Properties

Label 2368.2.a.s.1.2
Level $2368$
Weight $2$
Character 2368.1
Self dual yes
Analytic conductor $18.909$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.9085751986\)
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 \(\chi\) \(=\) 2368.1

$q$-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 \(-0.594091\pi\)
−0.291309 + 0.956629i \(0.594091\pi\)
\(6\) 0 0
\(7\) 4.60555 1.74073 0.870367 0.492403i \(-0.163881\pi\)
0.870367 + 0.492403i \(0.163881\pi\)
\(8\) 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 \(-0.396540\pi\)
0.319338 + 0.947641i \(0.396540\pi\)
\(14\) 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 \(-0.755970\pi\)
−0.720243 + 0.693722i \(0.755970\pi\)
\(24\) 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 \(-0.721655\pi\)
−0.641422 + 0.767188i \(0.721655\pi\)
\(30\) 0 0
\(31\) 3.30278 0.593196 0.296598 0.955002i \(-0.404148\pi\)
0.296598 + 0.955002i \(0.404148\pi\)
\(32\) 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 \(-0.560858\pi\)
−0.190029 + 0.981778i \(0.560858\pi\)
\(48\) 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 \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 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 \(-0.264970\pi\)
0.673082 + 0.739568i \(0.264970\pi\)
\(62\) 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 \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 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 \(-0.861474\pi\)
−0.906789 + 0.421585i \(0.861474\pi\)
\(80\) 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 \(-0.804384\pi\)
−0.817035 + 0.576588i \(0.804384\pi\)
\(102\) 0 0
\(103\) 3.30278 0.325432 0.162716 0.986673i \(-0.447975\pi\)
0.162716 + 0.986673i \(0.447975\pi\)
\(104\) 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 \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 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 \(-0.568152\pi\)
−0.212473 + 0.977167i \(0.568152\pi\)
\(128\) 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 \(-0.476292\pi\)
0.0744130 + 0.997228i \(0.476292\pi\)
\(150\) 0 0
\(151\) −13.3944 −1.09002 −0.545012 0.838428i \(-0.683475\pi\)
−0.545012 + 0.838428i \(0.683475\pi\)
\(152\) 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 \(-0.592909\pi\)
−0.287754 + 0.957704i \(0.592909\pi\)
\(158\) 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 \(-0.339119\pi\)
0.484176 + 0.874970i \(0.339119\pi\)
\(168\) 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 \(-0.155958\pi\)
0.882354 + 0.470587i \(0.155958\pi\)
\(174\) 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 \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 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 \(-0.350448\pi\)
0.452736 + 0.891644i \(0.350448\pi\)
\(192\) 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 \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −2.42221 −0.171706 −0.0858528 0.996308i \(-0.527361\pi\)
−0.0858528 + 0.996308i \(0.527361\pi\)
\(200\) 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 \(-0.322349\pi\)
0.529581 + 0.848260i \(0.322349\pi\)
\(224\) 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 \(-0.694892\pi\)
−0.574729 + 0.818344i \(0.694892\pi\)
\(230\) 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 \(-0.494709\pi\)
0.0166200 + 0.999862i \(0.494709\pi\)
\(240\) 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 \(-0.577475\pi\)
−0.240998 + 0.970526i \(0.577475\pi\)
\(264\) 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 \(-0.433642\pi\)
0.206963 + 0.978349i \(0.433642\pi\)
\(270\) 0 0
\(271\) 6.42221 0.390121 0.195061 0.980791i \(-0.437510\pi\)
0.195061 + 0.980791i \(0.437510\pi\)
\(272\) 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 \(-0.227811\pi\)
0.754640 + 0.656139i \(0.227811\pi\)
\(278\) 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 \(-0.239022\pi\)
0.731069 + 0.682304i \(0.239022\pi\)
\(294\) 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 \(-0.453887\pi\)
0.144361 + 0.989525i \(0.453887\pi\)
\(312\) 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 \(-0.453250\pi\)
0.146342 + 0.989234i \(0.453250\pi\)
\(318\) 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 \(-0.297069\pi\)
0.595209 + 0.803571i \(0.297069\pi\)
\(350\) 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 \(-0.595603\pi\)
−0.295850 + 0.955235i \(0.595603\pi\)
\(360\) 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 \(-0.653950\pi\)
−0.465011 + 0.885305i \(0.653950\pi\)
\(368\) 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 \(-0.531503\pi\)
−0.0988094 + 0.995106i \(0.531503\pi\)
\(374\) 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 \(-0.321234\pi\)
0.532550 + 0.846399i \(0.321234\pi\)
\(384\) 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 \(-0.402618\pi\)
0.301184 + 0.953566i \(0.402618\pi\)
\(390\) 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 \(-0.745944\pi\)
−0.698039 + 0.716060i \(0.745944\pi\)
\(398\) 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 \(-0.746810\pi\)
−0.699985 + 0.714158i \(0.746810\pi\)
\(422\) 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 \(-0.540055\pi\)
−0.125505 + 0.992093i \(0.540055\pi\)
\(432\) 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 \(-0.571474\pi\)
−0.222661 + 0.974896i \(0.571474\pi\)
\(440\) 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 \(-0.375088\pi\)
0.382429 + 0.923985i \(0.375088\pi\)
