128478
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 6.10522i 0.259620i
\(554\) 0 0
\(555\) 3.78729 1.04383i 0.160762 0.0443081i
\(556\) 0 0
\(557\) 10.0304i 0.425002i 0.977161 + 0.212501i \(0.0681609\pi\)
−0.977161 + 0.212501i \(0.931839\pi\)
\(558\) 0 0
\(559\) −7.49966 −0.317202
\(560\) 0 0
\(561\) 0.798873 0.0337284
\(562\) 0 0
\(563\) 0.310125i 0.0130702i −0.999979 0.00653511i \(-0.997920\pi\)
0.999979 0.00653511i \(-0.00208021\pi\)
\(564\) 0 0
\(565\) −5.76973 20.9341i −0.242735 0.880706i
\(566\) 0 0
\(567\) 0.633776i 0.0266161i
\(568\) 0 0
\(569\) −12.2850 −0.515014 −0.257507 0.966276i \(-0.582901\pi\)
−0.257507 + 0.966276i \(0.582901\pi\)
\(570\) 0 0
\(571\) 31.6124 1.32294 0.661470 0.749972i \(-0.269933\pi\)
0.661470 + 0.749972i \(0.269933\pi\)
\(572\) 0 0
\(573\) 13.7318i 0.573652i
\(574\) 0 0
\(575\) 16.6228 + 27.8652i 0.693217 + 1.16206i
\(576\) 0 0
\(577\) 6.36691i 0.265058i 0.991179 + 0.132529i \(0.0423098\pi\)
−0.991179 + 0.132529i \(0.957690\pi\)
\(578\) 0 0
\(579\) 6.34488 0.263684
\(580\) 0 0
\(581\) 1.09224 0.0453136
\(582\) 0 0
\(583\) 0.312636i 0.0129481i
\(584\) 0 0
\(585\) −0.594137 2.15569i −0.0245646 0.0891268i
\(586\) 0 0
\(587\) 14.8103i 0.611288i −0.952146 0.305644i \(-0.901128\pi\)
0.952146 0.305644i \(-0.0988718\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −21.2966 −0.876024
\(592\) 0 0
\(593\) 33.9864i 1.39565i −0.716267 0.697826i \(-0.754151\pi\)
0.716267 0.697826i \(-0.245849\pi\)
\(594\) 0 0
\(595\) −6.13343 + 1.69046i −0.251446 + 0.0693020i
\(596\) 0 0
\(597\) 11.2108i 0.458826i
\(598\) 0 0
\(599\) −1.10900 −0.0453124 −0.0226562 0.999743i \(-0.507212\pi\)
−0.0226562 + 0.999743i \(0.507212\pi\)
\(600\) 0 0
\(601\) −47.9670 −1.95661 −0.978306 0.207163i \(-0.933577\pi\)
−0.978306 + 0.207163i \(0.933577\pi\)
\(602\) 0 0
\(603\) 1.62345i 0.0661120i
\(604\) 0 0
\(605\) −23.6443 + 6.51670i −0.961279 + 0.264941i
\(606\) 0 0
\(607\) 35.5319i 1.44220i −0.692833 0.721098i \(-0.743638\pi\)
0.692833 0.721098i \(-0.256362\pi\)
\(608\) 0 0
\(609\) 2.21800 0.0898778
\(610\) 0 0
\(611\) 3.18827 0.128984
\(612\) 0 0
\(613\) 20.4955i 0.827806i −0.910321 0.413903i \(-0.864165\pi\)
0.910321 0.413903i \(-0.135835\pi\)
\(614\) 0 0
\(615\) 4.56155 + 16.5505i 0.183940 + 0.667382i
\(616\) 0 0
\(617\) 10.6119i 0.427218i 0.976919 + 0.213609i \(0.0685218\pi\)
−0.976919 + 0.213609i \(0.931478\pi\)
\(618\) 0 0
\(619\) −28.5312 −1.14677 −0.573383 0.819287i \(-0.694369\pi\)
−0.573383 + 0.819287i \(0.694369\pi\)
\(620\) 0 0
\(621\) 6.48933 0.260408
\(622\) 0 0
\(623\) 0.373457i 0.0149623i
\(624\) 0 0
\(625\) 11.8769 21.9986i 0.475076 0.879945i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −7.88722 −0.314484
\(630\) 0 0
