Properties

Label 308.2.a.b.1.1
Level 308
Weight 2
Character 308.1
Self dual yes
Analytic conductor 2.459
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 308.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.45939238226\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\)
Character \(\chi\) = 308.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.44949 q^{3} +2.00000 q^{5} -1.00000 q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-2.44949 q^{3} +2.00000 q^{5} -1.00000 q^{7} +3.00000 q^{9} -1.00000 q^{11} +4.44949 q^{13} -4.89898 q^{15} +4.44949 q^{17} +4.89898 q^{19} +2.44949 q^{21} +8.89898 q^{23} -1.00000 q^{25} -6.89898 q^{29} +1.55051 q^{31} +2.44949 q^{33} -2.00000 q^{35} +4.00000 q^{37} -10.8990 q^{39} +5.34847 q^{41} -6.89898 q^{43} +6.00000 q^{45} -1.55051 q^{47} +1.00000 q^{49} -10.8990 q^{51} -9.79796 q^{53} -2.00000 q^{55} -12.0000 q^{57} -7.34847 q^{59} -9.34847 q^{61} -3.00000 q^{63} +8.89898 q^{65} +14.6969 q^{67} -21.7980 q^{69} -3.10102 q^{71} +7.55051 q^{73} +2.44949 q^{75} +1.00000 q^{77} -1.10102 q^{79} -9.00000 q^{81} -7.10102 q^{83} +8.89898 q^{85} +16.8990 q^{87} +6.00000 q^{89} -4.44949 q^{91} -3.79796 q^{93} +9.79796 q^{95} +14.8990 q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} - 2q^{7} + 6q^{9} + O(q^{10}) \) \( 2q + 4q^{5} - 2q^{7} + 6q^{9} - 2q^{11} + 4q^{13} + 4q^{17} + 8q^{23} - 2q^{25} - 4q^{29} + 8q^{31} - 4q^{35} + 8q^{37} - 12q^{39} - 4q^{41} - 4q^{43} + 12q^{45} - 8q^{47} + 2q^{49} - 12q^{51} - 4q^{55} - 24q^{57} - 4q^{61} - 6q^{63} + 8q^{65} - 24q^{69} - 16q^{71} + 20q^{73} + 2q^{77} - 12q^{79} - 18q^{81} - 24q^{83} + 8q^{85} + 24q^{87} + 12q^{89} - 4q^{91} + 12q^{93} + 20q^{97} - 6q^{99} + O(q^{100}) \)

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
\(3\) −2.44949 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 4.44949 1.23407 0.617033 0.786937i \(-0.288334\pi\)
0.617033 + 0.786937i \(0.288334\pi\)
\(14\) 0 0
\(15\) −4.89898 −1.26491
\(16\) 0 0
\(17\) 4.44949 1.07916 0.539580 0.841934i \(-0.318583\pi\)
0.539580 + 0.841934i \(0.318583\pi\)
\(18\) 0 0
\(19\) 4.89898 1.12390 0.561951 0.827170i \(-0.310051\pi\)
0.561951 + 0.827170i \(0.310051\pi\)
\(20\) 0 0
\(21\) 2.44949 0.534522
\(22\) 0 0
\(23\) 8.89898 1.85557 0.927783 0.373121i \(-0.121712\pi\)
0.927783 + 0.373121i \(0.121712\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.89898 −1.28111 −0.640554 0.767913i \(-0.721295\pi\)
−0.640554 + 0.767913i \(0.721295\pi\)
\(30\) 0 0
\(31\) 1.55051 0.278480 0.139240 0.990259i \(-0.455534\pi\)
0.139240 + 0.990259i \(0.455534\pi\)
\(32\) 0 0
\(33\) 2.44949 0.426401
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) −10.8990 −1.74523
\(40\) 0 0
\(41\) 5.34847 0.835291 0.417645 0.908610i \(-0.362855\pi\)
0.417645 + 0.908610i \(0.362855\pi\)
\(42\) 0 0
\(43\) −6.89898 −1.05208 −0.526042 0.850458i \(-0.676325\pi\)
−0.526042 + 0.850458i \(0.676325\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) 0 0
\(47\) −1.55051 −0.226165 −0.113083 0.993586i \(-0.536072\pi\)
−0.113083 + 0.993586i \(0.536072\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −10.8990 −1.52616
\(52\) 0 0
\(53\) −9.79796 −1.34585 −0.672927 0.739709i \(-0.734963\pi\)
−0.672927 + 0.739709i \(0.734963\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) −12.0000 −1.58944
\(58\) 0 0
\(59\) −7.34847 −0.956689 −0.478345 0.878172i \(-0.658763\pi\)
−0.478345 + 0.878172i \(0.658763\pi\)
\(60\) 0 0
\(61\) −9.34847 −1.19695 −0.598474 0.801142i \(-0.704226\pi\)
−0.598474 + 0.801142i \(0.704226\pi\)
\(62\) 0 0
\(63\) −3.00000 −0.377964
\(64\) 0 0
\(65\) 8.89898 1.10378
\(66\) 0 0
\(67\) 14.6969 1.79552 0.897758 0.440488i \(-0.145195\pi\)
0.897758 + 0.440488i \(0.145195\pi\)
\(68\) 0 0
\(69\) −21.7980 −2.62417
\(70\) 0 0
\(71\) −3.10102 −0.368023 −0.184012 0.982924i \(-0.558908\pi\)
−0.184012 + 0.982924i \(0.558908\pi\)
\(72\) 0 0
\(73\) 7.55051 0.883720 0.441860 0.897084i \(-0.354319\pi\)
0.441860 + 0.897084i \(0.354319\pi\)
\(74\) 0 0
\(75\) 2.44949 0.282843
\(76\) 0 0
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −1.10102 −0.123874 −0.0619372 0.998080i \(-0.519728\pi\)
