Properties

Label 666.2.a.j.1.1
Level $666$
Weight $2$
Character 666.1
Self dual yes
Analytic conductor $5.318$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 1.1
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 666.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.30278 q^{5} +4.60555 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.30278 q^{5} +4.60555 q^{7} +1.00000 q^{8} -1.30278 q^{10} -1.30278 q^{11} -2.30278 q^{13} +4.60555 q^{14} +1.00000 q^{16} +6.00000 q^{17} +2.00000 q^{19} -1.30278 q^{20} -1.30278 q^{22} +6.90833 q^{23} -3.30278 q^{25} -2.30278 q^{26} +4.60555 q^{28} -6.90833 q^{29} +3.30278 q^{31} +1.00000 q^{32} +6.00000 q^{34} -6.00000 q^{35} +1.00000 q^{37} +2.00000 q^{38} -1.30278 q^{40} +0.908327 q^{41} -6.60555 q^{43} -1.30278 q^{44} +6.90833 q^{46} +2.60555 q^{47} +14.2111 q^{49} -3.30278 q^{50} -2.30278 q^{52} +6.00000 q^{53} +1.69722 q^{55} +4.60555 q^{56} -6.90833 q^{58} -3.39445 q^{59} -10.5139 q^{61} +3.30278 q^{62} +1.00000 q^{64} +3.00000 q^{65} +14.5139 q^{67} +6.00000 q^{68} -6.00000 q^{70} -6.00000 q^{71} -8.69722 q^{73} +1.00000 q^{74} +2.00000 q^{76} -6.00000 q^{77} -16.1194 q^{79} -1.30278 q^{80} +0.908327 q^{82} -17.2111 q^{83} -7.81665 q^{85} -6.60555 q^{86} -1.30278 q^{88} -5.21110 q^{89} -10.6056 q^{91} +6.90833 q^{92} +2.60555 q^{94} -2.60555 q^{95} +12.4222 q^{97} +14.2111 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} + 2 q^{8} + q^{10} + q^{11} - q^{13} + 2 q^{14} + 2 q^{16} + 12 q^{17} + 4 q^{19} + q^{20} + q^{22} + 3 q^{23} - 3 q^{25} - q^{26} + 2 q^{28} - 3 q^{29} + 3 q^{31} + 2 q^{32} + 12 q^{34} - 12 q^{35} + 2 q^{37} + 4 q^{38} + q^{40} - 9 q^{41} - 6 q^{43} + q^{44} + 3 q^{46} - 2 q^{47} + 14 q^{49} - 3 q^{50} - q^{52} + 12 q^{53} + 7 q^{55} + 2 q^{56} - 3 q^{58} - 14 q^{59} - 3 q^{61} + 3 q^{62} + 2 q^{64} + 6 q^{65} + 11 q^{67} + 12 q^{68} - 12 q^{70} - 12 q^{71} - 21 q^{73} + 2 q^{74} + 4 q^{76} - 12 q^{77} - 7 q^{79} + q^{80} - 9 q^{82} - 20 q^{83} + 6 q^{85} - 6 q^{86} + q^{88} + 4 q^{89} - 14 q^{91} + 3 q^{92} - 2 q^{94} + 2 q^{95} - 4 q^{97} + 14 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −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 \(0.163881\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.30278 −0.411974
\(11\) −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 \(0.603460\pi\)
\(14\) 4.60555 1.23089
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −1.30278 −0.291309
\(21\) 0 0
\(22\) −1.30278 −0.277753
\(23\) 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
\(26\) −2.30278 −0.451611
\(27\) 0 0
\(28\) 4.60555 0.870367
\(29\) −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 \(0.404148\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −6.00000 −1.01419
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) −1.30278 −0.205987
\(41\) 0.908327 0.141857 0.0709284 0.997481i \(-0.477404\pi\)
0.0709284 + 0.997481i \(0.477404\pi\)
\(42\) 0 0
\(43\) −6.60555 −1.00734 −0.503669 0.863897i \(-0.668017\pi\)
−0.503669 + 0.863897i \(0.668017\pi\)
\(44\) −1.30278 −0.196401
\(45\) 0 0
\(46\) 6.90833 1.01858
\(47\) 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
\(50\) −3.30278 −0.467083
\(51\) 0 0
\(52\) −2.30278 −0.319338
\(53\) 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
\(56\) 4.60555 0.615443
\(57\) 0 0
\(58\) −6.90833 −0.907108
\(59\) −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 \(0.735030\pi\)
\(62\) 3.30278 0.419453
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.00000 0.372104
\(66\) 0 0
\(67\) 14.5139 1.77315 0.886576 0.462583i \(-0.153077\pi\)
0.886576 + 0.462583i \(0.153077\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) −6.00000 −0.717137
