Properties

Label 3920.2.a.bq.1.1
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3920.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} +O(q^{10})\) \(q-2.41421 q^{3} +1.00000 q^{5} +2.82843 q^{9} -4.82843 q^{11} +0.828427 q^{13} -2.41421 q^{15} -0.828427 q^{17} +2.82843 q^{19} +2.41421 q^{23} +1.00000 q^{25} +0.414214 q^{27} -1.00000 q^{29} +6.00000 q^{31} +11.6569 q^{33} -2.00000 q^{39} -2.17157 q^{41} -6.41421 q^{43} +2.82843 q^{45} -2.00000 q^{47} +2.00000 q^{51} -6.82843 q^{53} -4.82843 q^{55} -6.82843 q^{57} +12.4853 q^{59} -11.4853 q^{61} +0.828427 q^{65} -12.4142 q^{67} -5.82843 q^{69} +12.4853 q^{71} +4.82843 q^{73} -2.41421 q^{75} -9.17157 q^{79} -9.48528 q^{81} +11.7279 q^{83} -0.828427 q^{85} +2.41421 q^{87} +2.65685 q^{89} -14.4853 q^{93} +2.82843 q^{95} +0.343146 q^{97} -13.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{5} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{5} - 4q^{11} - 4q^{13} - 2q^{15} + 4q^{17} + 2q^{23} + 2q^{25} - 2q^{27} - 2q^{29} + 12q^{31} + 12q^{33} - 4q^{39} - 10q^{41} - 10q^{43} - 4q^{47} + 4q^{51} - 8q^{53} - 4q^{55} - 8q^{57} + 8q^{59} - 6q^{61} - 4q^{65} - 22q^{67} - 6q^{69} + 8q^{71} + 4q^{73} - 2q^{75} - 24q^{79} - 2q^{81} - 2q^{83} + 4q^{85} + 2q^{87} - 6q^{89} - 12q^{93} + 12q^{97} - 16q^{99} + O(q^{100}) \)

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
\(3\) −2.41421 −1.39385 −0.696923 0.717146i \(-0.745448\pi\)
−0.696923 + 0.717146i \(0.745448\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) −4.82843 −1.45583 −0.727913 0.685670i \(-0.759509\pi\)
−0.727913 + 0.685670i \(0.759509\pi\)
\(12\) 0 0
\(13\) 0.828427 0.229764 0.114882 0.993379i \(-0.463351\pi\)
0.114882 + 0.993379i \(0.463351\pi\)
\(14\) 0 0
\(15\) −2.41421 −0.623347
\(16\) 0 0
\(17\) −0.828427 −0.200923 −0.100462 0.994941i \(-0.532032\pi\)
−0.100462 + 0.994941i \(0.532032\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.41421 0.503398 0.251699 0.967806i \(-0.419011\pi\)
0.251699 + 0.967806i \(0.419011\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0.414214 0.0797154
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 0 0
\(33\) 11.6569 2.02920
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −2.17157 −0.339143 −0.169571 0.985518i \(-0.554238\pi\)
−0.169571 + 0.985518i \(0.554238\pi\)
\(42\) 0 0
\(43\) −6.41421 −0.978158 −0.489079 0.872239i \(-0.662667\pi\)
−0.489079 + 0.872239i \(0.662667\pi\)
\(44\) 0 0
\(45\) 2.82843 0.421637
\(46\) 0 0
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −6.82843 −0.937957 −0.468978 0.883210i \(-0.655378\pi\)
−0.468978 + 0.883210i \(0.655378\pi\)
\(54\) 0 0
\(55\) −4.82843 −0.651065
\(56\) 0 0
\(57\) −6.82843 −0.904447
\(58\) 0 0
\(59\) 12.4853 1.62545 0.812723 0.582651i \(-0.197984\pi\)
0.812723 + 0.582651i \(0.197984\pi\)
\(60\) 0 0
\(61\) −11.4853 −1.47054 −0.735270 0.677775i \(-0.762945\pi\)
−0.735270 + 0.677775i \(0.762945\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.828427 0.102754
\(66\) 0 0
\(67\) −12.4142 −1.51664 −0.758319 0.651884i \(-0.773979\pi\)
−0.758319 + 0.651884i \(0.773979\pi\)
\(68\) 0 0
\(69\) −5.82843 −0.701660
\(70\) 0 0
\(71\) 12.4853 1.48173 0.740865 0.671654i \(-0.234416\pi\)
0.740865 + 0.671654i \(0.234416\pi\)
\(72\) 0 0
\(73\) 4.82843 0.565125 0.282562 0.959249i \(-0.408816\pi\)
0.282562 + 0.959249i \(0.408816\pi\)
\(74\) 0 0
\(75\) −2.41421 −0.278769
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −9.17157 −1.03188 −0.515941 0.856624i \(-0.672558\pi\)
−0.515941 + 0.856624i \(0.672558\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) 11.7279 1.28731 0.643653 0.765317i \(-0.277418\pi\)
0.643653 + 0.765317i \(0.277418\pi\)
\(84\) 0 0
\(85\) −0.828427 −0.0898555
\(86\) 0 0
\(87\) 2.41421 0.258831
\(88\) 0 0
\(89\) 2.65685 0.281626 0.140813 0.990036i \(-0.455028\pi\)
0.140813 + 0.990036i \(0.455028\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −14.4853 −1.50205
\(94\) 0 0
\(95\) 2.82843 0.290191
\(96\) 0 0
\(97\) 0.343146 0.0348412 0.0174206 0.999848i \(-0.494455\pi\)