−0.0619372 + 0.998080i \(0.519728\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) −7.10102 −0.779438 −0.389719 0.920934i \(-0.627428\pi\)
−0.389719 + 0.920934i \(0.627428\pi\)
\(84\) 0 0
\(85\) 8.89898 0.965230
\(86\) 0 0
\(87\) 16.8990 1.81176
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −4.44949 −0.466433
\(92\) 0 0
\(93\) −3.79796 −0.393830
\(94\) 0 0
\(95\) 9.79796 1.00525
\(96\) 0 0
\(97\) 14.8990 1.51276 0.756381 0.654131i \(-0.226966\pi\)
0.756381 + 0.654131i \(0.226966\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −3.55051 −0.353289 −0.176644 0.984275i \(-0.556524\pi\)
−0.176644 + 0.984275i \(0.556524\pi\)
\(102\) 0 0
\(103\) 9.55051 0.941040 0.470520 0.882389i \(-0.344066\pi\)
0.470520 + 0.882389i \(0.344066\pi\)
\(104\) 0 0
\(105\) 4.89898 0.478091
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 5.10102 0.488589 0.244295 0.969701i \(-0.421444\pi\)
0.244295 + 0.969701i \(0.421444\pi\)
\(110\) 0 0
\(111\) −9.79796 −0.929981
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 17.7980 1.65967
\(116\) 0 0
\(117\) 13.3485 1.23407
\(118\) 0 0
\(119\) −4.44949 −0.407884
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −13.1010 −1.18128
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 17.7980 1.57931 0.789657 0.613549i \(-0.210259\pi\)
0.789657 + 0.613549i \(0.210259\pi\)
\(128\) 0 0
\(129\) 16.8990 1.48787
\(130\) 0 0
\(131\) −9.79796 −0.856052 −0.428026 0.903767i \(-0.640791\pi\)
−0.428026 + 0.903767i \(0.640791\pi\)
\(132\) 0 0
\(133\) −4.89898 −0.424795
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.79796 −0.837096 −0.418548 0.908195i \(-0.637461\pi\)
−0.418548 + 0.908195i \(0.637461\pi\)
\(138\) 0 0
\(139\) −6.69694 −0.568027 −0.284013 0.958820i \(-0.591666\pi\)
−0.284013 + 0.958820i \(0.591666\pi\)
\(140\) 0 0
\(141\) 3.79796 0.319846
\(142\) 0 0
\(143\) −4.44949 −0.372085
\(144\) 0 0
\(145\) −13.7980 −1.14586
\(146\) 0 0
\(147\) −2.44949 −0.202031
\(148\) 0 0
\(149\) −7.79796 −0.638834 −0.319417 0.947614i \(-0.603487\pi\)
−0.319417 + 0.947614i \(0.603487\pi\)
\(150\) 0 0
\(151\) −20.6969 −1.68429 −0.842146 0.539249i \(-0.818708\pi\)
−0.842146 + 0.539249i \(0.818708\pi\)
\(152\) 0 0
\(153\) 13.3485 1.07916
\(154\) 0 0
\(155\) 3.10102 0.249080
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0 0
\(159\) 24.0000 1.90332
\(160\) 0 0
\(161\) −8.89898 −0.701338
\(162\) 0 0
\(163\) −0.898979 −0.0704135 −0.0352068 0.999380i \(-0.511209\pi\)
−0.0352068 + 0.999380i \(0.511209\pi\)
\(164\) 0 0
\(165\) 4.89898 0.381385
\(166\) 0 0
\(167\) −5.79796 −0.448660 −0.224330 0.974513i \(-0.572019\pi\)
−0.224330 + 0.974513i \(0.572019\pi\)
\(168\) 0 0
\(169\) 6.79796 0.522920
\(170\) 0 0
\(171\) 14.6969 1.12390
\(172\) 0 0
\(173\) −14.2474 −1.08321 −0.541607 0.840632i \(-0.682184\pi\)
−0.541607 + 0.840632i \(0.682184\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 18.0000 1.35296
\(178\) 0 0
\(179\) 23.5959 1.76364 0.881821 0.471585i \(-0.156318\pi\)
0.881821 + 0.471585i \(0.156318\pi\)
\(180\) 0 0
\(181\) 18.8990 1.40475 0.702375 0.711807i \(-0.252123\pi\)
0.702375 + 0.711807i \(0.252123\pi\)
\(182\) 0 0
\(183\) 22.8990 1.69274
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) −4.44949 −0.325379
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 17.7980 1.28782 0.643908 0.765103i \(-0.277312\pi\)
0.643908 + 0.765103i \(0.277312\pi\)
\(192\) 0 0
\(193\) −17.5959 −1.26658 −0.633291 0.773914i \(-0.718296\pi\)
−0.633291 + 0.773914i \(0.718296\pi\)
\(194\) 0 0
\(195\) −21.7980 −1.56098
\(196\) 0 0
\(197\) 13.5959 0.968669 0.484335 0.874883i \(-0.339062\pi\)
0.484335 + 0.874883i \(0.339062\pi\)
\(198\) 0 0
\(199\) −18.0454 −1.27921 −0.639603 0.768706i \(-0.720901\pi\)
−0.639603 + 0.768706i \(0.720901\pi\)
\(200\) 0 0
\(201\) −36.0000 −2.53924
\(202\) 0 0
\(203\) 6.89898 0.484213
\(204\) 0 0
\(205\) 10.6969 0.747107
\(206\) 0 0
\(207\) 26.6969 1.85557
\(208\) 0 0
\(209\) −4.89898 −0.338869
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 0 0
\(213\) 7.59592 0.520464
