Properties

Label 3072.2.d.i.1537.1
Level $3072$
Weight $2$
Character 3072.1537
Analytic conductor $24.530$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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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: \(8\)
Coefficient field: 8.0.18939904.2
Defining polynomial: \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1537.1
Root \(0.500000 - 0.691860i\) of defining polynomial
Character \(\chi\) \(=\) 3072.1537
Dual form 3072.2.d.i.1537.8

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{3} -3.79793i q^{5} +2.15894 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -3.79793i q^{5} +2.15894 q^{7} -1.00000 q^{9} +2.54266i q^{11} -1.95687i q^{13} -3.79793 q^{15} +0.224777 q^{17} +0.224777i q^{19} -2.15894i q^{21} -2.82843 q^{23} -9.42429 q^{25} +1.00000i q^{27} -2.62636i q^{29} -1.84106 q^{31} +2.54266 q^{33} -8.19951i q^{35} -5.18944i q^{37} -1.95687 q^{39} -5.88163 q^{41} -10.9670i q^{43} +3.79793i q^{45} -2.82843 q^{47} -2.33897 q^{49} -0.224777i q^{51} -10.6264i q^{53} +9.65685 q^{55} +0.224777 q^{57} -5.65685i q^{59} +8.46742i q^{61} -2.15894 q^{63} -7.43208 q^{65} +14.7422i q^{67} +2.82843i q^{69} -4.31788 q^{71} -5.97474 q^{73} +9.42429i q^{75} +5.48946i q^{77} +15.0075 q^{79} +1.00000 q^{81} -14.3059i q^{83} -0.853690i q^{85} -2.62636 q^{87} +1.42847 q^{89} -4.22478i q^{91} +1.84106i q^{93} +0.853690 q^{95} -16.3990 q^{97} -2.54266i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{7} - 8q^{9} + O(q^{10}) \) \( 8q + 8q^{7} - 8q^{9} - 8q^{15} - 8q^{25} - 24q^{31} + 16q^{39} + 8q^{49} + 32q^{55} - 8q^{63} - 16q^{65} - 16q^{71} + 16q^{73} - 24q^{79} + 8q^{81} + 24q^{87} + 16q^{89} + 48q^{95} + O(q^{100}) \)

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
\(5\) − 3.79793i − 1.69849i −0.528001 0.849244i \(-0.677058\pi\)
0.528001 0.849244i \(-0.322942\pi\)
\(6\) 0 0
\(7\) 2.15894 0.816003 0.408002 0.912981i \(-0.366226\pi\)
0.408002 + 0.912981i \(0.366226\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.54266i 0.766641i 0.923615 + 0.383321i \(0.125220\pi\)
−0.923615 + 0.383321i \(0.874780\pi\)
\(12\) 0 0
\(13\) − 1.95687i − 0.542739i −0.962475 0.271370i \(-0.912523\pi\)
0.962475 0.271370i \(-0.0874766\pi\)
\(14\) 0 0
\(15\) −3.79793 −0.980622
\(16\) 0 0
\(17\) 0.224777 0.0545165 0.0272583 0.999628i \(-0.491322\pi\)
0.0272583 + 0.999628i \(0.491322\pi\)
\(18\) 0 0
\(19\) 0.224777i 0.0515675i 0.999668 + 0.0257837i \(0.00820813\pi\)
−0.999668 + 0.0257837i \(0.991792\pi\)
\(20\) 0 0
\(21\) − 2.15894i − 0.471120i
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) −9.42429 −1.88486
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) − 2.62636i − 0.487703i −0.969813 0.243851i \(-0.921589\pi\)
0.969813 0.243851i \(-0.0784109\pi\)
\(30\) 0 0
\(31\) −1.84106 −0.330664 −0.165332 0.986238i \(-0.552870\pi\)
−0.165332 + 0.986238i \(0.552870\pi\)
\(32\) 0 0
\(33\) 2.54266 0.442620
\(34\) 0 0
\(35\) − 8.19951i − 1.38597i
\(36\) 0 0
\(37\) − 5.18944i − 0.853138i −0.904455 0.426569i \(-0.859722\pi\)
0.904455 0.426569i \(-0.140278\pi\)
\(38\) 0 0
\(39\) −1.95687 −0.313351
\(40\) 0 0
\(41\) −5.88163 −0.918557 −0.459278 0.888292i \(-0.651892\pi\)
−0.459278 + 0.888292i \(0.651892\pi\)
\(42\) 0 0
\(43\) − 10.9670i − 1.67244i −0.548391 0.836222i \(-0.684759\pi\)
0.548391 0.836222i \(-0.315241\pi\)
\(44\) 0 0
\(45\) 3.79793i 0.566162i
\(46\) 0 0
\(47\) −2.82843 −0.412568 −0.206284 0.978492i \(-0.566137\pi\)
−0.206284 + 0.978492i \(0.566137\pi\)
\(48\) 0 0
\(49\) −2.33897 −0.334139
\(50\) 0 0
\(51\) − 0.224777i − 0.0314751i
\(52\) 0 0
\(53\) − 10.6264i − 1.45964i −0.683638 0.729821i \(-0.739603\pi\)
0.683638 0.729821i \(-0.260397\pi\)
\(54\) 0 0
\(55\) 9.65685 1.30213
\(56\) 0 0
\(57\) 0.224777 0.0297725
\(58\) 0 0
\(59\) − 5.65685i − 0.736460i −0.929735 0.368230i \(-0.879964\pi\)
0.929735 0.368230i \(-0.120036\pi\)
\(60\) 0 0
\(61\) 8.46742i 1.08414i 0.840333 + 0.542071i \(0.182360\pi\)
−0.840333 + 0.542071i \(0.817640\pi\)
\(62\) 0 0
\(63\) −2.15894 −0.272001
\(64\) 0 0
\(65\) −7.43208 −0.921836
\(66\) 0 0
\(67\) 14.7422i 1.80104i 0.434811 + 0.900522i \(0.356815\pi\)
−0.434811 + 0.900522i \(0.643185\pi\)
\(68\) 0 0
\(69\) 2.82843i 0.340503i
\(70\) 0 0
\(71\) −4.31788 −0.512438 −0.256219 0.966619i \(-0.582477\pi\)
−0.256219 + 0.966619i \(0.582477\pi\)
\(72\) 0 0
\(73\) −5.97474 −0.699290 −0.349645 0.936882i \(-0.613698\pi\)
−0.349645 + 0.936882i \(0.613698\pi\)
\(74\) 0 0
