Properties

Label 3072.2.d.c.1537.4
Level $3072$
Weight $2$
Character 3072.1537
Analytic conductor $24.530$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.5300435009\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1536)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1537.4
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 3072.1537
Dual form 3072.2.d.c.1537.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{3} +1.41421i q^{5} -2.82843 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{3} +1.41421i q^{5} -2.82843 q^{7} -1.00000 q^{9} +4.24264i q^{13} -1.41421 q^{15} +4.00000 q^{17} +8.00000i q^{19} -2.82843i q^{21} +5.65685 q^{23} +3.00000 q^{25} -1.00000i q^{27} +1.41421i q^{29} -2.82843 q^{31} -4.00000i q^{35} -4.24264i q^{37} -4.24264 q^{39} +4.00000 q^{41} -1.41421i q^{45} -11.3137 q^{47} +1.00000 q^{49} +4.00000i q^{51} +7.07107i q^{53} -8.00000 q^{57} -12.0000i q^{59} +1.41421i q^{61} +2.82843 q^{63} -6.00000 q^{65} -4.00000i q^{67} +5.65685i q^{69} -11.3137 q^{71} -14.0000 q^{73} +3.00000i q^{75} -8.48528 q^{79} +1.00000 q^{81} +16.0000i q^{83} +5.65685i q^{85} -1.41421 q^{87} -6.00000 q^{89} -12.0000i q^{91} -2.82843i q^{93} -11.3137 q^{95} +16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} + 16q^{17} + 12q^{25} + 16q^{41} + 4q^{49} - 32q^{57} - 24q^{65} - 56q^{73} + 4q^{81} - 24q^{89} + 64q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3072\mathbb{Z}\right)^\times\).

\(n\) \(1025\) \(2047\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 1.41421i 0.632456i 0.948683 + 0.316228i \(0.102416\pi\)
−0.948683 + 0.316228i \(0.897584\pi\)
\(6\) 0 0
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 4.24264i 1.17670i 0.808608 + 0.588348i \(0.200222\pi\)
−0.808608 + 0.588348i \(0.799778\pi\)
\(14\) 0 0
\(15\) −1.41421 −0.365148
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 8.00000i 1.83533i 0.397360 + 0.917663i \(0.369927\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 0 0
\(21\) − 2.82843i − 0.617213i
\(22\) 0 0
\(23\) 5.65685 1.17954 0.589768 0.807573i \(-0.299219\pi\)
0.589768 + 0.807573i \(0.299219\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 1.41421i 0.262613i 0.991342 + 0.131306i \(0.0419172\pi\)
−0.991342 + 0.131306i \(0.958083\pi\)
\(30\) 0 0
\(31\) −2.82843 −0.508001 −0.254000 0.967204i \(-0.581746\pi\)
−0.254000 + 0.967204i \(0.581746\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 4.00000i − 0.676123i
\(36\) 0 0
\(37\) − 4.24264i − 0.697486i −0.937218 0.348743i \(-0.886609\pi\)
0.937218 0.348743i \(-0.113391\pi\)
\(38\) 0 0
\(39\) −4.24264 −0.679366
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) − 1.41421i − 0.210819i
\(46\) 0 0
\(47\) −11.3137 −1.65027 −0.825137 0.564933i \(-0.808902\pi\)
−0.825137 + 0.564933i \(0.808902\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.00000i 0.560112i
\(52\) 0 0
\(53\) 7.07107i 0.971286i 0.874157 + 0.485643i \(0.161414\pi\)
−0.874157 + 0.485643i \(0.838586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) − 12.0000i − 1.56227i −0.624364 0.781133i \(-0.714642\pi\)
0.624364 0.781133i \(-0.285358\pi\)
\(60\) 0 0
\(61\) 1.41421i 0.181071i 0.995893 + 0.0905357i \(0.0288579\pi\)
−0.995893 + 0.0905357i \(0.971142\pi\)
\(62\) 0 0
\(63\) 2.82843 0.356348
\(64\) 0 0
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 5.65685i 0.681005i
\(70\) 0 0
\(71\) −11.3137 −1.34269 −0.671345 0.741145i \(-0.734283\pi\)
−0.671345 + 0.741145i \(0.734283\pi\)
\(72\) 0 0
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 0 0
\(75\) 3.00000i 0.346410i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.48528 −0.954669 −0.477334 0.878722i \(-0.658397\pi\)
−0.477334 + 0.878722i \(0.658397\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 16.0000i 1.75623i 0.478451 + 0.878114i \(0.341198\pi\)
−0.478451 + 0.878114i \(0.658802\pi\)
\(84\) 0 0
\(85\) 5.65685i 0.613572i
\(86\) 0 0
\(87\) −1.41421 −0.151620
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) − 12.0000i − 1.25794i
