Properties

Label 592.2.a.f.1.2
Level $592$
Weight $2$
Character 592.1
Self dual yes
Analytic conductor $4.727$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.72714379966\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 592.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.302776 q^{3} +1.30278 q^{5} -4.60555 q^{7} -2.90833 q^{9} +O(q^{10})\) \(q+0.302776 q^{3} +1.30278 q^{5} -4.60555 q^{7} -2.90833 q^{9} -1.30278 q^{11} -2.30278 q^{13} +0.394449 q^{15} -6.00000 q^{17} -2.00000 q^{19} -1.39445 q^{21} +6.90833 q^{23} -3.30278 q^{25} -1.78890 q^{27} +6.90833 q^{29} -3.30278 q^{31} -0.394449 q^{33} -6.00000 q^{35} +1.00000 q^{37} -0.697224 q^{39} -0.908327 q^{41} +6.60555 q^{43} -3.78890 q^{45} +2.60555 q^{47} +14.2111 q^{49} -1.81665 q^{51} -6.00000 q^{53} -1.69722 q^{55} -0.605551 q^{57} -3.39445 q^{59} -10.5139 q^{61} +13.3944 q^{63} -3.00000 q^{65} -14.5139 q^{67} +2.09167 q^{69} -6.00000 q^{71} -8.69722 q^{73} -1.00000 q^{75} +6.00000 q^{77} +16.1194 q^{79} +8.18335 q^{81} -17.2111 q^{83} -7.81665 q^{85} +2.09167 q^{87} +5.21110 q^{89} +10.6056 q^{91} -1.00000 q^{93} -2.60555 q^{95} +12.4222 q^{97} +3.78890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - q^{5} - 2 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} - q^{5} - 2 q^{7} + 5 q^{9} + q^{11} - q^{13} + 8 q^{15} - 12 q^{17} - 4 q^{19} - 10 q^{21} + 3 q^{23} - 3 q^{25} - 18 q^{27} + 3 q^{29} - 3 q^{31} - 8 q^{33} - 12 q^{35} + 2 q^{37} - 5 q^{39} + 9 q^{41} + 6 q^{43} - 22 q^{45} - 2 q^{47} + 14 q^{49} + 18 q^{51} - 12 q^{53} - 7 q^{55} + 6 q^{57} - 14 q^{59} - 3 q^{61} + 34 q^{63} - 6 q^{65} - 11 q^{67} + 15 q^{69} - 12 q^{71} - 21 q^{73} - 2 q^{75} + 12 q^{77} + 7 q^{79} + 38 q^{81} - 20 q^{83} + 6 q^{85} + 15 q^{87} - 4 q^{89} + 14 q^{91} - 2 q^{93} + 2 q^{95} - 4 q^{97} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0.302776 0.174808 0.0874038 0.996173i \(-0.472143\pi\)
0.0874038 + 0.996173i \(0.472143\pi\)
\(4\) 0 0
\(5\) 1.30278 0.582619 0.291309 0.956629i \(-0.405909\pi\)
0.291309 + 0.956629i \(0.405909\pi\)
\(6\) 0 0
\(7\) −4.60555 −1.74073 −0.870367 0.492403i \(-0.836119\pi\)
−0.870367 + 0.492403i \(0.836119\pi\)
\(8\) 0 0
\(9\) −2.90833 −0.969442
\(10\) 0 0
\(11\) −1.30278 −0.392802 −0.196401 0.980524i \(-0.562925\pi\)
−0.196401 + 0.980524i \(0.562925\pi\)
\(12\) 0 0
\(13\) −2.30278 −0.638675 −0.319338 0.947641i \(-0.603460\pi\)
−0.319338 + 0.947641i \(0.603460\pi\)
\(14\) 0 0
\(15\) 0.394449 0.101846
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) −1.39445 −0.304294
\(22\) 0 0
\(23\) 6.90833 1.44049 0.720243 0.693722i \(-0.244030\pi\)
0.720243 + 0.693722i \(0.244030\pi\)
\(24\) 0 0
\(25\) −3.30278 −0.660555
\(26\) 0 0
\(27\) −1.78890 −0.344273
\(28\) 0 0
\(29\) 6.90833 1.28284 0.641422 0.767188i \(-0.278345\pi\)
0.641422 + 0.767188i \(0.278345\pi\)
\(30\) 0 0
\(31\) −3.30278 −0.593196 −0.296598 0.955002i \(-0.595852\pi\)
−0.296598 + 0.955002i \(0.595852\pi\)
\(32\) 0 0
\(33\) −0.394449 −0.0686647
\(34\) 0 0
\(35\) −6.00000 −1.01419
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) −0.697224 −0.111645
\(40\) 0 0
\(41\) −0.908327 −0.141857 −0.0709284 0.997481i \(-0.522596\pi\)
−0.0709284 + 0.997481i \(0.522596\pi\)
\(42\) 0 0
\(43\) 6.60555 1.00734 0.503669 0.863897i \(-0.331983\pi\)
0.503669 + 0.863897i \(0.331983\pi\)
\(44\) 0 0
\(45\) −3.78890 −0.564815
\(46\) 0 0
\(47\) 2.60555 0.380059 0.190029 0.981778i \(-0.439142\pi\)
0.190029 + 0.981778i \(0.439142\pi\)
\(48\) 0 0
\(49\) 14.2111 2.03016
\(50\) 0 0
\(51\) −1.81665 −0.254382
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) −1.69722 −0.228854
\(56\) 0 0
\(57\) −0.605551 −0.0802072
\(58\) 0 0
\(59\) −3.39445 −0.441920 −0.220960 0.975283i \(-0.570919\pi\)
−0.220960 + 0.975283i \(0.570919\pi\)
\(60\) 0 0
\(61\) −10.5139 −1.34616 −0.673082 0.739568i \(-0.735030\pi\)
−0.673082 + 0.739568i \(0.735030\pi\)
\(62\) 0 0
\(63\) 13.3944 1.68754
\(64\) 0 0
\(65\) −3.00000 −0.372104
\(66\) 0 0
\(67\) −14.5139 −1.77315 −0.886576 0.462583i \(-0.846923\pi\)
−0.886576 + 0.462583i \(0.846923\pi\)
\(68\) 0 0
\(69\) 2.09167 0.251808