\(75\) 9.42429i 1.08822i
\(76\) 0 0
\(77\) 5.48946i 0.625582i
\(78\) 0 0
\(79\) 15.0075 1.68848 0.844239 0.535966i \(-0.180053\pi\)
0.844239 + 0.535966i \(0.180053\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 14.3059i − 1.57028i −0.619319 0.785140i \(-0.712591\pi\)
0.619319 0.785140i \(-0.287409\pi\)
\(84\) 0 0
\(85\) − 0.853690i − 0.0925956i
\(86\) 0 0
\(87\) −2.62636 −0.281575
\(88\) 0 0
\(89\) 1.42847 0.151417 0.0757086 0.997130i \(-0.475878\pi\)
0.0757086 + 0.997130i \(0.475878\pi\)
\(90\) 0 0
\(91\) − 4.22478i − 0.442877i
\(92\) 0 0
\(93\) 1.84106i 0.190909i
\(94\) 0 0
\(95\) 0.853690 0.0875867
\(96\) 0 0
\(97\) −16.3990 −1.66507 −0.832535 0.553973i \(-0.813111\pi\)
−0.832535 + 0.553973i \(0.813111\pi\)
\(98\) 0 0
\(99\) − 2.54266i − 0.255547i
\(100\) 0 0
\(101\) 0.115816i 0.0115241i 0.999983 + 0.00576206i \(0.00183413\pi\)
−0.999983 + 0.00576206i \(0.998166\pi\)
\(102\) 0 0
\(103\) 13.3507 1.31548 0.657740 0.753245i \(-0.271512\pi\)
0.657740 + 0.753245i \(0.271512\pi\)
\(104\) 0 0
\(105\) −8.19951 −0.800191
\(106\) 0 0
\(107\) − 10.2926i − 0.995025i −0.867457 0.497513i \(-0.834247\pi\)
0.867457 0.497513i \(-0.165753\pi\)
\(108\) 0 0
\(109\) 9.95687i 0.953696i 0.878986 + 0.476848i \(0.158221\pi\)
−0.878986 + 0.476848i \(0.841779\pi\)
\(110\) 0 0
\(111\) −5.18944 −0.492559
\(112\) 0 0
\(113\) −18.8486 −1.77313 −0.886563 0.462608i \(-0.846914\pi\)
−0.886563 + 0.462608i \(0.846914\pi\)
\(114\) 0 0
\(115\) 10.7422i 1.00171i
\(116\) 0 0
\(117\) 1.95687i 0.180913i
\(118\) 0 0
\(119\) 0.485281 0.0444857
\(120\) 0 0
\(121\) 4.53488 0.412261
\(122\) 0 0
\(123\) 5.88163i 0.530329i
\(124\) 0 0
\(125\) 16.8032i 1.50292i
\(126\) 0 0
\(127\) 3.81580 0.338597 0.169299 0.985565i \(-0.445850\pi\)
0.169299 + 0.985565i \(0.445850\pi\)
\(128\) 0 0
\(129\) −10.9670 −0.965586
\(130\) 0 0
\(131\) 1.08532i 0.0948250i 0.998875 + 0.0474125i \(0.0150975\pi\)
−0.998875 + 0.0474125i \(0.984902\pi\)
\(132\) 0 0
\(133\) 0.485281i 0.0420792i
\(134\) 0 0
\(135\) 3.79793 0.326874
\(136\) 0 0
\(137\) 5.31010 0.453672 0.226836 0.973933i \(-0.427162\pi\)
0.226836 + 0.973933i \(0.427162\pi\)
\(138\) 0 0
\(139\) 12.3990i 1.05167i 0.850586 + 0.525836i \(0.176247\pi\)
−0.850586 + 0.525836i \(0.823753\pi\)
\(140\) 0 0
\(141\) 2.82843i 0.238197i
\(142\) 0 0
\(143\) 4.97567 0.416086
\(144\) 0 0
\(145\) −9.97474 −0.828357
\(146\) 0 0
\(147\) 2.33897i 0.192915i
\(148\) 0 0
\(149\) 1.45479i 0.119181i 0.998223 + 0.0595904i \(0.0189795\pi\)
−0.998223 + 0.0595904i \(0.981021\pi\)
\(150\) 0 0
\(151\) 2.03696 0.165766 0.0828829 0.996559i \(-0.473587\pi\)
0.0828829 + 0.996559i \(0.473587\pi\)
\(152\) 0 0
\(153\) −0.224777 −0.0181722
\(154\) 0 0
\(155\) 6.99222i 0.561628i
\(156\) 0 0
\(157\) 8.61790i 0.687784i 0.939009 + 0.343892i \(0.111745\pi\)
−0.939009 + 0.343892i \(0.888255\pi\)
\(158\) 0 0
\(159\) −10.6264 −0.842725
\(160\) 0 0
\(161\) −6.10641 −0.481252
\(162\) 0 0
\(163\) − 4.86054i − 0.380707i −0.981716 0.190354i \(-0.939037\pi\)
0.981716 0.190354i \(-0.0609634\pi\)
\(164\) 0 0
\(165\) − 9.65685i − 0.751785i
\(166\) 0 0
\(167\) −21.7023 −1.67937 −0.839686 0.543072i \(-0.817261\pi\)
−0.839686 + 0.543072i \(0.817261\pi\)
\(168\) 0 0
\(169\) 9.17064 0.705434
\(170\) 0 0
\(171\) − 0.224777i − 0.0171892i
\(172\) 0 0
\(173\) 12.3695i 0.940433i 0.882551 + 0.470217i \(0.155824\pi\)
−0.882551 + 0.470217i \(0.844176\pi\)
\(174\) 0 0
\(175\) −20.3465 −1.53805
\(176\) 0 0
\(177\) −5.65685 −0.425195
\(178\) 0 0
\(179\) 11.6413i 0.870111i 0.900404 + 0.435055i \(0.143271\pi\)
−0.900404 + 0.435055i \(0.856729\pi\)
\(180\) 0 0
\(181\) − 9.50732i − 0.706673i −0.935496 0.353337i \(-0.885047\pi\)
0.935496 0.353337i \(-0.114953\pi\)
\(182\) 0 0
\(183\) 8.46742 0.625930
\(184\) 0 0
\(185\) −19.7091 −1.44904
\(186\) 0 0
\(187\) 0.571533i 0.0417946i
\(188\) 0 0
\(189\) 2.15894i 0.157040i
\(190\) 0 0
\(191\) 20.8032 1.50526 0.752632 0.658441i \(-0.228784\pi\)
0.752632 + 0.658441i \(0.228784\pi\)
\(192\) 0 0
\(193\) 14.1454 1.01821 0.509103 0.860705i \(-0.329977\pi\)
0.509103 + 0.860705i \(0.329977\pi\)
\(194\) 0 0
\(195\) 7.43208i 0.532222i
\(196\) 0 0
\(197\) − 3.43463i − 0.244707i −0.992487 0.122354i \(-0.960956\pi\)
0.992487 0.122354i \(-0.0390442\pi\)
\(198\) 0 0
\(199\) 0.306182 0.0217047 0.0108523 0.999941i \(-0.496546\pi\)
0.0108523 + 0.999941i \(0.496546\pi\)
\(200\) 0 0
\(201\) 14.7422 1.03983
\(202\) 0 0
\(203\) − 5.67016i − 0.397967i
\(204\) 0 0
\(205\) 22.3380i 1.56016i
\(206\) 0 0