\(92\) 0 0
\(93\) − 2.82843i − 0.293294i
\(94\) 0 0
\(95\) −11.3137 −1.16076
\(96\) 0 0
\(97\) 16.0000 1.62455 0.812277 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.5563i 1.54791i 0.633238 + 0.773957i \(0.281726\pi\)
−0.633238 + 0.773957i \(0.718274\pi\)
\(102\) 0 0
\(103\) −19.7990 −1.95085 −0.975426 0.220326i \(-0.929288\pi\)
−0.975426 + 0.220326i \(0.929288\pi\)
\(104\) 0 0
\(105\) 4.00000 0.390360
\(106\) 0 0
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 0 0
\(109\) − 4.24264i − 0.406371i −0.979140 0.203186i \(-0.934871\pi\)
0.979140 0.203186i \(-0.0651295\pi\)
\(110\) 0 0
\(111\) 4.24264 0.402694
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 8.00000i 0.746004i
\(116\) 0 0
\(117\) − 4.24264i − 0.392232i
\(118\) 0 0
\(119\) −11.3137 −1.03713
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 4.00000i 0.360668i
\(124\) 0 0
\(125\) 11.3137i 1.01193i
\(126\) 0 0
\(127\) 8.48528 0.752947 0.376473 0.926427i \(-0.377137\pi\)
0.376473 + 0.926427i \(0.377137\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.00000i 0.349482i 0.984614 + 0.174741i \(0.0559088\pi\)
−0.984614 + 0.174741i \(0.944091\pi\)
\(132\) 0 0
\(133\) − 22.6274i − 1.96205i
\(134\) 0 0
\(135\) 1.41421 0.121716
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) − 4.00000i − 0.339276i −0.985506 0.169638i \(-0.945740\pi\)
0.985506 0.169638i \(-0.0542598\pi\)
\(140\) 0 0
\(141\) − 11.3137i − 0.952786i
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) 1.00000i 0.0824786i
\(148\) 0 0
\(149\) − 12.7279i − 1.04271i −0.853339 0.521356i \(-0.825426\pi\)
0.853339 0.521356i \(-0.174574\pi\)
\(150\) 0 0
\(151\) 2.82843 0.230174 0.115087 0.993355i \(-0.463285\pi\)
0.115087 + 0.993355i \(0.463285\pi\)
\(152\) 0 0
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) − 4.00000i − 0.321288i
\(156\) 0 0
\(157\) − 21.2132i − 1.69300i −0.532390 0.846499i \(-0.678706\pi\)
0.532390 0.846499i \(-0.321294\pi\)
\(158\) 0 0
\(159\) −7.07107 −0.560772
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −11.3137 −0.875481 −0.437741 0.899101i \(-0.644221\pi\)
−0.437741 + 0.899101i \(0.644221\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 0 0
\(171\) − 8.00000i − 0.611775i
\(172\) 0 0
\(173\) 7.07107i 0.537603i 0.963196 + 0.268802i \(0.0866276\pi\)
−0.963196 + 0.268802i \(0.913372\pi\)
\(174\) 0 0
\(175\) −8.48528 −0.641427
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) − 4.00000i − 0.298974i −0.988764 0.149487i \(-0.952238\pi\)
0.988764 0.149487i \(-0.0477622\pi\)
\(180\) 0 0
\(181\) 12.7279i 0.946059i 0.881047 + 0.473029i \(0.156840\pi\)
−0.881047 + 0.473029i \(0.843160\pi\)
\(182\) 0 0
\(183\) −1.41421 −0.104542
\(184\) 0 0
\(185\) 6.00000 0.441129
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.82843i 0.205738i
\(190\) 0 0
\(191\) 5.65685 0.409316 0.204658 0.978834i \(-0.434392\pi\)
0.204658 + 0.978834i \(0.434392\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) − 6.00000i − 0.429669i
\(196\) 0 0
\(197\) 7.07107i 0.503793i 0.967754 + 0.251896i \(0.0810542\pi\)
−0.967754 + 0.251896i \(0.918946\pi\)
\(198\) 0 0
\(199\) 8.48528 0.601506 0.300753 0.953702i \(-0.402762\pi\)
0.300753 + 0.953702i \(0.402762\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) − 4.00000i − 0.280745i
\(204\) 0 0
\(205\) 5.65685i 0.395092i
\(206\) 0 0
\(207\) −5.65685 −0.393179
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) − 12.0000i − 0.826114i −0.910705 0.413057i \(-0.864461\pi\)
0.910705 0.413057i \(-0.135539\pi\)
\(212\) 0 0
\(213\) − 11.3137i − 0.775203i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 0 0
\(219\) − 14.0000i − 0.946032i
\(220\) 0 0
\(221\) 16.9706i 1.14156i
\(222\) 0 0
\(223\) −25.4558 −1.70465 −0.852325 0.523013i \(-0.824808\pi\)
−0.852325 + 0.523013i \(0.824808\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) 8.00000i 0.530979i 0.964114 + 0.265489i \(0.0855335\pi\)
−0.964114 + 0.265489i \(0.914466\pi\)
