Properties

Label 384.3.b.c.319.8
Level $384$
Weight $3$
Character 384.319
Analytic conductor $10.463$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 319.8
Root \(0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 384.319
Dual form 384.3.b.c.319.5

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205 q^{3} +8.89898i q^{5} +2.82843i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q+1.73205 q^{3} +8.89898i q^{5} +2.82843i q^{7} +3.00000 q^{9} -18.2419 q^{11} +5.79796i q^{13} +15.4135i q^{15} -21.5959 q^{17} +18.2419 q^{19} +4.89898i q^{21} -33.3697i q^{23} -54.1918 q^{25} +5.19615 q^{27} +4.49490i q^{29} +2.25697i q^{31} -31.5959 q^{33} -25.1701 q^{35} +43.1918i q^{37} +10.0424i q^{39} -1.59592 q^{41} +63.4967 q^{43} +26.6969i q^{45} +72.3962i q^{47} +41.0000 q^{49} -37.4052 q^{51} +70.2929i q^{53} -162.334i q^{55} +31.5959 q^{57} +34.6410 q^{59} +63.5959i q^{61} +8.48528i q^{63} -51.5959 q^{65} -3.24259 q^{67} -57.7980i q^{69} +68.4537i q^{71} -10.0000 q^{73} -93.8630 q^{75} -51.5959i q^{77} +35.0552i q^{79} +9.00000 q^{81} +42.2691 q^{83} -192.182i q^{85} +7.78539i q^{87} +5.19184 q^{89} -16.3991 q^{91} +3.90918i q^{93} +162.334i q^{95} -26.8082 q^{97} -54.7257 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9} + O(q^{10}) \) \( 8 q + 24 q^{9} - 16 q^{17} - 120 q^{25} - 96 q^{33} + 144 q^{41} + 328 q^{49} + 96 q^{57} - 256 q^{65} - 80 q^{73} + 72 q^{81} - 272 q^{89} - 528 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 1.73205 0.577350
\(4\) 0 0
\(5\) 8.89898i 1.77980i 0.456160 + 0.889898i \(0.349225\pi\)
−0.456160 + 0.889898i \(0.650775\pi\)
\(6\) 0 0
\(7\) 2.82843i 0.404061i 0.979379 + 0.202031i \(0.0647540\pi\)
−0.979379 + 0.202031i \(0.935246\pi\)
\(8\) 0 0
\(9\) 3.00000 0.333333
\(10\) 0 0
\(11\) −18.2419 −1.65836 −0.829178 0.558985i \(-0.811191\pi\)
−0.829178 + 0.558985i \(0.811191\pi\)
\(12\) 0 0
\(13\) 5.79796i 0.445997i 0.974819 + 0.222998i \(0.0715845\pi\)
−0.974819 + 0.222998i \(0.928416\pi\)
\(14\) 0 0
\(15\) 15.4135i 1.02757i
\(16\) 0 0
\(17\) −21.5959 −1.27035 −0.635174 0.772369i \(-0.719072\pi\)
−0.635174 + 0.772369i \(0.719072\pi\)
\(18\) 0 0
\(19\) 18.2419 0.960101 0.480050 0.877241i \(-0.340618\pi\)
0.480050 + 0.877241i \(0.340618\pi\)
\(20\) 0 0
\(21\) 4.89898i 0.233285i
\(22\) 0 0
\(23\) − 33.3697i − 1.45086i −0.688299 0.725428i \(-0.741642\pi\)
0.688299 0.725428i \(-0.258358\pi\)
\(24\) 0 0
\(25\) −54.1918 −2.16767
\(26\) 0 0
\(27\) 5.19615 0.192450
\(28\) 0 0
\(29\) 4.49490i 0.154996i 0.996992 + 0.0774982i \(0.0246932\pi\)
−0.996992 + 0.0774982i \(0.975307\pi\)
\(30\) 0 0
\(31\) 2.25697i 0.0728054i 0.999337 + 0.0364027i \(0.0115899\pi\)
−0.999337 + 0.0364027i \(0.988410\pi\)
\(32\) 0 0
\(33\) −31.5959 −0.957452
\(34\) 0 0
\(35\) −25.1701 −0.719146
\(36\) 0 0
\(37\) 43.1918i 1.16735i 0.811988 + 0.583673i \(0.198385\pi\)
−0.811988 + 0.583673i \(0.801615\pi\)
\(38\) 0 0
\(39\) 10.0424i 0.257496i
\(40\) 0 0
\(41\) −1.59592 −0.0389248 −0.0194624 0.999811i \(-0.506195\pi\)
−0.0194624 + 0.999811i \(0.506195\pi\)
\(42\) 0 0
\(43\) 63.4967 1.47667 0.738334 0.674435i \(-0.235613\pi\)
0.738334 + 0.674435i \(0.235613\pi\)
\(44\) 0 0
\(45\) 26.6969i 0.593265i
\(46\) 0 0
\(47\) 72.3962i 1.54034i 0.637836 + 0.770172i \(0.279830\pi\)
−0.637836 + 0.770172i \(0.720170\pi\)
\(48\) 0 0
\(49\) 41.0000 0.836735
\(50\) 0 0
\(51\) −37.4052 −0.733436
\(52\) 0 0
\(53\) 70.2929i 1.32628i 0.748495 + 0.663140i \(0.230777\pi\)
−0.748495 + 0.663140i \(0.769223\pi\)
\(54\) 0 0
\(55\) − 162.334i − 2.95153i
\(56\) 0 0
\(57\) 31.5959 0.554314
\(58\) 0 0
\(59\) 34.6410 0.587136 0.293568 0.955938i \(-0.405157\pi\)
0.293568 + 0.955938i \(0.405157\pi\)
\(60\) 0 0
\(61\) 63.5959i 1.04256i 0.853387 + 0.521278i \(0.174545\pi\)
−0.853387 + 0.521278i \(0.825455\pi\)
\(62\) 0 0
\(63\) 8.48528i 0.134687i
\(64\) 0 0
\(65\) −51.5959 −0.793783
\(66\) 0 0
\(67\) −3.24259 −0.0483968 −0.0241984 0.999707i \(-0.507703\pi\)
−0.0241984 + 0.999707i \(0.507703\pi\)
\(68\) 0 0
\(69\) − 57.7980i − 0.837652i
\(70\) 0 0
\(71\) 68.4537i 0.964137i 0.876134 + 0.482068i \(0.160114\pi\)
−0.876134 + 0.482068i \(0.839886\pi\)
\(72\) 0 0
\(73\) −10.0000 −0.136986 −0.0684932 0.997652i \(-0.521819\pi\)
−0.0684932 + 0.997652i \(0.521819\pi\)
\(74\) 0 0
\(75\) −93.8630 −1.25151
\(76\) 0 0
\(77\) − 51.5959i − 0.670077i
\(78\) 0 0