\(207\) 2.82843 0.196589
\(208\) 0 0
\(209\) −0.571533 −0.0395337
\(210\) 0 0
\(211\) 10.2284i 0.704151i 0.935972 + 0.352076i \(0.114524\pi\)
−0.935972 + 0.352076i \(0.885476\pi\)
\(212\) 0 0
\(213\) 4.31788i 0.295856i
\(214\) 0 0
\(215\) −41.6517 −2.84063
\(216\) 0 0
\(217\) −3.97474 −0.269823
\(218\) 0 0
\(219\) 5.97474i 0.403735i
\(220\) 0 0
\(221\) − 0.439861i − 0.0295883i
\(222\) 0 0
\(223\) 1.71908 0.115118 0.0575591 0.998342i \(-0.481668\pi\)
0.0575591 + 0.998342i \(0.481668\pi\)
\(224\) 0 0
\(225\) 9.42429 0.628286
\(226\) 0 0
\(227\) − 14.3059i − 0.949518i −0.880116 0.474759i \(-0.842535\pi\)
0.880116 0.474759i \(-0.157465\pi\)
\(228\) 0 0
\(229\) − 16.9981i − 1.12327i −0.827386 0.561634i \(-0.810173\pi\)
0.827386 0.561634i \(-0.189827\pi\)
\(230\) 0 0
\(231\) 5.48946 0.361180
\(232\) 0 0
\(233\) −13.3779 −0.876418 −0.438209 0.898873i \(-0.644387\pi\)
−0.438209 + 0.898873i \(0.644387\pi\)
\(234\) 0 0
\(235\) 10.7422i 0.700742i
\(236\) 0 0
\(237\) − 15.0075i − 0.974844i
\(238\) 0 0
\(239\) 13.3675 0.864670 0.432335 0.901713i \(-0.357690\pi\)
0.432335 + 0.901713i \(0.357690\pi\)
\(240\) 0 0
\(241\) −0.211474 −0.0136222 −0.00681112 0.999977i \(-0.502168\pi\)
−0.00681112 + 0.999977i \(0.502168\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 8.88325i 0.567530i
\(246\) 0 0
\(247\) 0.439861 0.0279877
\(248\) 0 0
\(249\) −14.3059 −0.906601
\(250\) 0 0
\(251\) − 14.7555i − 0.931358i −0.884954 0.465679i \(-0.845810\pi\)
0.884954 0.465679i \(-0.154190\pi\)
\(252\) 0 0
\(253\) − 7.19173i − 0.452140i
\(254\) 0 0
\(255\) −0.853690 −0.0534601
\(256\) 0 0
\(257\) −0.742176 −0.0462957 −0.0231478 0.999732i \(-0.507369\pi\)
−0.0231478 + 0.999732i \(0.507369\pi\)
\(258\) 0 0
\(259\) − 11.2037i − 0.696163i
\(260\) 0 0
\(261\) 2.62636i 0.162568i
\(262\) 0 0
\(263\) 5.48435 0.338180 0.169090 0.985601i \(-0.445917\pi\)
0.169090 + 0.985601i \(0.445917\pi\)
\(264\) 0 0
\(265\) −40.3582 −2.47918
\(266\) 0 0
\(267\) − 1.42847i − 0.0874208i
\(268\) 0 0
\(269\) − 20.4694i − 1.24804i −0.781407 0.624021i \(-0.785498\pi\)
0.781407 0.624021i \(-0.214502\pi\)
\(270\) 0 0
\(271\) −14.0370 −0.852685 −0.426342 0.904562i \(-0.640198\pi\)
−0.426342 + 0.904562i \(0.640198\pi\)
\(272\) 0 0
\(273\) −4.22478 −0.255695
\(274\) 0 0
\(275\) − 23.9628i − 1.44501i
\(276\) 0 0
\(277\) − 13.4211i − 0.806394i −0.915113 0.403197i \(-0.867899\pi\)
0.915113 0.403197i \(-0.132101\pi\)
\(278\) 0 0
\(279\) 1.84106 0.110221
\(280\) 0 0
\(281\) −3.89359 −0.232272 −0.116136 0.993233i \(-0.537051\pi\)
−0.116136 + 0.993233i \(0.537051\pi\)
\(282\) 0 0
\(283\) 17.6569i 1.04959i 0.851228 + 0.524796i \(0.175858\pi\)
−0.851228 + 0.524796i \(0.824142\pi\)
\(284\) 0 0
\(285\) − 0.853690i − 0.0505682i
\(286\) 0 0
\(287\) −12.6981 −0.749545
\(288\) 0 0
\(289\) −16.9495 −0.997028
\(290\) 0 0
\(291\) 16.3990i 0.961328i
\(292\) 0 0
\(293\) − 15.7759i − 0.921639i −0.887494 0.460819i \(-0.847556\pi\)
0.887494 0.460819i \(-0.152444\pi\)
\(294\) 0 0
\(295\) −21.4844 −1.25087
\(296\) 0 0
\(297\) −2.54266 −0.147540
\(298\) 0 0
\(299\) 5.53488i 0.320090i
\(300\) 0 0
\(301\) − 23.6770i − 1.36472i
\(302\) 0 0
\(303\) 0.115816 0.00665345
\(304\) 0 0
\(305\) 32.1587 1.84140
\(306\) 0 0
\(307\) − 7.64129i − 0.436111i −0.975936 0.218056i \(-0.930029\pi\)
0.975936 0.218056i \(-0.0699714\pi\)
\(308\) 0 0
\(309\) − 13.3507i − 0.759493i
\(310\) 0 0
\(311\) 24.1623 1.37012 0.685059 0.728488i \(-0.259776\pi\)
0.685059 + 0.728488i \(0.259776\pi\)
\(312\) 0 0
\(313\) 16.6105 0.938881 0.469441 0.882964i \(-0.344456\pi\)
0.469441 + 0.882964i \(0.344456\pi\)
\(314\) 0 0
\(315\) 8.19951i 0.461990i
\(316\) 0 0
\(317\) 2.56213i 0.143903i 0.997408 + 0.0719517i \(0.0229227\pi\)
−0.997408 + 0.0719517i \(0.977077\pi\)
\(318\) 0 0
\(319\) 6.67794 0.373893
\(320\) 0 0
\(321\) −10.2926 −0.574478
\(322\) 0 0
\(323\) 0.0505249i 0.00281128i
\(324\) 0 0
\(325\) 18.4422i 1.02299i
\(326\) 0 0
\(327\) 9.95687 0.550616
\(328\) 0 0
\(329\) −6.10641 −0.336657
\(330\) 0 0
\(331\) 19.1275i 1.05134i 0.850688 + 0.525671i \(0.176186\pi\)
−0.850688 + 0.525671i \(0.823814\pi\)
\(332\) 0 0
\(333\) 5.18944i 0.284379i
\(334\) 0 0
\(335\) 55.9898 3.05905
\(336\) 0 0
\(337\) 1.12615 0.0613454 0.0306727 0.999529i \(-0.490235\pi\)
0.0306727 + 0.999529i \(0.490235\pi\)
\(338\) 0 0
\(339\) 18.8486i 1.02371i
\(340\) 0 0
\(341\) − 4.68119i − 0.253500i
\(342\) 0 0
\(343\) −20.1623 −1.08866
\(344\) 0 0
\(345\) 10.7422 0.578339
\(346\) 0 0