\(228\) 0 0
\(229\) − 24.0416i − 1.58872i −0.607450 0.794358i \(-0.707808\pi\)
0.607450 0.794358i \(-0.292192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) − 16.0000i − 1.04372i
\(236\) 0 0
\(237\) − 8.48528i − 0.551178i
\(238\) 0 0
\(239\) 5.65685 0.365911 0.182956 0.983121i \(-0.441433\pi\)
0.182956 + 0.983121i \(0.441433\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 1.41421i 0.0903508i
\(246\) 0 0
\(247\) −33.9411 −2.15962
\(248\) 0 0
\(249\) −16.0000 −1.01396
\(250\) 0 0
\(251\) 8.00000i 0.504956i 0.967603 + 0.252478i \(0.0812455\pi\)
−0.967603 + 0.252478i \(0.918755\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −5.65685 −0.354246
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) 12.0000i 0.745644i
\(260\) 0 0
\(261\) − 1.41421i − 0.0875376i
\(262\) 0 0
\(263\) 28.2843 1.74408 0.872041 0.489432i \(-0.162796\pi\)
0.872041 + 0.489432i \(0.162796\pi\)
\(264\) 0 0
\(265\) −10.0000 −0.614295
\(266\) 0 0
\(267\) − 6.00000i − 0.367194i
\(268\) 0 0
\(269\) 21.2132i 1.29339i 0.762748 + 0.646696i \(0.223850\pi\)
−0.762748 + 0.646696i \(0.776150\pi\)
\(270\) 0 0
\(271\) 8.48528 0.515444 0.257722 0.966219i \(-0.417028\pi\)
0.257722 + 0.966219i \(0.417028\pi\)
\(272\) 0 0
\(273\) 12.0000 0.726273
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 1.41421i − 0.0849719i −0.999097 0.0424859i \(-0.986472\pi\)
0.999097 0.0424859i \(-0.0135278\pi\)
\(278\) 0 0
\(279\) 2.82843 0.169334
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) − 11.3137i − 0.670166i
\(286\) 0 0
\(287\) −11.3137 −0.667827
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 16.0000i 0.937937i
\(292\) 0 0
\(293\) 18.3848i 1.07405i 0.843566 + 0.537025i \(0.180452\pi\)
−0.843566 + 0.537025i \(0.819548\pi\)
\(294\) 0 0
\(295\) 16.9706 0.988064
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 24.0000i 1.38796i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −15.5563 −0.893689
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) − 19.7990i − 1.12633i
\(310\) 0 0
\(311\) −28.2843 −1.60385 −0.801927 0.597422i \(-0.796192\pi\)
−0.801927 + 0.597422i \(0.796192\pi\)
\(312\) 0 0
\(313\) 16.0000 0.904373 0.452187 0.891923i \(-0.350644\pi\)
0.452187 + 0.891923i \(0.350644\pi\)
\(314\) 0 0
\(315\) 4.00000i 0.225374i
\(316\) 0 0
\(317\) − 21.2132i − 1.19145i −0.803188 0.595726i \(-0.796864\pi\)
0.803188 0.595726i \(-0.203136\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 32.0000i 1.78053i
\(324\) 0 0
\(325\) 12.7279i 0.706018i
\(326\) 0 0
\(327\) 4.24264 0.234619
\(328\) 0 0
\(329\) 32.0000 1.76422
\(330\) 0 0
\(331\) 4.00000i 0.219860i 0.993939 + 0.109930i \(0.0350627\pi\)
−0.993939 + 0.109930i \(0.964937\pi\)
\(332\) 0 0
\(333\) 4.24264i 0.232495i
\(334\) 0 0
\(335\) 5.65685 0.309067
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) − 18.0000i − 0.977626i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) −8.00000 −0.430706
\(346\) 0 0
\(347\) 32.0000i 1.71785i 0.512101 + 0.858925i \(0.328867\pi\)
−0.512101 + 0.858925i \(0.671133\pi\)
\(348\) 0 0
\(349\) − 12.7279i − 0.681310i −0.940188 0.340655i \(-0.889351\pi\)
0.940188 0.340655i \(-0.110649\pi\)
\(350\) 0 0
\(351\) 4.24264 0.226455
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) − 16.0000i − 0.849192i
\(356\) 0 0
\(357\) − 11.3137i − 0.598785i
\(358\) 0 0
\(359\) −11.3137 −0.597115 −0.298557 0.954392i \(-0.596505\pi\)
−0.298557 + 0.954392i \(0.596505\pi\)
\(360\) 0 0
\(361\) −45.0000 −2.36842
\(362\) 0 0
\(363\) 11.0000i 0.577350i
\(364\) 0 0
\(365\) − 19.7990i − 1.03633i
\(366\) 0 0
\(367\) −2.82843 −0.147643 −0.0738213 0.997271i \(-0.523519\pi\)
−0.0738213 + 0.997271i \(0.523519\pi\)
\(368\) 0 0
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) − 20.0000i − 1.03835i
\(372\) 0 0
\(373\) 4.24264i 0.219676i 0.993950 + 0.109838i \(0.0350331\pi\)
−0.993950 + 0.109838i \(0.964967\pi\)
\(374\) 0 0
\(375\) −11.3137 −0.584237
\(376\) 0 0