\(214\) 0 0
\(215\) −13.7980 −0.941013
\(216\) 0 0
\(217\) −1.55051 −0.105255
\(218\) 0 0
\(219\) −18.4949 −1.24977
\(220\) 0 0
\(221\) 19.7980 1.33175
\(222\) 0 0
\(223\) 21.1464 1.41607 0.708035 0.706178i \(-0.249582\pi\)
0.708035 + 0.706178i \(0.249582\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) −17.7980 −1.18129 −0.590646 0.806931i \(-0.701127\pi\)
−0.590646 + 0.806931i \(0.701127\pi\)
\(228\) 0 0
\(229\) −14.8990 −0.984552 −0.492276 0.870439i \(-0.663835\pi\)
−0.492276 + 0.870439i \(0.663835\pi\)
\(230\) 0 0
\(231\) −2.44949 −0.161165
\(232\) 0 0
\(233\) −10.8990 −0.714016 −0.357008 0.934101i \(-0.616203\pi\)
−0.357008 + 0.934101i \(0.616203\pi\)
\(234\) 0 0
\(235\) −3.10102 −0.202288
\(236\) 0 0
\(237\) 2.69694 0.175185
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 13.3485 0.859850 0.429925 0.902864i \(-0.358540\pi\)
0.429925 + 0.902864i \(0.358540\pi\)
\(242\) 0 0
\(243\) 22.0454 1.41421
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 21.7980 1.38697
\(248\) 0 0
\(249\) 17.3939 1.10229
\(250\) 0 0
\(251\) −2.44949 −0.154610 −0.0773052 0.997007i \(-0.524632\pi\)
−0.0773052 + 0.997007i \(0.524632\pi\)
\(252\) 0 0
\(253\) −8.89898 −0.559474
\(254\) 0 0
\(255\) −21.7980 −1.36504
\(256\) 0 0
\(257\) 15.7980 0.985450 0.492725 0.870185i \(-0.336001\pi\)
0.492725 + 0.870185i \(0.336001\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) −20.6969 −1.28111
\(262\) 0 0
\(263\) −19.5959 −1.20834 −0.604168 0.796857i \(-0.706494\pi\)
−0.604168 + 0.796857i \(0.706494\pi\)
\(264\) 0 0
\(265\) −19.5959 −1.20377
\(266\) 0 0
\(267\) −14.6969 −0.899438
\(268\) 0 0
\(269\) −22.8990 −1.39618 −0.698088 0.716012i \(-0.745965\pi\)
−0.698088 + 0.716012i \(0.745965\pi\)
\(270\) 0 0
\(271\) 30.6969 1.86471 0.932353 0.361549i \(-0.117752\pi\)
0.932353 + 0.361549i \(0.117752\pi\)
\(272\) 0 0
\(273\) 10.8990 0.659636
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −3.79796 −0.228197 −0.114099 0.993469i \(-0.536398\pi\)
−0.114099 + 0.993469i \(0.536398\pi\)
\(278\) 0 0
\(279\) 4.65153 0.278480
\(280\) 0 0
\(281\) −14.8990 −0.888799 −0.444399 0.895829i \(-0.646583\pi\)
−0.444399 + 0.895829i \(0.646583\pi\)
\(282\) 0 0
\(283\) −24.0000 −1.42665 −0.713326 0.700832i \(-0.752812\pi\)
−0.713326 + 0.700832i \(0.752812\pi\)
\(284\) 0 0
\(285\) −24.0000 −1.42164
\(286\) 0 0
\(287\) −5.34847 −0.315710
\(288\) 0 0
\(289\) 2.79796 0.164586
\(290\) 0 0
\(291\) −36.4949 −2.13937
\(292\) 0 0
\(293\) −25.3485 −1.48087 −0.740437 0.672126i \(-0.765381\pi\)
−0.740437 + 0.672126i \(0.765381\pi\)
\(294\) 0 0
\(295\) −14.6969 −0.855689
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 39.5959 2.28989
\(300\) 0 0
\(301\) 6.89898 0.397651
\(302\) 0 0
\(303\) 8.69694 0.499626
\(304\) 0 0
\(305\) −18.6969 −1.07058
\(306\) 0 0
\(307\) 17.7980 1.01578 0.507892 0.861421i \(-0.330425\pi\)
0.507892 + 0.861421i \(0.330425\pi\)
\(308\) 0 0
\(309\) −23.3939 −1.33083
\(310\) 0 0
\(311\) −6.44949 −0.365717 −0.182859 0.983139i \(-0.558535\pi\)
−0.182859 + 0.983139i \(0.558535\pi\)
\(312\) 0 0
\(313\) −33.5959 −1.89895 −0.949477 0.313837i \(-0.898385\pi\)
−0.949477 + 0.313837i \(0.898385\pi\)
\(314\) 0 0
\(315\) −6.00000 −0.338062
\(316\) 0 0
\(317\) 25.5959 1.43761 0.718805 0.695212i \(-0.244689\pi\)
0.718805 + 0.695212i \(0.244689\pi\)
\(318\) 0 0
\(319\) 6.89898 0.386269
\(320\) 0 0
\(321\) 9.79796 0.546869
\(322\) 0 0
\(323\) 21.7980 1.21287
\(324\) 0 0
\(325\) −4.44949 −0.246813
\(326\) 0 0
\(327\) −12.4949 −0.690969
\(328\) 0 0
\(329\) 1.55051 0.0854824
\(330\) 0 0
\(331\) −32.0000 −1.75888 −0.879440 0.476011i \(-0.842082\pi\)
−0.879440 + 0.476011i \(0.842082\pi\)
\(332\) 0 0
\(333\) 12.0000 0.657596
\(334\) 0 0
\(335\) 29.3939 1.60596
\(336\) 0 0
\(337\) −8.69694 −0.473752 −0.236876 0.971540i \(-0.576124\pi\)
−0.236876 + 0.971540i \(0.576124\pi\)
\(338\) 0 0
\(339\) 24.4949 1.33038
\(340\) 0 0
\(341\) −1.55051 −0.0839648
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −43.5959 −2.34713
\(346\) 0 0
\(347\) −22.4949 −1.20759 −0.603795 0.797140i \(-0.706345\pi\)