0.0174206 + 0.999848i \(0.494455\pi\)
\(98\) 0 0
\(99\) −13.6569 −1.37257
\(100\) 0 0
\(101\) 12.3137 1.22526 0.612630 0.790370i \(-0.290112\pi\)
0.612630 + 0.790370i \(0.290112\pi\)
\(102\) 0 0
\(103\) −0.414214 −0.0408137 −0.0204068 0.999792i \(-0.506496\pi\)
−0.0204068 + 0.999792i \(0.506496\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.75736 −0.266564 −0.133282 0.991078i \(-0.542552\pi\)
−0.133282 + 0.991078i \(0.542552\pi\)
\(108\) 0 0
\(109\) 3.48528 0.333829 0.166915 0.985971i \(-0.446620\pi\)
0.166915 + 0.985971i \(0.446620\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.4853 1.17452 0.587258 0.809400i \(-0.300207\pi\)
0.587258 + 0.809400i \(0.300207\pi\)
\(114\) 0 0
\(115\) 2.41421 0.225127
\(116\) 0 0
\(117\) 2.34315 0.216624
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 12.3137 1.11943
\(122\) 0 0
\(123\) 5.24264 0.472713
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −13.3137 −1.18140 −0.590700 0.806891i \(-0.701148\pi\)
−0.590700 + 0.806891i \(0.701148\pi\)
\(128\) 0 0
\(129\) 15.4853 1.36340
\(130\) 0 0
\(131\) −3.31371 −0.289520 −0.144760 0.989467i \(-0.546241\pi\)
−0.144760 + 0.989467i \(0.546241\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.414214 0.0356498
\(136\) 0 0
\(137\) 1.65685 0.141555 0.0707773 0.997492i \(-0.477452\pi\)
0.0707773 + 0.997492i \(0.477452\pi\)
\(138\) 0 0
\(139\) −12.1421 −1.02988 −0.514941 0.857225i \(-0.672186\pi\)
−0.514941 + 0.857225i \(0.672186\pi\)
\(140\) 0 0
\(141\) 4.82843 0.406627
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.82843 −0.641330 −0.320665 0.947193i \(-0.603906\pi\)
−0.320665 + 0.947193i \(0.603906\pi\)
\(150\) 0 0
\(151\) −0.343146 −0.0279248 −0.0139624 0.999903i \(-0.504445\pi\)
−0.0139624 + 0.999903i \(0.504445\pi\)
\(152\) 0 0
\(153\) −2.34315 −0.189432
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −5.31371 −0.424080 −0.212040 0.977261i \(-0.568011\pi\)
−0.212040 + 0.977261i \(0.568011\pi\)
\(158\) 0 0
\(159\) 16.4853 1.30737
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −23.6569 −1.85295 −0.926474 0.376359i \(-0.877176\pi\)
−0.926474 + 0.376359i \(0.877176\pi\)
\(164\) 0 0
\(165\) 11.6569 0.907485
\(166\) 0 0
\(167\) −19.5858 −1.51559 −0.757797 0.652491i \(-0.773724\pi\)
−0.757797 + 0.652491i \(0.773724\pi\)
\(168\) 0 0
\(169\) −12.3137 −0.947208
\(170\) 0 0
\(171\) 8.00000 0.611775
\(172\) 0 0
\(173\) −19.3137 −1.46839 −0.734197 0.678936i \(-0.762441\pi\)
−0.734197 + 0.678936i \(0.762441\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −30.1421 −2.26562
\(178\) 0 0
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −8.65685 −0.643459 −0.321729 0.946832i \(-0.604264\pi\)
−0.321729 + 0.946832i \(0.604264\pi\)
\(182\) 0 0
\(183\) 27.7279 2.04971
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.17157 0.518917 0.259458 0.965754i \(-0.416456\pi\)
0.259458 + 0.965754i \(0.416456\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 0 0
\(195\) −2.00000 −0.143223
\(196\) 0 0
\(197\) −23.6569 −1.68548 −0.842741 0.538320i \(-0.819059\pi\)
−0.842741 + 0.538320i \(0.819059\pi\)
\(198\) 0 0
\(199\) −1.65685 −0.117451 −0.0587256 0.998274i \(-0.518704\pi\)
−0.0587256 + 0.998274i \(0.518704\pi\)
\(200\) 0 0
\(201\) 29.9706 2.11396
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2.17157 −0.151669
\(206\) 0 0
\(207\) 6.82843 0.474608
\(208\) 0 0
\(209\) −13.6569 −0.944664
\(210\) 0 0
\(211\) −3.51472 −0.241963 −0.120982 0.992655i \(-0.538604\pi\)
−0.120982 + 0.992655i \(0.538604\pi\)
\(212\) 0 0
\(213\) −30.1421 −2.06531
\(214\) 0 0
\(215\) −6.41421 −0.437446
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −11.6569 −0.787697
\(220\) 0 0
\(221\) −0.686292 −0.0461650
\(222\) 0 0
\(223\) −11.6569 −0.780601 −0.390300 0.920688i \(-0.627629\pi\)
−0.390300 + 0.920688i \(0.627629\pi\)
\(224\) 0 0
\(225\) 2.82843 0.188562
\(226\) 0 0
\(227\) −26.9706 −1.79010 −0.895050 0.445967i \(-0.852860\pi\)
−0.895050 + 0.445967i \(0.852860\pi\)
\(228\) 0 0
\(229\) −0.343146 −0.0226757 −0.0113379 0.999936i \(-0.503609\pi\)