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 16.0000i 0.821865i 0.911666 + 0.410932i \(0.134797\pi\)
−0.911666 + 0.410932i \(0.865203\pi\)
\(380\) 0 0
\(381\) 8.48528i 0.434714i
\(382\) 0 0
\(383\) 28.2843 1.44526 0.722629 0.691236i \(-0.242933\pi\)
0.722629 + 0.691236i \(0.242933\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 9.89949i 0.501924i 0.967997 + 0.250962i \(0.0807470\pi\)
−0.967997 + 0.250962i \(0.919253\pi\)
\(390\) 0 0
\(391\) 22.6274 1.14432
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 0 0
\(395\) − 12.0000i − 0.603786i
\(396\) 0 0
\(397\) − 21.2132i − 1.06466i −0.846537 0.532330i \(-0.821317\pi\)
0.846537 0.532330i \(-0.178683\pi\)
\(398\) 0 0
\(399\) 22.6274 1.13279
\(400\) 0 0
\(401\) −20.0000 −0.998752 −0.499376 0.866385i \(-0.666437\pi\)
−0.499376 + 0.866385i \(0.666437\pi\)
\(402\) 0 0
\(403\) − 12.0000i − 0.597763i
\(404\) 0 0
\(405\) 1.41421i 0.0702728i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −32.0000 −1.58230 −0.791149 0.611623i \(-0.790517\pi\)
−0.791149 + 0.611623i \(0.790517\pi\)
\(410\) 0 0
\(411\) 12.0000i 0.591916i
\(412\) 0 0
\(413\) 33.9411i 1.67013i
\(414\) 0 0
\(415\) −22.6274 −1.11074
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) 8.00000i 0.390826i 0.980721 + 0.195413i \(0.0626047\pi\)
−0.980721 + 0.195413i \(0.937395\pi\)
\(420\) 0 0
\(421\) − 12.7279i − 0.620321i −0.950684 0.310160i \(-0.899617\pi\)
0.950684 0.310160i \(-0.100383\pi\)
\(422\) 0 0
\(423\) 11.3137 0.550091
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) − 4.00000i − 0.193574i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 11.3137 0.544962 0.272481 0.962161i \(-0.412156\pi\)
0.272481 + 0.962161i \(0.412156\pi\)
\(432\) 0 0
\(433\) 8.00000 0.384455 0.192228 0.981350i \(-0.438429\pi\)
0.192228 + 0.981350i \(0.438429\pi\)
\(434\) 0 0
\(435\) − 2.00000i − 0.0958927i
\(436\) 0 0
\(437\) 45.2548i 2.16483i
\(438\) 0 0
\(439\) −2.82843 −0.134993 −0.0674967 0.997719i \(-0.521501\pi\)
−0.0674967 + 0.997719i \(0.521501\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) − 24.0000i − 1.14027i −0.821549 0.570137i \(-0.806890\pi\)
0.821549 0.570137i \(-0.193110\pi\)
\(444\) 0 0
\(445\) − 8.48528i − 0.402241i
\(446\) 0 0
\(447\) 12.7279 0.602010
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 2.82843i 0.132891i
\(454\) 0 0
\(455\) 16.9706 0.795592
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) − 4.00000i − 0.186704i
\(460\) 0 0
\(461\) 15.5563i 0.724531i 0.932075 + 0.362266i \(0.117997\pi\)
−0.932075 + 0.362266i \(0.882003\pi\)
\(462\) 0 0
\(463\) 31.1127 1.44593 0.722965 0.690885i \(-0.242779\pi\)
0.722965 + 0.690885i \(0.242779\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) 0 0
\(467\) − 8.00000i − 0.370196i −0.982720 0.185098i \(-0.940740\pi\)
0.982720 0.185098i \(-0.0592602\pi\)
\(468\) 0 0
\(469\) 11.3137i 0.522419i
\(470\) 0 0
\(471\) 21.2132 0.977453
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 24.0000i 1.10120i
\(476\) 0 0
\(477\) − 7.07107i − 0.323762i
\(478\) 0 0
\(479\) −28.2843 −1.29234 −0.646171 0.763193i \(-0.723631\pi\)
−0.646171 + 0.763193i \(0.723631\pi\)
\(480\) 0 0
\(481\) 18.0000 0.820729
\(482\) 0 0
\(483\) − 16.0000i − 0.728025i
\(484\) 0 0
\(485\) 22.6274i 1.02746i
\(486\) 0 0
\(487\) 2.82843 0.128168 0.0640841 0.997944i \(-0.479587\pi\)
0.0640841 + 0.997944i \(0.479587\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) − 12.0000i − 0.541552i −0.962642 0.270776i \(-0.912720\pi\)
0.962642 0.270776i \(-0.0872803\pi\)
\(492\) 0 0
\(493\) 5.65685i 0.254772i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 32.0000 1.43540
\(498\) 0 0
\(499\) − 28.0000i − 1.25345i −0.779240 0.626726i \(-0.784395\pi\)
0.779240 0.626726i \(-0.215605\pi\)
\(500\) 0 0
\(501\) − 11.3137i − 0.505459i
\(502\) 0 0
\(503\) 11.3137 0.504453 0.252227 0.967668i \(-0.418837\pi\)
0.252227 + 0.967668i \(0.418837\pi\)
\(504\) 0 0
\(505\) −22.0000 −0.978987
\(506\) 0 0
\(507\) − 5.00000i − 0.222058i
\(508\) 0 0
\(509\) 32.5269i 1.44173i 0.693075 + 0.720865i \(0.256255\pi\)
