Properties

Label 3360.2.a.ba.1.2
Level $3360$
Weight $2$
Character 3360.1
Self dual yes
Analytic conductor $26.830$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.8297350792\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 3360.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +3.12311 q^{11} +5.12311 q^{13} +1.00000 q^{15} +2.00000 q^{17} -7.12311 q^{19} +1.00000 q^{21} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} +3.12311 q^{31} -3.12311 q^{33} +1.00000 q^{35} -2.00000 q^{37} -5.12311 q^{39} +2.00000 q^{41} +6.24621 q^{43} -1.00000 q^{45} +1.00000 q^{49} -2.00000 q^{51} -11.3693 q^{53} -3.12311 q^{55} +7.12311 q^{57} +4.00000 q^{59} +8.24621 q^{61} -1.00000 q^{63} -5.12311 q^{65} -14.2462 q^{67} -3.12311 q^{71} +6.87689 q^{73} -1.00000 q^{75} -3.12311 q^{77} -14.2462 q^{79} +1.00000 q^{81} +4.00000 q^{83} -2.00000 q^{85} +2.00000 q^{87} +16.2462 q^{89} -5.12311 q^{91} -3.12311 q^{93} +7.12311 q^{95} +13.1231 q^{97} +3.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} - 6 q^{19} + 2 q^{21} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 2 q^{31} + 2 q^{33} + 2 q^{35} - 4 q^{37} - 2 q^{39} + 4 q^{41} - 4 q^{43} - 2 q^{45} + 2 q^{49} - 4 q^{51} + 2 q^{53} + 2 q^{55} + 6 q^{57} + 8 q^{59} - 2 q^{63} - 2 q^{65} - 12 q^{67} + 2 q^{71} + 22 q^{73} - 2 q^{75} + 2 q^{77} - 12 q^{79} + 2 q^{81} + 8 q^{83} - 4 q^{85} + 4 q^{87} + 16 q^{89} - 2 q^{91} + 2 q^{93} + 6 q^{95} + 18 q^{97} - 2 q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.12311 0.941652 0.470826 0.882226i \(-0.343956\pi\)
0.470826 + 0.882226i \(0.343956\pi\)
\(12\) 0 0
\(13\) 5.12311 1.42089 0.710447 0.703751i \(-0.248493\pi\)
0.710447 + 0.703751i \(0.248493\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −7.12311 −1.63415 −0.817076 0.576530i \(-0.804407\pi\)
−0.817076 + 0.576530i \(0.804407\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 3.12311 0.560926 0.280463 0.959865i \(-0.409512\pi\)
0.280463 + 0.959865i \(0.409512\pi\)
\(32\) 0 0
\(33\) −3.12311 −0.543663
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) −5.12311 −0.820353
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 6.24621 0.952538 0.476269 0.879300i \(-0.341989\pi\)
0.476269 + 0.879300i \(0.341989\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) −11.3693 −1.56170 −0.780848 0.624721i \(-0.785213\pi\)
−0.780848 + 0.624721i \(0.785213\pi\)
\(54\) 0 0
\(55\) −3.12311 −0.421119
\(56\) 0 0
\(57\) 7.12311 0.943478
\(58\) 0 0
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 8.24621 1.05582 0.527910 0.849301i \(-0.322976\pi\)
0.527910 + 0.849301i \(0.322976\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −5.12311 −0.635443
\(66\) 0 0
\(67\) −14.2462 −1.74045 −0.870226 0.492653i \(-0.836027\pi\)
−0.870226 + 0.492653i \(0.836027\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.12311 −0.370644 −0.185322 0.982678i \(-0.559333\pi\)
−0.185322 + 0.982678i \(0.559333\pi\)
\(72\) 0 0
\(73\) 6.87689 0.804880 0.402440 0.915446i \(-0.368162\pi\)
0.402440 + 0.915446i \(0.368162\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −3.12311 −0.355911
\(78\) 0 0
\(79\) −14.2462 −1.60282 −0.801412 0.598113i \(-0.795918\pi\)
−0.801412 + 0.598113i \(0.795918\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) 16.2462 1.72209 0.861047 0.508525i \(-0.169809\pi\)
0.861047 + 0.508525i \(0.169809\pi\)
\(90\) 0 0
\(91\) −5.12311 −0.537047
\(92\) 0 0
\(93\) −3.12311 −0.323851
\(94\) 0 0
\(95\) 7.12311 0.730815
\(96\) 0 0
\(97\) 13.1231 1.33245 0.666225 0.745751i \(-0.267909\pi\)
0.666225 + 0.745751i \(0.267909\pi\)
\(98\) 0 0
\(99\) 3.12311 0.313884
\(100\) 0 0
\(101\) 16.2462 1.61656 0.808279 0.588799i \(-0.200399\pi\)