−0.603795 + 0.797140i \(0.706345\pi\)
\(348\) 0 0
\(349\) 26.2474 1.40499 0.702497 0.711687i \(-0.252068\pi\)
0.702497 + 0.711687i \(0.252068\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −13.5959 −0.723638 −0.361819 0.932248i \(-0.617844\pi\)
−0.361819 + 0.932248i \(0.617844\pi\)
\(354\) 0 0
\(355\) −6.20204 −0.329170
\(356\) 0 0
\(357\) 10.8990 0.576835
\(358\) 0 0
\(359\) 4.69694 0.247895 0.123947 0.992289i \(-0.460445\pi\)
0.123947 + 0.992289i \(0.460445\pi\)
\(360\) 0 0
\(361\) 5.00000 0.263158
\(362\) 0 0
\(363\) −2.44949 −0.128565
\(364\) 0 0
\(365\) 15.1010 0.790424
\(366\) 0 0
\(367\) −13.1464 −0.686238 −0.343119 0.939292i \(-0.611483\pi\)
−0.343119 + 0.939292i \(0.611483\pi\)
\(368\) 0 0
\(369\) 16.0454 0.835291
\(370\) 0 0
\(371\) 9.79796 0.508685
\(372\) 0 0
\(373\) 23.7980 1.23221 0.616106 0.787663i \(-0.288709\pi\)
0.616106 + 0.787663i \(0.288709\pi\)
\(374\) 0 0
\(375\) 29.3939 1.51789
\(376\) 0 0
\(377\) −30.6969 −1.58097
\(378\) 0 0
\(379\) −12.8990 −0.662576 −0.331288 0.943530i \(-0.607483\pi\)
−0.331288 + 0.943530i \(0.607483\pi\)
\(380\) 0 0
\(381\) −43.5959 −2.23349
\(382\) 0 0
\(383\) 12.6515 0.646463 0.323232 0.946320i \(-0.395231\pi\)
0.323232 + 0.946320i \(0.395231\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) 0 0
\(387\) −20.6969 −1.05208
\(388\) 0 0
\(389\) 25.5959 1.29776 0.648882 0.760889i \(-0.275237\pi\)
0.648882 + 0.760889i \(0.275237\pi\)
\(390\) 0 0
\(391\) 39.5959 2.00245
\(392\) 0 0
\(393\) 24.0000 1.21064
\(394\) 0 0
\(395\) −2.20204 −0.110797
\(396\) 0 0
\(397\) 13.5959 0.682360 0.341180 0.939998i \(-0.389174\pi\)
0.341180 + 0.939998i \(0.389174\pi\)
\(398\) 0 0
\(399\) 12.0000 0.600751
\(400\) 0 0
\(401\) −8.20204 −0.409590 −0.204795 0.978805i \(-0.565653\pi\)
−0.204795 + 0.978805i \(0.565653\pi\)
\(402\) 0 0
\(403\) 6.89898 0.343663
\(404\) 0 0
\(405\) −18.0000 −0.894427
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) −16.4495 −0.813375 −0.406687 0.913567i \(-0.633316\pi\)
−0.406687 + 0.913567i \(0.633316\pi\)
\(410\) 0 0
\(411\) 24.0000 1.18383
\(412\) 0 0
\(413\) 7.34847 0.361595
\(414\) 0 0
\(415\) −14.2020 −0.697151
\(416\) 0 0
\(417\) 16.4041 0.803311
\(418\) 0 0
\(419\) 13.5505 0.661986 0.330993 0.943633i \(-0.392616\pi\)
0.330993 + 0.943633i \(0.392616\pi\)
\(420\) 0 0
\(421\) 13.7980 0.672471 0.336236 0.941778i \(-0.390846\pi\)
0.336236 + 0.941778i \(0.390846\pi\)
\(422\) 0 0
\(423\) −4.65153 −0.226165
\(424\) 0 0
\(425\) −4.44949 −0.215832
\(426\) 0 0
\(427\) 9.34847 0.452404
\(428\) 0 0
\(429\) 10.8990 0.526208
\(430\) 0 0
\(431\) −18.8990 −0.910332 −0.455166 0.890407i \(-0.650420\pi\)
−0.455166 + 0.890407i \(0.650420\pi\)
\(432\) 0 0
\(433\) 30.8990 1.48491 0.742455 0.669896i \(-0.233661\pi\)
0.742455 + 0.669896i \(0.233661\pi\)
\(434\) 0 0
\(435\) 33.7980 1.62049
\(436\) 0 0
\(437\) 43.5959 2.08548
\(438\) 0 0
\(439\) −15.1010 −0.720732 −0.360366 0.932811i \(-0.617348\pi\)
−0.360366 + 0.932811i \(0.617348\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) 0 0
\(447\) 19.1010 0.903447
\(448\) 0 0
\(449\) −19.5959 −0.924789 −0.462394 0.886674i \(-0.653010\pi\)
−0.462394 + 0.886674i \(0.653010\pi\)
\(450\) 0 0
\(451\) −5.34847 −0.251850
\(452\) 0 0
\(453\) 50.6969 2.38195
\(454\) 0 0
\(455\) −8.89898 −0.417191
\(456\) 0 0
\(457\) −39.3939 −1.84277 −0.921384 0.388654i \(-0.872940\pi\)
−0.921384 + 0.388654i \(0.872940\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −22.6515 −1.05499 −0.527493 0.849559i \(-0.676868\pi\)
−0.527493 + 0.849559i \(0.676868\pi\)
\(462\) 0 0
\(463\) −15.1010 −0.701804 −0.350902 0.936412i \(-0.614125\pi\)
−0.350902 + 0.936412i \(0.614125\pi\)
\(464\) 0 0
\(465\) −7.59592 −0.352252
\(466\) 0 0
\(467\) −7.34847 −0.340047 −0.170023 0.985440i \(-0.554384\pi\)
−0.170023 + 0.985440i \(0.554384\pi\)
\(468\) 0 0
\(469\) −14.6969 −0.678642
\(470\) 0 0
\(471\) −24.4949 −1.12867
\(472\) 0 0
\(473\) 6.89898 0.317215
\(474\) 0 0
\(475\) −4.89898 −0.224781
\(476\) 0 0
\(477\) −29.3939 −1.34585
\(478\) 0 0
\(479\) −8.89898 −0.406605 −0.203302 0.979116i \(-0.565167\pi\)