\(79\) 35.0552i 0.443736i 0.975077 + 0.221868i \(0.0712155\pi\)
−0.975077 + 0.221868i \(0.928785\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 42.2691 0.509266 0.254633 0.967038i \(-0.418045\pi\)
0.254633 + 0.967038i \(0.418045\pi\)
\(84\) 0 0
\(85\) − 192.182i − 2.26096i
\(86\) 0 0
\(87\) 7.78539i 0.0894872i
\(88\) 0 0
\(89\) 5.19184 0.0583352 0.0291676 0.999575i \(-0.490714\pi\)
0.0291676 + 0.999575i \(0.490714\pi\)
\(90\) 0 0
\(91\) −16.3991 −0.180210
\(92\) 0 0
\(93\) 3.90918i 0.0420342i
\(94\) 0 0
\(95\) 162.334i 1.70878i
\(96\) 0 0
\(97\) −26.8082 −0.276373 −0.138186 0.990406i \(-0.544127\pi\)
−0.138186 + 0.990406i \(0.544127\pi\)
\(98\) 0 0
\(99\) −54.7257 −0.552785
\(100\) 0 0
\(101\) − 50.8786i − 0.503748i −0.967760 0.251874i \(-0.918953\pi\)
0.967760 0.251874i \(-0.0810469\pi\)
\(102\) 0 0
\(103\) − 148.764i − 1.44431i −0.691732 0.722154i \(-0.743152\pi\)
0.691732 0.722154i \(-0.256848\pi\)
\(104\) 0 0
\(105\) −43.5959 −0.415199
\(106\) 0 0
\(107\) −116.380 −1.08766 −0.543830 0.839195i \(-0.683026\pi\)
−0.543830 + 0.839195i \(0.683026\pi\)
\(108\) 0 0
\(109\) − 44.1816i − 0.405336i −0.979248 0.202668i \(-0.935039\pi\)
0.979248 0.202668i \(-0.0649612\pi\)
\(110\) 0 0
\(111\) 74.8105i 0.673968i
\(112\) 0 0
\(113\) 199.576 1.76615 0.883077 0.469227i \(-0.155467\pi\)
0.883077 + 0.469227i \(0.155467\pi\)
\(114\) 0 0
\(115\) 296.956 2.58223
\(116\) 0 0
\(117\) 17.3939i 0.148666i
\(118\) 0 0
\(119\) − 61.0825i − 0.513298i
\(120\) 0 0
\(121\) 211.767 1.75014
\(122\) 0 0
\(123\) −2.76421 −0.0224733
\(124\) 0 0
\(125\) − 259.778i − 2.07822i
\(126\) 0 0
\(127\) − 183.276i − 1.44312i −0.692352 0.721560i \(-0.743425\pi\)
0.692352 0.721560i \(-0.256575\pi\)
\(128\) 0 0
\(129\) 109.980 0.852555
\(130\) 0 0
\(131\) −176.891 −1.35031 −0.675155 0.737676i \(-0.735923\pi\)
−0.675155 + 0.737676i \(0.735923\pi\)
\(132\) 0 0
\(133\) 51.5959i 0.387939i
\(134\) 0 0
\(135\) 46.2405i 0.342522i
\(136\) 0 0
\(137\) 147.980 1.08014 0.540071 0.841619i \(-0.318397\pi\)
0.540071 + 0.841619i \(0.318397\pi\)
\(138\) 0 0
\(139\) 114.980 0.827193 0.413597 0.910460i \(-0.364272\pi\)
0.413597 + 0.910460i \(0.364272\pi\)
\(140\) 0 0
\(141\) 125.394i 0.889318i
\(142\) 0 0
\(143\) − 105.766i − 0.739621i
\(144\) 0 0
\(145\) −40.0000 −0.275862
\(146\) 0 0
\(147\) 71.0141 0.483089
\(148\) 0 0
\(149\) 229.303i 1.53895i 0.638679 + 0.769473i \(0.279481\pi\)
−0.638679 + 0.769473i \(0.720519\pi\)
\(150\) 0 0
\(151\) 225.674i 1.49453i 0.664527 + 0.747264i \(0.268633\pi\)
−0.664527 + 0.747264i \(0.731367\pi\)
\(152\) 0 0
\(153\) −64.7878 −0.423449
\(154\) 0 0
\(155\) −20.0847 −0.129579
\(156\) 0 0
\(157\) 36.8082i 0.234447i 0.993106 + 0.117223i \(0.0373994\pi\)
−0.993106 + 0.117223i \(0.962601\pi\)
\(158\) 0 0
\(159\) 121.751i 0.765728i
\(160\) 0 0
\(161\) 94.3837 0.586234
\(162\) 0 0
\(163\) −180.833 −1.10941 −0.554703 0.832048i \(-0.687168\pi\)
−0.554703 + 0.832048i \(0.687168\pi\)
\(164\) 0 0
\(165\) − 281.171i − 1.70407i
\(166\) 0 0
\(167\) − 174.791i − 1.04665i −0.852132 0.523326i \(-0.824691\pi\)
0.852132 0.523326i \(-0.175309\pi\)
\(168\) 0 0
\(169\) 135.384 0.801087
\(170\) 0 0
\(171\) 54.7257 0.320034
\(172\) 0 0
\(173\) − 158.111i − 0.913938i −0.889483 0.456969i \(-0.848935\pi\)
0.889483 0.456969i \(-0.151065\pi\)
\(174\) 0 0
\(175\) − 153.278i − 0.875872i
\(176\) 0 0
\(177\) 60.0000 0.338983
\(178\) 0 0
\(179\) −45.6979 −0.255295 −0.127648 0.991820i \(-0.540743\pi\)
−0.127648 + 0.991820i \(0.540743\pi\)
\(180\) 0 0
\(181\) − 263.778i − 1.45733i −0.684868 0.728667i \(-0.740140\pi\)
0.684868 0.728667i \(-0.259860\pi\)
\(182\) 0 0
\(183\) 110.151i 0.601920i
\(184\) 0 0
\(185\) −384.363 −2.07764
\(186\) 0 0
\(187\) 393.951 2.10669
\(188\) 0 0
\(189\) 14.6969i 0.0777616i
\(190\) 0 0
\(191\) − 154.963i − 0.811325i −0.914023 0.405663i \(-0.867041\pi\)
0.914023 0.405663i \(-0.132959\pi\)
\(192\) 0 0
\(193\) −182.767 −0.946981 −0.473491 0.880799i \(-0.657006\pi\)
−0.473491 + 0.880799i \(0.657006\pi\)
\(194\) 0 0
\(195\) −89.3668 −0.458291
\(196\) 0 0
\(197\) 178.697i 0.907091i 0.891233 + 0.453546i \(0.149841\pi\)
−0.891233 + 0.453546i \(0.850159\pi\)
\(198\) 0 0
\(199\) − 37.9125i − 0.190515i −0.995453 0.0952575i \(-0.969633\pi\)
0.995453 0.0952575i \(-0.0303674\pi\)
\(200\) 0 0
\(201\) −5.61633 −0.0279419
\(202\) 0 0
\(203\) −12.7135 −0.0626280
\(204\) 0 0
\(205\) − 14.2020i − 0.0692782i