\(71\) −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 \(0.669972\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) −16.1194 −1.81358 −0.906789 0.421585i \(-0.861474\pi\)
−0.906789 + 0.421585i \(0.861474\pi\)
\(80\) −1.30278 −0.145655
\(81\) 0 0
\(82\) 0.908327 0.100308
\(83\) −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
\(86\) −6.60555 −0.712295
\(87\) 0 0
\(88\) −1.30278 −0.138876
\(89\) −5.21110 −0.552376 −0.276188 0.961104i \(-0.589071\pi\)
−0.276188 + 0.961104i \(0.589071\pi\)
\(90\) 0 0
\(91\) −10.6056 −1.11176
\(92\) 6.90833 0.720243
\(93\) 0 0
\(94\) 2.60555 0.268742
\(95\) −2.60555 −0.267324
\(96\) 0 0
\(97\) 12.4222 1.26128 0.630642 0.776074i \(-0.282792\pi\)
0.630642 + 0.776074i \(0.282792\pi\)
\(98\) 14.2111 1.43554
\(99\) 0 0
\(100\) −3.30278 −0.330278
\(101\) −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 \(0.447975\pi\)
\(104\) −2.30278 −0.225806
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −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 \(0.469465\pi\)
\(110\) 1.69722 0.161824
\(111\) 0 0
\(112\) 4.60555 0.435184
\(113\) −11.2111 −1.05465 −0.527326 0.849663i \(-0.676805\pi\)
−0.527326 + 0.849663i \(0.676805\pi\)
\(114\) 0 0
\(115\) −9.00000 −0.839254
\(116\) −6.90833 −0.641422
\(117\) 0 0
\(118\) −3.39445 −0.312484
\(119\) 27.6333 2.53314
\(120\) 0 0
\(121\) −9.30278 −0.845707
\(122\) −10.5139 −0.951882
\(123\) 0 0
\(124\) 3.30278 0.296598
\(125\) 10.8167 0.967471
\(126\) 0 0
\(127\) −4.78890 −0.424946 −0.212473 0.977167i \(-0.568152\pi\)
−0.212473 + 0.977167i \(0.568152\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 3.00000 0.263117
\(131\) −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
\(134\) 14.5139 1.25381
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 9.90833 0.846525 0.423263 0.906007i \(-0.360885\pi\)
0.423263 + 0.906007i \(0.360885\pi\)
\(138\) 0 0
\(139\) 8.90833 0.755594 0.377797 0.925888i \(-0.376682\pi\)
0.377797 + 0.925888i \(0.376682\pi\)
\(140\) −6.00000 −0.507093
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) 3.00000 0.250873
\(144\) 0 0
\(145\) 9.00000 0.747409
\(146\) −8.69722 −0.719787
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) 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 \(0.683475\pi\)
\(152\) 2.00000 0.162221
\(153\) 0 0
\(154\) −6.00000 −0.483494
\(155\) −4.30278 −0.345607
\(156\) 0 0
\(157\) 7.21110 0.575509 0.287754 0.957704i \(-0.407091\pi\)
0.287754 + 0.957704i \(0.407091\pi\)
\(158\) −16.1194 −1.28239
\(159\) 0 0
\(160\) −1.30278 −0.102993
\(161\) 31.8167 2.50750
\(162\) 0 0
\(163\) −20.4222 −1.59959 −0.799795 0.600273i \(-0.795059\pi\)
−0.799795 + 0.600273i \(0.795059\pi\)
\(164\) 0.908327 0.0709284
\(165\) 0 0
\(166\) −17.2111 −1.33584
\(167\) −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
\(170\) −7.81665 −0.599510
\(171\) 0 0
\(172\) −6.60555 −0.503669
\(173\) 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
\(176\) −1.30278 −0.0982004
\(177\) 0 0
\(178\) −5.21110 −0.390589
\(179\) −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 \(0.233262\pi\)
\(182\) −10.6056 −0.786136
\(183\) 0 0
\(184\) 6.90833 0.509289
\(185\) −1.30278 −0.0957820
\(186\) 0 0
\(187\) −7.81665 −0.571610
\(188\) 2.60555 0.190029
\(189\) 0 0
\(190\) −2.60555 −0.189027
\(191\) −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 \(0.545985\pi\)
\(194\) 12.4222 0.891862
\(195\) 0 0
\(196\) 14.2111 1.01508
\(197\) 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 \(0.527361\pi\)
\(200\) −3.30278 −0.233542
\(201\) 0 0
\(202\) −16.4222 −1.15546
\(203\) −31.8167 −2.23309
\(204\) 0 0
\(205\) −1.18335 −0.0826485
\(206\) 3.30278 0.230115
\(207\) 0 0
\(208\) −2.30278 −0.159669
\(209\) −2.60555 −0.180230
\(210\) 0 0
\(211\) 6.69722 0.461056 0.230528 0.973066i \(-0.425955\pi\)