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −8.69722 −1.01793 −0.508967 0.860786i \(-0.669972\pi\)
−0.508967 + 0.860786i \(0.669972\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) 0 0
\(79\) 16.1194 1.81358 0.906789 0.421585i \(-0.138526\pi\)
0.906789 + 0.421585i \(0.138526\pi\)
\(80\) 0 0
\(81\) 8.18335 0.909261
\(82\) 0 0
\(83\) −17.2111 −1.88916 −0.944582 0.328276i \(-0.893533\pi\)
−0.944582 + 0.328276i \(0.893533\pi\)
\(84\) 0 0
\(85\) −7.81665 −0.847835
\(86\) 0 0
\(87\) 2.09167 0.224251
\(88\) 0 0
\(89\) 5.21110 0.552376 0.276188 0.961104i \(-0.410929\pi\)
0.276188 + 0.961104i \(0.410929\pi\)
\(90\) 0 0
\(91\) 10.6056 1.11176
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) −2.60555 −0.267324
\(96\) 0 0
\(97\) 12.4222 1.26128 0.630642 0.776074i \(-0.282792\pi\)
0.630642 + 0.776074i \(0.282792\pi\)
\(98\) 0 0
\(99\) 3.78890 0.380799
\(100\) 0 0
\(101\) 16.4222 1.63407 0.817035 0.576588i \(-0.195616\pi\)
0.817035 + 0.576588i \(0.195616\pi\)
\(102\) 0 0
\(103\) −3.30278 −0.325432 −0.162716 0.986673i \(-0.552025\pi\)
−0.162716 + 0.986673i \(0.552025\pi\)
\(104\) 0 0
\(105\) −1.81665 −0.177287
\(106\) 0 0
\(107\) −4.30278 −0.415965 −0.207983 0.978133i \(-0.566690\pi\)
−0.207983 + 0.978133i \(0.566690\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0.302776 0.0287382
\(112\) 0 0
\(113\) 11.2111 1.05465 0.527326 0.849663i \(-0.323195\pi\)
0.527326 + 0.849663i \(0.323195\pi\)
\(114\) 0 0
\(115\) 9.00000 0.839254
\(116\) 0 0
\(117\) 6.69722 0.619159
\(118\) 0 0
\(119\) 27.6333 2.53314
\(120\) 0 0
\(121\) −9.30278 −0.845707
\(122\) 0 0
\(123\) −0.275019 −0.0247977
\(124\) 0 0
\(125\) −10.8167 −0.967471
\(126\) 0 0
\(127\) 4.78890 0.424946 0.212473 0.977167i \(-0.431848\pi\)
0.212473 + 0.977167i \(0.431848\pi\)
\(128\) 0 0
\(129\) 2.00000 0.176090
\(130\) 0 0
\(131\) −3.39445 −0.296574 −0.148287 0.988944i \(-0.547376\pi\)
−0.148287 + 0.988944i \(0.547376\pi\)
\(132\) 0 0
\(133\) 9.21110 0.798704
\(134\) 0 0
\(135\) −2.33053 −0.200580
\(136\) 0 0
\(137\) −9.90833 −0.846525 −0.423263 0.906007i \(-0.639115\pi\)
−0.423263 + 0.906007i \(0.639115\pi\)
\(138\) 0 0
\(139\) −8.90833 −0.755594 −0.377797 0.925888i \(-0.623318\pi\)
−0.377797 + 0.925888i \(0.623318\pi\)
\(140\) 0 0
\(141\) 0.788897 0.0664372
\(142\) 0 0
\(143\) 3.00000 0.250873
\(144\) 0 0
\(145\) 9.00000 0.747409
\(146\) 0 0
\(147\) 4.30278 0.354887
\(148\) 0 0
\(149\) −1.81665 −0.148826 −0.0744130 0.997228i \(-0.523708\pi\)
−0.0744130 + 0.997228i \(0.523708\pi\)
\(150\) 0 0
\(151\) 13.3944 1.09002 0.545012 0.838428i \(-0.316525\pi\)
0.545012 + 0.838428i \(0.316525\pi\)
\(152\) 0 0
\(153\) 17.4500 1.41075
\(154\) 0 0
\(155\) −4.30278 −0.345607
\(156\) 0 0
\(157\) 7.21110 0.575509 0.287754 0.957704i \(-0.407091\pi\)
0.287754 + 0.957704i \(0.407091\pi\)
\(158\) 0 0
\(159\) −1.81665 −0.144070
\(160\) 0 0
\(161\) −31.8167 −2.50750
\(162\) 0 0
\(163\) 20.4222 1.59959 0.799795 0.600273i \(-0.204941\pi\)
0.799795 + 0.600273i \(0.204941\pi\)
\(164\) 0 0
\(165\) −0.513878 −0.0400054
\(166\) 0 0
\(167\) −12.5139 −0.968353 −0.484176 0.874970i \(-0.660881\pi\)
−0.484176 + 0.874970i \(0.660881\pi\)
\(168\) 0 0
\(169\) −7.69722 −0.592094
\(170\) 0 0
\(171\) 5.81665 0.444811
\(172\) 0 0
\(173\) −23.2111 −1.76471 −0.882354 0.470587i \(-0.844042\pi\)
−0.882354 + 0.470587i \(0.844042\pi\)
\(174\) 0 0
\(175\) 15.2111 1.14985
\(176\) 0 0
\(177\) −1.02776 −0.0772509
\(178\) 0 0
\(179\) −7.81665 −0.584244 −0.292122 0.956381i \(-0.594361\pi\)
−0.292122 + 0.956381i \(0.594361\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) −3.18335 −0.235320
\(184\) 0 0
\(185\) 1.30278 0.0957820
\(186\) 0 0
\(187\) 7.81665 0.571610
\(188\) 0 0
\(189\) 8.23886 0.599289
\(190\) 0 0
\(191\) −12.5139 −0.905472 −0.452736 0.891644i \(-0.649552\pi\)
−0.452736 + 0.891644i \(0.649552\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) −0.908327 −0.0650466
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 2.42221 0.171706 0.0858528 0.996308i \(-0.472639\pi\)
0.0858528 + 0.996308i \(0.472639\pi\)
\(200\) 0 0
\(201\) −4.39445 −0.309961