\(206\) 0 0
\(207\) − 100.109i − 0.483618i
\(208\) 0 0
\(209\) −332.767 −1.59219
\(210\) 0 0
\(211\) 19.8986 0.0943060 0.0471530 0.998888i \(-0.484985\pi\)
0.0471530 + 0.998888i \(0.484985\pi\)
\(212\) 0 0
\(213\) 118.565i 0.556645i
\(214\) 0 0
\(215\) 565.056i 2.62817i
\(216\) 0 0
\(217\) −6.38367 −0.0294178
\(218\) 0 0
\(219\) −17.3205 −0.0790891
\(220\) 0 0
\(221\) − 125.212i − 0.566571i
\(222\) 0 0
\(223\) 212.703i 0.953827i 0.878950 + 0.476914i \(0.158245\pi\)
−0.878950 + 0.476914i \(0.841755\pi\)
\(224\) 0 0
\(225\) −162.576 −0.722558
\(226\) 0 0
\(227\) −104.180 −0.458942 −0.229471 0.973315i \(-0.573700\pi\)
−0.229471 + 0.973315i \(0.573700\pi\)
\(228\) 0 0
\(229\) 316.545i 1.38229i 0.722715 + 0.691146i \(0.242894\pi\)
−0.722715 + 0.691146i \(0.757106\pi\)
\(230\) 0 0
\(231\) − 89.3668i − 0.386869i
\(232\) 0 0
\(233\) 120.424 0.516843 0.258422 0.966032i \(-0.416798\pi\)
0.258422 + 0.966032i \(0.416798\pi\)
\(234\) 0 0
\(235\) −644.252 −2.74150
\(236\) 0 0
\(237\) 60.7173i 0.256191i
\(238\) 0 0
\(239\) − 178.734i − 0.747839i −0.927461 0.373919i \(-0.878014\pi\)
0.927461 0.373919i \(-0.121986\pi\)
\(240\) 0 0
\(241\) 46.8082 0.194225 0.0971124 0.995273i \(-0.469039\pi\)
0.0971124 + 0.995273i \(0.469039\pi\)
\(242\) 0 0
\(243\) 15.5885 0.0641500
\(244\) 0 0
\(245\) 364.858i 1.48922i
\(246\) 0 0
\(247\) 105.766i 0.428202i
\(248\) 0 0
\(249\) 73.2122 0.294025
\(250\) 0 0
\(251\) 39.9833 0.159296 0.0796480 0.996823i \(-0.474620\pi\)
0.0796480 + 0.996823i \(0.474620\pi\)
\(252\) 0 0
\(253\) 608.727i 2.40603i
\(254\) 0 0
\(255\) − 332.868i − 1.30537i
\(256\) 0 0
\(257\) 290.000 1.12840 0.564202 0.825637i \(-0.309184\pi\)
0.564202 + 0.825637i \(0.309184\pi\)
\(258\) 0 0
\(259\) −122.165 −0.471679
\(260\) 0 0
\(261\) 13.4847i 0.0516655i
\(262\) 0 0
\(263\) 249.415i 0.948347i 0.880431 + 0.474174i \(0.157253\pi\)
−0.880431 + 0.474174i \(0.842747\pi\)
\(264\) 0 0
\(265\) −625.535 −2.36051
\(266\) 0 0
\(267\) 8.99252 0.0336799
\(268\) 0 0
\(269\) 100.858i 0.374937i 0.982271 + 0.187469i \(0.0600283\pi\)
−0.982271 + 0.187469i \(0.939972\pi\)
\(270\) 0 0
\(271\) − 222.817i − 0.822201i −0.911590 0.411101i \(-0.865144\pi\)
0.911590 0.411101i \(-0.134856\pi\)
\(272\) 0 0
\(273\) −28.4041 −0.104044
\(274\) 0 0
\(275\) 988.563 3.59477
\(276\) 0 0
\(277\) − 194.202i − 0.701090i −0.936546 0.350545i \(-0.885996\pi\)
0.936546 0.350545i \(-0.114004\pi\)
\(278\) 0 0
\(279\) 6.77091i 0.0242685i
\(280\) 0 0
\(281\) 202.767 0.721592 0.360796 0.932645i \(-0.382505\pi\)
0.360796 + 0.932645i \(0.382505\pi\)
\(282\) 0 0
\(283\) −156.549 −0.553177 −0.276589 0.960988i \(-0.589204\pi\)
−0.276589 + 0.960988i \(0.589204\pi\)
\(284\) 0 0
\(285\) 281.171i 0.986566i
\(286\) 0 0
\(287\) − 4.51394i − 0.0157280i
\(288\) 0 0
\(289\) 177.384 0.613784
\(290\) 0 0
\(291\) −46.4331 −0.159564
\(292\) 0 0
\(293\) 116.677i 0.398213i 0.979978 + 0.199107i \(0.0638040\pi\)
−0.979978 + 0.199107i \(0.936196\pi\)
\(294\) 0 0
\(295\) 308.270i 1.04498i
\(296\) 0 0
\(297\) −94.7878 −0.319151
\(298\) 0 0
\(299\) 193.476 0.647077
\(300\) 0 0
\(301\) 179.596i 0.596664i
\(302\) 0 0
\(303\) − 88.1243i − 0.290839i
\(304\) 0 0
\(305\) −565.939 −1.85554
\(306\) 0 0
\(307\) −17.9850 −0.0585832 −0.0292916 0.999571i \(-0.509325\pi\)
−0.0292916 + 0.999571i \(0.509325\pi\)
\(308\) 0 0
\(309\) − 257.666i − 0.833872i
\(310\) 0 0
\(311\) − 290.214i − 0.933164i −0.884478 0.466582i \(-0.845485\pi\)
0.884478 0.466582i \(-0.154515\pi\)
\(312\) 0 0
\(313\) −255.535 −0.816405 −0.408202 0.912891i \(-0.633844\pi\)
−0.408202 + 0.912891i \(0.633844\pi\)
\(314\) 0 0
\(315\) −75.5103 −0.239715
\(316\) 0 0
\(317\) − 93.4847i − 0.294904i −0.989069 0.147452i \(-0.952893\pi\)
0.989069 0.147452i \(-0.0471073\pi\)
\(318\) 0 0
\(319\) − 81.9955i − 0.257039i
\(320\) 0 0
\(321\) −201.576 −0.627961
\(322\) 0 0
\(323\) −393.951 −1.21966
\(324\) 0 0
\(325\) − 314.202i − 0.966776i
\(326\) 0 0
\(327\) − 76.5248i − 0.234021i
\(328\) 0 0
\(329\) −204.767 −0.622393
\(330\) 0 0
\(331\) −151.464 −0.457594 −0.228797 0.973474i \(-0.573479\pi\)
−0.228797 + 0.973474i \(0.573479\pi\)
\(332\) 0 0
\(333\) 129.576i 0.389116i
\(334\) 0 0
\(335\) − 28.8557i − 0.0861365i
\(336\) 0 0
\(337\) 102.767 0.304948 0.152474 0.988308i \(-0.451276\pi\)
0.152474 + 0.988308i \(0.451276\pi\)
\(338\) 0 0
\(339\) 345.675 1.01969
\(340\) 0 0