0.808279 + 0.588799i \(0.200399\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) −8.24621 −0.789844 −0.394922 0.918715i \(-0.629228\pi\)
−0.394922 + 0.918715i \(0.629228\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 19.3693 1.82211 0.911056 0.412283i \(-0.135268\pi\)
0.911056 + 0.412283i \(0.135268\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 5.12311 0.473631
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −1.24621 −0.113292
\(122\) 0 0
\(123\) −2.00000 −0.180334
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 12.4924 1.10852 0.554262 0.832343i \(-0.313001\pi\)
0.554262 + 0.832343i \(0.313001\pi\)
\(128\) 0 0
\(129\) −6.24621 −0.549948
\(130\) 0 0
\(131\) −18.2462 −1.59418 −0.797089 0.603861i \(-0.793628\pi\)
−0.797089 + 0.603861i \(0.793628\pi\)
\(132\) 0 0
\(133\) 7.12311 0.617652
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 21.1231 1.80467 0.902334 0.431037i \(-0.141852\pi\)
0.902334 + 0.431037i \(0.141852\pi\)
\(138\) 0 0
\(139\) 0.876894 0.0743772 0.0371886 0.999308i \(-0.488160\pi\)
0.0371886 + 0.999308i \(0.488160\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 16.0000 1.33799
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 4.24621 0.347863 0.173932 0.984758i \(-0.444353\pi\)
0.173932 + 0.984758i \(0.444353\pi\)
\(150\) 0 0
\(151\) 12.4924 1.01662 0.508309 0.861174i \(-0.330271\pi\)
0.508309 + 0.861174i \(0.330271\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) −3.12311 −0.250854
\(156\) 0 0
\(157\) −1.12311 −0.0896336 −0.0448168 0.998995i \(-0.514270\pi\)
−0.0448168 + 0.998995i \(0.514270\pi\)
\(158\) 0 0
\(159\) 11.3693 0.901645
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 6.24621 0.489241 0.244621 0.969619i \(-0.421337\pi\)
0.244621 + 0.969619i \(0.421337\pi\)
\(164\) 0 0
\(165\) 3.12311 0.243133
\(166\) 0 0
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) 13.2462 1.01894
\(170\) 0 0
\(171\) −7.12311 −0.544718
\(172\) 0 0
\(173\) 16.2462 1.23518 0.617588 0.786502i \(-0.288110\pi\)
0.617588 + 0.786502i \(0.288110\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) 11.1231 0.831380 0.415690 0.909506i \(-0.363540\pi\)
0.415690 + 0.909506i \(0.363540\pi\)
\(180\) 0 0
\(181\) −10.4924 −0.779896 −0.389948 0.920837i \(-0.627507\pi\)
−0.389948 + 0.920837i \(0.627507\pi\)
\(182\) 0 0
\(183\) −8.24621 −0.609577
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) 6.24621 0.456768
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 3.12311 0.225980 0.112990 0.993596i \(-0.463957\pi\)
0.112990 + 0.993596i \(0.463957\pi\)
\(192\) 0 0
\(193\) 18.0000 1.29567 0.647834 0.761781i \(-0.275675\pi\)
0.647834 + 0.761781i \(0.275675\pi\)
\(194\) 0 0
\(195\) 5.12311 0.366873
\(196\) 0 0
\(197\) −17.6155 −1.25505 −0.627527 0.778595i \(-0.715933\pi\)
−0.627527 + 0.778595i \(0.715933\pi\)
\(198\) 0 0
\(199\) 3.12311 0.221391 0.110696 0.993854i \(-0.464692\pi\)
0.110696 + 0.993854i \(0.464692\pi\)
\(200\) 0 0
\(201\) 14.2462 1.00485
\(202\) 0 0
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −22.2462 −1.53880
\(210\) 0 0
\(211\) −22.2462 −1.53149 −0.765746 0.643143i \(-0.777630\pi\)
−0.765746 + 0.643143i \(0.777630\pi\)
\(212\) 0 0
\(213\) 3.12311 0.213992
\(214\) 0 0
\(215\) −6.24621 −0.425988
\(216\) 0 0
\(217\) −3.12311 −0.212010
\(218\) 0 0
\(219\) −6.87689 −0.464697
\(220\) 0 0
\(221\) 10.2462 0.689235
\(222\) 0 0
\(223\) 1.75379 0.117442 0.0587212 0.998274i \(-0.481298\pi\)
0.0587212 + 0.998274i \(0.481298\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 4.00000 0.265489 0.132745 0.991150i \(-0.457621\pi\)
0.132745 + 0.991150i \(0.457621\pi\)
\(228\) 0 0
\(229\) −20.2462 −1.33791 −0.668954 0.743304i \(-0.733258\pi\)
−0.668954 + 0.743304i \(0.733258\pi\)