0.230528 + 0.973066i \(0.425955\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −4.30278 −0.294132
\(215\) 8.60555 0.586894
\(216\) 0 0
\(217\) 15.2111 1.03260
\(218\) 2.00000 0.135457
\(219\) 0 0
\(220\) 1.69722 0.114427
\(221\) −13.8167 −0.929409
\(222\) 0 0
\(223\) 15.8167 1.05916 0.529581 0.848260i \(-0.322349\pi\)
0.529581 + 0.848260i \(0.322349\pi\)
\(224\) 4.60555 0.307721
\(225\) 0 0
\(226\) −11.2111 −0.745751
\(227\) 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 \(0.305108\pi\)
\(230\) −9.00000 −0.593442
\(231\) 0 0
\(232\) −6.90833 −0.453554
\(233\) 9.51388 0.623275 0.311637 0.950201i \(-0.399123\pi\)
0.311637 + 0.950201i \(0.399123\pi\)
\(234\) 0 0
\(235\) −3.39445 −0.221429
\(236\) −3.39445 −0.220960
\(237\) 0 0
\(238\) 27.6333 1.79120
\(239\) −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 \(0.417048\pi\)
\(242\) −9.30278 −0.598005
\(243\) 0 0
\(244\) −10.5139 −0.673082
\(245\) −18.5139 −1.18281
\(246\) 0 0
\(247\) −4.60555 −0.293044
\(248\) 3.30278 0.209726
\(249\) 0 0
\(250\) 10.8167 0.684105
\(251\) 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
\(254\) −4.78890 −0.300482
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 11.2111 0.699329 0.349665 0.936875i \(-0.386296\pi\)
0.349665 + 0.936875i \(0.386296\pi\)
\(258\) 0 0
\(259\) 4.60555 0.286175
\(260\) 3.00000 0.186052
\(261\) 0 0
\(262\) −3.39445 −0.209710
\(263\) 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
\(266\) 9.21110 0.564769
\(267\) 0 0
\(268\) 14.5139 0.886576
\(269\) 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 \(0.437510\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 9.90833 0.598584
\(275\) 4.30278 0.259467
\(276\) 0 0
\(277\) −25.1194 −1.50928 −0.754640 0.656139i \(-0.772189\pi\)
−0.754640 + 0.656139i \(0.772189\pi\)
\(278\) 8.90833 0.534286
\(279\) 0 0
\(280\) −6.00000 −0.358569
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) 17.3944 1.03399 0.516996 0.855988i \(-0.327050\pi\)
0.516996 + 0.855988i \(0.327050\pi\)
\(284\) −6.00000 −0.356034
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) 4.18335 0.246935
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 9.00000 0.528498
\(291\) 0 0
\(292\) −8.69722 −0.508967
\(293\) 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
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) 1.81665 0.105236
\(299\) −15.9083 −0.920002
\(300\) 0 0
\(301\) −30.4222 −1.75351
\(302\) −13.3944 −0.770764
\(303\) 0 0
\(304\) 2.00000 0.114708
\(305\) 13.6972 0.784301
\(306\) 0 0
\(307\) 7.09167 0.404743 0.202372 0.979309i \(-0.435135\pi\)
0.202372 + 0.979309i \(0.435135\pi\)
\(308\) −6.00000 −0.341882
\(309\) 0 0
\(310\) −4.30278 −0.244381
\(311\) −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 \(0.223307\pi\)
\(314\) 7.21110 0.406946
\(315\) 0 0
\(316\) −16.1194 −0.906789
\(317\) 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
\(320\) −1.30278 −0.0728274
\(321\) 0 0
\(322\) 31.8167 1.77307
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) 7.60555 0.421880
\(326\) −20.4222 −1.13108
\(327\) 0 0
\(328\) 0.908327 0.0501540
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 1.21110 0.0665682 0.0332841 0.999446i \(-0.489403\pi\)
0.0332841 + 0.999446i \(0.489403\pi\)
\(332\) −17.2111 −0.944582
\(333\) 0 0
\(334\) −12.5139 −0.684729
\(335\) −18.9083 −1.03307
\(336\) 0 0
\(337\) −19.1194 −1.04150 −0.520751 0.853709i \(-0.674348\pi\)
−0.520751 + 0.853709i \(0.674348\pi\)
\(338\) −7.69722 −0.418674
\(339\) 0 0
\(340\) −7.81665 −0.423918
\(341\) −4.30278 −0.233008
\(342\) 0 0
\(343\) 33.2111 1.79323
\(344\) −6.60555 −0.356147
\(345\) 0 0
\(346\) 23.2111 1.24784
\(347\) −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 \(0.702931\pi\)