\(341\) − 41.1714i − 0.120737i
\(342\) 0 0
\(343\) 254.558i 0.742153i
\(344\) 0 0
\(345\) 514.343 1.49085
\(346\) 0 0
\(347\) 333.626 0.961458 0.480729 0.876869i \(-0.340372\pi\)
0.480729 + 0.876869i \(0.340372\pi\)
\(348\) 0 0
\(349\) − 553.939i − 1.58722i −0.608429 0.793609i \(-0.708200\pi\)
0.608429 0.793609i \(-0.291800\pi\)
\(350\) 0 0
\(351\) 30.1271i 0.0858321i
\(352\) 0 0
\(353\) −398.727 −1.12954 −0.564768 0.825249i \(-0.691035\pi\)
−0.564768 + 0.825249i \(0.691035\pi\)
\(354\) 0 0
\(355\) −609.168 −1.71597
\(356\) 0 0
\(357\) − 105.798i − 0.296353i
\(358\) 0 0
\(359\) − 110.280i − 0.307186i −0.988134 0.153593i \(-0.950916\pi\)
0.988134 0.153593i \(-0.0490845\pi\)
\(360\) 0 0
\(361\) −28.2327 −0.0782068
\(362\) 0 0
\(363\) 366.792 1.01045
\(364\) 0 0
\(365\) − 88.9898i − 0.243808i
\(366\) 0 0
\(367\) 372.181i 1.01412i 0.861912 + 0.507058i \(0.169267\pi\)
−0.861912 + 0.507058i \(0.830733\pi\)
\(368\) 0 0
\(369\) −4.78775 −0.0129749
\(370\) 0 0
\(371\) −198.818 −0.535898
\(372\) 0 0
\(373\) 277.980i 0.745254i 0.927981 + 0.372627i \(0.121543\pi\)
−0.927981 + 0.372627i \(0.878457\pi\)
\(374\) 0 0
\(375\) − 449.948i − 1.19986i
\(376\) 0 0
\(377\) −26.0612 −0.0691279
\(378\) 0 0
\(379\) 320.797 0.846430 0.423215 0.906029i \(-0.360901\pi\)
0.423215 + 0.906029i \(0.360901\pi\)
\(380\) 0 0
\(381\) − 317.444i − 0.833186i
\(382\) 0 0
\(383\) − 609.797i − 1.59216i −0.605191 0.796080i \(-0.706903\pi\)
0.605191 0.796080i \(-0.293097\pi\)
\(384\) 0 0
\(385\) 459.151 1.19260
\(386\) 0 0
\(387\) 190.490 0.492223
\(388\) 0 0
\(389\) 126.111i 0.324193i 0.986775 + 0.162097i \(0.0518256\pi\)
−0.986775 + 0.162097i \(0.948174\pi\)
\(390\) 0 0
\(391\) 720.649i 1.84309i
\(392\) 0 0
\(393\) −306.384 −0.779602
\(394\) 0 0
\(395\) −311.955 −0.789760
\(396\) 0 0
\(397\) − 4.36326i − 0.0109906i −0.999985 0.00549529i \(-0.998251\pi\)
0.999985 0.00549529i \(-0.00174921\pi\)
\(398\) 0 0
\(399\) 89.3668i 0.223977i
\(400\) 0 0
\(401\) 691.049 1.72331 0.861657 0.507491i \(-0.169427\pi\)
0.861657 + 0.507491i \(0.169427\pi\)
\(402\) 0 0
\(403\) −13.0858 −0.0324710
\(404\) 0 0
\(405\) 80.0908i 0.197755i
\(406\) 0 0
\(407\) − 787.902i − 1.93588i
\(408\) 0 0
\(409\) 485.959 1.18816 0.594082 0.804404i \(-0.297515\pi\)
0.594082 + 0.804404i \(0.297515\pi\)
\(410\) 0 0
\(411\) 256.308 0.623621
\(412\) 0 0
\(413\) 97.9796i 0.237239i
\(414\) 0 0
\(415\) 376.152i 0.906390i
\(416\) 0 0
\(417\) 199.151 0.477580
\(418\) 0 0
\(419\) −499.531 −1.19220 −0.596098 0.802911i \(-0.703283\pi\)
−0.596098 + 0.802911i \(0.703283\pi\)
\(420\) 0 0
\(421\) 21.8796i 0.0519705i 0.999662 + 0.0259853i \(0.00827230\pi\)
−0.999662 + 0.0259853i \(0.991728\pi\)
\(422\) 0 0
\(423\) 217.189i 0.513448i
\(424\) 0 0
\(425\) 1170.32 2.75370
\(426\) 0 0
\(427\) −179.876 −0.421256
\(428\) 0 0
\(429\) − 183.192i − 0.427021i
\(430\) 0 0
\(431\) 513.631i 1.19172i 0.803089 + 0.595859i \(0.203188\pi\)
−0.803089 + 0.595859i \(0.796812\pi\)
\(432\) 0 0
\(433\) −16.3837 −0.0378376 −0.0189188 0.999821i \(-0.506022\pi\)
−0.0189188 + 0.999821i \(0.506022\pi\)
\(434\) 0 0
\(435\) −69.2820 −0.159269
\(436\) 0 0
\(437\) − 608.727i − 1.39297i
\(438\) 0 0
\(439\) − 324.068i − 0.738197i −0.929390 0.369098i \(-0.879667\pi\)
0.929390 0.369098i \(-0.120333\pi\)
\(440\) 0 0
\(441\) 123.000 0.278912
\(442\) 0 0
\(443\) 221.003 0.498877 0.249439 0.968391i \(-0.419754\pi\)
0.249439 + 0.968391i \(0.419754\pi\)
\(444\) 0 0
\(445\) 46.2020i 0.103825i
\(446\) 0 0
\(447\) 397.165i 0.888511i
\(448\) 0 0
\(449\) 690.322 1.53747 0.768733 0.639570i \(-0.220887\pi\)
0.768733 + 0.639570i \(0.220887\pi\)
\(450\) 0 0
\(451\) 29.1126 0.0645512
\(452\) 0 0
\(453\) 390.879i 0.862867i
\(454\) 0 0
\(455\) − 145.935i − 0.320737i
\(456\) 0 0
\(457\) 397.878 0.870629 0.435315 0.900278i \(-0.356637\pi\)
0.435315 + 0.900278i \(0.356637\pi\)
\(458\) 0 0
\(459\) −112.216 −0.244479
\(460\) 0 0
\(461\) − 707.160i − 1.53397i −0.641665 0.766985i \(-0.721756\pi\)
0.641665 0.766985i \(-0.278244\pi\)
\(462\) 0 0
\(463\) 869.926i 1.87889i 0.342700 + 0.939445i \(0.388659\pi\)
−0.342700 + 0.939445i \(0.611341\pi\)
\(464\) 0 0
\(465\) −34.7878 −0.0748124
\(466\) 0 0
\(467\) −75.4396 −0.161541 −0.0807705 0.996733i \(-0.525738\pi\)
−0.0807705 + 0.996733i \(0.525738\pi\)
\(468\) 0 0
\(469\) − 9.17143i − 0.0195553i
\(470\) 0 0
\(471\) 63.7536i 0.135358i
\(472\) 0 0
\(473\) −1158.30 −2.44884