\(230\) 0 0
\(231\) 3.12311 0.205485
\(232\) 0 0
\(233\) 14.8769 0.974618 0.487309 0.873230i \(-0.337979\pi\)
0.487309 + 0.873230i \(0.337979\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 14.2462 0.925391
\(238\) 0 0
\(239\) 4.87689 0.315460 0.157730 0.987482i \(-0.449582\pi\)
0.157730 + 0.987482i \(0.449582\pi\)
\(240\) 0 0
\(241\) 3.75379 0.241803 0.120901 0.992665i \(-0.461422\pi\)
0.120901 + 0.992665i \(0.461422\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −36.4924 −2.32196
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 2.00000 0.125245
\(256\) 0 0
\(257\) 3.75379 0.234155 0.117077 0.993123i \(-0.462647\pi\)
0.117077 + 0.993123i \(0.462647\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 0 0
\(263\) 1.75379 0.108143 0.0540716 0.998537i \(-0.482780\pi\)
0.0540716 + 0.998537i \(0.482780\pi\)
\(264\) 0 0
\(265\) 11.3693 0.698412
\(266\) 0 0
\(267\) −16.2462 −0.994252
\(268\) 0 0
\(269\) 0.246211 0.0150118 0.00750588 0.999972i \(-0.497611\pi\)
0.00750588 + 0.999972i \(0.497611\pi\)
\(270\) 0 0
\(271\) −23.6155 −1.43454 −0.717271 0.696795i \(-0.754609\pi\)
−0.717271 + 0.696795i \(0.754609\pi\)
\(272\) 0 0
\(273\) 5.12311 0.310064
\(274\) 0 0
\(275\) 3.12311 0.188330
\(276\) 0 0
\(277\) 18.4924 1.11110 0.555551 0.831482i \(-0.312507\pi\)
0.555551 + 0.831482i \(0.312507\pi\)
\(278\) 0 0
\(279\) 3.12311 0.186975
\(280\) 0 0
\(281\) −24.7386 −1.47578 −0.737892 0.674919i \(-0.764178\pi\)
−0.737892 + 0.674919i \(0.764178\pi\)
\(282\) 0 0
\(283\) 20.0000 1.18888 0.594438 0.804141i \(-0.297374\pi\)
0.594438 + 0.804141i \(0.297374\pi\)
\(284\) 0 0
\(285\) −7.12311 −0.421936
\(286\) 0 0
\(287\) −2.00000 −0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) −13.1231 −0.769290
\(292\) 0 0
\(293\) 0.246211 0.0143838 0.00719191 0.999974i \(-0.497711\pi\)
0.00719191 + 0.999974i \(0.497711\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) −3.12311 −0.181221
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −6.24621 −0.360026
\(302\) 0 0
\(303\) −16.2462 −0.933320
\(304\) 0 0
\(305\) −8.24621 −0.472177
\(306\) 0 0
\(307\) 26.2462 1.49795 0.748975 0.662598i \(-0.230546\pi\)
0.748975 + 0.662598i \(0.230546\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) 0 0
\(313\) 8.63068 0.487835 0.243918 0.969796i \(-0.421567\pi\)
0.243918 + 0.969796i \(0.421567\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) −6.87689 −0.386245 −0.193122 0.981175i \(-0.561861\pi\)
−0.193122 + 0.981175i \(0.561861\pi\)
\(318\) 0 0
\(319\) −6.24621 −0.349721
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) −14.2462 −0.792680
\(324\) 0 0
\(325\) 5.12311 0.284179
\(326\) 0 0
\(327\) 8.24621 0.456017
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 14.2462 0.783043 0.391521 0.920169i \(-0.371949\pi\)
0.391521 + 0.920169i \(0.371949\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 14.2462 0.778354
\(336\) 0 0
\(337\) −14.0000 −0.762629 −0.381314 0.924445i \(-0.624528\pi\)
−0.381314 + 0.924445i \(0.624528\pi\)
\(338\) 0 0
\(339\) −19.3693 −1.05200
\(340\) 0 0
\(341\) 9.75379 0.528197
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.75379 0.523611 0.261805 0.965121i \(-0.415682\pi\)
0.261805 + 0.965121i \(0.415682\pi\)
\(348\) 0 0
\(349\) −28.2462 −1.51199 −0.755993 0.654580i \(-0.772845\pi\)
−0.755993 + 0.654580i \(0.772845\pi\)
\(350\) 0 0
\(351\) −5.12311 −0.273451
\(352\) 0 0
\(353\) 22.4924 1.19715 0.598575 0.801066i \(-0.295734\pi\)
0.598575 + 0.801066i \(0.295734\pi\)
\(354\) 0 0
\(355\) 3.12311 0.165757
\(356\) 0 0
\(357\) 2.00000 0.105851
\(358\) 0 0
\(359\) 7.61553 0.401932 0.200966 0.979598i \(-0.435592\pi\)
0.200966 + 0.979598i \(0.435592\pi\)
\(360\) 0 0
\(361\) 31.7386 1.67045