−0.203302 + 0.979116i \(0.565167\pi\)
\(480\) 0 0
\(481\) 17.7980 0.811517
\(482\) 0 0
\(483\) 21.7980 0.991841
\(484\) 0 0
\(485\) 29.7980 1.35306
\(486\) 0 0
\(487\) 27.1010 1.22806 0.614032 0.789281i \(-0.289546\pi\)
0.614032 + 0.789281i \(0.289546\pi\)
\(488\) 0 0
\(489\) 2.20204 0.0995797
\(490\) 0 0
\(491\) 3.30306 0.149065 0.0745325 0.997219i \(-0.476254\pi\)
0.0745325 + 0.997219i \(0.476254\pi\)
\(492\) 0 0
\(493\) −30.6969 −1.38252
\(494\) 0 0
\(495\) −6.00000 −0.269680
\(496\) 0 0
\(497\) 3.10102 0.139100
\(498\) 0 0
\(499\) −13.3031 −0.595527 −0.297763 0.954640i \(-0.596241\pi\)
−0.297763 + 0.954640i \(0.596241\pi\)
\(500\) 0 0
\(501\) 14.2020 0.634500
\(502\) 0 0
\(503\) 24.4949 1.09217 0.546087 0.837729i \(-0.316117\pi\)
0.546087 + 0.837729i \(0.316117\pi\)
\(504\) 0 0
\(505\) −7.10102 −0.315991
\(506\) 0 0
\(507\) −16.6515 −0.739520
\(508\) 0 0
\(509\) 4.69694 0.208188 0.104094 0.994567i \(-0.466806\pi\)
0.104094 + 0.994567i \(0.466806\pi\)
\(510\) 0 0
\(511\) −7.55051 −0.334015
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 19.1010 0.841692
\(516\) 0 0
\(517\) 1.55051 0.0681914
\(518\) 0 0
\(519\) 34.8990 1.53190
\(520\) 0 0
\(521\) −18.8990 −0.827979 −0.413990 0.910282i \(-0.635865\pi\)
−0.413990 + 0.910282i \(0.635865\pi\)
\(522\) 0 0
\(523\) −33.3939 −1.46021 −0.730106 0.683334i \(-0.760529\pi\)
−0.730106 + 0.683334i \(0.760529\pi\)
\(524\) 0 0
\(525\) −2.44949 −0.106904
\(526\) 0 0
\(527\) 6.89898 0.300524
\(528\) 0 0
\(529\) 56.1918 2.44312
\(530\) 0 0
\(531\) −22.0454 −0.956689
\(532\) 0 0
\(533\) 23.7980 1.03080
\(534\) 0 0
\(535\) −8.00000 −0.345870
\(536\) 0 0
\(537\) −57.7980 −2.49417
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 21.5959 0.928481 0.464241 0.885709i \(-0.346327\pi\)
0.464241 + 0.885709i \(0.346327\pi\)
\(542\) 0 0
\(543\) −46.2929 −1.98662
\(544\) 0 0
\(545\) 10.2020 0.437007
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) 0 0
\(549\) −28.0454 −1.19695
\(550\) 0 0
\(551\) −33.7980 −1.43984
\(552\) 0 0
\(553\) 1.10102 0.0468202
\(554\) 0 0
\(555\) −19.5959 −0.831800
\(556\) 0 0
\(557\) −24.2020 −1.02547 −0.512737 0.858546i \(-0.671368\pi\)
−0.512737 + 0.858546i \(0.671368\pi\)
\(558\) 0 0
\(559\) −30.6969 −1.29834
\(560\) 0 0
\(561\) 10.8990 0.460155
\(562\) 0 0
\(563\) 42.2929 1.78243 0.891216 0.453580i \(-0.149853\pi\)
0.891216 + 0.453580i \(0.149853\pi\)
\(564\) 0 0
\(565\) −20.0000 −0.841406
\(566\) 0 0
\(567\) 9.00000 0.377964
\(568\) 0 0
\(569\) 20.6969 0.867661 0.433830 0.900995i \(-0.357162\pi\)
0.433830 + 0.900995i \(0.357162\pi\)
\(570\) 0 0
\(571\) −29.7980 −1.24701 −0.623503 0.781821i \(-0.714291\pi\)
−0.623503 + 0.781821i \(0.714291\pi\)
\(572\) 0 0
\(573\) −43.5959 −1.82125
\(574\) 0 0
\(575\) −8.89898 −0.371113
\(576\) 0 0
\(577\) 26.4949 1.10300 0.551499 0.834176i \(-0.314056\pi\)
0.551499 + 0.834176i \(0.314056\pi\)
\(578\) 0 0
\(579\) 43.1010 1.79122
\(580\) 0 0
\(581\) 7.10102 0.294600
\(582\) 0 0
\(583\) 9.79796 0.405790
\(584\) 0 0
\(585\) 26.6969 1.10378
\(586\) 0 0
\(587\) 26.4495 1.09169 0.545844 0.837887i \(-0.316209\pi\)
0.545844 + 0.837887i \(0.316209\pi\)
\(588\) 0 0
\(589\) 7.59592 0.312984
\(590\) 0 0
\(591\) −33.3031 −1.36990
\(592\) 0 0
\(593\) −35.1464 −1.44329 −0.721645 0.692263i \(-0.756614\pi\)
−0.721645 + 0.692263i \(0.756614\pi\)
\(594\) 0 0
\(595\) −8.89898 −0.364823
\(596\) 0 0
\(597\) 44.2020 1.80907
\(598\) 0 0
\(599\) −16.8990 −0.690474 −0.345237 0.938516i \(-0.612201\pi\)
−0.345237 + 0.938516i \(0.612201\pi\)
\(600\) 0 0
\(601\) 21.3485 0.870822 0.435411 0.900232i \(-0.356603\pi\)
0.435411 + 0.900232i \(0.356603\pi\)
\(602\) 0 0
\(603\) 44.0908 1.79552
\(604\) 0 0
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −23.5959 −0.957729 −0.478864 0.877889i \(-0.658951\pi\)
−0.478864 + 0.877889i \(0.658951\pi\)
\(608\) 0 0
\(609\) −16.8990 −0.684781
\(610\) 0 0
\(611\) −6.89898 −0.279103
\(612\) 0 0
\(613\) 12.6969 0.512825 0.256412 0.966568i \(-0.417460\pi\)
0.256412 + 0.966568i \(0.417460\pi\)
\(614\) 0 0
\(615\) −26.2020 −1.05657
\(616\) 0 0