\(350\) −15.2111 −0.813068
\(351\) 0 0
\(352\) −1.30278 −0.0694382
\(353\) −31.8167 −1.69343 −0.846715 0.532047i \(-0.821423\pi\)
−0.846715 + 0.532047i \(0.821423\pi\)
\(354\) 0 0
\(355\) 7.81665 0.414865
\(356\) −5.21110 −0.276188
\(357\) 0 0
\(358\) −7.81665 −0.413123
\(359\) 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
\(362\) 20.0000 1.05118
\(363\) 0 0
\(364\) −10.6056 −0.555882
\(365\) 11.3305 0.593067
\(366\) 0 0
\(367\) −17.8167 −0.930022 −0.465011 0.885305i \(-0.653950\pi\)
−0.465011 + 0.885305i \(0.653950\pi\)
\(368\) 6.90833 0.360121
\(369\) 0 0
\(370\) −1.30278 −0.0677281
\(371\) 27.6333 1.43465
\(372\) 0 0
\(373\) 3.81665 0.197619 0.0988094 0.995106i \(-0.468497\pi\)
0.0988094 + 0.995106i \(0.468497\pi\)
\(374\) −7.81665 −0.404190
\(375\) 0 0
\(376\) 2.60555 0.134371
\(377\) 15.9083 0.819321
\(378\) 0 0
\(379\) −15.3305 −0.787477 −0.393738 0.919223i \(-0.628818\pi\)
−0.393738 + 0.919223i \(0.628818\pi\)
\(380\) −2.60555 −0.133662
\(381\) 0 0
\(382\) −12.5139 −0.640266
\(383\) −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
\(386\) −4.00000 −0.203595
\(387\) 0 0
\(388\) 12.4222 0.630642
\(389\) 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
\(392\) 14.2111 0.717769
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) 21.0000 1.05662
\(396\) 0 0
\(397\) 27.8167 1.39608 0.698039 0.716060i \(-0.254056\pi\)
0.698039 + 0.716060i \(0.254056\pi\)
\(398\) −2.42221 −0.121414
\(399\) 0 0
\(400\) −3.30278 −0.165139
\(401\) −13.8167 −0.689971 −0.344985 0.938608i \(-0.612116\pi\)
−0.344985 + 0.938608i \(0.612116\pi\)
\(402\) 0 0
\(403\) −7.60555 −0.378859
\(404\) −16.4222 −0.817035
\(405\) 0 0
\(406\) −31.8167 −1.57903
\(407\) −1.30278 −0.0645762
\(408\) 0 0
\(409\) −5.02776 −0.248607 −0.124303 0.992244i \(-0.539670\pi\)
−0.124303 + 0.992244i \(0.539670\pi\)
\(410\) −1.18335 −0.0584413
\(411\) 0 0
\(412\) 3.30278 0.162716
\(413\) −15.6333 −0.769265
\(414\) 0 0
\(415\) 22.4222 1.10066
\(416\) −2.30278 −0.112903
\(417\) 0 0
\(418\) −2.60555 −0.127442
\(419\) 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 \(0.253190\pi\)
\(422\) 6.69722 0.326016
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) −19.8167 −0.961249
\(426\) 0 0
\(427\) −48.4222 −2.34331
\(428\) −4.30278 −0.207983
\(429\) 0 0
\(430\) 8.60555 0.414997
\(431\) 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 \(0.592593\pi\)
\(434\) 15.2111 0.730156
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 13.8167 0.660940
\(438\) 0 0
\(439\) −9.33053 −0.445322 −0.222661 0.974896i \(-0.571474\pi\)
−0.222661 + 0.974896i \(0.571474\pi\)
\(440\) 1.69722 0.0809120
\(441\) 0 0
\(442\) −13.8167 −0.657191
\(443\) −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
\(446\) 15.8167 0.748940
\(447\) 0 0
\(448\) 4.60555 0.217592
\(449\) 0.788897 0.0372304 0.0186152 0.999827i \(-0.494074\pi\)
0.0186152 + 0.999827i \(0.494074\pi\)
\(450\) 0 0
\(451\) −1.18335 −0.0557216
\(452\) −11.2111 −0.527326
\(453\) 0 0
\(454\) 7.81665 0.366854
\(455\) 13.8167 0.647735
\(456\) 0 0
\(457\) 4.60555 0.215439 0.107719 0.994181i \(-0.465645\pi\)
0.107719 + 0.994181i \(0.465645\pi\)
\(458\) 17.3944 0.812789
\(459\) 0 0
\(460\) −9.00000 −0.419627
\(461\) 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 \(0.251331\pi\)
\(464\) −6.90833 −0.320711
\(465\) 0 0
\(466\) 9.51388 0.440722
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 66.8444 3.08659
\(470\) −3.39445 −0.156574
\(471\) 0 0
\(472\) −3.39445 −0.156242
\(473\) 8.60555 0.395684
\(474\) 0 0
\(475\) −6.60555 −0.303083
\(476\) 27.6333 1.26657
\(477\) 0 0
\(478\) −0.513878 −0.0235042
\(479\) −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
\(482\) 8.00000 0.364390
\(483\) 0 0
\(484\) −9.30278 −0.422853