\(362\) 0 0
\(363\) 1.24621 0.0654091
\(364\) 0 0
\(365\) −6.87689 −0.359953
\(366\) 0 0
\(367\) −30.2462 −1.57884 −0.789420 0.613854i \(-0.789618\pi\)
−0.789420 + 0.613854i \(0.789618\pi\)
\(368\) 0 0
\(369\) 2.00000 0.104116
\(370\) 0 0
\(371\) 11.3693 0.590266
\(372\) 0 0
\(373\) −3.75379 −0.194364 −0.0971819 0.995267i \(-0.530983\pi\)
−0.0971819 + 0.995267i \(0.530983\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −10.2462 −0.527707
\(378\) 0 0
\(379\) 20.4924 1.05263 0.526313 0.850291i \(-0.323574\pi\)
0.526313 + 0.850291i \(0.323574\pi\)
\(380\) 0 0
\(381\) −12.4924 −0.640006
\(382\) 0 0
\(383\) 28.4924 1.45589 0.727947 0.685633i \(-0.240474\pi\)
0.727947 + 0.685633i \(0.240474\pi\)
\(384\) 0 0
\(385\) 3.12311 0.159168
\(386\) 0 0
\(387\) 6.24621 0.317513
\(388\) 0 0
\(389\) −3.75379 −0.190325 −0.0951623 0.995462i \(-0.530337\pi\)
−0.0951623 + 0.995462i \(0.530337\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 18.2462 0.920400
\(394\) 0 0
\(395\) 14.2462 0.716805
\(396\) 0 0
\(397\) −10.8769 −0.545896 −0.272948 0.962029i \(-0.587999\pi\)
−0.272948 + 0.962029i \(0.587999\pi\)
\(398\) 0 0
\(399\) −7.12311 −0.356601
\(400\) 0 0
\(401\) 26.9848 1.34756 0.673779 0.738933i \(-0.264670\pi\)
0.673779 + 0.738933i \(0.264670\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −6.24621 −0.309613
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) −21.1231 −1.04193
\(412\) 0 0
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) −0.876894 −0.0429417
\(418\) 0 0
\(419\) 34.2462 1.67304 0.836518 0.547939i \(-0.184587\pi\)
0.836518 + 0.547939i \(0.184587\pi\)
\(420\) 0 0
\(421\) 26.4924 1.29116 0.645581 0.763692i \(-0.276615\pi\)
0.645581 + 0.763692i \(0.276615\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.00000 0.0970143
\(426\) 0 0
\(427\) −8.24621 −0.399062
\(428\) 0 0
\(429\) −16.0000 −0.772487
\(430\) 0 0
\(431\) −7.61553 −0.366827 −0.183414 0.983036i \(-0.558715\pi\)
−0.183414 + 0.983036i \(0.558715\pi\)
\(432\) 0 0
\(433\) 13.1231 0.630656 0.315328 0.948983i \(-0.397885\pi\)
0.315328 + 0.948983i \(0.397885\pi\)
\(434\) 0 0
\(435\) −2.00000 −0.0958927
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 7.61553 0.363469 0.181735 0.983348i \(-0.441829\pi\)
0.181735 + 0.983348i \(0.441829\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −26.7386 −1.27039 −0.635195 0.772351i \(-0.719080\pi\)
−0.635195 + 0.772351i \(0.719080\pi\)
\(444\) 0 0
\(445\) −16.2462 −0.770144
\(446\) 0 0
\(447\) −4.24621 −0.200839
\(448\) 0 0
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) 6.24621 0.294123
\(452\) 0 0
\(453\) −12.4924 −0.586945
\(454\) 0 0
\(455\) 5.12311 0.240175
\(456\) 0 0
\(457\) −12.2462 −0.572854 −0.286427 0.958102i \(-0.592468\pi\)
−0.286427 + 0.958102i \(0.592468\pi\)
\(458\) 0 0
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −18.4924 −0.861278 −0.430639 0.902524i \(-0.641712\pi\)
−0.430639 + 0.902524i \(0.641712\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 3.12311 0.144831
\(466\) 0 0
\(467\) 28.9848 1.34126 0.670629 0.741793i \(-0.266024\pi\)
0.670629 + 0.741793i \(0.266024\pi\)
\(468\) 0 0
\(469\) 14.2462 0.657829
\(470\) 0 0
\(471\) 1.12311 0.0517500
\(472\) 0 0
\(473\) 19.5076 0.896959
\(474\) 0 0
\(475\) −7.12311 −0.326831
\(476\) 0 0
\(477\) −11.3693 −0.520565
\(478\) 0 0
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) −10.2462 −0.467187
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −13.1231 −0.595890
\(486\) 0 0
\(487\) −32.0000 −1.45006 −0.725029 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) 0 0
\(489\) −6.24621 −0.282463
\(490\) 0 0
\(491\) −4.87689 −0.220091 −0.110046 0.993927i \(-0.535100\pi\)