\(474\) 0 0
\(475\) −988.563 −2.08118
\(476\) 0 0
\(477\) 210.879i 0.442093i
\(478\) 0 0
\(479\) − 90.5674i − 0.189076i −0.995521 0.0945380i \(-0.969863\pi\)
0.995521 0.0945380i \(-0.0301374\pi\)
\(480\) 0 0
\(481\) −250.424 −0.520633
\(482\) 0 0
\(483\) 163.477 0.338462
\(484\) 0 0
\(485\) − 238.565i − 0.491887i
\(486\) 0 0
\(487\) − 137.392i − 0.282120i −0.990001 0.141060i \(-0.954949\pi\)
0.990001 0.141060i \(-0.0450510\pi\)
\(488\) 0 0
\(489\) −313.212 −0.640516
\(490\) 0 0
\(491\) 89.5529 0.182389 0.0911944 0.995833i \(-0.470932\pi\)
0.0911944 + 0.995833i \(0.470932\pi\)
\(492\) 0 0
\(493\) − 97.0714i − 0.196899i
\(494\) 0 0
\(495\) − 487.003i − 0.983845i
\(496\) 0 0
\(497\) −193.616 −0.389570
\(498\) 0 0
\(499\) 268.286 0.537648 0.268824 0.963189i \(-0.413365\pi\)
0.268824 + 0.963189i \(0.413365\pi\)
\(500\) 0 0
\(501\) − 302.747i − 0.604285i
\(502\) 0 0
\(503\) − 93.2515i − 0.185391i −0.995695 0.0926953i \(-0.970452\pi\)
0.995695 0.0926953i \(-0.0295483\pi\)
\(504\) 0 0
\(505\) 452.767 0.896569
\(506\) 0 0
\(507\) 234.491 0.462508
\(508\) 0 0
\(509\) − 168.272i − 0.330594i −0.986244 0.165297i \(-0.947142\pi\)
0.986244 0.165297i \(-0.0528583\pi\)
\(510\) 0 0
\(511\) − 28.2843i − 0.0553508i
\(512\) 0 0
\(513\) 94.7878 0.184771
\(514\) 0 0
\(515\) 1323.85 2.57057
\(516\) 0 0
\(517\) − 1320.64i − 2.55444i
\(518\) 0 0
\(519\) − 273.857i − 0.527662i
\(520\) 0 0
\(521\) 316.788 0.608038 0.304019 0.952666i \(-0.401671\pi\)
0.304019 + 0.952666i \(0.401671\pi\)
\(522\) 0 0
\(523\) 443.591 0.848167 0.424083 0.905623i \(-0.360596\pi\)
0.424083 + 0.905623i \(0.360596\pi\)
\(524\) 0 0
\(525\) − 265.485i − 0.505685i
\(526\) 0 0
\(527\) − 48.7413i − 0.0924883i
\(528\) 0 0
\(529\) −584.535 −1.10498
\(530\) 0 0
\(531\) 103.923 0.195712
\(532\) 0 0
\(533\) − 9.25307i − 0.0173604i
\(534\) 0 0
\(535\) − 1035.66i − 1.93581i
\(536\) 0 0
\(537\) −79.1510 −0.147395
\(538\) 0 0
\(539\) −747.918 −1.38760
\(540\) 0 0
\(541\) − 966.443i − 1.78640i −0.449659 0.893200i \(-0.648454\pi\)
0.449659 0.893200i \(-0.351546\pi\)
\(542\) 0 0
\(543\) − 456.876i − 0.841392i
\(544\) 0 0
\(545\) 393.171 0.721415
\(546\) 0 0
\(547\) −578.470 −1.05753 −0.528766 0.848768i \(-0.677345\pi\)
−0.528766 + 0.848768i \(0.677345\pi\)
\(548\) 0 0
\(549\) 190.788i 0.347519i
\(550\) 0 0
\(551\) 81.9955i 0.148812i
\(552\) 0 0
\(553\) −99.1510 −0.179297
\(554\) 0 0
\(555\) −665.737 −1.19953
\(556\) 0 0
\(557\) 590.293i 1.05977i 0.848069 + 0.529886i \(0.177765\pi\)
−0.848069 + 0.529886i \(0.822235\pi\)
\(558\) 0 0
\(559\) 368.152i 0.658589i
\(560\) 0 0
\(561\) 682.343 1.21630
\(562\) 0 0
\(563\) 468.877 0.832818 0.416409 0.909177i \(-0.363288\pi\)
0.416409 + 0.909177i \(0.363288\pi\)
\(564\) 0 0
\(565\) 1776.02i 3.14340i
\(566\) 0 0
\(567\) 25.4558i 0.0448957i
\(568\) 0 0
\(569\) −548.829 −0.964549 −0.482275 0.876020i \(-0.660189\pi\)
−0.482275 + 0.876020i \(0.660189\pi\)
\(570\) 0 0
\(571\) 133.036 0.232987 0.116494 0.993191i \(-0.462835\pi\)
0.116494 + 0.993191i \(0.462835\pi\)
\(572\) 0 0
\(573\) − 268.404i − 0.468419i
\(574\) 0 0
\(575\) 1808.36i 3.14498i
\(576\) 0 0
\(577\) −735.535 −1.27476 −0.637378 0.770551i \(-0.719981\pi\)
−0.637378 + 0.770551i \(0.719981\pi\)
\(578\) 0 0
\(579\) −316.562 −0.546740
\(580\) 0 0
\(581\) 119.555i 0.205775i
\(582\) 0 0
\(583\) − 1282.28i − 2.19944i
\(584\) 0 0
\(585\) −154.788 −0.264594
\(586\) 0 0
\(587\) 178.290 0.303732 0.151866 0.988401i \(-0.451472\pi\)
0.151866 + 0.988401i \(0.451472\pi\)
\(588\) 0 0
\(589\) 41.1714i 0.0699006i
\(590\) 0 0
\(591\) 309.512i 0.523709i
\(592\) 0 0
\(593\) −584.261 −0.985263 −0.492632 0.870238i \(-0.663965\pi\)
−0.492632 + 0.870238i \(0.663965\pi\)
\(594\) 0 0
\(595\) 543.572 0.913566
\(596\) 0 0
\(597\) − 65.6663i − 0.109994i
\(598\) 0 0
\(599\) 134.050i 0.223790i 0.993720 + 0.111895i \(0.0356920\pi\)
−0.993720 + 0.111895i \(0.964308\pi\)
\(600\) 0 0
\(601\) 621.918 1.03481 0.517403 0.855742i \(-0.326899\pi\)
0.517403 + 0.855742i \(0.326899\pi\)
\(602\) 0 0
\(603\) −9.72777 −0.0161323
\(604\) 0 0
\(605\) 1884.51i 3.11490i
\(606\) 0 0
\(607\) 36.1404i 0.0595393i 0.999557 + 0.0297697i \(0.00947738\pi\)
−0.999557 + 0.0297697i \(0.990523\pi\)
\(608\) 0 0
\(609\) −22.0204 −0.0361583
\(610\) 0 0
\(611\) −419.750 −0.686989
\(612\) 0 0
\(613\) 385.857i 0.629457i 0.949182 + 0.314728i \(0.101913\pi\)