\(347\) − 29.4068i − 1.57864i −0.613982 0.789320i \(-0.710433\pi\)
0.613982 0.789320i \(-0.289567\pi\)
\(348\) 0 0
\(349\) − 27.2738i − 1.45993i −0.683482 0.729967i \(-0.739535\pi\)
0.683482 0.729967i \(-0.260465\pi\)
\(350\) 0 0
\(351\) 1.95687 0.104450
\(352\) 0 0
\(353\) 25.5908 1.36206 0.681029 0.732256i \(-0.261533\pi\)
0.681029 + 0.732256i \(0.261533\pi\)
\(354\) 0 0
\(355\) 16.3990i 0.870370i
\(356\) 0 0
\(357\) − 0.485281i − 0.0256838i
\(358\) 0 0
\(359\) 3.77296 0.199129 0.0995645 0.995031i \(-0.468255\pi\)
0.0995645 + 0.995031i \(0.468255\pi\)
\(360\) 0 0
\(361\) 18.9495 0.997341
\(362\) 0 0
\(363\) − 4.53488i − 0.238019i
\(364\) 0 0
\(365\) 22.6917i 1.18774i
\(366\) 0 0
\(367\) −27.4474 −1.43274 −0.716371 0.697720i \(-0.754198\pi\)
−0.716371 + 0.697720i \(0.754198\pi\)
\(368\) 0 0
\(369\) 5.88163 0.306186
\(370\) 0 0
\(371\) − 22.9417i − 1.19107i
\(372\) 0 0
\(373\) − 17.8518i − 0.924332i −0.886794 0.462166i \(-0.847072\pi\)
0.886794 0.462166i \(-0.152928\pi\)
\(374\) 0 0
\(375\) 16.8032 0.867712
\(376\) 0 0
\(377\) −5.13946 −0.264695
\(378\) 0 0
\(379\) 16.5018i 0.847642i 0.905746 + 0.423821i \(0.139311\pi\)
−0.905746 + 0.423821i \(0.860689\pi\)
\(380\) 0 0
\(381\) − 3.81580i − 0.195489i
\(382\) 0 0
\(383\) 17.1885 0.878291 0.439145 0.898416i \(-0.355281\pi\)
0.439145 + 0.898416i \(0.355281\pi\)
\(384\) 0 0
\(385\) 20.8486 1.06254
\(386\) 0 0
\(387\) 10.9670i 0.557482i
\(388\) 0 0
\(389\) − 2.66209i − 0.134973i −0.997720 0.0674866i \(-0.978502\pi\)
0.997720 0.0674866i \(-0.0214980\pi\)
\(390\) 0 0
\(391\) −0.635767 −0.0321521
\(392\) 0 0
\(393\) 1.08532 0.0547472
\(394\) 0 0
\(395\) − 56.9976i − 2.86786i
\(396\) 0 0
\(397\) − 11.8959i − 0.597037i −0.954404 0.298519i \(-0.903507\pi\)
0.954404 0.298519i \(-0.0964925\pi\)
\(398\) 0 0
\(399\) 0.485281 0.0242945
\(400\) 0 0
\(401\) −1.12389 −0.0561242 −0.0280621 0.999606i \(-0.508934\pi\)
−0.0280621 + 0.999606i \(0.508934\pi\)
\(402\) 0 0
\(403\) 3.60272i 0.179464i
\(404\) 0 0
\(405\) − 3.79793i − 0.188721i
\(406\) 0 0
\(407\) 13.1950 0.654051
\(408\) 0 0
\(409\) −13.7211 −0.678464 −0.339232 0.940703i \(-0.610167\pi\)
−0.339232 + 0.940703i \(0.610167\pi\)
\(410\) 0 0
\(411\) − 5.31010i − 0.261928i
\(412\) 0 0
\(413\) − 12.2128i − 0.600953i
\(414\) 0 0
\(415\) −54.3329 −2.66710
\(416\) 0 0
\(417\) 12.3990 0.607183
\(418\) 0 0
\(419\) 13.1629i 0.643048i 0.946901 + 0.321524i \(0.104195\pi\)
−0.946901 + 0.321524i \(0.895805\pi\)
\(420\) 0 0
\(421\) 11.9413i 0.581984i 0.956726 + 0.290992i \(0.0939852\pi\)
−0.956726 + 0.290992i \(0.906015\pi\)
\(422\) 0 0
\(423\) 2.82843 0.137523
\(424\) 0 0
\(425\) −2.11837 −0.102756
\(426\) 0 0
\(427\) 18.2807i 0.884663i
\(428\) 0 0
\(429\) − 4.97567i − 0.240227i
\(430\) 0 0
\(431\) −30.6054 −1.47421 −0.737105 0.675778i \(-0.763808\pi\)
−0.737105 + 0.675778i \(0.763808\pi\)
\(432\) 0 0
\(433\) 15.3137 0.735930 0.367965 0.929840i \(-0.380055\pi\)
0.367965 + 0.929840i \(0.380055\pi\)
\(434\) 0 0
\(435\) 9.97474i 0.478252i
\(436\) 0 0
\(437\) − 0.635767i − 0.0304128i
\(438\) 0 0
\(439\) 33.3676 1.59255 0.796274 0.604936i \(-0.206801\pi\)
0.796274 + 0.604936i \(0.206801\pi\)
\(440\) 0 0
\(441\) 2.33897 0.111380
\(442\) 0 0
\(443\) − 3.23617i − 0.153755i −0.997041 0.0768776i \(-0.975505\pi\)
0.997041 0.0768776i \(-0.0244951\pi\)
\(444\) 0 0
\(445\) − 5.42522i − 0.257180i
\(446\) 0 0
\(447\) 1.45479 0.0688091
\(448\) 0 0
\(449\) −27.4165 −1.29387 −0.646933 0.762547i \(-0.723948\pi\)
−0.646933 + 0.762547i \(0.723948\pi\)
\(450\) 0 0
\(451\) − 14.9550i − 0.704203i
\(452\) 0 0
\(453\) − 2.03696i − 0.0957049i
\(454\) 0 0
\(455\) −16.0454 −0.752221
\(456\) 0 0
\(457\) 10.9147 0.510567 0.255284 0.966866i \(-0.417831\pi\)
0.255284 + 0.966866i \(0.417831\pi\)
\(458\) 0 0
\(459\) 0.224777i 0.0104917i
\(460\) 0 0
\(461\) − 25.2181i − 1.17452i −0.809398 0.587261i \(-0.800206\pi\)
0.809398 0.587261i \(-0.199794\pi\)
\(462\) 0 0
\(463\) 22.4937 1.04537 0.522686 0.852525i \(-0.324930\pi\)
0.522686 + 0.852525i \(0.324930\pi\)
\(464\) 0 0
\(465\) 6.99222 0.324256
\(466\) 0 0
\(467\) 34.2482i 1.58482i 0.609991 + 0.792408i \(0.291173\pi\)
−0.609991 + 0.792408i \(0.708827\pi\)
\(468\) 0 0
\(469\) 31.8275i 1.46966i
\(470\) 0 0
\(471\) 8.61790 0.397092
\(472\) 0 0
\(473\) 27.8852 1.28216
\(474\) 0 0
\(475\) − 2.11837i − 0.0971974i
\(476\) 0 0
\(477\) 10.6264i 0.486548i
\(478\) 0 0
\(479\) 36.2362 1.65568 0.827838 0.560968i \(-0.189571\pi\)
0.827838 + 0.560968i \(0.189571\pi\)
\(480\) 0 0
\(481\) −10.1551 −0.463032