\(202\) 0 0
\(203\) −31.8167 −2.23309
\(204\) 0 0
\(205\) −1.18335 −0.0826485
\(206\) 0 0
\(207\) −20.0917 −1.39647
\(208\) 0 0
\(209\) 2.60555 0.180230
\(210\) 0 0
\(211\) −6.69722 −0.461056 −0.230528 0.973066i \(-0.574045\pi\)
−0.230528 + 0.973066i \(0.574045\pi\)
\(212\) 0 0
\(213\) −1.81665 −0.124475
\(214\) 0 0
\(215\) 8.60555 0.586894
\(216\) 0 0
\(217\) 15.2111 1.03260
\(218\) 0 0
\(219\) −2.63331 −0.177942
\(220\) 0 0
\(221\) 13.8167 0.929409
\(222\) 0 0
\(223\) −15.8167 −1.05916 −0.529581 0.848260i \(-0.677651\pi\)
−0.529581 + 0.848260i \(0.677651\pi\)
\(224\) 0 0
\(225\) 9.60555 0.640370
\(226\) 0 0
\(227\) 7.81665 0.518810 0.259405 0.965769i \(-0.416474\pi\)
0.259405 + 0.965769i \(0.416474\pi\)
\(228\) 0 0
\(229\) 17.3944 1.14946 0.574729 0.818344i \(-0.305108\pi\)
0.574729 + 0.818344i \(0.305108\pi\)
\(230\) 0 0
\(231\) 1.81665 0.119527
\(232\) 0 0
\(233\) −9.51388 −0.623275 −0.311637 0.950201i \(-0.600877\pi\)
−0.311637 + 0.950201i \(0.600877\pi\)
\(234\) 0 0
\(235\) 3.39445 0.221429
\(236\) 0 0
\(237\) 4.88057 0.317027
\(238\) 0 0
\(239\) −0.513878 −0.0332400 −0.0166200 0.999862i \(-0.505291\pi\)
−0.0166200 + 0.999862i \(0.505291\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 7.84441 0.503219
\(244\) 0 0
\(245\) 18.5139 1.18281
\(246\) 0 0
\(247\) 4.60555 0.293044
\(248\) 0 0
\(249\) −5.21110 −0.330240
\(250\) 0 0
\(251\) 6.78890 0.428511 0.214256 0.976778i \(-0.431267\pi\)
0.214256 + 0.976778i \(0.431267\pi\)
\(252\) 0 0
\(253\) −9.00000 −0.565825
\(254\) 0 0
\(255\) −2.36669 −0.148208
\(256\) 0 0
\(257\) −11.2111 −0.699329 −0.349665 0.936875i \(-0.613704\pi\)
−0.349665 + 0.936875i \(0.613704\pi\)
\(258\) 0 0
\(259\) −4.60555 −0.286175
\(260\) 0 0
\(261\) −20.0917 −1.24364
\(262\) 0 0
\(263\) 7.81665 0.481996 0.240998 0.970526i \(-0.422525\pi\)
0.240998 + 0.970526i \(0.422525\pi\)
\(264\) 0 0
\(265\) −7.81665 −0.480173
\(266\) 0 0
\(267\) 1.57779 0.0965595
\(268\) 0 0
\(269\) −6.78890 −0.413926 −0.206963 0.978349i \(-0.566358\pi\)
−0.206963 + 0.978349i \(0.566358\pi\)
\(270\) 0 0
\(271\) −6.42221 −0.390121 −0.195061 0.980791i \(-0.562490\pi\)
−0.195061 + 0.980791i \(0.562490\pi\)
\(272\) 0 0
\(273\) 3.21110 0.194345
\(274\) 0 0
\(275\) 4.30278 0.259467
\(276\) 0 0
\(277\) −25.1194 −1.50928 −0.754640 0.656139i \(-0.772189\pi\)
−0.754640 + 0.656139i \(0.772189\pi\)
\(278\) 0 0
\(279\) 9.60555 0.575069
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) −17.3944 −1.03399 −0.516996 0.855988i \(-0.672950\pi\)
−0.516996 + 0.855988i \(0.672950\pi\)
\(284\) 0 0
\(285\) −0.788897 −0.0467303
\(286\) 0 0
\(287\) 4.18335 0.246935
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 3.76114 0.220482
\(292\) 0 0
\(293\) −25.0278 −1.46214 −0.731069 0.682304i \(-0.760978\pi\)
−0.731069 + 0.682304i \(0.760978\pi\)
\(294\) 0 0
\(295\) −4.42221 −0.257471
\(296\) 0 0
\(297\) 2.33053 0.135231
\(298\) 0 0
\(299\) −15.9083 −0.920002
\(300\) 0 0
\(301\) −30.4222 −1.75351
\(302\) 0 0
\(303\) 4.97224 0.285648
\(304\) 0 0
\(305\) −13.6972 −0.784301
\(306\) 0 0
\(307\) −7.09167 −0.404743 −0.202372 0.979309i \(-0.564865\pi\)
−0.202372 + 0.979309i \(0.564865\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) 0 0
\(311\) −5.09167 −0.288722 −0.144361 0.989525i \(-0.546113\pi\)
−0.144361 + 0.989525i \(0.546113\pi\)
\(312\) 0 0
\(313\) 27.0278 1.52770 0.763850 0.645394i \(-0.223307\pi\)
0.763850 + 0.645394i \(0.223307\pi\)
\(314\) 0 0
\(315\) 17.4500 0.983194
\(316\) 0 0
\(317\) −5.21110 −0.292685 −0.146342 0.989234i \(-0.546750\pi\)
−0.146342 + 0.989234i \(0.546750\pi\)
\(318\) 0 0
\(319\) −9.00000 −0.503903
\(320\) 0 0
\(321\) −1.30278 −0.0727138
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) 7.60555 0.421880
\(326\) 0 0
\(327\) 0.605551 0.0334871
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −1.21110 −0.0665682 −0.0332841 0.999446i \(-0.510597\pi\)
−0.0332841 + 0.999446i \(0.510597\pi\)
\(332\) 0 0
\(333\) −2.90833 −0.159375
\(334\) 0 0
\(335\) −18.9083 −1.03307
\(336\) 0 0
\(337\) −19.1194 −1.04150 −0.520751 0.853709i \(-0.674348\pi\)
−0.520751 + 0.853709i \(0.674348\pi\)