\(617\) 29.3939 1.18335 0.591676 0.806176i \(-0.298466\pi\)
0.591676 + 0.806176i \(0.298466\pi\)
\(618\) 0 0
\(619\) 10.4495 0.420000 0.210000 0.977701i \(-0.432654\pi\)
0.210000 + 0.977701i \(0.432654\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.00000 −0.240385
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 12.0000 0.479234
\(628\) 0 0
\(629\) 17.7980 0.709651
\(630\) 0 0
\(631\) −21.7980 −0.867763 −0.433882 0.900970i \(-0.642856\pi\)
−0.433882 + 0.900970i \(0.642856\pi\)
\(632\) 0 0
\(633\) 48.9898 1.94717
\(634\) 0 0
\(635\) 35.5959 1.41258
\(636\) 0 0
\(637\) 4.44949 0.176295
\(638\) 0 0
\(639\) −9.30306 −0.368023
\(640\) 0 0
\(641\) 31.5959 1.24796 0.623982 0.781439i \(-0.285514\pi\)
0.623982 + 0.781439i \(0.285514\pi\)
\(642\) 0 0
\(643\) −20.2474 −0.798481 −0.399241 0.916846i \(-0.630726\pi\)
−0.399241 + 0.916846i \(0.630726\pi\)
\(644\) 0 0
\(645\) 33.7980 1.33079
\(646\) 0 0
\(647\) −24.2474 −0.953266 −0.476633 0.879102i \(-0.658143\pi\)
−0.476633 + 0.879102i \(0.658143\pi\)
\(648\) 0 0
\(649\) 7.34847 0.288453
\(650\) 0 0
\(651\) 3.79796 0.148854
\(652\) 0 0
\(653\) −43.3939 −1.69813 −0.849067 0.528285i \(-0.822835\pi\)
−0.849067 + 0.528285i \(0.822835\pi\)
\(654\) 0 0
\(655\) −19.5959 −0.765676
\(656\) 0 0
\(657\) 22.6515 0.883720
\(658\) 0 0
\(659\) 14.8990 0.580382 0.290191 0.956969i \(-0.406281\pi\)
0.290191 + 0.956969i \(0.406281\pi\)
\(660\) 0 0
\(661\) 9.59592 0.373238 0.186619 0.982432i \(-0.440247\pi\)
0.186619 + 0.982432i \(0.440247\pi\)
\(662\) 0 0
\(663\) −48.4949 −1.88339
\(664\) 0 0
\(665\) −9.79796 −0.379949
\(666\) 0 0
\(667\) −61.3939 −2.37718
\(668\) 0 0
\(669\) −51.7980 −2.00262
\(670\) 0 0
\(671\) 9.34847 0.360894
\(672\) 0 0
\(673\) 36.6969 1.41456 0.707282 0.706932i \(-0.249921\pi\)
0.707282 + 0.706932i \(0.249921\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −24.9444 −0.958691 −0.479345 0.877626i \(-0.659126\pi\)
−0.479345 + 0.877626i \(0.659126\pi\)
\(678\) 0 0
\(679\) −14.8990 −0.571770
\(680\) 0 0
\(681\) 43.5959 1.67060
\(682\) 0 0
\(683\) −45.7980 −1.75241 −0.876205 0.481938i \(-0.839933\pi\)
−0.876205 + 0.481938i \(0.839933\pi\)
\(684\) 0 0
\(685\) −19.5959 −0.748722
\(686\) 0 0
\(687\) 36.4949 1.39237
\(688\) 0 0
\(689\) −43.5959 −1.66087
\(690\) 0 0
\(691\) −6.85357 −0.260722 −0.130361 0.991467i \(-0.541614\pi\)
−0.130361 + 0.991467i \(0.541614\pi\)
\(692\) 0 0
\(693\) 3.00000 0.113961
\(694\) 0 0
\(695\) −13.3939 −0.508059
\(696\) 0 0
\(697\) 23.7980 0.901412
\(698\) 0 0
\(699\) 26.6969 1.00977
\(700\) 0 0
\(701\) 8.69694 0.328479 0.164239 0.986421i \(-0.447483\pi\)
0.164239 + 0.986421i \(0.447483\pi\)
\(702\) 0 0
\(703\) 19.5959 0.739074
\(704\) 0 0
\(705\) 7.59592 0.286079
\(706\) 0 0
\(707\) 3.55051 0.133531
\(708\) 0 0
\(709\) −15.7980 −0.593305 −0.296652 0.954986i \(-0.595870\pi\)
−0.296652 + 0.954986i \(0.595870\pi\)
\(710\) 0 0
\(711\) −3.30306 −0.123874
\(712\) 0 0
\(713\) 13.7980 0.516738
\(714\) 0 0
\(715\) −8.89898 −0.332803
\(716\) 0 0
\(717\) −39.1918 −1.46365
\(718\) 0 0
\(719\) 10.8536 0.404770 0.202385 0.979306i \(-0.435131\pi\)
0.202385 + 0.979306i \(0.435131\pi\)
\(720\) 0 0
\(721\) −9.55051 −0.355680
\(722\) 0 0
\(723\) −32.6969 −1.21601
\(724\) 0 0
\(725\) 6.89898 0.256222
\(726\) 0 0
\(727\) −6.44949 −0.239198 −0.119599 0.992822i \(-0.538161\pi\)
−0.119599 + 0.992822i \(0.538161\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −30.6969 −1.13537
\(732\) 0 0
\(733\) 30.6515 1.13214 0.566070 0.824357i \(-0.308463\pi\)
0.566070 + 0.824357i \(0.308463\pi\)
\(734\) 0 0
\(735\) −4.89898 −0.180702
\(736\) 0 0
\(737\) −14.6969 −0.541369
\(738\) 0 0
\(739\) 17.3939 0.639844 0.319922 0.947444i \(-0.396343\pi\)
0.319922 + 0.947444i \(0.396343\pi\)
\(740\) 0 0
\(741\) −53.3939 −1.96147
\(742\) 0 0
\(743\) 19.5959 0.718905 0.359452 0.933163i \(-0.382964\pi\)
0.359452 + 0.933163i \(0.382964\pi\)
\(744\) 0 0
\(745\) −15.5959 −0.571390
\(746\) 0 0
\(747\) −21.3031 −0.779438
\(748\) 0 0
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −3.10102 −0.113158 −0.0565789 0.998398i \(-0.518019\pi\)
−0.0565789 + 0.998398i \(0.518019\pi\)