\(482\) 0 0
\(483\) 6.10641i 0.277851i
\(484\) 0 0
\(485\) 62.2824i 2.82810i
\(486\) 0 0
\(487\) 16.8200 0.762186 0.381093 0.924537i \(-0.375548\pi\)
0.381093 + 0.924537i \(0.375548\pi\)
\(488\) 0 0
\(489\) −4.86054 −0.219801
\(490\) 0 0
\(491\) 8.63577i 0.389727i 0.980830 + 0.194863i \(0.0624263\pi\)
−0.980830 + 0.194863i \(0.937574\pi\)
\(492\) 0 0
\(493\) − 0.590346i − 0.0265879i
\(494\) 0 0
\(495\) −9.65685 −0.434043
\(496\) 0 0
\(497\) −9.32206 −0.418151
\(498\) 0 0
\(499\) − 27.8275i − 1.24573i −0.782329 0.622865i \(-0.785969\pi\)
0.782329 0.622865i \(-0.214031\pi\)
\(500\) 0 0
\(501\) 21.7023i 0.969586i
\(502\) 0 0
\(503\) −25.7308 −1.14728 −0.573639 0.819108i \(-0.694469\pi\)
−0.573639 + 0.819108i \(0.694469\pi\)
\(504\) 0 0
\(505\) 0.439861 0.0195736
\(506\) 0 0
\(507\) − 9.17064i − 0.407283i
\(508\) 0 0
\(509\) 2.45386i 0.108765i 0.998520 + 0.0543826i \(0.0173191\pi\)
−0.998520 + 0.0543826i \(0.982681\pi\)
\(510\) 0 0
\(511\) −12.8991 −0.570623
\(512\) 0 0
\(513\) −0.224777 −0.00992417
\(514\) 0 0
\(515\) − 50.7050i − 2.23433i
\(516\) 0 0
\(517\) − 7.19173i − 0.316292i
\(518\) 0 0
\(519\) 12.3695 0.542959
\(520\) 0 0
\(521\) 33.5944 1.47180 0.735898 0.677092i \(-0.236760\pi\)
0.735898 + 0.677092i \(0.236760\pi\)
\(522\) 0 0
\(523\) − 30.8522i − 1.34907i −0.738242 0.674536i \(-0.764344\pi\)
0.738242 0.674536i \(-0.235656\pi\)
\(524\) 0 0
\(525\) 20.3465i 0.887994i
\(526\) 0 0
\(527\) −0.413828 −0.0180266
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 5.65685i 0.245487i
\(532\) 0 0
\(533\) 11.5096i 0.498537i
\(534\) 0 0
\(535\) −39.0907 −1.69004
\(536\) 0 0
\(537\) 11.6413 0.502359
\(538\) 0 0
\(539\) − 5.94721i − 0.256164i
\(540\) 0 0
\(541\) 38.4825i 1.65449i 0.561841 + 0.827245i \(0.310093\pi\)
−0.561841 + 0.827245i \(0.689907\pi\)
\(542\) 0 0
\(543\) −9.50732 −0.407998
\(544\) 0 0
\(545\) 37.8155 1.61984
\(546\) 0 0
\(547\) 9.61829i 0.411248i 0.978631 + 0.205624i \(0.0659224\pi\)
−0.978631 + 0.205624i \(0.934078\pi\)
\(548\) 0 0
\(549\) − 8.46742i − 0.361381i
\(550\) 0 0
\(551\) 0.590346 0.0251496
\(552\) 0 0
\(553\) 32.4004 1.37780
\(554\) 0 0
\(555\) 19.7091i 0.836606i
\(556\) 0 0
\(557\) 6.07174i 0.257268i 0.991692 + 0.128634i \(0.0410592\pi\)
−0.991692 + 0.128634i \(0.958941\pi\)
\(558\) 0 0
\(559\) −21.4609 −0.907701
\(560\) 0 0
\(561\) 0.571533 0.0241301
\(562\) 0 0
\(563\) − 14.2554i − 0.600793i −0.953814 0.300397i \(-0.902881\pi\)
0.953814 0.300397i \(-0.0971191\pi\)
\(564\) 0 0
\(565\) 71.5857i 3.01163i
\(566\) 0 0
\(567\) 2.15894 0.0906670
\(568\) 0 0
\(569\) 32.5018 1.36255 0.681274 0.732029i \(-0.261426\pi\)
0.681274 + 0.732029i \(0.261426\pi\)
\(570\) 0 0
\(571\) 12.9706i 0.542801i 0.962466 + 0.271401i \(0.0874868\pi\)
−0.962466 + 0.271401i \(0.912513\pi\)
\(572\) 0 0
\(573\) − 20.8032i − 0.869065i
\(574\) 0 0
\(575\) 26.6559 1.11163
\(576\) 0 0
\(577\) 11.7536 0.489308 0.244654 0.969611i \(-0.421326\pi\)
0.244654 + 0.969611i \(0.421326\pi\)
\(578\) 0 0
\(579\) − 14.1454i − 0.587862i
\(580\) 0 0
\(581\) − 30.8857i − 1.28135i
\(582\) 0 0
\(583\) 27.0192 1.11902
\(584\) 0 0
\(585\) 7.43208 0.307279
\(586\) 0 0
\(587\) 9.13585i 0.377077i 0.982066 + 0.188538i \(0.0603750\pi\)
−0.982066 + 0.188538i \(0.939625\pi\)
\(588\) 0 0
\(589\) − 0.413828i − 0.0170515i
\(590\) 0 0
\(591\) −3.43463 −0.141282
\(592\) 0 0
\(593\) −5.49270 −0.225558 −0.112779 0.993620i \(-0.535975\pi\)
−0.112779 + 0.993620i \(0.535975\pi\)
\(594\) 0 0
\(595\) − 1.84307i − 0.0755583i
\(596\) 0 0
\(597\) − 0.306182i − 0.0125312i
\(598\) 0 0
\(599\) −36.4348 −1.48868 −0.744342 0.667799i \(-0.767237\pi\)
−0.744342 + 0.667799i \(0.767237\pi\)
\(600\) 0 0
\(601\) 9.97474 0.406878 0.203439 0.979088i \(-0.434788\pi\)
0.203439 + 0.979088i \(0.434788\pi\)
\(602\) 0 0
\(603\) − 14.7422i − 0.600348i
\(604\) 0 0
\(605\) − 17.2232i − 0.700221i
\(606\) 0 0
\(607\) −4.51900 −0.183421 −0.0917103 0.995786i \(-0.529233\pi\)
−0.0917103 + 0.995786i \(0.529233\pi\)
\(608\) 0 0
\(609\) −5.67016 −0.229766
\(610\) 0 0
\(611\) 5.53488i 0.223917i
\(612\) 0 0
\(613\) − 11.9316i − 0.481913i −0.970536 0.240957i \(-0.922539\pi\)
0.970536 0.240957i \(-0.0774612\pi\)
\(614\) 0 0
\(615\) 22.3380 0.900757
\(616\) 0 0
\(617\) 32.1201 1.29311 0.646554 0.762869i \(-0.276210\pi\)
0.646554 + 0.762869i \(0.276210\pi\)
\(618\) 0 0
\(619\) 21.2715i 0.854975i 0.904021 + 0.427488i \(0.140601\pi\)
−0.904021 + 0.427488i \(0.859399\pi\)
\(620\) 0 0
\(621\) − 2.82843i − 0.113501i