\(338\) 0 0
\(339\) 3.39445 0.184361
\(340\) 0 0
\(341\) 4.30278 0.233008
\(342\) 0 0
\(343\) −33.2111 −1.79323
\(344\) 0 0
\(345\) 2.72498 0.146708
\(346\) 0 0
\(347\) −31.8167 −1.70801 −0.854004 0.520267i \(-0.825832\pi\)
−0.854004 + 0.520267i \(0.825832\pi\)
\(348\) 0 0
\(349\) −22.2389 −1.19042 −0.595209 0.803571i \(-0.702931\pi\)
−0.595209 + 0.803571i \(0.702931\pi\)
\(350\) 0 0
\(351\) 4.11943 0.219879
\(352\) 0 0
\(353\) 31.8167 1.69343 0.846715 0.532047i \(-0.178577\pi\)
0.846715 + 0.532047i \(0.178577\pi\)
\(354\) 0 0
\(355\) −7.81665 −0.414865
\(356\) 0 0
\(357\) 8.36669 0.442812
\(358\) 0 0
\(359\) 11.2111 0.591699 0.295850 0.955235i \(-0.404397\pi\)
0.295850 + 0.955235i \(0.404397\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) −2.81665 −0.147836
\(364\) 0 0
\(365\) −11.3305 −0.593067
\(366\) 0 0
\(367\) 17.8167 0.930022 0.465011 0.885305i \(-0.346050\pi\)
0.465011 + 0.885305i \(0.346050\pi\)
\(368\) 0 0
\(369\) 2.64171 0.137522
\(370\) 0 0
\(371\) 27.6333 1.43465
\(372\) 0 0
\(373\) 3.81665 0.197619 0.0988094 0.995106i \(-0.468497\pi\)
0.0988094 + 0.995106i \(0.468497\pi\)
\(374\) 0 0
\(375\) −3.27502 −0.169121
\(376\) 0 0
\(377\) −15.9083 −0.819321
\(378\) 0 0
\(379\) 15.3305 0.787477 0.393738 0.919223i \(-0.371182\pi\)
0.393738 + 0.919223i \(0.371182\pi\)
\(380\) 0 0
\(381\) 1.44996 0.0742838
\(382\) 0 0
\(383\) −20.8444 −1.06510 −0.532550 0.846399i \(-0.678766\pi\)
−0.532550 + 0.846399i \(0.678766\pi\)
\(384\) 0 0
\(385\) 7.81665 0.398374
\(386\) 0 0
\(387\) −19.2111 −0.976555
\(388\) 0 0
\(389\) −11.8806 −0.602369 −0.301184 0.953566i \(-0.597382\pi\)
−0.301184 + 0.953566i \(0.597382\pi\)
\(390\) 0 0
\(391\) −41.4500 −2.09621
\(392\) 0 0
\(393\) −1.02776 −0.0518435
\(394\) 0 0
\(395\) 21.0000 1.05662
\(396\) 0 0
\(397\) 27.8167 1.39608 0.698039 0.716060i \(-0.254056\pi\)
0.698039 + 0.716060i \(0.254056\pi\)
\(398\) 0 0
\(399\) 2.78890 0.139620
\(400\) 0 0
\(401\) 13.8167 0.689971 0.344985 0.938608i \(-0.387884\pi\)
0.344985 + 0.938608i \(0.387884\pi\)
\(402\) 0 0
\(403\) 7.60555 0.378859
\(404\) 0 0
\(405\) 10.6611 0.529753
\(406\) 0 0
\(407\) −1.30278 −0.0645762
\(408\) 0 0
\(409\) −5.02776 −0.248607 −0.124303 0.992244i \(-0.539670\pi\)
−0.124303 + 0.992244i \(0.539670\pi\)
\(410\) 0 0
\(411\) −3.00000 −0.147979
\(412\) 0 0
\(413\) 15.6333 0.769265
\(414\) 0 0
\(415\) −22.4222 −1.10066
\(416\) 0 0
\(417\) −2.69722 −0.132084
\(418\) 0 0
\(419\) 25.1472 1.22852 0.614260 0.789104i \(-0.289455\pi\)
0.614260 + 0.789104i \(0.289455\pi\)
\(420\) 0 0
\(421\) 28.7250 1.39997 0.699985 0.714158i \(-0.253190\pi\)
0.699985 + 0.714158i \(0.253190\pi\)
\(422\) 0 0
\(423\) −7.57779 −0.368445
\(424\) 0 0
\(425\) 19.8167 0.961249
\(426\) 0 0
\(427\) 48.4222 2.34331
\(428\) 0 0
\(429\) 0.908327 0.0438544
\(430\) 0 0
\(431\) 5.21110 0.251010 0.125505 0.992093i \(-0.459945\pi\)
0.125505 + 0.992093i \(0.459945\pi\)
\(432\) 0 0
\(433\) −11.9361 −0.573612 −0.286806 0.957989i \(-0.592593\pi\)
−0.286806 + 0.957989i \(0.592593\pi\)
\(434\) 0 0
\(435\) 2.72498 0.130653
\(436\) 0 0
\(437\) −13.8167 −0.660940
\(438\) 0 0
\(439\) 9.33053 0.445322 0.222661 0.974896i \(-0.428526\pi\)
0.222661 + 0.974896i \(0.428526\pi\)
\(440\) 0 0
\(441\) −41.3305 −1.96812
\(442\) 0 0
\(443\) −0.275019 −0.0130666 −0.00653328 0.999979i \(-0.502080\pi\)
−0.00653328 + 0.999979i \(0.502080\pi\)
\(444\) 0 0
\(445\) 6.78890 0.321825
\(446\) 0 0
\(447\) −0.550039 −0.0260159
\(448\) 0 0
\(449\) −0.788897 −0.0372304 −0.0186152 0.999827i \(-0.505926\pi\)
−0.0186152 + 0.999827i \(0.505926\pi\)
\(450\) 0 0
\(451\) 1.18335 0.0557216
\(452\) 0 0
\(453\) 4.05551 0.190545
\(454\) 0 0
\(455\) 13.8167 0.647735
\(456\) 0 0
\(457\) 4.60555 0.215439 0.107719 0.994181i \(-0.465645\pi\)
0.107719 + 0.994181i \(0.465645\pi\)
\(458\) 0 0
\(459\) 10.7334 0.500991
\(460\) 0 0
\(461\) −16.4222 −0.764858 −0.382429 0.923985i \(-0.624912\pi\)
−0.382429 + 0.923985i \(0.624912\pi\)
\(462\) 0 0
\(463\) −30.3028 −1.40829 −0.704145 0.710056i \(-0.748669\pi\)
−0.704145 + 0.710056i \(0.748669\pi\)
\(464\) 0 0
\(465\) −1.30278 −0.0604148