\(485\) −16.1833 −0.734848
\(486\) 0 0
\(487\) −22.7889 −1.03266 −0.516332 0.856389i \(-0.672703\pi\)
−0.516332 + 0.856389i \(0.672703\pi\)
\(488\) −10.5139 −0.475941
\(489\) 0 0
\(490\) −18.5139 −0.836372
\(491\) 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
\(494\) −4.60555 −0.207214
\(495\) 0 0
\(496\) 3.30278 0.148299
\(497\) −27.6333 −1.23952
\(498\) 0 0
\(499\) 8.23886 0.368822 0.184411 0.982849i \(-0.440962\pi\)
0.184411 + 0.982849i \(0.440962\pi\)
\(500\) 10.8167 0.483735
\(501\) 0 0
\(502\) 6.78890 0.303003
\(503\) 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
\(506\) −9.00000 −0.400099
\(507\) 0 0
\(508\) −4.78890 −0.212473
\(509\) 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
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 11.2111 0.494501
\(515\) −4.30278 −0.189603
\(516\) 0 0
\(517\) −3.39445 −0.149288
\(518\) 4.60555 0.202356
\(519\) 0 0
\(520\) 3.00000 0.131559
\(521\) 9.63331 0.422043 0.211021 0.977481i \(-0.432321\pi\)
0.211021 + 0.977481i \(0.432321\pi\)
\(522\) 0 0
\(523\) 32.2389 1.40971 0.704853 0.709353i \(-0.251013\pi\)
0.704853 + 0.709353i \(0.251013\pi\)
\(524\) −3.39445 −0.148287
\(525\) 0 0
\(526\) 7.81665 0.340822
\(527\) 19.8167 0.863227
\(528\) 0 0
\(529\) 24.7250 1.07500
\(530\) −7.81665 −0.339534
\(531\) 0 0
\(532\) 9.21110 0.399352
\(533\) −2.09167 −0.0906004
\(534\) 0 0
\(535\) 5.60555 0.242349
\(536\) 14.5139 0.626904
\(537\) 0 0
\(538\) 6.78890 0.292690
\(539\) −18.5139 −0.797449
\(540\) 0 0
\(541\) −20.9361 −0.900113 −0.450056 0.893000i \(-0.648596\pi\)
−0.450056 + 0.893000i \(0.648596\pi\)
\(542\) 6.42221 0.275857
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) −2.60555 −0.111610
\(546\) 0 0
\(547\) −13.3944 −0.572705 −0.286353 0.958124i \(-0.592443\pi\)
−0.286353 + 0.958124i \(0.592443\pi\)
\(548\) 9.90833 0.423263
\(549\) 0 0
\(550\) 4.30278 0.183471
\(551\) −13.8167 −0.588609
\(552\) 0 0
\(553\) −74.2389 −3.15696
\(554\) −25.1194 −1.06722
\(555\) 0 0
\(556\) 8.90833 0.377797
\(557\) 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
\(560\) −6.00000 −0.253546
\(561\) 0 0
\(562\) 12.0000 0.506189
\(563\) −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
\(566\) 17.3944 0.731143
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) 10.4222 0.436922 0.218461 0.975846i \(-0.429896\pi\)
0.218461 + 0.975846i \(0.429896\pi\)
\(570\) 0 0
\(571\) −20.3028 −0.849645 −0.424822 0.905277i \(-0.639663\pi\)
−0.424822 + 0.905277i \(0.639663\pi\)
\(572\) 3.00000 0.125436
\(573\) 0 0
\(574\) 4.18335 0.174609
\(575\) −22.8167 −0.951520
\(576\) 0 0
\(577\) −28.2389 −1.17560 −0.587800 0.809007i \(-0.700006\pi\)
−0.587800 + 0.809007i \(0.700006\pi\)
\(578\) 19.0000 0.790296
\(579\) 0 0
\(580\) 9.00000 0.373705
\(581\) −79.2666 −3.28853
\(582\) 0 0
\(583\) −7.81665 −0.323733
\(584\) −8.69722 −0.359894
\(585\) 0 0
\(586\) 25.0278 1.03389
\(587\) 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
\(590\) 4.42221 0.182059
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) −36.5139 −1.49945 −0.749723 0.661752i \(-0.769813\pi\)
−0.749723 + 0.661752i \(0.769813\pi\)
\(594\) 0 0
\(595\) −36.0000 −1.47586
\(596\) 1.81665 0.0744130
\(597\) 0 0
\(598\) −15.9083 −0.650540
\(599\) 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 \(0.638717\pi\)
\(602\) −30.4222 −1.23992
\(603\) 0 0
\(604\) −13.3944 −0.545012
\(605\) 12.1194 0.492725
\(606\) 0 0
\(607\) −31.5139 −1.27911 −0.639554 0.768746i \(-0.720881\pi\)
−0.639554 + 0.768746i \(0.720881\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) 13.6972 0.554584
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) −8.18335 −0.330522 −0.165261 0.986250i \(-0.552847\pi\)
−0.165261 + 0.986250i \(0.552847\pi\)
\(614\) 7.09167 0.286197