−0.110046 + 0.993927i \(0.535100\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) 0 0
\(495\) −3.12311 −0.140373
\(496\) 0 0
\(497\) 3.12311 0.140090
\(498\) 0 0
\(499\) −12.4924 −0.559238 −0.279619 0.960111i \(-0.590208\pi\)
−0.279619 + 0.960111i \(0.590208\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) 0 0
\(503\) 36.4924 1.62712 0.813558 0.581483i \(-0.197527\pi\)
0.813558 + 0.581483i \(0.197527\pi\)
\(504\) 0 0
\(505\) −16.2462 −0.722947
\(506\) 0 0
\(507\) −13.2462 −0.588285
\(508\) 0 0
\(509\) −28.2462 −1.25199 −0.625996 0.779827i \(-0.715307\pi\)
−0.625996 + 0.779827i \(0.715307\pi\)
\(510\) 0 0
\(511\) −6.87689 −0.304216
\(512\) 0 0
\(513\) 7.12311 0.314493
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −16.2462 −0.713130
\(520\) 0 0
\(521\) −10.4924 −0.459681 −0.229841 0.973228i \(-0.573821\pi\)
−0.229841 + 0.973228i \(0.573821\pi\)
\(522\) 0 0
\(523\) −14.7386 −0.644475 −0.322238 0.946659i \(-0.604435\pi\)
−0.322238 + 0.946659i \(0.604435\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) 6.24621 0.272089
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 10.2462 0.443813
\(534\) 0 0
\(535\) −8.00000 −0.345870
\(536\) 0 0
\(537\) −11.1231 −0.479997
\(538\) 0 0
\(539\) 3.12311 0.134522
\(540\) 0 0
\(541\) 18.4924 0.795051 0.397526 0.917591i \(-0.369869\pi\)
0.397526 + 0.917591i \(0.369869\pi\)
\(542\) 0 0
\(543\) 10.4924 0.450273
\(544\) 0 0
\(545\) 8.24621 0.353229
\(546\) 0 0
\(547\) −38.2462 −1.63529 −0.817645 0.575723i \(-0.804721\pi\)
−0.817645 + 0.575723i \(0.804721\pi\)
\(548\) 0 0
\(549\) 8.24621 0.351940
\(550\) 0 0
\(551\) 14.2462 0.606909
\(552\) 0 0
\(553\) 14.2462 0.605811
\(554\) 0 0
\(555\) −2.00000 −0.0848953
\(556\) 0 0
\(557\) −6.87689 −0.291383 −0.145692 0.989330i \(-0.546541\pi\)
−0.145692 + 0.989330i \(0.546541\pi\)
\(558\) 0 0
\(559\) 32.0000 1.35346
\(560\) 0 0
\(561\) −6.24621 −0.263715
\(562\) 0 0
\(563\) −16.4924 −0.695073 −0.347536 0.937667i \(-0.612982\pi\)
−0.347536 + 0.937667i \(0.612982\pi\)
\(564\) 0 0
\(565\) −19.3693 −0.814873
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 36.4924 1.52716 0.763580 0.645713i \(-0.223440\pi\)
0.763580 + 0.645713i \(0.223440\pi\)
\(572\) 0 0
\(573\) −3.12311 −0.130470
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 17.6155 0.733344 0.366672 0.930350i \(-0.380497\pi\)
0.366672 + 0.930350i \(0.380497\pi\)
\(578\) 0 0
\(579\) −18.0000 −0.748054
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 0 0
\(583\) −35.5076 −1.47057
\(584\) 0 0
\(585\) −5.12311 −0.211814
\(586\) 0 0
\(587\) −0.492423 −0.0203245 −0.0101622 0.999948i \(-0.503235\pi\)
−0.0101622 + 0.999948i \(0.503235\pi\)
\(588\) 0 0
\(589\) −22.2462 −0.916639
\(590\) 0 0
\(591\) 17.6155 0.724606
\(592\) 0 0
\(593\) −2.49242 −0.102352 −0.0511758 0.998690i \(-0.516297\pi\)
−0.0511758 + 0.998690i \(0.516297\pi\)
\(594\) 0 0
\(595\) 2.00000 0.0819920
\(596\) 0 0
\(597\) −3.12311 −0.127820
\(598\) 0 0
\(599\) 3.12311 0.127607 0.0638033 0.997962i \(-0.479677\pi\)
0.0638033 + 0.997962i \(0.479677\pi\)
\(600\) 0 0
\(601\) 0.246211 0.0100432 0.00502158 0.999987i \(-0.498402\pi\)
0.00502158 + 0.999987i \(0.498402\pi\)
\(602\) 0 0
\(603\) −14.2462 −0.580151
\(604\) 0 0
\(605\) 1.24621 0.0506657
\(606\) 0 0
\(607\) −30.2462 −1.22766 −0.613828 0.789440i \(-0.710371\pi\)
−0.613828 + 0.789440i \(0.710371\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −42.9848 −1.73614 −0.868071 0.496440i \(-0.834640\pi\)
−0.868071 + 0.496440i \(0.834640\pi\)
\(614\) 0 0
\(615\) 2.00000 0.0806478
\(616\) 0 0
\(617\) 11.3693 0.457711 0.228856 0.973460i \(-0.426502\pi\)
0.228856 + 0.973460i \(0.426502\pi\)
\(618\) 0 0
\(619\) 8.87689 0.356793 0.178396 0.983959i \(-0.442909\pi\)