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 66.8444 3.08659
\(470\) 0 0
\(471\) 2.18335 0.100603
\(472\) 0 0
\(473\) −8.60555 −0.395684
\(474\) 0 0
\(475\) 6.60555 0.303083
\(476\) 0 0
\(477\) 17.4500 0.798979
\(478\) 0 0
\(479\) −12.1194 −0.553751 −0.276875 0.960906i \(-0.589299\pi\)
−0.276875 + 0.960906i \(0.589299\pi\)
\(480\) 0 0
\(481\) −2.30278 −0.104998
\(482\) 0 0
\(483\) −9.63331 −0.438331
\(484\) 0 0
\(485\) 16.1833 0.734848
\(486\) 0 0
\(487\) 22.7889 1.03266 0.516332 0.856389i \(-0.327297\pi\)
0.516332 + 0.856389i \(0.327297\pi\)
\(488\) 0 0
\(489\) 6.18335 0.279621
\(490\) 0 0
\(491\) 14.7250 0.664529 0.332265 0.943186i \(-0.392187\pi\)
0.332265 + 0.943186i \(0.392187\pi\)
\(492\) 0 0
\(493\) −41.4500 −1.86681
\(494\) 0 0
\(495\) 4.93608 0.221860
\(496\) 0 0
\(497\) 27.6333 1.23952
\(498\) 0 0
\(499\) −8.23886 −0.368822 −0.184411 0.982849i \(-0.559038\pi\)
−0.184411 + 0.982849i \(0.559038\pi\)
\(500\) 0 0
\(501\) −3.78890 −0.169275
\(502\) 0 0
\(503\) 24.5139 1.09302 0.546510 0.837453i \(-0.315956\pi\)
0.546510 + 0.837453i \(0.315956\pi\)
\(504\) 0 0
\(505\) 21.3944 0.952040
\(506\) 0 0
\(507\) −2.33053 −0.103503
\(508\) 0 0
\(509\) −25.8167 −1.14430 −0.572152 0.820148i \(-0.693891\pi\)
−0.572152 + 0.820148i \(0.693891\pi\)
\(510\) 0 0
\(511\) 40.0555 1.77195
\(512\) 0 0
\(513\) 3.57779 0.157964
\(514\) 0 0
\(515\) −4.30278 −0.189603
\(516\) 0 0
\(517\) −3.39445 −0.149288
\(518\) 0 0
\(519\) −7.02776 −0.308484
\(520\) 0 0
\(521\) −9.63331 −0.422043 −0.211021 0.977481i \(-0.567679\pi\)
−0.211021 + 0.977481i \(0.567679\pi\)
\(522\) 0 0
\(523\) −32.2389 −1.40971 −0.704853 0.709353i \(-0.748987\pi\)
−0.704853 + 0.709353i \(0.748987\pi\)
\(524\) 0 0
\(525\) 4.60555 0.201003
\(526\) 0 0
\(527\) 19.8167 0.863227
\(528\) 0 0
\(529\) 24.7250 1.07500
\(530\) 0 0
\(531\) 9.87217 0.428416
\(532\) 0 0
\(533\) 2.09167 0.0906004
\(534\) 0 0
\(535\) −5.60555 −0.242349
\(536\) 0 0
\(537\) −2.36669 −0.102130
\(538\) 0 0
\(539\) −18.5139 −0.797449
\(540\) 0 0
\(541\) −20.9361 −0.900113 −0.450056 0.893000i \(-0.648596\pi\)
−0.450056 + 0.893000i \(0.648596\pi\)
\(542\) 0 0
\(543\) 6.05551 0.259867
\(544\) 0 0
\(545\) 2.60555 0.111610
\(546\) 0 0
\(547\) 13.3944 0.572705 0.286353 0.958124i \(-0.407557\pi\)
0.286353 + 0.958124i \(0.407557\pi\)
\(548\) 0 0
\(549\) 30.5778 1.30503
\(550\) 0 0
\(551\) −13.8167 −0.588609
\(552\) 0 0
\(553\) −74.2389 −3.15696
\(554\) 0 0
\(555\) 0.394449 0.0167434
\(556\) 0 0
\(557\) −6.51388 −0.276002 −0.138001 0.990432i \(-0.544068\pi\)
−0.138001 + 0.990432i \(0.544068\pi\)
\(558\) 0 0
\(559\) −15.2111 −0.643361
\(560\) 0 0
\(561\) 2.36669 0.0999218
\(562\) 0 0
\(563\) −44.0555 −1.85672 −0.928359 0.371684i \(-0.878780\pi\)
−0.928359 + 0.371684i \(0.878780\pi\)
\(564\) 0 0
\(565\) 14.6056 0.614460
\(566\) 0 0
\(567\) −37.6888 −1.58278
\(568\) 0 0
\(569\) −10.4222 −0.436922 −0.218461 0.975846i \(-0.570104\pi\)
−0.218461 + 0.975846i \(0.570104\pi\)
\(570\) 0 0
\(571\) 20.3028 0.849645 0.424822 0.905277i \(-0.360337\pi\)
0.424822 + 0.905277i \(0.360337\pi\)
\(572\) 0 0
\(573\) −3.78890 −0.158283
\(574\) 0 0
\(575\) −22.8167 −0.951520
\(576\) 0 0
\(577\) −28.2389 −1.17560 −0.587800 0.809007i \(-0.700006\pi\)
−0.587800 + 0.809007i \(0.700006\pi\)
\(578\) 0 0
\(579\) −1.21110 −0.0503317
\(580\) 0 0
\(581\) 79.2666 3.28853
\(582\) 0 0
\(583\) 7.81665 0.323733
\(584\) 0 0
\(585\) 8.72498 0.360734
\(586\) 0 0
\(587\) 2.36669 0.0976838 0.0488419 0.998807i \(-0.484447\pi\)
0.0488419 + 0.998807i \(0.484447\pi\)
\(588\) 0 0
\(589\) 6.60555 0.272177
\(590\) 0 0
\(591\) −1.81665 −0.0747272
\(592\) 0 0
\(593\) 36.5139 1.49945 0.749723 0.661752i \(-0.230187\pi\)
0.749723 + 0.661752i \(0.230187\pi\)
\(594\) 0 0
\(595\) 36.0000 1.47586
\(596\) 0 0
\(597\) 0.733385 0.0300154
\(598\) 0 0
\(599\) 35.2111 1.43869 0.719343 0.694655i \(-0.244443\pi\)
0.719343 + 0.694655i \(0.244443\pi\)
\(600\) 0 0
\(601\) −20.6972 −0.844257 −0.422129 0.906536i \(-0.638717\pi\)
−0.422129 + 0.906536i \(0.638717\pi\)
\(602\) 0 0
\(603\) 42.2111 1.71897
\(604\) 0 0
\(605\) −12.1194 −0.492725