\(622\) 0 0
\(623\) 3.08398 0.123557
\(624\) 0 0
\(625\) 16.6958 0.667833
\(626\) 0 0
\(627\) 0.571533i 0.0228248i
\(628\) 0 0
\(629\) − 1.16647i − 0.0465101i
\(630\) 0 0
\(631\) 36.4685 1.45179 0.725894 0.687807i \(-0.241426\pi\)
0.725894 + 0.687807i \(0.241426\pi\)
\(632\) 0 0
\(633\) 10.2284 0.406542
\(634\) 0 0
\(635\) − 14.4921i − 0.575103i
\(636\) 0 0
\(637\) 4.57707i 0.181350i
\(638\) 0 0
\(639\) 4.31788 0.170813
\(640\) 0 0
\(641\) 14.0036 0.553109 0.276555 0.960998i \(-0.410807\pi\)
0.276555 + 0.960998i \(0.410807\pi\)
\(642\) 0 0
\(643\) − 23.4807i − 0.925990i −0.886361 0.462995i \(-0.846775\pi\)
0.886361 0.462995i \(-0.153225\pi\)
\(644\) 0 0
\(645\) 41.6517i 1.64004i
\(646\) 0 0
\(647\) −12.1908 −0.479270 −0.239635 0.970863i \(-0.577028\pi\)
−0.239635 + 0.970863i \(0.577028\pi\)
\(648\) 0 0
\(649\) 14.3835 0.564600
\(650\) 0 0
\(651\) 3.97474i 0.155782i
\(652\) 0 0
\(653\) − 1.39055i − 0.0544165i −0.999630 0.0272083i \(-0.991338\pi\)
0.999630 0.0272083i \(-0.00866173\pi\)
\(654\) 0 0
\(655\) 4.12198 0.161059
\(656\) 0 0
\(657\) 5.97474 0.233097
\(658\) 0 0
\(659\) − 25.5349i − 0.994698i −0.867551 0.497349i \(-0.834307\pi\)
0.867551 0.497349i \(-0.165693\pi\)
\(660\) 0 0
\(661\) 6.44726i 0.250769i 0.992108 + 0.125385i \(0.0400165\pi\)
−0.992108 + 0.125385i \(0.959983\pi\)
\(662\) 0 0
\(663\) −0.439861 −0.0170828
\(664\) 0 0
\(665\) 1.84307 0.0714710
\(666\) 0 0
\(667\) 7.42847i 0.287631i
\(668\) 0 0
\(669\) − 1.71908i − 0.0664635i
\(670\) 0 0
\(671\) −21.5298 −0.831148
\(672\) 0 0
\(673\) −10.8569 −0.418504 −0.209252 0.977862i \(-0.567103\pi\)
−0.209252 + 0.977862i \(0.567103\pi\)
\(674\) 0 0
\(675\) − 9.42429i − 0.362741i
\(676\) 0 0
\(677\) 33.5262i 1.28852i 0.764807 + 0.644259i \(0.222834\pi\)
−0.764807 + 0.644259i \(0.777166\pi\)
\(678\) 0 0
\(679\) −35.4045 −1.35870
\(680\) 0 0
\(681\) −14.3059 −0.548204
\(682\) 0 0
\(683\) − 25.2206i − 0.965040i −0.875885 0.482520i \(-0.839722\pi\)
0.875885 0.482520i \(-0.160278\pi\)
\(684\) 0 0
\(685\) − 20.1674i − 0.770557i
\(686\) 0 0
\(687\) −16.9981 −0.648519
\(688\) 0 0
\(689\) −20.7945 −0.792205
\(690\) 0 0
\(691\) 15.3523i 0.584028i 0.956414 + 0.292014i \(0.0943254\pi\)
−0.956414 + 0.292014i \(0.905675\pi\)
\(692\) 0 0
\(693\) − 5.48946i − 0.208527i
\(694\) 0 0
\(695\) 47.0907 1.78625
\(696\) 0 0
\(697\) −1.32206 −0.0500765
\(698\) 0 0
\(699\) 13.3779i 0.506000i
\(700\) 0 0
\(701\) − 8.61079i − 0.325225i −0.986690 0.162613i \(-0.948008\pi\)
0.986690 0.162613i \(-0.0519921\pi\)
\(702\) 0 0
\(703\) 1.16647 0.0439942
\(704\) 0 0
\(705\) 10.7422 0.404574
\(706\) 0 0
\(707\) 0.250040i 0.00940372i
\(708\) 0 0
\(709\) 32.3624i 1.21539i 0.794169 + 0.607697i \(0.207906\pi\)
−0.794169 + 0.607697i \(0.792094\pi\)
\(710\) 0 0
\(711\) −15.0075 −0.562826
\(712\) 0 0
\(713\) 5.20730 0.195015
\(714\) 0 0
\(715\) − 18.8972i − 0.706717i
\(716\) 0 0
\(717\) − 13.3675i − 0.499218i
\(718\) 0 0
\(719\) −1.46744 −0.0547262 −0.0273631 0.999626i \(-0.508711\pi\)
−0.0273631 + 0.999626i \(0.508711\pi\)
\(720\) 0 0
\(721\) 28.8233 1.07344
\(722\) 0 0
\(723\) 0.211474i 0.00786481i
\(724\) 0 0
\(725\) 24.7516i 0.919251i
\(726\) 0 0
\(727\) −15.3928 −0.570889 −0.285445 0.958395i \(-0.592141\pi\)
−0.285445 + 0.958395i \(0.592141\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 2.46512i − 0.0911759i
\(732\) 0 0
\(733\) − 17.5624i − 0.648683i −0.945940 0.324342i \(-0.894857\pi\)
0.945940 0.324342i \(-0.105143\pi\)
\(734\) 0 0
\(735\) 8.88325 0.327664
\(736\) 0 0
\(737\) −37.4844 −1.38075
\(738\) 0 0
\(739\) 20.7266i 0.762441i 0.924484 + 0.381220i \(0.124496\pi\)
−0.924484 + 0.381220i \(0.875504\pi\)
\(740\) 0 0
\(741\) − 0.439861i − 0.0161587i
\(742\) 0 0
\(743\) −31.7821 −1.16597 −0.582986 0.812482i \(-0.698116\pi\)
−0.582986 + 0.812482i \(0.698116\pi\)
\(744\) 0 0
\(745\) 5.52518 0.202427
\(746\) 0 0
\(747\) 14.3059i 0.523426i
\(748\) 0 0
\(749\) − 22.2212i − 0.811944i
\(750\) 0 0
\(751\) 29.7594 1.08594 0.542968 0.839753i \(-0.317301\pi\)
0.542968 + 0.839753i \(0.317301\pi\)
\(752\) 0 0
\(753\) −14.7555 −0.537720
\(754\) 0 0
\(755\) − 7.73625i − 0.281551i
\(756\) 0 0
\(757\) − 22.1119i − 0.803672i −0.915712 0.401836i \(-0.868372\pi\)
0.915712 0.401836i \(-0.131628\pi\)
\(758\) 0 0
\(759\) −7.19173 −0.261043
\(760\) 0 0
\(761\) 4.55957 0.165284 0.0826422 0.996579i \(-0.473664\pi\)
0.0826422 + 0.996579i \(0.473664\pi\)
\(762\) 0 0
\(763\) 21.4963i 0.778219i
\(764\) 0 0
\(765\) 0.853690i 0.0308652i
\(766\) 0 0