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) −47.5694 −1.91507 −0.957536 0.288314i \(-0.906905\pi\)
−0.957536 + 0.288314i \(0.906905\pi\)
\(618\) 0 0
\(619\) −2.69722 −0.108411 −0.0542053 0.998530i \(-0.517263\pi\)
−0.0542053 + 0.998530i \(0.517263\pi\)
\(620\) −4.30278 −0.172804
\(621\) 0 0
\(622\) −5.09167 −0.204157
\(623\) −24.0000 −0.961540
\(624\) 0 0
\(625\) 2.42221 0.0968882
\(626\) 27.0278 1.08025
\(627\) 0 0
\(628\) 7.21110 0.287754
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 18.3028 0.728622 0.364311 0.931277i \(-0.381305\pi\)
0.364311 + 0.931277i \(0.381305\pi\)
\(632\) −16.1194 −0.641196
\(633\) 0 0
\(634\) 5.21110 0.206959
\(635\) 6.23886 0.247582
\(636\) 0 0
\(637\) −32.7250 −1.29661
\(638\) 9.00000 0.356313
\(639\) 0 0
\(640\) −1.30278 −0.0514967
\(641\) 2.48612 0.0981959 0.0490980 0.998794i \(-0.484365\pi\)
0.0490980 + 0.998794i \(0.484365\pi\)
\(642\) 0 0
\(643\) −29.8167 −1.17585 −0.587927 0.808914i \(-0.700056\pi\)
−0.587927 + 0.808914i \(0.700056\pi\)
\(644\) 31.8167 1.25375
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 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
\(650\) 7.60555 0.298314
\(651\) 0 0
\(652\) −20.4222 −0.799795
\(653\) −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
\(656\) 0.908327 0.0354642
\(657\) 0 0
\(658\) 12.0000 0.467809
\(659\) 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 \(0.578075\pi\)
\(662\) 1.21110 0.0470708
\(663\) 0 0
\(664\) −17.2111 −0.667920
\(665\) −12.0000 −0.465340
\(666\) 0 0
\(667\) −47.7250 −1.84792
\(668\) −12.5139 −0.484176
\(669\) 0 0
\(670\) −18.9083 −0.730492
\(671\) 13.6972 0.528775
\(672\) 0 0
\(673\) 24.3028 0.936803 0.468402 0.883516i \(-0.344830\pi\)
0.468402 + 0.883516i \(0.344830\pi\)
\(674\) −19.1194 −0.736453
\(675\) 0 0
\(676\) −7.69722 −0.296047
\(677\) 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
\(680\) −7.81665 −0.299755
\(681\) 0 0
\(682\) −4.30278 −0.164762
\(683\) −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
\(686\) 33.2111 1.26801
\(687\) 0 0
\(688\) −6.60555 −0.251834
\(689\) −13.8167 −0.526373
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 23.2111 0.882354
\(693\) 0 0
\(694\) −31.8167 −1.20774
\(695\) −11.6056 −0.440224
\(696\) 0 0
\(697\) 5.44996 0.206432
\(698\) −22.2389 −0.841753
\(699\) 0 0
\(700\) −15.2111 −0.574926
\(701\) −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
\(704\) −1.30278 −0.0491002
\(705\) 0 0
\(706\) −31.8167 −1.19744
\(707\) −75.6333 −2.84448
\(708\) 0 0
\(709\) −1.66947 −0.0626982 −0.0313491 0.999508i \(-0.509980\pi\)
−0.0313491 + 0.999508i \(0.509980\pi\)
\(710\) 7.81665 0.293354
\(711\) 0 0
\(712\) −5.21110 −0.195294
\(713\) 22.8167 0.854490
\(714\) 0 0
\(715\) −3.90833 −0.146163
\(716\) −7.81665 −0.292122
\(717\) 0 0
\(718\) 11.2111 0.418395
\(719\) 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
\(722\) −15.0000 −0.558242
\(723\) 0 0
\(724\) 20.0000 0.743294
\(725\) 22.8167 0.847389
\(726\) 0 0
\(727\) 29.9083 1.10924 0.554619 0.832104i \(-0.312864\pi\)
0.554619 + 0.832104i \(0.312864\pi\)
\(728\) −10.6056 −0.393068
\(729\) 0 0
\(730\) 11.3305 0.419362
\(731\) −39.6333 −1.46589
\(732\) 0 0
\(733\) 29.6333 1.09453 0.547266 0.836959i \(-0.315669\pi\)
0.547266 + 0.836959i \(0.315669\pi\)
\(734\) −17.8167 −0.657625
\(735\) 0 0
\(736\) 6.90833 0.254644
\(737\) −18.9083 −0.696497
\(738\) 0 0
\(739\) −42.3305 −1.55715 −0.778577 0.627549i \(-0.784058\pi\)
−0.778577 + 0.627549i \(0.784058\pi\)
\(740\) −1.30278 −0.0478910
\(741\) 0 0
\(742\) 27.6333 1.01445
\(743\) −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
\(746\) 3.81665 0.139738
\(747\) 0 0
\(748\) −7.81665 −0.285805
\(749\) −19.8167 −0.724085
\(750\) 0 0