0.178396 + 0.983959i \(0.442909\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −16.2462 −0.650891
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 22.2462 0.888428
\(628\) 0 0
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) −14.2462 −0.567133 −0.283566 0.958953i \(-0.591518\pi\)
−0.283566 + 0.958953i \(0.591518\pi\)
\(632\) 0 0
\(633\) 22.2462 0.884208
\(634\) 0 0
\(635\) −12.4924 −0.495747
\(636\) 0 0
\(637\) 5.12311 0.202985
\(638\) 0 0
\(639\) −3.12311 −0.123548
\(640\) 0 0
\(641\) 8.24621 0.325706 0.162853 0.986650i \(-0.447930\pi\)
0.162853 + 0.986650i \(0.447930\pi\)
\(642\) 0 0
\(643\) 5.75379 0.226907 0.113454 0.993543i \(-0.463809\pi\)
0.113454 + 0.993543i \(0.463809\pi\)
\(644\) 0 0
\(645\) 6.24621 0.245944
\(646\) 0 0
\(647\) −24.9848 −0.982256 −0.491128 0.871088i \(-0.663415\pi\)
−0.491128 + 0.871088i \(0.663415\pi\)
\(648\) 0 0
\(649\) 12.4924 0.490370
\(650\) 0 0
\(651\) 3.12311 0.122404
\(652\) 0 0
\(653\) −33.6155 −1.31548 −0.657739 0.753246i \(-0.728487\pi\)
−0.657739 + 0.753246i \(0.728487\pi\)
\(654\) 0 0
\(655\) 18.2462 0.712938
\(656\) 0 0
\(657\) 6.87689 0.268293
\(658\) 0 0
\(659\) −31.6155 −1.23157 −0.615783 0.787916i \(-0.711160\pi\)
−0.615783 + 0.787916i \(0.711160\pi\)
\(660\) 0 0
\(661\) 48.2462 1.87656 0.938280 0.345876i \(-0.112418\pi\)
0.938280 + 0.345876i \(0.112418\pi\)
\(662\) 0 0
\(663\) −10.2462 −0.397930
\(664\) 0 0
\(665\) −7.12311 −0.276222
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −1.75379 −0.0678054
\(670\) 0 0
\(671\) 25.7538 0.994214
\(672\) 0 0
\(673\) −45.2311 −1.74353 −0.871765 0.489925i \(-0.837024\pi\)
−0.871765 + 0.489925i \(0.837024\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 24.2462 0.931858 0.465929 0.884822i \(-0.345720\pi\)
0.465929 + 0.884822i \(0.345720\pi\)
\(678\) 0 0
\(679\) −13.1231 −0.503619
\(680\) 0 0
\(681\) −4.00000 −0.153280
\(682\) 0 0
\(683\) 30.2462 1.15734 0.578670 0.815562i \(-0.303572\pi\)
0.578670 + 0.815562i \(0.303572\pi\)
\(684\) 0 0
\(685\) −21.1231 −0.807072
\(686\) 0 0
\(687\) 20.2462 0.772441
\(688\) 0 0
\(689\) −58.2462 −2.21900
\(690\) 0 0
\(691\) −45.3693 −1.72593 −0.862965 0.505264i \(-0.831395\pi\)
−0.862965 + 0.505264i \(0.831395\pi\)
\(692\) 0 0
\(693\) −3.12311 −0.118637
\(694\) 0 0
\(695\) −0.876894 −0.0332625
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) −14.8769 −0.562696
\(700\) 0 0
\(701\) −5.50758 −0.208018 −0.104009 0.994576i \(-0.533167\pi\)
−0.104009 + 0.994576i \(0.533167\pi\)
\(702\) 0 0
\(703\) 14.2462 0.537306
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −16.2462 −0.611002
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) −14.2462 −0.534275
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) 0 0
\(717\) −4.87689 −0.182131
\(718\) 0 0
\(719\) −39.2311 −1.46307 −0.731536 0.681803i \(-0.761196\pi\)
−0.731536 + 0.681803i \(0.761196\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −3.75379 −0.139605
\(724\) 0 0
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) 18.7386 0.694977 0.347489 0.937684i \(-0.387034\pi\)
0.347489 + 0.937684i \(0.387034\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.4924 0.462049
\(732\) 0 0
\(733\) −6.38447 −0.235816 −0.117908 0.993025i \(-0.537619\pi\)
−0.117908 + 0.993025i \(0.537619\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) −44.4924 −1.63890
\(738\) 0 0
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 0 0
\(741\) 36.4924 1.34058
\(742\) 0 0
\(743\) 34.7386 1.27444 0.637218 0.770683i \(-0.280085\pi\)
0.637218 + 0.770683i \(0.280085\pi\)
\(744\) 0 0
\(745\) −4.24621 −0.155569
\(746\) 0 0
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) −8.00000 −0.292314
\(750\) 0 0