\(752\) 0 0
\(753\) 6.00000 0.218652
\(754\) 0 0
\(755\) −41.3939 −1.50648
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 0 0
\(759\) 21.7980 0.791216
\(760\) 0 0
\(761\) −13.8434 −0.501822 −0.250911 0.968010i \(-0.580730\pi\)
−0.250911 + 0.968010i \(0.580730\pi\)
\(762\) 0 0
\(763\) −5.10102 −0.184669
\(764\) 0 0
\(765\) 26.6969 0.965230
\(766\) 0 0
\(767\) −32.6969 −1.18062
\(768\) 0 0
\(769\) 4.85357 0.175024 0.0875121 0.996163i \(-0.472108\pi\)
0.0875121 + 0.996163i \(0.472108\pi\)
\(770\) 0 0
\(771\) −38.6969 −1.39364
\(772\) 0 0
\(773\) 32.6969 1.17603 0.588014 0.808851i \(-0.299910\pi\)
0.588014 + 0.808851i \(0.299910\pi\)
\(774\) 0 0
\(775\) −1.55051 −0.0556960
\(776\) 0 0
\(777\) 9.79796 0.351500
\(778\) 0 0
\(779\) 26.2020 0.938786
\(780\) 0 0
\(781\) 3.10102 0.110963
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 20.0000 0.713831
\(786\) 0 0
\(787\) −18.6969 −0.666474 −0.333237 0.942843i \(-0.608141\pi\)
−0.333237 + 0.942843i \(0.608141\pi\)
\(788\) 0 0
\(789\) 48.0000 1.70885
\(790\) 0 0
\(791\) 10.0000 0.355559
\(792\) 0 0
\(793\) −41.5959 −1.47711
\(794\) 0 0
\(795\) 48.0000 1.70238
\(796\) 0 0
\(797\) −12.2020 −0.432218 −0.216109 0.976369i \(-0.569337\pi\)
−0.216109 + 0.976369i \(0.569337\pi\)
\(798\) 0 0
\(799\) −6.89898 −0.244068
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 0 0
\(803\) −7.55051 −0.266452
\(804\) 0 0
\(805\) −17.7980 −0.627296
\(806\) 0 0
\(807\) 56.0908 1.97449
\(808\) 0 0
\(809\) 13.5959 0.478007 0.239004 0.971019i \(-0.423179\pi\)
0.239004 + 0.971019i \(0.423179\pi\)
\(810\) 0 0
\(811\) 15.5959 0.547647 0.273823 0.961780i \(-0.411712\pi\)
0.273823 + 0.961780i \(0.411712\pi\)
\(812\) 0 0
\(813\) −75.1918 −2.63709
\(814\) 0 0
\(815\) −1.79796 −0.0629798
\(816\) 0 0
\(817\) −33.7980 −1.18244
\(818\) 0 0
\(819\) −13.3485 −0.466433
\(820\) 0 0
\(821\) 3.79796 0.132550 0.0662748 0.997801i \(-0.478889\pi\)
0.0662748 + 0.997801i \(0.478889\pi\)
\(822\) 0 0
\(823\) −43.1918 −1.50557 −0.752786 0.658265i \(-0.771291\pi\)
−0.752786 + 0.658265i \(0.771291\pi\)
\(824\) 0 0
\(825\) −2.44949 −0.0852803
\(826\) 0 0
\(827\) 23.3031 0.810327 0.405163 0.914244i \(-0.367215\pi\)
0.405163 + 0.914244i \(0.367215\pi\)
\(828\) 0 0
\(829\) −14.0000 −0.486240 −0.243120 0.969996i \(-0.578171\pi\)
−0.243120 + 0.969996i \(0.578171\pi\)
\(830\) 0 0
\(831\) 9.30306 0.322720
\(832\) 0 0
\(833\) 4.44949 0.154166
\(834\) 0 0
\(835\) −11.5959 −0.401293
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 14.4495 0.498852 0.249426 0.968394i \(-0.419758\pi\)
0.249426 + 0.968394i \(0.419758\pi\)
\(840\) 0 0
\(841\) 18.5959 0.641239
\(842\) 0 0
\(843\) 36.4949 1.25695
\(844\) 0 0
\(845\) 13.5959 0.467714
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 0 0
\(849\) 58.7878 2.01759
\(850\) 0 0
\(851\) 35.5959 1.22021
\(852\) 0 0
\(853\) 8.85357 0.303141 0.151570 0.988446i \(-0.451567\pi\)
0.151570 + 0.988446i \(0.451567\pi\)
\(854\) 0 0
\(855\) 29.3939 1.00525
\(856\) 0 0
\(857\) 43.1464 1.47385 0.736927 0.675972i \(-0.236276\pi\)
0.736927 + 0.675972i \(0.236276\pi\)
\(858\) 0 0
\(859\) −39.8434 −1.35944 −0.679719 0.733473i \(-0.737898\pi\)
−0.679719 + 0.733473i \(0.737898\pi\)
\(860\) 0 0
\(861\) 13.1010 0.446482
\(862\) 0 0
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 0 0
\(865\) −28.4949 −0.968856
\(866\) 0 0
\(867\) −6.85357 −0.232760
\(868\) 0 0
\(869\) 1.10102 0.0373496
\(870\) 0 0
\(871\) 65.3939 2.21579
\(872\) 0 0
\(873\) 44.6969 1.51276
\(874\) 0 0
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) 7.30306 0.246607 0.123303 0.992369i \(-0.460651\pi\)
0.123303 + 0.992369i \(0.460651\pi\)
\(878\) 0 0
\(879\) 62.0908 2.09427
\(880\) 0 0
\(881\) 8.69694 0.293007 0.146504 0.989210i \(-0.453198\pi\)
0.146504 + 0.989210i \(0.453198\pi\)
\(882\) 0 0
\(883\) −10.2020 −0.343326 −0.171663 0.985156i \(-0.554914\pi\)
−0.171663 + 0.985156i \(0.554914\pi\)
\(884\) 0 0
\(885\) 36.0000 1.21013
\(886\) 0 0
\(887\) 38.2929 1.28575 0.642874 0.765972i \(-0.277742\pi\)
0.642874 + 0.765972i \(0.277742\pi\)
\(888\) 0 0