\(606\) 0 0
\(607\) 31.5139 1.27911 0.639554 0.768746i \(-0.279119\pi\)
0.639554 + 0.768746i \(0.279119\pi\)
\(608\) 0 0
\(609\) −9.63331 −0.390361
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) −8.18335 −0.330522 −0.165261 0.986250i \(-0.552847\pi\)
−0.165261 + 0.986250i \(0.552847\pi\)
\(614\) 0 0
\(615\) −0.358288 −0.0144476
\(616\) 0 0
\(617\) 47.5694 1.91507 0.957536 0.288314i \(-0.0930948\pi\)
0.957536 + 0.288314i \(0.0930948\pi\)
\(618\) 0 0
\(619\) 2.69722 0.108411 0.0542053 0.998530i \(-0.482737\pi\)
0.0542053 + 0.998530i \(0.482737\pi\)
\(620\) 0 0
\(621\) −12.3583 −0.495921
\(622\) 0 0
\(623\) −24.0000 −0.961540
\(624\) 0 0
\(625\) 2.42221 0.0968882
\(626\) 0 0
\(627\) 0.788897 0.0315055
\(628\) 0 0
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) −18.3028 −0.728622 −0.364311 0.931277i \(-0.618695\pi\)
−0.364311 + 0.931277i \(0.618695\pi\)
\(632\) 0 0
\(633\) −2.02776 −0.0805961
\(634\) 0 0
\(635\) 6.23886 0.247582
\(636\) 0 0
\(637\) −32.7250 −1.29661
\(638\) 0 0
\(639\) 17.4500 0.690310
\(640\) 0 0
\(641\) −2.48612 −0.0981959 −0.0490980 0.998794i \(-0.515635\pi\)
−0.0490980 + 0.998794i \(0.515635\pi\)
\(642\) 0 0
\(643\) 29.8167 1.17585 0.587927 0.808914i \(-0.299944\pi\)
0.587927 + 0.808914i \(0.299944\pi\)
\(644\) 0 0
\(645\) 2.60555 0.102593
\(646\) 0 0
\(647\) 25.9361 1.01965 0.509826 0.860277i \(-0.329710\pi\)
0.509826 + 0.860277i \(0.329710\pi\)
\(648\) 0 0
\(649\) 4.42221 0.173587
\(650\) 0 0
\(651\) 4.60555 0.180506
\(652\) 0 0
\(653\) 6.90833 0.270344 0.135172 0.990822i \(-0.456841\pi\)
0.135172 + 0.990822i \(0.456841\pi\)
\(654\) 0 0
\(655\) −4.42221 −0.172790
\(656\) 0 0
\(657\) 25.2944 0.986827
\(658\) 0 0
\(659\) 42.1194 1.64074 0.820370 0.571833i \(-0.193767\pi\)
0.820370 + 0.571833i \(0.193767\pi\)
\(660\) 0 0
\(661\) −12.4861 −0.485654 −0.242827 0.970070i \(-0.578075\pi\)
−0.242827 + 0.970070i \(0.578075\pi\)
\(662\) 0 0
\(663\) 4.18335 0.162468
\(664\) 0 0
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) 47.7250 1.84792
\(668\) 0 0
\(669\) −4.78890 −0.185149
\(670\) 0 0
\(671\) 13.6972 0.528775
\(672\) 0 0
\(673\) 24.3028 0.936803 0.468402 0.883516i \(-0.344830\pi\)
0.468402 + 0.883516i \(0.344830\pi\)
\(674\) 0 0
\(675\) 5.90833 0.227412
\(676\) 0 0
\(677\) −36.2389 −1.39277 −0.696386 0.717667i \(-0.745210\pi\)
−0.696386 + 0.717667i \(0.745210\pi\)
\(678\) 0 0
\(679\) −57.2111 −2.19556
\(680\) 0 0
\(681\) 2.36669 0.0906918
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) −12.9083 −0.493202
\(686\) 0 0
\(687\) 5.26662 0.200934
\(688\) 0 0
\(689\) 13.8167 0.526373
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) −17.4500 −0.662869
\(694\) 0 0
\(695\) −11.6056 −0.440224
\(696\) 0 0
\(697\) 5.44996 0.206432
\(698\) 0 0
\(699\) −2.88057 −0.108953
\(700\) 0 0
\(701\) 14.8806 0.562031 0.281016 0.959703i \(-0.409329\pi\)
0.281016 + 0.959703i \(0.409329\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) 1.02776 0.0387075
\(706\) 0 0
\(707\) −75.6333 −2.84448
\(708\) 0 0
\(709\) −1.66947 −0.0626982 −0.0313491 0.999508i \(-0.509980\pi\)
−0.0313491 + 0.999508i \(0.509980\pi\)
\(710\) 0 0
\(711\) −46.8806 −1.75816
\(712\) 0 0
\(713\) −22.8167 −0.854490
\(714\) 0 0
\(715\) 3.90833 0.146163
\(716\) 0 0
\(717\) −0.155590 −0.00581061
\(718\) 0 0
\(719\) 8.36669 0.312025 0.156012 0.987755i \(-0.450136\pi\)
0.156012 + 0.987755i \(0.450136\pi\)
\(720\) 0 0
\(721\) 15.2111 0.566491
\(722\) 0 0
\(723\) 2.42221 0.0900828
\(724\) 0 0
\(725\) −22.8167 −0.847389
\(726\) 0 0
\(727\) −29.9083 −1.10924 −0.554619 0.832104i \(-0.687136\pi\)
−0.554619 + 0.832104i \(0.687136\pi\)
\(728\) 0 0
\(729\) −22.1749 −0.821294
\(730\) 0 0
\(731\) −39.6333 −1.46589
\(732\) 0 0
\(733\) 29.6333 1.09453 0.547266 0.836959i \(-0.315669\pi\)
0.547266 + 0.836959i \(0.315669\pi\)
\(734\) 0 0
\(735\) 5.60555 0.206764
\(736\) 0 0
\(737\) 18.9083 0.696497
\(738\) 0 0
\(739\) 42.3305 1.55715 0.778577 0.627549i \(-0.215942\pi\)
0.778577 + 0.627549i \(0.215942\pi\)
\(740\) 0 0
\(741\) 1.39445 0.0512264
\(742\) 0 0
\(743\) −35.4500 −1.30053 −0.650266 0.759706i \(-0.725343\pi\)