−0.693075 + 0.720865i \(0.743745\pi\)
\(510\) 0 0
\(511\) 39.5980 1.75171
\(512\) 0 0
\(513\) 8.00000 0.353209
\(514\) 0 0
\(515\) − 28.0000i − 1.23383i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −7.07107 −0.310385
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) 8.00000i 0.349816i 0.984585 + 0.174908i \(0.0559627\pi\)
−0.984585 + 0.174908i \(0.944037\pi\)
\(524\) 0 0
\(525\) − 8.48528i − 0.370328i
\(526\) 0 0
\(527\) −11.3137 −0.492833
\(528\) 0 0
\(529\) 9.00000 0.391304
\(530\) 0 0
\(531\) 12.0000i 0.520756i
\(532\) 0 0
\(533\) 16.9706i 0.735077i
\(534\) 0 0
\(535\) 16.9706 0.733701
\(536\) 0 0
\(537\) 4.00000 0.172613
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) − 4.24264i − 0.182405i −0.995832 0.0912027i \(-0.970929\pi\)
0.995832 0.0912027i \(-0.0290711\pi\)
\(542\) 0 0
\(543\) −12.7279 −0.546207
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) 0 0
\(549\) − 1.41421i − 0.0603572i
\(550\) 0 0
\(551\) −11.3137 −0.481980
\(552\) 0 0
\(553\) 24.0000 1.02058
\(554\) 0 0
\(555\) 6.00000i 0.254686i
\(556\) 0 0
\(557\) 29.6985i 1.25837i 0.777258 + 0.629183i \(0.216610\pi\)
−0.777258 + 0.629183i \(0.783390\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.00000i 0.337160i 0.985688 + 0.168580i \(0.0539181\pi\)
−0.985688 + 0.168580i \(0.946082\pi\)
\(564\) 0 0
\(565\) − 25.4558i − 1.07094i
\(566\) 0 0
\(567\) −2.82843 −0.118783
\(568\) 0 0
\(569\) 20.0000 0.838444 0.419222 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) − 20.0000i − 0.836974i −0.908223 0.418487i \(-0.862561\pi\)
0.908223 0.418487i \(-0.137439\pi\)
\(572\) 0 0
\(573\) 5.65685i 0.236318i
\(574\) 0 0
\(575\) 16.9706 0.707721
\(576\) 0 0
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 45.2548i − 1.87749i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 6.00000 0.248069
\(586\) 0 0
\(587\) 4.00000i 0.165098i 0.996587 + 0.0825488i \(0.0263060\pi\)
−0.996587 + 0.0825488i \(0.973694\pi\)
\(588\) 0 0
\(589\) − 22.6274i − 0.932346i
\(590\) 0 0
\(591\) −7.07107 −0.290865
\(592\) 0 0
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) − 16.0000i − 0.655936i
\(596\) 0 0
\(597\) 8.48528i 0.347279i
\(598\) 0 0
\(599\) 39.5980 1.61793 0.808965 0.587857i \(-0.200028\pi\)
0.808965 + 0.587857i \(0.200028\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) 4.00000i 0.162893i
\(604\) 0 0
\(605\) 15.5563i 0.632456i
\(606\) 0 0
\(607\) 42.4264 1.72203 0.861017 0.508576i \(-0.169828\pi\)
0.861017 + 0.508576i \(0.169828\pi\)
\(608\) 0 0
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) − 48.0000i − 1.94187i
\(612\) 0 0
\(613\) 15.5563i 0.628315i 0.949371 + 0.314158i \(0.101722\pi\)
−0.949371 + 0.314158i \(0.898278\pi\)
\(614\) 0 0
\(615\) −5.65685 −0.228106
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 44.0000i 1.76851i 0.467005 + 0.884255i \(0.345333\pi\)
−0.467005 + 0.884255i \(0.654667\pi\)
\(620\) 0 0
\(621\) − 5.65685i − 0.227002i
\(622\) 0 0
\(623\) 16.9706 0.679911
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 16.9706i − 0.676661i
\(630\) 0 0
\(631\) −8.48528 −0.337794 −0.168897 0.985634i \(-0.554020\pi\)
−0.168897 + 0.985634i \(0.554020\pi\)
\(632\) 0 0
\(633\) 12.0000 0.476957
\(634\) 0 0
\(635\) 12.0000i 0.476205i
\(636\) 0 0
\(637\) 4.24264i 0.168100i
\(638\) 0 0
\(639\) 11.3137 0.447563
\(640\) 0 0
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 0 0
\(643\) − 16.0000i − 0.630978i −0.948929 0.315489i \(-0.897831\pi\)
0.948929 0.315489i \(-0.102169\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 39.5980 1.55676 0.778379 0.627795i \(-0.216042\pi\)
0.778379 + 0.627795i \(0.216042\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 8.00000i 0.313545i
\(652\) 0 0
\(653\) 4.24264i 0.166027i 0.996548 + 0.0830137i \(0.0264545\pi\)
−0.996548 + 0.0830137i \(0.973545\pi\)
\(654\) 0 0
\(655\) −5.65685 −0.221032
\(656\) 0 0
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) − 20.0000i − 0.779089i −0.921008 0.389545i \(-0.872632\pi\)