−0.0113379 + 0.999936i \(0.503609\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −11.1716 −0.731874 −0.365937 0.930640i \(-0.619251\pi\)
−0.365937 + 0.930640i \(0.619251\pi\)
\(234\) 0 0
\(235\) −2.00000 −0.130466
\(236\) 0 0
\(237\) 22.1421 1.43829
\(238\) 0 0
\(239\) −1.31371 −0.0849767 −0.0424884 0.999097i \(-0.513529\pi\)
−0.0424884 + 0.999097i \(0.513529\pi\)
\(240\) 0 0
\(241\) 16.3431 1.05275 0.526377 0.850251i \(-0.323550\pi\)
0.526377 + 0.850251i \(0.323550\pi\)
\(242\) 0 0
\(243\) 21.6569 1.38929
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.34315 0.149091
\(248\) 0 0
\(249\) −28.3137 −1.79431
\(250\) 0 0
\(251\) −13.3137 −0.840354 −0.420177 0.907442i \(-0.638032\pi\)
−0.420177 + 0.907442i \(0.638032\pi\)
\(252\) 0 0
\(253\) −11.6569 −0.732860
\(254\) 0 0
\(255\) 2.00000 0.125245
\(256\) 0 0
\(257\) 17.6569 1.10140 0.550702 0.834702i \(-0.314360\pi\)
0.550702 + 0.834702i \(0.314360\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2.82843 −0.175075
\(262\) 0 0
\(263\) −19.0416 −1.17416 −0.587079 0.809530i \(-0.699722\pi\)
−0.587079 + 0.809530i \(0.699722\pi\)
\(264\) 0 0
\(265\) −6.82843 −0.419467
\(266\) 0 0
\(267\) −6.41421 −0.392543
\(268\) 0 0
\(269\) −30.4558 −1.85693 −0.928463 0.371425i \(-0.878869\pi\)
−0.928463 + 0.371425i \(0.878869\pi\)
\(270\) 0 0
\(271\) 0.485281 0.0294787 0.0147394 0.999891i \(-0.495308\pi\)
0.0147394 + 0.999891i \(0.495308\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.82843 −0.291165
\(276\) 0 0
\(277\) −12.1421 −0.729550 −0.364775 0.931096i \(-0.618854\pi\)
−0.364775 + 0.931096i \(0.618854\pi\)
\(278\) 0 0
\(279\) 16.9706 1.01600
\(280\) 0 0
\(281\) 26.2843 1.56799 0.783994 0.620768i \(-0.213179\pi\)
0.783994 + 0.620768i \(0.213179\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) 0 0
\(285\) −6.82843 −0.404481
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −16.3137 −0.959630
\(290\) 0 0
\(291\) −0.828427 −0.0485633
\(292\) 0 0
\(293\) −16.0000 −0.934730 −0.467365 0.884064i \(-0.654797\pi\)
−0.467365 + 0.884064i \(0.654797\pi\)
\(294\) 0 0
\(295\) 12.4853 0.726921
\(296\) 0 0
\(297\) −2.00000 −0.116052
\(298\) 0 0
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −29.7279 −1.70782
\(304\) 0 0
\(305\) −11.4853 −0.657645
\(306\) 0 0
\(307\) 13.2426 0.755797 0.377899 0.925847i \(-0.376647\pi\)
0.377899 + 0.925847i \(0.376647\pi\)
\(308\) 0 0
\(309\) 1.00000 0.0568880
\(310\) 0 0
\(311\) 18.8284 1.06766 0.533831 0.845591i \(-0.320752\pi\)
0.533831 + 0.845591i \(0.320752\pi\)
\(312\) 0 0
\(313\) −17.6569 −0.998024 −0.499012 0.866595i \(-0.666304\pi\)
−0.499012 + 0.866595i \(0.666304\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 25.7990 1.44902 0.724508 0.689267i \(-0.242067\pi\)
0.724508 + 0.689267i \(0.242067\pi\)
\(318\) 0 0
\(319\) 4.82843 0.270340
\(320\) 0 0
\(321\) 6.65685 0.371549
\(322\) 0 0
\(323\) −2.34315 −0.130376
\(324\) 0 0
\(325\) 0.828427 0.0459529
\(326\) 0 0
\(327\) −8.41421 −0.465307
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 10.9706 0.602997 0.301498 0.953467i \(-0.402513\pi\)
0.301498 + 0.953467i \(0.402513\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.4142 −0.678261
\(336\) 0 0
\(337\) 14.8284 0.807756 0.403878 0.914813i \(-0.367662\pi\)
0.403878 + 0.914813i \(0.367662\pi\)
\(338\) 0 0
\(339\) −30.1421 −1.63710
\(340\) 0 0
\(341\) −28.9706 −1.56884
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −5.82843 −0.313792
\(346\) 0 0
\(347\) −22.0711 −1.18484 −0.592418 0.805630i \(-0.701827\pi\)
−0.592418 + 0.805630i \(0.701827\pi\)
\(348\) 0 0
\(349\) −26.6569 −1.42691 −0.713454 0.700702i \(-0.752870\pi\)
−0.713454 + 0.700702i \(0.752870\pi\)
\(350\) 0 0
\(351\) 0.343146 0.0183158
\(352\) 0 0
\(353\) −21.1716 −1.12685 −0.563425 0.826168i \(-0.690516\pi\)
−0.563425 + 0.826168i \(0.690516\pi\)
\(354\) 0 0
\(355\) 12.4853 0.662650
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) −29.7279 −1.56031
\(364\) 0 0
\(365\) 4.82843 0.252731
\(366\) 0 0
\(367\) 11.2426 0.586861 0.293431 0.955980i \(-0.405203\pi\)