\(751\) 6.24621 0.227927 0.113964 0.993485i \(-0.463645\pi\)
0.113964 + 0.993485i \(0.463645\pi\)
\(752\) 0 0
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) −12.4924 −0.454646
\(756\) 0 0
\(757\) 52.2462 1.89892 0.949460 0.313887i \(-0.101631\pi\)
0.949460 + 0.313887i \(0.101631\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 20.7386 0.751775 0.375887 0.926665i \(-0.377338\pi\)
0.375887 + 0.926665i \(0.377338\pi\)
\(762\) 0 0
\(763\) 8.24621 0.298533
\(764\) 0 0
\(765\) −2.00000 −0.0723102
\(766\) 0 0
\(767\) 20.4924 0.739938
\(768\) 0 0
\(769\) −15.7538 −0.568096 −0.284048 0.958810i \(-0.591678\pi\)
−0.284048 + 0.958810i \(0.591678\pi\)
\(770\) 0 0
\(771\) −3.75379 −0.135189
\(772\) 0 0
\(773\) 0.246211 0.00885560 0.00442780 0.999990i \(-0.498591\pi\)
0.00442780 + 0.999990i \(0.498591\pi\)
\(774\) 0 0
\(775\) 3.12311 0.112185
\(776\) 0 0
\(777\) −2.00000 −0.0717496
\(778\) 0 0
\(779\) −14.2462 −0.510423
\(780\) 0 0
\(781\) −9.75379 −0.349018
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) 1.12311 0.0400854
\(786\) 0 0
\(787\) 51.2311 1.82619 0.913095 0.407747i \(-0.133685\pi\)
0.913095 + 0.407747i \(0.133685\pi\)
\(788\) 0 0
\(789\) −1.75379 −0.0624365
\(790\) 0 0
\(791\) −19.3693 −0.688694
\(792\) 0 0
\(793\) 42.2462 1.50021
\(794\) 0 0
\(795\) −11.3693 −0.403228
\(796\) 0 0
\(797\) 12.7386 0.451226 0.225613 0.974217i \(-0.427562\pi\)
0.225613 + 0.974217i \(0.427562\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 16.2462 0.574032
\(802\) 0 0
\(803\) 21.4773 0.757916
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −0.246211 −0.00866705
\(808\) 0 0
\(809\) 3.75379 0.131976 0.0659881 0.997820i \(-0.478980\pi\)
0.0659881 + 0.997820i \(0.478980\pi\)
\(810\) 0 0
\(811\) 12.3845 0.434878 0.217439 0.976074i \(-0.430230\pi\)
0.217439 + 0.976074i \(0.430230\pi\)
\(812\) 0 0
\(813\) 23.6155 0.828233
\(814\) 0 0
\(815\) −6.24621 −0.218795
\(816\) 0 0
\(817\) −44.4924 −1.55659
\(818\) 0 0
\(819\) −5.12311 −0.179016
\(820\) 0 0
\(821\) −10.9848 −0.383374 −0.191687 0.981456i \(-0.561396\pi\)
−0.191687 + 0.981456i \(0.561396\pi\)
\(822\) 0 0
\(823\) −20.4924 −0.714321 −0.357160 0.934043i \(-0.616255\pi\)
−0.357160 + 0.934043i \(0.616255\pi\)
\(824\) 0 0
\(825\) −3.12311 −0.108733
\(826\) 0 0
\(827\) 16.9848 0.590621 0.295310 0.955401i \(-0.404577\pi\)
0.295310 + 0.955401i \(0.404577\pi\)
\(828\) 0 0
\(829\) −26.4924 −0.920120 −0.460060 0.887888i \(-0.652172\pi\)
−0.460060 + 0.887888i \(0.652172\pi\)
\(830\) 0 0
\(831\) −18.4924 −0.641495
\(832\) 0 0
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 0 0
\(837\) −3.12311 −0.107950
\(838\) 0 0
\(839\) −10.7386 −0.370739 −0.185369 0.982669i \(-0.559348\pi\)
−0.185369 + 0.982669i \(0.559348\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 24.7386 0.852044
\(844\) 0 0
\(845\) −13.2462 −0.455684
\(846\) 0 0
\(847\) 1.24621 0.0428203
\(848\) 0 0
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 41.6155 1.42489 0.712444 0.701729i \(-0.247588\pi\)
0.712444 + 0.701729i \(0.247588\pi\)
\(854\) 0 0
\(855\) 7.12311 0.243605
\(856\) 0 0
\(857\) −31.7538 −1.08469 −0.542344 0.840156i \(-0.682463\pi\)
−0.542344 + 0.840156i \(0.682463\pi\)
\(858\) 0 0
\(859\) 19.6155 0.669273 0.334637 0.942347i \(-0.391386\pi\)
0.334637 + 0.942347i \(0.391386\pi\)
\(860\) 0 0
\(861\) 2.00000 0.0681598
\(862\) 0 0
\(863\) −40.0000 −1.36162 −0.680808 0.732462i \(-0.738371\pi\)
−0.680808 + 0.732462i \(0.738371\pi\)
\(864\) 0 0
\(865\) −16.2462 −0.552388
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −44.4924 −1.50930
\(870\) 0 0
\(871\) −72.9848 −2.47300
\(872\) 0 0
\(873\) 13.1231 0.444150
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −22.4924 −0.759515 −0.379758 0.925086i \(-0.623993\pi\)