\(631\) 7.71111 0.306974 0.153487 0.988151i \(-0.450950\pi\)
0.153487 + 0.988151i \(0.450950\pi\)
\(632\) 0 0
\(633\) 20.2810i 0.806096i
\(634\) 0 0
\(635\) −6.60865 23.9780i −0.262256 0.951537i
\(636\) 0 0
\(637\) 6.59833i 0.261435i
\(638\) 0 0
\(639\) −4.41005 −0.174459
\(640\) 0 0
\(641\) −27.5124 −1.08667 −0.543337 0.839515i \(-0.682839\pi\)
−0.543337 + 0.839515i \(0.682839\pi\)
\(642\) 0 0
\(643\) 24.1798i 0.953558i −0.879023 0.476779i \(-0.841804\pi\)
0.879023 0.476779i \(-0.158196\pi\)
\(644\) 0 0
\(645\) −16.1669 + 4.45583i −0.636572 + 0.175448i
\(646\) 0 0
\(647\) 27.9347i 1.09823i 0.835748 + 0.549113i \(0.185034\pi\)
−0.835748 + 0.549113i \(0.814966\pi\)
\(648\) 0 0
\(649\) 0.700200 0.0274853
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 27.2868i 1.06782i −0.845543 0.533908i \(-0.820723\pi\)
0.845543 0.533908i \(-0.179277\pi\)
\(654\) 0 0
\(655\) −27.9030 + 7.69046i −1.09026 + 0.300491i
\(656\) 0 0
\(657\) 3.85555i 0.150419i
\(658\) 0 0
\(659\) −30.3198 −1.18109 −0.590545 0.807005i \(-0.701087\pi\)
−0.590545 + 0.807005i \(0.701087\pi\)
\(660\) 0 0
\(661\) 16.7104 0.649960 0.324980 0.945721i \(-0.394642\pi\)
0.324980 + 0.945721i \(0.394642\pi\)
\(662\) 0 0
\(663\) 4.48933i 0.174351i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 22.7104i 0.879351i
\(668\) 0 0
\(669\) 11.3900 0.440362
\(670\) 0 0
\(671\) 0.535685 0.0206799
\(672\) 0 0
\(673\) 0.950445i 0.0366370i 0.999832 + 0.0183185i \(0.00583128\pi\)
−0.999832 + 0.0183185i \(0.994169\pi\)
\(674\) 0 0
\(675\) −2.56155 4.29400i −0.0985942 0.165276i
\(676\) 0 0
\(677\) 34.1991i 1.31438i −0.753726 0.657188i \(-0.771746\pi\)
0.753726 0.657188i \(-0.228254\pi\)
\(678\) 0 0
\(679\) 2.82320 0.108344
\(680\) 0 0
\(681\) 14.7020 0.563383
\(682\) 0 0
\(683\) 5.68208i 0.217419i −0.994074 0.108709i \(-0.965328\pi\)
0.994074 0.108709i \(-0.0346718\pi\)
\(684\) 0 0
\(685\) 7.00511 + 25.4164i 0.267651 + 0.971111i
\(686\) 0 0
\(687\) 9.84145i 0.375475i
\(688\) 0 0
\(689\) 1.75688 0.0669319
\(690\) 0 0
\(691\) 13.7795 0.524197 0.262098 0.965041i \(-0.415586\pi\)
0.262098 + 0.965041i \(0.415586\pi\)
\(692\) 0 0
\(693\) 0.112780i 0.00428416i
\(694\) 0 0
\(695\) 0.243119 0.0670068i 0.00922201 0.00254171i
\(696\) 0 0
\(697\) 34.4673i 1.30554i
\(698\) 0 0
\(699\) −22.8439 −0.864034
\(700\) 0 0
\(701\) 20.4113 0.770924 0.385462 0.922724i \(-0.374042\pi\)
0.385462 + 0.922724i \(0.374042\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 6.87293 1.89427i 0.258850 0.0713425i
\(706\) 0 0
\(707\) 10.4436i 0.392770i
\(708\) 0 0
\(709\) −3.17852 −0.119372 −0.0596858 0.998217i \(-0.519010\pi\)
−0.0596858 + 0.998217i \(0.519010\pi\)
\(710\) 0 0
\(711\) −9.63309 −0.361269
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0.105726 + 0.383604i 0.00395394 + 0.0143460i