\(767\) −11.0698 −0.399706
\(768\) 0 0
\(769\) 36.5794 1.31909 0.659543 0.751667i \(-0.270750\pi\)
0.659543 + 0.751667i \(0.270750\pi\)
\(770\) 0 0
\(771\) 0.742176i 0.0267288i
\(772\) 0 0
\(773\) − 26.4611i − 0.951739i −0.879516 0.475869i \(-0.842133\pi\)
0.879516 0.475869i \(-0.157867\pi\)
\(774\) 0 0
\(775\) 17.3507 0.623255
\(776\) 0 0
\(777\) −11.2037 −0.401930
\(778\) 0 0
\(779\) − 1.32206i − 0.0473676i
\(780\) 0 0
\(781\) − 10.9789i − 0.392856i
\(782\) 0 0
\(783\) 2.62636 0.0938584
\(784\) 0 0
\(785\) 32.7302 1.16819
\(786\) 0 0
\(787\) 18.9164i 0.674298i 0.941451 + 0.337149i \(0.109463\pi\)
−0.941451 + 0.337149i \(0.890537\pi\)
\(788\) 0 0
\(789\) − 5.48435i − 0.195248i
\(790\) 0 0
\(791\) −40.6930 −1.44688
\(792\) 0 0
\(793\) 16.5697 0.588406
\(794\) 0 0
\(795\) 40.3582i 1.43136i
\(796\) 0 0
\(797\) − 47.8065i − 1.69339i −0.532075 0.846697i \(-0.678588\pi\)
0.532075 0.846697i \(-0.321412\pi\)
\(798\) 0 0
\(799\) −0.635767 −0.0224918
\(800\) 0 0
\(801\) −1.42847 −0.0504724
\(802\) 0 0
\(803\) − 15.1917i − 0.536105i
\(804\) 0 0
\(805\) 23.1917i 0.817401i
\(806\) 0 0
\(807\) −20.4694 −0.720558
\(808\) 0 0
\(809\) −29.9862 −1.05426 −0.527129 0.849785i \(-0.676732\pi\)
−0.527129 + 0.849785i \(0.676732\pi\)
\(810\) 0 0
\(811\) − 11.3899i − 0.399954i −0.979801 0.199977i \(-0.935913\pi\)
0.979801 0.199977i \(-0.0640867\pi\)
\(812\) 0 0
\(813\) 14.0370i 0.492298i
\(814\) 0 0
\(815\) −18.4600 −0.646626
\(816\) 0 0
\(817\) 2.46512 0.0862438
\(818\) 0 0
\(819\) 4.22478i 0.147626i
\(820\) 0 0
\(821\) − 19.6929i − 0.687286i −0.939100 0.343643i \(-0.888339\pi\)
0.939100 0.343643i \(-0.111661\pi\)
\(822\) 0 0
\(823\) 22.4666 0.783137 0.391568 0.920149i \(-0.371933\pi\)
0.391568 + 0.920149i \(0.371933\pi\)
\(824\) 0 0
\(825\) −23.9628 −0.834277
\(826\) 0 0
\(827\) − 10.7927i − 0.375299i −0.982236 0.187649i \(-0.939913\pi\)
0.982236 0.187649i \(-0.0600869\pi\)
\(828\) 0 0
\(829\) − 45.9421i − 1.59563i −0.602900 0.797817i \(-0.705988\pi\)
0.602900 0.797817i \(-0.294012\pi\)
\(830\) 0 0
\(831\) −13.4211 −0.465572
\(832\) 0 0
\(833\) −0.525748 −0.0182161
\(834\) 0 0
\(835\) 82.4238i 2.85239i
\(836\) 0 0
\(837\) − 1.84106i − 0.0636363i
\(838\) 0 0
\(839\) 22.9142 0.791085 0.395542 0.918448i \(-0.370557\pi\)
0.395542 + 0.918448i \(0.370557\pi\)
\(840\) 0 0
\(841\) 22.1022 0.762146
\(842\) 0 0
\(843\) 3.89359i 0.134102i
\(844\) 0 0
\(845\) − 34.8295i − 1.19817i
\(846\) 0 0
\(847\) 9.79053 0.336407
\(848\) 0 0
\(849\) 17.6569 0.605982
\(850\) 0 0
\(851\) 14.6779i 0.503153i
\(852\) 0 0
\(853\) − 55.0728i − 1.88566i −0.333278 0.942829i \(-0.608155\pi\)
0.333278 0.942829i \(-0.391845\pi\)
\(854\) 0 0
\(855\) −0.853690 −0.0291956
\(856\) 0 0
\(857\) 1.79079 0.0611723 0.0305861 0.999532i \(-0.490263\pi\)
0.0305861 + 0.999532i \(0.490263\pi\)
\(858\) 0 0
\(859\) − 25.5468i − 0.871647i −0.900032 0.435823i \(-0.856457\pi\)
0.900032 0.435823i \(-0.143543\pi\)
\(860\) 0 0
\(861\) 12.6981i 0.432750i
\(862\) 0 0
\(863\) −42.1150 −1.43361 −0.716806 0.697273i \(-0.754397\pi\)
−0.716806 + 0.697273i \(0.754397\pi\)
\(864\) 0 0
\(865\) 46.9784 1.59731
\(866\) 0 0
\(867\) 16.9495i 0.575634i
\(868\) 0 0
\(869\) 38.1590i 1.29446i
\(870\) 0 0
\(871\) 28.8486 0.977497
\(872\) 0 0
\(873\) 16.3990 0.555023
\(874\) 0 0
\(875\) 36.2771i 1.22639i
\(876\) 0 0
\(877\) − 1.01044i − 0.0341203i −0.999854 0.0170601i \(-0.994569\pi\)
0.999854 0.0170601i \(-0.00543067\pi\)
\(878\) 0 0
\(879\) −15.7759 −0.532108
\(880\) 0 0
\(881\) 44.3972 1.49578 0.747889 0.663823i \(-0.231067\pi\)
0.747889 + 0.663823i \(0.231067\pi\)
\(882\) 0 0
\(883\) 1.82389i 0.0613787i 0.999529 + 0.0306894i \(0.00977026\pi\)
−0.999529 + 0.0306894i \(0.990230\pi\)
\(884\) 0 0
\(885\) 21.4844i 0.722189i
\(886\) 0 0
\(887\) 4.38532 0.147245 0.0736223 0.997286i \(-0.476544\pi\)
0.0736223 + 0.997286i \(0.476544\pi\)
\(888\) 0 0
\(889\) 8.23808 0.276296
\(890\) 0 0
\(891\) 2.54266i 0.0851823i
\(892\) 0 0
\(893\) − 0.635767i − 0.0212751i
\(894\) 0 0
\(895\) 44.2128 1.47787
\(896\) 0 0
\(897\) 5.53488 0.184804
\(898\) 0 0
\(899\) 4.83528i 0.161266i
\(900\) 0 0
\(901\) − 2.38857i − 0.0795747i
\(902\) 0 0
\(903\) −23.6770 −0.787922
\(904\) 0 0
\(905\) −36.1082 −1.20028
\(906\) 0 0
\(907\) − 4.06248i − 0.134893i −0.997723 0.0674463i \(-0.978515\pi\)
0.997723 0.0674463i \(-0.0214851\pi\)
\(908\) 0 0
\(909\) − 0.115816i − 0.00384137i
\(910\) 0 0
\(911\) 42.3784 1.40406 0.702029 0.712149i \(-0.252278\pi\)
0.702029 + 0.712149i \(0.252278\pi\)