−0.379758 + 0.925086i \(0.623993\pi\)
\(878\) 0 0
\(879\) −0.246211 −0.00830450
\(880\) 0 0
\(881\) −23.7538 −0.800285 −0.400143 0.916453i \(-0.631039\pi\)
−0.400143 + 0.916453i \(0.631039\pi\)
\(882\) 0 0
\(883\) −1.75379 −0.0590197 −0.0295098 0.999564i \(-0.509395\pi\)
−0.0295098 + 0.999564i \(0.509395\pi\)
\(884\) 0 0
\(885\) 4.00000 0.134459
\(886\) 0 0
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) −12.4924 −0.418982
\(890\) 0 0
\(891\) 3.12311 0.104628
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −11.1231 −0.371804
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −6.24621 −0.208323
\(900\) 0 0
\(901\) −22.7386 −0.757534
\(902\) 0 0
\(903\) 6.24621 0.207861
\(904\) 0 0
\(905\) 10.4924 0.348780
\(906\) 0 0
\(907\) −10.7386 −0.356570 −0.178285 0.983979i \(-0.557055\pi\)
−0.178285 + 0.983979i \(0.557055\pi\)
\(908\) 0 0
\(909\) 16.2462 0.538853
\(910\) 0 0
\(911\) 39.6155 1.31252 0.656261 0.754534i \(-0.272137\pi\)
0.656261 + 0.754534i \(0.272137\pi\)
\(912\) 0 0
\(913\) 12.4924 0.413439
\(914\) 0 0
\(915\) 8.24621 0.272611
\(916\) 0 0
\(917\) 18.2462 0.602543
\(918\) 0 0
\(919\) 1.75379 0.0578522 0.0289261 0.999582i \(-0.490791\pi\)
0.0289261 + 0.999582i \(0.490791\pi\)
\(920\) 0 0
\(921\) −26.2462 −0.864842
\(922\) 0 0
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −28.2462 −0.926728 −0.463364 0.886168i \(-0.653358\pi\)
−0.463364 + 0.886168i \(0.653358\pi\)
\(930\) 0 0
\(931\) −7.12311 −0.233450
\(932\) 0 0
\(933\) −16.0000 −0.523816
\(934\) 0 0
\(935\) −6.24621 −0.204273
\(936\) 0 0
\(937\) −34.1080 −1.11426 −0.557129 0.830426i \(-0.688097\pi\)
−0.557129 + 0.830426i \(0.688097\pi\)
\(938\) 0 0
\(939\) −8.63068 −0.281652
\(940\) 0 0
\(941\) −22.9848 −0.749285 −0.374642 0.927169i \(-0.622234\pi\)
−0.374642 + 0.927169i \(0.622234\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) −31.2311 −1.01487 −0.507436 0.861689i \(-0.669407\pi\)
−0.507436 + 0.861689i \(0.669407\pi\)
\(948\) 0 0
\(949\) 35.2311 1.14365
\(950\) 0 0
\(951\) 6.87689 0.222999
\(952\) 0 0
\(953\) 27.3693 0.886579 0.443290 0.896378i \(-0.353811\pi\)
0.443290 + 0.896378i \(0.353811\pi\)
\(954\) 0 0
\(955\) −3.12311 −0.101061
\(956\) 0 0
\(957\) 6.24621 0.201911
\(958\) 0 0
\(959\) −21.1231 −0.682101
\(960\) 0 0
\(961\) −21.2462 −0.685362
\(962\) 0 0
\(963\) 8.00000 0.257796
\(964\) 0 0
\(965\) −18.0000 −0.579441
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 0 0
\(969\) 14.2462 0.457654
\(970\) 0 0
\(971\) −44.0000 −1.41203 −0.706014 0.708198i \(-0.749508\pi\)
−0.706014 + 0.708198i \(0.749508\pi\)
\(972\) 0 0
\(973\) −0.876894 −0.0281119
\(974\) 0 0
\(975\) −5.12311 −0.164071
\(976\) 0 0
\(977\) 35.3693 1.13156 0.565782 0.824555i \(-0.308574\pi\)
0.565782 + 0.824555i \(0.308574\pi\)
\(978\) 0 0
\(979\) 50.7386 1.62161
\(980\) 0 0
\(981\) −8.24621 −0.263281
\(982\) 0 0
\(983\) 44.4924 1.41909 0.709544 0.704661i \(-0.248901\pi\)
0.709544 + 0.704661i \(0.248901\pi\)
\(984\) 0 0
\(985\) 17.6155 0.561277
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 58.7386 1.86589 0.932947 0.360013i \(-0.117228\pi\)
0.932947 + 0.360013i \(0.117228\pi\)
\(992\) 0 0
\(993\) −14.2462 −0.452090
\(994\) 0 0
\(995\) −3.12311 −0.0990091
\(996\) 0 0
\(997\) −19.8617 −0.629028 −0.314514 0.949253i \(-0.601841\pi\)
−0.314514 + 0.949253i \(0.601841\pi\)
\(998\) 0 0
\(999\) 2.00000 0.0632772
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 3360.2.a.ba.1.2 2
4.3 odd 2 3360.2.a.bg.1.1 yes 2
8.3 odd 2 6720.2.a.ct.1.2 2
8.5 even 2 6720.2.a.cw.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3360.2.a.ba.1.2 2 1.1 even 1 trivial
3360.2.a.bg.1.1 yes 2 4.3 odd 2
6720.2.a.ct.1.2 2 8.3 odd 2
6720.2.a.cw.1.1 2 8.5 even 2