\(716\) 0 0
\(717\) 24.1907i 0.903417i
\(718\) 0 0
\(719\) −40.5312 −1.51156 −0.755780 0.654826i \(-0.772742\pi\)
−0.755780 + 0.654826i \(0.772742\pi\)
\(720\) 0 0
\(721\) −3.72005 −0.138542
\(722\) 0 0
\(723\) 22.8896i 0.851274i
\(724\) 0 0
\(725\) 15.0275 8.96455i 0.558108 0.332935i
\(726\) 0 0
\(727\) 4.39066i 0.162840i −0.996680 0.0814202i \(-0.974054\pi\)
0.996680 0.0814202i \(-0.0259456\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 33.6684 1.24527
\(732\) 0 0
\(733\) 30.1823i 1.11481i −0.830241 0.557404i \(-0.811797\pi\)
0.830241 0.557404i \(-0.188203\pi\)
\(734\) 0 0
\(735\) 3.92031 + 14.2240i 0.144603 + 0.524658i
\(736\) 0 0
\(737\) 0.288892i 0.0106415i
\(738\) 0 0
\(739\) −49.1565 −1.80825 −0.904125 0.427267i \(-0.859476\pi\)
−0.904125 + 0.427267i \(0.859476\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 21.8317i 0.800927i −0.916313 0.400463i \(-0.868849\pi\)
0.916313 0.400463i \(-0.131151\pi\)
\(744\) 0 0
\(745\) −4.97595 + 1.37144i −0.182305 + 0.0502457i
\(746\) 0 0
\(747\) 1.72338i 0.0630551i
\(748\) 0 0
\(749\) 0.323214 0.0118100
\(750\) 0 0
\(751\) 22.8170 0.832605 0.416302 0.909226i \(-0.363326\pi\)
0.416302 + 0.909226i \(0.363326\pi\)
\(752\) 0 0
\(753\) 11.6763i 0.425510i
\(754\) 0 0
\(755\) 38.2242 10.5351i 1.39112 0.383412i
\(756\) 0 0
\(757\) 11.4649i 0.416699i 0.978054 + 0.208349i \(0.0668091\pi\)
−0.978054 + 0.208349i \(0.933191\pi\)
\(758\) 0 0
\(759\) −1.15477 −0.0419156
\(760\) 0 0
\(761\) −26.9005 −0.975143 −0.487571 0.873083i \(-0.662117\pi\)
−0.487571 + 0.873083i \(0.662117\pi\)
\(762\) 0 0
\(763\) 4.02821i 0.145831i
\(764\) 0 0
\(765\) 2.66728 + 9.67760i 0.0964357 + 0.349895i
\(766\) 0 0
\(767\) 3.93483i 0.142079i
\(768\) 0 0
\(769\) −12.9298 −0.466260 −0.233130 0.972446i \(-0.574897\pi\)
−0.233130 + 0.972446i \(0.574897\pi\)
\(770\) 0 0
\(771\) −24.9993 −0.900328
\(772\) 0 0
\(773\) 7.56231i 0.271998i −0.990709 0.135999i \(-0.956576\pi\)
0.990709 0.135999i \(-0.0434243\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.11347i 0.0399455i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0.784766 0.0280811
\(782\) 0 0
\(783\) 3.49966i 0.125067i
\(784\) 0 0
\(785\) −7.63756 27.7111i −0.272596 0.989052i
\(786\) 0 0
\(787\) 53.3318i 1.90107i 0.310612 + 0.950537i \(0.399466\pi\)
−0.310612 + 0.950537i \(0.600534\pi\)
\(788\) 0 0
\(789\) −7.12965 −0.253822
\(790\) 0 0
\(791\) −6.15466 −0.218835
\(792\) 0 0
\(793\) 3.01033i 0.106900i
\(794\) 0 0
\(795\) 3.78729 1.04383i 0.134321 0.0370208i
\(796\) 0 0
\(797\) 46.9391i 1.66267i 0.555774 + 0.831334i \(0.312422\pi\)
−0.555774 + 0.831334i \(0.687578\pi\)
\(798\) 0 0
\(799\) −14.3132 −0.506365
\(800\) 0 0
\(801\) −0.589258 −0.0208204
\(802\) 0 0