0.921008 0.389545i \(-0.127368\pi\)
\(660\) 0 0
\(661\) − 26.8701i − 1.04512i −0.852601 0.522562i \(-0.824976\pi\)
0.852601 0.522562i \(-0.175024\pi\)
\(662\) 0 0
\(663\) −16.9706 −0.659082
\(664\) 0 0
\(665\) 32.0000 1.24091
\(666\) 0 0
\(667\) 8.00000i 0.309761i
\(668\) 0 0
\(669\) − 25.4558i − 0.984180i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −8.00000 −0.308377 −0.154189 0.988041i \(-0.549276\pi\)
−0.154189 + 0.988041i \(0.549276\pi\)
\(674\) 0 0
\(675\) − 3.00000i − 0.115470i
\(676\) 0 0
\(677\) 32.5269i 1.25011i 0.780580 + 0.625055i \(0.214924\pi\)
−0.780580 + 0.625055i \(0.785076\pi\)
\(678\) 0 0
\(679\) −45.2548 −1.73672
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) 0 0
\(683\) 32.0000i 1.22445i 0.790685 + 0.612223i \(0.209725\pi\)
−0.790685 + 0.612223i \(0.790275\pi\)
\(684\) 0 0
\(685\) 16.9706i 0.648412i
\(686\) 0 0
\(687\) 24.0416 0.917245
\(688\) 0 0
\(689\) −30.0000 −1.14291
\(690\) 0 0
\(691\) 40.0000i 1.52167i 0.648944 + 0.760836i \(0.275211\pi\)
−0.648944 + 0.760836i \(0.724789\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.65685 0.214577
\(696\) 0 0
\(697\) 16.0000 0.606043
\(698\) 0 0
\(699\) 10.0000i 0.378235i
\(700\) 0 0
\(701\) − 46.6690i − 1.76267i −0.472496 0.881333i \(-0.656647\pi\)
0.472496 0.881333i \(-0.343353\pi\)
\(702\) 0 0
\(703\) 33.9411 1.28011
\(704\) 0 0
\(705\) 16.0000 0.602595
\(706\) 0 0
\(707\) − 44.0000i − 1.65479i
\(708\) 0 0
\(709\) 35.3553i 1.32780i 0.747822 + 0.663899i \(0.231099\pi\)
−0.747822 + 0.663899i \(0.768901\pi\)
\(710\) 0 0
\(711\) 8.48528 0.318223
\(712\) 0 0
\(713\) −16.0000 −0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 5.65685i 0.211259i
\(718\) 0 0
\(719\) 16.9706 0.632895 0.316448 0.948610i \(-0.397510\pi\)
0.316448 + 0.948610i \(0.397510\pi\)
\(720\) 0 0
\(721\) 56.0000 2.08555
\(722\) 0 0
\(723\) 8.00000i 0.297523i
\(724\) 0 0
\(725\) 4.24264i 0.157568i
\(726\) 0 0
\(727\) 42.4264 1.57351 0.786754 0.617266i \(-0.211760\pi\)
0.786754 + 0.617266i \(0.211760\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 4.24264i 0.156706i 0.996926 + 0.0783528i \(0.0249660\pi\)
−0.996926 + 0.0783528i \(0.975034\pi\)
\(734\) 0 0
\(735\) −1.41421 −0.0521641
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 36.0000i − 1.32428i −0.749380 0.662141i \(-0.769648\pi\)
0.749380 0.662141i \(-0.230352\pi\)
\(740\) 0 0
\(741\) − 33.9411i − 1.24686i
\(742\) 0 0
\(743\) 11.3137 0.415060 0.207530 0.978229i \(-0.433458\pi\)
0.207530 + 0.978229i \(0.433458\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 0 0
\(747\) − 16.0000i − 0.585409i
\(748\) 0 0
\(749\) 33.9411i 1.24018i
\(750\) 0 0
\(751\) 8.48528 0.309632 0.154816 0.987943i \(-0.450521\pi\)
0.154816 + 0.987943i \(0.450521\pi\)
\(752\) 0 0
\(753\) −8.00000 −0.291536
\(754\) 0 0
\(755\) 4.00000i 0.145575i
\(756\) 0 0
\(757\) − 21.2132i − 0.771007i −0.922706 0.385503i \(-0.874028\pi\)
0.922706 0.385503i \(-0.125972\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −36.0000 −1.30500 −0.652499 0.757789i \(-0.726280\pi\)
−0.652499 + 0.757789i \(0.726280\pi\)
\(762\) 0 0
\(763\) 12.0000i 0.434429i
\(764\) 0 0
\(765\) − 5.65685i − 0.204524i
\(766\) 0 0
\(767\) 50.9117 1.83831
\(768\) 0 0
\(769\) −32.0000 −1.15395 −0.576975 0.816762i \(-0.695767\pi\)
−0.576975 + 0.816762i \(0.695767\pi\)
\(770\) 0 0
\(771\) 2.00000i 0.0720282i
\(772\) 0 0
\(773\) − 18.3848i − 0.661254i −0.943761 0.330627i \(-0.892740\pi\)
0.943761 0.330627i \(-0.107260\pi\)
\(774\) 0 0
\(775\) −8.48528 −0.304800
\(776\) 0 0
\(777\) −12.0000 −0.430498
\(778\) 0 0
\(779\) 32.0000i 1.14652i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 1.41421 0.0505399
\(784\) 0 0
\(785\) 30.0000 1.07075
\(786\) 0 0
\(787\) 32.0000i 1.14068i 0.821410 + 0.570338i \(0.193188\pi\)
−0.821410 + 0.570338i \(0.806812\pi\)
\(788\) 0 0
\(789\) 28.2843i 1.00695i
\(790\) 0 0
\(791\) 50.9117 1.81021
\(792\) 0 0
\(793\) −6.00000 −0.213066
\(794\) 0 0