\(462\) 0 0
\(463\) 30.3028 1.40829 0.704145 0.710056i \(-0.251331\pi\)
0.704145 + 0.710056i \(0.251331\pi\)
\(464\) 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 \(-0.410701\pi\)
0.276875 + 0.960906i \(0.410701\pi\)
\(480\) 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 \(-0.672703\pi\)
−0.516332 + 0.856389i \(0.672703\pi\)
\(488\) 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 \(-0.684044\pi\)
−0.546510 + 0.837453i \(0.684044\pi\)
\(504\) 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 \(-0.306109\pi\)
0.572152 + 0.820148i \(0.306109\pi\)
\(510\) 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 \(-0.351404\pi\)
0.450056 + 0.893000i \(0.351404\pi\)
\(542\) 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 \(-0.455932\pi\)
0.138001 + 0.990432i \(0.455932\pi\)
\(558\) 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 \(-0.755557\pi\)
−0.719343 + 0.694655i \(0.755557\pi\)
\(600\) 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 \(-0.720881\pi\)
−0.639554 + 0.768746i \(0.720881\pi\)
\(608\) 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 \(-0.447153\pi\)
0.165261 + 0.986250i \(0.447153\pi\)
\(614\) 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 \(-0.381305\pi\)
0.364311 + 0.931277i \(0.381305\pi\)
\(632\) 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 \(-0.670290\pi\)
−0.509826 + 0.860277i \(0.670290\pi\)
\(648\) 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 \(-0.543159\pi\)
−0.135172 + 0.990822i \(0.543159\pi\)
\(654\) 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 \(-0.421925\pi\)
0.242827 + 0.970070i \(0.421925\pi\)
\(662\) 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 \(-0.254790\pi\)
0.696386 + 0.717667i \(0.254790\pi\)
\(678\) 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 \(-0.590671\pi\)
−0.281016 + 0.959703i \(0.590671\pi\)
\(702\) 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 \(-0.490020\pi\)
0.0313491 + 0.999508i \(0.490020\pi\)
\(710\) 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 \(-0.549864\pi\)
−0.156012 + 0.987755i \(0.549864\pi\)
\(720\) 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 \(-0.312864\pi\)
0.554619 + 0.832104i \(0.312864\pi\)
\(728\) 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 \(-0.684331\pi\)
−0.547266 + 0.836959i \(0.684331\pi\)
\(734\) 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 \(-0.274657\pi\)
0.650266 + 0.759706i \(0.274657\pi\)
\(744\) 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 \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) 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 \(-0.554072\pi\)
−0.169058 + 0.985606i \(0.554072\pi\)
\(758\) 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 \(-0.143431\pi\)
0.900186 + 0.435506i \(0.143431\pi\)
\(774\) 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 \(-0.400695\pi\)
0.306939 + 0.951729i \(0.400695\pi\)
\(798\) 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 \(-0.562677\pi\)
−0.195635 + 0.980677i \(0.562677\pi\)
\(822\) 0 0
\(823\) −12.8444 −0.447728 −0.223864 0.974620i \(-0.571867\pi\)
−0.223864 + 0.974620i \(0.571867\pi\)
\(824\) 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 \(-0.526148\pi\)
−0.0820527 + 0.996628i \(0.526148\pi\)
\(830\) 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 \(-0.821183\pi\)
−0.846313 + 0.532686i \(0.821183\pi\)
\(840\) 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 \(-0.436689\pi\)
0.197589 + 0.980285i \(0.436689\pi\)
\(854\) 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 \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 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 \(-0.538851\pi\)
−0.121751 + 0.992561i \(0.538851\pi\)
\(878\) 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 \(-0.341665\pi\)
0.477162 + 0.878815i \(0.341665\pi\)
\(888\) 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 \(-0.779254\pi\)
−0.769018 + 0.639227i \(0.779254\pi\)
\(912\) 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 \(-0.718471\pi\)
−0.633716 + 0.773566i \(0.718471\pi\)
\(920\) 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 \(-0.459334\pi\)
0.127408 + 0.991850i \(0.459334\pi\)
\(942\) 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 \(-0.364261\pi\)
0.413630 + 0.910445i \(0.364261\pi\)
\(968\) 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 \(-0.561293\pi\)
−0.191370 + 0.981518i \(0.561293\pi\)
\(984\) 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 \(-0.168902\pi\)
0.862492 + 0.506070i \(0.168902\pi\)
\(992\) 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 \(-0.378206\pi\)
0.373358 + 0.927687i \(0.378206\pi\)
\(998\) 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 2368.2.a.s.1.2 2
4.3 odd 2 2368.2.a.ba.1.1 2
8.3 odd 2 592.2.a.f.1.2 2
8.5 even 2 74.2.a.a.1.1 2
24.5 odd 2 666.2.a.j.1.1 2
24.11 even 2 5328.2.a.bf.1.1 2
40.13 odd 4 1850.2.b.i.149.3 4
40.29 even 2 1850.2.a.u.1.2 2
40.37 odd 4 1850.2.b.i.149.2 4
56.13 odd 2 3626.2.a.a.1.2 2
88.21 odd 2 8954.2.a.p.1.1 2
296.221 even 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 8.5 even 2
592.2.a.f.1.2 2 8.3 odd 2
666.2.a.j.1.1 2 24.5 odd 2
1850.2.a.u.1.2 2 40.29 even 2
1850.2.b.i.149.2 4 40.37 odd 4
1850.2.b.i.149.3 4 40.13 odd 4
2368.2.a.s.1.2 2 1.1 even 1 trivial
2368.2.a.ba.1.1 2 4.3 odd 2
2738.2.a.l.1.1 2 296.221 even 2
3626.2.a.a.1.2 2 56.13 odd 2
5328.2.a.bf.1.1 2 24.11 even 2
8954.2.a.p.1.1 2 88.21 odd 2