−0.949182 + 0.314728i \(0.898087\pi\)
\(614\) 0 0
\(615\) − 24.5987i − 0.0399978i
\(616\) 0 0
\(617\) 290.849 0.471392 0.235696 0.971827i \(-0.424263\pi\)
0.235696 + 0.971827i \(0.424263\pi\)
\(618\) 0 0
\(619\) 275.658 0.445327 0.222664 0.974895i \(-0.428525\pi\)
0.222664 + 0.974895i \(0.428525\pi\)
\(620\) 0 0
\(621\) − 173.394i − 0.279217i
\(622\) 0 0
\(623\) 14.6847i 0.0235710i
\(624\) 0 0
\(625\) 956.959 1.53113
\(626\) 0 0
\(627\) −576.370 −0.919250
\(628\) 0 0
\(629\) − 932.767i − 1.48294i
\(630\) 0 0
\(631\) 940.608i 1.49066i 0.666694 + 0.745331i \(0.267709\pi\)
−0.666694 + 0.745331i \(0.732291\pi\)
\(632\) 0 0
\(633\) 34.4653 0.0544476
\(634\) 0 0
\(635\) 1630.97 2.56846
\(636\) 0 0
\(637\) 237.716i 0.373181i
\(638\) 0 0
\(639\) 205.361i 0.321379i
\(640\) 0 0
\(641\) −34.3633 −0.0536088 −0.0268044 0.999641i \(-0.508533\pi\)
−0.0268044 + 0.999641i \(0.508533\pi\)
\(642\) 0 0
\(643\) 308.340 0.479534 0.239767 0.970830i \(-0.422929\pi\)
0.239767 + 0.970830i \(0.422929\pi\)
\(644\) 0 0
\(645\) 978.706i 1.51737i
\(646\) 0 0
\(647\) − 190.503i − 0.294441i −0.989104 0.147220i \(-0.952967\pi\)
0.989104 0.147220i \(-0.0470327\pi\)
\(648\) 0 0
\(649\) −631.918 −0.973680
\(650\) 0 0
\(651\) −11.0568 −0.0169844
\(652\) 0 0
\(653\) 1103.38i 1.68971i 0.534993 + 0.844857i \(0.320314\pi\)
−0.534993 + 0.844857i \(0.679686\pi\)
\(654\) 0 0
\(655\) − 1574.15i − 2.40328i
\(656\) 0 0
\(657\) −30.0000 −0.0456621
\(658\) 0 0
\(659\) −815.544 −1.23755 −0.618774 0.785569i \(-0.712370\pi\)
−0.618774 + 0.785569i \(0.712370\pi\)
\(660\) 0 0
\(661\) 121.576i 0.183927i 0.995762 + 0.0919633i \(0.0293143\pi\)
−0.995762 + 0.0919633i \(0.970686\pi\)
\(662\) 0 0
\(663\) − 216.874i − 0.327110i
\(664\) 0 0
\(665\) −459.151 −0.690453
\(666\) 0 0
\(667\) 149.993 0.224877
\(668\) 0 0
\(669\) 368.413i 0.550692i
\(670\) 0 0
\(671\) − 1160.11i − 1.72893i
\(672\) 0 0
\(673\) −1326.60 −1.97118 −0.985590 0.169152i \(-0.945897\pi\)
−0.985590 + 0.169152i \(0.945897\pi\)
\(674\) 0 0
\(675\) −281.589 −0.417169
\(676\) 0 0
\(677\) − 657.526i − 0.971234i −0.874172 0.485617i \(-0.838595\pi\)
0.874172 0.485617i \(-0.161405\pi\)
\(678\) 0 0
\(679\) − 75.8249i − 0.111671i
\(680\) 0 0
\(681\) −180.445 −0.264970
\(682\) 0 0
\(683\) −772.318 −1.13077 −0.565386 0.824826i \(-0.691273\pi\)
−0.565386 + 0.824826i \(0.691273\pi\)
\(684\) 0 0
\(685\) 1316.87i 1.92243i
\(686\) 0 0
\(687\) 548.272i 0.798067i
\(688\) 0 0
\(689\) −407.555 −0.591517
\(690\) 0 0
\(691\) −1269.00 −1.83647 −0.918237 0.396030i \(-0.870387\pi\)
−0.918237 + 0.396030i \(0.870387\pi\)
\(692\) 0 0
\(693\) − 154.788i − 0.223359i
\(694\) 0 0
\(695\) 1023.20i 1.47224i
\(696\) 0 0
\(697\) 34.4653 0.0494481
\(698\) 0 0
\(699\) 208.581 0.298400
\(700\) 0 0
\(701\) − 535.383i − 0.763741i −0.924216 0.381871i \(-0.875280\pi\)
0.924216 0.381871i \(-0.124720\pi\)
\(702\) 0 0
\(703\) 787.902i 1.12077i
\(704\) 0 0
\(705\) −1115.88 −1.58281
\(706\) 0 0
\(707\) 143.906 0.203545
\(708\) 0 0
\(709\) − 56.8673i − 0.0802078i −0.999196 0.0401039i \(-0.987231\pi\)
0.999196 0.0401039i \(-0.0127689\pi\)
\(710\) 0 0
\(711\) 105.166i 0.147912i
\(712\) 0 0
\(713\) 75.3143 0.105630
\(714\) 0 0
\(715\) 941.208 1.31638
\(716\) 0 0
\(717\) − 309.576i − 0.431765i
\(718\) 0 0
\(719\) 980.922i 1.36429i 0.731219 + 0.682143i \(0.238952\pi\)
−0.731219 + 0.682143i \(0.761048\pi\)
\(720\) 0 0
\(721\) 420.767 0.583589
\(722\) 0 0
\(723\) 81.0741 0.112136
\(724\) 0 0
\(725\) − 243.587i − 0.335982i
\(726\) 0 0
\(727\) 83.1673i 0.114398i 0.998363 + 0.0571990i \(0.0182169\pi\)
−0.998363 + 0.0571990i \(0.981783\pi\)
\(728\) 0 0
\(729\) 27.0000 0.0370370
\(730\) 0 0
\(731\) −1371.27 −1.87588
\(732\) 0 0
\(733\) − 345.476i − 0.471317i −0.971836 0.235659i \(-0.924275\pi\)
0.971836 0.235659i \(-0.0757247\pi\)
\(734\) 0 0
\(735\) 631.953i 0.859800i
\(736\) 0 0
\(737\) 59.1510 0.0802592
\(738\) 0 0
\(739\) 1388.23 1.87852 0.939261 0.343203i \(-0.111512\pi\)
0.939261 + 0.343203i \(0.111512\pi\)
\(740\) 0 0
\(741\) 183.192i 0.247222i
\(742\) 0 0
\(743\) − 885.269i − 1.19148i −0.803178 0.595739i \(-0.796859\pi\)
0.803178 0.595739i \(-0.203141\pi\)
\(744\) 0 0
\(745\) −2040.56 −2.73901
\(746\) 0 0
\(747\) 126.807 0.169755
\(748\) 0 0
\(749\) − 329.171i − 0.439481i
\(750\) 0 0
\(751\) − 1225.11i − 1.63130i −0.578545 0.815651i \(-0.696379\pi\)
0.578545 0.815651i \(-0.303621\pi\)