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) 2.60555 0.0950147
\(753\) 0 0
\(754\) 15.9083 0.579347
\(755\) 17.4500 0.635069
\(756\) 0 0
\(757\) 9.30278 0.338115 0.169058 0.985606i \(-0.445928\pi\)
0.169058 + 0.985606i \(0.445928\pi\)
\(758\) −15.3305 −0.556830
\(759\) 0 0
\(760\) −2.60555 −0.0945133
\(761\) −42.1194 −1.52683 −0.763414 0.645909i \(-0.776478\pi\)
−0.763414 + 0.645909i \(0.776478\pi\)
\(762\) 0 0
\(763\) 9.21110 0.333464
\(764\) −12.5139 −0.452736
\(765\) 0 0
\(766\) −20.8444 −0.753139
\(767\) 7.81665 0.282243
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 7.81665 0.281693
\(771\) 0 0
\(772\) −4.00000 −0.143963
\(773\) 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
\(776\) 12.4222 0.445931
\(777\) 0 0
\(778\) 11.8806 0.425939
\(779\) 1.81665 0.0650884
\(780\) 0 0
\(781\) 7.81665 0.279702
\(782\) 41.4500 1.48225
\(783\) 0 0
\(784\) 14.2111 0.507539
\(785\) −9.39445 −0.335302
\(786\) 0 0
\(787\) 25.2111 0.898679 0.449339 0.893361i \(-0.351659\pi\)
0.449339 + 0.893361i \(0.351659\pi\)
\(788\) 6.00000 0.213741
\(789\) 0 0
\(790\) 21.0000 0.747146
\(791\) −51.6333 −1.83587
\(792\) 0 0
\(793\) 24.2111 0.859761
\(794\) 27.8167 0.987176
\(795\) 0 0
\(796\) −2.42221 −0.0858528
\(797\) 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
\(800\) −3.30278 −0.116771
\(801\) 0 0
\(802\) −13.8167 −0.487883
\(803\) 11.3305 0.399846
\(804\) 0 0
\(805\) −41.4500 −1.46092
\(806\) −7.60555 −0.267894
\(807\) 0 0
\(808\) −16.4222 −0.577731
\(809\) −29.4500 −1.03541 −0.517703 0.855561i \(-0.673213\pi\)
−0.517703 + 0.855561i \(0.673213\pi\)
\(810\) 0 0
\(811\) 54.1472 1.90136 0.950682 0.310166i \(-0.100385\pi\)
0.950682 + 0.310166i \(0.100385\pi\)
\(812\) −31.8167 −1.11655
\(813\) 0 0
\(814\) −1.30278 −0.0456623
\(815\) 26.6056 0.931952
\(816\) 0 0
\(817\) −13.2111 −0.462198
\(818\) −5.02776 −0.175791
\(819\) 0 0
\(820\) −1.18335 −0.0413242
\(821\) −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 \(0.571867\pi\)
\(824\) 3.30278 0.115058
\(825\) 0 0
\(826\) −15.6333 −0.543952
\(827\) −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 \(0.473852\pi\)
\(830\) 22.4222 0.778286
\(831\) 0 0
\(832\) −2.30278 −0.0798344
\(833\) 85.2666 2.95431
\(834\) 0 0
\(835\) 16.3028 0.564181
\(836\) −2.60555 −0.0901149
\(837\) 0 0
\(838\) 25.1472 0.868695
\(839\) 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
\(842\) 28.7250 0.989928
\(843\) 0 0
\(844\) 6.69722 0.230528
\(845\) 10.0278 0.344965
\(846\) 0 0
\(847\) −42.8444 −1.47215
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) −19.8167 −0.679706
\(851\) 6.90833 0.236814
\(852\) 0 0
\(853\) −11.5416 −0.395178 −0.197589 0.980285i \(-0.563311\pi\)
−0.197589 + 0.980285i \(0.563311\pi\)
\(854\) −48.4222 −1.65697
\(855\) 0 0
\(856\) −4.30278 −0.147066
\(857\) 14.8444 0.507075 0.253538 0.967326i \(-0.418406\pi\)
0.253538 + 0.967326i \(0.418406\pi\)
\(858\) 0 0
\(859\) −24.0555 −0.820764 −0.410382 0.911914i \(-0.634605\pi\)
−0.410382 + 0.911914i \(0.634605\pi\)
\(860\) 8.60555 0.293447
\(861\) 0 0
\(862\) 5.21110 0.177491
\(863\) −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
\(866\) −11.9361 −0.405605
\(867\) 0 0
\(868\) 15.2111 0.516298
\(869\) 21.0000 0.712376
\(870\) 0 0
\(871\) −33.4222 −1.13247
\(872\) 2.00000 0.0677285
\(873\) 0 0
\(874\) 13.8167 0.467355
\(875\) 49.8167 1.68411
\(876\) 0 0
\(877\) 7.21110 0.243502 0.121751 0.992561i \(-0.461149\pi\)
0.121751 + 0.992561i \(0.461149\pi\)
\(878\) −9.33053 −0.314890
\(879\) 0 0
\(880\) 1.69722 0.0572134
\(881\) −25.5416 −0.860520 −0.430260 0.902705i \(-0.641578\pi\)
−0.430260 + 0.902705i \(0.641578\pi\)
\(882\) 0 0
\(883\) −2.42221 −0.0815137 −0.0407568 0.999169i \(-0.512977\pi\)
−0.0407568 + 0.999169i \(0.512977\pi\)
\(884\) −13.8167 −0.464704