0.293431 + 0.955980i \(0.405203\pi\)
\(368\) 0 0
\(369\) −6.14214 −0.319747
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 12.9706 0.671590 0.335795 0.941935i \(-0.390995\pi\)
0.335795 + 0.941935i \(0.390995\pi\)
\(374\) 0 0
\(375\) −2.41421 −0.124669
\(376\) 0 0
\(377\) −0.828427 −0.0426662
\(378\) 0 0
\(379\) −21.1716 −1.08751 −0.543755 0.839244i \(-0.682998\pi\)
−0.543755 + 0.839244i \(0.682998\pi\)
\(380\) 0 0
\(381\) 32.1421 1.64669
\(382\) 0 0
\(383\) −16.8995 −0.863524 −0.431762 0.901988i \(-0.642108\pi\)
−0.431762 + 0.901988i \(0.642108\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −18.1421 −0.922217
\(388\) 0 0
\(389\) 12.3431 0.625822 0.312911 0.949782i \(-0.398696\pi\)
0.312911 + 0.949782i \(0.398696\pi\)
\(390\) 0 0
\(391\) −2.00000 −0.101144
\(392\) 0 0
\(393\) 8.00000 0.403547
\(394\) 0 0
\(395\) −9.17157 −0.461472
\(396\) 0 0
\(397\) 28.6274 1.43677 0.718384 0.695646i \(-0.244882\pi\)
0.718384 + 0.695646i \(0.244882\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.68629 0.383835 0.191918 0.981411i \(-0.438529\pi\)
0.191918 + 0.981411i \(0.438529\pi\)
\(402\) 0 0
\(403\) 4.97056 0.247601
\(404\) 0 0
\(405\) −9.48528 −0.471327
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 24.7990 1.22623 0.613116 0.789993i \(-0.289916\pi\)
0.613116 + 0.789993i \(0.289916\pi\)
\(410\) 0 0
\(411\) −4.00000 −0.197305
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 11.7279 0.575701
\(416\) 0 0
\(417\) 29.3137 1.43550
\(418\) 0 0
\(419\) 23.3137 1.13895 0.569475 0.822009i \(-0.307147\pi\)
0.569475 + 0.822009i \(0.307147\pi\)
\(420\) 0 0
\(421\) −3.48528 −0.169862 −0.0849311 0.996387i \(-0.527067\pi\)
−0.0849311 + 0.996387i \(0.527067\pi\)
\(422\) 0 0
\(423\) −5.65685 −0.275046
\(424\) 0 0
\(425\) −0.828427 −0.0401846
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 9.65685 0.466237
\(430\) 0 0
\(431\) 21.7990 1.05002 0.525010 0.851096i \(-0.324062\pi\)
0.525010 + 0.851096i \(0.324062\pi\)
\(432\) 0 0
\(433\) −31.7990 −1.52816 −0.764081 0.645120i \(-0.776807\pi\)
−0.764081 + 0.645120i \(0.776807\pi\)
\(434\) 0 0
\(435\) 2.41421 0.115753
\(436\) 0 0
\(437\) 6.82843 0.326648
\(438\) 0 0
\(439\) −33.9411 −1.61992 −0.809961 0.586484i \(-0.800512\pi\)
−0.809961 + 0.586484i \(0.800512\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.2132 0.580267 0.290133 0.956986i \(-0.406300\pi\)
0.290133 + 0.956986i \(0.406300\pi\)
\(444\) 0 0
\(445\) 2.65685 0.125947
\(446\) 0 0
\(447\) 18.8995 0.893915
\(448\) 0 0
\(449\) −1.82843 −0.0862888 −0.0431444 0.999069i \(-0.513738\pi\)
−0.0431444 + 0.999069i \(0.513738\pi\)
\(450\) 0 0
\(451\) 10.4853 0.493733
\(452\) 0 0
\(453\) 0.828427 0.0389229
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −32.2843 −1.51019 −0.755097 0.655613i \(-0.772410\pi\)
−0.755097 + 0.655613i \(0.772410\pi\)
\(458\) 0 0
\(459\) −0.343146 −0.0160167
\(460\) 0 0
\(461\) 18.6863 0.870307 0.435154 0.900356i \(-0.356694\pi\)
0.435154 + 0.900356i \(0.356694\pi\)
\(462\) 0 0
\(463\) −11.0416 −0.513148 −0.256574 0.966525i \(-0.582594\pi\)
−0.256574 + 0.966525i \(0.582594\pi\)
\(464\) 0 0
\(465\) −14.4853 −0.671739
\(466\) 0 0
\(467\) 22.8995 1.05966 0.529831 0.848103i \(-0.322255\pi\)
0.529831 + 0.848103i \(0.322255\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 12.8284 0.591103
\(472\) 0 0
\(473\) 30.9706 1.42403
\(474\) 0 0
\(475\) 2.82843 0.129777
\(476\) 0 0
\(477\) −19.3137 −0.884314
\(478\) 0 0
\(479\) 24.3431 1.11227 0.556133 0.831093i \(-0.312284\pi\)
0.556133 + 0.831093i \(0.312284\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.343146 0.0155814
\(486\) 0 0
\(487\) −15.6569 −0.709480 −0.354740 0.934965i \(-0.615431\pi\)
−0.354740 + 0.934965i \(0.615431\pi\)
\(488\) 0 0
\(489\) 57.1127 2.58273
\(490\) 0 0
\(491\) 13.3137 0.600839 0.300420 0.953807i \(-0.402873\pi\)
0.300420 + 0.953807i \(0.402873\pi\)
\(492\) 0 0
\(493\) 0.828427 0.0373105
\(494\) 0 0
\(495\) −13.6569 −0.613830
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −4.82843 −0.216150 −0.108075 0.994143i \(-0.534469\pi\)