\(752\) 0 0
\(753\) 69.2531 0.0919695
\(754\) 0 0
\(755\) −2008.27 −2.65996
\(756\) 0 0
\(757\) 86.9694i 0.114887i 0.998349 + 0.0574435i \(0.0182949\pi\)
−0.998349 + 0.0574435i \(0.981705\pi\)
\(758\) 0 0
\(759\) 1054.35i 1.38912i
\(760\) 0 0
\(761\) −764.020 −1.00397 −0.501985 0.864877i \(-0.667397\pi\)
−0.501985 + 0.864877i \(0.667397\pi\)
\(762\) 0 0
\(763\) 124.965 0.163781
\(764\) 0 0
\(765\) − 576.545i − 0.753653i
\(766\) 0 0
\(767\) 200.847i 0.261861i
\(768\) 0 0
\(769\) 269.918 0.350999 0.175500 0.984480i \(-0.443846\pi\)
0.175500 + 0.984480i \(0.443846\pi\)
\(770\) 0 0
\(771\) 502.295 0.651485
\(772\) 0 0
\(773\) 1109.24i 1.43498i 0.696569 + 0.717490i \(0.254709\pi\)
−0.696569 + 0.717490i \(0.745291\pi\)
\(774\) 0 0
\(775\) − 122.309i − 0.157818i
\(776\) 0 0
\(777\) −211.596 −0.272324
\(778\) 0 0
\(779\) −29.1126 −0.0373718
\(780\) 0 0
\(781\) − 1248.73i − 1.59888i
\(782\) 0 0
\(783\) 23.3562i 0.0298291i
\(784\) 0 0
\(785\) −327.555 −0.417268
\(786\) 0 0
\(787\) 1260.98 1.60226 0.801129 0.598491i \(-0.204233\pi\)
0.801129 + 0.598491i \(0.204233\pi\)
\(788\) 0 0
\(789\) 432.000i 0.547529i
\(790\) 0 0
\(791\) 564.485i 0.713634i
\(792\) 0 0
\(793\) −368.727 −0.464977
\(794\) 0 0
\(795\) −1083.46 −1.36284
\(796\) 0 0
\(797\) 386.779i 0.485293i 0.970115 + 0.242647i \(0.0780155\pi\)
−0.970115 + 0.242647i \(0.921984\pi\)
\(798\) 0 0
\(799\) − 1563.46i − 1.95677i
\(800\) 0 0
\(801\) 15.5755 0.0194451
\(802\) 0 0
\(803\) 182.419 0.227172
\(804\) 0 0
\(805\) 839.918i 1.04338i
\(806\) 0 0
\(807\) 174.691i 0.216470i
\(808\) 0 0
\(809\) 1139.98 1.40912 0.704561 0.709643i \(-0.251144\pi\)
0.704561 + 0.709643i \(0.251144\pi\)
\(810\) 0 0
\(811\) 959.450 1.18305 0.591523 0.806288i \(-0.298527\pi\)
0.591523 + 0.806288i \(0.298527\pi\)
\(812\) 0 0
\(813\) − 385.930i − 0.474698i
\(814\) 0 0
\(815\) − 1609.23i − 1.97452i
\(816\) 0 0
\(817\) 1158.30 1.41775
\(818\) 0 0
\(819\) −49.1973 −0.0600700
\(820\) 0 0
\(821\) − 918.674i − 1.11897i −0.828840 0.559485i \(-0.810999\pi\)
0.828840 0.559485i \(-0.189001\pi\)
\(822\) 0 0
\(823\) 678.107i 0.823945i 0.911196 + 0.411972i \(0.135160\pi\)
−0.911196 + 0.411972i \(0.864840\pi\)
\(824\) 0 0
\(825\) 1712.24 2.07544
\(826\) 0 0
\(827\) −1237.58 −1.49647 −0.748234 0.663434i \(-0.769098\pi\)
−0.748234 + 0.663434i \(0.769098\pi\)
\(828\) 0 0
\(829\) 778.120i 0.938625i 0.883032 + 0.469313i \(0.155498\pi\)
−0.883032 + 0.469313i \(0.844502\pi\)
\(830\) 0 0
\(831\) − 336.368i − 0.404775i
\(832\) 0 0
\(833\) −885.433 −1.06294
\(834\) 0 0
\(835\) 1555.46 1.86283
\(836\) 0 0
\(837\) 11.7276i 0.0140114i
\(838\) 0 0
\(839\) − 345.582i − 0.411897i −0.978563 0.205949i \(-0.933972\pi\)
0.978563 0.205949i \(-0.0660280\pi\)
\(840\) 0 0
\(841\) 820.796 0.975976
\(842\) 0 0
\(843\) 351.203 0.416611
\(844\) 0 0
\(845\) 1204.78i 1.42577i
\(846\) 0 0
\(847\) 598.968i 0.707165i
\(848\) 0 0
\(849\) −271.151 −0.319377
\(850\) 0 0
\(851\) 1441.30 1.69365
\(852\) 0 0
\(853\) 1026.38i 1.20326i 0.798774 + 0.601632i \(0.205482\pi\)
−0.798774 + 0.601632i \(0.794518\pi\)
\(854\) 0 0
\(855\) 487.003i 0.569594i
\(856\) 0 0
\(857\) −296.061 −0.345462 −0.172731 0.984969i \(-0.555259\pi\)
−0.172731 + 0.984969i \(0.555259\pi\)
\(858\) 0 0
\(859\) −167.118 −0.194550 −0.0972748 0.995258i \(-0.531013\pi\)
−0.0972748 + 0.995258i \(0.531013\pi\)
\(860\) 0 0
\(861\) − 7.81837i − 0.00908057i
\(862\) 0 0
\(863\) − 10.0553i − 0.0116516i −0.999983 0.00582580i \(-0.998146\pi\)
0.999983 0.00582580i \(-0.00185442\pi\)
\(864\) 0 0
\(865\) 1407.03 1.62662
\(866\) 0 0
\(867\) 307.238 0.354369
\(868\) 0 0
\(869\) − 639.473i − 0.735873i
\(870\) 0 0
\(871\) − 18.8004i − 0.0215848i
\(872\) 0 0
\(873\) −80.4245 −0.0921243
\(874\) 0 0
\(875\) 734.762 0.839728
\(876\) 0 0
\(877\) − 1154.58i − 1.31651i −0.752793 0.658257i \(-0.771294\pi\)
0.752793 0.658257i \(-0.228706\pi\)
\(878\) 0 0
\(879\) 202.090i 0.229909i
\(880\) 0 0
\(881\) 160.220 0.181862 0.0909310 0.995857i \(-0.471016\pi\)
0.0909310 + 0.995857i \(0.471016\pi\)
\(882\) 0 0
\(883\) −1311.46 −1.48523 −0.742616 0.669718i \(-0.766415\pi\)
−0.742616 + 0.669718i \(0.766415\pi\)
\(884\) 0 0
\(885\) 533.939i 0.603321i
\(886\) 0 0
\(887\) 234.673i 0.264569i 0.991212 + 0.132285i \(0.0422313\pi\)
−0.991212 + 0.132285i \(0.957769\pi\)
\(888\) 0 0
\(889\) 518.384 0.583109
\(890\) 0 0
\(891\) −164.177 −0.184262