\(795\) − 10.0000i − 0.354663i
\(796\) 0 0
\(797\) 49.4975i 1.75329i 0.481137 + 0.876645i \(0.340224\pi\)
−0.481137 + 0.876645i \(0.659776\pi\)
\(798\) 0 0
\(799\) −45.2548 −1.60100
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) − 22.6274i − 0.797512i
\(806\) 0 0
\(807\) −21.2132 −0.746740
\(808\) 0 0
\(809\) −4.00000 −0.140633 −0.0703163 0.997525i \(-0.522401\pi\)
−0.0703163 + 0.997525i \(0.522401\pi\)
\(810\) 0 0
\(811\) 40.0000i 1.40459i 0.711886 + 0.702295i \(0.247841\pi\)
−0.711886 + 0.702295i \(0.752159\pi\)
\(812\) 0 0
\(813\) 8.48528i 0.297592i
\(814\) 0 0
\(815\) −22.6274 −0.792604
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 12.0000i 0.419314i
\(820\) 0 0
\(821\) 7.07107i 0.246782i 0.992358 + 0.123391i \(0.0393769\pi\)
−0.992358 + 0.123391i \(0.960623\pi\)
\(822\) 0 0
\(823\) 25.4558 0.887335 0.443667 0.896191i \(-0.353677\pi\)
0.443667 + 0.896191i \(0.353677\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 20.0000i 0.695468i 0.937593 + 0.347734i \(0.113049\pi\)
−0.937593 + 0.347734i \(0.886951\pi\)
\(828\) 0 0
\(829\) 52.3259i 1.81735i 0.417500 + 0.908677i \(0.362906\pi\)
−0.417500 + 0.908677i \(0.637094\pi\)
\(830\) 0 0
\(831\) 1.41421 0.0490585
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) − 16.0000i − 0.553703i
\(836\) 0 0
\(837\) 2.82843i 0.0977647i
\(838\) 0 0
\(839\) −45.2548 −1.56237 −0.781185 0.624299i \(-0.785385\pi\)
−0.781185 + 0.624299i \(0.785385\pi\)
\(840\) 0 0
\(841\) 27.0000 0.931034
\(842\) 0 0
\(843\) 22.0000i 0.757720i
\(844\) 0 0
\(845\) − 7.07107i − 0.243252i
\(846\) 0 0
\(847\) −31.1127 −1.06904
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) − 24.0000i − 0.822709i
\(852\) 0 0
\(853\) 15.5563i 0.532639i 0.963885 + 0.266320i \(0.0858077\pi\)
−0.963885 + 0.266320i \(0.914192\pi\)
\(854\) 0 0
\(855\) 11.3137 0.386921
\(856\) 0 0
\(857\) 20.0000 0.683187 0.341593 0.939848i \(-0.389033\pi\)
0.341593 + 0.939848i \(0.389033\pi\)
\(858\) 0 0
\(859\) − 8.00000i − 0.272956i −0.990643 0.136478i \(-0.956422\pi\)
0.990643 0.136478i \(-0.0435784\pi\)
\(860\) 0 0
\(861\) − 11.3137i − 0.385570i
\(862\) 0 0
\(863\) −5.65685 −0.192562 −0.0962808 0.995354i \(-0.530695\pi\)
−0.0962808 + 0.995354i \(0.530695\pi\)
\(864\) 0 0
\(865\) −10.0000 −0.340010
\(866\) 0 0
\(867\) − 1.00000i − 0.0339618i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 16.9706 0.575026
\(872\) 0 0
\(873\) −16.0000 −0.541518
\(874\) 0 0
\(875\) − 32.0000i − 1.08180i
\(876\) 0 0
\(877\) − 32.5269i − 1.09836i −0.835705 0.549178i \(-0.814941\pi\)
0.835705 0.549178i \(-0.185059\pi\)
\(878\) 0 0
\(879\) −18.3848 −0.620103
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) − 24.0000i − 0.807664i −0.914833 0.403832i \(-0.867678\pi\)
0.914833 0.403832i \(-0.132322\pi\)
\(884\) 0 0
\(885\) 16.9706i 0.570459i
\(886\) 0 0
\(887\) 16.9706 0.569816 0.284908 0.958555i \(-0.408037\pi\)
0.284908 + 0.958555i \(0.408037\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 90.5097i − 3.02879i
\(894\) 0 0
\(895\) 5.65685 0.189088
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) 0 0
\(899\) − 4.00000i − 0.133407i
\(900\) 0 0
\(901\) 28.2843i 0.942286i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −18.0000 −0.598340
\(906\) 0 0
\(907\) − 32.0000i − 1.06254i −0.847202 0.531271i \(-0.821714\pi\)
0.847202 0.531271i \(-0.178286\pi\)
\(908\) 0 0
\(909\) − 15.5563i − 0.515972i
\(910\) 0 0
\(911\) 5.65685 0.187420 0.0937100 0.995600i \(-0.470127\pi\)
0.0937100 + 0.995600i \(0.470127\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) − 2.00000i − 0.0661180i
\(916\) 0 0
\(917\) − 11.3137i − 0.373612i
\(918\) 0 0
\(919\) 2.82843 0.0933012 0.0466506 0.998911i \(-0.485145\pi\)
0.0466506 + 0.998911i \(0.485145\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 0 0
\(923\) − 48.0000i − 1.57994i
\(924\) 0 0
\(925\) − 12.7279i − 0.418491i
\(926\) 0 0
\(927\) 19.7990 0.650284
\(928\) 0 0
\(929\) −20.0000 −0.656179 −0.328089 0.944647i \(-0.606405\pi\)