−0.108075 + 0.994143i \(0.534469\pi\)
\(500\) 0 0
\(501\) 47.2843 2.11251
\(502\) 0 0
\(503\) −37.8701 −1.68854 −0.844271 0.535916i \(-0.819966\pi\)
−0.844271 + 0.535916i \(0.819966\pi\)
\(504\) 0 0
\(505\) 12.3137 0.547953
\(506\) 0 0
\(507\) 29.7279 1.32026
\(508\) 0 0
\(509\) 24.6569 1.09290 0.546448 0.837493i \(-0.315980\pi\)
0.546448 + 0.837493i \(0.315980\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 1.17157 0.0517262
\(514\) 0 0
\(515\) −0.414214 −0.0182524
\(516\) 0 0
\(517\) 9.65685 0.424708
\(518\) 0 0
\(519\) 46.6274 2.04672
\(520\) 0 0
\(521\) −18.9706 −0.831115 −0.415558 0.909567i \(-0.636414\pi\)
−0.415558 + 0.909567i \(0.636414\pi\)
\(522\) 0 0
\(523\) −24.3431 −1.06445 −0.532226 0.846602i \(-0.678644\pi\)
−0.532226 + 0.846602i \(0.678644\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.97056 −0.216521
\(528\) 0 0
\(529\) −17.1716 −0.746590
\(530\) 0 0
\(531\) 35.3137 1.53248
\(532\) 0 0
\(533\) −1.79899 −0.0779229
\(534\) 0 0
\(535\) −2.75736 −0.119211
\(536\) 0 0
\(537\) −24.1421 −1.04181
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 18.6569 0.802121 0.401060 0.916052i \(-0.368642\pi\)
0.401060 + 0.916052i \(0.368642\pi\)
\(542\) 0 0
\(543\) 20.8995 0.896883
\(544\) 0 0
\(545\) 3.48528 0.149293
\(546\) 0 0
\(547\) −5.10051 −0.218082 −0.109041 0.994037i \(-0.534778\pi\)
−0.109041 + 0.994037i \(0.534778\pi\)
\(548\) 0 0
\(549\) −32.4853 −1.38644
\(550\) 0 0
\(551\) −2.82843 −0.120495
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −34.2843 −1.45267 −0.726336 0.687340i \(-0.758778\pi\)
−0.726336 + 0.687340i \(0.758778\pi\)
\(558\) 0 0
\(559\) −5.31371 −0.224746
\(560\) 0 0
\(561\) −9.65685 −0.407713
\(562\) 0 0
\(563\) −16.2721 −0.685786 −0.342893 0.939374i \(-0.611407\pi\)
−0.342893 + 0.939374i \(0.611407\pi\)
\(564\) 0 0
\(565\) 12.4853 0.525260
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.65685 −0.153303 −0.0766517 0.997058i \(-0.524423\pi\)
−0.0766517 + 0.997058i \(0.524423\pi\)
\(570\) 0 0
\(571\) −14.8284 −0.620550 −0.310275 0.950647i \(-0.600421\pi\)
−0.310275 + 0.950647i \(0.600421\pi\)
\(572\) 0 0
\(573\) −17.3137 −0.723291
\(574\) 0 0
\(575\) 2.41421 0.100680
\(576\) 0 0
\(577\) 23.9411 0.996682 0.498341 0.866981i \(-0.333943\pi\)
0.498341 + 0.866981i \(0.333943\pi\)
\(578\) 0 0
\(579\) 4.82843 0.200663
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 32.9706 1.36550
\(584\) 0 0
\(585\) 2.34315 0.0968772
\(586\) 0 0
\(587\) −22.2843 −0.919770 −0.459885 0.887978i \(-0.652109\pi\)
−0.459885 + 0.887978i \(0.652109\pi\)
\(588\) 0 0
\(589\) 16.9706 0.699260
\(590\) 0 0
\(591\) 57.1127 2.34930
\(592\) 0 0
\(593\) 43.7990 1.79861 0.899304 0.437323i \(-0.144073\pi\)
0.899304 + 0.437323i \(0.144073\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.00000 0.163709
\(598\) 0 0
\(599\) −17.6569 −0.721440 −0.360720 0.932674i \(-0.617469\pi\)
−0.360720 + 0.932674i \(0.617469\pi\)
\(600\) 0 0
\(601\) 8.34315 0.340324 0.170162 0.985416i \(-0.445571\pi\)
0.170162 + 0.985416i \(0.445571\pi\)
\(602\) 0 0
\(603\) −35.1127 −1.42990
\(604\) 0 0
\(605\) 12.3137 0.500623
\(606\) 0 0
\(607\) 4.21320 0.171009 0.0855043 0.996338i \(-0.472750\pi\)
0.0855043 + 0.996338i \(0.472750\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.65685 −0.0670291
\(612\) 0 0
\(613\) −15.4558 −0.624256 −0.312128 0.950040i \(-0.601042\pi\)
−0.312128 + 0.950040i \(0.601042\pi\)
\(614\) 0 0
\(615\) 5.24264 0.211404
\(616\) 0 0
\(617\) −11.3137 −0.455473 −0.227736 0.973723i \(-0.573132\pi\)
−0.227736 + 0.973723i \(0.573132\pi\)
\(618\) 0 0
\(619\) −42.4853 −1.70763 −0.853814 0.520578i \(-0.825716\pi\)
−0.853814 + 0.520578i \(0.825716\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 32.9706 1.31672
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −8.14214 −0.324133 −0.162067 0.986780i \(-0.551816\pi\)
−0.162067 + 0.986780i \(0.551816\pi\)
\(632\) 0 0
\(633\) 8.48528 0.337260
\(634\) 0 0
\(635\) −13.3137 −0.528338
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 35.3137 1.39699