\(889\) −17.7980 −0.596924
\(890\) 0 0
\(891\) 9.00000 0.301511
\(892\) 0 0
\(893\) −7.59592 −0.254188
\(894\) 0 0
\(895\) 47.1918 1.57745
\(896\) 0 0
\(897\) −96.9898 −3.23839
\(898\) 0 0
\(899\) −10.6969 −0.356763
\(900\) 0 0
\(901\) −43.5959 −1.45239
\(902\) 0 0
\(903\) −16.8990 −0.562363
\(904\) 0 0
\(905\) 37.7980 1.25645
\(906\) 0 0
\(907\) 12.8990 0.428304 0.214152 0.976800i \(-0.431301\pi\)
0.214152 + 0.976800i \(0.431301\pi\)
\(908\) 0 0
\(909\) −10.6515 −0.353289
\(910\) 0 0
\(911\) −12.4949 −0.413974 −0.206987 0.978344i \(-0.566366\pi\)
−0.206987 + 0.978344i \(0.566366\pi\)
\(912\) 0 0
\(913\) 7.10102 0.235009
\(914\) 0 0
\(915\) 45.7980 1.51403
\(916\) 0 0
\(917\) 9.79796 0.323557
\(918\) 0 0
\(919\) 43.5959 1.43810 0.719048 0.694960i \(-0.244578\pi\)
0.719048 + 0.694960i \(0.244578\pi\)
\(920\) 0 0
\(921\) −43.5959 −1.43653
\(922\) 0 0
\(923\) −13.7980 −0.454165
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) 28.6515 0.941040
\(928\) 0 0
\(929\) 16.2929 0.534551 0.267276 0.963620i \(-0.413877\pi\)
0.267276 + 0.963620i \(0.413877\pi\)
\(930\) 0 0
\(931\) 4.89898 0.160558
\(932\) 0 0
\(933\) 15.7980 0.517202
\(934\) 0 0
\(935\) −8.89898 −0.291028
\(936\) 0 0
\(937\) −20.9444 −0.684223 −0.342112 0.939659i \(-0.611142\pi\)
−0.342112 + 0.939659i \(0.611142\pi\)
\(938\) 0 0
\(939\) 82.2929 2.68553
\(940\) 0 0
\(941\) 48.4495 1.57941 0.789704 0.613488i \(-0.210234\pi\)
0.789704 + 0.613488i \(0.210234\pi\)
\(942\) 0 0
\(943\) 47.5959 1.54994
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 39.5959 1.28669 0.643347 0.765575i \(-0.277545\pi\)
0.643347 + 0.765575i \(0.277545\pi\)
\(948\) 0 0
\(949\) 33.5959 1.09057
\(950\) 0 0
\(951\) −62.6969 −2.03309
\(952\) 0 0
\(953\) −24.2020 −0.783981 −0.391991 0.919969i \(-0.628213\pi\)
−0.391991 + 0.919969i \(0.628213\pi\)
\(954\) 0 0
\(955\) 35.5959 1.15186
\(956\) 0 0
\(957\) −16.8990 −0.546266
\(958\) 0 0
\(959\) 9.79796 0.316393
\(960\) 0 0
\(961\) −28.5959 −0.922449
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) −35.1918 −1.13287
\(966\) 0 0
\(967\) −29.1010 −0.935826 −0.467913 0.883775i \(-0.654994\pi\)
−0.467913 + 0.883775i \(0.654994\pi\)
\(968\) 0 0
\(969\) −53.3939 −1.71526
\(970\) 0 0
\(971\) 6.04541 0.194006 0.0970032 0.995284i \(-0.469074\pi\)
0.0970032 + 0.995284i \(0.469074\pi\)
\(972\) 0 0
\(973\) 6.69694 0.214694
\(974\) 0 0
\(975\) 10.8990 0.349047
\(976\) 0 0
\(977\) 23.7980 0.761364 0.380682 0.924706i \(-0.375689\pi\)
0.380682 + 0.924706i \(0.375689\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 15.3031 0.488589
\(982\) 0 0
\(983\) 4.65153 0.148361 0.0741804 0.997245i \(-0.476366\pi\)
0.0741804 + 0.997245i \(0.476366\pi\)
\(984\) 0 0
\(985\) 27.1918 0.866404
\(986\) 0 0
\(987\) −3.79796 −0.120890
\(988\) 0 0
\(989\) −61.3939 −1.95221
\(990\) 0 0
\(991\) −28.0908 −0.892334 −0.446167 0.894950i \(-0.647211\pi\)
−0.446167 + 0.894950i \(0.647211\pi\)
\(992\) 0 0
\(993\) 78.3837 2.48743
\(994\) 0 0
\(995\) −36.0908 −1.14416
\(996\) 0 0
\(997\) 32.9444 1.04336 0.521680 0.853141i \(-0.325306\pi\)
0.521680 + 0.853141i \(0.325306\pi\)
\(998\) 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 308.2.a.b.1.1 2
3.2 odd 2 2772.2.a.n.1.2 2
4.3 odd 2 1232.2.a.n.1.2 2
5.2 odd 4 7700.2.e.j.1849.3 4
5.3 odd 4 7700.2.e.j.1849.2 4
5.4 even 2 7700.2.a.s.1.2 2
7.2 even 3 2156.2.i.e.1145.2 4
7.3 odd 6 2156.2.i.i.177.1 4
7.4 even 3 2156.2.i.e.177.2 4
7.5 odd 6 2156.2.i.i.1145.1 4
7.6 odd 2 2156.2.a.c.1.2 2
8.3 odd 2 4928.2.a.bq.1.1 2
8.5 even 2 4928.2.a.bp.1.2 2
11.10 odd 2 3388.2.a.h.1.1 2
28.27 even 2 8624.2.a.bj.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
308.2.a.b.1.1 2 1.1 even 1 trivial
1232.2.a.n.1.2 2 4.3 odd 2
2156.2.a.c.1.2 2 7.6 odd 2
2156.2.i.e.177.2 4 7.4 even 3
2156.2.i.e.1145.2 4 7.2 even 3
2156.2.i.i.177.1 4 7.3 odd 6
2156.2.i.i.1145.1 4 7.5 odd 6
2772.2.a.n.1.2 2 3.2 odd 2
3388.2.a.h.1.1 2 11.10 odd 2
4928.2.a.bp.1.2 2 8.5 even 2
4928.2.a.bq.1.1 2 8.3 odd 2
7700.2.a.s.1.2 2 5.4 even 2
7700.2.e.j.1849.2 4 5.3 odd 4
7700.2.e.j.1849.3 4 5.2 odd 4
8624.2.a.bj.1.1 2 28.27 even 2