−0.650266 + 0.759706i \(0.725343\pi\)
\(744\) 0 0
\(745\) −2.36669 −0.0867089
\(746\) 0 0
\(747\) 50.0555 1.83144
\(748\) 0 0
\(749\) 19.8167 0.724085
\(750\) 0 0
\(751\) −14.0000 −0.510867 −0.255434 0.966827i \(-0.582218\pi\)
−0.255434 + 0.966827i \(0.582218\pi\)
\(752\) 0 0
\(753\) 2.05551 0.0749070
\(754\) 0 0
\(755\) 17.4500 0.635069
\(756\) 0 0
\(757\) 9.30278 0.338115 0.169058 0.985606i \(-0.445928\pi\)
0.169058 + 0.985606i \(0.445928\pi\)
\(758\) 0 0
\(759\) −2.72498 −0.0989105
\(760\) 0 0
\(761\) 42.1194 1.52683 0.763414 0.645909i \(-0.223522\pi\)
0.763414 + 0.645909i \(0.223522\pi\)
\(762\) 0 0
\(763\) −9.21110 −0.333464
\(764\) 0 0
\(765\) 22.7334 0.821927
\(766\) 0 0
\(767\) 7.81665 0.282243
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) −3.39445 −0.122248
\(772\) 0 0
\(773\) −50.0555 −1.80037 −0.900186 0.435506i \(-0.856569\pi\)
−0.900186 + 0.435506i \(0.856569\pi\)
\(774\) 0 0
\(775\) 10.9083 0.391839
\(776\) 0 0
\(777\) −1.39445 −0.0500256
\(778\) 0 0
\(779\) 1.81665 0.0650884
\(780\) 0 0
\(781\) 7.81665 0.279702
\(782\) 0 0
\(783\) −12.3583 −0.441649
\(784\) 0 0
\(785\) 9.39445 0.335302
\(786\) 0 0
\(787\) −25.2111 −0.898679 −0.449339 0.893361i \(-0.648341\pi\)
−0.449339 + 0.893361i \(0.648341\pi\)
\(788\) 0 0
\(789\) 2.36669 0.0842565
\(790\) 0 0
\(791\) −51.6333 −1.83587
\(792\) 0 0
\(793\) 24.2111 0.859761
\(794\) 0 0
\(795\) −2.36669 −0.0839379
\(796\) 0 0
\(797\) −17.3305 −0.613879 −0.306939 0.951729i \(-0.599305\pi\)
−0.306939 + 0.951729i \(0.599305\pi\)
\(798\) 0 0
\(799\) −15.6333 −0.553067
\(800\) 0 0
\(801\) −15.1556 −0.535496
\(802\) 0 0
\(803\) 11.3305 0.399846
\(804\) 0 0
\(805\) −41.4500 −1.46092
\(806\) 0 0
\(807\) −2.05551 −0.0723575
\(808\) 0 0
\(809\) 29.4500 1.03541 0.517703 0.855561i \(-0.326787\pi\)
0.517703 + 0.855561i \(0.326787\pi\)
\(810\) 0 0
\(811\) −54.1472 −1.90136 −0.950682 0.310166i \(-0.899615\pi\)
−0.950682 + 0.310166i \(0.899615\pi\)
\(812\) 0 0
\(813\) −1.94449 −0.0681961
\(814\) 0 0
\(815\) 26.6056 0.931952
\(816\) 0 0
\(817\) −13.2111 −0.462198
\(818\) 0 0
\(819\) −30.8444 −1.07779
\(820\) 0 0
\(821\) 11.2111 0.391270 0.195635 0.980677i \(-0.437323\pi\)
0.195635 + 0.980677i \(0.437323\pi\)
\(822\) 0 0
\(823\) 12.8444 0.447728 0.223864 0.974620i \(-0.428133\pi\)
0.223864 + 0.974620i \(0.428133\pi\)
\(824\) 0 0
\(825\) 1.30278 0.0453568
\(826\) 0 0
\(827\) −27.3944 −0.952598 −0.476299 0.879283i \(-0.658022\pi\)
−0.476299 + 0.879283i \(0.658022\pi\)
\(828\) 0 0
\(829\) 4.72498 0.164105 0.0820527 0.996628i \(-0.473852\pi\)
0.0820527 + 0.996628i \(0.473852\pi\)
\(830\) 0 0
\(831\) −7.60555 −0.263834
\(832\) 0 0
\(833\) −85.2666 −2.95431
\(834\) 0 0
\(835\) −16.3028 −0.564181
\(836\) 0 0
\(837\) 5.90833 0.204222
\(838\) 0 0
\(839\) 49.0278 1.69263 0.846313 0.532686i \(-0.178817\pi\)
0.846313 + 0.532686i \(0.178817\pi\)
\(840\) 0 0
\(841\) 18.7250 0.645689
\(842\) 0 0
\(843\) −3.63331 −0.125138
\(844\) 0 0
\(845\) −10.0278 −0.344965
\(846\) 0 0
\(847\) 42.8444 1.47215
\(848\) 0 0
\(849\) −5.26662 −0.180750
\(850\) 0 0
\(851\) 6.90833 0.236814
\(852\) 0 0
\(853\) −11.5416 −0.395178 −0.197589 0.980285i \(-0.563311\pi\)
−0.197589 + 0.980285i \(0.563311\pi\)
\(854\) 0 0
\(855\) 7.57779 0.259155
\(856\) 0 0
\(857\) −14.8444 −0.507075 −0.253538 0.967326i \(-0.581594\pi\)
−0.253538 + 0.967326i \(0.581594\pi\)
\(858\) 0 0
\(859\) 24.0555 0.820764 0.410382 0.911914i \(-0.365395\pi\)
0.410382 + 0.911914i \(0.365395\pi\)
\(860\) 0 0
\(861\) 1.26662 0.0431661
\(862\) 0 0
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 0 0
\(865\) −30.2389 −1.02815
\(866\) 0 0
\(867\) 5.75274 0.195373
\(868\) 0 0
\(869\) −21.0000 −0.712376
\(870\) 0 0
\(871\) 33.4222 1.13247
\(872\) 0 0
\(873\) −36.1278 −1.22274
\(874\) 0 0
\(875\) 49.8167 1.68411
\(876\) 0 0
\(877\) 7.21110 0.243502 0.121751 0.992561i \(-0.461149\pi\)
0.121751 + 0.992561i \(0.461149\pi\)
\(878\) 0 0
\(879\) −7.57779 −0.255593
\(880\) 0 0
\(881\) 25.5416 0.860520 0.430260 0.902705i \(-0.358422\pi\)
0.430260 + 0.902705i \(0.358422\pi\)
\(882\) 0 0