\(640\) 0 0
\(641\) 14.5147 0.573297 0.286648 0.958036i \(-0.407459\pi\)
0.286648 + 0.958036i \(0.407459\pi\)
\(642\) 0 0
\(643\) 30.2843 1.19430 0.597148 0.802131i \(-0.296301\pi\)
0.597148 + 0.802131i \(0.296301\pi\)
\(644\) 0 0
\(645\) 15.4853 0.609732
\(646\) 0 0
\(647\) −17.0416 −0.669976 −0.334988 0.942222i \(-0.608732\pi\)
−0.334988 + 0.942222i \(0.608732\pi\)
\(648\) 0 0
\(649\) −60.2843 −2.36636
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −24.8284 −0.971611 −0.485806 0.874067i \(-0.661474\pi\)
−0.485806 + 0.874067i \(0.661474\pi\)
\(654\) 0 0
\(655\) −3.31371 −0.129477
\(656\) 0 0
\(657\) 13.6569 0.532805
\(658\) 0 0
\(659\) −26.8284 −1.04509 −0.522544 0.852613i \(-0.675017\pi\)
−0.522544 + 0.852613i \(0.675017\pi\)
\(660\) 0 0
\(661\) 26.1716 1.01796 0.508978 0.860779i \(-0.330023\pi\)
0.508978 + 0.860779i \(0.330023\pi\)
\(662\) 0 0
\(663\) 1.65685 0.0643469
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2.41421 −0.0934787
\(668\) 0 0
\(669\) 28.1421 1.08804
\(670\) 0 0
\(671\) 55.4558 2.14085
\(672\) 0 0
\(673\) 18.3431 0.707076 0.353538 0.935420i \(-0.384978\pi\)
0.353538 + 0.935420i \(0.384978\pi\)
\(674\) 0 0
\(675\) 0.414214 0.0159431
\(676\) 0 0
\(677\) 0.142136 0.00546272 0.00273136 0.999996i \(-0.499131\pi\)
0.00273136 + 0.999996i \(0.499131\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 65.1127 2.49512
\(682\) 0 0
\(683\) 43.2426 1.65463 0.827317 0.561736i \(-0.189866\pi\)
0.827317 + 0.561736i \(0.189866\pi\)
\(684\) 0 0
\(685\) 1.65685 0.0633051
\(686\) 0 0
\(687\) 0.828427 0.0316065
\(688\) 0 0
\(689\) −5.65685 −0.215509
\(690\) 0 0
\(691\) 4.82843 0.183682 0.0918410 0.995774i \(-0.470725\pi\)
0.0918410 + 0.995774i \(0.470725\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12.1421 −0.460577
\(696\) 0 0
\(697\) 1.79899 0.0681416
\(698\) 0 0
\(699\) 26.9706 1.02012
\(700\) 0 0
\(701\) −42.7990 −1.61650 −0.808248 0.588843i \(-0.799584\pi\)
−0.808248 + 0.588843i \(0.799584\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 4.82843 0.181849
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 38.3137 1.43890 0.719451 0.694543i \(-0.244394\pi\)
0.719451 + 0.694543i \(0.244394\pi\)
\(710\) 0 0
\(711\) −25.9411 −0.972868
\(712\) 0 0
\(713\) 14.4853 0.542478
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 0 0
\(717\) 3.17157 0.118445
\(718\) 0 0
\(719\) 41.1127 1.53324 0.766622 0.642098i \(-0.221936\pi\)
0.766622 + 0.642098i \(0.221936\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −39.4558 −1.46738
\(724\) 0 0
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) 40.4142 1.49888 0.749440 0.662072i \(-0.230323\pi\)
0.749440 + 0.662072i \(0.230323\pi\)
\(728\) 0 0
\(729\) −23.8284 −0.882534
\(730\) 0 0
\(731\) 5.31371 0.196535
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 59.9411 2.20796
\(738\) 0 0
\(739\) −41.1127 −1.51236 −0.756178 0.654367i \(-0.772935\pi\)
−0.756178 + 0.654367i \(0.772935\pi\)
\(740\) 0 0
\(741\) −5.65685 −0.207810
\(742\) 0 0
\(743\) 1.92893 0.0707657 0.0353828 0.999374i \(-0.488735\pi\)
0.0353828 + 0.999374i \(0.488735\pi\)
\(744\) 0 0
\(745\) −7.82843 −0.286811
\(746\) 0 0
\(747\) 33.1716 1.21368
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 41.6569 1.52008 0.760040 0.649876i \(-0.225179\pi\)
0.760040 + 0.649876i \(0.225179\pi\)
\(752\) 0 0
\(753\) 32.1421 1.17132
\(754\) 0 0
\(755\) −0.343146 −0.0124884
\(756\) 0 0
\(757\) 19.4558 0.707135 0.353567 0.935409i \(-0.384969\pi\)
0.353567 + 0.935409i \(0.384969\pi\)
\(758\) 0 0
\(759\) 28.1421 1.02149
\(760\) 0 0
\(761\) −13.3137 −0.482622 −0.241311 0.970448i \(-0.577577\pi\)
−0.241311 + 0.970448i \(0.577577\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2.34315 −0.0847166
\(766\) 0 0
\(767\) 10.3431 0.373469
\(768\) 0 0
\(769\) 44.6274 1.60931 0.804653 0.593745i \(-0.202351\pi\)
0.804653 + 0.593745i \(0.202351\pi\)
\(770\) 0 0
\(771\) −42.6274 −1.53519
\(772\) 0 0
\(773\) 25.1127 0.903241 0.451620 0.892210i \(-0.350846\pi\)
0.451620 + 0.892210i \(0.350846\pi\)
\(774\) 0 0