−0.328089 + 0.944647i \(0.606405\pi\)
\(930\) 0 0
\(931\) 8.00000i 0.262189i
\(932\) 0 0
\(933\) − 28.2843i − 0.925985i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 42.0000 1.37208 0.686040 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 0 0
\(939\) 16.0000i 0.522140i
\(940\) 0 0
\(941\) − 52.3259i − 1.70578i −0.522094 0.852888i \(-0.674849\pi\)
0.522094 0.852888i \(-0.325151\pi\)
\(942\) 0 0
\(943\) 22.6274 0.736850
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) 52.0000i 1.68977i 0.534946 + 0.844886i \(0.320332\pi\)
−0.534946 + 0.844886i \(0.679668\pi\)
\(948\) 0 0
\(949\) − 59.3970i − 1.92811i
\(950\) 0 0
\(951\) 21.2132 0.687885
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) 8.00000i 0.258874i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −33.9411 −1.09602
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 0 0
\(963\) 12.0000i 0.386695i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 42.4264 1.36434 0.682171 0.731193i \(-0.261036\pi\)
0.682171 + 0.731193i \(0.261036\pi\)
\(968\) 0 0
\(969\) −32.0000 −1.02799
\(970\) 0 0
\(971\) 24.0000i 0.770197i 0.922876 + 0.385098i \(0.125832\pi\)
−0.922876 + 0.385098i \(0.874168\pi\)
\(972\) 0 0
\(973\) 11.3137i 0.362701i
\(974\) 0 0
\(975\) −12.7279 −0.407620
\(976\) 0 0
\(977\) −36.0000 −1.15174 −0.575871 0.817541i \(-0.695337\pi\)
−0.575871 + 0.817541i \(0.695337\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 4.24264i 0.135457i
\(982\) 0 0
\(983\) 16.9706 0.541277 0.270638 0.962681i \(-0.412765\pi\)
0.270638 + 0.962681i \(0.412765\pi\)
\(984\) 0 0
\(985\) −10.0000 −0.318626
\(986\) 0 0
\(987\) 32.0000i 1.01857i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −25.4558 −0.808632 −0.404316 0.914619i \(-0.632490\pi\)
−0.404316 + 0.914619i \(0.632490\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) 12.0000i 0.380426i
\(996\) 0 0
\(997\) 7.07107i 0.223943i 0.993711 + 0.111971i \(0.0357165\pi\)
−0.993711 + 0.111971i \(0.964283\pi\)
\(998\) 0 0
\(999\) −4.24264 −0.134231
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 3072.2.d.c.1537.4 4
4.3 odd 2 inner 3072.2.d.c.1537.2 4
8.3 odd 2 inner 3072.2.d.c.1537.3 4
8.5 even 2 inner 3072.2.d.c.1537.1 4
16.3 odd 4 3072.2.a.h.1.2 2
16.5 even 4 3072.2.a.h.1.1 2
16.11 odd 4 3072.2.a.b.1.1 2
16.13 even 4 3072.2.a.b.1.2 2
32.3 odd 8 1536.2.j.b.385.2 yes 4
32.5 even 8 1536.2.j.b.1153.1 yes 4
32.11 odd 8 1536.2.j.c.1153.1 yes 4
32.13 even 8 1536.2.j.c.385.2 yes 4
32.19 odd 8 1536.2.j.c.385.1 yes 4
32.21 even 8 1536.2.j.c.1153.2 yes 4
32.27 odd 8 1536.2.j.b.1153.2 yes 4
32.29 even 8 1536.2.j.b.385.1 4
48.5 odd 4 9216.2.a.i.1.2 2
48.11 even 4 9216.2.a.h.1.2 2
48.29 odd 4 9216.2.a.h.1.1 2
48.35 even 4 9216.2.a.i.1.1 2
96.5 odd 8 4608.2.k.bb.1153.1 4
96.11 even 8 4608.2.k.y.1153.2 4
96.29 odd 8 4608.2.k.bb.3457.2 4
96.35 even 8 4608.2.k.bb.3457.1 4
96.53 odd 8 4608.2.k.y.1153.1 4
96.59 even 8 4608.2.k.bb.1153.2 4
96.77 odd 8 4608.2.k.y.3457.2 4
96.83 even 8 4608.2.k.y.3457.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.j.b.385.1 4 32.29 even 8
1536.2.j.b.385.2 yes 4 32.3 odd 8
1536.2.j.b.1153.1 yes 4 32.5 even 8
1536.2.j.b.1153.2 yes 4 32.27 odd 8
1536.2.j.c.385.1 yes 4 32.19 odd 8
1536.2.j.c.385.2 yes 4 32.13 even 8
1536.2.j.c.1153.1 yes 4 32.11 odd 8
1536.2.j.c.1153.2 yes 4 32.21 even 8
3072.2.a.b.1.1 2 16.11 odd 4
3072.2.a.b.1.2 2 16.13 even 4
3072.2.a.h.1.1 2 16.5 even 4
3072.2.a.h.1.2 2 16.3 odd 4
3072.2.d.c.1537.1 4 8.5 even 2 inner
3072.2.d.c.1537.2 4 4.3 odd 2 inner
3072.2.d.c.1537.3 4 8.3 odd 2 inner
3072.2.d.c.1537.4 4 1.1 even 1 trivial
4608.2.k.y.1153.1 4 96.53 odd 8
4608.2.k.y.1153.2 4 96.11 even 8
4608.2.k.y.3457.1 4 96.83 even 8
4608.2.k.y.3457.2 4 96.77 odd 8
4608.2.k.bb.1153.1 4 96.5 odd 8
4608.2.k.bb.1153.2 4 96.59 even 8
4608.2.k.bb.3457.1 4 96.35 even 8
4608.2.k.bb.3457.2 4 96.29 odd 8
9216.2.a.h.1.1 2 48.29 odd 4
9216.2.a.h.1.2 2 48.11 even 4
9216.2.a.i.1.1 2 48.35 even 4
9216.2.a.i.1.2 2 48.5 odd 4