\(912\) 0 0
\(913\) 36.3751 1.20384
\(914\) 0 0
\(915\) − 32.1587i − 1.06313i
\(916\) 0 0
\(917\) 2.34315i 0.0773775i
\(918\) 0 0
\(919\) 44.8603 1.47980 0.739902 0.672715i \(-0.234872\pi\)
0.739902 + 0.672715i \(0.234872\pi\)
\(920\) 0 0
\(921\) −7.64129 −0.251789
\(922\) 0 0
\(923\) 8.44955i 0.278120i
\(924\) 0 0
\(925\) 48.9068i 1.60804i
\(926\) 0 0
\(927\) −13.3507 −0.438494
\(928\) 0 0
\(929\) −2.96695 −0.0973426 −0.0486713 0.998815i \(-0.515499\pi\)
−0.0486713 + 0.998815i \(0.515499\pi\)
\(930\) 0 0
\(931\) − 0.525748i − 0.0172307i
\(932\) 0 0
\(933\) − 24.1623i − 0.791038i
\(934\) 0 0
\(935\) 2.17064 0.0709876
\(936\) 0 0
\(937\) −54.7669 −1.78916 −0.894579 0.446910i \(-0.852524\pi\)
−0.894579 + 0.446910i \(0.852524\pi\)
\(938\) 0 0
\(939\) − 16.6105i − 0.542063i
\(940\) 0 0
\(941\) 8.89558i 0.289988i 0.989433 + 0.144994i \(0.0463162\pi\)
−0.989433 + 0.144994i \(0.953684\pi\)
\(942\) 0 0
\(943\) 16.6358 0.541735
\(944\) 0 0
\(945\) 8.19951 0.266730
\(946\) 0 0
\(947\) − 15.8541i − 0.515189i −0.966253 0.257595i \(-0.917070\pi\)
0.966253 0.257595i \(-0.0829299\pi\)
\(948\) 0 0
\(949\) 11.6918i 0.379532i
\(950\) 0 0
\(951\) 2.56213 0.0830826
\(952\) 0 0
\(953\) 30.2807 0.980887 0.490443 0.871473i \(-0.336835\pi\)
0.490443 + 0.871473i \(0.336835\pi\)
\(954\) 0 0
\(955\) − 79.0090i − 2.55667i
\(956\) 0 0
\(957\) − 6.67794i − 0.215867i
\(958\) 0 0
\(959\) 11.4642 0.370198
\(960\) 0 0
\(961\) −27.6105 −0.890661
\(962\) 0 0
\(963\) 10.2926i 0.331675i
\(964\) 0 0
\(965\) − 53.7232i − 1.72941i
\(966\) 0 0
\(967\) −10.5273 −0.338537 −0.169268 0.985570i \(-0.554140\pi\)
−0.169268 + 0.985570i \(0.554140\pi\)
\(968\) 0 0
\(969\) 0.0505249 0.00162309
\(970\) 0 0
\(971\) 38.7050i 1.24210i 0.783771 + 0.621051i \(0.213294\pi\)
−0.783771 + 0.621051i \(0.786706\pi\)
\(972\) 0 0
\(973\) 26.7688i 0.858168i
\(974\) 0 0
\(975\) 18.4422 0.590622
\(976\) 0 0
\(977\) 16.9009 0.540706 0.270353 0.962761i \(-0.412860\pi\)
0.270353 + 0.962761i \(0.412860\pi\)
\(978\) 0 0
\(979\) 3.63211i 0.116083i
\(980\) 0 0
\(981\) − 9.95687i − 0.317899i
\(982\) 0 0
\(983\) −10.0798 −0.321496 −0.160748 0.986995i \(-0.551391\pi\)
−0.160748 + 0.986995i \(0.551391\pi\)
\(984\) 0 0
\(985\) −13.0445 −0.415632
\(986\) 0 0
\(987\) 6.10641i 0.194369i
\(988\) 0 0
\(989\) 31.0192i 0.986354i
\(990\) 0 0
\(991\) 42.3446 1.34512 0.672561 0.740042i \(-0.265194\pi\)
0.672561 + 0.740042i \(0.265194\pi\)
\(992\) 0 0
\(993\) 19.1275 0.606993
\(994\) 0 0
\(995\) − 1.16286i − 0.0368651i
\(996\) 0 0
\(997\) − 47.6132i − 1.50793i −0.656917 0.753963i \(-0.728140\pi\)
0.656917 0.753963i \(-0.271860\pi\)
\(998\) 0 0
\(999\) 5.18944 0.164186
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.i.1537.1 8
4.3 odd 2 3072.2.d.f.1537.5 8
8.3 odd 2 3072.2.d.f.1537.4 8
8.5 even 2 inner 3072.2.d.i.1537.8 8
16.3 odd 4 3072.2.a.i.1.1 4
16.5 even 4 3072.2.a.n.1.4 4
16.11 odd 4 3072.2.a.t.1.4 4
16.13 even 4 3072.2.a.o.1.1 4
32.3 odd 8 384.2.j.b.97.2 8
32.5 even 8 384.2.j.a.289.4 8
32.11 odd 8 48.2.j.a.13.3 8
32.13 even 8 192.2.j.a.49.1 8
32.19 odd 8 48.2.j.a.37.3 yes 8
32.21 even 8 192.2.j.a.145.1 8
32.27 odd 8 384.2.j.b.289.2 8
32.29 even 8 384.2.j.a.97.4 8
48.5 odd 4 9216.2.a.x.1.1 4
48.11 even 4 9216.2.a.y.1.1 4
48.29 odd 4 9216.2.a.bn.1.4 4
48.35 even 4 9216.2.a.bo.1.4 4
96.5 odd 8 1152.2.k.f.289.1 8
96.11 even 8 144.2.k.b.109.2 8
96.29 odd 8 1152.2.k.f.865.1 8
96.35 even 8 1152.2.k.c.865.1 8
96.53 odd 8 576.2.k.b.145.4 8
96.59 even 8 1152.2.k.c.289.1 8
96.77 odd 8 576.2.k.b.433.4 8
96.83 even 8 144.2.k.b.37.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.2.j.a.13.3 8 32.11 odd 8
48.2.j.a.37.3 yes 8 32.19 odd 8
144.2.k.b.37.2 8 96.83 even 8
144.2.k.b.109.2 8 96.11 even 8
192.2.j.a.49.1 8 32.13 even 8
192.2.j.a.145.1 8 32.21 even 8
384.2.j.a.97.4 8 32.29 even 8
384.2.j.a.289.4 8 32.5 even 8
384.2.j.b.97.2 8 32.3 odd 8
384.2.j.b.289.2 8 32.27 odd 8
576.2.k.b.145.4 8 96.53 odd 8
576.2.k.b.433.4 8 96.77 odd 8
1152.2.k.c.289.1 8 96.59 even 8
1152.2.k.c.865.1 8 96.35 even 8
1152.2.k.f.289.1 8 96.5 odd 8
1152.2.k.f.865.1 8 96.29 odd 8
3072.2.a.i.1.1 4 16.3 odd 4
3072.2.a.n.1.4 4 16.5 even 4
3072.2.a.o.1.1 4 16.13 even 4
3072.2.a.t.1.4 4 16.11 odd 4
3072.2.d.f.1537.4 8 8.3 odd 2
3072.2.d.f.1537.5 8 4.3 odd 2
3072.2.d.i.1537.1 8 1.1 even 1 trivial
3072.2.d.i.1537.8 8 8.5 even 2 inner
9216.2.a.x.1.1 4 48.5 odd 4
9216.2.a.y.1.1 4 48.11 even 4
9216.2.a.bn.1.4 4 48.29 odd 4
9216.2.a.bo.1.4 4 48.35 even 4