\(775\) 6.00000 0.215526
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.14214 −0.220065
\(780\) 0 0
\(781\) −60.2843 −2.15714
\(782\) 0 0
\(783\) −0.414214 −0.0148028
\(784\) 0 0
\(785\) −5.31371 −0.189654
\(786\) 0 0
\(787\) 28.5563 1.01792 0.508962 0.860789i \(-0.330029\pi\)
0.508962 + 0.860789i \(0.330029\pi\)
\(788\) 0 0
\(789\) 45.9706 1.63660
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −9.51472 −0.337878
\(794\) 0 0
\(795\) 16.4853 0.584673
\(796\) 0 0
\(797\) −8.00000 −0.283375 −0.141687 0.989911i \(-0.545253\pi\)
−0.141687 + 0.989911i \(0.545253\pi\)
\(798\) 0 0
\(799\) 1.65685 0.0586153
\(800\) 0 0
\(801\) 7.51472 0.265520
\(802\) 0 0
\(803\) −23.3137 −0.822723
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 73.5269 2.58827
\(808\) 0 0
\(809\) −9.62742 −0.338482 −0.169241 0.985575i \(-0.554132\pi\)
−0.169241 + 0.985575i \(0.554132\pi\)
\(810\) 0 0
\(811\) −24.6274 −0.864786 −0.432393 0.901685i \(-0.642331\pi\)
−0.432393 + 0.901685i \(0.642331\pi\)
\(812\) 0 0
\(813\) −1.17157 −0.0410889
\(814\) 0 0
\(815\) −23.6569 −0.828663
\(816\) 0 0
\(817\) −18.1421 −0.634713
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −19.9411 −0.695950 −0.347975 0.937504i \(-0.613131\pi\)
−0.347975 + 0.937504i \(0.613131\pi\)
\(822\) 0 0
\(823\) 12.0711 0.420771 0.210385 0.977619i \(-0.432528\pi\)
0.210385 + 0.977619i \(0.432528\pi\)
\(824\) 0 0
\(825\) 11.6569 0.405840
\(826\) 0 0
\(827\) −16.2132 −0.563788 −0.281894 0.959446i \(-0.590963\pi\)
−0.281894 + 0.959446i \(0.590963\pi\)
\(828\) 0 0
\(829\) 6.68629 0.232225 0.116112 0.993236i \(-0.462957\pi\)
0.116112 + 0.993236i \(0.462957\pi\)
\(830\) 0 0
\(831\) 29.3137 1.01688
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −19.5858 −0.677794
\(836\) 0 0
\(837\) 2.48528 0.0859039
\(838\) 0 0
\(839\) −20.8284 −0.719077 −0.359539 0.933130i \(-0.617066\pi\)
−0.359539 + 0.933130i \(0.617066\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) −63.4558 −2.18554
\(844\) 0 0
\(845\) −12.3137 −0.423604
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 33.7990 1.15998
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 53.4558 1.83029 0.915147 0.403121i \(-0.132075\pi\)
0.915147 + 0.403121i \(0.132075\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) 0 0
\(857\) −22.2843 −0.761216 −0.380608 0.924736i \(-0.624285\pi\)
−0.380608 + 0.924736i \(0.624285\pi\)
\(858\) 0 0
\(859\) −46.6274 −1.59091 −0.795453 0.606015i \(-0.792767\pi\)
−0.795453 + 0.606015i \(0.792767\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16.5563 0.563585 0.281792 0.959475i \(-0.409071\pi\)
0.281792 + 0.959475i \(0.409071\pi\)
\(864\) 0 0
\(865\) −19.3137 −0.656686
\(866\) 0 0
\(867\) 39.3848 1.33758
\(868\) 0 0
\(869\) 44.2843 1.50224
\(870\) 0 0
\(871\) −10.2843 −0.348469
\(872\) 0 0
\(873\) 0.970563 0.0328486
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 30.8284 1.04100 0.520501 0.853861i \(-0.325745\pi\)
0.520501 + 0.853861i \(0.325745\pi\)
\(878\) 0 0
\(879\) 38.6274 1.30287
\(880\) 0 0
\(881\) −3.82843 −0.128983 −0.0644915 0.997918i \(-0.520543\pi\)
−0.0644915 + 0.997918i \(0.520543\pi\)
\(882\) 0 0
\(883\) 38.2843 1.28837 0.644184 0.764870i \(-0.277197\pi\)
0.644184 + 0.764870i \(0.277197\pi\)
\(884\) 0 0
\(885\) −30.1421 −1.01322
\(886\) 0 0
\(887\) −44.0711 −1.47976 −0.739881 0.672738i \(-0.765118\pi\)
−0.739881 + 0.672738i \(0.765118\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 45.7990 1.53432
\(892\) 0 0
\(893\) −5.65685 −0.189299
\(894\) 0 0
\(895\) 10.0000 0.334263
\(896\) 0 0
\(897\) −4.82843 −0.161216
\(898\) 0 0
\(899\) −6.00000 −0.200111
\(900\) 0 0
\(901\) 5.65685 0.188457
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8.65685 −0.287764
\(906\) 0 0
\(907\) −28.2132 −0.936804 −0.468402 0.883515i \(-0.655170\pi\)
−0.468402 + 0.883515i \(0.655170\pi\)
\(908\) 0 0
\(909\) 34.8284 1.15519
\(910\) 0 0
\(911\) 49.7990 1.64991 0.824957 0.565195i \(-0.191199\pi\)
0.824957 + 0.565195i \(0.191199\pi\)
\(912\) 0 0
\(913\) −56.6274 −1.87409
\(914\) 0 0