\(883\) 2.42221 0.0815137 0.0407568 0.999169i \(-0.487023\pi\)
0.0407568 + 0.999169i \(0.487023\pi\)
\(884\) 0 0
\(885\) −1.33894 −0.0450078
\(886\) 0 0
\(887\) −28.4222 −0.954324 −0.477162 0.878815i \(-0.658335\pi\)
−0.477162 + 0.878815i \(0.658335\pi\)
\(888\) 0 0
\(889\) −22.0555 −0.739718
\(890\) 0 0
\(891\) −10.6611 −0.357159
\(892\) 0 0
\(893\) −5.21110 −0.174383
\(894\) 0 0
\(895\) −10.1833 −0.340392
\(896\) 0 0
\(897\) −4.81665 −0.160823
\(898\) 0 0
\(899\) −22.8167 −0.760978
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) −9.21110 −0.306526
\(904\) 0 0
\(905\) 26.0555 0.866115
\(906\) 0 0
\(907\) −26.0000 −0.863316 −0.431658 0.902037i \(-0.642071\pi\)
−0.431658 + 0.902037i \(0.642071\pi\)
\(908\) 0 0
\(909\) −47.7611 −1.58414
\(910\) 0 0
\(911\) 46.4222 1.53804 0.769018 0.639227i \(-0.220746\pi\)
0.769018 + 0.639227i \(0.220746\pi\)
\(912\) 0 0
\(913\) 22.4222 0.742067
\(914\) 0 0
\(915\) −4.14719 −0.137102
\(916\) 0 0
\(917\) 15.6333 0.516257
\(918\) 0 0
\(919\) 38.4222 1.26743 0.633716 0.773566i \(-0.281529\pi\)
0.633716 + 0.773566i \(0.281529\pi\)
\(920\) 0 0
\(921\) −2.14719 −0.0707522
\(922\) 0 0
\(923\) 13.8167 0.454781
\(924\) 0 0
\(925\) −3.30278 −0.108595
\(926\) 0 0
\(927\) 9.60555 0.315488
\(928\) 0 0
\(929\) −36.5139 −1.19798 −0.598991 0.800756i \(-0.704431\pi\)
−0.598991 + 0.800756i \(0.704431\pi\)
\(930\) 0 0
\(931\) −28.4222 −0.931500
\(932\) 0 0
\(933\) −1.54163 −0.0504708
\(934\) 0 0
\(935\) 10.1833 0.333031
\(936\) 0 0
\(937\) −28.9083 −0.944394 −0.472197 0.881493i \(-0.656539\pi\)
−0.472197 + 0.881493i \(0.656539\pi\)
\(938\) 0 0
\(939\) 8.18335 0.267053
\(940\) 0 0
\(941\) −7.81665 −0.254816 −0.127408 0.991850i \(-0.540666\pi\)
−0.127408 + 0.991850i \(0.540666\pi\)
\(942\) 0 0
\(943\) −6.27502 −0.204343
\(944\) 0 0
\(945\) 10.7334 0.349157
\(946\) 0 0
\(947\) −39.6333 −1.28791 −0.643955 0.765064i \(-0.722707\pi\)
−0.643955 + 0.765064i \(0.722707\pi\)
\(948\) 0 0
\(949\) 20.0278 0.650128
\(950\) 0 0
\(951\) −1.57779 −0.0511635
\(952\) 0 0
\(953\) −18.7527 −0.607461 −0.303730 0.952758i \(-0.598232\pi\)
−0.303730 + 0.952758i \(0.598232\pi\)
\(954\) 0 0
\(955\) −16.3028 −0.527545
\(956\) 0 0
\(957\) −2.72498 −0.0880861
\(958\) 0 0
\(959\) 45.6333 1.47358
\(960\) 0 0
\(961\) −20.0917 −0.648118
\(962\) 0 0
\(963\) 12.5139 0.403254
\(964\) 0 0
\(965\) −5.21110 −0.167751
\(966\) 0 0
\(967\) −25.7250 −0.827260 −0.413630 0.910445i \(-0.635739\pi\)
−0.413630 + 0.910445i \(0.635739\pi\)
\(968\) 0 0
\(969\) 3.63331 0.116719
\(970\) 0 0
\(971\) −31.5416 −1.01222 −0.506110 0.862469i \(-0.668917\pi\)
−0.506110 + 0.862469i \(0.668917\pi\)
\(972\) 0 0
\(973\) 41.0278 1.31529
\(974\) 0 0
\(975\) 2.30278 0.0737478
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) −6.78890 −0.216974
\(980\) 0 0
\(981\) −5.81665 −0.185711
\(982\) 0 0
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) 0 0
\(985\) −7.81665 −0.249059
\(986\) 0 0
\(987\) −3.63331 −0.115649
\(988\) 0 0
\(989\) 45.6333 1.45105
\(990\) 0 0
\(991\) −54.3028 −1.72498 −0.862492 0.506070i \(-0.831098\pi\)
−0.862492 + 0.506070i \(0.831098\pi\)
\(992\) 0 0
\(993\) −0.366692 −0.0116366
\(994\) 0 0
\(995\) 3.15559 0.100039
\(996\) 0 0
\(997\) −23.5778 −0.746716 −0.373358 0.927687i \(-0.621794\pi\)
−0.373358 + 0.927687i \(0.621794\pi\)
\(998\) 0 0
\(999\) −1.78890 −0.0565982
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 592.2.a.f.1.2 2
3.2 odd 2 5328.2.a.bf.1.1 2
4.3 odd 2 74.2.a.a.1.1 2
8.3 odd 2 2368.2.a.s.1.2 2
8.5 even 2 2368.2.a.ba.1.1 2
12.11 even 2 666.2.a.j.1.1 2
20.3 even 4 1850.2.b.i.149.3 4
20.7 even 4 1850.2.b.i.149.2 4
20.19 odd 2 1850.2.a.u.1.2 2
28.27 even 2 3626.2.a.a.1.2 2
44.43 even 2 8954.2.a.p.1.1 2
148.147 odd 2 2738.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.1 2 4.3 odd 2
592.2.a.f.1.2 2 1.1 even 1 trivial
666.2.a.j.1.1 2 12.11 even 2
1850.2.a.u.1.2 2 20.19 odd 2
1850.2.b.i.149.2 4 20.7 even 4
1850.2.b.i.149.3 4 20.3 even 4
2368.2.a.s.1.2 2 8.3 odd 2
2368.2.a.ba.1.1 2 8.5 even 2
2738.2.a.l.1.1 2 148.147 odd 2
3626.2.a.a.1.2 2 28.27 even 2
5328.2.a.bf.1.1 2 3.2 odd 2
8954.2.a.p.1.1 2 44.43 even 2