\(885\) 0 0
\(886\) −0.275019 −0.00923945
\(887\) −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
\(890\) 6.78890 0.227564
\(891\) 0 0
\(892\) 15.8167 0.529581
\(893\) 5.21110 0.174383
\(894\) 0 0
\(895\) 10.1833 0.340392
\(896\) 4.60555 0.153861
\(897\) 0 0
\(898\) 0.788897 0.0263258
\(899\) −22.8167 −0.760978
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) −1.18335 −0.0394011
\(903\) 0 0
\(904\) −11.2111 −0.372876
\(905\) −26.0555 −0.866115
\(906\) 0 0
\(907\) 26.0000 0.863316 0.431658 0.902037i \(-0.357929\pi\)
0.431658 + 0.902037i \(0.357929\pi\)
\(908\) 7.81665 0.259405
\(909\) 0 0
\(910\) 13.8167 0.458018
\(911\) 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
\(914\) 4.60555 0.152338
\(915\) 0 0
\(916\) 17.3944 0.574729
\(917\) −15.6333 −0.516257
\(918\) 0 0
\(919\) −38.4222 −1.26743 −0.633716 0.773566i \(-0.718471\pi\)
−0.633716 + 0.773566i \(0.718471\pi\)
\(920\) −9.00000 −0.296721
\(921\) 0 0
\(922\) 16.4222 0.540837
\(923\) 13.8167 0.454781
\(924\) 0 0
\(925\) −3.30278 −0.108595
\(926\) 30.3028 0.995811
\(927\) 0 0
\(928\) −6.90833 −0.226777
\(929\) 36.5139 1.19798 0.598991 0.800756i \(-0.295569\pi\)
0.598991 + 0.800756i \(0.295569\pi\)
\(930\) 0 0
\(931\) 28.4222 0.931500
\(932\) 9.51388 0.311637
\(933\) 0 0
\(934\) 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 \(0.656539\pi\)
\(938\) 66.8444 2.18255
\(939\) 0 0
\(940\) −3.39445 −0.110715
\(941\) 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
\(944\) −3.39445 −0.110480
\(945\) 0 0
\(946\) 8.60555 0.279791
\(947\) −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
\(950\) −6.60555 −0.214312
\(951\) 0 0
\(952\) 27.6333 0.895601
\(953\) 18.7527 0.607461 0.303730 0.952758i \(-0.401768\pi\)
0.303730 + 0.952758i \(0.401768\pi\)
\(954\) 0 0
\(955\) 16.3028 0.527545
\(956\) −0.513878 −0.0166200
\(957\) 0 0
\(958\) −12.1194 −0.391561
\(959\) 45.6333 1.47358
\(960\) 0 0
\(961\) −20.0917 −0.648118
\(962\) −2.30278 −0.0742445
\(963\) 0 0
\(964\) 8.00000 0.257663
\(965\) 5.21110 0.167751
\(966\) 0 0
\(967\) 25.7250 0.827260 0.413630 0.910445i \(-0.364261\pi\)
0.413630 + 0.910445i \(0.364261\pi\)
\(968\) −9.30278 −0.299003
\(969\) 0 0
\(970\) −16.1833 −0.519616
\(971\) −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
\(974\) −22.7889 −0.730203
\(975\) 0 0
\(976\) −10.5139 −0.336541
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) 6.78890 0.216974
\(980\) −18.5139 −0.591404
\(981\) 0 0
\(982\) 14.7250 0.469893
\(983\) 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
\(986\) −41.4500 −1.32004
\(987\) 0 0
\(988\) −4.60555 −0.146522
\(989\) −45.6333 −1.45105
\(990\) 0 0
\(991\) 54.3028 1.72498 0.862492 0.506070i \(-0.168902\pi\)
0.862492 + 0.506070i \(0.168902\pi\)
\(992\) 3.30278 0.104863
\(993\) 0 0
\(994\) −27.6333 −0.876475
\(995\) 3.15559 0.100039
\(996\) 0 0
\(997\) −23.5778 −0.746716 −0.373358 0.927687i \(-0.621794\pi\)
−0.373358 + 0.927687i \(0.621794\pi\)
\(998\) 8.23886 0.260797
\(999\) 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 666.2.a.j.1.1 2
3.2 odd 2 74.2.a.a.1.1 2
4.3 odd 2 5328.2.a.bf.1.1 2
12.11 even 2 592.2.a.f.1.2 2
15.2 even 4 1850.2.b.i.149.2 4
15.8 even 4 1850.2.b.i.149.3 4
15.14 odd 2 1850.2.a.u.1.2 2
21.20 even 2 3626.2.a.a.1.2 2
24.5 odd 2 2368.2.a.s.1.2 2
24.11 even 2 2368.2.a.ba.1.1 2
33.32 even 2 8954.2.a.p.1.1 2
111.110 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 3.2 odd 2
592.2.a.f.1.2 2 12.11 even 2
666.2.a.j.1.1 2 1.1 even 1 trivial
1850.2.a.u.1.2 2 15.14 odd 2
1850.2.b.i.149.2 4 15.2 even 4
1850.2.b.i.149.3 4 15.8 even 4
2368.2.a.s.1.2 2 24.5 odd 2
2368.2.a.ba.1.1 2 24.11 even 2
2738.2.a.l.1.1 2 111.110 odd 2
3626.2.a.a.1.2 2 21.20 even 2
5328.2.a.bf.1.1 2 4.3 odd 2
8954.2.a.p.1.1 2 33.32 even 2