\(892\) 0 0
\(893\) 1320.64i 1.47889i
\(894\) 0 0
\(895\) − 406.664i − 0.454374i
\(896\) 0 0
\(897\) 335.110 0.373590
\(898\) 0 0
\(899\) −10.1448 −0.0112846
\(900\) 0 0
\(901\) − 1518.04i − 1.68484i
\(902\) 0 0
\(903\) 311.069i 0.344484i
\(904\) 0 0
\(905\) 2347.35 2.59376
\(906\) 0 0
\(907\) −1161.70 −1.28081 −0.640406 0.768036i \(-0.721234\pi\)
−0.640406 + 0.768036i \(0.721234\pi\)
\(908\) 0 0
\(909\) − 152.636i − 0.167916i
\(910\) 0 0
\(911\) − 1382.33i − 1.51737i −0.651456 0.758687i \(-0.725841\pi\)
0.651456 0.758687i \(-0.274159\pi\)
\(912\) 0 0
\(913\) −771.069 −0.844545
\(914\) 0 0
\(915\) −980.235 −1.07129
\(916\) 0 0
\(917\) − 500.322i − 0.545608i
\(918\) 0 0
\(919\) − 1206.60i − 1.31294i −0.754350 0.656472i \(-0.772048\pi\)
0.754350 0.656472i \(-0.227952\pi\)
\(920\) 0 0
\(921\) −31.1510 −0.0338230
\(922\) 0 0
\(923\) −396.892 −0.430002
\(924\) 0 0
\(925\) − 2340.64i − 2.53043i
\(926\) 0 0
\(927\) − 446.291i − 0.481436i
\(928\) 0 0
\(929\) 1159.90 1.24854 0.624272 0.781207i \(-0.285396\pi\)
0.624272 + 0.781207i \(0.285396\pi\)
\(930\) 0 0
\(931\) 747.918 0.803349
\(932\) 0 0
\(933\) − 502.665i − 0.538762i
\(934\) 0 0
\(935\) 3505.76i 3.74948i
\(936\) 0 0
\(937\) 173.837 0.185525 0.0927624 0.995688i \(-0.470430\pi\)
0.0927624 + 0.995688i \(0.470430\pi\)
\(938\) 0 0
\(939\) −442.599 −0.471352
\(940\) 0 0
\(941\) − 603.968i − 0.641837i −0.947107 0.320918i \(-0.896008\pi\)
0.947107 0.320918i \(-0.103992\pi\)
\(942\) 0 0
\(943\) 53.2553i 0.0564743i
\(944\) 0 0
\(945\) −130.788 −0.138400
\(946\) 0 0
\(947\) −951.077 −1.00431 −0.502153 0.864779i \(-0.667458\pi\)
−0.502153 + 0.864779i \(0.667458\pi\)
\(948\) 0 0
\(949\) − 57.9796i − 0.0610955i
\(950\) 0 0
\(951\) − 161.920i − 0.170263i
\(952\) 0 0
\(953\) −1407.98 −1.47742 −0.738709 0.674024i \(-0.764564\pi\)
−0.738709 + 0.674024i \(0.764564\pi\)
\(954\) 0 0
\(955\) 1379.01 1.44399
\(956\) 0 0
\(957\) − 142.020i − 0.148402i
\(958\) 0 0
\(959\) 418.549i 0.436444i
\(960\) 0 0
\(961\) 955.906 0.994699
\(962\) 0 0
\(963\) −349.139 −0.362554
\(964\) 0 0
\(965\) − 1626.44i − 1.68543i
\(966\) 0 0
\(967\) − 467.718i − 0.483679i −0.970316 0.241840i \(-0.922249\pi\)
0.970316 0.241840i \(-0.0777508\pi\)
\(968\) 0 0
\(969\) −682.343 −0.704172
\(970\) 0 0
\(971\) 1115.70 1.14902 0.574510 0.818498i \(-0.305193\pi\)
0.574510 + 0.818498i \(0.305193\pi\)
\(972\) 0 0
\(973\) 325.212i 0.334237i
\(974\) 0 0
\(975\) − 544.214i − 0.558168i
\(976\) 0 0
\(977\) 494.363 0.506001 0.253001 0.967466i \(-0.418583\pi\)
0.253001 + 0.967466i \(0.418583\pi\)
\(978\) 0 0
\(979\) −94.7090 −0.0967406
\(980\) 0 0
\(981\) − 132.545i − 0.135112i
\(982\) 0 0
\(983\) − 188.391i − 0.191649i −0.995398 0.0958243i \(-0.969451\pi\)
0.995398 0.0958243i \(-0.0305487\pi\)
\(984\) 0 0
\(985\) −1590.22 −1.61444
\(986\) 0 0
\(987\) −354.667 −0.359339
\(988\) 0 0
\(989\) − 2118.87i − 2.14243i
\(990\) 0 0
\(991\) − 1895.81i − 1.91303i −0.291681 0.956516i \(-0.594215\pi\)
0.291681 0.956516i \(-0.405785\pi\)
\(992\) 0 0
\(993\) −262.343 −0.264192
\(994\) 0 0
\(995\) 337.382 0.339078
\(996\) 0 0
\(997\) 1317.98i 1.32195i 0.750410 + 0.660973i \(0.229856\pi\)
−0.750410 + 0.660973i \(0.770144\pi\)
\(998\) 0 0
\(999\) 224.431i 0.224656i
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 384.3.b.c.319.8 yes 8
3.2 odd 2 1152.3.b.j.703.2 8
4.3 odd 2 inner 384.3.b.c.319.4 yes 8
8.3 odd 2 inner 384.3.b.c.319.5 yes 8
8.5 even 2 inner 384.3.b.c.319.1 8
12.11 even 2 1152.3.b.j.703.1 8
16.3 odd 4 768.3.g.g.511.2 4
16.5 even 4 768.3.g.c.511.1 4
16.11 odd 4 768.3.g.c.511.3 4
16.13 even 4 768.3.g.g.511.4 4
24.5 odd 2 1152.3.b.j.703.8 8
24.11 even 2 1152.3.b.j.703.7 8
48.5 odd 4 2304.3.g.x.1279.3 4
48.11 even 4 2304.3.g.x.1279.4 4
48.29 odd 4 2304.3.g.o.1279.1 4
48.35 even 4 2304.3.g.o.1279.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.b.c.319.1 8 8.5 even 2 inner
384.3.b.c.319.4 yes 8 4.3 odd 2 inner
384.3.b.c.319.5 yes 8 8.3 odd 2 inner
384.3.b.c.319.8 yes 8 1.1 even 1 trivial
768.3.g.c.511.1 4 16.5 even 4
768.3.g.c.511.3 4 16.11 odd 4
768.3.g.g.511.2 4 16.3 odd 4
768.3.g.g.511.4 4 16.13 even 4
1152.3.b.j.703.1 8 12.11 even 2
1152.3.b.j.703.2 8 3.2 odd 2
1152.3.b.j.703.7 8 24.11 even 2
1152.3.b.j.703.8 8 24.5 odd 2
2304.3.g.o.1279.1 4 48.29 odd 4
2304.3.g.o.1279.2 4 48.35 even 4
2304.3.g.x.1279.3 4 48.5 odd 4
2304.3.g.x.1279.4 4 48.11 even 4