\(915\) 27.7279 0.916657
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −19.1127 −0.630470 −0.315235 0.949014i \(-0.602083\pi\)
−0.315235 + 0.949014i \(0.602083\pi\)
\(920\) 0 0
\(921\) −31.9706 −1.05347
\(922\) 0 0
\(923\) 10.3431 0.340449
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1.17157 −0.0384795
\(928\) 0 0
\(929\) −11.4853 −0.376820 −0.188410 0.982090i \(-0.560333\pi\)
−0.188410 + 0.982090i \(0.560333\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −45.4558 −1.48816
\(934\) 0 0
\(935\) 4.00000 0.130814
\(936\) 0 0
\(937\) −10.6274 −0.347183 −0.173591 0.984818i \(-0.555537\pi\)
−0.173591 + 0.984818i \(0.555537\pi\)
\(938\) 0 0
\(939\) 42.6274 1.39109
\(940\) 0 0
\(941\) 10.2843 0.335258 0.167629 0.985850i \(-0.446389\pi\)
0.167629 + 0.985850i \(0.446389\pi\)
\(942\) 0 0
\(943\) −5.24264 −0.170724
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 43.1838 1.40328 0.701642 0.712530i \(-0.252451\pi\)
0.701642 + 0.712530i \(0.252451\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) 0 0
\(951\) −62.2843 −2.01971
\(952\) 0 0
\(953\) −2.34315 −0.0759019 −0.0379510 0.999280i \(-0.512083\pi\)
−0.0379510 + 0.999280i \(0.512083\pi\)
\(954\) 0 0
\(955\) 7.17157 0.232067
\(956\) 0 0
\(957\) −11.6569 −0.376813
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) −7.79899 −0.251319
\(964\) 0 0
\(965\) −2.00000 −0.0643823
\(966\) 0 0
\(967\) 27.5269 0.885206 0.442603 0.896718i \(-0.354055\pi\)
0.442603 + 0.896718i \(0.354055\pi\)
\(968\) 0 0
\(969\) 5.65685 0.181724
\(970\) 0 0
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −2.00000 −0.0640513
\(976\) 0 0
\(977\) 21.3137 0.681886 0.340943 0.940084i \(-0.389254\pi\)
0.340943 + 0.940084i \(0.389254\pi\)
\(978\) 0 0
\(979\) −12.8284 −0.409998
\(980\) 0 0
\(981\) 9.85786 0.314737
\(982\) 0 0
\(983\) 14.2132 0.453331 0.226665 0.973973i \(-0.427218\pi\)
0.226665 + 0.973973i \(0.427218\pi\)
\(984\) 0 0
\(985\) −23.6569 −0.753770
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −15.4853 −0.492403
\(990\) 0 0
\(991\) 15.6569 0.497356 0.248678 0.968586i \(-0.420004\pi\)
0.248678 + 0.968586i \(0.420004\pi\)
\(992\) 0 0
\(993\) −26.4853 −0.840485
\(994\) 0 0
\(995\) −1.65685 −0.0525258
\(996\) 0 0
\(997\) −17.4558 −0.552832 −0.276416 0.961038i \(-0.589147\pi\)
−0.276416 + 0.961038i \(0.589147\pi\)
\(998\) 0 0
\(999\) 0 0
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 3920.2.a.bq.1.1 2
4.3 odd 2 245.2.a.h.1.1 2
7.2 even 3 560.2.q.k.81.2 4
7.4 even 3 560.2.q.k.401.2 4
7.6 odd 2 3920.2.a.bv.1.2 2
12.11 even 2 2205.2.a.n.1.2 2
20.3 even 4 1225.2.b.g.99.3 4
20.7 even 4 1225.2.b.g.99.2 4
20.19 odd 2 1225.2.a.k.1.2 2
28.3 even 6 245.2.e.e.226.2 4
28.11 odd 6 35.2.e.a.16.2 yes 4
28.19 even 6 245.2.e.e.116.2 4
28.23 odd 6 35.2.e.a.11.2 4
28.27 even 2 245.2.a.g.1.1 2
84.11 even 6 315.2.j.e.226.1 4
84.23 even 6 315.2.j.e.46.1 4
84.83 odd 2 2205.2.a.q.1.2 2
140.23 even 12 175.2.k.a.74.3 8
140.27 odd 4 1225.2.b.h.99.2 4
140.39 odd 6 175.2.e.c.51.1 4
140.67 even 12 175.2.k.a.149.3 8
140.79 odd 6 175.2.e.c.151.1 4
140.83 odd 4 1225.2.b.h.99.3 4
140.107 even 12 175.2.k.a.74.2 8
140.123 even 12 175.2.k.a.149.2 8
140.139 even 2 1225.2.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.e.a.11.2 4 28.23 odd 6
35.2.e.a.16.2 yes 4 28.11 odd 6
175.2.e.c.51.1 4 140.39 odd 6
175.2.e.c.151.1 4 140.79 odd 6
175.2.k.a.74.2 8 140.107 even 12
175.2.k.a.74.3 8 140.23 even 12
175.2.k.a.149.2 8 140.123 even 12
175.2.k.a.149.3 8 140.67 even 12
245.2.a.g.1.1 2 28.27 even 2
245.2.a.h.1.1 2 4.3 odd 2
245.2.e.e.116.2 4 28.19 even 6
245.2.e.e.226.2 4 28.3 even 6
315.2.j.e.46.1 4 84.23 even 6
315.2.j.e.226.1 4 84.11 even 6
560.2.q.k.81.2 4 7.2 even 3
560.2.q.k.401.2 4 7.4 even 3
1225.2.a.k.1.2 2 20.19 odd 2
1225.2.a.m.1.2 2 140.139 even 2
1225.2.b.g.99.2 4 20.7 even 4
1225.2.b.g.99.3 4 20.3 even 4
1225.2.b.h.99.2 4 140.27 odd 4
1225.2.b.h.99.3 4 140.83 odd 4
2205.2.a.n.1.2 2 12.11 even 2
2205.2.a.q.1.2 2 84.83 odd 2
3920.2.a.bq.1.1 2 1.1 even 1 trivial
3920.2.a.bv.1.2 2 7.6 odd 2